1 Introductory Chapter: Microstructure and Mechanical Properties Hael Mughrabi Institut fur Werkstoffwissenschaften, Universitat Erlangen-Niirnberg, Erlangen, Federal Republic of Germany
List 1.1 1.2 1.3 1.4
of Symbols and Abbreviations Introduction Materials and Materials Development The Strength of Materials, Plasticity and Fracture The Stability of Plastic Flow in a Tensile Test, Failure by Necking, Superplasticity 1.5 Deformation Geometry, Crystallography of Slip, Orientation Factors 1.6 Dislocation Concepts of Plasticity and Work Hardening 1.7 Microstructure and Strength: The Major Hardening Mechanisms 1.8 Deformation Mechanisms at Low and High Temperatures, Deformation-Mechanism Maps 1.9 Microstructural Heterogeneities and Instabilities, Localisation of Strain, Damage and Failure 1.10 Microstructure-Based Modelling 1.11 References
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
2 3 3 4 6 7 7 9 10 12 15 16
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1 Introductory Chapter: Microstructure and Mechanical Properties
List of Symbols and Abbreviations a a0 b E G ^ K Kc KIc m T Tm
half crack length interatomic distance modulus of Burgers vector Young's modulus shear modulus elastic energy release rate stress-intensity factor critical stress-intensity factor, fracture toughness plane strain fracture toughness strain-rate sensitivity temperature melting temperature
a y y yeff ys s s 0 X Q a ac af cry T T* xG r th Ty <j)
geometrical constant (resolved) shear strain (resolved) shear strain rate specific fracture energy surface energy tensile strain tensile strain rate work hardening rate angle between load axis and slip direction dislocation density tensile stress theoretical cohesive strength fracture stress tensile yield stress (resolved) shear stress effective or thermal component of shear yield stress athermal component of shear yield stress theoretical shear stress, ideal shear strength (resolved) shear yield stress angle between specimen (load) axis and normal to the slip plane
b.c.c. DS f.c.c. ODS PFZ PSB TEM
body-centred cubic directionally solidified face-centred cubic oxide dispersion strengthened precipitate-free zone persistent slip band transmission electron microscopy
1.2 Materials and Materials Development
1.1 Introduction The mechanical behaviour of materials can be described largely in terms of the material properties that govern plastic deformation and fracture. Macroscopically, these properties can be expressed in terms of material parameters, which can usually be measured without detailed knowledge of the microscopic origin of these properties. Microscopically, these properties are related to processes on a microscale ranging from atomic dimensions to typical dimensions of the microstructure of the material, such as the grain size in the case of a polycrystalline material. One important goal of materials scientists is therefore to establish experimentally and theoretically the microscopic mechanisms responsible for a particular material property and to relate the microscopic behaviour to the property that can usually be measured macroscopically. The knowledge and understanding of the relevant material properties is the first step toward improving these properties and/or developing new materials with superior properties. The appropriate consideration of material properties in materials selection during the design of components is one major task of the discipline of materials science and technologyThis volume is devoted to some important aspects of plasticity and fracture. The topics of the individual chapters are related and form a unit, as outlined in the Preface. Together, they represent the effort to establish, based on experiment and theory, the microscopic mechanisms responsible for the mechanical behaviour of different materials and to provide a framework to relate the microscopic behaviour to macroscopic properties that can be measured readily and used by engineers.
In order to prepare the ground for the subsequent chapters which address in detail specific topics in the light of most recent research, it is appropriate to survey first at an elementary level some of the basic concepts of plasticity and fracture. While this chapter addresses mainly crystalline materials, many of the aspects discussed apply, with some modifications, also to glassy solids, a class of materials of increasing importance which is covered in more detail in Chap. 10. In this introductory approach, reference will be made to some of the historical developments that have led to our present understanding. Explicit reference to the vast literature will be confined to a few cases. For further details, the reader is referred to the literature recommended under the heading "General Reading" and to the listed "Proceedings of Conferences".
1.2 Materials and Materials Development A large part of the science of materials stems from early metallurgical developments. Referring to a classical textbook such as CottrelFs Theoretical Structural Metallurgy (1955), one can conclude that most of the concepts that were developed for metals and alloys in that era are still valid today. On the other hand, it is true that these concepts had to be developed further and that they had to be adjusted continually in order to do justice to the growing importance of new and more complex metallic and non-metallic materials. Today's materials can be classified broadly into metallic materials, ceramics, glasses and polymers and include also the growing family of different kinds of composite materials.
1 Introductory Chapter: Microstructure and Mechanical Properties TEMPERATURE CAPABILITY (1000h at 150 MPa)
11UU-
OF SUPERALLOYS Engineering ceramics -1500 °C • ODS ALLOYS IN-SITU COMPOSITES^
SINGLE CRYSTAL ALLOYS
1000 DSMarM002 DS Mar M20Q^« • I N 1 0 0 -*^ 900
^^T """. c ^^X^Nimonic115
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WROUGHT ALLOYS
^ ^ ^ ^ ^ " * * N imonic105 800
•
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^^Nimonic 80 750
1940
1950
1960
1970 YEAR
1980
Turbine-blade materials provide an interesting example of materials development. These materials must withstand fairly high stresses at the turbine inlet temperatures. The latter should be as high as possible in order to increase the efficiency and performance of the engine. Figure 1-1 shows how the development of nickel-base superalloys as turbine-blade materials over a period of about 50 years improved the temperatute capability by more than 200 °C. The figure shows the route of development from wrought to cast alloys (DS: directionally solidified, single crystals) and to mechanically alloyed powder metallurgical products (ODS: oxide dispersion strengthened). In addition, in situ composites are included and the potential use of ceramics, combined with a further substantial increase of temperature capability, is indicated. Intermetallics are another class of promising new engineering materials for high-temperature applications. These developments came about by careful control and modification of alloy chemistry and also by the introduction of new processing techniques such as vacuum melting and mechanical alloying. In par-
I
i
1990
2000
Figure 1-1. Increased temperature capability of nickel-base superalloys, shown as the temperature for 1000 h life with a stress of 150 MPa, as a function of the year of introduction into service. From McLean (1987). Reproduced with the author's permission.
allel, new theoretical concepts relating strength and microstructure had to be developed. This example shows that the development of new materials is a long process that requires vigorous efforts in technology and science.
1.3 The Strength of Materials, Plasticity and Fracture It has been established by early observations that plastic deformation occurs by shear. The upper limit of the shear resistance of a material is given by the so-called theoretical shear stress, which refers to the shear deformation of a perfect crystal in such a way that all atomic bonds across the shear plane are broken at the same time. In a simple model, one obtains a rather high value for the theoretical shear stress G Hh '
2TC
(1-1)
where G is the shear modulus. In general, crystalline materials yield plastically at much lower shear yield stresses ty (or tensile yield stresses ay) as a result of disloca-
1.3 The Strength of Materials, Plasticity and Fracture
tion glide (see Sec. 1.6 and Chaps. 2-8). Fortunately, a number of potent strengthening mechanisms exist, whereby the yield strength or flow stress of ductile materials can be enhanced considerably (see Sec. 1.7 and Chaps. 2 and 5-7). In a similar manner, the stress required to cause fracture of a perfect crystal by decohesion can be estimated. We consider first the case of brittle (cleavage) fracture in the absence of plastic deformation. Assuming that fracture occurs in a plane perpendicular to a tensile stress by simultaneous bond breaking, one obtains in a crude model a rather high value for the theoretical cohesive strength oc, namely £
(1-2)
where E is Young's modulus. Taking into account that two new fracture surfaces with a specific surface energy of 2 ys (for the two surfaces) must be created, it can be shown that oc can be expressed more specifically as (1-3) where a0 is the interatomic spacing. There is an obvious resemblance of this expression to the famous Griffith condition (1920) for brittle fracture: a>
h Ey / \l n a
(for plane stress)
(1-4)
In deriving this expression, Griffith assumed that the material contained a flaw from the beginning in the form of a crack of length 2 a lying perpendicular to the tensile stress a. He obtained Eq. (1-4) for plane stress conditions by balancing the decrease in elastic potential energy against the expenditure of surface energy in forming new crack surfaces. Comparison of Eqs. (1-3)
and (1-4) reveals why brittle fracture generally occurs at stresses that are much lower than the theoretical cohesive strength, which represents an upper limit of the fracture strength. The cause lies in the prior presence of small cracks which are inevitably found in most brittle materials. There is an intimate relationship between the Griffith expression and the stress distribution around the tip of a sharp crack. An important result of the stress analysis of cracks is that the occurrence of fracture is governed by the so-called stressintensity factor K. For an infinitely large cracked plate one has K=
(1-5)
At the point of fracture the stress a attains the value of the fracture stress of and the corresponding critical value Kc of the stress-intensity factor is called the fracture toughness. In ductile fracture, plastic deformation occurs at the crack tip. In this case, the energy necessary to create unit crack area is larger than 2ys (twice the specific surface energy) and it is appropriate to replace 2 ys by a specific fracture energy 2y eff , which also includes the work of plastic deformation in a global manner. The so-called elastic energy release rate & for plane stress has been shown by Irwin to be equal to K2/E, or K = ^E^
(1-6)
Under conditions of plane strain one finds K=
(1-7)
At the point of instability (fracture), ^ reaches a critical value ^ c = 2y eff . The critical stress-intensity factor or fracture toughness Kc is obtained when ^ = ^ c ; the plane strain fracture toughness is denoted by Klc. The latter is measurable in speci-
6
1 Introductory Chapter: Microstructure and Mechanical Properties
K,e 1
gantly (Hart, 1967) in terms of the work hardening rate 0 and the strain-rate sensitivity m, which are defined as
1 fda
(1-8)
and (1-9)
Figure 1-2. Schematic illustration of fracture toughness Klc as a function of yield stress a .
mens which are sufficiently thick that plane strain conditions prevail at the crack tip. Klc has been shown to be an absolute lower limit of the fracture toughness and a true material parameter. To conclude this section, it should be noted that materials of high strength (tensile yield stress ay) usually have a low fracture toughness Klc and vice versa. The reason is simply that in high-strength materials little plastic deformation occurs at the crack tip (y eff « ys). Hence the fracture toughness is lower than in less strong but more ductile materials. The relationship between Klc and oy is shown schematically in Fig. 1-2. This behaviour reflects the interrelationship between strength, ductility and toughness. Many more details on fracture phenomena are dealt with in Chap. 12.
1.4 The Stability of Plastic Flow in a Tensile Test, Failure by Necking, Superplasticity In tensile deformation of ductile materials, failure sets in when localised deformation leads to necking and, subsequently, to fracture. The condition for stable uniform tensile deformation can be formulated ele-
Here a and s denote the true stress and the true strain, respectively, and e is the true strain rate. Then the condition for stable tensile plastic deformation is derived as 0 + m>l
(1-10)
Here, two limiting cases are of interest: Material with low strain-rate sensitivity In this case the condition for plastic stability is the well-known Considere criterion m>\
or
-— > a As
(1-11)
Material with negligible work hardening This situation applies in high-temperature deformation (see also Sec. 1.8). Then the condition for stable plastic flow is diner m = —dine
(1-12)
This is the requirement for superplasticity (see Chap. 9) with Newtonian flow (seca) in the limiting case (m = 1). In the first case [Eq. (1-11)] the material resists progressive necking by work hardening; in the second case [Eq. (1-12)], by strain-rate hardening.
1.6 Dislocation Concepts of Plasticity and Work Hardening
1.5 Deformation Geometry, Crystallography of Slip, Orientation Factors It is well established from slip line studies, combined with X-ray analysis of lattice orientations, that plastic deformation of crystalline solids occurs by shear on densely packed erystallographic slip planes {hkl} in densely packed erystallographic directions (wvw). The pioneering studies which were performed in the 1920s and 1930s are treated in detail in the classical monograph by Schmid and Boas (1935, 1950). Perhaps the most notable result of these studies was Schmid's law of resolved shear stress: The onset of plastic deformation occurs when the shear stress acting on the incipient slip plane and in the slip direction reaches a critical value. This law which is obeyed in good approximation in most cases implies that the magnitude of the stress acting normal to the slip plane has no effect on the process of slip. In a uniaxial test, the resolved shear stress T is related to the axial stress o via an orientation factor: T=
a cos (f) cos X
(1-13)
an average orientation factor can be defined for a polycrystalline aggregate in order to relate the axial stress and strain to a mean shear stress and strain, respectively. Further details are treated in Chaps. 2 and 3 of this Volume and in several books dealing with deformation geometry (for example, Cottrell, 1964; Backofen, 1972; Reid, 1973).
1.6 Dislocation Concepts of Plasticity and Work Hardening With the introduction of the concept of dislocations in three independent papers by Orowan, Polanyi and Taylor (1934), the discrepancy between the theoretical shear stress and the experimentally determined values could be resolved. It is remarkable that, in his paper, Taylor not only derived in a dislocation concept an expression for the yield or flow stress in shear iy that is still valid today but also put forward the first dislocation-based work hardening theory. Taylor's result for the yield stress can be written in a form that is still used today: (1-14)
Here b is the modulus of the Burgers vecwhere (j) is the angle between the specimen tor, Q the dislocation density and a a geometrical constant that lies, typically, in the axis and the normal to the slip plane and X range 0.1 < a < 0.5. In general terms, this the angle between the load axis and the slip direction. The orientation factor cos (j) cos A, expression relates the yield stress to the inter-obstacle spacing, which, for an ensemcalled the Schmid factor, has its maximum value of 0.5 when cj) = X = 45°. In this case ble of dislocations of density Q, is approximately equal to 1/>/(?. Assuming that disT = or/2. Plastic flow sets in when x reaches locations travel a constant slip path, a critical value (critical resolved shear Taylor was also able to formulate his wellstress). known "parabolic" work-hardening law. These considerations, which were first In subsequent years much effort was put verified for single crystals, can also be apinto experimental and theoretical studies plied to individual grains of polycrystals. of work hardening. In particular, the soThen, by a suitable averaging procedure called three-stage work hardening of faceand taking into account multiple slip with centred cubic (f.c.c.) and, subsequently, appropriate compatibility requirements,
8
1 Introductory Chapter: Microstructure and Mechanical Properties
body-centred cubic (b.c.c.) single crystals with the three hardening stages I, II and III, and, in more recent years, also stage IV, received much attention. While much progress and many refinements of older concepts have been made in the past decades, as can be judged from Chap. 2, in which these topics are presented and discussed in detail, a number of issues are still under discussion. On the other hand, it is remarkable how advanced the understanding of dislocations, flow stress and work hardening already was in the 1950s, before the observations of dislocation distributions in deformed materials by transmission electron microscopy (TEM) had become common practice. The progress made is illustrated best by reference to some of the pertinent literature. The situation in the 1950s is well documented in some of the early standard monographs on dislocation theory and plasticity (Read, 1953; Cottrell, 1953; Seeger, 1958) and also, for example, in the proceedings of the historical Lake Placid Conference on Dislocations and Mechanical Properties of Matter (Fisher et al., 1957). The impact TEM had within a few years after having become available is reflected impressively in subsequent conference proceedings such as Electron Microscopy and Strength of Crystals (Thomas and Washburn, 1963) or in the series of international conferences on Strength of Metals and Alloys. Among other more recent publications, the collections of papers published in the proceedings of two symposia held in England (Dislocations and Properties of Real Materials, 1985) and in the United States (TMS-AIME Symposium, 1985) on the occasion of the 50th anniversary of the discovery of dislocations provide vivid evidence of the history and progress of dislocation science. Finally, attention is drawn to a notable series of con-
Figure 1-3. Examples of TEM observations of dislocation distributions in deformed metals, (a) Dislocation arrangement in the primary glide plane of a copper single crystal, deformed at 78 K to a shear flow stress of 12 MPa in stage II. (b) Dislocation wall structure in the primary glide plane of a copper single crystal fatigued at room temperature.
ferences that have been held in Aussois since 1981 on current topics of plasticity. The proceedings of these meetings reflect the shift in emphasis to more specialised areas such as the atomistics of dislocation structure and processes, dislocation modelling of more and more complex loading modes and dislocations in new materials. In all problems of crystal plasticity, the distribution of dislocations plays an important role. Thus, in a di- or multipolar dislocation array, the stress fields of the dislocations are mutually screened, whereas a group of dislocations piling up against an
1.7 Microstructure and Strength: The Major Hardening Mechanisms
obstacle gives rise to high internal stresses. Now, TEM observations have shown that, in general, dislocations in deformed materials are not distributed randomly or homogeneously but, instead, show a strong tendency to cluster locally. In particular, under conditions of multiple slip, dislocations cluster in dislocation walls and networks which build up so-called cell structures and subgrain structures at low and high temperatures, respectively. Some of these dislocation distributions can be quite regular and, as a general rule, the wall spacings vary inversely as the yield stress. Examples of typical dislocation distributions are shown in Fig. 1-3. In the past, the heterogeneity of the dislocation distributions has only rarely been taken into account explicitly in dislocation models of plasticity. On the other hand, in recent years, there has been an increasing interdisciplinary interest in finding a theoretical basis for understanding the evolution and the scaling of heterogeneous dislocation patterns as a function of stress and in incorporating these features in models of plasticity. Chapter 4 reviews the current situation of this new field of research.
1.7 Microstructure and Strength: The Major Hardening Mechanisms During plastic deformation the dislocation density Q increases as a result of a variety of dislocation source or generation mechanisms. It is then clear and in accord with Eq. (1-14) that the yield stress will increase as a consequence of plastic straining. Thus, strain or work hardening represents an obvious means of strengthening any ductile material by enhancing the intensity of the elastic interaction of dislocations simply by increasing their density (see
also Chap. 2). On the other hand, one can also increase the lattice resistance against dislocation glide by alloying, as is common practice in technical materials. The interaction between glide dislocations and solute atoms is a complex subject and is treated in detail in Chap. 5. In addition, in alloys with a miscibility gap (see Chap. 4 in Vol. 5 of this Series), precipitates of a second phase can be introduced by a suitable heat treatment and ageing. These precipitates provide strong obstacles to dislocation glide by a variety of mechanisms. Instead of or in addition to precipitates, foreign particles may be introduced to provide so-called dispersion hardening. In Chap. 7, current dislocation theories of particle hardening and the pertinent experimental data are reviewed comprehensively. Finally, the strength can be enhanced considerably by reducing the grain size (grain refinement), since grain boundaries also represent effective barriers to dislocation glide. This strengthening is well described by the Hall-Petch relation (Chap. 2). The superposition of all mentioned strengthening mechanisms can be exploited in practice. While a linear superposition can be assumed for crude purposes, the rigorous treatment of the superposition of different strengthening mechanisms is a formidable task. Besides these possibilities of strengthening different materials, one can, of course, search for materials with an intrinsically high lattice resistance to dislocation glide. Intermetallic compounds are an example of materials of high intrinsic strength. In spite of their low-temperature brittleness, these materials have received much attention in recent years, in particular as potential high-temperature materials. Chapter 6 reviews the current state and also gives an indication of the complexity of dislocation structures in such materials with a superlattice crystal structure.
10
1 Introductory Chapter: Microstructure and Mechanical Properties
1.8 Deformation Mechanisms at Low and High Temperatures, Deformation-Mechanism Maps The yield stress of a given material usually depends on temperature and on strain rate through specific thermally activated dislocation mechanisms. In the range of low and intermediate temperatures at which diffusion and dislocation climb are negligible the (shear) yield stress xy can be regarded according to Seeger (1954, 1958) as the sum of two contributions: T y ^ T G + T *(y ? T)
Ty/Gl
T*(0)/G
\ \ 'o.2
T
Figure 1-4. Schematic illustration of the dependence of the yield stress ry, normalised with respect to the shear modulus G, on temperature T for two different shear strain rates y.
(1-15)
The term xG represents the stress component due to the long-range elastic interaction of dislocations and depends on temperature only through the temperature dependence of the shear modulus G. Hence TG is called the athermal stress component in contrast to the thermal or effective stress component T*, which is required in order to permit the glide dislocations to overcome short-range barriers by thermal activation. The thermal stress component T* depends characteristically on temperature T and on the shear strain rate y. Figure 1-4 shows schematically the expected dependence of xy, normalised with respect to G, on temperature for two strain rates. At 0 K, the short-range barriers must be overcome without any thermal activation entirely by mechanical work with the stress TG + T * ( 0 K ) . The low-temperature increase of the yield stress is most pronounced for materials with a high Peierls energy, such as b.c.c. metals and the elemental covalent semiconductors germanium and silicon, and is paired with a loss of ductility to the extent of low-temperature brittleness. In these materials, the ratecontrolling thermally activated process of dislocation motion is the nucleation and
propagation of double kinks, see e.g. Seeger (1981). Above a characteristic temperature To, which increases with increasing strain rate, thermal activation is so large that T* vanishes. At still higher temperatures, above about 0.4 Tm (Tm: melting temperature), typical high-temperature deformation mechanisms characterised by dislocation glide combined with dynamic recovery through dislocation climb operate. These deformation mechanisms are typical of dislocation creep. As a consequence, the yield stress drops, as indicated in Fig. 1-4. In addition to mechanisms of dislocation creep, plastic deformation by diffusional creep processes in the absence of dislocation motion can also occur at high temperatures and at low stresses. Depending on whether lattice or grain boundary diffusion dominates, a distinction is made between Nabarro-Herring and Coble creep. So far the implicit assumption, also with respect to Fig. 1-4, was that a given microstructure prevails. Strictly speaking, this is only the case in steady-state deformation, which is characterised by a dynamic equilibrium between dislocation multiplication and annihilation, i.e. be-
1.8 Deformation Mechanisms at Low and High Temperatures, Deformation-Mechanism Maps
TEMPERATURE -200
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.5 .6 .? .8 TEMPERATURE T / T M
Figure 1-5. Dislocation-mechanism map for nickel with a grain size of 1 mm. From Frost and Ashby (1982). Reproduced with the authors' permission.
tween work hardening and softening by dynamic recovery. In this case, which is best fulfilled in high-temperature creep, a unique relationship exists between ry and y at a given temperature T. A detailed account of high-temperature dislocation mechanisms is presented in Chap. 8. The regimes of temperature and stress in which the different deformation mechanisms discussed above dominate are displayed conveniently in the form of deformation-mechanism maps introduced by Ashby (1972, 1973) and presented in the form of a monograph by Frost and Ashby (1982). In these maps, the ordinate represents the shear stress normalised with respect to the shear modulus, and the abscissa the homologous temperature T/Tm. The
upper limit of stress is given by the theoretical shear stress (ideal strength). Then, using appropriate constitutive laws to describe the different deformation mechanisms, one can determine the regimes of stress and temperature in which a particular mechanism is dominant, in the sense that it operates at a higher strain rate than all others. The maps are completed by the inclusion of contours of constant strain rate which cover, typically, the range from I s " 1 down to 10~ l o s~ 1 . The latter value is considered to lie at the limit of detectability. Figure 1-5 shows as an example a typical deformation-mechanism map for polycrystalline nickel with a rather large grain size of 1 mm. We can distinguish the regimes
12
1 Introductory Chapter: Microstructure and Mechanical Properties
of (thermally activated) dislocation glide, high-temperature dislocation creep, represented by so-called power-law creep (see Chap. 8) and creep by diffusional flow (Nabarro-Herring and Coble creep). The regime of power-law creep has been divided into two sub-regimes according to whether lattice diffusion or diffusion along dislocation cores is dominant. These mcips show a very characteristic dependence on grain size. Thus, in Fig. 1-5, the dominant high-temperature deformation mechanism is seen to be power-law creep, whereas the field of diffusional flow in which only negligible strain rates occur is relatively insignificant. On the other hand, for a realistic smaller grain size, the importance of diffusional creep increases at the expense of the field of power-law creep. This is because diffusional creep rates are larger in fine- than in coarse-grained material. In other words, at high temperatures, materials with a fine grain size are mechanically softer than those with a coarse grain size. This behavior is opposite to the situation at low temperatures, where grain refinement is used to strengthen the material (Sec. 1-7). In practice, important consequences result with respect to the use of high-temperature materials. Thus, as demonstrated by Ashby (1973) for a nickel-base superalloy, the performance of this alloy under typical turbine service conditions is considerably better for a grain size of 1 cm than for a grain size of 100 Jim. While Coble creep is the dominant mechanism in both cases, the strain rates are considerably smaller for the larger grain size. For these reasons, monocrystalline superalloys are used increasingly, with the advantages that, with the elimination of the grain boundaries, diffusional Coble creep and, at the same time, damage due to grain boundary sliding can be avoided, see Fig. 1-1.
1.9 Microstructural Heterogeneities and Instabilities, Localisation of Strain, Damage and Failure The homogeneity of microstructure is a prerequisite for homogeneous deformation. Heterogeneities can be present in the material in the fabricated state. Alternatively, microstructural heterogeneities can develop during straining, in particular at high temperatures, as a consequence of microstructural instabilities. The mechanical properties can deteriorate dramatically due to the presence of microstructural heterogeneities. The latter frequently give rise to more or less severe localisation of strain, associated with crack initiation and, ultimately, failure. Precipitate coarsening during high-temperature deformation is another form of microstructural instability. In this case, the adverse effects on the strength properties are not a consequence of microstructural heterogeneity but a result of the morphological changes of the precipitates. In the following, some examples of the behaviour due to the effects described above will be given. Precipitate-free zones (PFZs) at grain boundaries are a frequently encountered feature in precipitation-hardened alloys. An example is shown in Fig. 1-6. The reason for the occurrence of PFZs can lie in the depletion of either vacancies or solute atoms near the grain boundaries, see e.g. Hornbogen and Kreye (1982). By reducing the fracture strain strongly, PFZs have a deleterious effect on the mechanical properties. Fracture occurs in an intercrystalline fashion and, when inspected superficially at low magnification, the fracture surface is strongly reminiscent of that produced by brittle fracture, as shown in Fig. 1-7 a taken from the work of Vasudevan and Doherty (1987) on an Al-Mg-Zn-Cu
1.9 Microstructural Hetereogeneities and Instabilities
Figure 1-6. TEM micrograph of precipitate distribution in an A l - Z n - M g - C u alloy in the peak-aged condition. Note the precipitate-free zones at the grain boundaries. From Volkl (1987). Reproduced with the author's permission.
alloy. At higher magnification (Fig. l-7b), however, it becomes obvious that fracture is preceded by extensive deformation, localised at the grain boundaries, and that the fracture surface exhibits fine dimples, typical of ductile fracture. For these reasons, Hornbogen and Kreye (1982) introduced the term intercrystalline pseudo-brittleness to characterise this type of grain boundary ductile fracture behaviour. This
13
phenomenon is a non-trivial problem in many alloys and, in particular, in the lowdensity Al-Li alloys, which are currently of considerable interest. Next, we discuss an example of localised strain in extensively deformed metals and alloys. High-strain deformation is typical of forming operations, such as rolling and drawing, and is treated in more detail in Chap. 3. In addition to textures, frequently microstructural instabilities and localised deformation in micro-shear bands occur. The crystallographic orientation relationships between these shear bands and the surrounding matrix are complex. In metal fatigue, during repeated cyclic deformation to large cumulative strains, a simpler type of shear band is commonly observed, namely persistent slip bands (PSBs). PSBs are thin lamellae (thicknesses of about 1 jim in metals and 0.1 |um in precipitationhardened alloys) that lie parallel to the activated crystallographic glide planes. Examples of TEM micrographs of PSBs in a pure metal and in an alloy are shown in Figs. 1-8 a and c, respectively. The origin of PSBs lies in local deformation-induced microstructural instabilities of the cyclically hardened dislocation structure in the case of pure metals and of the precipitate structure in the case of precipitation-hardened alloys. Strain localisation in PSBs in fatigue usually leads to crack initiation in the
Figure 1-7. Scanning electron micrographs of the fracture surface of tensile-deformed, slightly over-aged Al-Zn-Mg-Cu. (a) Low magnification. (b) Higher magnification. From Vasudevan and Doherty (1987). Reproduced with the permission of the authors and the publishers (Pergamon Press).
14
1 Introductory Chapter: Microstructure and Mechanical Properties
PSB
M
PSB
Figure 1-8. Examples of persistent slip bands in fatigued metals and alloys, (a) TEM micrograph of the characteristic "ladder" dislocation structure of PSBs in the bulk of fatigued copper. M: matrix, (b) Scanning electron micrograph of surface relief of emerging PSBs in fatigued copper, (c) TEM micrograph of PSBs in a fatigued nickel-base superalloy Nimonic 80A containing a bi-modal distribution of y'-particles. Part (c) is from Gerold et al. (1984) and is reproduced with permission of the authors and the publisher (Riederer Verlag).
valleys of the rough surface profiles of emerging PSBs (Fig. 1-8 b) and, ultimately, to failure. Quite generally, strain localisation is important in fatigue, which is treated in more detail in Chap. 11. Other examples of microstructure-controlled damage, with strain localisation, leading to failure, are found in Chaps. 6, 9, 10,12 and 13. Referring to the last chapter, on friction and wear, it is interesting to note that many of the aspects of plastic deformation and fracture dealt with in the earlier chapters now come into play simultaneously, superimposed on the specific tribological features of friction and wear at the contact interfaces between solids. As another example of microstructural instability, we shall now discuss a special case of precipitate coarsening during high-temperature creep deformation of monocrystalline precipitation-hardened nickel-base superalloys. These alloys represent currently the most advanced class of turbine-blade materials. They are hardened primarily by a high volume fraction (about 70%) of ordered y'-precipitates of cuboidal shape in a y-matrix, as shown in Fig. 1-9 a. Usually, these monocrystals have orientations close to <001 > in the direction of solidification. When creep-deformed in tension in this direction, referred to as [001], an interesting change in the morphology of the y'-predpitates can occur, as shown in Fig. 1.9 b, by a process called y'-rafting. Under the given circumstances, lateral coarsening of the initially cuboidal y'-precipitates has led to the formation of the plate-like y/y'-rafts, which lie roughly perpendicular to the stress axis. The process of raft formation is complex and involves a combination of creep and stress-induced diffusion. Raft formation is favoured by long-time exposure to high temperatures at relatively low stresses (and creep rates). Hence, the cuboidal y'-precip-
1.10 Microstructure-Based Modelling
if» ff •*» ^«f is «« *rj;
9 * 1
15
itates can be preserved at lower temperatures and higher stresses (Fig. 1.9 a), whereas rafts form at higher temperatures and lower stresses (Fig. 1.9 b). Raft formation can have adverse effects on the creep strength, causing an enhancement of the creep rate (Schneider et al., 1992). In other cases, when the crystals have such orientations that y/y'-rafting occurs on a plane under an inclined angle to the stress axis, localised shear strain can develop along the softer y-lamellae.
1.10 Microstructure-Based Modelling
Figure 1-9. Scanning electron micrograph of the yfmorphology in tensile creep-deformed specimens of the monocrystalline nickel-base superalloy CMSX-4. The stress axis is vertical. The contrast of the y'-phase is dark and of the y-matrix light, (a) Specimen crept to rupture at T=800°C, a = 654 MPa. (b) Specimen crept to rupture at T = 950°C, a = 301 MPa. From Schneider et al. (1992). Reproduced with permission of the authors and the publisher (The Minerals, Metals and Materials Society).
As a result of the increased understanding of the fundamental processes of plasticity and fracture, it has become possible to perform more quantitative computations of the mechanical behaviour of materials based on the actual details of the microstructural processes. Thus, microstructure-based modelling is gaining importance rapidly. Progress in this area has been facilitated greatly by the increasing use of computers. These developments are reflected in the proceedings of several conferences that have been dedicated largely to modelling. The basic procedure for the modelling of a strength property is illustrated schematically in a flowchart in Fig. 1-10. The experimental input data are obtained from a detailed quantitative metallographic characterisation of the microstructure of the material and from comprehensive mechanical measurements. At the heart of the procedure there must be a realistic theoretical model into which the experimental data are fed so that the result of the computation can be compared with the experimentally measured strength property under consideration. So-called constitutive equa-
16
1 Introductory Chapter: Microstructure and Mechanical Properties
QUANTITATIVE MICROSTRUCTURAL ANALYSIS
MEASUREMENT OF MECHANICAL PROPERTIES
MODELLING
THEORETICAL MODEL
—
COMPUTATION OF STRENGTH PROPERTIES
tions are essential ingredients of the theoretical concept whose soundness must be checked and perhaps improved by repetition of the modelling cycle until a satisfactory result is obtained. In this volume, the reader will find many examples of such modelling. In many cases, microstructure-based modelling goes beyond the mere computational description of material behaviour and has been applied to simulate the response of whole components subjected to complex states of stress and temperature similar to those encountered under service conditions. This kind of approach, ranging from fundamentals to technological applications, will make microstructure-based modelling an indispensable tool in materials engineering.
1.11 References Ashby, M. F. (1972), Ada Metall 20, 887-897. Ashby, M. F. (1973), in: The Microstructure and Design of Alloys, Proc. 3rd Int. Conf. on the Strength of Metals and Alloys (ICSMA3), Cambridge. Sponsored by the Institute of Metals and the Iron and Steel Institute. Vol. 2, pp. 8-42. Backofen, W. A. (1964), Deformation Processing. Reading, Mass.: Addison-Wesley. Cottrell, A. H. (1953), Dislocations and Plastic Flow in Crystals. Oxford: Clarendon. Cottrell, A. H. (1955), Theoretical Structural Metallurgy. London: Edward Arnold.
Figure 1-10. Schematic flowchart, illustrating the strategy and procedure followed in microstructurebased modelling.
Cottrell, A. H. (1964), The Mechanical Properties of Matter. New York: Wiley. Dislocations and Properties of Real Materials (1985), Proc. of the Conference to Celebrate the 50th Anniversary of the Concept of Dislocation in Crystals, Book 323. London: Institute of Metals. Fisher, I C , Johnston, W. G., Thomson, R., Vreeland, Jr., T. (Eds.) (1957), Dislocations and Mechanical Properties of Crystals, Proc. of the Lake Placid Conf New York: Wiley. Frost, H. X, Ashby, M. F. (1982), Deformation-Mechanism Maps. Oxford: Pergamon. Gerold, V., Lerch, B. A., Steiner, D. (1984), Z. Metallkd. 75, 546-553. Griffith, A. A. (1920), Philos. Trans. R. Soc. London A221, 163-198. Hart, E. W. (1967), Ada Metall. 15, 351-355. Hornbogen, E., Kreye, H. (1982), J. Mater. Sci. 17, 979-988. McLean, M. (1987), in: Proc. 1st ASM European Conf on Advanced Materials and Processing Techniques for Structural Applications: Khan, T., Lasalmonie, A. (Eds.). Chatillon: ONERA, pp. 31-47. Read, Jr., W. T. (1953), Dislocations in Crystals. New York: McGraw-Hill. Reid, C. N. (1973), Deformation Geometry for Materials Scientists. Oxford: Pergamon. Schmid, E., Boas, W. (1935), Kristallplastizitdt. Berlin: Springer (in German). Schmid, E., Boas, W. (1950), Plasticity of Crystals. London: F. A. Hughes (English translation). Schneider, W., Hammer, I , Mughrabi, H. (1992), in: Super alloys 1992, Proc. 7 th Int. Symp. on Super alloys, Seven Springs: Antolovich, S. D. et al. (Eds.). Warrendale, Pa: The Minerals, Metals and Materials Society, pp. 589-598. Seeger, A. (1954), Philos. Mag. Ser. 7, 45, 111-173. Seeger, A. (1958), "Kristallplastizitat", in: Handbuch der Physik, Vol. VII/2. Berlin: Springer. Seeger, A. (1981), Z. Metallkd. 72, 369-380.
1.11 References
Taylor, G. I. (1934), Proc. R. Soc. London 66 B, 362387. Thomas, G., Washburn, J. (Eds.) (1963), Electron Microscopy and Strength of Crystals. New York: Interscience. TMS-AIME Symposium (1985), "50th Anniversary of the Introduction of Dislocations". Metall. Trans. 16A, December. Vasudevan, A. K., Doherty, R. D. (1987), Acta Metall. 35, 1193-1219. Volkl, L. (1987), Diplomarbeit, Universitat Erlangen-Niirnberg.
General Reading Books Argon, A. S. (Ed.) (1975), Constitutive Equations in Plasticity. Cambridge, Mass.: MIT Press. Ashby, M. R, Jones, D. R. H. (1980), Engineering Materials. Oxford: Pergamon. Embury, J. D., Thompson, A. W. (Eds.) (1988), Modeling of Material Behaviour and its Design. Warrendale, Pa: The Minerals, Metals and Materials Society. Hertzberg, R. W. (1983), Deformation and Fracture Mechanics of Engineering Materials, 2nd ed. New York: Wiley. Kocks, U. R, Argon, A. S., Ashby, M. F. (1975), Thermodynamics and Kinetics of Slip. Prog. Mater. Sci. 19. Latanision, R. M., Jones, R. H. (Eds.) (1987), Chemistry and Physics of Fracture. Dordrecht: Martinus Nijhoff. Lowe, T, Rollett, R., Follansbee, P., Daehn, G. (Eds.) (1991), Modeling the Deformation of Crystalline Solids. Warrendale, Pa: The Minerals, Metals and Materials Society. McClintock, R A., Argon, A. S. (Eds.) (1966), Mechanical Behaviour of Materials. Reading, Mass.: Addison-Wesley. Reid, C. N. (1973), Deformation Geometry for Materials Scientists, Oxford: Pergamon.
17
Proceedings of Conferences, Conference Series Andersen, S. L, Bilde-Sorensen, J. B., Hansen, N., Juul Jensen, D., Leffers, T, Lilholt, H., Lorentzen, T, Pedersen, O. B., Ralph, B. (Eds.) (1992), Modelling of Plastic Deformation and its Engineering Applications, Proc. 13th Riso Int. Symp. on Materials Science. Roskilde, Denmark: Riso National Laboratory. Andersen, S. I., Bilde-Sorensen, J. B., Hansen, N., Leffers, T., Lilholt, H., Pedersen, O. B., Ralph, B. (Eds.) (1987), Constitutive Relations and Their Physical Basis, Proc. 8th Int. Symp. on Metallurgy and Materials Science. Roskilde, Denmark: Ris0 National Laboratory. Antolovich, S. D., Stusrud, R. W, MacKay, R. A., Anton, D. L., Khan, T, Kissinger, R. D., Klarstrom, D. L. (Eds.) (1992), Superalloys 1992, Proc. 7th Int. Symp. on Superalloys. Warrendale, Pa: The Minerals, Metals and Materials Society. Bilde-Sorensen, J. B., Hansen, N., Juul Jensen, D., Leffers, T., Lilholt, H., Pederson, O. B. (Eds.) (1989), Materials Architecture, Proc. 10th Int. Symp. on Metallurgy and Materials Science. Roskilde, Denmark: Riso National Laboratory. Brandon, D. G., Chaim, R., Rosen, A. (Eds.) (1991), Strength of Metals and Alloys, Proc. 9th Int. Conf. (ICSMA9), Haifa, Vols. 1 and 2. London: Freund. Caillard, D., Thibault, X, Veyssiere, P. (Eds.) (1991), Mechanisms of Deformation and Strength of Advanced Materials, Proc. Int. C.N.R.S. Meeting, Aussois, 1990. J. de Phys. Ill 1, 829-1166. Castaing, X, Strudel, X L., Zaoui, A. (Eds.) (1988), Mechanisms and Mechanics of Plasticity, Proc. Int. C.N.R.S. Meeting, Aussois, 1987. Rev. Phys. Appl. 23. Dislocations and Properties of Real Materials (1985), Proc. of the Conf. to Celebrate the 50th Anniversary of the Concept of Dislocation in Crystals, Book 323. London: Institute of Metals. Kettunen, P. O., Lepisto, T. K., Lehtonen, M. E. (Eds.) (1988), Strength of Metals and Alloys, Proc. 8th Int. Conf (ICSMA 8), Tampere, Vols. 1 and 2. Oxford: Pergamon. Yan, M. G., Zhang, S. H., Zheng, Z. M. (Eds.) (1988), Mechanical Behaviour of Materials- V, Proc. 5th Int. Conf, Beijing, Vols. 1 and 2-. Oxford: Pergamon.
2 Flow Stress and Work Hardening Javier Gil Sevillano Escuela Superior de Ingenieros Industrials (Universidad de Navarra) and CEIT (Centro de Estudios e Investigaciones Tecnicas de Guipuzcoa), San Sebastian (Spain)
List of 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.6.1 2.2.6.2 2.2.6.3 2.2.6.4 2.2.7 2.2.8 2.2.8.1 2.2.8.2 2.2.8.3 2.2.8.4 2.2.9 2.3 2.3.1 2.3.2 2.3.3 2.3.3.1 2.3.3.2 2.3.3.3
Symbols and Abbreviations Introduction Scope of the Chapter The Theoretical Interpretation of Flow Stress and Work Hardening Phenomenology of Plastic Stress-Strain Curves: Deformation Stages . . . . The Flow Stress • The Ideal Shear Strength: Dislocationless Flow Obstacle-Limited Dislocation Glide Kinetics The Peierls-Nabarro Stress The Background Internal Stress Point-Like Obstacles The Dislocation Forest Contribution to the Flow Stress The F C C . Forest Other Lattice Structures Latent Hardening and Related Observations Thermally Activated Glide Through Dislocation Substructures Solid Solution Hardening and the Strengthening Effects of Precipitates or Dispersed Phases Extended Obstacles. Two-Phase Materials and Grain Size Effects Plastic Flow Transmission in Heterogeneous Materials with Strong Tendency to Planar Slip Compatible Plastic Flow in Heterogeneous Crystals Without Pileup Formation The Composite Approach to the Flow Stress of Crystals Containing Heterogeneous Dislocation Distributions Grain-Size Effects in the Flow Stress of Polycrystals Superposition of Different Contributions to the Flow Stress Work Hardening The Semi-Phenomenological Approach The Athermal Work Hardening Limit of Self-Similar Dislocation Structures The Experimental Work Hardening Behavior Stage I Stage II Stage III
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.
21 23 23 24 25 27 28 28 29 32 33 36 37 40 41 44 47 47 48 49 50 51 56 56 56 58 59 59 59 59
20
2 Flow Stress and Work Hardening
2.3.3.4 Stage IV 2.3.3.5 Synthesis of Behavior: From Stage I to the True Saturation Stress, T*Y . . . . 2.3.4 A Work Hardening Model Consistent with the Micromechanisms Determining the Flow Stress 2.3.4.1 The Athermal Hardening Term, 9h 2.3.4.2 The Athermal Saturation Stress, (T™)0 2.3.4.3 Dynamic Recovery 2.3.4.4 Stage II 2.3.4.5 Stage I Paradox 2.3.4.6 Stage IV and Its Connection with Stage I 2.3.5 Influence of Crystal Structure 2.4 Acknowledgements 2.5 References
63 64 67 67 70 71 73 76 78 81 83 83
List of Symbols and Abbreviations
21
List of Symbols and Abbreviations aVl terms of interaction matrix between different slip systems b magnitude of Burgers vector D grain size Ed dislocation line energy / factor or fraction / h , / s respectively, volume fraction of hard and soft zones of a crystal F force (on obstacles) ^obs obstacle-strength AF free-energy increment g geometric factor G elastic shear modulus h spacing (between slip planes) he critical passing distance for edge dislocations htj terms of work hardening matrix I, II, III, IV successive work hardening stages k Boltzmann constant K ratio of tree spacing to tree length Knp Hall-Petch constant / length of dislocation segment L average interobstacle length or distance M orientation factor n number of dislocations in pileup NA areal density p, q exponents in kinetic equation r, R radius (r 0 , R, respectively, inner and outer cut-off radius) s surface s{ characteristic structural parameters S size t thickness T, 7^i respectively, absolute temperature and absolute melting temperature Td line tension of dislocation u displacement v velocity of dislocation glide displacement V volume (F a : activation volume) w dislocation width W work x distance or size X obstacle size ye,ys respectively, mutual annihilation distance for edge and screw dislocation segments a
proportionality constant relating normalized flow stress, T C /G, with the inverse of normalized mean interdislocation distance, b J^
22
2 Flow Stress and Work Hardening
j? an orientation factor /?cl angle between primary and cross slip plane £, 8 respectively, macroscopic strain and strain rate r\ correction factor accounting for stress-screening distance of dislocation cp angle subtended by two dislocation segments cpc critical cp, characteristic of obstacle strength , T cylindrical coordinates F, f respectively, microscopic or intrinsic (slip) shear strain and strain rate X structural wavelength A mean free path v Poisson modulus va attack frequency vD Debye frequency Q dislocation density Qf, Qg, Qm, Qt respectively, forest, geometrically necessary, mobile and total dislocation density QD extra dislocation density associated with the presence of grain boundaries or, a respectively, macroscopic stress and macroscopic flow stress t, TC respectively, microscopic (resolved) shear stress and microscopic or intrinsic flow stress (critical resolved shear stress, CRSS) TB background level of TC TJ ideal shear strength T,0 friction stress TP Peierls stress TS saturation stress 0, 9 respectively, macroscopic and microscopic work hardening rate 6f forest term of work hardening #h, 9S respectively, hardening and softening (recovery) terms of work hardening 9T work hardening rate at transition from stage III to stage IV 90 extrapolation of work hardening rate to zero stress b.c.c. CRSS C-S DPM f.c.c. h.c.p. LHR RT SFE TEM
body-centered cubic critical resolved shear stress Cottrell-Stokes dislocation percolation model face-centered cubic hexagonal close-packed latent hardening ratio room temperature stacking-fault energy transmission electron microscopy
2.1 Introduction
relationship
2.1 Introduction 2.1.1
dd/ds = M 2 (di c /dr) + Tc(dM/de)
Scope of the Chapter
The plastic response of crystalline materials at low temperatures is highly sensitive to the applied stress and only weakly sensitive to the strain rate (Frost and Ashby, 1982, Kocks et al., 1975). For a given material with its current structure, a small change of the applied stress level induces an enormous plastic strain rate change, so that a small microplastic transient strain separates the elastic from the elasto-plastic regime. If such transient strain is neglected, a definite yield or flow stress can be ascribed to the material with its current structure. Moreover, under the same circumstances, a marked strengthening of the material is observed by accumulation of plastic strain. Such irreversible change of the flow stress constitutes the work or strain hardening phenomenon. This chapter will be devoted to the flow stress and work hardening of crystalline materials in the slip-dominated plastic regime, i.e., low temperatures and quasistatic conditions (T/TM < 0.3 x 10" 5 , e < lO^" 1 ). If not stated to the contrary, only the "microscopic" or intrinsic components of the flow stress and work hardening will be considered, excluding the orientation or geometrical component of their macroscopic counterparts. For a volume element of a single crystal, the macroscopic flow stress, a, and plastic strain, £, are related to the critical resolved shear stress (CRSS) for current microstructure state, TC, and amount of cyrstallographic slip, T, via an orientation factor, M,
= dr/M
23
(2-1)
so that the macroscopic work hardening rate is related to the intrinsic one by the
(2-2)
where 9 =d
(2-3)
represents what could be called the microscopic or intrinsic work hardening rate of the crystalline element. The second term of Eq. (2-2) is the geometrical component of macroscopic work hardening, arising from the orientation change of the crystalline element associated with the increment of strain. This analysis is unambiguously valid for the simplest case of a test on a single-glide crystal (see Basinski and Basinski, 1979, for some uncertainties in the derivation of 9 for duplex slip situations). For a volume element from a polycrystal grain undergoing a particular uniform local strain increment or, in general, for any crystal under polyslip conditions, it is often satisfactory to use df =^dri9 the total amount of slip in the active systems. When several different families of slip systems are active and a rigorous analysis of the behavior of multiple glide is needed, the definition of intrinsic work hardening becomes much more complicated. The hardening effect on one slip system arising from its activity or from the activity of the other systems has to be tackled with a "work hardening matrix" involving hardening rate terms htj which, in their simplest form are hij = 8(Tc)i/brj
(2-4)
where i and j are any pair of slip systems (Mandel, 1965; Hill, 1966; Hutchinson, 1970; Asaro, 1979; Peirce et al., 1982; Franciosi, 1988). When the macroscopic behavior of a polycrystal is being measured, the analysis for deriving 9 is complex (Kocks, 1970 and 1987; Gil Sevillano et al., 1980; Aernoudt
24
2 Flow Stress and Work Hardening
et al., 1987 and Chap. 3 of this book), but in most cases it is satisfactory to consider an average behavior, = (d(j/de)/M:
(2-5)
where M, the average orientation factor of the aggregate is, of course, changing with progressing macroscopic strain. Because of the intrinsic heterogeneous nature of the plastic strain of crystals at a micro- or submicroscopic level, the assumption of uniformity of strain in the crystal element under study can never be strictly true. However those phenomena associated with work hardening that involve gross departures of strain homogeneity - such as intense shear bands - will not be discussed here. They often involve an essential component of geometrical work hardening or softening, so they are more logically placed in Chap. 3 of this Volume. 2.1.2 The Theoretical Interpretation of Flow Stress and Work Hardening "The current behavior of a material depends only on its current state, which is embodied in its current structure (microstructure, substructure ...). The current rate of evolution (of the state) is one aspect of current behavior". This rather obvious statement for a fixed set of external variables is what Kocks (1987) has called humorously an "article of faith" in relation to the flow stress and the work hardening of a material. Since the advent of the theory of dislocations, it has been known that low temperature plastic deformation (i.e., crystallographic slip) results from the long range movement of dislocations by glide under the action of the applied stress. It is also common knowledge that moving dislocations are lost at the crystal surface or at internal sinks (grain boundaries, interfaces) or by mutual annihilation, but the
loss is often incomplete, giving way to some storage of dislocation segments or loops which are immobile under the current flow stress level. In other words, strain modifies the current structure, increasing the dislocation density. The dislocations, with their surrounding elastic stress-strain fields, partially screen other mobile dislocations from the applied stress, and act as local obstacles to the non-coplanar mobile dislocations that contact them, i.e., the flow stress depends on the current dislocation density. Therefore, what is needed to build up a complete theory of the strength of a material is a flow-stress theory for the current structure (micro- and substructure), coupled with a theory of structural evolution for a given slip increment, i.e., a work hardening theory. This chapter intends to deal with such a coherent understanding of both flow stress and work hardening behavior. However, the most important contributions to the flow stress other than dislocation-dislocation interactions are treated separately in other chapters of this volume. Therefore this chapter - including its section on flow stress - is mainly focused on work hardening behavior. The phenomenon of "work hardening" has drawn the attention of man from very early times (e.g. see Tylecote, 1976). When applied at low temperatures (T<0.3 7^), work hardening can change the strength of ductile materials by two or even three orders of magnitude (Cottrell, 1953; Basinski and Basinski, 1979) and in some instances raise it to absolute values up to one half of the "ideal strength", a{ ~ G/5, where G represents the shear modulus (Gil Sevillano, 1988). It is therefore no wonder that work hardening has been thoroughly studied by mechanical and metallurgical engineers and by physicists. The amount of literature
2.1 Introduction
on the subject is consequently extremely large but, fortunately, extensive reviews are available (see General Reading), as well as compilations of stress-strain curves in handbooks for engineering use (e.g., Boyer, 1986). The reader is referred especially to those works by Nabarro et al. (1964), Hirsch (1975) and to Basinski and Basinski (1979) for a very detailed account of the subject, particularly with reference to the behavior of single crystals, and to acquire a good historical perspective. The number of internal and external parameters influencing the work hardening behavior is also large: crystal structure, crystal orientation or texture, stacking fault energy (SFE), chemical composition, geometry and size of the microstructure and current dislocation substructure are amongst the former, and temperature, strain rate, deformation mode, previous deformation history, size and surface to volume ratio of the sample amongst the latter. Such long lists, and the large quantity and variety of literature can be rather discouraging. It is, however, to be recognized that the phenomenon of work hardening is almost universal with a general pattern common to very different materials. This chapter will mainly deal with the general pattern of behavior, only underlining, when necessary, the particularities of specific families of materials. 2.1.3 Phenomenology of Plastic Stress-Strain Curves: Deformation Stages
Macroscopic work hardening curves of technical materials are normally fitted, for engineering use, to simple phenomenological equations. Extrapolations beyond the experimental strain range used for fitting should be made with great care, as it has been well known since the fifties that crys-
25
tals show several deformation or hardening stages. A deformation stage is a strain range where the flow stress or the substructure evolution are controlled by a particular set of micromechanistic processes. In fact, as the activity of plastic mechanisms varies with temperature, strain rate, etc., every stage constitutes a regime of work hardening behavior in the strain/strain rate/temperature field. The existence of three successive stages (Fig. 2-1), mainly inferred from the huge amount of previous work on f.c.c. single crystal plasticity, was acknowledged around 1955 (Diehl et al., 1955; Friedel, 1955; Diehl, 1956; Seeger et al., 1957). The classic scheme of Fig. 2-1 is representative of the behavior of a single crystal of high purity material with medium or high stacking fault energy (SFE), starting its deformation under geometrical conditions of single glide and rotating towards multiple glide. After a short microplastic transient strain, stage I or the "easy glide" stage, which is only present under single glide conditions in high purity crystals, sets in. During a strain interval influenced by a large number of internal and external variables, the crystal shows a weak, approximately constant work hardening rate, 91 which is rather insensitive to the external variables T, f; 2xl(T4<0I/G<2xl(r3 Before reaching a double or multiple slip orientation, the strain hardening rate dramatically increases and the crystal enters stage II, characterized by a linear stressstrain curve whose relative slope 3xl(T3< 0n/G<6xl(T3 is almost insensitive to the material, its internal variables or the external ones of the test. This athermal hardening stage ends when a critical stress, Tm, which is T, f and
26
2 Flow Stress and Work Hardening (a)
STRAIN-
0-05
0-1
0-15
0-2
0-25
0-3
0-35
Figure 2-1. (a) The stages of work hardening after Diehl (1956) (schematic), (b) Actual stress-strain curves of copper single crystals tested in tension (Diehl, 1956).
SFE dependent, is reached. For high enough T and SFE the linear stage II can be totally absent. Beyond TIII? 9 steadily decreases (at a faster rate for higher T or lower f) during "stage III", which used to be termed the "parabolic hardening stage", in the belief that the physical work hardening law was parabolic (Taylor, 1934; Bell, 1964, 1965; Kuhlmann-Wilsdorf, 1968). (2-6)
In fact, neither stage II is perfectly linear nor stage III truly parabolic. It was later realized that the strain hardenability of the materials in stage III decreases faster than parabolically (Kocks et al., 1968) and that in most cases a better fit to the whole stage III stress-strain curve was achieved by a Voce-type equation (Voce, 1948; Kocks, 1976) _Ti)
=
e x p [ - ( r -F
(2-7)
2.2 The Flow Stress
which contains four material parameters zx or Fx, F2 and T"1 (only three of which are independent). This equation has the feature of leading asymptotically to a saturation stress, TSHI, in contrast to the assumption of parabolic behavior, Eq. (2-6). The grains in a polycrystal deform under the constraints imposed by the elastoplastic compatibility with their neighbors. As a consequence, the individual crystallites activate more than one slip system immediately beyond the microplastic regime (polyslip conditions, see Chap. 3 of this Volume), and stage I is always absent from their stress-strain curves. The overwhelming majority of information on work hardening has been obtained from the results of tension tests. Interest in the large strain behavior, well beyond the maximum useful strains provided by tensile tests, strongly increased late in the 1960s, and several seminal works of this period combined testing techniques obviating the shortcomings of tensile and compressive tests and extending the stressstrain curves to very large strains, with substructural observations (e.g., Embury et al, 1966; Langford and Cohen, 1969). At the end of the next decade it was clear that a new work hardening regime, "stage IV" should be added to the three classical ones (Gil Sevillano et al., 1980). Stage IV is characterized by a thermally dependent, very low but persistent work hardening rate that delays the attainment of a saturation stress to much higher values than T™. The actual functional form of the stage IV stress-strain curve is in some cases nearly linear, but 6lY is in general a decreasing function of strain that could well be described by a Voce equation (Aernoudt et al., 1987) apart from the final tail, where other new stages are not excluded (Brion
27
etal., 1981, 1986; Siethoff and Schroeter, 1984).
2.2 The Flow Stress The process of plastic deformation by dislocation glide is self-sustaining, because dislocations are able to multiply under the application of the flow stress. Only in some well characterized cases (e.g., yield point or Portevin/Le Chatelier phenomena, see Neuhauser and Schwink, Chap. 5 of this Volume) is the release or multiplication of dislocations critical for the flow stress. In most cases, the flow stress is controlled by the propagation of dislocations through the structure of the material. Structure is taken here in its broad sense, including the crystallographic lattice with its periodicity, its point defect contents, extraneous atoms in solution, dislocation density and spatial distribution (the "substructure"), grain and twin boundaries, and second phases with their associated interfaces (the "microstructure"). The CRSS or "microscopic flow stress", xc, is the resolved shear stress to be applied in order to sustain the long-range propagation of dislocations under the superposed effects of all coexisting structural features, which represent obstacles of differing strenghts resisting dislocation glide. Actual values of the flow stress of real materials range between two absolute bounds. The upper bound is represented by the "ideal strength" for a shear collapse of the perfect lattice, x{. The lower bound, by the Peierls-Nabarro stress, TP, the stress for dislocation glide in the absence of any other lattice imperfection. Any kind of obstacle to dislocation movement raises the flow stress above the unavoidable PeierlsNabarro stress, which well deserves its denomination of lattice friction stress.
28
2 Flow Stress and Work Hardening
2.2.1 The Ideal Shear Strength: Dislocationless Flow
overcome before reaching any other, the strain increment per obstacle is:
The resolved shear stress for homogeneous crystallographic slip - i.e., elastic collapse of the lattice - is a high fraction of the elastic shear modulus, G. According to theoretical estimations of its value (Macmillan, 1983; Kelly and Macmillan, 1986),
dFobs « bLbu
G/30
G/10
(2-8)
Values close to the upper level have been experimentally observed in single-crystalline whiskers of b.c.c. (Fe, W), high SFE f.c.c. (Ni, Al) and some h.c.p. metals (Cd). The lowest xi value given by (2-8) corresponds to medium and low SFE f.c.c. metals. The ideal strength for homogeneous lattice instability is essentially athermal, scaling only with the temperature dependence of the shear modulus. The whiskers' real strength may show a higher sensitivity to temperature, because of the possibility of failure from the thermal nucleation of dislocation loops from the free surface (Friedel, 1965; Kelly and Macmillan, 1986). 2.2.2 Obstacle-Limited Dislocation Glide Kinetics A surface increase, ds, of a dislocation loop gliding on its slip plane in an element of unit volume gives way to a slip increment: = bds
(2-9)
The presence of structural obstacles hinders the dislocation motion, controlling its velocity, v. Let us assume that under the applied resolved shear stress, T, there is a length Qm of mobile dislocation lines per unit volume which, on the average, are restrained by one obstacle every L units of length. If 8u is the distance moved by a dislocation segment when one obstacle is
(2-10)
There are Qm/L obstacles per unit volume in contact with the dislocation. In quasistatic conditions, the time spent by the dislocation in reaching a new obstacle when freed from a previous one can be neglected if compared to the mean waiting time at the obstacles, 8t. Accordingly, v ~ 8u/8t is the effective dislocation velocity which produces a slip strain rate, f = bgmv
(2-11)
i.e., the well known Orowan equation. Above 0 K, the energy barrier that each obstacle represents for the dislocation can be overcome with the help of thermal activation. In the simplest case, the frequency of obstacle passing is given by v = va exp [- (AF - T Va)/(kT)]
(2-12)
where va is a characteristic attack frequency, AF the free-energy increment for attaining the critical configuration of releasing the dislocation from the obstacle and (T Va) the fraction of that energy provided by the applied stress. The "activation volume", Va, is an effective volume equal to Va~bL5ua
(2-13)
where 8wa, the activation distance, is the dislocation displacement for reaching the activated state. The attack frequency is given by va = vDb/L
(2-14)
where vD is the Debye frequency (1012 to 10 13 s~ 1 ). Introducing reasonable values for the different parameters which enter in Eqs. (2-11) to (2-14), it can be seen that the thermal motion contribution in the activated glide of dislocations will be about 20 kT to 30 kT (-0.6 eV at RT).
2.2 The Flow Stress
The contribution of the external stress to the activation energy, T Fa, is expected (and observed) to be approximately independent of strain or stress in a deformation regime at constant temperature and strain rate (Nabarro, 1986). This stems from the proportionality between the flow stress and the inverse of the interobstacle distance L (Sec. 2.2.5). Thus the value x Va represents a sort of signature of the micromechanism responsible for the flow stress which can be derived from strain rate changes at constant structure and temperature, = kTd(lnr)/d(lnT)
(2-15)
Figure 2-2 shows an example of x Fa vs. F curves of copper single crystals with indications of the three conventional stages. The transition from stage I to stage II is apparent in these curves, but no noticeable deviation is observed at the transition from stage II to stage III. It appears that the flow stress is determined by the same mechanism in stages II and III. An extrapolation of the kinetic equation to 0 K leads to the athermal limit of the flow stress for the current structure, where the obstacles have to be overcome by an exclusively mechanical contribution (2-16)
29
The kinetics of obstacle-controlled glide (low temperature plasticity) have been only briefly sketched here. The reader can find detailed accounts of this subject in Kocks etal. (1975), Cagnon (1979) and Schoeck (1980). It is to be noted that Eq. (2-12) is strictly valid only for a regular array of equal size circular obstacles of constant adverse internal stress. For other strength profiles of obstacles, and allowing for nonrandom distributions on the slip plane, the general equation is (Kocks et al., 1975):
t = f0 exp
kr
(2-17)
0
473 K
Figure 2-2. Curves of x Va vs. F for a copper crystal at various temperatures. The conventional work hardening stages are indicated. Adapted from Nabarro (1986).
30
2 Flow Stress and Work Hardening
overcome the potential barrier between two neighboring stable equilibrium positions, energy must be provided as work under the action of the applied stress. Such energy per interatomic displacement, determinant of the Peierls-Nabarro stress, TP, depends obviously on the type of atomic bonding in the crystal, responsible for the interatomic forces, and in the atomic structure at the dislocation core. The value of TP increases strongly as directionality of the atomic bonding increases, being minimum for f.c.c. metals and maximum for covalent crystals. Experimental order of magnitude values of t p (extrapolated to OK) are shown in Table 2-1 (Frost and Ashby, 1982; Takeuchi and Suzuki, 1989). The lattice resistance of f.c.c. metals is negligible in the macroplastic regime. The TP level for b.c.c. and for h.c.p. prismatic slip can represent a high fraction of their flow stress at low temperatures. The observed activation energy for their lattice barriers (~0.5 eV) and the activation distance (~ b/2) are low, so that TP is strongly temperature and strain rate dependent. On the other hand, the Peierls stress of covalent crystals is so high that long range dislocation glide in their lattices can only take
place at high temperatures (AF ~ 5 eV). Ionic crystals constitute an intermediate case. Above 0 K, the dislocation does not overcome the lattice barrier simultaneously all along its line. Thermal fluctuations help to nucleate local bulges that place short dislocation segments into the next lattice energy valley. The segments are bounded by a pair of opposite sign kinks that can move sideways under the applied stress, away one from another. Kinks of opposite sign from neighboring bulges mutually annihilate when they meet along the line. The nucleation and propagation of such kink pairs has been thoroughly discussed by Kocks et al. (1975). The PeierlsNabarro stress component of the flow stress appears to be controlled by the nucleation of these double kinks, and its temperature and strain rate dependence is well described by the phenomenological Eq. (2-17) with p = 3/4, 4 = 4/3. The classical semi-discrete interpretation of the Peierls-Nabarro stress (Nabarro, 1967; Hirth and Lothe, 1968) highlighted, at least semi-quantitatively, many of the characteristics of the core effects on dislocation behavior. Their calculation showed r p /G = [2/(1 -v)] exp(- 2TI w/b)
Table 2-1. Experimental order of magnitude values of T (extrapolated to 0 K) System
Approximate
f.c.c. metals and basal slip in h.c.p. metals
<10~ 5
b.c.c. metals, prismatic slip in h.c.p. metals and slip in noncompact planes in f.c.c. metals 5 x 10" 3 Ionic crystals: Alkali halides Oxides
10~2->2xl0~2 10~2 -+ 3xlO~ 2
Covalent solids
2x10"
•5x10"
(2-18)
with w the dislocation width, w = h/(i — v), h being the lattice spacing of the glide plane. Equation (2-18) can explain the selection of active slip planes, the high sensitivity of TP to w, the high mobility of partial dislocations when their Burgers vectors remain in the slip plane (e.g., Shockley dislocations in f.c.c), or the tendency of dislocations with high TP to remain straight in the potential valley along the more dense directions. Experimental (T P ) 0 values obey Eq. (2-18) within one order of magnitude range (Fig. 2-3). The classical treatment is
2.2 The Flow Stress
31
10°
10'
2
a.
Figure 2-3. Experimentally obtained i p /G values vs. h/b values. NaCl(l) and NaCl(2) denote respectively the primary and the secondary slip system of NaCltype ionic crystals. Solid line is Eq. (2-18) with v = 0.3. From Takeuchi and Suzuki (1989).
MgT^-Csl.CsBr
10
s
io- 6
0
0.2
0.4
0.6
0.8 h/b
1.0
1.2
1.4
1.6
however limited to a two dimensional core spread on the slip plane of the dislocation, and cannot be applied to other important situations such as the three fold symmetry of the core spread on planes intersecting the <111> zone of the screw dislocations in b.c.c. metals. During the last twenty years, dislocation core effects on plasticity have received much attention and have been tackled by atomistic computer simulations. Extensive reviews are already available (Puls, 1981; Christian, 1983; Vitek, 1985; Veyssiere, 1988; Duesbery, 1989).
ex: fee metals, hep on basal plane.
G
c
\
ex: bcc, prism., Te, S Si Pt3AL, garnet
1 (a)
ex: (a) Be, spinel, (P)B2,L1 2 ,L1 0
6
c3
(b)
S
A
; \
IVJ
As could be anticipated, core effects have a strong influence on work hardening behavior as well as on flow stress of crystals (Sec. 2.3.5). Symptoms of core-controlled plasticity are: pronounced dependence upon temperature of the CRSS, and deviations from the Schmid law (anomalous choice of active slip planes, orientation influence on CRSS, influence of non-glide stress components, slip asymmetry). Figure 2-4 summarizes schematically the distinct dependences of the microscopic flow stress upon temperature that can be found in
\
y
(c)
•
Figure 2-4. Schematic view of the distinct dependences of the CRSS upon temperature that can be found in crystals and the corresponding dislocation energy diagrams, (a) CRSS is temperature insensitive, (b) The sessile core (S) is stable and there is no metastable glissile core (G): the CRSS shows a pronounced negative sensitivity to temperature, (c) There are two metastable cores and two different CRSS regimes depending on the nature of the stable core at low temperature (a) sessile core, (p) glissile core: in both cases, the CRSS reaches a peak value at intermediate temperature (Veyssiere, 1988).
32
2 Flow Stress and Work Hardening
crystals depending on their dislocation core spread possibilities: glissile (in the glide plane) or sessile (out of the glide plane). A glissile-sessile transition in the core spread behavior can produce an anomalous temperature dependence of the flow stress (Fig. 2-4 c).
nates, r, (j)) Tij = Gbf(cl))/(2nr)
(2-19)
In particular, the maximum interaction stress between two parallel edge dislocations gliding on parallel planes separated a distance h, is
= Gb/[Sn(l-v)he]
(2-20)
2.2.4 The Background Internal Stress
A structured material undergoing plastic strain develops a non-uniform field of long range internal stresses that will influence the motion of mobile dislocations. The internal stress level will oscillate with a wavelength linked to the structural feature responsible. In fact, the internal stress pattern will be the result of the interference of the fields associated with all the structural features present in the material, giving a background with a wavelength k{ that can be very large compared with that of other much more localized obstacles, i.e., L<^X{. The stress barriers from localized obstacles may be overcome by thermal activation, and the background stress pattern will behave as quasi-athermal. The background will then contribute with an additive term to the flow stress, i.e., with an effective back stress experienced by the mobile dislocations. The magnitude of such apparent back stress levels has been discussed by several authors (Chen et al., 1964; Li, 1968; Argon, 1968; Kocks et al., 1975). It will always be smaller than the peak level of the background internal stress and is usually estimated to be very low. A particular source of internal stresses in a crystal is the presence of stored and mobile dislocations. In fact, this contribution was assumed to be the fundamental component of flow stress of work hardened crystals in the early days of dislocation theory (Taylor, 1934). The stress field around a dislocation line is (cylindrical coordi-
When the dislocation density is Q, the distance between any moving dislocation and the nearest one is, on the average, hec~ 1/(2 y/g), with the dislocation only moving through the crystal under an applied stress over (2-21) The above equation is valid on the assumption of a screening of the stress field of an individual dislocation by the rest of the dislocation density beyond the average dislocation spacing £~ 1/2 , a correct assumption only for perfect random dislocation distribution (Wilkens, M., 1969; Kocks and Scattergood, 1969). Many variants, more or less elaborate, of Taylor's original theory are possible. If it is assumed that dislocations are grouped into high energy configurations [e.g., pileups of n dislocations separated, on the average, (Q/n)~i/2], the strengthening contribution of the interaction stress will be higher than that given by Eq. (2-21), by a factor yfn for pileups. Pileup based flowstress theories were developed by Mott (1952), Friedel (1955), and with particular detail by Seeger and coworkers (Seeger, 1957, 1958, 1968; Seeger et al., 1957). They were debated at length (Nabarro et al., 1964; Hirsch and Mitchell, 1967, 1968; Hazzledine, 1967; Hazzledine and Hirsch, 1967). From the theoretical point of view, the long-range stress contribution of pileup groups to flow stress was shown to
2.2 The Flow Stress
be much lower than the previously assumed passing stress, so that pileup based flow stress theories have now lost much of their previous support. Besides theoretical criticisms, pileup based flow stress theories do not find the necessary support from TEM observations of dislocation pileups for a quantitative agreement with the experimental flow stress level (Mughrabi, 1975). The dislocation arrangements were shown to be rather complex, even for stage II when single glide conditions were still prevailing. The long range stresses of large planar arrays of primary dislocations were partially relaxed by secondary slip, and when unrelaxed primary pileups were observed the number of dislocations in them was not sufficient to explain the flow stress according to the long-range stress theories. Furthermore, the density of secondary dislocations was found to be comparable to that of primary dislocations, suggesting that the forest contribution to flow stress should not be overlooked. For the cellular dislocation structures of stage III, the pileup based theories seemed hardly applicable. On the other hand, internal stresses were shown to be always present except in stage I. However (see Sec. 2.2.8.3) recent developments have shown that such stresses can be ascribed to other sources much more plausible than long pileups: plastic strain gradients of different types (Kocks et al., 1980; Mughrabi, 1983; Ungar et al., 1984; Pedersen et al., 1981; Jackson, 1985 a) or even subgrain boundaries (Nix et al., 1982; Gibeling and Nix, 1980; Argon and Takeuchi, 1981; Morris and Martin, 1984). For low-energy dislocation arrays the associated internal stresses will be correspondingly weaker. For example, for an edge dipole of height h, the maximum interaction stress with a similar edge dislocation gliding on a plane at a distance r > h
33
from the dipole is T~Gbh/[$n(l-v)r2]
(2-22)
On the average, a gliding dislocation would find the nearest dipoles at a distance 1/(2^/^/2) if all the stored dislocation density was present as randomly distributed edge dipoles, so that /~ 93) TB^{(h/r)/[4n^2(l-v)]}G i.e., a very low contribution compared with Eq. (2-21). Negligible effects are expected (and indeed observed) when dislocations are arranged in dipolar clusters or multipoles, as in stage I substructures (see Sec. 2.3.4.5). Effective interaction stresses occur then only for r~h where the short range field, Eq. (2-19) prevails. 2.2.5 Point-Like Obstacles Even if the applied resolved shear stress overcomes the lattice friction stress and the possible internal stress level in the crystal volume element under consideration, the long range stress movement of a dislocation can be hindered by the presence of discrete obstacles distributed on its slip plane. Such obstacles can be of great variety. Any disturbance of the perfect crystal lattice - extraneous atoms, other dislocations pinching-off the slip plane, boundaries of any type, precipitates, dispersoids or second phases, etc. - will represent an energy barrier for the dislocation line. Energy must be supplied from the applied mechanical stress helped by thermal fluctuations. When the interaction distance of the obstacles with the dislocation line (the effective obstacle size) is small compared with the distance between neighboring obstacles, they can be treated approximately as point-like. When the dislocation line is pinned by point obstacles, it must be bent between
34
2 Flow Stress and Work Hardening
them - by raising the applied resolved stress - so that the dislocation segments become curved with a radius R, R = TJ(T b)~G 6/(2 T)
(2-24)
where Td is the line tension of the dislocation, equal to the dislocation line energy £ d , in the simplest isotropic fixed value approximation. The dislocation then exerts an effective force on the obstacle, (2-25)
F = 2Edcos((p/2)
cp being the angle subtended by the two segments. The dislocation escapes from the obstacle when the force F equals the obstacle strength, Fobs. At the critical point, 2) = Fobs (2-26) so that the obstacles in a crystal can be characterized by a critical angle cpc. The maximum force on the obstacles is Fmax = 2Ed^Gb2
(cp = O)
(2-27)
The strength of real extensive obstacles is size dependent [Fobs ~ (i c ) obs b t, with (i c ) obs the mean effective obstacle shear strength and t the obstacle size] so that some obstacles are non-penetrable for the dislocations, i.e., Fmax < Fobs. For 0 < cpc < n, the obstacles are sheared. For Fmax < Fobs5 the dislocation circumvents the obstacle, leaving around it a dislocation loop (the Orowan mechanism). Therefore strong obstacles (cpc = 0) always contribute to work hardening. The flow stress level for point-like obstacle controlled glide depends not only on their strength, but also on their spatial distribution. If they were regularly distributed in a square array of side L on the slip plane surface, a very unlikely situation, the flow stress from their exclusive contribution would be, from Eqs. (2-24) to (2-26) Tc = (Gb/L)cos(cpJ2) = Gb ,/A^cos(q>J2) (2-28)
where JVA is the obstacle areal density. The obstacles are rarely distributed in regular arrays. Departure from regularity means non-homogeneities of areal density, with softer areas of lower than the mean density. Under stress, the dislocation line can take advantage of such density fluctuations, finding percolation paths through the slip plane under stresses lower than those given by Eq. (2-28). The problem has been tackled repeatedly, both analytically (Kocks, 1966, 1967; Hanson and Morris, 1975; Dorn etal., 1969; Labusch, 1977) and by numerical techniques (Foreman and Makin, 1966, 1967; Morris and Klahn, 1974; Hanson and Morris, 1975, 1978; Labusch and Schwartz, 1976; Altintas, 1978) for random distributions of point obstacles, both for the athermal limit of the flow stress and for thermally activated slip. Deviations from Eq. (2-28) can be very important and are sensitive to obstacle strength. Weak obstacles (cpc ~ n) can be overcome by a dislocation line by successive release of consecutive obstacles along the only slightly curved line ("unzipping"), a mechanism foreseen by Friedel (1956) and confirmed by computer experiments (Fig. 2-5). For endless glide of the dislocation through the glide plane (the percolation threshold) the dislocation has to find (on the average) no more than one obstacle in the area swept after being released from a previous obstacle, i.e., (2-29) Computer simulations show this equation to be valid for q>c > 100°. On the other hand, with strong obstacles (cpc ~ 0), the severely curved dislocation emits fingers searching for low density gates of easy passage (Fig. 2-6) up to the percolation point allowing for long-range slip. Computer simulation results for strong
2.2 The Flow Stress
Figure 2-5. Computer simulation. Some successive equilibrium positions of a dislocation line under increasing stress. Random array of 9000 point obstacles, cpc = 160°. (From Jimenez, 1982.)
obstacles are very well described by T=0.8G6
V
fAcos(
An important consequence of the penetration of the dislocation along easy paths of
35
lower than the mean obstacle density is that hard areas or islands on the slip plane are surrounded by Orowan loops once the percolation threshold has been attained. This means that the density of obstacles of the structure increases after the passage of dislocations, i.e., non-regular arrays of strong obstacles contribute to both the flow stress [through Eq. (2-30)] and to work hardening. For randomly distributed points, the critical obstacle strength for storing concave dislocation loops along hard areas is cpc~90o (Vetter et al., 1982; Jimenez Ibanez, 1982). In many instances, the strength of the obstacles is not uniform and frequently several types of obstacles (i.e., solid solution atoms, fine precipitates, forest dislocations) are simultaneously present on the slip plane. The problem of superposition has been frequently debated. There is agreement on two extreme cases (Kocks et al., 1975). When strong obstacles are intermingled with a higher density of weak obstacles, additivity of the two individual contributions is observed; 7. weak
strong
(2-31)
In other situations and particularly for mixtures of obstacles of intermediate strength, a quadratic superposition rule is more convenient, T? = T? + T | + ...
Figure 2-6. Computer simulation of dislocation glide through a random distribution of 9000 strong point obstacles, cpc = 0 (Jimenez, 1982).
(2-32)
An excellent review of the strengthening effects of point-like obstacles was published by Brown and Ham (1971) and the reader is referred to it, as only a simple sketch of the subject has been presented here (see also Ardell, 1985). Particularly when a broad range of obstacle densities is compared, a more precise value of the dislocation line tension has to be used, leading to a deviation of the TC OC y/N^ proportionality. For the isotropic, mid edge/screw
36
2 Flow Stress and Work Hardening
approximation,
(r>
~~,
2
Ed - {Gb (l-v/2)/[47i(lwith appropriate cut-off radius, ro = b R = L = L[cos(cpc)]-1/2,
("Friedel distance") (2-34) o X
to the formation of a pair of jogs or kinks, each requiring an energy supply of the order of Gb3/(4n). The energy supply during the cutting of repulsive trees is estimated to be more or less of this order of magnitude. The work done when cutting the tree is ^T^Vy/^, with )S>1 on account of the non-regularity of the location of trees and of the actual interaction distance, a small multiple of b; gf is the forest density (gf = Q/2 for a random distribution of dislocation lines in the volume, Q being the total dislocation density). This estimation (Nabarro et al., 1964) leads to an equation of the general form Tc = OiGb ^JQ
2.2.6 The Dislocation Forest Contribution to the Flow Stress
Long-distance glide of a dislocation through a crystal implies encounters with a "forest" of other non-coplanar dislocations piercing the slip plane. Depending on the Burgers vectors of the mobile dislocation and the trees, the moving dislocation line will be repelled or attracted at the proximity of the trees. The dislocation may only continue to glide freely if the forest is cut, overcoming local repulsions or detaching the line from local attractions. Repulsive trees are cut more easily than attractive ones (Carrington et al., 1960; McLean, 1962). One type of interaction is obtained from the other by simply reversing the sign of one of the dislocations. If their lines were absolutely rigid, equal strength should be shown by the two types of interaction. However, the dislocations bend as they approach each other. For repulsive interaction they bend away, decreasing the interaction. For attractive interaction they bend closer, increasing the attraction, and join forming two threefold nodes connected by an extended dislocation segment (Fig. 2-7). The cutting leads
(2-35)
similar to that derived for the effect of diffuse long-range stress fields, Eq. (2-22), or for interaction with point-like obstacles, now with a < 0.1 (equivalent to q>c ~ 146° on the Friedel approximation), i.e., weak obstacles. With the interaction distance being also short, cutting of repulsive trees will be a thermally activated process. Detailed calculations of repulsive tree strengths are not easy and the value Gb2/ (2%) for their point-like obstacle force
Figure 2-7. Attractive dislocation intersection (schematic).
2.2 The Flow Stress
could be an underestimation (Shetty and Hirth, 1971). However, there is a general consensus on the minor importance of their contribution to the flow stress in most situations. On the other hand, the energy saved by forming the attractive junction can amount to hundreds of eV and the junction has to be undone by the applied stress. Consequently, most of the attractive trees of the forest give a quasi-athermal contribution to the flow stress that, as it will be seen, represents the main part of dislocation strengthening in most situations. An exact calculation of the strength of the attractive junctions is quite complicated. There are twelve parameters describing a specific function (Hirth and Lothe, 1968) so that simplifications are unavoidable. For any spatial position of the pinning points of the tree and the mobile dislocation segment, the strength is calculated determining the equilibrium configuration of the junction by using virtual displacements of the nodes. At any moment, the increment of line energy of the configuration, d£ d , for a displacement increment, dx, is equated to the work done by the stress, dW
37
(a < 1). Also, the energy for jog or kink formation only increases the junction resistance for this case, where the peak stress occurs when the two nodes come together (Fig. 2-8). Only such attractive junctions will contribute to the thermal dependence of the flow stress, together with the repulsive trees (see Sec. 2.2.6.4). The individual process of attractive interaction, first suggested by Hirsch (1958) as a possible strengthening mechanism, was studied by Saada (1960 a, 1960 b, 1963), Carrington et al. (1960), Baird and Gale (1965) and later on, thoroughly, by Schoeck and Frydman (1972) and Piischl et al. (1982). The strengthening ability of a forest depends strongly on the crystallographic structure of the material and on the geometrical characteristics of the configuration of the meeting dislocations. 2.2.6.1 The F.C.C. Forest
Schoeck and Frydman (1972) explored in detail the tree strengths of f.c.c. forests, and their work was complemented by Piischl et al. (1982). For a random f.c.c. forest with tree length equal to tree separation, only about 20% of the trees
(
The flow stress follows a relationship with the total dislocation density similar to Eq. (2-22) or (2-35). As stressed by Nabarro etal. (1964), almost any dislocation-dislocation interaction mechanism leads to the proportionality T OC Gb ^JQ, that stems from dimensional analysis. The junction rupture occurs by reabsorption of the segment which links the two threefold nodes, or when one of the mobile dislocation arms becomes unstable under the applied stress. Only in the former case does the attractive junction behave as a point-like obstacle in terms of strength
jog or kink
Figure 2-8. Attractive dislocation intersection. Stress vs. node separation (including the final effect of creating a jog or kink) for two junctions of different strength (schematic).
38
2 Flow Stress and Work Hardening
form junctions (15% with mobile edge dislocations, 24% with mobile screws). The density function of reactive trees versus strength of such forest is reproduced in Fig. 2-9. The calculations assume a line energy E edge = 0.82 Gb 2 , corresponding to an outer cut-off radius R = 103b in Eq. (2-33), equivalent to a total dislocation density 1.5 x 10 13 m~ 2 for the random forest. Accordingly, the calculated strengths have to be multiplied by a factor r]=
(2-37)
-
for large departures from the reference density, with R = 1/^/Q = L (Foreman, 1967; Brown and Ham, 1971). The flow stress contribution of the random f.c.c. forest was estimated by Schoeck and Frydman, using a quadratic averaging procedure as an extension of Eq. (2-31), as a = 0.3. After the complementary calculations of Puschl et al. (1982), the final estimation is
Equation (2-38) compares very well with experimental results for f.c.c. metals, particularly when strained under polyslip conditions, i.e., with dislocation structures containing a multiplicity of Burgers vectors comparable in that respect to the random forest assumed by Schoeck and Frydman. Figure 2-10 reproduces a compilation of measures of Q VS. TC in Cu and Ag for different structures going from stages I to III and spanning 6 orders of magnitude of dislocation density. The linear regression of the experimental data corresponds to 0.425
(2-39)
T OC Q
equivalent to Eq. (2-38) with a = 0.40 (Basinski and Basinski, 1979). Accordingly,
TC (random f.c.c. forest) ~ 0.36 77 Gb with rj given by Eq. (2-37).
a)
Edge Screw FCC, random forest
1
,1 0.3
i \ ,' \
I 11 0.2
/ /
1 1 1 »
» /
/
0.1
> /
k=1
\
!
\ \
0.1
1
10
Resolved Shear Stress kg/mm 2
Figure 2-9. Density of reactive trees versus strength for a random f.c.c. forest with tree length equal to tree separation (K = 1) and Eedge = 0.82 Gb2. From Schoeck and Frydman (1972). The critical stress for edge and screw Frank-Read sources are indicated in the abscissa.
Figure 2-10. Compilation of forest dislocation density vs. flow stress data of Cu and Ag (TC corrected for comparison with Cu) reproduced from Basinski and Basinski (1979). The regression line corresponds to Eq. (2-39). Different symbols refer to different references, see original paper for a complete explanation.
2.2 The Flow Stress
the theoretical estimation based on the attractive forest appears to explain 90% of the observed flow stress and the agreement is even better if only multiple slip data is considered (stages II and III, TC > 10 MPa). The deviation for £ > 1 0 1 4 m ~ 2 could be due to a bias in TEM measurements due to dislocation overlapping. Another independent extensive averaging of TEM data (Mecking and Kocks, 1981) for copper yields a similar agreement with Eq. (2-38) if the proper transformation from areal to total dislocation density is performed. However, individually measured a values are frequently observed to deviate from those average values. For instance, precision a measurements of different f.c.c. metals deformed under polyslip conditions (symmetrically oriented single crystals or polycrystals) range from 0.2 to 0.4 (AlHaidari et al., 1983; Doner et al., 1974; Essmann et al, 1968; Gottler, 1973; Hansen, 1977; Staker and Holt, 1972; Ambrosi etal., 1980). Some deviations from Eq. (2-38) can be theoretically predicted. This equation corresponds to a random forest with a ratio of tree spacing to tree length K~\. The strength of the junctions is sensitive to X, particularly for K <^ 1 (very strong forest) and to a lesser extent for K > 1. Strong changes of K can be imagined to occur in latent-hardening experiments when a secondary slip system is activated after a previous stage I deformation of the crystal (see Sec. 2.2.6.3). Another prominent source of numerical discrepancy with Eq. (2-38) could arise from the non-homogeneous spatial distribution of dislocation density. Dislocation structures tend to be clustered, forming lamellar, cellular or subgrain arrays, their heterogeneity increasing with increasing strain. The random forest hypothesis may hold locally (in dislocation
39
walls or in cell interiors), but the local dislocation density may be significantly larger or smaller than the average dislocation density. The deformed crystal can be considered as a composite and its strength calculated according to the "rule-of-mixtures" using Eq. (2-38) with the appropriate parameters for each composite region (see Sec. 2.2.9.3). Alternatively, instead of considering independently the deformation of the low and high dislocation density region of the composite, the long-range glide of a dislocation on a slip plane with non-homogeneous density of obstacles can be considered. Whatever approach is taken, the effective a value calculated for the average dislocation density will differ from that given by Eq. (2-38). It is easy to show that, for heterogeneous dislocation distributions, the flow stress calculated with the "rule of mixtures" assuming the same a for each local region is always lower than the flow stress calculated from the mean dislocation density, i.e., assuming a homogeneous dislocation distribution (Takeuchi, 1970; Mughrabi, 1987). On the other hand, computer experiments of dislocation glide through a large number of point and linear obstacles, mimicking the strength spectrum of Schoeck and Frydman's random forest but with non-homogeneous spatial distributions representative of the different types of clustered dislocation substructures produced by work hardening, have also been performed (Hernandez and Gil Sevillano, 1987). They predict 0.20 < a/rj < 0.37 in the ordered sequence random forest -> cellular arrays -> subgrains. However, the correction factor rj decreases as clustering increases, so that it will tend to counteract the effect of clustering in the computed a/rj sequence. The discrepancy with the random forest a value estimated by Schoeck and coworkers can be ascribed to their ap-
40
2 Flow Stress and Work Hardening
proximate averaging procedure and to the inexact reproduction of the strength spectrum of the f.c.c. forest in the computer experiments. For very large strains (stage IV, Q > 1015 m~ 2 ), direct measurement of dislocation densities in TEM micrographs is no longer possible. As dislocations are then mainly arranged in low-angle boundaries, a lower-bound estimation of Q can be obtained from measures of their surface per unit volume and average misorientation, i.e., neglecting the redundant dislocation density (Gil Sevillano et al., 1980; Gil Sevillano and Aernoudt, 1987). Thus values about a ~ 0.25 have been obtained for Al or Cu subjected to large strains, in close agreement with the prediction of Eq. (2-38) for £~10 1 5 cm~ 2 . To summarize: a large amount of experimental data from f.c.c. metals supports the validity of Eq. (2-38) both qualitatively and semiquantitatively throughout six orders of magnitude of increase in dislocation density, and without distinction of deformation stages. Deviations from the average a value can be attributed to anisotropy and non homogeneity of the dislocation substructure; differences in the composition of the forest; and the participation of other contributions to the flow stress, particularly the background long-range stress component, Eq. (2-21) and the weak contribution of repulsive trees (predictions of forest hardening usually underestimate the measured flow stress). The importance of the forest contribution to the flow stress, championed by Basinski (1974, Basinski and Basinski, 1979) is reinforced by other arguments: quantitative agreement between theoretical and experimental a values is also found for lattice structures other than f.c.c. on the basis of the strengthening effect of attractive junctions, which also explains convinc-
ingly the results of latent-hardening experiments as well as several aspects of the observed temperature and strain rate dependence of the flow stress. These aspects are treated in the following subsections of this chapter. 2.2.6.2 Other Lattice Structures
Theoretical or numerical simulations of the strength of the attractive part of the dislocation forest have been performed for crystallographic structures other than f.c.c. with a similar conclusion: the strength of the forest accounts for a large fraction of the flow stress arising from dislocation interactions. Lavrentev (1980) has published a review of theoretical predictions and experimental values of a for different lattices and forest compositions. Despite the difficulties of consistent comparison between values from different sources (choice of effective G modulus, use of areal density of forest or total (volume) density, choice of orientation factor when dealing with polycrystals, averaging procedure for the theoretical a value), the universality of the T-Q1/2 relation and the essential contribution of forest dislocations to the flow stress of crystals with different lattice structures is clearly confirmed. More recently, Michel and Champier (1981) have performed a detailed analysis of the effect of pyramidal forests on the flow stress for basal slip in Zn, on the same premises as Schoeck and Frydman (1972) for f.c.c. metals. Their predicted a values (0.71 and 0.84 corrected for volume density) agree very well with the experimental ones, 0.80 and 0.87 to 1.03, for two particular types of forest. Loginov and Predvoditelev (1982) have performed numerical simulations of dislocation glide through a random forest of NaCl (predicted a = 0.74
2.2 The Flow Stress
vs. experimental a = 0.76) and Mg (two types of pyramidal forest for basal slip, 1.06-1.34 vs. 1.28 and 0.32-0.35 vs. 0.77). If the last case is excepted, the agreement is again very good. Qualitatively, b.c.c. forest junctions are predicted to be less stable than those possible in f.c.c. forests. Baird and Gale (1965) calculated - under a more simple hypothesis than Schoeck and Frydman - a = 0.18 for a b.c.c. random forest, but this figure is an underestimation. Frydman (1974) has made a detailed analysis of the strength of a transition b.c.c. metal random forest, accounting for the character of the dislocations and for the elastic anisotropy. The fraction of the forest making attractive junctions with screw mobile dislocations is very low (5 to 10%) and the a value computed for a random forest is then 0.28 to 0.31. Mobile edges make attractive junctions with approximately 1/3 of the forest, their a value amounting to 0.39 to 0.49. Accordingly, the average a value expected to b.c.c. metals at intermediate temperatures is about 0.4, similar to that predicted for f.c.c. metals. Experimental values do not differ from f.c.c. ones, e.g., see Lavrentev (1980) or Doner et al. (1975). The latter authors made a detailed compilation of measurements in Fe polycrystals and multiple-slip single crystals at RT, concluding with a value oc/rj = 0.33 - with G = 70.7 GPa -, in reasonable agreement with the theoretical prediction. After very large strains (Q > 1011 cm" 2 ), an estimation of the non-redundant dislocation density from the size and misorientation of subgrains (Gil Sevillano et al., 1980, measurements of Langford and Cohen, 1969,1975) leads to a lower figure, a ~ 0.20, again in agreement with the expected influence of the decrease of the outer cut-off radius in dislocation line energy as dislocation density increases.
41
2.2.6.3 Latent Hardening and Related Observations
The very good quantitative agreement of the flow stress of near isotropic dislocation structures induced by multiple slip with the predictions of hardening from a random dislocation forest, implying the irrelevance of any long-range stress component, does not seem very surprising. A much more crucial test for the general relevance of the forest intersection contribution to flow stress is constituted by the comparison of the CRSS of different slip systems in spatially very anisotropic dislocation structures. Single slip in stage I mainly induces dislocation accumulation corresponding to the macroscopically active system, as evidenced by TEM observations and etch-pit countings (Sec. 2.3.4.5). The dislocation density corresponding to secondary systems is only about one tenth of the primary density during stage I. The gliding primary dislocations encounter a weak forest of short trees and low density relative to the total dislocation density contained in the crystal. The transition from stage I to stage II is marked by a rapid increase of secondary dislocation density. In stage II, the forest in the primary system attains an areal density close to 1/2 of the total volume density, as in multiple slip of random forest situations, see Fig. 2-11 (Basinski and Basinski, 1979; Essmann, 1965, 1966; Mecking and Bulian, 1976). As referred to above and evidenced in Figs. 2-10 and 2-11, the same proportionality constant between the flow stress and the square root of forest density in the active system is found in single glide (primary system) or in multiple glide, and its value agrees with or slightly underestimates that predicted theoretically for crossing the forest.
42
2 Flow Stress and Work Hardening
0.1 1 Resolved Shear Stress kg/mm2
Figure 2-11. Forest densities (measured by etch-pit density) as a function of the microscopic flow stress (CRSS) at room temperature in Cu and Ag single crystals. Straight line, least-squares fit to forest density on primary plane (single-glide crystals) and multiple glide data. Curved line, least-squares fit (5th degree polynomial) to forest density on the cross-slip plane of single-glide crystals. From Basinski and Basinski (1979), compilation from different sources.
If a crystal strained in early stage I is retested so that single glide is established in a new system coplanar with the primary one, the new gliding dislocations should encounter the same forest previously found by dislocations of the primary system. However, if the new active system is noncoplanar, its gliding dislocations should find a forest about ten times more dense. If the density of the forest is the main parameter for the flow stress, roughly the same CRSS should be observed in coplanar nonactive (latent) systems as that measured in the primary one, and about three times
higher CRSS in the non-coplanar latent systems. Moreover, near isotropic CRSS should be expected in multislipped crystals, without distinction of active or latent systems. All this is essentially borne out by "latent hardening" and related experiments in f.c.c. crystals, reviewed extensively by Basinski and Basinski (1979), whose conclusions have been subsequently corroborated (Franciosi et al., 1980; Franciosi and Zaoui, 1982; Franciosi, 1985; Franciosi etal., 1987; Kocks et al., 1991). All experimental observations so far have shown that the flow stress of latent systems is equal or greater than that of the active system. The ratio of their CRSS, the "latent hardening ratio", LHR, tends to be near one for latent systems coplanar with the previously active one and significantly greater for non-coplanar systems, between 2 and 4 for very small stage I prestrains. It drops rapidly with increasing prestrains (Fig. 2-12) in parallel with the relative increase of dislocation density corresponding to secondary systems as the transition stage I-stage II proceeds (Fig. 2-13). It is also clear from Fig. 2-12 that forest density alone does not completely characterize the flow stress. Both Basinski and Franciosi as well as other authors (Miyamoto et al., 1977) have shown that the effect of the forest density depends on the type of reaction of the dislocations of the new active system with the trees, in qualitative agreement with the theoretical predictions of the strength of dislocation pair interactions referred to in the previous section. For instance, in Fig. 2-12 the flow stresses for non-coplanar systems crossing the previously active glide plane can be separated into two families, one for systems forming strong attractive junctions with the primary dislocations (i.e., Lomer-Cottrell locks) upper curve - and the other for systems forming attractive but weaker junctions
2.2 The Flow Stress
43
(Hirth locks). Franciosi and coworkers have performed a more detailed analysis of their results in terms of a generalization of Eq. (2-35) accounting for the strengthening differences among forest systems: (Tc)g = GfeQ>ijej) 1/2
1.0 1000 o-a 0 g/mm 2
Figure 2-12. Latent hardening ratio, LHR, as a function of primary hardening in single slip, (T — T 0 ), where T 0 is the CRSS of the unstrained crystal and T the final CRSS after the prestrain in the primary system (labelled o, a0 in the figure). Room temperature of 77 K data, Cu and Cu-Al crystals. Lower line: fit to data corresponding to coplanar latent systems. Upper lines: intersecting latent systems. From Basinski and Basinski (1979), compilation from several sources.
200
300
(2-40)
where the summation is extended to all the dislocation systems of the material. Four different values for the terms of the interaction matrix, aij? are enough to achieve a good description of the f.c.c. behavior. They represent the main interaction types among the 12{lll}<110> f.c.c. slip systems, in order of increasing value: diagonal or self-hardening terms, interaction terms involving no junctions, glissile junctions and sessile junctions. An effect attributable to the SFE of the material is observed on the absolute value of the interaction terms. The strength of attractive junctions increases as SFE decreases, an effect already discussed by Sastry et al. (1974). The shape of the LHRs vs. primary glide flow stress displayed by f.c.c. crystals (i.e., a rather sudden jump followed by a long decreasing period after the maximal value)
400
x [g /mm 2 ]
Figure 2-13. The square root of the ratio of primary to secondary dislocation density in several near [011] Cu crystals as a function of the flow stress of the primary system. Stage I ends approximately for T ~ 100gmm" 2 . From Mecking and Bulian (1976).
44
2 Flow Stress and Work Hardening
appears to be a general feature of slip-controlled plastic behavior. Results are available for b.c.c. and for ionic crystals (Nakada and Keh, 1966, 1969; Franciosi, 1983). For b.c.c. crystals, an analysis of the LHRs in terms of interactions of the mobile and forest dislocations has been performed by Franciosi (1983). Dislocation pair-wise interactions emerge again as the main contribution to the flow stress. As for h.c.p. metals, interaction coefficients for different basal/pyramidal combination situations in Mg and Zn have been calculated in a way akin to that developed by Saada, Schoeck and co-workers and successfully compared with experimental values, as reviewed by Lavrentev (1980). To finish this section, it is important to note that LHRs are very difficult to reconcile with flow stress models relying on a strong contribution from long-range stresses, particularly from long range back stresses associated with coplanar dislocation groups (Seeger's school). As has been repeatedly remarked (Hirsch, 1975; Basinski and Basinski 1979), Stroh (1953) showed that the LHR of a coplanar system on the exclusive basis of the long-range stresses of the primary one would be 0.5 instead of > 1 , as observed. 2.2.6.4 Thermally Activated Glide Through Dislocation Substructures
Section 2.2.2 briefly sketched the glide kinetics and its importance as a tool for analyzing the mechanisms underlying the dislocation glide plasticity was stressed. The temperature and strain rate sensitivity of the flow stress is diagnosed by studying its reversible changes (at a given structure, s) with small changes of temperature and strain rate. The definition of the new flow stress level after the change can be the object of some controversy (see Basinski
and Basinski, 1979) as some degree of back extrapolation to the initial structural state is necessary (the change is only seen after a finite strain, i.e., after some unavoidable structural change). However, since the classic work of Cottrell and Stokes (1955) it is generally accepted that at low stresses - stages I and II - and if a short transient at the beginning of stage I is neglected, the reversible changes, [AT C /AT]^ S or [Azc/Alnf]Ts are proportional to the current flow stress level, TC(S, T,f). This is the so-called "Cottrell-Stokes law". The low-stress deviation from the C-S law is convincingly ascribed to soluble impurities (Basinski and Basinski, 1979) but systematically increasing deviations are observed at large stresses (stage III and beyond) in the form of an upward curvature in the ATC — TC plots ("Haasen plots", Haasen, 1958), Fig. 2-14. The deviation is still more pronounced at very large strains (stage IV), Fig. 2-15 (Korbel et al, 1979; Korbel and Szczerba, 1982; Christodoulou etal., 1982; Alberdi, 1984; Hughes, 1986; Rollet, 1988), the rate sensitivity increasing with strain (stress) and temperature. Solution hardening enhances the rate sensitivity at very low stresses, as referred to above, but decreases it at large strains, when dislocation-dislocation interactions predominate (Kocks, 1979; Mulford, 1979). The latter effect is evident in Fig. 2-15. Two possible explanations were advanced by Cottrell and Stokes for the ATC — TC proportionality in reversible flowstress changes. The first one is strict similitude of the dislocation structure throughout the stress range where the C-S law holds. Only the scale changes, the distribution and proportion of the various kinds of obstacles remains the same. The second one is to assume that the same obstacles are responsible for the long range interactions (quasi-athermal obstacles) and short
2.2 The Flow Stress
45
a a. A g
-A
— 210~5
•U)
o<1A5>
T=295K
D
x
Polycr
0.2 0.6
-^^
10
60
90
20
30
120 40
range interactions (obstacles which may be overcome by thermal activation). Similitude is assumed by most work hardening models, at least inside a particular deformation stage (Nabarro et al, 1964). It has been given the category of "principle" (Kuhlmann-Wilsdorf, 1968; Hansen and Kuhlmann-Wilsdorf, 1986)
1
•
I
'
I
'
I
'
I
-
298 K
-
Ni+60% Co—v -5o
_
rfB "
Ni+30% Co-x \
/B
1
A e• A
i
\ 2 Nickel
^c/^V /
-
< 1 0
1
0
m = 300
0
0.1 0.3
100 200 300 400 500 600 SHEAR STRESS, 0 8z
(MPa)
Figure 2-15. Haasen plot of the rate sensitivity of the torsional shear stress (macroscopic) of Ni-Co alloys vs. stress at 298 K. From Hughes (1986).
0 [MPal X iMPa)
Figure 2-14. The rate sensitivity of the flow stress vs. the flow stress, both as a function of prestrain for silver single and polycrystals. From Mecking and Kocks (1981). The rate sensitivity parameter m = [5 \nf/bxc]TS ranges from 50 to 500 in the glidecontrolled plastic regime.
and, although strict similitude of dislocation substructures is not observed, an amazing constancy of the ratio between some characteristic structural dimensions (e.g., cell size and interdislocation distance) has been proved to hold over a broad strain or dislocation density range. As will be further discussed (Sec. 2.3.3), similitude is clearly not obeyed in stage III, where departures from the C-S law become increasingly evident. However, it is not necessary to invoke similitude to justify the C-S behavior. Basinski and Basinski (1979) have gathered much evidence showing that, for the very anisotropic single slip dislocation structures in stages I and II, the C-S ratios, (1 + ATC/TC) are independent of strain (structure) or crystal orientation (flow stress in the primary or in latent hardening systems). Similitude of the structures that the gliding dislocations "see" cannot be invoked to explain here the observed C-S behavior. This strongly points to the interaction of the mobile dislocations with the forest as responsible both for the flow stress level and for its reversible changes, i.e., the second explanation offered by Cottrell and Stokes.
46
2 Flow Stress and Work Hardening
As already described above, the mobile dislocation experiences attractive interactions (which can represent very strong and long obstacles) and weak repulsive interactions with the trees. The thermal activation for freeing the mobile dislocation from junctions formed by attractive interactions with the forest was studied by Guyot (1966). As could be qualitatively anticipated, the activation energies in the glidecontrolled regime (T < 0.3 TM, f < 102 s~\ TC given approximately by Eq. (2-38)) are so high that the attractive part of the forest can be considered athermal but for its weakest trees (less than 2% according to the simplified calculation of Guyot). By contrast, the thermally activated contribution of repulsive trees to the flow stress naturally explains the operation of the Cottrell-Stokes law at low dislocation density levels (constriction of partial dislocations and jog and kink formation). At large stresses or dislocation densities, the systematic departures from the C-S behavior could be interpreted as an argument against the forest dominance of the flow stress, or as an increasing intervention of a new, thermally activated mechanism in the process of dislocation glide through the substructures of work hardening. However, Basinski (1974) argued that deviations from the Cottrell-Stokes law with increasing deformation are in fact a required condition of the forest intersection mechanism. The flow stress, Eqs. (2-38) and (2-37) for f.c.c. metals, is affected by the logarithmic term of the dislocation line energy, as evidenced by the proportionality (2-39). As has been discussed already, it is mainly dictated by the strongest trees, those giving strong junctions (about 20% of a random forest), requiring severe bowouts of the dislocation line. On the other hand, the rest of the trees (~80%), which only contribute with a small fraction of the
flow stress, represent weak obstacles requiring only shallow dislocation bow-outs. Their (independent) contribution would be affected by a logarithmic term with a larger outer cut-off radius [Eq. (2-34)] and their strength, mainly associated with dislocation core effects, would be independent of the forest spacing, in contrast with the junctions' strength. Therefore, for a reversible flow stress change, the following parametric equations can be written: ATC~CI£1/2
^
(2-41
2 Q
^
2
) ]
a)
(2-41 b)
Accordingly, the Cottrell-Stokes ratio should slowly increase with increasing dislocation density, as observed. Figure 2-16 reproduces an example given by Basinski (1974), where the agreement with Eqs. (241) is obvious. It is to be noted, however, that Basinski's opinion is not unanimously accepted. For instance, Mecking and Kocks (1981) have criticized his approach. Without denying the capability of Eqs. (2-41) for fitting the observed deviations from C-S behavior, they have criticized the validity of Eq. (241 a), which should probably include a logarithmic factor formally similar to that of Eq. (2-41 b), as the line tension of the dislocation is involved in crossing the weak as well as the strong trees. The observed deviation should be ascribed to another physical origin and they noted a correlation between the deviations from Cottrell-Stokes behavior and the deviation from athermal work hardening behavior. Saimoto and coworkers (Saimoto and Sang, 1983; Saimoto and Duesbery, 1984), who performed very precise measurements of the strain rate sensitivity and its strain rate and temperature dependence, attribute the observed deviation from the C-S law to the plastic generation of point defects, point defect clusters and small loops (i.e., debris genera-
2.2 The Flow Stress
47
Figure 2-16. Change in flow stress of 99.999% Cu single crystal at RT for strain rate changes from 2 x 10 ~4 to 2 x l O ~ 5 s ~ 1 a s a function of the flow changes during a tensile test. Lower curve, continuous test; upper curve (offset by 20gmm~ 2 ), with intermittent twists. Inset: stress-strain curves (offset of 1 kg mm ~ 2 for the twisted crystal). Continuous lines: calculated according to Eq. (2-41). From Basinski (1974). 1
2
3 4 5 Resolved Shear Stress kg/mm2
tion) which act as weak obstacles (Fleischer, 1962; Kroupa and Hirsch, 1964; Foreman, 1968) and consequently are more rate sensitive than interactions with monopole dislocations of the forest. Their origin is ascribed to dynamic recovery processes (annihilation of edge dipoles). Indirect evidence for the increasing presence of a weak defect population additional to forest interactions beyond stage II have been advanced by Basinski et al. (1977). It could well be that an eclectic position incorporating both Basinski's and Mecking and Kocks' or Saimoto's positions is the closest to reality. The undeniable presence of a constant proportion of weak repulsive trees in a forest could be responsible for deviations from C-S law detectable even in stage II. The deviation would be enhanced by the dynamic recovery debris accumulating from stage III.
2.2.7 Solid Solution Hardening and the Strengthening Effects of Precipitates or Dispersed Phases
The strengthening effect of foreign atoms in solid solution as well as their influence on work hardening are dealt with by Neuhauser and Schwink in Chap. 5 of this Volume. Similarly, Reppich reviews in detail in Chap. 7 the flow stress and work hardening of crystals containing small precipitates or dispersed second phases. 2.2.8 Extended Obstacles. Two-Phase Materials and Grain Size Effects
A broad variety of factors, often superimposed, contribute to the resistance offered by isolated particles to dislocation glide (see index of Chap. 7). All of them with the exception of the contribution arising from the creation of new interfaces -
48
2 Flow Stress and Work Hardening
furnish a resisting force proportional to the particle size. Accordingly, although extraneous particles dispersed in a crystal can be classified as soft or hard, they can act as strong or weak obstacles depending on their size. Obstacles over the critical size are not penetrable by isolated gliding dislocations, which will try to circumvent them. Any crystal with such finite-size, nonpenetrable obstacles intersecting its slip planes behaves as a composite material. Strictly speaking, the situation described is that of two-phase materials, but it also applies to two circumstances common to single-phase crystals. The first one occurs when weak or medium strength point-like obstacles are non-uniformly distributed in the crystal, and the slip plane (or the volume) can be divided in soft, penetrable, areas and hard, non-penetrable regions (or volumes). This is the ordinary situation of strained crystals where the stored dislocations form clustered arrays of increasing heterogeneity as strain and temperature increase. The second, more specific, is the situation of grains in polycrystals, where the grain boundaries define a totally closed domain for dislocation glide, and the orientation dependence of the resolved shear stress makes impossible the immediate fulfilment of the conditions for uniform plastic strain in every grain for some uniform critical value of the macroscopic applied stress. All these cases have in common the preferred yielding, at an early microstrain stage of, respectively, the matrix, the softer part of the crystal or the softer grains, leading to the development of strain gradients with a characteristic wavelength related to the structural heterogeneity (Ashby, 1970, 1971). Long range stresses are developed as a consequence of the strain gradients. Their accommodation is the key to under-
standing further plastic deformation of the "composite". In ductile two-phase composites, relaxation of the long range stress level occurs by plastic flow of the harder phase and/or by extra (redundant) plastic flow of the softer one (creating "geometrically necessary dislocation" arrays). The particular solution for compatible flow depends on the ability of dislocations to cross-slip, i.e., on the stacking-fault energy or, more generally, on the dislocation core properties of the crystal (extent of tendency to slip planarity) and, of course, on the flow stress ratio between the hard and soft phases or zones of the crystalline material. 2.2.8.1 Plastic Flow Transmission in Heterogeneous Materials with Strong Tendency to Planar Slip
In plastically heterogeneous materials in which cross-slip is difficult, the repeated activity of dislocation sources promotes the building up of pileups in the softer phase, held up at the frontier of the hard phase, which needs a higher stress level for yielding. If the hard phase is discontinuous in the crystal, the hard islands will be surrounded by pileups of Orowan loops. The piling up of dislocations will continue until the induction of plastic flow (or, eventually, fracture) in the hard phase when some critical stress, T* is reached at the head of the pileup. If the pileup has accummulated n dislocations up to then, for a planar barrier (straight dislocations), *? = *(Tc)s
(2-42)
where (TC)S is the flow stress of the soft phase, the stress necessary to overcome the back stress from the pileup (ignoring any other possible strengthening contributions). The pileup size is equal to the interphase distance L measured on the soft-phase slip plane (or half of it when the sources are not
2.2 The Flow Stress
located at the interfaces). Then, the number of dislocations in equilibrium under (t c ) s is,
which gives, combining both equations, (TC)S - [GbT*l{nL)Y'2 oc L ^ 2
(2-44)
It is clear from Eq. (2-42) that strong barriers are needed to form pileups containing a large number of dislocations. They can be observed in low SFE metal matrix composites or, in the form of piled-up Orowan loops around hard particles, in weakly deformed dispersion hardened materials. In the latter case (diluted small isolated particles), for a concentric array of n loops around a particle of radius r, the stress at the interface is (Hazzledine and Hirsch, 1974),
duce some redundant flow of the matrix modifying the geometrically necessary array - and plastic flow of the hard phase (eventually, its debonding or its fracture). As deformation proceeds, the necessary interface dislocation array is automatically maintained to assure the compatible deformation of the two phases. The interface acts as a selective partial dislocation sink for the matrix, transmitting the plastic flow but retaining the dislocations needed for building the critical stress level which, with the help of the applied stress, maintains the plastic flow of the hard phase. The flow stress, if a load transient attributable to the internal stress state is neglected, is given by the law of mixtures (on the upper-bound assumption of elastoplastic isostrain of the two phases), xc = fs ( T j + fh (Tc)h
The high stress concentration at the particle interface will very soon induce prismatic dislocation punching from the interface, or some kind of redundant plastic flow in the matrix relaxing the local stress; this is not possible for highly concentrated large or elongated particles, as in fibered or lamellar in situ composites, where relaxation must come from plastic flow of the hard phase or fracture will soon occur. 2.2.8.2 Compatible Plastic Flow in Heterogeneous Crystals Without Pileup Formation
When screw dislocations can easily depart from their original slip planes, the formation of long, unrelaxed pileups is rather improbable. However, the preferred flow in the softer phase leads to dislocation retention at the hard phase boundary. These dislocations accommodate plastic mismatch and provide local internal stresses, which, beyond some strain level, will in-
49
(2-46)
where the suffixes refer to soft and hard volumes. The density of geometrically necessary dislocations can represent an important contribution to the flow stress in the softer phase. If a plastic mismatch SF (in terms of resolved shear strain) is to be absorbed by geometrically necessary dislocations stored in the soft phase, they represent an average density (Ashby, 1970, 1971), bQ=gtdr/(bL2)
(2-47)
where t is the size of the hard phase and L the interphase distance in the soft phase, t/L ~ fh (two dimensional approximation), and g a geometrical factor. On account of the general relationship between flow stress and dislocation density, Eq. (2-35), including in g1 the geometric factor and the relative mismatch and excluding any other possible hardening contribution, (2_48)
50
2 Flow Stress and Work Hardening
where g± may be strain dependent. Equation (2-48) is in fact a work hardening contribution. For a given strain level, it is formally similar to (2-44). This strengthening effect of £g must be superposed on that of bowing between hard particles, approximately proportional to L"1 [e.g., Eq. (2-30) with the appropriate correction for finite size of the particles].
2.2.8.3 The Composite Approach to the Flow Stress of Crystals Containing Heterogeneous Dislocation Distributions
Following earlier attempts (e.g., the mixed pileup/forest model of Hirsch, 1964, 1975 and particularly the model of Takeuchi, 1970), thorough analyzes explicitly using composite mechanics have been developed in the last ten years by Mughrabi (Mughrabi, 1983, 1986, 1987, 1988; Mughrabi et al., 1986; Ungar et al., 1984,1991) and Pedersen groups (Pedersen et al., 1981; Pedersen, 1987, 1990) for the flow stress of work hardened crystals, taking into account explicitly the ubiquitous heterogeneity of the spatial dislocation distribution. Dislocation mats or bundles of stage II and dislocation cell walls of stages III or IV constitute hard but deformable inclusions of high dislocation density embedded in a soft matrix of lower dislocation density. A local flow stress can be ascribed to each region of the crystal, which can then be treated as a composite, as sketched above. Under the macroscopic flow stress, glide occurs preferentially in the soft region. A plastic mismatch between hard and soft regions is established which is accommodated by glide dislocations held up at the "interfaces". An internal stress pattern immediately develops, helping plastic deformation inside hard regions and opposing the dislocation glide in the soft matrix.
From the equilibrium condition, Wi = 0
(2-49)
where / and x{ represent respectively the area or volume fraction and the internal stress level of each region. In the plastic isostrain simplification (i.e., neglecting the elastic accommodation), the macroscopic flow stress is given by . n
where TC are the virtual local flow stresses in the absence of internal stresses. On account of Eq. (2-49), Eq. (2-50) reduces to the rule of mixtures, Eq. (2-46). Internal stresses must exist in heterogeneous dislocation structures (and indeed they are observed), but the flow stress of the composite crystal is only controlled by local virtual flow stresses (i.e., in accordance with all the reviewed evidence, by the strength of the local forest plus a possible superposed friction effect). The internal stresses, independent of their origin (pileups or dislocation arrays smeared on the interface), have no net effect on the flow stress of crystals with work hardened dislocation structures. This is a very interesting conclusion from the "composite" models of crystal plasticity, clarifying much of the past controversy between long-range stress and forest models of the flow stress. The quasi-composite behavior with its corollary of polarized internal stress regions is implicit in the "dislocation percolation model" of the flow stress of Kocks (Kocks, 1966, 1985; Kocks et al, 1980; Hasegawa et al, 1982) and related developments (Torrealdea and Gil Sevillano, 1983; Martin Meizoso et al, 1984; Hernandez Olivares and Gil Sevillano, 1987; Gil Sevillano and Aernoudt, 1987). Detailed analysis of the internal stresses built up in the hard regions have been made by Jackson (Jackson and Siedersleben, 1984; Jack-
2.2 The Flow Stress
son, 1985 a) accounting for crystallography and slip activity implications. However, up to now, the "dislocation percolation model" (DPM) approaches which, in fact, make a rigorous calculation of the flow stress in the soft matrix region, defined as the slipped area at the percolation threshold, have not paid attention to the virtual local flow stress of the areas not invaded by the gliding dislocation. The flow stress of the matrix is taken to be the mean flow stress of the whole crystal, although it is recognized that the hard regions yield plastically under the internal stresses generated by the accumulating interface dislocations. Accordingly, DPM underestimates the flow stress by an amount fh(Th~TsX which perhaps is not always negligible. On the other hand the composite models of Mughrabi or Pedersen do not consider in much detail the processes of dislocation glide occurring through the heterogeneously distributed dislocation structure. The validity of these interpretations of the flow stress of heterogeneous dislocation structures is supported by direct or indirect observations and measurements using different experimental techniques: surface observations of slip-line behavior, TEM observations and TEM in-situ experiments, long-range internal stress evaluations using X-ray diffraction analysis, stored energy measurements, Bauschinger effects or dimensional changes during recovery of deformed crystals (see references in the previous paragraph). Of particular interest are the results of Mughrabi's group, combining TEM and long-range stress investigations by X-ray line-broadening measurements. In [001] copper single crystals deformed in tension (multiple glide) a weakly oriented cellular structure develops. The broadened line profiles of {001} Bragg reflections are asymmetric and can be decomposed into two mutual-
51
ly displaced symmetric subprofiles. They correspond to volumes of material under long-range internal stresses that help or oppose the applied tensile stress. The magnitude of the stresses and the volume fraction of the two regions can be evaluated from the profiles. The agreement of the volume fractions measured by X-rays or by TEM is satisfactory, as is the comparison of local flow stresses estimated in the two regions from the measured internal stresses (X-rays) - using the composite assumption - with estimates based on the local dislocation densities (TEM). Figure 2-17 gives an example of their results. 2.2.8.4 Grain-Size Effects in the Flow Stress of Polycrystals The polycrystal volume is built up of regions displaying a differing elastic and plastic behavior (the grains) surrounded by two-dimensional crystallographic defects (the grain boundaries). The heterogeneity generates an elastoplastic mismatch from the microplastic regime, with its associated pattern of fluctuating internal stresses. Such a stress field, whose wavelength is equal to the grain size, D, induces - after a microplastic transient strain marking the passage from single to polyslip - the polyslip situation prevailing everywhere in the polycrystal (see Chap. 3 of this Volume for a detailed description). Polyslip in itself does not make polycrystal grains any harder (in the intrinsic sense of changing their TC) although, by suppressing stage I (requiring single slip) it automatically raises their work hardening rate with respect to their unconstrained behavior. Moreover, as in other composites, the admissible plastic mismatch is resolved by some kind of dislocation accumulation containing geometrically necessary dislocations - which simultaneously is responsi-
52
2 Flow Stress and Work Hardening
same plastic strain tensor, implies the existence of a true polycrystalline strengthening effect (Ashby, 1970, 1971; Mecking, 1980, 1981). The effect is indeed observed and it is grain-size dependent, constituting the well known Hall-Petch relationship,
(a)
T-
0.5
-
100
Figure 2-17. (a) TEM micrograph of dislocation cell ;tructure in Cu crystal deformed in tension along 001] to TC = 75.6 MPa. From Mughrabi et al. (1986). b) Normalized local flow stresses, (ic)h/Tc and (TC)S/TC of, respectively, hard (walls) and soft (cell interiors) regions of tensile strained [001] copper (open circles) and copper-manganese (full circles) single crystals as a function of the measured macroscopic flow stress (a constant (TC)0 value - the solid solution strengthening contribution - has been subtracted from the Cu-Mn values). The volume fraction of the hard regions is also indicated in the figure. Plotted from data of Hilscher and Wilkens (1989) and Ungar et al. (1991).
ble for the building up of the necessary internal stress state. The presence of those dislocations in the polycrystal grain, in contrast with their absence in a single crystal of similar orientation submitted to the
/_
\
\
j{
r\-l/2
(^
C4\
where (T C ) 0 collects any grain size independent contribution to the flow stress and KHP is an empirical strain-dependent factor. It is formally similar to Eqs. (2-44) or (2-48) for general non-homogeneous materials. The influence of grain size on the yield or flow stress of polycrystals has been extensively reviewed during the past twenty years (Armstrong, 1970 a, 1970b, 1983; Embury, 1971; Hansen, 1983, 1985; Hirth, 1972; Kocks, 1970; Li and Chou, 1970; Thompson, 1977). Equation (2-51) is rather universal, but the Hall-Petch slope displays a wide variation. For the yield stress of materials in which dislocation generation or unlocking is difficult (i.e., materials showing a yield drop), KHP has a much higher value than the slope corresponding to their subsequent flow stress. The KHP constant of materials not affected by solute pinning is, on the other hand, small, increasing as the tendency to slip planarity increases (Fig. 2-18). The original interpretations of the HallPetch behavior (Hall, 1951; Petch, 1953; Cottrell, 1958) relied on the propagation of slip from a grain to its neighbor with the help of the stress concentration of pileups of dislocations held up by the grain boundaries. This leads to Eq. (2-44) or one of its many variants on account of the assumed geometry of the grain, position of the pileup source, existence of other pileups, etc., i.e., *
[g2(T*/G)/n]V2
(2-52)
2.2 The Flow Stress
2
U
6
d~ 1/2 -(GRAIN
8
10
12
U
SIZE)" 1 / 2 (mm- 1 / 2 )
Figure 2-18. Yield stress-grain size relationship at room temperature for aluminum, copper, 70-30 brass and iron. From Hansen (1985).
where g2 is a numerical factor of the order of unity. An upper bound for (T*/G) is provided by the athermal limit for the homogeneous nucleation of dislocations, i.e., the ideal shear strength, Eq. (2-8); introducing it into Eq. (2-52) a normalized value 0.18 is obtained for the Hall-Petch slope, which compares well with the highest measured values for the yield stress of materials whose dislocations are blocked by strong solute pinning (see Dingley and McLean, 1967; Wilson, 1967, 1968; and Embury, 1971 for more detailed discussion on this and related topics). Extensive calculations on pileup stresses aimed at explaining the Hall-Petch behavior are available (Li and Chou, 1970, offered an excellent review). However, their limited applicability, restricted to alloys where dislocations are strongly locked by solutes, or to low SFE or long range ordered alloys was early recognized. Pure metals or alloys, where pileups of the required size are not directly observed, still show a Hall-Petch relation between their
53
yield or flow stress and grain size. This consideration motivated the development of non-pileup theories (Li, 1963; Li and Chou, 1970; Hirth, 1972). Non-pileup theories of Hall-Petch grain boundary strengthening rely on the enhanced dislocation accumulation in the deforming polycrystal relative to the single crystal because of the presence of grain boundaries. The enhancement is attributed to the ability of boundaries for either emission of dislocations (source theory of Li) or trapping of gliding dislocations ("work hardening" theories). The source theory of Li was based on the frequent observation of an enhanced slip activity at the vicinity of grain boundaries and on the plausibility of easy dislocation emission from stress concentrations associated with ledges (grain boundary dislocations). At yield, Li assumed the emission of a constant dislocation length per unit boundary area (i.e., per source if the density of ledges is a characteristic of the boundaries) and its permanent trapping in the grain, which, on account of Eq. (2-35), leads to the Hall-Petch relation. The "work hardening" models are based on the trapping of gliding dislocations at or in the vicinity of grain boundaries, limiting their mean free path. The extra storage of dislocations would be responsible for the yield or flow stress enhancement of polycrystals relative to single crystals. After some amount of plastic strain, F, a grain boundary has received « FD/b gliding dislocations from the grain interior. Those dislocations find their mean free path interrupted by the boundary and remain in the grain volume except for some fraction transmitted through the boundary. An excess dislocation density is then stored in the polycrystal grains, = g3F/(bD)
(2-53)
2 Flow Stress and Work Hardening
where g3 is a numerical factor including geometry, loss of dislocation length by transmission and possible gain by emission of dislocations from boundary sources excited by the stress fields of the incoming dislocations (i.e., the "work hardening" model includes the "source" model). On account of the general relationship (2-35) and for low strain levels where the dislocation storage from work hardening at the grain interior is still negligible, (2-54)
1
,
0.0A6
JJ
£ 3
/ \ /
X
a.
For plastic strains of the order of the conventional yield strain, Fy = M sy ~ 0.006, if transmission or extra slip compensate each other or their balance is assumed to be negligible, the normalized yield slope predicted by Eq. (2-54) is, for g3 ~ 3n/2 (spherical grain), KH?/(G^/b) ^ 0.06, in good agreement with the experimental values. The extra density of dislocations accumulating at grain boundaries provides the geometrically necessary dislocation array - in Ashby's sense - for a compatible deformation of the polycrystal. However, as stressed by Mecking (1981), most of the density of the array is probably redundant. The "work hardening" model has known many developments in recent years. For various f.c.c, b.c.c. and h.c.p. materials it has been shown that the average dislocation density for a given strain increases as grain size decreases and that the flow stress is only dictated by the average dislocation density, e.g., Fig. 2-19, although some discrepant results have also been reported (e.g., Al-Haidary et al., 1983). The linear Q — D relation or the parabolic dependence of KHP on strain predicted by Eqs. (2-53) and (2-54) are most frequently observed only as an asymptotic limit at low strains or low temperatures, but this is easily accounted for, at least qualitatively, considering that the factor g3 is changing with
i
grain size, mm
- dislocat ion density
54
/
1
0.130
V /
/
_
/A
rL. (U9
¥^ r I
I
5 10 15 e - true strain (%) Figure 2-19. Dislocation density as a function of grain size and tensile strain for 99.99% aluminum polycrystals. From Hansen (1977). Equivalent results have been obtained by Conrad (1982) for a-Ti of various grain sizes deformed between 78 and 650 K.
strain as the mutual orientation of neighboring grains changes, as well as their shape, and because of the increasing activity of dynamic recovery. Actually, beyond the yield strain, superposition of the virtual work hardening of the grain volume, as if grain boundaries were not present, and the extra work hardening because of the presence of the grain boundary barriers cannot be ignored. It appears reasonable to assume that the extra stored dislocation density induced by the presence of grain boundaries is concentrated in their neighborhood so that polycrystals have to be treated as composite materials. Let X be the thickness of the dislocation-rich zone close to the boundary and gwh the dislocation density that would have been stored by work hardening in the absence of grain boundaries. Then, under the simple "rule of mixtures" approximation,
2.2 The Flow Stress
aGb D >-2X). which can evolve in a complicated manner as £wh, X/D and g3 change with plastic strain. The influence of strain on KHP is usually observed to be rather weak, at least for moderate strains (Fig. 2-20). A moderate initial increase is very frequently quoted and a decrease at large strains, but purity or temperature changes can alter this pattern rather drastically (Hansen, 1977, 1983; Al-Haidary et al, 1983). In some cases a strengthening effect of grain size has been quoted to persist up to very large strains (Gil Sevillano et al, 1980). The composite model (Kocks, 1970; Hirth, 1972) has been further developed by Thompson and coworkers (Thompson, 1977; Thompson and Baskes, 1973; Thompson, Baskes and Flanagan, 1973). Hansen and coworkers (Hansen, 1985; Barlow etal, 1985) have tried to characterize the grain boundary zone of deformed polycrystals from the substructural point of view. The microstructures at the grain boundary zone vary from boundary to boundary and
15
!
i
I
70/30 Brass
Cu _
5 k
Ai 0 0.0
I
0.05
I i 0.10 0.15 e - T R U E STRAIN
I 0.20
Figure 2-20. Values of the Hall-Petch constant k for the tensile flow stress (o ~ 3 TC) as a function of tensile strain (e ~ T/3) at room temperature for 99.99% Al, 99.999% Cu and 70-30 brass. From Hansen (1985).
even along the same boundary and they conclude that the grain boundary region in general cannot be classified as a welldefined structural element. However, at a number of grain boundaries a relatively well-defined deformation zone of higher dislocation density and smaller cellular or subgrain size is observed. Therefore, when present, this zone (one to ten micrometers thick) will be stronger than the matrix, in agreement with some microhardness measurements (Thompson, 1977). These observations justify the composite model of the flow stress of polycrystals, but real structures are far more complicated than simple geometrical idealizations. Recent work in this area has focused on the observation and understanding of the detailed interactions of gliding dislocations with grain boundaries (Martinez-Hernandez et al., 1987; Shen et al., 1988; Kirchner, 1988; George, 1988; Baillin et al., 1990; Lee et al., 1990). The mechanisms reviewed by Hirth (1972) and some others (dislocation emission and absorption, slip transfer, multiple slip and cross slip close to the boundary, stress concentrations) have been confirmed and even observed operating in situ. In relation to the subject of grain boundary strengthening and its rationalization through "work hardening" models, it is an interesting conclusion from such studies that dislocation transmission across grain boundaries is in general a difficult mechanism requiring large pileup stresses to operate. Therefore, it is likely that such a mechanism is, in most situations, bypassed by lower energy processes such as the activation of nearby sources, cross slip, etc., leading to the formation of a grain-boundary zone in an advanced stage of hardening relative to the grain interior.
56
2 Flow Stress and Work Hardening
2.2.9 Superposition of Different Contributions to the Flow Stress
Previous Sees. 2.2.3 to 2.2.8 have given an idea of the wide variety of possible strengthening mechanisms contributing to the flow stress in the dislocation glide regime. The observed flow stress is a result of some complex statistical average of the varying local resistance to glide found by mobile dislocation lines. Achieving a rigorous analysis of such an averaging process is a formidable task even for relatively simple cases such as mixtures of point obstacles (see references in Sec. 2.2.5). Moreover, some of the strengthening mechanisms are not sufficiently well understood even if taken separately. Superposition laws have been reviewed and discussed by Kocks and coworkers (Kocks etal., 1975; Kocks, 1979). For discrete obstacles the two simple rules given in Sec. 2.2.5, Eqs. (2-31) and (2-32) can be applied. As discussed in detail by Kocks et al. (1975), linear superposition is plausible when one of the flow stress contributions is the lattice resistance and for a limited range of solid solution and precipitation strengthening contributions. A very sensitive tool for analysing superposition is provided by the "Haasen plot", described in Sec. 2.2.6.4, a display of the strain-rate sensitivity of the flow stress at constant structure versus the flow stress when the latter is only increased by work hardening. As previously noted, when the flow stress is determined solely by dislocation-dislocation contributions, the Cottrell-Stokes behavior is observed, at least asymptotically, i.e., a linear Haasen plot passing through the origin. Any deviation from this behavior representing a negative intercept with the abscissa in the Haasen plot indicates the presence of a contribution to the flow stress more strain-rate sen-
sitive than the forest dislocation contribution. Alternatively, any positive intercept indicates the presence of a quasi-athermal contribution of a level given by the intercept. This very practical diagnostic tool for superposition effects is easily derived for an additive superposition law (Kocks, 1979). Let (T C ) 0 superpose with the dislocationdislocation contribution, (TC)Q: *c = (Tc)o + (*c), ">
I2"56)
Figure 2-21 (a) shows this behavior schematically and Fig. 2-21 (b) shows actual experimental results. The application of the Haasen plot for assessing non-additive superposition behavior is further discussed by Kocks (1979, 1987).
2.3 Work Hardening 2.3.1 The Semi-Phenomenological Approach Section 2.1.2 in the introduction to this chapter started with a quotation from Kocks (1987). It was a reminder that all current response of a material - including the current rate of evolution of such response or the rate of evolution of the structure - is embodied in its current structure. Translating this reflection to the response of crystalline materials to plastic behavior it means that "history variables such as time and strain can enter the constitutive relations only in differential form: describing the evolution of the state", which completes the aforementioned quotation. Accordingly, the differential constitutive equation for plasticity is the expression of the work hardening rate, 0, in terms of the external variables and the necessary number of structural parameters, si9 providing
2.3 Work Hardening
la) drc/d\nT
d(Tc)p/d\nT
(b) T=300K
A A
57
polycrystal or a component of a two phase material. The mainframe of Sec. 2.2 of this chapter has been an overview of the present understanding of the contribution of dislocation substructures to the flow stress. Most other contributions can either be linearly superposed (lattice potential, solution effects), or constitute a hidden dislocation substructural contribution in the form of a particular extra work hardening of the crystal (grain size or two phase strengthening). The most important conclusion from this overview is that the dislocation contribution to the flow stress is essentially dependent on a single structural parameter, the forest or - for most cases of technical importance - the total dislocation density, through the proportionality
INCONEL 600
I 2.
INCONEL X 750 (aged to d=20nm)
500
1000
1500
FLOW STRESS x [MPa]
Figure 2-21. (a) Schematic "Haasen plot" for a basic material with flow stress from dislocation-dislocation interactions (1), with an additive contribution more sensitive to the strain-rate (2), and with an additive quasi-athermal contribution (3). (b) The Haasen plot for pure Ni and two of its commercial alloys: Inconel 600 (solution hardened) and Inconel X750 (precipitation hardened, size of precipitates quoted in the figure). It may be seen that solution hardening decreases the strain-rate sensitivity of dislocationdislocation interactions of Ni, probably affected by SFE changes. From Mulford (1979).
a complete description of the structure: 9Si)
(2-57)
This equation can be integrated along any history path for a given crystalline volume element from a single crystal, a grain of a
arising from a dislocation percolation model of glide (without neglecting the possible correction noted in Sec. 2.2.8.3). For f.c.c. materials in the dislocation glide regime, where only one family of dislocations is present, this equation holds, as noted, all along the four deformation stages, with the same value of the proportionality factor a, i.e., a is rather insensitive to the forest composition and to the spatial distribution of the forest density. Moreover, it is also rather insensitive to strain rate or temperature (Sec. 2.2.6.4). Thus it makes full sense to compare the evolution of the work hardening rate, 0, related to the evolution of the structure, with TC, equivalent to one structural parameter. This can be done in a 9 — TC plot, as advocated by Mecking and Kocks (Kocks etal., 1968; Kocks, 1976; Mecking and Liicke, 1969; Lixcke and Mecking, 1973; Mecking, 1977), which has proved to be very useful in a variety of applications. The conclusion about the flow stress is also
58
2 Flow Stress and Work Hardening
valid for other non f.c.c. crystal structures, but the analysis of their work hardening behavior in multiple slip becomes much more complicated (the notion of amount of crystallographic slip, F as the algebraic addition of individual slip loses its meaning, and the use of a work hardening matrix, Eq. (2-5) becomes unavoidable). Another related possible plot, already recommended by Nabarro et al. (1964), is the TC 9 vs. TC plot. Noticing that = [(aGb)2/(2Tc)]dQ/dr
(2-58)
the TC 9 vs. TC diagram is equivalent to plotting the rate of change of dislocation density with strain versus the square root of the current dislocation density level. The potential interest of such graphs is in allowing a better discrimination between the work hardening stages than from merely inspecting stress-strain curves. They can also be linked with some basic limits of work hardening behavior and provide a simple characterization of the stages with the use of a minimum of structural parameters (one if possible, the dislocation density), without entering into detailed descriptions of the complex dislocation substructures observed.
merely strain-induced change of scale. Alternatively, any clear qualitative strain-induced substructural change represents a stage change (e.g., the passage from isolated primary edge multipole bundles in stage I towards the layered structure of misoriented low dislocation level slabs separated by dislocation-rich layers slightly deviated from the primary slip plane, typical of stage II), although the opposite assertion is not necessarily true (e.g., no clear structural change marks the transition from stage II to stage III, as noted by Basinski and Basinski, 1979). Such a tendency can be justified from the scale-invariance of the equilibrium conditions of dislocation configurations and from the necessary shrinking of their characteristic size to remain stable under the increasing flow stress, i.e., the "principle of similitude" invoked by Kuhlmann-Wilsdorf (1968, 1985). If similitude holds, the rate of athermal accumulation of dislocation length during plastic straining, dg+/dF, is, from purely geometrical considerations, and whatever micromechanism is taking place, inversely proportional to the characteristic size of the structure, i.e., dQ+/dFocy/gazTc
2.3.2 The Athermal Work Hardening Limit of Self-Similar Dislocation Structures
As noted by Nabarro et al. (1964), common to most work hardening theories is the assumption of geometrical self-similarity of dislocation structures in the range of a particular deformation stage. This is of course very convenient for the theoretician but, to some extent, is also supported by experimental observations: during one particular stage, the substructure of dislocations, along with any other dimensionally characterizable magnitude related to it (slip line length, etc.), mainly undergoes a
(2-59)
which means, on account of Eq. (2-58), that the athermal extrapolation of the work hardening rate, 90, is a constant, independent of the current value of the flow stress. Any change in the athermal work hardening limit in the 9 — TC plot means that similitude does not hold. Moreover, the net dislocation storage at finite temperature, dg/dF, can be understood as the result of the competition of two dynamic processes: dislocation accumulation and dynamic recovery. Dislocation accumulation is generally admitted explicitly or implicitly - to be an athermal
59
2.3 Work Hardening
process, dictated by the geometry and strength of the obstacles encountered during the long-range gliding of mobile dislocations, i.e., dg+/dF can always be obtained from the current 90 TC value. All the temperature and rate sensitivity is ascribed to the dynamic recovery term, dg~/dF. As TC can be considered only very weakly temperature and rate dependent (we are dealing here with the dislocation-dislocation flow stress contribution), Eq. (2-57) takes the form 9 = 9h (st) - 9S (F, T, st)
(2-60)
190K
3p
single slip orientation 7=1.6 to 4-10-3s1
o CD
20
40 x[MPa]
80
which in the case of similitude will simply be: 9 = 9h-9s(f,T,rc)
(2-61)
2.3.3 The Experimental Work Hardening Behavior Several examples of work hardening behavior in f.c.c. metals are presented in Figs. 2-22-2-24. The different work hardening stages can be clearly distinguished with the help of the combination of TC 9 — TC and 9 — TC plots. The stages cannot be perfectly formulated by simple mathematical expressions, but the essence of the observed behavior can be synthesized as follows. 2.3.3.1 Stage I Only present in single crystals initially oriented for single slip, stage I occupies a range of initial TC values so narrow that it cannot be conveniently studied in these plots. It is almost reduced to a starting base-line point in the 9 — TC plot [Fig. 2-22 (b)] and is merely indicated by a small positive shift in the abscissa of the TC 9 — TC plot [Fig. 2-22 (a)]. 2.3.3.2 Stage II The presence of stage II is enhanced using 9 — TC plots. It can be observed in single
20
(b)
20
40 x[MPa]
Figure2-22. (a) xc0 vs. TC for Ag single crystals at different temperatures. From Mecking (1977); (b) 6 vs. TC for the curves of (a). The heavy curve corresponds to the straight line of (a). From Mecking (1977).
crystals and in coarse grain polycrystals (Fig. 2-23) as an athermal plateau of a level raging from 3 x 1(T 3 G to 6 x 10" 3 G depending on crystal orientation, i.e., of about ten times the stage I work hardening rate. 2.3.3.3 Stage III Stage III is triggered by a critical stress, TIII? strain rate and temperature dependent, related to the SFE of the material, as is well known from the early works on deformation stages. The athermal limit for 3 411 is about 3 x 10 ~ G and its value decreases exponentially with temperature
60
2 Flow Stress and Work Hardening (a)
Cu D= 450/im = 2 - I 0 " 4 sec"1
Figure 2-23. As Fig. 2-22 but for coarse grained Cu polycrystals. From Mecking (1977). 200
300
400
- [MPo]
(e.g., Mitchell, 1964). Stage III is sufficiently well described by a linear region of negative slope after the plateau in the 9 — TC plots of single crystals or as a symmetric parabola in TC 6 — TC plots. Polycrystals show an initial stage of very high but rapidly decreasing work hardening rate roughly in agreement with the parabolic work hardening rate predicted by Eq. (2-54) - seen as a plateau in TC 9 — TC plots - which bypasses stage II, apart from coarse grain sizes.
Stage III curves focus to a common value at zero stress, Of}1, higher than but of the same order of magnitude as 011, and their rapidly decreasing slope leads them to a virtual saturation stress T™, which is
Figure 2-24. As Figs. 2-22 and 2-23, for Cu polycrystals strained in torsion up to large strains (torsional (macroscopic) stresses and strains). Normalized by the temperature-dependent shear modulus (labelled \i in the figure). From Alberdi (1984).
2.3 Work Hardening
473K.0.35
i
373K.0.28 ,
Cu. 7=1.4x10- 2 s- 1
10
77K, I_ = 0.057 473K, 0.35 373K, 0.28 293K, 0.22
61
62
2 Flow Stress and Work Hardening
ature limit), (TSIH)O- I t s value is approximately 5 x 10" 3 G for f.c.c. metals. The latter assertion means that at absolute zero temperature, when 9r = 0, the extrapolated work hardening rate still obeys a Voce equation:
temperature and strain rate dependent, i.e., 0 = C(1-T c /T s m )
(2-62)
which is the Voce equation (2-7) in differential form. Of course actual experimental curves do not exactly conform to Eq. (2-62) and more subtle phenomenological analysis of the curves are available (Estrin and Mecking, 1984; Mecking and Estrin, 1987; Follansbee and Kocks, 1988; Follansbee etal., 1990). The temperature dependence of the work hardening rate in stage III is contained in that of the saturation stress, Tsm. Its functional dependence (Fig. 2-25) is similar to that of the stress triggering stage III, Tm,
0{T = 0K) = C f l -T C /(T S IU )O]
The constant 0" is a common athermal work hardening rate limit for stage III structures extrapolated to TC = 0 (or Q — 0), but the athermal limit of the current work hardening rate (for TC > 0) at constant structure is structure dependent during stage III. That is to say (Sec. 2.3.2) that similitude is broken during stage III. The behavior is shown schematically in Fig. 2-26, where a Voce behavior has been assumed for 9h. To confirm this conclusion, the required data on how strain hardening depends on strain rate or temperature changes at constant structure are lacking. Apparently, only Mecking and Kocks (1976, 1981), Fig. 2-27, have made an attempt to get such data. Clearly more experimental work is needed to definitively settle this matter.
ln(Tsm/G) = ln[(Tsm)0/G0] +
+ kT/(Gb3gs)ln(f/f0)
(2-64)
1
(2-63)
where, as in the case of Tm, gs is a scaling parameter that is correlated with the normalized SFE (Kocks, 1976, 1987; Alberdi, 1984; Mecking et al., 1986). Equation (2-63) implies that there exists an absolute maximum saturation stress (in the zero temper-
-2
10
y-- 410" 4 Cs" 1 ]
10
A (Mi
8
Cu
10
Al 10
0.00
00]
0.02
003
0.04
Figure 2-25. Saturation stress TSIH (normalized by the shear modulus G labelled JU in the figure), vs. normalized temperature, kT/(Gb3), for <111> single crystals of Al, Ni, Cu and Ag. From Mecking et al. (1986).
2.3 Work Hardening
63
e/G
0(T,f,rc)/G e
9(T.f)/G \ r
c/G
7
^rc)/Go
JVG
(rs
Figure 2-26. Stage III ideal behavior. The lower Yoce curve represents current behavior at finite temperature and strain rate. The upper curve is the Voce behavior extrapolated to zero temperature (finite saturation stress) and the intermediate curve is the athermal limit of the current behavior for temperature of strain rate changes at constant structure, i.e., the hardening term of 9(T, f).
an asymptotic athermal work hardening rate for zero flow stress, 3 x 10~ 4 < 9™/G < 5 x 10~4, and leading towards a steady state with a saturation stress, TSIV, which is strain-rate and temperature dependent:
2.3.3.4 Stage IV
Before the virtual saturation stress Tsm is attained, the transition towards stage IV takes place. There is still no complete agreement on the phenomenology of stage IV It is characterized by a temperature and strain rate dependent work hardening rate that, at least for pure f.c.c. metals, admits an idealized Voce-type description with
120
0-^V[1-T/TSIV]
(2-65)
Frequently stage IV is idealized as a stage of a constant but temperature dependent work hardening rate (Hughes, 1986; Rol-
Al<145> T= 453K
80
initial slope 8"
I
transient (~ 3%)
yA = 1.6x10" 5 sec' 1
2. 40
-40
0
50
100
150
200
1/ 7 [10 3 sec]
250
300
Figure 2-27. Effect of strain rate changes on stress-strain curve of an <145> aluminum single crystal at T/Tu = 0.5. Instantaneous work hardening rate after strain rate changes to f from the rate fx which was used in prestraining to 10 MPa (stage III) (strain rate labelled y in the figure). From Mecking and Kocks (1981).
64
2 Flow Stress and Work Hardening
lett, 1988; Hughes and Nix, 1988), but for pure f.c.c. metals or its dilute alloys the slope of the stress-strain curves steadily decreases at large strains (the transition III -> IV takes place roughly around e ~ 1). b.c.c. metals or ocTi alloys deformed by axisymmetric elongation (wire drawing) do effectively show a linear stage IV up to very high strains, but their particular behavior is explained by an extra work hardening term due to the continuous accumulation of geometrically necessary dislocations to cope with the grain "curling" induced by their respective <110> or <1100> textures (Gil Sevillano etal., 1980, Gil Sevillano, 1991). Solutes can enhance f.c.c. work hardening besides their additive increase of the flow stress (see Neuhauser and Schwink, Chap. 5 of this Volume). This enhancement could be responsible for the constant work hardening rate observed in some f.c.c. alloys. Some linear stress-strain curves of f.c.c. alloys at large strains have been reported by Gil Sevillano et al. (1980). Moreover, other functional forms different from the Voce equation can reproduce the curvature shown by most of the experimental 9 — TC plots during stage IV, e.g., the parabolic 9 — TC relationship 9 = 91Y[1-(TJT™)2]
(2-66)
that corresponds to a hyperbolic tangent work hardening curve. However, there are some indications that the end tail of stage IV 9 — TC curves, where the loss of work hardening ability accelerates relative to the linear decay, could be the result of structural damage (Aernoudt et al., 1987) or it could be, in some cases, a genuine new stage V (Brion etal., 1981; Siethoff and Schroeter, 1984; Brion etal., 1986). Mecking and Embury (quoted by Rollett etal., 1987) have remarked that, at the transition from stage III to IV, the work hardening rate 9T scales with the transition
stress, TT, i.e., ^ n
III
: CT,
(2-67)
Mecking and Estrin (1987) have confirmed the general validity of Eq. (2-67) for a great variety of materials, temperatures and strain rates. The constant c has a value ranging from 0.05 to 0.1 (Fig. 2-28). Equation (2-67) shows that the temperature and strain rate dependent processes underlying 0IV, i.e., its dynamic recovery processes, are probably the same that determine TSHI, i.e., the dynamic recovery during stage III. Alberdi (1985) has also shown that the saturation stress after stage IV, TSIY, has near the same temperature dependence as TSIH [Eq. (2-63)] as evidenced in Fig. 2-29. Accordingly, the same conclusions derived in Sec. 2.3.3.3 for stage III apply for stage IV A true saturation stress exists for pure f.c.c. metals strengthened by plastic deformation, and it has an absolute limit at zero temperature, TQV ~ 7.5 x 10~ 3 G. 2.3.3.5 Synthesis of Behavior: From Stage I to the True Saturation Stress, TSIV Figure 2-30 represents an idealized 9-xc plot of an f.c.c. single crystal initially deforming by single slip, from its easy glide stage I till its steady state, TSIV, when its athermal hardening ability is entirely balanced by dynamic recovery. A fine grain polycrystal of the same material directly starts with multiple slip behavior (stage III): the observation of stage II in coarse grain polycrystals could be due to the existence in their grains of a core initially deforming by near single-slip conditions, or from the contribution of surface grains. Considering the whole work hardening history, it is apparent that in most general cases, the crystals suffer a burst of strong work hardening (stages II and III) between two periods of very weak hardening ability,
2.3 Work Hardening
H
d
65
Y
Figure 2-28. Enlarged view of the stage IV region of (9 T plots of aluminum, copper and austenitic stainless steel (decreasing SFE). Torsional work hardening rate vs. torsional shear stress. An orientation factor M ~ 1.5 is to be used for converting in 6 = &/M2, TC = xjM. From Alberdi (1985).
ending - if fracture does not intervene - in a steady state of saturation. Moreover, a fact that apparently has been overlooked up to now, the work hardening rate of the final stage of weak work hardening ability, stage IV, extrapolates back to the initial stage I. Quantitatively, the available values of the athermal work hardening limit of stage IV, #oV> a r e °f the same order of the mid or lower-bound values of 9l. Stage I is considered to be the intrinsic baseline of the work hardening of a crystal, only perceptible for the near-perfect lattice, although quasi independent of its initial structural imperfection level (Nabarro et al, 1964; Nabarro, 1986). It is suggested here that this contribution is
masked in the early stage (in terms of stress, not in terms of strain) by the work hardening burst of stage II and only emerges again after stage III is completed. In polycrystals, the remaining contribution to work hardening from the presence of the grain boundaries (Eq. (2-54) with proper factor) after large strains is presumably superposed on the near-perfect lattice contribution. Actually, theoretical models developed up to now for stage IV are in good harmony with this view. All of them consider the simultaneous action of two work hardening processes, one of them mainly responsible for stages II and III and the other, a residual one [debris from recovery (Rollet et al., 1987), interplanar capture of
66
2 Flow Stress and Work Hardening
log (TIV G )
-2.2
-
-2.4 -
-2.8
-
-3
0
0.2
0.4
0.6 2
0.8 3
1(T kT / Gb
O
Al
•
Cu
A
SS 321
Figure 2-29. (a) Temperature dependence of the saturation stress TSIY of Al, Cu
0.2
0.4
0.6
1(T2kT / Gb3
0.8
1.2
and AISI 321 stainless steel. Data taken from Alberdi (1985). Compare with Fig. 2-35. (b) Ratio TsIV/xsni vs. normalized temperature, as in part (a).
2.3 Work Hardening
67
0[IO-3G1
6
o
.. Mll
.-• — •
f -~
1
™~
2
3
4
r c [io" 3 G] Figure 2-30. Idealized complete work hardening behavior of f.c.c. single crystals (sc) or polycrystals (p).
dipoles at the near-clean cell interiors (Prinz and Argon, 1984)], evolving further than stage III, responsible for the weak but persisting stage IV. In fact, in the Prinz and Argon model, the mechanism of dislocation accumulation assumed for stage IV is the same mechanism accepted today for stage I dislocation accumulation (e.g., Nabarro, 1986). The parallelism between stages I and IV is still higher. The terminal stage IV substructure is constituted by subgrains (Gil Sevillano et al., 1980; Gil Sevillano and Aernoudt, 1987). A grown-in dislocation substructure in Zn or Cu single crystals in the form of sub-boundaries has been shown to affect the flow stress level of the crystals, but not their stage I work hardening slope (Edwards et al., 1953; Nabarro, 1986). This further supports the previous conclusion on the substructural evolution
during stage III towards substructures contributing to the strength by the forest mechanism, but unable to accumulate Orowan loops anymore, i.e., 9h = 0. 2.3.4 A Work Hardening Model Consistent with the Micromechanisms Determining the Flow Stress 2.3.4.1 The Athermal Hardening Term, 0 h
The flow stress, as explained (Sec. 2.2.6), is controlled by the interaction of the gliding dislocations with the forest dislocation density all through stages I to IV. Backstresses are not considered relevant for the macroscopic stress level (Sec. 2.2.8.3) and the coplanar interaction stresses between dislocations gliding in parallel planes can only make, at most, a contribution to the flow stress much weaker than the forest contribution. A new problem arises now:
68
2 Flow Stress and Work Hardening
the athermal envelope of the 9 — % curves makes a strong temporary jump between two periods of extremely low work hardening ability. However, in principle, forest hardening is accompanied - but for absolutely regular dislocation distributions, which is not the case in the slip-controlled regime - by a high rate dislocation storage leading to an athermal work hardening term, 0h, which can be very important. This can explain stages II and III, as set out below. Further explanations have to be sought for the near absence of work hardening during stages I and IV. In the absence of regularity in the position and strength distribution of the dislocation trees, the mobile dislocation penetrates preferentially the softer areas of the slip plane, circumventing the harder spots where dislocation density or tree strength are higher than average. This occurs because (Fig. 2-9) a large portion of the forest consists of hard obstacles (Sec. 2.2.5). At the critical stress when the mobile dislocation glides over its slip plane (the percolation threshold), unsheared islands are always left behind - even for a random distribution - encircled by concave dislocation loops. These loops contribute to the forest density of the other non-coplanar slip systems. Thus, under multiple slip conditions, the flow stress from the forest hardening implies a particular athermal hardening rate, 9f. This hardening term and other related implications arising from the forest hardening model were early pointed out by Kocks (1966) in his "statistical work hardening model", recently developed and extended (Kocks, 1985). Kocks' estimates of df and computer simulations of the dislocation percolation process through an f.c.c. random forest (Hernandez Olivares and Gil Sevillano, 1987) give work hardening rates lower than but of similar order of magnitude to 611 or 0"1
values for random or cellular areal distributions of the trees (Fig. 2-31). Some of the discrepancy between calculated and experimental values arises from an imperfect simulation of the f.c.c. forest (the forest used was not as strong as the real one judging by the calculated a values). Moreover, only the stored length corresponding to the first-arriving dislocations gliding through the softer areas - the concave loops - has been considered for computing 6f. In fact, these loops arising from glide in near-by parallel planes stack in the third dimension, around the denser bundles of the forest, developing the polar distribution of internal stresses typical of a composite (Sec. 2.2.8.3): positive (in the sense of favouring glide) long-range stresses inside the hard volume not penetrated by the gliding dislocation, and negative, shortrange back stresses in the slipped volume. The exact stress tensor inside the hard zone depends on the composition of dislocation loops captured around its boundary, but, as long as the hard zone remains unrelaxed by plastic deformation, the stress level increases elastically in proportion to the mismatch. Only a very small strain difference must be reached between hard and soft zones to produce dislocation glide inside the hard volumes. This difference is continuously adjusted and, from the plastic point of view, the hard zones are deforming like the softer part of the crystal, although by the activation of a set of slip systems not necessarily coincident with the macroscopic one. Accordingly, some dislocation storage can occur inside the hard spot and, roughly, the same number of convex loops reaches the boundary of the hard spot coming from its inside as the number of concave loops coming from the matrix are stopped at the outside. In general - and particularly when the hard volumes form layers nearly parallel to slip planes, as
2.3 Work Hardening
found in stage II in single crystals or early in stage III in multiple glide single or polycrystals - the slip systems activated inside do not coincide with or are not limited to the matrix slip system (Torrealdea and Gil Sevillano, 1983; Martin Meizoso et al, 1984; Jackson, 1983) and some stabilizing reactions between the two loop groups will take place, perhaps increasing 0f over the value computed purely from the Orowan loops. It is to be noted that dislocations not pertaining to the macroscopic active set cannot escape from the hard volume to the matrix. Under polyslip conditions, the active set of slip systems inside the hard spot is not expected to be very different from the macroscopic one, but this is only a guess. In this case annihilation reactions can occur. The previous discussion offers a plausible explanation for the disagreement between experimental and calculated 9f values, but, more importantly, it is the basis for a qualitative understanding of the building up of heterogeneous cellular structures with an associated internal stress pattern and local misorientations, in very good
(a) 'y\
•*>.
• * "\ •
* •*''
4m -•' f
• .
»•"
69
'.v
J7
•
1 (b) A
3
a
a b
-Q-
7~
-
X
C2 .
O d, • d2
a -C
< 2
-i t-
-
o Figure 2-31. (a) Some geometrical arrangements of obstacles simulating a random f.c.c. forest, (b) Athermal work hardening rate
2 1 )—
0.10
I
i
i
0.20
0.30 a
040
vs. strength [oc = xc/(Gby>/Q)] for 0.50
the six spatial distributions of the f.c.c. forest of (a).
70
2 Flow Stress and Work Hardening
agreement with direct and indirect experimental observations as described in detail by Kocks (1985) (see also Kubin, in Chap. 4 of this Volume). As could easily be anticipated, computer simulations show that the athermal work hardening depends on the spatial distribution of the trees. As indicated in Fig. 2-33, 0f is maximum for near-uniform distributions of dislocations with narrow channels of lower than the average dislocation density, and decreases as the clustering of the dislocations increases, from cells of thick walls to subgrain-like structures with twodimensional boundaries. In fact, an ideal equilibrium sub-boundary with a regular distribution of dislocations retains no dislocation length at all when crossed by a gliding dislocation. Thus, the athermal Voce-type 9[T = 0,si(T)]-zc[si(T)] evolution during stage III is - at a qualitative level - explained by the strain-induced structural evolution from near-uniform spatial dislocation distributions to cells of increasing wall sharpness and progressive cleaning to cell interiors, leading towards ideal subgrains unable to accumulate any additional dislocation length. Such evolution is indeed experimentally observed (Gil Sevillano et al., 1980). 2.3.4.2 The Athermal Saturation Stress,
If it is accepted that the main dislocation accumulation mechanism for work hardening is based on the formation of Orowan loops around impenetrable volumes (cell walls) for dislocations gliding in the softer matrix of the crystal, the athermal saturation stress, (Tsm)0, must be set by the exhaustion of this accumulation process. As mentioned, the dislocation length stored is not exactly the Orowan loop, but the result of its reaction with the dislocations resist-
ing its glide and with those corresponding to the slip systems activated inside the hard volume and expanding from its interior. As described above, the transition from stage III to IV is accompanied by an evolution towards subgrain structures that ideally should not be able to retain any Orowan loop if their dislocations were located with perfect regularity. This is hardly the case and, although the transition III -» IV can be roughly ascribed to a structural transition from cells to subgrains (Gil Sevillano et al., 1980) the structures at the transition itself are still mixed in character, combining sub-boundaries with loose tangled cellular walls (Alberdi, 1985; Hughes, 1986). Then, the saturation stress TSIH - even its athermal limit (Tsm)o - is better explained by a total annihilation of the arriving Orowan loop by interaction with dislocation loops of the same slip system expanding from the inside of non slipped areas, than by assuming that no loops at all are held up by the denser spots. Of course, the structural evolution makes such annihilation easier. The internal stresses are built up by retaining the necessary number of loops around the hard volume; the internal stress level in the hard volume (the cell wall) is inversely proportional to the loop spacing, h. For instance, in an f.c.c. crystal, inside a planar cell wall of finite thickness parallel to the B slip plane, (111), receiving loops from the CI system, (TTl)<001>, (Taylor and Elam nomenclature), four slip systems are equally stressed, among them the CI system. The stress level is (T.)h = 0.6 G b/[2 h(l~ v)]
(2-68)
In general, the numerical factor of the proportionality will vary depending on the orientation of the cell wall relative to the slip system concerned. Beyond some critical value of/?, loops of the CI system will be
2.3 Work Hardening
activated inside the wall and they will pass at a distance < h/2 of a corresponding loop of opposite sign. For multiple slip, the complex effect of the net of dislocations collected by the hard spot is expected to lead to relationships formally similar to (2-68) for every slip system. Essmann and Mughrabi (1979) have discussed the annihilation of both screw and non-screw closely spaced dipoles during gliding at low temperatures, and the contribution of such processes to dynamic recovery and saturation stress. From the observation of structures after very low temperature deformation, they estimate that both screw and edge dislocation dipoles annihilate athermally for dipole heights below, respectively, 50 and 1.6 nm in copper, i.e., in terms of the Burgers vector, 1956 and 6 b. Then, in a multislip situation, the saturation stress will be reached when h/2 <6b for the different components of the grid of geometrically necessary dislocations surrounding the cell walls. Its value will be determined by the lowest of the resolved internal stresses in the systems correspondent to those components, i.e., (h/2)max<6b. Thus (*.ffl)o = fx
[(^iUmin *
fPG/[24(l-v)]
where / is a factor easily derived from the composite model of the flow stress (Sec. 2.2.8.3) and 4 > p > 0 is an orientation factor (for copper at room temperature, according to Fig. 2.17, / ^ 2 ) . The experimental value for f.c.c. polycrystals, (Tsm)0 ~ 5 x 10" 3 G is explained if ( / P) ~ 0.1. The criterion (h/2) < 6 b is equivalent to a critical misorientation across the non-penetrable boundaries. 2.3.4.3 Dynamic Recovery
For finite temperatures, thermal activation can help the dislocation loops of op-
71
posite sign to mutually annihilate even if they are passing each other at hypercritical distances. The screw dipoles will annihilate by cross-slip. The boundaries of the hard spots, where the gliding dislocations are constricted against the strongest trees, will constitute the preferred site for dynamic recovery processes. The critical distance for self-annihilation of edge dipoles has been observed to be athermal (Essmann and Mughrabi, 1979). The saturation stress of stage III, tsm, is however temperature dependent. This has led some authors to propose diffusion processes as responsible for the dynamic recovery events taking place in the cell walls (Prinz and Argon, 1984; Nix et al., 1985; Nabarro, 1989). The first two groups of authors favor the operation of a static recovery process, involving climb of edge dislocations. However, the inadequacy of an approach to finite-temperature work hardening as a combination of strain hardening and static recovery (the "BaileyOrowan" approach) when confronted with experimental results has repeatedly been discussed by Mecking and Kocks (1976, 1981). By contrast, Nabarro (1989), after reviewing the operation of five possible temperature-dependent mechanisms for stage III dynamic recovery, suggests that it is controlled by the climb of edge dislocations mediated by vacancies produced by plastic deformation, a truly dynamic recovery process. At the cellular walls, dislocations are retained by attractive junctions that cannot be directly broken under the flow stress with the help of thermal activation. The attraction between dislocations of opposite sign on either side of the polarized cell wall offers the driving force for recovery (Kocks, 1985), but the undoing of the strong junctions requires some indirect mechanism capable of being thermally activated. The
72
2 Flow Stress and Work Hardening
mechanism could start by the freeing of dislocation segments linked to nodes, where the dislocation is constricted, by thermally activated cross-slip according to the mechanism proposed by Friedel and Escaig (Escaig, 1968; Bonneville and Escaig, 1979). Caillard and Martin (1983) have observed the operation of a dynamic recovery mechanism of this type in aluminum at intermediate temperatures. It involves the double cross-slip of the screw part of loops of sign opposite to that of the loops captured against the walls, so that their non-screw segments can be mutually annihilated without climb (Fig. 2-32). Pris-
©
matic loops will be left as debris, which will need diffusion for complete annihilation (Nabarro, 1989). The operation of this mechanism appears most plausible in the conditions prevalent at the cellular arrays developed by the process of percolation of dislocations, probably the essential process of substructure formation during stage III. In conclusion, the exact nature of dislocation reactions involved in the recovery term of finite temperature work hardening during low temperature stage III is not completely clear. The process is dynamic in nature and probably relies strongly on cross-slip of screw dislocations, although every mechanism proposed leaves debris which would require diffusion for complete annealing. Dynamic recovery by diffusion at low temperatures cannot be excluded because of the high vacancy concentration generated by forest cutting zones of high local dislocation density, where the recovery events presumably take place. Phenomenologically, the Voce equation (2-62) was interpreted by Kocks (1976) as the balance from an accumulation term (assuming similitude) C = 6h
(2-59)
and an annihilation term that should vary as /dF
® Figure 2-32. Annihilation of edge dislocation lines through cross slip of screw segments from a small dislocation loop of opposite sign expanding on a parallel slip plane. Mechanism of Caillard and Martin (1983), scheme adapted from Weertman and Weertman (1987). The two edge dislocation lines could be products of rearrangement of an Orowan loop around an impenetrable zone. A thick cell wall and the expanding loop are a result of the internal stresses inside the wall volume.
CCQ - *
9TOCTC
(2-70)
in order to arrive at the Voce equation. The proportionality (2-70) is difficult to explain. Kocks (see also Prinz and Argon, 1984) considers that dislocation annihilation processes take place in localized "recovery sites" of the structure of a number proportional to Q per unit area of glide plane. Then, a constant length must be assumed to be annihilated per site in order to obtain (2-70). It appears much more logical to assume that annihilated length scales with the characteristic dimension of the disloca-
2.3 Work Hardening
tion structure, Q 1/2 , but then 6Toc6h. This is compatible with Eq. (2-64), i.e., with an athermal current work hardening rate obeying a Voce law, as extrapolations to zero temperature suggest. But this athermal non-self-similar evolution of the dislocation structure with strain remains to be fully explained. As Nix et al. (1985) remark, it is necessary to have a clear picture of strain hardening without recovery before claiming to understand dynamic recovery. 2.3.44 Stage II Stage II has been considered to be the backbone of work hardening behavior of crystals (Nabarro et al., 1964) and most of the theoretical efforts devoted to work hardening have been directed towards its rationalization. In fact, it constitutes a particular case of athermal stage III type hardening shown by crystals macroscopically deforming by single glide. What is most striking is that multiple glide oriented single crystals or polycrystals only show, asymptotically, an athermal hardening rate, 6Q1 for negligible stresses (zero stress back extrapolation), whereas stage II is a pure athermal hardening stage lasting up to stress levels Tm, at which dynamic recovery is already very active in stage III deforming crystals (compare Fig. 2-24 with Tm values in Fig. 2-35, below). The end of stage II, Tm, has been linked to cross-slip activity from early studies of dislocation plasticity. For a long time, the most widely accepted model for the onset of stage III considered Tm to be the critical stress for cross-slip at the head of pileups held up by obstacles. The critical point was assumed to be the compression of the stacking-fault width of the leading dislocation up to a limit where the activation energy for the total constriction needed for slip plane change could be thermally supplied (Seeger, 1957). The model has been
73
used to deduce SFE values in good agreement with other more direct SFE measurement methods (Haasen, 1967; Ahlers, 1970). However, the model is confronted by serious objections (Nabarro, 1986; Bonneville etal., 1988): cross-slip is abundant at stresses lower than t m , pileups with the necessary number of dislocations are not observed and even the t m method for deducing cross-slip activation energy is questionned, as it compares stresses not corresponding to constant structure. The observation that dynamic recovery is active in multiple glide crystals below Tm is rather puzzling. Stage II structures are layered structures (Mughrabi, 1975; Basinski and Basinski, 1979). They are formed by slabs of low dislocation density separated by sheets of interacting primary and secondary dislocations lying almost parallel to the primary plane. The slabs are rotated with alternating sign and periodic internal stresses are also observed, varying from 0 to TC. The dense sheets of high dislocation density are composed of bundles of dipoles - inherited from stage I - and networks of interacting primary and secondary dislocations with a predominance of Lomer-Cottrell locks for decreasing SFE or temperature. Formally, most multiple glide structures - typical of stage III are relatively similar: layered structures formed by slabs of high dislocation density misoriented relative to the "matrix" of lower dislocation density, and initially near parallel to the active slip planes (Kawasaki, 1974,1979; Kawasaki and Takeucki, 1980; Torrealdea and Gil Sevillano, 1982; Gil Sevillano and Aernoudt, 1987). However, the activity of several slip systems leads to the coexistence of two or in general three families of slabs, leading to development of checkerboard structures (e.g., see Hughes, 1986 or Hughes and Nix, 1990 for very good examples), as in Fig. 2-33.
74
2 Flow Stress and Work Hardening
1
Figure 2-33. Double layered structure of copper, early stage III, room temperature rolling, s = 0.10 (r ~ 0.30), o = 255 MPa D.I.
The origin of these planar dislocation formations has been thoroughly reviewed by Jackson (1985 b), and appears to be the same in both cases. Groups of dislocations in lightly strained crystals can glide through large areas. Glide is punctuated by crossslip at existing defects and the cross-slip of a dislocation group produces box-like, planar multipole arrays parallel to the glide plane of the travelling dislocation group (Fig. 2-34). The internal stresses of the multipolar array are relaxed by internal secondary slip. Secondary dislocations react at the multipolar boundary, stabilizing the double-walled planar array.
Longitudinal section, rolling direction labelled D.L. From Torrealdea (1981).
Thereafter, the layered dislocation patterns evolve differently in single glide and multiple glide crystals. Relaxed multipoles are only slightly rotated from the primary plane. They begin to trap dislocations coming from outside (gliding dislocations) and from inside (activated by means of the internal stresses). In the single-glide case, similitude is approximately maintained. The contribution of gliding dislocations remains the same (the plastic strain is still carried almost entirely by primary dislocations) and inside the multipolar array the internal stresses do not favor activation of the primary system; the macroscopic stress,
2.3 Work Hardening
75
(a)
(b)
\
V
Figure 2-34. (a) Sketch of formation of multipolar arrays by cross-slip of a processing of gliding dislocations, (b) Sketch of the plastic relaxation of a multipolar array by secondary slip (dotted) activated by the internal stresses of the dipolar array. In fee. crystals, the primary and secondary interaction forms Lomer-Cottrell locks LL. After Jackson (1985 b).
76
2 Flow Stress and Work Hardening
on the other hand, does not favor the glide of secondary systems apart from in nearly symmetrical crystals, so that dislocations generated inside the multipole cannot escape from it. Accordingly, for singleglide crystals, dynamic recovery by annihilation of gliding dislocations of the matrix with dislocations from the interior volume of the multipole can hardly occur, and this offers an explanation for the athermal character of stage II. By contrast, in the multiple-glide case, encounters of arriving matrix dislocations (or their reaction products) with dislocations of opposite sign activated inside the cell-wall embryo can always occur - Eq. (2-69) with /? > 0 - and the interface of the double-walled array will act as a preferred dynamic recovery site right from the start of deformation, as observed (for polycrystals, TC > 0.5 x 10" 3 G, Fig. 2-24). The temperature dependence of the stress for the onset of stage III is different from that of the saturation stress at the end of this stage (Fig. 2-35). The negative slope of the log(i/G) versus T plot is approximately double for TIH than for Tsm, i.e., the activation energy for the onset of dynamic recovery - the first annihilation process, involving perhaps only the cross-slip of screw components - is about half the activation energy for saturation, requiring a process for annihilation of edge components. The factor of two could be speculatively linked to a cross-slip event from one or two constrictions for the first and second processes, according to the cross-slip model of Friedel-Escaig (Bonneville et al., 1988). 2.3.4.5 Stage I Paradox The phenomenology of stage I is apparently paradoxical. Its flow stress is controlled by the density of forest dislocations,
-3.7 0
0.5 102kT/Gb3
Figure 2-35. Temperature dependence of TIII5 stress for the onset of stage III in copper single crystals (band of results compiled by Mitchell, 1964), compared with that of ism, the saturation stress of the same stage measured in torsion of copper polycrystals (from data of Alberdi, 1985, transformed with
as in stages II, III and beyond. It is most likely that the same interaction mechanism between gliding (primary) dislocations and the attractive part of the forest (secondary dislocations) is determinant of the flow stress in all deformation stages. Such interaction is responsible for the high athermal work hardening rates of stage II or early stage III. However, the work hardening rate during stage I is extremely weak. The athermal hardening rate associated with the forest is dependent on its composition and spatial structure and it has been observed to decrease linearly throughout stage III as the clustering of dislocation structures increases (Sees. 2.3.3.3 and 2.3.3.5). However, the forest dislocation structure of stage I, as evidenced by etch pits on the primary plane, is uniformly distributed, more akin to the structure of stage II than to low-energy configurations,
2.3 Work Hardening
yet it is unable to retain dislocation loops that would generate internal stresses and secondary slip. It remains self-similar during stage I, with a ratio of primary to secondary dislocations of about ten. Rather suddenly, an intense storage of secondary dislocations starts, leading to stage II. There is no doubt that such sudden activity of secondary slip is due to the development of internal stresses (Basinski and Basinski, 1979). It seems that for some reason the very effective process of dislocation storage operative in stage II begins to work. A factor contributing to this transition is the size effect associated with the virtual mean free path of gliding dislocations. For any possible athermal hardening mechanism, 0 h , a virtual mean free path can be defined as the distance, A, to be moved by a dislocation of unit length in order to store a unit length of dislocation (Nabarro etal, 1964; Ashby, 1970, 1971; Kocks, 1976). According to this definition and on account of Eq. (2-61), A=
77
suggest that size effects could be a determinant for the observation of stage I in single crystals and the onset of stage II, without resort to any structural transition or exhaustion of the structural pattern that makes possible stage I as invoked by most models. The mechanisms of stage I have been recently reviewed in detail by Nabarro (1986). Qualitatively, the foundations of stage I behavior are well agreed and a most detailed statistical theory has been developed by Argon and coworkers (Argon and East, 1968; Argon, 1969). All dislocation configurations observable during stage I are resolvable into dipolar groupings ("multipoles" or dipolar "braids"), although the friction stress level, SFE and particularly short-range order influence the aspect of the dipolar arrays (Gerald and Karnthaler, 1989). No internal stresses with a range longer than the dislocation spacing are observed (Mughrabi, 1975).
(2-7.)
2(9JG)(TJG)
If the size of the crystal is S > A, the virtual mean free path can be considered undisturbed and the hardening mechanism operative in the crystal volume. If S <^ A, its contribution will be negligible. For stage II, 9h = 6n a2b
16b
(2-72)
The end of stage I at room temperature occurs in copper at x c ^ 5 x 10~5 G (Nabarro, 1986). This means S > 80 fim for stage II to be observed. Basinski and Basinski (1979) note that 6U barely decreases for decreasing crystal size until a critical value S ~ 200 jim, below which there is a sharp drop (Fig. 2-36). The reasonable agreement between both figures
0.1
1 Lemm
10
Figure 2-36. Rate of hardening in stage II, 0 n , as a function of size (glide path for edge dislocations Le) for Cu crystals deformed at room temperature. Plotted by Basinski and Basinski (1979). Data from Fourie (1967, 1968).
78
2 Flow Stress and Work Hardening
Dislocation glide distances appear to be comparable to the crystal size and almost no size effect is observed for the stage I work hardening rate 9Y (Basinski and Basinski, 1979). The processes occurring during stage I arise from the interaction of primary dislocation loops expanding under the influence of the applied stress in neighboring parallel slip planes. When screw-oriented segments of opposite type pass closer than a critical spacing, ys9 their mutual attraction stress is sufficient to overcome their lattice friction stress (plus other - solid solution, etc. components) on the cross-slip plane and mutual annihilation follows. The value of Gb sin/?c s
(2-73)
2TI(TO-STC)
where /?cl is the angle between the primary and the cross-slip plane, T 0 the effective "friction" stress on the cross-slip plane and s the ratio of Schmid factors for the crossslip and primary systems (Taylor and Christian, 1967). This process probably does not require thermal activation and leads to the formation of segmented long edge dislocations predominating in stage I structures with forest segments of maximum height ys. The forest density fixes the flow stress value, i c , but, as argued above, it is unable to store any significant dislocation length through the percolation process. However, mutual capture of edge dislocation segments gliding at a distance lower than the passing distance, /ze, occurs, forming dipoles [Eq. (2-20)],
K=
Gb 8TI(1-V)(T C -T 0 )
(2-20)
Primary edge dipoles are swept by free dislocations, forming dipole bundles against any present obstacle (grown-in dislocations or sub-boundaries). The dipolar ar-
rays do not represent an obstacle to free dislocations, at least for low friction stress materials (Hazzledine, 1971). Capture of gliding dislocations can occur at multipoles or by mutual interaction of gliding edge segments. The critical distances ys and he are related through the flow stress level, TC. Applying statistical arguments, other characteristic lengths of the structure can be scaled with them (Nabarro, 1986). The elaborate theory of Argon and East (later modified by Argon, 1969) analyzes the statistics of capture to calculate the rate of growth of multipoles and accounts also for the probability of formation of dislocation sources in multipoles and for their exhaustion. Imposing the condition of constant dislocation flux through the crystal section, the stress-strain curve corresponding to constant strain rate, f, is derived by Argon and East in reasonable agreement with experimental measurements. In concentrated solid solutions with short range order, slip induces a softening effect. Then extensive deformation during stage I occurs non-homogeneously by propagation of Liiders bands, with negligible work hardening rate (Gerold and Karnthaler, 1989; see also Neuhauser and Schwink, Chap. 5 of this Volume). 2.3.4.6 Stage IV and Its Connection with Stage I
The mechanisms of annihilation of screw segments and trapping of edge ones in the form of dipoles must be operative as long as traffic of dislocations of opposite sign exists in the lattice, i.e., as long as plastic deformation by crystallographic slip is going on. After the burst of hardening represented by stage II or during early stage III, the contribution of stage I type accumulation mechanisms to total work hardening
2.3 Work Hardening
is of little relevance, but once the saturation of stage III is attained, they are again the only available direct dislocation accumulation processes. However, only the mutual trapping of gliding dislocations themselves is now possible, the dipolar bundles constituting the basis of stage I structure being absent. Therefore, quantitatively, a work hardening rate lower than 0, is expected from this origin. In fact, this contribution has been considered by Prinz and Argon (1984) as responsible for stage IV work hardening. A much simpler evaluation of this contribution than that elaborated by the last authors is presented below. Once Tsm is attained, the virtual mean free path corresponding to stage III processes can be considered unlimited. The gliding dislocations will however still interact in the cell interior. Given the stress level, let us assume that cross-slip is almost immediate for screw segments and that strain is carried out entirely by edge components. If hc is the passing distance for edge dipoles [Eq. (2-20)] and ye the mutual annihilation distance for edges (~6b according to Essmann and Mughrabi, 1979), the dislocation accumulation rate is, (2-74)
where Qm is the mobile dislocation density and P > 1 a factor accounting for the simultaneous presence of several dislocation families. On account of Eqs. (2-58), (2-38) and (2-20) and for y, ~ 6fc, C / G * (QjQ) [3 x 10- 2 -
3(T C /G)]
(2-75)
Equation (2-75) is a Voce function with a work hardening limit for TC = 0, {0™)o/G ~3xlO~2(Qm/g) and a saturation stress for the steady state corresponding to he = ye, i.e., for the athermal limit of ye ~ 6 ft, (0SIV)O ~ 10" 2 G. The latter value
79
is comparable to the experimental one (7.5xlO~ 3 G for copper, Fig. 2-37). The back-extrapolated work hardening rate is proportional to the ratio of mobile to total dislocation density. For copper, Follansbee and Weertman (1982) estimate it to be in the range 3 x 10~ 4 to 3 x 10" 3 (r ~ 0.3, stage III). Follansbee et al. (1975), at high strain rates, found it to decrease slightly with increasing strain, from 10 ~2 to 2 x 10" 3 for r ^ 0.15 to T - 0.60. Taking it as 3 x l 0 " 3 (the same value obtained by Prinz and Argon in their model through rather intricate procedures), an athermal limit 0 Q V ~ 1 O ~ 4 G is predicted, which is lower than but of the same order of magnitude as the experimental values (3 to 4xl0"4G). The saturation stress for stage IV is limited by ye9 the self-annihilation distance for edges, and it was also this parameter which fixed the saturation stress for stage III according to our description. This explains in a very natural way why both saturation stresses have the same temperature dependence, because the same recovery process is expected to be active to reduce the stored length of dislocations in the cell walls at finite temperatures. In the case of stage IV, however, dislocations are incorporated as dipole pairs swept by other mobile dislocations, the individual Orowan loops (or their reaction products) being immediately annihilated beyond TSIH. This implies also that no net Burgers vector increment is incorporated in the cell boundaries and misorientation increments are only possible through recovery rearrangements of the wall structure. Indeed, a noteworthy decrease of the rate of misorientation growth has been observed at the onset of stage IV (Torrealdea, 1981; Alberdi, 1985). Rollett and coworkers (Rollett, 1988; Rollett et al, 1987, 1988) have proposed a different explanation for stage IV, although
80
2 Flow Stress and Work Hardening [0001]
(a)
100 Shear strain,
200
300
%
Figure 2-37. Work hardening curves of single crystals of different materials, initially deforming by single slip. To be compared with Fig. 2-1. (a) Zn, room temperature, influence of orientation (Liicke et al., 1955), (b) NaCl (Hesse, 1964). 10 15 Shear strain a (%)
also based on the coexistence of two work hardening processes characterized by two structural parameters evolving differently with strain, one of them temporarily masked by the other. They assume that, all along stage III, every recovery event is not perfect, leaving some residual fraction, / , of difficult to annihilate debris. In particular, this debris is assumed to consist of prismatic loops arising from the collapse of
25
edge dipoles. Accordingly, all along stage III a density of debris accumulates,
dr
=f
dr ' (2-76)
which gives way to an increasing additional hardening contribution. At the end of stage III and beyond, (0h)dOCT III
2.3 Work Hardening
proportionality explains in a very simple and elegant way the empirical law (2-67). Beyond TSIH, debris is generated at a constant rate and needs diffusional processes for recovery. The accumulation of debris and its contribution to the flow stress was claimed by Saimoto and coworkers to explaining the increasing deviation from the Cottrell-Stokes Law at increasing strains, as mentioned in Sec. 2.2.6.4. The debris contribution should be seen as a complement more than an alternative to hardening by dipole trapping as suggested above. The debris in the form of small prismatic loops, as remarked in Sec. 2.2.6.4 only constitutes weak obstacles for gliding dislocations, i.e., penetrable obstacles, with no ability to increase the dislocation density by the Orowan looping mechanism. Dislocation density has been, however, shown to increase beyond TSIH (Gil Sevillano et al., 1980, Zehetbauer et al., 1988). Finally, another interpretation of stages III and IV based on a two-parameter description of cellular structures much like that of Prinz and Argon (1984) was presented by Nix et al. (1985) and has been recently reappraised by Haasen (1989). 2.3.5 Influence of Crystal Structure
In their classic paper, Nabarro et al. (1964) reviewed the plasticity of single crystals of many different lattice structures. More recently, Sestak (1979) has published a new survey on this subject, based on the influence of the interaction of the dislocation core with the crystal lattice on dislocation mobilities and resulting differences in plastic behavior of f.c.c, h.c.p. and b.c.c. metals. Haasen (1985) has reviewed the plasticity of ionic crystals, where the earlier work of Argon et al. (1972) in NaCl showed similarities with f.c.c. behavior and underlying hardening processes. Finally,
81
the analogies and differences of high temperature plasticity of diamond cubic silicon and germanium with cubic metals have been studied by Brion et al. (1981), Siethoff (1983) and Siethoff and Schroter (1983, 1984). The first conclusion from such surveys is that stress-strain curves and the temperature dependence of work hardening behavior are strikingly similar for different crystal structures in regimes of relatively high dislocation mobility for glide and crossslip. Moreover, remarkable similarities have been found between the underlying dislocation structures of different crystals for equivalent hardening stages. The comparison of work hardening curves of different materials (e.g., Fig. 2-37) exhibiting the same successive stages described for f.c.c. metals is definitive. Quantitative differences are understood in terms of their different Peierls stress and stacking-fault energies, and in their different possibilities for dislocation-dislocation interaction. Below some critical temperature, the mobility of screw dislocations is severely restricted in b.c.c. metals and they display a unique low-temperature regime of parabolic hardening. TEM observations show straight dislocations parallel to the screw orientation. Plastic flow occurs initially ("stage 0") by exhaustion of edge segments, whose t p is about one order of magnitude lower than the Peierls stress of screw segments. This leads to an increase of the density of screw dislocations which then remain straight whilst moving. The transition to the intermediate-temperature regime occurs when thermal activation is able to suppress the differences between the mobilities of screw and non-screw dislocations and the interaction of dislocations with localized obstacles is like that described in Sec. 2.2.5. Below the critical
2 Flow Stress and Work Hardening
82
temperature, the passing of the obstacles is conditional on the building up of mixed dislocation segments by accumulation of kinks up to a length critical for edge gliding, Xc, i.e., the passing of the obstacles is controlled by the lattice friction stress of the screw dislocations, Fig. 2-38 (Louchet et al., 1979). Work hardening must be due to the interaction of screws of different systems, i.e., to forest interactions among screws, which appear uniformly distributed in TEM observations (secondary slip is always operative at low temperatures). The proportionality TC = a G b *J~Q is still valid but, as shown by Louchet and coworkers, a is now strongly temperature dependent. An interesting point about the peculiar way of obstacle passing by the mechanism of nucleation of double kinks and their mi-
gration and side accumulation at the obstacles, is that the critical temperature (fixed by XC2(T) ^ gf 1 ) decreases for increasing obstacle density, i.e., a transition from "low-temperature" to "normal" behavior is expected (and observed) to take place at sufficiently large strains in b.c.c. metals. Banded and subsequently checkerboard structures very similar to stage II and stage III f.c.c. dislocation structures develop (Taylor and Christian, 1967; Garrat-Reed and Taylor, 1979). Remarkable differences between athermal work hardening rates of different b.c.c. metals are however noticed (10~ 2 G for W, Mo and <100>Fe, 10" 3 G for Nb, Ta or <110>Fe, Takeuchi, 1969). Differences are attributed to different slip behavior and ease of crossslip.
V Lmirv LQ
V .D
Figure 2-38. Interaction of a screw dislocation with localized obstacles during low-temperature deformation of b.c.c. metals. Successive positions of the dislocation are presented. The critical configuration for passing a uniform distribution of obstacles, characterized by a critical angle or by a critical distance, Xc (the obstacle strength) is assumed to be bS = LOXC = gf1 (Louchet et al., 1979).
2.5 References
2.4 Acknowledgements The encouragement and patience of Prof. H. Mughrabi is sincerely acknowledged. Particular thanks are given to Prof. C M . Sellars who started the revision of the manuscript and to Dr. J. C. Zubillaga who thoroughly completed it. Miss Alicia de Miguel typed and retyped the text and Mrs. Segunda Perez prepared the figures. To all of them, my gratitude.
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Soc. 38, 49-55. Kuhlmann-Wilsdorf, D. (1968), in: Work Hardening: Hirth, J. P., Weertman, J. (Eds.). New York: Gordon and Breach, pp. 97-139. Kuhlmann-Wilsdorf, D. (1985), Metall. Trans. A 16A, 2091. Labusch, R. (1977), /. Appl Phys. 48, 4550. Labusch, R., Schwartz, R. B. (1976), Nucl. Metall. 20, 650. Langford, G., Cohen, M. (1969), Trans. Am. Soc. Met. 62, 623. Langford, G., Cohen, M. (1975), Metall. Trans. A 6A, 901. Lavrentev, F. F. (1980), Mater. Sci. Eng. 46,191 -208. Lee, T. C , Robertson, I. M., Bimbaum, H. K. (1990), Metall Trans. A 21A, 2437-2447. Li, J. C. M. (1963), Trans. AIME 227, 239-247. Li, J. C. M. (1968), in: Dislocation Dynamics: Rosenfield, A. R., Hahn, G. T, Bement, A. L., Jr., Jaffee, R. I. (Eds.). New York: McGraw-Hill. Li, J. C. M., Chou, Y. T. (1970), Metall Trans. 1, 1145-1159. Loginov, B. M., Predvoditelev, A. A. (1982), Phys. Status Solidi (a) 72, 69-11. Louchet, R, Kubin, L., Vesely, D. (1979), Philos. Mag. A 39, 433. Liicke, K., Mecking, H. (1973), in: The Inhomogeneity of Plastic Deformation: Reed-Hill, R. E. (Ed.). Metals Park, Ohio: American Society for Metals, pp. 223-250. Liicke, K., Masing, G., Schroder, K. (1955), Z. Metallkd. 46, 792. . Macmillan, N. H. (1983), in: Atomistics of Fracture: Latanision, R. M., Pickens, J. R. (Eds.). New York: Plenum, p. 95. Mandel, J. (1965), Int. J. Solids Struct. 1,213. Martin Meizoso, A., Gil Sevillano, I , Fuentes Perez, M. (1984), Rev. Metal Madrid 20, 205. Martinez-Hernandez, M., Kirchner, H. O. K., Korner, A., George, A., Michel, J. P. (1987), Philos. Mag. A56, 641-658. McLean, D. (1962), Mechanical Properties of Metals. New York: John Wiley.
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Mecking, H. (1977), in: Work Hardening in Tension and Fatigue: Thompson, A. W. (Ed.)- New York: TMS-AIME, pp. 67-88. Mecking, H. (1980), Strength of Metals and Alloys, in: Proc. 5th Int. Conf. on the Strength of Metals and Alloys. Haasen, P., Gerold, V., Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1573-1594. Mecking, H. (1981), in: Deformation of Poly crystals: Mechanisms and Microstructures: Hansen, N., Horsewell, A., Leffers, T, Lilholt, H. (Eds.). Roskilde, Denmark: Ris0 National Laboratory, pp. 73-86. Mecking, H., Bulian, G. (1976), Acta Metall 24, 249-260. Mecking, H., Estrin, Y. (1987), in: Constitutive Relations and their Physical Basis: Andersen, S. I., Bilde-Sorenson, I B., Hansen, N., Leffers, X, Lilholt, H., Pedersen, O. B., Ralph, B. (Eds.). Roskilde, Denmark: Ris0 National Laboratory, pp. 123-145. Mecking, H., Kocks, U. F. (1976), Proc. 4th Int. Conf on the Strength of Metals and Alloys. Nancy, France: Laboratoire de Physique du Solide, ENSMIM, INPL, pp. 334-339. Mecking, H., Kocks, U. F. (1981), Acta Metall 29, 1865-1875. Mecking, H., Lixcke, K. (1969), Z. Metallkd. 60,185195 and Acta Metall. 17, 279-289. Mecking, H., Nicklas, B., Zarubova, N., Kocks, U. F. (1986), Acta Metall. 34, 527-535. Michel, X P., Champier, G. (1981), Philos. Mag.A43, 1139-1155. Mitchell, T. E. (1964), Prog. Appl. Mater. Res. 6,119. Miyamoto, H., Funami, K., Uehara, M. (1977), /. Fac. Eng. Univ. Tokyo (B) 34, 349-369. Morris, X W, Klahn, D. H. (1974), /. Appl. Phys. 45, 2027. Morris, M. A., Martin, J. L. (1984), Acta Metall. 32, 549-561. Mott, N. F. (1952), Philos, Mag. 43, 1151. Mughrabi, H. (1975), in: Constitutive Equations in Plasticity: Argon, A. S. (Ed.). Cambridge, Mass: MIT Press, pp. 199-250. Mughrabi, H. (1983), Acta Metall. 31, 1367-1379. Mughrabi, H. (1986), S. Afr. J. Phys. 9, 62-68. Mughrabi, H. (1987), Mater. Sci. Eng. 85, 15-31. Mughrabi, H. (1988), Rev. Phys. Appl. 23, 367-379. Mughrabi, H., Ungar, T, Kienle, W., Wilkens, M. (1986), Philos. Mag. 53, 793-813. Mulford, R. A. (1979), Acta Metall. 27, 1115-1124. Nabarro, F. R. N. (1967), Theory of Crystal Dislocations. Oxford: Oxford University Press. Nabarro, F. R. N. (1986), in: Strength of Metals and Alloys, Proc. Int. Conf on the Strength of Metals and Alloys 7, Vol. 3: McQueen, H. I, Bailon, I P., Dickson, J. I., Jonas, J. J., Akben, M. G. (Eds.). Oxford: Pergamon, p. 1667. Nabarro, F. R. N. (1989), Acta Metall. 37, 15211546.
Nabarro, F. R. N., Basinski, Z. S., Holt, D. B. (1964), Adv. Phys. 13, 193-323. Nadgorny, E. (1988), Prog. Mater. Sci. 31, 1-536. Nakada, Y, Keh, A. S. (1966), Acta Metall. 14, 961. Nakada, Y, Keh, A. S. (1969), Phys. Status Solidi (a) 32, 715. Nix, W. D., Gibeling, J. C , Fuchs, K. P. (1982), in: Mechanical Testing for Deformation Model Development: ASTM, Special Technical Publication 765, Rohde, R. W, Swearengen, J. C. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 301-321. Nix, W. D., Gibeling, J. C , Hughes, D. A. (1985), Metall. Trans. A 16A, 2215-2226. Pedersen, O. B. (1987), Acta Metall. 35, 2567-2581. Pedersen, O. B. (1990), Acta Metall. Mater. 38,1221 1239. Pedersen, O. B., Brown, L. M., Stobbs, W. M. (1981), Acta Metall. 29, 1843-1850. Peirce, D., Asaro, R. X, Needleman, A. (1982), Acta Metall. 30, 1087. Petch, N. J. (1953), J. Iron Steel Inst. 174, 25-28. Prinz, F. B., Argon, A. S. (1984), Acta Metall. 32, 1021-1028. Puls, M. P. (1981), in: Dislocation Modelling in Physical Systems: Ashby, M. F , Bullough, R. Hartley, C. S., Hirth, J. P. (Eds.). Oxford: Pergamon, pp. 249-268. Puschl, W, Frydman, R., Schoek, G. (1982), Phys. Status Solidi (a) 74, 211-216. Rollett, A. D. (1988), Ph.D. Thesis, Drexel University, Department of Mater. Eng., Philadelphia. Edited by Los Alamos National Laboratory, LA11202-T. Los Alamos, NM. Rollett, A. D., Kocks, U. F , Doherty, R. D. (1987), in: Formability and Metallurgical Structure. Philadelphia: TMS-AIME, p. 211. Rollett, A. D., Kocks, U. F , Embury, J. D. Stout, M. G., Doherty, R. D. (1988), in: Strength of Metals and Alloys, Proc. Int. Conf. on the Strength of Metals and Alloys 8, Vol. 1: Kettunen, P. O., Lepisto, T. K., Lehtonen, M. E. (Eds.). Oxford: Pergamon, pp. 433-438. Saada, G. (1960a), Acta Metall. 8, 841-847. Saada, G. (1960 b), Met. Corros. 35,311 and 419. Saada, G. (1963), in: Electron Microscopy and Strength of Crystals: Thomas, G., Washburn, J. (Eds.). New York: Interscience, pp. 651-662. Saimoto, S., Duesbery, M. S. (1984), Acta Metall. 32, 147-155. Saimoto, S., Sang, H. (1983), Acta Metall. 31, 18731881. Sastry, D. H., Luton, M. X, Jonas, X X (1974), Philos. Mag. 30, 115-127. Schoek, G. (1980), in: Dislocations in Solids, Vol. 3: Nabarro, F. R. N. (Ed.). Amsterdam, North Holland, pp. 63-163. Schoeck, G., Frydman, R. (1972), Phys. Status Solidi (a) 53, 661-673.
2.5 References
Seeger, A. (1957), Dislocations and Mechanical Properties of Crystals. New York: John Wiley, pp. 232329. Seeger, A. (1958), Handbuch der Physik, Vol. VII-2: Fliigge, S. (Ed.). Berlin: Springer, p. 1. Seeger, A. (1968), in: Work Hardening: Hirth, J. P., Weertman, J. (Eds.). New York: Gordon and Breach, pp. 27-63. Seeger, A., Diehl, X, Mader, S., Rebstock, H. (1957), Philos. Mag. 2, 323. Sestak, B. (1979), in: Strength of Metals and Alloys, Proc. Int. Conf. on the Strength of Metals and Alloys 5, Vol. 3: Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon, pp. 1461-1482. Shen, Z., Wagoner, R. H., Clark, W. A. T. (1988), ActaMetall. 36, 3231. Shetty, M. N., Hirth, J. P. (1971), Met. Trans. 2, 297302. Siethoff, H. (1983), Philos. Mag. A47, 657-669. Siethoff, H., Schroeter, W. (1983), Scr. Metall. 17, 393-398. Siethoff, H., Schroeter, W. (1984), Z. Metallkd. 75, 475-481 and 482-491. Staker, M. R., Holt, D. L. (1972), Ada Metall 20, 569. Stroh, A. N. (1953), Proc. Phys. Soc. London B66, 2. Takeuchi, S., Suzuki, T. (1989), in: Strength of Metals and Alloys, Proc. Int. Conf on the Strength of Metals and Alloys 8, Vol. 1: Kettunen, P. O., Lepisto, T. K., Lehtonen, M. E. (Eds.). Oxford: Pergamon, pp. 161-166. Takeuchi, T. (1969), J. Phys. Soc. Jpn. 27, 436-443. Takeuchi, T. (1970), /. Phys. Soc. Jpn. 28, 855-964. Taylor, G. I. (1934), Proc. R. Soc. London 145, 362 and 388. Taylor, G., Christian, J. W. (1967), Philos. Mag. 15, 873-892 and 893-929. Thompson, A. W. (1977), in: Work Hardening in Tension and Fatigue: Thompson, A. W. (Ed.). New York: TMS-AIME, pp. 89-126. Thompson, A. W, Baskes, M. I. (1973), Philos. Mag. 28, 301-308. Thompson, A. W, Baskes, M. I., Flanagan, W. F. (1973), Acta Metall. 21, 1017-1028. Torrealdea, F. I (1981), Ph.D. Thesis, Faculty of Sciences, University of Navarra, San Sebastian, Spain. Torrealdea, F. J., Gil Sevillano, J. (1983), Strength of Metals and Alloys, Vol. I: Gifkins, R. C. (Ed.). Oxford: Pergamon, p. 547. Tylecote, R. F. (1976), A History of Metallurgy. London: The Metals Society, p. 8. Ungar, T, Mughrabi, H., Ronnpagel, D., Wilkens, M. (1984), Acta Metall. 32, 333-342. Ungar, X, Mughrabi, H., Wilkens, M., Hilscher, A. (1991), Philos. Mag. A64, 495-496. Vetter, R., Fastenau, R. H. X, Van den Beukel, A. (1982), Phys. Status Solidi (a) 70, 177-182. Veyssiere, P. (1988), Rev. Phys. Appl 23, 431-443.
87
Vitek, V. (1985), in: Dislocations and Properties of Real Materials. London: The Institute of Metals, p. 30-50. Voce, E. (1948), J. Inst. Met. 74, 537-562. Weertman, X, Weertman, X R. (1987), in: Constitutive Relations and their Physical Basis: Andersen, S. L, Bilde-Sorensen, X B., Hansen, N., Leffers, X, Lilholt, H., Pedersen, O. B., Ralph, B. (Eds.). Roskilde, Denmark: Ris0 National Laboratory, pp. 191-203. Wilkens, M. (1969), Acta Metall. 17, 1155. Wilson, D. V (1967), Met. Sci. J. 1, 40-47. Wilson, D. V. (1968), Acta Metall. 16, 743. Zehetbauer, M., Seumer, V, Witzel, W (1988), in: Strength of Metals and Alloys, Proc. Int. Conf on the Strength of Metals and Alloys 8, Vol. 1: Kettunen, P.O. (Ed.). Oxford: Pergamon, pp. 451456.
General Reading On the flow stress: Frost, H. X, Ashby, M. F. (1982), Deformation-Mechanism Maps. The Plasticity and Creep of Metals and Ceramics. Oxford: Pergamon. Kelly, A., Macmillan, N. H. (1986), Strong Solids, 3rd. ed. Oxford: Oxford University Press. Kelly, A., Nicholson, R. B. (Eds.) (1971), Strengthening Methods in Crystals. London: Applied Science. Kocks, U. R, Argon, A. S., Ashby, M. F. (1975), Prog. Mater. Sci. 19. On work hardening: Basinski, S. X, Basinski, Z. S. (1979), in: Dislocations in Solids, Vol. 4: Nabarro, F. R. N. (Ed.). Amsterdam, North Holland, pp. 261-362. Clarebrough, L. M., Hargreaves, M. E. (1959), Prog. Met. Phys. 8, 1-103. Hirsch, P. B. (1975), in: The Physics of Metals, Part 2, Defects: Hirsch, P. B. (Ed.). Cambridge: Cambridge University Press, pp. 189-246. Honeycombe, R. W K. (1968), The Plastic Deformation of Metals. London: Edward Arnold Ltd. Kuhlmann-Wilsdorf, D. (1985), Metall. Trans. A 16A, 2091. Mitchell, T. E. (1964), Prog. Appl. Mater. Res. 6, 119. Mughrabi, H. (1975), in: Constitutive Equations in Plasticity: Argon, A. S. (Ed.). Cambridge, Mass: MIT Press, pp. 199-250. Nabarro, F. R. N. (1986), in: Strength of Metals and Alloys, Proc. Int. Conf on the Strength of Metals and Alloys 7, Vol. 3: McQueen, H. X, Bailon, X P., Dickson, X L, Jonas, X X, Akben, M. G. (Eds.). Oxford: Pergamon, p. 1667.
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2 Flow Stress and Work Hardening
Nabarro, F. R. N., Basinski, Z. S., Holt, D. B. (1964), Adv. Phys. 13, 193-323. c A /<mc-7\ r»- * .• J njf i in Seeger, A. (1957), Dislocations and Mechanical Prop. r > , . , XT v i T u AI7-1 o-5o erties of Crystals. New York: John Wiley, J pp. 232^^g ^ ' ^^ o A wn i> i* i* A X/r / 1 Q ^ • Seeger, A., Wilkens, M. (1967), in: Realstruktur und Ewenschaften von Reinstoffen II., Int. Symp. uber 5 , / . „,. , Ji * „ j .j V> o n . Reinstoffe in Wissenschaft und Technik, Part+ 3. Berlin: Akademie-Verlag, p. 29.
On the influence of crystal structure: ,,, o / i n - m , r ,^ , , .„ o^ Sestak, B. K(1979), y in: Strength of / Metals and Alloys, „ ' . * •. .. . j ^ » D T Proc. Int. Conf. on the Strength of Metals and Aln ^ u v ^ ± ^ c ^ i o Yr /7o ^ 5, Vol. 3: Haasen, P., Gerold, V., Kostorz, G. (Eds.). Oxford: Pergamon, pp. 1461-1482. v & ' ^F ~ , , , ^ . A , On work hardening at large strains: ° Gil Sevillano, X, Van Houtte, P., Aernoudt, E. (1980), Prog. Mat. Sci. 25, 69-412. c
3 Deformation and Textures of Metals at Large Strain Etienne Aernoudt1, Paul Van Houtte1, and Torben Leffers 2 1
2
Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, Leuven, Belgium Materials Department, Ris0 National Laboratory, Roskilde, Denmark
List of 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.6
Symbols and Abbreviations 90 Introduction 92 Methods for the Description of Crystallite Orientations and Texture 92 The Extreme Models 94 The Deformed State 96 Dislocation Accumulation in the Grain Boundaries 97 Dislocation Accumulation Within the Grains 99 The f.c.c. Materials 101 Heterogeneous Flow and Microstructure 102 Heterogeneities at Grain Scale 103 Twin Lamellae and Bundles 105 Organized Structures in Materials with Higher Stacking Fault Energies . 106 Shear Bands 109 Grains at Very High Strains 111 Modelling of the Plastic Deformation of Single-Phase Polycrystals at Large Strains 112 3.6.1 A Few Definitions 113 3.6.2 The Classical Taylor Theory 113 3.6.3 A Few Variants on the Same Theme 115 3.6.3.1 The Bishop-Hill Theory 115 3.6.3.2 The Strain Rate Sensitivity (Viscoplastic) Method 116 3.6.4 Relaxed Constraint Theories (Mixed Boundary Conditions) and Sachs Theory 117 3.6.5 Other Methods 119 3.7 Application and Evaluation of the Models 120 3.7.1 Prediction of Deformation Textures 121 3.7.2 The Case of Materials with Mechanical Twinning 123 3.7.3 Stress-Strain Curves 125 3.7.4 Conclusions 126 3.8 Deformation of Multiphase Materials 127 3.8.1 Two Ductile Phases: Strain Partitioning 127 3.8.2 The Case of Lamellar Pearlite 129 3.9 Change of Strain Path 130 3.10 Concluding Remarks 133 3.11 References 133 Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
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3 Deformation and Textures of Metals at Large Strain
List of Symbols and Abbreviations b b\ c D E,Ei} £, Etj £EQ f(g) fij Ftj g G Gijkl h, k, I
v x1 x2 x3 x X x i 2 l
Burgers vector unit vector on slip direction (slip system s) lattice constant grain size macroscopic strain macroscopic strain rate equivalent strain rate (usually according to the von Mises convention) orientation distribution function local velocity gradient macroscopic velocity gradient shorthand for the three Euler angles that describe a crystalline orientation shear modulus accommodation tensor Miller indices of the shear plane (slip system), or of a crystal plane parallel to the rolling plane (texture component) dimensionless parameter or constant strain rate sensitivity exponent Taylor factor tensor that characterizes the strain caused by unit slip on slip system s number of available slip systems number of accumulated dislocations per unit length of boundary in grain i unit vector normal to slip plane (slip system s) rate of plastic work per unit volume averages of P and M for all crystallites macroscopic stress absolute temperature time increment Miller indices of the shear direction (slip system), or of a crystal direction parallel to the rolling direction (texture component) volume fraction of twins reference system of a polycrystalline sample crystal reference system
a a a, a 0 ocs Aoc y F ys yx y
grain orientation (in Sec. 3.4) coupling factor between inclusion and matrix (in Sec. 3.6) instantaneous, initial angle between lamellae and wire axis ratio of T£ t o a reference value orientation difference between two grains shear strain total slip in a crystallite slip rate on the slip system s shear magnitude macroscopic shear rate
k m M Mstj n nt n\ P P, M S, Stj T At u, v, w
List of Symbols and Abbreviations
8 £tj Eij Q QG Gij o'ip S'ij T TC TC xcs zf T0 q>l9$9cp2 d>tj cofj Qtj
tensile strain local strain local strain rate dislocation density density of geometrically necessary dislocations local stress deviatoric stresses (local, macroscopic) macroscopic shear stress reference value of TCS critical resolved shear stress critical resolved shear stress on the slip system s resolved shear stress on the slip system s lattice friction stress Euler angles (used to describe a crystallite orientation) local rate of rotation rate of rotation of the crystal lattice macroscopic rate of rotation
b.c.c. Bs CB CRSS DDW FC f.c.c. FEM h.c.p. MB1 MB2 ND ODF OFHC RC RD TD TEM 2,3,6-D
body-centered cubic brass cell block critical resolved shear stress dense dislocation wall full constraints Taylor model face-centered cubic finite-element method hexagonal close-packed first generation microband second generation microband normal direction orientation distribution function oxygen free, highly conductive relaxed constraints Taylor model rolling direction transverse direction transmission electron microscopy 2,3,6-dimensional
91
92
3 Deformation and Textures of Metals at Large Strain
3.1 Introduction In this chapter, we shall deal with a number of topics relating to large plastic deformation of polycrystalline materials. Large plastic deformation means: true strains above about 5%, with no upper limit except of course the maximum strains that can be achieved in forming processes such as rolling and wire drawing, executed at room temperature. Polycrystals differ from single crystals in several ways. There are grain boundaries, and the lattices of the grains have more or less different crystallographic orientations. In general there is a crystallographic texture, which means that some crystallographic orientations occur more frequently than others. The mechanical properties of polycrystals are not as anisotropic as those of single crystals, but they are seldom isotropic. During plastic deformation, internal stresses of various kinds develop, some of which are inconceivable in single crystals. Strain heterogeneities of various kinds also develop, some of them linked to particular deformation mechanisms dominant in certain ranges of strain. They strongly influence the microstructure and even the crystallographic texture that develop during deformation. The table of contents of this chapter reflects these features. After an introductory note on crystal orientations and textures, there are four sections in which the general principles are explained; they mainly refer to single phase materials. The first two describe the dislocation accumulation in polycrystals and the strain heterogeneities that develop, and discuss these features mainly on the basis of dislocation interactions. The next two focus on mathematical models for the plastic deformation of polycrystals and deformation texture prediction.
The next section leaves the realm of single-phase materials and discusses some typical aspects of the plastic deformation when two phases are present. The final section has the task of demonstrating the limits of our present understanding by describing what happens when changes of strain path occur during plastic deformation (e.g., cross rolling or deep drawing followed by biaxial stretching as may occur in industrial sheet drawing operations).
3.2. Methods for the Description of Crystallite Orientations and Texture Crystallographic texture will be dealt with in Chap. 10 of Vol. 15 (Cahn, 1992). However, it was felt necessary to say a few words about it here. In a polycrystalline material, each "grain" is a crystallite which has its own crystal lattice. By "crystallite orientation" we mean the orientation of the crystal lattice with respect to the external axes of the sample. Let the latter be defined by the "sample reference system", denoted by x1x2x2). For rolled material, xx usually is the rolling direction (RD), x2 the transverse direction (TD) and x 3 the direction normal to the rolling plane (ND). A "crystal reference system" x\ xc2 xc3 is placed on 3 characteristic directions of the crystal lattice that are normal to each other, for example the three <100> directions in the case of a cubic metal. It is now possible to describe the crystallite orientation by describing the orientation of x\x\x\ with respect to x±x2x3. This description is often done by means of the three Euler angles
3.2 Methods for the Description of Crystallite Orientations and Texture
93
rolling: (hkl) [uvw] means: the rolling plane is parallel to the (hkl) plane of the crystallite's lattice, and the rolling direction is parallel with the [uvw] direction. A drawback is that normally crystallite orientations are not low-index orientations (cannot be described by low Miller indices). Hansen et al. (1978) have published tables that enable the conversion from Miller-index notation to Euler angles and vice versa. The symbol g is often used as a shorthand notation for q>l9 x around x 3 . Let of all crystallites for which the crystal orithe rotated axes be x\ x'2 x3. They are in entation is within the range dg. f(g) is alturn rotated by an angle $ around x\. By ways non-negative. It would be a Dirac now, x'3 has already become x\9 around which a third rotation by an angle q>2 is function if all crystallites of the polycrystal would have the same orientation. On the carried out. So three successive rotations other hand, f(g) would be equal to unity bring the sample system into the angular for all g if all crystal orientations are position of the crystal system. Note that equally represented in the material. Both other definitions for the Euler angles have cases are extremely rare. In most polycrysbeen proposed as well [also described by talline materials, fig) is a function which Bunge (1982)], but those explained above has maxima and minima. It may be zero in are by far the most widely used in texture a large range of Euler space (which is the literature. Theoretically the range of the orientation space). For more details we Euler angles to be considered is 0-360° for cp1 and q>2, and 0-180° for . For rolled refer to the literature (Bunge, 1982) and to Chap. 10 of Vol. 15. cubic metals (also when subsequently recrystallized), symmetry allows a reduction Figure 3-2 shows a kind of "tube" in of this range to 0-90° for each of the three Euler space that surrounds most crystallite angles. Even within this reduced range, orientations with a high density [high value each crystallite orientation has at least 3 of/(#)] for the typical deformation texture equivalent Euler angle representations of face-centered cubic (fee.) metals with (Hansen et al., 1978). medium or high stacking fault energy ("copper type rolling texture"). The orienA widely used alternative way to detation symbols are explained in the subtitle scribe the orientation of a crystallite is of Fig. 3-3. The "fibers" are the central based on Miller indices. Example for
94
3 Deformation and Textures of Metals at Large Strain
the structure of rolled and/or recrystallized body-centered cubic (b.c.c.) metals. Note that in Figs. 3-2 and 3-5, not all crystallographically equivalent Euler angle representations are shown for each orientation.
3.3 The Extreme Models
Goss
Figure 3-2. Structure of the ODF of the rolling texture of f.c.c. metals in Euler space. See Fig. 3-3 for the Miller index descriptions of the ideal orientations. (By courtesy of J. Hirsch.)
axes of the tube (which changes direction at Bs). The "ideal orientations" Cu, S, Bs, G etc. (see caption to Fig. 3-3) are low-index orientations in terms of which these textures are often described. Figure 3-4 shows a very common representation of such ODF. The deformation textures of f.c.c. metals with low stacking fault energy ("brass type rolling texture") are similar, but strongly dominated by the Bs (brass) component. In a similar way, Figs. 3-5 and 3-6 describe
90
In Sec. 3.6 we shall describe the different models suggested for the plastic deformation of polycrystals in terms of mathematics. However, the reader without prior knowledge of the field will need a basic physical understanding of the models already in Sees. 3.4 and 3.5. In the present section we provide the necessary background by describing the physics behind the two "extreme" models (cf. Sec. 3.6), the Taylor model and the Sachs model. For simplicity, we limit ourselves to the case of macroscopically homogeneous strain (no spatial strain variation apart from possible fluctuations at the microstructural scale). In the Taylor model (Taylor, 1938) it is assumed that the microscopic strain is identical to the macroscopic strain. With the above limitation that means that the strains are identical throughout the specimen. This is obtained by multiple slip: in each crystallite (which is normally taken to be a grain) there are at least five active slip systems. With the homogeneous strain
0°
1 T Cu 90°
q>2=45°
Figure 3-3. Position of the ideal orientations and the a-fiber of cold rolled f.c.c. metals in some frequently used sections of Euler space. C (cube): (100) [001]; G (goss): (110) [001]; Bs (brass): (11 0) [1 T 2]; Cu (copper): (112) [111]; T (Taylor): (4 411) [1111 8]; S: (1 2 3) [6 3 4].
95
3.3 The Extreme Models
0-70 0 00
1-00
1.40 90 0
2.00
2.80
4-00 90 0 L
5-60
8-00
11-0
90 0
5^
IL
16.0 —*
90.
90
Figure 3-4. Representation of an ODF by means of
96
3 Deformation and Textures of Metals at Large Strain
has the highest resolved shear stress from the applied tensile stress. The single-slip pattern leads to different strains in the different grains, but the average strain will be an axisymmetric tensile strain (unless there are special texture effects which we shall disregard). Because of the heterogeneous strain pattern, strain continuity can only be accounted for by additional assumptions, as for instance provided by the "modified Sachs model" (e.g., Pedersen and Leffers, 1987).
3.4 The Deformed State
Figure 3-5. Structure of the ODF of the rolling texture of b.c.c. metals. See Fig. 3-6 for the Miller index description of the ideal orientations. (After Emren et al., 1986.)
pattern strain continuity is automatically accounted for. In the Sachs model (Sachs, 1928) it is assumed that there is single slip in the individual grains (until they reach an orientation where one more slip system is activated). In a tensile experiment, for instance, slip will take place on the one slip system which
6
-O
The deformed state is characterized by and produced by - accumulated dislocations (and in some cases other accumulated defects). During plastic deformation dislocations are generated, dislocations move, dislocations interact with pre-existing structures (in single-phase materials primarily the grain boundaries), dislocations interact with each other and thereby partly annihilate. At a given strain all the dislocation segments which have not annihilated make up the structure of the deformed state - with one addition: the change in the grain morphology, e.g. the flattening of the grains during compression, is not produced by accumulated dis-
Q
Figure 3-6. Position of the ideal orientations and the a- and y-fibers of cold rolled and/or recrystallized b.c.c. metals in some frequently used sections of Euler space. C (cube): (100) [001]; H (rotated cube): (1 00) [011]; G (goss): (110)[001]; T* (b.c.c. Taylor): (1111 8)[4411]; I: (211)[011]; E: (111)[011]; F: (111)[11 2]; L: (110) [11 0].
3.4 The Deformed State
locations (or any other accumulated defects) but by accumulated strain. The grain morphology is a structural component of great significance in the deformation models to be described in Sec. 3.6. A complete mapping of the dislocation distribution, including the distribution of Burgers vectors, would take us very far in the direction of a complete description of the deformed state and hence an advanced understanding of the deformation processes. In practice such a mapping goes far beyond our present experimental capacities. Therefore, we have to subdivide the accumulated dislocations in different groups which we deal with separately in various qualitative or, at the very best, semiquantitative ways. The accumulated dislocations may (i) be absorbed in the grain boundaries, (ii) be stored as parts of deformation-induced dislocation complexes (e.g., dislocation walls and subgrain boundaries) or (iii) be stored as individual dislocation segments in the dislocation network. 3.4.1 Dislocation Accumulation in the Grain Boundaries Dislocation accumulation process (i) is illustrated in Fig. 3-7. Two grains, 1 and 2, in a material subjected to a tensile strain in the horizontal direction meet at a grain boundary. The active slip planes in the two grains are indicated (we imagine that both families of slip planes are perpendicular to the plane of the paper and that the slip directions are in the plane of the paper). Slip is performed by edge dislocations which end up in the boundary where the dislocation from the two grains partially annihilate because the Burgers vectors have opposite components. The Burgers vector components which do not annihilate produce a change in the
(a)
97
(b)
Figure 3-7. Sketch showing the orientation change between two grains (1 and 2) and the disclocation accumulation in the boundary between them during tensile deformation (tensile axis horizontal), (a) Before and (b) after deformation.
orientation relation between the crystal lattices in the two grains - the change from Fig. 3-7 a to Fig. 3-7 b. The change in orientation difference, Aoc, between grains 1 and 2 for shear strains y1 and y2 may (with an ad hoc sign convention adjusted to Fig. 3-7) be expressed as Aa = A (ax + a 2 ) = Aax + Aa2 = = ?! sin2
sin2 a2
(3-1)
Each grain behaves like a pack of sliding cards touching the boundary: when for instance (Xi^O 0 we get A a ^ y ^ and when a ^ O w e get Aa x ~ 0 . The number of accumulated dislocations per unit length of boundary in grain 1 is n1 = —^
(3-2)
b being the magnitude of the Burgers vector (and mutual reactions between the dislocations from the two grains being neglected). When the orientation relation between the two grains changes, at least one of the two grains must be subjected to a change in lattice orientation relative to a coordinate system referring to the sample geometry. In practice the multiple interactions between the individual grains and their many neighbors makes the lattice ori-
98
3 Deformation and Textures of Metals at Large Strain
entations of all the grains (except those with "stable end orientations") change - as also indicated in Fig. 3-7. Thus, process (i) leads to the formation of a deformation texture. One should notice that texture formation via dislocation accumulation produces a change in the internal state of the material by changing the orientation relations between the grains. A simple orientation change without any changes in the internal stage - as one observes during deformation of single crystals - does not involve dislocation accumulation. In Fig. 3-7 the boundary between the two grains is shortened during deformation (from a to b). If the slip processes in the two grains are perfectly coupled, they will impose the same shortening on the boundary, and the dislocation accumulated in the boundary will form a wall with no long-range stresses associated. Applying the same ad hoc sign convention as we used for Eqs. (3-1) and (3-2) we get the condition for identical imposed shortenings on the boundary from the two grains: A/
y2
(3-3)
In Fig. 3-7 a the angles between the slip planes and the tensile direction in the two crystals are 45°, i.e. a1 + ot2 = 90°. Thus, the condition for identical imposed shortenings (and hence for the absence of longrange stresses) is y1==y2- If the slip processes are not perfectly coupled, they will try to impose different shortenings, and consequently there will be long-range stresses associated with the dislocations accumulated in the boundary (we refer to temperatures where grain-boundary sliding can be neglected). Thus, intergranular stresses may develop as a consequence of accumulation process (i). These intergranular stresses are essential in the models for
plastic deformation of polycrystals to be discussed in Sec. 3.6 - even though their origin is not necessarily considered explicitly in the models. Equation (3-3) is the condition for zero misfit stress across a boundary. This implies that it is not fulfilled when there is a finite misfit stress. However, there are narrow limits for the deviation from the equation: the misfit stress is equal to the difference in A/// between the two grains multiplied by a factor of the order of the elastic moduli, and the misfit stress has an upper limit of the order of the flow stress. Thus, for substantial strains Eq. (3-3) has to be fulfilled approximately. Dislocation accumulation in the grain boundaries [process (i)] may be said to be the fundamental dislocation accumulation process in deformed polycrystals. Conceptually one can imagine plastic deformation without the intragranular accumulation processes (ii) and (iii) but not without process (i): the Taylor model does not require accumulation according to (ii) and (iii) but of course it does not exclude it either (e.g., Sec. 3.5.1). The dislocations accumulated in the grain boundaries are "geometrically necessary" (Cottrell, 1964) because of the necessary difference in slip pattern (as referred to a common sample coordinate system) between the different grains. The fundamental nature of process (i) is also reflected in the fundamental structural components which it leads to/may lead to: texture and intergranular stresses. The detailed mechanisms for the absorption of lattice dislocations in grain boundaries are not known. An absorbed dislocation may, under appropriate diffraction conditions, be visible in transmission electron microscopy (TEM) as an extrinsic dislocation (e.g., Randle and Ralph, 1988), but even with the best viewing conditions one can only see a small fraction
3.4 The Deformed State
of the dislocations accumulated in the boundaries after sizeable strains. The dislocations accumulated in accordance with process (i) are mainly visible via their effects, viz. the texture and the intergranular stress. "Visible" in this connection refers to visibility in microscopical investigations as well as visibility in macroscopic/statistical measurements, with diffraction methods for instance. 3.4.2 Dislocation Accumulation Within the Grains
The dislocations accumulated in accordance with processes (ii) and (iii) are directly visible in TEM - as dislocation complexes (ii), individual dislocations (iii) or complexes with visible individual dislocations [(ii) plus (iii)]. It is difficult to draw a sharp borderline between processes (ii) and (in). If the geometrically necessary dislocations accumulated according to (i) represent one extreme, then the dislocations accumulated at very fine scale, in cells or subgrains with process (ii) or homogeneously with process (iii), represent the other extreme. These dislocations are geometrically redundant, i.e. they do not accumulate because it is required by some heterogeneity at coarser scale; they accumulate because dislocation generation and motion cannot, in practice, take place without dislocation accumulation. They are the main cause of work hardening - in accordance with the fundamental expression for the critical resolved shear stress, T.
(3-4)
where u0 is a lattice friction stress, k is a dimensionless parameter, G is the shear modulus, b is the magnitude of the Burgers vector, and Q is the dislocation density
99
which, for most conditions, particularly at large strains, is dominated by the redundant dislocations. The dislocations accumulated in the grain boundaries are not to be included in Q in Eq. (3-4). In the present chapter we shall not go into a detailed discussion of work hardening - which is dealt with in Chaps. 2 and 4. However, we shall make a few remarks about Eq. (3-4). The simple interpretation is that it describes TC as the sum of a lattice friction term and a (dominant) friction term from the dislocations, but the dislocation friction is an anisotropic friction, k is normally taken to be in the range ~ 0.1 - 1 , depending on the geometry of the dislocation arrangement, and the orientation relation to the applied stress and strain is one of the important geometrical parameters. For instance, the concept of latent hardening (Chap. 2) requires a directionality. Christodoulou et al. (1986) have provided a direct experimental demonstration of the directionality of the dislocations accumulated in the subgrains in polycrystalline copper: by strain reversal the subgrains formed during forward strain are dissolved, and new subgrains form after a finite reverse strain. In Sec. 3.4.1 we ascribed texture formation to dislocation accumulation in the grain boundaries. However, this is to be taken with the modification that all accumulation of geometrically necessary dislocations, also intragranular accumulation, takes part in texture formation. If, for instance, there are different slip patterns in the two halves of a grain, geometrically necessary dislocations will form a wall inside the grain [accumulation process (ii)]. This will split the grain in two differently oriented parts, which will have an effect on texture. The mechanism is basically the same as that for dislocation accumulation in the grain boundaries - with a starting
100
3 Deformation and Textures of Metals at Large Strain
point at zero orientation difference instead of a finite orientation difference. With a very high degree of intragranular heterogeneity the effect on texture will be strong. For strains (and grains) which are not very large texture formation is mainly governed by the dislocations accumulated in the grain boundaries; the effect of the geometrically necessary dislocations accumulated inside the grains is an added orientation spread. This is illustrated in Fig. 3-8 for a two-dimensional orientation distribution. The orientation distribution for a finite number of grains before deformation is symbolized by the filled circles. In Fig. 38 a dislocation accumulation in the grain boundaries [process (i)] is accompanied by accumulation of geometrically necessary dislocations in discrete walls which split the grains [process (ii)]. In Fig. 3-8b process (i) is accompanied by accumulation of geometrically necessary dislocations in a continuous distribution [process (iii)] which spreads the orientations. The accumulation of redundant dislocations has no direct effect on texture. As just described for the textures, the internal (intergranular) stresses associated with the dislocations accumulated in the grain boundaries are supplemented by (intragranular) internal stresses associated with the accumulation of geometrically necessary dislocations in the grains. There is a two-way relation between the internal stresses and the dislocations accumulated in the grains: the geometrically necessary dislocations in the grains may accumulate as a result of variations in the intergranular stresses, and the accumulation of dislocations, in walls for instance, may create internal stresses which change the slip pattern between the walls. In practice it is difficult to make an experimental distinction between the effects of the intra- and the intergranular stresses. The complex pattern
(a)
(b) Figure 3-8. Sketch of the orientation changes during deformation of 6 grains (in two dimensions). The arrows indicate the average orientation changes corresponding to process (i). There are added orientation changes corresponding to (a) process (ii) and (b) process (iii).
of constantly changing internal stresses quoted by Pedersen and Leffers (1987) is a combination of intra- and intergranular stresses associated with dislocation interaction across the grain boundaries. Even though we are mainly dealing with single-phase materials, it should be mentioned that geometrically necessary dislocations accumulate around hard particles. This accumulation was actually the main subject of Ashby's (1970) paper about geometrically necessary dislocations. The accumulation is also accompanied by orientation changes, internal stresses and stored energy. Recently Humphreys and Kalu (1990) have presented extensive experimental observations on dislocation accu-
3.4 The Deformed State
mulation around particles and a theoretical interpretation. 3.4.3 The f.c.c. Materials So far we have described the deformed state in general terms which may apply to any ductile material. The actual manifestation of these general principles in specific microstructures and textures is subjected to great variations depending on materials properties, or more specifically dislocation properties, in the material in question. The important parameters are: the number of families of physically identical slip systems, the number of slip systems in each family, the relative mobility of the dislocations belonging to the different families (by ordinary slip, by cross slip, by climb), the relative mobility of edge and screw dislocations in the different families, the dissociation of the dislocations and the mobility of the partial dislocations (formation of stacking faults and twins). These parameters are determined by the core structure of the dislocations, but in most practical cases they are empirical parameters which are more or less known (particularly less). In Sec. 3.5 we shall describe heterogeneous flow and its microstructural manifestations on the basis of observations on f.c.c. materials. The present general remarks about the deformed state of f.c.c. materials are to be seen in connection with Sec. 3.5. There are scattered indications in the literature that planes others than {111} may act as slip planes in f.c.c. materials. This has lead Haessner (1963) and Bacroix and Jonas (1988) to suggest that such non-octahedral slip plays a role in texture formation (at ambient temperature and elevated temperature, respectively). In Sec. 3.5, referring to deformation at ambient temperature, we shall assume that {111} is the
101
only significant slip plane. The mobile dislocations may be perfect (Burgers vector Vi <110» or partial (Burgers vector 1/6 <211», the latter type leaving stacking faults or twinned material behind them. Two types of behavior are observed in f.c.c. materials: a "copper-type" and a "brass-type" behavior. The difference is revealed most clearly in the deformation textures, particularly in the rolling textures which may be of the copper type or the brass type, e.g. Sec. 3.2 and Hirsch and Lucke (1988) and Leffers and Juul Jensen (1991). The fully developed copper-type rolling texture is characterized by orientations in an "orientation tube" from the orientation {211}<111> to the orientation {110}<112>, while the fully developed brass-type texture is characterized by the main orientation {110}<112> with spread to {110}<001>, e.g. Sec. 3.2. At ambient temperature copper-type behavior is characteristic for materials with high to intermediate stacking fault energy, and brass-type behavior is characteristic for materials with low stacking fault energy. All pure f.c.c. metals except silver show copper-type behavior at ambient temperature,while silver and many alloys including brass and austenitic steel show brass-type behavior. An increase in rolling temperature and a decrease in strain rate tend to change the texture in the direction of the copper type, while a decrease in rolling temperature and an increase in strain rate tend to change the texture in the direction of the brass type. Temperature changes may change the texture from one type to the other (Hu and Cline, 1961; Hu and Goodman, 1963), whereas the practically possible change in strain rate only produces slight modifications (Leffers, 1968 a). At the dislocation level the brass-type behavior is characterized by "planar slip".
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3 Deformation and Textures of Metals at Large Strain
In the early stage of deformation this leads to the formation of pile-ups at the grain boundaries. In the later stages the brasstype behavior is characterized by slip on one single slip plane (Sec. 3.5.2) and by the effects of partial dislocations (stacking faults and twin lamellae). These features characteristic of planar slip are absent in materials with copper-type behavior. At the very fine scale the redundant dislocations are arranged in cells or subgrains for copper-type behavior, whereas they are distributed rather homogeneously for brass-type behavior. In Sec. 3.5 the detailed differences between the microstructures for copper-type and brass-type behavior and the derived differences in deformation pattern are discussed - together with the variations within the range of materials with copper-type behavior. The difference between the copper-type and the brass-type behavior may be ascribed to a difference in cross-slip frequency (e.g., Dillamore, 1970). With a low cross-slip frequency, as favored by low stacking fault energy, low deformation temperature and high strain rate, the dislocations are tightly bound to their slip plane, and we get planar slip. With a high cross-slip frequency, as favored by the opposite set of conditions, the dislocations may escape from their slip plane. At ambient temperature the type of behavior follows the stacking fault energy as described above. It should be emphasized that the change in the type of behavior with change in deformation temperature is caused primarily by the temperature change and not by a concurrent change in stacking fault energy. Silver, for instance, changes from brass-type to copper-type behavior when the deformation temperature is increased (Hu and Cline, 1961) in spite of the fact that the stacking fault energy decreases (Sakaetal., 1978).
Recently a number of authors, e.g. Gerold and Karnthaler (1989), have argued (without specific reference to texture formation and microstructures at large strains) that short-range order and shortrange clustering are the main causes of planar slip (rather than low stacking fault energy). However, in the present context the exact atomic-scale reason for the difference between the copper-type and the brass-type behavior is not particularly important, and therefore this point is not to be discussed.
3.5 Heterogeneous Flow and Microstructure In terms of Sec. 3.4 heterogeneous flow is reflected in heterogeneities in the population of dislocations accumulated within the grains [processes (ii) and (hi)]. The formation of cells or subgrains per se does not count in this connection; these heterogeneities may be said to represent the "atomic scale" of the microstructure but variations in the cell/subgrain structure are relevant. In this section we describe the specific types of heterogeneities observed - which are subject to very large variations depending on the material and the deformation conditions. Therefore it is necessary to limit the scope. We have limited the scope to f.c.c. materials deformed at room temperature, with special emphasis on rolling. With this limitation we cannot claim to cover all types of heterogeneities, but we do cover a large variety. We are going to characterize the different materials by their stacking fault energy - with the qualification quoted in Sec. 3.4.3. We shall deal with grain-scale heterogeneities (Sec. 3.5.1), "organized structures" (Sees. 3.5.2 and 3.5.3), large-scale hetero-
3.5 Heterogeneous Flow and Microstructure
geneities, shear bands (Sec. 3.5.4) and the degeneration of the concept of grains at very large strains (Sec. 3.5.5). The term organized structures (e.g., Leffers and Juul Jensen, 1991) covers structural features which repeat themselves several or many times within a grain. The organized structures may be of grain scale in one or two dimensions (but obviously never in three). They are repetitive in the dimension^) in which they are not of grain scale. We are to present a detailed discussion of the organized structure in brass (and other f.c.c. materials with low stacking fault energy), the "bundle" structure - because its role in the overall deformation pattern is fairly well understood. The roles of the organized structures in f.c.c. materials with higher stacking fault energies are less known; therefore we only present these structures in a shorter summary. 3.5.1 Heterogeneities at Grain Scale These heterogeneities are related to the grain structure. They may be throughgrain variations, or they may be concentrated close to the grain boundaries. And they may be continuous variations or localized heterogeneities. They may be observed as variations in lattice orientation, variations in the dislocation arrangement (including the cell/subgrain size), variations in the slip-line pattern on the surface, variations in strain as measured on the surface. To a first approximation one may summarize all the observations reported in the literature with a very simple statement: everything that may vary does vary [e.g., the reviews by Thompson (1977) and Leffers (1981)]. Observations of through-grain variations are common, but normally the variations are most pronounced near the grain boundaries. One of the classical ob-
103
servations is that of Boas and Hargreaves (1948) on strain variations in tensile-deformed very coarse-grained aluminum. Large variations were observed within the grains and from grain to grain. Bretheau and Caldemaison (1981) have reported similar observations with similar results for copper with grain size 100 |im (and iron with grain size 200 |im). Bay and Hansen (1981) and Barlow etal. (1985) have studied the grain-scale heterogeneities in aluminum with grain sizes in the range 20-400 [im rolled to reductions up to 30%. The main observations may be summarized as follows: there are banded structures along many grain boundaries; there are cumulative changes in lattice orientation across these banded structures (over distances of ~ 10 (im from the boundaries); close to some grain boundaries the cell size is different from that in the grain interior (indicating a difference in strain and/or slip pattern); the pattern of surface slip markings in the vicinity of many grain boundaries deviates from that in the grain interior. Hansen et al. (1985) also observed large grain-scale heterogeneities in the surface slip markings on aluminum rolled up to 90% reduction. In aluminum compressed to reductions in the range 20-80% Bellier and Doherty (1977) and Faire and Doherty (1979) observed that many grains were subdivided in "deformation bands" [using the terminology of Hu (1962)]. It was suggested that the deformation bands were separated by transition bands (microbands in Hu's terminology). It is a general observation for aluminum that grain-scale heterogeneities become increasingly more pronounced with increasing grain size. In copper (grain size ~ 30 jim) rolled to reductions up to 50% Leffers (1975 a) observed a high level of grain-scale heterogeneities: variations in lattice orientation
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3 Deformation and Textures of Metals at Large Strain
which may be cumulative throughout the grains even though they are most pronounced near the grain boundaries; occasional subdivision in deformation bands; banded structures along many grain boundaries; deviations in the slip-line pattern in the vicinity of many grain boundaries. Hansen and Ralph (1981) observed that the average subgrain size is smaller close to the grain boundaries than in the grain interior in copper tensile-deformed to a strain of 0.2. In rolled brass the twin lamellae (see Sec. 3.5.2) act as built-in markers which very clearly reveal the intragranular orientation variation by bending. Such bent twins are common at reductions above - 2 0 % in brass with 15% zinc (Leffers, 1975 a, b; Leffers and Bilde-S0rensen, 1990). In some grains the radius of curvature is approximately constant while, in others, it is particularly small close to the grain boundaries. Observations on surface slip markings reveal through-grain variations in the slip-line pattern in a large fraction of the grains. One may conceptually make a distinction between two types of grain-scale heterogeneities: (i) heterogeneities in lattice orientation and slip-line pattern without significant strain heterogeneities and (ii) heterogeneities in lattice orientation, slipline pattern and microstructure with significant strain heterogeneities. Heterogeneities of type (i) may develop within a Taylor deformation pattern, and we do not exclude that some of the observed grainscale heterogeneities are of this type, for instance the subdivision in deformation bands (e.g., Dillamore etal., 1972). However, most observed grain-scale heterogeneities seem to be of type (ii), i.e. they reflect a significant deviation from the Taylor deformation pattern. The theoretical implications of these deviations from
the Taylor model are to be discussed in Sec. 3.6. Here we shall concentrate on the microstructural aspects, including the grain-size effect on flow stress, the HallPetch effect (e.g., Hansen, 1985). As already mentioned heterogeneous strain implies intragranular accumulation of geometrically necessary dislocations as reflected in the variations in lattice orientation for instance. Ashby (1970) suggested a homogeneous distribution of the intragranular geometrically necessary dislocations as a first crude approximation. The microstructural observations quoted above show that the geometrically necessary dislocations may be concentrated in the vicinity of the grain boundaries. For otherwise identical conditions the density of geometrically necessary dislocations increases with decreasing grain size - which provides the standard explanation of the grain-size effect on flow stress, e.g. Ashby (1970) and Thompson (1977). In Ashby's simple approach the density of geometrically necessary dislocations, QG, is determined by kds
(3-5)
where e is the (tensile) strain, b is the magnitude of the Burgers vector, D is the grain size and A: is a constant. In practice it is rather difficult to provide a theoretical estimate of the grain-size dependent part of QG. The average value of k is proportional to the deformation heterogeneity, for which the magnitude is a debatable point. k is subject to large local variations - both intra- and intergranular. For large changes in Z), k may change as a result of fundamental changes in the deformation pattern. Finally, the organized structures (e.g., Sec. 3.5.3) contribute to QG, and this contribution may also depend on the grain size (e.g., Hansen, 1985). Leffers etal.
3.5 Heterogeneous Flow and Microstructure
(1988 b) have summarized the observations on grain size effect on texture (with due consideration to the problem of possible differences in initial texture). They concluded that the texture development depends on grain size which shows that the slip pattern does, to some extent, change with large changes in grain size (i.e. there is not simple scaling as suggested by Ashby). It should be underlined, however, that the difficulties encountered when one attempts to derive a quantitative description of the grain-size effect on flow stress do not question Ashby's suggestion about the basic origin of the effect: the genometrically necessary dislocations introduced by grainscale heterogeneities in the deformation pattern, i.e. by deviation from the homogeneous deformation pattern of the Taylor model. The fact that a grain-size effect on flow stress is a very general observation (e.g., Hansen, 1985) confirms the earlier statement that some level of deviation is the rule. One normally observes that the grain-size effect is strongest in materials with low stacking fault energy (e.g., Li, 1969); this is consistent with a number of other observations indicating that the deviation from the Taylor model is most pronounced in these materials (e.g., Sec. 3.5.2), i.e. for brass-type behavior.
105
is well established at ~40% reduction, which is the level of reduction we shall concentrate on. We refer to material with an initial grain size of ~40 |im. Materials with very fine grains (Yeung etal., 1988) and with very coarse grains (unpublished work by Leffers) have different microstructural evolutions. There is a significant number of twins in about 50% of the grains. The other 50% have a rather homogeneous distribution of dislocations (as directly viewed by TEM) without any distinct microstructural features; there are no quantitative investigations of the dislocation distribution, so it is not necessarily homogeneous in a quantitative sense. In some of the twinned grains the twins are scattered, but in most cases the twins are arranged in bundles, consisting of a composite of twin lamellae and matrix material, separated by matrix without twins. An example is shown in Fig. 3-9. Normally there is only one system of parallel bundles in a grain, but grains with two intersecting systems are also observed. Even though the twin lamellae are the predominant microstructural features, the volume fraction of twinned material is low, of the order of few percents. The type of
3.5.2 Twin Lamellae and Bundles
In f.c.c. materials with low stacking fault energy rolled to reductions up to ~ 6 0 80% the microstructure is dominated by twin lamellae and "bundles" of twin lamellae. As an example we shall use brass with 15% zinc - which seems to be fairly representative of materials with low stacking fault energy and hence with brass-type behavior (Leffers and Bilde-S0rensen, 1990). The first scattered twin lamellae appear at about 15% reduction, and the structure
Figure 3-9. Bundles in brass with 15% zinc rolled to 37% reduction viewed in longitudinal section (with rolling direction indicated).
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3 Deformation and Textures of Metals at Large Strain
structure developed in a given grain (bundles, scattered twins, no twins) is determined by the orientation of the grain (Duggan and Lee, 1989; Leffers and Ananthan, 1991). For instance, grains with orientations near {211} <111> have bundles, grains with orientations near {110} <112> have scattered twins or no twins, and grains with orientations near {110} <001> have no twins. In the grains with bundles shear is concentrated in the bundles (shear on the {111} plane parallel to the bundle) as shown by Leffers and Bilde-S0rensen (1990). This means that these grains predominantly deform by shear on one single slip plane. Because of the low volume fraction of twins, the shear is mainly produced by normal <110> slip, but there is also some contribution from the slip by partial dislocations by which the twins are formed. The predominance of one single slip plane is revealed by direct experimental observations: for slip on one single slip plane (parallel to the plane of the twin lamellae) the twin lamellae will remain parallel to the {111} plane on which they formed originally, and they actually do remain approximately parallel to this plane. With reference to the composite structure of the bundles, it is not surprising that slip on intersecting slip planes is difficult. Leffers and Bilde-S0rensen actually suggest that the preference for one single slip plane is built into the preexisting dislocation structure, i.e. that the bundles form as a result of the preference for one slip plane. The bundles obviously reflect a plastic instability, but the mechanisms behind this instability are not known. In spite of the obvious formal similarity between the bundles and the second-generation microbands in copper (they are both plates parallel to {111} with concentrated shear), there appears to be no similarity between
the crystallographic orientation of the grains in which they form (Leffers and Ananthan, 1991). The deformation pattern in brass is as follows: there is slip on one single slip plane in the grains with bundles, and the grains without bundles maintain strain continuity by a heterogeneous pattern of multiple slip (with accumulation of geometrically necessary dislocations as described in Sec. 3.3.1). As revealed by the very frequent observation of bent twins (Leffers, 1975 b; Leffers and BildeS0rensen, 1990), the single-plane slip is heterogeneous in many grains. Altogether this means that neither the grains with nor the grains without bundles follow the Taylor deformation pattern. As a consequence of slip on one single slip system in the grains with bundles, these grains continue to rotate towards an orientation with {111} (and the twin lamellae) parallel to the rolling plane. At the microscopic level this orientation is observed already at ~40% reduction, and at a reduction of ~60% it is a significant component of the texture. The rotation towards this orientation is logical, but it is not understood how the orientation is actually reached (e.g., Leffers and Juul Jensen, 1991). With increasing reduction most grains fill up with twin lamellae. This makes continued slip increasingly difficult - because the active slip planes parallel to the twin lamellae and approximately parallel to the rolling plane have a decreasing resolved shear stress (Duggan et al., 1978) and/or because there are few non-twinned grains left to maintain strain continuity (Leffers and Bilde-Sorensen, 1990). 3.5.3 Organized Structures in Materials with Higher Stacking Fault Energies
There is a wealth of recent observations which demonstrate the complexity of the
3.5 Heterogeneous Flow and Microstructure
microstructures in f.c.c. materials with high and intermediate stacking fault energies - as for instance summarized by Bay etal. (1992) and Leffers and Juul Jensen (1991). The picture of the microstructures as it emerges today is thus far removed from the traditional picture of a simple cell or subgrain structure. In aluminum, nickel and copper rolled to moderate reductions a large fraction of the grains develop "dense dislocation walls", DDWs, which are approximately parallel to the transverse direction, making angles of about 45° with the rolling plane. The DDWs have no specific crystallographic orientation. They may partly transform to "first generation microbands", MBls. The formation of DDWs and MBls was first described in details by Bay et al. (1989) for aluminum. In aluminum there is a cell structure at a finer scale superimposed on the DDW/MB1 structure. At 30% rolling reduction the typical orientation change across the cell boundaries is ~ 1° whereas there are orientation changes of -10° across the DDWs/ MBls. At low reductions there is only one system of parallel DDWs/MBls; from about 30% reduction two crossing (approximately symmetrical) systems are observed in a number of grains. Bay et al. suggest that the DDWs/MBls subdivide the grains into "cell blocks", CBs, each CB having its specific combination of active slip systems. This suggestion is further elaborated by Hansen (1990) and Bay et al. (1992). It is suggested that there is, in each CB, less than the five slip systems required by the Taylor model. Thus, the strain in the individual CB is different from the macroscopic strain, but it is suggested that the individual grains, with their combination of different CBs, approximately follow the macroscopic strain. The
107
driving force for the subdivision into CBs is suggested to be a reduction in work hardening rate relative to that for a Taylor deformation pattern. With subdivision there is less than five slip systems active at any given point as compared with five or more at any given point for the Taylor pattern, and the work hardening rate increases with the number of interacting slip systems. There is a clear trend that neighboring DDWs/MBls have approximately the same absolute magnitude of the associated orientation changes but opposite signs. This indicates that there are, in the simple case with one family of parallel DDWs/ MBls, two families of CBs each with their own specific combination of active slip systems. In nickel (Hughes and Hansen, 1991) and in copper (Ananthan et al., 1991 a, b) one also observes a subdivision of a large fraction of the grains by DDWs/MBls. There are certain differences from the pattern in aluminum as described above. For instance there are two crossing systems of DDWs/MBls from the very early stage of deformation, and the MBls have a doublewall structure whereas the MBls in aluminum consist of "small pancake-shaped cells". But the interpretation is that it is basically the same type of subdivision with basically the same slip pattern as in aluminum (Hansen, 1990; Leffers et al., 1991; Bay etal., 1992). At higher strains (starting already at 40% reduction in copper) another type of subdivision becomes increasingly important in all three metals: a subdivision by dislocation walls approximately parallel to the rolling plane. At very high strains (reductions of 90% or more) the dislocation walls parallel to the rolling plane are the one predominant microstructural feature as first observed by Hu (1969). It is sug-
108
3 Deformation and Textures of Metals at Large Strain
gested that this subdivision also represents a subdivision into zones with different combinations of less than five slip systems (Leffers et al., 1991; Bay et al., 1992), but obviously the detailed mechanism (actual combinations of slip systems, dislocation reactions to form the walls) is different from that for the DDWs/MBls since the orientation of the dislocation walls is different. As first described in details for copper by Malin and Hatherly (1979), there is another type of organized structure with a micromechanical function completely different from that of the grain subdivisions described above, viz. "second generation microbands", MB2s. They are thin platelets, approximately parallel to a {111} plane, with concentrated slip parallel to the {111} habit plane. Malin and Hatherly used the simple term "microband", but Hansen (1990) suggested the designation second generation microbands to distinguish them from first generation microbands. The numbering refers to the historical fact that the term microband was first used by Hu (1962) for at type of bands (in rolled single crystals) with a micromechanical function similar to that of the MBls, viz. to separate zones with different slip patterns. The great majority of MB2 observations refer to copper, but MB2s have also been observed in nickel (Hughes and Hansen, 1991) and in Al-Mg (Korbel and Martin, 1986). MB2s in copper have recently been investigated by Ananthan et al. (1991 a) and Leffers et al. (1991). They were found mainly to appear in grains with an equiaxed cell structure, i.e. not (or rarely) in the grains with a high density of DDWs/ MBls. The MB2s first appear after rolling to 5% reduction, and they are very common in material rolled to reductions in the range 20-40%. At higher strains they are
less frequent, but they are observed up to rolling reductions of 80%. At reductions up to about 20% there is normally only one system of parallel MB2s in each grain, but at higher reductions there may be two or three intersecting systems. MB2s are not very stable structures; for instance Ananthan et al. (1991a, b) and Leffers etal. (1991) quote direct observation of vanishing or vanished MB2s. This structural instability makes it difficult to assess the contribution of the MB2s to the overall strain. Ananthan et al. (1991 a), on the basis of the actually observed MB2s, estimated their contribution (in grains with relatively high density of visible MB2s) to be less than 10% of the total shear, but this is certainly an underestimate, as argued by Leffers etal. (1991). Leffers et al. also discussed the possible effects of the MB2s on the overall deformation pattern. The micromechanical function of MB2s is identical to that of bundles in f.c.c. materials with low stacking fault energy (Sec. 3.5.2), viz. that of concentrated slip parallel to their habit plane. This inspired Leffers and Ananthan (1991) to look for a similarity in crystallographic orientation between the grains with a high density of MB2s in copper and the grains with bundles in brass. However, the result was negative. For instance, grains in copper with orientations close to {110} <001> had a high density of MB2s, while grains in brass with this orientation had no bundles. When it comes to understanding the effects of the organized structures in f.c.c. materials with higher stacking fault energies on the overall deformation pattern, the problem is that the Taylor model seems to work quite well, e.g. Sec. 3.7 and Leffers and Juul Jensen (1991) - even though the microstructure, as described above, indicates several trends of non-Taylor behav-
3.5 Heterogeneous Flow and Microstructure
ior. It is obvious that we cannot claim to understand what really goes on unless we can reconcile microstructure and the overall deformation pattern (as for instance reflected in the texture). Thus further work is needed. For aluminum Juul Jensen and Hansen (1990) have suggested a specific macroscopic role of the DDWs/MBls, namely that they are the cause of the fraction of the mechanical anisotropy which cannot be explained in terms of the texture. 3.5.4 Shear Bands At high rolling reductions shear bands become a significant component of the microstructure in the whole range of f.c.c. materials. In materials with low stacking fault energy the formation of shear bands is the predominant deformation mode in a range of reductions. Shear bands are plateshaped zones with concentrated shear - as the second-generation microbands in copper and the bundles in brass. However, the shear bands are not crystallographic (they are approximately parallel to the transverse direction, making angles of ~35° with the rolling plane), and they are not necessarily limited to the individual grains; they may extend through many grains or even the whole specimen without any change in direction. With reference to the formal similarity between (second-generation) microbands and shear bands Korbel and Martin (1986) have suggested that the microbands are the first stage in the formation of shear bands, a point which Ananthan et al. (1991 a) have questioned. From the description below it will be evident that there is no reason to believe that the bundles in materials with low stacking fault energy are the direct initiators of shear bands. Since the shear bands are macroscopic features with no direct relation to the crys-
109
tal lattice of the individual grain, the theoretical description of shear bands must to a large extent be based on continuum mechanics - even though the elementary deformation process must be crystallographic slip. A number of papers by various authors dealing with the theory of shear banding were presented as "viewpoints" in Scripta Metallurgica, introduced by Hutchinson (1984). The macroscopic nature of the shear bands does certainly not mean that the microstructure is irrelevant: the microstructure provides the constitutive conditions for shear-band formation. In this section we shall describe the phenomenology of shear bands and the microstructural conditions for their formation - with special emphasis on materials with low stacking fault energy for which these conditions are fairly well understood. Duggan et al. (1978) were the first to provide a systematic description of the formation of shear bands in materials with low stacking fault energy, in their case rolled brass with 30% zinc and a grain size of 25 |im, which seems to be fairly representative for materials with low stacking fault energy. The observations of Duggan et al. have been confirmed in various later works, e.g. Hirsch et al. (1988). At 50% reduction, where the first shear bands appear, high densities of twin lamellae were observed in most grains, and many of the grains had the twin lamellae nearly parallel to the rolling plane. This was the result of a microstructural evolution similar to that described in detail in Sec. 3.5.2 for brass with 15% zinc - apparently with a difference in evolution rate: the microstructure in brass with 30% zinc as described by Duggan et al. seems to correspond to that in brass with 15% zinc rolled to 60-70% reduction. This difference is consistent with the appearance of shear bands in
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3 Deformation and Textures of Metals at Large Strain
brass with 15% zinc: they first appear at about 70% reduction. The shear bands observed by Duggan et al. had the typical shear-band orientation quoted above. They had thicknesses in the range 0.1-1 jim and associated shears in the range 1-10 (with average values ~3-4). They always appeared in heavily twinned regions. They cut through the twin lamellae, eroding the twin/matrix composite structure. The transformed structure in the shear bands consists of elongated "crystallites" in the size range 0.02-0.1 jim with very large orientation differences (up to ~30°). At 50 % reduction the grains with shear bands only contain relatively few parallel bands. At higher reductions two intersecting systems are normally observed and the total volume of the shear bands increases. At 90% reduction most of the twinned structure has been eroded by shear bands (three is less than 25 % left), which means that shear banding has been the predominant deformation mode in a range of reductions. If, for instance, the material is 75% transformed to shear bands and if we take the average shear in the shear bands to be 3.5, this would, simplistically, account for the strain from 65% to 90% reduction. After 90 % reduction it appears that shear-band formation ceases and more homogeneous slip processes take over. All the observations quoted above agree with the qualitative explanation of the reason for the formation of the shear bands suggested by Duggan et al. As argued in Sec. 3.5.2 the twinned structure has locked itself in a situation where continued deformation by glide is becoming increasingly more difficult - because of the many grains with twin lamellae nearly parallel to the rolling plane and/or the few grains left to deform by slip on more than one slip
plane. In order to release the structural lock we need a process which can destroy the twinned structure, and the shear bands obviously have this capacity. Once the twinned structure has been destroyed at ~ 9 0 % reduction, shear bands are not needed any more, and normal slip takes over again. Thus, the succession of events in materials with low stacking fault energy may be seen as a clear demonstration of the importance of microstructure for the formation of shear bands. Low workhardening rate is normally quoted as a general condition for shear bands. In materials with low stacking fault energy it obviously makes no sense to talk about one general work-hardening rate for shear bands and slip. The relevant condition is the high effective work-hardening rate for continued slip which makes shear banding a favorable alternative. It should be mentioned that the microstructural conditions under which shear bands form in brass with very fine grains ( ~ 2 - 5 Jim) deviate from those described by Duggan etal.. This has led Yeung et al. (1988) to question the idea that twin lamellae parallel to the rolling plane are the important condition for shear bands in materials with low stacking fault energy. In rolled aluminum and copper shear bands are observed at reductions of - 6 0 % or more, e.g. Brown (1972), Malin and Hatherly (1979). Brown quotes observations of shear bands in pure aluminum (without specifying the purity), but it should be noticed that other workers (e.g., Lloyd, 1987) only observe shear bands in aluminum alloys. The shear bands have the typical plate shape and the typical orientation; they are about 0.1-1 |im thick. There are two sets of shear bands, either present simultaneously in one grain/area (aluminum) or spatially separated (copper). Brown quotes that the two sets in
3.5 Heterogeneous Flow and Microstructure
aluminum are not quite symmetrical. The shear bands are clearly less frequent and take up a clearly smaller volume fraction than in brass. Malin and Hatherly quote that the overall shear contribution from the shear bands is small in copper, while Brown suggests a considerable contribution in aluminum. The shear bands always form in areas with well developed dislocation walls parallel to the rolling plane. This observation raises the obvious question whether there are microstructural reasons for shear-band formation similar to that suggested for brass (with twin lamellae parallel to the rolling plane). However, as described in Sec. 3.5.3 the functional role in the deformation process of the walls parallel to the rolling plane in aluminum and copper is unknown, and therefore the question about a simple microstructural reason for shear-band formation cannot be answered at present. It is obvious that deformation by shear bands must follow a set of rules which is radically different from that for deformation by normal slip. Models which aim at describing strain ranges where shear bands are important/predominant must therefore include (or be based on) shear-band formation. 3.5.5 Grains at Very High Strains
In a non-deformed single-phase material, a recrystallized material for instance, the grain boundaries are, apart from some stray dislocations, the only interruptions in the regular crystal lattice and the only obstacles to dislocation motion. It is therefore logical to describe the early deformation processes in terms of the grain structure. As we have described, dislocations and other defects (stacking faults in the case of twin lamellae) accumulate during deformation to form various types of sec-
111
ondary boundaries within the grains. This means that the role of the original grains becomes increasingly less obvious with increasing strain. There is no simple answer to the question when the original grains cease to be relevant. If we take the bundles in brass (Sec. 3.5.2) as one example, then the grains certainly split up (in'bundles and matrix between the bundles, the bundles themselves consisting of a composite of twin and matrix lamellae). Nevertheless, each grain behaves as one entity with one specific slip pattern. If we, on the other hand, consider a grain which splits up in two deformation bands (Sec. 3.5.1), then the crystal lattices of the two deformation bands will rotate away from one another, and at a certain strain the two deformation bands will have taken over the role of the original grain. In brass the twin/bundle structure is eventually transformed by shear bands to a structure of fine crystallites with large orientation differences (Sec. 3.5.4); at this stage the crystallites and not the original grains must be seen as representing the relevant subdivision. Altogether, the shear bands at high strains, going through many grains, will make the original grain structure irrelevant. As an extreme example Korbel et al. (1981) have subjected aluminum to very high cyclic strains (cumulated strain ~90). Extensive shear banding produced a stable structure of equiaxed subgrains of a size of ~0.5 jim with absolutely no trace left of the original grain structure. Gotthardt et al. (1972) have mapped the spatial orientation distribution in copper rolled to 95% reduction. They found that the material consisted of "discs" with approximately constant lattice orientation (less than 10° variation). There were large orientation differences between the discs. The discs were parallel to the rolling plane;
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3 Deformation and Textures of Metals at Large Strain
the thickness was ~0.5-1 jam, and the diameter was in the range 1-10 jim. The original grain size was ~10-15 jum, which means that a grain deformed as one entity would have the approximate dimensions 0.5 pm x 10 jim x 200 Jim. Thus, the original grains remain intact in the thickness dimension as far as major orientation changes are concerned (other observations, not to be discussed in details, indicate some splitting in the thickness dimension), whereas the grains split up in several or many differently oriented regions (discs) in the other two dimensions. The subdivision in discs with internal orientation variations less than 10° has no clear correlation to the visible microstructure, which is dominated by dislocation walls parallel to the rolling plane (cf. Sec. 3.5.3) with a spacing of ~0.1 |im. As a crude rule one may quote that the original grain structure is easily recognizable in the microstructure up to rolling reductions of ~ 50 % (or, more general, up to strains of ~1). With increasing strain the original grain structure becomes increasingly hidden by the deformation-induced structures, and by —90% reduction (strain ~2) it is invisible. It should be noticed that the degeneration of the grains at high strains does not necessarily mean that all information related to the original grain structure is lost. The initial grain structure may push the deformation processes in a particular direction which leads to particular structures up to very high strains even though the original grains are not visible any more. For instance, copper with very large initial grain size has a texture after 95 % rolling reduction which is clearly different from the ordinary copper texture (Leffers et al., 1988 b, Chung e t a l , 1988).
3.6 Modelling of the Plastic Deformation of Single-Phase Polycrystals at Large Strains Note: The summation convention will be used in the mathematical expressions of this section, except when specified otherwise or when summation symbols are explicitly used. For many engineering applications, it is sufficient to use the mathematical theory of plasticity from continuum mechanics (Hill, 1950) to treat problems related to the plastic deformation of metals at room temperature. For large strains it will, at least, be necessary to introduce suitable work hardening laws into these models. Although Hill proposed an anisotropic yield criterion in addition to the isotropic von Mises and Tresca criteria, a general anisotropic analysis will prove impossible on this basis because of the unavailability of suitable values for the coefficients of the yield criterion. Only in sheet metal forming it is, to a certain extent, possible to apply this criterion to anisotropic cases by assuming plane stress. The necessary coefficients can then be obtained by measuring the r-value [see for example Backofen (1972)]. The r-value is the ratio between the transverse strain and the thickness strain in a uniaxial tensile test, at a longitudinal strain of 10% or 15%. Although quite useful, the continuum theory of plasticity (e.g., Hill, 1950; or Backofen, 1972) is incapable of explaining the interaction between plastic deformation and microstructure or crystallographic texture. It is the purpose of the present section to explain various theories that are capable of doing so. They are based on the laws that govern the plastic behavior of the individual crystallites. A
3.6 Modelling of the Plastic Deformation of Single-Phase Polycrystals at Large Strains
critical evaluation (including a comparison of the predictions of the models with experimental results) will be given in Sec. 3.7. Most of the discussions in the present section only apply to single-phase polycrystalline materials. The models presented here are usually rigid-plastic. So the strains which are mentioned are plastic strains unless stated otherwise. 3.6.1 A Few Definitions
We will use quantities such as stresses, plastic strains, plastic strain rates, rates of rigid body rotation and velocity gradients in order to describe the state that exist in the polycrystalline material during the plastic deformation. Two kinds of quantities will be used: local and macroscopic. The local quantities (denoted otj, eij9 sij9 cay, and ftj) describe the situation at a given point in the microstructure. Fluctuations on an atomic scale are however neglected. Detailed experimental investigations as well as theoretical studies have shown that they vary from point to point. The corresponding macroscopic quantities
113
sumed that the macroscopic velocity gradient Fij9 is known, e.g. from a finite element analysis or, in simple cases, from macroscopic observation. 3.6.2 The Classical Taylor Theory
One of the oldest and most popular theories for the plasic deformation of polycrystals was proposed by Taylor (1938). For general reading: see e.g. Kocks (1970) or Gil Sevillano et al. (1980). This theory, which is also called the "full constraints" (FC) theory, will be treated here in a slightly generalized form. Its basic assumptions are: (i) The macroscopic velocity gradient tensor Ftj is supposed to be known. (ii) The local quantities defined in Sec. 3.6.1 are homogeneous within each crystallite. (iii) The distortion is homogeneously distributed over the considered "control volume". This means that the local velocity gradients ftj of the various crystallites are all equal to the macroscopic velocity gradient
are denoted Sij9 Eij9 Etj Qtj, and Fijm They
are equal to the averages of the local quantities over a "control volume" that is large enough to count thousands of crystallites but still small compared to the deformed body as a whole. The latter can usually be subdivided in a large number of such volumes. These could all have different values for their "macroscopic" quantities in case of complex forming processes such as forging or rolling. In a continuum-mechanical analysis (e.g., a finite element calculation), the distribution of Sij9 Etj etc. throughout the workpiece would be computed as if they were quantities that have a continuous variation in space. Such variation will not be considered further in the discussions below. In most cases, it will be as-
*lj=ftj
(3"6)
(iv) The distortion of a given crystal is accomplished by multiple slip, in such a way thatfij takes the desired value. It will be explained in the next section how ftj depends on the slip activity. (v) If there are several different combinations of active slip systems that can achieve (iv), then the plastic work that is internally dissipated by friction along the glide planes must be calculated for each combination. The one that leads to the smallest plastic work is to be preferred. We will now use the above assumptions to find the active slip systems and the corresponding slip rates ys for a particular crystallite with orientation g. The devia-
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3 Deformation and Textures of Metals at Large Strain
toric part of the stress <jftj and (btj, the rate of rotation of the crystal lattice, will also be found. The activation of a slip system s is governed by the generalized Schmid law (3-7)
with (3-8)
as the "resolved shear stress on slip system ,?". The coefficients Mstj are geometrical factors given by (no summation over s)
in which b\ and n\ are the components of the unit vectors in the slip direction and in the direction normal to the slip plane of slip system s. The M\j are the independent of the crystal orientation g when expressed in the crystal reference system x*. Of course they do depend on the crystal orientation when expressed in the external reference system. xcs is the critical resolved shear stress valid for slip system s. It has a positive value. Although it depends on many parameters (temperature, strain rate, strain history, impurities ...), it will be treated here either as a material constant or as a function of the slip rate on the same slip system (see below). Slip takes place when in Eq. (3-7) one of the equality signs hold. The material is undeformed or only elastically deformed if none of the two equalities holds. In the Taylor theory, the local stress is initially unknown. So Eqs. (3-7) to (3-8) cannot be used to identify the active slip systems. At the moment, we will simply assume that some slip systems are active and others are not. Let ys be a set of slip rates on the n available slip systems. Some of them may be zero, in which case the system is not active. The distortion caused by this slip activity would be described by
a local velocity gradient given by (3-10) s=l
in which cb^j is the (unknown) rate of rotation of the crystal lattice. Note that the "slips" ys have the nature of a shear, and the product (without summation) b\ n] is nothing else than the distortion tensor that describes a unit strain on slip system s (assuming that the lattice does not rotate). So the second term of the right hand side describes the distortion that the slip activity causes in a reference system which is co-rotational with the crystal lattice. In Eq. (3-6), ftj is now substituted for its value given by Eq. (3-10), after which the symmetric part of the tensorial equations is taken. This makes cbfj vanish, since it is skew-symmetrical. The following result is obtained [see also Eq. (3-9)]: Eij = Msijys
(3-11)
(Etj is the symmetrical part of Ftj). Equation (3-11) stands for 6 linear equations with n unknowns. Of those 6 equations, only 5 are independent from each other because the property of volume conservation during plastic deformation is implemented as well in the Etj as in the Ms(j tensors. For f.c.c. metals with {lll}<110> slip systems, n = 12. n = 24 for b.c.c. metals if {110} <111> and {112} <111> slip systems are adopted. So, in these cases, Eq. (3-11) stands for a set of 5 linear equations with much more than 5 unknowns. It can be solved if n — 5 of the n unknown slip rates ys are deliberately set equal to zero. The solution would have the aspect of 5 or less non-zero slip rates, and n — 5 or more zero slip rates. Unfortunately it is possible to make many different choices of the n — 5 slip systems out of n that should be non-active. Each of these choices leads to a solution.
3.6 Modelling of the Plastic Deformation of Single-Phase Polycrystals at Large Strains
Taylor (1938) proposed the following criterion to choose among these solutions [assumption (v) given above]: P = rcs\ys\ =
(3-12)
It means that the "right" solution should be the one that leads to the smallest value of the internally dissipated plastic power per unit volume. By "dissipated plastic power" we mean: energy dissipation per unit time. Equation (3-11) to (3-12) can be solved elegantly and efficiently by means of linear programming. This method does not only yield the slip rates, but also the deviatoric part of the local stress tensor atj (e.g., Van Houtte, 1988). Once the slip rates are known, it is possible to calculate the rate of lattice rotation cbfj from Eq. (3-10) (after substitution of ft j by F(j). The rate of lattice rotation is used to predict the evolution of the crystallite orientation g, which allows the prediction of deformation textures. The solutions of Eqs. (3-11) to (3-12) depend on the crystal orientation g since the quantities Mstj must be transformed from the crystal reference system (in which they are constants) to the sample reference system. For the expression of the transformation matrix, see Gil Sevillano et al. (1980) or Bunge (1982). The critical resolved shear stresses xcs are often characterized by means of their ratio to a reference value TC: «. = ^
(3-13)
A quantity that is often used is the Taylor factor (3-14)
M = ^EQ
in which EEQ is an equivalent strain rate, e.g. according to the von Mises conven-
115
tion. The Taylor factor is to be regarded as a function of the crystal orientation g and of the local strain mode S^/E^Q. It is a particularly useful quantity when the as are all equal to 1, i.e. when the critical resolved shear stresses all have the same value rc. In such cases, however, a special problem arises. It turns out that even when considering Eq. (3-12), there are multiple solutions, i.e. several sets of 5 active slip systems exist that satisfy the equations. It is mathematically allowed to make positive linear combinations between the different solutions. Such combined solutions may activate up to 8 slip systems in the case of f.c.c. metals. Note that although multiple solutions are found for the slip rates and for the rate of lattice rotation, a unique solution is found for P, the plastic power dissipation per unit volume, and for the Taylor factor M. The local (deviatoric) stress atj is also unique, except in some very rare cases. Further discussion on this problem can be found elsewhere (Van Houtte, 1988; Leffersetal., 1988a). 3.6.3 A Few Variants on the Same Theme 3.6.3.1 The Bishop-Hill Theory
Taylor's assumptions (i) through (iv) (see previous section) are also adopted. Taking the symmetric part of Eq. (3-6) shows that the local strain rate then is known, because it is equal to Etj, the symmetric part of the macroscopic velocity gradient tensor Ftj. For each slip system, Eq. (3-7) defines two hyperplanes in 6-D stress space. The points on the planes represent stress states for which the slip system is activated; it is not activated by stresses represented by points between the two planes. Other stresses are forbidden. Combining this for all slip systems leads to a hypersurface that contains all admissible stress states of the
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3 Deformation and Textures of Metals at Large Strain
crystallite. This is the single crystal yield locus. It can be shown that it is a hyperprism which is parallel to the Tt-vector in stress space. The latter is a unit vector that represents a hydrostatic stress state. So the yield locus is fully known when its intersection with the 7i-plane is known. The 71plane is the hyperplane through the origin and perpendicular to the 7i-vector. It contains the representations of all possible deviatoric stress states. The intersection has the shape of a hyperpolyhedron or a faceteye in 5 dimensions (Fig. 3-10). Once the yield locus is known, the maximum work principle (Hill, 1950) can be invoked to find the plastic stress that corresponds to a known plastic strain rate. It must be on or inside the yield locus and fulfil the condition that the plastic power dissipation per unit volume is maximal: (3-15) = max It is also possible to represent the plastic strain rate 8tj as a vector in stress space. Proper tensor-vector conversion formulas should be used [see for instance Van Houtte (1988)] so that the scalar product between a stress "vector" and a strain rate "vector" is equal to P. stj is known and
P =
(a)
(b)
Figure 3-10. A schematical representation of a portion of the yield locus of a single crystal in stress space, (a) Case where the plastic strain rate vector stj happens to be normal to a facet of the yield locus. All stresses that belong to the facet satisfy the maximum work principle, (b) Case where stj is not normal to any facet or to any rib. Only one stress satisfies the maximum work principle.
equal to Etj, a-j is sought. If stj happens to be normal to one of the facets of the yield locus (Fig. 3-10 a), then any stress vector pointing to a point of that facet would satisfy Eq. (3-15). But this will rarely happen. The most common case is that stj is not normal to any of the facets, not even to any of the edges. In that case, Eq. (3-15) will be satisfied for a stress represented by a vertex of the yield locus (Fig. 3-10b). The number of such vertices is limited, so a list of them can be made for a given yield locus. Equation (3-15) can be checked for each vertex on the list which allows a rapid identification of the "active vertex", directly giving the local deviatoric stress (j-j but also identifying the active slip systems. In order to find the slip rates ys, Eq. (3-11) must still be solved, assuming that the ys of the non-active slip systems are zero. This procedure was first proposed by Bishop and Hill (1951 a, b). It can be shown that the Taylor theory and the Bishop-Hill theory are mathematically equivalent, and that they always find the same solutions (see for example Kocks, 1970; Van Houtte, 1988). One way of demonstrating this is to look at them as linear programming problems. The Taylor Eqs. (3-11) and (3-12) can be regarded as a linear programming problem. So can Eqs. (3-7) and (3-15), which form the basis of the Bishop-Hill theory. Both problems form a primal-dual set as defined in linear programming theory (see e.g., Gass, 1969), and must lead to the same value of P and to the same set of active slip systems. They are both plagued by the same problem of a multiplicity of solutions. 3.6.3.2 The Strain Rate Sensitivity (Viscoplastic) Method Some authors (e.g., Canova and Kocks, 1984; Asaro and Needleman, 1985; Toth
3.6 Modelling of the Plastic Deformation of Single-Phase Polycrystals at Large Strains
et al., 1988) do not solve the Taylor theory by means of linear programming. Instead, they assume that a certain slip activity would take place on all slip systems in the presence of a stress, even if the resolved shear stress is small. According to this theory, the slip rate on slip system s is given by (no summation over s):
117
solution is chosen by a real crystallite depends on secondary effects, such as strain rate sensitivity (quite small at room temperature), small variations in the CRSS on the individual slip systems, local stress fluctuations etc. (Leffers et al., 1988 a). The particular choice made by the strain rate sensitivity method is based on a single secondary effect, neglecting all other.
(3-16) T 0 and y0 are constants that render the arguments of the exponents dimensionless. m is the strain rate sensitivity exponent. At room temperature, values of the order of 0.01 are used. At elevated temperatures, much higher values are used. It can be shown that for small values of m (such as 0.01), this model only predicts significant slip activity on slip systems for which the resolved shear stress is approximately equal to T 0 . In such case, the predictions are not very different from those of the generalized Schmid law [Eq. (3-7)]. However, the mathematical treatment is entirely different from those which were presented in the two previous sections. In Eq. (3-16) T* is substituted by the value given by Eq. (3-8). Expressions for ys which are functions of atj are then obtained. These are in turn used to replace the ys in Eq. (3-11). A set of non-linear equations in which only the atj are unknown is then obtained. It can be solved by an iterative technique (Toth et al., 1988). Unlike the classical Taylor theory and the Bishop-Hill theory, this method always finds a unique solution for the slip rates and the lattice rotation. It should be remembered however that the classical Taylor theory and the Bishop-Hill theory take the primary effects into account which lead to a particular vertex in stress space and to a set of possible solutions for the slip rates and the lattice rotation. Which particular
3.6.4 Relaxed Constraint Theories (Mixed Boundary Conditions) and Sachs Theory
The Taylor theory emphasises on the homogeneity of deformation [Eq. (3-6)]. The "relaxed constraints" (RC) theories or theories with mixed boundary conditions partly sacrifice homogeneity of deformation in favor of homogeneity of stress. Some of the 9 equations represented by Eq. (3-6) are "relaxed", i.e. dropped. Note that the classical Taylor-Bishop-Hill theory is now often called the "full constraints" (FC) theory. We shall explain this theory by means of one of its variants, the "lath" theory for rolling. Let the rolling direction be x± and the rolling plane normal x3. Suppose that for some reason, it is not necessary to require that the local shear/ 13 is equal to the macroscopic shear F13 (Fig. 3-11)./ 13 has become a free parameter, and the equation for the suffix 13 is left out of the set of Eq. (3-6). Of the 9 equations, 8 are left.
Figure 3-11. The simple shear which is described by the component / 1 3 of the velocity gradient.
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3 Deformation and Textures of Metals at Large Strain
The same holds for Eq. (3-10). When the symmetrical part of that equation is taken, the following expression is found for the suffices 13 or 31: f ( / 1 3 + / 3 1 ) = Ms13ys
(3-17)
This is one of the 6 equations of the set of Eq. (3-11). But it cannot be used to find the slip rates, since/ 13 has become a free parameter, to be found itself. The 5 remaining equations [of which only 4 are independent, see the comments about Eq. (311)] can be used to find the slip rates. In a completely similar way as for the classical Taylor theory (Sec. 3.6.2), this is done by adopting Eq. (3-12) as an additional condition. The equations can then be solved by linear programming, as in the classical Taylor theory. The deviatoric stress tensor aftj is found as a by-product (Van Houtte, 1982, 1988). Only 4 slip systems will be found active instead of 5, because Eq. (3-11) has one usable equation less. Once the slip rates ys found, Eq. (3-17) can be used to find/13: /i3=-/3i+2Ms13ys
(3-18)
Once the ys and / 1 3 quantities have been found, Eq. (3-10) can be used as usual for the calculation of the lattice rotation (b\}. It can be shown that the deviatoric stress tensor which was obtained has the following property (Van Houtte, 1982, 1988): '13
= <7 31 =0
(3-19)
for all crystallites, which partially achieves stress homogeneity throughout the material. This is a general property: for each component/^- that is relaxed, the corresponding stress component otj vanishes. This is the reason why RC models are sometimes called models with mixed boundary conditions: one can enforce stress conditions such as Eq. (3-19) by adopting a RC model. Sometimes the con-
ditions given by Eq. (3-19) are used as the basic assumption of the RC model (Kocks and Canova, 1981). According to Van Houtte (1982), the lath model would be suitable for elongated grains, which during rolling retained their original width, but became much longer and thinner. The reasoning is the following. The fact that one of the shears (/13) is not imposed and thus will not normally be the same for neighboring grains, will cause strain misfits at the two faces that are normal to x1 (Fig. 3-11). These misfits must be accommodated by complex deformation patterns. These patterns are only necessary in the vicinity of the affected parts of the grain boundary. It is seen in Fig. 3-11 that for long, elongated grains, these only represent a small fraction of the total grain boundary surface. So in such case, the freedom of/13 could be tolerated. The "pancake model" is another RC model for rolling. Not only/ 1 3 , but also / 2 3 is relaxed. Accordingly, a31 and
3.6 Modelling of the Plastic Deformation of Single-Phase Polycrystals at Large Strains
Canova etal. (1984) and Van Houtte (1984, 1986). An extreme example of a RC theory is obtained when (for tensile deformation in the direction x±) Eq. (3-19) is extended for all local stress components except a11. Hence the latter is the only non-zero stress component. The "subspace Bishop-Hill" theory described above can be used. In practice this means that one sets all stress components equal to zero except all9 and then uses Eq. (3-7) and Eq. (3-8), which leads to - ^ ^ F ^ T :
(3-20)
So each slip system sets a limit for a11. The lowest of these limits will determine crll9 and identify the active slip system. In case all slip systems have the same critical resolved shear stress TC, the lowest limit is for the slip system s which has the highest value for \M\^\ [see Eq. (3-9)]. Normally only one slip system will be active. For special crystal orientations, it is possible that 2 slip systems or more are simultaneously active. Such extreme RC model is equivalent the Sachs (1928) theory for the plastic deformation of polycrystals. Of all RC variants, this is the one that achieves the largest stress homogeneity and the smallest strain homogeneity. It should not be confounded with the "modified Sachs theory" (Pedersen and Leffers, 1987), which will be mentioned in the next subsection. 3.6.5 Other Methods Although relaxed constraints models form a useful departure from the too rigid Taylor assumptions, they are insatisfactory from a theoretical point of view. The choice of the relaxations is indeed always somewhat arbitrary. Self consistent methods and finite element methods offer ways
119
of relaxing the rigid Taylor assumptions in a more rigorous way. Consider a small portion of a grain boundary which can be considered flat. Let us call the normal on it x 3 . Then stress continuity requires that on both sides of the grain boundary, the components of the local stress tensor are the same except
120
3 Deformation and Textures of Metals at Large Strain
macroscopic strain rate. The equation has the following aspect (the dots that denote "rates" have been deliberately omitted): 7
ij -
s
tj
=
a G
tjki (Eki
-
hi)
(3-21)
The accommodation tensor Gim depends on the shape of the inclusion and on the properties of the matrix. Eshelby calculated it for the elastic case, a is called the "coupling factor" (see below). Some plastic models (Kroner, 1961; Budiansky and Wu, 1962) assume that the misfit between inclusion and matrix is accommodated by additional elastic deformation of the matrix (a = l), whereas other models (Hihi et al, 1985; Molinari et al., 1987) allow for plastic accommodation in the matrix ( 0 < a < l ) . With a = l, the results are nearly identical to those of the Taylor theory. Leffers (1975 b) has tried to use experimental observations of the degree of strain heterogeneity in polycrystals in order to make estimations of the coupling factor a. For brass it definitely was much lower than 1. It is obvious that some model for crystal plasticity must be added to Eq. (3-21) before it can be solved. Pedersen and Leffers (1987) used a model of this type with the purpose to simulate a deformation mode by intermittent avalanches of many dislocations on few glide planes, belonging to a family of primary slip planes. The resulting pile-ups are then assumed to cause the activation of secondary slip systems in the neighboring crystallites. This model is called "modified Sachs theory". Finite element models could be used in order to calculate the stress and the strain distribution in a given microstructure. Harren and Asaro (1989) reported such a study for a hypothetical 2-D polycrystal. Although the method and the results are quite interesting, it is also clear that a 3-D
method would be needed for applications on a real material. Adding a third dimension to the mesh that was used by Harren and Asaro would lead to prohibitive calculation times, even for a supercomputer. But some day it may become possible. Actually, some 3-D attempts have already been undertaken by Havlicek et al. (1989).
3.7 Application and Evaluation of the Models The models described in Sec. 3.6 are normally used for the calculation of the response of all crystallites of a polycrystal to an increment of macroscopic deformation. The procedure can be repeated for a series of such increments. Such increment is best specified by a macroscopic velocity gradient tensor Ftj and a time increment At. One of the results is the rate of lattice rotation. It is used to estimate the lattice orientations of the crystallites at the end of the time increment. The change of the rate of lattice rotation during the increment is usually neglected. Because of this, the time increment At must be sufficiently small. Its optimal size depends on the magnitude of the rate of lattice rotation. It is possible to change the macroscopic strain mode (described by the velocity gradient Ftj) from one increment to another. Experimental strain histories (Aernoudt and Van Houtte, 1981) or strain histories obtained by finite element simulations of a forming process (Mathur and Dawson, 1989) have been used. Nevertheless, for most applications, velocity gradients that do not change during the deformation have been used. Usually all elements are zero except: - F 3 3 = -2F11 = -2F22 in the case of axisymmetric tension or compression (in the direction x 3 );
3.7 Application and Evaluation of the Models 11 in the case of rolling , x 3 :ND); - F12 in the case of simple shear (x±:shear direction, x2:shear plane normal). In addition to the applications that we are going to discuss (prediction of deformation textures, calculation of stressstrain curves), it is also possible to use the polycrystal models for the calculation of the plastic anisotropy of materials with a known crystallographic texture, as for example reviewed by Van Houtte et al. (1989). All these applications require the knowledge of a set of crystallites (typically several hundreds) that is representative for the initial condition of polycrystalline sample. The orientation g and the volume fraction of each of the crystallites of the representative set must be known. It is possible to generate such representative set by discretization of the measured or assumed ODF of the sample (see e.g., Van Houtte, 1990).
3.7.1 Prediction of Deformation Textures
If the lattice orientations of the grains before deformation were statistically random, deformation changes a texture-free material to a material with a deformation texture. In practice ideally texture-free materials are very rare, and therefore one normally observes that deformation changes a preexisting texture to another texture. At large strains the texture approaches a specific type of texture characteristic of the material and the deformation mode, although the effect of strong initial texture may remain visible even after very large strains. We will not give an exhaustive review of all observed or predicted deformation textures here. Textures for their own sake are treated in Chap. 10, Vol. 15. We will only
121
compare observed and predicted deformation textures in order to evaluate the models explained in Sec. 3.6. Such work only makes sense if the initial textures of the simulations were equal to those of the samples that have really been deformed. Either this is done on the basis of individual crystallite orientation data (measured and predicted), or on the basis of quantitatively determined ODF diagrams (measured and predicted). It is also required that various models are tried in the same study. Otherwise a confrontation with experimental data would not allow the models to be compared with each other. No evaluation will be given on a model unless it was tested in such test. It was mentioned above (Sees. 3.6.2 and 3.6.3.1) that the Taylor or the Bishop-Hill theories lead to a set of multiple solutions for the lattice rotation if one uses exactly the same value for the critical shear stresses (CRSS) for all slip systems. Slight perturbations on these values cause the models to select a particular solution from the entire set (Gil Sevillano etal., 1980). Such perturbations can be expected to exist in a real material. However it is impossible to measure them and/or to predict them. Several methods exist for making the necessary choice of a particular solution among the total set (Van Houtte, 1988). Some of these are based on qualitative estimations of the perturbations. There are some differences in the textures predictions with the different methods. One of these methods selects the solution which (i) activates no more than 5 slip systems, and (ii) leads to the smallest value of the plastic power dissipation, recalculated by means of Eq. (3-12) but without minimization, simply using the already known values of the slip rates ys, and new values of the TCS which are proportional to y™. m is the strain rate sensitivity exponent.
122
3 Deformation and Textures of Metals at Large Strain
Strain rate sensitivity is hence taken into account in a certain way (but not in the same way as by the "strain rate sensitivity method" described in Sec. 3.6.3.2). This method gave the best results in a study in which the experimentally observed texture evolution of a Al 3004-alloy was quantitatively compared to the theoretical predictions of various RC and FC models (Dezillie et al., 1988). Figure 3-4 shows the experimentally observed texture, Fig. 3-12 the predicted one. The initial texture of the material was taken into account. ODF techniques were used for the comparison of experimental and simulated textures. Skalli et al. (1985) reported tests on multicrystals of 99.993% pure aluminum. The lattice rotations of 19 grains with 15 different crystal orientations were measured before and after rolling and compared to model predictions. 6 grains followed the predictions of a RC model (pancake); 3 rotated as predicted by a FC model in which the ambiguity problem is resolved in the following way: of all lattice rotations, the one whch brings the crystallite into the softest orientation (after the same time increment) is chosen. For the remaining grains, the predictions of the two models were similar. It can be concluded from all of this that the best way to resolve the ambiguity problem may be different from one material to another. While the "copper-type" rolling textures which were described in Sec. 3.2 are more
PHI2 = 65
or less satisfactorily explained by the usual models (Taylor-Bishop-Hill, self consistent), the "brass-type" texture (also described in Sec. 3.2) and the intermediary "transition-type textures" are not. This point will be discussed again in the next section. Wagner et al. (1991) have compared the texture evolution observed in low carbon steel samples to predictions by an FC, an RC (pancake) and a visco-plastic self consistent model for three different initial textures. Each time the ODF of the initial texture was used as initial texture of the simulations. It was concluded that the selfconsistent model leads to the best predictions, closely followed by the pancake model. These predictions are even quantitatively very good. Figure 3-13 shows the experimental texture and the simulated textures for one of the three cases (65 % rolling reduction, weak initial texture). The fact that the pancake model and the self-consistent model both predict the correct end orientations is ascribed to the fact that, on average, these models activate only three slip systems in the crystallites, which is more realistic than 5, as predicted by FC models. The good quantitative agreement of the self-consistent prediction is ascribed to the fact that in such a model there is more scatter in the strain mode of each crystallite than in FC Taylor simulations, for which the predictions are usually far too sharp.
Figure 3-12. Some sections of a predicted ODF of a cold rolled f.c.c. metal. Full constraints Taylor model, 86% rolling reduction. See Fig. 3-3 for interpretation.
3,7 Application and Evaluation of the Models
, = 45°
a) Experimental
123
(1988) found that the calculated FC rolling textures obtained with different solutions of the ambiguity problem are not very different, a point which was also demonstrated by Leffers (1988). A much stronger influence was found for axisymmetric deformation (e.g., Leffers, 1988).
3.7.2 The Case of Materials with Mechanical Twinning b) L A P
c) Taylor RC pancake
d)
VPSC
Figure 3-13. Comparison of observed and simulated rolling textures for steel. 65% rolling reduction. The starting texture was taken into account. See Fig. 3-6 for interpretation, (a) Experimental result, (b) Result of a FC model (LAP: Los Alamos Polycrystal Code), (c) Result of a RC (pancake) model, (d) Result of a viscoplastic self-consistent (VPSC) model. From Wagner et al. (1991).
To summarize, we conclude that for rolling of b.c.c. metals and f.c.c. metals with high or medium stacking fault energy, sometimes FC, sometimes RC and sometimes self-consistent models gave the best results. The predicted textures are often too sharp when compared to experimental ones. The differences between the predictions by the various models should however not be exaggerated. Dezillie et al.
In Sec. 3.5.2, it was discussed how shear localization in bundles associated with mechanical twinning promotes single slip in brass (and other f.c.c. metals with low stacking fault energy) and hence affects the development of the deformation texture. In hexagonal close-packed (h.c.p.) metals and in many minerals with a low lattice symmetry, mechanical twinning is also an important deformation mechanism, but in a different way. Figure 3-14 shows schematically the formation of a mechanical twin. AB is the twinning plane. In f.c.c. metals it must be a {111} plane; in b.c.c. metals it must be a {211} plane. Various type of twinning planes have been observed in h.c.p. metals (Partridge, 1967). Figures 3-14a and b show, that the formation of a twin in the region ABCD causes a shape change which can be described by a simple shear. The twinning plane AB is the shear plane. The shear direction and the shear magnitude / are determined by the nature of the lattice and of the twinning plane. Suppose now that the twinned zones are thin lamellae, and that they are more or less homogeneously distributed within the parent crystal. Let v be the volume fraction of these twins. An increase of v due to deformation will then cause an apparent shear of the entire crystallite, given by: dy = yl(lV
(3-22)
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3 Deformation and Textures of Metals at Large Strain
A
D
(a)
A/
'
><
(b) Figure 3-14. Mechanical Twin, (a) Before twinning, (b) After twinning. AB (or DC) is the trace of the twinning plane.
mere coincidence, since the Taylor theory does not apply to this material, as can for example be verified from Fig. 3-15. In brass, single slip is the predominant deformation mechanism for those grains that develop "bundles" (Leffers and BildeSorensen, 1990). See also Sec. 3.5.2. The situation for brass cannot be compared to that for h.c.p. metals or for most low-symmetry minerals. In such materials, the number of slip systems readily available for plastic deformation in a particular temperature and strain rate range is usually very low. Very often the number of slip systems available is insufficient to accommodate a deformation described by a general strain rate tensor. In mathematical terms this means that Eq. (3-11) cannot always be solved, the number of independent unknowns being smaller than the number 0.35 Experimental Calculated
0.30
It is therefore possible to describe a "twinning system" in the same way as a slip system: (hkl) [uvw], in which the (hkl) is the shear plane (which is the twinning plane) and [uvw] are the Miller indices of the shear direction (in the parent lattice). Mechanical twinning systems can thus be incorporated in the Taylor theory. This was first proposed by Chin et al. (1969); they also have listed all mechanical twinning systems of f.c.c. metals. Carpay et al. (1975) have listed those of b.c.c. metals. Partridge (1967) gives lists for various h.c.p. metals, as do Thornburg and Piehler (1975) for Ti-alloys. For more details, we refer to Van Houtte (1978), Gil Sevillano et al. (1980) and Tome et al. (1988). This method does predict a brass type texture when applied to f.c.c. metals (Van Houtte, 1978), but this is believed to be a
0.25
0.00 0
2
4
6 8 10 DEVIATION (°)
14
Figure 3-15. Distributions of the angular deviation of twin lamellae from the nearest {111} plane. The experimental distribution represents TEM measurements on the grains with bundles in brass with 15% zinc. The calculated distribution is based on the Taylor theory with mechanical twinning incorporated. Both distributions correspond to 37% rolling reduction. From Leffers and Juul Jensen (1991).
3.7 Application and Evaluation of the Models
of independent equations. For example, in many (not all) h.c.p. metals, there are only 3 slip systems readily available, namely the 3 {0001}<1120> systems. They have a much lower critical resolved shear stress than other types of slip systems. With only these 3 slip systems, it is not possible to accommodate any strain that involves elongation or compression of the oaxis. Experimental observations have shown, that such strains are achieved by the activation of mechanical twinning systems. The mechanical twins observed in crystals that underwent compression of the oaxis did not belong to the same twinning systems than those observed in crystals of which the oaxis was elongated (Philippe et al., 1988). Obviously the stress necessary to activate these twinning systems is less than the stress needed to activate additional types of slip systems. It is clear that for these materials the mechanical twinning shear itself has a significant contribution to the strain. It can then be expected that a Taylor model in which mechanical twinning is incorporated as described above would be more realistic than in the brass case. Satisfactory deformation texture predictions have been reported by Philippe et al. (1988) using the Taylor theory.
125
macroscopic tensile or compression stress and E the corresponding macroscopic strain rate. The latter is known, since it can be derived from the velocity gradient Ftj, which serves as input for the Taylor theory. So S is equal to P/E. In the same way the macroscopic shear stress T can be obtained as P/y for a "simple shear" test. Here y is the macroscopic shear rate. Simple shear can be achieved by means of a torsion test. All this shows that in principle the evolution of the macroscopic stress as a function of macroscopic strain can be predicted for axisymmetric deformation and for simple shear. If, however, the critical resolved shear stresses zcs are kept constant (as is often done during a deformation texture simulation), then the obtained curves will only reflect the influence of texture evolution. If the objective is to make a quantitative prediction of a stress-strain curve, then the change of the TCS during a strain increment must be modelled as well. This requires the implementation of realistic work-hardening models for individual slip systems (see Chap. 2 of this Volume). It is often assumed that the values ots and TC as defined by Eq. (3-13) do not vary from one crystallite to another. It is then possible to calculate P as follows (using the Taylor or the Bishop-Hill theory):
3.7.3 Stress-Strain Curves Equation (3-12) of the Taylor theory gives P(g), the average power dissipation per unit volume for a crystallite with lattice orientation g. Hence it is possible to calculate P, the average of P{g) for the whole polycrystal, for each strain step of a deformation texture simulation. All models discussed in Sec. 3.6 can be used for this purpose. Note that if axisymmetric deformation (tensile or compression test) is simulated, P is equal to SE, in which S is the
in which M is the average of the Taylor factor defined by Eq. (3-14). EEQ is a scalar measure of the macroscopic strain rate tensor, for example the von Mises equivalent strain rate. In this formula, M is only a function of the crystallographic texture, whereas the reference critical resolved shear stress TC is only a function of the dislocation configuration. At the next level of sophistication, one could allow for different hardening rates in
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3 Deformation and Textures of Metals at Large Strain
each crystallite. TC would then be different for every crystallite, except for the first strain increment after annealing. So Eq. (3-14) can still be used, but Eq. (3-23) not. Furthermore a model for the evolution of TC is required (see Chap. 2 of this Volume). TC is often assumed to be a function of F only, where F is the sum of the absolute values of the slips that have taken place in the crystallite after the last annealing treatment. It can be calculated by the models of Sec. 3.6. Several review papers have been devoted to this method of predicting polycrystal stress-strain curves (Kocks, 1970; Gil Sevillano et al., 1980; Tome etal., 1984; Aernoudt et al., 1987). Some drawbacks were found. Kocks (1970) reported that T°(F) relationships of single crystals are not unique but depend on crystal orientation, number of active slip systems etc. Tome etal. (1984) have tried to use the method to explain the differences observed between measured equivalent stress-equivalent strain curves of OFHC copper in tension, compression and torsion (Fig. 3-16). Only about half of the observed differences could be explained, which probably means that the other half of the differences cannot be ascribed to differences in texture evolution only. It would have been interesting to try to explain this "other half" by means of more sophisticated models for hardening (Hihi et al., 1985; Asaro and Needleman, 1985), but no such results are known to the present authors. 3.7.4 Conclusions Most of the conclusions presented here have also been drawn by an expert's panel discussion at ICOTOM 8 (Leffers et al., 1988 a). The models discussed in Sees. 3.6 and 3.7 can predict the development (or change) of deformation textures and can
estimate the influence of the latter on polycrystal stress-strain curves. Physical models as those discussed in Chap. 2 of this Volume are also necessary for this last application in order to come to know the value of the critical resolved shear stresses. It appears that the existing models can achieve reasonable simulations of the change of the texture due to deformation, at least for b.c.c. metals and f.c.c. metals with medium or high stacking fault energy. Note that a quantitative agreement between predicted and real deformation texture is only possible if the initial texture is taken into account in a quantitative way. If the latter is omitted, errors are introduced that overshadow the differences between the results of the various models. The different types of heterogeneities that have been discussed in Sec. 3.5 have a major influence on the deformation texture in the case of brass and other f.c.c. metals with low stacking fault energy. At present no existing model can predict these heterogeneities in a quantitative way. Brass-type textures [rolling, simple shear, wire drawing (Chin, 1969)] cannot be simulated by the classical methods used for materials with higher stacking fault energy, whether sophisticated or not. The rolling textures at least are fairly well simulated with the modified Sachs model, e.g. Leffers and Juul Jensen (1991), and so is the orientation distribution of the twin lamellae (Leffers and Van Houtte, 1989). None of the existing models is capable of explaining the effects of grain size on the deformations texture which were reviewed by Leffers et al. (1988 b). These effects are probably related to the more subtle implications of the microstructures on the overall deformation pattern (Sees. 3.5.1., 3.5.2, 3.5.3). It is expected that the difference between predictions by the full constraint Taylor
3.8 Deformation of Multiphase Materials
127
400 -
_
300 -
S. 200 -
100
(a) 05
10
15
100 |
(b)
theory and by self consistent models may increase in the case of more heterogeneous materials such as two-phase materials with a hard and a soft phase. Such "heterogeneity" may as a matter of fact merely be due to the differences in crystallite orientations for single-phase materials with very anisotropic crystallites such as h.c.p. metals or most geological materials.
Figure 3-16. Comparison of stress-strain curves of OFHC copper for different deformation modes, (a) Macroscopic curves for tension, compression and torsion; T = 293 K, von Mises equivalent strain rate = 10" 2 s~1. (b) Average microscopic hardening behavior obtained from the data represented in (a). Dashed line: by dividing the macroscopic stress by the average Taylor factor at zero strain. Full line: by a more sophisticated procedure (by dividing the stress by the average Taylor factor that corresponds to each F valve). In each category, the curves for tension, compression and torsion should be identical if the differences between the curves of (a) were only due to texture. From Tome etal. (1984).
3.8 Deformation of Multiphase Materials 3.8.1 Two Ductile Phases: Strain Partitioning When structures consist of two ductile phases of comparable volume fractions but with different flow stress or hardening
128
3 Deformation and Textures of Metals at Large Strain
behavior, it is normal to find different amounts of strain being taken up by the two constituents. This was first reported by Honeycombe and Boas (1948) in an oc-P brass with 40% Zn. The "strain partitioning" has been quantitatively determined in a number of cases, for dual-phase steels (Balliger and Gladman, 1981; Yegneswaren uand Tangri, 1983), for a silvermagnesium alloy (Clarebrough, 1949) and for an a-p' copper-zinc-aluminum alloy (Adnyana, 1982). As an example, Fig. 3-17 shows the plastic strain distribution during wire drawing of a copper-zinc-aluminum alloy (Adnyana, 1982), quenched from the high-temperature two-phase region. The aspect ratio of the phase constituents was taken as a measure of strain. It was found that the matrix a-phase, being softer than the martensitic P'-phase, takes up the larger part of the strain during the first drawing passes. However, due to the rapid strain hardening of the low-stacking fault a-phase, the flow stress difference decreases and a more homogeneous distribution of strain results after a total strain of
b .6
Figure 3-17. Plastic strain distribution between a and /T-phase in a copper-zinc-aluminum alloy (Adnyana, 1982). s^: strain in a-phase, s^: strain in /T-phase.
8&1. Similar behavior has been found in ferritic-martensitic and in ferritic-bainitic structures tensile tested to a true strain of e = 0.40 (Balliger and Gladman, 1981). Inhomogeneous strain distribution is strongest after the onset of large scale yielding and decreases with increasing strain as the strain hardening rates gradually equalize. Strain partitioning means that the softer phase flows around the harder phase. Although this requires a very complex material flow, attempts are being made to rationalize the stress-strain behavior of two-phase alloys on the basis of simple assumptions. So the model of Ashby (1970) is based on the observation of the steep rise in dislocation density with strain in dispersion-hardened alloys with non-deformable particles. That rise results from the geometrically necessary dislocations (see Sec. 3.4.2), being injected into the matrix to relax the high stresses generated by the primary glide loops around the particles. The corresponding constituent behavior is treated in Chap. 7 of this Volume on particle strengthening and in Chap. 2 on work hardening. In the case of two ductile phases, a similar reasoning can be made, but the number of geometrically necessary dislocations will now reflect the difference in plastic strain between the two phases and the model has to be adjusted correspondingly: at comparable strains, the number of compatibility dislocations will be much smaller. The Ashby model was applied to dual-phase steels by Balliger and Gladman (1981) and Sarosiek and Owen (1984). Furthermore, for a better description of the flow curve, dynamic recovery phenomena which take place in the softer phase as soon as the dislocation density reaches a critical value have to be taken into account. This is known to occur in an early
3.8 Deformation of Multiphase Materials
stage of polycrystal deformation, namely when entering stage III (Gil Sevillano e t a l , 1980). Other authors prefer to model flow curves of two-phase materials by using a rule of mixtures for either strains or stresses, taking the flow curves of the unconstrained individual phases as basis (Fischmeister and Karlsson, 1977). From what has been said above it is clear that none of these approaches is physically justified, certainly not in the case of equiaxed phase morphologies hitherto treated. The inhomogeneity of plastic deformation of two-phase materials can also be modelled by means of continuum-mechanical methods, such as self consistent models, or by elasto-plastic finite-element analysis. Early FEM studies were reported by Sundstrom (1973), Karlsson and Sundstrom (1974) and Fischmeister and Karlsson (1977). An interesting qualitative agreement with measured flow curves was found. However, also here the constitutive behavior of the elements has to be introduced and further refinement is needed e.g. by taking Ashby-type interaction processes into account. When materials with two highly ductile phases are deformed into the large strain region - and assuming that structural damage is negligible - deformation textures are being developed in the fibrous two-phase structure. However, results hitherto obtained (Wassermann, 1969) indicate that texture development in the softer matrix phase is usually inhibited by the presence of a harder phase. The harder phase, on the other hand, tends to develop a stronger texture, corresponding to the texture that is calculated for straining under unconstrained Sachs-type conditions instead of fully constrained Taylor-conditions. The flow curves of two-phase materials drawn into a fibrous substructure
129
evolve from a parabolic into an exponential relationship (Wassermann, 1973), a behavior that is also found in heavily drawn pearlitic steel, as will be discussed in the next paragraph. 3.8.2 The Case of Lamellar Pearlite Ultra high strength steel with a tensile strength up to 4000 MPa and beyond is produced by severely cold drawing material with fine lamellar pearlitic structure. Lamellar pearlite with an interlamellar spacing of ^ 8 0 n m is formed by the isothermal decomposition of eutectoid austenite at a temperature of about 550 °C. It consists of about 88 volume percent of ferrite and about 12 percent of cementite in alternating layers, thus forming a colony of pearlite. Several, differently oriented pearlite colonies are usually formed out of one former austenite grain. During wire drawing, there is a gradual reorientation of the lamellar structure into an orientation parallel to the wire axis. If the two phases are assumed to deform homogeneously, the reorientation with strain is given by Eq. (3-24) (Gil Sevillano, 1974): a = arctan (e(3/2)£ tan oc0)
(3-24)
with a0 being the initial and a the instantaneous angle of the lamellae with the wire axis. Figure 3-18 shows how, after a strain of e = 1, about 70 % of the colonies already have an orientation within 30° of the wire axis; at a strain of s = 2, the same colonies are found within 10° of the wire axis. In reality, the orthorhombic cementite, being brittle in massive form and in coarse lamellar size, turns out to be deformable to nearly the same extent as ferrite when patented into the abovementioned fine size. However the deformation of the lamellar aggregate does not occur as ho-
130
3 Deformation and Textures of Metals at Large Strain
In spite of all these effects, the micro structural scale of the phases decreases nearly proportional to the wire diameter, at least from deformations of about s = 1.5 (Langford and Cohen, 1969). If substructural spacing determines the mean free path of dislocations and hence flow stress, an exponential relationship (with a positive exponent) between stress and strain results. For example, Gil Sevillano (1974) found for eutectoidic steel:
a o
Figure 3-18. Reorientation with strain of an initially randomly oriented lamellar structure under the assumption of homogeneous strain (Gil Sevillano, 1974).
mogeneously as was assumed when deriving Eq. (3-24). In the initial phases of drawing, colonies at large angles to the drawing axis tend to buckle (Gil Sevillano, 1974). Other colonies deform by intense shear banding (Aernoudt, 1987). The transverse sections of the drawn wire reveal severe curling effects - see Fig. 3-19 reflecting plane strain deformation behavior of the pearlite colonies and concomitant bending in order to achieve overall axisymmetric behavior of the aggregate (Hosford, 1964).
Figure 3-19. Curling of microstructure in transverse section of a drawn high carbon steel wire (Langford, 1977).
1304e£/4(MPa)
(3-25)
The <110> drawing texture in pearlitic steel wire is at least as strong as in heavily drawn ferritic steel wire (Watte, 1990).
3.9 Change of Strain Path We know from the previous sections that property changes during low temperature deformation are being governed by two concomitant features: the development of a dislocation substructure on one hand, and the development of texture on the other hand. Both features are interrelated: the crystal orientation influences the substructure in a given strain mode and the strain hardening matrix connected with a given substructure influences the selection of active slip systems, hence also the resulting orientation changes. When, in a subsequent operation, the strain mode changes, the mechanical response of the material will be determined by the nature and the evolution of both the substructure and the texture generated in the first deformation mode. Three different phenomena can occur on subsequent straining: a Bauschinger effect, a cross effect and a strain softening effect (Aernoudt etal., 1987). The three phenomena are schematically represented in Fig. 3-20. Each of the two first effects is occurring in
3.9 Change of Strain Path
a A
Figure 3-20. Schematic representation of the (a) Bauschinger-effect, (b) cross-effect, (c) strain-softening effect, three features characteristic of changing strain paths.
the initial stage of subsequent yielding, whereas strain softening only becomes apparent after a subsequent strain of several percent. The Bauschinger effect is a - sometimes very large - drop in yield stress on strain reversal, like in tensile-compressive, in bending-unbending or in reverse torsion tests sequences (Chap. 11 of this Volume). It is linked to the ease of dislocation motion in the opposite sense compared to that of the forward strain mode. Mainly, longrange back-stresses in the cell structure of a predeformed single-phase material (Pedersen et al, 1981; Mughrabi, 1983), or back-stresses exerted by Orowan-loops in a predeformed particle-hardened structure (Orowan, 1959; Brown et al., 1971) are responsible for the effect. In practice the superimposed influence of intergranular stresses is not very important (Strauven, 1982; Leffers and Pedersen, 1982). One should notice that the Bauschinger effect is a consequence of the internal stresses associated with dislocation accumulation as described in Sees. 3.4.1 and 3.4.2. When we look in a general way on strain path changes, it can be shown that the
131
Bauschinger effect is rather the exception than the rule. Indeed, as soon as the change in strain mode vector, as represented in stress space, deviates some degrees from perfect strain reversal, a cross effect takes place, which means that the subsequent yield stress is higher than the stress needed for further forward deformation. Examples of non-reversal strain path changes, for which the cross effect has been reported, are for instance plane strain of a sheet followed by uniaxial tension in the previous zero strain direction (Wagoner and Laukonis, 1983), two successive mutually perpendicular uniaxial tensions (Schmitt, 1987) etc. The cross effect may be linked to the increased stress needed to operate new slip systems in the subsequent strain mode. The new mobile dislocations have to cross the former forest substructure, causing a latent hardening effect (Aernoudt etal., 1987). For completeness, it has to be mentioned that the yield stresses under consideration are in fact the "back extrapolated stresses" after the microyield region, see Fig. 3-21. At a more advanced stage of subsequent deformation, annihilating reactions between the new dislocations and the former substructure may lead to a marked de-
<* A
Figure 3-21. Back-extrapolated yield stress.
132
3 Deformation and Textures of Metals at Large Strain
crease of strain hardening and even to strain softening. This often remains so even after filtering out the possible influence of texture (Strauven and Aernoudt, 1987). It was found that the strongly elongated cigar-like subgrain structure in a heavily predrawn ferritic steel wire was thus gradually relaxed into an equiaxed cell-subgrain structure in a subsequent reverse compression test (Strauven and Aernoudt, 1987). The minimum flow stress in the reverse flow curve corresponds indeed to the strain at which the "collapse" of the former substructure was complete. Further reverse straining then gradually transforms the structure into a deformation substructure with pancake morphology, which is a structure typical for the new strain mode. The "collapsed" substructure of Fig. 3-22 a has all the characteristics of a dynamically recovered, lowenergy dislocation substructure. It is similar to the structure of the same material after a thermal recovery treatment [Fig. 3-22b (Strauven, 1982)]. Also in non-reversing strain path changes, a strain softening can become apparent. This is illustrated for the case of predrawn steel wire material, Fig. 3-23, which was subsequently compressed in transverse direction. It was shown how material flow is enhanced in directions along which the flow stress decreased by changing the strain path (Aernoudt et al., 1987). The dynamic recovery phenomena responsible for strain softening in single phase materials are also believed to play an important role in two-phase and in particle-hardened materials. Indeed similar effects as the abovementioned were measured in ferritic-pearlitic and in fullpearlitic steels. The effect even increases with increasing pearlite content, see Fig. 3-24 (De Bleser, 1979).
(a)
(b) Figure 3-22. (a) Equiaxed cell structure in the transverse section of a ferritic steel wire (0.07 C, 0.32 Mn, 0.03 Si, 0.006 P, 0.025 S) drawn to a true strain of s = 2.50 and subsequently compressed. The true strain in compression was e = 0.40 (Strauven, 1982). (b) Equiaxed cell structure in the same material drawn to ed = 2.50 and subsequently recovery annealed for 60min at 500 °C (Strauven, 1982).
An obvious question that arises concerns the possible contribution of changes in the orientation distribution to the effects discussed so far. It is true that texture softening could in principle explain transient softening phenomena when changing strain mode. But several examples of plastically predeformed materials have shown that in these cases the influence of texture on the transients is not of primary importance. So it was demonstrated (Strauven and
3.11 References
Aernoudt, 1987) that the expected texture change when compressing wire drawn ferritic steel should lead to hardening rather than to softening. Other authors (Raphanel et aL, 1986) found that the influence of texture on the back-extrapolated flow stress of sheet steel material amounts only to 30 to 50 % of the influence of structural changes.
HC *
—<
|
/ //
/
— — •— - - - ^ * r
1
1°
/ |
»MC
/ /
.7 1,0
0,8
1/ II II l! II ll
3.10 Concluding Remarks
|l
.LC
| 1 -^" ' 1 /
0,6
133
f
0,1
0,2
0,3
0,4
0,5
£
Figure 3-23. Monotonic (dashed) and subsequent (solid) flow curves of ferritic-pearlitic steels. Previous strain mode: wire drawing. Subsequent strain mode: transverse compression on carefully machined square specimens (Aernoudt et aL, 1990). HC: high carbon, MC: medium carbon, LC: low carbon.
500
Figure 3-24. Reverse compressional flow curves of ferritic and pearlitic steels after different drawing strains. The dashed curve is the forward flow curve, estimated by means of tensile tests on samples taken after different wire drawing passes (De Bleser, 1979).
The topics of this chapter have not been treated exhaustively. The development of dislocation patterns and deformation structures could have been described for more materials. Some variants of the models for texture development exist that were not mentioned. The field of multiphase materials could have been treated much more extensively. The authors nevertheless hope that the interested reader has got a general idea of the kind of phenomena that take place in a polycrystalline material during large plastic deformation, and that he came to the insight that there is a concurrent development of deformation texture and deformation microstructure which interact with each other. Together they control the evolution of the mechanical properties. It is also hoped that the reader has gained some understanding of how the effect of large deformations of polycrystals on texture and flow stress can be modelled, and, most importantly, what the limitations of these modelling attempts are.
3.11 References Adnyana, D. N. (1982), Ph. D. Thesis, Katholieke Universiteit Leuven, Belgium. Aernoudt, E. (1966), Ph. D. Thesis, Technical University of Aachen, Germany.
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3 Deformation and Textures of Metals at Large Strain
Aernoudt, E. (1987), in: Proc. DGM Symposium on "Ziehen von Stangen, Drdhten, Rohren": Funke, P. (Ed.). Oberursel, FRG: Deutsche Gesellschaft fur Metallkunde, pp. 7-32. Aernoudt, E., Van Houtte, P. (1981), in: Ziehen von Drdhten, Stangen und Rohren. Oberursel, FRG: Deutsche Gesellschaft fur Metallkunde, pp. 71-84. Aernoudt, E., Gil Sevillano, I , Van Houtte, P. (1987), in: Constitutive Relations and Their Physical Basis: Andersen, S. I., Bilde-Sorensen, I B., Hansen, N., Leffers, T., Lilholt, H., Pedersen, O. B., Ralph, B. (Eds.). Roskilde: Rise National Laboratory, pp. 1-38. Aernoudt, E., Strauven, Y, Lu, L., Van Houtte, P. (1990), in: Proc. 23th Century Aristotle Conference on the Mechanics, Physics and Structure of Materials, Tessaloniki (in press). Ananthan, V. S., Leffers, T, Hansen, N. (1991a), Scripta Metall. Mater. 25, 137. Ananthan, V S., Leffers, X, Hansen, N. (1991b), Mater. Sci. Techn. 7, 1069. Asaro, R. X, Needleman, A. (1985), Ada metall. 33, 923. Ashby, M. F. (1970), Phil Mag. 21, 399-424. Backofen, W. A. (1972), Deformation Processing. Reading, MA: Addison-Wesley. Bacroix, B., Jonas, J. X (1988), in: Proc. ICOTOM 8: Kallend, J. S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 403-421. Balliger, N. G., Gladman, T. (1981), Metals Sci. 15, 95-108. Barlow, C. Y, Bay, B., Hansen, N. (1985), Phil Mag. 51, 253. Bay, B., Hansen, N. (1981), in: Deformation of Polycrystals: Hansen, N., Horsewell, A., Leffers, T., Lilholt, H. (Eds.). Roskilde: Riso National Laboratory, pp. 137-144. Bay, B., Hansen, N., Kuhlmann-Wilsdorf, D. (1989), Mater. Sci. Eng. A113, 385. Bay, B., Hansen, N., Hughes, D. A., KuhlmannWilsdorf, D. (1992), Acta Metall. Mater. 40, 205. Bellier, S. P., Doherty, R. D. (1977), Acta Metall. 25, 521. Bishop, X F. W, Hill, R. (1951 a), Phil. Mag. 42, 414. Bishop, X F. W, Hill, R. (1951 b), Phil. Mag. 42,1298. Boas, W, Hargreaves, M. F. (1948), Proc. Roy. Soc. 193 A, 89. Bretheau, X, Caldemaison, D. (1981), in: Deformation of Polycrystals: Hansen, N., Horsewell, A., Leffers, X, Lilholt, H. (Eds.). Roskilde: Riso National Laboratory, pp. 157-161. Brown, K. (1972), /. Inst. Metals 100, 341. Brown, L. M., Stobbs, W. M. (1971), Phil Mag. 23, 1185 and 1201. Budiansky, B., Wu, X X (1962), in: Proc. 4th U.S. National Congress of Applied Mechanics. New York: ASME, pp. 1175-1185. Bunge, H. J. (1982), Texture Analysis in Materials Science. London: Butterworth.
Cahn, R. W. (1992), in: Materials Science and Technology, Vol. 15: Chan, R. W, Haasen, P., Kramer, E. X (Eds.). Weinheim: VCH Publishers, 429-480. Canova, G. R., Kocks, U. F. (1984), in: Proc. ICOTOM 7: Brakman, C. M., Jongenburger, P., Mittemeijer, E. X (Eds.). Zwijndrecht: Netherlands Society for Material Science, p. 573. Canova, G. R., Kocks, U. F , Jonas, X X (1984), Acta
Metall. 32,211. Carpay, F. M. A., Chin, G. Y, Mahajan, S., Rubin, X X (1975), Acta Metall. 23, 1473. Chin, G. Y. (1969), in: Textures in Research in Practice: Grewen, X, Wassermann, G. (Eds.). Berlin: Springer, p. 51. Chin, G. Y, Hosford, W. F , Mendorf, D. R. (1969), Proc. Roy. Soc. A309, 433. Christodoulou, N., Woo, O. X, McEwen, S. R. (1986), Acta Metall. 34, 1553-1562. Chung, C. Y, Duggan, B. X, Bingley, M. S., Hutchinson, W. B. (1988), in: Proc. ICSMA 8: Kettunen, P. O., Lepisto, X K., Lehtonen, M. E. (Eds.). Oxford: Pergamon Press, pp. 319-324. Clarebrough, L. M. (1949), Aust. J. Sci. Res. A. 2, 72-90. Cottrell, A. H. (1964). The Mechanical Properties of Matter. New York: Wiley, pp. 277-278. De Bleser, W (1979), Engineering Thesis, Katholieke Universiteit Leuven, Belgium. Dezillie, L., Van Houtte, P., Aernoudt, E. (1988), in: Proc. ICOTOM 8: Kallend, X S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 357-368. Dillamore, I. L. (1970), Met. Trans. 1, 2464. Dillamore, I. L., Morris, P. L., Smith, C. X E., Hutchinson, W. B. (1972), Proc. Roy. Soc. A 329, 405. Duggan, B. X, Lee, C. S. (1989), in: Materials Architecture: Bilde-Sorensen, X B., Hansen, N., Juul Jensen, D., Leffers, X, Lilholt, H., Pedersen, O. B. (Eds.). Roskilde: Ris0 National Laboratory, pp. 325-329. Duggan, B. X, Hatherly, M., Hutchinson, W. B., Wakefield, P. X (1978), Metals Sci. 12, 343. Emren, F , von Schlippenbach, U., Liicke, K. (1986), Acta Metall. 34, 2105. Eshelby, X D. (1957), Proc. Roy. Soc. A241, 376. Faire, P. Doherty, R. D. (1979), J. Mater. Sci. 14, 897. Fischmeister, H., Karlsson, B. (1977), Z. Metallkde, 68, 311-327. Gass, S. (1969), Linear Programming Methods and Applications. New York: McGraw-Hill. Gerold, V, Karnthaler, H. P. (1989), Acta Metall. 37, 2177. Gil Sevillano, X (1974), Ph.D. Xhesis, Katholieke Universiteit Leuven, Belgium. Gil Sevillano, X, Van Houtte, P., Aernoudt, E. (1977), Scripta Metall. 11, 581-585. Gil Sevillano, X, Van Houtte, P., Aernoudt, E. (1980), Progress in Materials Science 25, 69—412.
3.11 References Gotthardt, R., Hoschek, G., Reimold, O., Haessner, F. (1972), Texture 1, 99. Haessner, F. (1963), Z. Metallkde. 54, 98. Hansen, N. (1985), Met. Trans. 16A, 2167. Hansen, N. (1990), Mater. Sci. Tech. 6, 1039. Hansen, N., Ralph, B. (1981), Acta Metall. 30, 411. Hansen, I, Pospiech, X, Lucke, K. (1978), Tables for Texture Analysis of Cubic Crystals. Berlin: Springer. Hansen, N., Bay, B., Juul Jensen, D., Leffers, T. (1985), in: Proc. ICSMA 7: McQueen, H. X, Bailon, J.-P., Dickson, X I., Jonas, I X, Akben, M. G. (Eds.). Oxford: Pergamon Press, 317-322. Harren, S. V., Asaro, R. J. (1989), J Mech. Phys. Solids 37, 191-232. Havlicek, F , Tokuda, M., Hino, S. (1989), in: Advances in Plasticity (Proc. Plasticity 1989), Khan, A. S., Tokuda, M. (Eds.), Oxford: Pergamon Press, pp. 613-616. Hihi, A., Berveiller, M., Zaoui, A. (1985), Journal de Mecanique Theorique et Appliquee 4, 201. Hill, R. (1950), The Mathematical Theory of Plasticity. Oxford: Clarendon Press. Hill, R. (1965), J. Mech. Phys. Solids 13, 89. Hirsch, X, Lucke, K. (1988), Acta Metall. 36, 2863. Hirsch, X, Virnich, K. H., Lucke, K. (1981), in: Proc. ICOTOM 6, Vol. 1; Nagashima, S. (Ed.). Tokyo: The Iron and Steel Institute of Japan, p. 375. Hirsch, X, Lucke, K., Hatherly, M. (1988), Acta Metall. 36, 2905. Honeff, H., Mecking, H. (1981), in: Proc. ICOTOM 6, Vol. 1: Nagashima, S. (Ed.). Tokyo: The Iron and Steel Institute of Japan, p. 347. Honeycombe, R. W. K., Boas, W. (1948), Aust. J. Sci. Res. A. 1, 70-84. Hosford, W. F (1964), Trans. Met. Soc. AIME 230, 12-15. Hu, H. (1962), Acta Metall. 10, 1112. Hu, H. (1969), in: Textures in Research and Practice: Grewen, X, Wassermann, G. (Eds.). Berlin: Springer-Verlag, pp. 200-226. Hu, H., Cline, R. S. (1961), J. Appl. Phys. 32, 760. Hu, H., Goodman, S. R. (1963), Trans. Met. Soc. AIME 227, 627. Hughes, D. A., Hansen, N. (1991), Mater. Sci. Tech. 7, 544. Humphreys, F. X, Kalu, P. N. (1990), Acta Metall. Mater. 38, 917. Hutchinson, X W. (1984), Scripta Metall. 18, 421; 423-458. Juul Jensen, D., Hansen, N. (1990), Acta Metall Mater. 38, 1369.
Karlsson, B., Sundstrom, B. O. (1974), Mater. Sci. Eng. 16, 161-168. Kocks, U. F. (1970), Met. Trans. 1, 1121-1143. Kocks, U. F , Canova, G. R. (1981), in: Deformation of Poly crystals'. Hansen, N., Horsewell, A., Leffers, T, Lilholt, H. (Eds.). Roskilde: Ris0 National Laboratory, pp. 35-44.
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Kocks, U. F., Chandra, H. (1982), Acta Metall. 30, 695. Korbel, A., Martin, P. (1986), Acta Metall. 34, 1905. Korbel, A., Richert, M., Richert, X (1981), in: Deformation of Poly crystals'. Hansen, N., Horsewell, A., Leffers, T, Lilholt, H. (Eds.). Roskilde: Riso National Laboratory, pp. 445-450. Kroner, E. (1961), Acta Metall. 9, 155. Langford, G. (1977), Metall. Trans. 8A, 861-875. Langford, G., Cohen, M. (1969), Trans. ASM 62, 623-628. Leffers, T. (1968 a), Scripta Metall. 2, 447. Leffers, T. (1968 b), Riso Report No. 182. Roskilde: Ris0 National Laboratory. Leffers, T. (1975 a), Riso Report No. 302. Roskilde: Ris0 National Laboratory. Leffers, T. (1975 b), Scripta Metall. 9, 261. Leffers, T. (1981), in: Deformation of Poly crystals: Hansen, N., Horsewell, A., Leffers, T, Lilholt, H. (Eds.). Roskilde: Riso National Laboratory, pp. 55-71. Leffers, T. (1988), in: Proc. ICOTOM8: Kallend, X S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 273-284. Leffers, T, Ananthan, V. S. (1991), Textures and Microstructures 14-18, 971-976. Leffers, T, Bilde-S0rensen, J. B. (1990), Acta Metall. Mater. 38, 1917. Leffers, T, Juul Jensen, D. (1991), Textures and Microstructures 14-18, 933-952. Leffers, T, Pedersen, O. B. (1982), in: Proc. ICSMA 6: Gifkins, R. C. (Ed.). Oxford: Pergamon Press, pp. 75-82. Leffers, T, Van Houtte, P. (1989), Acta Metall. 37, 1191. Leffers, T, Asaro, R. X, Driver, X H., Kocks, U. K, Mecking, H., Tome, C , Van Houtte, P. (1988 a), In: Proc. ICOTOM 8: Kallend, J. S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 265-272. Leffers, T, Juul Jensen, D., Hansen, N. (1988 b), in: Proc. ICOTOM 8: Kallend, X S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 449-454. Leffers, T, Ananthan, V. S., Hansen, N. (1991), in: Proc. ICSMA 9: Brandon, D. G., Chaim, R., Rosen, A. (Eds.); London: Freund Publishing House, pp. 725-732. Li, X C. M. (1969), Phil. Mag. 19, 189. Lloyd, D. X (1987), in: Formability and Metallurgical Structure: Sachdev, A. K., Embury, X D. (Eds.). Warrendale: The Metallurgical Society, pp. 193209. Malin, S., Hatherly, M. (1979), Metal. Sci. 13, 463. Mathur, K. K., Dawson, P. R. (1989), International Journal of Plasticity 5, 67-94. Mecking, H. (1981), in: Deformation of Poly crystals: Hansen, N., Horsewell, A., Leffers, T., Lilholt, H. (Eds.). Roskilde: Ris0 National Laboratory, pp. 73-86.
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3 Deformation and Textures of Metals at Large Strain
Molinari, A., Canova, G. R., Ahzi, S. (1987), Acta MetalL 35, 2983. Mughrabi, H. (1983), Acta MetalL 31, 1367. Orowan, E. (1959), in: Internal Stress and Fatigue in Metals. Elsevier, p. 1959. Partridge, P. G. (1967), Met. Rev. 2, 169. Pedersen, O. B. (1987), Ada MetalL 35, 2567. Pedersen, O. B., Leffers, T. (1987), in: Constitutive Relations and Their Physical Basis: Andersen, S. I., Bilde-Sorensen, J. B., Hansen, N., Leffers, T., Lilholt, H., Pedersen, O. B., Ralph, B. (Eds.). Roskilde: Ris0 National Laboratory, pp. 147-172. Pedersen, O. B., Brown, L. M., Stobbs, W. M. (1981), Acta MetalL 29, 1843. Philippe, M. X, Esling, C , Hocheid, B. (1988), Textures and Microstructures 7, 265. Randle, V., Ralph, B. (1988), Revue Phys. Appl. 23, 501. Raphanel, J. L., Schmitt, J.-H., Baudelet, B. (1986), Intl. J. Plasticity 2, 371-378. Sachs, G. (1928), Z. Verein Deut. Ing. 72, 734. Saka, H., Iwata, X, Imura, T. (1978), Phil. Mag. 37, 291. Sarosiek, A. M., Owen, W. (1984), Mater. Sci. Eng. 66, 13-34. Schmitt, J. H. (1987), Ph. D. Thesis, University of Grenoble, France. Skalli, A., Fortunier, R., Fillit, R., Driver, J. H. (1985), Acta MetalL 33, 997-1007. Strauven, Y. (1982), Ph. D. Thesis, University of Leuven, Belgium. Strauven, Y., Aernoudt, E. (1987), Acta MetalL 35, 1029-1036. Sundstrom, B. O. (1973), Mater. Sci. Eng. 12, 265276. Taylor, G. I. (1938), J. Inst. Metals 62, 307. Thompson, A. W. (1977), in: Work Hardening in Tension and Fatigue: Thompson, A. W. (Ed.). New York: AIME, pp. 89-126. Thornburg, D. R., Piehler, H. R. (1975), Met. Trans. 6A, 1511. Tome, C , Canova, G. R., Kocks, U. R, Christodoulou, N., Jonas, J. J. (1984), Acta MetalL 32, 16371653. Tome, C , Canova, G. R., Ahzi, S., Molinari, A. (1988), in: Proc. ICOTOM 8: Kallend, J. S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 395-400. Toth, L. S., Gilormini, P., Jonas, J. J. (1988), Acta MetalL 36, 3077. Van Houtte, P. (1978), Acta MetalL 26, 591-604. Van Houtte, P. (1982), Mater. Sci. Eng. 55, 69-77. Van Houtte, P. (1984), in: Proc. ICOTOM 7: Brakman, C. M., Jongenburger, P., Mittemeijer, E. X (Eds.). Zwijndrecht: Netherlands Society for Material Science, pp. 7-23.
Van Houtte, P. (1986), in: Strength of Metals and Alloys (Proc. ICSMA 7), Vol. Ill: McQueen, H. X, Bailor, X P., Dickinson, X L., Jonas, X X, Akben, M. G. (Eds.). Toronto: Pergamon Press, pp. 1701 — 1725. Van Houtte, P. (1988), Textures and Microstructures 8-9, 313-350. Van Houtte, P. (1990), Manual of the MTM-QTA Software System, Internal Report. Leuven: Dept. MTM, K. U. Leuven. Van Houtte, P., Mols, K., Van Bael, A., Aernoudt, E. (1989), Textures and Microstructures 11, 23-39. Wagner, R, Canova, G., Van Houtte, P., Molinari, A. (1991), Textures and Microstructrures 14-18, 1135-1140. Wagoner, R. H., Laukonis, X V. (1983), Met. Trans. 14A, 1487. Wassermann, G. (1969), Metall 25, 414-417. Wassermann, G. (1973), Z. Metallk. 64, 844-848. Watte, P. (1990), unpublished results. Leuven: Dept. MTM, K. U. Leuven. Yegneswaran, A. H., Tangri, K. (1983), Z. Metallk. 7, 521-524. Yeung, W Y, Hirsch, X, Hatherly, M. (1988), in: Proc. ICOTOM 8: Kallend, X S., Gottstein, G. (Eds.). Warrendale: The Metallurgical Society, pp. 467-472.
General Reading Ashby, M. F. (1970), Phil. Mag. 21, 399-424. Backofen, W. A. (1972), Deformation Processing. Reading, MA: Addison-Wesley. Bunge, H. X (1982), Texture Analysis in Materials Science. London: Butterworth. Fischmeister, H., Karlsson, B. (1977), Z. Metallkde. 68, 311-327. Gil Sevillano, X, Van Houtte, P., Aernoudt, E. (1980), Progress in Materials Science 25, 69-412. Hill, R. (1950), The Mathematical Theory of Plasticity. Oxford: Clarendon Press. Hutchinson, X W, Anand, L., Asaro, R. X, Needleman, A., Canova, G. R., Kocks, U. F, Stout, M. G., Clifton, R. X, Duffy, X, Hartley, K. A., Shawki, T. G., Hatherly, M., Malin, A. S., Onyewuenyi, O. A. (1984), Scripta MetalL 18, 421-458. Kocks, U. F (1970), Met. Trans. 1, 1121-1143. Leffers, X, Bilde-Sorensen, X B. (1990), Acta MetalL Mater. 38, 1917. Leffers, T., Juul Jensen, D. (1991), Textures and Microstructures 14-18, 933-952. Wenk, H. R. (1985), Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modern Texture Analysis. Orlando, FL: Academic Press.
4 Dislocation Patterning Ladislas P. Kubin L.E.M., CNRS-ONERA (OM), Chatillon Cedex, France
List of 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6
Symbols and Abbreviations Introduction Dislocation Theory and Collective Effects The Internal Stress Dislocation Populations - Mobile and Forest Densities Slip Patterning Forest Patterning Phenomenological Scaling Laws Classical Models of Dislocation Patterning Holt's Model Kocks' Statistical Model Energetic Models and Thermodynamic Approaches Composite Models Self-Organization - Dynamic Approaches The Reaction-Diffusion Approach Reaction-Diffusion Dynamics Dislocation Populations and Reaction Terms Internal Stresses and the Diffusion Term Simple Dynamic Models The Model of Walgraef and Aifantis Kratochvil's Model
138 140 140 141 143 144 146 148 150 150 151 153 155 156 159 159 160 162 164 165 167
4.3.7
Discussion and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
] 7(1
4.4 4.4.1 4.4.2 4.4.3 4.4.3.1 4.4.3.2 4.4.3.3 4.43 A 4.4.4 4.4.4.1 4.4.4.2 4.5 4.6 4.7
Approach by Numerical Simulations Introduction - Methodology Simulation Techniques Results of Two-Dimensional Simulations Annealing Experiments Slip Patterning Slip Propagation Patterning of Immobile Dislocations Three-Dimensional Simulation Method Examples of Patterns Appendix Acknowledgements References
171 171 172 175 175 175 177 179 181 181 184 185 187 187
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
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4 Dislocation Patterning
List of Symbols and Abbreviations a A, b c{ d d0 D E /c 9 /w h /, J kB K / p q Ra Rc T v v y yc y e , ys
lattice parameter Burgers vector of a dislocation and its modulus concentration of element i cell size or distance between dislocation walls spacing of slip planes diffusion coefficient elastic interaction energy between dislocations volume fraction of cell interiors, volume fraction of cell walls dipole width dislocation flux and its modulus Boltzmann constant coefficient relating the reduced stress to the reduced cell size or wall spacing mean dislocation spacing control parameter wave number Rayleigh number cut-off radius for long range elastic stresses temperature dislocation velocity average dislocation velocity characteristic annihilation distance characteristic distance associated with cross-slip critical annihilation distances for edge and screw dislocations
a
coefficient relating the reduced stress to the mean dislocation spacing
ft
rate of blocking of a mobile dislocation bj a slow dipole
s e y y Fo, F I fjt v II Q Q{ Q{ £m G
plastic strain plastic strain rate shear strain shear strain rate line tension of a dislocation wavelength of a spatial structure shear modulus Poisson ratio cross slip probability per unit glide area dislocation density density of immobile dislocations density of forest dislocations density of mobile dislocations stress applied stress internal stress
List of Symbols and Abbreviations
t ia TC i het TW 0 co
resolved shear stress resolved applied stress stress of the cell phase mean macroscopic flow stress stress of the wall phase Schmid factor amplification factor
LEDS PSB TEM
low energy dislocation structures persistent slip b a n d transmission electron microscopy
139
140
4 Dislocation Patterning
If the properties of individual dislocations are, on the whole, mastered and understood in the simpler crystalline structures, one is very far indeed from this happy state of affairs for the large dislocation densities involved in straining processes. (XFriedel, 1979)
4.1 Introduction 4.1.1 Dislocation Theory and Collective Effects The main object of dislocation theory is the prediction of the mechanical properties of bulk materials, or at least their understanding, in terms of the physical properties of the material considered and of its structural defects. To date this remains a distant aim despite many detailed studies performed in carefully controlled conditions, in particular on single crystals. A major achievement of dislocation theory is concerned with the interaction of a mobile dislocation segment with a localized defect which is assumed to be distributed either periodically or at random. The plastic strain rate is obtained by summing up over the total number of mobile segments the individual probabilities for cutting through such obstacles, which is equivalent to assume that all events are independent. This leads within a good numerical accuracy to the prediction of the thermal component of the yield stress. However, experimental observations on strained single or polycrystals reveal that plastic activity is heterogenous from the very beginning and is confined within a small fraction (a few percent) of the total volume of the sample. The value of this "active volume" (Schwink, 1966; Neuhauser, 1983), its distribution along the axis of the specimen, its dependence on
the initial microstructure, on deformation conditions, and its evolution with strain are not predictable from theory. Models which assume plastic flow to be homogenous assimilate the local strain rate to the average strain rate and, therefore, identify two quantities which differ by one to two orders of magnitude, at least at moderate strains. More generally, strain nonuniformities may occur at various scales of observation, macroscopic, mesoscopic (i.e., optical or metallographic) and microscopic. Structural instabilities can in principle be investigated in terms of the dislocation theory, while geometrical instabilities (for instance necking or several types of shear banding) are better described within the framework of micromechanics. In both cases very basic problems such as the conditions for the occurrence of strain nonuniformities, their scaling laws and the kinetics of their further development are still not well understood. At moderate and large strains an increasing density of immobile or slow dislocations gets stored in the deforming crystal, leading to an increased glide resistance, possibly to long-range internal stresses, and to strain hardening. As revealed by transmission electron microscopy these dislocations form various of bi- or three-dimensional arrangements, e.g. wall and channel structures or cell structures, where dislocation-rich regions alternate more or less regularly with dislocation-poor regions (for recent reviews on this subject, see Louchet and Brechet, 1988; Amodeo, 1988; Hansen and Kuhlmann-Wilsdorf, 1986 and other contributions to the same conference). The relation between microstructure and the strain hardening properties of materials has been the object of many phenomenological models. In the simplest ones the dislocation densities are consid-
4.1 Introduction
ered uniform and their evolution with time or strain is estimated through a balance beween production and storage or annihilation events. More elaborated models assume a particular type of spatial distribution for the mobile and stored dislocations, consistent with experimental observations, and consider it as the smallest elementary volume whose mechanical properties are identical to those of the bulk crystal. The underlying questions, which are still a matter of debate are as follows: what are the conditions of occurrence for such types of ordered or semi-ordered microstructures, which we further refer to as "patterns", what are the material parameters governing their geometry, their evolution, their transformation during straining, how can they be related to strain hardening properties, is the formation of a field of long-range internal stresses necessarily associated with patterning? Although this brief description of dislocation theory and of its limitations may seem a bit schematic, the need for a consistent modelling of collective dislocation effects and of their consequences regarding strain hardening and strain localizations has been recognized very early. The tools, both conceptual and numerical, needed to develop such an approach have only been made available in the last two decades through the investigation of the dynamic behavior of ensembles of interactingj>opulations. The application to dislocation theory was initiated in the mid-eighties and, as will be shown below, it has yielded a few promising preliminary results. Only a few idealized situations have been investigated to date and this domain is still in its infancy. The connection between collective dislocation effects and patterning phenomena occurring in other systems brought far from thermodynamic equilibrium is, how-
141
ever, firmly established. It can be illustrated by recent achievements (Brechet, 1987; Lepinoux, 1987; Amodeo, 1988; Schiller, 1989) and by the proceedings of three recent conferences (cf. General Reading). The aim of the present chapter is to present a critical survey of this new domain, of its basic concepts and of its potential for further development. In the remaining of this section, the main ingredients of dislocation patterning are introduced, viz. internal stresses, dislocation populations and dislocation interactions. Section 4.2 briefly recalls a few current models involving ordered dislocation microstructures and emphasizes their limitations. The theoretical approach of dislocation patterning through the framework of reaction-diffusion is reviewed in Sec. 4.3 and the complementary approach by numerical simulations is discussed in Sec. 4.4. 4.1,2 The Internal Stress
Each infinitesimal fragment of dislocation line interacts via its long-range elastic stress field with all the other fragments present in the crystal. Some of the latter belong to the same segment or loop, and these contributions coherently add up to produce a resulting force which opposes the bowing out of the segment or the expansion of the loop. This defines the line tension Fo, In current elaborations of the line tension it is assumed that contributions from other segments cancel each other beyond a certain critical radius of the order of the mean distance between dislocation segments. Thus, within this simplified scheme, the line tension can be written as:
where ji is the shear modulus, x a coefficient which depends on the character of the
142
4 Dislocation Patterning
segment and Q is the dislocation density. For a perfect dislocation, the core radius is taken of the order of the magnitude of the Burgers vector, b. It is not easy to define the line tension of a segment immersed into a dislocation microstructure (Hirth and Lothe, 1968) but, as far as line tension effects are concerned, the exact value of the outer cut-off radius is not too critical since it enters a logarithmic term. In practice, however, the sum of the contributions from all remaining parts of the dislocation microstructure may result in a non-zero tensorial stress contribution, the internal stress, and the value of the related cut-off distance is of prominent importance as will be seen in the following. If we assume that, in average, each infinitesimal segment of dislocation line is sensitive to its surrounding up to a maximum distance i? c , the line tension writes:
The calculation of the internal stresses produced by elementary configurations such as the ones of Fig. 4-1 is a typical exercise in dislocation theory. In the absence of any applied stress, the internal stresses produced by these arrangements can be either long-ranged, as is the case for the dislocation pile-up, or short-ranged for the dipolar and multipolar arrangements. Figure 4-2 shows three general types of configurations: random (a), self-screening (b), where a dislocation of given sign tends to be preferentially surrounded by dislocations of the opposite sign, and polarized (c), where there is an excess of dislocations of one sign on one side and of the other sign on the other side. In the case of a random
(a)
(4-2)
Whether or not one has to split the elastic interactions into a local contribution (with cut-off radius Q~ 1/2) and a long-range (up to Rc) contribution is not a matter of academic debates because dislocation velocities are strongly stress-dependent and the stress components entering the stress vs. velocity law have to be consistently defined. Seeger (1955) defined the driving force for plastic flow, the effective stress, as the difference between the applied stress and the internal stress. This is at the basis of all the microscopic approaches of plasticity, including the ones considered here: at some point of the elaboration one necessarily has to estimate the sum of the (longrange) pair interactions of dislocation segments within the microstructure and the way they affect the stability of the latter and its further evolution with time or strain.
(b)
-(c)
Figure 4-1. Elementary dislocation configurations: (a) Piled-up group, (b) tilt wall, (c) dipolar wall.
(a)
(b)
(0 Figure 4-2. Three possible arrangements for a small group of dislocations: (a) Random, (b) multipolar, (c) polarized.
4.1 Introduction
distribution extending all through the crystal, the effective outer cut-off radius is of the order of the size of the crystal and it decreases to values of the order of the mean spacing between dislocations for periodic or nearly periodic configurations (Wilkens, 1969; Wilkens and Kronmiiller, 1975). In the configuration of Fig. 4-2 (b), the inner dislocations are effectively screened from applied stresses, in compression as well as in tension. This type of structure is formed during fatigue and is stable provided the applied stress is not too large. In uniaxial deformation, the polarized configuration of Fig. 4-2 (c) produces an internal stress of same sign as the applied stress in its interior and a back stress outside (Hasegawa etal, 1986). The tendency of dislocations to form self-screening or polarized arrangements stems from their response to mutual elastic interactions and it effectively decreases their total interaction energy. This points out towards an energetic or thermodynamic approach of patterning but, as discussed in Sec. 4.2.3, only static configurations can be dealt with within such a frame. Real configurations are less perfect and more complex than those pictured on Figs. 4-1 and 4-2. This is due to a permanent competition between relaxation or recovery effects inside the dense regions and the disturbing influence of incoming mobile dislocations at the interface between dislocation-poor and dislocation-dense regions. The formation of the dislocation rich regions through dislocation-dislocation interactions and, further, the interaction of mobile dislocations with the dislocation walls and tangles and their rate of crossing, storage, incorporation, lead to strain hardening. It follows that understanding strain hardening implies that we understand how dislocation patterns are formed, what are their geometrical characteristics (wall di-
143
mensions, cell dimensions, dislocation densities) and how they evolve with e.g. strain and temperature. Indeed, the study of strain hardening (cf. Gil Sevillano, Chap. 2 of this Volume) makes it clear that a different type of microstructure can be associated with every deformation stage. If every dislocation segment is sensitive to the configuration which surrounds it up to a certain distance materialized by the cut-off distance Rc, its motion will, on its turn, affect the other segments contained within this interaction distance. Therefore, each dislocation segment is a source of internal stress and simultaneously reacts to the internal stress. As a consequence, the internal stress is a non-local quantity, which means that its value at a given position in space depends on the distribution of dislocations within an extended neighbourhood. The mathematical treatment of this many-body problem is a major source of difficulty in all theories of dislocation patterning. Different approaches emphasize either the occurrence of long-range stresses in the stressed state or the tendency to self-screening. They also differ in the way they analyze the type of cooperative mechanism by which both patterns and internal stresses emerge from a random initial dislocation distribution. 4.1.3 Dislocation Populations Mobile and Forest Densities We write Orowan's law in the following form which, although approximate (Neuhauser, 1980), is commonly used: fi = Z &t Qmt bt vt
(4-3)
i
e is the total strain rate, (P the Schmid factor, gm the average density of mobile dislocations and v their average velocity, the sum being carried out over all the active systems. The values of both gm and v are
144
4 Dislocation Patterning
time-dependent since they evolve during a deformation test. Actually, these two quantities are also space-dependent, so that the strain rate involved in Eq. (4-3) is a local strain rate. Ideally, the total strain rate could be obtained by integrating Eq. (4-3) over the volume of the specimen, yielding a macroscopic constitutive form which would contain an information about slip heterogeneity in the form of length scales related to intrinsic material properties. For this purpose, however, one has to work out the spatial interactions of the various volume elements in which the local strain rate is defined. Here, it seems natural to take account two densities, those of the mobile and of the forest dislocations. The mobile dislocations carry the strain rate while forest dislocations, which can be mobile or immobile, have to be intersected by the mobiles. Each of these two populations has its own creation and annihilation rates; the mobile dislocations multiply through the operation of Frank-Read sources or through the expansion of single loops and they can either mutually annihilate or become immobile through their interaction with the forest or by forming stable sessile configurations. The immobile dislocations are subject to recovery mechanisms which tend to limit their density to a maximum value. According to the situation which has to be modelled, several types of dislocation populations have to be defined and the simple distinction between mobile and forest dislocations might not be sufficient for a discussion of patterning. A population is an ensemble of dislocations or of elementary configurations which share the same properties of motion, creation and annihilation. It follows that forest dislocations should be subdivided into e.g. dipoles and multipolar ensembles, sessile junctions, mobile dislocations on intersecting slip
systems and tangled dislocations. In principle, one should also distinguish between dislocations of same or opposite sign, of different characters, of different Burgers vectors, all these populations permanently exchanging densities via their mutual interactions. To date, no model is able to account for such a complexity and most of the attempts presented in Sec. 4.3 essentially deal with two populations at most, mobiles and forest. 4.1.4 Slip Patterning
A vast amount of experimental data has been collected on the slip line patterns formed at the surface of strained single crystals of various materials. The review of Neuhauser (1983) summarizes the results of earlier studies and focusses on more recent developments involving the dynamic aspects of slip pattern formation. A number of studies have dealt with surface observations of f.c.c. crystals, in particular copper, during deformation stages I and II (see e.g., Mader, 1957). Fine primary slip lines are left by the sudden emergence of a small number (10 to 20) dislocations and their length is one order of magnitude larger that the distance between obstacles in the slip plane which are multipolar bundles, somehow permeable to the mobile dislocations. This lead to the early models identifying the flow stress as the stress necessary to overcome the long-range internal stress of piled-up groups. The distance between fine slip lines is inversely proportional to the flow stress, which can be understood as resulting either from the passing stresses between pile-ups or from the annihilation of screw dislocations of opposite sign at the head of interacting pile-ups. However, TEM observations did not provide strong evidence for the existence of dislocation pile-ups and compet-
145
ing models assume that the flow stress is rather controlled by a forest mechanism. As discussed by Basinski and Basinski (1979), Nabarro (1986) and Takamura (1987) in their review articles, it follows that the connection between the spatial distribution of fine slip lines and workhardening is far from being clearly understood, all the more as there is no consensus about the detail of the work-hardening processes. It has been noticed very early that slip markings actually have a complex structure which extends over several scales of observation, macroscopic, mesoscopic and microscopic. The following quotation of Read is reproduced from Basinski and Basinski (1979): "What appears to the eye as a single line is observed to be a cluster of fine lines when viewed on a light microscope, and these, in turn, may be revealed as clusters of lines by the electron microscope". This, as well as Fig. 4-3, which illustrates the hierarchy of surface markings in neutron-irradiated copper (Neuhauser, 1983), strongly suggests that this spatial structure has a fractal dimension. Preliminary attempts to measure this dimension by analyzing the distribution of spacings or the number of slip steps found at a certain magnification were performed on deformed Cd single crystals (Sprusil and Hnilica, 1985), by Kleiser and Bocek (1986), on previously published micrographs of Cu and Co deformed crystals, and by Neuhauser (1988 a, b) on neutronirradiated copper. Self-similar distributions with a fractal dimension of the order of 0.5 extending over two orders of magnitude in scale were obtained for Cd and Cu and a fractal dimension of about 0.7 was measured on the neutron-irradiated copper specimens, the evidence being less clear for Cd. As emphasized by Hornbogen (1989) in his review of fractals in the mi-
(b)
(0
Figure 4-3. Hierarchical structure of slip patterns in neutron irradiated copper crystals, (a) Liiders band (macroscale), (b) slip band bundle (mesoscale), (c) slip bands consisting of slip lines (microscale). After Neuhauser (1983).
crostructure of metals, these results have still to be confirmed and their physical meaning is for the moment quite unclear. There is an increasing tendency to treat the formation of slip lines as an instability phenomenon, caused either by a fast initial multiplication rate (Estrin and Kubin, 1986) or by a softening of the microstructure (cf. the extended review by Luft, 1991), for instance through the destruction of short-range order in dilute alloys (Olfe and Neuhauser, 1988). In every case a slip line has a limited period of activity, at the end of which strain hardening and/or backstresses block the activity of dislocation sources. The plastic strain rate must be carried out through the initiation of fresh dislocation lines as old ones cease to be active. New slip lines are activated either at random along the specimen gauge length or in a spatially correlated sequence, leading to Liiders-like slip propagation. This last phenomenon has been examined in detail
146
4 Dislocation Patterning
during the early stages of the deformation of copper alloys (Neuhauser, 1983). The occurrence of two types of spatial distributions for the active slip volume, one ordered, the other disordered, is not understood. Together with the inherent tendency of slip lines and bundles to cluster, it suggests that different types of interactions may be effective with very different length scales. In addition, grip effects (Shinohara et al., 1986) or surface effects in thin crystals certainly influence slip patterning. Further, the length scales associated with slip propagation, viz. the width of Liiders-like bands (or true Luders bands) and their average propagation velocity, appear as free parameters in all the models proposed to date (Kocks, 1981). The assumptions that slip propagates by a cross-slip mechanism, through the influence of long-range dynamic stress fields emanating from pile-ups or as a result of geometrical constraints may lead to a solution of this problem at mesoscale (Estrin and Kubin, 1991), but not at a finer scale. In summary, none of the main features associated with slip patterns is consistently understood. This includes the hierarchical structure of slip markings, its relation to strain hardening, the transition between orderly and random slip initiation and more fundamentally the nature of the interactions between slip lines and the mechanisms by which slip activity is transferred from one to the other. 4.1.5 Forest Patterning It is largely believed that at strains of the order of a few percent the density of mobile dislocations saturates, the saturation value being dictated by an equilibrium between the production rate and the annihilation or storage rate of dislocations. As a consequence of storage, an increased density of
immobile or almost immobile dislocations is accumulated in the crystal. Dislocations can be trapped into dipolar or multipolar configurations or blocked by sessile junctions or locks, forming a variety of structures which have received various denominations: tangles, braids, veins, clusters, walls... At some critical value of strain, a more or less ordered pattern invariably emerges. The persistent slip bands (PSBs), the wall-and-channel and the labyrinth structures formed during the fatigue of copper single crystals, offer the best known examples of a nearly perfect spatial periodicity. The cell structures formed in creep at medium and high temperatures or during low temperature deformation although less perfect are also well-documented. For a discussion of the characteristic dislocation microstructures developed during plastic straining and of their relation to strain hardening properties, the reader is referred to the reviews of Gil Sevillano (this Volume), Basinski and Basinski (1979) and Nabarro (1986). The way material properties (cross-slip, alloying effects) and experimental conditions (temperature, mode of testing) affect forest patterning are qualitatively understood (Louchet and Brechet, 1988; Kubin et al., 1989). Here we only focus on two points. A necessary condition for the obtention of a pattern is that dislocation-dislocation interactions must significantly contribute to the flow stress. It follows that materials with a low yield stress and a high strain hardening rate, e.g. f.c.c. metals, will exhibit patterning at small strains, while materials with high yield stresses and moderate strain hardening rate like b.c.c. metals at low temperatures will tend to exhibit a rather homogeneous microstructure up to larger strains. Dislocation-dislocation interactions can be of two kinds: longranged, i.e. acting at distances larger than
4.1 Introduction
the mean distance between dislocations, Q~1/2, or short-ranged as is the case when two dislocation segments interact to form a dipole or a Lomer-Cottrell lock. The models discussed in Sec. 4.2 reflect this distinction by identifying either local or longrange interactions as the cause
147
(a) 1
r—
T
1
I"' •
1
1 1
O L A T
_
10 9
7
1 T3
6
3_
—
A
* *
4
-
A
Q 0
1
8
a o A
_
A O
A
• g
O 0 A
i O
A
O
o
2 5
6 7
Figure 4-4. Mean distance between dislocation walls dN in cold-rolled copper specimens, as a function of magnification M. L and T denote thin foils with normals parallel to the long and transverse directions of the sheet, respectively, (a) e = 0.57 (stage III), (b) e = 1.28 (stage IV). After Gil Sevillano et al. (1991).
cell structure developed during stages II and III (cf. Gil Sevillano, Chap. 2 of this Volume). This suggests a possible correlation between the fractal dimension of dislocation arrangements and their distance to equilibrium. The quasi-periodic structure of PSBs could, then, be considered as a
148
4 Dislocation Patterning
structure close to equilibrium, while low temperature cell structures would not be adequately described as periodic ones. While most of the theoretical attempts have, up to now, focussed on the search for periodic structures, it is not excluded that in the future chaotic and turbulent flow patterns (Ananthakrishna and Valsakumar, 1983; Ananthakrishna, 1988) could be obtained through experimental or theoretical studies. 4.1.6 Phenomenological Scaling Laws
The inverse of the square root of a dislocation density is a length scale related to a mean distance between dislocation segments or to an average segment length. The reduced forms bg1/2 and aj[i, where a is a flow stress, are often found to be related by a/fi = abQ1/2
(4-4)
where a is a constant of the order of 0.3 which appears to be rather insensitive to the type of microstructure examined. A survey of literature shows, however, that experimental values range between 1/5 and 1/2 (Embury, 1971), going occasionally up to 1 (Lavrentev, 1980). Because of its non dimensional character, Eq. (4-4) can be derived or used within very different contexts (Nabarro et al., 1964). The only situation where this expression does not hold is, perhaps, easy glide and the density Q [Eq. (4-4)] is, presumably, a forest density. Indeed, Eq. (4-4) has originally been derived from the consideration of forest intersection mechanisms. In such a case, the coefficient a should contain a logarithmic term stemming from line tension effects during the bowing out of dislocation segments. This point is discussed in detail by Basinski and Basinski (1979) who show that the logarithmic factor induces small deviations
from the quadratic relation between forest density and the flow stress. Their conclusion is that forest intersection processes reasonably account for this scaling law. In a heterogeneous dislocation configuration, however, the mean spacing between forest obstacles is space-dependent and so is the flow stress. In static models such as the composite model, which is discussed in more detail in Sec. 4.2.4, Eq. (4-4) is applied to the high-density walls as well as to the less dense cell interiors and the flow stress of the composite material turns out to be again given by Eq. (4-4) with a constant coefficient a' = 2 a (/ c / w ) 1/2 , where fc and / w are the volume fractions of the cell interiors and of the cell walls respectively (Mughrabi, 1987). Typically, with 0.1
{[1/(4KX)]
ln(RJb)}
(4-5)
The outer cut-off radius JRC appears here as one global parameter representing the arrangement of the microstructure (cf. Sec. 4.1.2). As will be repeatedly emphasized in this chapter it is not always justified to assume that the interaction forces between dislocations are screened out within short cut-off distances of the order of the mean spacing Q~1/2. This conclusion was reached by Wilkens (1969) through an argument which does not involve the forest mechanism but the long-range pair interactions within a conditional statistical arrangement of dislocations, i.e. an arrange-
4.1 Introduction
are even some notable exceptions of materials which do not obey Eq. (4-6), in particular stainless steels where the the microstructure is planar rather than threedimensional. Raj and Pharr (1986) have compiled the available data on the simultaneous measurement of flow stresses and cell sizes in various materials. As shown in Fig. 4-5, which reproduces their results on aluminum and a-iron, the correlation is reasonably good over a wide range of stresses, at least in logarithmic scales. As noticed by Mughrabi (1986), Eq. (4-6) has been derived by many authors, on the basis of very different models. It can for instance result from the fact that dislocation stress fields decrease as the inverse of distance (Gil Sevillano and Aernoudt, 1987) or that the flow stress associated with the forest mechanism is inversely proportional to the average distance between dislocations. Finally, by combining Eqs. (4-5) and (4-6) we obtain a third scaling law, according to which the cell size should be proportional to some average dislocation spacing Q~~112. The value of the characteristic length d, its stress dependence as given by Eq. (4-6) and the physical content of the parameter
ment which consists of a two-dimensional periodic array of cells in which dislocations are statistically distributed. The flow stress, according to this view could be a sum of two terms, one stemming from the forest mechanism and the other from the longrange stress field (Mughrabi, 1975; Saada, 1990). Unfortunatelly, it will perhaps be difficult to draw reliable data on Rc from experimental checks of Eq. (4-5) because of its weak, logarithmic dependence on the arrangement of dislocations in the microstructure. As a consequence, dynamic models for dislocation patterning may make use of Eq. (4-4) as a starting point. Alternatively, this expression can be derived from theory or from numerical simulations. Consistent values should be obtained for the constant a but this is not expected to be a critical check for the validity of any approach. A second expression scales the distance between walls or the cell diameter d/b, to the reduced stress, oj\x\ a/n = K(b/d)
(4-6)
K is very often assumed to be a universal constant of the order of 20, but this is perhaps an oversimplification. Indeed, there
106
149
108
106 10*
102
102 10-5
10-3
10-1
10-7
10-1
10-5
a/ju
a/ju
(a)
(b)
Figure 4-5. The relation between reduced cell size d/b and reduced flow stress ol\i in (a) iron and (b) aluminum. After Raj and Pharr (1986).
150
4 Dislocation Patterning
K are the major unknowns which are to be derived from the models or the simulations to be discussed in the following.
4.2 Classical Models of Dislocation Patterning
cut-off radius is scaled by the mean distance between dislocations: Kc OC Q
'
(4- /)
The modification of the total interaction energy 5£, due to a small periodic spatial perturbation bg of wavelength X (Fig. 4-6),
4.2.1 Holt's Model The aim of Holt's (1970) model is to explain the formation of cell structures by analogy with the spinodal decomposition of solid solutions. We discuss this early attempt in some detail because, although imperfect, it makes use of some essential features to be examined in relation with the reaction-diffusion approach (Sec. 4.3) and the simulations of annealed configurations (Sec. 4.4.3.1). A two-dimensional system is constructed which contains an array of parallel screw dislocations, with both signs in equal density and with an uniform initial distribution. There is no applied stress, no annihilation and multiplication, and the constant initial density only evolves under the influence of mutual interaction stresses. The driving force for this evolution is the reduction in total elastic interaction energy, whose gradient is the local force experienced by dislocations. To estimate the elastic interaction energy of a dislocation at a given position it is necessary to know how dislocations of both signs are distributed around it. Holt postulates that a distribution function can be defined, which has two properties: (i) There is short range order, i.e., a dislocation of given sign is preferentially surrounded at short distances by dislocations of opposite sign. This correlation vanishes at large distances and elastic interactions need not be calculated beyond some cut-off radius JRC since, then, contributions from attractive and repulsive interactions cancel out. (ii) This
0
X
x,y
Figure 4-6. A periodic fluctuation, bg (x, y) of a uniform density Q0.
is calculated through a Taylor expansion of the energy with respect to its initial value. bE = — Fx bg — F2 V 2 (5@)
(4-8)
where Fx and F2 are numerical coefficients. Kinetics are introduced by expressing that dislocation velocities, v, are proportional to the interaction forces. The latter are, by definition, given by the gradient of the energy fluctuation, — V5£. The dislocation flux, / = QV can, then, be written as: J=-QDV5E
(4-9)
where D is the dislocation mobility. The last step consists of writing the balance equation which expresses that the local change in density within a small volume element is due to the difference of incoming and outcoming fluxes ddg/dt = — div/
(4-10)
The equation which governs the stability of the dislocation distribution is finally obtained by combining Eqs. (4-8) to (4-10): (4-11)
4.2 Classical Models of Dislocation Patterning
This equation has solutions which are periodic in space and whose evolution in time is governed by an amplitude factor of the form Qxp[co(X)t]. All fluctuations leading to a negative value of the amplification factor co decay with increasing time while those leading to a positive value exponentially grow in amplitude. Among the latter, one maximizes co and becomes dominant. It corresponds to a wavelength of the order of Rc and by extrapolating this result to fluctuations of large amplitude one expects to obtain the spontaneous formation of a periodic structure with a spatial period of the order of: A OC Kc OC Q
(4-1Z)
In the absence of any applied stress and of creation and annihilation events, the thermodynamic system considered here is a closed system. It is, therefore, justified to say that the solution obtained for infinitesimally small dislocation movements is a configuration of lower energy than the initial one. The extrapolated result of Eq. (412) is nevertheless surprising because a set of parallel dislocations of same Burgers vector is not thought to spontaneously decompose into a periodic structure. Indeed, simulations confirm that this does not occur (cf. the discussion in Sec. 4.4.3.4). Several objections can be raised against the procedure described above and some underlying hypotheses. The assumption of local order involved in the definition of the distribution function for the dislocation densities is not compatible with the randomness of the initial configuration. In addition, there is no obvious reason for introducing a constant cut-off i.e., for taking a value that does not depend on the distribution of dislocations. The hypothesis that the mean distance between dislocations scales the cut-off radius is plausible, but needs a justification. In a further study,
151
Staker and Holt (1972) proposed that Rc be defined as the distance at which the interaction of two dislocations no longer influence their mobility in the presence of a friction force. As noted by Nabarro (1986), this destroys the final result and the wavelength of the periodic structure is no longer scaled by Q " 1 / 2 . A last difficulty arises from the expansion of the total energy. Assuming a screening effect, it would seem natural to take i? c , rather than the dimension of the crystal, as the outer cut-off radius in the logarithmic term of the line energy. In such a case, the successive terms in the expansion of the fluctuation of interaction energy may not converge. In summary, the crucial problem in Holt's model, which has also been met with all subsequent approaches, is related to the treatment of the interaction forces. The related form necessarily contains an information about the distribution of dislocations, which is precisely the quantity searched for. Introducing self-screening ab initio is, then, equivalent to artificially driving the solutions of the problem to periodic ones. Once the distance between dislocation walls and clusters becomes larger than the cut-off distance, these ensembles no longer mutually interact and an effective way to minimize energy is, as a consequence, to have a periodicity of the order of Rc. 4.2.2 Kocks' Statistical Model
Several models implicitly or explicitly postulate that long-range stresses do not play a significant role in the hardening and flow properties of materials. This is the case for the mesh-length theory (Sec. 4.2.3) and for Kocks' statistical model. In their work on the thermodynamics and kinetics of slip, Kocks et al. (1975) assume, for instance, that the internal stress and the local glide resistance can be lumped into one
152
4 Dislocation Patterning
glide resistance which is, in practice, treated as a local quantity. The implicit postulate is that the internal stress is a local stress, i.e. that it does not depend on what occurs beyond the nearest dislocation. Plastic flow properties can, then, be established on a phenomenological basis, in particular with help of Eq. (4-4), taking advantage of its insensitivity to the arrangement of the microstructure. Thus, no information about pattern formation can be obtained within this framework. Kock's (1985) statistical model (see also Kocks et al., 1980) essentially aims at rationalizing the strain hardening rate during stage II in terms of dislocation interactions; it also attempts to explain the succession of events through which a cell structure is formed. Figure 4-7, after Jimenez (1982), shows the shape of a dislocation line gliding through a random array of infinitely hard obstacles. Two features characterize the critical shape obtained when the applied stress reaches the critical value at which the gliding dislocation line can
percolate through the obstacles (Kocks, 1985; see also Gil Sevillano et al., 1991): there are impenetrable clusters around which the gliding dislocation has left sessile debris. - There is a "fingering" effect, i.e., the dislocation line preferentially moves through the most penetrable areas in the random distribution. In Kocks' model, the impenetrable obstacles represent attractive junctions formed by reaction with forest dislocations. As a consequence the flow stress is defined as the stress associated with the forest mechanism in "the hardest of the soft regions that all dislocations must go through" and its value is given by Eq. (4-4). During further deformation, slip remains concentrated in the open channels while more debris accumulate around the "hard spots". The latter grow in size, both in the slip plane and in the direction perpendicular to it, until they become connected in three dimensions and form a cell structure. The emphasis is put on the mechanisms occurring in these dense regions, particularly on dynamic recovery, in order to explain why the walls connecting the hard regions can be penetrated by the mobile dislocations and why plastic activity is not rapidly choked by the increase in stored density. The cell size is taken to be the initial spacing, d, between the hard spots on which it is anchored. An empirical correlation is found between d and the mean dislocation spacing, 1 = Q~X/2: d«10Z
(4-13)
Combining this expression with Eq. (4-4), we have G= 10a jib/d Figure 4-7. Critical shape of a dislocation line which has moved from bottom to top across a random array of infinitely hard point obstacles. Notice the islands left by the gliding dislocation. After Jimenez (1982).
(4-14)
i.e., Eq. (4-6) is recovered with K& 10 a. As far as patterning is concerned, the main conclusion is that the connection between cell size and flow stress is indirect, since
4.2 Classical Models of Dislocation Patterning
both quantities are independently derived from the statistics of forest obstacles. The average distance between obstacles, /, determines on the one hand the flow stress in the open channels and on the other hand the average distance between the hard spots from which the cell structure has emerged. The weak point in this description is as follows. The fact that forest obstacles are certainly not distributed at random all through stage II does not influence much the value of the constant a in Eq. (4-4) but it may destroy the empirical correlation of Eq. (4-13). Actually, this model privileges the role of a single slip system during stage II. Although the mean free-path of the secondary dislocations is small during this stage, their density can, however, be as large as that of the primary dislocations (Basinski and Basinski, 1979). Finally, the distribution of the impenetrable clusters can be thought of being three-dimensional, rather than two-dimensional, from the very beginning, all the more as slip patterning effects should be taken into account. The statistical model contains, however, the interesting suggestion that the initial stage of development of cell structures is controlled by the formation of sessile anchoring points provided by interactions with the forest. 4.2.3 Energetic Models and Thermodynamic Approaches
It has often been thought that the minimization of some potential function associated with plastically deformed states may be helpful in raising indeterminacies at microscopic or macroscopic scales. In this respect it is instructive to compare various models proposed to describe the yield point and the deformation curve of covalent semiconductors. A given strain rate
153
can be achieved either by a large number of slow dislocations or by a small number of fast dislocations. In Sumino's model (1974; see also Takeuchi, 1973) this indeterminacy is solved through an extremum principle, involving the definition of a steady state of deformation and of a free-energy. Noticing that the application of equilibrium thermodynamics to such an irreversible process as plasticity might be incorrect, Nishioka and Ohsaka (1978) attempted to justify Sumino's result through a consideration of the rate of entropy production. However, in the earlier model of Alexander and Haasen (1968) there is no indeterminacy and no need for a thermodynamic elaboration, simply because a multiplication rule is explicitely specified for the mobile dislocations. The question of what kind of thermodynamics is applicable to plastic flow is perhaps better understood today. Before coming back to this subject we examine briefly how patterning is treated in a thermodynamic context by the mesh-length theory of work-hardening (Kuhlmann-Wilsdorf, 1962; Kuhlmann-Wilsdorf, 1989 and references therein; Hansen and KuhlmannWilsdorf, 1986). This theory has been expanded to cover almost all aspects of patterning and of its relation to mechanical properties in various materials and experimental conditions. Within this frame, lowenergy dislocation structures (LEDS) i.e., perfectly self-screened structures which produce no significant long-range internal stresses are considered to occur almost universally on the basis of the second law of thermodynamics. Two basic principles serve to estimate the spatial arrangements of dislocation patterns (Kuhlmann-Wilsdorf, 1989). The first one states that the effective stress of every dislocation cannot exceed the friction stress, including the bowing out stress.
154
4 Dislocation Patterning
For crystals where the friction stress is small, this is equivalent to a hypothesis of static equilibrium. The shrinking of length scales with increasing stress [cf. Eq. (4-6)] is, then, considered as a succession of equilibrium states. (This is a similitude argument; similitude should not, however, be confused with self-similarity, cf. Sec. 4.1.5.) The kind of dislocation structure formed is determined by assuming that it is, among those accessible to the dislocations, the one which minimizes (or nearly minimizes) the stored energy per unit length of dislocation line. This argument is clearly of thermodynamic nature, even if this is not stated explicitly. The type of cell structure which is discussed consists of a three-dimensional checkerboard of planar small angle boundaries with alternating misorientations (cf. Fig. 4-8), like those formed at high temperatures or during deformation stage IV at low temperatures (Gil Sevillano and Aernoudt, 1987). In such cases, recovery by cross-slip and climb can occur at sufficient rates to allow for fast rearrangement when the structure is locally disturbed by the emission or incorporation of mobile dislo-
A
- -* — —
sA
1f *y
A
A
/ A
A
/ A A
A
Figure 4-8. Three-dimensional checkerboard pattern of cubic dislocation cells. The alternating misorientations are represented by arrows. After KuhlmannWilsdorf and van der Merwe (1982).
cations. As in the kinematic model, the flow stress is given by Eq. (4-4) and, although the background is quite different, the value of the coefficient a is found within the usual range. The coefficient K in Eq. (4-6) is estimated by computing the cell size which minimizes the energy. It is found that K&5Ja (Hansen and KuhlmannWilsdorf, 1986), a value not so different from that obtained by Kocks (cf. Sec. 4.2.2). The mesh-length theory and its limitations have been the object of several critical discussions (Mughrabi, 1983; Nabarro, 1986; Kuhlmann-Wilsdorf, 1988). Its basic principles may appear to be in conflict with the very intimate nature of plastic flow since a succession of thermodynamic equilibrium states is necessarily reversible, while plastic deformation is inherently an irreversible process. A useful consequence of this irreversibility is that dislocation patterns can be observed by transmission electron microscopy in the unloaded state. This is, indeed, an evidence for the occurrence of metastable local energy minima (Seeger, 1988). The concept of LEDS undoubtedly applies to situations which imply thermal equilibrium, like epitaxy or the formation of fine polygonisation (mosaic structures) in slightly cold-worked and subsequently annealed materials. On the other hand a specimen undergoing plastic deformation dissipates the mechanical work produced by the motion of its external boundaries under the influence of applied forces. In thermodynamic terms, this defines an open and dissipative system brought far from equilibrium by its continuous exchanges with the external world. It has been suggested the distance the thermal equilibrium can be estimated through the measure of the ratio (4-15)
4.2 Classical Models of Dislocation Patterning
155
of the stored energy Ws to the mechanical common belief that long-range internal work WE during plastic flow (Seeger, 1988; stresses are not large enough to contribute Seeger and Frank, 1988).
(a)
Figure 4-9. (a) A cell structure with two symmetric active slip systems under axial stressing, (b) Interfacial dislocations are replaced by dislocations of resultant Burgers vector. The arrows indicate the sense of the resulting long range elastic stresses parallel to the stress axis. After Mughrabi (1983).
156
4 Dislocation Patterning
4-9 a). The two phases are strained in parallel and the mean macroscopic flow stress i het is the spatial average of the local flow stresses Jw ^w "•" / c
(4-16)
Strain compatibility requires that the total, i.e., elastic (el) plus plastic (pi) shear strain is continuous at the interface between walls (w) and cells (c): 7tot = Vel.w + 7pl,w = Vel.c + 7pl, c
(4-17)
Plastic flow is mainly carried by the soft cell interiors while the deformation of the wall phase is predominantly elastic. As a result, the elastic strains and stresses are larger in the walls than in the cells and there is an elastic strain mismatch at the interfaces. The heterogeneity of the structure thus results in an internal redistribution of the external stress in such a fashion that long-range elastic internal stresses result with a wavelength of the order of the cell diameter, much larger than the mean spacing between dislocations. The plastic mismatch at the interfaces must be accommodated by a certain density of glide dislocations, as shown in Fig. 4-9 a. This does not mean that the walls are infinitely hard obstacles and mobile dislocations in excess to the ones maintaining the continuity of plastic strains may penetrate them. Indeed, walls are necessarily strained plastically since their structure evolves during plastic flow. The interface dislocations produce an internal stress field which is conveniently visualized on Fig. 4-9 b, where dislocations with the resultant Burgers vector are sketched. We see that the resultant elastic stress field in the vertical walls is a dilatation which adds up to the applied stress, while the cell interior is subject to an elastic compression. These elastic stresses are precisely those required to obtain, under an applied stress, the local
flow stresses needed for the simultaneous plastic straining of the two phases. The composite models show that longrange internal stresses exist in the absence of dislocation pile-ups, even if the wall or cell structures undergo dynamic recovery and consist of self-screening arrangements. In the unloaded state, an internal stress is frozen-in since the interface dislocations are not removed. Experimental evidences obtained by transmission electron microscopy in wall structures formed during fatigue are reviewed by Mughrabi (1987). Within a discrete approach of patterning, like that used in numerical simulations (Sec. 4.4), stress equilibrium and strain continuity automatically follow from the additivity of the stress and strain fields of individual dislocations. Within a continuum approach, as in the reaction-diffusion schemes (Sec. 4.3), the conditions of strain continuity and stress equilibrium have to be taken into account, which has been realized only recently. In this respect, the classical models described in the preceding sections cannot be fully consistent since they mix continuous and discrete approaches and rely on a postulate about the importance of long-range internal stresses. 4.2.5 Self-Organization Dynamic Approaches
Open systems, i.e., systems which may exchange energy and/or mass with their environment, appear characteristically prone to spontaneous forms of organization. We have seen in Sec. 4.2.3 that plasticity is an irreversible phenomenon, characterized by a dissipation of energy within the specimen and a large distance from thermodynamic equilibrium. These are precisely the ingredients needed to obtain self-organization in system elements containing a large number of interacting individual objects (Haken, 1987).
4.2 Classical Models of Dislocation Patterning
Self-organization was already known to occur in hydrodynamic systems at the turn of this century and it was later on observed in a complex chemical reaction called the Belouzov-Zhabotinskii reaction, in the form of spatio-temporal chemical oscillations. This last effect was, at first, rejected as contrary to the laws of thermodynamics but, today, the relation between various types of instabilities and the distance to thermal equilibrium is recognized and actively investigated in a large number of physical, chemical and biological systems. In the last two decades a considerable understanding of pattern-forming systems has been reached through a synergy between experimentation in an increasingly large amount of systems, the advent of powerful simulation methods and new theoretical approaches based on bifurcation analysis and instability theory. Not too much detail can be included in the present review about the theoretical tools and for more detail the reader is referred to the book of Nicolis and Prigogine (1977), which emphasizes patterning associated with chemical reactions within the reaction-diffusion approach and to that of Haken (1987) which covers all fundamental aspects. The important point to be kept in mind is that the distance to thermal equilibrium necessarily implies non-linear dynamics and that both features are at the origin of spatio-temporal organization. The impact on materials science has been considerable in the recent past and it has been summarized in the proceedings of several conferences (cf. General Reading). As an example, we briefly describe a typical hydrodynamic instability, the Rayleigh-Benard instability. A viscous fluid is confined between two horizontal plates and heated from below so that it is subject to an external driving constraint represented by a temperature gradient (Fig.
o
O: O
157
o
Figure 4-10. Fluid flow pattern in the RayleighBenard convection (compare with Fig. 4-8).
4-10). Molecules tend to flow upwards because the hot fluid is less dense than the cold one and a downwards flux is induced by gravity. Heat is dissipated at molecular scale through viscosity effects. The respective influences of heat diffusion, gravity and viscosity are combined into a single dimensionless coefficient, the Rayleigh number JRa, which characterizes the state of the system. When ,Ra = 0, i.e., in the absence of temperature gradient, the system is in thermodynamic equilibrium and its stable state consists of a homogeneous spatial distribution of fluid molecules. Then, the spatial average of the instantaneous velocities of the molecules, v, is equal to zero. In the presence of a temperature gradient, the macroscopic state of the system can be represented through the bifurcation diagram of Fig. 4-11, where v is plotted as a function of Ra. For small values of Ra, the system is close to thermal equilibrium and the uniform solution remains stable. In this domain a linear expansion with respect to the uniform state can be performed yielding the Onsager relations, according to which fluxes are proportional to forces. At molecular scale, stability means that any local fluctuation in velocity dies out with time because of the dissipation due to viscous drag and thermal diffusion. With increasing values of the Rayleigh number, the behavior drastically changes beyond a threshold value Rac. Fluctua-
158
4 Dislocation Patterning
Figure 4-11. Bifurcation diagram associated with the Rayleigh-Benard convection, v is the average velocity of the fluid molecules and Ra is the Rayleigh number. Further bifurcations occur for large values of Ra.
tions are no longer damped and they may grow (cf. Sees. 4.2.1 and 4.5 where a similar type of analysis is performed), destabilizing the uniform solution and leading to a nonuniform convective pattern. This new structure is characterized by large-scale collective motion of fluid molecules during macroscopic times, forming a fluid flow pattern which consists of convection rolls with alternating sense of rotation (Fig. 4-10). The translational symmetry of the system has disappeared, which is equivalent to an ordering process. In the diagram of Fig. 4-11, the average velocities split into two branches according to the sense of rotation. When the value of the Rayleigh number further increases this non-uniform state undergoes further bifurcations towards other organized states and, finally, to chaos. When such systems are driven far from equilibrium, it is not always possible to define a function whose minimum represents a dynamic stable state. One has therefore to study the stability of the solutions of the evolutionary equations describing the dynamics, in the present case the Navier-Stokes equations or simpli-
fied equivalent forms. Patterning occurs through the destabilization of uniform states by fluctuations and the transitions between uniform and non-uniform states (or between various non-uniform states) occur at critical values of a control parameter, here the Rayleigh number Ra. Once one, or a few, predominant fluctuations have emerged from the uniform background, the non-linearities of the system fully enter into play to limit their growth and the new structure gets stabilized. Despite the complexities of the dynamics, there exists a restricted number of bifurcation types and this explains why generic forms of patterns may occur in very different types of systems. For instance, there is some analogy between the roll pattern of Fig. 4-10 and the checkerboard pattern of cubic dislocation cells depicted in Fig. 4-8. Indeed, it has been suggested long ago that the transition between stages I and II of the deformation curves in uniaxial deformation resembles the transition between laminar and convective flow in a fluid (Cottrell, 1958). These similitudes cannot, however, be further elaborated in numerical form since no evolution equation similar to the Navier-Stokes equation exists in the domain of dislocation dynamics. As shown in the next section, the reaction-diffusion approach to spatial ordering in certain types of chemical systems has provided a practical frame for the study of dislocation patterns. It consists of studying the coupled evolution of several species or populations which react at short distances and are able to diffuse or move over large distances. The local competition between diffusion and non-linearities yields a variety of instabilities and types of patterns.
4.3 The Reaction-Diffusion Approach
4,3 The Reaction-Diffusion Approach
where Dt is a diffusion coefficient. Then, divJ^- = — Dt Acf, so that Eq. (4-18) can be rewritten in the reaction-diffusion form
In this section we first outline the various steps leading to evolutionary laws of the reaction-diffusion type for the dislocation populations, emphasizing a few difficulties or controversial points. A critical review of the available results is then presented. 4.3.1 Reaction-Diffusion Dynamics This approach derives from the study of particular sets of coupled chemical reactions which are characterized by temporal and/or spatial organization of the concentration of the final products (cf. Nicolis and Prigogine, 1977). For each constituent in concentration c/? a balance equation is written in the form of a reaction-transport equation = gi{ci9cj9p)
(4-18)
Jt = ci vi is the flux of the species i and g{ is a function which takes into account its reactions with the other constituents, i.e., which includes couplings between the various chemical species, p is a control parameter representing the external constraints, for instance the rate at which fresh products are introduced into the reaction vessel or the rate at which reaction products are removed. This equation simply expresses that the net increase in local concentration within a small volume element is due to the difference between incoming and outcoming fluxes and to the local excess of production to loss of the considered species through reactions. Under a concentration gradient the mobility of each species, vt, is given by Fick's law v^-D.VCi
159
(4-19)
dcjdt = Dt Act + gt(ci9 Cj, p)
(4-20)
This system has a uniform steady state corresponding to thermal equilibrium, {cf}, which is easily obtained through the condition {gt} = 0. When the external constraints or driving forces increase, this uniform state may become unstable. An analysis of the possible non-uniform solutions and of the corresponding critical points is given by Walgraef (1988; see also Sec. 4.5 and Schiller, 1989; Dewel and Borckmanns, 1990). They are of several kinds: a transition between homogeneous states, a transition to spatially periodic structures (Turing structures), temporal oscillations (via the so called Hopf bifurcation), travelling or standing waves with well-defined spatiotemporal periodicities. These systems undoubtedly have some analogy with plastic flow. Local dislocation-dislocation interactions are equivalent to chemical reactions, while longrange motion can possibly be treated as having a diffusive character. The application to dislocation patterning necessarily involves, as a first step, writing coupled balance equations for several dislocation populations g{. By analogy with Eq. (4-18), we have: div/ ; = gi(Qi9Qj,
(4-21)
j . = Q.i>. is now a dislocation flux, proportional to the associated strain rate [cf. Eq. (4-3)]. The term gt contains all the local mechanisms such as multiplication, annihilation, trapping ... which modify the density gt. It also contains couplings with other densities (gj), which represent local interactions of segments of different Burgers vectors.
160
4 Dislocation Patterning
As noticed by Aifantis (1986; see also Bottani, 1989), Eq. (4-21) is a compromise between discrete and continuous descriptions of the dislocation densities. Its lefthand side is the familiar continuity equation used in micromechanical approaches (see e.g., Kosevich, 1979), which applies to a continuum of dislocations of infinitesimal Burgers vectors, assuming conservation of the total Burgers vector. The righthand side, on the contrary, attempts to represent discrete source and sink mechanisms, the dislocation densities being averaged within an elementary volume of unspecified size. This size must, however, be smaller than the wavelength of the investigated dislocation pattern, typically 1 jim. On the other hand it must be larger than the average distance between dislocations to allow for a meaningful averaging of the local densities. A simple argument shows that a discrete description of dislocation densities, not attempted as yet, should certainly be more rigorous. To have at least one segment of length 0.1 jim every (0.1 jLim)3, one needs a rather high density, larger than 10 1 4 m" 2 . Then, every local process having an intrinsic length scale smaller than 0.1 jum has to be treated as a reaction, while all other mechanisms with length scales larger than 0.1 jim will enter the transport term in Eq. (4-20). As illustrated by the example of cross-slip in Sec. 4.3.2, some arbitrary assumptions have to be made when deciding, in a specific model, which mechanisms are smeared out by the averaging process and which ones are not. Within a continuous description of the densities, three elements are worth being discussed: the selection of the relevant populations, the form of the reaction terms and the treatment of the flux terms leading to the reaction-diffusion forms.
4.3.2 Dislocation Populations and Reaction Terms As seen in Sec. 4.1.3 a general treatment of patterning would require an unreasonable number of coupled species and only one or two populations are usually considered, viz. mobile and forest dislocations or mobile dislocations with Burgers vectors of opposite sign. This reduces not only the number of coupled equations but also the number of coupling mechanisms and reaction terms. Expressing the reaction terms is not a problem in principle since the corresponding local mechanisms have been the object of many models describing the evolution with time of average dislocation densities. The various creation, storage and annihilation processes can be expressed through different forms which have been discussed by Kocks et al. (1975) and Kubin and Lepinoux (1988). Here, we briefly reproduce a few arguments through a typical set of two equations describing the coupled evolution with time of a mobile density gm and a forest density gf dgj(vdt)
= (4-22) 12
dgt/{v dt) = (k Q} - LR 6f) Q
(4-23)
v is now the instantaneous spatial average of the dislocation velocities and k is a constant. Time dependences can easily be transformed into strain dependences by noticing that the common factor Qmvdt is in fact a strain increment. The first term at the right-hand side of Eq. (4-22) describes multiplication of mobile dislocations which, assuming a fixed density of multiplication events, is proportional to the area swept per unit time, gm v. A length scale L s is introduced which, according to the assumptions made, may have the following
4.3 The Reaction-Diffusion Approach
forms: L s = const. (Kubin and Estrin, 1990), L s oce" 1/2 (Kocks et al. 1975), or Lsoccr"1 (Alexander and Haasen, 1968). In the same way the quadratic term in the right-hand side of Eq. (4-22) describes events where the encounter of two mobile segments removes them from the mobile density (annihilation, formation of a dipole). The length y, which will be met again in the following, is called the characteristic annihilation or trapping distance. It is the critical distance between gliding planes within which two mobile dislocations will interact strongly enough to annihilate or form a dipole. Dislocations can also be removed from the mobile density by being blocked at obstacles with average spacing L o . In the present context the obstacles are forest obstacles, so that L0 = gf1/2 in the last term at the righthand side of Eq. (4-22). This introduces a coupling between the two densities. In the equation describing the evolution of forest dislocations, the storage term discussed above comes in as a creation term, while the second term at the right-hand side of Eq. (4-23) describes dynamic recovery, LR being the average recovered length per event (Kocks, 1976). Well-known evolutionary laws for single populations are obtained by decoupling these two expressions. Equation (4-23) was used by Kocks (1976) with a saturated (constant) value of gm in the discussion of low temperature work hardening and creep. In the absence of forest dislocations . (Q{ = 0) and with a constant average dislocation velocity, Eq. (4-22) reduces to the classical form used by Johnston and Gilman (1959). By discretizing this equation one recovers the so-called Verhulst equation for population dynamics (Grosbras, 1988) which has been applied recently by Bocek (1988 a, b) to the study of dynamic recrystallization.
161
According to the specific situation to be modelled, the length scales defined above may be constants or they may depend on the local densities or effectives stresses, as is certainly the case for the critical annihilation distance for cross-slip. This provides a wealth of possible systems which has not yet been explored in detail. In fact, whereas Eqs. (4-22) and (4-23) look reasonable as far as the time evolution of homogeneous densities are concerned, three types of difficulties arise when they are brought into reaction-transport or reaction-diffusion forms. The first one is concerned with the fact that, as discussed in the previous section, reactions are described both within a continuum framework and through discrete length scales. For instance, we consider the exchange of mobile dislocations between two slip planes by a double cross-slip mechanism (Fig. 4-12). The net increase in dislocation density in the plane y = 0 is the difference between the gain in dislocations having cross-slipped into it from neighbouring slip planes and the losses to these other slip planes. If 17 is the cross-slip probability per unit glide area and yc the maximum cross-slip distance, we have:
= nv J [em(y)-em(O)]dy
(4-24)
/ /c"
o --
-7c
"
Figure 4-12. Exchange of dislocation densities between neighboring slip planes via double cross-slip processes. yc is the maximum cross-slip distance.
162
4 Dislocation Patterning
Expanding the term between square brackets and retaining the first non-zero term, we obtain (dQm/dt)y
=0
= (4-25)
This expression clearly has a diffusive character, with a diffusion coefficient D = JJvyl/'i. A a more refined treatment has been proposed by Maligyn (1988,1989; see also Vladimirov and Kusov, 1976) and similar arguments have been used to justify the diffusive character of slip propagation at mesoscale (Brechet, 1987; Estrin, 1988; Kubin and Lepinoux, 1988; Aifantis, 1988). For such a description to be valid, the length scale beyond which the dislocation density is made continuous has to be small compared to yc. If this is not the case, patterning arising from double cross-slip mechanisms is smoothed out by the averaging process and a term similar to the second term at the right-hand side of Eq. (4-22) will appear on the reaction side of the balance equation. It follows that the diffusive character of cross-slip mechanisms is easier to justify at mesoscale than at microscale. A second difficulty stems from the fact that one has to define the stress dependence of the average dislocation velocity which appears in both the reaction terms and in the flux term of the reaction-transport equation. This can be done through an Arrhenius form or a power law but, at some step, one necessarily has to define the internal stress. A simple way to by-pass the problem is to make use of the phenomenological form of Eq. (4-4). As shown in Sec. 4.1.6, this is perhaps acceptable but certainly at the expense of a loss of physical insight into patterning mechanisms. Alternatively, a full treatment of the internal stress can be attempted (cf. next section).
The third difficulty arises from the fact that the system of Eqs. (4-22) and (4-23) is perhaps not sophisticated enough to account for patterning. This system is more particularly suited for dealing with uniform densities (Estrin and Kubin, 1986), but the study of reaction-diffusion forms shows that interesting instabilities and/or patterns appear as soon as cubic non-linearities are involved (Nicolis and Prigogine, 1977), or when more coupled equations are taken into consideration. Indeed, it seems difficult to describe the patterning of mobile dislocations with only one density gm9 neglecting the relative distribution of dislocations of both signs. In the same way, forest patterning involves at least the interaction of two slip systems and it could be necessary to introduce a new population consisting of sessile junctions. As will be shown below, the technique of adiabatic elimination can be helpful, when its use is justified, in reducing the number of interacting populations while increasing the non-linearity. 4.3.3 Internal Stresses and the Diffusion Term
The internal stress, which is the most controversial subject in classical models, is also the weakest point of all theoretical elaborations based on the reaction-diffusion approach. It can be estimated either by taking at a given position r in the crystal the gradient of the total interaction energy, E{(r\ or by directly computing the sum of the pair interaction stresses. Walgraef and Aifantis (1985 a) make use of the following expression for the interaction energy, which is similar to that adopted by Holt (1970): Ei(r) = $Q(r')f(r')I(W-r\)dr'
(4-26)
f(r') is a distribution function accounting for the relative arrangement of dislocations
4.3 The Reaction-Diffusion Approach
163
of opposite sign and / is the pair interacdensity g should be thought of as a density tion. As already emphasized, this expresof dipoles rather than a density of isolated sion is non-local and it cannot be treated dislocations. This is implicit in the model consistently. The simplest possible approxof Walgraef and Aifantis (1985 a) and Kraimation consists of assuming that selftochvil and Libovicky (1986) have further screening configurations are formed and of noticed that, then, the applied stress term introducing a cut-off. This restricts the must be removed from the expression of range of practical situations which can the flux since a dipole is only sensitive to safely be investigated to those involving stress gradients. For a quanlitative examimultipolar configurations. As a consenation of reaction-diffusion forms it may quence, reaction-diffusion forms have be useful, although this is not truly necesmainly been applied to the study of fatigue sary, to write the divergence of the flux J in patterning. Within these limits, which are the form certainly worth further discussion, a Taylor div/=-DA£ (4-29) expansion of the energy can be performed around r' = r. The non-local form of Eq. where D is a matrix of positive diffusion (4-26) is then transformed into a local one. coefficients which depend on both the moAs in Holt's model [cf. Eq. (4-8)], the first bilities and the dislocation densities. This term in the expansion is proportional to last step brings the reaction-transport the local density g (r). The internal stress is, equations [Eq. (4-21)] into the form of reaction-diffusion equations, provided that therefore, proportional to the gradient in some constant average value is attributed dislocation density and the local disloto the diffusion coefficients. Here, we have cation velocity can be taken proportional schematically reproduced the initial steps to the sum of the applied stress aa and the internal stress GX OC VE{ (r) = — rj VQ (r). of the model of Walgraef and Aifantis (1985 a). As shown in Sec. 4.3.6, the reacThen, tion-diffusion form associated with the (4-27) dipolar density can be reached through a v = M [<xa - r\ VQ (r)] somewhat different, perhaps more conwhere M is a mobility tensor. Under zero vincing, procedure (Kratochvil, 1988 b). applied stress, the driving force for the evolution of the structure is a reduction in the We have examined in Sec. 4.3.2 the propinteraction energy under the influence of agation of slip along the specimen axis; the the internal stresses. We see here that this propagation of deformation inside a crosseffect is counteracted by the dissipative section is more difficult to treat. One can term containing the applied stress, which either consider that it has a diffusive charmaintains the system far from thermodyacter (Walgraef and Aifantis, 1985 a) or asnamic equilibrium. The velocities entering sume that mobile dislocations are unithe reaction terms [cf. Eqs. (4-22) and (4-23)] formly distributed in the active slip planes can be developed and the fluxes can be (Kratochvil, 1988 b). written as: This discussion shows that several obstacles have to be overcome before reliable (4-28) =Qv = M[g
164
4 Dislocation Patterning
the reaction terms still deserve further examination. It appears possible to treat the dislocations as having a diffusive type of dynamics, although this implies some physical assumptions and a restricted domain of application. Despite these weaknesses, the various models published to date have yielded a number of interesting results. We now review these achievements, leaving out technical aspects which can be quite complex. 4.3.4 Simple Dynamic Models To the author's knowledge, the earliest article devoted to the dynamic aspects of dislocation patterning is that of Vladimirov and Pegel (1973). This phenomenological model consists of a set of three coupled equations describing the evolution of one mobile population in one dimension. One equation accounts for conditions of straining at a constant plastic strain rate. It contains an Arrhenius form where the internal stress is proportional to Q1/2. The second equation is basically a reaction-transport form with one reaction term describing multiplication at a rate proportional to some power of the dislocation density. The last equation connects the dislocation flux to the strain rate through a particular form of Orowan's law which is perhaps questionable. A linear stability analysis of this system shows that heterogeneities in the dislocation density develop under the influence of the creation term. They are favoured by a low initial density, a high temperature and a strong work hardening rate. The main outcome of this study is to show in a qualitative manner that it is rather easy to construct simple dynamic systems having spatially modulated solutions. The same conclusion holds for the simplified treatment of two interacting populations which was developed by Malygin
(1989), in an attempt to connect the formation of cell structures with the occurrence of non planar slip. The two populations are mobile in two intersecting slip planes and their diffusive character stems from double cross-slip events (Valdimirov and Kusov, 1976; Malygin, 1988; see also Sec. 4.3.2). Each population sees an internal stress proportional to the square root of the density of the other one. Although periodic solutions are also obtained in this case, their physical meaning is uncertain since the forest dislocations investigated in this model appear to behave essentially like mobile dislocations. The model of Burmeister and Hermann (1979) deals with an idealized situation in order to examine the stability of multipolar ensembles. Edge dislocations are distributed on a set of parallel slip planes and all dislocations have the same sign in each slip plane, this sign being different for two neighboring planes. Three equations are written. One expresses the internal stress in one slip plane, due to the long-range stresses from all other planes. A non-local form analogous to that of Eq. (4-26) is used and no approximation is made on it, which is the most original part of this model. The balance equation for the dislocation density is written in the reaction-transport form and the dislocation velocities are taken proportional to the effective stresses. Figure 4-13 shows a typical result of stability analysis for a set of 33 parallel slip planes with interplanar spacing d0. A uniform dislocation density is unstable with respect to fluctuations of wavelength X in two domains of the plane (2ndo/A,N), where N is the index of the slip planes. The transition between domain A and domain B, of smaller wavelength, is tentatively identified with the early stage of formation of PSBs (see also Schwab, 1990; Schwab and Burmeister, 1991).
4.3 The Reaction-Diffusion Approach
165
In one dimension, the coupled system reads (Walgraef et al., 1987; Schiller, 1989): dQi/dt-Did2Qi/dx2
= (4-30)
= PQi-PQmQ? Figure 4-13. Linear stability analysis of the system of Burmeister and Hermann (1979). N is the index of the parallel slip planes of spacing d0, and X is the wavelength of the periodic perturbation. In the two hatched areas, the uniform distribution of dislocations is unstable.
Finally, a simplified and preliminary treatment of the formation of wall structures in PSBs has recently been proposed by Differt and Essmann (1988), who consider the coupled evolution of single edge dislocations and dislocation dipoles. 4.3.5 The Model of Walgraef and Aifantis
The reaction-diffusion approach to fatigue patterning (Walgraef and Aifantis, 1985 a) has been developed in a series of articles (Walgraef and Aifantis, 1985 b, c, d; Walgraef, 1986; Aifantis, 1985, 1986, 1987, 1988, 1990; Walgraef et al, 1987) and has further been complemented by numerical simulations of the solutions of the model equations (Schiller and Walgraef, 1988). A full account of the various applications of reaction-diffusion schemes to dislocation patterning, including Neumann's instabilities (Neumann, 1986) and propagation effects can be found in the remarkable work of Schiller (1989). For the investigation of fatigue patterning, two populations are considered: the mobile dislocations in density gm and immobile, or rather slow dislocations in density Q{ , which are supposed to be arranged in form of edge dipoles or dipolar clusters.
(4-31)
Dx and Dm are the diffusion coefficients of the slow and mobile dislocations respectively and g(Q{) is a function describing the stress-assisted generation and annihilation of slow dislocations. The freing rate of immobile dislocations is determined by the parameter p which is proportional to the plastic strain rate. This quantity plays the role of a control parameter and its value governs the nature of the solutions of the system. /? is a constant associated with the rate of blocking of a mobile dislocation by a "slow" dipole. For the sake of simplicity we have only reproduced here the uni-dimensional version of the model. Bi- and three-dimensional versions, which contain anisotropic mobilities and diffusion coefficients, have been discussed by Walgraef and Aifantis (1985 c, d). The diffusive character of the mobile dislocation flux is justified by considering time scales larger than the time of a half-cycle in fatigue (Aifantis, 1986; Walgraef and Aifantis, 1988; Walgraef, 1990). The diffusion coefficient is estimated by considering the balance equations for two coupled mobile populations g+ and Q~, or equivalently Qm = Q+ + Q~ and 8 = Q + — Q~. As-
suming that 8 is a fast variable, the equation for the time evolution of this quantity is solved with constant gm and the result is inserted into the equation for Qm. The variable 8 is thus "adiabatically eliminated" and it follows that Dm = v2/(pQf0) (Schiller, 1989), where v is the average dislocation velocity during a half-cycle and gi0 the steady state value of the slow den-
166
4 Dislocation Patterning
sity. For the slow dislocations, the longrange internal stresses are supposed to be effectively screened and the diffusion coefficient D{ is estimated following the procedure outlined in Sec. 4.3.3. As a result, one obtains D{PDS, a necessary condition for obtaining Turing (i.e., patterning) instabilities in such systems (Dewel and Borckmans, 1990; see also Sec. 4.5). In Eqs. (4-30) and (4-31), the physical meaning of some of the reaction terms is perhaps uncertain. The crucial one is certainly that containing the cubic nonlinearity (Nabarro, 1990; unpublished communication), which critically governs the type of the heterogeneous solutions. Indeed, if the slow population is of dipolar nature, the encounter of a mobile dislocation with a dipole should instead be represented by a quadratic form. On the other hand, if the slow species represents individual dislocations the expansion of the internal stress leading to a diffusion-like form for the flux of slow dislocations no longer holds in the absence of self-screening. The occurrence of high-order non-linearities has been justified through the consideration of a larger number of interacting populations and adiabatic elimination procedures (Walgraef, 1990). Despite this, the model captures the essence of fatigue patterning in f.c.c. crystals. For the reader not familiar with such procedures, we reproduce in the Appendix (Sec. 4.5) the starting point of the analysis of the set of Eqs. (4-30) and (4-31). The pattern wavelength is a function of the rate constants and of the diffusion coefficients (cf. Appendix) and it is found proportional to ft"1/2 (Schiller and Walgraef, 1988; Schiller, 1989), although the proportionality constant is not estimated. The numerical values for all the coefficients being approximately defined, the formation of PSBs at the surface of single
crystals and their propagation inside the crystal have been computed. Figure 4-14 reproduces a result where this development is triggered by a heterogeneity in dislocation density at the surface of the specimen. Three stages of the development of the density of slow dislocations are shown for (p — pc)/pc = 3.5, where pc is the critical value of the control parameter (cf. Appendix). The respective widths of the dislocation rich-walls and of the channels are quite reasonable, the spatial periodicity
2
4 6 8 10 Position (\im)
12
--20
-•10
2
4 6 8 10 Position (pm)
12
2
4 6 8 10 Position ( p m )
12
Figure 4-14. Three stages of the development of a simulated PSB from a local heterogeneity of the dislocation density located at the left border. After Schiller (1989).
4.3 The Reaction-Diffusion Approach
being of the order of 1 jim while the values of the dislocation densities in the rich regions are of the order of 10 15 m~ 2 . The results of the bi- and three-dimensional versions of the model are summed up in Fig. 4-15, where a bifurcation diagram for fatigue patterning is presented (compare with Fig. 4-11). The amplitude of the dislocation density modulations is plotted as a function of the control parameter which is, in this version of the model, the absolute value of the maximum stress during one fatigue cycle. For low stress values the homogeneous solution is stable. For higher stresses, there is a transition towards a polygonal rod-like structure which gradually fills the whole material and which represents the so-called matrix structure. With increasing stress level, a layered ladder-like structure, akin to the PSB structure, emerges. 4.3.6 Kratochvil's Model
The question of the stability of an assembly of dipoles has been treated by Kratochvil and coworkers in a series of papers (Kratochvil and Libovicky, 1986; Kratochvil, 1988 a, b). This work was further extended in form of a global framework able to account for both dislocation patterning, in particular the formation of cell structures (Kratochvil, 1990 a, b), and geometrical or mechanical instabilities (Franek et al., 1991; Kratochvil, 1988 c, 1989, 1991). This model may be seen as an improved version of earlier approaches in two respects. First, it does not attempt to reproduce exactly the classical reactiondiffusion scheme. As a consequence, the evolutionary law for the dipolar density is more accurate in terms of elementary dislocation properties, while the flux of the mobile dislocations is simply treated as a homogeneous, non diffusive, background.
Mixed 0,1
167
Mixed 1,2
Figure 4-15. Bifurcation diagram for patterning in fatigue after Walgraef and Aifantis (1985). rf is the amplitude of the modulation of the spatial pattern and a is the absolute value of the maximum stress per cycle. The preferred stable states are represented by heavy lines (compare with Fig. 4-11).
In addition, the stress and strain tensors are consistently defined, which allows a connection with the mechanical approach of plastic instabilities. The type of microstructure investigated consists of clusters of prismatic dipolar loops in form of braids (bundles) or veins (patches). Such arrangements are formed in uniaxial deformation as well as in cyclic deformation in b.c.c. crystals like Fe-Si and in f.c.c. crystals. An example of dislocation patches in Fe —0.9 wt.% Si, in a section parallel to the slip direction and parallel to the slip plane, is shown in Fig. 4-16. The mechanism by which such clusters are formed is described as follows (cf. e.g., Franek et al., 1991). Dipoles are essentially produced by the dragging of jogs formed at dislocation intersections. An edge dipole or a dipolar loop has two stable positions and two possible signs (cf. Fig. 4-17), hence four possible configurations. Under a uniform stress, it remains stationary but two configurations are stretched and two are compressed. Under a stress gradient, there is a non-zero force which tends to bring the dipoles to the positions of maximum or minimum internal stress. Because of dis-
4 Dislocation Patterning
168
B is a constant stemming from the stress vs. velocity relation, Q± is the density of dipoles (cf. Fig. 4-17), h is the dipole width, axy is the shear stress and q = An(1 — v)hj{\xb\ v being Poisson's coefficient. Figure 4-18 shows how a pattern of clusters emerges through the up-hill diffusion of dipoles. A small initial heterogeneity attracts neighboring dipoles and a cluster grows, producing depleted zones in its vicinity. The edges of these depleted
Figure 4-16. A section of the vein structure in a plane perpendicular to the slip plane and parallel to the slip direction in a Fe-0.9wt.%Si single crystal cycled with a plastic strain amplitude of 2x 10 ~3 up to a cumulated strain of 1.6. After S. Libovicky, reproduced from Franek et al. (1988).
symetries between the four configurations the net result is, during cyclic deformation, an up-hill drift towards the maxima of internal stresses. In the absence of climb, the resulting flux is related to the shear stress gradient in the slip plane through a positive diffusion coefficient for dipoles of plus and minus sign whose value is given by (Kratochvil, 1988 a): ±
=
±(BQ±h/2)[l±qaxy]
J_
J_
T T
7 (b)
x
Figure 4-18. The formation of a periodic pattern of dipolar clusters through up-hill dipole diffusion (represented by arrows), (a) Initial cluster and the depleted zones, (b) Formation of new clusters at the edge of the depleted zones. After Kratochvil and Libovicky (1986).
(4-32)
± T
(a)
_L
Figure 4-17. The four possible stable equilibrium configurations of a dipole. Under a positive shear stress, the positive configurations at right are stretched (h increases) and the two negative configurations at left are compressed. After Kratochvil (1988 b).
zones block some of the mobile dislocations, thus increasing the local stress gradients. With continued up-hill diffusion, new germs of clusters are formed at some distance of the first one and the propagation of this mechanism leads to a periodic pattern. We now focus on the application of the model to cyclic deformation. The rate at which dipoles are produced is proportional to the absolute value of the shear strain rate y. At high stresses dipoles can be destroyed and the cluster structure may un-
4.3 The Reaction-Diffusion Approach
169
dergo a catastrophic collapse. This is described by a destruction term depending on the strain rate and which contains a nonlinear stress contribution with a threshold ac. As a result, the balance equation for the total dipolar density writes: y
(4-33)
where A and m are constants. A second ingredient is a constitutive form, relating the shear strain rate to a power of the stress reduced by the glide resistance. The glide resistance increases with strain through strain hardening mechanisms: as dipole clustering occurs, the width of the channels left for the operation of dislocation sources decreases so that the bowing stresses increase while the stress needed to pass through a cluster also increases. This hardening is phenomenologically represented by a non-local form, similar to that of Eq. (4-26) which is truncated not in real space but in Fourier space. The advantage of this procedure is that it leads to periodic solutions without selecting a priori the value of the wavelength. The system of equations is closed with the conditions of stress equilibrium, strain compatibility and the Hooke's law. As emphasized in the discussion of the composite model (Sec. 4.2.4), this ensures the consistency of the solutions within a continuous description of the dislocation densities. At small applied stresses, the destruction term in the balance equation may be neglected and a simple solution is drawn from linear stability analysis. It consists of periodic patterns which reproduce well the three-dimensional geometry, in particular the preferred directions which appear at approximately ±45° of the trace of the slip plane in a section perpendicular to the primary slip plane (cf. Fig. 4-19 which simulates the arrangement of Fig. 4-16).
Figure 4-19. Simulated dipolar pattern viewed in a section perpendicular to the slip plane and parallel to the slip direction (compare with Fig. 4-16). After Franek et al. (1988).
The interest of these results is that they are based on a careful justification of the diffusive character of the dipolar loops in a non uniform stress field. The pattern geometry is reasonably well described and it is understood as a competition between uphill dipole drift and non local hardening. In the present stage of the model, the amplitude and the wavelength of the patterns cannot, however, be unambiguously defined because the hardening term is not expressed in terms of dislocation mechanisms. At larger stresses, the collapse of the vein structure in the plateau region of the cyclic stress strain curve is attributed to the onset of dipole destruction, and the last term at the right-hand side of the balance equation, Eq. (4-33) comes into play. The type of instability that occurs is, however, considered to be geometrical rather than structural and the linear equations of equilibrium, compatibility and Hooke's law are replaced by non-linear forms, by analogy with the mechanical approach of shear banding at large strains. Localization has, therefore, a geometrical origin within this model but the formation of a dislocation
170
4 Dislocation Patterning
pattern within PSBs is further treated as a structural instability. The expressions describing the dipolar density are solved with a different type of boundary conditions and periodic solutions representing the ladder-like structure are obtained with walls perpendicular to the slip direction. This duality of origin also characterizes the application of the model to cell structure formation. The monotonic deformation of a crystal oriented for symmetrical double slip has been examined along the same lines. The emergence of a cellular pattern, analogous to that described in Mughrabi's composite model (Sec. 4.2.4) is predicted from the linear stability analysis (Kratochvil, 1990 a) of a uniform distribution of dislocations. The cell misorientations do not follow, however, from this approach and they are treated separately through an argument of mechanical origin. An instability of the internal bending type (Kratochvil, 1990 b; Kratochvil and Orlova, 1990) is, then, thought to occur in synergy with the patterning instability. Kratochvil's model potentially leads towards a unified description of dislocation patterning and plastic instabilities but only partial solutions could be obtained to date, owing to the mathematical complexity of the differential equation systems. Further progress may help to elucidate the origin of various types of strain localizations. For instance, length scales for the development of plastic heterogeneities at mesoscale can be derived by introducing strain or strain rate gradients in the constitutive equations. This has been done recently for shear bands (Coleman and Hodgdon, 1985; Triantafyllidis and Aifantis, 1986; Zbib and Aifantis, 1988 a), as well as for PortevinLe Chatelier bands (Zbib and Aifantis, 1988 b; Estrin, 1988; Fressengeas and JeanClaude, 1991) and Luders bands (Hahner, 1991). These gradients may be thought to
stem from structural instabilities, i.e., from long-range stresses or from the diffusive character of the mobile density, or from a geometrical origin. 4.3.7 Discussion and Prospects
The non-linear and dynamic approach of dislocation patterning appears quite attractive but still needs clarification. The reaction-diffusion approach seems naturally suited for problems involving point defectinduced dislocation patterning (Murphy, 1988; Estrin et al., 1990). Its adaptation to dislocation patterning is due to Walgraef and Aifantis, who emphasize the analogy with chemical patterning and whose work marks the entry of non-linear dynamics into dislocation theory. Some inaccuracies of previous elaborations are corrected in Kratochvil's work. The diffusive character of the dipolar flux is thoroughly established and a procedure avoiding the truncation of long-range stresses in real space is presented. Still, a few basic questions remain unsolved. The evolutionary equations for the dislocation densities are in fact a compromise between continuous and discrete descriptions, and the conditions for mixing these two approaches in a compatible manner have not yet been fully investigated. It follows from Kratochvil's work that a link can be established between micro- and mesoscales by taking into account stress equilibrium and strain compatibility. The relation between this frame and the composite models is not clear, however, since the storage of mobile dislocations in the dense dipolar regions is not explicitly treated as a source of internal stresses in Kratochvil's work. There is no agreement between the main two models outlined above about the nature of the instability leading to the formation of PSBs. This is because these two
4.4 Approach by Numerical Simulations
models, in order to obtain a closed set of equations, complement the behavior of the dipole population with either a mobile population, hence a structural origin for the PSBs, or with conditions on stresses and strains, hence a mixed origin. In addition, not all materials exhibit PSBs so that effects which may govern their occurrence, like slip geometry, cross-slip ability or short-range ordering in dilute alloys, should appear in the bifurcation conditions. Nevertheless, the reaction-diffusion approach appears potentially able to treat all the aspects of dislocation patterning related to dipolar and multipolar clusters. As far as cell structures are concerned, cell misorientations must, indeed, be taken into account at large strains. On the other hand, at low temperatures and low strains cell walls are not dipolar and, in addition, they do not necessarily introduce misorientations. Kock's statistical model (cf. Sec. 4.2.2) as well as three-dimensional simulations (cf. Sec. 4.4.4) suggest that, in every case, reaction junctions should play a central role in the formation of patterns in multiple slip. Other situations worth being explored are related to the mobile population and to problems associated with slip patterning and slip propagation (cf. Walgraef, 1990). In these last two cases, selfscreening effects can no longer be invoked to simplify the treatment of the non local internal stress or hardening rate. To date, one does not see well how to approach these questions in a manner which is neither too simple nor too complicated. Undoubtedly, reaction-diffusion forms are difficult to treat and going beyond the bifurcation towards non uniform solutions involves either a complex analysis or a numerical computation of the solutions. A phenomenological approach, which is becoming popular for the treatment of instabilities at micro- and mesoscale, consists of
171
transforming classical kinematic expressions by introducing an additional diffusion term. Such simplified models may become quite useful once the underlying physical justifications have been elaborated. It is rather easy to construct simple differential forms exhibiting a variety of instabilities but it appears at some steps that a detailed understanding of the basic dislocation mechanisms is still missing. This is quite paradoxical after so many years of development of dislocation theory and so many experimental observations, unfortunately static ones in most of the cases. The main motivation for performing "numerical experiments" is, as a consequence, to identify, among many possible ones, the relevant elementary mechanisms and to establish a basis for their modelling.
4.4 Approach by Numerical Simulations 4.4.1 Introduction - Methodology
Numerical simulations have appeared in the recent years as a useful complement to other investigations involving either experiment or theoretical modelling. Progress in computer technology has allowed, in particular, the study of collective effects in systems containing a large number of particles which interact through a force field. The application to dislocation patterning is, however, not straightforward and it necessarily involves adapting classical methods to the problem of the anisotropic motion of linear objects, with long-range interactions and several types of local interactions. The first two-dimensional simulations appeared in the mid-eighties and paved the way for the three-dimensional
172
4 Dislocation Patterning
simulations which are now being developed. The first objective of a mesoscopic simulation of plastic flow based on dislocation dynamics is to reproduce, within a reasonable numerical accuracy, a few well-defined types of collective behavior and the corresponding mechanical responses. The evolution rules must account for a minimum number of basic dislocation properties, mechanisms and mutual interactions, of which some define the material considered and the deformation temperature. Only those microscopic features which are relevant at macroscopic scale should ideally be retained and their treatment has necessarily to be simplified because technical constraints do not allow to start from atomic scale. This holds particularly for properties like climb, cross-slip and the intersection of two dislocations of different Burgers vectors. In this last case, for example, the formation of jogs can safely be ignored since it does not contribute significantly to the flow stress at and above room temperature, but the formation of reaction junctions has necessarily to be taken into account since it is at the basis of the forest mechanism. Therefore, the preliminary step which consists of selecting the mechanisms to be implemented, of translating them in a simplified code without altering their essence, and of defining the smallest length and time scales is a very critical one. Once the output of a simulation can reasonably be considered as reliable, a large variety of "experiments" can be performed and a check of existing models or the elaboration of new ones becomes possible. Up to now, the results obtained have been more qualitative than quantitative. Progress in both simulation techniques and computer technology will undoubtedly allow a large development in the years to come. It is hoped that a few typical pat-
terns will be obtained and examined numerically and that constitutive equations, valid at a scale on the order of ten micrometers, will be established on a physical basis for a few model materials. In parallel, some very basic questions like the existence of long-range stresses, the self-screening properties of dislocation ensembles, the origin of the phenomenological scaling laws and the possible diffusional nature of plastic flow will be approached on a rather neutral basis, not involving any postulate. 4.4.2 Simulation Techniques
In the past, simulation methods have been applied to rather simple situations, e.g. to the motion of one dislocation line past an array of randomly distributed obstacles (Foreman and Makin, 1966, 1967). Several attempts to derive equilibrium shapes or to study the dynamics of small dislocation ensembles are reviewed by Neuhauser (1980) and Amodeo (1988). Such simulations can still be of a great interest for the study of small groups of dislocations which do not necessitate heavy computations (cf. e.g., Neumann, 1986). For example, Fig. 4-20 reproduces several types of multipolar configurations formed by the interaction of pile-ups gliding in neighboring planes, in an alloy where dislocation motion reduces the friction stress by destroying short-range order (after Olfe and Neuhauser, 1988). For large dislocation ensembles, one has to solve the equations of motion of all the segments, under the influence of longrange and local interactions, within a small time step, either in a continuous (molecular dynamics) or discretized space (cellular automata). As emphasized by Ghoniem and Amodeo (1990), the practical implementation is a serious challenge in the context of numerical simulations of particle systems
4.4 Approach by Numerical Simulations
x
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Figure 4-20. Simulated final configurations of multipoles on slip planes separated by a distance of 20 nm, under an applied shear stress of 20 MPa. After Olfe and Neuhauser (1988), (a) The alloy friction stress was initially 25 MPa and the destruction of short range order reduced it to a final value of 6 MPa. (b) For both groups the alloy friction stress has already been cleared to the value 6 MPa. (c) The upper group sees the initial friction stress (25 MPa) and destroys it, while the lower one moves on an already cleared slip plane.
and the peculiarities of dislocation dynamics introduce a few more difficulties. Three two-dimensional simulations have been published to date. Two derive from molecular dynamics (Amodeo, 1988; Ghoniem and Amodeo, 1988, 1989a,b, 1990; Gulluoglu et al, 1989, 1990) while the third one (Lepinoux, 1987, 1988; Lepinoux and Kubin, 1987; see also Steck and Hesselbarth, 1991) is adapted from lattice gas automata applied to fluid dynamics (D'Humieres and Lallemand, 1986; Lepinoux, 1987). In this section we discuss some similarities and differences between these two methods; the three-dimensional simulation is described in Sec. 4.4.4. In two dimensions sets of infinite, parallel and straight dislocations with parallel or antiparallel Burgers vector and of same character (edge or screw) are considered and the configurations are viewed in a direction parallel to the dislocation lines. For each dislocation, the internal stress is the sum of its pair interaction stresses
173
with all other dislocations within an extended neighborhood. It is computed using the stress fields of infinite dislocations in isotropic elasticity and converted into a resolved shear stress with help of the Peach-Koehler formula. The mathematical difficulty stemming from the non-local character of internal stresses reappears, therefore, in simulations through the fact that a large number of pair interactions has to be computed. Ghoniem and Amodeo take into account all interactions taking place within a simulated area of the order of 2 x 4 j^im2, which typically contains 400 dislocations. To reduce computing time, Gulluoglu et al. make use of a smaller simulated area ( l x l jam2), and sum the interactions due to all the replicas produced by periodic boundary conditions. The method used by Lepinoux consists of truncating the longrange interaction forces at a distance such that adopting a larger value does not significantly modify the configurations obtained, nor the average interaction energy (cf. Sec. 4.4.3.4). For screw dislocations the cut-off value is about 5 jim. The sum of the resolved internal and applied stress defines the local effective stress and a dislocation is mobile if the latter is larger than the friction stress. Relations between effective stresses and velocities are defined for dislocation glide, cross-slip and climb, which can be either linear or in form of a power law. Situations of pure glide and pure climb have been examined by Ghoniem and Amodeo, while the simultaneous occurrence of glide and cross-slip or climb was considered by Lepinoux. Because of the large variations in local stresses, the velocity spectrum of dislocations extends from zero to several m s" 1 and it is necessary to define a time step for the motion of the dislocations such that their mean free path is small compared to
174
4 Dislocation Patterning
the average distance between dislocations. The minimum time step is defined by Ghoniem and Amodeo as the smallest among several time intervals relative to various local interactions. According to the situation considered and to the evolution of the simulated microstructure, this time step may vary by several orders of magnitude, e.g. from a few nanoseconds to a few milliseconds in the simulation of fatigue. In the method deriving from cellular automata, space is discretized and an elementary cell is defined which cannot contain more than one dislocation. Its size is taken as twice the critical annihilation distance, y, for a dipole. Numerical values for this quantity have been measured by Essmann and Mughrabi (1979) for copper at room temperature during the saturation stage of low cycle fatigue: ys = 50 nm for the annihilation of a screw dipole by cross-slip and ye = 1.5nm for the destruction of an edge dipole, presumably under the effect of its self-stresses. The simulated area contains 2 to 4 x l 0 4 sites and its maximum size is 31x13 jim2 for screws and 1 x 0.4 jim2 for edges, with periodic boundary conditions. A step of the simulation consists of defining, in terms of a number of elementary cells, the distance moved by each dislocation. Then, the positions of all the dislocations are simultaneously updated to produce a new generation. This procedure has the advantage of being based on the physical definition of an elementary length scale, while the discretization of space leads to a better computational efficiency. This gain in simplicity has, however, a counterpart and the output of this simulation allows for a qualitative description of the configurations but not for a quantitative analysis. Local interactions are treated like reactions or topological events. Without enter-
ing here into the detail, the immobilization, mutual trapping, annihilation, cross-slip and climb of mobile dislocations occur either spontaneously within the simulations or through local criteria based on local effective stresses. The modelling of dislocation multiplication is more troublesome and must be treated through ad hoc assumptions since the formation and operation of sources cannot be represented in two dimensions. Cross-slip is another three-dimensional feature, whose occurrence is related to the core structure of the dislocations or to their width of splitting. In the discretized method, a local deterministic criterion is probably sufficient for a qualitative description of the annihilation of screw dislocations, since the critical annihilation distance already scales all the patterns. However, its thermally activated character should be taken into account in a more quantitative description. Two-dimensional simulations are restricted to the study of somehow idealized situations because they cannot treat line tension effects, nor the interactions betwen the screw and edge portions of an expanding loop (the motion of the edge parts creates screw portions and vice-versa). Even easy glide is, in practice, a three-dimensional phenomenon since it involves the interaction of dislocation loops gliding in neighbouring slip planes and the propagation of slip along the axis of the specimen. Despite these shortcomings, two-dimensional simulations have yielded interesting results as will be illustrated in the following. A new methodology has been established and model materials and generic types of behavior have been investigated. Similar types of non-random configurations were systematically found to emerge from initially random ones.
4.4 Approach by Numerical Simulations
4.4.3 Results of Two-Dimensional Simulations 4.4.3.1 Annealing Experiments
This first example deals with a generalization of the situation first considered by Holt (1970) and discussed in Sec. 4.2.1. In the annealing experiments performed by Lepinoux (1987, 1988), an initially random distribution of edge or screw dislocations evolves under the influence of its internal stresses, the friction stress being of the order of a few MPa. There is no multiplication but the direct annihilation of two dislocations of opposite sign gliding in the same row of cells of the discretized space is permitted. After a few tens of steps, all the remaining dislocations are immobilized and a stable configuration is reached. A typical result obtained with edge dislocations is reproduced in Fig. 4-21 a. About one half of the initial density has been annihilated and multipolar clusters have formed with preferential directions reflect-
175
ing the equilibrium position of elementary dipoles. As expected, no periodic pattern is obtained. Changing the values of the initial density, of the friction stress, or introducing a small applied stress of the order of the friction stress and allowing for dislocation climb modify the configurations within the clusters without altering the global landscape (cf. Fig. 4-21 b). A measurement of the total elastic energy shows that it has, indeed, decreased. The multipolar clusters are clearly self-screening configurations trapped into local minima of energy. 4.4.3.2 Slip Patterning
The second example deals with slip organization in single glide. The evolution of an initially random configuration of screw dislocations is examined under the effect of an applied shear (Lepinoux and Kubin, 1987; Kubin and Lepinoux, 1988). Multiplication is accounted for by introducing pairs of dislocations of opposite sign in
Figure 4-21. Two annealed configurations of edge dislocations obtained from random initial conditions after 100 time steps. The slip planes are seen edge-on and the slip directions are horizontal. The initial density and the friction stress are larger in (b) than in (a). Notice the formation of self-screening multipolar clusters. After Lepinoux (1987).
176
4 Dislocation Patterning
open spaces within the slip lamellae of the discrete lattice. According to the value of the local stress, these pairs either recombine and annihilate, or they expand, leaving open spaces for further multiplication. The rate of increase of the total density being imposed, the new dislocations can be placed (i) at random in the available sites (random multiplication) or, (ii) preferentially in the active slip lamellae (autocatalytic multiplication). Then, the occurrence of dislocation pile-ups directly results from the way fresh dislocations are introduced. Case (ii) is supposed to represent both the propagation of dislocation pile-ups emitted by a source and the softening of the microstructure by passing dislocations, for example the destruction of short-range order in alloys (Olfe and Neuhauser, 1988; Gerold and Karnthaler, 1989). Figure 4-22 a reproduces a dynamic configuration obtained in case (i) after 100 steps performed under a constant applied stress of 15 MPa, the friction stress being
half this value. Although there is some tendency to form a lamellar slip structure, patterning is not much pronounced. Figure 4-22 b shows the configuration obtained in case (ii), all other conditions being the same as before. Slip appears to exhibit a fine structure, which results from the preferential annihilation of screw dislocations of opposite sign, gliding in neighboring rows of cells. The typical length scale of this fine slip structure, about 200 nm, is directly related to the underlying base of discretization i.e., to the critical distance for the annihilation of dislocations by cross-slip. Figure 4-23 is reproduced from the work of Ghoniem and Amodeo (1990) who simulated the easy glide of edge dislocations in copper with an autocatalytic multiplication rule. Transient slip localization and the formation of multipolar layers are observed which bear some resemblance with the configurations of Fig. 4-20. A typical distance between these layers is of the order of 1 jim but, because of the multipolar
Figure 4-22. Patterning of screw dislocations in easy glide, after 100 time steps under an applied shear stress of 15 MPa, from an initial random configuration, (a) Random multiplication, (b) Layered pattern obtained with autocatalytic multiplication. Notice cross-slip events occurring through the interaction of dislocation pile-ups. After Lepinoux (unpublished result).
4.4 Approach by Numerical Simulations
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character of these arrangements, it is not certain that these transient localizations are spatially correlated. This ensemble of results suggests that the smallest length scale associated with slip patterning is determined by the crossslip of screw dislocations, while a larger scale has to do with the passing stresses or long-range interactions between edge pileups. However, such an interpretation will remain speculative as long as the interaction of dislocation loops, having both screw and edge portions, has not been simulated. 4.4.3.3 Slip Propagation
Two-dimensional simulations also yield an insight into the problem of slip propagation. The initial configuration leading to the arrangements of Fig. 4-24 consists of screw dislocations introduced at random
Figure 4-23. Edge dislocations in copper. The applied stress is 30 MPa and the friction stress 5 MPa. The initial configuration (top, left) is random. Planar arrays with a high concentration of dipoles form and disappear; their spacing is of the order of 1 jam. After Ghoniem and Amodeo (1990). (Courtesy of Kluwer Academic Publishers.)
within a slice of thickness three elementary glide rows. The multiplication rule is of autocatalytic type and dislocations are able to cross-slip into the vertical plane. The objective of this study is to determine whether or not slip activity will spread out along the direction normal to the slip planes. The result is that, under a constant applied shear stress, a critical condition is met which depends on the value of the constant controlling the rate at which fresh dislocations are introduced (Lepinoux, 1987). For small values, the slice remains active (cf. Fig. 4-24 a), with a fine structure identical to that of Fig. 4-22 b. Dislocations are eliminated by cross-slip from one out of every two elementary slip rows. In such a configuration, the passing stresses between the pile-ups are not sufficient to induce cross-slip and multiplication outside the active lamella and the thickness of the active band remains constant. When the mul-
178
4 Dislocation Patterning
Figure 4-24. Transition between static and propagating patterns. The initial configuration consists of screw dislocations randomly distributed within three glide rows. The applied stress is 15 MPa, the friction stress 7.5 MPa and the multiplication rule is of autocatalytic type. After Lepinoux (unpublished result), (a) Small multiplication rate: the glide band does not expand, (b) Large multiplication rate: band propagation takes place.
tiplication factor is large, the configurations are less well-organized, as shown in Fig. 4-24 b, but slip activity can spread out. The strain rate per row can, then, be plotted as a function of the row number, i.e., of a coordinate parallel to the specimen axis, and its evolution with time can be investigated. This is done in Fig. 4-25, where a constant strain hardening rate has artificially been introduced to limit slip activity behind the propagating front. The rate at which the thickness of the active slip lamella increases, vb, is found to be of the form: vh = vd(l-K*/Km),
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where vd is the velocity of dislocations at the front of the propagating band, Km is the multiplication coefficient and K^ its critical value. No propagation occurs if
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stresses of dynamic pile-ups (Kocks, 1981). To simulate this situation, a random distribution of edge dislocations was introduced all through the simulated crystal. To simulate pinning by interstitial impurities, the stress vs. velocity relationship was such that some overstress was needed before any motion could occur. Several active germs are initiated at positions where internal stresses are favourable, but not all of them are able to expand further. For those which do, the band velocity follows a relation similar to that of Eq. (4-34). These results are probably related to the so-called "Liiders-like" slip propagation observed in copper alloys during stage I (Neuhauser, 1990) and the following conclusions can be drawn. There is a transition between static slip patterning and slip propagation. This transition is controlled by multiplication mechanisms, and probably by other factors, like the applied stress and temperature. The band velocity is proportional to the velocity of dislocations at
4.4 Approach by Numerical Simulations
179
4.4.3.4 Patterning of Immobile Dislocations
8
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Figure 4-25. Slip propagation by cross-slip of screw dislocations. The strain rate within each glide row, s, is plotted as a function of row number, N. Step numbers are circled and increase from bottom to top. The initial configuration consists of a random distribution within an active slice 0.3 jam thick. The active slip band expands with an approximatively constant velocity. After Lepinoux (1987).
the front, for edge as well as for screw ensembles. It is not, possible, however, to discuss the respective roles of the two competing mechanisms, cross-slip and the long-range stresses emanating from dynamic pile-ups.
Although two-dimensional simulations appear more suited for the study of slip patterning, they can yield some information about the organization of immobile dislocations. Here we will not discuss the simulation of PSBs (Lepinoux, 1987; Amodeo, 1988; Ghoniem and Amodeo, 1990), which are perhaps not too convincing since the dynamic behavior involves, in that case, a strong coupling between edge and screw segments (Mughrabi, 1980). An interesting quantity to study is the cut-off distance for long-range interactions, in particular in annealed structures like the one of Fig. 4-21 where self-screening has certainly occurred. Figure 4-26 has been obtained as follows. For each (screw) dislocation of an annealed configuration, the elastic energy of interaction, £(r), with its neighbors situated within a circle of radius r has been computed. The average of E(r) over the whole configuration E(r% is then plotted as a function of r. Figure 4-26 shows that this average value decreases and saturates beyond a cut-off value of the order of 5 Q ~1/2 ( « 5 jim in the present case). It is emphasized that the configuration on which this estimate has been performed was obtained by computing long-range stresses up to
Figure 4-26. Average elastic interaction energy, E(r) vs. reduced interaction distance n = rg112. The average was made on an annealed configuration of screw dislocations. Notice the cut-off at Rc = 5Q'1/2.
180
4 Dislocation Patterning
values of 10 jim, so that the effective cut-off determined on Fig. 4-26 appeared spontaneously and does not result from the imposed truncation. The value 5Q~1/2 is obviously too large to justify a treatment of the internal stress through a local expansion as in the reaction-diffusion approach or in Holt's model. Figure 4-27, which is reproduced from Gulluoglou and coworkers (1989), illustrates what happens when elastic interactions are truncated at a too small value, either in simulations or in theoretical mod-
(a)
(b)
Figure 4-27. Two annealed configurations of edge dislocations after Gulluoglu et al. (1989). (Courtesy of Pergamon Press.) (a) Without truncation of the long range stresses (compare with Fig. 4-21 b). (b) With a cut-off distance of 0.5 urn.
els. Figure 4-27 a, is an annealed configuration of edge dislocations obtained without any truncation of the long-range stress fields, the latter being calculated through a summation on the simulated cell and on an infinite set of periodic replicas. Although the densities are different, the result is rather similar to that of Fig. 4-21 b, with a loose organization of multipolar clusters. When a cut-off radius of 0.5 }im is introduced, a periodic pattern of sharp walls is formed (Fig. 4-27 b). Dislocations segregate into tilt walls of alternating sign and the period (twice the wall distance) is equal to 0.5 jam, i.e., precisely the distance at which elastic interactions are truncated. Under an applied stress, walls are formed which have a dipolar character (Kubin and Lepinoux, 1988) and this is due to multiplication effects: a wall incorporates mobile dislocations which change sign when going from one side to the other. Such walls are only transiently stable, except if they reach such a height that they intercept the upper and lower boundaries of the simulation, as in Fig. 4-27 b. Then, they are stabilized by the periodic boundary conditions. Thus, although no periodic wall structures are expected in easy glide, there seem to be two artificial manners to produce them: by truncating long-range stresses, by introducing effects which "pin" the evolving walls and increase their stability. Such mechanisms are thought to be available in three dimensions; reaction junctions in the walls and the triple points between different types of walls within a cell structure should act as stabilizing factors. A structure of sharp and periodic walls has also been obtained by Lubarda et al. (1992) in a simulation dealing with a fixed number of edge dislocations, whose equilibrium positions are defined by finding a configuration which minimizes the potential energy.
4.4 Approach by Numerical Simulations
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Finally, bidimensional cell structures have been obtained by Ghoniem and Amodeo (1989 b, 1990) in the case of two sets of infinite dislocations of edge character with orthogonal glide directions. Intersecting dislocations can become sessile if suitable conditions for making a junction are met. A stress of 100 MPa is applied in duplex condition and recovery by climb is allowed. Figure 4-28 shows the initial random configuration and the successive steps of the formation of dislocation cells. The cell size is about 0.5 \xm and the boundaries of the cells are essentially composed of dipoles and reaction junctions. As shown by Ghoniem and Amodeo (1990), the occurrence of a cellular configuration can also be detected by monitoring the evolu-
Figure 4-28. Successive
steps of the simulated formation of a cell structure in iron at 600 °C, under an applied stress of 100 MPa. Edge dislocations are mobile in two slip planes with orthogonal slip directions. After Ghoniem and Amodeo (1990). (Reprinted by permission of Kluwer Academic Publishers.)
tion with time of the internal and effective stresses. 4.4.4 Three-Dimensional Simulation 4.4.4.1 Method
While two-dimensional simulations constituted a necessary preliminary step, three-dimensional simulations will bring more quantitative information and a deeper understanding of patterning and of its relation to macroscopic behavior. These simulations incorporate slip crystallography and treat stresses and strains as tensorial quantities. Conceptually, the situation is simpler than in two dimensions since all elementary processes either occur naturally within the simulation or can be repro-
182
4 Dislocation Patterning
duced within reasonable approximations. On the other hand the practice is fairly complicated and the development is limited in the present stage by the capacities, in speed more than in memory, of the available computers. At present, the strains which can be attained are of the order of one, perhaps a few, percent but the feasibility of such numerical simulations has been demonstrated (Kubin and Canova, 1990; Kubin et al., 1992; Canova et al., 1992). The principles of the three-dimensional simulation are as follows. Dislocation loops or segments of arbitrary shapes are fragmented into a succession of edge and screw segments. These segments perform elementary jumps on a three-dimensional lattice which has the symmetries of the crystal lattice of the material considered. Figure 4-29 shows the case of a f.c.c. lattice: <110> segments joining a summit to the centre of an adjacent face define the possible sites for screw segments and <112> segments which join a summit to the centre of an opposite face define the possible sites for edge segments. The elementary translations of these segments are also defined in this lattice. For instance in a {111} slip plane, the elementary translation of an edge segment is a <110> translation and that of a screw segment a <112> translation. As illustrated by Fig. 4-30, the continuity of an expanding dislocation loop is maintained by creating two new segments every time one elementary segment moves. The elementary length scale is the smallest of the two critical distances for the annihilation of dipoles (cf. Sec. 4.4.2), i.e., the critical annihilation distance for edge dipoles, ye. The parameter of the three-dimensional lattice, a, is defined through the requirement that the elementary translation along <111), d m , is equal to 2ye. With yQ«1.6 nm (Essmann and Mughrabi, 1979), we have dllt = a/^/3 = 2y e , hence
[001]
[ 001)
Figure 4-29. The f.c.c. lattice underlying the three-dimensional simulation and a portion of dislocation loop (bold lines) lying in the (111) slip plane. The screw segment is along [101] and the edge one along [121].
a = 5.54nm. It should be noted that this critical annihilation distance does not depend on applied stress and remains constant during a simulation. This is not the case for the critical annihilation distance for screw dislocations which does not appear here as a scaling factor. This quantity
i
b
Figure 4-30. The expansion of a gliding dislocation loop made of elementary screw and edge segments. Every time one segment of given character moves, here a screw segment, two new segments of the other character are created to maintain the continuity of the loop.
4.4 Approach by Numerical Simulations
is reproduced by the simulation as a byproduct of cross-slip processes. A three-dimensional net of elementary lattices is built, forming a cube of side 10|im which contains about 6 x 109 elementary cells. Actually, the simulation does not follow all the possible dislocation sites but only those, in much smaller number, which are occupied. No periodic boundary conditions are imposed and no image forces are introduced. The surfaces of the simulated specimen are subject to prescribed external constraints, in general constant stresses or constant stress or strain rates. To compute long-range interactions, it is necessary to make use of the stress fields of segments of finite length, which differ from those of infinite segments. This is conveniently done by using reasonably compact formulae (Devincre and Condat, 1992) derived from the work of de Wit (1967 a, b). Although this leads to some numerical complexity, an interesting consequence is that dislocation loops have a line tension via the mutual interaction of the various edge and screw segments of which they are formed. According to dislocation theory, the shear stress x needed to maintain a dislocation loop of diameter D in equilibrium is x = 2F/D, where the line energy, T, is derived from Eq. (4-2) with Rc = D/2. Figure 4-31 shows that the relation between t and D measured on simulated single loops is in good agreement with this theoretical prediction. A rigorous treatment of the line tension of discretized loops is presented by Devincre and Condat (1992). It is shown that, in addition to dislocation interactions, a force deriving from the modification in self-energies has to be taken into account. Local interactions are treated like in the two-dimensional simulations except that three-dimensional mechanisms can be simulated in a more consistent manner. The
183
Figure 4-31. The stress T needed to maintain a dislocation loop of diameter D in equilibrium vs. D. The full line was calculated as indicated in the text and the solid circles represent measurements on simulated loops. After Kubin and Canova (1990).
initial configurations are reminiscent of a Frank net, i.e., they consist of an array of segments pinned at their ends, in density g, with their lengths statistically distributed around the value Q ~1/2. At the beginning of plastic flow, some of these segments pass the critical bowed-out configuration and act as dislocation sources. Other sources spontaneously form during further straining, the pinning points being provided either by edge segments left on the cross-slip planes or by sessile junctions. When two attractive dislocations of different Burgers vectors cross each other, a sessile junction is formed if the reaction is energetically favorable. Other local interactions such as direct annihilation or trapping naturally occur within the simulation. The local stress is the sum of the local applied and internal stresses, minus a small friction stress ( 3 X 1 0 ~ 5 J I ^ 1 . 7 M P a for copper). The long-range interactions of the considered elementary segment are computed up to a distance of 5jim. From the results obtained in two dimensions (cf. Sec. 4.4.3.4 and Fig. 4-26), this cut-off distance is sufficient to avoid artificial forms of organization. The local effective stress tensor
184
4 Dislocation Patterning
being computed, the velocity of the segment considered can be defined. The stress vs. velocity law describes the motion of a dislocation in a perfect lattice; it is of viscous type and it includes a drag coefficient (5 x 10" 5 P a s " 1 for copper at room temperature) which takes into account the interactions with electrons and phonons. For each segment a maximum free-path of the order of 1 jum is allowed. It is divided into several substeps during which the possibility for further motion is reconsidered. Since dislocations can move at velocities as high as 10 2 ms~ 1 , this defines an elementary time scale of the order of 10" 7 s. Once every dislocation has moved to its new position, the corresponding free-flight and waiting times (for those dislocations which are blocked) are computed. The thermally activated nature of crossslip and climb is then accounted for by introducing jump probabilities based on local stresses. For cross-slip, this probability is fitted in such a way that its value is 1 during a time step when the local internal stress, resolved in the slip plane, is equal to % , the critical stress for the onset of stage III. Tm is typically of the order of 30 Mpa for copper and a few MPa for aluminum at room temperature. Finally, short-range order and its destruction by moving dislocations can be simulated through an alloy friction force which decreases locally with the passage of mobile dislocations. Precipitates are represented by zones with high local friction stress. According to the value of the latter, either cutting or Orowan by-passing occur. The number of segments increases with increasing strain as dislocations bow-out and multiply. This causes an increase in the number of interactions to be computed and a decrease in numerical efficiency, so that the maximum strain reached to date is about 10" 2 . At such strains the internal
stress is still a small fraction of the applied stress and the effective cut-off radius is slightly below the imposed one. A critical step is expected to be reached at some higher strain, as the effective cut-off distance for long-range elastic interactions should substantially decrease, indicating the presence of well-formed patterns and allowing for a reduction of the imposed cut-off radius to its effective value. 4.4.4.2 Examples of Patterns Figure 4-32 shows the results of a simulation of multiple glide in copper at room temperature. A constant stress of 20 MPa was applied along [011] until a strain of £ = 4 x l O " 3 was obtained and the total density increased from an initial value of lO^mT 2 to 6 x l 0 1 2 m " 2 . Four slip systems were active and Fig. 4-32 a shows a view of the simulated crystal where only dislocations of one of the active slip systems are imaged. The view is taken parallel to the screw direction of this system, so that the slip planes are seen edge-on. A slip pattern has developed, from the initial random distribution of sources. A few crossslip events are imaged through the edge segments connecting the "glide bands". Figure 4-32 b shows a "thin slice" of thickness 3 jim, cut parallel to one of the active slip planes. Although a cell structure has not yet formed at such small strains, there are already regions of high dislocation density and a kind of layered structure has appeared. Figure 4-32 c shows the distribution of reaction junctions in the same conditions as Fig. 4-32 b. The comparison of these last two figures shows that the regions of larger dislocation density are also those which contain the largest density of junctions. This suggests a possible scenario for the formation of dislocation patterns. Initially, slip patterns form on
4.5 Appendix
185
each system and the encounter of mobile dislocations of different Burgers vectors leads to the formation of a pattern of junctions. Mobile dislocations further accumulate around the junctions, as in Kocks' model (cf. Sec. 4.2.2), forming germs for the dislocation-dense regions. The arrangement of forest dislocations may, thus, inherit its structure from that of slip patterns, via the arrangement of the junctions. At mesoscopic and macroscopic scales, several checks have been performed which indicate that the global properties measured on the simulation are coherent with experimental results. For instance, (i) the flow stress increases as the square root of the dislocation density, (ii) the slopes of the simulated stress-strain curves exhibit reasonable values and the expected orientation dependence in multiple slip and (iii) in multiple slip, the energy stored is about 20% of the mechanical work.
4.5 Appendix Linear stability analysis of the set of reaction-diffusion forms, Eqs. (4-30) and (431), after Schiller (1989; see also Misbah, 1988). The homogeneous steady state of the system, g i0 , gm0 is obtained by setting 9@m/8t = 0. The solutions are: ) = 0 , Qmo = P/PQio Figure 4-32. Simulated dislocation configurations at £ = 4 x l O ~ 3 i n a copper single crystal at room temperature. A constant stress of 20 MPa was applied along the [Oil] axis. The orientation of the simulated crystal is indicated by its three [100] directions of length 10 urn. (a) [101] view of one active slip system, parallel to its screw direction, showing cross-slip events, (b) {111} "thin slice" of thickness 3 jam showing dislocation density modulations, (c) Same as (b) with only the reaction junctions in contrast.
(4-35)
The stability of this solution is tested by introducing temporal and spatial deviations from this steady state in the form 10
cot iqx
(4~36)
where the constants 5@i0 and 5^ m0 are small.
186
4 Dislocation Patterning
The basic set of equations is linearized near the steady state. For instance, omitting the exponential forms and taking into account Eq. (4-35), the cubic term yields: *- q O+
P QiO
2
P Q\0 QmO
The linearized system eventually writes i0
= Wo + P-Q2D{]
bQi0 + pQ?0 dQm0 (4-37)
= -p5^i0
2
- [Pgf0 + q Dm] (4-38)
where gro = (dg/dQi)o is negative (cf. for a justification, Walgraef, 1987). This linear system has non-trivial solutions provided that its discriminant vanishes, which gives the condition
-g'oPQ?o = 0
(4-39)
For small values of the control parameter (p~0) and with help of the condition D{<^Dm, one can show that the discriminant of this characteristic equation is always positive, so that the two roots co1 and co2 are real. Whatever the value of q, the sum of these roots is negative and their product is positive. The two solutions are real and negative and, as a consequence, all fluctuations [Eq. (4-36)] are exponentially damped out with time. Therefore, the uniform steady-state solution is stable. For non-zero values of the control parameter, critical conditions are reached when the real part of one solution becomes positive for a particular value of the wavenumber q, while the real part of the other solution remains negative for all q's. This is illustrated by Fig. 4-33 which shows that with increasing values of the control parameter the uniform steady state becomes unstable beyong a critical value pc and for
Figure 4-33. Schematic dependence of the real part of the amplification factor co on the wave number q (dispersion relation). When the control parameter p is larger than the critical value pc, a band of unstable modes (q1, q2) appears.
a critical wave number qc, leading to unstable modes within a band of wave numbers (qx,q2). It can be seen from Eq. (4-39) that a purely temporal instability (# = 0) can occur when —g'o—pc + PQfo = 0' Then, the sum of the real parts of the roots vanishes and, since the discriminant of the characteristic equation is negative, the roots become purely imaginary. This oscillatory instability, the so-called Hopf bifurcation, occurs when the control parameter reaches the critical value PHopf = - 0 0 + ^ 0
(4-40)
The spatio-temporal or Turing instability depicted in Fig. 4-33 occurs when one of the roots of the characteristic equation is such that 1
=0
(4-41)
while the other root has a negative real part. Assuming for simplicity that the two roots are real, the condition cOi=0 is equivalent to P = co1 co2 = 0, or: (4-42)
4.7 References
187
Expressing the characteristic equation in 4.7 References the form CL>2 + SCL> + P = O, we get (dco/dq)1 = (co i dS/dq - dP/Qq)/(2 co ± - S). Hence, with a>1 = 09 the second condition in Eq. Aifantis, E. C. (1985): in: Dislocations in Solids, Yamada Science foundation. Tokyo: University of (4-41) is equivalent to 9P/9g = 0, or: Tokyo Press, pp. 41-47. (4-43)
From the last two expressions we obtain the critical condition for the onset of the patterning instability, p = pc, and the corresponding wave number qc: i°4
'
(4-44)
The Turing instability is, in the present case, reached before the Hopf instability (Pc < Pnopf) because of the large value of the ratio Dm/D{. The initial uniform density is, thus, expected to become unstable, leading to a pattern of wavelength Xc = 2n/qc whose value is defined in terms of the rate constants and the diffusion coefficients. To obtain more information about the developed pattern, it is necessary to work out solutions valid beyond the bifurcation point. In this domain the linear approximation breaks down but a weakly non-linear analysis can be performed (Misbah, 1988).
4,6 Acknowledgements It is a pleasure to thank all those who contributed to this chapter either through stimulating discussions and/or by authorizing reproduction of their published or unpublished works. Particular gratitude goes to E. C. Aifantis, Y. Brechet, N. M. Ghoniem, P. Hahner, H. Kirchner, J. Kratochvil, H. Mughrabi, I Lepinoux, H. Neuhauser and G. Saada.
Aifantis, E. C. (1986), Mat. Sci. Eng. 81, 563-574. Aifantis, E. C. (1987), in: Constitutive Relations and their Physical Basis, 8th Riso Int. Symposium: Anderson et al. (Eds.). Roskilde, Dk: Riso Nat. Lab., pp. 205-213. Aifantis, E. C. (1988), Solid State Phenomena Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 397-406. Aifantis, E. C. (1990), in: Patterns, Defects and Materials Instabilities, NATO ASI Series: Walgraef, D., Ghoniem, N. M. (Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 221-239. Alexander, H., Haasen, P. (1968), Solid State Physics 22, 27. Amodeo, R. J. (1988), Thesis, University of California, Los Angeles. Ananthakrishna, G. (1988), Solid State Phenomena Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 357-368. Ananthakrishna, G., Valsakumar, M. C. (1983), /. Phys. D (Appl. Phys.) 15, L171-L175. Basinski, Z. S. (1974), Scripta Met. 8, 1301. Basinski, S. X, Basinski, Z. S. (1979), in: Dislocation in Solids, Vol. 4: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, pp. 261-362. Bocek, M. (1988 a), Solid State Phenomena Vol. 3 and 4. Aedermannsdrof, CH: Trans Tech Publications, pp. 369-376. Bocek, M. (1988 b), Z. Metallkde 79, 132-144. Bottani, C. E. (1989), Europhys. Lett. 9, 785-790. Brechet, Y. (1987), Doctorate Thesis, University of Grenoble. Burmeister, H.-J., Hermann, H. (1979), Phys. Stat.
Sol. (a) 54, K59-K61. Canova, G., Kubin, L. P., Brechet, Y. (1992), in: Mecamat' 91, A.A. Balkema Publishers, in press. Coleman, B. D., Hodgdon, M. L. (1985), Archive Rat. Mech. Anal. 90,219. Cottrell, A. H. (1958), Dislocations and Strength of Crystals, New York: J. Wiley and Sons. Devincre, B., Condat, M. (1992), to be published. Dewel, G., Borckmans, P. (1990), in: Patterns, Defects and Materials Instabilities: Walgraef, D., Ghoniem, N. M. (Eds.). NATO ASI Series. Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 63-72. Differt, K., Essmann, U. (1988), Scripta Met. 22, 1337-1342. Embury, J. D. (1971), in: Strengthening Methods in Crystals: Kelly, A., Nicholson, R. B. (Eds.). London: Applied Science Publishers, pp. 331-402. Essmann, U., Mughrabi, H. (1979), Phil. Mag. A, 40, 731-756.
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Estrin, Y. (1988), Solid State Phenomena Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 417-428. Estrin, Y, Kubin, L. P. (1986), Acta Met. 34, 24552464. Estrin, Y, Kubin, L. P. (1991), in: 5th Int. Symp. on Plasticity of Metals and Alloys, Prague, Aug. 1990, Mat. Sci. Eng. A137, 125-134. Estrin, Y, Abromeit, Ch., Aifantis, E. C. (1990), Phys. Stat. Sol. (b) 157, 117-128. Foreman, A. J. E., Makin, J. E. (1966), Phil. Mag. 14, 911. Foreman, A. J. E., Makin, J. E. (1967), Can. J. Phys. 45, 511. Franek, A., Kalus, R., Kratochvil, J. (1991), Phil. Mag. A64, 497-511. Fressengeas, C , Jean-Claude, V. (1991), in: Non Linear Phenomena in Materials Science II: Kubin, L., Martin, G. (Eds.). Aedermannsdorf, CH: Trans Tech Publications, to be published. Friedel, J. (1979), in: Dislocations in Solids Vol. 1: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, pp. 1-32. Gerold, V, Karnthaler, H. P. (1989), Acta Met. 37, 2177-2183. Ghoniem, N. M., Amodeo, R. J. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 377-388. Ghoniem, N. M., Amodeo, R. J. (1989a), Phys. Rev. B41, 6958-6967. Ghoniem, N. M., Amodeo, R. J. (1989b), Phys. Rev. B41, 6968-6976. Ghoniem, N. M., Amodeo, R. J. (1990), in: Patterns, Defects and Materials Instabilities, NATO ASI Series: Walgraef, D., Ghoniem, N. M. (Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 303-329. Gil Sevillano, X, Aernoudt, E. (1987), Mat. Sci. Eng. 86, 35. Gil Sevillano, X, Bouchaud, E., Kubin, L. P. (1991), Scripta Met. 25, 355-360. Grosbras, M. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 17-28. Gulluoglu, A. N., Srolovitz, D. X, LeSar, R., Lomdahl, P. S. (1989), Scripta Met. 23, 1347-1352. Gulluoglu, A. N., Srolovitz, D. X, LeSar, R., Lomdahl, P. S. (1990), in: Simulation and Theory of Evolving Microstructures: Anderson, M. P., Rollett, A. D. (Eds.). Warrendale, PA: The Minerals, Metall. and Mat. Soc, in press. Hahner, P. (1991), in: Non Linear Phenomena in Materials Science II: Kubin, L., Martin, G. (Eds.). Aedermannsdorf, CH: Trans Tech Publications, to be published. Hansen, N., Kuhlmann-Wilsdorf, D. (1986), Mat. Sci. Eng. 81, 141-161. Hasegawa, T, Yakou, T, Kocks, U. F. (1986), Mat. Sci. Eng. 81, 189-199.
Hirth, X P., Lothe, X (1968), Theory of Dislocations. New York: MacGraw-Hill, p. 153. Holt, D. L. (1970), J. Appl. Phys. 41, 3197-3201. Hornbogen, E. (1989), Intern. Mat. Reviews 34, 277296. D'Humieres, D., Lallemand, P. (1986), Physica 140A, 326-335. Jimenez, J. (1982), Master Thesis, ESII of San Sebastian, University of Navarra, Spain. Johnston, W. G., Gilman, X X (1959), /. Appl. Phys. 30, 129. Kleiser, T, Bocek, M. (1986), Z. Metallkde 77, 582587. Kocks, U. F. (1976), J. Eng. Mat. and Technology 98, 76-85. Kocks, U. F. (1981), in: Chalmers Anniversary Volume, Progress in Materials Science. Oxford: Pergamon Press, pp. 185-241. Kocks, U. F. (1985), in: Dislocations and Properties of Real Materials. London: The Institute of Physics, pp. 125-143. Kocks, U. F , Argon, A. S., Ashby, M. F. (1975), Thermodynamics and Kinetics of Slip, Progress in Materials Science, Vol. 19: Chalmers, B., Christian, X W, Massalski, T. B. (Eds.). Oxford: Pergamon Press. Kocks, U. F, Hasegawa, T, Scattergood, R. O. (1980), Scripta Met. 14, 449. Kosevich, A. M. (1979), in: Dislocations in Solids, Vol. 1: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, pp. 33-141. Kratochvil, X (1988 a), Czech. J. Phys. B38, 421-424. Kratochvil, X (1988 b), Rev. Phys. Appl. 23, 419-429. Kratochvil, X (1988 c), in: Basic Mechanisms in Fatigue of Metals: Lukas, P., Polak, X (Eds.). Prague: Academia Elsevier, pp. 15-25. Kratochvil, X (1989), in: Advances in Plasticity 1899: Khan, A. S., Tokuda, M. (Eds.). Oxford: Pergamon Press, pp. 565-568. Kratochvil, X (1990 a), Scripta Met. Mater. 24, 891894. Kratochvil, X (1990 b), Scripta Met. Mater. 24, 12251228. Kratochvil, X (1991), /. Mech. Behaviour of Materials, in press. Kratochvil, X, Libovicky, S. (1986), Scripta Met. 20, 1625-1630. Kratochvil, X, Orlova, A. (1990), Phil. Mag. A61, 281-290. Kubin, L. P., Canova, G. (1990), in: Electron Microscopy in Plasticity and Fracture Research of Materials: Messerschmidt, U. et al. (Eds.). Berlin: Akademie Verlag, pp. 23-32. Kubin, L. P., Estrin, Y (1990), Acta Met. 38, 697708. Kubin, L. P., Lepinoux, X (1988), in: Strength of Metals and Alloys (ICSMA 8) Vol. 1: Kettunen, P. O., Lepisto, T. K., Lehtonen, M. E. (Eds.). Oxford: Pergamon Press, pp. 35-59.
4.7 References
Kubin, L. P., Estrin, Y, Canova, G. (1990), in: Patterns, Defects and Materials Instabilities, NATO ASI Series: Walgraef, D., Ghoniem, N. M. (Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 277-302. Kubin, L. P., Canova, G., Condat, M., Devincre, B., Pontikis, V., Brechet, Y (1992), in: Non-Linear Phenomena in Materials Science, Vol. 2. Aedermannsdorf, CH: Trans Tech Publications. Kuhlmann-Wilsdorf, D. (1962), Trans. Metall Soc. AIME 224, 1047. Kuhlmann-Wilsdorf, D. (1988), in: Strength of Metals and Alloys (ICSMA 8), Vol. 1: Kettunen, P. O., Lepisto, T. K., Lehtonen, M. E. (Eds.). Oxford: Pergamon Press, pp. 221-226. Kuhlmann-Wilsdorf, D. (1989), Mat. Sci. Eng. A113, 1-41. Kuhlmann-Wilsdorf, D., van der Merwe, J. H. (1982), Mat. Sci. Eng. 35, 79. Lavrentev, F. F. (1980), Mat. Sci. Eng. 46, 191. Lepinoux, J. (1987), Doctoral Thesis, University of Poitiers. Lepinoux, J. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 389-396. Lepinoux, I, Kubin, L. P. (1987), Scripta Met. 21, 833-838. Louchet, R, Brechet, Y (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 335-346. Lubarda, V, Blume, I, Needleman, A. (1992), to be published. Luft, A. (1991), Progress in Materials Science 35, 97-204. Mader, S. (1957), Z. Physik 149, 73. Malygin, G. A. (1988), Sov. Phys. Solid State 30, 1828-1829. Malygin, G. A. (1989), Sov. Phys. Solid State 31, 96-98. Masing, G. (1923), Wissenschaftl. Veroffentl. a. d. Siemens-Konzern 3, 231—239. Misbah, C. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 29-56. Mughrabi, H. (1975), in: Constitutive Equations in Plasticity: Argon, A. S. (Ed.). Cambridge, MA: MIT Press, pp. 199-250. Mughrabi, H. (1980), in: Strength of Metals and Alloys (ICSMA 5), Vol. 3: Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1615-1638. Mughrabi, H. (1981), in: Continuum Models of Discrete Systems 4: Brulin, O., Hsieh, R. K. T. (Eds.). Amsterdam: North-Holland, pp. 241-257. Mughrabi, H. (1983), Acta Metall. 31, 1367-1379. Mughrabi, H. (1986), S. Afr. J. Phys. 9, 62-68. Mughrabi, H. (1987), Mat. Sci. Eng. 85, 15-31. Murphy, S. M. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 295-302.
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Nabarro, F. R. N. (1986), in: Strength of Metals and Alloys (ICSMA-CIRMA 7), Vol. 1: Me Queen, H. J. et al. (Eds.). Oxford: Pergamon Press, pp. 1667-1700. Nabarro, F. R. N., Basinski, Z. S., Holt, D. B. (1964), Adv. in Physics 13, 193. Neuhauser, H. (1980), in: Strength of Metals and Alloys (ICSMA 5), Vol. 3: Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1531-1550. Neuhauser, H. (1983), in: Dislocations in Solids, Vol. 6: Nabarro, F. R. N. (Ed.). Amsterdam: NorthHolland; pp. 319-440. Neuhauser, H. (1988a), Res. Mechanica23,113-135. Neuhauser, H. (1988 b), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 279-294. Neuhauser, H. (1990), in: Patterns, Defects and Materials Instabilities, NATO ASI Series: Walgraef, D., Ghoniem, N. M. (Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 241—276. Neumann, P. (1986), Mat. Sci. Eng. 81, 465-475. Nishioka, K., Ohsaka, K. (1978), Phil Mag. A37, 561-566. Olfe, I , Neuhauser, H. (1988), Phys. Stat. Sol. (a) 109, 149-160. Pedersen, O. B., Brown, L. M., Stobbs, W M. (1981), Acta Metall. 29, 1843. Prinz, F. B., Argon, A. S. (1984), Acta Metall. 32, 1021. Raj, S. V, Pharr, G. M. (1986), Mat. Sci. Eng. 81, 217-237. Saada, G. (1991), in: 5th Int. Symp. on Plasticity of Metals and Alloys, Prague, Aug. 1990, Mat. Sci. Eng. A137, 177-183. Schwab, A. (1990), Thesis A, Padagogische Hochschule, Dresden. Schwab, A., Burmeister, H. J. (1991), to be published. Seeger, A. (1955), Phil. Mag. 46, 1194. Seeger, A. (1988), in: Strength of Metals and Alloys (ICSMA 8), Vol. 1: Kettunen, P.O., Lepisto, T. K., Lehtonen, M. E. (Eds.). Oxford: Pergamon Press, pp. 463-468. Seeger, A., Frank, W. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 125-138. Schiller, C. (1989), Thesis, Universite Libre de Bruxelles (in English). Schiller, C , Walgraef, D. (1988), Acta Met. 36, 563574. Shinohara, K., Kitajima, S., Kutsuwada, M. (1986), Acta Met. 12, 2335-2342. Schwink, Ch. (1966), Phys. Stat. Sol. 18, 557-567. Sprusil, B., Hnilica, F. (1985), Czech. J. Phys. B35, 897-900. Staker, M. R., Holt, D. L. (1972), Acta Metall. 20, 569-579. Steck, B., Hesselbarth, H. W. (1991), in: Plasticity 91, Grenoble (France), Aug. 1991, to be published in Int. J. of Plasticity.
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Sumino, K. (1974), Mat. Sci. Eng. 13, 269-275. Takamura, I (1987), Trans. Jap. Inst. of Metals 28, 165-181. Takeuchi, S. (1973), J. Phys. Soc. Japan 35, 188-193. Triantafyllidis, N., Aifantis, E. C. (1986), J Elasticity 16, 225. Vladimirov, V. I., Kusov, A. A. (1976), Sov. Phys. Solid State 18, 886-892. Vladimirov, V. I., Pegel, B. (1973), Phys. Stat. Sol. (b) 56, K105-108. Walgraef, D. (1986), in: Mechanical Properties and Behaviour of Solids: Plastic Instabilities: Balakrishnan, V., Bottani, C. E. (Eds.). Singapore: World Scientific, pp. 354-395. Walgraef, D. (1988), Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications, pp. 77-96. Walgraef, D. (1990), in: Patterns, Defects and Materials Instabilities, NATO ASI Series: Walgraef, D., Ghoniem, N. M. (Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 73-87. Walgraef, D., Aifantis, E. C. (1985a), J Appl. Phys. 58, 688-691. Walgraef, D., Aifantis, E. C. (1985 b), Int. J. Eng. Sci. 23, 1351-1358. Walgraef, D., Aifantis, E. C. (1985c), Int. J. Eng. Sci. 23, 1359-1364. Walgraef, D., Aifantis, E. C. (1985d), Int. J. Eng. Sci. 23, 1365-1372. Walgraef, D., Aifantis, E. C. (1988), Res. Mechanica 23, 161. Walgraef, D., Schiller, C , Aifantis, E. C. (1987), in: Patterns, Defects and Microstructures in Nonequilibrium systems, NATO ASI Series: Walgraef, D. (Ed.). Dordrecht, Netherlands: Martinus Nijhoff Publishers.
Wilkens, M. (1969), Ada Met. 17, 1155-1159. Wilkens, M., Kronmiiller, H. (1975), Cryst. Latt. Defects 6, 41-49. de Wit, R. (1967a), Phys. Stat. Sol. 20, 567-573. de Wit, R. (1967b), Phys. Stat. Sol. 20, 575-580. Zbib, H. M., Aifantis, E. C. (1988 a), Scripta Met. 22, 703-708. Zbib, H. M., Aifantis, E. C. (1988 b), Scripta Met. 22, 1331-1336.
General Reading Haken, H. (1987), Advanced Synergetics, 2nd ed. Berlin: Springer Verlag. Kubin, L., Martin, G. (Eds.) (1988), Non Linear Phenomena in Materials Science - Solid State Phenomena, Vol. 3 and 4. Aedermannsdorf, CH: Trans Tech Publications. Kubin, L., Martin, G. (Eds.) (1992), Non-Linear Phenomena in Materials Science, Vol. 2. Aedermannsdorf, CH: Trans Tech Publications. Nicolis, G., Prigogine, I. (1977), Self-Organization in Nonequilibrium Systems. New York: J. Wiley and Sons. Walgraef, D. (Ed.) (1987), Patterns Defects and Microstructures in Nonequilibrium Systems, NATO ASI Series. Dordrecht, Netherlands: Martinus Nijhoff Publishers. Walgraef, D., Ghoniem, N. M. (Eds.), Patterns, Defects and Materials Instabilities, NATO ASI Series. Dordrecht, Netherlands: Kluwer Academic Publishers.
5 Solid Solution Strengthening Hartmut Neuhauser and Christoph Schwink
Institut fur Metallphysik und Nukleare Festkorperphysik, Technische Universitat, Braunschweig, Federal Republic of Germany
List of Symbols and Abbreviations 5.1 Introduction 5.2 Elementary Interactions and Threshold Stress for the Motion of a Single Dislocation in a Solid Solution 5.2.1 Lattice Modifications Induced by Solute Atoms 5.2.2 Resulting Elementary Interactions of a Single Dislocation with Solutes .. 5.2.2.1 General Effects 5.2.2.2 Special Effects in Different Lattices 5.2.2.3 Drag Processes on Moving Dislocations 5.2.2.4 Superposition of Various Effects 5.2.3 The Threshold Stress for the Motion of a Single Dislocation at Zero Temperature 5.2.3.1 True Point Obstacles Overcome by the Dislocation in Overdamped Motion 5.2.3.2 Solute Atom Obstacles at Zero Temperature (T = 0) 5.2.4 The Effect of Temperature on the Threshold Stress 5.2.4.1 Thermally Activated Overcoming of Fixed Obstacles 5.2.4.2 Diffusive Processes (Moving Obstacles) 5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of F.C.C. Alloys .. 5.3.1 Survey of the Macroscopic Behavior of F.C.C. Alloys 5.3.1.1 Stress-Strain Curves of Mono- and Polycrystals 5.3.1.2 Temperature and Concentration Dependence of the CRSS TO(T9C) 5.3.1.3 Strain Rate Dependence of the CRSS 5.3.2 Survey of Microscopic Observations 5.3.2.1 Structure and Formation of Slip Lines 5.3.2.2 Dislocation Structure 5.3.2.3 Dislocation Motion 5.3.3 Analysis of the CRSS for Dislocations Moving Independently 5.3.3.1 Analysis for 80 < T < 300 K (Overdamped Thermally Activated Motion of Single Dislocations) 5.3.3.2 Analysis for the Low Temperature Regime (T < 80 K) (Underdamped Dislocation Motion) 5.3.3.3 Analysis for Higher Temperatures (T > 300 K) (Mobile Solute Atoms) . . . 5.3.4 Effect of Correlated Dislocation Motion 5.3.4.1 Inhomogeneity of Deformation: The Active Slip Volume Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
193 196 196 196 198 198 199 200 201 201 201 202 204 205 206 207 208 208 209 210 211 211 212 213 214 214 217 218 223 223
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5.3.4.2 Moving Dislocation Groups 5.3.4.3 Slip Transfer Process 5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals 5.4.1 F.C.C. Alloys at Low Temperatures (T < 300 K) 5.4.1.1 Macroscopic Behavior 5.4.1.2 Evolution of Microstructure 5.4.1.3 Superposition of Solid Solution and Strain Hardening 5.4.2 F.C.C. Alloys at Elevated Temperatures (T > 300 K) 5.4.2.1 Static and Dynamic Strain Ageing 5.4.2.2 Static and Dynamic Recovery 5.4.3 H.C.P. Alloys 5.4.3.1 The Critical Resolved Shear Stress 5.4.3.2 Work Hardening and High Temperature Behavior 5.5 Solid Solution Effects in B.C.C. Alloys 5.5.1 Peculiarities of Plasticity of B.C.C. Structures 5.5.2 Solute Hardening and Softening of B.C.C. Crystals by Substitutional Atoms 5.5.3 Solute Hardening and Softening of B.C.C. Crystals by Interstitials 5.5.4 Models of Solid Solution Hardening and Softening in B.C.C. Crystals . . . 5.6 Solid Solution Effects in Nonmetallic Systems 5.6.1 Systems with Alkali Halide Structure 5.6.1.1 Dilute Solid Solutions (c < 1000 ppm) 5.6.1.2 Concentrated Solid Solutions 5.6.2 Systems with Diamond Cubic Structure 5.7 Acknowledgements 5.8 References
223 224 225 225 225 226 226 228 228 230 231 231 233 234 234 235 237 238 238 239 239 240 241 242 243
List of Symbols and Abbreviations
193
List of Symbols and Abbreviations a ak b B Be Br Bph c ch cm c0 cx d D Ew Ed f(tw) Fm /w / cr fd /p /s f° AG (T) AG0 AG00 k K Ka ky /a lh LF Zo Lo / Zloc ms n naB S t
lattice constant height of a kink magnitude of the Burgers vector viscous drag coefficient electron damping coefficient phonon radiation damping coefficient phonon scattering damping coefficient atomic solute concentration barrier concentration maximum concentration in the dislocation core region matrix concentration concentration in the stacking fault grain size diffusion constant interaction energy between solute atom and dislocation dielastic interaction energy of a dislocation a n d solute atom strain ageing function maximum interaction force between a dislocation and a single obstacle volume fraction of cell walls critical value of increase of the obstacle strength by strain ageing dielastic interaction force parelastic interaction force interaction force due to Suzuki segregation maximum force due to SRS activation free enthalpy activation free enthalpy at zero stress original obstacle strength before strain ageing Boltzmann constant bulk modulus effectivity factor for strain ageing Hall-Petch parameter active crystal length barrier distance along the dislocation line Friedel length length of the crystal average solute atom distance in the slip plane external extension rate local extension rate Schmidt orientation factor number of dislocations in a group number of active slip band bundles strain rate sensitivity time
194
5 Solid Solution Strengthening
T tc TD TK TL Tm 7^i tw t^h v V Va F eff Vo w Wd Wm Ws
absolute temperature characteristic time constant for solute redistribution Debye temperature critical temperature of T O ( T ) behavior for b.c.c. crystals dislocation line tension melting temperature temperature at the TO(T) maximum for f.c.c. alloys waiting time of a dislocation at an obstacle critical waiting time for dislocation recapture by solute cloud average dislocation velocity activation volume active slip volume effective activation volume total crystal volume range of interaction between obstacle and dislocation dilatation energy density free binding enthalpy of solute atoms in a cloud around the dislocation shear energy density
awf jS L y ys y y0 y true 3 s eL es eu (c) rj 9 6X 9C 9D 9max x \i v vD v0 £m £t £w
interaction parameter of forest dislocations in cell walls Labusch's parameter resolved shear strain stacking fault energy average shear strain rate preexponential factor in shear strain rate true shear strain rate size misfit parameter tensile strain Labusch effective interaction parameter Fleischer effective interaction parameter tensile strain extent of stage I misfit parameter for the shear modulus work hardening rate angle of turn of dislocation at extended obstacle critical breaking angle D e b y e temperature of a dislocation maximum slope of the stress-strain curve misfit parameter for the bulk modulus shear modulus transition rate of the dislocation across the obstacle (Arrhenius equation) D e b y e frequency attack frequency average mobile dislocation density average total dislocation density density of dislocations inside the cell walls
List of Symbols and Abbreviations
Qml gwf a ai
local mobile dislocation density forest density in the cell walls tensile stress stress parameter in the Hall-Petch relation yield stress resolved shear stress resolved shear stress a t the e n d of stage A (h.c.p.) critical m i n i m u m o r threshold stress (single point obstacle) stress contribution from dislocation interaction stress contribution from solute interaction reduced resolved shear stress m a x i m u m a n d m i n i m u m values of stress in serrated yielding athermal stress component critical resolved shear stress (CRSS) maximum of T 0 at TM CRSS stress at T = 0 effective stress, thermally activated component of x Friedel-Fleischer threshold stress CRSS at T = 0 calculated by Friedel statistics CRSS at T = 0 calculated by Mott-Labusch statistics atomic volume
b.c.c. CRSS f.c.c. h.c.p. LRO PLC SRO SRS TEM
body-centered cubic critical resolved shear stress face-centered cubic hexagonal close-packed long-range order Portevin - LeChatelier (effect) short-range order short-range segregation transmission electron microscopy
195
196
5 Solid Solution Strengthening
5.1 Introduction In this chapter we will give a review of the principles of strengthening by solving foreign atoms into the pure matrix. Solid solution strengthening is also important in dispersion strengthened materials as the matrix contribution (cf. Chap. 7 by Reppich in this Volume). We will start out in Sec. 5.2 with elementary interactions of solute atoms and dislocations in situations where their nucleation and motion is responsible for plastic flow. In particular the motion of dislocations across solute obstacles, supported by thermal activation and modified by the possible mobility of the solutes at elevated temperatures will be considered in some detail. These principles will then be applied to the explanation of the critical resolved shear stress T 0 of f.c.c. alloys (Sec. 5.3) whose dependence on various deformation conditions has been studied most widely in the literature. In particular we show for one example a complete analysis of T 0 measurements which permits conclusions on the nature of the solute obstacles. Nevertheless, there are still unsolved or only partly solved problems in this field, mainly connected with the analysis for higher temperatures and with the correlated motion of dislocations. Section 5.4 extends the considerations to the more practically interesting case when solid solution hardening is superimposed on strain hardening produced by increasing deformation in the densely packed face-centred cubic (f.c.c.) and hexagonal close packed (h.c.p.) structures. This superposition has recently attracted most interest, after the basic understanding of each single process has been advanced sufficiently. The remaining Sees. 5.5 and 5.6 deal with solid solution effects in the bodycentred cubic (b.c.c.) lattice and in nonmetallic systems (such as alkali halides and
semiconductors), emphasizing the few differing and many common features discussed before. As the field of solid solution hardening has been covered earlier in several extended reviews (Haasen, 1965, 1979, 1983; Kocks, 1985 a; Nabarro, 1985 a; Suzuki, 1979) we will refer in particular to citations of more recent articles omitting many classical original references which can be found in the reviews mentioned before and in standard text books.
5.2 Elementary Interactions and Threshold Stress for the Motion of a Single Dislocation in a Solid Solution The basic problem of plasticity of a solid solution at ordinary temperatures can be traced back to the interactions between solute atoms and the dislocations. These interactions arise from local solute-induced changes of the lattice which are felt energetically by the distortion field of the dislocation. Therefore we start out with a list of the most important reasons for lattice changes. Then the resulting interaction forces (or energies) between a dislocation and solute atoms are inferred and an estimate for some examples is given. It should be noted, however, that several of these interactions are interrelated and superposed on each other in reality. 5.2.1 Lattice Modifications Induced by Solute Atoms
Several kinds of local modifications of the lattice induced by single solute atoms occur in a similar fashion for all types of lattices and materials. These effects are closely interrelated as they are a consequence of electronic binding potentials.
197
5.2 Elementary Interactions and Threshold Stress
Table 5-1. (a) Elastic interaction parameters for Cu alloys at room temperature*. 1 da a dc
Solute
in % Al Ge Mn Ni Si Zn
6.7 + 0.2 9.2 + 0.2 10.5 + 0.3 -2.9 + 0.2 1.8 + 0.2 6.0 + 0.2
C
Ref.
lim
in at.% a b
10 10 15 30 10 15
a b a b c a b a b a b
1 d/i /i dc
n-
Ref.
poly (p)-crystals single (s)-crystals
P
0.58 1.2 0.55 -0.60 0.70 0.48
d
e
d
e
d
f
d
g
d
e
d
d
0.68 2.0 0.65 + 0.02 -0.63 + 0.02 1.2 0.51+0.02
s
a
Pearson (1958); b Pearson (1964); c Pearson (1967); d Hopkin et al. (1970); e Neighbours and Smith (1954); Waldorf, (1960); g Schmunk and Smith (1960). * 3 and r\ values have been re-evaluated from the data on lattice spacings a and elastic constants {with ^ = [c44 (cll — c12)/2]1/2; cik are the elastic stiffness constants}, which are the most reliable ones to our knowledge. Limits of error are indicated only where it seems fully justified. This is possible for r\ in just a few cases. clim is the approximate limiting solute concentration for linear variation of 3. Only data below clim have been used for <5 and r\. f
Table 5-1. (b) Elastic interaction parameters for Fe alloys at room temperature. Solute
3 in %
-n
B P Si V Al Mo
-7.7 -3.34 -2.3 3.9 6.3 10.3
1.84 1.31 1.39 -0.12 1.32 2.27
a
Ref. a b a a a a
(2) Modulus Change
Suzuki (1979); b Hattendorf (1986).
Nevertheless the following separation is common and useful to indicate their relative importance (cf. Haasen, 1979; Schwink, 1988): (1) Dilatation The size difference of matrix and solute atoms is characterized in the isotropic case by the size misfit parameter Ida adc
f.c.c. and b.c.c. substitutional solutes). Recent ab initio calculations (Rupp and Gilmore, 1989), however, question the assumed isotropy, which also does not apply for h.c.p. structures with axis ratio # 1 , for interstitial solutes in b.c.c. structures and for concentrated f.c.c. substitutional solid solutions [cf. Sec. 5.2.2.1, (li, P)].
(5-1)
(a is the lattice constant, c the atomic solute concentration; cf. Table 5-1 for some
The modified binding forces around the foreign atoms are characterized by the "modulus misfit" parameter ldfi - —
fi dc
J
and
1 dK x=— — K dc
(5-2)
for the shear modulus \i and the bulk modulus K, respectively. [See Table 5-1, and Masudo-Jindo and Terakura (1989) for recent ab initio electronic structure calculations.] (3) Atomic Ordering or Segregation This occurs according to the minimum of the configurational free enthalpy with
198
5 Solid Solution Strengthening
negative or positive "exchange energy", respectively [cf. the recent considerations and simulations by Ackland and Vitek (1990)]. Thus, if the concentration is high enough, a tendency to form larger obstacles than single solute atoms is expected. (4) Chemical Effects They summarize the effects of segregation of solutes in the stacking fault ribbon between partial dislocations (Suzuki effect). For modern treatments cf. Simon et al. (1981), Simon and Papon (1984) and Shinodaetal. (1987). (5) Additional Changes of the Local Electronic Structure Apart from the binding forces mentioned above, there are (i) in metals: long-range Friedel charge oscillations due to foreign atoms of different valency leaving a negatively charged dislocation core; (ii) in ionic crystals: space charges and tetragonal distortions resulting from pairs of foreign atom and vacancy in case of different valency of impurity and host atoms; (iii) in semiconductor crystals: changes of the electronic structure and the effective Fermi level by impurities. 5.2.2 Resulting Elementary Interactions of a Single Dislocation with Solutes 5.2.2.1 General Effects (1) Dilatational Effects (i) Parelastic interaction: Usually, the most important part of the interaction energy arises from the strain field of the dislocation and the locally modified lattice by the size misfit [Sec. 5.2.1 (1)]. The interaction is called parelastic because the distor-
tion is present already without the dislocation's stress (strain) field. (oc) In case of spherical distortions a maximum force of /max (f.c.c.) = jub2 8
(5-3)
is obtained (cf. Haasen, 1979) for an edge dislocation, which is in the order of l x l O ~ 1 0 N . For screw dislocations just a higher order effect is expected, which is of similar order as the dielastic interaction (see below), cf., however, Sec. 5.2.2.2 (1). (p) For tetragonal distortions around interstitials in b.c.c, around atom pairs in substitutional f.c.c. alloys, the calculation is more involved because of the tensorial deformation field of the defect (e.g., Saxlova, 1969; Frank, 1967 a, b; Barnett and Nix, 1973; Parameswaran and Bapna, 1980). The interaction force amounts typically to 5 x 10 ~ 10 N. (ii) Diffusive interactions: The longrange stress field of the dislocation tends to induce a redistribution of the solute atoms in a kind of "cloud" around the dislocation at high enough temperatures by diffusive atomic motions. Such diffusive interactions will contribute markedly if the concentration of the solutes is not too low. Although all of the interaction effects shown below may be involved, the main effects result from the parelastic interaction discussed above, forming "Cottrell clouds" with spherical defects, or clouds with elastic dipoles (Zener pairs in substitutional, Snoek pairs in interstitial solid solutions). (2) Dielastic Interaction The dielastic interaction energy arises from the modulus misfit [Sec. 5.2.1 (2)] because the strain field of the dislocation is proportional to the shear modulus. The
5.2 Elementary Interactions and Threshold Stress
change in the dielastic interaction energy of the dislocation in the defected crystal, Ed is, using Eq. (5-2) and the variations of energy densities by shears Ws and dilatations Wd
Q is the atomic volume. Taking Cu-Ge as an example, the dielastic interaction for an edge dislocation gives a maximum force /^ ax (edge) = aju6 2 |f/|«0.27xl(T 1 0 N, and for a screw /^ ax (screw) « 0.21 x 10" 1 0 N, taking rj and % for polycrystals (Saxl, 1964), a = 16 for edge, a = 3 for screw dislocation. (3) Interaction by Short-Range Order and Short-Range Segregation In the strain field of a dislocation shortrange order (SRO) or short-range segregation (SRS) [Sec. 5.2.1 (3)] will occur more readily than in the ideal lattice because of a higher driving force on thermal equilibrium. The corresponding effects of stress induced order [Sec. 5.2.2.1. (l,ii)] produce a frictional force to a moving dislocation. The maximum force per unit length required to pull the dislocation free from its SRO or SRS atmosphere can be roughly estimated (Nabarro, 1967, p. 420) to be /°^/ ai uZ?/2, where fJ2 is the strain produced by ordering or segregation an atomic fraction / a of solute atoms. The importance of this interaction will increase with temperature. Contrary to the parelastic and dielastic interactions, the cutting of a dislocation through SRO or SRS obstacles at lower temperatures is an energy storing interaction, and the interaction force with a following dislocation moving on the same slip plane will be strongly reduced (e.g., Biichner and Pitsch, 1985; Schwander etal., 1992).
199
(4) Chemical Interaction In the stacking fault ribbon of a split dislocation a segregation of solute atoms is expected (Suzuki, 1957). This changes the width of the fault and locks the dislocation. The effect should be stronger for edges than for screws and may be important in high temperature deformation. TEM observations of the change of the dislocation width with temperature by Saka (1984,1985) indicate that Suzuki segregation occurs in noble metal base alloys with solutes of the IVb and Vb groups (e.g., Cu-Si, -Ge), not with those of the Illb group (e.g., Cu-Al): IVb and Vb atoms have a large binding energy of vacancy-solute pairs which favors diffusion. The resulting interaction force per unit length (Suzuki, 1962) is given by the change in stacking fault energy ys fs = ys(c0)-ys(c1)
(5-5)
where c0 is the matrix concentration and c± the concentration in the stacking fault, cf. Paninetal. (1963).
5.2.2.2 Special Effects in Different Lattices (1) Effects of Anisotropy and Dislocation Splitting in EC.C. Crystals The splitting of dislocations into partials also affects some of the interactions discussed earlier: The parelastic interaction with a screw will be considerably enhanced by the edge components of its partials such that the variation with dislocation character is smaller for the dissociated than for the undissociated dislocation (Hazzledine, 1968). Similar results were obtained by Bapna and Parameswaran (1980) for the elastic interaction between spherical and tetragonal misfit defects and extended screw dislocations.
200
5 Solid Solution Strengthening
(2) Influence of Solutes on Kink Creation and Motion in B.C.C. Crystals In b.c.c. crystals the "threefold split" screw dislocations (e.g., Vitek, 1985) propagate by the creation of double kinks and their sidewise movement along the screw dislocation line. Edge dislocations move much more easily than screws, therefore the latter control the macroscopic flow stress. Solutes may influence the processes of kink nucleation and motion (cf. Nadgornyi, 1989), and in addition, the motion of jogs which may arise if double kinks along the dislocations are ejected into different slip planes (Suzuki, 1971, 1979). In special cases, a solute may even facilitate kink nucleation resulting in "solid solution softening" in the low temperature range (cf. Sees. 5.5.2, 5.5.3, 5.6.1). (3) Electrostatic Interactions in Ionic Crystals The interaction between a charged dislocation core and the electrostatic potential produced by an impurity will be especially strong in the case of ionic crystals doped with solutes of different valency. The electrostatic interaction superposed on the parelastic one [cf. Sec. 5.2.1 (5, iii)] is particularly strong for edge dislocations in the {100} slip planes (Haasen, 1983). The different mobility of charged edge and screw dislocations has been discussed by Hirth and Lothe (1968). At sufficiently high temperatures, atmospheres of charged point defects are formed around the dislocations (Frank, 1967 a, b; Haasen, 1985; Sakamoto, 1987). (4) Interaction Effects in Semiconductor Crystals Crystals of diamond cubic structure like Si or Ge (and similarly III-V and II-VI
compounds with partially ionic binding) become ductile at elevated temperatures. The effect of dopants (impurities) on the ductility may be rather diverse. The dislocation mobility is enhanced by addition of > 1 0 1 9 c m " 3 n-dopants ("Patel effect"; Haasen, 1983; Hirsch, 1979, 1985; Nikitenko and Farber, 1985). Different types of solutes may produce hardening as well as softening effects (Alexander and Haasen, 1968; Sumino etal., 1980). 5.2.2.3 Drag Processes on Moving Dislocations Contrary to the pure crystal (Nabarro, 1967), very little work has been done on dynamic drag mechanisms of moving dislocations in solid solutions (see Alshits and Indenbom, 1986). In principle, the same reasons for lattice changes listed in Sec. 5.2 may cause an increased transfer of energy from the moving dislocation to the phonon and electron systems in a solid solution compared with the pure crystal. Classes of dynamic dragging involve the emission of elastic waves during the deceleration and acceleration periods of dislocation lines at obstacles (Ookawa and Yazu, 1963), the excitation of local or quasilocal vibrations of impurity atoms (Takamura and Morimoto, 1963; Kosevich and Natsik, 1967, 1968), and the radiation of phonons by dislocations vibrating like a string after passage of an obstacle (Kaneda, 1970). At ordinary temperatures the drag by phonon processes is large enough to cause an overdamped dislocation motion [damping constant B = Tb/v = 3 x 10" 5 to 10" 4 Ns/m 2 , Kaneda (1970)]. At very low temperatures (< 20 K) the phonon degrees of freedom freeze out and only the drag by electrons (Motowidlo et al., 1981) remains. The dislocations then move in an under-
5.2 Elementary Interactions and Threshold Stress
damped manner with damping constants typically around B = 2 x 10" 7 Ns/m2 (Galligan et al., 1986). Much more experimental results than for normal-state alloys (e.g., Motowidlo et al., 1981) are available for superconducting (Pb and Nb based) alloys (Kostorz, 1973; Startsev, 1983). 5.2.2.4 Superposition of Various Effects
Many of the various interactions listed separately, in real cases occur simultaneously and not simply additive, so that an experimental separation is difficult. We just mention two cases: In Sec. 5.2.1 (2) we indicated already the close relation of dilatational and modulus effects, as both originate from the electronic bonding forces between the atoms. The different characteristics of the interaction forces of parelastic/* and dielastic/1 origin lead to a nonadditive superposition of/5 and/ 1 for statistical obstacle distributions. In the case of Mott-Labusch statistics (cf. Sec. 5.2.3.2) the most preferable superposition is (Labusch, 1970) / ^ / i & 2 e L = / ^ 2 0 2 + «V)* (5-6) In a more recent analysis Gypen and Deruyttere (1981) find that Eq. (5-6) is valid for edges [with rj replaced by ri' = (ri + A5)/(l + \ri +A5\/2) with a material constant A], while for screw dislocations Fleischer's (1964) linear combination 8s=\d + a?i'\ is preferable. Another example of superposition (Shtremel, 1966) considers the kinetics of the formation of Suzuki segregation and of a Cottrell atmosphere (Sec. 5.2.2.1). The former is determined by short range interactions, the latter by the long range stress field of the dislocation. Accordingly, in the initial stage of Suzuki segregation its kinetics is not influenced. However, its equilibrium saturation concentration is attained,
201
by assistance of the long range Cottrell field, in a finite time instead of asymptotically. 5.2.3 The Threshold Stress for the Motion of a Single Dislocation at Zero Temperature
In a first step towards a physical understanding of the critical resolved shear stress (CRSS T0) of solid solutions, a single dislocation is considered which is forced by the external shear stress T to move across a field of randomly distributed obstacles at zero temperature. It has been realized early that for a regular array this stress is overestimated considerably, because a dislocation line may break, by means of an "unzipping effect", rows of neighboring obstacles (Argon, 1972). The threshold stress is defined as the minimum stress which is necessary for glide across the whole slip plane (Kocks, 1985 a). We will first consider true point obstacles (Sec. 5.2.3.1) and then realistic solute atoms (Sec. 5.2.3.2). As many competent detailed reviews exist (e.g., Friedel, 1964; Kocks etal, 1975; Haasen, 1979, 1983; Kocks, 1985 a; Nadgornyi, 1989), we just give a brief account of the main issues. 5.2.3.1 True Point Obstacles Overcome by the Dislocation in Overdamped Motion
To calculate the threshold stress in the most simple case, identical and pointlike obstacles (Fig. 5-1 a) without spatial extension, and a quasistationary overdamped movement of the dislocation are considered (cf. Sec. 5.2.2.3). The maximum interaction force Fm between a dislocation and a single obstacle is conveniently characterized by a critical breaking angle 9C (Fig. 5-1 b): Fm = 2TLsin
6C&
fib2sin9C
(5-7)
202
5 Solid Solution Strengthening
critical minimum stress TC (threshold stress) is obtained applying Fleischer's average balance of forces, Fm = xcbLF, which with Eqs. (5-7) and (5-8) yields the Friedel-Fleischer threshold stress
(a)
(b)
(c)
(d)
o
o
(5-9)
o
ZB-
Figure 5-1. (a) Dislocation propagation across the slip plane containing pointlike obstacles ("Friedel limit"). Definition of the Friedel length L F [Eq. (5-8)]. (b) Balance of forces at a point obstacle, (c) Dislocation propagation in case of extended penetrable obstacles with interaction distance w ("Mott-Labusch limit"), (d) Interpretation of Labusch's parameter /?L [Eq. (5-10)]: it measures the ratio of the angle Q± to that of breakthrough, 6 = 6C in (b). /b is the barrier distance.
where the line tension TL is approximated by fib2/!. For example, taking the size effect interaction with an edge dislocation in Cu-Ge [Table 5-1 and Eq. (5-3)], Fm « 0.085 \i b2 yields 6C « 4°, i.e., solid solution obstacles are rather weak and the whole dislocation remains rather straight. The mean spacing between neighboring anchoring points along the dislocation under stress T has been estimated by Friedel (1964) in a heuristic approximation which proved to be very powerful: (5-8) where the average obstacle distance in the slip plane L o = b/yc is expressed by the obstacle concentration per atom, c. The
The approximations made are good enough for breaking angles of 0° < 6C < 40°. A slight modification of Eq. (5-9), TC = 0.8 TCF, can be used for the range of breaking angles of 4O°<0C<9O°, i.e. for strong point obstacles. Computer simulations proved here very useful (e.g., Morris and Klahn, 1973, 1974; Altintas and Morris, 1986; Schwarz and Labusch, 1978). The above treatment is applicable for the case of easily flexible dislocations in a slip plane of very low Peierls potential, i.e. for f.c.c. and h.c.p. (basal glide) crystals. For the more complicated case of a high Peierls potential, where the dislocation propagates by creation and motion of double kinks, we refer to the extensive discussion by Nadgornyi (1989). 5.2.3.2 Solute Atom Obstacles at Zero Temperature (J=0) In the case of realistic obstacles in solid solutions, e.g. pairs of foreign atoms, with their average distance /b in the glide plane, we have to take account of the range w of interaction between obstacle and dislocation (Fig. 5-1 c). Typically assumed values are w&2b to 3b (Schwarz and Labusch, 1978; Kocks, 1985 a). The glide obstacles formed by foreign atoms can always be considered to be weak, i.e. Fm<^2 TL, 9C is small [Eq. (5-7)]. Fm9 w and /b characterizing these barriers are interrelated in their effect on the
5.2 Elementary Interactions and Threshold Stress
threshold stress. Labusch (1972), Nabarro (1972) and Schwarz and Labusch (1978) have shown that the value of the dimensionless parameter (5-10) decides whether the glide barriers can be taken as pointlike. (1) Friedel Versus Mott-Labusch Statistics for Overdamped Dislocation Motion The first factor in pL in Eq. (5-10) is a measure of the localization of the defects, the second one of their strength [Nabarro (1972, 1985 a)]: w/lh = sin (9J2) gives the angle 9X indicated in Fig. 5-1 d; F m /(2r L ) = sin 9C gives the critical angle 6C according to Eq. (5-7) (Fig. 5-1 b). 2 TJFm usually takes values between 3-10 for realistic solid solutions. Two extreme configurations can be distinguished for weak obstacles with symmetrical interaction profile according to whether the parameter j8L is much smaller or larger than unity. The case Ph<^l means that the solid solution is very dilute, lh>w, and each obstacle opposes the dislocation motion (Fig. 5-1 a). The threshold stress at T=0, T 00 , calculated by "Friedel statistics" is given by Eq. (5-9) which may be rewritten as (5-11) The factor 0.95 improves a little the Friedel approach (Hanson and Morris, 1975; Labusch, 1977) and Lo is replaced by lh = b/cl12 with the barrier concentration ch. This can be distinctly smaller than the atomic solute concentration c, if solute pairs or larger units act as effective obstacles. Equation (5-11) holds well up to at least F m / ( 2 r L ) ^ 0 . 5 (Morris and Klahn, 1974;Kocks, 1985a).
203
The case /?L > 1 applies for more concentrated solid solutions, though still / b >w, and additionally for weak, more diffuse obstacles. Here the dislocation in the equilibrium configuration cannot avoid interacting with the obstacles at their front and back flanks with forces opposing and assisting the external stress (Fig. 5-1 c). The statistics for this case is called "MottLabusch statistics" (Nabarro, 1972,1985 a). The critical threshold stress at zero temperature is found to be (cf. Haasen, 1979, 1983) _L
_
(5-12)
As pointed out by Hilzinger (1977) (and partly by Labusch, 1972) from very general arguments the exponents in too oc(cZ'F£'Wy') have to fulfil the scaling relations a' + /?' = 2 and a'—•/?' — Y = — 1, yielding the powers a' = 1 /2, £' = 3/2, y' = 0 in Eq. (5-11), and a'= 2/3, j8' = 4/3, and y' = l/3 in Eq. (5-12), respectively. Equations (5-11) and (5-12) give slightly different dependences of T 00 on c (and on Fm). However, recent computer simulations by Arsenault et al. (1989) indicate a c1/2 dependence even for large solute concentrations. In fact, the regimes of Friedel and Mott-Labusch statistics overlap for 0.5
204
5 Solid Solution Strengthening
tions. Various relations for the superposition effect of two kinds of obstacles have been proposed (e.g., Kocks, 1980, 1985 a; Schlipf, 1982). A general rule follows from the scaling relation mentioned above (Hilzinger, 1977). (3) Linear Obstacles (Trough Models) In the concepts discussed so far the extension of the obstacle in the direction of the dislocation line is negligibly small (w <4 /b)- The trough model worked out for solid solution hardening by Suzuki (1980, 1986) and Kocks (1985 a), considers the obstacles smeared out along the dislocation. We will discuss this model in Sec. 5.3.3.1. Further, linear barriers play a role in kink-mode crystals (Nadgornyi, 1989) which will be considered briefly in Sees. 5.5 and 5.6. (4) Inertial and Quantum Softening Effects The dislocation motion is viscous only at high enough temperatures due to phonon interactions (cf. Sec. 5.2.2.3). However, at sufficiently low temperature T (<20 K), and in particular at T=0 (as considered here), in dilute alloys the dislocation motion may become underdamped involving "inertial effects". (i) Underdamped motion: Inertial effects have been considered first independently by Granato (1971), Suenaga and Galligan (1971) and Kamada and Yoshizawa (1971). They realized that a dislocation which approaches an obstacle with sufficient velocity may overshoot the next static equilibrium position exerting a force on the obstacle which exceeds the maximum force in the equilibrium case. Thus the critical threshold stress is reduced in comparison to the T 00 value determined in the preceding sections. The lowest solute concentration required for inertial effects to ap-
pear in dilute Cu alloys is c> 10 6 . Clear evidence for inertial effects has also been found in superconducting metals and alloys (Kostorz, 1973; Startsev, 1983). For further experimental and theoretical work see Kocks et al. (1975), Schwarz and Labusch (1978), Schwarz and Granato (1977), Isaac etal. (1978), and Landau (1980, 1981). (ii) Tunneling and zero point vibrations: At extremely low temperatures (in the order of 1 K) and at a high stress level, obstacles may be crossed by the dislocation with a lower stress than T 00 even if it is moving quasistatically. The latter may happen if the dislocation motion is governed essentially by the laws of quantum mechanics. Two possible quantum effects have been discussed, i.e. tunneling of the dislocation through the obstacle (Mott, 1956), and overcoming of the obstacle by zero point vibrations (Leibfried, 1957), see also Natsik (1979), Petukhov and Sukharev (1983), Startsev (1983), and Labusch (1984).
5.2.4 The Effect of Temperature on the Threshold Stress
One of the main characteristics of the mechanical behavior of solid solutions is a strong decrease of the critical resolved shear stress from its value T O O~ T o ( ^ = 0 ) to that at ambient temperature T 0 (J). This is clearly due to the effect of thermal activation. The extensive work on this process has been reviewed by many authors, e.g. Seeger (1955, 1958), Kocks etal. (1975), Schoeck (1980), Nadgornyi (1989). In Sec. 5.2.4.1 we will explain just the main modifications of the theories for T=0 discussed before with the obstacles at fixed positions. The effects of diffusive processes at higher temperature on the threshold stress will be reviewed in Sec. 5.2.4.2.
5.2 Elementary Interactions and Threshold Stress
5.2.4.1 Thermally Activated Overcoming of Fixed Obstacles
Schwarz and Labusch (1978) starting from their simulations at T=0 discuss two main characteristics of the changes of dislocation motion induced by temperature in the case of dilute alloys with fixed solute obstacles: (i) The effect of damping: For T>30 K the strongly increasing viscous damping induces permanently quasistationary dislocation motion [Sec. 5.2.4.1 (1)]. Below about 20 or 30 K the viscous damping becomes too small to keep overdamped conditions of dislocation motion. Then inertial effects help in overcoming the obstacles [Sec. 5.2.4.1 (2)]. (ii) The effect of thermal activation: Its main effect is the lowering of the effective height of all barrier potentials along the dislocation line in such a way that the decreased waiting times of the dislocation in front of the obstacles remain largely the same for all activation events on an average (Schwarz, 1980). For concentrated solid solutions the situation becomes even more complex (Arsenault et al., 1989; Arsenault and Li, 1989; Labusch, 1988; Nabarro, 1985 b; Schwink and Wille, 1980; Schwink, 1988) as no longer single atoms but certain groups of them will act as effective local obstacles with a kind of increased friction by the remaining solute atoms in between. (1) Modification of Theories for T=0 for Overdamped Motion The effect of thermal activation at a single obstacle can be safely described (Labusch, 1984) by the Arrhenius equation for the transition rate v (or for the shear strain rate y) of the dislocation across the obstacle
= v o exp
[
-
AG(T)~| AG(kT J
f or y = 7 0 exp
205
(5-13)
AG(T)I)
L kT J/
with yo = blhQmvo (cf. Sec. 5.4.3.1). The "attack frequency" v0 is some fraction of the Debye frequency vD (Granato et al., 1964), vo = ( 4 x l 0 ~ 2 to 4xlO" 3 )v D « 10 ~1 x s ~x, independent of segment length and varying at most slightly with stress (cf. Kocks et al., 1975; Schoeck, 1980; Nadgornyi, 1989). Qm is the mobile dislocation density and lh the barrier distance along the dislocation line. AG is the activation free enthalpy of the barrier which is usually taken as a linear expansion in the effective stress T*, AG(T*) = AG0 -V-T*
(5-14)
Here AG0 is the activation free enthalpy necessary at zero effective stress, V is the activation volume and T* = T — T^ accounts for a possible athermal stress TM. The exact path of a dislocation driven by the applied stress across the obstacle field at T>0 with a prescribed average velocity may be drastically changed by thermal fluctuations in comparison with the case for T=0. This makes it much more difficult to obtain an analytical solution of the case for T>0 than for T=0 (Sec. 5.2.3). Attempts to perform calculations in the Friedel regime have been made, e.g. by Landau (1975, 1983), Strunin (1976), Louat (1978), Suzuki (1981), Schlipf (1982), Schoeck (1985), and for the Mott-Labusch regime by Labusch et al. (1975) (cf. Haasen, 1979,1983; Nabarro, 1985 a), and several others. The overall shape of the dislocation is expected to remain straight in the average. This has been confirmed in all computer simulations of the problem, for dilute sys-
206
5 Solid Solution Strengthening
terns (e.g., Morris and Klahn, 1973, 1974; Zaitsev and Nadgornyi, 1976; Schwarz, 1980; Labusch, 1988; Nadgornyi, 1989) as well as for concentrated solid solutions (Arsenault and Li, 1989). The simulations also reveal a jerky dislocation motion across the obstacle field: the average area swept out during an activation event is much larger than the average area per obstacle, l\ (Labusch, 1988; Nadgornyi, 1989). This implies that the effective activation energy and the effective activation volume may be up to an order of magnitude higher than that of a single obstacle. Due to limited computer time all simulations are carried out for average dislocation velocities much higher than those in usual experiments. The effect of a dispersion of obstacle strengths has been considered in various simulations (e.g., Arsenault and Cadman, 1974; Hanson and Morris, 1975) and analytical approaches (Diehl etal., 1965; Frank, 1968). While the transition from a regular to a random distribution reduces the effective A G, the effect of obstacle dispersion is to provide some additional "hard" configurations which raises effectively A G, as though an additional temperature dependent long-range internal stress exists (Vydashenko and Landau, 1981). (2) Combination with Inertial Effects A theoretical description which intends to combine both inertial effects and thermal activation was proposed by Isaac and Granato (1979, 1988) using the common concept of stochastic motion for both effects. They have shown that the process of overcoming the obstacle depends critically on the initial conditions and only slightly on the potential shape. Calculating the probability for the dislocation to overshoot beyond the barrier, they obtain the
overall average dislocation velocity v which, together with the mobile dislocation density Qm, determines the external shear strain rate y = gmbv. 5.2.4.2 Diffusive Processes (Moving Obstacles) In this section we relax the condition assumed up to now, that the pinning obstacles remain fixed in the array of the slip plane and in particular along the dislocation. As any motion of the solute atoms is strongly temperature dependent, we will discuss in subsections the ranges of lower, intermediate and high temperatures. (1) Relaxation of Solute Atoms in the Dislocation Core Region (Low T) Macroscopic evidence from stress relaxation as well as from internal friction [see Sec. 5.3.3.3 (1)] suggests that already at rather low temperatures (T< TJ5, Tm is the melting temperature) effects of mobility of solute atoms occur. This cannot be long-range diffusive motion, but consists probably just in a switching of the solute atoms to neighboring favorable positions in the dislocation core region forming Zener pairs for substitutional, or Snoek pairs for interstitial solid solutions [cf. Sec. 5.2.2.1 (1, ii)]. At T< TJ2 pipe diffusion along the dislocation core (cf. e.g., Atkinson and LeClaire, 1984; Balluffi and Granato, 1984) may operate. According to Sleeswyk (1958) and Kocks (1985 a) this mechanism may also increase the strength of obstacles. Zaitsev (1983) considered the joint motion of the dislocation and the obstacles during the waiting time in a kind of directional diffusion under the driving force of the dislocation on the obstacle and calculated the resulting dislocation velocity.
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of RC.C. Alloys
The competing processes of impurity diffusion along the dislocation and the diffusional exchange of impurity atoms between dislocation core and bulk for a dislocation under external stress have been considered in detail by Suprun (1981). Above a critical value of stress the number of solute atoms pinning the dislocation drops sharply. All of these processes already occur in a temperature range which is clearly dominated by the thermally activated unpinning of dislocations. Their existence is noticeable only in appropriate experiments with low stress or strain rates (long waiting times) or very high sensitivity. With increasing temperature there is a gradual transition to much stronger effects considered in the next section. (2) Formation of and Breakaway from Impurity Clouds (Intermediate T) If the temperature is high enough and the waiting time tw of a dislocation is long enough, the processes in Sec. 5.2.2.1 (1, ii), (3), (4) will produce a cloud of solute atoms (Cottrell or Snoek atmospheres) around the dislocation (e.g., Friedel, 1964; Yoshinaga and Morozumi, 1971; Paidar, 1978; Louat, 1981; Malygin, 1982; Barnett etal., 1982). The kinetics of this process may be described by a semiempirical equation for the solute concentration in the dislocation core region (Schwarz, 1982; McCormick, 1988)
207
tial stages of segregation. It plays an important role in the processes of dynamic strain ageing and deformation instabilities discussed in Sees. 5.3.1.2, 5.3.3.3, and 5.4.2.1. (3) Dragging of Impurity Clouds (High T) At high enough temperatures the diffusional processes are fast enough to establish the solute cloud at dislocations immediately after breakaway from their obstacles. Then the dislocations move in a quasiviscous manner dragging their impurity cloud with them, and the deformation curve is smooth. In this case (e.g., Hirth and Lothe, 1968; Takeuchi and Argon, 1979; Yoshinaga etal., 1985), the dislocation velocity follows a linear stress dependence, V = T b/B. The effective viscous drag coefficient B is strongly temperature dependent and governed by the thermally activated process controlling the dislocation motion, e.g. kink diffusion or nucleation, or diffusion of solutes. As the cloud will decrease with increasing temperature, the resistance to dislocation motion by the solutes will decrease and vanish at a sufficiently high temperature [cf. Sec. 5.3.3.3 (3), Kurishita etal. (1989), and Chap. 8 on high temperature creep by Blum in this Volume].
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of EC.C. Alloys
(5-15) with the maximum concentration cm = cm(T) in the core region (Louat, 1981) and a characteristic time tc which depends essentially on a diffusion constant D(T). Equation (5-15) approaches the wellknown t2/3 law (Friedel, 1964) for the ini-
In the following sections we will specify the elementary processes reviewed in Sec. 5.2. for various systems in an attempt to understand different observations on macro- and microscale of substitutional and interstitial solid solutions in comparison with those of pure metal crystals. In view
208
5 Solid Solution Strengthening
of the earlier extended reviews of the subject (cf., e.g., for f.c.c. by Haasen, 1979, 1983; Kocks, 1985 a; Nabarro, 1985 a) we will focus in particular to recently achieved evidence and understanding.
1 =C 2 = Cu u 60 . 3 = C u u ,50 . 5 = C
70
/ 2 5 10 15
at.% Al at.% at.%
Al Al
. 4
j
/
/ J
at.%
;/>o
5.3.1 Survey of the Macroscopic Behavior of F.C.C. Alloys 5.3.1.1 Stress-Strain Curves of Monoand Polycrystals We start out with the comparison of complete stress-strain curves of two typical f.c.c. solid solutions, concentrating the discussion here to the CRSS T 0 and to stage I; the later stages of deformation will be considered in the next Sec. 5.4. We select two alloy systems with quite different behavior of their stacking fault energy ys: CuMn where ys remains virtually unchanged with alloying (Steffens et al., 1987) up to at least 10 at.%, and CuAl, where ys decreases rapidly with increasing Al content (Carter and Ray, 1977). Figure 5-2 a gives some stress-strain curves for CuAl alloy crystals oriented for single glide (cf. also Rosi, 1973; Takeuchi, 1976). For the polycrystals (Fig. 5-2b) the tensile stress o and strain s are plotted, while for single crystals (Fig. 5-2 a) the corresponding quantitites x and y, respectively, are resolved to the active slip system. The curves for CuMn crystals (not shown) are quite similar (Wille and Schwink, 1980; Steffens and Schwink 1983) in spite of a different ys value. The curve of a [100] oriented crystal practically coincides with that of a polycrystal. While the single glide crystals show the well-known stages I, II, and III of work hardening, stage I is lacking in the multiple glide (and poly-) crystals due to the interaction of several slip systems from the beginning of deformation.
"30
20 10
(a)
e in %
Figure 5-2. Stress-strain curves for (a) CuAl crystals oriented for single glide with indicated concentrations (in at.%). (b) CuAl polycrystals deformed in tension at room temperature; strain rate 8 = 6.4 •10" 5 s~ 1 [(a) Koeppen and Neuhauser, 1989; (b) Rindfleisch et al., 1990)].
We note without going into detail that twinning deformation modes occur in CuAl at higher stresses and in single crystals of appropriate orientations (Demirskii and Komnik, 1982; Galligan and Haasen, 1986; Tranchant et al., 1988; Korbel and Szczerba, 1988), and that microtwins have been observed in low stacking fault energy CuSi (Coujou, 1983 a, b) and CuGe (Gryziecki, 1988) alloys [cf. Sec. 5.2.2.1.(4)].
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of F.C.C. Alloys
For crystals oriented for single glide the extent % of stage I, which is known to be determined by the onset of slip on a certain secondary system (e.g., Haasen, 1979, 1983), shows a complicated dependence on solute concentration. A maximum in su(c) around c = 3 at.% coincides with the transition from homogeneous to inhomogeneous mode of slip (Sec. 5.3.2). The later increase of sn(c) for c> 7 at.% is accompanied by Luders slip (see Sees. 5.3.2 and 5.3.4). 5.3.1.2 Temperature and Concentration Dependence of the CRSS T O ( T, C)
The CRSS T 0 , defined as the extrapolated value to plastic strain y = 0, excluding an initial yield point, is the same for multiple glide crystals as for single glide crystals. The dependences of the CRSS on T and c are shown in Fig. 5-3 again for the example of CuAl which has been investigated in a large temperature range (Koppenaal and Fine, 1962; Suzuki and Kuramoto, 1968; Startsev etal., 1979; Nixon and Mitchell,
1981). CuMn alloys with a nearly constant stacking fault energy show similar behavior (Wille and Schwink, 1986; Wille et al., 1987). The rapid decrease of T 0 in the low T region due to thermal activation (Sec. 5.3.3.1) is followed by the "plateau region" which, as indicated by the broken line in Fig. 5-3, is largely covered by serrated yielding (Sec. 5.3.3.3). In single glide crystals the end of this "Portevin-LeChatelier regime" often coincides with a maximum of the CRSS at the temperature TM, T0M = T 0 (r M ). This has been found also in other Cu base alloys like CuZn and CuGe (Traub et al., 1977), not in CuNi (Suzuki, 1988). Beyond the maximum (T> TM) the load trace is smooth again and decreases rapidly with increasing T approaching a common curve for all alloys. For a discussion of the depencence T0 (r, c) in terms of the theoretical concepts up to about 300 K in Sec. 5.3.3 a compilation of CRSS data is given in Fig. 5-4 for different Cu-based alloys comparing the concentration dependence at room tern-
35 30-
1000
in K
209
Figure 5-3. Temperature dependence of the critical resolved shear stress (CRSS) T0 of CuAl crystals with various solute concentrations [lines and (•): Suzuki and Kuramoto (1968), Startsev et al. (1979), Nixon and Mitchell (1981); (•) and bars: Neuhauser et al. (1990) with corrected T values]. All data have been extrapolated to a common shear strain rate ofy = 3.6 1 0 - 5 s - 1 . The Porte vin - LeChatelier (PLC) regime (cf. Sec. 5.3.3.3) is indicated by the dotted line and bars connecting average maximum and minimum stresses of serrations.
210
5 Solid Solution Strengthening
DK
yl
10-
-
Figure 5-4. Comparison of the CRSS T0 of various Cubased alloy crystals over a wide range of solute concentrations at room temperature. Note the log-log scale and the parallel behavior of all alloys. (+) CuMn (Nagata and Yoshida, 1972; Wille and Schwink, 1980), (A) CuGe (Peissker, 1965; Rosi, 1973; Traub et al., 1977; Wille et al., 1987), (o) CuAl (Basinski et al., 1972), (*) CuZn (Neuhauser et al., 1983), (x)CuNi (Suzuki and Ishii, 1968; Neuhauser et al., 1979).
.+-^A"cu-Ge
x /Qu-Ni
< s> '
T"
—
*xx
Cu!i
xc
0.10.01
0.1
1 c in %
perature near to the beginning of the plateau region. A striking similarity of different systems is observed; the same is found in the low temperature region. However, no simple power dependence appears to be valid for the whole range of concentrations, this dependence varies with temperature. In a special plot suggested by Basinski etal. (1972) the similarity becomes quite evident. It is called rule of "stress equivalence" (Nabarro, 1982, 1985 b). Deviations from this rule have been discussed by Schwink and Wille (1980). 5.3.1.3 Strain Rate Dependence of the CRSS One of the important experimental methods to gain insight into the process of thermal activation of dislocations at discrete obstacles is to measure, in addition to the temperature dependence of the CRSS, its strain rate sensitivity S at various fixed temperatures: S=
8T
9 In y
(5-16)
where x1 and T 2 are the shear stresses measured at strain rates yx and y2> respectively. S is connected directly with the effective activation volume V in the Arrhenius equation [Eq. (5-13)], if y0 = blhQmv0 and AG0 are constant: V=
-
8AG 8T*
kT
(5-17)
To justify the Arrhenius Eq. (5-13), the dependence of S, Eq. (5-16), on the magnitude of the strain rate change as well as its dependence on the base strain rate has to be checked. A slight dependence on the strain rate factor y2h\ has indeed been found (Traub etal., 1977; Flor and Neuhauser, 1982), increasing with higher temperature (see Sees. 5.3.3.3 and 5.4.2.1). For a fixed temperature, again a remarkable similarity for different alloys with various solute concentrations is noticed corresponding to a second kind of stress equivalence (Basinski etal., 1972; Nabarro, 1985 a, b; Schwink and Wille, 1980), if V is plotted versus the CRSS at this temperature.
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of F.C.C. Alloys
In Fig. 5-5 the dependence of S (in normalized scale) on temperature is shown for CuAl crystals deformed in stage I. At very low T the anomalies due to inertial effects are clearly recognized in the S(T) behavior. At higher T, after the normal behavior in the thermally activated overdamped region, a gap with anomalous behavior exists in the range of serrated yielding where S attains negative values [cf. Engelke et al. (1992b) for CuAl]. The high temperature maximum of the CRSS (Fig. 5-3) is accompanied by a pronounced maximum in S (Fig. 5-5) after which the different alloys approach a common decreasing curve.
211
information on the microprocesses occurring on the dislocation level. Therefore much effort has been spent in gathering more direct information (cf. Haasen, 1983). In particular, the micro-observations will show a characteristic transition of the mode of slip from "homogeneous" to "inhomogeneous" and "planar" with increasing concentration. 5.3.2.1 Structure and Formation of Slip Lines
A prominent feature revealed by slip lines is the coarsening of the slip line pattern on alloying (cf. reviews by Seeger, 1958; Haasen, 1983). Electron microscopic 5.3.2 Survey of Microscopic Observations replica show that the slip lines in stage I The macroscopic stress-strain curves remain rather similar to those in Cu (i.e., and their variation with the available very faint, long and nearly randomly disparameters naturally give widely averaged tributed) in Cu alloys with up to a few % of solute (Fig. 5-6a), e.g. up to about 3 at.% in CuAl (Jackson et al., 1977) and CuGe (Heege and Neuhauser, 1973), 6-8 at.% in CuZn (Schmidt-Hohagen and Neuhauser, 1981), about 5 at.% in CuMn (Steffens and Schwink, 1983), and CuNi (Arkan and Neuhauser, 1987 a, b). This will be called "homogeneous slip" (although, strictly speaking, slip is by nature heterogeneous, cf. Sprusil and Hnilica, 1988; Kleiser and Bocek, 1986) and is sometimes referred to as "wavy slip". For higher concentration, 200 400 600 800 T in K an increasing tendency of clustering of the fine lines is observed, which we will call Figure 5-5. Temperature dependence of the normalized strain rate sensitivity S • [b3/(k T)] = [AT/(A In y)] • "inhomogeneous slip" (cf. Fig. 5-6 b). At [b3/(k T)] (determined from stationary stresses during still higher concentrations, e.g. > 6 at.% in strain rate changes by a factor of 10 from the base CuAl (Mitchell et al., 1967; Charsley and 1 value of n = 1.3 • l O ^ s " ) for Cu-15 at.% Al crysDesvaux, 1969; Rosi, 1973) and CuGe, tals oriented for single glide. (•) Koppenaal and Fine > 12 at.% in CuZn (Neuhauser et al., (1962), (x) Komnik and Demirskii (1981), (*) Neuhauser et al. (1990) with corrected T values. Note the 1975), > 11 at.% in CuMn (Steffens et al., negative values at T > 450 K coinciding with the PLC 1987), clusters of strong lines, called slip regime with load serrations (J, J,), and the peak at bands appear. Here a considerable shear elevated T (cf. Fig. 5-3). The negative value [5 PLC , cf. occurs on virtually single slip planes. This Eq. (5-22)] is determined from the breakaway average situation is called "planar slip" since exstresses i max of serrations.
212
5 Solid Solution Strengthening
5.2.2.1 (3)] favor the formation of extended dislocation groups. They move very rapidly until the interaction with closely spaced neighboring groups slows down the motion of further dislocations in the slip band (Neuhauser, 1988). 5.3.2.2 Dislocation Structure
ftp1 urn
Figure 5-6. Comparison of slip line patterns (EM replica) on CuGe crystals oriented for single glide, after deformation at room temperature into stage I, for (a) Cu-2at.%Ge, (b) Cu-3 at.% Ge, and (c) Cu-7.6 at.% Ge. The transition from Cu-like "homogeneous" (or "wavy") slip (a), to inhomogeneous (b), and to planar slip (c) is shown (Heege and Neuhauser, 1973).
tended groups of dislocations on virtually single slip planes are observed in transmission electron microscopy (TEM). The formation of these slip bands has been studied by light microscopy (cf. reviews by Pond, 1972; Neuhauser, 1983, 1988). They appear to evolve in a succession of slip lines. Provided the first slip line originates in a distance far enough from other slip bands, it is formed in a very short time (10 jas), corresponding to dislocation velocities up to the m/s range. However, the rate slows down rapidly by orders of magnitude when the number of lines per band increases. From this it has been inferred that a local overstress and an obstacle destruction mechanism [e.g., Sec.
For strains shortly after reaching the CRSS, in crystals oriented for single glide one finds with increasing solute concentration a transition from a homogeneous dislocation structure as in Cu, to a more inhomogeneous one (involving still many activated, but clearly correlated neighboring slip planes), up to a planar arrangement (extended multipoles on few single slip planes), see e.g. for CuAl (Pande and Hazzledine, 1971), for CuZn (Olfe and Neuhauser, 1988) and for CuGe (Monchoux and Neuhauser, 1987). This transition is also observed in alloys in which ys does not decrease, e.g. NiFe (Karnthaler and Schiigerl, 1979), and NiCr (Clement e t a l , 1984), CuMn (Steffens etal., 1987; Steffens and Schwink, 1983), CuNi (Arkan and Neuhauser, 1987 a, b), in the latter case observed by etch pits. Gerold and Karnthaler (1989) also arrive at the conclusion (Neuhauser etal., 1986; Steffens etal., 1987) that destruction of short-range order or segregation may be the reason for this phenomenon rather than obstruction to cross slip (cf. Sec. 5.3.4). Recently Hong and Laird (1990) discussed this transition in terms of the friction forces in alloys on the partial inhibiting easy cross slip. We will restrict ourselves to the simpler case of homogeneous glide in the further discussions. For CuMn crystals oriented for single slip the dislocation structure (Heinrichs et al., 1992) develops quite similarly to that of neutron-irradiated Cu crystals (Ess-
213
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of F.C.C. Alloys
mann and Rapp, 1973). In crystals oriented for multiple slip, on the other hand, a cell structure of dislocations is found to develop from the beginning of deformation (Neuhaus and Schwink, 1992), cf. Sec. 5.4.1.2. The dislocation structures developed during high temperature deformation in solid solution alloys have been reviewed recently by Orlova and Cadek (1986). They are characterized by the formation of sharp subboundaries by "condensation" of the cell walls. 5.3.2.3 Dislocation Motion Recent TEM in situ investigations on a series of CuZn alloys (Schmidt-Hohagen and Neuhauser, 1988) with concentrations between 5 and 30 at.% Zn have confirmed the correlated rather smooth dislocation movement in extended groups for > 16 at.% Zn (cf. also Fujita etal., 1985; Tabata et aL, 1977). For < 12 at.% Zn single dislocations move in jerky manner: mostly only one dislocation at a time runs across the field of view often exceeding the resolution of recording, and, moreover, showing a lot of cross-slip events in spite of the lowered stacking fault energy. Even if such cross-slip events in TEM in situ deformation might be partly due to a surface effect (Hazzledine et aL, 1975), the lack of dependence of the observed frequency of cross-slip on increasing solute concentration seems to indicate that the cross-slip event is less dependent on stacking fault energy than is generally accepted. The transition from jerky motion of dislocations at ambient temperatures to viscous motion at temperatures beyond the PLC effect ( r > TM) has been observed directly by in situ TEM (AlMg: Nohara etal., 1980; CuGe: Monchoux and Neuhauser, 1987).
Dislocation velocity measurements by the stress pulse technique were made using etch pits [Fig. 5-7 for CuAl and CuNi; for a review, cf. Nadgornyi (1989)]. Kleintges and Haasen (1980) found that only measurements with freshly produced moving dislocation groups are reliable. The corresponding results yield a stress dependence of the velocity with a rather high power of the stress (which may be alternatively expressed by an Arrhenius type behavior). For Cu-5 at.% Ni the velocity measured by etch pits agrees quite satisfactorily with the results of slip line cinematography (Arkan and Neuhauser, 1987 a, b) if the very first rapid period of growth for single isolated slip bands is compared, avoiding interactions with other dislocations on
102 ,0.35'at.% Ni
10°
/ / 4"!
2 at.% Ni '-"
tj
10"
.
/
- 55atat.% Ni
f
/ o . 6 at.% Al
icr )' 10"
0.5
1.0
2.0 3.0 r in MPa
5.0
10.0
Figure 5-7. Dislocation velocities v measured by etch pitting before and after a stress pulse (pulse height T) for several CuNi and CuAl single crystals at room temperature. The values of the CRSS at room temperature for a shear strain rate of/y = 4-10~ 4 s~ 1 are indicated by arrows. (CuNi: Suzuki and Ishii, 1968; Arkan and Neuhauser, 1987 a, b; CuAl: Kleintges and Haasen, 1980; Friedrichs and Haasen, 1981.)
214
5 Solid Solution Strengthening
neighboring slip planes. The etch pit results, in particular, show that the average dislocation velocity at the macroscopic CRSS takes a value of several cm/s for all CuNi alloys with solute concentrations between 0.35 and 5 at.% at room temperature. For CuAl Kleintges and Haasen (1980) found the edge dislocations to move more slowly than screws at a given stress, as expected from the size effect interaction and in qualitative agreement with slip line measurements on CuZn alloys (Flor and Neuhauser, 1982). It also agrees with the higher friction stress for edges compared to screws found by TEM from the curvature of dislocation loops in Cu-5 to 15 at.% Al by Prinz et al. (1981). 5.3.3 Analysis of the CRSS for Dislocations Moving Independently 5.3.3.1 Analysis for 80 < J<300 K (Overdamped Thermally Activated Motion of Single Dislocations) At present a quantitative analytical description of the dependence of the CRSS T 0 on strain rate y and temperature T between 80 and 300 K is only possible, if (i) the solute concentration c is low enough such that homogeneous deformation with "wavy glide" occurs, and if (ii) the condition for applying the Friedel statistics is fulfilled, pL
(1) Discrete obstacle model (Fleischer, 1964; Friedel, 1964; and many others): Here the obstacles are considered to be the single solute atoms. (2) Discrete barrier model (Wille et al., 1987): Here the obstacles are considered to be solute atom pairs (doublets) or triplets, as occurring statistically in random alloys (Behringer, 1958; Freyen and Herring, 1981; Sprusil, 1983). (3) Cluster model (Schwink and Traub, 1968; Labusch, 1972; Kratochvil e t a l , 1973): Here the obstacles are clusters of solute atoms small enough for thermal activation, but not specified further, with a possible wide distribution of activation energies. Consequently, with increasing Tthe average effective cluster size and activation energy increases. This is the essential difference to the discrete obstacle model. (4) Trough model (Suzuki, 1980, 1986; Kocks, 1985 a; Feltham, 1968): Here the obstacles are considered as short rows of solute atoms segregated at the dislocation core (linear barriers). The earlier distinction between source hardening and friction hardening (Suzuki, 1980; Haasen, 1979, 1983; cf. also Bengus etal., 1984) is of less importance in the connection discussed here: both mechanisms yield a "stationary" CRSS T 0 , the former with an upper and lower yield point, the latter without yield points. By the following procedure a decision can be achieved which of the 4 models, if any, would describe a given alloy system: (i) The concepts of thermally activated glide common to all theoretical descriptions are applied in a way that a realistic barrier profile is chosen. It must not be simply parabolic which certainly is a good approximation for low enough T (Kocks etal., 1975), but it has to be bell-shaped, because the equilibrium configuration of the moving dislocation in front of the bar-
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of F.C.C. Alloys
rier is shifting with increasing T to and beyond the point of inflection of the barrier potential. Examples for potentials which, additionally, have the advantage to permit analytical treatment, are the wellknown Cottrell-Bilby potential (Cottrell and Bilby, 1949) (a non-energy-storing potential which is not changed by the passing dislocation) and the Seeger potential (Seeger, 1958) (an energy-storing potential). The detailed shape of the potential is not of importance for the analysis. For the Cottrell-Bilby potential the activation free enthalpy in Eq. (5-13) is found (Kocks et al., 1975; Wille et al., 1987)to be
t
/
T
* \ 0.46-12
i - (—• J
r
(5-i8)
where the effectively applied stress (5-19) is the CRSS T O ( 7 ) , corrected for the temperature dependence of the shear modulus (cf. Diehl et al., 1965) and reduced by an athermal part T^ which may not be negligible. (ii) The experiments have to be performed in a way that the temperature dependence of the CRSS T O ( J ) at constant strain rate y (e.g., Fig. 5-3), and the temperature dependence of the strain rate sensitivity S(T) [Eq. (5-16)] (e.g., Fig. 5-5) are measured on one and the same single crystal during stage I, to obtain a sufficient precision of the data. Instead of evaluating T 0 (T) directly, the derivative dT 0 /dTcan be used for further calculations avoiding any assumption on the athermal part x^ of T 0 . A joint analysis of the two (independent) functions of T, d t o / d r and S(T), yields quite reliable values for two parameters, the average obstacle strength A Go and the
215
concentration of effective glide barriers cb (or, equivalently, for their average distance / b ). If now, for a given concentration, both parameters AG0 and ch show no or a relatively weak variation with T and if, in addition, the athermal part TM calculated from these parameters turns out to be independent of T, then the assumptions underlying the analysis are fulfilled. Otherwise, such variations would immediately show that any of the models (1) to (4) fails and is not applicable to the system examined. (iii) A distinction between the 4 models can be achieved by a few further steps. At first, the dependence of S on the CRSS T 0 at constant temperature is considered, i.e. effectively the dependence on solute concentration. For the models (1) to (3) the function S(T 0 ) has to be a straight line through the origin (Kocks et al., 1975; Kocks, 1985 a), while this is not expected for the trough model (4). In the first case, only the models (1) to (3) remain. Among these, the cluster model (3) can be excluded, if A Go has been proven as really independent of T. In the cluster model the average activation enthalpy AG 0 should increase slightly with increasing T. Finally, the value of A Go itself must not exceed a value of at most 0.4 eV (Saxl, 1964), if single solute atoms are the glide obstacles. This gives readily a distinction between the models (1) and (2). The procedure described has been applied till now only for CuMn and CuGe alloys (Wille etal., 1987). For both systems the discrete barrier model [model (2)] describes the measurements successfully. The effective obstacles are identified to be probably Mn-doublets and Ge-triplets, respectively. The parameters determined are listed in Table 5-2 for CuMn alloys of two series (series 1: c Mn >2.0at.%, series 2: c Mn <2.0 at.%). The barrier concentration
216
5 Solid Solution Strengthening
Table 5-2. Parameters of effective glide barriers for dislocation motion in some CuMn and CuGe alloys, obtained from a consistent fit of T O (T) and S(T) (Willeetal, 1987)a. AG0 in eV
TM in MPa
(±0.1 eV) 0.4 1.2 2.0 3.8 7.6
at.% at.% at.% at.% at.%
Mn Mn Mn Mn Mn
1.34 1.36 1.21 1.28 1.23
0.5 1.0 2.0 3.3
at.% at.% at.% at.%
Ge Ge Ge Ge
1.14 1.35 1.23 1.13
c b -10- 4
(±10%) 2.0 + 0.2 5.7 + 0.3 10.3 + 0.6 14 ± 1 20.4 + 1.5 2.2 + 0.2 3.3 + 0.25 6.1 ±0.3 9.2 ±0.5
1.33 5.1 10.8 23.5 53.8 0.7 1.04 3.15 6.47
a
AG0 is the average barrier strength; the athermal stress contribution TM of TO(T) *S smaller than the plateau stress tOp > TM « 0.7 TOp; ch is the barrier concentration.
ch in the glide plane shows the expected dependence on solute concentration c. The parameter /?L [cf. Eq. (5-10)] turns out to be smaller than 0.2 for all alloys, justifying the application of Friedel statistics up to the highest solute concentration of 7.6 at.% Mn. In a log-log plot of T oo = To (T= 0) versus solute concentration c a dependence T 0 0 OC C051 is found (Fig. 5-8), i.e. with an exponent between that expected for Friedel and Mott-Labusch statistics (cf., Haasen 1983). A replot of T 00 versus the barrier concentration ch (Fig. 5-8) yields T 00 OC C%5 as expected. The conclusion that also in other Cu base alloys not single solute atoms but certain larger units constitute the effective obstacles was arrived at long before by different authors (e.g., Schwink and Traub, 1968; Traub et al. 1977; Basinski etal., 1972; Feltham, 1968; Butt and Feltham, 1978). Solute pairing was also recently established for the ternary system CuNiPd (Ray and Mitchell, 1989; Wong etal., 1989). The success of the above analysis
seems to be due to several favorable conditions in this system, namely strong size effect (-• pointlike strong obstacles), a constant stacking fault energy, and a weak tendency for short-range ordering (-> "homogeneous" glide of single quasi-independent dislocations). The described analysis, if applied to other systems with strong pointlike obstacles, might well arrive at another of the 4 models quoted above. For CuAl alloys e.g., as measured by Basinski et al. (1972), quite good agreement with predictions of the trough model is reported (Kocks, 1985 a), whereas a special version of it (Suzuki, 1980, 1986) seems to give satisfactory agreement for CuAl as well as for CuNi (Suzuki, 1988). For CuSi alloys strong indications for the Suzuki effect in the stacking fault ribbons have been found from macro- and micro-observations by Coujou (1983 a, b). For systems with weak interaction and, consequently, smeared-out obstacles, the Mott-Labusch statistics may give the appropriate description. However, for this statistics a safe theory involving the ther-
Figure5-8. Plots of the CRSS TOO = T O ( T - > 0 ) for CuMn alloys versus solute concentration c and versus barrier concentration cb, respectively. Diagram in log-log scales to demonstrate the different powers for CuMn alloys (see text) (Wille et al., 1987).
217
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of RC.C. Alloys
mal activation is still missing. The problem is being attacked by computer simulations (Labusch, 1988; Labusch and Schwarz, 1992; Arsenault et al., 1989). First results indicate the simultaneous activation of several weak obstacles as one effective obstacle with considerably enlarged activation energy and activation volume [Sec. 5.2.4.1 (1)]. The similarities to the description with the discrete barrier model (Wille etal., 1987) are apparent. A continuous transition from systems with strong, pointlike barriers to others with weak, smearedout ones as effective obstacles is certainly conceivable. Further theories developed for TO(T, C) as a whole (e.g., Feltham, 1968; Feltham and Kausar, 1990; Butt and Feltham, 1978; Butt etal., 1986; Suzuki, 1981) or only for the plateau region (Schoeck, 1985), fail to explain one or the other of the experimental observations. They are therefore not considered further. We finally remark that in alloys with a higher solute concentration, where glide occurs inhomogeneously by large trains of dislocations of the same sign (Sec. 5.3.2), the basis of the foregoing discussion is violated. The dislocations essentially act no longer independently with the glide obstacles which may be strong or weak, pointlike or smeared out (see Sec. 5.3.4).
18). The anomalies appear even more drastically in the dependence of the strain rate sensitivity 5(7) or activation volume Fon temperature (cf. Wille etal., 1987). They begin to occur for T<80 K (Fig. 5-9). In fact, the analysis of the low temperature regime shows that there are two different types of anomalies of probably different physical origin (Wille and Schwink, 1986). For the range 20 < T< 80 K the possibilities of the self-heating temperature instability (Basinski, 1960; Estrin and Tangri, 1981; Kubin etal., 1982) and the inertial effects of moving dislocations have to be considered, while quantum effects (zeropoint vibrations, tunneling) are expected only at still lower temperature (Startsev, 1983). For CuMn (Wille and Schwink, 1986) as well as for CuAl (Schwarz and Mitchell, 1974) temperature instabilities as reason for the anomalies can be excluded. For T< 80 K they are therefore attributed to inertial effects [cf. Sec. 5.2.3.2 (4)]. As-
o Cu 1.0 at.% At • Cu 0.5 at.% Al
5.3.3.2 Analysis for the Low Temperature Regime ( J < 8 0 K) (Underdamped Dislocation Motion) At low temperatures, a slight deviation in the T O (7) plots (Fig. 5-3) from the expected monotonous curvature is observed. Such anomalies have been found in many cases, cf. Parkomenko and Pustovalov (1982), Startsev (1983). They prevent a safe extrapolation of high temperature data of TO(T) to the value at T=0, T 0 0 in Eq. (5-
0
50
100
150 Tin K
200
250
Figure 5-9. Low temperature part of the temperature dependence of the CRSS for two concentrations of CuAl single crystals (measurements of Basinski et al., 1972), to demonstrate the low temperature anomalies, i.e., the deviation from simple Arrhenius behavior [Eqs. (5-13), (5-18)], which is indicated by broken lines (Wille et al., 1987). The full lines are calculated with Eq.(5-20).
218
5 Solid Solution Strengthening
suming similar behavior of the phonon system for the alloys as in pure Cu, the damping coefficient B = Br + Be + Bph(T) drops rapidly with decreasing T (Alshits and Indenbom, 1986) and the temperature dependent scattering contribution Bph(T) vanishes around 30 K. The remaining radiation damping Br and electron damping Be are independent of r a n d only a small fraction of £ ph (300 K). At 80 K the damping amounts to about 30% of its value at 300 K. Applying the treatment of Schwarz and Labusch (1978) for the underdamped case, a heuristic formula has been derived by Wille et al. (1987) for T O (7) in the underdamped situation: (5-20)
('-(01
17
= T 00
with T0 = AG0/[k'ln(y0/y)]. The factor r [B(T)/B(TD)] accounts for the inertial effect (Tu is the Debye temperature). This relation describes quite well the measured behavior for CuGe (Wille etal., 1987), CuMn [data of Wille and Schwink (1986)], CuAl [data of Basinski et al. (1972)], and CuNi [data of Kamada and Yoshizawa (1971)] as shown in Fig. 5-9 for the example of CuAl. For CuMn in the derivative dxo/dTvs. T [Fig. 5-9 in Wille and Schwink (1986)], the second low temperature anomaly becomes obvious for T<20 K. For the inertial effect an upward curvature of the TO(T) curves towards lower T( < 20 K) would be expected, whereas the observed CRSS remains at considerably lower values for c Mn >3.8at.%. This effect has been ascribed (Wille and Schwink 1986) to zero point vibrations (Natsik, 1979): The inertial effect may be suppressed at higher con-
centrations (c>2 at.% for CuMn) according to the shorter lengths of dislocation segments between obstacles, whereas the effect by zero point vibrations occurs in all alloys below about 20 K. According to Natsik, this effect can be described by introducing a "Debye temperature of the dislocation" 9D appearing as a scaling factor and being related to the usual Debye temperature TD of the crystal by 9D&TDI10. A closer analysis shows that the zero point vibration effect can be detected the better, the higher the stress level at low temperature. Therefore the strongly hardened CuMn system is especially favorable in this respect [cf. also recent work on PbSn by Shepel et al. (1991)], as are also b.c.c. alloys. In this connection we mention recent precise low temperature measurements of the 0.2% proof stress of heavily predeformed polycrystals (Obst and Bauriedl, 1988; Obst and Nyilas, 1991) showing a monotonous increase of T O ( J ) with decreasing T in high stacking fault alloys, while a transition to a smaller slope at a characteristic low temperature (7\ < 35 K) occurs in low stacking fault alloys. Due to the very high dislocation density from predeformation, in this case the direct effect of the solute obstacles (and accordingly inertial effects) seem to be masked and the solute atoms enter indirectly through the change of the stacking fault energy ys. According to early ideas by Seeger (1958) this could cause the edge and screw dislocations to govern the flow stress for T< Tx and T>Tl9 respectively. 5.3.3.3 Analysis for Higher Temperatures (J>300 K) (Mobile Solute Atoms) With increasing temperature (T> 300 K) the CRSS of solid solutions approaches a value which is only slightly temperature
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of RC.C. Alloys
dependent and increases with solute concentration (Kostorz and Haasen, 1969). In the greater part of this "plateau" region the CRSS is not well defined because of fluctuations and serrations in the load course (cf. for CuAl: Fig. 5-3, for CuZn and CuGe: Traub e t a l , 1977). The latter phenomenon is the Portevin-LeChatelier (PLC) effect (Portevin and LeChatelier, 1923) which was already introduced in Sec. 5.2.4.2. For polycrystals it will be discussed in Sec. 5.4.2. In the following the transition to the plateau region (1), to the PLC regime (2), and to the smooth high temperature region (3) will be considered for single crystals. In these regimes the mobility of the solute atoms plays an important role (cf. Sec. 5.2.4.2), summarized in the term "dynamic strain ageing" because it produces an ageing effect during deformation. (For "static strain ageing" cf. Sec. 5.4.2.1.) (1) "Plateau Stress" and Dynamic Strain Ageing In spite of a smooth load trace, in the transition region indications for solute mobility can be detected. A very sensitive method which clearly shows such rearrangements already at rather low temperatures is internal friction (Schwarz and
T =375 338 296 r0 = 11.60 11.86 12.63
Funk, 1983) where changes of solute configurations at the dislocations at about T/Tm «1/5 (Tm is the melting temperature) for waiting times of 0.6 ms were observed. If the solutes are mobile, they will increase the obstacle strength (cf. Sec. 5.2.4.2) during the waiting time /w of the dislocation at the obstacle: AG =AG
r in MPa
+ K 'fit )
(5-21)
AG 00 characterizes the original obstacle strength, f(tw) is a function which increases monotonically with tw, e.g., f(tw) oc c(tw) [Eq. (5-15)], and Ka is a factor assumed to be constant. An increase of A Go has been observed earlier to occur during stress relaxation experiments on CuZn (Neuhauser and Flor, 1978) even well below room temperature. These changes can be qualitatively recognized in Fig. 5-10 as the curvature in the stress relaxation curves. They have been quantitatively described by a diffusion type equation (Flor and Neuhauser, 1980) indicating that rearrangements in the dislocation core occur with activation energies similar to those for pipe diffusion (Oren and Bauer, 1967; Atkinson and LeClaire, 1984; Balluffi and Granato, 1984). Of course, at the considered temperatures no real diffusional motion can occur, but rather a switching of atomic positions to-
248 193 148 77 (K) 14.6017.48 18.73 30.0 (MPa)
10°
219
Figure 5-10. Stress relaxation for various temperatures in stage I of Cu-30 at.% Zn single crystals, plotted as log of the stress rate (— f, which is proportional to the plastic shear strain rate y) versus stress T. T0 indicates the starting stresses (at 7 = 1.4 • 10 ~4 s"1) for stress relaxations at temperatures T. The initial slope of the curves corresponds to the effective activation volume Ve{{ (Eq. 5-24) (Flor and Neuhauser, 1980).
220
5 Solid Solution Strengthening
wards an energetically more favorable configuration resulting in an increase of AG00 by only a few %. Summing up, the plateau region arises as a superposition of a thermally activated stress contribution which decreases continuously with increasing T7, and another contribution originating from (slight) solute motions and increasing continuously with T [cf. the early suggestion by Gleiter (1968)]. Explanations with fixed obstacles (e.g., Labusch etal., 1975) have not been successful (see also Sec. 5.4.2). (2) Serrated Yielding and Dynamic Strain Ageing There is general agreement about the mobility of solutes as the reason for the prominent fluctuations and even serrations in the load curve at intermediate temperatures. They are caused by the locking of dislocations mainly by the size misfit interaction [cf. Sees. 5.2.2.1, (1, ii), 5.2.4.2]. This explains why serrations are observed in CuAl, CuGe, CuMn (large size misfit) but not in CuNi, CoNi (small size misfit). As the locking force for split edge dislocations exceeds that for screws, it is plausible to assume (Suzuki, 1986) that the breakaway process starts at screw dislocations. If the dislocation breaks away from a "solute atmosphere" and if diffusional motion is not rapid enough to reestablish the cloud during the average waiting times, the released dislocation can readily expand over a considerable area, because the breakaway stress of the aged dislocation exceeds that for motion of a fresh dislocation of any character through the fixed obstacle arrangement. The dislocations are therefore able to multiply quickly occupying a narrow region of plasticity in the crystal. The resulting plastic instability belongs to the "type S" as introduced by Es-
trin and Kubin (1988). The microprocess was observed directly by slip line cinematography (Neuhauser etal., 1990) performed simultaneously with recording of the load (Fig. 5-11). A one to one correlation of each serration with the rapid development of a bundle of slip bands could be established. With increasing temperature there is a gradual transition in the load trace from a smooth course to slight load fluctuations, and to rather regular PLC serrations (Neuhauser etal., 1990). The fluctuations correspond to a variation of the number of active slip band bundles naB = ///loc (/ is the external extension rate, /loc the local extension rate produced by all active slip bands in one slip line bundle). They become visible in the load trace if naB is less than about 3 to 10. Serrations (drops in stress T) occur if /loc exceeds the externally applied rate /, such that in the average naB < 1. The stress %{t) drops from the breakaway value Tmax (point B in Fig. 5-11) to a minimum value i min (R in Fig. 5-11) which is determined by the competition between the production of further dislocations and the recapture by the mobile solute atoms: The waiting times at ordinary obstacles in the alloy increase during the drop of stress x{t). Thus rearrangements and diffusive motions in the core and around the mobile dislocations occur with increasing probability. As soon as the source dislocation is arrested for a waiting time longer than the critical waiting time corresponding to a critical value of the increase of the obstacle strength by solute rearrangement,/ cr =/(^ rit ), it will be completely fixed by further solute rearrangement and the plastic deformation stops suddenly (point R in Fig. 5-11). A change in obstacle strength of about 2-3 % occurring between points B (=fB) and R (=/ c r ) can be estimated (Engelke et al., 1992b).
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of F.C.C. Alloys
The serrations like Fig. 5-11 belong to the so-called type B serrations (see Sec. 5.4.2.1) which are associated with the rapid localized plastic strain in a narrow band. This plastic front usually propagates discontinuously along the specimen, as observed directly in CuAl crystals (Hampel etal., 1988) and described in a model by Schwarz (1985). However, this simple Lixdersband-like propagation is no longer observed in the high-temperature PLC range (Neuhauser etal., 1990), where the diffusional ageing of dislocations occurs very rapidly. The increasing solute atmosphere at the source dislocation with higher T is reflected in the first slightly, then rapidly increasing peak stresses in the ''plateau" (PLC) region up to the CRSS maximum (cf. Fig. 5-3). For the strain rate sensitivity S [Eq. (5-16)], taking the breakaway (peak) stresses and using Eqs. (5-21) and (5-15) with y oc t'1 and/oc c(tw), the following
t in s
Figure 5-11. Stress drop in extended time scale recorded during an observed rapid development of a slip band bundle during deformation (Cu-10 at.% Al single crystal, shear strain rate y = 3.6 • 10~5 s" 1 , T = 500K in the Portevin-LeChatelier regime) (Neuhauser et al., 1990).
221
relation can be derived (cf. Schwarz, 1982) di
dlny kT K
eff
PLC
1— kT
(5-22)
Here x = K2(Dtw)2/3 is strongly T-dependent (mainly through the diffusion constant D), and Kl9 K2 are weakly T-dependent quantitities. Equation (5-22) not only explains the increase of the effective activation volume Veff = kT/S [see Eq. (5-24) below] for large ageing times and thus the curvature of stress relaxations in Fig. 5-10. It also shows that SPhC may attain negative values which for increasing T are followed by an abrupt increase, as in fact observed in Fig. 5-5 (Paidar, 1978; Hong, 1985 a, b; Balik and Lukac, 1989; Neuhauser, 1990; Engelke et al., 1992b). A very sensitive tool to monitor plastic instabilities, even those in a scale much finer than the load trace, has been used by Schaarwachter and Ebener (1990) and taseres and Rodriguez (1987) who recorded the acoustic emission during deformation (see also Chmelik et al., 1992). The second authors found indications which partially support the alternative explanation of the PLC effect suggested by Korbel etal. (1976), Korbel and Dybiec (1981), who consider this effect as being the consequence of the correlated motion of dislocations interacting through their stress fields (cf. Pawelek, 1989, and Sec. 5.3.4.2). However, the approach of Kubin et al. (1988) based on the diffusional concepts of McCormick (1972), van den Beukel (1975, 1980, 1983), van den Beukel and Kocks (1982) models quite satisfactorily the appearance of the PLC effect and its dependences on T, c, e, and e, and is supported by the micro-observations during straining in the electron microscope (Tabata et al.,
222
5 Solid Solution Strengthening
1980; Nohara et al., 1980; Monchoux and Neuhauser, 1987), by the results of internal friction (Schwarz and Funk, 1983) and by high resolution recordings of serrations (Schwarz and Funk, 1985). Even rather fine details like the transient flow behavior of dynamic strain ageing (McCormick, 1988; McCormick and Estrin, 1989; Estrin and McCormick, 1991) can be satisfactorily treated on this basis. In view of many open questions (cf. also Sec. 5.4.2.1) existing complete kinetic models for the PLC effect (e.g., Ananthakrishna et al., 1981,1983) represent just a first step. The phenomenological models of the PLC effect based on a hypothetical multivalued force-strain rate law with negative gradient dr/ds give still the most concise description of serrated yielding (Wilcox and Rosenfield, 1966; Penning, 1972; Kocks, 1981; van den Beukel etal., 1983; Schlipf, 1986, 1987; Zbib and Aifantis, 1988a, b; Kubin etal., 1988).
o
15
Figure 5-12. Check of Eq. (5-23) (Friedel, 1964) for non-saturated clouds by plotting the concentration dependence of the product of the maximum of the CRSS TOM at the end of the PLC regime and its location on the temperature scale TM for CuAl alloy crystals (Engelke et al., 1992 b).
(3) Smooth High Temperature Region For crystals oriented for single glide, the PLC serrations suddenly disappear on increasing the temperature beyond the value TM of the maximum TOM in the CRSS T 0 (T) curve (cf. Fig. 5-3). With increasing solute concentration the maximum becomes more pronounced and shifts towards lower temperatures (Engelke etal., 1992b): The nearly linear behavior in Fig. 5-12 can be explained according to a relation proposed by Friedel (1964) for the transition from breakaway of the dislocation from nonsaturated atmospheres to dragging of the atmospheres,
Wl c_
(5-23)
Here Wm is the free enthalpy of binding of a solute atom to the dislocation and Ah is
a structural factor ^0.1 according to Endo etal. (1984). From Fig. 5-12 Wm is found to be about 0.14 to 0.11 eV for the Cu-2 to 15 at.% Al alloys, which appears quite reasonable (cf. Saxl, 1964; Oren etal., 1966). Above TM the curves T O (7) approach a common curve for all concentrations (Fig. 5-3): The alloying effect gradually disappears at elevated T due to the high mobility of solutes such that the interaction of dislocations remains as the controlling process [Sec. 5.2.4.2 (3)]. Comparison of Figs. 5-3 and 5-5 shows an intimate connection of the TO(T) curve with the dependence of the strain rate sensitivity S(T). Both reach a maximum at the same temperature, i.e. when the serrations disappear. In fact, also the slip bands and even fine slip traces disappear at this tern-
5.3 The Critical Resolved Shear Stress (CRSS) and Stage I of RC.C. Alloys
perature. The smooth and slow dislocation movement on many slip planes in this temperature regime has been observed directly in TEM in situ studies on Al-Mg (Nohara etal., 1980) and Cu-Ge (Monchoux and Neuhauser, 1987). The viscous nature of glide for T> TM suggests that now the flow stress is controlled by the edge dislocations with their stronger misfit interaction (Suzuki, 1986). The relation v = xb\B [cf. Sec. 5.2.4.2 (3)] has been checked by recent measurements of the strain rate sensitivity of the CRSS and stress relaxation for Cu-Al alloys (Engelke etal., 1992a,b). They indicate a strong dependence of the mobile dislocation density on stress. 5.3.4 Effect of Correlated Dislocation Motion In this section we will not consider all the consequences of the effects of correlated dislocation motion [cf. e.g., Gilman (1969), Feltham (1976), Neuhauser (1980, 1983), Bengus (1984), and Chap. 4 by Kubin, this Volume], but give just a few indications where, in particular in concentrated f.c.c. solid solutions (cf. Sees. 5.3.2.1 and 5.3.2.2), problems are to be expected. Many of these have not yet been solved to date. 5.3.4.1 Inhomogeneity of Deformation: The Active Slip Volume The commonly used quantities of strain e or resolved strain y assume homogeneous slip on all crystallographic planes of the crystal. In reality, however, only a very small fraction of these is actually sheared, as can be seen immediately by the occurrence of separated slip lines (Fig. 5-6). This was realized already by Orowan (1934), but has been neglected afterwards on the assumption that the relation between the so-called active slip volume Va to the total
223
crystal volume Vo should remain constant. Schwink (1966) pointed out that this assumption must not be fulfilled in general, and for several examples it was shown (e.g., Traub et al, 1977) that F a varies with strain rate and temperature T. As Va is only a small fraction of F o , the true shear strain rate ytrue = l/(mslj exceeds the macroscopic average shear strain rate y = l/(msl0) = e/ms by several orders of magnitude (/a = /0 VJV0 *s the active crystal length, /0 the length of the crystal, / the extension rate, and ms the Schmid orientation factor). The mobile dislocation density which enters the preexponential in the Arrhenius Eq. (5-13) as a local quantity Qml in microscopic modeling, is connected with the average mobile density Qm by Qm = Qml(VJV0). The activation volume Keff measured by strain rate changes [or stress relaxation, Eq. (5-16)] is connected with the true microscopic activation volume V= —(dAG/dz)T according to dln(y/y0) dx (5-24) The correction usually amounts to a few 10% and may reach a factor of 2; it leads to smaller activation free enthalpies determined from thermal activation analysis than without correction. 5.3.4.2 Moving Dislocation Groups As the stress field of a dislocation decreases slowly with distance, dislocations moving in groups interact with each other over considerable distances. Although the large stresses around dislocation groups are compensated as far as possible by the formation of multipoles, the possible syn-
224
5 Solid Solution Strengthening
ergistic effect by movement of the dislocations in a group may produce a smaller macroscopic yield stress than expected for a single dislocation. As revealed by simulations (e.g., Zaitsev and Nadgornyi, 1971; Arsenault and Cadman, 1978; Olfe and Neuhauser, 1988) the simple familiar picture of static pile-ups with their strong stress concentration ( « n - T, n is the number of dislocations in the group) at the front dislocation does not apply for the moving groups: In a field of fixed obstacles of finite height which is not changed by the dislocation, the moving group will always spread out. A dynamic pile-up occurs, however, in all cases where the dislocation diminishes the obstacle strength in the cutting process (energy storing obstacle), such as in short-range ordered or segregated solid solutions (Arkan and Neuhauser, 1987 a, b; Olfe and Neuhauser, 1988; Schwander etal., 1992) or in situations of strain softening (Luft, 1991; Korbel and Szczerba, 1988; Pawelek and Korbel, 1990). Depending on the efficiency of obstacle destruction, this mechanism will produce a stress increase by a factor of about 2 - 3 on the leading dislocation, as found also by the analysis of dislocation groups after in situ deformation in TEM (Clement etal., 1984; Monchoux and Neuhauser, 1987). This means that the group can move through an obstacle field at a stress which is 2 - 3 times lower than a single dislocation would need, because the succeeding dislocations in the group, feeling diminished friction, push the leading one by their interaction stresses. It is well possible that the stress dependence of the observed CRSS is influenced by this effect: In the concentration range around the transition from homogeneous to inhomogeneous and in particular to planar slip, the CRSS increases less strongly (Olfe and Neuhauser,
1988) than expected for the c2/3 or the c1'2 laws which are suggested from theoretical considerations for independently moving dislocations [Sees. 5.2.3.2 (1), 5.2.4.1 (1)]. To explain the observed low yield stress by this mechanism, it is necessary to produce a first group at this low external stress. This may happen during the yield point region (cf. Koppenaal and Fine, 1961; Jackson and Nathanson, 1978; Sec. 5.4.2.1), in particular near the grips, where stress concentrations are unavoidable in usual experiments. The formation of a Liiders band and its propagation for crystals in this range of high concentrations appears to support this view. 5.3.4.3 Slip Transfer Process For low concentrations with independently moving dislocations (Sees. 5.2.3, 5.2.4, and 5.3.3) the threshold stress for a single dislocation, including source stresses, should give the measured CRSS. In the case of higher c, on the other hand, at the transition to planar slip, either the motion of dynamic dislocation groups across the obstacle field, or the local transfer process of slip from previously active slip zones to new ones [e.g., by double cross-slip (Jackson, 1985)] controls the measured CRSS. The latter is suggested from several observations by etch pits, electron microscopy and cinematography of slip lines (Arkan and Neuhauser, 1987 a, b). Theories of the CRSS based on the dislocation motion incorporating the elastic interaction within or between dislocation groups, or based on the slip transfer processes of glide do not yet exist.
5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals
5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals In deformed samples, in addition to the strengthening by solutes, work hardening as another strengthening mechanism is present. This situation will be considered in the following section, again by dividing the discussion into the cases of low T (fixed obstacles) and elevated T (moving obstacles, diffusive processes). In addition, in this section a short review of the relatively small knowledge on alloys with the hexagonal close packed structure will be given. 5,4.1 F.C.C. Alloys at Low Temperatures (T<300K) 5.4.1.1 Macroscopic Behavior The stress-strain curves of f.c.c. single and polycrystals shown in Fig. 5-2 indicate already that, except of stage I in crystals oriented for single glide (Sec. 5.3.1.1), the work hardening rates are only slightly affected by alloying. In a very first approximation, the curves are shifted in the scale of the resolved shear stress T just by the CRSS T 0 , i.e. i(y, c) = T 0 (C) + r r (y), where Tr denotes the contribution by strain hardening. The simple additivity of the two hardening mechanisms is to be expected only for low solute concentrations, when slip is still "homogeneous", otherwise, the averaged strain y is no longer a safe measure of the microscopic events (cf. Sec. 5.3.4.1). Crystals oriented for multiple glide, e.g. CuMn with c < 5 at. % (Neuhaus and Schwink, 1992) are more appropriate for comparisons than single glide because of the lack of stage I. The plot of the work hardening rate 0 = di/dy versus "reduced stress" Tr = T — T 0 for [100] and [111] orient-
225
ed pure Cu and CuMn alloys in Fig. 5-13 shows that for xr > 20 MPa the common behavior of pure Cu is approached by the alloy of each orientation which means a transition to the simple additive behavior. The initial hardening rate of the alloys exceeds that of pure Cu by about 60-70%. Polycrystals [cf. Fig. 5-2 and e.g., Vohringer and Macherauch (1967 a, b); Vohringer (1975); Mohamed et al. (1988)] exhibit a similar stress-strain curve as the multiple slip crystals. Nakanishi and Suzuki (1974) studied the dependence on grain size d according to the Hall-Petch relation for the yield stress <7y = Gi + ky-d~1/2 (withconstants o{ and ky independent of d) for CuAl and CuNi alloys and found characteristic differences in the Hall-Petch parameter ky. Calculations of the stress redistribution among grains in deformed polycrystalline alloys have been performed by Weng (1984), Zhu and Weng (1987).
700-
600-
(a) [111] Cu 0.8 at.% Mn (b) [100] Cu 1.2 at.% Mn
70 Figure 5-13. Work hardening rate 0 = dz/dy for CuMn crystals oriented for multiple slip with [111] and [100] axes, respectively, versus reduced stress Tr = T — T 0 (T0 is the CRSS) at room temperature. Arrows indicate initial 9 values for pure Cu crystals for comparison; dotted lines extrapolate the high stress curves to lower stresses (Neuhaus and Schwink, 1992).
226
5 Solid Solution Strengthening
5.4.1.2 Evolution of Microstructure
In multiple glide Cu crystals (Gottler, 1973; Ambrosi et al., 1974) a cell structure develops already in the very first stages of deformation, and shows a nearly equal distribution of dislocations of all characters and equivalent slip systems. In single glide Cu crystals (Mughrabi, 1975) the cell structure can develop only gradually during stage II and in particular after the transition to stage III. Therefore the multiple glide case is better suited in order to understand the important strengthening mechanisms and their superposition. The same behavior as in pure Cu has been observed in crystals of solid solutions (Neuhaus and Schwink, 1992) with the important difference that, at a given reduced stress Tr = T — T 0 , the width of the cell walls is larger than in pure Cu, while the density of dislocations inside the walls QW is practically the same. As the cell interior is nearly free of dislocations, the average total dislocation density gt is connected with the volume fraction of walls/ w by Qt = Qw'fw. The behavior of the total dislocation density Qt and of the fraction / w with increasing reduced stress xr is shown for Cu and CuMn crystals for comparison in Figs. 5-14 and 5-15, respectively (see also Neuhaus et al., 1989; Hilscher and Wilkens, 1989). They show that, in the range of reduced stresses > 20 MPa, the hardening process in the alloys is the same as in the pure metal. With the forest density in the cell walls, @wf, the total flow stress of all samples coincides on a common line (Fig. 5-16) according to ~~
(5-25)
The experimentally found value of ocwf = 0.26 + 0.03 compares favorably (Neuhaus and Schwink, 1992) with the lowest a value calculated for attractive junction reactions
0
10
20
30
40
T-TQ
50
60
70
in MPa
Figure 5-14. Square root of the total dislocation density, y/()t, determined after deformation to various stress levels from TEM micrographs of large specimen areas (> 3000 um2) plotted versus reduced stress T — T 0 . (o) Pure [100] oriented Cu single crystals (Gottler, 1973); (A) pure [111] oriented Cu crystals (Ambrosi et al., 1974); (•) [100] oriented Cu-3 at.% Mn crystals; (A) [111] oriented Cu-0.8 at.% Mn crystals (Neuhaus and Schwink, 1992).
between cutting dislocations (Schoeck and Frydman, 1972; Piischl et al., 1982).
5.4.1.3 Superposition of Solid Solution and Strain Hardening
The results shown in Sees. 5.4.1.1 and 5.4.1.2 indicate already a linear superposition of these two contributions in a wide range of work hardening, at least in "simple" solid solution crystals with multiple slip orientation and homogeneous slip. (For pure work hardening cf. Chap. 2 by
5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals
1.0-
0.5-
10
20
30 T-TQ
40 50 in MPa
60
70
Figure 5-15. Evolution with reduced shear stress (T — T0) of the areal ratio covered with dislocations in cell walls, / w (equal to the volume fraction), for multiple slip [100] single crystals of Cu-3 at.% Mn (•) (Neuhaus and Schwink, 1992) in comparison to pure Cu (o) (Schwink and Gottler, 1976).
Sevillano, this Volume.) Such additivity is expected for a mixture of many weak (i.e., the solute barriers) and few strong obstacles (i.e., the forest dislocations) (Kocks et al., 1975). A quite sensitive check for the
10
20
30 r-rQ
40 50 in MPa
60
70
227
superposition law is the stress dependence of the strain rate sensitivity (Kocks, 1980), the socalled Haasen plot: S (or 1/V) versus T. A linear behavior with the same slope as for the pure solvent metal indicates a linear superposition of the stress contributions. Examples for CuNi alloys in Fig. 5-17 show approximately this behavior, the better the lower the deformation temperature [for details, cf. Neuhauser et al. (1979); cf. also similar results by Saitou and Hikage (1985) and, for polycrystals, den Otter and Vetter (1978)]. Den Otter and van den Beukel (1979) found this simple relation to be valid only for polycrystalline CuNi alloys, whereas for CuAu and CuZn it has to be modified. Experiments on series of Cubased polycrystalline alloys by Vohringer and Macherauch (1967 a, b), Vohringer (1975) show similar behavior as single crystals. Constitutive modeling of solute and strain hardening was attempted e.g. by Schmidt and Miller (1982) and Henshall and Miller (1990).
Figure 5-16. Square root of the forest dislocation density in the cell walls, / plotted versus reduced shear stress T — T 0 , for various pure and alloyed multiple slip crystals: (•) [100] Cu, (A) [111] Cu, (A) [111] Cu-0.8 at.% Mn, (•) [100] Cu-1.4 at.% Mn, (•) [100] Cu-3 at.% Mn (Neuhaus and Schwink, 1992).
228
5 Solid Solution Strengthening
0.40
Figure 5-17. "Haasen" plot of strain rate sensitivity S = Ax/ A\ny = kT/V(determined from stress relaxation experiments) along the strain hardening curve (characterized by the flow stress T), for single glide oriented CuNi alloys, in comparison with pure Cu, at T = 78 K (Neuhauser et al., 1979). 10
20
30
40
50
60
5.4.2 EC.C. Alloys at Elevated Temperatures (J>300 K) 5.4.2.1 Static and Dynamic Strain Ageing Static ageing means the occurrence of a yield point phenomenon (upper and lower yield stress) on loading (or reloading) after a definite annealing treatment, during which the solute atoms have time to concentrate around the dislocations and pin them, so that a higher stress is needed to activate the first sources than necessary for further dislocation multiplication by "fresh" dislocations. A recent theoretical treatment of static strain ageing has been given by Kubin et al. (1992). An especially large difference between the yield stress (upper yield point) and the flow stress (lower yield point) has been attained by Ray and Mitchell (1989) for a specially designed alloy (CuNiPd) forming tightly bound pairs of solute atoms. Dynamic strain ageing denotes the analogous pinning mechanisms occurring during straining, in competition with the movement of dislocations, i.e. occurring essentially during the waiting times of the dislocations at obstacles. The considerations in Sec. 5.3.3.3 on the influence of mobile solute atoms will now be extended to the case of increasing deformation. We
70
use again as a distinct and sensitive indication of the mobility of solutes the "Haasen plot" S(x) [cf. Sec. 5.4.1.3, Kocks (1980)]. The extrapolation (T-»0) and the slope should be positive in case of stationary thermally activatable obstacles with some athermal component, as in Fig. 5-17. The values Socl/V for T->T 0 increase with increasing solute concentration as expected. A decreasing slope of the curves with increasing temperature is generally attributed (at least in part) to the mobility of the solute atoms. Examples for this are shown in Fig. 5-18 a, b, for [100] CuMn alloy crystals where the slope of S(T —T 0 ) becomes negative for solute concentrations > 5 at.% at room temperature, or for higher T if 2 at.% Mn is chosen (Diehl et al., 1988). Correspondingly Fig. 5-18 c shows that, the lower the base strain rate e l5 i.e. the longer the waiting times, the more pronounced is the negative slope in the S vs. (T — T 0 ) diagram. If the concentration or temperature are high enough, S may even attain negative values (cf. Figs. 5-5 and 5-18 a, b). This has often been taken as a criterion for the onset of serrated yielding [Wilcox and Rosenfield (1966), see e.g. Kubin and Estrin (1985) and earlier work cited there]. However, there is experimental evidence (Mul-
229
5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals
2.0 10-3'
X at.% Mn , 296 K, [100] 3- 10"6 1/s
Cu \ \ yr=5.0
^ 1.5-10"3
\
•to
3.3
A 4?,/
^ 1.0 10"3 X -
0.5 10"3
X-
2.0\ _O8 .
**< •«-^^ 40 50
X-
n 20
10 -0.5 •io-
Cu 3.3 at.% Mn, 293 K, [100]
• ^ .
30
"
—
-
"
60
in MPa
3
70 (a)
10
Cu 2 at.% Mn [100]
2.0-10"3 . 1.5-10"3 _
1.0-10"3
y
0.5-10"3
296 K
^\350 0 -0.5-10"
3
x20
450 Kx
^
30
K
/
40 , 50
" ~400 K
D) D
60 70 in MPa (b)
ford and Kocks, 1979 a, b; Ling and McCormick, 1990) that 5*<0 is only a necessary, not a sufficient condition for serrations (Schlipf, 1991, 1992). Much more work than for single crystals (e.g., Brindley and Worthington, 1970; Tensi et al., 1972; Tensi and Pless, 1972, on AlMg; Engelke etal., 1992b; Neuhauser etal., 1990, on CuAl) has been done for polycrystals which show serrations along the stress-strain curve usually only after a critical prestrain (e.g., McCormick, 1972, 1974; Rauchle etal., 1973a, b; Mayer etal., 1978; Miura etal., 1987). Various typical characteristics of these serrations depending on the deformation conditions (s, s, T, c), as originally observed by Caisso (1959) have been described (Munz and
20
30 40 50 (r-r o j in MPa
60
70
Figure 5-18. Normalized strain rate sensitivity S • b3/ (kT) = b3/V plotted versus reduced stress T — T0 for [100] oriented crystals of (a) Cu-Mn alloys with various concentrations at room temperature, determined from strain rate changes (fi2/fii) by a factor of 2; (b) Cu-2 at.% Mn for increasing temperatures (^/^ = 2) (serrations occur in the dashed parts of the curves); (c) Cu-3.3 at.% Mn at room temperature for increasing base strain rates e1, determined by changes (£2 by stress relaxations (s2 = 0) (Diehl et al., 1988).
Macherauch, 1966) and attributed to various modes of strain localization and propagation. Most popular in the literature is the classification by Brindley and Worthington (1970) (cf. Wijler and Schade van Westrum, 1971, 1973; Cuddy and Leslie, 1972; Pink and Grinberg, 1982; Fellner et al., 1991) as types A, B and C serrations. They were more recently investigated by Chihab etal. (1987) on Al-5at.% Mg polycrystals with respect to the conditions (e, T) and their appearance applying video recordings. They have been discussed in the context of the phenomenological approach to the PLC effect [cf. Sees. 5.2.4.2 (2), 5.3.3.3 (2)] by Kubin et al. (1988): type B is similar to the PLC phenomenon in single crystals shown in Sec. 5.3.3.3 and
230
5 Solid Solution Strengthening
corresponds to an intermittent propagation of a narrow plastic zone; type A is associated with a nearly smooth continuous propagation of a deformation band over the specimen length with a sharp serration whenever a new band is initiated at one end of the specimen. This indicates a stress concentration in the propagating plastic front (cf. Sec. 5.3.4.2). In type C serrations extremely large stress drops occur (sometimes accompanied by audible acoustic emission) with plastic bands appearing at random within the specimen and sometimes with some plasticity on reloading. The investigations of the conditions for onset, dis- and reappearance of the PLC serrations began with early experimental studies (e.g., Munz and Macherauch, 1966; Rauchle et al., 1973 a, b) and are presently continued from the theoretical as well as from the experimental side (Kubin and Estrin, 1990,1992; Kalk and Schwink, 1992). We note that the discussion so far does not permit one to differentiate conclusively between two proposed possibilities of diffusive effects on the flow stress % which can be considered as composed of a stress Td originating from dislocation interaction and rf from solute interaction (Wycliffe et al., 1980; Mulford and Kocks, 1979 a, b; Schwink and Wille, 1980; Schwarz, 1982; Hong, 1984,1985). While Kocks favors the view that the solutes increase the dislocation interaction by pipe diffusion from the forest to the glide dislocation (Sleeswyk, 1958) increasing x d , other authors (e.g., Schwarz, 1982; Louat, 1981; Balik and Lukac, 1989) consider as more important the diffusive effects to increase r f , either again by pipe diffusion [cf. Sec. 5.2.4.2.(1)], or by bulk diffusion (e.g., Mayer et al., 1978; van den Beukel, 1975, 1980, 1983) assisted by deformation induced vacan-
cies. Recent experiments allowing to discern between pipe and bulk diffusion indicate that a pipe diffusion mechanism (Springer and Schwink, 1991) as well as bulk diffusion mechanisms (Kalk et al., 1993) may occur on the same system depending on the deformation conditions. Arguments against the vacancy hypothesis in every case have been inferred from results on the interstitial f.c.c. solid solution Ni-C (Kocks etal., 1985) while it is strongly supported again by Ling and McCormick (1990) for AlMgSi. As indicated in Sec. 5.3.4 the question arises whether the externally measured stress is the correct quantity for the analysis if effects of correlated dislocation movement and obstacle destruction are relevant. Then local stress concentrations and their relaxation by local plasticity (Korbel et al., 1976; Korbel and Dybiec, 1981) contribute to the PLC serrations. An attempt to combine both the diffusional and cooperative dislocation aspects was proposed by Schoeck (1984). The localization of flow in the deformation mode of serrated yielding was considered by Ling and McCormick (1990) and Estrin etal. (1991), and a dynamic view of the PLC effect and the propagation of PLC bands was suggested by Pawelek (1989). The dislocation dynamics in strain ageing alloys has been very recently treated by Schlipf (1992). 5.4.2.2 Static and Dynamic Recovery The main microprocesses at elevated temperatures are the annihilation of dipoles and shortening of segments, often supported by cross-slip of screw and climb of edge components, as well as the polygonization of dislocations by rearranging into energetically more favorable configurations (Hansen and Kuhlmann-Wilsdorf, 1986). They are in principle the same in pure and
5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals
alloyed metals: First a sharpening of the cell walls into subboundaries, then a coarsening of the subboundary structure (e.g., Hasegawa and Kocks, 1979; Langdon, 1983; Orlova and Cadek, 1986). According to the interaction of solutes with dislocations the static (i.e., at 8 = 0) as well as the dynamic (i.e., at e>0) recovery processes will be retarded due to the "fractional stress" of the solutes. For more details see e.g. McQueen and Evangelista (1988), Soliman and El-Dahsham (1988), Oikawa and Yoshinaga (1988), and Kocks (1985 b) (see also the contributions by Blum, Chap. 8, and Mukherjee, Chap. 9, in this Volume). The onset of recovery processes can be most easily checked in a plot of 9x versus T (Liicke and Mecking, 1972), cf. Fig. 5-19, which has been selected for an interstitial
231
f.c.c. solid solution. Recovery leads to complicated transients if a change in temperature or strain rate is made in a tensile test or a stress change in a creep test (Kocks and Mecking, 1981). Takeuchi (1977 a, b) found for CuAl single crystals a pronounced dependence of the work hardening rate on the initial orientation of the crystal: While no dramatic effect occurs in single slip crystals, the maximum slope of the stress-strain curve 9max has a sharp maximum around 600 °C for the multiple glide crystals. It is supposed to be connected with the intersection of dislocations moving on different slip planes and dragging their solute atmospheres. Recovery is probably superposed in particular in the sharp decrease of 0max with increasing temperature. 5.4.3 H.C.P. Alloys The special features of solid solutions hardening in the other close packed structure, the hexagonal densely packed structure, are connected with the possible variation of the axial ratio with composition and the different behavior of basal and nonbasal slip systems changing with alloying (Haasen, 1983). In the literature, which is relatively sparse up to date, the systems based on Mg, Cd, and Zn have been mostly investigated, cf. the recent review by Lukac (1981). 5.4.3.1 The Critical Resolved Shear Stress
77>x10 3
Figure 5-19. Product of work hardening rate and flow stress versus reduced flow stress, all normalized, for interstitial f.c.c. NiC solid solutions at the indicated temperatures of 295 K and 555 K. The curves indicated by "N" are for "pure" Ni (<0.05 at%C), those indicated by "C" and "D" are for NiC alloys. ("C": 0.28 at.% C before and 0.15 at.% C after the test; "D": 0.9 at.% C both before and after the test.) The arrows mark the beginning of dynamic recovery. (From Kocks et al., 1985.)
Mg alloys have been studied e.g., by Scharf et al. (1968) and Lukac (1992) MgCd single crystals) and by Akhtar and Teghtsoonian (1969) (MgZn single crystals), crossing the ideal axial ratio of ^/S/3 with the possibility of basal as well as pyramidal or prismatic slip. The temperature dependence of the CRSS for basal slip is similar to the f.c.c.
232
5 Solid Solution Strengthening
case with a strong decrease with increasing T in the range below room temperature and a "plateau" at higher T, and can be interpreted accordingly, with the special feature that size effect and modulus effect appear to be additive. For basal slip (Fig. 5-20 a) the addition of solutes increases the CRSS for all temperatures and decreases the apparent activation volume. Up to 0.025 at.% there is only an increase of short range obstacles, i.e. of the thermally activated component T* of the CRSS, while at higher c both T* and TM (athermal component) increase. Quantitative estimates indicate a transition of the rate controlling mechanism from the intersection of forest dislocations at very low c [where the forest dislocation density increases with alloying, cf. Seeger (1958)], to the dislocation pinning by solute atoms at higher c. For prismatic slip, which is important in addition to basal slip in polycrystalline aggregates (Akthar and Teghtsoonian, 1968), a solution softening effect has been observed at low temperatures in systems with axial ratios smaller (MgLi: Ahmadieh e t a l , 1965), equal to (MgZn, Fig. 5-20) and larger (MgAl: Akthar and Teghtsoonian, 1969) than the ideal value. For higher r t h e CRSS increases at low c and decreases for higher c levels. This is explained by an increasing x^ and decreasing T* with solute content, and the latter by a decrease of the Peierls stress, which is supposed to control prismatic slip contrary to basal slip. Probably the nucleation of a kink pair becomes easier in the presence of a small amount of solute (Mitchell and Raffo, 1967), similar as in b.c.c. materials (Sec. 5.5). The temperature dependence of the CRSS of Cd-base alloys (Lukac and Stulikova, 1974; Rojko and Lukac, 1973; Lukac and Trojanova, 1979) shows quite
0.20
Figure 5-20. Stress-strain curves of MgZn single crystals (a) for basal glide orientation at T = 295 K, for the following concentrations (in at.% Zn): (1) 0, (2) 0.019, (3) 0.054, (4) 0.15, (5) 0.268, (6) 0.45; (b) for prismatic slip orientation at T = 78 K with concentrations (in at.% Zn): (1) 0, (2) 0.006, (3) 0.256, (4) 0.45. (From Akthar and Teghtsoonian, 1969.)
similar simple thermally activated behavior as found for f.c.c. crystals in the low temperature range. The interaction parameter £L determined at very low T (Navratil et al., 1983) has been found to be in good agreement with Labusch's theory (cf. Sec. 5.2.2.4) if the interaction of solutes with edge dislocations is taken to be rate controlling, with sL = (S2 + oc2ri2)1/2.
Furthermore, for Zn alloy crystals, in the range of very low T (e.g., Mikulowski and Wielke, 1988) quite similar anomalous behavior as found in f.c.c. crystals (Sec. 5.3.1.2) has been detected. This anomaly increases with deformation and is sup-
233
5.4 Solid Solution Hardening and Strain Hardening in Densely Packed Crystals
posed to originate from underdamped dislocation motion (Sec. 5.3.3.2), as confirmed in a recent analysis by Zagorniko etal. (1986). They showed that the rate controlling obstacles in fact are not single impurity atoms, but pair complexes (cf. Sec. 5.3.3.1), which below T < 3 0 K are surmounted as a result of the quantum motion of dislocation segments.
Cd-Zn
10
0
5.4.3.2 Work Hardening and High Temperature Behavior
Because of the low melting temperature and corresponding rapid recovery processes in h.c.p. metals even around room temperature, the measured results on work hardening are difficult to interpret. Recovery is supposed to be the reason for the observed maximum of the work hardening rate in the easy glide region (stage A) at T/Tm = 03 found for Zn and Cd (Wielke, 1976) and CdZn (Hamersky etal., 1990). The observed concentration and temperature dependence of the work hardening rate of stage A resembles that of the CRSS for prismatic slip (Akthar and Teghtsoonian, 1969) and is therefore explained by the cross-slipping of dislocation segments in prismatic planes (Lukac, 1981). This on the one hand produces short pieces of forest dislocations, but on the other hand makes annihilation more probable with increasing temperature, causing a sharp maximum of the stress at the end of stage A (Fig. 5-21), similar to the maximum of the ratio of the critical stresses for prismatic and basal slip. The length of stage A is determined by the formation of sessile dislocations by a reaction of basal dislocations with dislocations on the pyramidal slip system. Such reactions are much more difficult than analogous Lomer-Cottrell reactions in f e e . crystals, because in h.c.p. usually the
200 7" in K
300
Figure 5-21. Temperature dependence of the ratio of the stress at the end of the easy glide stage (TA) and the CRSS (T0) for CdZn crystals with the indicated concentrations. (From Lukac, 1981.)
CRSS for basal glide is much smaller than for nonbasal glide. For higher temperatures, similar diffusive processes of solutes as discussed in Sec. 5.2.4.2 will occur, resulting in serrated yielding during deformation which is often observed in the "plateau" region of the T O (7) curve. A recent investigation of the PLC phenomenon in Zn-0.2 at.% Ga and Zn-0.2at.% Ag alloys was performed by Mikulowski and Korbel (1982). For these alloys a definite temperature interval was identified where typical load serrations occurred, which are connected with localized deformation in coarse slip bands, quite similar to the case of f e e . alloys (Sees. 5.3.3.3, 5.4.2.1). The authors explained these observations without diffusive processes, as ageing effects were not found (Mikulowski et al., 1982). Another typical high temperature phenomenon, superplasticity (e.g., Langdon, 1991), was studied in h.c.p. alloys e.g. by Vostry et al. (1985,1988) for Zn-0.25 wt.% Cd, and found to occur even for rather large grain sizes in Zn-1.1 wt.% Al (Malek, 1988). Extensive investigations of the temperature dependence and of the strain rate
234
5 Solid Solution Strengthening
sensitivity have been performed by Malek et al. (1988) on Zn-0.35 wt.% Al-0.25 wt.% Cd showing the best condition for superplasticity at T - 3 7 0 K ( = 0.53 T J at a strain rate of s = 1.7-10' 3 s" 1 .
5.5 Solid Solution Effects in B.C.C. Alloys Solid solutions with the b.c.c. structure certainly have most importance in practical applications because of their outstanding strength and variability. The understanding of the hardening effects in these systems is difficult for several reasons which distinguish the plastic deformation in b.c.c. structures.
0
0.2 ( U 0.6 0.8 1.0
5.5.1 Peculiarities of Plasticity of B.C.C. Structures The peculiarities of the plasticity of b.c.c. metals are (cf., Sestak and Seeger, 1978; Brunner and Diehl, 1991a, b): (i) A strong temperature dependence of the work hardening curve. (ii) The existence of a critical temperature TK below which the CRSS increases very strongly with decreasing temperature. (hi) The appearance of "noncrystallographic" slip planes observed by slip lines at the crystal surface. (iv) The strong dependence of the CRSS on the crystallographic orientation of the deformation axis and a pronounced asymmetry for loading in tension and compression. (v) The especially strong effect of small amounts of interstitial solute atoms (cf. Fig. 5-22). These features have been traced back to the following peculiarities of the b.c.c. structure and its dislocations: (1) Contrary to the close packed structures (f.c.c, h.c.p.), the b.c.c. structure per-
Figure 5-22. Stress-strain curves for tensile deformation at various temperatures for (a) middle oriented FeSi crystals (substitutional alloys) with concentrations indicated in at.% at T = 113K and 295 K (s = 5.5- l O ^ s " 1 ) (from Sestak and Seeger, 1978); (b) N-doped Fe single crystals (interstitial alloys) with concentrations indicated in at.-ppm at 77 K (e = 1.7- 10" 4 s" 1 )(from Aono et al., 1981).
mits not only substitutional but also a variety of interstitial solid solutions. Due to the tetragonality of the distortion field of interstitials, their interaction with dislocations is much stronger than that with substitutional atoms. Special effects result from the strong interaction between interstitial and substitutional foreign atoms in
5.5 Solid Solution Effects in B.C.C. Alloys
the b.c.c, structure, implying that their effects are not additive (cf. Haasen, 1983). (2) Dislocations in the b.c.c. structure show different core splitting depending on their character: While the edge dislocation is extended in its slip plane and therefore moves rather easily, the screw dislocation shows a complicated core structure with a threefold splitting (Vitek, 1985). The existence of two equivalent configurations of the core characterized by the "polarity" (Seeger and Wuthrich, 1976) implies an especially high Peierls potential for the propagation of screws and accounts for the "noncrystallographic" slip direction. The screw dislocations control the macroscopic flow stress [after an extended microplastic region due to the movement of edges and kinks on mixed dislocations, cf. Seeger and Sestak (1971)]. (3) The Peierls stress of screws varies with the type of the applied stress (violation of Schmid's law for the CRSS) and there is a stress-induced asymmetry in the motion of screws. This accounts for the observed complicated dependence of the flow stress on crystal orientation (Sestak, 1972). The sensitivity of the Peierls stress to details of the atomic potentials (Vitek, 1985; Duesbery, 1984) is responsible for the differences between various b.c.c. metals. (4) The effect of solutes on the movement of screws may be twofold: On one hand, the propagation of the dislocation by kink nucleation and lateral motion is retarded by the solute obstacles in the slip plane, with strong parelastic effects also on screws because of the tetragonal distortion. On the other hand, by its stress field, the solute may help in the generation of a new kink pair, resulting in a reduction of the resistance to dislocation motion if the nucleation is rate controlling ("solid solution softening").
235
In comparison with f.c.c. alloys, there are the following important differences (cf. Suzuki, 1979): (a) dr o /d T of b.c.c. alloys for T< TK is smaller than that for the pure metal. (b) The concentration dependence is linear and solution softening at low T is observed in many b.c.c. alloys. (c) The hardening rate ATO/AC for b.c.c. alloys depends mainly on the modulus misfit. (d) In b.c.c. alloys the dislocation density increases rapidly in the initial stage of plastic deformation and reaches an equilibrium density which gives the lowest stress at a given strain rate. In view of the wide variability of the effects in different b.c.c. solid solutions (cf. Pink and Arsenault, 1979), we will restrict the following discussion to a few typical examples taken from recent results on the Fe system. 5.5.2 Solute Hardening and Softening of B.C.C. Crystals by Substitutional Atoms
The peculiarities of b.c.c. plasticity are generally suppressed by the addition of substitutional atoms, i.e. the temperature interval with 3-stage work hardening is extended to lower temperatures with increasing solute content; the flow stress, work hardening rate in stage I and length of stage I increase (Sestak and Seeger, 1978; cf. Fig. 5-22 a). For > 6 a t . % Si the work hardening curves become parabolic independent of T. The orientation dependence also decreases with increasing solute concentration. The solutes cause a strong change of the slip line structure: instead of nearly homogeneous wavy slip in pure crystals, sharp and only slightly wavy slip steps are observed in the alloys; they are clustered and oriented in the average along the (Oil)
236
5 Solid Solution Strengthening
plane of maximum stress. The slip steps are stronger at 195 K than at room temperature (Hattendorf and Biichner, 1990). The propagation of slip in FeSi has been investigated by Zarubova and Kadeckova (1972), Sestak and Novak (1974) and Sestak and Arnold (1974). For a review of dislocation mobility data see Nadgornyi (1989). The temperature dependence of the CRSS shows an increase of T 0 at ambient temperature and of the critical temperature TK with increasing solute concentration. For instance, for Fe alloys, the substitutional solutes V, Cr, Co produce weak, the solutes Si (Fig. 5-23), Mo, and in particular P produce a strong hardening effect corresponding to the large modulus interaction (cf. Table 5-1). At low T in many systems (see the review by Pink and Arsenault, 1979) a slight decrease of the yield stress with increasing solute concentration in the range of a few percent is observed (Fig. 5-24) and called "alloy softening". It disappears at both high and very low T. The complicated interference of hardening and softening in-
250 -
Fe
v
200 150
^
\
100 50 n 0
^
V
.% Si
^ \ 2 a t . % Si 1 at o/o Si
o^T^" -
100 200 300 400 500 Tin K
Figure 5-23. Temperature dependence of the CRSS (lower yield point) of substitutional solid solutions of FeSi single crystals of middle orientation (tensile tests, e = 5.5 • 10~5 s"1) (from Sestak and Seeger, 1978).
3.0 at.% Si U at.% Ni 3.0 at.% Ni
100
200 r in K
300
Figure 5-24. Low temperature solid solution softening effect in substitutional Fe alloys with indicated concentrations in a plot of CRSS versus temperature. (From Pink and Arsenault, 1979.)
duced by different solute species in the various solvents and the problems to extract the interstitial impurities from the b.c.c. structures for long time have prevented from establishing the mechanism for solute softening (cf. Sec. 5.5.4). In some systems even the interaction between substitutional and interstitial constituents is necessary for the occurrence of alloy softening. At very low T anomalies in the temperature and strain rate dependence of the CRSS have been observed (Takeuchi et al., 1982, 1985) which are explained either by the quantum mechanical vibrations of dislocations [cf. Sec. 5.2.3.2 (4, ii)] or very recently by the quantum statistics of kink nucleation (Regelmann etal., 1991). While in most other substitutional Fe alloys (FeSi, -Mo, -Ge, -Be, -Ni, -Pt, -Cr) T 0 at room temperature increases linearly with concentration, in FeP a definite deviation from linearity has been observed for higher concentration [Hattendorf and Buchner (1990) for single crystals, Spitzig (1974) for polycrystals], see Sec. 5.5.4. Further a dependence of the CRSS on the final heat treatment was noticed, indicating an influence of short-range order (SRO).
5.5 Solid Solution Effects in B.C.C. Alloys
In alloys with high solute concentrations like Fe-7 to 20at.% Si, ordering effects give a considerable contribution to the mechanical strength. Biichner and Pitsch (1985) determined the configurational energies in SRO solid solutions, and Falk and Biichner (1990) considered the SRO and LRO parameters including the destruction of order by plastic deformation [cf. Sec. 5.2.2.1 (3)]. In the high temperature region, serrated flow has been observed [e.g., for Fe-6 at.% Si: Zarubova and Kadeckova (1979) and for Fe with 0.15 at.% Ti and 1.5 to 3 at. % Ni and Si: Cuddy and Leslie (1972)], with type B and C serrations and the propagation of deformation zones, similar to f.c.c. alloys (Sec. 5.4.2.1). Accordingly, they are commonly discussed in terms of diffusive effects. From the serrations an activation energy has been deduced which is half of that for diffusion of the substitutional atom in Fe. Annealing experiments contradict the vacancy hypothesis as reason for the high diffusivity (Cuddy and Leslie, 1972). 5.5.3 Solute Hardening and Softening of B.C.C. Crystals by Interstitials Only relatively few reliable investigations exist on the solid solution hardening by interstitials because of the great difficulty of removing all unwanted traces of them (C, N, O, H) from the b.c.c. lattice. The examples in Fig. 5-22 b indicate the big hardening effect produced by only a comparatively small amount of interstitial atoms, as expected from their tetragonal distortions (Sec. 5.2.2.1). For T> TK the CRSS, the work hardening rate in stage I, and its length increase with solute content, while the maximum work hardening rate 6n decreases. The few slip line studies indicate an increasing ten-
237
dency to coarse slip with preferential single slip approximately along the plane of maximum shear stress. For T
238
5 Solid Solution Strengthening
5.5.4 Models of Solid Solution Hardening and Softening in B.C.C. Crystals The model of double kink nucleation and propagation along screw dislocations worked out by Seeger (1981, 1984) proved to be successful in describing details of the temperature and strain rate dependence of the CRSS of pure b.c.c. crystals in the range of TK/20.1 at.% (Hirth and Lothe, 1968). The propagation of kinks along the dislocation is retarded by solutes only slightly, as the interaction energy is estimated to be in the range of 0.01 to 0.1 eV (Takeuchi, 1968, 1969). However, a considerable hardening occurs from a statistical interaction. It has been estimated from simple arguments by Nabarro (1985 a) to result in the dependence 2
b2roz
cF
(5-26)
(Ew is the interaction energy between solute atom and dislocation, ak the height of the kink). A further contribution of resistance arises from the possibility that kink pairs in the screw dislocation may be formed in three distinct {110} planes. Thus, when kinks on different planes meet each other, a sessile jog will be formed sep-
arated by about twice the mean free path of a kink. Depending on their mobility, this contribution may amount to 0.08-0.4 of the total flow stress (Suzuki, 1979). The detailed theory worked out by Suzuki (1971, 1979, 1984), which accounts for all the peculiarities of b.c.c. alloy plasticity mentioned in Sees. 5.5.1 to 5.5.3, has recently been checked by Hattendorf and Buchner (1990, 1992) improving Suzuki's approximations for the strengthening by substitutional atoms. They found very good agreement for FeSi, FeMo and FeGe data taken from literature. Discrepancies observed for FeP are interpreted as the consequence of grouping of P atoms to doublets or triplets which act as effective obstacles and whose presence was checked by MoBbauer spectroscopy (Hattendorf, 1986) (cf. analogous situations in f.c.c. alloys in Sec. 5.3.3.1). The analysis yields an interaction energy of £ w ^0.1 eV for the FeSi, -Mo, and -Ge alloys, and £ w ^0.34eV for FeP. The modifications of the theory for application to hardening by interstitials are indicated by Suzuki (1979, 1984), cf. also the review by Pink and Arsenault (1979).
5.6 Solid Solution Effects in Nonmetallic Systems Simple nonmetallic crystalline solid solutions are not only excellent test systems for dislocation theory because of easy production and handling of very perfect crystals, but are also of interest in many applications. We will consider in the following a few aspects of the mechanical properties of solid solutions of these systems, where the interactions of dislocations with solutes are important and in particular electronic interactions [Sees. 5.2.1 (5), 5.2.2.2 (3, 4)] come into play [cf. the reviews by Haasen
5.6 Solid Solution Effects in Nonmetallic Systems
239
(1985) on alkali halides and by Hirsch (1985) on semiconductors and the recent comprehensive review by Nadgornyi (1989)]. 5.6.1 Systems with Alkali Halide Structure 5.6.1.1 Dilute Solid Solutions (c< 1000 ppm) While NaCl single crystals are quite ductile at room temperature and below, polycrystalline NaCl shows a transition from ductile to brittle behavior at a temperature depending on strain rate [such that, e.g. at room temperature plasticity is possible only for e < 1 0 " 1 2 s" 1 , cf. Haasen (1985)]. This transition is only slightly dependent on solute content for dilute solutions. It is supposed to occur because the fracture stress may be lower than the stress to activate all slip systems in the grains of the polycrystal necessary to accommodate deformation of neighboring grains (Skrotzki etal., 1981). With increasing solute content up to about 1000 ppm, which produces a transition from "homogeneous" wavy slip to inhomogeneous localized slip (see, e.g. Bengus and Kovalenko, 1979), due to its influence on cross-slip (Vladimirov and Kusov, 1978; Tokii, 1979) and resulting stress concentrations in pileups, the transition to brittle behavior occurs at a somewhat higher temperature for the same strain rate. It is well established that slip occurs in the alkali halide crystals on two different slip systems, i.e. {110} [110] with low, and {001} [110] with high Peierls stresses. The difference in the CRSS T 0 (Fig. 5-25) depends on the ionicities of the components. Double kink nucleation and propagation controls the CRSS at low temperatures. Correspondingly at low T and small c solute softening has been observed (Sakamo-
600
Figure 5-25. Temperature dependence of the CRSS of undoped and doped KC1 single crystals, (o, •) {110} slip plane, (•, •) {100} slip plane, open symbols: undoped, closed symbols: doped with 56 ppm of Sr2 + . (From Haasen, 1985.)
to, 1984) (cf. Sec. 5.5.2), and at very low T inertial effects (Suzuki and Koizumi, 1985) and quantum effects (Fomenko et al., 1987) [cf. Sees. 5.2.3.2 (4) and 5.3.3.2] have been found. The solute atoms tend to form dipole complexes with vacancies [Sec. 5.2.2.2 (3)], if their valency does not coincide with that of the host atoms. The resulting metal ionvacancy dipole pair with its tetragonal distortion provides effective interaction with the dislocations which in ionic crystals furthermore carry an effective charge. For instance, the maximum interaction force can be estimated to be (Haasen, 1985) 6 to 10 x 10" 1 0 N (cf. Sec. 5.2.2) for various alkali halides with divalent impurities. Therefore an extended range of the CRSS at low to intermediate temperatures is characterized by this interaction of single dislocations with the dipole complexes. With increasing T, these obstacles become
240
5 Solid Solution Strengthening
mobile and tend to reorient in the distortion field of the dislocation according to the Snoek-Schoeck interaction [Sec. 5.2.2.1 (l,ii)] resulting in an extended plateau region in the temperature dependence of the CRSS. At still higher T9 T 0 decreases rapidly because of easy mobility of the dipoles such that only the interaction with forest dislocations remains. Thus, as can be seen in Fig. 5-25 for the temperature dependence of the CRSS, we have quite similar processes and dependences for the CRSS as those in f.c.c. and b.c.c. metal alloys (Sees. 5.3, 5.4, and 5.5). There are similar three-stage work hardening curves (Bengus et al., 1966) and localization of glide in slip bands (Komnik et al., 1967). The cooperative nature of dislocation glide in alkali halides (Bengus, 1984) has recently been established by measurements of electrical noise (Golovin and Orlov, 1988) produced by the movement of charged dislocations. One important difference to the metallic alloys is the high density of deformation-produced point defects. It is a consequence of the dragging of charged jogs (Gilman, 1962; Appel et al., 1977), and of the pronounced cross-slip activity which can be recognized in the fine structure of the wide slip bands (Smirnov, 1969; Bengus and Kovalenko, 1977) and was attributed to the high stacking-fault energy (Haasen, 1985). Recent investigations using the Blaha effect (Ohgaku and Takeuchi, 1989, 1990) succeeded in separating the effective from the total stress and the effects of two kinds of obstacles (forest dislocations and tetragonal defects). Recently, Vesna et al. (1990) pointed out that screw dislocations in NaCl show different ageing behavior near the surface and in the volume of the crystals.
5.6.1.2 Concentrated Solid Solutions We consider now the deformation of concentrated alloys, which in fact are mixtures of two types of alkali halides, e.g. KC1 and KBr, over the whole concentration range. Investigations of this system over the temperature range of 1.6 to 923 K have been performed by Sakamoto and Yamada (1980). They interpreted their results on the basis of computer simulations of the glissile movement of a dislocation, assuming a size effect for both Cl~ and Br~ ions and adding a Peierls stress with periodic variation along the dislocation path. Figure 5-26 a shows the CRSS in dependence on concentration and temperature, indicating the solute softening effect in the 4.2 K curve at low concentrations (Sakamoto, 1984). The stress-strain curves show the PLC effect at intermediate to elevated temperature, and a smooth course at lower as well as at higher temperatures. The model calculations (Sakamoto, 1986, 1987) result in a maximum of the dragging stress versus dislocation velocity, which explains well the TO(T) behavior (Fig. 5-26b) with its plateau region. The calculations and the thermal activation analysis (Kataoka etal., 1978) indicate the rate controlling role of the edge dislocation movement by double kink production and propagation through the random arrangement of obstacles arising from concentration fluctuations in the alloy. The observed solute softening effect (see Fig. 5-26 a) at low and high cBr values is attributed (Sakamoto, 1984) to the solute atoms facilitating nucleation of double kinks over the Peierls barrier (cf. Sec. 5.5.2), here on edge dislocations. Deviations from the Arrhenius type behavior for T< 15 K have been found in concentrated KCl-KBr single crystals (Kataoka et al.,
5.6 Solid Solution Effects in Nonmetallic Systems
1989) and interpreted by fluctuations of the activation enthalpy by the random distribution of solute atoms. In summary, despite of important peculiarities a striking similarity of the results on alkali halides with those in f.c.c. and b.c.c. alloys can be observed, supporting the view that the alkali halides provide a very suitable model system to study details of solid solution hardening.
KC!
20
200
60 c in mol %
400
80
600
7" in K
241
5.6.2 Systems with Diamond Cubic Structure Practical interest in materials with diamond cubic structure is mainly focussed on the semiconductors Ge and Si, as well as on semiconducting compounds. In our context of solid solutions we will consider briefly some features of doped simple semiconductors, in particular with respect to the dislocation behavior and to mechanical properties. While at room temperature these materials appear brittle, they become quite plastic for T> Tm/2. For details we refer to reviews by Alexander and Haasen (1968), Labusch and Schroter (1980), and Hirsch (1985), where the knowledge about dislocation types (glide and shuffle set), reconstruction of bonds in the dislocation core, and the effects of dislocations on the electronic energy levels is compiled. We just note that it is now well established that the glissile dislocations on {111} planes are dissociated, and that the cores tend to be reconstructed to avoid dangling bonds for 30° partials, probably also for 90°, but less likely for 60° partials (Alexander, 1984; Alexander et al., 1985). The dislocation motion in these substances is certainly controlled by the high Peierls potential, at least for relatively low temperatures, thus
KBr
1000
Figure 5-26. (a) Concentration dependence of the CRSS for deformation by compression at 7 = 4.2, 77, and 293 K with indicated deformation rates, for various KCl-KBr solid solution crystals. Note the solid solution softening effect at the lowest temperature for low concentrations of either KC1 or KBr. (From Kataoka et al., 1978.) (b) Temperature dependence of the CRSS of KC1-17 mol% KBr single crystals deformed with e = 8.3 • 10" 4 s~ 1 . (From Sakamoto, 1987.)
242
5 Solid Solution Strengthening
the nucleation and propagation of double kinks in the two partials is most important. Kinks can be reconstructed or be associated with dangling bonds with a possible transformation into the latter, and their motion along the dislocations is obstructed by a considerable secondary Peierls potential. The dopants needed to obtain the desired electronic properties of the material interact specifically with the dislocations. As indicated in Sec. 5.2.2.2 (4) electronic effects [change of Fermi level, change of dislocation or kink charge state, cf. Hirsch (1979, 1980, 1985)] influence the dislocation mobility in different ways: e.g. doping of Ge and Si with group V elements (P, As) increases the dislocation velocity for a given stress and temperature, whereas doping with group III elements (Al, Ga) decreases the velocity (e.g., Patel and Chaudhari, 1966; George and Champier, 1979). Accordingly the yield stress and creep behavior are differently affected by the different dopants. The upper yield stress strongly depends on dislocation nucleation and multiplication (cf. Alexander and Haasen, 1968), the lower yield stress represents the effect of the dopant on the dislocation mobility. At about 500 °C its ratio for doped material to that for intrinsic one is about 0.3 in n-type Si, i.e. there is a considerable softening due to electronic effects (Rabier et al., 1983). Quite spectacular effects also result e.g. in changes of indentation hardness and fracture (Roberts et al., 1983). An extended review of the experimental results on dislocation velocities (Louchet and George, 1983) and their interpretation by mixed kink-obstacle mode of dislocation motion has been given by Nadgornyi (1989). The obvious similarities to the materials discussed in Sees. 5.4 and 5.5 are supported by the tendency to inhomogeneity of
30
Figure 5-27. Stress-strain curves of pure and doped Si single crystals (n = 1020 P atoms • cm" 3 ), deformed in compression at T = 1000 °C with a shear strain rate of 7 = 2.4 • 10~4 s" 1 . (From Siethoff, 1969.)
deformation with increasing concentration of the dopant. For instance, Siethoff (1969, 1973) observed in Si highly doped with P the formation and propagation of Liiders bands of two kinds: The type K bands with their band front perpendicular to the primary Burgers vector (accompanied by PLC-effect, cf. Sees. 5.2.4.2 and 5.3.3.3), and the type G bands with their band front parallel to the primary glide plane (without serrations). These observations indicate that in addition to the nonlocal doping effect on the Peierls potential, a local solute effect must be taken into account for the highly doped materials. This is supported by the similar deformation curves for pure and doped Si as those for pure and alloyed f.c.c. metals (Fig. 5-27).
5.7 Acknowledgements The authors wish to thank Mrs. I. Benzel for her invaluable help in preparing the figures, and Dr. D. Brunner, MPI fur Metallforschung in Stuttgart, for his informations on works about solid solution softening. Much careful work of many of our coworkers over the years has entered this review. Our work was supported continuously by the Deutsche Forschungs-
5.8 References
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General Reading Friedel, J. (1964), Dislocations. Oxford: Pergamon Press. Gilman, JJ. (1969), Micromechanics of Plastic Flow. New York: McGraw-Hill. Haasen, P. (1983), in: Physical Metallurgy: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: Elsevier, pp. 1341-1409. Haasen, P. (1985), in: Dislocations and Properties of Real Materials. London: The Institute of Metals, pp. 312-332. Hirsch, P. B. (1985), in: Dislocations and Properties of Real Materials. London: The Institute of Metals, pp. 333-348. Hirth, J. P., Lothe, J. (1968), Theory of Dislocations. New York: McGraw-Hill. Kocks, U. F. (1985), Metall. Trans. A 116, 21092129. Kocks, U. R, Argon, A. S., Ashby, M. F. (1975), Progr. Mat. Sci. 19, 1-278. Nadgornyi, E. M. (1989), Progr. Mat. Sci. 31,1-510. Neuhauser, H. (1983), in: Dislocations in Solids, Vol. 6: Nabarro, F. R. N. (Ed.). Amsterdam: NorthHolland, pp. 319-440. Pink, E., Arsenault, R. J. (1974), Progr. Mat. Sci. 24, 1-50. Suzuki, H. (1979), in: Dislocations in Solids, Vol. 4: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, pp. 191-217. Suzuki, T, Takeuchi, S., Yoshinaga, H. (1989), Dislocation Dynamics and Plasticity. Berlin: Springer.
6 Deformation of IntermetaUic Compounds Yukichi Umakoshi Department of Materials Science and Engineering, Osaka University, Osaka, Japan
List of 6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.3.2.1 6.3.2.2 6.3.2.3 6.3.3 6.3.4 6.4 6.4.1 6.4.2 6A.2A 6.4.2.2 6.4.2.3 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.6 6.6.1 6.6.1.1 6.6.1.2 6.6.2 6.6.2.1 6.6.2.2 6.7 6.8 6.9
Symbols and Abbreviations Introduction Crystal Structure and Types of Intermetallics Crystal Structure Types of Intermetallics Planar Faults and Dislocation Dissociations Stability of Planar Faults The F.C.C.-Based Ordered Structure L l 2 Structure L l 0 Structure D022 Structure £2, D03 and L2X Structures D019 Structure Deformation Behavior of Intermetallics Slip Systems and the Effect of the Dislocation Core Structure Strength and Flow Stress Anomalous Strengthening and Strengthening Mechanisms High-Temperature Strength and Creep Effect of Off-Stoichiometry Ductility and Embrittlement Effect of Deformation Mode Grain Boundary Embrittlement Hydrogen Embrittlement Attempts at Ductility Improvement Advanced Intermetallics as High-Temperature Structural Materials High-Strength and Low-Density Aluminides TiAl Based Aluminides Al3Ti Based Aluminides Intermetallics for Use at Extremely High Temperatures ,415 Compounds Refractory Metal Silicides Summary Acknowledgements References
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
252 254 254 254 256 257 257 258 258 259 260 261 263 263 263 269 269 274 275 276 276 277 278 278 280 280 280 291 295 296 298 305 306 306
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List of Symbols and Abbreviations A a, b, c b eja £APB £CSF E SISF / f G h, k, I AH fcy K n N Q r R S T Tc Tm V V(rk)
constant lattice parameters Burgers vector electron concentration fault energy of anti-phase boundary fault energy of complex stacking fault fault energy of supperlattice intrinsic stacking fault fault vector of planar faults based on the hard sphere model additional fault vector to stabilize planar faults shear modulus Miller indices for crystallographic planes and directions activation enthalpy constant tension/compression asymmetry stress exponent ratio of Schmid factor for (010) [101]-slip to that for (111) [101]-slip ratio of Schmid factor for (111) [121]-slip to that for (111) [101]-slip separation distance between two superpartials on (h k I) plane gas constant degree of long range order temperature critical ordering temperature melting temperature ordering energy ordering energy at fe-th nearest neighbor atoms
a phase a2 phase y phase y (f) e 9 x
Ti 3 Al phase with disordered h.c.p. structure Ti 3 Al phase with D0l9 structure TiAl phase with L l 0 structure energy of stacking fault as a function of fault vector steady state creep rate angle between dislocation line and Burgers vector parameter for relative importance of core constriction effect before and after cross-slip spacing between lamellae in TiAl crystals containing numerous twins Poisson ratio yield stress lattice resistance to dislocation slip in the y phase resolved shear stress for (111)[121] slip resolved shear stress for (111)[101] slip angle between loading axis and lamellar plane in polysynthetically twinned TiAl crystals pair-wise interaction energy between fc-th nearest neighbor atoms A and B
X v oy a0 ie TP 4> ^AB (rk)
List of Symbols and Abbreviations
253
X xj/
angle between m a x i m u m resolved shear stress plane a n d reference (101) plane angle between observed slip plane a n d reference (T01) plane
APB CRSS CSF MRSS PST SESF SISF
anti-phase boundary critical resolved shear stress complex stacking fault maximum resolved shear stress polysynthetically twinned crystal of TiAl intermetallic compound superlattice extrinsic stacking fault superlattice intrinsic stacking fault
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6 Deformation of Intermetallic Compounds
6.1 Introduction In recent years, as improvements in gas turbine performance at increasing operating temperatures have been achieved, the requirements for material strength and oxidation resistance at high temperatures have become more severe. In intermetallic compounds different atomic species occupy different regular lattice sites and the strong bonds between their atoms result in attractive properties such as an increase in the melting and softening temperatures, superior shear modulus and high stability against chemical reactions compared to conventional metals and disordered alloys. However, the strong bonds and low symmetry of the deformation behavior, on the basis of the ordered structure, give rise to low temperature embrittlement of the intermetallic compounds, which is a disadvantage as far as their application and processing for practical purposes is concerned. Plastic behavior and motion of dislocations in intermetallic compounds is governed by the dislocation core structure on the basis of the crystal structure, together with ordered faults such as the anti-phase boundary (APB). Superlattice dislocations travelling together in pairs or groups in the intermetallic compounds result in interesting mechanical properties such as anomalous strengthening at high temperatures, which has been discovered in several L l 2 , B2, L l 0 and D019 compounds. The dissociation and motion of superlattice dislocations and the attractive mechanical properties of intermetallic compounds were reviewed by Marcinkowski (1963), Stoloff and Davies (1966), Pope and Ezz (1984), Yamaguchi and Umakoshi (1990), and by the many contributors to the two books, "Mechanical Properties of Intermetallic Compounds" and "Intermetallic Compounds" edited by Westbrook (1960,1967)
and also to the Proceedings of the Conference on Ordered Alloys, edited by Kear et al. (1970). Many international symposiums and conferences have been held recently, and the resultant volumes are listed in the references. In this chapter attention is first focused on dislocations and planar faults in intermetallic compounds in order to understand their plastic behavior. Then the experimental and theoretical approaches to the study of attractive mechanical properties and interesting deformation characteristics are presented. Ductility improvement of several intermetallic compounds has been established through micro- and macro-alloying additions, grain refinement and microstructure control by thermo-mechanical processing. For example, it has been discovered that small additions of boron result in a more than 50 % elongation of polycrystalline Ni3Al at room temperature due to the suppression of grain boundary embrittlement. Ductility and embrittlement will be discussed in Sec. 6.5. Some new compounds such as TiAl and Al3Ti are of great interest for automotive and aerospace applications because of their low density and superior strength-toweight ratio. The aluminides and silicides of refractory metals such as Mo, Nb and W are also of interest in the development of new, extremely high-temperature materials for aircraft gas turbines and airframes operating around 1500°C. Finally, recent studies on such advanced intermetallics as high-temperature structural materials are presented.
6.2 Crystal Structure and Types of Intermetallics 6.2.1 Crystal Structure
Intermetallic compounds crystallize in geometrical close packed structures such
6.2 Crystal Structure and Types of Intermetallics
as Laves and a phase, and in superlattice structures which are constructed on the basis of body-centered cubic (b.c.c), facecentered cubic (f.c.c.) and hexagonal-close packed (h.c.p.) structures (see Chapter 3 "Structure of Intermetallic Compounds and Phases" of Volume 1 in this Series). Since the nature of dislocations and mechanical properties of compounds are reflected by their principal crystal structure, their plastic behavior should be discussed and understood in consideration of basic crystal lattices. In this section superlattice and several other ordered structures, with which compounds are of importance in plastic deformation, are shown. Figure 6-1 a shows a general description of the b.c.c.-based superlattice structures of B2, ZX)3 and L2X superlattices. In AB compounds with the B2 structure, A atoms regularly occupy a sites and B atoms occu-
255
py both ft and y sites. This structure can be observed in CuZn, FeCo, NiAl, CoAl, FeAl, AgMg, AuZn, NiTi and AgCd. In A 3 B compounds with the Z>03 structure /? sites are occupied by B atoms but the other two sites are occupied by A atoms. Fe3Al and Fe3Si crystallize in this structure. If a, j8 and y sites are regularly occupied by A, B and C atoms, respectively, A2BC compounds form the L2X structure. Ag2MgZn, Cu 2 MnAl, Co2AlTi, Ni2AlTi and Ni 2 AlNb belong to this group. The Cllb structure shown in Fig. 6-1 b is one of the long period ordered structures derived by stacking three b.c.c. lattices and then compressing them along the oaxis. MoSi 2 , WSi2 and ReSi2 have this structure and are of interest as potential new high temperature materials for use above 1500°C. Figure 6-1 c shows the atomic arrangement in the f.c.c.-based superlattice struc-
Figure6-1. Atomic arrangement of (a) b.c.c. based £2, D0 3 and L2 l 9 (b) Cll fc , (c) f.c.c. based L l 2 and Ll 0 , (d) D0 22 , (e) D0 19 , (f) C40, and (g) A15 structure.
256
6 Deformation of Intermetallic Compounds
ture. When A 3 B compounds crystallize in the LI 2 structure B atoms occupy the corner a sites and A atoms occupy both face centered /? and y sites. Many attractive intermetallic compounds such as Ni 3 Al, Ni 3 Mn, Ni 3 Fe, Ni 3 Ge, Co3Ti, Zr 3 Al, Pt3Al, Pt 3 Fe and Pt 3 Ge have the L l 2 structure. In the L l 0 structure with AB composition a and /? sites are occupied by one atomic species and y sites are occupied by different species. In this structure the aand /?-axes are no longer identical to the oaxis. TiAl, CuAu, NiMn and FePd have the L l 0 structure. When an APB with a displacement vector of the type l/2[110] is introduced on every (001) plane, the D0 22 structure, as shown in Fig. 6-1 d, can be derived. By introducing such APBs on every two (001) planes, the D0 23 structure is derived from the L\2 structure. The D022 structure is observed in Ni 3 V and new compounds with high strength/density ratio and good oxidation resistance such as Al3Ti, A13V, Al 3 Nb and Al3Ta. Al3Zr and Al3Hf crystallize in the D023 structure. The phase stability of the D022 against the D023 and L\2 structures has important consequences for the plastic behavior of these compounds; details will be given in the corresponding section. The D019 structure, as shown in Fig. 6-1 e, is based on the h.c.p. lattice. Plastic characteristics of Ti3Al, Ti3Sn and Mn 3 Sn with the D019 structure have been of interest. The C40 structure in the h.c.p. based structure in Fig. 6-1 f is not a superlattice structure but is of interest for high temperature silicides such as TaSi2, CrSi 2 , VSi2 and NbSi 2 . The ^415 structure shown in Fig. 6-1 g is also important for superconducting and ultra-high temperature materials such as V3Si, Nb 3 Al and Nb 3 Sn. In this chapter Miller indices for crystallographic planes and directions are given using notations based on the superlattice
structure except for the Ll2-derivative structures such as D022. In the LI 2-derivative structures notations based on the f.c.c. lattice are used for the sake of simplicity and for comparison with the LI 2 structure. The mixed notations {hkl) and (h k I] are used for the L l 0 , D0 22 and C\\h structures to differentiate the first two indices from the third one, which does not play the same role as the first two, because in these structures the a- and 6-axes are no longer identical to the c-axis. 6.2.2 Types of Intermetallics In a phase diagram intermediate phases which are formed by different atomic species at an appropriate simple ratio are called intermetallic compounds. The intermetallic compounds are divided into the three groups of Kurnakov-type, Berthollide-type and Daltonide-type compounds. The Kurnakov-type compounds have a wide composition range on both sides of stoichiometry and the order-disorder transformation exists below their melting points, as shown in Fig. 6-2 a. Cu 3 Au, Fe3Al, Ti3Al, CuZn and FeCo belong to this type and can deform to a certain degree because there are no strong bonds between the different atomic species. The intermetallic compounds, such as FeAl, NiAl, CoAl and Ni3Al, whose ordered structures are stable up to their melting points are Berthollide-type compounds as shown in Fig. 6-2 b. They are harder than the Kurnakov-type compounds. The Daltonide-type compounds have no solubility off-stoichiometry, as shown in Fig. 6-2 c, and the strong bonds leading to ordered structures with lower symmetry cause lowtemperature embrittlement. Possible candidates for structural materials should be Kurnakov- or Berthollide-type compounds; it is unlikely that the Daltonide-
6.3 Planar Faults and Dislocation Dissociations
Composition (a)
B
257
A (b)
Figure 6-2. Phase diagram of (a) Kurunakov-type, (b) Berthollide-type and (c) Daltonide-type intermetallic compounds.
type compounds will be deformable at low temperatures.
6.3 Planar Faults and Dislocation Dissociations 6.3.1 Stability of Planar Faults A generalized planar fault is formed by cutting the crystal along a crystallographic plane and displacing one half of the crystal with respect to the other one by an appropriate displacement. There are three types of important planar faults, as shown in Fig. 6-3 a. In the figure, when the layer A is displaced by the vector fA with respect to the layer B, the configuration of atom neighbors across the fault is changed. This is the anti-phase boundary (APB). The APB only exists in ordered lattices, since the atomic arrangements do not change after shifting by/ A when the lattice sites are randomly occupied by A and B atoms. The superlattice intrinsic stacking fault (SISF) is produced after a displacement of / S F . The SISF does not involve any wrong bonds between unlike atoms. The complex stacking fault (CSF) is formed when the layer A is displaced by the vector fc with respect to the layer B. The CSF is also a
F-Surface c 0 CO Li.
Figure 6-3. (a) The atomic arrangement on the (111) plane in an A3B compound with the L l 2 structure. Large, medium and small circles represent atoms on A, B and C layers, respectively. The stacking sequence of (111) planes is ABCABC. (b) Variation of the planar fault energy with the fault vector.
258
6 Deformation of Intermetallic Compounds
stacking fault but involves the effect of wrong bonds. These planar faults are not always stable at positions predicted on the basis of a hard sphere model. The stability of planar faults in intermetallic compounds, which affects the possible dislocation dissociation schemes, can be determined by calculating the energy of stacking faults, y ( / ) , as a function of the fault vector / , i.e. the y-surface (see the review by Vitek, 1974). The stability of such a planar fault depends on whether a local minimum exists in the y-surface in the vicinity of the corresponding vector/ A (see Fig. 6-3 b). Sometimes the y-surface may have a stationary point for the vector fA+f, which requires an additional vector / ' to the vector / A determined by a hard-sphere model. The existence of/' is closely related to the stability of the planar fault. The stable position can also be estimated on the basis of the crystal symmetry. If there are mirror planes in the unfaulted crystal perpendicular to the plane of the fault, which remain on displacing one half of the crystal relative to the other by the vector / , the stability of the planar fault can be deduced using the following rules, obtained on the basis of symmetry considerations by Yamaguchi et al. (1981 b): (1) If at least two non-parallel mirror planes are maintained on displacement by the fault vector / , the y-surface has a stationary point for/. If this stationary point for/is minimum, the fault for/is guaranteed to be stable on symmetry grounds. For example, if there are two mirror planes of 1-1' and 2-2' perpendicular to the fault of vector/ SF , as shown in Fig. 6-3 a, a local minimum of the y-surface can exist forfSF. (2) When the displacement by / A maintains only one mirror plane 2-2', the y-surface has a vanishing first derivative along the 2-2' mirror plane normal and may
have a stationary point for the vector where/' is a vector parallel to the mirror plane. It is possible that there are no stable faults with displacements close to/A.
/A+/'>
6.3.2 The F.C.C.-Based Ordered Structure In the L l 2 , Ll0 and D022 structures based on the f.c.c. lattice there are three different types of planar fault: APB, CSF and SISF. On the basis of the hard sphere model and geometrical considerations, the energy of planar faults on possible slip planes is calculated by counting the number of atomic bonds across the fault plane. The energy of planar faults is calculated assuming central pair-wise interactions and taking into account interactions up to the second-nearest neighbors. The result is shown in Table 6-1. The APB energy on {001} is much lower than that on {111}. 6.3.2.1 L\2 Structure In the LI 2 structure slip always occurs along <110> directions. These are the three planar faults of APB, CSF and SISF on {111} planes with fault vectors o f / A , / c a n d / j F respectively, as shown in Fig. 6-3. Table 6-1. The energy of planar faults in L l 2 structure. The ordering energy V(?k) = [$AA(rk) + + ^BB(^) - 2AB(rfc)]/2, r' = (2/^/3) a, where a is a lattice constant and
Fault
Energy
(111) (111)
APB
[2V(r i ) - 6F(r 2 )]/( v /3 a2) [2V(r][) (r 2 ) + 4 (r) +
(111) (001)
SISF
CSF
APB
+ r r ' /f^
2 [3*,4A(0 + ^BB(rO]/(V3fl2)-2F(r 2 )/a 2
6.3 Planar Faults and Dislocation Dissociations
259
[112]
[110] (11D
(a)
b6
b3
b5
CSF APB
SISF Cb)
b2
b1
APB CSF
b6
b5
b2
CSF APB
CSF
(c)
b,
b2
b
APB (d)
;
bi'
SISF (e)
Figure6-4. Atomic arrangement and possible dissociation schemes for [110] dislocation on (111) in the L l 2 structure, (a) Types of dissociation, (b) six-fold, (c) four-fold, (d) two-fold (APB-splitting) and (e) two-fold (SISF-splitting) dissociations (Marcinkowski etal, 1961; Kear et al., 1968; Yamaguchi et al., 1981b).
Considering the deviation from fault vectors predicted on the basis of the hardsphere model, six-fold dissociation involving two CSFs, two APBs and an SISF, four-fold dissociation involving two CSFs and an APB, and two-fold dissociation involving an APB or an SISF are proposed for a <110> superlattice dislocation, as indicated in Fig. 6-4 (Marcinkowski et al., 1961; Kear et al., 1968). The energy of these planar faults varies in the order ECSF>EAPB>EslSF. From the y-surface calculation and the crystal symmetry consideration, the APB and CSF are not always stable for the displacement vectors derived from the hard-sphere model, and additional displacements for the stationary fault vectors of APB and CSF may be needed along the directions perpendicular and parallel to their fault vectors (see Fig. 6-4). On the other hand, an APB on (001) and an SISF are always stable at the
displacement vectors derived from the hard sphere model (Yamaguchi et al., 1981 b). The relative stability and energy of these planar faults, which depend on the materials, affect the dissociation form of <110> superlattice dislocations and the deformation behavior of the materials. Sixfold dissociation is strongly favored over four-fold dissociation when the SISF energy is extremely low. However, in this case two-fold dissociation involving an SISF may occur and therefore the six-fold dissociation is very unlikely. The four-fold dissociation is confirmed in Cu 3 Au (Cockayne et al., 1969; Sastry and Ramaswami, 1976), Ni 3 Fe (Korner et al., 1987) and Ni3Al (Veyssiere et al., 1985). 6.3.2.2 Ll0 Structure Figure 6-5 shows the atomic arrangements on (111) planes in the L\Q struc-
260
6 Deformation of Intermetallic Compounds
tor of l/2[011] is always stable on the (010) plane and its energy is lower than that on (111), similar to the APB in the L\2 structure. The APB energy on the {111} plane can be roughly estimated to be twice that in the LI 2 structure. When the APB energy is much higher than the SISF energy the dissociation shown in Fig. 6-5 c is unlikely, and the dissociation of the [Oil] dislocation changes from that in Fig. 6-5 c to that in Fig. 6-5 d (Amelinckx, 1979). In L l 0 intermetallic compounds the motion of b31 plays an important role accompanied by the creation of the SISF which is left behind. The propagation of Shockley partials of the type b31 on several adjacent (111) planes produces twins, which are the major mode of deformation in L\Q compounds. Numerous twins were observed in TiAl (Lipsitt et al. 1975). If the splitting of b4 occurs on (111) instead of on (111), nonplanar splitting of <101] dislocations can occur on two {111} planes intersecting along the dislocation line (Greenberg, 1989).
ture (see Fig. 6-1 c). There are three planar faults, on the basis of the hardsphere model: APB(A3), CSF(6 n ) and SISF(A31), similar to the L\2 structure. Slip occurs along <110> directions but [110]- and [Oll]-slip are not identical because of the tetragonality of the Ll0 structure. [110] dislocation (Ax) can dissociate into two Shockley partials (bxl and b12, similar to the case of f.c.c. but in this instance a CSF is formed between the two Shockley partials (Fig. 6-5b). [Oil] dislocation (b2) can be dissociated into four superpartials, as shown in Fig. 6-5 c, but the four-fold dissociation is different from that in the L\2 structure (see Fig. 6-5c). In this case the motion of a superpartial with b31 Burgers vector creates the SISF with no disturbance of the ordered arrangements of different atomic species and therefore the dislocation dissociation has an asymmetry. The APB with a fault vec-
6.3.2.3 D022 Structure
The D022 structure is derived from the LI 2 structure and for the sake of simplicity and comparison with L l 0 and L\2 structures, the Miller indices for crystallographic planes and directions are given using f.c.c. notations. In the D022 structure 1/2<112] is the shortest, <110] is the next, and < 101] is the third shortest lattice translation vector on the most closely packed {111} planes.
1112] 10111. ,[1011 ... ,i l-o [121] (111)
[110]
(a)
\ SISF
(c)
\SISF
(d)
Figure 6-5. Atomic arrangement and possible dissociation schemes for [110] and [011] dislocations on (111) in the LI 0 structure (Amelinckx, 1979; Greenberg, 1989).
6.3 Planar Faults and Dislocation Dissociations
Since cja is nearly equal to 2 in Al3Ti, A13V and Ni3V, <101] is a much larger translation vector than the others and in fact <101] slip has not been observed. Possible planar faults on (111) are APB-I (l/2<110]), CSF (1/6<211]) and APB-II (l/2<101]). The APB-I can be created on the (001) plane and the energy is expected to be much lower than that on the (111) plane because there is no contribution for atomic interaction with the first nearest neighbors, similar to L\2 and L\o structure. Therefore, <110] dislocations splitting on (001) play a major role in the deformation mode of the D022 structure at high temperatures because of the sessile nature of dissociated <110] dislocations on the {111} plane. The 1/2<112] dislocation shows fourfold dissociation involving an SISF, an APB-II and a CSF, quite similar to that in the L\Q structure. The energy of APB-II is slightly lower than that of APB-I. Since the energy of the SISF is relatively lower than that of other faults, the 1/6<112] Shockley partial moves independently, trailing the stacking faults and resulting in twinning, which does not disturb the ordered symmetry. In fact numerous {111}<112] twins were observed in Ni3V (Vanderschaeve et al., 1979; Vanderschaeve and Escaig, 1983). 6.3.3 Bl, Z)03 and L21 Structures
In the B2, D0 3 and L2X structures based on the b.c.c. lattice (see Fig. 6-1 a), no stable intrinsic stacking faults except APBs are believed to exist, similar to the underlying b.c.c. structure (Vitek, 1968, 1974). In the B2 structure only one type of APB with a displacement vector of a/2 exists on possible slip planes of {110} and {112}. The a/2-APB is always stable at the position expected from a hard-sphere
261
model on both planes and two-fold dissociation of a superlattice dislocation is expected to occur. In the D0 3 and L2X structures two different types of APB-I and APB-II are thought to exist at the fault vectors in the vicinity of f± = a/4( 111 >, and / 2 and / 2 ' = a/2, respectively. As can be seen from the result of the y-surface calculations on (110) in Fig. 6-6 a, Paidar (1976) has found that the position of local minimum corresponding to the APB-I deviates from/; to (/; +//), while the APB-II exists at a/2, as suggested by a hard-sphere model. The additional fault vector / / is required to stabilize the APB-I. The / / must be along <001> for the APB-I on (110) and <111> for the APB-I on (112). The magnitude of// varies from material to material and the APB-I may be destabilized on increasing / / . Considering the shift of the stationary point for APBs, the dissociation of <111> superlattice dislocations in the ZX)3 and L2t structure is schematically drawn in Fig. 6-6b and c (Yamaguchi et al., 1981 a). Two- or four-fold dissociations which depend on the stability of APBs may possibly occur and they have been observed in Fe3Al (Crawford and Ray, 1977) and Fe3Si (Hanada, 1984). In the L2t structure <111>, <001> and <110> slip directions may exist, and the subsequent transition in slip occurs with increasing energy and instability of APBs. 4xa/4 (6-1) a<001> dislocation can dissociate into two superpartials bound by a/ 2<001>-APB which is always stable. a<110> and a<001> dislocations have been observed in strongly ordered L2t structures, such as Ni2AlTi and Co2AlTi, which are thought to be high-temperature struc-
262
6 Deformation of Intermetallic Compounds
(110) [110]
[1111-
(112)
(C)
Figure 6-6. (a) The (110) y-surface and possible dissociations of <111 > superlattice dislocation in the D0 3 structure (Paidar, 1976), (b) on (110), and (c) on (112) (Yamaguchi et al., 1981 a). The fault energy increases on increasing the number in (a).
110101
1/6112101 on ( 1 0 1 1 ) 112101 (c)
on (0001) 121101
(a)
Figure 6-7. Possible dissociation schemes for 1/3 <1120> superlattice dislocations in the D019 structure, (a) on (0001), (b) (10T0) and (c) (1011) plane (Umakoshi and Yamaguchi, 1981b).
6.4 Deformation Behavior of Intermetallics
tural materials with good oxidation resistance (Strutt et al., 1976a, b). 6.3.4 D0 19 Structure Since (0001), {1011}, {1011} and {1121} slips have been observed in several compounds with the D0 19 structure, 1/3<1120> dislocation dissociation and planar faults on these planes are indicated in Fig. 6-7. The stability of the planar faults APB, CSF and SISF, and the dislocation dissociation can be understood from the similarity of atomic arrangements on (0001) planes in the D019 structure, and on {111} planes in the L\2 structure. Figure 6-7 shows that the SISF on the basal plane, the 1/3<112O>-APB on {10T0} planes and the 1/2<1O1O>-APB on {1120} planes are always stable, but the others are either unstable or stable, with fault vectors deviating from those deduced purely from geometrical considerations (Umakoshi and Yamaguchi, 1981b). The character of the dissociation of the l/3<1120> superlattice dislocation on the basal plane depends on the stability and energy of the APB and CSF; similar to that of a <110> superlattice dislocation on {111} planes in the L\2 structure. When both the APB and CSF are stable, (a) b = b1+b2 + b5 + b6 or When the CSF is unstable and the APB is stable, (b) b = bf1 + b2 When both the APB and CSF are unstable, (c) b = bf[ + b^ The cores of superlattice partial dislocations with large Burgers vectors, such as bf[ and b2, are likely to be dissociated further into fractional dislocations, and such fractional dislocations form immobile nonplanar cores. When dissociation (b) or (c) occurs the basal slip would be expected to
263
less favored. On the {1010} planes where the 1/6<112O>-APB is always stable, the l/3<1120> superlattice dislocation can be dissociated into two l/6<1120> superpartials. Therefore, depending on the type of dislocation dissociation on the basal plane and the core structure of the corresponding superpartials, slip on the {1010} prismatic planes could be favored over basal slip. In fact {1010}<1120> slip was observed in Mg3Cd (Stoloff and Davies, 1964a, b), Ti3Sn (Jones and Edington, 1973) and Ti3Al (Minonishi and Yoo, 1990a,b; Minonishi, 1991; Umakoshi et al., 1991 e). Since no contributions to the energy of APB on that plane come from first-nearest neighbor interaction, the APB energy is expected to be lowest on the basal plane. The l/3<1120> superlattice dislocation on the {1011} and {1012} planes can be dissociated into two superpartials, although the stationary point possibly deviates along the direction perpendicular to <1120>. 1/3<1126> superlattice dislocations can be dissociated into four superpartials involving the corresponding faults and the 1/6APB on {1011} and {1121} planes and may be mobile. {1121}<1126> slip has been observed in Mg3Cd (Davies and Stoloff, 1964; Stoloff and Davies, 1964 a, b) and Ti3Al (Minonishi and Yoo, 1990a,b; Umakoshi et al., 1991 e).
6.4 Deformation Behavior of Intermetallics 6.4.1 Slip Systems and the Effect of the Dislocation Core Structure The plastic properties of crystals are known to be characterized by their crystal structure. For example, f.c.c. crystals such as Al and Cu are always deformed by
264
6 Deformation of Intermetallic Compounds
{lll}<110> slip systems and their flow stress does not depend strongly on the deformation temperature, while in b.c.c. crystals slip occurs on {110}, {112} and sometimes non-crystallographic planes depending on crystal orientations, and the flow stress rapidly increases with decreasing temperature. Such a marked difference in the plastic properties can be explained by the dislocation core structures [see the review by Vitek (1974) and Chap. 7 in Vol. 1 of this Series]. Atomic ordering produces new, specific properties in addition to common plastic behaviors due to the based-lattices. In B2 intermetallic compounds the slip direction is known to be either <111> or <001>. The first theoretical prediction of the slip direction in B2 intermetallic compounds was made by Rachinger and Cottrell (1956) based on ordering energy. A <111> superlattice dislocation can be dissociated into two 1/2<111> superpartials bound by an APB. To maintain a balance between the surface tension of a ribbon of the APB and the elastic repulsive force, the separation distance (r) between two superpartials on (hkt) is given by Stoloff and Davies (1964 a).
4VhS2
2TI '
• [cos2 9 + sin2 9/(1 -v)]
(6-2)
where G is the shear modulus, b the Burgers vector of the superpartial, a the lattice parameter, v the Poisson ratio, 9 the angle between the dislocation line and the Burgers vector, Fthe ordering energy, and S the degree of long range order, respectively. When the separation distance approaches a lattice parameter with increasing ordering energy, it rarely represents a true dissociation and <001> dislocation may exist (Rachinger and Cottrell, 1956). From the
viewpoint of the energy of dislocation, the energy of an extended dislocation is the sum of the self-energies of the two superpartials, the elastic energy and the energy of the APB ribbon. In strongly ordered B2 compounds such as NiAl, CoAl, AgMg and AuZn, the difference between the energies of an <001> dislocation and a dissociated <111> superlattice dislocation is smaller than that in b.c.c. crystals and hence, the relative mobility of various dislocations may be an important factor determining the operating slip systems (Ball and Smallman, 1966; Potter, 1970). Transition of the slip direction from <111> to <001> is observed at low temperatures in AgMg (Aboelfotoh, 1972), AuZn (Schulson and Teghtsoonian, 1969) and NiAl (Ball and Smallman, 1966) depending on the crystal orientation. Atomistic studies of <001> dislocation core structures provide an answer as to why <001>-slip is preferential at low temperatures in B2 compounds compared to b.c.c. crystals. Figure 6-8 shows the schematic dislocation core structure of <001> screw dislocations in b.c.c. and B2 lattices. The elastic center of the dislocation is placed in the position S and the atoms are relaxed to new equilibrium positions [see the review by Vitek (1974) for the details]. In the figure the shadow shows the region where large relative displacements between atoms parallel to the Burgers vector (the continuous distribution of Burgers vector) are distributed around the elastic center of the dislocation. Since the core of a <001> screw dislocation in b.c.c. crystals is nonplanar (cylindrical core), as shown schematically in Fig. 6-8 a, a high applied stress is required for the core to be transformed into a glissile planar configuration prior to the net movement. Therefore, <001>-slip is never observed due to the high Peierls
6.4 Deformation Behavior of Intermetallics
265
o
J-s
(110)
J-s
Slip plane
(11D [110]
(111) [110]
(a) stress. The atomic ordering changes the core structure of the <001> screw dislocation. The large inelastic displacements of the <001> screw dislocation in E2 compounds are mostly confined to a single (110) plane (planar core) and therefore, the dislocation can easily move on the (110) plane as shown in Fig. 6-8 b (Yamaguchi and Umakoshi, 1975 a, b; 1976). In B2 compounds with <111 >-slip directions, it is expected that slip occurs on {110} planes where the APB energy is the lowest. However, the observed slip planes are found to vary with crystal orientation and are often non-crystallographic planes. To investigate the orientation dependence of the operative slip plane, the variation of the observed slip plane with orientation may be plotted as a curve of \jj versus x i*1 a similar way to that used for the <111>slip in b.c.c. metals and alloys as shown in Fig. 6-9. Taking [111] as the slip direction, for example, x *s the angle between the maximum resolved shear stress (MRSS) plane and the reference (T01) plane, and \j/ is the angle between the observed slip plane and the reference (101) plane. In compression, the shear in the [111] direction on the (211) plane (^ = 30°) is in the twinning sense, while that on the (TT2)
(b)
Figure 6-8. Schematic dislocation core structure of <001> screw dislocation in (a) b.c.c. and (b) B2 lattices. The shadow shows the region where large displacements parallel to the [001] direction exist (Yamaguchi and Umakoshi, 1975 a).
plane (x= — 30°) is in the anti-twinning sense. Although the i/j-x behavior varies from one material to another, Fig. 6-10 a indicates that \jf tends to degenerate to 30° and the most fundamental slip planes of <111 > superlattice dislocations in almost all compounds with the B2 structure, except for FeCo, are {112} planes at low temperature (Yamaguchi, 1982). Thus the strong tendency for {112}-slip at low temperatures cannot be directly interpreted from the effect of the APB but probably stems from dislocation core effects. Observed slip plane
-30
011
Observed slip plane Figure 6-9. A [111] stereographic projection showing the definition of x (stress axis) and ij/ (observed slip plane) for -type slip.
266
6 Deformation of Intermetallic Compounds
A
30
I
I
I
T
O AgMg at 4.2K
0 AgMg at 570K
AgMgat77K
A CuZn at 473K
A CuZn at 77K
• FeCo at 600K
• FeCo at 77K
Q FeAl at 623K
ty FeAl at 77K J
-30
-20
-10
0
L
J_
10
20
1 30 -30
1 -20
i 1 -10
0
X (deg.)
X
(a)
(b)
10
20
30
Figure 6-10. \j/-x relations for -slip of B2 compounds, (a) at low temperature, (b) at high temperature (Yamaguchi, 1982).
Atomistic studies of the core structure of <111> superlattice screw dislocations have been carried out by Takeuchi (1980), Umakoshi et al. (1983) and Yamaguchi and Umakoshi (1984). As seen from Fig. 6-11, the core structure of 1/2<111> superpartials in the B2 lattice is very similar to that of 1/2<111> screw dislocations in b.c.c. lattices. The cores of the superpartials are non-planar and the large inelastic displacements parallel to the <111> direction spread mainly on the three principal intersecting {110} planes, but they never overlap on the APB. The three layer faults also extend on the {112} planes with a twinning sense at the tip of the faults on the {110} planes. The core moves first along the (101) plane, converting the configuration, and the second jump occurs along the (llO) plane. The observed slip plane stems from a combination of two equivalent {110}-type alternative slips. The multi-layer faults created on the {112}
planes in front of the core have an important function in the strong tendency for {112}-slip of B2 intermetallic compounds at low temperatures. As the deformation temperature increases, there is a greater tendency for | \j/ \ to degenerate to 0° as shown in Fig. 6-1 Ob. This indicates that {110}-slip is favored. Although superpartials have a strong tendency to slip on the twinning {112} plane, the first jump always occurs along a {110} plane. Therefore it may be possible for the trailing partial to follow the leading one by means of a sequence of elementary jumps with the help of thermal activation. At high temperatures the Peierls stress decreases rapidly and the effect of the APB becomes important for the motion of the <111 > superlattice dislocation. The L2t structure of Ni2AlTi and Co2AlTi is so stable and the ordering forces are so large that dislocation dissociations involving ordered faults, such as the
6.4 Deformation Behavior of Intermetaliics
267
(101)
1101]
(111)
Figure 6-11. Schematic dislocation core structure of two dissociated 1/2<111> superpartials in the B2 lattice. The elastic center is placed in position S. The shadow shows the region where the inelastic displacements parallel to the Burgers vector are mainly distributed (Umakoshi et al., 1983).
APB, rarely occur. In fact a/2 perfect dislocations, which are unusual in b.c.cbased lattices, are activated at high temperature instead of <111 > and <001> superlattice dislocations (Strutt et al., 1972, 1976 a). The observed slip planes of the <110>-slip in Ni2AlTi and Co2AlTi, which show high strength and good creep resistance at high temperatures, are non-crystallographic {hhk} planes which are thought to be made up of composite slip on fundamental {110} and {100} planes in the zone of the <110>-slip direction (Yamaguchi et al., 1984; Umakoshi et al., 1985b; 1986). The slip behavior can also be explained by the core structure of the a/2<110> screw dislocation which has a planar core spread on {110} and splits into two well separated non-screw a/2 superpartials (Umakoshi et al, 1985 a). One of the most striking features discovered in LI 2 intermetallic compounds like Ni3Al is the anomalous strengthening which is closely related to the peculiar slip behavior of <110> superlattice disloca-
tions. Regarding slip systems, {001}<110>slip appears in the high temperature range above the anomalous strengthening peak temperature, while the L\2 intermetallic compounds can deform by {lll}<110>slip below the peak temperature, similar to the conventional f.c.c. metals. In general, a <110> superlattice dislocation has two- or four-fold dissociation involving an APB as described in Sec. 6.3.2.1. Since the APB energy on {001} is lower than that on {111}, the dissociation is mostly favored on the {001} but the dislocation dissociated on the {001} is non-glissile at low temperatures. The anisotropy of APB energies was first pointed out by Flinn (1960) and was subsequently used in the cross slip model by Kear and Wilsdorf (1962), Thornton et al. (1970) and Takeuchi and Kuramoto (1973) for the anomalous strengthening mechanisms (see Sec. 6.4.2.1). An answer as to why the dislocation becomes non-glissile on the {001} was found by atomistic calculations of the dislocation
268
6 Deformation of Intermetallic Compounds
1989). The temperature dependence of the yield stress of Pt 3 Al is quite similar to that in b.c.c. metals at low temperatures. The high Peierls stress for the <110> superlattice dislocation dissociated into two 1/3<112> superpartials, involving an SISF on {111}, is also due to the non-planar core structure. The extended core configuration depends on the based crystal structure and in addition it is also influenced by the atomic ordering. Therefore, a unique slip behavior has been observed in several intermetallic compounds. In the D019 intermetallic compounds slip can occur on basal and prismatic planes, and very occasionally on pyramidal planes, as shown in Table 6-2. The operative slip structures and slip systems are closely related to the stability and energy of planar faults on the possible slip planes, such as APB, CSF and SISF (see Sec. 6.3.4). One example is Mg 3 Cd. The atomic ordering activates non-basal slips in addition to the basal slip which is dominant in the disordered state, since APBs on the non-basal planes are seen to have energies lower than those on the basal plane (Davies and Stoloff, 1964; Stoloff and Davies, 1964b). The increase in the num-
core structure (Yamaguchi et al., 1982; Tichy et al., 1986; Farkas and Savino, 1988). A l/2<110> superpartial trailing the APB on {111} has the entire planar core dissociated into two overlapping Shockley partials on the {111} plane, and they can easily move on the {111} plane under an applied stress, similar to f.c.c. metals (see Fig. 6-12 a). On the other hand, the inelastic displacements of the l/2<110> superpartial with the APB on {001} never spread on the {001} plane, and they are confined on one or another {111} intersecting along the dislocation line (see Fig. 6-12 b). Therefore, the superpartial becomes immobile on the {001} plane and the parts of dislocations which have cross slipped from {111} onto {001} act as dragging points for the motion of the whole dislocation. Since the transition from the sessile to the glissile configuration of the core requires thermal activation, {001}slip occurs only at high temperatures. A sharp increase of the yield stress at low temperatures has been observed in Pt 3 Al in which the dissociation with an SISF is expected to occur since the SISF energy on {111} is low and the APB energy is high (Wee et al., 1984; Heredia et al.,
After cross slip
APB (111)"
(111)
or
(a)
(b)
Figure 6-12. Schematic representation of inelastic displacements parallel to the Burgers vector around the <110> screw dislocations in the LI 2 structure. Sx and S2 show the elastic center of l/2<110> superpartials (Yamaguchi et al., 1982).
6.4 Deformation Behavior of Intermetallics
Table 6-2, Observed slip systems of various D019type intermetallic compounds. Compound
Dominant slip
Secondary slip
Mg3Cd Ti3Sn
(0001)<1120>, {1010}<1210> {1010}<1210>
Ti3Al Mn3Sn
{1010}<1210> (0001)<1210>,
{1122}, {10T1} <1210> (0001)<1210>, {1100} [0001] {1121} {1011}
Ti6AlGa
(0001)<1210>,
{1121}
ber of operative slip systems makes Mg3Cd more ductile in the ordered state than in the disordered state. In the binary Mn3Sn (Takeuchi and Kuramoto, 1974) and Ti3Al (Sastry and Lipsitt, 1977; Lipsitt et al., 1980; Yang, 1982; Reran, 1984; Thomas et al., 1987; Court et al., 1989), even the combination of the primary and secondary slip modes along <1120> and [0001] directions gives only four independent slip systems since the (0001)<1120> slip system produces the same strain as the {1120}[0001] system. Therefore, the von Mises criterion for plastic deformation of polycrystals is not satisfied, explaining the poor ductility of these compounds. Although several observations of 1/3<1126> dislocations with Burgers vectors containing a c component have been reported in binary Ti3Al, the density of these dislocations is too low to contribute to the plastic behavior. According to recent results using the Ti3Al single crystals, {1121} slip can be observed only in the limited orientation near the [0001] corner, since the critical resolved shear stress (CRSS) for {1121} slip is about 8 times as large as that for {1010}<1210> slip in other orientations (Minonishi, 1991; Umakoshi et al., 1991 e). The CRSS exhibits an anomalous strengthening peak. High ac-
269
tivity of climb mobility of <1120> dislocations in Ti3Al induces an improvement in the ductility accompanied by a rapid decrease in the yield stress above 600 °C. In order to improve the ductility of D019 intermetallic compounds non-basal slip by dislocations with Burgers vectors containing a c component is necessary. The most attractive alloying design is with the niobium-modified Ti3Al compounds. An addition of Nb decreases the planar nature of the slip and increases the occurrence of non-basal slip, and results in an improvement in the ductility (Sastry and Lipsitt, 1977). 6.4.2 Strength and Flow Stress 6.4.2.1 Anomalous Strengthening and Strengthening Mechanisms The motion of superlattice dislocations traveling in pairs or in groups in ordered lattices gives rise to various attractive and interesting mechanical properties and plastic behavior which have been observed in intermetallic compounds. Among them there is the anomalous strength increase with increasing temperature, sometimes called a positive temperature dependence, which has been found at intermediate temperatures in various compounds, such as Ni3Al, Co3Ti, Ni 3 Ge (Ll 2 ), FeCo, CoTi, CuZn (52), Fe3Al, Fe3(Al,Si) (2X)3), Ti3Al, Mn 3 Sn (Z)019), and TiAl (Ll 0 ). The anomalous strengthening behavior may be classified into different groups in terms of the mechanism responsible for the strength anomaly. Effect of Transition in Types of Dislocations One possible mechanism for the anomalous strengthening, which was first proposed by Stoloff and Davies (1966) for FeCo, is based on the transition from glide
270
6 Deformation of Intermetallic Compounds
motion of superlattice dislocations as pairs or groups to that of unit dislocations. In the fully ordered B2 structure the deformation is known to be controlled by the motion of <111> superlattice dislocations, composed of two superpartials combined with an APB. The separation width between the superpartials depends on the ordering energy and the degree of long range order (S). When the separation width exceeds about 400A (b is the Burgers vector) with decreasing S, each superpartial with a wide dissociation begins to glide independently, and an additional stress is required for the motion of the superpartial leaving behind an APB trail. As long as unit dislocations move independently, the decrease in yield stress with increasing temperature is attributable to thermal activation. A rapid increase of S below the ordering temperature (Tc) with decreasing temperature will increase the number of paired dislocations for deformation, causing a decrease in the yield stress. According to this mechanism, the anomalous strength peak should occur at a temperature just below I
CO
g
four-fold dissociation
-
Cross-Slip Model Anomalous strengthening has also been found in nickel-based L\2 compounds, such as Ni3Al and Ni 3 Ga, which are fully I
I
I
Tc, where a large portion of dislocations begin to move as superdislocations. It is also important that no strong orientation dependence is anticipated. This mechanism may also be applicable for the anomalous strengthening of Ag2MgZn, in which there is a transition from the L2 1 to the B2 structure. This is strongly supported by the fact that the peak temperature, which lies just below Tc (~0.92r c ), and the shape of the CRSS versus temperature curve are only weakly dependent not only on the crystal orientation but also on the operative slip systems, as shown in Fig. 6-13 (Yamaguchi and Umakoshi, 1980). In the compound Ag2MgZn the transition from four-fold to two-fold dissociation should be considered to correspond to the transition from the L21 to the B2 structure (see Sec. 6.3.3).
two-fold dissociation
"GO
i
A £ 200-
# O'
/D
y {1101-slip
l
D
;= 150H v
CO
g
• \
\
o
: / _ 100 h-
°
\A
\ \\
011
,
:
A
001
I
:
v\ D \\
I
I
100 200 Temperature (°C)
I
A
300
:
Figure 6-13. Temperature dependence of the CRSS for {110}- and {112}-slipin Ag2MgZn single crystals (Yamaguchi and Umakoshi, 1980). The symbols in this figure correspond to those in a unit triangle [001]-[011]-[Tll].
6.4 Deformation Behavior of Intermetallics
ordered up to their melting temperature. A noticeable feature in these compounds is that the anomalous stress increase starts early from room temperature, and the peak temperature and flow stress depend strongly on the crystal orientation. A widely accepted model for anomalous strengthening is the cross-slip mechanism which is based on the difference in the core configurations and in the mobilities between dislocations on {111} and {001} planes, and on the anisotropy of APB energies. This model was first proposed by Kear and Wilsdorf (1962). As described in Sec. 6.4.1, the core of a <110> superlattice screw dislocation on {001} never spreads on the {001} plane but forms sessile configurations, while the dislocations on {111} planes exhibit the glissile planar core structure (see Fig. 612). From the viewpoint of APB energies, the dissociation of the dislocation is most favorable on the {001} plane. Therefore, some parts of moving screw dislocations on the {111} plane undergo thermally activated cross-slip on the {001} plane and act as dragging points for the motion of the entire dislocation. The driving force to produce the dragging points is provided by the stress component of applied stress on the {001} cross-slip plane, production of which is due to a thermally activated process. Takeuchi and Kuramoto (1973) have demonstrated for Ni 3 Ga that the CRSS for {lll}<110> slip is larger for specimens having a large stress component on the {001} cross-slip plane and that cross-slip of screw dislocations onto the {001} plane is responsible for the deformation mechanism below the peak temperature. According to the results of core calculations for <110> superlattice dislocations in the LI 2 structure, the core of l/2[101] superpartials on the (111) plane exhibits a split configuration composed of two Shockley partials with Burgers vectors of
271
1/6 [211] and 1/6 [112]. After the cross-slip the superpartials never spread on the {001} plane, but on the (111) or (111) plane (see Fig. 6-12). When cross-slip happens, the separation width between two Shockley partials on primary (111) and secondary (111) planes is important. The two partials should contract on the primary (111) plane before cross-slip but after the cross-slip they are required to extend on the secondary (111) plane. As a result of the effect of the edge components of the two Shockley partials under an applied stress, constriction and extension of the partials occurs. Since the applied stress in tension or compression affects the separation width between two Shockley partials (the core width) in different ways, a tension/compression asymmetry of yield stress appears. Paidar et al. (1984) have given the activation enthalpy for cross-slip using a double kink formation process and evaluated the tension/compression asymmetry for the yield stress focusing on the separation width of the two Shockley partials. The asymmetry is given by K. K=[l-x(N+\/3)(2N-}/3)/ N=TJTP
(6-3)
where Te, t p and TC are the resolved shear stress for the (lll)[12l], (111) [101] and (010) [T01] slip systems, respectively. N and Q represent the effect of the stress component on the ease of cross-slip of 1/2 [T01] superpartials from (111) onto (010) and that of the applied stress in the constricting cores of the two superpartials on the primary (111) plane, respectively, x is a parameter for the relative importance of the core constriction effect, before and after cross-slip, and depends on the material.
272
6 Deformation of Intermetallic Compounds
The asymmetry in Ni3(Al, Ta) single crystals, obtained by Umakoshi et al. (1984 a), is shown schematically in Fig. 6-14. Samples near [001] are stronger in tension than in compression, while the reverse is true for samples near [Til] and [Oil]. The asymmetry is very large for samples oriented near [Oil], and disappears somewhere between [001] and the [012]—[T13] great circle. A similar asymmetry was found in Ni3(Al, Nb) (Ezz et al., 1982) and Ni 3 Ge compounds (Ezz et al., 1987). In intermetallic compounds with the D0 19 structure, l/3<1120> superlattice dislocations are thought to be dissociated into superpartials combined with the APB on the basal and prism planes. The anisotropy of the APB energy and the difference in the mobility of dislocations on the two planes are analogous to those on {111} and {001} planes in L\2 compounds. In this case the core of a superlattice dislocation on the basal plane forms a glissile configuration with high energy, while on the prism plane it has a sessile configuration with low energy. Therefore the anomalous stress in-
113
001 011 012 Figure 6-14. Orientation dependence of the asymmetry of tensile (T) and compressive (C) yield stress of Ni3(Al, Ta) single crystals at 400 K (Umakoshi et al., 1984 a).
crease, which occurs above 427 °C in Mn3Sn with the D019 structure, can be explained by cross-slip of the dislocation from the glissile configuration on the basal plane onto the sessile one on the prism plane, since the larger the stress component for cross-slip onto the prism plane, the higher the CRSS for (0001)<1120> slip becomes (Takeuchi and Kuramoto, 1972, 1974). An analogous stress peak was found in MoSi2 with the Cllb structure, and in this case cross-slip from {110) onto {013), corresponding to that from {111} onto {001} in LI 2 compounds, should be considered (Umakoshi et al., 1990c). A peak in the yield stress-temperature curves has been found in CuZn at around 200 °C. Brown (1959) explained this in terms of the difference in atomic order on the slip plane between the APB generated by slip of a dislocation and that formed under thermal equilibrium. However the peak temperature, which depends on crystal orientation and is in the range 0.58 to 0.69 Tc, and the deviation from the CRSS law in Fig. 6-15 suggest that another mechanism involving cross-slip is applicable (Umakoshi et al., 1976). The CRSS increases with increasing x on the twinning side, while it varies insignificantly with x on the anti-twinning side. <111> superlattice screw dislocations move as pairs on {110} planes on which the APB energy is lowest. This is supported by the fact that the observed slip plane is {110} for all orientations, at least those below 200 °C. The cross-slip parts of superpartials on the twinning {112} plane act as dragging points and give rise to a CRSS increase with increasing temperature. Since in this case the driving force for the cross-slip cannot be given by the anisotropy of the APB energy, the core structure of the superpartials probably has some bearing on the tendency for {112}-slip.
6.4 Deformation Behavior of Intermetallics
273
70
~ |
60
Q.
50
10 CO CC
100 200 Temperature (°C)
40
o
300
30 —
-30
-20
-10
10
20
_r 30
Figure6-15. Orientation dependence of the CRSS for {110} slip at 150°C in CuZn single crystals (Umakoshi et al., 1976). The symbols in this figure correspond to those in a unit triangle [001]—[011]—[Tl 1].
Effect of Climb Dissociation and Elastic Anisotropy Veyssiere (1984) proposed the importance of a glide/climb dissociation of parts of moving dislocations in relation to the anomalous strengthening in L\2 intermetallic compounds, and direct evidence of this has been reported in Ni 3 Al from weak beam studies. When a superpartial of a paired dislocation on a glide plane undergoes a positive climb, extra atoms must be emitted from the superpartial but the negative climb motion of another superpartial can absorb the extra atoms. Since the simultaneous positive and negative climb motion form a closed circuit for mutual transformation of point defects between two superpartials, diffusion of the climb dissociation from the bulk is not nec-
essary. Once climb dissociation occurs, the non-screw component of the climb dissociation is sessile for the motion of the whole dislocations. A related configuration was observed on the {013} planes which would be favorable APB planes (Veyssiere et al., 1985; Douin et al., 1986). Such non-conservative climb dissociation plays an important role in the anomalous strengthening. The previously mentioned cross-slip mechanism is based on the assumption that the APB and dislocation dissociations are of shear-type and that they are planar lying on {111} and {001} planes in L\2 intermetallic compounds (Yoo, 1986, 1987). However, the APB interface must be bent due to the force couples caused by the non-radial components of the elastic interaction forces between two superpar-
274
6 Deformation of Intermetallic Compounds
tials. The elastic anisotropy plays a decisive role in the non-planar dissociation which is related to the glide and climb dissociation of fractional edge dislocations with a pair of opposite sign. As the elastic anisotropy increases with increasing temperature, the glide/climb dissociation occurs more easily. However, to explain the strong orientation dependence of the CRSS in LI 2 compounds, the cross-slip mechanism with a double kink formation process is also needed. At an intermediate temperature the climb dissociation may superimpose its effect on the cross-slip mechanism. 6.4.2.2 High-Temperature Strength and Creep
Strong bonds between different atomic species make the intermetallic compounds stable up to high temperatures, and result in high strength at high temperatures. Some b.c.c.-based intermetallic compounds such as NiAl, CoAl and FeAl, which have high-melting temperatures (Tm and high APB energies, are expected to be useful as high-temperature structural materials with good oxidation resistance. Although these compounds retain their high strength at intermediate temperatures where deformation is still by slip, the deformation is mainly controlled by a diffusion process above 0.6 7^. Rudy and Sauthoff (1986) found that the creep of binary FeAl was controlled by the viscous glide motion of <001> dislocations. Therefore, an effective approach for the improvement of high-temperature strength is to suppress the diffusion (see the reviews by Sauthoff, 1986, 1989, 1990). The ternary B2 intermetallic compounds with a composition lying on a line of the pseudobinary FeAl-CoAl and NiAl-CoAl compounds (Jung et al., 1987) exhibit maxi-
mum creep resistance due to a decrease in the diffusion coefficient, as shown in Fig. 6-16 (Sauthoff, 1991). A highly ordered structure is also effective for obtaining high strength at high temperatures. Substitutional replacement of Al by Ti or Nb in binary NiAl and CoAl promotes a strongly long range ordered structure, such as the L2± structure. Co2AlTi, Ni2AlTi and Ni 2 AlNb with the L21 structure, in which the shear modulus increases and the diffusion coefficient decreases, exhibit good strength and creep resistance at high temperatures. In the ternary L2± compounds <110>-slip becomes dominant instead of the <001>-slip observed in binary NiAl and CoAl. The yield stress and creep strength of ternary Ni2AlTi and Co2AlTi are about two to
I
I at 900°C
6 =1x10~ 7 /s
100 \
a.
O
50
CoAl
:
(Ni 0 2 Fe 0 8 )AI O NiAl
20
-
CD
(Ni 08 Fe 02 )AI
6
-
10
5
o FeAl \ I
I 10
10
10 2
Diffusion coefficient (m /s) Figure 6-16. Creep resistance with the secondary creep rate of 1 x 10" 7 s" 1 for binary and pseudo-binary B2 compounds as a function of diffusion coefficient at 900 °C (Sauthoff, 1991).
275
6.4 Deformation Behavior of Intermetallics
three times larger than those of binary NiAl and CoAl (Strutt et al, 1976a, b). 6.4.2.3 Effect of Off-Stoichiometry The strength of intermetallic compounds depends strongly on compositional deviations from stoichiometry, in general showing a minimum at a stoichiometric composition, and significant defect hardening on both sides of stoichiometry (see Fig. 6-17). The magnitude of the strength increase has an asymmetry on both sides and varies from material to material. It is believed that Ni-rich deviations from stoichiometry in NiAl induce partial replacement of Al-occupied sites by Ni atoms (anti-structure defects), while Ni-deficient compositions are accommodated by vacant Ni sites (constitutional vacancies) (Pascoe and Newey, 1968). The constitutional vacancies may be induced to hold a constant electron concentration in NiAl. A linear relationship exists between the strength and the square root of the defect concentration, although the slopes of the straight lines differ with concentration. Such defect structures in B2 compounds may be responsible for the asymmetry of defect hardening on both sides of stoichiometry. One exception is FeAl, in which the strength decreases continuously with increasing Fe content (Westbrook, 1956). This cannot be understood by a mechanism based on the defect structure since the existence of anti-structure and constitutional vacancies depends on the composition. It is known that NiAl is deformed by <100>-slip but that FeAl is deformed by -slip. In FeAl, the decrease in the APB energy with increasing Fe content would cause <111> superlattice dislocations involving the APB to become more mobile. The ductility improvement in Fe-rich FeAl is also attractive for industrial applications.
42
300
44
46 48 50 52 54 56 Al, Mg, Zn or Mn (at%) (a)
\V
\Ni3Si Ni3Ge\
/
/
200 -
"
d RT
/
D /
/
^
\
t
Ni 3 Ga " " ~ " ~ - - o ^
J /
/ *
77K
/
•/ /
100h
Ni3AI
22
"-Cjj^n
1
I
23
24
I
1
1
25
26
27
28
AI.Ga.Ge or Si (at%) (b)
Figure 6-17. Variation of strength with composition for (a) B2 and (b) Ll 2 intermetallic compounds (Pascoe and Newey, 1968; Westbrook, 1956; Suzuki et al., 1989).
There is a similar compositional dependence of yield stress in L l 2 intermetallic compounds. In Ni 3 Al no constitutional vacancies are believed to be formed and only anti-structure defects exist since the lattice parameters continuously change
276
6 Deformation of Intermetallic Compounds
throughout the stoichiometric composition and there is no difference in the compositional dependence of the diffusion rate. The fact that the lattice parameter of Ni 3 Ga changes with composition, similar to that of Ni 3 Al, suggests the presence of anti-structure defects in Ni 3 Ga (Noguchi et al., 1981). The increase in content of minority elements, such as Al and Ga, leads to a sharp strength increase which is more pronounced at high temperatures. This tendency shows that the strength anomaly becomes more considerable and the activation energy for the formation of dragging points decreases with increasing content of a minority element. In this case, the decrease in stability of the LI 2 structure compared with the D019 and D022 structures lowers the APB energy on cube planes and results in high activity for cross-slip onto the cube planes [see the review by Suzuki et al. (1989) for details].
6.5 Ductility and Embrittlement Strong bonds between different atomic species in intermetallic compounds promote attractive properties such as a high elastic modulus and superior high-temperature strength accompanied by good structural and chemical stabilities. However, at the same time strong bonding gives rise to low temperature embrittlement which has often been a severe hindrance to the application and processing of intermetallic compounds. The causes of intergranular fracture and/or transgranular cleavage in a brittle manner, and an approach to suppressing the brittleness will be discussed later. 6.5.1 Effect of Deformation Mode When a critical stress for nucleation of a crack is lower than the Peierls stress for the
motion of dislocations, transgranular cleavage occurs in a brittle manner. To avoid brittle cleavage fracture, it is necessary to raise the fracture stress and/or lower the flow stress. In general, strongly ordered bonds give rise to a high Peierls stress and from the viewpoint of a phase diagram, Daltonide-, Berthollide- and Kurnakov-type compounds become less favorable for ductility in sequence (see Sec. 6.2.2). CuZn and Ni 3 Fe with low ordering energy, for example, are deformable at room temperature. An addition of Cr to Fe3Al decreases the transition temperature, where the D03 changes to the B2 structure, and results in an improvement in ductility (McKamey et al., 1988). The ductility depends on the deformation mode and is reduced by planar and coarse slip. In Cu 2 MnAl single crystals with the L2X structure, brittle fracture at temperatures below 240 K occurs in the orientation range where slip is expected to take place on {110} planes, while good ductility is observed at all temperatures investigated when slip is activated on {112} planes, as shown in Fig. 6-18. Brittle fracture is triggered by the sudden split of coarse {110} slip bands whose occurrence is associated with the dissociated dislocation configurations (Umakoshi et al., 1984a). Thus deformation behavior and activated slip modes are related to the fracture behavior. An addition of alloying elements which would affect the phase stability against an analogous crystal structure would decrease the energy of planar faults on a certain plane and activate an additional slip and/ or twin which can be operative under a lower applied stress, and result in ductility improvement (see Sec. 6.6.1.2). Impurity atoms often cause a loss in ductility but little attention has been paid to this aspect in intermetallic compounds. The decrease in impurity content is expect-
6.5 Ductility and Embrittlement 111
deformed at 77K
\
/
(211)1111] slip
\
'(Ti2)\( 101)1111]slip\ 'M11] slip \ \ 001
011
Figure 6-18. Orientation dependence of the observed slip planes and deformability of Cu2MnAl single crystals deformed at 77 K. Micrographs show the specimens after a compression test (Umakoshi et al., 1984 b).
ed to lower the flow stress and to suppress the nucleation and growth of cracks. Great success in ductility improvement has been reported for TiAl (Kawabata et al., 1988) and NiAl (Hahn and Vedula, 1989). Thus high purity Ti-48 at.% Al exhibits 2.7% elongation at room temperature, which is the highest value among binary TiAl compounds.
277
ture, a schematic structure among various possible configurations is given in Fig. 6-19. The majority of atom bond pairs near the grain boundary are A-A bonds accompanied by a decreasing number of A-B bonds at the first nearest neighbors and B-B bonds are scarcely seen (Takasugi and Izumi, 1985 a). The concept of bond directionality and charge polarizability is often useful to describe the elastic properties and to understand the fracture on an atomic scale. The directionality of A-B bonds due to the off-center contribution by the first nearest neighbor's interaction becomes stronger as the covalent nature of the A-B bonds in A3B-type intermetallic compounds increases. The cohesive strength at grain boundaries may decrease because the A-B covalent bonds will be allowed only at limited angles across the grain boundaries. The effect of the heteropolarity of the electron charge density should also be considered. Strongly electro-negative atoms (B atoms) draw charge out of metal-metal bonds (A-A bonds) at grain boundaries and form het-
O o,
B atom
6.5.2 Grain Boundary Embrittlement
The low ductility of polycrystalline intermetallic compounds, in particular Ni3Al, stems from the weakness of grain boundaries, and the fracture is controlled by intergranular fracture. While the segregation of impurities of grain boundaries and the limited number of operative slip systems certainly reduce the deformability, the main cause of grain boundary embrittlement should, nevertheless, be related to the structure and chemical bonding at the grain boundary itself. With regard to the grain boundary structure of A3B-type intermetallic compounds with the L\2 struc-
Figure 6-19. Schematic grain boundary structure in the LI 2 structure (Takasugi and Izumi, 1985 a).
278
6 Deformation of Intermetallic Compounds
eropolar bonds. The effects of charge polarization and charge directionality must be more apparent near grain boundaries where a heterogeneous distribution and combination of atomic bonds exists, as seen from Fig. 6-19. Strong heteropolarity of the electron charge density of A-B bonds and discharging from A-A bonds at grain boundaries lead to the formation of a cavity between A-A atoms. Effects of the combination of constituent elements on grain boundary embrittlement have been experimentally investigated in Li 2 compounds. Takasugi and Izumi (1985 b) estimated the cohesive strength of grain boundaries in Ni-based L\2 compounds taking into account the combined effects of the valency difference and the atomic size ratio between component atoms. The degree of grain boundary fragility is ranked as Ni 3 Ge>Ni 3 Si> Ni 3 Ga > Ni 3 Al > Ni 3 Mn > Ni 3 Fe. Taub and Briant (1987) have suggested a significant contribution by electro-negativity, and obtained a threshold value of about 1.8 for the average electro-negativity in Nibased LI 2 compounds, in achieving the ductility improvement. 6.5.3 Hydrogen Embrittlement
Since the appearance of the first report on hydrogen embrittlement in an (Fe, Ni)3V compound by Kuruvilla et al. (1982), the effect of hydrogen on the mechanical properties of intermetallic compounds has been investigated in Ni 3 Fe (Camus et al., 1989), Co 3 Ti (Takasugi and Izumi, 1985c), Ni 3 Al (Kuruvilla and Stoloff, 1985), Fe3Al and FeAl (Liu et al., 1990). The ductility and tensile strength are lowered remarkably when tested in precharging and/or simultaneously charging conditions of hydrogen, but even in air the hydrogen effect occurs. The reaction of
metal atoms with a trace of water vapor in air produces atomic hydrogen which drives into the intermetallic compounds and causes hydrogen embrittlement in a similar fashion as it occurs in many Al alloys (Liu et al., 1989, 1990; Liu and George, 1990, 1991). The increase in the strain rate enhances the effect of hydrogen on the elongation and the tensile strength, while the yield stress is independent of test environment and strain rate. This suggests that the hydrogen embrittlement is caused by dynamic effects, and by the mobility of hydrogen near the crack tip, rather than by static effects such as the segregation of hydrogen into grain boundaries. Atomic hydrogen is transported by the moving dislocations and through grain boundaries aggregates, and forms severe stress concentrations which can develop into cracks. Not only grain boundaries but also dislocations and/or other lattice defects could contribute to hydrogen embrittlement since hydrogen embrittlement was observed even in Co3Ti single crystals (Takasugi and Izumi, 1986). The susceptibility for hydrogen embrittlement is enhanced by atomic ordering since transport of hydrogen by moving dislocations can be effectively accompanied by planar slip in the ordered state. The detailed mechanisms responsible for environmental embrittlement of intermetallic compounds have been reviewed by Liu (1991). 6.5.4 Attempts at Ductility Improvement
Ductility improvement has been attempted through additions of micro- and macro-alloying elements, grain refinement, transformation into a highly symmetrical crystal structure and microstructure control. One of the most encouraging results is the discovery of good ductility for polycrystalline Ni3Al by the addition of
6.5 Ductility and Embrittlement
small amounts of boron (Aoki and Izumi, 1979). The addition of a few hundred ppm wt.% of boron as a dopant is remarkably effective at suppressing intergranular brittle fracture. The effect of boron has been explained on the basis of several assumptions, such as (1) the increment of cohesive strength across grain boundaries due to the formation of additional, strong covalent bonds (Liu et al., 1985; White et al., 1984), (2) activation of the motion of dislocations within grain boundaries and the accommodation of slip (Baker et al., 1987), and (3) disordering in the vicinity of grain boundaries (King and Yoo, 1987). High ductility, exceeding 50% tensile elongation, has been accomplished by grain refinement and control of boron and aluminum contents in Ni 3 Al by an appropriate thermo-mechanical process, as shown
279
in Fig. 6-20 (Liu et al., 1985). The addition of 500 ppm of boron can make Ni3Al able to be deep-drawn even at room temperature (Liu and White, 1985; Liu and Stiegler, 1984). Additions of substitutional elements such as Fe, Cr, Hf and Zr together with boron are also effective for the development of new Ni3Al-based intermetallic compounds with high strength and good ductility. Schulson and Barker (1983) have suggested that the stress required for a crack to propagate may increase with decreasing grain size and fracture will be delayed in compounds with fine grains below a critical grain size. They have reported that high ductility exceeding 40 % elongation can be achieved at 400 °C in Ni-rich NiAl with small grain sizes of less than 20 |im. Grain refinement of FeAl with compositional de-
600
o
0.20 0.12 0.06 0.08 0.10 Boron concentration (wt%) Figure 6-20. Effect of boron additions on the yield stress and tensile elongation of Ni3Al (24at% Al) deformed at room temperature (Liu et al., 1985). The symbols • and o represent the tensile elongation and yield stress, respectively. 0.02
0.04
280
6 Deformation of Intermetallic Compounds
viations on the Fe-rich side from stoichiometry improved the ductility (Crimp et al., 1987). The beneficial effect of grain refinement is also found in a stoichiometric NiAl with a boron addition (Hahn and Vedula, 1989) and low contents of carbon and oxygen. The purification of 2?2-type intermetallic compounds effectively improves ductility since the B2 structure is a b.c.c.-based structure and interstitial atoms harden alloys with b.c.c. lattices. Optimization of the relative stability between analogous types of crystal structure may also be exploited as a possible method for improving ductility. In A3B-type intermetallic compounds the cubic ordered structure may be stabilized with respect to the hexagonal structure as the electron concentration decreases. The partial replacement of Co with Fe in Co3Ti with a D019 structure alters the electron concentration and ductile (Fe, Co)3V compounds with an L\2 structure are obtained (Liu and Inouye, 1979). Deformation of polycrystalline crystals must satisfy the von Mises criterion concerning slip systems. The addition of Nb to Ti3Al with a 5 0 1 9 structure activates (c + a) dislocations and non-basal slip, and results in better ductility. Control of microstructure is also important for ductility and an example can be seen in Ti-rich TiAl with a unique lamellar structure. The role and control of lamellae will be described in Sec. 6.6.1.
6.6 Advanced Intermetallics as High-Temperature Structural Materials 6.6.1 High-Strength and Low-Density Aluminides As the requirements for high-temperature structural materials of high-strength-
low-density ratio are becoming increasingly severe, intermetallic compounds in the Al-Ti system are becoming of much more interest. Although there are three aluminides based on the compositions Al3Ti, TiAl and Ti3Al in the Al-Ti system, recent research and development activities have been focused on TiAl and Al3Ti because of their better oxidation resistance and superior-strength-weight ratio at elevated temperatures. In this section the plastic behavior of TiAl and Al3Ti based aluminides will be described. 6.6.1.1 TiAl Based Aluminides Since McAndrew and Kessler (1956) pointed out the attractive characteristics of TiAl for low density structural materials, much attention has been given to the deformation behavior of this compound. However, serious problems such as poor tensile elongation and low impact toughness at room temperature, exist for industrial applications. This section will cover the microstructure, the deformation modes and the mechanical properties of TiAl. Lamellar Structure and Twins The TiAl phase with an Ll0 structure has a wide composition range on the Alrich side of off-stoichiometric composition. There are two types of peritectic reactions connected with the TiAl(y) phase. During solidification the peritectic reaction between the b.c.c. p-Ti phase and the liquid phase produces the h.c.p. Ti3Al(a) phase and at lower temperatures the L + a->y peritectic reaction occurs (McCullough et al., 1989). After the peritectic reactions, the phase transformation from a phase to y phase proceeds. Finally, a twophase structure composed of a 2 (ordered Ti3Al with a D019 structure) and y phases can be obtained but the morphology, such
6.6 Advanced Intermetallics as High-Temperature Structural Materials
as a lath- or a cell-like structure, depends on the composition and the solidification path. By annealing at 1200°-1400°C, a ysingle phase with equiaxial grains can be obtained, even near stoichiometric compositions. The oc-+Y phase transformation begins as the a phase cools. During the phase transformation the y plates nucleate and become thicker at the expense of the a phase, maintaining the following orientation relationship between the two phases, since the atomic arrangements on (0001) planes in the h.c.p. lattice are analogous to
281
those on the {111} plane in the L l 0 structure through the stacking faults. {lll} y //(0001) a2 ,
<110>y//<1120]a2 (6-4)
The a phase orders to the oc2 phase at the eutectoid and finally a lamellar structure composed of a 2 and y phases is formed. At the interface between the oc2 and the y phase twinned TiAl, which are closely related to the accommodation of strains created during the phase transformation, can very often be observed. Figure 6-21 shows an example of the twinned structure and
01Distance (b)
101
+o
+
'+
0001
T010+ T01T+T +
+
o
OOOT +
+
+
o
+H +10T0
OTT
+
Figure 6-21. Results of analytical electron microscopy examinations of PST crystals of TiAl. (a) Scanning transmission image of lamellae, (b) distribution of Al concentration across the lamellae, (c) micro diffraction patterns from region B in (a). The dark bars in (b) show the aluminum content (Fujiwara et al., 1990).
282
6 Deformation of Intermetallic Compounds
the distribution of aluminum concentration across the lamellar boundaries obtained by Fujiwara et al. (1990). The bright region A is mainly composed of twin lamellae of Ti Al, while the darker region B is composed of thin twins and Ti3Al plates. Only the [112] twin (ordered twin) among three possible shear vectors on the (111) plane can preserve the symmetry of the L l 0 ordered structure. The [121] and [211] twinning shear vectors destroy the ordered symmetry and are thought to be forbidden. Therefore, only four <112] {111} ordered twinning systems exist in TiAl. Furthermore, the number of operative twinning systems decrease from 4 to 0 through 2, depending on the loading modes and orientations, similar to the D022 structure (see Fig. 6-32). As can be seen from the atomic arrangements in Fig. 6-22, three <1120] directions in the D019 structure are identical, while the <110] and <101] directions in the L l 0 structure are not equivalent. When considering nucleation of the y phase from the oc2 phase, three configurations are possible for superimposing the y phase onto the oc2 phase. For example, the directions [TlO]M (A), [OI1]M (B) or [10l]M (C) in the y phase can lie parallel to the [1210](A') direction on the (0001) plane in the oc2 phase. Therefore, the existence of three types of do-
mains is possible and the formation of these domains is expected to play an important role in the accommodation of strains created during the phase transformation from oc2 to y phase. Combinations of such domains form ordered twinning and pseudo-twinning relations between two neighboring TiAl domains (Yamaguchi et al., 1990). In the absence of ordered twins, stacking faults near the TiAl/TiAl domain interface may develop into thin Ti3Al plates to decrease the energy of the interface and form a lamellar structure. Under an applied stress, the activity of deformation twins depends on the angle between the odirection of the domain and the loading axis. The volume fraction, size and distribution of domains of TiAl play an important role in the plastic behavior of TiAl, Fujiwara et al. (1990) and Yamaguchi et al. (1990), using the floating zone method, first succeeded in controlling the lamellar structure and obtaining TiAl crystals with a single set of unidirectionally aligned lamellae containing numerous twins and thin Ti3Al plates. They called these crystals "polysynthetically twinned crystals" (PST crystals). More recently Umakoshi et al. (1991 a, b) controlled the lamellar spacing and the thickness of Ti3Al plates in PST crystals of TiAl containing
[110]
[1011
[011J.
(111)
(a)
[21101
Figure 6-22. Atomic arrangement and orientation relationship between L l 0 and D019 structures, (a) (111) in TiAl with the L l 0 structure, (b) (0001) in Ti3Al with the D0 19 structure (Yamaguchi et al., 1990).
6.6 Advanced Intermetallics as High-Temperature Structural Materials
283
rate, as shown in Fig. 6-24 (Umakoshi et al., 1991a). Strength and Deformation Mode
(a)
10
(b) Figure 6-23. Optical micrographs of PST crystals of Ti-48.lat.%A1, (a) growth rate, 2.5 mm h" 1 , (b) growth rate, 20 mm r T 1 (Umakoshi et al., 1991 b).
48.1-51.6 at. %A1 using the same method at various crystal growth rates. The lamellar spacing is very sensitive to the crystal growth rate and a PST crystal with extremely fine and uniformly distributed lamellae is obtained at the growth rate of 2.5 mm h " 1 (see Fig. 6-23). Since the formation of domains and twins is closely related to the accommodation of strains created during the phase transformation from a to y, the volume fraction of the oc phase for the transformation, which depends on the chemical compositions of the TiAl compounds may affect the number of nucleation sites for twins. Therefore, the spacings between lamellae can be changed depending on their aluminum compositions even at a constant crystal growth
The mechanical behavior and the deformation structure of TiAl compounds depend strongly on their compositions and, in particular, there is considerable difference between single phase and two phase compounds. Single-phase (y) TiAl, which has a composition on the Al-rich side of off-stoichiometry, shows a slight decrease in yield stress and no useful tensile elongation up to 600 °C (see Fig. 6-25; Lipsitt et al., 1975). Ductility improvement starts at 700 °C. The high strain rate sensitivity of the ductile-brittle transition temperature suggests poor impact toughness for the Alrich TiAl compound. Two-phase (a 2 + y) TiAl with Ti-rich compositions shows higher strength and better ductility than the single-phase TiAl over the whole temperature range (see Fig. 6-25; Hall and Huang, 1989). The lamellar structure in two-phase TiAl is believed to yield good ductility and high strength. A fine and homogeneous distribution of lamellae composed of TiAl domains and thin Ti3Al plates improves the mechanical behavior; details will be discussed in Sec. 6.6.6.1. Slip of TiAl compounds always occurs on {111} planes and the Burgers vectors of possibly activated dislocations are, in order of increasing energy, l/2<110], 1/2<112] and <101]. Twinning is another important deformation mode and only ordered twins are allowed, as described previously. The 1/2<110] unit dislocation is expected to dissociate into two Shockley partials, in a similar way as it occurs in f.c.c. crystals, but the dissociation is not observed because a CSF of high-fault energy is created. <101] superlattice dislocations may be dissociated by the following reactions and
284
6 Deformation of Intermetallic Compounds
2.0 |— 1.9 1.8 1.7 ~ 1.6 -
Crystal growth rate: 5mm/h
1.5 —
1 1-4£ o
1.3-
I
1.1 -
i
1.0 — 0.9 0.8 0.7 0.6 0.5 — 0.4 0.3 48
49
50 Al (at%)
51
often form sessile dissociations involving SISF and superlattice extrinsic stacking faults (SESF): [011]->l/2[110] + 1/2[T12] (Greenberg, 1973). 1/2[112] can be dissociated further. (SISF) + 1/6[T12]. Another type of dissociation is
Decomposition of the 1/2<112] unit dislocation is apparent from the above reaction. The major deformation mode for Alrich single phase TiAl is that l/2<110] easy slip occurs as well as <101]- and 1/2<112]slip but few deformation twins are observed at room temperature (Shechtman et al., 1974; Lipsitt et al., 1975; Hug et al.,
52
Figure 6-24. The lamellar spacings of PST crystals as a function of compositions of TiAl compounds at a growth rate of 5 mm h~ 1 (Umakoshi etal, 1991b).
1986,1988). <101] superlattice dislocations may be pinned by the trailing Shockley partial and become less mobile than l/2<110] dislocations below 600 °C. At elevated temperatures, where the ductility is rapidly improved with decreasing strength, formation of twins and extended faults on the {111} plane becomes more frequent, and at the same time the faulted dipoles vanish. Twinning deformation becomes dominant in Ti-rich two phase compounds even at room temperature. Such different twinning activity in Ti-rich and in Al-rich compounds is one of the reasons why the former compounds are more ductile than the latter. One of the most effective alloying elements for increasing the twinning activity is Mn. Addition of Mn decreases the stack-
6.6 Advanced Intermetallics as High-Temperature Structural Materials
285
600
- 40 500
400 CO Q_
300
A A Ti-48at%AI O • Ti-52at%AI 200
100
k j
200
400
600
___• ™
m strain rate i^
800
.
1000
Temperature(°C) Figure 6-25. Temperature dependence of yield stress and tensile ductility of binary TiAl compounds (Lipsitt et al., 1975; Hall and Huang, 1989).
ing fault energy and activates the formation of deformation twins (Hanamura et al., 1988; Hanamura and Tanino, 1989). Addition of 1 at. % Mn to Ti-rich Ti-Al gives a 2% tensile elongation (Tsujimoto and Hashimoto, 1989). Tetragonal distortions are unfavorable for the uniformity of deformation, and alloying to decrease the tetragonality will enhance the plastic deformation. A decrease in the Al content, and an addition of small amounts of V, Cr, and Ga reduces the electron concentration (e/a). V, in particular, is an effective element for ductility improvement. However, the effects of Mn and V additions require compositions on the Tirich side of off-stoichiometry. In Al-rich TiAl single crystals, an anomalous stress increase is observed at
intermediate temperatures, as shown in Fig. 6-26 (Kawabata et al., 1985). In region A l/2<110] dislocations are dominantly operative. In the dissociation reaction of the dislocations, the anisotropy of the fault energy of the APB and the SISF provides a driving force for forming their faults on (100) and (111), planes, respectively (Hug et al., 1988). The anomalous strengthening of Al-rich TiAl would then be caused by the locking due to the located faults. Another explanation for the anomalous strengthening is proposed by Greenberg (1989); however, the fact that the anomalous strengthening occurs similarly for both <101] and <110] slip complicates the hypothesis.
286
6 Deformation of Intermetallic Compounds
T
400 —
300
CL CO CO
0
200 —
100
Figure 6-26. Temperature dependence of the CRSS of Alrich TiAl single crystals (Kawabata et al., 1985). The symbols in this figure correspond to those in a unit triangle 200
400
600
800
1000
1200
Temperature (°C)
Deformation of PST Crystals PST crystals of TiAl which contain a single set of lamellae with parallel lamellar planes have been grown by Fujiwara et al. (1990) and Umakoshi et al. (1991 b, c) (see Fig. 6-23). The PST crystals have no grain boundaries but thin a 2 plates and y phase ordered domains of three different types, related by a 120°-rotation round the normal to the lamellar planes. Using these crystals the effects of the lamellar structure on the deformation of TiAl have been investigated (Fujiwara et al., 1990; Yamaguchi et al., 1990; Umakoshi et al.,
1991 b, c). The yield stress of the PST crystals is given as a function of the angle () between the loading axis and the lamellar plane, in Fig. 6-27. The values of the yield stress for specimens loaded in compression parallel or perpendicular to the lamellar planes are much higher than those for specimens loaded at intermediate angles of 0°<4><90°. The yield stress for specimens with the same value of (j> exhibits no significant orientation dependence. Therefore, the yield stress for PST crystals is mainly determined by the angle between the loading axis and the lamellar plane. For specimens with 0°<<90°, shear deformation
6.6 Advanced Intermetallics as High-Temperature Structural Materials
287
600
8 £ 400 45 Angle
70 <j> (deg.)
200Ti-49.3at%AI
_L 30 Angle (deg.)
60
90
Figure 6-27. Yield stress of PST crystals of TiAl compounds as a function of the angle (0) between the compression axis and the lamellar planes (Fujiwara et al., 1990; Umakoshi et al., 1991b). The symbols in this figure correspond to data of specimens with orientations indicated in a standard projection.
on {111} planes parallel to the lamellar boundaries (easy deformation mode) is always preferred to that across the lamellae. Loading parallel or perpendicular to the lamellar planes causes slip and/or deformation twinning to occur on {111} planes across the lamellae. When the slip and/or deformation twin proceeds across twin boundaries the twin boundaries act as an effective barrier against propagation of the slip and/or twin through the twin boundaries, similar to grain boundaries in polycrystals (hard deformation mode). However, the large difference in yield stress between specimens with
(j) = 90° is caused not only by the lamellar boundaries but also by a difference in the mode of deformation of the oc2 phase in the lamellae. Since the oc2 and the y phase forming the lamellae satisfy the orientation relations, as shown in Eq. (6-4), the compression axes of the a 2 and the y phase are given as the following for specimens loaded parallel (0 = 0°) and perpendicular (0 = 90°) to the lamellar planes. [10T0]B2. and
Y
for
0 = 0° (6-5)
[0001]a2 and <111>Y for (/> = 90° The mode of deformation in the a 2 phase depends strongly on the loading ax-
288
6 Deformation of Intermetallic Compounds
es and the specimens with 4> = 0° and with (f) = 90° are deformed by prismatic {1010}slip and pyramidal {1121}-slip, respectively, while for both orientations the y phase is deformed similarly by {lll}-slip, as shown in Fig. 6-28 (Umakoshi et al., 1991c). Since the CRSS for {1121}-slip, which occurs only for orientations near the [0001] corner, is about 8 times higher than that for {10T0}-slip, according to the results for ordered Ti3Al single crystals (Minonishi, 1991; Umakoshi et al., 1991 e), and the thin oc2 plates act as an effective barrier for the motion of dislocations, the specimens with 0 = 90° are strengthened considerably. Ductility is also related to the deformation mode. The hard mode of deformation across lamellar boundaries strengthens PST crystals but reduces the ductility, while the easy mode of deformation parallel to the lamellar planes enhances the duc-
- Compressive axis
tility appreciably. A tensile strain for Ti49.3 at.% Al PST crystals of as much as 6-7% was obtained at room temperature (Nakamura et al., 1990). When the rolling direction is controlled in the range of 45°60° to the lamellar planes, the thickness of PST crystals can be reduced by as much as 50% without intermediate annealing (Nishitani et al., 1990). It is known that a fine microstructure is effective for ductility improvement of materials because of its uniformity of deformation. In PST crystals the thickness, size and volume fraction of domains and lamellae of TiAl and Ti3Al thin plates are thought to play an important role in relation to ductility. Umakoshi et al. (1991b) have accomplished an improvement in ductility by controlling the average lamellar spacing, as shown in Fig. 6-29. PST crystals with 0 = 45° can be deformed into a dented shape by the easy deformation mode although the compres-
"Compressive axis -
TiAl
Ti3AI
(a2)
X
10Mm
* =0°
T i A I ( Y ) X T i 3 A I (a 2 )
(111) C2111) (111) \ \
TiAl\ (Y) \ i3AI(a2)
\
T~^(1H)
y
\
Y
s
— (1100)
IT-
TiAl (Y)
[1210]a
\ \V
2
(a)
TiAl (Y)
Ti3AI (a 2 )
TiAl (Y)
[0001] a
(b)
Figure 6-28. Slip markings of a2 and y phase in the lamellae of PST crystals of TiAl compounds deformed at room temperature. The compressive stress was applied (a) parallel or (b) perpendicular to the lamellar planes (Umakoshi et al, 1991c).
6.6 Advanced Intermetallics as High-Temperature Structural Materials
289
:UUU
Crystal growth r a t e : 2 . 5 m m / h ^ ^ ^ - ^ ^ ^ Lamellar spacing:0.37Mm^^^^ ^
^
^
^
= 9 0 °
1500 f
Load
/
Ti-48.1at%AI
r
1000
4>=9O°
_T
S
*
Crystal growth rate:5mm/h Lamellar spacing:0.53Mm <^>=70.5o
500 -
I
1% i
i—i
-
^^^^
i
/ ^ ^
4>=45°
n
Figure 6-29. Stress-strain curves of PST crystals of Ti-48.1 at.% Al deformed at room temperature (Umakoshi et al., 1991 b).
Compressive strain (%)
sion test was stopped at several ten percent of strain. On the other hand, the hard deformation mode makes plastic deformation more difficult. Cracks were often initiated at thick Ti3Al plates. Fine and uniformly distributed lamellae of submicron spacing produce strikingly good ductility and high strength (see Fig. 6-29). Lamellar boundaries are thought to constitute an effective barrier against dislocation motion through the boundaries when the compression axis for specimens is fixed perpendicular to the lamellar planes, as previously described. The change in composition of TiAl compounds over a wide range promotes growth of PST crystals containing different average lamellar spacings at a constant rate of crystal growth (see Fig. 6-24; Umakoshi et al., 1991 c). The chemical compositions of the y phase equivalent to the a 2 phase of the
PST crystals are almost the same. Figure 6-30 shows the variation of the yield stress of Ti-51.6-48.1 at. % Al with the reciprocal value of the square root of the lamellar spacings (Umakoshi et al., 1991a). The yield stress apparently follows a law similar to the Hall-Petch relationship, o
=
/T 1 / 2
(6-6)
where a0 is the lattice resistance to dislocation slip in the y phase and X is the lamellar spacing. If the slope ky can be interpreted by a dislocation pile-up model, the stress concentration at the tip of pile-up dislocations on slip planes acts at lamellar boundaries to propagate the plastic flow to the next domain, and ky is related to the critical stress necessary to generate a dislocation in the next domain. The values of ky for inter/metallic compounds are, in general, much larger than those for convention-
290
6 Deformation of Intermetallic Compounds
1500 1400 1300 1200 1100 -1000
k y (MPa.m 1/2 )
p / —
Ti-48.1at%AI 900
8 800 £ 700 2 600 500
400
y—i.y
ky=0.50 Ti-50.8at%AI
CO
>•
Ti-48.1at%AI
Ni3Fe:1.24 Ni3AI:0.73 AI:0.07 Ni:0.16
CO
|
I
* - Ti~49.1at%AI
Ti-50.3at%AI Ti-50.3at%AI Ti-51.6at%AI
300
o Crystal growth rate:2.5mm/h • Crystal growth rate:5mm/h _ I I I I I I I I 100 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1-6 A reciprocal value of square root of lamellar spacing (xio3m 200
al metals and alloys, as shown in Fig. 6-30. The reason for this is that in intermetallic compounds a dislocation should be generated as a superlattice dislocation involving an APB which will require additional energy, and therefore an additional stress is necessary to create the superlattice dislocation. The high value of ky for PST crystals suggests that the lamellar boundaries act in a similar manner to grain boundaries in the intermetallic compounds. The sharp increase in ky for Ti-48.1 at. % Al may be due to the effects of a critical, fine lamellar spacing and also the existence of thin oc2 plates, which can be deformed by {1121}-slip along this compression axis. Although the volume fraction of thin a 2 plates increases with decreasing aluminum content, the volume fraction of oc2-plates in Ti-48.1 at. % Al is not enough as a stress component to obtain a high yield stress. Since dislocations can be piled up at the interface between the y phase and the oc2 plates, owing to the
1/2
)
Figure 6-30. Variation of yield stress for PST crystals of TiAl compounds as a function of reciprocal values of the square root of the lamellar spacing (Umakoshi et al., 1991b).
high Peierls stress for {1121}-slip, the thin oc2 plates rather than the domain boundaries act as an effective barrier to stop the motion of the dislocations. The effect of lamellar spacings on the yield stress depends strongly on the mode of deformation, such as {1121}-slip or {1010}slip, which occurs in specimens loaded perpendicular or parallel to the lamellar planes, respectively. Next, it is necessary to distinguish the effects of the thin a 2 plates and the domain boundaries in the lamellae as a dragging barrier for the motion of dislocations. When the most highly stressed plane is parallel to the lamellar boundaries (0 = 45°)? slip only occurs on the {111} planes of the y phase parallel to the lamellar planes. In this case the effect of the size of the domains in the TiAl lamellae on the yield stress can only be obtained using PST crystals containing different domain sizes, as shown in Fig. 6-31 (Umakoshi et al., 1991c). Slip deformation is also impinged
6.6 Advanced Intermetallics as High-Temperature Structural Materials
I
I
Load
t CO Q_
291
I
O
yS
=0.27MPa.m1/2
CO CO
0
«
100
0)
\ ' • ' • ' • .
•
]
•
:
"•'••' ;
•
•
'-'
.
'
<
.
.
.
"
' - •
50 . . • '
• " • • ' • • •
. ' • ' '
• '
•'
10|jm 0
I
i
i
0 0.1 0.2 0.3 0.4 3 1/2 A reciprocal value of square root of domain size (x10 rrf )
at domain boundaries and the refinement of domains is effective in strengthening the TiAl crystals. However, the domain boundaries make less contribution to the strengthening than the thin oc2 plates/y phase boundaries because the value of ky in Fig. 6-31 is smaller than in Fig. 6-30. The above results strongly suggest that the lamellar structure is one of the most important factors in achieving a good combination of strength and ductility in TiAl compounds, and the existence of fine lamellae with submicron spacing gives high strength and good ductility to these intermetallic compounds.
Figure 6-31. Optical micrographs of domains and variation of yield stress for PST crystals of TiAl compounds as a function of reciprocal values of square root of domain size.
6.6.1.2 Al 3 Ti Based Aluminides
TiAl has poor oxidation resistance at temperatures above 800 °C because the external layer of these intermetallic compounds is not covered with protective A12O3 film but with TiO 2 or a mixture of TiO2 and A12O3 (Umakoshi et al., 1989d). A13X compounds of aluminum with elements of groups IVA and VA, such as Al3Ti, AI3V and Al 3 Nb, have a lower density than TiAl and exhibit excellent oxidation resistance even at 1000 °C. Such attractive characteristics make Al3Ti a potential candidate for a low density, high
292
6 Deformation of Intermetallic Compounds
(001)<110]-slip can be activated, and this has also been confirmed by slip trace analysis (Yamaguchi et al., 1988). The <110] superlattice dislocation is dissociated into l/2<110] superpartials separated by a ribbon of APB. The energy of the APB on (001) is lower than that on {111}, since no contribution to the energy of the APB on the (001) plane comes from the first nearest neighbor atoms. In L\2 compounds the slip plane for <110>-slip is fixed to the {111} planes, but at high temperatures {001}<110]-slip becomes the major deformation mode. The fact that <110] dislocations show a high activity in the high-temperature deformation of Al3Ti seems to indicate that (001)<110]-slip may be a common high-temperature deformation mode in intermetallic compounds with the LI 2 structure, and also in those with Ll 2 derivative, long-period ordered structures, such as the D0?7 and D0 99 structures. The
temperature structural material regardless of its lower melting point (1340°C) and much narrower composition range than TiAl. The major mode of deformation of Al3Ti with a D022 structure (see Fig. 6-1 d) is twinning of the type {111}<112], which is likely to result from the propagation of 1/6<112] Shockley partials on several adjacent {111} planes, and does not disturb the DO22 symmetry of the lattice by virtue of ordered twinning. Only four ordered twinning systems are available in the D022 structure, although twelve twinning systems are possible in f.c.c. metals. The possible ordered twinning systems depend on orientation and loading mode, as shown in Fig. 6-32, since the motion of Shockley partials is sensitive to the crystal orientation and loading mode. This has been confirmed for several oriented samples by slip trace analysis. At high temperatures
110
112
001
011
Twinning system
010 Orientation region A B C
(111X1121
C
C
T
(7i1)[112]
C
C
T
(111)11121
C
T
T
(111X112]
C
T
T
T: Tension, C: Compression
Figure 6-32. Orientation dependence of the operative twinning systems in the £>022 structure (Yamaguchi et al., 1988). Deformation markings and trace analysis show the result of an Al3(V0#95Ti0 05) single crystal deformed at20°C.
293
6.6 Advanced Intermetallics as High-Temperature Structural Materials
better ductility at high temperatures may be attributable to the augmentation of slip modes by <100]- and <110]-slip. When different types of twins encounter each other, dislocations for accommodation should be emitted from the intersection of the twins without crack nucleation. The dislocations proceeding from the intersection are thought to be l/2<110] superpartials trailing an APB. The crack-free accommodation is an attractive feature and encouraging for ductility improvement of Al3Ti. To improve the room temperature ductility of Al3Ti, an addition of alloying elements which decrease the energy of the APB on (001) and increase the ease of movement of <110] superlattice dislocations would be effective. When considering the relative phase stability between structures analogous to the LI 2 structure, the D022 structure cna be derived from the LI2 structure by introducing a l/2[110]APB on every (001) plane, and the D023 structure by introducing it on every two (001) planes. Therefore, introducing l/2[110]-APB on the (001) plane in the D0 22 structure is equivalent to creating four (001) layers of DO23-type stacking in the D022 structure (see Fig. 6-33). This suggests that the addition of alloying elements which form a stable D023 structure of Al3X-type compounds (X:Zr, Hf) would decrease the APB energy on the (001) plane. Since the introduction of an APB on every (001) plane in the Z>022 structure again transforms it into the LI 2 structure, addition of alloying elements (Li, Sc, Er, Yb) which destabilize the D022 structure in relation to the L l 2 structure decreases the APB energy on (001) in Al3Ti, and therefore makes the dissociation of dislocations of the type <110] on (001) in the Z>022 planes well defined. Figure 6-34 shows the effects of alloying elements on the yield stress and fracture
APB
APB
Jk
D0 22 (b)
o
APB
L DOo
/ 1 (d)
(c)
>
Figure 6-33. Structural relationship between Ll 2 and Ll2-derivative, long-period superlattice structures, (a) Ll 2 , (b) D0 22 , (c) D023, and (d) D0 22 structure with an APB on (001) (Yamaguchi et al, 1990).
strain of Al3Ti based compounds. Li, Hf and Zr additions are effective in improving the ductility of Al3Ti, while Ti, Nb and Ta additions are harmful to ductility because of their solution hardening effect (Umakoshi et al., 1987). More significant improvements in the ductility of ternary Al3(Ti, X) (X:Li, Zr, Hf) compounds at high temperatures strongly suggest that the activity of (001)<110]-slip is enhanced. The addition of alloying elements which decrease the energy of pure stacking faults led by Shockley partials of the 1/6<112] type increases the activity of ordered twinning and improves the ductility. This is demonstrated impressively in A13V in which the deformation is mainly controlled by the motion of <110]-type dislocations without twinning and therefore only a small fracture strain is obtained at low
294
6 Deformation of Intermetallic Compounds
400
AI3(Ti0 99X001)
(Ni) /
(Ta) /
_
(Hf) 300
(Zr) S
(Nb) /
j
U3Ti /
200 CO
iI
100 —
11
n
(Li) ^ •*
/
/
-
j
t
2%<
-
Compressive Strain (a)
100
200
300
400
500
600
Figure 6-34. Effect of additional elements on (a) stressstrain curves and (b) compressive fracture strains of Al3Ti based compounds (Umakoshi et al., 1987).
Temperature (°C) (b)
temperature. The ductility of A13V has been found to be increased by the partial replacement of V by Ti, as shown in Fig. 635 (Umakoshi et al., 1988). A substantial improvement in compression ductility is seen to occur by replacing about 5 at. % of V in A13V by Ti. In this ternary compound deformed at room temperature, not only <110] dislocations but also numerous deformation twins and stacking faults are observed. Therefore, the improvement in ductility of ternary A13(V, Ti) compounds is obviously due to a substantial increase in
the activity of ordered twins which are not operative in binary A13V. Partial replacement of V by Ti may decrease the ordering energy and increase the ease of movement of <110] dislocations, since the A13(V, Ti) phase with the D022 structure possesses a much wider composition range in the Al 3 V-Al 3 Ti quasi-binary system than A13V and Al3Ti binary compounds. In fact the separation between two l/2<110] superpartials in A1(VO 95 Ti 0 05 ) deformed at 503 °C was observed to be 5.7 nm wide on (100) (Umakoshi et al., 1988).
6.6 Advanced Intermetallics as High-Temperature Structural Materials T
15
r
i D
, I
r
^"i
I
AI
295
i
3 ( V 0.99 T i 0.0i)
10
CD D O
D AI3CV0.93Ti0.07)
2
7 '0 03'
• / £
\
'
AI 3 (Vo.75Tio. 25 )
1^ 0
100
200
300
400
500
600
700
800
900
1000
TemperatureC°C) Figure 6-35. Temperature dependence of compressive fracture strain of A13(V, Ti) compounds (Umakoshi et al., 1988).
An alternative approach to improvement of the ductility consists of macro-alloying, by which the D022 structure changes into a high symmetry L\2 structure. Such a transformation of crystal structure augments the number of available slip systems and may result in an improvement in the ductility. The L\2 structure is found to be stabilized by alloying with Ni (Raman and Schubert, 1965 b), Pd (Powers and Wert, 1990), V (Raman, 1966), Mn (Mabuchi et al., 1989), Cu (Raman, 1966), Zn (Raman and Schubert, 1965 a), Cr (Zhang et al., 1990), Ag (Mabuchi et al., 1990) or Fe (Seibold, 1981). Ternary L\2 compounds in the Al-Ti-Ni, Al-Ti-Fe and Al-Ti-Cu do not seem to be as brittle as Al3Ti and exhibit more than 10% fracture strains in compression at room temperature (George et al., 1989; Mabuchi et al., 1989; Kumar
and Pickens, 1988; Vasudevan et al., 1989). A beneficial effect on ductility is found in a composition of Al 66 Fe 9 Ti 24 in which the deformation is controlled by the motion of a <110> superlattice dislocation combined with an SISF. However, this compound does not show adequate room temperature tensile ductility which may be due to grain boundary embrittlement. A search must be made for a dopant to suppress the grain boundary embrittlement, as boron did for Ni 3 Al. 6.6.2 Intermetallics for Use at Extremely High Temperatures
New, extremely high temperature materials are required for use at more than 1500°C in aircraft gas turbines and spacecraft airframes. A compilation of about 300 binary metallic and metal-metalloid
296
6 Deformation of Intermetallic Compounds
compounds that melt above 1500°C was made by displaying melting temperature, specific gravity, elastic modulus and hot hardness in search of the candidates for ultra-high temperature materials (Fleischer, 1985, 1987; Fleischer and Taub, 1989). From the viewpoint of oxidation resistance and high-temperature strength, transition metal aluminides and silicides, whose melting temperatures are around 2000 °C or higher, are under consideration. Quite recently Nb 3 Al with the A15 structure and MoSi2 with the C\\h structure evoked great interest as potential candidates. This section deals with the deformation characteristics of ^415 compounds and refractory metal silicides. 6.6.2.1 A\5 Compounds The A\5 structure can be found in several intermetallic compounds such as Nb 3 Al (1960°C), W3Si (2164°C), V3Si (2050°C), Cr3Si (1730 °C), V3Sn (1300 °C), Nb 3 Sn (1550°C), V 3 Ge (1920°C), V 3 Ga (1300°C) and Mo 3 Al (2150°C). The number in parentheses is the melting point of the intermetallic compound. Many of these compounds have a high melting temperature and exhibit superconductivity. Their plastic deformation has been investigated from the view point of fabrication and construction of superconductive wires. The compounds are extremely brittle at ambient temperature and have not been amenable to conventional deformation processing procedures. They become ductile at elevated temperatures, and V3Si and Nb 3 Sn can be deformed up to strains of 10 to 20% in the temperature range 1520°C to 1650°C. Under high hydrostatic pressure even cold working is possible. The use of encapsulated Nb 3 Sn powders which induce grain refinement is effective in improving the deformability (Wright, 1977).
Figure 6-36 shows the temperature dependence of the yield stress of some ^415 compounds (Mahajan et al., 1978; Eisenstatt and Wright, 1980; Soscia and Wright, 1986; Umakoshi, 1991). Smooth stressstrain curves with a very large work hardening rate were obtained at homologous temperatures around 0.6 Tm (Tm: the melting temperature). At temperatures above 1300 °C these materials exhibit a yield drop followed by a very gradual work hardening or sometimes work softening, and an improvement in ductility. A rapid decrease in yield stress starts at temperatures where the samples become deformable. The strong temperature sensitivity to yield stress for V3Si in the temperature range of 1200 °C to 1500°C exhibits similar trends to covalently bonded Si crystals. This suggests that covalent bonding also exists in V3Si, and that probably V atoms in V-V chains exhibit a strong covalent bonding nature. The high strength and high softening temperature of Nb 3 Al are attractive properties for high-temperature materials. As seen from the crystal structure in Fig. 6-1 f, the most favorable slip direction is <001> which corresponds to the shortest translational vector of the A15 structure. The Burgers vector of moving dislocations, which control the high-temperature deformation, has been determined as <001> in V3Si (Kramer, 1983) and Nb 3 Al (Umakoshi, 1991). In V3Si deformed at 1200 °C to 1500°C, edge and mixed dislocations can be observed but <001> screw dislocations are rarely observed. In this temperature range screw dislocations may have a higher mobility than edge dislocations and can easily cross-slip between two slip planes of the {100}-type. The ratio of the maximum displacement of atoms perpendicular to the slip plane for {010}<001> to that for {110}<001> is 1.24, and from a geometrical point of view
6.6 Advanced Intermetallics as High-Temperature Structural Materials
1200
1300
1400
1500
297
1600
Temperature (°C)
Figure 6-36. Yield stress of ^415 compounds as a function of temperature (Mahajan et al., 1978; Eisenstatt and Wright, 1980; Soscia and Wright, 1986; Umakoshi et al., 1990 c).
the {110} plane may be favorable as the slip plane. On the other hand, the number of covalent V-V bonds cut per unit area during slip on {110} is 44% larger than that on {100}. The deviation of observed slip planes from {100} is less than 10°, so that {100} is the fundamental slip plane in V3Si. Therefore, the choice of slip plane in V3Si seems to be responsible for the strong covalent bonds. In Nb 3 Al the operative slip system is also {010}<001>. As seen from Fig. 6-37, dislocation nodes can be observed here and there, but numerous dislocations are straight and of screw character (Umakoshi, 1991). The climb mobility of an <001 > screw dislocation is not high even at 1300°C and the high strength of Nb 3 Al would be attributable to the high Peierls stress of <001> screw dislocations.
The main problem which should be overcome before the industrial application of high-temperature materials is the improvement in ductility. The major deformation mode of Nb 3 Al is {010}<001> slip so that only three independent slip systems are available. For improving the ductility of Nb 3 Al, we should look for elements which change the A15 structure into one with high geometrical symmetry, such as the LI 2 structure. The structure maps for the binary compounds presented by Pettifor (1988) suggest the suitable addition of a third or more alloying element to change the phase stability. A pseudo-plasticity owing to combined ductile/brittle phases should be taken into consideration for ductility improvement. Two-phase alloys with average compositions of Nb 8 5 2 1 Al 1 4 79 and Nb 8 6 3 8 Al 1 3 62
298
6 Deformation of Intermetallic Compounds
Figure 6-37. Dislocation structure in Nb 8 6 4 A1 13 6 deformed at 1300°C Burgers vectors of dislocations A and B are [010] and [001], respectively. (Umakoshi et al., 1989 c.)
200nm
have been deformed in compression, and their yield stress at 1100°C and 1300°C and the ductile/brittle transition temperatures are shown in Fig. 6-38 (Umakoshi, 1991). Alloys containing Nb-rich solid solution as a dispersed secondary phase show better ductility accompanied by inferior strength, and this is further enhanced with increasing fraction of the Nb-rich phase. Refining of Nb 3 Al grains and a dispersed Nb-rich phase would be required to achieve a higher ductility.
6.6.2.2 Refractory Metal Silicides In general, silicides have an intricate crystal structure and can hardly be deformed. From crystal symmetry considerations cubic silicides with B2, Z>03, L2t and C\ structures are favorable for slip deformation, but they are unlikely to be used in extremely high temperature technology because of their melting points and high-temperature strength. We should look for a silicide with the C l l b or C40
6.6 Advanced Intermetallics as High-Temperature Structural Materials
299
900 1050
6
800
\
o
0 D
!\
2 950
3
Nb 3 AI/Nb 2 AI| Nb3AI
600
\ \ \
^ \
500
850
400 \ \
300
\
750
200 at 1300°C I
650 vol%Nb2AI
I 10
Nb3AI
Cllb-Type Silicides MoSi2 forms SiO2 surface films which are viscous and protective against oxidation. The thermal expansion coefficient of MoSi2 is similar to that of a commercial
Table 6-3. Melting temperatures of C\\b- and C40type silicides.
MoSi2 WSi2 ReSi2 VSi2 CrSi2 TaSi2 NbSi2
Crystal structure 2080 2165 1980 1750 1550 2200 1930
""^ I 15
100 I 20
25
P
Figure 6-38. Effect of secondary phase on ductile/ brittle transition temperature and yield stress of Nb3Al based compounds (Umakoshi et al., 1989 c). The symbol • shows a ductile/brittle transition temperature.
vol%Nb
structures which are constructed on the basis of the b.c.c. and h.c.p. lattices, respectively. The melting points of silicides which are of interest for their mechanical behavior at extremely high temperatures are given in Table 6-3.
Silicides
a.
-
Nb3AI/Nb
\ CO
700
at11OO°C
K
I
.I
\
C\\h
cnb cnb C40 C40 C40 C40
refractory niobium alloy, Cb 752. Therefore, it has been used as a heating element and a coating material for not only Mobased alloys but also Nb-based alloys. As shown in Fig. 6-39, the yield stress depends strongly on the deformation temperature and crystal orientation. The yield stress for the sample near the [001] corner is much higher than that for other orientations. It exceeds 700 MPa around 1100°C and is about 250 MPa near 1500 °C (Umakoshi et al., 1990a, b). Such high strength at high temperatures makes it most attractive as a new, high temperature material. A slight peak or gradual decrease appears in the yield stress-temperature curves. Since the Cllb structure is basically derived by stacking three b.c.c. lattices, C\\h compounds should possess the deformation characteristics of b.c.c. crystals. For most orientations slip occurs on the most densely packed {110) plane. (013)slip, which corresponds to the (Oll)-slip in b.c.c. lattices, is also observed but is limited to orientations near the [001] corner.
300
6 Deformation of Intermetallic Compounds
800
1000
1100
1200 1300 Temperature (°C)
1400
1500
Figure 6-39. Yield stress of MoSi2 single crystals as a function of temperature (Umakoshi et al., 1990 c). The symbols in this figure indicate data for specimens with various orientations, indicated in a standard triangle.
40 -
Strain rate: 1.7x10~ /sec
—
Strain rate: 1.7x10 /sec
1000
1100
1200
1300
1400
Temperature (° C)
Figure 6-40. Temperature dependence of fracture strain for MoSi2 single crystals (Umakoshi et al., 1990 b).
301
6.6 Advanced Intermetallics as High-Temperature Structural Materials
Slip in MoSi2 occurs at around 1000°C along <331] corresponding to <111> in the b.c.c. lattice, and with increasing temperature <100]- and <110]-slips are activated. At low temperatures MoSi2 is extremely brittle, and below 900 °C fracture often occurs before observable slip strain is recorded. Naturally the increase in the deformation temperature induces a ductility improvement, and a remarkable improvement in ductility occurs above 1200 °C due to a substantial increase in the activity of <100]- and <110]-slip, as shown in Fig. 6-40. Since the fracture strain does not exhibit a strong sensitivity to strain rate there would be no severe problem in the fracture toughness above 1200 °C. A 1/2<331] superlattice dislocation can be dissociated into three equivalent 1/6<331] superpartials bound by two APBs, and the Burgers vector of each superpartial becomes smaller than that of <100] and <110] ordinary dislocations, which do not disturb the ordered atomic arrangements (see Fig. 6-41 a). The energy and stability of planar faults are very important when considering the dissociation and the motion of dislocations in ordered
£ I (0001)
(110)
(a)
crystals. Since a 1/2<331] superlattice dislocation moves on (110) or on {013), it is necessary to calculate the APB energy on both planes. The energy can be calculated by counting the number of atomic bonds on the fault plane. In MoSi2 with the Cllb structure, the APB energy can be described as a function of V(rk) where V{rk) = [<£MoMoW + ^siSi(^)]/2 - <£MoSi(r,), and $MoSi(rk) is the pair-wise interaction energy between the &-th nearest-neighbor atoms Mo and Si. When interaction up to the fifth nearest neighbors is taken into account, the APB energy on {110) and {013] is given by £ APB on {110)=
(6-7)
= [4 V(r±) - 8 V(r3) + 8 V(r5)]/(yfi<*c) EAPB on {013) = [4V(r2)-4V(r3) +16
V ( r
4
) - j
+ 2
where a and c are the lattice constants of the C\\b structure. The APB energy on {110) may be much larger than that on {013) since the effect of the first nearestneighbor bonds on the APB energy on the {013) is absent and the effect of the interaction energy at the first nearest neigh-
/
•
/
M
O
/
O
»- [1100] l ' (b)
Figure 6-41. (a) The atomic arrangement on the (110) plane in MoSi2 with the C l l b structure. Large and small circles represent atoms on the adjacent A and B layers, (b) The atomic arrangement on the (0001) plane in CrSi2 with the C40 structure.
302
6 Deformation of Intermetallic Compounds
bor is, in general, much stronger than that of further neighbors. The CRSS for <331]{013}-slip is much higher than that for <331]{110)-slip at all temperatures. Therefore, from the viewpoint of the anisotropy of APB energies and the difference of the CRSS on {110) and {013), the relation between {110)- and {013)-slip in MoSi2 is quite similar to that between {111}- and {001}-slip in L\2 intermetallic compounds showing anomalous strengthening. A deviation from the CRSS law at 1060 °C was observed: the higher the stress component for cross-slip onto {013), the larger the CRSS for {110)-slip becomes. A slight anomalous peak is observed around 1050 °C, as shown in Fig. 6-39. The part which has cross-slipped onto the {013) plane will act as a dragging point for the motion of the whole dislocation on the {110) plane (Umakoshi et al., 1990b). At high temperatures where rapid softening and improvement in ductility begin, <100] and <110] dislocations are activated, and they have a strong tendency to form dislocation nodes because of their high climb mobility as shown in Fig. 6-42. The dislocation reaction at the node is carried
out with [010] + [100] = [110]. Attention should be given to the formation of stacking faults created by the 1/4[111] partial dislocation, as indicated by D in Fig. 6-42. Figure 6-41 a shows the atomic arrangements on (110), in which large circles atomic arrangements on (110), in which large circles represent atoms in the A-layer and small circles represent atoms in the B-layer, which is immediately above and below the A-layer. If atoms of the A-layer are displaced with respect to the B-layer by a vector of 1/4[111] (/ SF ), the atomic arrangements are as given in Fig. 6-41 b and the stacking sequence changes from ABAB to ABC. The atomic arrangements and the stacking sequence are equivalent to those on the (0001) plane in the C40 structure. The energy of the stacking fault is represented by ^ S F = [4 ^MOMO (
where a and c are the lattice constants. Therfore, the formation of the stacking fault is closely related to the phase stability of the Cllb structure in relation to the C40 structure (Umakoshi et al., 1989b). A transition from the Cllb to the C40 struc-
0.5|jm
Figure 6-42. Burgers vector analysis of dislocations forming nodes in MoSi2 single crystals deformed at 1300°C, usingJTive different g vectors of 002, 112, 011, 110 and 101. The Burgers vectors of A, B, C, and D dislocations are [100], [010], [110] and 1/4[11 Irrespectively (Umakoshi et al., 1990 b).
6.6 Advanced Intermetallics as High-Temperature Structural Materials
303
Figure 6-43. Dislocation structure of (Mo0 97 Cr 0 03)Si2 single crystals deformed at 1150°C (Umakoshi et al., 1990a) (g = 011 or 002).
ture occurs in MoSi2 above 1900 °C, where the free energy of the stacking fault should fall to zero and the fault can easily be formed. The increase in formation of stacking faults is also responsible for the ductility improvement of MoSi2 at high temperatures. The poor ductility of MoSi2 at low temperatures should be a consequence of the high Peierls stress of 1/2<331], <100] and <110] dislocations, although enough slip systems are provided to satisfy the von Mises criterion. Addition of alloying elements which destabilize the Cllb structure of MoSi2 with respect to the C40 structure decreases the stacking fault energy and may activate the motion of 1/4 [111] partial dislocations trailing the stacking fault. Then the ductility will be improved. The addition of Cr to MoSi2 is expected to decrease the stability of the Cll b structure relative to the C40 structure because CrSi2 crystallizes in the C40 structure, and small amounts of Cr are soluble in MoSi2 with the Cllb structure in the MoSi 2 -CrSi 2 pseudo-binary system. As expected, numerous 1/2<111> dislocations combined with a SF on (110) are observed in Fig. 6-43 (Umakoshi et al., 1990a). The stack-
ing faults cannot be created on the {013) plane. The ductility improvement obtained by a Cr addition is not marked, possibly due to solution hardening by the Cr, but a sign of ductility improvement is noticed in the stress-strain curves for (Mo 0 97 Cr 0 03 ) Si2 single crystals. The effect of Cr on ductility is indirectly supported by the fact that the difference in the yield stress for {110)- and {013)-slip is enlarged by a Cr addition. A systematic search for proper amounts and alloying elements should be made taking into account that Mo, W and Re form XSi2 silicides with the Cllb structure and Cr, Ta, V and Nb form those with the C40 structure. For high-temperature structural applications creep strength is very important. At temperatures between 1200 °C and 1400 °C, creep curves for MoSi2 single crystals showed steady-state creep after normal primary creep. The steady-state creep rate, 8, is represented by the empirical power law creep equation, e = A (a/Gf exp [ - AH/(R T)]
(6-8)
where A is a constant, a is the applied stress, G is the shear modulus, n is the stress exponent, AH is the activation en-
304
6 Deformation of Intermetallic Compounds
thalpy, T is the temperature and R is the gas constant. The stress exponent n, which is dependent on the creep mechanism, can be derived from the relation between the steady-state creep rate and the applied stress at constant temperature. The value of n is nearly 3 for MoSi2 single crystals after creep at 1200 °C, and at 1400 °C under applied stresses higher than 50 MPa (Umakoshi et al, 1990a). The creep of MoSi2 may be controlled by a viscous glide motion of dislocations. As the applied stress decreases at 1400 °C, the value of n approaches 1, which suggests the operation of a bulk diffusion mechanism. Strongly ordered silicides make diffusion difficult. The activation energy obtained for creep of MoSi2 at 50 MPa is 520 kJ/ mol which is higher than the value of 350 kJ/mol for Nb 3 Al. Therefore, MoSi2 exhibits a high creep resistance even at 1400°C.
1
I
I
I
1
'
1
C40-Type Silicides From the crystal structure in Fig. 6-1 g, possible slip planes are (0001) and {1120} planes. On the basal plane (see Fig. 6-41 b), the shortest lattice translational vector is l/2<2110>. A l/2<2110> dislocation can be dissociated into two l/4<2110> partials combined with a SF, and the motion of l/4<2110> partial dislocations trailing the SF locally recreates the atomic arrangements on the {110) planes of the Cllb structure in the C40 structure. The SF cannot be formed on the prismatic {1120} planes. The deformation characteristics of C40type silicides will be described taking CrSi2 as an example. CrSi2 single crystals are deformable above 700 °C. With increasing temperature smooth stress-strain curves were obtained, and below 900 °C, they exhibited considerable work hardening after
'
1
1
V
120 o 100 -
/A 2110
V
•
o
Q.
v
0001
"co
6
-
\ _
fioo
A V
—
I
—
IC\I
V
>
60 -
AV
40 --
-
o o
20 -
% \ O
I
|
800
900
1
|
Ad O i
1000 Temperature (°C)
1
1100
,
1200
1300
Figure 6-44. Yield stress of CrSi2 single crystals as a function of temperature (Umakoshi et al., 1990d).
6.7 Summary
yielding. At high temperatures above 900 °C, the samples exhibit a yield drop, followed by a very gradual work hardening or sometimes work softening, and an improvement in ductility. Slip always occurs on (0001) regardless of crystal orientations. From slip trace analysis no evidence of activation of the superlattice dislocations has been obtained and slip occurs by (0001)<1120> slip systems. The CRSS for (0001)<1120> slip does not depend on crystal orientation and it decreases continuously with increasing temperature (see Fig. 6-44; Umakoshi et al., 1991 d).
6.7 Summary It was known as early as the end of the last century that intermetallic compounds had remarkably varied mechanical characteristics, such as higher strength and yet greater brittleness than solid solutions. These peculiar mechanical properties are, of course, due to the strong bonds between the different component atoms. The ordered atomic arrangement forms various types of ordered structures in which a superlattice dislocation can be dissociated into several superpartials. Dislocation dissociation depends on the stability and energy of possible planar faults, which may not be stable at the fault vectors, predicted on the basis of a hard-sphere model. The motion of dissociated dislocations in a group and the characteristic core structure in the ordered state have important implications for understanding the attractive plastic behavior of intermetallic compounds, such as anomalous strengthening, although the principal plastic characteristics depend on crystal structures such as the f.c.c, b.c.c. or h.c.p. lattice.
305
From the viewpoint of structural applications an improvement in ductility is needed. The discovery of boron as a dopant for suppressing the grain boundary embrittlement of Ni3Al was a notable step forward in the development and applications of high-temperature materials based on Ni3Al, but a problem still exists in the weakness of the grain boundary at high temperatures. TiAl is a new material expected to be applicable in airframes and car bodies because of its low density and superior strength at high temperature. This attractive mechanical property has been demonstrated in Ti-rich TiAl with a lamellar structure containing numerous twins and thin a 2 plates. The strength and ductility of TiAl crystals is known to be closely related to the lamellar structure, and reduction of the lamellar spacing to a submicron scale provides a good combination of high strength and good ductility. Attention has only been given to the deformation behavior of the y phase matrix in the lamellae. However, the thin oc2 plates with the D019 structure are found to act as an effective barrier against the motion of dislocations passing through lamellar boundaries and to strengthen the TiAl crystals. Attempts to activate more slip systems should be made, since the oc2 phase shows large plastic anisotropy and poor ductility at low temperature because there is at present an insufficient number. To accelerate the structural applications of TiAl, more intensive and systematic work is necessary in searching for alloying elements and fabrication processes to control the microstructure, particularly the thickness and distribution of the oc2 plates in lamellae, focusing on the deformation mode of the oc2 phase. Refractory materials which operate above 1500°C should be developed. Some candidates can be found in refractory met-
306
6 Deformation of Intermetallic Compounds
al silicides such as MoSi 2 . It is very difficult to pursue good ductility at low temperature, as well as conventional alloys and low-temperature intermetallics, although a hopeful indication was noted in the ductility improvement for MoSi2 with a small addition of Cr, which was chosen from the viewpoint or phase stability of the Clib structure in relation to the C40 structure. It is a fact that refractory metal silicides are extremely brittle at low temperature, but, in sharp contrast to ceramics, the silicides can be deformed by the motion of dislocations and become ductile at the high temperatures at which they will be used. Many difficult barriers have yet to be overcome for ultra-high temperature intermetallics before they can be structurally applied but their advantage over ceramics encourages us to continue in their development.
6.8 Acknowledgements The author would like to thank Prof. M. Yamaguchi for supplying valuable data and photographs. He is also grateful to Prof. H. Mughrabi for his encouragement and critical reading. This work was supported by the Mazda Foundation, Iketani Science and Technology Foundation, and a Grant-in-Aid for Scientific Research and Development from the Ministry of Education, Science and Culture of Japan.
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(Eds.). New York: Elsevier Science Publishers, BV, p. 131. Yamaguchi, M., Umakoshi, Y (1990), Prog. Mater. Sci. 34 (1), 1. Yamaguchi, M., Pope, D. P., Vitek, V, Umakoshi, Y (1981a), Phil. Mag. A 43, 1265. Yamaguchi, M., Vitek, V, Pope, D. P. (1981 b), Phil. Mag. A 43, 1027. Yamaguchi, M., Paidar, V, Pope, D. P., Vitek, V (1982), Phil. Mag. A 45, 867. Yamaguchi, M., Umakoshi, Y, Yamane, T. (1984), Phil. Mag. 50, 205. Yamaguchi, M., Shirai, Y, Umakoshi, Y (1988), in: Dispersion Strengthened Aluminum Alloys: Kim, YW, Griffith, W M. (Eds.). Warrendale: The Minerals, Metals and Materials Society, p. 721. Yamaguchi, M., Nishitani, S. R., Shirai, Y (1990), in: High Temperature Aluminides and Intermetallics: Whang, S. H., Liu, C. T., Pope, D. P, Stiegler, J. O. (Eds.). Warrendale, PA: AIME, p. 63. Yang, W. J. S. (1982), Metall. Trans. 13A, 324. Yoo, M. H. (1986), Scripta Metall. 20, 915. Yoo, M. H. (1987), Acta Metall. 35, 1559. Zhang, S., Nic, J. P., Mikkola, D. E. (1990), Scripta Metall 24, 57.
General Reading (Books and Proceedings of Symposia on Intermetallic Compounds) Izumi, O. (Ed.) (1991), Intermetallic CompoundsStructure and Mechanical Properties. Proc. Int. Symp. (JIMIS-6). Sendai: The Jap. Inst. Metals. Johnson, L., Pope, D. P., Stiegler, J. O. (Eds.) (1991), High-Temperature Ordered Intermetallic Alloys IV, MRS Symp. Proc, Vol. 213. Pittsburg, PA: MRS. Kear, B. H., Sims, C. T., Stoloff, N. S., Westbrook, J. H. (1970), Ordered Alloys; Structural Applications and Physical Metallurgy. Baton Rouge, LA: Claitor's Publ. Kim, Y M., Griffith, W M. (Eds.) (1988), Dispersion Strengthened Aluminum Alloys. Warrendale, PA: AIME. Koch, C. C , Liu, C. X, Stoloff, N. S. (Eds.) (1985), High-Temperature Ordered Intermetallic Alloys I, MRS Sympos. Proc, Vol. 39. Pittsburgh, PA: MRS. Liu, C. T., Taub, A. L, Stoloff, N. S., Koch, C. C. (Eds.) (1989), High-Temperature Ordered Intermetallic Alloys III, MRS Symp. Proc, Vol. 133. Pittsburgh, PA: MRS.
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Stoloff, N. S., Davies, R. G. (1966), Prog. Mater. Sci. 13, 1. Stoloff, N. S., Koch, C. C , Liu, C. T., Izumi, O. (Eds.) (1987), High-Temperature Ordered Intermetallic Alloys II, MRS Symp. Proc, Vol. 81. Pittsburgh, PA: MRS. Westbrook, J. H. (1960), Mechanical Properties of Intermetallic Compounds. New York: John Wiley.
Westbrook, J. W. (1967), Intermetallic Compounds. New York: John Wiley. Whang, S. H., Liu, C. T., Pope, D. P., Stiegler, J. O. (Eds.) (1991), High-Temperature Ordered Intermetallics. Warrendale, PA: AIME. Yamaguchi, M., Umakoshi, Y. (1990), The Deformation Behaviour of Intermetallic Superlattice Compounds, in: Prog. Mater. Sci. 34 (1).
7 Particle Strengthening Bernd Reppich Institut fur Werkstoffwissenschaften, Lehrstuhl I, Universitat Erlangen-Niirnberg, Erlangen, Federal Republic of Germany
List of 7.1 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.2 7.2.2.1 7.2.2.2 7.2.2.3 7.2.2.4 7.2.2.5 7.2.2.6 7.2.3 7.2.3.1 7.2.3.2 7.3 7.3.1 7.3.2 7.3.3 7.3.3.1 7.3.3.2 7.3.4 7.4
Symbols and Abbreviations Introduction Yielding at Low Temperatures Elementary Dislocation-Particle Interaction The Fleischer-Friedel Point-Obstacle Approximation Particles as Extended Obstacles Particle-Strengthening Mechanisms Chemical Strengthening Modulus-Mismatch Strengthening Coherency Strengthening Stacking-Fault Strengthening Atomic-Order Strengthening Summary Duplex-Particle Strengthening Theory: Mixtures of Particles Experimental Results Yielding at High Temperatures The Threshold-Stress Concept The Climb Threshold Climb Models Local Climb General Climb Interfacial Pinning References
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
312 315 316 316 316 321 324 324 326 328 331 334 343 343 345 346 348 348 349 351 351 352 353 355
312
7 Particle Strengthening
List of Symbols and Abbreviations A±,A2,A3 am, a p b C1,C2,C3, C SL ct,c2 Dv d d1,d2 E Eel Fm FR / Gm, Gp K KEDGE k kB L Lmax Lp / ZF /SL ll912 m ns R ftp0.2 A^po.2 r rs s m , sp
C4
coefficients lattice parameter of the matrix, particle Burgers vector coefficients empirical constant in the theory of Schwarz and Labusch relative obstacle concentration of type 1, type 2 volume diffusion coefficient length of dislocation lying inside the particle at the critical configuration length of the leading (1) and trailing (2) dislocations of a pair inside the ordered particle total energy of the system energy due to elastic interaction between dislocation segments mean maximum interaction force between a dislocation and one obstacle (particle) repulsive elastic interaction force between pairwise-coupled dislocations in order strengthening volume fraction of particles shear modulus of the matrix (particle) force of interaction between two partial dislocations of Burgers vector bp pre-logarithmic line tension factor for an initially straight edge dislocation in the given anisotropic material relaxation factor Boltzmann's constant mean planar square lattice spacing mean planar particle spacing along a random straight line planar inter-particle spacing effective mean planar obstacle spacing along the bowing dislocation at the critical configuration Friedel spacing (length) Schwarz-Labusch spacing (length) effective planar spacings of the leading (1) and trailing (2) dislocations of a pair in order strengthening exponent appearing in Eqs. (7-28) and (7-29) for the maximum force and CRSS due to modulus strengthening number of obstacles (particles) per unit area in the glide plane; areal obstacle density dislocation climb resistance total 0.2% yield stress 0.2% yield stress increment due to particles mean radius of spherical particles mean particle radius of circular section in any plane ribbon width of the stacking-fault of the dissociated dislocation in the matrix, precipitate
List of Symbols and Abbreviations
T TL, Tp V1,V2 w Y
a
Pu y ?APB
A
,yspF
e s 80 rjSL 0 £, Q a Gm
RSBH>THR
x matrix , TY,
313
temperature dislocation line tension of the matrix, particle areal fractions of regionally distributed obstacles (particles) of type 1, type 2 in the glide plane adjustable parameter in the Hiither and Reppich theory of order strengthening by strong pair coupling range of interaction force between a dislocation and one obstacle range of interaction force of one extended obstacle (particle) perpendicular to the dislocation in the theory of Schwarz-Labusch superposition exponent in the empirical addition rule coefficients parameters in Eqs. (7-38) and (7-39) y-phase (matrix) y' (Ni 3 Al/Ti)-phase (precipitation) antiphase boundary energy on the glide plane of an ordered precipitate energy of a matrix-precipitate interface created by particle shearing stacking-fault energy of the matrix (precipitate) separation of the leading and trailing dislocations of a pair in order strengthening lattice misfit parameter strain rate; creep rate reference strain rate normalized obstacle (particle) depth in the theory of Schwarz and Labusch angle between the Burgers vector of a dislocation and its direction dimensionless parameter characterizing the range of interaction between particle and dislocation in units of the mean planar particle radius dislocation density total applied (engineering) stress creep strength of a particle-free reference matrix creep threshold stress d u e to particles normal stress component acting on a climbing dislocation segment critical resolved shear stress (CRSS); particle strengthening contribution of particles to the CRSS determined experimentally CRSS derived within the Fleischer-Friedel point-obstacle approximation CRSS in the theory of Schwarz and Labusch theoretically predicted Orowan stress theoretically predicted cutting stress theoretically predicted CRSS due to the antiphase boundary energy in order strengthening theoretically predicted CRSS due to order strengthening in the model of Raynor and Silcock, and Brown and Ham; model of Huther and Reppich CRSS contribution of the solid solution matrix, ordered y' precipitates
314
7 Particle Strengthening
tth
individual CRSS contribution of class 1, class 2 particles in the case of bimodal particle dispersions theoretically predicted CRSS allowing the particle to be overcome by climb theoretically predicted detachment threshold stress due to interfacial pinning theoretically predicted threshold stress due to particles
APB CRSS FF HR HVEM MA RSBH SL TEM
antiphase domain boundary critical resolved shear stress Fleischer-Friedel theory Hiither-Reppich model high-voltage electron microscopy mechanical alloying model of Raynor and Silcock, and Brown and Ham Schwarz-Labusch theory transmission electron microscopy
T1?T2
TC Td
7.1 Introduction
IX Introduction The high strength of two-phase alloy systems is essentially due to the interaction between dislocations and metallurgical obstacles. Precipitated and dispersoid particles act as the most effective obstacles. Precipitation is usually produced by thermal pretreatment of a two- or multicomponent system resulting in age-hardening. Nonmetallic dispersoids are introduced, for example, by internal oxidation or by powder metallurgical processes such as mechanical alloying (MA). For the understanding of the phenomenon of "particle strengthening", which is the topic of this chapter, it is, however, relatively unimportant how the particle microstructure was produced. Therefore, precipitation hardening and dispersion hardening will be treated together. In the last two decades, remarkable advances have been made in the comprehen sive understanding of the operating dislocation-particle interaction mechanisms, which can be seen by comparing the famous reviews of Kelly and Nicholson (1963), Brown and Ham (1971), Haasen (1977), Kocks (1977), and Gerold (1979), with the recent excellent articles of Ardell (1985) and Nembach and Neite (1985). In fact, the theoretical approach reached a new level of quality with the new theory of Schwarz and Labusch (1978) - referred to as SL - simulating the movement of a dislocation through a random array of extended obstacles with the aid of a digital computer. The importance of this procedure is evident: it signals the departure from the Fleischer-Friedel point-obstacle approximation and introduces as a new element the finite range of dislocation-particle interaction forces. It is, however, perplexing that the acceptance and the practical impact of the SL-theory still seems to
315
be limited. In recent years, there has been a tendency, started by Haasen and Labusch (1979) and Reppich (1982), to apply the SL-theory exclusively to order hardening of yjy' superalloys. Another interesting aspect has not been considered in the reviews mentioned. While the principles of particle strengthening have been successfully applied to materials for use at high temperatures, the details of the governing high-temperature creep processes in particle-containing alloys are still a matter of controversy (Blum and Reppich, 1985; Wilshire and Evans, 1990). With this situation as the starting point, this chapter reviews the best current hardening models available, which survived the test of time (including high-temperature yielding mechanisms). Furthermore, a rigorous application of the SL-theory is discussed. The purpose is partly didactic and partly critical with particular emphasis on the standpoint of the experimentally oriented researcher. The chapter is not intended to be comprehensive. Rather, it is an attempt to fill some gaps left by recent reviews. At the same time, this chapter overlaps with Chaps. 2 and 5 of this volume as well as with the articles of Brown and Ham (1971), Ardell (1985) and Nembach and Neite (1985). The approach will focus on the increase in yield stress or critical resolved shear stress (CRSS), as a particularly suitable measure of obstacle-controlled strength, which can be obtained easily by common deformation experiments. However, the effects of secondphase particles upon ductility, fracture toughness, electrical and magnetic properties, stress corrosion cracking, etc. lie outside the scope of the review presented and will be excluded. The chapter is structured as follows: Sec. 7.2.1 surveys the elementary interaction between dislocations and monodispersed
316
7 Particle Strengthening
individual particles from a phenomenological viewpoint. In Sec. 7.2.2 we deal with the origin of the diverse interaction forces, derive formulas for the CRSS as a function of the relevant particle parameters, and apply the basic models to experimental data in order to correlate the particle microstructure with the macroscopic strength. In Sec. 7.2.3 we extend the discussion to hardening by two distinct simultaneously existing populations of particles. Section 7.3 describes diffusion-controlled yielding at high temperatures, while the effect of secondphase particles upon the creep response will be presented in Chap. 8 of this volume.
7.2 Yielding at Low Temperatures 7.2.1 Elementary Dislocation-Particle Interaction In the light of the dislocation theory, dislocation-particle interaction may be subdivided into several categories. On the one hand, Nabarro (1972) distinguished between localized and diffuse interaction. (a)
Second-phase particles, on the other hand, can be treated either as point-like obstacles or as extended (finite) obstacles. The former are considered to have direct physical contact with the dislocations, while the latter interact with dislocations over a finite distance characterized by their interaction range Y (Fig. 7-1). In terms of the specific obstacle strength, however, particles can be classified either as "weak", penetrable obstacles or as "strong", non-penetrable ones. Kelly and Nicholson (1963) therefore distinguished clearly between "cutting" and "looping" mechanisms for the CRSS. From a more phenomenological viewpoint, particles can be specified depending upon the nature of the dispersed phase, and also on their crystallographic relationships to the matrix, i.e., on the degree of coherency. This question will be tackled at the end of the next section. 7.2.1.1 The Fleischer-Friedel Point-Obstacle Approximation Consider (overdamped) motion of a dislocation encountering monodispersed par(b)
Figure 7-1. Dislocation moving over a field of randomly spaced "weak" point-obstacles by the Friedel process (a), and over "strong" point-obstacles by the Orowan process (b). Computer simulation by Foreman and Makin (1966).
7.2 Yielding at Low Temperatures
tides in an isotropic medium acting as individual point-like obstacles of identical strength, i.e., 7 = 0. Then the dislocation can overcome these particles at the stress (Fig. 7-1): T =
bl
(7-1 a) (7-1 b)
where b is the magnitude of the Burgers vector, TL the dislocation line tension, and / the effective mean planar particle-spacing along the bending dislocation. The maximum force, Fm, is determined by the force balance depicted in Fig. 7-1 a, showing the dislocation configuration at the critical breaking angle 0. Thus, a complete modelling of T according to Eq. (7-1) involves two steps: - The specification of Fm9 or, more precisely, of the specific obstacle strength, [FJ(2 TL)] = cos 0 , which in turns is derived from the detailed mechanisms of the operating dislocation-particle interaction considered in Sec. 7.2.2. - The estimation of /, taking into account its dependence upon the applied stress, by statistical methods. This will be considered next. The bending of the dislocation around obstacles influences strongly the effective distance /. Depending on the dislocation's flexibility, / will stay within the two limiting distances: L max , i.e., the mean obstacle spacing along an ideally random straight line; L, i.e., the minimum obstacle spacing in the glide plane along a completely flexible line if the obstacles form a square array, related to the areal obstacle density ns by If, for simplicity, / is set proportional to the average planar square lattice spacing, L in
317
Fig. 7-1, this already leads to the most important result of the dislocation theory: the CRSS is proportional to the square root of the obstacle concentration (and not to the concentration itself, as in the rule of mixing, see Sec. 7.2.3.1). A flexible dislocation under an applied stress is bent between the touched obstacles, so that it comes in contact with more obstacles than in the initially quasistraight position (Fig. 7-1). Thus, / is reduced and consequently the increase in T with Fm is greater than linear (Eq. (7-1)). The earliest approach to solving this problem was taken by Friedel (1956, 1964) and by Fleischer and Hibbard (1963). The result for widely-spaced weak obstacles is: 1
(?2)
1
With Eq. (7-1), the Fleischer-Friedel flow stress (index FF) becomes: TFF
=
2TJ
\bL
(7-3)
With respect to Eq. (7-3), obstacles of different strength, FJ(2 TL), can be classified as (see Fig. 7-3): weak obstacles, Fm < 2 TL, FF cutting; strong obstacles, Fm~2TL, Orowan bypassing. Then Eq. (7-2) gives / = L, and Eq. (7-3) becomes TOR = 2 TL/(b L), which is the classical expression for the Orowan stress (Orowan, 1948). Particles are usually not arranged in a periodic array. A random array may be a considerably better representation of reality. This has been achieved by computer simulations which started with the graphical methods of Kocks (1966) and Foreman and Makin (1966), and were extended by Morris and Klahn (1974) and Hanson and Morris (1975 a). The first important result of such computer experiments is that there is, indeed, a well-defined stress at which the
318
7 Particle Strengthening
dislocation moves over large distances through the array (Fig. 7-1). This stress is identified as the flow stress of the array, and obviously corresponds to the yielding in a deformation experiment. The second remarkable result concerns the way in which the dislocation moves through the array of obstacles. When the obstacles are weak, the dislocation remains relatively straight and "cuts" through the array by the FF process (Fig. 7-1 a). When the obstacles are strong, the dislocation frequently penetrates the array deeply along paths of easy movement (regions where the obstacle spacing is larger than L in Fig. 7-1 b), surrounding difficult groups (where the obstacle spacing is substantially lower than L) and leaving Orowan loops behind. The third important result concerns the absolute value of the critical flow-stress, required to break a row of obstacles with spacing L = n~1/2. It is shown that the FF formula (Eq. (7-3)) overestimates the CRSS for point obstacles of all strengths (Fig. 7-2). All current estimates, however, yield a pre-factor in Eq. (7-3) of about 0.9 (within a few percent). A proper approximation for weak obstacles is therefore: = 0.9
2Tj
\bL
(7-4)
At large values of specific particle-strength, the deviations from Eq. (7-3) are substantial (see Fig. 7-2). However, = 0.8
2TL ~bh
(7-5)
predicts the correct behavior for the CRSS in the Orowan limit [FJ(2 TL) ~ 1]. The effect of a non-random array of point-obstacles, intensively discussed by Brown and Ham (1971) on the basis of the results of the computer experiments of Foreman and Makin (1966) can be sum-
1.0
- — (Fm/2rL)=cos(0/2) 0.97 0.9 0.8 0.6 0.4
\
^
0.2
0
Eq.(7-1b) with /=/_ Eq.(7-3)
20
60
100
140
180
Critical breaking-angle 0 — •
Figure 7-2. Yield stress as a function of the breaking angle 0 and specific obstacle strength cos 0/2 = Fm/(2 TL). Computed data points of Foreman and Makin (1966) for random spacing, for a square lattice of obstacles (see Eq. (7-1 b), and for the Friedel relationship (see Eq. (7-3)). Taken from Brown and Ham (1971).
marized as follows: The FF-cutting process is not markedly affected by the regularity of the array and thus the FF formula, Eq. (7-4), is a proper approximation. When the Orowan process dominates, the CRSS exceeds the value of 0.8 in Eq. (7-5) and increases towards the value 1 as the obstacle array becomes more regular (periodic). We expect the above considerations to apply to those age-hardened alloys where overlapping diffusion-fields or alignment during coarsening prevents the distribution of precipitates from being random (see Sec. 7.2.2.5). Some further complications which affect the evaluation of experimental data considerably are the - modelling of the line tension, TL, - distribution of obstacle strengths,
7.2 Yielding at Low Temperatures
- dislocation damping, - thermal activation. These have been treated by Brown and Ham (1971), Kocks (1977), Ardell (1985), and Nembach and Neite (1985). Despite uncertainties and limitations, the FF theory enables one to derive essential features of the behavior of real materials containing second-phase particles in small volume fractions, which can be approximated as point obstacles. For applications to experiment it is useful to express Eqs. (7-3) to (7-5) in terms of metallurgically controllable variables, such as particle size and volume fraction. It will be assumed throughout the remainder of this paper that particles are spherical in shape; this simplifies the discussion. Under certain circumstances, when particles deviate from spherical morphology, special modifications of equations derived for spherical particles are necessary. Characteristic parameters of spherical particles and appropriate relationships among these are summarized below: r: mean radius. n 4
(7-6)
mean radius of circular section in any plane. ns: mean number of particles intersecting unit area of the glide plane (areal obstacle density). (7-7)
/ = ns7i(2/3)r2
volume fraction. /2
<7 8)
-
mean planar square lattice spacing.
V f
f
(7-9)
319
mean planar particle spacing along a random straight line. (7-10) planar inter-particle spacing. When a dislocation is moved by an applied stress on one of the slip planes of the matrix, it will encounter second-phase particles and interact with them. Depending upon the degree of coherency, two different reactions may occur (see Fig. 7-3): - Incoherent particles are impenetrable for the matrix dislocations, i.e., the particle will not be sheared along with the matrix. Rather, the moving dislocation is forced by the applied stress to bend around the particle and bypass it by the Orowan mechanism depicted on the right-hand side of Fig. 7-3. Around the particles, the bypassing dislocation leaves concentric dislocation loops. Either during or after bypassing, some special cross-slip may occur which alters the final configuration of the Orowan loops (Brown and Ham, 1971; Gleiter, 1967 a, b). - Particles having coherent interfaces are penetrable for the moving dislocation, i.e., the particles will be sheared along with the matrix by the FF-cutting mechanism depicted on the left-hand side of Fig. 7-3. In order to calculate the cutting stress according to Eq. (7-4), we have to specify the maximum interaction force, Fm. Whatever the origin of the specific interaction mechanism is (which will be dealt with in detail in Sec. 7.2.2), one can safely assume that Fm will depend upon the length of the dislocation segment just lying inside the particle. Thus Fm is proportional to the particle size, and we write (C±: coefficient): (7-11)
320
7 Particle Strengthening
bl "weak" obstacles
Eq.(7-1) "strong" obstacles
Fm<2TL=C,h2rs
/=Eq.(7-4)
Friedel cutting
^L)
Eq.(7-5)
•*•* //// Orowan looping :%
7
iI-
Eq.(7-13)
Figure 7-3. Scheme of particle strengthening for alloys containing small volumefractions according to the FleischerFriedel approximation.
The "hardening parameter" h reflects the specific hardening mechanism which will be derived within the point-obstacle approximation in Sec. 7.2.2. Substitution of Fm and L in Eqs. (7-4) and (7-5) by the expressions (7-8) and (7-11), respectively, leads to the two formulas (7-12) and (7-13) in Fig. 7-3 for the FF-cutting stress and the Orowan stress. They are represented in Fig. 7-3 showing the CRSS normalized with respect to the volume fraction / versus mean particle size, r. In principle, that process operates which requires the lower stress. For alloys containing non-shearable, hard particles in form of incoherent precipitates or dispersoids, the prediction is rather trivial. Over the entire particle-
size range dispersion strengthening is exclusively due to Orowan looping. The CRSS decreases hyperbolically with increasing particle size, as described by Eq. (7-13) (dashed curve in Fig. 7-3). The behavior of alloys containing shearable coherent precipitates is a little more complex. Underaged fine-dispersed particles exhibit a parabolic CRSS increase due to FF cutting, as described by Eq. (7-12) in Fig. 7-3, curve 1. However, above the critical size, rc, the Orowan mechanism (Fig. 73, curve 2) requires the lower stress and therefore the dislocations prefer Orowan looping leading to overaging behavior. Thus peak-aging of coherent particles is the result of the transition in the interac-
7.2 Yielding at Low Temperatures
tion mechanism (the particular behavior of coherent, ordered precipitates will be treated in Sec. 7.2.2.5). The results of mechanical measurements confirm convincingly the theoretically predicted functional dependencies as documented in a very earlier representation of Gerold (1974) (Fig. 7-4). Electron-microscopic observations, on the other hand, provided direct evidence for particle cutting and Orowan looping (Figs. 7-10 b and 7-11). 7.2.1.2 Particles as Extended Obstacles
The Orowan Stress Metallurgical real-life obstacles such as second-phase particles have a finite extent, which may have a substantial influence on the CRSS in a number of different ways. It is obvious that the point-obstacle approximation is justified as long as the mean-planar spacing is very large compared to the dimensions of the particles. This condition is fulfilled for small volume fractions / only. The oldest known method to describe extended impenetrable hard particles is to use Eq. (7-5), but to replace the center-tocenter square lattice spacing L by the interparticle spacing, L p = L - 2r s (Eq. (7-10)).
All further effects of the extended width of impenetrable obstacles, such as - the statistical distribution in the glide plane (Kocks, 1967; Foreman and Makin, 1966), - the elastic self-interaction of the dislocation segments near the particles (Ashby, 1969; Foreman et al, 1970; Bacon et al, 1973), - the elastic anisotropy (Scattergood and Bacon, 1975), entering the Orowan stress via the outer cut-off radius of the dislocation dipole line tension have now been extensively discussed. A critical review of Kocks (1977) suggested the following best formula for the anisotropic Orowan-stress: C
OR
= 0.9
[ln(8r s /b)] 3 / 2 /
KEDGE
\b(L-
2rs)
which replaces Eq. (7-5). X EDGE is the prelogarithmic line tension factor for an initially straight edge dislocation in the given anisotropic material. The Theory of Schwarz and Labusch If the obstacles are extended, have a finite range of interaction, and are not so widely-spaced, the dislocation touches
Orowan Eq.(7-5)
\ Al-Zn-Mg
321
^
^
Fm/2TL=0.-\
CD d
10"
F m /2r L =0.01 refined Eq.(7-14)
2
CM
NiAI
10-3
Cu-AI2O3
10 100 Relative particle radius rib -
1000
Figure 7-4. Normalized CRSS vs. particle radius. Theoretical estimates (FF limit) and experimental results (Gerold, 1974).
322
7 Particle Strengthening
many obstacles with all strengths, F < Fm, and only very few of them with the maximum force Fm. Schwarz and Labusch (1978) were the first to re-analyze successfully the problem of finite obstacle extent for more concentrated particle arrays within the weak-obstacle limit with the aid of a digital computer. They eliminated the two essential assumptions of the FF theory, namely the point-obstacle condition and the squared-obstacle array. The authors simulated the movement of a dislocation through a random array of obstacles of interaction range 1^L perpendicular to the dislocation. The analysis shows that the normalized stress, (T SL /T FF ), is a unique function of the "normalized obstacle depth", rjSL, and that the FF theory can only be applied if the criterion (index SL for Schwarz-Labusch): Y holds. The nature of the solution depends upon the interaction force-distance profile.
1.0 0.8
elastic (energy conserving) 0.4
0.8
1.2
SL treat two different types (see Fig. 7-5). For elastically interacting obstacles with a symmetrical force-profile, SL's computer simulated data in Fig. 7-5 can be best represented by the interpolation formula: = T F F X 0.94(1+ 2.5
(7-16)
For obstacles with an asymmetric force profile, representing energy storing interaction, the data follow the relationship: (7-17) with CSL = 2/3, for rjSL <^ 1 (FF hardening), CSL > 1, for rjSL > 1 (SL hardening). The term T FF in these equations is merely an abbreviation for the FF flow-stress given by Eq. (7-3). The numerical factor 0.94 agrees well with the value 0.96 obtained by Foreman and Makin (1966) and Hanson and Morris (1975 a). It can be seen in Fig. 7-5 that, at small values of rjSL (< 0.4), the behavior of both obstacle types is nearly identical. For a steeper front
Figure 7-5. Reduced yield-stress as a function of the normalized obstacle depth rjSh, obtained from simulation for the two obstacle profiles as schematically shown in the inset (Schwarz and Labusch, 1978).
7.2 Yielding at Low Temperatures
flank of the force profile (dashed in the insert of Fig. 7-5), the increase of TSL with rjSL is significantly weaker than plotted in Fig. 7-5. Further, Eqs. (7-16) and (7-17) can be reduced to Eq. (7-4) if rjSL vanishes, providing a continuous transition between SL and FF statistics via the parameter rjSL. A comparison of Eqs. (7-16) and (7-17) with Eqs. (7-1), (7-2), and (7-3) leads to the new Schwarz-Labusch spacing (index SL) replacing the earlier Friedel spacing, JF: For elastic obstacles with symmetrical interaction force profile: /el
_
0.94(1 +2.5*/ s
323
and 1.2, respectively. Haasen and Labusch (1979) calculated the interaction range from the energy stored in the particle, 71 r l 7 = J F (y) dy, w ith the special form of the profile used by SL (see Fig. 7-5), to Following Haasen and Labusch (1979), we prefer to deduce YSL from the certain characteristics of actual force-distance profiles. Figure 7-6 shows concrete examples. Fe represents parelastic interaction due to
(7-18)
holds, while for energy storing obstacles with an asymmetrical interaction force profile the Schwarz-Labusch spacing is given by: rstor
*F
0.94(1
(7-19)
The new parameter rjSh may be interpreted as the relative change in the applied force on an obstacle as the dislocation segment moves forward by an amount, 7SL, at the obstacle, while it is pivoted at two neighboring obstacles one Friedel spacing (Eq. 7-2)) away (Kocks, 1977). For applications to experiment, it is necessary to express rjSL in terms of particle parameters. In Eq. (7-15), the particle's interaction range, 7SL, appears. In correlations of 1^L with the particle extent, however, fairly large inconsistencies exist in the literature. Reppich et al. (1982) and Ardell (1985) chose 7SL to be the half of the particle size in the glide plane, i.e., rs. Nembach and Neite (1985) considered YSL to be an adjustable parameter and write: YSL = £rs
(7-20)
From the fit of experimental data of twophase superalloys they obtained £ = 0.6
-20b 0 20b y Figure 7-6. Interaction-force profiles between a straight undissociated edge dislocation and a y' particle of radius r = 20 b embedded in a f.c.c. y-matrix (Nembach and Neite, 1985). Fe: coherency lattice misfit force; FG: modulus misfit force; Fy: force due to the antiphase boundary energy. Material: Ni-based superalloy Nimonic PE 16.
324
7 Particle Strengthening
the particle lattice mismatch (see Sec. 7.2.2.3), FG represents dielastic interaction due to the modulus mismatch (see Sec. 7.2.2.2), and Fy represents energy storing interaction, here due to order hardening (see Sec. 7.2.2.5). Evidently, the profiles are indeed either symmetrical for a (nonsplit) dislocation interacting elastically, or asymmetrical if the (non-paired) dislocation experiences an energy storing interaction. It is therefore obvious that the earlier assumptions of Reppich et al. (1982), Ardell (1985) and Nembach and Neite (1985) are quite inadequate. Rather we suggest that YSh ~ 2 rs in the case discussed, provided the definition of YSL given by SL itself in Fig. 7-5 is correctly applied. Specifically, for order hardening depicted in Fig. 7-6, YSL equals exactly Ir^. The modifications necessary for pairwise-coupled dislocations as well as for stacking-fault hardening will be treated in Sec. 7.2.2.4 and 7.2.2.5, respectively. For elastically interacting obstacles, the force profile in Fig. 7-6 is not limited exactly by the particle's periphery. Rather, the force flanks decay hyperbolically with increasing distance from the particle interface. However, for both modulus mismatch as well as for coherency lattice mismatch interaction, YSL ~ 2r s , as plotted in Fig. 76, seems quite an appropriate choice. After substitution of Eq. (7-8) and (7-20) into Eq. (7-15), rjSL becomes:
for both symmetrical and asymmetrical force profiles. For higher specific obstacle strengths, jF m ~2T L , rjSL attains its minimum value:
•» = 0.54 e / 1
(7-22)
which is constant for a given /, independent of particle size. The parameter £, describing the interaction range YSL in units of rs, (Eq. (7-20)) will be specified for the diverse operating hardening-mechanisms discussed in Sec. 7.2.2. We conclude this section by emphasizing that the SL theory becomes most important for two-phase materials containing weak (small) particles at large values of / (Eq. (7-21)). Its judicious application will have great utility and general validity in the quantitative description of particle strengthening presented subsequently. 7.2.2 Particle-Strengthening Mechanisms 7.2.2.1 Chemical Strengthening An edge (or screw) dislocation shearing through a coherent spherical particle creates new precipitate-matrix interface of specific surface energy ys (per unit area) as two lunar edges of width b (hatched in Fig. 7-7). Then the maximum interaction force which can be exerted is: Fm = 2 ys b
(7-23)
We emphasize that jFm essentially does not depend upon particle size. Accordingly, 37lV -1/2 chemical strengthening is an example (7-21) 32J < FmJ which cannot be fitted into the general scheme of Eq. (7-3) (and, as we will see below, it is contrary to Eq. (7-12) in Fig. 7-3). Nevertheless, this mechanism is one of the 1 Approximating the profile of Fyin Fig. 7-6 by an ellipsoid, the stored energy, n r^ y — J Fy (y) dy (y: stored earliest treated theoretically. The original anti-phase domain ^boundary energy) corresponds to model of Kelly and Fine (1957) has been the area under the F-y curve, (l/2)rcFmax(ySL/2). We modified by Brown and Ham (1971) in obtain rSL = 4rs2y/Fmax. Setting FBua = XSLy, where terms of FF statistics. The authors obXSL = 2 rs is the "particle width" parallel to the dislocation line, we obtain YSL = 2rs. tained, with JFm according to Eq. (7-23) and /2
325
7.2 Yielding at Low Temperatures
7s
Figure 7-7. Chemical (surface) strengthening: edge dislocation shearing a spherical particle creates the (hatched) new particle-matrix interphase of area 2 rs b.
utilizing Eqs. (7-4) and (7-8) (ignoring the factor 0.9): (7-24)
An alternative model was presented by Harkness and Hren (1970) to interpret hardening of some binary Al-Zn alloys by GP zones. It predicts opposite behavior to that described by Eq. (7-24), namely increasing CRSS with increasing r. The authors reported an interfacial energy of 0.320 Jm~ 2 , which is clearly too high by about an order of magnitude. Therefore Gerold (1979) has questioned their analysis. Let us treat chemical strengthening within the SL-theory next. Surface energy hardening is commonly called chemical strengthening because it is governed by the chemical bonding between matrix and precipitate. The surface energy of coherent particles is low; values should cover the range ys = 0.01 to 0.10 J m " 2 . With G f c - l O J n r 2 Eq. (7-23) gives extremely small specific particle strengths, F m /(2T L )~10~ 2 to 10" 3 . Consequently, chemical strengthening is the weakest interaction occurring, and thus no significant amount of strengthening should be expected according to Eq. (7-24), even for
small r. On the other hand, as mentioned in Sec. 7.2.1.2, the SL correction becomes more important with decreasing Fm/(2 7jJ. In the following, we attempt to demonstrate that, even in the particular case of chemical strengthening, the SL correction cannot be ignored. First we have to specify the range of dislocation-particle interaction, YSL. Figure 7-7 suggests that here, typically YSL = 2r s , or £ = 2. Secondly, the left-hand front flank of the force-distance profile is somewhat steep, like the dashed curve in the insert of Fig. 7-5. In this case, the calculations of SL provide low values of the parameter CSL. An appropriate choice may be CSL = 1: Substituting £ = 2 into Eq. (721) and utilizing Eq. (7-23), we find for the reduced particle depth: rjSL = 1.085
Tyi2
(7-25)
which is independent of r, and a function of ys and / only. Insertion into Eq. (7-17) for energy-storing interaction leads to the new cutting stress: zzSL = 0.94 SL
~—)\
(7-26a)
where xFF stands for the result from Brown and Ham given by Eq. (7-24). Rearranging Eq. (7-26 a) yields the CRSS in an alternative form: (?_26b) 1/2
= 0.94-
«/rJ
+1
Inspection of Eq. (7-26) reveals the following: - For fixed /, the CRSS decreases as r increases. This "age-softening" response is contrary to all the other shearing mechanisms treated subsequently for which Eq. (7-12) in Fig. 7-3 predicts normal parabolic age-hardening.
326
7 Particle Strengthening
- Equation (7-26) predicts maximum strength at minimum particle size (which Brown and Ham (1971) termed "molecular hardening"), e.g., when underaged particles are extremely small or when at least one dimension of the precipitate is of atomic size. Thus chemical strengthening should not be significant in real alloys containing nearly equiaxed precipitates usually larger than 10 nm. Rather the most likely candidates for the effect are systems containing atomically thin rods or plate-like precipitates such as Al-Cu (0f or ©") or Cu-Be. But reliable experimental data ideally suited to identify chemical strengthening as the only operating mechanism are not yet available. Moreover, we should expect that in underaged alloys chemical strengthening occurs in combination with other cutting mechanisms. Reppich (1975) and Knoch and Reppich (1975), for instance, interpreted peak-aging of MgO single crystals containing finely dispersed coherent MgO-ferrite spinel precipitates by superposition of chemical and order hardening. - Finally, let us briefly discuss the particularly important role of the SL correction in the present case. The first term in the brackets in Eq. (7-26) represents the conventional part of the CRSS, as described by Eq. (7-24), while the second term reflects the SL correction. Using CSL = 1 and ys < 10~2 (TJb) as above, the latter becomes
After insertion into Eq. (7-26 a), we obtain TsL = (2-4) T FF for / = 0.01 -0.10. In other words, the extent of strengthening predicted by Eq. (7-26) amounts to several times the conventional value given by Eq. (7-24). Thus, correct application of Eq. (7-26) can essentially raise the CRSS to a level comparable to other shearing mech-
anisms. Convincing experimental evidence for that, however, is still missing. 7.2.2.2 Modulus-Mismatch Strengthening
This type of dislocation-particle interaction is the analogue to the well-known modulus difference solid solution hardening treated in Chap. 5 of this volume. The stress field of a moving dislocation will interact dielastically with a large elastic inhomogeneity like the coherent secondphase particle in Fig. 7-6 consisting of a phase with shear modulus Gp embedded in a matrix with shear modulus Gm. The interaction force exerted is then proportional to AG = \Gm — Gp|. A satisfactory general solution for the problem of deriving the maximum interaction force from the interaction energy has not yet been given. Approximate calculations, on the other hand, have been refined over the years. The first was presented by Weeks et al. (1969). Knowles and Kelly (1971) proposed a model which essentially describes overaging because they assumed a fixed obstacle spacing along the dislocation to estimate the CRSS. Russell and Brown (1972) introduced the interesting idea that a dislocation is "refracted" when it penetrates the particle (note that the physical problem is similiar to that treated by Hiither and Reppich (1978) with respect to order hardening, see Sec. 7.2.2.5). Their estimate predicts maximum strength at a very small particle size, and thus also describes overaging behavior. The model of Melander and Persson (1978), which combines modulus hardening with the theory of Hanson and Morris (1975 b), was the first one which predicts normal age-hardening at small particle size according to: (7-27) AG/G 2r 2bln( ll2
\f b
3/2
7.2 Yielding at Low Temperatures
Nembach (1983) derived the interaction force for different models of dislocation core and presented the result of his approach as: (7-28) The parameters CG and m are sensitive to the core model used and amount roughly to 0.05 and 0.85, respectively. For the example depicted in Fig. 7-6, Fm reaches its maximum value of about AG b2, close to the particle's periphery. Substitution of Eq. (7-28) into Eq. (7-4) and (7-8) gives Nembach's FF-version of the cutting stress T F F = a G AG 3 / 2
(7-29)
where the numerical factor aG = 0.055 (ignoring the factor 0.9 in Eq. (7-4)). Turning now to the application of the SL-theory, we conclude from the interaction force profile in Fig. 7-6 that the range of interaction can again be chosen to be 7SL = 2r s , i.e., £ = 2. Using Fm^ AGb2 (following Nembach and Neite (1985)) and inserting into Eq. (7-21) we obtain for the reduced particle depth: 1/2
rjSL = 1.085
(7-30)
.(AG/G)J
The interaction force profile in Fig. 7-6 is symmetrical like the one at the bottom in Fig. 7-5. Consequently, the SL interpolation, Eq. (7-16), for elastic interaction has to be applied. Now the CRSS in the new SL version is:
nt
1/2-11/3
i.94 1 + 2.71 = T F F x 0.94
(-^)
where T FF is Nembach's result after Eq. (7-29). Modulus mismatch hardening seems to cover a broad spectrum of specific obstacle
327
strengths. For y'-precipitating Ni-based superalloys Nembach and Neite (1985) reported modulus differences of several MPa. Thus, the specific particle strength, Fm/(2 TL) = AG/G, is of the order of magnitude of 10~2. In this class of alloys, modulus-mismatch hardening is indeed a very weak mechanism. Nembach and Neite (1985) claimed that under these circumstances rjSL is small and, therefore, T FF (Eq. (7-29)) and TSL (Eq. (7-31)) are nearly identical. But this conclusion is incorrect. With their value (AG/G) ~ 0.64 AG b2, Eq.(7-30) gives f/SL — 3.2 for / = 0.20, which is clearly far outside the FF theory. With this result, the SL correction term in Eq. (7-31) becomes 2. This means that the SL-modified CRSS is twice that estimated conventionally. Nevertheless, it must be stressed in accord with Nembach and Neite (1985) that the measured extent of hardening in y'/y alloys cannot be explained by modulus differences. Rather, it is generally accepted that order hardening is, in essence, the dominant mechanism. The crux with modulus mismatch strengthening is that there are no ideally suited alloy systems to test the predictions of the theory. In most cases, modulus mismatch hardening will be outweighed by other mechanisms like in the aforementioned y'/y superalloys. In Cu-Co, Cu-Fe and Cu 3 Au-Co, massive coherency-strain hardening occurs, see next section. Another major difficulty in comparing theory with experiment is that GP is not known with certainty because independent measurements of GP from bulk phases are often impossible. Gp may then be considered as an adjustable parameter which leads to agreement between experiment and theoretical estimates. The AG/G values between 0.88 and 1.45, respectively, reported by Ardell (1985) for Cu-Co, Cu-Fe and Cu 3 Au-Co are unrealistically high. It
328
7 Particle Strengthening
must be stressed that those AG/G values which correspond to the specific particle strength would be equal to or even higher than the theoretical upper limit, FJ(2 TL) < 0.8, attributed to the Orowan stress. Available experimental data analyzed in the literature are of alloy systems in the peak-aged and overaged condition. As already pointed out by Ardell (1985), several research groups (Knowles and Kelly, 1971; Russell and Brown, 1972; Melander and Persson, 1978) attempt to demonstrate that in Cu-Fe, Fe-Cu and A l - Z n - M g alloys modulus mismatch hardening is the determining mechanism and claim that the measured CRSS can be best explained by their own version of the hardening model. Ibrahim and Ardell (1978) reported that the data of overaged Cu 3 Au-Co disagree with the model of Russell and Brown (1972) and can be interpreted by the model of Knowles and Kelly (1971). Ardell (1985) concluded further that the overaging found in these alloys is definitely not the result of the Orowan mechanism, which is supported by the low work-hardening rates observed and by the absence of any Orowan loops. This interpretation must, however, be doubted particularly in the light of the special cross-slip mechanism proposed by Gleiter (1967 a, b). Accordingly, a dislocation which encounters the strain field of a coherent precipitate undergoes a series of complex manoeuvres and thus stable concentric Orowan loops cannot be observed. 7.2.2.3 Coherency Strengthening
A coherent precipitate shown in Fig. 7-6 whose lattice parameter, a p , may differ from that of the matrix, am, distorts its environment by producing a strain field by which it interacts parelastically with dislocations. This mechanism is the direct coun-
terpart of the atomic size difference solid solution hardening (see Chap. 5 of this volume), and could well be the oldest source of age-hardening known (Wilm, 1911). Theory At least five research groups, Gerold and Haberkorn (1966), Gleiter (1967 c), Brown and Ham (1971), Jansson and Melander (1978), and Nembach (1984), have treated lattice mismatch interaction within the framework of the FF theory. By far the most thoroughly modeled case is that of an undissociated, straight, pure edge-dislocation in an isotropic linear-elastic medium interacting with a spherical precipitate of radius r with a misfit: £ =
The maximum interfaction force, Fm, depends on the distance of the operating glide plane z from the particle's center (Fig. 7-6). Fm is zero when the slip plane contains the particle's center, and has its maximum value: (7-32)
at the matrix-particle interface for z0 = / In the particular case shown in Fig. 7-6, the interaction comes from the strain field outside the particle. So far the lattice mismatch hardening is the only cutting mechanism for which the interaction force does not derive from the properties of the dispersed phase itself. The dislocation feels the strain fields of the precipitates that physically intersect its slip plane as well as the strain fields of precipitates that do not. Thus, the effective size of the obstacles, i.e., their interaction range, is not identical with the physical size of particles.
329
7.2 Yielding at Low Temperatures
A further problem arises. Statistically, both repulsive and attractive particles contribute an identical maximum interactionforce. The only difference is that the former experience the maximum force before and the latter after passing the particle center. Consequently, in order to derive the CRSS with Eq. (7-1) it is necessary to find some means of properly averaging the effects of all particles with respect to both Fm and /. The details of the various averaging procedures applied have been discussed by Gerold (1968) and Brown and Ham (1971). The approaches of Gerold and Haberkorn (1966), Brown and Ham (1971), and Janson and Melander (1978), unlike the estimate by Gleiter (1967 c), provide a cutting stress for underaged alloys given by: I
1
/
,1/2
, p
\-l/2
(7-35 a) (7-35 b)
= 0.54
which, for Fm/(2 TL) ~ 1 acquires its lowerlimit value: 1/2
(7-35 c)
The force-distance profile for misfitting interaction in Fig. 7-6 is symmetrical. Therefore the SL spacing according to Eq. (7-18) has to be used. Replacing the Friedel spacing, /F, in the derivation of the CRSS by /SL, given by Eq. (7-18), the cutting stress for underaged misfitting particles in the new SL version becomes
2
(7-33 a) or, using TL = G b2/2:
V'2
(7-33 b)
The numerical factor a£ differs from one estimate to the other, varying between 3 and 4.1. A useful average value is a£ = 3.7. The maximum CRSS for larger, peakaged misfitting precipitates follows by combining Eqs. (7-32) and (7-33 a) with the criterion FJ(2 TL) ~ 1 (Ardell, 1985): l
PEAK
= 1.8Gs/1/2
(7-34)
Evidently, Eqs. (7-33) and (7-34) are FF relationships of the type of Eq. (7-3) or Eq. (7-12), derived within the point-obstacle approximation. As already pointed out in Sec. 7.2.1.2, this is not strictly justified because of the finite range of the misfitting interaction. Specifically, as shown in Fig. 7-6, we have 7 S L >2r s , i.e., £ ~ 2. Thus, using Eq. (7-32) the reduced particle depth given by Eq. (7-21) becomes:
with ?jSL given by Eq. (7-35 b). Correspondingly, using Eq. (7-35 c), the peak stress (Eq. (7-34)) can be rewritten: 1 + 2 . 7 /
1
^
3
(7-37)
Comparison with Experiments Figure 7-8 shows data for different binary alloys for which coherency strengthening is claimed to be the predominant mechanism, namely Cu-Co, Cu-Fe and Cu 3 Au-Co single crystals as well as Cu-Mn polycrystals [taken from a compilation of Ardell (1985)]. In particular, Cu-Co seems to be an ideal system. The f.c.c. Co-rich precipitates show distortion contrasts, and / and r can be estimated from magnetic measurements. The experimental values are presented in Fig. 7-8 in the normalized form according to Eq. (7-36) as Ai/Ge 3/2 vs. (rf/b)1/2. AT is the CRSS increment due to the misfitting precipitates alone, obtained from the total ex-
330
7 Particle Strengthening
2 Eq.(7-36) / Eq.(7-39) /
t
C
•
/°
•
#1 £x10 2
: in
1--
0 Cu-1.40Co D Cu-1.36Fe
X
^
-1.49 -0.97
A"Cu3Au-1.5Co V Cu-38Mn
-4.28 +1.05
i
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
Figure 7-8. Reduced plot of coherency strengthening data of a variety of Cu-based alloys tested at room temperature. Data taken from a compilation of Ardell (1985).
1/2
perimentally measured CRSS by linear subtraction of the solid solution matrix CRSS 2 . All the alloys in Fig. 7-8 exhibit the general features derived in Sec. 7.2.1.1 for coherent particles (see Figs. 7-3 and 7-4), namely parabolic underaging with fine dispersions, sharp peak-aging followed by overaging with coarse dispersions. For underaged alloys, the predicted linear increase of the CRSS proportional to (r/) 1 / 2 is found throughout with one exception. In reviewing the data of Livingston (1959) for polycrystalline Cu-Co (not plotted in Fig. 7-8), Brown and Ham (1971) concluded that a / 2 / 3 behavior gives a better fit. 2 The evaluation of experimental CRSS data, i.e., the separation of the particle CRSS from the measured total CRSS of the two-phase material involves two general problems: The first question is related to the rule of superposition of strengthening processes affected by the interaction of dislocations with different types of obstacles - with solute atoms randomly distributed in the matrix on the one hand, and with second-phase particles on the other hand. The second problem is concerned with the estimation of the absolute amount of the (depleted) matrix CRSS. These topics have been intensively discussed by Reppich et al. (1982), Nembach and Neite (1985), and Ardell (1985).
However, it has long been recognized that Eq. (7-33) seriously overestimates the observed strengthening (Gerold, 1979; Nembach, 1984; Ardell, 1985). According to Eq. (7-36), the plot in Fig. 7-8 should give straight lines with slope 4-4.5 3 . But only the data of underaged Cu-Fe approach the theoretical straight line (Fig. 7-8 dashed line), as already reported in the literature by Wendt and Wagner (1980). In contrast, the data of all other alloy systems are several times lower than the theoretical ones, far outside the limits of all the possible experimental errors. For Al-Zn, Gerold (1979) reported quite similar discrepancies. The analogous situation was found by Ardell (1985) when analyzing the data of the same alloys in peak-aged condition with respect to Eq. (7-34). The discussion of further possible effects and refinements in the literature, such as the anisotropy of the matrix, interaction with screw dislocations, cross-slip, modulus hardening etc. were unsuccessful and not 3 The theoretical straight line in Fig. 15 of Ardell's paper (1985) seems too steep by a factor of about 2.
7.2 Yielding at Low Temperatures
helpful in solving the discrepancy (Gerold, 1979; Gerold and Pham, 1980; Ardell, 1985). Dislocation Splitting Nembach (1984) pointed out that the chronic discrepancies observed are a consequence of the approximation quoted above, namely the neglect of dislocation splitting into two Shockley partials. If the particle radius r is of the same order of magnitude as the width of the stacking fault ribbon between the two partials, s, splitting must be taken into account. Dissociation of dislocations reduces the maximum interaction force by a factor of P± {r/bY2. After Nembach (1984) Eq. (7-32) becomes: (7-38) Equation (7-38) holds for 30b>r>6b, i.e., for r > 1.5 nm. The coefficients f}± and P2 depend on the splitting widths and are given by Nembach (1984). For Cu, f}1 and j82 equal 0.328 and 0.239, respectively. Consequently, the right-hand side of Eq. (7-36) has to be multiplied by the factor [Piir/bY2]312, so that the cutting stress for dissociated partials can be rewritten as: <-SL
f r\1/2
— ry* Gp3/2t —
(Xc
KJ b
1 | —:
I
b
)
(7-39 a)
/?2~|3/2
=3.71/^1-
[0.94(1+ 2.5 n
J
(7-39 b)
rjSh is given by Eq. (7-35 b). For undissociated dislocations, i.e., 5 = 0, px = 1, /?2 = 0, Eq. (7-39) leads to Eq. (7-36). In the extreme case where 5 is much larger than r, Fm is reduced to one-half of the value given by Eq. (7-32). Correspondingly, % calculated with Eq. (7-39) would be lowered by the factor of 2.8.
331
In Cu, the separation s of the partials is about 4 nm (Cockayne et al., 1971). A recent in-situ HVEM deformation study of Nembach et al. (1988) provides convincing evidences that in Cu-Co the dissociation for interacting partials is of the same order of magnitude. The radius of the misfitting particles in underaged Cu-Co investigated (r = 2 - 3 nm) was indeed of the same order of magnitude as the splitting widths. Application of Eq. (7-39) reduces x by about 50%. The factor a8 • 0.94 (1 + 2.5 7?SL)1/3 quoted from experimental AT 0 values now ranges from 3.5 to 4.6, in excellent agreement with the theoretically predicted value 4.1. Therefore, the (dotted) theoretical straight line in Fig. 7-14 with slope a* = 1.73 fits the experimental data points of underaged Cu-Co very well. In conclusion, allowing for dislocation spitting and SL theory, theory and experiment can indeed by reconciled for Cu-Co age-hardened by small spherical misfitting precipitates. In Cu-1.15 at.% Fe foils, however, Nembach et al. (1988) found neither evidence for the occurrence of dislocation splitting nor for stacking fault fringes. Although this surprising observation is not yet understood, it explains the quite reasonable agreement, within 25%, of the measured AT 0 values for underaged material with Eq. (7-36), (represented in Fig. 7-8 by the straight line through the origin with slope 3). In C u - M n and Cu 3 Au-Co, on the other hand, one would suggest that dissociated partials interact. But direct electron microscopic support for this interpretation is still missing. 7.2.2.4 Stacking-Fault Strengthening
Hirsch and Kelly (1965) first suggested that a strong interaction occurs between a dislocation and precipitate in which the stacking fault energy, ygF, is substantially
332
7 Particle Strengthening
different from that of the matrix, yfF. The interaction force arises due to the different stacking-fault ribbon widths of the dissociated dislocation inside and outside the particle, sm and sp, respectively (Fig. 7-9). Hirsch and Kelly (1965) and Gerold and Hartmann (1968) analyzed the maximum interaction force for various situations reflecting the relative ratios of rs and splitting width s as depicted in Fig. 7-9. Hirsch and Kelly (1965) estimated the CRSS by taking the effective obstacle spacing along the dislocation as given by the Mott spacing. Gerold and Hartmann (1968) used the more appropriate Friedel spacing ZF (Eq. (7-2)) in their approach. The maximum force experienced by the split dislocation is Fm = AySF d (ygp, r)
(a)
(7-40) an
= ITSF ~~ 7SFI> d d is the effective chord length of the stacking fault inside the particle when the dislocation is just breaking away from it. The functional dependence of d upon ySF and rs is rather complex. After Gerold and Hartmann (1968), who used the model depicted in Fig. 7-9 a, the commonly expected, linear proportionality between Fm and rs is observed only for small rs. The deviation increases with increasing AySF. When rs is comparable to or less than sm9 d is equal to 2r s (Fig. 7-9). Substituting this result into Eq. (7-40) and combining it with the FF formula, Eq. (7-4) (ignoring the factor 0.9), we expect the increase in the CRSS for underaged alloys to vary as: A?SF
JAML
,1/2/
fv
(c)
Figure 7-9. Stacking fault strengthening: (a) theoretically assumed configuration of the split partials; (b) interaction-force profiles and range of interaction 1^L involved; (c) experimental data of Gerold and Hartmann (1968) on age hardened Al-1.8 at.% Ag taken from Ardell (1985).
tion of Hirsch and Kelly (1965) as: (7-42)
\l/2
(7-41) For increasing particle size, d increases less than linearly and thus strengthening falls below the value predicted by Eq. (7-41). For overaged particles, 2 r s > s m , Ardell (1985) presented the respective approxima-
where K oc G b^ and y*F = (yfF + ygF)/2. Note the analogy to the model of Hiither and Reppich (1978) who deal with similarly overaged ordered particles (see Sec. 7.2.2.5). Again, in all the cases discussed, the SL spacing ZSL applies instead of the Friedel
7.2 Yielding at Low Temperatures
spacing /F. For underaged particles, the force-distance profile in Fig. 7-9 is non-symmetrical. Therefore Eqs. (7-17) and (7-19) hold. For Fm = AySF 2 rs the reduced particle depth is now according to Eq. (7-21):
333
these data in light of the conventional relation Eq. (7-41). Accordingly, for underaged particles a straight line with slope 0.96 [Ay|F/(TL b)]1/2 is expected. The data of an ageing series at an ageing temperature Ta = 140 °C (open symbols) indeed exhibit 1/2 l/2 the predicted behavior. Gerold and Hart(7-43 a) AySF mann used the isotropic constant line tension approximation, TL = (G b2)/2, and As shown in Fig. 7-9, the range of stacking assumed that the ratio AySF/G is temperafault interaction becomes 7SL = rs + (sm/2) ture-independent. The authors could fit the = 2 rs for sm > 2 rs, or £ = 2. This leads to: data with AySF = 0.180 Jm~ 2 . Assuming 7SF — 0-200 J m ~ 2 as a most likely value for (7-43 b) pure Al, the stacking fault energy inside the silver-rich precipitates was calculated Insertion into Eq. (7-17) yields the new to be ylF = 0.020 J m " 2 . Ardell (1985) used CRSS for underaged alloys: ^ ..*. the line tension for edge dislocations estimated to be 0.133 (Gb2) and obtained = T F F X 0 . 9 4 1 + 1.22CSL ypS¥ = 0.109 J m ~ 2 and 0.093 J m " 2 at Ay,S F' 295 K and 77 K, respectively, which seems In the case of overaged particles, the force a little high. profile in Fig. 7-9 is symmetrical. For However, let us reinterpret the data for rjSL < 0.4, on the other hand, Eqs. (7-16) underaged material on the basis of the and (7-17) can be used alternatively. For rather more relevant Eq. (7-44). According overaged particles, Fm ~ 2 TL, f?SL tends to to this, the straight lines in Fig. 7-9 acquire its minimum value, rj™ln, given by now have the slope 0.94[Ay|F/(TLb)]1/2 Eq. (7-22). Now the range of interaction • [0.94 (1 + CSL rjSI)]. Therefore the above 5k = rs + (sJ2) * rs (Fig. 7-9); { ~ 1 may values of AySF have to be divided by the be a useful compromise. Then the reduced term [0.94 (1 + CSL rjsl)]2/3. In Eq. (7-21) for particle depth becomes: rjSL we set £ = 2 (see above) and [Fm/(2 TL)] ~ 1. Taking CSL = 1, this (minimum) cor= *?SL V-3*r J ^/-HOJ rection yields [0.94(1 + 1.09/ 1/2 )] 2/3 . With which is indeed < 0.4 even for large volthe volume fractions reported by Gerold ume fraction, / < 0.55. Combining Eqs. and Hartmann (1968), / = 2% and 2.9% (7-45) and (7-42) with Eq. (7-16) gives the for the 140 °C and 225 °C ageing series, reCRSS for overaged alloys: spectively, and with the quoted cor(7 46) rection factor of 1.07, the above value (KfVr • " of Gerold and Hartmann reduces to AySF = 0.168 J m " 2 which leads to the higher value y£F = 0.032 J m " 2 . Stacking fault strengthening seems to be The data of Al-Ag aged at 225 °C (full the most important mechanism in Al-Ag symbols in Fig. 7-9) exhibit smaller absoalloys. The experimental data of Gerold lute amounts of strengthening and a deparand Hartmann (1968) are plotted in Fig. 7-9 1/2 ture from the behavior predicted by Eqs. as AT VS. [(r f)/b] . Gerold and Hartmann (7-41) and (7-44). The former is due to the (1968) as well as Ardell (1985) analyzed
"-"Hi)
334
7 Particle Strengthening
smaller AySF. Gerold and Hartmann (1968) found AySF = 0.140 Jm~ 2 , which results from the larger value of y£F = 0.060 J m" 2 . The same SL correction as above, applying Eq. (7-44) instead of Eq. (7-41), leads to the slightly higher value of y|F = 0.069 J m~ 2 . This value reflects the expected increase of 7SF w i ^ decreasing Ag content of the precipitated phase at 225 °C. The higher value of 7SF> o n the other hand, is consistent with smaller splitting width, sp, so that the interaction length d of the leading partial now becomes smaller than 2r s , Fig. 7-9 a. Consequently, the interaction force decreases and is no longer linearly proportional to rs. Thus, the CRSS drops below the value predicted by Eq. (7-44) and the dependence of T on rs becomes substantially weaker than rs1/2, which is in fact seen in Fig. 7-9 c.
Ni
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7.2.2.5 Atomic-Order Strengthening Theory
(b)
When a matrix dislocation shears a coherent ordered particle (Fig. 7-10 a), it creates disorder in the form of an antiphase domain boundary (APB) of surface energy on 7APB the slip plane within the precipitate phase. The APB energy per unit area represents the force per unit length opposing the dislocation as it penetrates the particle: Fm = y A P h d
(7-47 a)
d is the length of the dislocation lying in the ordered particle. For spherical particles d = 2rs = 2 (n/4) r; the mean maximum force is given by F=
(7-47 b)
The second, trailing dislocation (symbol cOziin Fig. 7-10 a), on the other hand, removes the APB created by the first, leading one. The APB thus causes pair-coupling.
Figure 7-10. Order strengthening: (a) dislocation shearing an ordered particle creates an anti-phase domain boundary APB (bold-dashed line); (b) HREM micrograph showing a strongly coupled dislocation pair just shearing a 5' (Al3Li) particle in a deformed binary Al-Li single crystal (Lendvai et al., 1989).
Therefore, in alloys containing ordered particles, the dislocations travel in pairs (Figs. 7-10b, 7-11 a) 4 . The first complete analysis of order hardening including the effect of paired dislocations was presented by Gleiter and Hornbogen (1965 a, b). The dislocations of the pair are controlled by the following forces (Fig. 7-12 a): - the forward force T b lla, due to the applied shear stress; 4 More generally, the dislocations move in groups. The number of dislocations in the group is determined by the number required to restore perfect order within the ordered precipitate.
7.2 Yielding at Low Temperatures
- the repulsive force between the paired dislocation, FR; - the attractive force of the APB, yAPB d1 a, according to Eq. (7-47). Consequently, the force balance established in the critical moment when the first dislocation cuts is:
335
(a)
dislocation 1: (7-48 a) dislocation 2: =0
(7-48 b)
Eliminating FR in Eq. (7-48 a, b), one obtains the forward stress upon the first dislocation:
H]
(7-49)
As essential parameters in Eq. (7-49), the lengths of the dislocation segments inside the ordered particles, d1 and d29 as well as the effective spacings along the cutting dislocations, lx and l2, appear. Hence, the dislocation configuration must be discussed. Indeed, the crux of the current models relating xAPB to the particle microstructure is the different assumptions upon the ratios djlx and d2/l2. In particular, neither d2 nor l2 are easy to specify. Electron microscopy of deformed specimens, however, will help in this regard. Depending on the dispersion of the ordered particles the following situations can be distinguished.
(c)
• Figure 7-11. TEM micrographs illustrating typical dislocation y'-particle interaction in Ni-based superalloys deformed at room temperature, (a) Weak-beam dark field HVEM micrograph of an in-situ deformed underaged single crystal Nimonic PE16 foil showing a procession of weakly coupled dislocation pairs (Nembach and Neite, 1985); (b) shearing of y' in peak aged Nimonic 105; (c) shearing of coarse cuboidal y' in IN-100; (d) concentric Orowan loops around overaged / in Nimonic 105. From Reppich et al. (1982) and Reppich (1989).
^<'\ .•»
(d)
336
7 Particle Strengthening d, = 2rs
The numerical factors, A± and A2, are listed in Table 7-1. Ardell and coworkers (1976) extended the RSBH model to higher volume fractions. In Eq. (7-49) lx is, roughly speaking, replaced by the interparticle spacing {I i—2 rs) reflecting that the dislocation can bend outwards only between the precipitates. Further, the variation of TL due to the change of the dislocation character during the bending-out of the initially straight segments is included. Combined Theories
o Figure 7-12. A pair of dislocations shearing coherent ordered particles. Schematic illustration of the modelled dislocation configuration in the case of (a) weak pair-coupling, and (b) strong pair-coupling.
(a) Underaged Small Particles (Weak Pair-Coupling)
First Haasen and Labusch (1979), and later Reppich (1982) and Nembach and Neite (1985) applied the SL theory to order hardening. It could be demonstrated that the criterion of Eq. (7-15) does not hold. Rather the Labusch length of Eq. (7-19) instead of the Friedel length of Eq. (7-2) has to be taken, reflecting the extended range of the operating APB interaction. Indeed, the approaches discussed agree that:
Conventional APB Theories Raynor and Silcock (1970) and Brown and Ham (1971) (referred to as RSBH in the following) considered the classical case depicted in Fig. 7-12 a. The dislocations of the pair are loosely coupled and lie in different particles. Dislocation 2 is nearly straight; one has simply d2/l2 = / (Eq. (7-9)). Slight bending and the particle spacing along dislocation 1 are determined by the Friedel condition, Eq. (7-2), i.e., RSBH use: dl=2rs
d2/l2 = f
Substitution into Eq. (7-49) gives immediately the cutting stress: /7 ^ r
RSBH
However, the obvious dissent in the different combined models is the treatment of the force that the precipitates exert on dislocation 2. On the other hand, the influence of the second, trailing dislocation on the CRSS via l2 and d2 is profound. Haasen and Labusch (1979) assumed that the term d2/l2 in Eq. (7-49) is small compared to the term dx/ll9 and can therefore be ignored because the first dislocation interacts with many more precipitates than the second one. This situation is depicted in Fig. 7-12 a where the dislocation 2 (dashed line) avoids all the particles, i.e., d2 vanishes. Nembach and Neite (1985) adapted the straight shape of the second dislocation from the RSBH model (Fig. 7-12 a). The
337
7.2 Yielding at Low Temperatures Table 7-1. Parameters Al9 A2 and A3 of Eq. (7-54) using
CSL
= 1 and £ = 2 (A2 values in brackets according to
Nembach and Neite, 1985, using CSL = 0.82 and f = 1). Model
A
A2
A, formula
value
formula
value
-1.00
RSBH (1970/71) Eq. (7-50)
0.98
Haasen & Labusch (1979)
0.90
Nembach & Neite (1985)
0.90
0.55 0.55
- 1
1.10 (0.45) 0.111(-0.55)
Eq. (7-54)
0.90
0.55
-0.94
0.17
authors argued that only lx, but not l2 has to be modified in terms of the SL theory and set d2/l2 = f (Eq. (7-9)). Both models lead again to a relationship of the RSBH type, Eq. (7-50), with the numerical factors Ax and A2 given in Table 7-1. We suggest, however, that the procedure of Haasen and Labusch and of Nembach and Neite seems rather inadequate. First and quite generally - we may claim that even the particle spacing along dislocation 2, l2, is strictly governed by the operating Labusch interaction too. The inadequacy is accentuated by the direct in-situ HVEM observations of Nembach and Neite (Fig. 7-1 la) which documents in a convincing manner that it is no longer justified to assume the second dislocation straightens out, and that the particles thus acquire the possible maximum length, djf. Typically, the trailing, second dislocations follow the overall curvature of the leading dislocations, and bend more or less slightly forward between the particles encountered. In terms of the Friedel or Labusch criterion (Eqs. (7-2) and (7-19)), this is equivalent to a substantially smaller spacing l2 compared to the maximum value, d2/f. As a consequence, and employing a somewhat simplified way of specifying the reduction of l2 by Labusch statistics, we obtain: (7-51)
-0.51
-1
Now we have the same coefficients in front of the ratios d1j\1 and d2/l2 in both, Eq. (7-48 a) and (7-48 b). This leads to the type of relationship already proposed by Reppich (1982) (Eq. (7-17): (7-52)
[0.94(1
where TRSBH stands for the RSBH result given by Eq. (7-50). With the new hardening parameter, rjSL in Eq. (7-21), and substituting Fm from Eq. (7-47 b), rjSL now becomes: 1/2
1/2
(7-53) To sum up, one recognizes that, evidently, all of the resulting CRSS estimates of the different models exhibit a similiar structure, which can be expressed in the following final form: 1/2 T
APB ~ ^ I
(7-54)
Note that Eq. (7-54), which is the counterpart of Eq. (7-52), is composed of three terms in which / appears with increasing exponents. The parameters Au A2 and A3 have been compiled in Table 7-1. In practical cases, however, Eq. (7-52), together with Eq. (7-50) and (7-56 a), are
338
7 Particle Strengthening
relatively easy to handle and makes the SL correction more transparent. Therefore, let us now return to the new SL parameter rjSL (which enters Eq. (7-54) also via £ in A2 and A3). rjSL seems obviously to be a source of some confusion in the literature (Ardell, 1985). According to Eq. (7-15), rjSL is a function of the true particle interaction range YSL. In order to specify 7SL, we have to strictly correlate the actual shape of the force-distance profile with the real particle geometry. The situation is the analogue to that depicted in Fig. 7-9 a, b for stacking fault strengthening, with s replaced by A. Evidently, symmetry and range of the resulting force profile are significantly modified due to pairwise dislocation motion. In particular, 7SL depends upon the degree of pair-coupling and obeys the condition: A
(7-55)
The separation A between weakly coupled pairs, shearing underaged particles is always larger than the particle size in the glide plane, i.e., A>2rs. The resulting force profile is asymmetrical as in Fig. 7-9 b, left, and Fy in Fig. 7-6. It is exactly limited by the particle's surface, and therefore T^L attains its largest value 1^L = 2 rs; i.e., £ = 2 (which has been used for determining the parameters A2 and A3 in Tab. 7-1). Substitution of this result into Eq. (7-53) leads to 5 : 1/2
i.e., if the condition Fm ~ 2 TL is fulfilled. In this case, pair-coupling is extremely strong. Figure 7-9 a, b, right, shows that in this case, YSL approaches the lowest value, YSL ~ rs, or = 1. Insertion into Eq. (7-21) gives: (7-56b) which is very close to the earlier estimate of Reppich (1982). (b) Peak-Aged Particles (Medium Pair-Coupling) With coarser (stronger) dispersions, the pair-coupling becomes stronger so that the distance of the paired dislocations is smaller. But the dislocations may still lie in different particles (Fig. 1-12 a). The trailing dislocation 2 remains nearly straight, and again l2 is given by Eq. (7-51). However, with substantially increasing specific particle strength, FJ(2 TL), the leading dislocation 1 bends more foreward and aquires a nearly half-circular shape. Thus it may be no longer appropriate to use the Labusch spacing ZSL. For Fm ~ 2 TL, Eq. (7-19) is reduced to its minimum value 6 : L h =
0.94(1+CSL^n)
(? 57)
"
where ^ is given by Eq. (7-56 b). Substitution into Eq. (7-49) yields: "-PEAK
(7-56 a) (7-58) According to Eq. (7-21), rjSL has its lowest value for high specific particle strengths, 5 For comparison, earlier estimates yielded the following numerical factors in Eq. (7-56 a): 1.27 (Haasen and Labusch, 1979, using 7SL = (7i/1.4)rs); 2.1 (Grohlich etal, 1982); 0.89 (Reppich, 1982 and Nembach and Neite, 1985; using 7SL = rs).
6 For strong pair-coupling, the force profile is symmetrical like in Fig. 7-9 b, right. Consequently, the respective interpolation formulas Eqs. (7-16) and (7-18), should hold instead of Eqs. (7-17) and (7-19). However, for stronger obstacles, r\ < 0.4, the shape of the force profile does not influence the SL correction (see Fig. 7-5). Therefore, formulas (7-17) and (7-19) for energy-storing obstacles will simply be applied, in analogy to the case of weak pair-coupling.
7.2 Yielding at Low Temperatures
This equation is equivalent to the corresponding formula of Brown and Ham (1971) except for the SL term, 0.94 [1 + (C SL / 1/2 )/2]. One recognizes that the peak stress due to pairwise particle cutting typically depends on yAPB and / but not on r and TL. (c) Overaged Particles (Strong Pair-Coupling) When the ordered particles become large, the coupling of the dislocations within the pair would be particularly strong and both dislocations might interact with the same particle as documented in the micrograph, Fig. 7-10 b. The Orowan process is supposed not to operate because it may require higher stresses than pairwise shearing (see next section). This case depicted in Fig. 7-12 b has been analyzed by Htither and Reppich (1978). The situation is quite similiar to the one described for peak-aged particles above. Again, dislocation 1 bends outward strongly, and /x can be approximated by Eq. (7-57). Since dislocation 2 is in contact with just the same particle as dislocation 1, l2 also approximately equals the "modified" L (Eq. (7-57)), so we have: L Hiither and Reppich (1978) derived the parameters dx and d2 in Eq. (7-49) from the balance of the forces acting on the pair. They showed that the maximum force on the whole pair, which results from the superposition of the force profiles of the individual dislocations, reaches its maximum, Fm = yApB^i(^X when the second trailing dislocation is just touching the particle. In this critical position, dislocation 1 may attain the length d1(A) while d2 = 0, Fig. 7-12 b. In order to find t, one has to specify d t (A) as a function of the separation A of
339
the strongly coupled pair inside the particle from which the leading dislocation 1 is breaking free. The physical problem is again the analogue to stacking fault strengthening treated in the previous section. Therefore, we expect that d1(A) will fall below its maximum value 2 rs in the same manner as d (s) in Fig. 7-9a, right. Consequently, Fm(dt) will now increase substantially less than linearly with r, as described by Eq. (7-47 b), so that a similiar l/r 1/2 dependence of the CRSS should result, as predicted in Eq. (7-42). The resulting cutting-stress according to Hiither and Reppich (index "HR") has the form:
(7-59) The constant w accounts for the elastic repulsion of the dislocations within the precipitate and is essentially an adjustable parameter which will be determined by fitting the experimental data (see Sec. 7.2.2.5). Equation (7-59) can be simplified for large particles r > [(4/TC2) W TL]/yAPB to: (7-60)
The CRSS according to Eq. (7-59) has its maximum at: r
ws
= [ ~ J IWI
(7-61)
rws indicates the transition from "weak" to "strong" pair-coupling (index "w s") at the peak stress: (7-62) Note that the term 0.94 [1 + (C SL / 1/2 )/2] allowing for the SL correction is missing in the original formulas of Hiither and Reppich (1978).
340
7 Particle Strengthening
Evaluation and Discussion of Experimental Data Currently there is a large and still growing number of studies in the literature dealing with age-hardening by monodispersed, coherent, ordered particles. Tables 7-2 and 7-3, taken from the excellent review article of Ardell (1985), give a survey of representative Ni-, Fe-, and Co-rich alloys strengthened by Ni3(Al/Ti)-type y' precipitates as well as of alloys strengthened by other precipitates (including non-nickel-containing LI 2 precipitates as well as a ceramic model system). These two tables identify those alloys for which reliable data exist on yield strength or CRSS as a function of ageing time and particle parameters such as r and / Besides their technical importance, commercial y'-precipitating superalloys (see Chap. 14, Vol. 7 of this Series) represent particularly interesting model systems because the models which serve as basis for order-hardening theories approximate their particle microstructure. The Ni3(Al/Ti) phase is long-range ordered, having f.c.c. L l 2 structure and it is completely coherent. Thus, the contributions of modulus and coherency strengthening to the CRSS may be neglected. Reppich et al. (1982) analyzed the room temperature yielding of the commercial Ni-based wrought alloys Nimonic PE16 and Nimonic 105 in detail. After suitable one-stage isothermal ageing in Nimonic PE16, the /-phase precipitates are monodispersed, randomly spaced, spherical particles. In Nimonic 105, the shape of the y' precipitates is pherical at small sizes also, but turns cuboidal after long-term ageing and tends to be aligned along [100]. Adequate separation of the solid-solution matrix CRSS yields the experimental CRSS increment due to the y'-particle strength-
Table 7-2. Nickel-, Iron-, and Cobalt-base alloys, strengthened by y' precipitates, for which data are available on yield strength or CRSS as a function of ageing time and/or particle size (m = monocrystalline, p = polycrystalline) after Ardell (1985). Alloy
Sample type
Reference
NiAl
m
P
Ardell etal. (1976) Munjal and Ardell (1975) Travina and Nosova (1970) Phillips (1966)
Ni-Ti
P
Dawance et al. (1964)
Co, Ni, Cr-Ti
P
Chaturvedi and Lloyd (1976)
Fe, Ni, Cr-Ti
P
Singhal and Martin (1968)
Ni, Cr-Ti, Al (Nimonic 80 A)a
P
Melander and Persson (1978)
Ni, Cr, Mo-Ti, Al
P
Castagne et al. (1968)
Fe, Ni, Cr-Ti, Al
P
Raynor and Silcock (1970)
Ni, Cr, Co, Mo-Ti, Al (Nimonic 105)a
p
Castagne (1966) Pineau (1969) Reppich et al. (1982)
Ni, Fe, Cr, Mo-Ti, Al (Nimonic PE-16)a
m
Nembach and Neite (1985) Raynor and Silcock (1970) Reppich etal. (1982)
P
Fe, Ni, Cr, Mn, Mo-Ti, Al (A-286)
P
Raynor and Silcock (1970) Thompson and Brooks (1982)
a
NIMONIC is a trademark of the INCO family of companies.
ening alone, AT, represented in Figs. 7-13 and 7-14 as a function of the mean y' radius, r. In fact, the course of the data points exhibits the functional dependencies predicted by the "conventional" theories, namely the parabolic increase with underaged dispersions in the RSBH regime,
7.2 Yielding at Low Temperatures
341
Table 7-3. Alloys strengthened by ordered precipitates that differ from y', and non-Ni-base binary alloys strengthened by L l 2 type precipitates for which data exist on yield strength or CRSS as a function of ageing time and/or particle size (m = monocrystalline, p = polycrystalline) after Ardell (1985). Alloy
Sample type
Precipitates
Structure
Reference
Al-Li Pb-Na Ni-Mo Cu-Ti MgO-Fe 2 O 3 Co, Ni, Fe, Cr-Nb Fe-Si, Ti
P m P m m P P
Al3Li(5') Pb 3 Na Ni4Mo(p) Cu4Ti(p') MgFe 2 O 4 Ni3Nb(y") Fe2TiSi
Ll2 Ll2
Fe-Ni, Al, Ti Fe, Cr-Ni, Al
P P
Ni(AlTi) NiAl
Noble et al. (1982) Rembges et al. (1976) Goodrum et al. (1977) Greggi and Soffa (1979) Huther and Reppich (1979) Chaturvedi and Chung (1981) Brown and Whiteman (1969) Jack and Honeycombe (1972) Smith and White (1976) Taillard and Pineau (1982)
500
V O D A • X
Dla Dla
spinel Do 22 L2X
B2 B2
f = 12.9% f = 10.8% f= 7.8% f= 7.1% f= 4.2% f~ single crystals
YAPB= 0.125
Jm"2
w= 1.2 CSL=3 25
50 75 100 125 Mean particle radius r (nm) —*•
150
170
500
100 200 300 Mean particle radius r (nm) —*•
400
Figures 7-13 and 7-14. CRSS increase due to y' particles as a function of the mean particle-radius. Experimental data associated with theoretical curves, both normalized with respect to volume fraction / Figure 7-13: Nimonic PE16; Fig. 7-14: Nimonic 105. Thin broken lines: theoretical values according to conventional APB hardening theories. Thick full lines: theoretical values including the theory of Schwarz and Labusch. Curve 1: cutting stress for weakly coupled dislocation pairs (Eqs. (7-50), (7-52)). Curve 2: cutting stress for strongly coupled dislocation pairs (Eq. (7-59)). Curve 3: Orowan stress (Eq. (7-14)). After Reppich et al. (1982).
342
7 Particle Strengthening
Eq. (7-50), which is followed by a distinct maximum, and then a gradual hyperbolic decrease for overaged particles in the HR regime, Eq. (7-59) excluding the SL term However, one recognizes at a glance that the theoretical values (thin broken lines) underestimate the experimental CRSS quite considerably over the entire range investigated. A detailed analysis revealed further substantial uncertainties (Reppich etal., 1982; Ardell, 1985; Nembach and Neite, 1985). For instance, the theoretical peak stress does not vary for different / as experimentally observed. The application of the SL criterion, Eq. (7-15), using Eq. (7-56 a) 0.1 < 7eT = 1.22
f—)
1/2
<0.96
rj shows that, e.g.,\7APB for Nimonic PE16, rjSL clearly does not lie within the FF regime, where rjSL<^4. Therefore the SL-modified model should be applied (Eqs. (7-52) or (7-54)). The SL analysis proceeds in the following fashion. For a reinterpretation of the data with respect to Eq. (7-52), the data for the underaged material are plotted in Fig. 7-15 as normalized critical stress A *ARSBH VS - particle depth rjSL, where T RSBH and rjSL were calculated with formulas (7-50) and (7-56a), respectively7. Equation (7-52) postulates a straight line of slope 0.94 CSL and an intercept of 0.94. Up to rjSL ~ 0.5, data points of Nimonic PE16 in Fig. 7-15 for different / indeed follow the same straight lines. For polycrystalline Nimonic PE16, the slope is CSL = 3, and for monocrystalline Nimonic PE 16 of Martens and Nembach (1975) the slope is CSL = 2.5. For Nimonic 105 CSL = 1.9. These esti7 Note that the »ySL value in Fig. 7-29 has been calculated by Reppich et al. (1982) using a relation slightly different from Eq. (7-56 a), in which the numerical factor was 0.89 instead of 1.22.
Nimonic PE 16:
f=10.8% f = 12.9%
37APB
= 0.125^2
/ oV/7 ^
•
-
0.94 * =
\"\~~~~ Jslope
2/3
n min
'SL
0.185 °^ 0.203 0.2
0.4
0.6
0.8
Figure 7-15. Normalized CRSS of underaged Nimonic PE 6 vs. reduced particle depth rjSL represented according to Eqs. (7-52) and (7-56 a). Data adopted from Fig. 7-13.
mates are distinctly larger than CSL = 2/3 for FF hardening (dashed line in Fig. 7-15) also exceeding considerably the theoretical minimum value, CSL = 1, for SL hardening. Hence it is obvious that estimates of x according to Eq. (7-54) (with A2 and A3 from Table 7-1, which are based on minimum values, CSL = 2/3 or CSL = 1, respectively), are too low. Figures 7-13 and 7-14 show the result of the correct data fitting procedure. The full curves 1 represent the theoretical CRSS for weakly coupled dislocation pairs in the RSBH regime calculated with Eqs. (7-52) and (7-56 a). The shaded curve bands 2 represent the CRSS for strongly coupled pairs according to the HR relationship (Eq. (7-59)). The values used for yAPB and for the parameters w and CSL are listed in the legends. In proof of the relevance of the SL correction we can state that the corrected CRSS (full lines) agree with the experimental data points satisfactorily and exceed the "conventional" FF
7.2 Yielding at Low Temperatures
curves (thin broken lines) significantly. Experimental data points are fairly well met in the parabolic increase, in the ageing peak, and at the beginning of the hyperbolic decrease. But with increasing r, the data points for overaged material tend to deviate slightly from the calculated CRSS for strongly coupled dislocation movement towards the higher theoretical Orowan stress. The Orowan stress is taken as the stress for screw dislocations defined by Eq. (7-14) and plotted in Figs. 7-13 and 7-14 as curve bands 3, which account for different volume fractions. Reppich et al. (1982) neglected the factor 0.9 because heavily overaged y' precipitates are no longer randomly distributed. Following the model of Hxither and Reppich (1978), the type of dislocation-particle interaction will change above the critical radius rc from pairwise shearing to Orowan looping. rc results from the intersection of the cutting stress (curves 2) with the Orowan stress (curves 3). As a consequence of varying /, and of the y' sizedistribution there is a wide range below and around rc where the data points leave the CRSS for strongly coupled pairs and finally coincide with the Orowan curves 3. Obviously, in slightly overaged material both mechanisms occur simultaneously. For heavily overaged spherical dispersions, r > rc on the other hand, the Orowan process may be favorized alone. This behavior predicted by the model of Hiither and Reppich (1978) has been impressively verified by direct TEM observations, Figs. (7-llb,d). For more complex particle morphology, especially for octahedrally shaped, irregular-globular and dendritic precipitates Hiither and Reppich extended their model appropriately and applied it to the ceramic model-system MgO-Fe 2 O 3 (Hiither and Reppich, 1979). The authors demonstrated
343
that the curvature of the particle-matrix interface is an essential feature for the condition under which coherent ordered precipitates are sheared by paired dislocations or bypassed in the Orowan process by non-coupled single dislocations. 7.2.2.6 Summary
It could be shown that in all circumstances the CRSS derived within the FF point-obstacle approximation has to be modified in terms of the theory of Schwarz and Labusch (1978). The correction term is proportional to the new variable rjSL, the reduced particle depth. This is the most important consequence of the foregoing sections. Table 7-4 gives a compilation of the "best" formulas for rjSL and for the CRSS according to the various dislocation-particle interaction mechanisms which may be used in evaluating experimental data. 7.2.3 Duplex-Particle Strengthening
For technical applications, commercial two-phase alloy systems often experience a multistage thermal pretreatment in order to optimize, or, at least, to improve their mechanical behavior. Under such conditions, however, the precipitated particles are never monodispersed. For example, isothermal double-ageing leads to a particle microstructure which, under specific circumstances, exhibits a bimodal particle size distribution as shown in Fig. 7-16. The resulting increase in the CRSS due to the presence of two distinct coexisting classes of particles (finely dispersed particles and coarsely dispersed ones), termed "duplexparticle hardening" (Reppich et al., 1986 b) will be considered in this section.
Table 7-4. Equations and parameters to analyze particle strengthening data (L = [2n/(3 f)]1/2r, CSL > 1). Type of dislocation-particle interaction
Strengthening Maximum mechanisms interaction force, Fm
AY'2 3 ' 2Tj
-1i2
= — I —— I
CRSS interpolation formula
\bL 1/2
elastic (energy conserving) i! Force
Modulus
AG b2
0.055 AG3/2
1.09
VAG/G CD
2
^
Coherency
rY
{4Gerb)xP1l-)
CO
1/2
12
afGe3l2[^ir
0.54
=
TFFX0.94(1+2.5);SL)1'3
3 CD 3
energy storing
Chemical
2 ys b
(6b
Stacking fault
( - I AySF r
0.96 &yl>\ „
vl/2
f\"
Force
1/2
Underaged
1.22
u2
1^0.96^)
-/
AySFr
12.2
^
Order Overaged
0.7
w TLfyAPB b2r
1/2
0.54/
1/2
7.2 Yielding at Low Temperatures
50 30 10 0.4 n
I r2=5.5 nm
1
r^=75 nm
11
i j i
25
50
75
100
50
100
100
200
A;=156 nm (c)
50 150 250 Particle radius (nm)
7.2.3.1 Theory: Mixtures of Particles The current literature proposes the following theoretical suggestions for the manner in which obstacles of different strengths, concentrations and spatial distributions combine to produce the total CRSS (Brown and Ham, 1971; Lilholt, 1983): Randomly Intermixed Distribution of Obstacles The first and most common assumption is simple linear addition: T= T1+T2
345
(7-63 a)
where TX and T 2 represent the individual critical resolved shear stress contributions of class 1 particles and class 2 particles, respectively. Koppenaal and Kuhlmann-Wilsdorf (1964) proposed the so-called Pythagorean
Figure 7-16. y' duplex microstructure in Nimonic PE16 after double ageing. TEM dark field micrographs are shown on the left-hand side, and y' sizedistribution histograms on the right-hand side. The mean radii, r1 and r2 are indicated (Reppich et al., 1986 b).
superposition: (7-63 b) Louat (1979) recognized that the theory of Hanson and Morris (1975 b) offers a physical foundation for Eq. (7-63 b). However, it must be emphasized that Eqs. (7-63 a, b) are developed within the framework of the Fleischer-Friedel pointobstacle approximation. If concentrated real-life metallurgical obstacles with a more extended range of interactions such as second-phase particles are involved, the critical resolved shear stress increment due to these obstacles should be described in terms of Labusch's (1970) theory. The rule for the additive particle-hardening treated here becomes, in Labusch's formulation: T 3/2
=
T 3/2
(7-63 c)
which should hold well with increasing volume fractions and increasingly better
346
7 Particle Strengthening
the weaker and more finely dispersed the two particle populations are. An appropriate ad hoc generalization of Eq.(7-67) is (Reppich et al, 1983; Neite etal, 1983):
mixing rule used as a weighted averaging of the glide resistances according to the areal fractions Vt and V2 of the slip planes in which they apply: T
T«
= x\ + x\
(7-64)
where the superposition exponent a may be used as an adjustable parameter in the analysis of experimental data (see below). For certain special cases, Brown and Ham (1971) introduced a generalized "rule of mixing" of the type: x = c\l2x1 + cy2x2
(7-65)
which works as a weighted averaging according to the relative obstacle concentrations cx and c 2 . Brown and Ham (1971) compared the previous superposition rules with the computer-simulation results of Foreman and Makin (1967). They concluded that linear superposition (Eq.(7-63a)) is a poor approximation, and can be expected only if few strong obstacles are introduced among many weak obstacles, such as incoherent non-shearable precipitates or dispersoids in a concentrated solid solution, usually referred to as dispersion strengthening. In this case, Eq. (7-63 a) describes the experimental results rather well (Ebeling and Ashby, 1966; Hirsch and Humphreys, 1970). Pythagorean addition has more general validity, except for the special circumstances mentioned above, and is quite exact when different obstacles have the same strengths. Therefore, Lilholt (1983) proposed to use Eq. (7-63 b) in all practical cases. The mixing rule in Eq. (7-65) is almost as good as Pythagorean addition, except for obstacles of very different strengths. Regionally Distributed Obstacles For obstacles localized in well-defined regions, Kocks (1979) discussed a special
=F1T
1
+ F2T2
(7-66)
Equation (7-66) implies, however, that the specimens may be considered to consist of two "phases" of flow stresses xx and T 2 , and that the actual flow stress in each phase equals the glide resistance there. Therefore, we expect the mixing rule in Eq. (7-66) to apply for materials which attain the typical characteristics of a "compound", or, eventually, for second-phase particles arranged in more or less localized groups. 7.2.3.2 Experimental Results
Remarkably, in contrast to the well-investigated monodispersed particle strengthening described in Sect. 7.2.2., there have been only very few relevant experimental studies addressing the specific question which superposition law describes strengthening by two coexisting classes of particles in real crystals. The findings were rather controversial. Chellman and Ardell (1976) interpreted experimental data of binary NiAl single crystals containing yf precipitates, using a modified linear superposition rule but replacing z± in Eq. (7-63 a) by half the measured value. Nembach and Chow (1978) and Nembach and Neite (1985) analyzed the superalloy Nimonic PE16 in peak-aged condition with bimodal size distribution of shearable y' precipitates and concluded the Pythagorean addition (7-63 b) holds. Reppich et al. (1986 b) investigated "duplex" y'-particle hardening in polycrystalline Nimonic PE16 more systematically. In contrast to the earlier studies cited above, each of the three CRSS values in Eqs. (7-63) to (7-66), x as well as xx and T 2 , were derived directly from mea-
7.2 Yielding at Low Temperatures
sured yield stresses. The required individual CRSS contributions of the two y' populations, Tx(rx,f1) and T 2 (r r ,/ 2 ), could be estimated separately and accurately from the experimental yield stress of suitable one-stage-aged specimens containing unimodal size distributions of y' particles within the same range of radii (rl9r2) and volume fractions (fl9f2) as those contained in the corresponding bimodal material. Hence, no questionable assumptions concerning their dependence upon r and / were necessary. As usual, the measurement of the total CRSS % was straightforward with suitable isothermally double-aged specimens containing the two particle classes simultaneously (Fig. 7-16). The detailed analysis of the experimental data reveals the following trends: - Evidently, none of the addition rules, Eqs. (7-63) to (7-66), describe the particle superposition satisfactorily over the entire range investigated. - If two classes of widely differing mixtures of impenetrable and shearable y' particles exist as randomly intermixed distributions (Fig. 7-16 a, b), then the superposition of their individual CRSS increments, t1 and T 2 , to the measured total CRSS T can be empirically described by Eq. (7-64). The superposition exponent a is not constant but varies with increasing particle radius r2 of the weaker, finely-dispersed y' population from a = 1 to a — 2, in remarkable agreement with the computer simulation results of Foreman and Makin (1967). However, it must be emphasized that, this time, there is no theoretical foundation for the observed fractional exponents. - For regionally distributed y' particles, forming a spatially size-modulated particle microstructure (Fig. 7-16 c), the experimental data exhibit a clear tendency
347
to follow a rule of mixing, Eq. (7-66); V1 and V2 are the areal fractions of the welldefined regions which contain predominantly one of the two classes of y' precipitates only. The situation is, however, rather more complex with the ODS superalloy MA 6000. This alloy receives its extraordinary strength from massive volume fractions of coarse y' precipitates (fy, = 57%) and finely dispersed oxide particles (/o = 3%), respectively, and can be considered as a "compound" (Fig. 7-17). The room temperature yield stress seems to be described best by the modified mixing rule (7-66). ^matrix appearing in Eq. (7-67) in Fig. 7-17 represents the dislocation glide resistance along the "dispersion-hardened NiCr matrix". ry/ represents the glide resistance along the / phase,which itself is hardened by the oxide dispersoids as well as by the Ti atoms in solid solution.
^ APB
Eq.(7-67) Dispersoid
Figure 7-17. Glide-resistance diagram of the ODS alloy MA 6000 leading to the "rule of mixing" behavior, Eq. (7-67).
348
7 Particle Strengthening
7.3 Yielding at High Temperatures (a)
7.3.1 The Threshold-Stress Concept
Stable second-phase particles introduce a yield strength or "threshold stress" t th for dislocation glide due to Friedel-cutting or Orowan bypassing. At low temperatures, no glide is possible at resolved stress x below Tth. The original concept of "true" threshold stress has been extended to high temperatures first by Brown and Ham (1971). Lund and Nix (1976) gave a first interpretation of the experimental threshold stress of dispersion hardened TD Nichrome as the Orowan stress, T OR . It is, on the other hand, a perplexing fundamental experimental observation that - at high temperatures - such a "true" threshold does not exist. Instead, the material definitively creeps even under the lowest stress applied (Ilschner, 1973; Wilshire and Evans, 1985). It is, therefore, also clear that with the phenomenon of creep deformation below the Orowan stress (or cutting stress in the case of coherent particles) this mechanism has lost its predominating microstructural significance. And thus, no peculiarities of creep deformation are to be expected at stresses of the order of the Orowan stress (or cutting stress). This is illustrated in Fig. 7-18, which shows the yield stress of oxide-dispersion strengthened (ODS) alloys containing non-shearable, hard particles as a function of temperature. At low temperatures a detailed analysis provides the following main results, (i) The matrix yield-stress and the increment of the yield stress due to the particles can be superimposed by linear addition (Eq. (7-63 a), (ii) The absolute amount of the measured yield-stress increments due to the disperoMATRIX\
0 2 ~ KPo.i h exactly with the Orowan stress calculated
RT 200 400 600 800 1000 1200
600 1000 1400 ~-200 0 200 Temperature (°C) — ^
Figure 7-18. Compressive 0.2% yield stress vs. temperature. Shaded: Orowan stress given as low-temperature yield-stress increment due to oxide dispersoids. (a) ODS Superalloy MA 754 (Reppich et al., 1986 a); (b) Pt-based ODS alloys (Reppich et al., 1990 a).
according to Eq. (7-14) with the particle parameters estimated by TEM (shaded bands in Fig. 7-18). This indicates that the Orowan process indeed controls dispersion strengthening in the ODS alloys in the low-temperature regime. This interpretation, however, does not hold at high temperatures. One recognizes in Fig. 7-18 that, above 800 °C (a) or above 1000 °C (b), respectively, the difference in the yield stress of the ODS alloys and the reference matrix alloys amounts roughly to one third or one half of the respective Orowan stress (shaded bands). This evidently suggests a change from glide-controlled to diffusional c//mfr-controlled overcoming of the oxide dispersoids (Brown and Ham, 1971; Blum and Reppich, 1985). This conclusion is supported and specified more precisely by recent results plotted in Fig. 7-19. Since yielding data from hightemperature tests conducted over many or-
7.3 Yielding at High Temperatures 100 80
required to realize the creep deformation rate s. In the light of dislocation motion, rth corresponds preferentially to the additive yield-flow stress contribution to overcome second-phase particles at given e (and temperature).
ODS Platinum, 1250 °C O 0.40%/ZrO2 V0.16%/ZrO,
60 40 A 20
I 0 <7th(MPa)
10'
10
349
10
10
7.3.2 The Climb Threshold
It is well-established that at high temperatures dislocations can undergo non-pla300 nar motion by climb, which allows the dislocation segment arrested at a particle to (b) bulge out of the slip plane and finally surmount the particle. Current theories of overcoming of particles by climb predict different values of the threshold stress Tth 1(T 10" 10 1 associated with this process, which depend Strain rate e (s" ) • sensitively on the assumptions and details Figure 7-19. Creep threshold stress ath due to partiof the model. Figure 7-20 shows the situacles vs. log strain-rate, (a) Pt-based ODS alloys; numbers next to the dotted area denote ath in units of the tion envisaged in various models. Blum Orowan stress, (b) y' precipitating Ni-based alloy Niand Reppich (1985) provided a simple monic 90. Data taken from Reppich et al. (1990 a, b) derivation of how dislocation climb leads and (1992). to a threshold stress for creep, and for the way in which it is influenced by modifications. ders of magnitude down to very low strain They considered a segment of an edge rates are not available, creep data are comdislocation located above the equator of a piled for ODS Pt-based alloys (a) and y'spherical particle (Fig. 7-21) and climbing hardened Nimonic 90 (b). r th is replaced by upwards. An increment of climb not only the engineering creep strength increment due to the particles,
7'-Nimonic90, 850°C Or = 20nm • Ar=110nrr
350
7 Particle Strengthening
(i) Local climb
Figure 7-20. Compilation by Blum and Reppich (1985) of models for dislocation climb over secondphase particles, (a) Brown and Ham (1971); (b) and (e) Shewfelt and Brown (1977), and Stevens and Flewitt (1981); (c) Lagneborg (1973); (d) and (e) HauBelt and Nix (1977); (f) Evans and Knowles (1980); (i) and (ii) Arzt and Ashby (1982).
(f) node
is due to the normal stress component an of the segment climbing over the area / c dz, where /c < /. This term is particularly important in the situation shown in Fig. 7-20 f where dy = 0. The fourth term accounts for the elastic interaction between the parts of the dislocation segment near the particle. Although relatively small, it may significantly influence the critical situation in the Climb direction
Slip direction
overcoming the particle, as shown in the case of the Orowan process (Bacon et al., 1973). Earlier climb models ignore the last two terms. Furthermore, it is assumed that the line tension is constant and equal to TL along the dislocation. Thus Eq. (7-68 a) can be simplified to: dE=TLdl-Tbldy
Climb occurs as long as dE/dy < 0. The yield stress, i.e., the critical stress allowing the particle to be overcome by climb (index c\ follows from the condition (d£/dj/)max = 0 as:
Glide plane xy
(a) Figure 7-21. (a) Climb of an edge dislocation over a spherical particle; (b) top view; see also Fig. 7-20.
(7-68 b)
bl J2
(7-69)
The important parameter R=
(7-70)
7.3 Yielding at High Temperatures
is called climb resistance (Arzt and Ashby, 1982). It describes the rate of increase of the line length / as the dislocation segment climbs over the particle. In the literature, TC has often been approximated by setting the length / of the gliding dislocation segment equal to the mean planar square lattice particle spacing L = [27i/(3/)] 1/2 r, Eq.(7-8). Then t c can be expressed in units of the classical Orowan stress, 2 TL/(b L), as: [2TJ(bL)]
2
K
}
However, the Friedel spacing /F for weak obstacles (Eq. (7-2)) is a more appropriate choice for /, because, in general, the spacing of particles along the dislocation line is larger than the mean planar spacing and depends on the force acting on the climbing segment. The mean maximum force Fm is given by the yield stress as TC b lF. Insertion into Eq. (7-2) and combination with (7-69) yields: [2 717(6L)]
(7-72)
For 0.5 R < 1 the value of TC calculated from Eq. (7-72) is less than that obtained from Eq. (7-71). Equations (7-71) and (7-72) characterize yielding under the condition that all particles must be overcome by climb. However, this condition is unrealistically strict as there is a statistical distribution of particle spacing, so that the dislocation can find "easy gates" through the particle structure. Statistical arguments are well established in the models of room temperature yielding (see Sec. 7.2.1.1). Arzt and Ashby (1982) used these arguments in their analysis of high temperature yielding. Blum and Reppich (1985) modified the approach of Arzt and Ashby (1982) by using the more appropriate Friedel approximation, Eq. (7-2),
351
and showed that Eq. (7-72) has to be replaced by: #3/2
4h,c
[2 717(6 L)]
(2 ^ 2 + * 3 ' 2 )
(7-73)
Figure 7-22 shows the threshold stress for climb-controlled overcoming of particles as a function of the climb resistance for the various approximations made according to Eqs. (7-71) to (7-73) and according to Arzt and Ashby (1982). One recognizes that the Friedel approximation for / is most important at low values of the climb resistance, whereas the influence of climb statistics becomes essential for R > 1. In other words, at very low values of i?, the dislocations overcome the particles individually. The treatment of Blum and Reppich (1985), which combines both modifications, yields the lowest values for the threshold. R = 2 is the maximum possible value; here the climb threshold reaches the Orowan stress in the classical treatment (with or without Friedel correction) as well as in the refined treatment. The reason is that the refinements lower both the climb threshold [from 1 to 0.5 in units of 2 TJ (bL)) and the Orowan stress [from 1 via Kock's value of 0.8, which accounts for the randomness of the particle array, to 0.53, which accounts for both randomness and the interaction term d£ el in Eq. (7-68 a), see Eq. (7-14)]. 7.3.3 Climb Models
The next problem is to estimate the climb resistance R. The various models shown in Fig. 7-20 a to f can be grouped into the two classes, local climb and general climb (Figs. 7-20 i and 7-20 ii). 7.3.3.1 Local Climb
Brown and Ham (1971) proposed a model later modified by Shewfelt and
352
7 Particle Strengthening
Brown (1977) and Stevens and Flewitt (1981) based on the assumption that the dislocation climbs only in the particle-matrix interface, while the dislocation segments between the particles remain in their slip plane. These models yielded constant R values depending only on the shape of the particles. For cuboidal particles, as considered by Brown and Ham (Fig. 7-20 a), R = yjl. A rough estimate of the average value of R in the case of a spherical particle of radius r can be obtained from the increase in line length of a dislocation segment arriving in the plane of the equator and climbing to the top of the particles with A//Aj/ = 2r(0.57i-l)/r = 1.2. A refined treatment (accounting for the statistics of a slip plane intersecting a particle (Shewfelt and Brown, 1977) and the specific particle strength (Dorn et al., 1969) yields a value of R, which is somewhat lower: R = 0.77 is a reasonable estimate (Arzt and Ashby, 1982). 7.3.3.2 General Climb
Lagneborg was the first to point out that the sharp bends in the dislocation line
where it leaves the particle interface (see Fig. 7-20 i) are not realistic. Rather, line tension will cause the dislocation to unravel from the particle interface by climb. General climb causes a considerable reduction of R compared to local climb. Moreover R is no longer constant but depends on the volume fraction with / 1 / 2 . The lowest value of R results if the dislocation has the zig-zag shape shown in Fig. 7-20 e. Then the line length increases by A/= /{[I - (2r//) 2 ] 1/2 - 1} ~ (2r)2/l as the dislocation moves forward by Ay = r. Thus the climb resistance can be approximated by R = 2r/l = (6n)1/2f1/2. For volume fractions between 1 and 10%, R varies from 0.14 to 0.45. Refinements of Arzt and Ashby accounting for the Shewfelt and Brown averaging procedure reduce R by a factor of 3 compared to the above estimates so that 0.047 < R < 0.14 for l % < / < 1 0 % , Fig. 7-22. Blum and Reppich (1985) used the model shown in Fig. 7-22 to convert R values into Tth values. The ranges of R for local and for general climb (1 % < / < 10%), respectively, are shown as shaded bands. The intersection of these bands with the respec-
0.05 0.14 0.45 0.77 1.2
General climb
I Local! 'climb i Orowan process 0 . 6 — Brown & Ham (1971)
).5s 0.32 — Shewfelt & Brown (1977) 0.28—Arzt & Ashby (1982) 0.19—-Blum & Reppich (1985) 0.07...0.03 - - Arzt & Ashby (1982) 0.02..0.004-- Blum & Reppich (1985) General climb
Figure 7-22. Threshold stress xth c for climb over particles as a function of climb resistance R (Blum and Reppich, 1985).
7.3 Yielding at High Temperatures
tive TthjC curves yields the corresponding threshold stresses. Each of the modifications for R and Tth c discussed above lowers the threshold stress (in units of 2 TL/(bL)) for local climb from the value 0.6 of Brown and Ham via R = 0.32 (Shewfelt and Brown) and R = 0.28 (Arzt and Ashby) to the value [2 717(6 L)]
- 0.2
derived by Blum and Reppich (1985). However, this is not a true threshold value as climb will become less and less localized when the stress is lowered. In Lagneborg's (1973) model, r th scales as the applied stress. General climb yields significantly lower R values. Consequently the threshold stress decreases by more than one order of magnitude to
353
traction causes the dislocation segment near the particle to maximize its length in the particle interface. Even dislocations whose slip plane lies above or below the particle are attracted and may end up in the interface after some climb and glide motion. First Nardone and Tien (1983) and later Schroder and Arzt (1985) observed that dislocations in the ODS superalloys MA 754 and MA 6000 were indeed pinned at the departure side of the oxide particles, as shown in Fig. 7-23, indicating that attractive interaction between dislocations and particle interfaces does exist. A threshold stress thus arises from the force necessary to unpin the dislocations from the interface. New results along similiar lines have been reported in a recent, more quantitative study by Herrick et al. (1988). What is apparent from the micrographs is
0.004 < tth/[2 TJ(bL)] < 0.02 for 1% < / < 1 0 % Although this is a true threshold value (unless diffusion creep by vacancy currents between grain boundaries is taken into account) it is extremely low compared to the Orowan stress, and therefore, well below the bulk of experimental data (compare Fig. 7-19). 7.3.4 Interfacial Pinning
Srolowitz et al. (1984) introduced an interesting new viewpoint that leads to higher values of the threshold stress. The authors suggested that the assumption of a slipping particle-matrix interface is more appropriate than that of an "adherent" interface on which all theories discussed so far are based. An essential result of the analysis of Srolowitz et al. is that there is an attractive interaction between dislocations and the interface of incoherent particles if the interface is free to slip. This at-
Figure 7-23. Interfacial pinning due to attractive interaction between incoherent particles and dislocations, (a) Schematic drawing illustrating the critical dislocation configuration in the moment of detachment (Blum and Reppich, 1985); (b) TEM micrograph of MA 6000 (Stiele, 1991).
354
7 Particle Strengthening
that the dislocations remain bound to the particle over which they have climbed. A first rough estimate of the dislocation detachment stress was given by Blum and Reppich (1985). The authors based their estimates on the idea that the physical origin of the attractive dislocation-particle interaction is that the line tension of the dislocation lying in the interface is lowered compared to the matrix value TL. In the limit, the dislocation line in the interface disappears. In other words: the dislocation line in the particle interface has a line energy of zero. In this case, the problem is not to create new dislocation length during climbing but to recreate a new dislocation segment of length s
(7-75)
The parameter k (0 < k < 1) can be thought of as the "relaxation factor". For k = 1 no attractive interaction exists. The resulting
detachment threshold (index "d") is [2TJ(bL)]
• = (1 - k 2 ) 1 1 2
(7-76)
Arzt and Wilkinson compared i d with the threshold for local climb, which the authors calculated to be i th = 0.4 k5/2 [2 TJ{b L)] (Fig. 7-24). For k = 1, r th it equals 0.4 [2 TL/(b L)], which is close to the estimates of Shewfelt and Brown (1977) but twice as large as the more appropriate value of Blum and Reppich (1985) (Fig. 7-22). However, the overall threshold stress for dislocation bypass (as a serial process consisting of the climb step and subsequent detachment) is simply the largest value of the two stresses depicted in Fig. 7-24. The critical k value below which dislocation bypass becomes detachment-controlled, is 0.94. In other words, only a small attractive interaction, corresponding to a relaxation of the dislocation line tension by about 6%, is necessary for interfacial pinning. The kinetics of combined general and local climb with a detachment threshold have been discussed by Rosier and Arzt (1988 a, b) assuming the dislocation shape in the vicinity of the incoherent particle is determined by the condition of constant
0.94
0.2 0.4 0.6 Relaxation Parameter k
0.8
1.0
Figure 7-24. Normalized threshold-stress for local climb and for detachment vs. relaxation parameter k (Arzt and Wilkinson, 1986).
7.4 References
chemical potential for vacancies along the climbing dislocation (Brown and Ham, 1971; Lagneborg, 1973). Their analysis shows: (i) The attractive interaction stabilizes local climb during certain parts of the climb process. Thus it is no longer necessary to postulate local climb below the Orowan stress a priori, (ii) The dislocation detachment may be thermally activated. Then dislocation creep well below the "athermal" threshold stress Td as given by Eq. (7-76) may be possible. The result of Rosier and Arzt (1990) is: -,2/3
feBTln
(7-77)
1 2
Gb r
(1-fc)
The reference strain rate is given as £0 = 3 DwLpg/b; Dy: volume diffusion coefficient, kB: Boltzman's constant, T: absolute temperature, Q: density of mobile dislocations. The obvious weak point of the approach of Arzt et al. is that the relaxation factor k was not modelled from first principles, but was introduced as an adjustable parameter only. The most recent fit of creep data of a dispersion-strengthened superalloy, two aluminium alloys and pore-strengthened tungsten yielded k values between 0.74 and 0.95 (Rosier and Arzt, 1990), which supports the above conclusion derived from Fig. 7-24. The results plotted in Fig. 7-19 a are qualitatively consistent with the predictions of Eq. (7-77) with respect to the functional dependencies upon e and the particle parameters. However, a detailed analysis reveals significant discrepancies between the experimental data points in Fig. 7-19 and the model of Rosier and Arzt (Heilmaier et al., 1992). Nevertheless, interfacial pinning seems to be a potential candidate for explaining the threshold behavior of alloy systems containing hard, non-penetrable particles.
355
7.4 References Ardell, A. X, Munjal, V., Chellman, D. J. (1976), Met. Trans. 7 A, 1263. Ardell, A. J. (1985), Met. Trans. 16 A, 2131. Arzt, E., Ashby, M. F. (1982), Scripta Met. 16, 1285. Arzt, E., Wilkinson, D. S. (1986), Acta Met. 34,1893. Ashby, M. (1969), in: Physics of Strength and Plasticity. Argon, A. S. (Ed.). Cambridge, USA: MIT Press, p. 113. Bacon, D. X, Kocks, U. R, Scattergood, R. O. (1973), Phil. Mag. 28, 1241. Blum, W, Reppich, B. (1985), in: Progress in Creep and Fracture, Vol. 3. Wilshire, B., Evans, R. W. (Eds.). Swansea: Pineridge Press, pp. 83-135. Brown, G. G., Whiteman, X A. (1969), J. Aust. Inst. Met. 14, 217. Brown, L. M., Ham, R. K. (1971), in: Strengthening Methods in Crystals. Kelly, A., Nicholson, R. B. (Eds.). London: Applied Science Publishers Ltd., pp. 9-135. Castagne, X L. (1966), J. Phys. (France) 27 C3, 233. Castagne, X L., Lecroisey, E, Pinea, A. (1968), C. R. Acad. ScL, Paris, 226, 510. Chaturvedi, M. C , Lloyd, D. X (1976), Met. Sci. J. 10, 373. Chaturvedi, M. C , Chung, D. W. (1981), Met. Trans. 12 A, 11. Chellman, D. X, Ardell, A. X (1976), in: Proc. 4th ICSMA, Vol. 1. Nancy: Laboratoire de Physique du Solid, Ecole Nationale Superieure de la Metallurgie et de l'lndustrie des Mines (Ed.). Nancy, pp. 219-223. Cockayne, D. X H., Jenkins, M. L., Ray, 1.1. (1971), Phil. Mag. 24, 1383. Dawance, M. M., Ben Israel, P. M., Fine, M. E. (1964), Acta Met. 12, 705. Dorn, I E . , Guyot, R., Stefansky, T. (1969), in: Physics of Strength and Plasticity. Argon, A. S. (Ed.). Cambridge, USA: MIT Press, p. 133. Ebeling, R., Ashby, M. F. (1966), Phil. Mag. 13, 805. Evans, H. E., Knowles, G. (1980), Met. ScL 14, 262. Fleischer, R. L., Hibbard, W. R. (1963), in: N.P.L. Symposium on the Relation between the Structure and Mechanical Properties of Metals. London: Her Majesty's Stationery Office, p. 261. Foreman, A. X E., Makin, M. X (1966), Phil. Mag. 14, 911. Foreman, A. X E., Makin, M. X (1967), Can. J. Phys. 45, 511. Foreman, A. X E., Hirsch, P. B., Humphreys, F. X (1970), in: Fundamental Aspects of Dislocation Theory, Vol. 2. Simmons, X A., de Witt, R., Bullough, R. (Eds.). Washington: N.B.S. pub. no. 317, p. 1083. Friedel, X (1956), in: Les Dislocations. Paris: GauthiersVillars, p. 205. Friedel, X (1964), in: Dislocations. Oxford: Pergamon Press, p. 225.
356
7 Particle Strengthening
Gerold, V., Haberkorn, H. (1966), Phys. Stat. Sol. 16, 675. Gerold, V. (1968), Acta Met. 16, 823. Gerold, V., Hartmann, K. (1968), Trans. Jpn. Inst. Met. 9, Supplement, 509. Gerold, V. (1974), Trans. Indian Inst. Met. 27, 317. Gerold, V. (1979), in: Dislocations in Solids, Vol. 4. Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland Publishing Company, pp. 219-260. Gerold, V., Pham, H. M. (1980), Z. Metallkd. 71, 286. Gleiter, H., Hornbogen, E. (1965 a), Phys. Stat. Sol. 12, 235. Gleiter, H., Hornbogen, E. (1965b), Phys. Stat. Sol. 12, 251. Gleiter, H. (1967 a), Acta Met. 15, 1213. Gleiter, H. (1967b), Acta Met. 15, 1223. Gleiter, H. (1967c), Z. Angew. Physik 23, 108. Goodrum, J. W, Le Fevre, B. G. (1977), Met. Trans. 8 A, 939. Greggi, X, Soffa, W. A. (1979), in: Proc. 5th ICSMA, Vol. 1. Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 651-656. Grohlich, M., Haasen, P., Frommeyer, G. (1982), Scripta Met. 16, 367-370. Haasen, P. (1977), Comtemp. Phys. 18, 373. Haasen, P., Labusch, R. (1979), in: Proc. 5th ICSMA, Vol. 1. Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 639-643. Hanson, K., Morris, J. W. (1975 a), J. Appl. Phys. 46, 983. Hanson, K., Morris, X W. (1975b), J. Appl. Phys. 46, 2378. Harkness, S. D., Hren, X X (1970), Met. Trans. 1, 43. HauBelt, J. H., Nix, W. D. (1977), Acta Met. 25,1491. Heilmaier, M., Schlegl, M., Reppich, B. (1992), Proc. Annu. Metg. Deutsche Gesellschaft f. Materialk., Hamburg, Germany: DGM e.V, p. 162. Herrick, R. S., Weertman, X R., Petkovic-Luton, R., Luton, M. X (1988), Scripta, Met. 22, 1879. Hirsch, P. B., Humphreys, F. X (1970), Proc. Roy. Soc. London, Ser. A 318, 45. Hirsch, P. B., Kelly, A. (1965), Phil. Mag. 12, 881. Hiither, W, Reppich, B. (1978), Z. Metallkd. 69, 628. Hiither, W, Reppich, B. (1979), Mater. Sci. Eng. 39, 247. Ibrahim, I. A., Ardell, A. X (1978), Mater. Sci. Eng. 36, 139. Ilschner, B. (1973), Hochtemperatur-Plastizitdt. Berlin: Springer. Jack, D. H., Honeycombe, W. K. (1972), Acta Met. 20, 787. Jansson, B., Melander, A. (1978), Scripta Met. 12, 497. Kelly, A., Fine, M. E. (1957), Acta Met. 5, 365. Kelly, A., Nicholson, R. B. (1963), Progr. Mater. Sci. 10, 151. Knoch, H., Reppich, B. (1975), Acta Met. 23, 1061. Knowles, G., Kelly, P. M. (1971), in: Effect of Second-Phase Particles on the Mechanical Properties of Steel. London: The Iron and Steel Institute, p. 9.
Kocks, U. F. (1966), Phil. Mag. 13, 541. Kocks, U. F (1967), Can. J. Phys. 45, 137. Kocks, U. F. (1977), Mater. Sci. Eng. 27, 291. Kocks, U. F. (1979), in: Proc. 5th ICSMA, Vol. 3. Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1661-1680. Koppenaal, T. X, Kuhlmann-Wilsdorf, D. (1964), /. Appl. Phys. Lett. 4, 59. Labusch, R. (1970), Phys. Stat. Sol. 41, 659. Lagneborg, R. (1973), Scripta Met. 7, 605. Lendvai, X, Gudladt, H. X, Wunderlich, W., Gerold, V (1989), Z. Metallkd. 80, 316. Lilholt, H. (1983), in: Proc. 4th Riso Int. Syrnp. on Metallurgy and Materials Science. Bilde-Sorenson, X B., Hansen, N., Horsewell, A., Leffers, X, Lilholt, H. (Eds.). Roskilde, Denmark: Riso National Laboratories, pp. 381-392. Livingston, X D. (1959), Trans. AIME 215, 566. Louat, N. (1979), in: Proc. 5th ICSMA, Vol.2. Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 941-946. Lund, R. W, Nix, W. D. (1976), Acta Met. 24, 469. Martens, V, Nembach, E. (1975), Acta Met. 23, 149. Melander, A., Persson, P. A. (1978), Acta Met. 26, 267. Morris, X W, Klahn, D. H. (1974), / Appl. Phys. 45, 2027. Munjal, V, Ardell, A. X (1975), Acta Met. 23, 513. Nabarro, F. R. N. (1972), J. Less Common Met. 28, 257. Nardone, V C , Tien, X K. (1983), Scripta Met. 17, 467. Neite, G., Sieve, M., Mrotzek, M., Nembach, E, (1983), in: Proc. 4th Riso Int. Symp. on Metallurgy and Materials Science. Bilde-Sorenson, X B., Hansen, N., Horsewell, A., Leffers, T, Lilholt, H. (Eds.). Roskilde, Denmark: Riso National Laboratories, p. 447-451. Nembach, E., Chow, C. (1978), Mater. Sci. Eng. 36, 271. Nembach, E. (1983), Phys. Stat. Sol. (A) 78, 571. Nembach, E. (1984), Scripta Met. 18, 105. Nembach, E., Neite, G. (1985), in: Progress in Materials Science, Vol. 29. Christian, X W, Haasen, P., Massalski, T. B. (Eds.). Oxford: Pergamon Press, pp. 177-319. Nembach, E., Suzuki, K., Ichihara, M., Takeuchi, S. (1988), Mater. Sci. Eng. A 101, 109. Noble, B., Harris, S. X, Dinsdale, K. (1982), Met. Sci. 16, 425. Orowan, E. (1948), in: Symp. on Internal Stresses in Metals and Alloys, Session III Discussion. London: Institute of Metals, p. 451. Phillips, V A. (1966), Acta Met. 14, 1533. Pineau, A., Lecroisey, F , Castagne, X L. (1969), Acta Met. 17, 905. Raynor, D., Silcock, X M. (1970), Met. Sci. 4, 121. Rembges, M., Haasen, P., Schulz, Z. (1976), Z. Metallkd. 67, 330. Reppich, B. (1975), Acta Met. 23, 1055.
7.4 References
Reppich, B. (1982), Acta Met, 30, 87. Reppich, B., Schepp, P., Wehner, G. (1982), Acta Met. 30, 95. Reppich, B., Kiihlein, W, Pillhofer, H. (1983), Proc. Annual Meeting Deutsche Gesellsch. f Metallkd. Erlangen, Germany: DGM e.V., p. 38. Reppich, B., Listl, W., Meyer, T. (1986a), in: Proc. Conf. High Temperature Alloys for Gas Turbines and Other Applications. Betz et al. (Eds.). Dordrecht, Belgium: D. Reidel Publ. Comp., pp. 10231035. Reppich, B., Kiihlein, W, Meyer, G., Puppel, D., Schumann, G. (1986b), Mater. Sci. Eng. 83, 45. Reppich, B. (1989), in: Festigkeit und Verformung bei hoher Temperatur. Schneider, K. (Ed.). Oberursel, Germany: DGM Informationsgesellschaft, pp. 139— 163. Reppich, B., Brungs, R, Hummer, G., Schmidt, H. (1990 a), in: Proc. 4th Int. Conf. Creep and Fracture of Eng. Mater, and Structures. Wilshire, B., Evans, R. W. (Eds.). London: The Institute of Metals, pp. 141-158. Reppich, B., Heilmaier, M., Liebig, K., Schumann, G., Stein, K. D., Woller, T. (1990b), Steel Res. 6, 251. Reppich, B., Driieke, K., Heilmaier, M., Schumann, G., Stein, K. D., Woller, T. (1992), in preparation. Rosier, X, Arzt, E. (1988 a), Acta Met. 36, 1043. Rosier, J., Arzt, E. (1988b), Acta Met. 36, 1053. Rosier, X, Arzt, E. (1990), Acta Met. 38, 671. Russell, K. C , Brown, L. M. (1972), Acta Met. 20, 969. Scattergood, R. O., Bacon, D. X (1975), Phil. Mag. 31, 179. Schroder, J . H , Arzt, E. (1985), Scripta Met. 19, 1129. Schwarz, R. B., Labusch, R. (1978), /. Appl Phys. 49, 5174. Shewfelt, R. S., Brown, L. M. (1977), Phil. Mag. 35, 945. Singhal, L. K., Martin, X W. (1968), Acta Met. 16, 947. Smith, I. O., White, M. G. (1976), Met. Trans. 7 A, 293.
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Srolowitz, D. X, Luton, M. X, Petkovic-Luton, R., Barnett, D. M., Nix, W D. (1984), Acta Met. 32, 1079. Stevens, R. A., Flewitt, P. E. X (1981), Acta Met. 29, 867. Stiele, H. (1991), Diploma Thesis, Univ. ErlangenNurnberg. Taillard, R., Pineau, A. (1982), Mater. Sci. Eng. 56, 219. Thompson, A. W, Brooks, X A. (1982), Acta Met. 30, 2197. Travina, N. T, Nosova, G.I. (1970), Phys. Met. Metallogr.29 (3) 119. Weeks, R. W, Pati, S. R., Ashby, M. R, Barrand, P. (1969), Acta Met. 17, 1403. Wilm, A. (1911), Metallurgie 8, 225. Wilshire, B., Evans, R. W (1990), Proc. 4th Int. Conf. Creep and Fracture of Eng. Mater and Structures. London: The Institute of Metals.
General Reading Ardell, A. X (1985), Met. Trans. 16 A, 2131. Blum, W, Reppich, B. (1985), in: Progress in Creep and Fracture, Vol.3. Wilshire, B., Evans, R. W (Eds.). Swansea: Pineridge Press, pp. 83-135. Brown, L. M., Ham, R. K. (1971), in: Strengthening Methods in Crystals. Kelly, A., Nicholson, R. B. (Eds.). London: Applied Science Publishers Ltd., pp. 9-135. Gerold, V. (1979), in: Dislocations in Solids, Vol. 4. Nabarro, R R. N. (Ed.). Amsterdam: North-Holland Publishing Company, pp. 219-260. Haasen, P. (1977), Comtemp. Phys. 18, 373. Kocks, U. P. (1977), Mater. Sci. Eng. 27, 291. Lilholt, H. (1983), in: Proc. 4th Riso Int. Symp. on Metallurgy and Materials Science. Bilde-Sorenson, X B., Hansen, N., Horsewell, A., Leffers, T, Lilholt, H. (Eds.). Roskilde, Denmark: Riso National Laboratories, pp. 381-392. Nembach, E., Neite, G. (1985), in: Progress in Materials Science, Vol. 29. Christian, X W, Haasen, P., Massalski, T. B. (Eds.). Oxford: Pergamon Press, pp. 177-319.
8 High-Temperature Deformation and Creep of Crystalline Solids Wolfgang Blum Institut fur Werkstoffwissenschaften, Lehrstuhl I, Universitat Erlangen-Niirnberg, Federal Republic of Germany
List of Symbols and Abbreviations 8.1 Introduction 8.2 General Considerations 8.2.1 Role of Lattice Defects 8.2.2 Evolution of Deformation Resistance 8.2.3 Evolution of Dislocation Structure 8.2.4 A Simple Model of Plastic Deformation 8.3 Pure Materials 8.3.1 Evolution of Deformation Resistance and Dislocation Structure 8.3.1.1 Deformation at Constant Strain-Rate 8.3.1.2 Deformation at Constant Stress 8.3.2 Steady-State Deformation 8.3.2.1 Equation of State 8.3.2.2 Dislocation Structure 8.3.3 Response to Change in Deformation Conditions 8.3.3.1 Single Change 8.3.3.2 Cyclic Changes 8.4 Solid Solutions 8.4.1 Evolution of Deformation Resistance and Dislocation Structure 8.4.1.1 Deformation at Constant Strain-Rate 8.4.1.2 Deformation at Constant Stress 8.4.2 Steady-State Deformation 8.4.2.1 Dislocation Structure 8.4.2.2 Equation of State 8.4.3 Response to Change in Deformation Conditions 8.5 Particle-Hardened Metals 8.5.1 Evolution of Deformation Resistance and Dislocation Structure 8.5.2 Steady-State Deformation 8.5.2.1 Dislocation Structure 8.5.2.2 Equation of State 8.5.3 Response to Change in Deformation Conditions 8.6 Ceramics 8.6.1 Evolution of Deformation Resistance 8.6.2 Steady-State Deformation 8.7 Modeling of Deformation 8.8 Acknowledgements 8.9 References Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
360 363 364 364 365 367 369 371 371 371 376 378 379 380 382 382 384 385 386 386 387 389 389 389 390 391 391 393 393 394 396 397 398 398 399 403 403
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8 High-Temperature Deformation and Creep of Crystalline Solids
List of Symbols and Abbreviations a ag Aa Aas Aah A, A b B cx to c 17 Cik dg dsl D Dv DGB E™ (...) /s /h fm / sub F F(...) G kB /crel /, /0 /b L+ m n,no,nmin p Pt q Q r R s S, So Sj1 S* S™ t U
width of hard regions (subgrain boundaries) width of glassy boundary phase operational activation area activation area in soft region activation area in hard region numerical constants magnitude of Burgers vector proportionality factor constants elastic constants grain size length of slip lines diffusion coefficient coefficient of bulk diffusion coefficient of grain boundary diffusion function describing evolution of S™ volume fraction of soft region volume fraction of hard region coupling factor between ef h and £m volume fraction of material containing subgrains factor of acceleration of creep by cyclic loading, force function describing kinetics of deformation elastic shear modulus at test temperature Boltzmann constant factor of relaxation of dislocation line energy at particles instantaneous and initial specimen length, respectively inverse of subgrain boundary dislocation length per subgrain boundary area slip distance of dislocations effective-stress exponent stress exponents stress exponent [Eq. (8-25)] material parameters stress exponent [Eq. (8-31)] activation energy radius of dispersed particles stress ratio in cyclic deformation spacing of dislocations in subgrain boundaries instantaneous and initial specimen cross section, respectively parameters of dislocation structure parameters of microstructure except dislocation structure microstructural parameters time lifetime
List of Symbols and Abbreviations
361
T Tm v ^mob vfs vfh vm V Va w
temperature melting point dislocation velocity velocity of mobile dislocations velocity of free dislocations in soft regions velocity of free dislocations in hard regions velocity of migrating subgrain boundaries volume apparent activation volume subgrain size
a
dislocation interaction constants relating dislocation spacings a n d athermal stress components shear strain stacking fault energy grain boundary width uniaxial (plastic, tensile or compressive) strain strain at end of life in creep plastic strain due to free dislocations local plastic strain due to free dislocations in soft regions local plastic strain due to free dislocations in hard regions strain at point of stress reduction plastic strain rate plastic strain rate due to free dislocations plastic strain rate due to migrating subgrain boundaries minimum creep rate average rate of cyclic creep expected average rate of cyclic creep creep rate before stress reduction viscosity of glassy boundary phase work-hardening rate at constant strain rate work-hardening rate at constant stress subgrain misorientation dislocation density mobile dislocation density total dislocation density dislocation density at point of minimum deformation resistance density of free dislocations inside subgrains density (length per total volume) of subgrain boundary dislocations uniaxial (tensile or compressive) normal stress effective stress effective stress in soft region thermal stress component due to dislocations effective stress due to solute drag critical effective stress for cloud formation
y ySF 5 8 8Y gf %s ef h 80 8 ef £m e min e cyc 8rcyc 80 rj 0 ^creep <9mis Q gmoh gtot @ extr Q{ gh a cr* Gf <j\ Gfol ^rit, i
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8 High-Temperature Deformation and Creep of Crystalline Solids
o
%\t,2 aG oG f (7G b crG p as <7 h cr0 x TG TG f $ Q
critical effective stress for break-away athermal stress athermal stress due to free dislocations athermal long-range back stress due to hard regions (subgrain boundaries) athermal stress due to particles local stress in soft region local stress in hard region stress before reduction resolved shear stress athermal shear stress component athermal shear stress due to free dislocations Schmid factor or reciprocal of Taylor factor atomic volume
CERT TEM
constant extension rate test transmission electron microscope
8.1 Introduction
8.1 Introduction Each engineering material has a limited lifespan of use as a component of a product. During its life it undergoes plastic deformation one or several times. This occurs in the course of forming operations where the component takes on the required shape, and it may also occur while the component is stressed during use. Ideally, a material should have a low resistance to forming and a high resistance to unwanted deformation during use. A huge amount of knowledge has been accumulated, mostly by trial and error, which allows the production of a component of a given shape with a high resistance to deformation and fracture. As materials are being modified continuously and as new materials with improved properties appear, the control of plastic deformation is a never ending task. It could be best fulfilled, if it were possible to calculate the plastic deformation of materials under stress on the basis of their microstructural properties. This requires the knowledge of the constitutive laws of plastic deformation describing the relation between the basic parameters of deformation at a given temperature, the material parameters, and the microstructure. The deformation parameters are the (uniaxial normal) stress a, the temperature T, the (true) plastic strain 8, and the plastic strain rate e = ds/dt (t: time). Relevant material parameters P{ are the melting point Tm, the elastic constants Cik, the coefficient of thermal expansion, the stacking fault energy ySF etc. The microstructure is characterized by a set of parameters which is of importance with respect to plastic deformation and fracture. As dislocations are the main carriers of plastic deformation, we differentiate between the parameters S/- of the dislocation structure and the remaining microstructural parameters S* describing
363
the grain structure, the phase structure etc. The constitutive laws have the general form (Mecking and Kocks, 1981; Frost and Ashby, 1982): (8-1)
ij,fc = 1,2,... ij,fc,/ = l , 2 , . . . ,
m = ±,R
(8-2)
Eq. (8-1) is called kinetic law. It relates the rate of plastic deformation s to stress a for a given material (Pf) with a certain microstructure (S/,Sf) at a given temperature. The tensorial character of stress, strain, and strain rate is neglected here 1 . We have also neglected the elastic strain. This is not a severe restriction: the total strain is obtained if the elastic strain is added to the plastic strain. The kinetic law, Eq. (8-1) must be supplemented by the structural evolution laws, Eq. (8-2), which tell how the microstructure changes with time. Due to the multitude of microstructural parameters, which may be of influence on plastic deformation, it is difficult to find a formulation of the constitutive laws which is both general and exact. The task is to simplify the laws to obtain an approximate description which is sufficiently exact within a restricted range of applications. Nevertheless, it appears to be of importance to establish a close connection between microstructure and macroscopic behaviour, because this is the only way to classify and structure the variety of material responses, in other words, to understand the material behaviour. It follows from the above that two types of tests have to be carried out. The first 1 Deformation in a multiaxial state of stress can often be reduced to the uniaxial case due to the fact that dislocation motion is rather insensitive to the hydrostatic component of the stress tensor.
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8 High-Temperature Deformation and Creep of Crystalline Solids
type is the deformation test, where the relations between s, cr, and T are determined. The second type of tests comprises the microstructural investigations during plastic deformation, aiming at assessing which microstructural quantities are relevant, and quantifying them.
8.2 General Considerations 8.2.1 Role of Lattice Defects For selecting useful microstructural parameters S™ it is helpful to recall which lattice defects are responsible for deformation. Plastic deformation results from motion of lattice dislocations, sliding of grain boundaries (motion of grain boundary dislocations), and diffusion of vacancies. Thus, lattice defects of different geometrical dimensions are involved and can be considered as active carriers of deformation. Diffusional flow, i.e., deformation of grains by vacancy currents under the influence of stress, is the simplest mechanism of plastic deformation. The size dg of a grain changes at a rate dg, which is proportional to the atomic mobility D/(kB T) (kB: Boltzmann constant, D: coefficient of self diffusion) and to the potential gradient Qojdg (Q: atomic volume & b3 for metals, where b is the magnitude of the Burgers vector) so that the strain rate e = dg/dg is (Frost and Ashby, 1982): 44 D oQ
GB/dg
(8-3)
The diffusion coefficient is the sum of the contributions from bulk diffusion (Dy: coefficient of bulk diffusion) and from diffusion via grain boundaries (DGB: coefficient of grain boundary diffusion, S: effective width of grain boundary). Diffusional flow
via bulk diffusion and grain boundary diffusion is called Nabarro-Herring creep and Coble creep, respectively. Due to its weak stress sensitivity and strong dependence on grain size, diffusional flow is usually of importance only when the grain size is small and the stress is low (i.e., the temperature is high). There is no doubt that diffusional flow does exist, e.g., in ceramics (see Sec. 8.6). In metals, however, it is difficult to find unequivocal evidence for diffusional flow being the dominant mechanism of plastic deformation (Wilshire, 1990). This is probably related to the relative ease of dislocation motion in metals and to the possible existence of a threshold stress for diffusional flow of particle-hardened materials (Frost and Ashby, 1982). Dislocations are the most important carriers of plastic deformation. Dislocation motion is identical to shearing of a crystal. This happens in so many ways, on so many slip planes, and in so many slip directions that plastic deformation can be achieved by dislocation motion alone (five linearly independent slip systems being sufficient for a general change of shape). The rate of plastic deformation is simply proportional to the density of the dislocation current. The basic law of plastic deformation, the Orowan equation, is often derived in an erroneous manner in standard textbooks. We present a correct derivation in the following. Consider a dislocation segment of length Vdg {Q: length of dislocation lines per volume V) moving at a velocity v (Fig. 8-1). In the time interval dt the segment creates a shear strain d2y equal to b times the sheared area, (v dt) (Vdg\ per volume V:
d2y = bvdgdt 2
(8-4)
Note that d y is a differential of second order. The differential shear of first order, i.e., the shear dy in the time dt, is obtained
8.2 General Considerations
Figure 8-1. A dislocation segment of length V dg moving a distance vdt at the velocity v in the time interval dt covers the hatched area vV dg dt.
by integrating over all dislocation segments: dy = b dt j v dg
(8-5)
The integral in Eq. (8-5) can either be expressed as vQ,tiv is the average dislocation velocity (= Q ~x J v dg) of the total dislocation density, or as ^ mob ^ mob , if the dislocations move in a stop and go manner, so that only the density gmob of dislocations is moving in the time interval dt at the average velocity vmoh. Thus: (8-6) moh
£
(8-7) dy Note that there is no additional term in Eq. (8-6) which is proportional to gmob; opposite statements in the literature are wrong. Sliding on grain boundaries is not sufficient for a general change of shape and therefore is not an independent mechanism
365
of deformation, but needs support by other mechanisms of plastic deformation to fulfill the requirements of grain compatibility. The formation and growth of cavities is part of the fracture process, but is at the same time intimately related to plastic deformation (see Chap. 12 of this Volume). The deformation associated with cavitation may make an important contribution to long term creep at very low stresses. We conclude that a consideration of the dislocation structure is essential. Often, the grain structure (grain size, texture) and the phase structure (volume fraction, size and spatial configuration of phases) can be regarded as being approximately constant in a limited range of plastic deformation, so that their structural evolution laws simplify to Sf = 0. The situation is quite different for dislocations. They move during deformation and interact with each other through their stress fields, which decrease inversely proportional to the distance from the dislocation. This interaction poses a limit on the density of individual dislocations which can be introduced into a crystal at a given stress. Each change in stress changes this limit and therefore causes the dislocation density to change with continuing strain. Thus it is necessary to pay close attention to the development of the characteristic parameters S/- of the dislocation structure with deformation. In addition the parameters S* characterizing the structure of grains, phases, cavities, and cracks need to be known for the formulation of the laws of plastic deformation. 8.2.2 Evolution of Deformation Resistance The tests in which the resistance of a material against plastic deformation is measured are usually carried out at constant temperature (except for thermal fatigue) and at either (approximately) con-
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8 High-Temperature Deformation and Creep of Crystalline Solids
stant strain rate s or stress a. As the two types of tests need different experimental equipment, many investigators are familiar with one type only. This has led to a division of the field of plastic deformation into "plastic deformation", in the narrow sense of deformation at constant imposed strain rate, and "creep", namely plastic deformation at constant imposed stress. One should keep in mind, however, that both kinds of tests are just different and complementary ways of investigating the kinetic law, Eq. (8-1). In the following we will give a brief comparison of the two types of tests and their results. The emphasis will be placed on those features which are typical for deformation at high temperatures, namely the effects of dynamic recovery of the dislocation structure 2 . In the "constant e" test (Fig. 8-2) (constant extension rate test (CERT), sometimes called dynamic test) the strain s increases in proportion to time t (Fig. 8-2 a). Often the test is done simply at constant cross head speed of the testing machine (unless there is sophisticated equipment to control s). In many experimental situations, but not after an abrupt change in cross head speed, the cross head speed equals the rate t at which the length / of a specimen of initial length Zo changes with time t. The initial strain rate s0 is l/l0 and for constant / the true strain rate s = s0 (lo/l) decreases gradually with increasing true tensile strain s = In (l/l0). The force F necessary to deform the specimen is measured and converted into the (true) stress a = F/S, 2 It is customary in the literature to use a homologous temperature of 0.5 (half of the melting point) as the boundary between low temperature and high temperature deformation. While this makes sense from a practical point of view, there is no such strict boundary in reality, because dynamic recovery is not only activated by temperature, but also by stress.
Figure 8-2. Deformation test at constant strain rate: (a) strain e, (b) strain rate e, (c) stress a as a function of time t. (d) Stress-strain curve.
where S is the instantaneous cross section of the specimen. Assuming that the volume SI of the plastically deformed specimen remains constant (note that cavity growth violates this assumption) and strain is homogeneous (no necking or local shearing), S equals So lo/l = So exp(— e). Figure 8-2 c shows the flow stress a as a function of time t. By combining Fig. 8-2 a and c one obtains the stress-strain curve (Fig. 8-2 d). The reason for using s as abscissa instead of t is that the development of the dislocation structure is governed primarily by e, not £, and that the strain range covered in tests at different s is similar, unlike the time range. The "creep" test (sometimes called static test) is shown in Fig. 8-3. Ideally, the stress a is held constant (Fig. 8-3 c). In creep rupture tests, where many specimens of different cross sections creep at the same time under the same load F9 obviously the condition of constant o cannot be realized; a changes gradually with e since F/S = (F/So) (So/S) = (F/So) (l/l0) = a0 exp (e).
8.2 General Considerations
--Tin
6
a e
V: .
I b
a c
Figure 8-3. Deformation test at constant stress: (a) strain e, (b) strain rate e, (c) stress a as a function of time t. (d) Strain rate-strain curve (semi-logarithmic). Roman numerals designate the primary (I), secondary (II), and tertiary (III) stages of creep.
The creep test consists of two different parts. In the first part, the stress is raised from 0 to the final value a. Although this part is short in time, there may be appreciable plastic strain occurring already during loading, so that the "instantaneous" strain 8ins is a sum of elastic and plastic components. As loading is generally done in a uncontrolled manner and the rate of plastic deformation during loading is not measured, information on the beginning of plastic deformation cannot be obtained from the simple creep test. However, in principle, the point of plastic yielding could be determined also in creep, provided that the deformation history is documented in detail. The second and essential part of the test comprises the creep, i.e., the plastic deformation at constant stress. The e-t curve (creep curve) in tension is generally S-shaped (Fig. 8-3 a). Figure 8-3 b shows the creep rate (i.e., the plastic strain
367
rate) e as a function of time. One divides the creep curve into the primary stage, where e decreases, the secondary stage, where & is approximately constant and close to its minimum value, and the tertiary stage, where e increases until the specimen fractures. The secondary stage of creep is most important from an engineering point of view, because it extends over most of the creep life, even though the strain in this stage may be rather small. Often the modeling of creep is done in the e-£-space. For example, the ©-projection concept (Wilshire and Evans, 1985) fits an S-shaped curve to the e-t curve. However, the constitutive laws, Eqs. (8-1) and (8-2), suggest that the e-t curves be differentiated in order to determine the creep rate e as a basic parameter of deformation. The e-t curve (Fig. 8-3 b) corresponds to the o-t curve (Fig. 8-2 c). For exactly the same reasons that the latter is replaced by the o-e curve, it is advisable to replace the e-t curve by an e-e curve (Fig. 8-3 d). The semi-logarithmic representation of the curves is advantageous, because it leads to similar shapes of curves for different stresses (Blum and Ilschner, 1967; compare Fig. 8-14). The work hardening in creep is reflected by a pronounced decrease in s with e at constant o. 8.2.3 Evolution of Dislocation Structure
With increasing strain, the rates of work hardening diminish in both the o-e and the e-e curves. This phenomenon is due to the fact that the dislocation structure (S/) approaches a steady state, which is based on dynamic equilibrium of generation and annihilation of dislocations and depends uniquely on the deformation conditions (a, T or e, T), and on the grain and phase structures (S*) of a given material (fj). If the steady-state dislocation structure remains the same, there is no change in deforma-
368
8 High-Temperature Deformation and Creep of Crystalline Solids
(a)
b \
a = const.
\
A
x
5 Boi
tion resistance with strain and both o and e are constant during deformation. This is indeed found in coarse grained, single phase materials. Being a state of dynamic equilibrium, the steady state of deformation is independent of the mode of testing, for constant Sf. However, in the transient period, i.e., the period of work hardening at constant e between the yield point and the steady state of deformation (also called saturation stage) or the transient range of creep at constant a preceding the steady-state range, there are significant differences between the two modes of testing. We illustrate them in Fig. 8-4 for the case of a coarse grained, pure material of low initial dislocation density. At constant e, there is a gradual refinement of the dislocation structure due to the continuous generation of dislocations, which leads to a continuous increase in flow stress. The spatial distribution of dislocations is not homogeneous. Due to the interaction of dislocations, a cellular pattern forms (see Chap. 4 of this Volume) by accumulation of dislocations in cell boundaries, which attain the character of subgrain boundaries with increasing stress and strain. The distinction between cell boundaries and subgrain boundaries is as follows: The term cell boundary is generally used if the boundary contains dipoles or multipoles (i.e., dislocations, which annihilate upon reaction) and has a finite thickness perpendicular to the boundary area. Subgrain boundaries (low-angle grain boundaries), on the other hand, are planar dislocation networks without dipoles. Cell boundaries can be converted into subgrain boundaries by an annealing treatment, during which all the dipoles are annihilated and only the geometrical dislocations causing a misorientation between the neighbouring subgrains remain. The trans-
lz > T,
subgrain
increase
V formation
of p
—•
11
(b) Figure 8-4. Schematic description of evolution of deformation resistance and dislocation structure for deformation at a low temperature Tt and a high temperature T2. (a) Constant strain-rate, (b) constant stress.
8.2 General Considerations
formation of cell boundaries into subgrain boundaries thus means that the relative dipole content of the boundaries decreases from a large value close to 1, to a small value near 0. The subgrain character of the boundaries increases as the steady state of deformation is approached. Therefore cell boundaries are more typical of low-temperature deformation (far from the steady state) and subgrain boundaries of hightemperature deformation (close to the steady state). The dislocation structure is characterized by three spacings: the size w (linear intercept) of cells or subgrains, the average spacing of free dislocations, which can be approximated as £f~0'5, where gf is the density (length per volume) of free dislocations, i.e., dislocations not part of subgrain boundaries or tangles of (geometrical) dislocations around particles, and the spacing s of dislocations in cell or subgrain boundaries. During work hardening at constant e both w and gf0'5 decrease simultaneously with increasing stress (Fig. 8-4 a). The sequence of events is different in the tests at constant a (Fig. 8-4 b). There is not enough time and strain for the subgrain structure to form during the initial increase in stress. In a short strain interval after loading, a rather homogeneous dislocation structure develops. Subgrain formation is comparatively slow. It starts only after Qf0'5 has attained a value close to its steady-state value and takes a strain interval which is about one order of magnitude larger than the strain connected with raising the density of free dislocations. As the stress is higher in the transient range of creep than in the transient range of deformation at constant e, the development of the dislocation structure towards the steady state occurs in a smaller strain interval in creep, than at constant L For the simple case considered in Fig. 8-4 the steady-state range of
369
creep is identical to the secondary stage (Fig. 8-4). Note, however, that this is generally not so for engineering materials with unstable microstructure (see Sec. 8.5.2). 8.2.4 A Simple Model of Plastic Deformation Many basic features of plastic deformation can be explained in terms of a simple model (Alexander and Haasen, 1968) using only a single parameter out of the whole set Sj1 of dislocation structural parameters, namely the density Q of dislocations. We will call it the @-model in the following. The derivation of the kinetic law starts from Eq. (8-6). The basic idea is that the interaction of dislocations causes an athermal stress component a G , which has to be subtracted from the applied stress o to obtain the so-called effective or thermal stress component
(8-8)
available for dislocation motion at a velocity v. The athermal stress aG varies with temperature only through the temperature dependence of the shear modulus G. The velocity v is nonlinearly related to a*: v = B <7*m, Pi9 Si = const.
(8-9)
X
aG =
(8-10)
where the constant a of dislocation interaction ranges between 0.1 and 1. By combining Eqs. (8-6) to (8-10), we obtain the following kinetic law: (8-11)
c2 = a
(8-12) c
iQ.
370
8 High-Temperature Deformation and Creep of Crystalline Solids
It is easy to see from the kinetic law that the variation of the deformation resistance with Q is non-monotonic in the test at constant s as well as in the test at constant o. At constant a the rate of creep e is zero for Q = 0 and for Q = [(r/(c2Gb)]2; in between these values there is a maximum of L At constant 4 the flow stress a is large for Q « 0 and for large values of Q, in between there is a minimum of a. At the point of minimum deformation resistance £ and o are related by: ,g . ^
and the dislocation spacing at this extremal point, £ext°r'5> *s inversely proportional to stress: G«t°r'5 = ci
G b
(1 + m/2)/°
(8"14)
with a proportionality constant, which increases strongly with increasing stress sensitivity m of the dislocation velocity. The deformation behaviour for Q~0'5 < £e~t°r'5, where the deformation resistance increases with decreasing dislocation spacing (refining dislocation structure), is called normal transient behaviour. The opposite behaviour for Q~0'5 > £e~t°r'5, where the deformation resistance decreases with decreasing dislocation spacing (refining dislocation structure), is called inverted transient behaviour. To make the model complete, the law of structural evolution must be formulated. The basic form of this law is:
The term (d@/de)+ has a simple geometrical meaning. From the definition of ds [$b times the area slipped per volume in dt, Eqs. (8-5) and (8-7)], one finds that it represents the increase in dislocation length per area slipped. For instance, if the disloca-
tions are circular loops which expand during deformation up to a certain maximum radius L + , one obtains
2nL+ de
(8-16)
L+ is the slip distance of dislocations (in the absence of recovery). It is reasonable to assume that L + is proportional to the average dislocation spacing: L+ =
-0.5
(8-17)
Similar to (dQ/ds) + the term (dg/ds) in Eq. (8-15) can be related to the slip associated with recovery of the dislocation density. It is responsible for attainment of a steady-state density of dislocations. Therefore it is bound to increase more strongly with Q than the generation term. Kocks (1976) assumes: &Q
de
= C5Q,
c5 = c5{s,T)
(8-18)
Thus the structural evolution law in the one-parameter model reads: dQ
de
= C6Q0-5
-C5Q,
(8-19)
The £-model has frequently been applied in the form described above or a slightly modified form, e.g., by Alexander and Haasen (1968) to semiconductors and by Estrin and Mecking (1984) to pure and particle-hardened metals. We will also use it in the following sections. However, it will become clear that a single dislocation structural parameter is insufficient as a basis for modeling. An extended model which differentiates between the influences of free dislocations and of subgrain boundaries on deformation will be presented in Sec. 8-7.
371
8.3 Pure Materials
8.3 Pure Materials 8.3.1 Evolution of Deformation Resistance and Dislocation Structure 8.3.1.1 Deformation at Constant Strain-Rate
Stress-Strain Curves Let us first consider the simple case of a pure monocrystalline material of low initial dislocation density. Figure 8-5 shows the T-y curves of InP with <123> orientation deformed in compression. After the onset of plastic deformation, the stress falls from the upper yield stress to the lower yield stress. The lower yield stress is the point of minimum deformation resistance explained above on the basis of Eq. (8-12) following Alexander and Haasen (1968). The dislocation density at this point has the value given by Eq. (8-14). After the lower yield stress the crystal hardens with increasing dislocation density (see Chap. 2 of this Volume). The x-y curves of semiconductors consist of a series of different stages labeled I to V (Siethoff and Schroter, 1978). A similar fine structure of hardening stages as in semiconductors has been found for f.c.c. metals (Schroter and Siethoff, 1984). Figure 8-6 shows results of Anongba (1990) for Cu with <112> orientation, where two closepacked slip planes are equally stressed. The distinction between the deformation stages becomes more pronounced by differentiating the stress-strain curve and plotting the work hardening coefficient 0 = 81/87 as a function of T (Kocks, 1976; Mecking, 1977; see also Chap. 2 of this Volume and Fig. 8-7). Usually, the point of minimum deformation resistance (lower yield stress) cannot be seen in metals, because the dislocation density at the onset of deformation is already too high. It can be seen that the
0.02
0.04
0.06
0.08
0.10
T Figure 8-5. Shear stress-shear strain curves for the beginning of plastic deformation in compression at constant homologous temperature T = 0.75Tm (Tm: melting point) of InP single crystals oriented (in <123» for single glide (after Reppich et al., 1990; the original curves have been sheared in an attempt to subtract the elastic component of strain). The circles mark the lower yield stress representing the point of minimum deformation resistance. 60
11
1"
•
•
1
1
t
50
1
1
1
I
I
•
i
•
=2- 10"4/s
1
'
29 4 K ^ - " * " "
40 Q_
30-
-
4K
20"
\845K -|935K
10-
-
— -
T
—
—
0.1
0.2
0.3
-.1064K
•
1 i
0
*
=
i
0.4
-
1133K i
i—"i
0.5
0.6
0.7
T Figure 8-6. Shear stress-shear strain curves of Cu single crystals oriented (in <112» for double glide in tension at constant shear strain rate in a large range of homologous temperatures from 0.22 Tm to 0.84 Tm. From Anongba (1990).
work hardening becomes less pronounced with increasing temperature (Figs. 8-6 and 8-7). At the highest temperatures, the difference between the yield stress and the maximum stress is quite small. The hard-
372
8 High-Temperature Deformation and Creep of Crystalline Solids
4U
T =2 •10
^ £
30
s
/
\ \
IV
294^
^
IV
10
-
\lll \
© 20 o
-
\v
>5K]837
771K
78K
n
0
1
2
3
4
5
6
7
8
9
10 11
104 T/G Figure 8-7. <9-T curves for Cu derived from the x-y curves of Fig. 8-6. 0 and T are normalized by the shear modulus G. Roman numerals designate the different hardening stages. Arrows indicate onset of instable deformation. From Anongba (1990).
ening stages I, II, and even III disappear with increasing temperature, and the steady state of deformation is reached within a small interval of strain. Dislocation Structure The increase in flow stress during deformation reflects the way that dislocations are accumulated. Basically, the dislocation structure evolves from the initial to the final structure, which is the steady-state structure at the given conditions of deformation. Microstructural observations show that the accumulation of dislocations occurs differently in the different stages of work hardening (see Chap. 2). In stage I, deformation is carried by dislocations of one slip system only (single slip on the primary slip system). Their interaction is rather weak so that the work hardening rate 0 is small. When dislocations of other than the primary slip system take part in deformation, new possibilities for a strong dislocation interaction arise, leading to a reduction of the average glide path and the
average velocity of dislocations, and consequently to accelerated growth in dislocation density and flow stress. Two basic observations must be noted. First, the existence of slip lines at the surface of single crystals documents that slip is a local event: dislocations are generated at certain sources and move groupwise over a glide zone of finite extension (Fig. 8-8 a). Second, the spatial arrangement of dislocations tends to become heterogeneous: the dislocations form a pattern (see Chap. 4); they accumulate in linearly (bundles) or two-dimensionally extended arrangements (sheets parallel to the slip plane, walls perpendicular to the slip plane, cell and subgrain boundaries). Sheets and walls are schematically depicted in Fig. 8-8 a. Pattern formation starts already in stage II and continues until a subgrain structure is reached in the final stages of deformation (Fig. 8-9). The characteristic dislocation spacings decrease with increasing strain and stress. While the strain dependence of the dislocation spacings differs for single- and polycrystals and depends on crystal orientation, there is a rather unique correlation with stress normalized by the shear modulus G. Figure 8-10 shows this correlation for the length of slip lines dsl, the cell size w and the average dislocation spacing £t~t0"5 (giot: total dislocation density) for Cu single crystals. The values of Q^?'5 determined from transmission electron microscopy (TEM) agree with those from etch pit observations at high stresses above « 10" 3 G within a factor of 2. For lower stresses TEM yields significantly lower values. This difference may mean that at low stresses there is a large fraction of dislocation dipoles arranged in dense bundles which are not resolved by the etch pit technique in contrast to TEM, so that the dislocation spacing determination from etch pits is closer to Qf~0-5 than to ^t;t0'5. The cell size
8.3 Pure Materials
(a)
373
(d)
(c)
Figure 8-8. Fundamentals of formation of dislocation structures: (a) Dislocations are emitted from a source (S) and glide over a limited distance. If they are held in a slip plane (due to low stacking-fault energy or short-range order) and react with dislocations from other slip systems, they form a sheet parallel to the slip plane. Formation of walls of edge dislocations with edges perpendicular to the slip plane is enhanced if dislocations can leave their slip planes by climb or by cross slip (of screw dislocations leaving edge components behind). The shear within a finite volume is connected with long-range internal stresses due to the change in shape of the sheared volume. The long-range stresses can be relaxed by some form of spatially correlated glide: If the slip distance is larger than the crystal dimension, the dislocations disappear from the crystal through the surface, leaving a band of stress-free sheared volume [marked by slip plane traces in (a)] with the same lattice orientation as the undeformed matrix behind. If the slip distance is smaller than the crystal dimension, relaxation of long internal stresses is associated with lattice rotation connected with single slip (b) or slip on two (or more) systems (c), (d). (b) The slipped volume extends over the whole crystal. The parallel dislocation walls of opposite sign form a kink band. The lattice rotation manifests itself by the polygonisation of the surface contour, (c) In between two sheared volume elements (A) with no misorientation relative to the matrix isolated stress-free subgrains (B) are formed by shear on the second system (B) and concurrent rotation, (d) Alternating shear on two slip systems coupled with alternating lattice rotation results in the so-called checkerboard structure.
w is about 14 times the average spacing of dislocations. It is interesting to note that the subgrain size determined from Fig. 8-9 for deformation at high temperature (0.84 Tm), close to the steady state of deformation, is in quantitative agreement with the cell size determined at the same value of ajG in the beginning of work hardening at room temperature. The slip line length dsl is about three times the cell size. All the spacings are (approximately) inversely proportional to the shear-modulus normal-
ized stress. In analogy to Eq. (8-10) we express these relations as: — = T
csb-, a
~1 — GG
(8-20) Qtot
=
GG
(8-21)
The factor c7 is of order 1, c8 equals 14.
374
8 High-Temperature Deformation and Creep of Crystalline Solids (a/G) G(300 K)/MPa
10"'
Figure 8-9. Subgrain structure observed by Anongba (1990) in stage IV of tensile deformation of [112]-oriented single crystals of pure Cu (99.97%) at 1145 K, y = 2 • KT 2 /s, T = 7 MPa, y = 0.38.
Figure 8-10. Length dsl of slip lines [faint slip lines due to homogeneous deformation (Ambrosi and Schwink, 1980)], size w of cells (Gottler, 1973) or subgrains (evaluated from Fig. 8-9), and average dislocation spacing £t~t0>5 [etch pit data from Livingston (1962), TEM-data from the review by Mughrabi (1975)] as a function of shear-modulus normalized stress
Interpretation of Hardening Stages In principle, work hardening can be understood as follows. Neglecting the a-term of Eq. (8-12) corresponding to the thermal component of the flow stress and using Eqs. (8-19) and (8-7), we have for O 0.5
-10
= 2/{a<Pabc5)
(8-22)
Neglecting the second term on the right hand side of Eq. (8-22) we obtain the maximum hardening coefficient in the absence of recovery to be G/c9. The hardening rate in stage II is experimentally found as 9U = G/300. This value equals the maximum hardening coefficient if c 4 = 100 and a = 1/3 (consistent with c7 « 1). This means that the slip distance L+ should be 100 times the average dislocation spacing
8.3 Pure Materials
[Eq. (8-17)]. This value of L+ is in reasonable agreement with the observed length of slip lines dsl and equal to about 7 times the cell size w (Fig. 8-10), indicating that cell boundaries are partly penetrable in stage II (see also Schwink and Gdttler, 1976). The advent of stage III, in which the work hardening coefficient 0 decreases with stress, shows that recombination and annihilation of dislocations must no longer be neglected, Eq. (8-22). It is generally accepted that for metals stage III is caused by annihilation of screw dislocations by cross slip. However, for diamond-cubic semiconductors which have a low normalized stacking fault energy [ySF/(G b)]9 cross slip is not so easy. It has been proposed that in these materials recovery in stage III is due to climb of edge dislocations (Schroter and Siethoff, 1984). The recovery processes operating in stage III are not sufficient to attain a steady state of deformation. This is because they affect one type of dislocation only, either screw or edge. The other type of dislocation continues to be stored. Consequently, the decrease of the work hardening coefficient & is interrupted in stage IV and continues in stage V only when both types of dislocations are able to recover. It follows that work hardening cannot be fully understood by the simple £-model. Rather, the heterogeneous nature of the dislocation structure (Fig. 8-11 a) must be taken into account. The physical basis for this was first established by Mughrabi (1980). He suggested that cell boundaries constitute hard regions, compared to the soft regions in between. The difference in resistance to plastic deformation leads to internal stresses, which reduce the stress as acting in the soft regions and enhance the stress
375
one obtains: (8-23) fs and fh are the volume fractions of the soft and the hard regions, respectively. The analysis of X-ray line profiles has proven that the composite model is applicable to the dislocation cell structure found in stage II of work hardening of Cu (Mughrabi, 1983). Haasen (1989) has developed a theory which explains stages II to V consistently in terms of: stage III: recovery due to cross slip in f.c.c. metals and climb in diamond-cubic semiconductors, in agreement with the proposal of Schroter and Siethoff (1984), stage IV: increase of fhah due to deposition of dislocations in the cell boundaries, stage V: additional recovery due to climb in metals and cross slip in semiconductors. However, the description of the dislocation structure in terms of cell boundaries is appropriate only as long as the dipole content of the boundaries outweighs the content of geometrical dislocations. This is not the case for high temperatures and high strains. Here the boundaries have subgrainboundary character (Fig. 8-11 b). As subgrain boundaries consist of families of dislocations of the same sign, they constitute sinks for intruding dislocations of opposite sign. Therefore, the question is not only how quickly the dipoles in the walls are
(a)
(b)
Figure 8-11. (a) Cell structure with dislocation walls with a high content of dislocation dipoles, (b) subgrain structure with planar boundaries (dislocation networks).
376
8 High-Temperature Deformation and Creep of Crystalline Solids
annihilated, but rather, how quickly dislocations are transported to their sinks. The tendency to form a subgrain structure is in principle easy to understand from an energetic argument. To come into a lowenergy position, dislocations have two choices. The first is to find a partner of opposite sign and form a dipole, which has the tendency to disappear by annihilation. However, due to the inhomogeneous distribution of dislocation sources (as revealed by the slip lines) many dislocations do not have a dipole partner nearby. They can form subgrain boundaries and thus relieve their long-range stresses as shown in Fig. 8-8 b to d. This is obviously what occurs. Geometrical dislocations leading to misorientations of the order of 1° are not only present in the steady state, but can be found already in stage II (Seeger and Wilkens, 1967; Anongba, 1990). However, the geometrical dislocations in stage II are hidden by the large number of dislocation dipoles which are also present at that stage. They can be made visible by an annealing treatment causing the dipoles to disappear. Although dislocation energy is the driving force for subgrain formation, the subgrains are not strictly low-energy arrangements during steady-state deformation, because they form in the course of a dynamic process, where new dislocations, which are not yet in a low-energy position, are arriving at a certain rate. Therefore, subgrain boundaries may carry long-range stresses during steady-state deformation (Blum and Reppich, 1985), as cell boundaries do (Mughrabi, 1983). So far, we have discussed only dislocation reactions which tend to lead into a steady state of deformation. However, in pure Cu, as in many other pure materials, a stable steady state of deformation is not attained at high temperatures. This is because in addition to recovery of the dislo-
cation density as expressed by Eq. (8-18), other mechanisms of restoration of the dislocation structure appear. They are related to long-range motion of grain boundaries, which are already present in polycrystalline material or are newly formed in single crystals by twinning or subgrain growth. The volume swept by the grain boundaries has a low dislocation density and deforms rapidly. This causes the sudden drop in stress which can be seen in Fig. 8-6 at high temperatures. 8.3.1.2 Deformation at Constant Stress Strain Rate-Strain Curves We start again by considering an InP single crystal in < 123) orientation and low initial dislocation density. Figure 8-12 shows a set of 8-s curves of long specimens deformed in compression at 1001 K = 0.75 7^. The minimum of the deformation resistance [Eq. (8-11)] is expressed as a maximum in the creep rate at constant stress. After the maximum, the loga-e curve consists of two stages. In the first stage, the absolute magnitude of the slope, Q creep _ | g jOg ^g g | ( ^ J Q ^ c a n b e regarded as the rate of work hardening in creep) decreases with strain. In this stage single slip dominates. It corresponds to the stage I of easy glide at constant L The subsequent steep decrease in 8 with 8 indicates that multiple slip has set in, leading to strong hardening. After reaching a maximum, the hardening rate 0 c r e e p decreases again. However, the steady state of creep with <9creep = 0 is not reached in the tests of Fig. 8-12 in spite of a decrease in creep rate of up to 6 orders of magnitude. The dislocations are distributed homogeneously in the first stage of work hardening, while in the second stage, a subgrain structure develops (Fig. 8-13).
8.3 Pure Materials
377
InP, undoped 1001K, air
CO
11/
0.20
0.10
Usually, the point of minimum deformation resistance is not seen in the g-e curves, either because the maximum creep rate is too large to be measured or because the initial dislocation density is too high [see Eq. (8-14)]. Figure 8-14 shows e-e curves for pure Al deformed at the same stress at different temperatures. It can be seen that the shape of the curves in the semi-logarithmic representation is nearly independent of temperature. This indicates that the dislocation structure which is formed in the material is similar at all temperatures and depends mainly on stress, consistent with the dislocation structural observations reported below. Dislocation Structure The development of the dislocation structure with strain at constant stress de-
0.30
Figure 8-12. e-s curves measured in compression at constant homologous temperature T = 0.75 Tm and different stresses a for InP single crystals oriented (in <123» for single glide (Geibel, 1990; unpublished). The circles mark the lower yield stress representing the point of minimum deformation resistance.
pends on the number of active slip systems. When only a single slip system is activated in the beginning of creep of a single crystal at high temperature, one observes formation of kink bands [e.g., in Cu (Hasegawa et al., 1971) and in Al deforming on a single, noncompact slip plane (Carrard and Martin, 1984)]. Kink bands consist of two, or two groups of dislocation walls of opposite sign. Walls form because the slip distances of edge dislocations emitted by the sources are limited (Fig. 8-8 a). Incipient walls have a tendency to grow due to the stresses at their free ends. When a wall reaches the surface of the crystal, the longrange stresses are relaxed and the surface becomes kinked (Fig. 8-8 b). This process is called polygonization. As shown by Hasegawa et al. (1971), the spacing of the dislocation walls in the kink bands increases during further creep. This is achieved by
(b)
(a)
O 10/im
Figure 8-13. Dislocation structure in InP made visible by etching after compression at 1068 K and constant stress (Geibel, 1990; unpublished), (a) Small strain (e = 0.07 at 8.7 MPa): homogeneous distribution of dislocations, (b) large strain (e = 0.52 at 33.5 MPa): subgrain structure.
378
8 High-Temperature Deformation and Creep of Crystalline Solids
0.30
Figure 8-14. e-e curves of polycrystalline Al measured in tension at constant stress in a range of homologous temperatures from 0.40 Tm to 0.59 Tm (Absenger, 1977; unpublished).
migration of walls: As has been shown in LiF (Biberger, 1989; Biberger and Blum, 1992) walls of opposite sign move towards each other and recombine upon meeting. In this way the average spacing between kink bands grows. The dynamic equilibrium between dissolution of boundaries by recombination and formation of new boundaries establishes the equiaxed steadystate subgrain structure. When slip occurs on two different slip systems, two interpenetrating systems of walls form a box structure with alternating signs of misorientation [e.g., in Al (Carrard and Martin, 1984), Mo (Clauer et al., 1970), LiF (Biberger, 1989)]. Again the final structure is an equiaxed subgrain structure. In polycrystals, subgrain formation begins near grain boundaries due to the larger number of slip systems, which are activated in this region for reasons of compatibility. If the strains are extremely large
(e.g., in torsion), the size of the grains perpendicular to the direction of elongation finally becomes smaller than the subgrain size. In this case an appreciable fraction of the subgrain interfaces actually consists of large-angle boundaries (Kassner and McMahon, 1987; McQueen et al., 1985). The quantitative development of the dislocation structure in pure polycrystalline materials is quite similar to that of single phase alloys of class M and will be described in Sec. 8.4.1.2. The high mobility of grain boundaries in materials of high purity leads to restoration of the dislocation structure by dynamic recrystallization or dynamic grain growth at high temperatures, before dynamic recovery has led into a stable steady state of deformation (McQueen and Jonas, 1975). The associated softening is visible in the &-8 curve as a sudden increase in 8 at constant a corresponding to the sudden decrease in flow stress at constant e. Pure Al is an exception among pure metals, because it does not recrystallize dynamically at constant stress or strain-rate due to its high stacking fault energy, which makes dynamic recovery a very competitive process, but shows only dynamic grain growth. But even in Al recrystallization during creep can be triggered by a moderate stress reduction (Straub and Blum, 1990). 8.3.2 Steady-State Deformation The occurrence of dynamic recrystallization means that a steady state of deformation, in the sense of a stable equilibrium between generation and annihilation of dislocations due to recovery, does not exist when the temperature and strain rate are high. In metals like Cu, Ni and Ag instability due to dynamic recrystallization can hardly be avoided (Frost and Ashby, 1982; see also Fig. 8-6). Therefore, the steady-
8.3 Pure Materials
state data for these materials are scarce or unreliable. Al, oc-Fe and alkali halides are not so prone to dynamic recrystallization and can be used in the present context. 8.3.2.1 Equation of State
In Sec. 8.2.4 we have presented the state of minimum deformation resistance in the beginning of deformation as a particularly simple case, where the kinetic law is independent of the dislocation structure. The steady state of deformation which is reached at large strains is of similar simplicity, because again the parameters S / of the dislocation structure are fixed by the condition Sj1 = 0 and therefore do not enter the kinetic law as independent variables. For instance in the g-model the steady-state dislocation spacing is simply 5 Q-°- = c5/c6 according to Eq. (8-19). It has been noted early by Dorn, Sherby and coworkers (see e.g., Sherby and Burke, 1967) that the rate of steady-state creep at high temperatures has an activation energy, which is very close to that of self-diffusion. This suggests that steady-state deformation is controlled by climb of edge dislocations. Different models of climbcontrolled creep lead to the so-called natural law of steady-state deformation. As climb during creep involves diffusion over distances of the order of the discloation spacing, the natural law can be obtained in a somewhat simplistic manner from the law of diffusive flow [Eq. (8-3)] by replacing the grain size dgby Q~0'5 & b G/a [see Eq. (8-21)]:
ion is firmly established that the natural law does not hold for pure materials because the stress exponent of the steadystate creep rate is larger than 3 (e.g., Nix and Ilschner, 1980). However, recently it has been reported that the natural law does in fact hold in the limiting case of low stresses for pure LiF (Fig. 8-15). Even for a pure metal like Al, the curve for the steadystate relation in Fig. 8-15 continuously approaches the line for the natural creep law with decreasing stress above 10~ 4 G. The increase in stress exponent with decreasing stress below 10" 4 G detected by Straub and Blum (1990) is not fully understood; it may be related to impurities. At very low stresses, below 5 x l O " 6 G , a range exists where the stress exponent drops to 1.
10" LiF Ai
6 = A'-.
Qa
^
(8-24)
Rigorous treatments of climb-controlled creep show that the numerical constants A' and A are of the order of 1 (see Nix and Ilschner, 1980). In the literature, the opin-
Straub, Blum literature
10"6 r
ekj DGb 10"1
10,-18 "
10"'
D
379
10"
a/G Figure 8-15. Steady-state relation between e normalized by kB T/(D G b\ and stress a, normalized by the shear modulus G(T), for pure materials: Al (Straub and Blum, 1990) and LiF (Biberger and Blum, 1989). The dashed reference line represents the natural creep law.
380
8 High-Temperature Deformation and Creep of Crystalline Solids
Creep in this range is called Harper-Dorn creep. However, due to the extremely low creep rates and achievable strains and in the absence of convincing stress change tests confirming the stress exponent of 1, it is not yet clear whether a new mechanism of dislocation motion is warranted in the Harper-Dorn range. In any case, the data in this range are not too far from the predictions of the natural creep law. Thus the results in the Harper-Dorn range may be explained in terms of Eq. (8-24) assuming that the dislocation density in Al remains constant ( ^ 3 x l 0 8 m ~ 2 ) below cr = 5xlO~ 6 G, because strain during creep is too small to produce a change in the grown-in dislocation density. With increasing stress there is a smoothly increasing deviation of the steady-state deformation rates from the prediction of the natural creep law and the stress dependence of s becomes exponential. This indicates that a process of thermally activated glide takes over as rate-controlling process. The whole range of steady-state deformation at low and at high stresses can be described in a phenomenological manner with a sinh-law of the type (Barrett and Nix, 1965; Blum and Reppich, 1969):
cording to Eq. (8-25) n increases when the range of stresses shifts to higher values (Blum and Reppich, 1969). 8.3.2.2 Dislocation Structure
Figure 8-16 shows the characteristic dislocation spacings in the steady state of deformation. The subgrain size w in Al varies inversely proportional to the shear-modulus-normalized stress according to Eq. (8-20) with c8 = 28. This type of relation is commonly found. The average value of c8 for a large number of materials was reported by Raj and Pharr (1986) to be 23. Equation (8-20) is derived not only from direct observations, where it is impossible to avoid an error of the order of 20%, due to the wide distribution of sizes and to systematic errors related to the uncertain distinction between cell boundaries and subgrain boundaries, but also by transient creep tests. Even a small change in the subgrain size w from the initial value to the steady-state value is reflected as a transient change in e. From the sign of the change of e with e, the direction in which w changes can be deduced. For pure materials the transient is normal. This means that ds/ds and dw/ds have opposite signs. Using the transients to sense changes in w, Konig and (8-25) e = Blum (1980 a) could show that w is a unique function of ajG. Changes of either a The temperature dependence of the steady or G (due to a change in temperature) by a state rate of deformation is mainly carried few percent cause corresponding changes by the diffusion coefficient D = D0 exp [ — Q/ in w. (kBT)]. Va is an apparent activation volDue to the insufficient stability of the ume, which may depend on temperature, p dislocation structure in the subgrain inteis a constant stress exponent of order 2 to rior to unloading and specimen prepara3. In a limited range of stresses Eq. (8-25) tion and handling, the spacing gf0'5 cancan be approximated by: not be determined reliably by transmission n electron microscopy in many pure metals, a = c12cr exp (8-26) kBT e.g., AL Therefore, in Fig. 8-16 we present with a constant stress exponent n and a data for £f~0'5, obtained from etch pit inconstant apparent activation energy Q. Ac- vestigations on NaCl, where the danger of
8.3 Pure Materials
(a/G) ^,(300 K)/MPa 10-1 •D
10°
_
10u
grains in Al after high temperature creep, and also reported for NaCI (Raj et al., 1989) does not represent the true dislocation arrangement under stress in steadystate creep (Vogler et al., 1991). The spacing of free dislocations obeys a relation of type (8-21):
10'
\
10'
n Al o NaCI
nuD \
• \
10b
D
10 2 L
"D
o
104 5'
Q. CO
10 3 10'
bG q o S
•
D c a •
10*
10" 101 10"
10"
a/G Figure 8-16. Steady-state dislocation structure as a function of shear modulus normalized stress a/G(T) for pure materials: Subgrain size w and dislocation spacing s in subgrain boundaries in polycrystalline Al from the review of Blum (1991), average spacing @f~0'5 of free dislocations in single crystalline NaCI oriented (in <001» for multiple glide from Eggeler and Blum (1981) recalculated with the G(7>data from Frost and Ashby (1982).
rearrangement of the free dislocations is less severe. The data for NaCI are considered to be representative for Al as well, because the subgrain size in NaCI also obeys Eq. (8-20) with c8 « 30 (Blum et al., 1991). The distribution of etch pits in the interior of subgrains is rather homogeneous with little tendency to form cellular structures. This implies that the pronounced cellular structure seen inside sub-
G
,-0.5
105
CD
381
c 1 3 = af
G,f
\-i
(8-27) 7
G,f
a1 a G f is the component of the athermal stress <JG, which is due to free dislocations. The constant c13 is about 1. This means that af < 1/3 for $ = 1/3 and aGtf/a < 1. It is not clear, however, whether Eq. (8-27) holds strictly for the whole range of stresses. The data in Fig. 8-16 indicate that the stress dependence of £f~0"5 may be weaker than given by the inverse proportionality of Eq. (8-27). The dislocation spacing s in subgrain boundaries is difficult to determine for a number of reasons (Blum, 1991). In this paper we have used the approximate relation s = b/0mis = 1.73 Zb, where s is the average of the minimum spacing of the dislocation families present in each boundary, 0 m i s is the misorientation, and Z^1 is the dislocation line length per subgrain boundary area. It can be seen clearly, that the stress sensitivity is much weaker for s than for the other two spacings, but a limited stress sensitivity of s cannot be excluded. Comparing the steady-state data in Fig. 8-16 with the data in Fig. 8-10 obtained during work hardening one recognizes a basic similarity. The spacings of subgrain and cell boundaries as well as the average dislocation spacing and the spacing of free dislocations are approximately the same at the same value of the shearmodulus-normalized stress. This shows the
8 High-Temperature Deformation and Creep of Crystalline Solids
8.3.3 Response to Change in Deformation Conditions
Certainly it is difficult to make a clear distinction between the last two modes of nonelastic deformation. However, the problem is relieved to some extent because anelasticity is often comparatively fast and ceases shortly after the change in stress. There are numerous efforts to derive a socalled internal back stress from that degree of unloading where anelastic back flow sets in, which is determined in the "dip test". However, detailed investigations have shown that not only the measurement, but also the interpretation of this internal stress is not a straightforward matter (see Cadek, 1988). We will not go into details here and rather confine ourselves to plastic deformation. Figure 8-17 shows the result of stress change tests with pure Al. (Strain-rate change tests yield analogous results.) The
8.3.3.1 Single Change
Al
=
523 K a0 = 20MPa
10*
o/o0:
E
0.90
=
0.74
E
CO
0.60
_
i
i i
i i
iii|
i
•u;
i i
i i i
10*
i
• the elastic reaction coupled with each change in stress, • the anelastic reaction connected with the back flow due to unbowding of dislocation segments and relaxation of long-range internal stresses after stress reduction or forward flow due to bowding and build up of long-range internal stresses after stress increase, • the plastic deformation occurring in addition to the anelastic one.
E
In111
A change in imposed conditions of deformation disturbs the deforming system and decouples the different mechanisms of dislocation motion taking part in the process. Thus, change tests give valuable information about the kinetics of deformation and enable the determination of parameters which are important for the quantitative modeling of deformation. We differentiate between
ii
close relationship of deformation at low and at high temperatures. In order to understand the kinetics of deformation at high temperatures, it is important to note that not only the free dislocations are moving, but the dislocations stored in subgrain boundaries also take part in deformation and contribute to strain. This is because most subgrain boundaries have a tilt component, which means that the material is sheared, when the boundary moves. There is general agreement that the relative contribution of migration of subgrain boundaries (more generally: collective motion of groups of dislocations of the same sign) is of the order of 10% (Poirier, 1985; Caillard and Martin, 1987; Biberger, 1989).
ii|
382
10-'
i
0.02
i
i
0.04
i
0.06
Figure 8-17. s-s curves of pure Al for tests (tension and compression) with reduction in stress from
383
8.3 Pure Materials
circles mark the value of s after each stress reduction which is found immediately after the anelastic reaction has ceased. It is called the strain rate at constant structure, ecs, because the strain at which it is measured is virtually the same as at the last point of deformation before stress reduction, so that the dislocation structure did not have the chance to change significantly. However, small scale changes cannot be exluded. From 8CS, the strain rate changes towards the new steady-state value ess at the new stress. The transient after the change in deformation conditions is due to the restructuring of the dislocation pattern. Figure 8-18 shows that the subgrain size of polycrystalline Al blows up with strain after a decrease in stress. The softening associated with subgrain growth confirms that the transient deformation behaviour of pure Al is normal in the sense described above. However, there is an initial range where s decreases with strain. Measurements of subgrain boundary migration in LiF have shown that the strain rate found in this range is predominantly due to the migration of subgrain boundaries (Biberger, 1989). Motion of free dislocations dominates only after a certain degree of recovery of the dislocation structure. Stress change tests and, in a more limited way, stress relaxation tests offer a unique opportunity to measure the relation between £cs and a at constant structure in a large 8-range. Hart (1970) and Hart et al. (1975) have used this relation as derived from stress relaxation to define a state of hardness of the material. Figure 8-19 shows scs as a function of a for pure Al and pure LiF in a semi-logarithmic plot. The data for the two materials are quite similar. The log(£cs)-(x curve can be explained as the sum of two branches (Blum et al., 1989 a, and 1989 b; Vogler and Blum, 1990).
10"'
E 15.0 MPa
573 K o a steady state
10"4
•So- 5
Al
6.23
MPa
3.44 MPa
10'6 10"7 100
_50
20
10
0.04
0.08
0.12
e-e Figure 8-18. Tensile creep rate e and subgrain size w in pure Al at 573 K as a function of strain after stress reduction from different initial stresses to the same final stress. The steady-state values are marked by open symbols. Replotted from Ferreira and Stang (1979).
Al 523Ktf o = 20.0 MPa LiF 773K^ 0 = 20.5 MPa
0.5
1
o/o0 Figure 8-19. ecs-cr relation at constant dislocation structure for pure LiF (Biberger, 1989) and pure Al (Blum et al., 1989 b). s cs and a have been normalized by the values of creep rate, £ 0 , and stress, G0, before the stress reduction at s 0 . scs can be split into the contributions from the free dislocations (• • • • highstress branch) and subgrain boundary dislocations ( • • - • • - • • low-stress branch).
384
8 High-Temperature Deformation and Creep of Crystalline Solids
The steep high stress-branch corresponds to the contribution of free dislocations. The flatter low-stress branch reflects subgrain boundary migration (see above). Note that there is slow forward flow even after full unloading as a consequence of the internal stresses inherited from prior deformation (Blum et al., 1989 b). The slope of the scs-(j curve at a = a0 (which is approximately equal to the slope of the high stress branch (dotted) in Fig. 8-19) can be used to define an operational activation area according to: Aa =
kBTdlns
(8-28)
There is again a close relation between work hardening at constant s at low temperatures, and steady-state deformation at high temperatures: The data of Aa/b measured in the two cases are quite similar, at the same stress, and show a similar stress dependence (Blum et al, 1989 b). The fact that Aa/b is of the same order as the average spacing of free dislocations indicates that forest dislocations are the obstacles for thermally activated flow through the subgrain interior during steady-state deformation at elevated temperatures. This mechanism of thermally activated flow is the same as invoked for low temperatures (see e.g., Zeyfang et al., 1974). 8.3.3.2 Cyclic Changes
Consider the case where the changes in imposed strain rate or stress are repeated cyclically in time with a period At. As in the case of monotonic deformation, the material reacts to cyclic deformation by an evolution of the deformation resistance, which is the macroscopic expression of the evolution of the dislocation structure. There is a transient period where the material hardens or softens, followed by a steady state.
The steady state of cyclic deformation is reached, when the histories of evolution of the dislocation structure and of the deformation resistance repeat themselves exactly from cycle to cycle. Note the difference between the steady states of monotonic and cyclic deformation: In the monotonic steady state dislocation structure is constant with strain at constant deformation conditions. In the cyclic steady state, on the other hand, the dislocation structure changes continuously during each cycle, i.e., at a given point of the cycle the dislocation structure is different from the steadystate structure which would be reached under conditions of monotonic deformation at the same stress. This difference in dislocation structure causes the difference in strain rate observed at the same stress in the steady state of monotonic deformation, compared to the steady state of cyclic deformation. As an example, we consider cyclic straining in opposite directions at constant average strain. The small positive and negative contributions of irreversible strain in each cycle cause fatigue (see Chap. 11 of this Volume). The dislocation structure tends to develop towards a subgrain structure with continued cyclic straining, which is quite similar, though not identical to the subgrain structure formed under monotonic conditions at the same a/G (Konig and Blum, 1980 b). Qualitatively speaking, one may say that the to and fro motion of dislocations enhances the recovery of the dislocation density, so that a steady state of cyclic deformation can be established even at low homologous temperatures. The cyclic deformation under conditions where the average strain changes from cycle to cycle is called cyclic creep. The effect of cycling on the rate of deformation is shown in Fig. 8-20 for pure Al. At a certain strain, the mode of deformation is switched
8.4 Solid Solutions -i—|—i
">
deformation is expected to occur at a rate equal to the rate of monotonic deformation at the same stress and strain. Consequently the predicted rate of cyclic creep in the fth cycle is:
r
€cyc 10"
385
o per cycle ~ smoothed
CO
• vb
\
U + At
(8-29) Iff* 52
monotonic
cyclic
mon.
51 «50 * . ' 2 1 0.26
At 2 = 5S ° At=10S R = 0.05
Al 295 K 0.27
0.28
0.29
0.30
0.31
0.32
€ Figure 8-20. s and a as a function of e for compressive creep of pure Al at room temperature with a change in deformation mode from monotonic creep at o2 to cyclic creep (R = ojo2 = 0.05, At2/At = 0.5). The change is connected with a sudden increase in creep rate (cyclic creep acceleration) by a factor F = £cyc/£cyc ~ 30 corresponding to an extra strain per cycle of 6 x 10" 4 (Vogler, 1988; unpublished).
from (monotonic) creep at a stress a2 to cyclic creep, where the stress varies between the levels o2 and o1 = Ro2> The variation of stress a with time t is rectangular: each period consists of a time interval Atx with a = a1 and an interval At2 = At — At± with a = a2. The cyclic
creep rate scyc is defined as the net rate of deformation per cycle. ecyc can be compared to the rate of creep e'cyc predicted from monotonic deformation. The prediction is made on the basis of a specific rule. The life fraction rule and the strain rule are often used and lead to similar results. According to the strain rule,
In the case considered in Fig. 8-20, a± is small compared with al9 which means that ax ^ 0 and £(cr2,£) can be treated as constant with each cycle, so that sfcyci = (At2/At)s((T2,8) = 0.5e(t72,e). However, the observed rate ecyc t differs from the predicted rate e'cyC)i by a factor Ft« 30. This means that creep of pure Al is significantly accelerated by cyclic stressing. This is typical for pure materials with high mobility of free dislocations. The acceleration corresponds to a small extra strain per cycle, equal to (scyc — £rcyc) At, which increases strongly with the amount of nonelastic back flow in the low stress period of the cycle (Blum et al., 1983). It seems that the cell and subgrain boundaries are weakened by the reverse motion of dislocations. This is consistent with the observation of Hasegawa et al. (1975), who have shown that cell boundaries can dissolve partially in the course of backward straining.
8.4 Solid Solutions Solute atoms are obstacles to dislocation motion and result in solute hardening. At low temperatures, the solute atoms can be regarded to be immobile. At high temperatures, on the other hand, where the diffusive mobility of solute atoms (through the lattice and/or via short circuit paths along the dislocations) is large, the solute atoms are attracted to the dislocations and form a cloud of solute atoms around the dislocations which is dragged along with
386
8 High-Temperature Deformation and Creep of Crystalline Solids
them. The configuration of the cloud is determined by the dynamic equilibrium of solute atoms, which are caught by the moving dislocation and which are left behind during motion. Cloud formation sets in, once the velocity of dislocations is of the same order as the diffusive velocity of solute atoms. The relation between dislocation velocity v and effective stress a* is schematically depicted in Fig. 8-21. It differs for dislocations with and without clouds. The motion of dislocations without clouds is described in terms of thermally activated glide. This is justified by the jerky nature of dislocation motion at low temperatures. The obstacles which are overcome by thermal activation are formed by forest dislocations and (groups of) solute atoms (see Chap. 5 of this Volume). Motion of dislocations with clouds occurs in a viscous manner: \
m« 1
(8-30)
The "effective stress" afol is necessary to drag the cloud along with the dislocations.
Figure 8-21. Schematic description of dislocation velocity v as a function of a* for solid solutions. Both axes are drawn on a logarithmic scale.
In addition, the effective stress o\ exists for thermally activated overcoming of forest dislocations (or other obstacles formed by the dislocations themselves). In a somewhat formal manner, we write the total effective stress as the sum of the two components: °*q = « i ) q + (°*±Y,
l
(8-31)
The exponent q means that the two effective stress components may not add up linearly. In Fig. 8-21 it is assumed for simplicity that o\ is negligible so that a*ol « cr*. At a certain critical value cr*rit 2 the stress is so high that the dislocations break away from their clouds. This means that the dislocation velocity jumps from the low value with clouds to the higher value without clouds. Such a sudden jump leads to instabilities in plastic deformation which are described below. 8.4.1 Evolution of Deformation Resistance and Dislocation Structure 8.4.1.1 Deformation at Constant Strain-Rate
Solute atoms enhance the yield stress and the work hardening rate during deformation at constant s at low temperatures (Kocks, 1985; see also Chap. 5 of this Volume). In an intermediate temperature range, fluctuations in flow stress develop during deformation. This so-called Portevin-LeChatelier effect is seen in Fig. 8-22 to set in at 673 K in the alloy NiCr22Col2Mo9, which is essentially solution hardened and can be explained on the basis of Fig. 8-21. Assume that the stress a* is larger than the critical stress °"?rit,2 s o that the dislocations move without clouds. As the dislocation density Q increases, the average velocity v decreases [Eq. (8-6)]. If v reaches i;crit at a critical effective stress crcrit 1? clouds form and the
8.4 Solid Solutions 2500
U3-1CTV T = 77K 2000 ^123 K
/
/
,
1500 Q.
K 1000 -
K
mK
/////0^ //wr
500 yw^
^973K ^1023 K 1073 K
'
1
40
H
80
120
160
e/10" 2 Figure 8-22, True stress-true strain curves for NiCr22Col2Mo at t/l0 = 3 x KT 4 /s in a large range of temperatures. Between 400 K and 900 K, the curves show fluctuations due to the Portevin-LeChatelier effect. The curves correspond to the nominal stressnominal strain curves published by Viereck et al. (1989).
velocity of the dislocations suddenly drops to the value for dislocations with clouds. In order to maintain the strain rate constant, the stress must rise steeply until a*tlX 2 is exceeded again. For a quantitative description of the instability underlying the Portevin-LeChatelier effect, the reader is referred to Kubin and Estrin (1985) and to Chap. 5 of this Volume. At temperatures above 900 K, the effective stress to move the dislocations always remains below cr*rit 2 so that no instabilities occur. 8.4.1.2 Deformation at Constant Stress If the effective stress exceeds cr*ritj 2 one observes fast plastic flow during and shortly after loading the sample. This flow
387
is associated with an increase in dislocation density [Eq. (8-19)], which means that a* and v decrease quickly. When v reaches t;crit at a*riu 1, clouds form, causing another steep decrease in v (Fig. 8-21). Subsequently the balance between generation and annihilation of dislocations is established at a certain steady-state value of
388
8 High-Temperature Deformation and Creep of Crystalline Solids
900
X6CrNiMo17 13 823 K
800 700 600
0.1
0.2
I
CO
I
ejns, stepwise loading a = const.
•
e = const.
0.4
0.3
0.5
0.6
0.7
I
AlZn : 523 K ; 16MPa -
10" •
10"
o r*
Figure 8-23. Relation between instantaneous tensile strain £ins and stress cr for the austenitic steel X6CrNiMo 1713, plotted as a stress-strain curve. There is no difference between increase in stress in large steps and continuous increase in stress at approximately constant rate a. Actual as data measured at constant (high) elongation rate fall onto the same curve. After Eckert (1986).
^
—
-
-
—
5
w
-
10
Pt5
O)
cb
0.1 CO
[
f
_
1.0
-
0.8 3 0.6 0.4
-
0.2 0.0
0.1
0.2
0.3
0.4
e (a) Figure 8-24. Strain rate and dislocation structure (subgrain size w, spacing of free dislocations gf~0>5, dislocation spacing s in subgrain boundaries, fraction / sub of volume with subgrains) as a function of strain at constant stress for (a) Al-5 at.% Zn and (b) Al-5 at.% Mg. The s-data show large scatteq the dashed line was estimated to represent the true variation of the average spacing s with strain. From Blum (1991).
8.4 Solid Solutions
of Zn and of Mg. The hardening of Zn is weak. Therefore AlZn shows a normal transient behaviour of the same type as pure Al (Fig. 8-14), which is called class M (pure metals) behaviour. Mg is a strong hardener, thus the behaviour of AlMg is inverted. AlMg is a prominent member of class A (strongly solute-hardened alloys) of materials. Interestingly, the development of the dislocation structure is quite similar in the two cases: A subgrain structure forms and the volume fraction / sub increases gradually with strain from 0 to 1. The local subgrain size w is fairly constant. The dislocation spacing in subgrain boundaries s decreases slowly with strain. The average spacing of free dislocations Qf0'5 changes rather quickly from its initial value to a value close to the steady-state value. Note that the decrease in creep rate in AlZn cannot be caused by refinement of the structure of free dislocations, but must be associated with subgrain formation. Frequent reports in the literature (see e.g., Oikawa and Langdon, 1985) claiming that the dislocation structures in classes M and A of alloys are qualitatively different may be due to the fact that subgrain formation in class A materials is overlooked, because the creep rate reaches an approximate steady-state value long before the microstructural development is complete. The striking difference in the transient creep behaviour can be easily understood in terms of the £-model. According to this model (Sec. 8.2.4) the type of transient behaviour depends on whether the dislocation spacings are larger or smaller than the spacing £e~t°r"5 at the point of minimum deformation resistance. In AlZn, the stress sensitivity m of the dislocation velocity [Eq. (8-9)] is quite large leading to a large critical spacing Q~xtr'5 according to Eq. (8-14), so that £~0-5 <ee";tor-5. This means normal transient behaviour. In AlMg, on
389
the other hand, m « 1, so that the critical -0.5 1.5bG/o (for a spacing Q( ) is smaller than the steady-state value of the dislocation spacing £f~0'5 (see Sec. 8.4.2.1). This means inverted transient behaviour. We conclude that the transient deformation behaviour of materials is determined by the kinetics of dislocation motion and not by the dislocation structure. 8.4.2 Steady-State Deformation 8.4.2.1 Dislocation Structure The steady-state dislocation structure in solid solutions is easily investigated, because instabilities due to grain boundary motion are suppressed and solute hardening (plus precipitation at dislocations) pins the dislocation structure during cooling under stress. Figure 8-25 shows the characteristic dislocation spacings of AlZn and AlMg. The data are similar to those of pure Al [Fig. 8-16 and Eqs. (8-27) and (8-20)]. This means that the stress is the most significant parameter in determining the dislocation spacings. However, in the case of strong solid-solution hardening (AlMg at large stress), the spacings tend to be larger than for pure material as well as for AlZn at the same stress. This can be understood on the basis of Eq. (8-27): the dragging stress
390
8 High-Temperature Deformation and Creep of Crystalline Solids
less, Fig. 8-26 shows that the natural creep law is a good approximation of the behaviour in the limiting case of AlMg.
(a/G) GA|(300 K)/MPa 102 10° • o
10"
AI-5at%Zn 1 10' AI-5at%Mg : 10b
bG
10° 10' I,
10
10-
8.4.3 Response to Change in Deformation Conditions
Figure 8-27 shows that the transient behaviour of AlMg is inverted after a stress reduction. The stress dependences of the constant structure strain rates of AlZn and AlMg are presented in Fig. 8-28. In the case of AlMg, the stress exponent of ecs (~ 2) is smaller than the steady state exponent («3.5), consistent with the inverted character of transient deformation. The separation into two branches as observed for pure Al (Fig. 8-19) is not observed, because there is no longer a qualitative difference between the kinetics of free
101
10" 6
10" 5
10"=
10" 2
a/G Figure 8-25. Steady-state dislocation structure as a function of shear modulus normalized stress G/G(T) for solid solutions: Subgrain size w, average spacing £f~0*5 of free dislocations, and dislocation spacing s in subgrain boundaries for AlZn and AlMg. From Blum (1991).
tion motion is diffusive (governed by the diffusive motion of the cloud of solute atoms) and the natural third power creep law results for the same reasons as described above (Sec. 8.3.2.1). Note, however, that D is now the diffusivity of the solutes and that, due to the overproportional increase in afoU the stress exponent of the density of free dislocations is smaller than 2, and the stress exponent of the dislocation velocity is larger than one, resulting in a total stress exponent which is somewhat larger than the ideal value of 3. Neverthe-
10" Figure 8-26. Steady-state relation between normalized s and normalized stress for solid-solution hardened Al-5 at.% Mg (after Blum, 1991). The dotted reference line represents the natural creep law.
8.5 Particle-Hardened Metals 10"2
AIMg O
10.-3
C cs
^
-
= o0 = 70.4 MPa
573 K : 584 ! < • =10* O/OQ. 1 0.92
10"
0.83
CO
0.68
10"I-5
10,-6
: -
I
X
i to-
:
= 10"4 E
0.51
^~~~-———2£Z__ ^
-10-
0.17
_
I =1(T
o.iT"^* = 10"'
0.02
0.04
0.06
€-€f
Figure 8-27. s-e curves of solid-solution hardened Al5 at.% Mg for tests (tension and compression) with reductions in stress from oQ to o. The tests were done at slightly different temperatures of 573 K (full lines, corresponding to left a-axis) and 584 K (dashed lines, corresponding to right e-axis). The creep rates £cs measured at constant dislocation structure (introduced by creep at a0 to the strain s 0 , where the creep rate is s0) after the elastic and anelastic back flow (not shown in the figure) are marked by circles. From Blum (1991) with additional tests by Vogler (1990, unpublished).
391
dislocations and of subgrain boundaries. AlZn lies between pure Al and AIMg. The high stress branch of scs, corresponding to the reaction of free dislocations to the stress reduction, can be characterized in terms of an activation area which is significantly smaller than that of pure Al, eventhough the steady-state creep rate as well as the low stress branch of ecs are not much influenced by Zn. This means that Zn has a strong influence on the motion of free dislocations, but only a weak influence on the processes governing steady-state deformation. From the weak transient response described above it follows that solute atoms damp the deviations of £ from the value to be expected from the strain rule. This means that there are limited influences of cyclic changes in stress on the expected strain rate as long as the strain in the high stress part of the cycle is large. For small strains, however, deceleration of creep must be expected. We will describe cyclic creep deceleration in Sec. 8.5.3.
8.5 Particle-Hardened Metals 8.5.1 Evolution of Deformation Resistance and Dislocation Structure
Figure 8-28. Normalized ecs-
Due to the existence of particles as dislocation obstacles limiting the slip distance of dislocations, the rate of dislocation generation is higher in particle-hardened metals than in single-phase metals. This shortens the strain interval necessary to form the steady-state dislocation structure. At the same time, the stresses are high due to particle hardening. This means that high densities of dislocations have to be moved so that the absolute value of the strain to steady state is quite large. We will present two examples in the following, which are
392
8 High-Temperature Deformation and Creep of Crystalline Solids
representative for two different classes of particle-hardened metals, namely those containing a high initial dislocation density due to martensitic phase transformation or thermomechanical treatment such as mechanical alloying, and those with a moderate initial dislocation density. The first example is the ferritic 12%Crsteel (X20CrMoV121). The martensitic transformation followed by tempering results in a fine subgrain structure with a subgrain size smaller than 1 |im. Figure 829 shows the material behaviour during deformation at constant stress in compression, and at constant load in tension. The strain-time curve of the tensile test is characterized by a minimum creep rate in the secondary stage of creep followed by a pronounced tertiary stage (Fig. 8-29 a). The corresponding e-s curve (Fig. 8-29 b) shows that the tertiary stage of creep covers the largest interval of strain during the test. Interestingly, the s-s curves at constant stress in compression and in tension have the same shape up to a fairly large strain of about 0.14. This indicates that formation of pores influences the rate of tensile creep only in the last part of the tertiary stage of creep at a > 0.14. Therefore the main increase of £ in this stage must have reasons other than pore formation, namely loss of subgrain hardening, of particle hardening, or of both. In fact, both the subgrain size w and the particle size dp grow during creep (Fig. 8-29 b). It has been shown (Straub and Blum, 1991) that the rate of growth of w with strain is of the order of magnitude expected from dynamic subgrain growth. We conclude that the primary, secondary, and tertiary stages of creep all fall into the transient range preceding the steady state of deformation. The transient takes a strain of about 0.4, while the tertiary stage sets in at about 0.02. The initial increase of s in the tertiary stage is due to microstructural
0.2 tension, constant load j o n = 160 MPa
400
Figure 8-29. Creep and microstructural development at 873 K of the ferritic (tempered martensite) steel X20CrMoV121: (a) Tensile strain as a function of time at constant load and an initial stress G0 = 160 MPa (Eggeler et al, 1986). (b) Upper part: Compressive creep rate at 196 MPa (Straub, 1991; unpublished) and tensile creep rate from constant load curve in (a) as a function of strain. The tensile curve at a constant stress of 160 MPa was calculated from the constant load curve using a stress exponent of E of 13. Lower part: Increase in subgrain size w and size dp of carbide particles with strain for creep at constant load (
softening other than pore formation, the details of which still have to be clarified. The second example is a /-hardened Nibase superalloy single crystal oriented in [001]. It has a rather low initial dislocation density. Initially, the /-particles are cuboidal, surrounded by the y-matrix. Figure 8-30 represents a strain-time curve, which exhibits a pronounced tertiary stage following the minimum creep rate, as in the
8.5 Particle-Hardened Metals
previous example. The variation of the strain-rate reveals that here, the minimum creep rate occurs also in the transient range of creep before s has reached its steady-state value. The final microstructure contains plate-like rafts of y surrounded by yf. At the phase boundaries, dislocation networks have formed. Figure 8-31 shows that the networks have the character of subgrain boundaries. They also carry long-range internal stresses, as has been unequivocally shown by X-ray line analysis (Biermann et al., 1991). The uniaxial internal stresses add to the applied stress in the hard y'-phase and counteract it in the softer y-phase, thus achieving compatible deformation of the two phases. We conclude that the secondary stage of creep, where the creep rate is at a minimum, usually occurs at the beginning of the transient range of creep and must not
393
;&;:
100
> f ,""'•»•' 010 S, . ./.-af
1
H
Figure 8-31. Rafts of y in y' with dislocation networks in the phase boundaries after deformation of SRR 99 by e = 0.027 at a = 305 MPa and 1173 K (see Fig. 8-30). Section perpendicular to [001] crystal axis. From Hammer (1990).
be misinterpreted as steady-state rate of deformation. 8.5.2 Steady-State Deformation 8.5.2.1 Dislocation Structure
600
0.20
Figure 8-30. Tensile deformation of Ni-base superalloy single crystal SRR 99 at constant stress: (a) Strain-time curve, (b) strain rate-strain curve. After Hammer (1990).
In the steady-state of deformation which is reached after a sufficient amount of strain (see last section), the dislocation structure of particle-hardened alloys is not very different from that of single-phase materials as far as the dependence of the characteristic dislocation spacings on stress is concerned. An example is given in Fig. 8-32 for NiCr22Col2Mo. Similar data for particlehardened Al-alloys have been published by Blum (1991). At a given shear-modulus-normalized stress, the subgrain sizes of the particlehardened alloys are similar to (or a little smaller than) those of the pure metals. In other words: The factor cs in Eq. (8-20) is not increased by particle hardening. A special situation is encountered in the y'hardened Ni-base superalloys, where the location of the subgrain boundaries is die-
394
8 High-Temperature Deformation and Creep of Crystalline Solids
in particle-hardened than in pure materials. This has been clearly proven by Singer et al. (1980). They showed that the spacing of free dislocations grows when the particlehardening term increases at constant applied stress. In other words: the factor c13 increases with particle hardening, indicating that the ratio oGijo entering c 13 [Eq. (8-27)] is reduced by particle hardening, because the particle hardening term crG p has to be subtracted from the athermal stress aG to arrive at oG f .
(a/G)G(1073K)/MPa
8.5.2.2 Equation of State
The drastic effect of particle hardening on the rate of deformation is shown in Fig. 8-33 for a mechanically alloyed Al. This material contains Cu and Mg in solid solution and is also hardened by a large
10-= Figure 8-32. Steady-state dislocation structure as a function of shear modulus normalized stress ojG (T) for the solute and carbide hardened Ni-base alloy NiCr22Col2Mo: Subgrain size w, average spacing £f~0'5 of free dislocations, and dislocation spacing s in subgrain boundaries. For most of the data the temperature of deformation was 1073 K. After An (1989).
tated by the y-y' phase boundaries (Fig. 8-31). But even here the estimated value of the factor 3 c8 « 24 is of the same order as for pure materials at the same (normalized) stress. The situation is somewhat different for the average spacing £f~0"5 of free dislocations (not entangled at particles and subgrain boundaries). gf0'5 tends to be larger 3
Calculated from w « 1 urn (Fig. 8-31) at 305 MPa, b = 0.254nm, G(1173 K) = [0.5(CX1 - C 1 2 )C 4 4 ] 0 5 = 57.2 GPa (Hammer, 1990).
10"' Figure 8-33. Relation between normalized minimum creep rate and normalized stress for mechanically alloyed (solute and particle hardened) AlCu4Mgl.5 (IN 9021) (after Blum, 1991). The dotted reference line represents the natural creep law.
395
8.5 Particle-Hardened Metals
number of fine carbide and oxide particles. There is a kind of threshold stress where the minimum creep rate decreases over several orders of magnitude with decreasing stress. The simplest way to account for this threshold effect of particles is to replace the stress in the phenomenological power law [Eq. (8-26)] by the difference G — GG p of the applied stress and the athermal stress due to particle hardening: Q
(8-32)
where n0 is the stress exponent of the steady-state deformation of a particle-free reference material having the same composition as the matrix between the particles. The problem is to obtain the appropriate value of oG p (see e.g., Blum and Reppich, 1985 and Chap. 7 of this Volume). Basically, the situation is as follows: At high temperatures GG p is smaller than the stress for bypassing incoherent particles (Orowan stress) or cutting coherent particles, because the dislocations can climb over the particles. General climb leads to very small values of GG p . A reason for the relatively large values of the threshold stress for incoherent particles is the attractive interaction between particles and dislocations due to the partial relaxation of the dislocation line energy in the particle interface. However, even this attractive interaction does not lead to an absolute threshold below which the strain rate is zero. This can be seen for instance in Fig. 8-33: the emin-cr curve is S-shaped, which means that some deformation continues to occur even below the threshold stress. This has been explained by thermally activated detachment of dislocations from particles. According to Rosier and Arzt (1990) the stress exponent nmin = dlogs min /dlogo- varies for G < <7G,p
(which in this case designates the athermal detachment stress) approximately as: 05
2/cBr (8-33) with stress, temperature, particle radius r and the factor ferel (< 1) of relaxation of the dislocation line energy at particles. This means that nmin is nearly constant in the interval 0.2 < G/GGP < 0.9. The detachment process is easier for small than for large particles. This not only confines the increase of particle hardening with decreasing particle size, but also has the beneficial effect of limiting the degradation of the particle hardening with growth of particles during service at high temperatures. The minimum creep rate smin arises as a result of hardening, due to the development of the dislocation structure, and softening, due to changes in the dislocation structure and degradation of the particle structure (e.g., change in shape and growth of particles). In addition, there may be a degrading influence by the growth of cavities, although this influence is usually of importance only in the later stages of tertiary creep shortly before fracture (see Sec. 8.5.1). It is clear that emin does not have a simple physical meaning. Nevertheless, it is an important quantity in estimating the creep life to a certain strain sY or to fracture, simply because the time to pass through the minimum of the creep rate represents the largest portion of the total life. It is therefore not surprising that the creep life tx and 8min are closely related (MonkmanGrant relation): 14
-
- f
min
dr (8-34) —j as h— c 14 is the Monkman-Grant constant. It depends on the shape of the log £-£ curves and t
396
8 High-Temperature Deformation and Creep of Crystalline Solids
Figure 8-34. o-oG,v a s a function of temperature-compensated time for Ni-base superalloy single crystal (Hammer, 1990). QSD: activation energy of self-diffusion. 10"1
10
10"
t f exp[-Q SD /(k B T)]/h is truly constant if the curves are congruent, i.e., if smin/s (e) is independent of stress. According to Eq. (8-32) smin can be replaced by cr — (7G p for a > aG p . Figure 834 shows an example of the MonkmanGrant relation, modified according to the threshold-stress concept, for Ni-base superalloy single crystals containing widely differenting initial y/yf and dislocation microstructures. 8.5.3 Response to Change in Deformation Conditions
There are few systematic studies of the transient response of particle-hardened materials to changes in s or a. As particlehardened alloys are usually additionally hardened by solute atoms, their transient deformation response lies somewhere between class M (normal transients) and class A materials (inverted transients). Figure 8-35 shows the transient response of the Ni-base alloy NiCr22Col2Mo to three different reductions in stress. The response is neither normal nor inverted, but mixed. From stress reduction tests of this kind, the parameters crs*ol and m [Eq. (8-30)] have been obtained. With these parameters, the deformation behaviour of NiCr22Col2Mo could be quantitatively described on the
basis of the one-parameter model (An etal., 1990). Cyclic repetition of stress changes has little effect on the rate of creep of NiCr22Col2Mo at 873 K, if the strain in the high-stress (a2) phase of the cycle is large compared to the anelastic back strain in the low-stress (0^) phase. This is because the transients are not very pronounced 10"
223 MPa -= 191 MPa 179 MPa
10-* c/)
141 MPa
10'
NiCr22Co12Mo 1073 K 0.05 < e o < 0.14
10- 5
0.02
0.04
0.06
Figure 8-35. Transient deformation during creep of NiCr22Col2Mo at 223 MPa after stress reduction by different amounts and after restoration of initial stress (after Wolf, 1990).
8.6 Ceramics
(Fig. 8-35) and accelerations of creep are more or less compensated by decelerations. The maximum observed values for the acceleration factor are around F = 1.3 (Wolf, 1990). However, if the nonelastic strain in the high-stress phase is of the same order as that of the anelastic back strain in the lowstress phase, deceleration of creep is commonly observed. Figure 8-36 shows a test where the (average) rate of cyclic creep scyc is smaller than expected from the strain rule by a factor of 40 (i.e., F = 0.025). Note that the strain rates in the high-stress and the low-stress phase of the cycle are not resolved in Fig. 3-36. The reason for this deceleration of creep is that the force on 10"
NiCr22Co12Mo 1073 K 170 MPa
179 MPa
CO
F=0.025
10"' Parameters for cyclic deformation: A t ^ 11.0s At 2 =11.5s <3>=180 MPa Rs 0 monotonic cyclic
10*
0.09
0.11
397
the obstacles to dislocation motion (e.g., particles, subgrain boundaries) is relaxed during anelastic back flow, so that the force is smaller in the high-stress phase than during monotonic deformation. In this sense, cyclic creep with small absolute magnitude of R = ojo2 can be regarded as a stress reduction test. In fact this analogy is quite realistic, because there is a transient response in the period of cyclic creep after the mode change from monotonic to cyclic loading, where the average rate of cyclic creep behaves like the monotonic rate of creep after a large stress reduction (compare Figs. 8-35 and 8-36). In addition, it has been found that the dislocation structure coarsens after the mode change although the high stress a2 during cyclic creep is the same as the stress during the former monotonic creep (An, 1988). Switching back from cyclic to monotonic creep at o = a2 causes a transient reaction, which is qualitatively the same as the response to an increase in stress during monotonic creep (Fig. 8-35). Cyclic-creep deceleration tends to prolong the creep life. However, part of this prolongation may be compensated by a reduction in fracture strain.
8.6 Ceramics
monotonic
0.13
0.15
0.17
e Figure 8-36. e-e curve for tensile creep of NiCr22Col2Mo at 1073 K with a change in deformation mode from monotonic creep to cyclic creep (R « 0, At2/At = 0.51) and back. The change from monotonic to cyclic creep is connected with a sudden decrease in creep rate (cyclic-creep deceleration) followed by a transient in e. In the steady state of cyclic creep after the transient, the rate factor F = £cyc/s'cyc « 0.025. This corresponds to a "missing strain" per cycle of (ecyc - s'cyc)At = 2.7 x 10" 5 . The change from cyclic back to monotonic creep is followed by a transient of the same type as the transient after an increase in stress. From An (1988).
There is a great deal of similarity in the high-temperature plastic deformation of metals and ceramics (Cannon and Langdon, 1983 and 1988). In the present description, we will follow the recent review of Hiibner (1991). When plastic deformation of ceramics is carried by dislocations, the microstructural processes are very much the same as described in the previous sections for metals. Dislocation motion is easy in materials like MgO and UO 2 . However, due to the differences in atomic bonding and crystal structure, the mobility
8 High-Temperature Deformation and Creep of Crystalline Solids
398
of dislocations is usually restricted by strong lattice friction (deep Peierls valleys). Therefore, alternative mechanisms of plastic deformation by transport of matter are much more important than in metals, especially at low stresses and small grain sizes.
104 HPSN + Y2O3 1673 K 90MPa
CO
8.6.1 Evolution of Deformation Resistance In the case of plastic deformation by dislocation motion, the evolution of the dislocation structure and the corresponding evolution of the deformation resistance is not different from that of metals. Transient ranges of deformation occur not only for deformation by dislocation motion, but also for deformation by transport of matter. An example is shown for the case of ceramics containing an amorphous (glassy) grain-boundary phase of width ag. According to Hiibner (1991), the deformation is due to a squeezing of the viscously flowing amorphous phase out of the boundary region between the facets of two grains, which are compressed against each other. The viscous flow is assumed to be controlled by the condition of maintaining a laminar flow through the channel between the grain facets. The model yields a transient creep rate & which decreases with time t according to: -1.5
6 r,\d, (8-35) Note the similarity of Eq. (8-35) to Eq. (8-3) for diffusional flow via grain boundaries with respect to stress and grain-size dependence. The diffusion coefficient is replaced by the inverse of the viscosity Y\. The decrease of s with time is due to the decrease in width of the channel between grain boundaries under compression. The maximum achievable strain emax « #g/dg is given by contact of the rigid grains. Figure 8-37
1CT
Iff 7 0.01
0.02
0.03
0.04
€ Figure 8-37. s as a function of s for hot-pressed Si3N4 with Y 2 O 3 (Messner, 1990; cited by Hubner, 1991).
shows a measured e-e curve for Si 3 N 4 . There is very good agreement between the measured points and the curve calculated from Eq. (8-35) with reasonable values for the viscosity and initial width of the amorphous grain boundary layer. The creep rate falls rapidly by nearly two orders of magnitude. The subsequent decrease with strain is very slow giving the impression of a steady state of deformation. The increase in 8 due to fracture is not described by the model. 8.6.2 Steady-State Deformation In the case of plastic deformation of crystalline ceramics by dislocations, the dislocation structure and the equation of state are the same as for metals, i.e., a subgrain structure develops and the empirical law, Eq. (8-26) holds. In most cases, however, deformation of grains by transport of matter from regions of compressive stress to regions with tensile stress is faster than by dislocation mo-
8.7 Modeling of Deformation
tion. In single-phase ceramics, matter transport is achieved by diffusional flow. The rate-controlling process is diffusion of the slower moving species via the fastest path. In ceramics with an amorphous grain-boundary phase, transport may occur through the amorphous or liquid film. The driving force is the same as for diffusive flow, namely a gradient of the chemical potential due to the applied stress. The potential gradient causes a change in the solubility of the grain-boundary phase, which in turn leads to dissolution of material in grain-boundary regions of compressive stress, and precipitation in grain-boundary regions of tensile stress. The rate-controlling step may be the diffusion through the grain-boundary phase or the dissolution of material at the interface. The kinetic law is similar to that of diffusive flow [linear stress-dependence, grain-size dependence with an exponent between 1 and 3 (Hiibner, 1991)]. In addition, the steadystate strain rate increases with increasing width ag of the grain-boundary layer. The growth of cavities is coupled to local plastic deformation in the vicinity of the cavities. This mechanism of plastic deformation is particularly pronounced in ceramics with glassy phases and causes the creep rate in tension to be distinctly higher than in compression of engineering ceramics of low ductility.
8.7 Modeling of Deformation The modeling of plastic deformation at elevated temperatures by transport of matter is essentially solved. Refinements may be necessary to incorporate the effect of the grain boundary structure (particles at grain boundaries and grain boundary dislocations) on diffusive flow (Frost and Ashby, 1982). However, in spite of signifi-
399
cant progress, the modeling of plastic deformation by dislocations connected with solute atoms and second phases is still unfinished. Below, we list some of the main experimental results described in the preceding sections, which will have to be taken care of by a model: (1) The steady-state dislocation structure is generally a subgrain structure. (2) The characteristic spacings of the dislocation structure depend mainly on shear modulus normalized stress and, to some extent, on particle hardening. (3) Subgrain formation is a hardening process in class M materials. (4) Subgrain boundary migration keeps the subgrain structure equiaxed and leads to annihilation of dislocations. (5) The steady-state strain rate is coupled with the rate of subgrain boundary migration. (6) The kinetics of motion of the free dislocations in the interior of subgrains is usually different from that of the subgrain boundaries. The simple £-model (Sec. 8.2.4), which uses the (total) dislocation density Q as the only microstructural parameter, does in fact explain some main points of the kinetics of plastic deformation, e.g., the normal and inverted transient behaviour. However, it is not satisfactory in describing both subgrain-boundary dislocations and free dislocations in terms of the same parameter Q, because most of the subgrain boundary dislocations are geometrical in nature, i.e., they form a grain boundary and do not necessarily carry long-range stresses as free dislocations do. In the following, we present a more detailed model which makes a clear distinction between free dislocations and subgrain boundaries. Although this approach is complicated, it may be useful in that it can explain more features of the
400
8 High-Temperature Deformation and Creep of Crystalline Solids
plastic deformation behaviour with a single set of equations. Let us start with the subgrain structure of Fig. 8-8 c, part of which is shown at higher magnification in Fig. 8-38. From the experimental data, we assume that the motion of subgrain boundaries is slow compared to the motion of free dislocations. This means that the subgrain boundaries can be regarded as fixed with respect to the glide motion of free dislocations. The total rate of plastic deformation is made up from the contributions of free dislocations (gf) and those of migrating boundary dislocations (sm): (8-36)
s = ef
The influence of subgrain boundaries on plastic deformation is described in terms of the composite model (Nix and Ilschner, 1980; Blum et al., 1989 a and b). Similar to a cell boundary (Sec. 8.3.1), a subgrain boundary is considered to constitute a hard region of width a, deforming parallel to the soft subgrain interior, so that the
local stress oh in the vicinity of a subgrain boundary is larger than the applied stress a, while the local stress as in the soft subgrain interior is smaller than a [Eq. (8-23)]. The idea of a stress concentration at subgrain boundaries is supported by direct electron microscopic observations of strongly curved dislocation segments at subgrain boundaries (Morris and Martin, 1984) and, indirectly, by the observation (Biberger, 1989) that the velocity of subgrain boundary migration decreases strongly after a decrease in stress (consistent with a decrease in concentrated stress (7h). A different view has been taken by Derby and Ashby (1987), who considered hard and soft subgrains. But their model does not explain the variation of the creep rate with subgrain size, in contrast to the approach presented here (see below). Previously, the composite model was used to calculate the flow stress (Mughrabi, 1980; Prinz and Argon, 1984; Nix et al., 1985) on the basis of Eq. (8-23). Blum et al. (1989a, b) and Vogler and Blum (1990) have applied the model to describe the kinetics of flow. It turns out that the rate of deformation by free dislocations is given by:
X X v
\ x /
x
/. + / h = l
xx
\
'
x
Figure 8-38. Magnified view of subgrain structure from Fig. 8-8 c. Free dislocations are held up at the boundaries causing the local plastic strain in the subgrain interior to be larger than the local plastic strain in the region adjacent to the subgrain boundaries. According to the composite model, this plastic strain inhomogeneity leads to internal stresses.
(8-37)
where effS and sf h are the local plastic strains in the soft and the hard regions, respectively. The volume fraction of the hard region is the product of the boundary area per volume, 2/w, and the width a of the hard region adjacent to the boundaries: = 2a/w
(8-38)
gf s and £m are expressed by the Orowan equation (Eq. (8-6)) as: Bfs = 0bgfv^s
(8-39)
&m=*bQbVm
(8-40)
8.7 Modeling of Deformation
where vfs and vm are the velocities of free dislocations in the subgrain interiors and of migrating subgrain boundaries, respectively, and gh is the density of subgrain boundary dislocations, which can be expressed as the product of the boundary area per volume, 2/w, and the dislocation length per boundary area, « 2/s: 4 sw
(8-41)
bw
(@mis: angle of subgrain misorientation). The local strain rate % h due to motion of free dislocations through the subgrain boundaries by the knitting process (insertion of dislocations into boundaries, extraction from boundaries, Blum, 1977; Caillard and Martin, 1987) is coupled to the velocity of subgrain boundary migration in order to take the experimentally observed coupling between boundary migration and "knitting out" (extraction) (Caillard, 1985) into account: sfh=f i, n
gm t/ m
(8-42)
in
v
/
The above equations constitute the kinetic laws of plastic deformation. The laws of evolution of the parameters of the dislocation structure can be formulated in a phenomenological way on the basis of the steady-state values of these parameters using differential equations of the type: = cx(xcc
(8-43)
-x)de 0>5
00
where x stands for w, gf , and / sub , x is the steady-state value of x, and cx is a constant which determines the rate at which x approaches x00 with strain in an exponential fashion. The model works as follows: The glide of free dislocations through the subgrain interiors contributes most of the strain, but is controlled by the motion of dislocations through the hard boundary regions, which is in turn coupled to the migration of sub-
401
grain boundaries. Note that the three processes are not independent of each other, but are related by internal stresses. Although subgrain migration is only responsible for about 10% of the strain, it determines the rate of steady-state deformation. Subgrain boundaries move either by diffusion control (Derby and Ashby, 1987) or by glide control (Caillard, 1985) depending on which process is faster. Glide has the lower activation energy and the higher stress sensitivity. Therefore it is the faster process at low temperatures. This explains the change in stress dependence of the steady-state deformation rate with stress, called powerlaw breakdown (see Fig. 8-15). Presently, we want to show that the model presented above is able to account for the steady-state deformation as well as for the transient deformation of pure Al, a representative of the class M of materials, as well as for AlMg, representative of class A. We will proceed as follows in three steps: (1) Fitting of the model to the observed stress and temperature dependence of the steady-state deformation behaviour of pure Al. (2) Calculation of the transient behaviour of Al. (3) Modification of the model to account for the steady-state and the transient behaviour of AlMg. Step (1), the fitting procedure, has been achieved simply by Vogler and Blum (1990) choosing: sinh
w
\MkBT \M kBT
(8-44)
(8-45)
with a constant stress concentration factor aja (= 5) in the steady state of deformation and a constant value of Aah at con-
402
8 High-Temperature Deformation and Creep of Crystalline Solids
stant temperature. It has been shown (Blum et al., 1991) that the activation area in the soft region, Aas, to varies with stress as required from experiment4. Figure 8-39 shows that the steady-state relation between e and a is in agreement with the observed one. In the second step the model was used to predict the transient deformation behaviour after a decrease in stress. The result is reproduced in Fig. 8-40 a. The deviations between calculation and experiment, in the beginning of the transient after large stress reduction, have been attributed to the neglection of that part of the anelastic back flow, which is due to short-range dislocation interactions. In the third step, we will now change the equations of the model consistent with the requirements imposed by the viscous nature of dislocation motion in AlMg. The most important change is to replace Eq. (8-44) by: ^ s = c17(T)<sol
(8-46)
where the factor c 17 depends on temperature and concentration of solute atoms. Certainly more changes such as replacement of <7h in Eq. (8-45) by a£ sol (in analogy to <7*sol) and a change in the dislocation spacings (see Sec. 8.4.2.1) would be necessary. However, we restrict the number of changes to one, by switching from Eq. (8-44) to Eq. (8-46), leaving all other aspects of the model unchanged in order to show that this change is the essential one. It changes the deformation characteristics from that typical for class M to that typical for class A. Figure 8-39 shows the calculated steadystate s — a relation of AlMg. The exponential stress dependence of the steady-state 4 The stress dependence of Aas presented by Vogler and Blum (1990) is incorrect; the corrected version was published by Blum et al. (1991).
10 Figure 8-39. Normalized creep rate as a function of normalized stress for Al and AlMg as calculated from the model. The data for Al from Fig. 8-15 and the natural creep law are shown for comparison. From Zhu et al. (1991), unpublished.
creep rate of Al has turned into a powerlaw dependence for AlMg with a stress exponent of 3, close to the observed value (Fig. 8-26). In contrast to Al, the steadystate stress concentration factor ojo is no longer constant for AlMg, but decreases with increasing stress from the Al-value of 5 (at a small stress chosen to be a = 3 MPa) to 1 (at a = 75 MPa), in order to allow G*SO1 to increase with a, so that the free dislocations in the interior of subgrains can keep up with the boundary dislocations. The calculated transient response of AlMg is shown in Fig. 8-40 b. Again it is seen that replacing Eq. (8-44) by Eq. (8-46) leads to a change from normal to inverted transient behaviour, as it is found experimentally.
8.9 References
10- 2
1
z
i
r
i
i
i
=
Al
:
523 K
I
_o0 = 20MPa :
~~ i
:
0.74
1
_i
0.90
Iiii i i
Iff 4 E
lO"5^
10,-6
-
-
i
i
403
This shows that the macroscopically observed behaviour depends sensitively on the kinetics of flow in the soft regions, although the hard regions are essential in controlling the rate of creep. The success of the composite model in describing not only the steady state, but also the transient deformation of single-phase materials is promising and opens the prospect that it will be possible in the future to calculate the deformation of more complicated, two-phase materials under complicated conditions of loading on a microstructural basis.
i
0.02
0.04
0.06
e-e c
8.8 Acknowledgements Thanks are due to Q. Zhu, M. Eng., for preparing the drawings, to Prof. O. Vohringer and Dr. P. Anongba for supplying Figs. 8-22 and 8-9, to Dipl.-Ing. A. Absenger, Dipl.-Ing. B. Geibel, Dipl.-Ing. S. Straub, Dipl.-Ing. S. Vogler, and Q. Zhu, M. Eng., for contributing unpublished results, and to Profs. E. Kramer, P. Lukac and H. Mughrabi for critical reading of the manuscript.
(a)
8,9 References
0.06
(b) Figure 8-40. Measured and calculated transient creep rate-strain curves after stress reduction for (a) Al (Vogler and Blum, 1990) and (b) AlMg (Zhu et al., 1991; unpublished). The thick dashed lines were calculated from the composite model. The thin lines represent measured curves (from Figs. 8-17 and 8-27).
Alexander, H., Haasen, P. (1968), Solid State Phys. 22, 27. Ambrosi, P., Schwink, Ch. (1980), in: Proc. 5th Int. Conf. on the Strength of Metals and Alloys (ICSMA 5). Vol. 1: Haasen, P., Gerold, V., Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 2933. An, S. U. (1988), Dr.-Ing. Thesis, Univ. of ErlangenNurnberg. An, S. U., Wolf, H., Vogler, S., Blum, W. (1990), in: Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Evans, R. W. (Eds.). London: The Institute of Metals, pp. 81-95. Anongba, P. (1990), Thesis, Ecole Poly technique Federale de Lausanne. Barrett, C. R., Nix, W. D. (1965), Ada Met. 13,1247. Biberger, M. (1989), Dr.-Ing. Thesis, University of Erlangen-Numberg to appear 1992 in Phil. Mag. A.
404
8 High-Temperature Deformation and Creep of Crystalline Solids
Biberger, M., Blum, W. (1989), Scripta Met. 23,1419. Biberger, M., Blum, W. (1992), Phil. Mag. A 65, 757. Biermann, H., Kuhn, H.-A., Ungar, T., Hammer, X, Mughrabi, H. (1991), in: Proc. of the 9th Intern. Conf. on the Strength of Metals and Alloys (ICSMA 9): Brandon, D. G., Chaim, R., Rosen, A. (Eds.). London: Freund Publishing House, pp. 421-428. Blum, W. (1977), Z. Metallkde. 68, 484. Blum, W. (1991), in: Hot Deformation of Aluminum Alloys: Langdon, T. G., Merchant, H. D., Morris, J. G., Zaidi, M. A. (Eds.). Warrendale: TMS, pp. 181-209. Blum, W, Ilschner, B. (1967), Phys. Stat. Sol. 20, 629. Blum, W, Reppich, B. (1969), Ada Met. 17, 955. Blum, W., Reppich, B. (1985), in: Creep Behaviour of Crystalline Solids: Wilshire, B., Evans, R. W. (Eds.). Swansea: Pineridge Press, pp. 83-135. Blum, W., Portella, P. D., Feilhauer, R. (1983), in: Festigkeit und Verformung bei hoher Temperatur: Ilschner, B. (Ed.). Oberursel: Deutsche Gesellschaft fur Metallkunde, pp. 41-59. Blum, W., Rosen, A., Cegielska, A., Martin, I L. (1989 a), Acta Met. 37, 2439. Blum, W, Yogler, S., Biberger, M., Mukherjee, A. K. (1989b), Mater. Sci. Eng. A 112, 93. Blum, W, Straub, S., Vogler, S. (1991), in: Proc. of the 9th Intern. Conf on the Strength of Metals and Alloys (ICSMA 9): Brandon, D. G., Chaim, R., Rosen, A. (Eds.). London: Freund Publishing House, pp. 111-126. Caillard, D. (1985), Phil. Mag. A 51, 157. Caillard, D., Martin, J. L. (1987), Rev. Phys. Appl. 22, 169. Cannon, W. R., Langdon, T. G. (1983), J. Mater. Sci. 18, 1. Cannon, W. R., Langdon, T. G. (1988), /. Mater. Sci. 23, 1. Carrard, M., Martin, J. L. (1984), in: Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Evans, R. W. (Eds.). Swansea: Pineridge Press, pp. 27-38. Clauer, A. H., Wilcox, B. A., Hirth, J. P. (1970), Acta Met. 18, 381. Cottrell, A. H. (1953), Dislocations and Plastic Flow in Crystals. London: Clarendon Press. Derby, B., Ashby, M. F. (1987), Acta Met. 35, 1349. Eckert, J. (1986), Thesis Univ. Erlangen-Nurnberg. Eggeler, G., Blum, W. (1981), Phil. Mag. A 44, 1065. Eggeler, G., Ilschner, B., Schepp, P., Zohner, R. (1986), Mater. Technik 14, 187. Estrin, Y, Mecking, H. (1984), Acta Met. 32, 57. Ferreira, L, Stang, R. G. (1979), Mater. Sci. Eng. 38, 169. Frost, H. X, Ashby, M. F. (1982), Deformation-Mechanism Maps. Oxford: Pergamon Press. Gottler, E. (1973), Phil. Mag. 28, 1057. Haasen, P. (1989), J. Phys. France 50, 43. Hammer, X (1990), Dr.-Ing. Thesis, Univ. ErlangenNurnberg.
Hart, E. W. (1970), Acta Met. 18, 599. Hart, E.W., Li, C. Y, Yamada, H., Wire, G. L. (1975), in: Constitutive Equations in Plasticity: Argon, A. S. (Ed.). MIT Press, Cambridge (Mass.), pp. 149-197. Hasegawa, T., Karashima, S., Hasegawa, R. (1971), Met. Trans. 2, 1449. Hasegawa, T., Yakou, T., Karashima, S. (1975), Mater. Sci. Eng. 20, 267. Hiibner, H. (1991), in: Micro structure and Mechanical Properties of Materials. Tenckhoff, E., Vohringer, O. (Eds.). Oberursel: DGM-Informationsgesellschaft, pp. 151-160. Kassner, M. E., McMahon, M. E. (1987), in: Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Evans, R. W. (Eds.). London: The Institute of Metals, pp. 29-40. Konig, G., Blum, W. (1980 a), in: Proc. 5th Int. Conf. on the Strength of Metals and Alloys (ICSMA 5). Vol. 1: Haasen, P., Gerold, V., Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 363-367. Konig, G., Blum, W. (1980b), Acta Met. 28, 519. Kocks, U. F. (1976), /. Eng. Mater. Tech. (ASME-H) 98, 76. Kocks, U. F. (1985), Met. Trans. 16 A, 2109. Kubin, L. P., Estrin, Y (1985), Acta Met. 33, 397. Livingston, X D. (1962), Acta Met. 10, 229. McQueen, H. X, Jonas, X X (1975), in: Treatise on Materials Science and Technology. Vol. 6: Plastic Deformation of Materials: Arsenault, X R. (Ed.). New York: Academic Press, pp. 393-493. McQueen, H. X, Knustad, O., Ryum, N., Solberg, X K. (1985), Scripta Met. 19, 73. Mecking, H. (1977), in: Work Hardening in Tension and Fatigue: Thompson, A. W. (Ed.). Warrendale: AIME, pp. 67-87. Mecking, H., Kocks, U. F. (1981), Acta Met. 29, 1865. Messner, E. (1990), Thesis, Tech. Univ. HamburgHarburg. Morris, M. A., Martin, X L. (1984), Acta Met. 32, 549. Mughrabi, H. (1975), in: Constitutive Equations in Plasticity: Argon, A. S. (Ed.). Cambridge (Mass.): MIT Press, pp. 199-250. Mughrabi, H. (1980), in: Proc. 5th Int. Conf on the Strength of Metals and Alloys (ICSMA 5). Vol. 3: Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1615-1633. Mughrabi, H. (1983), Acta Met. 31, 1367. Navratil, V, Hamersky, M., Blazkova, X (1974), Czech. J. Phys. B24, 907. Nilsvang, N. (1989), Thesis, Ecole Poly technique Federate de Lausanne. Nix, W. D., Ilschner, B. (1980), in: Proc. 5th Int. Conf. on the Strength of Metals and Alloys (ICSMA 5). Vol. 3: Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1503-1530. Nix, W D., Gibeling, X C , Hughes, D. A. (1985), Met. Trans. 16 A, 2215.
8.9 References
Oikawa, H., Langdon, T. G. (1985), in: Creep Behaviour of Crystalline Solids: Wilshire, B., Evans, R. W. (Eds.). Swansea: Pineridge Press, pp. 33-82. Poirier, J.-P. (1985), Creep of Crystals. Cambridge: Cambridge University Press. Prinz, F. B.? Argon, A. S. (1984), Acta Met. 32, 1021. Raj, S. V., Pharr, G. (1986), Mater. Sci. Eng. 81, 217. Raj, S. V., Pharr, G. M., Whittenberger, J. D. (1989), Mater. Sci. Eng. A 113, 161. Reppich, B., Rieger, K., Miiller, G. (1990), Z. Metallkde. 81, 166. Rosier, I, Arzt, E. (1990), Acta Met. 38, 671. Schroter, W, Siethoff, H. (1984), Z. Metallkde. 75, 482. Schwink, C , Gottler, E. (1976), Acta Met. 24, 173. Seeger, A., Wilkens, M. (1967), in: Reinststoffprobleme III, Realstruktur und Eigenschaften von Reinststoffen; Rexer, E. (Ed.). Berlin: Akademie-Verlag, pp. 29-122. Sherby, O. D., Burke, P. M. (1967), in: Progress in Materials Science. Vol. 13: Mechanical Behaviour of Crystalline Solids at Elevated Temperature: Chalmers, B., Hume-Rothery, W. (Eds.). Oxford: Pergamon Press, pp. 325-390. Siethoff, H., Schroter, W. (1978), Phil. Mag. A 37, 711. Singer, R., Blum, W, Nix, W. D. (1980), Scripta Met. 14, 755. Stejskalova, V., Hamersky, M., Lukac, P., Vostry, P., Sprusil, B. (1981), Czech. J. Phys. B31, 195. Straub, S., Blum, W. (1990), Scripta Met. Mater. 24, 1837. Straub, S., Blum, W. (1991), Steel Res. 62, 72. Viereck, D., Merckling, G., Lang, K. H., Eifler, D., Lohe, D. (1989), in: Festigkeit und Verformung bei hoher Temperatur: Schneider, K. (Ed.). Oberursel: DGM-Informationsgesellschaft, pp. 201-218. Vogler, S., Blum, W. (1990), in: Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Evans, R. W. (Eds.). London: The Institute of Metals, pp. 65-79. Vogler, S., MeBner, A., Blum, W. (1991), Mater. Sci. Eng. A 145, 13. Wilshire, B. (1990), in: Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Evans, R. W. (Eds.). London: The Institute of Metals, pp. 1-9. Wilshire, B., Evans, R. W. (1985), Creep of Metals and Alloys. London: The Institute of Metals.
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Wolf, H. (1990), Dr.-Ing. Thesis, Univ. ErlangenNurnberg. Zeyfang, R., Buck, O., Seeger (1974), Phys. Stat. Sol. (b) 61, 551.
General Reading Alexander, H. (1986), in: Dislocations in Solids. Vol. 7: Nabarro, F. R. N. (Ed.). Amsterdam: North Holland, pp. 115-234. Bendersky, L., Rosen, A., Mukherjee, A. K. (1985), Int. Met. Rev. 1,1. Cadek, J. (1988), Creep in Metallic Materials. Amsterdam: Elsevier. Caillard, D., Martin, J. L. (1987), Rev. Phys. Appl. 22, 169. Cannon, W R., Langdon, T. G. (1983), J. Mater. Sci. 18, 1. Cannon, W. R., Langdon, T. G. (1988), J. Mater. Sci. 23, 1. Dislocations and Properties of Real Materials, Proc. Conf. on the 50th Anniversary of the Concept of Dislocations in Crystals, Book 323 (1985). London: The Institute of Metals. Frost, H. I, Ashby, M. F. (1982): Deformation-Mechanism Maps: Oxford: Pergamon Press. Haasen, P. (1974), Physikalische Metallkunde. Berlin: Springer. Ilschner, B. (1973), Hochtemperaturplastizitdt. Berlin: Springer. McQueen, H. I, Jonas, J. J. (1975), in: Treatise on Materials Science and Technology. Vol. 6: Plastic Deformation of Materials: Arsenault, R. J. (Ed.). New York: Academic Press, pp. 393-493. Nix, W. D., Ilschner, B. (1980), in: Proc. 5th Int. Conf on the Strength of Metals and Alloys (ICSMA 5). Vol. 3: Haasen, P., Gerold, V, Kostorz, G. (Eds.). Oxford: Pergamon Press, pp. 1503-1530. Poirier, J.-P. (1985), Creep of Crystals. Cambridge: Cambridge University Press. Riedel, H. (1987), Fracture at High Temperatures. Berlin: Springer. Wilshire, B., Evans, R. W. (Eds.) (1981, 1984), Creep and Fracture of Engineering Materials and Structures. Swansea: Pineridge Press (1987, 1990), London: The Institute of Metals.
9 Superplasticity in Metals, Ceramics and Intermetallics Amiya K. Mukherjee Department of Mechanical, Aeronautical and Materials Engineering, University of California, Davis, CA, U.S.A.
List of 9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.5 9.5.1 9.5.2 9.5.2.1 9.5.2.2 9.5.3 9.5.4 9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.6.5 9.7 9.7.1 9.7.2 9.7.3 9.7.3.1 9.7.4 9.8 9.9 9.10 9.11 9.12
Symbols and Abbreviations Introduction General Requirements for Superplasticity Mechanical Aspects Region III Region II Region I Microstructural Aspects of Superplasticity Grain Shape and Grain Size Grain Boundary Migration and Sliding Grain Rotation and Grain Rearrangement Dislocation Activity Characteristics of Models and Constitutive Relations Diffusion-Accommodation Model by Ashby and Verrall Dislocation-Accommodation Mechanisms Pile-Ups Within the Grains Dislocation Pile-Up in Grain Boundary Plane Accommodation by Both Diffusional and Dislocation Motion Reflections on Microstructure and Mechanisms Cavitation and Failure in Superplasticity Flow Localization During Superplastic Deformation Cavitation Failure in Superplastic Alloys The Nucleation of Cavities The Growth of Cavities Influence of Hydrostatic Pressure on Cavitation Recent Advances in Superplasticity High Strain Rate Superplasticity "Low-Temperature" Superplasticity Superplasticity in Ceramics Reflections Superplasticity in Ordered Intermetallics Unconventional Approaches Superplastic Forming Technology Conclusion Acknowledgements References
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
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List of Symbols and Abbreviations A A b c d Do £gb Dv G k K lo Lo m n V P P Q r rc dr/ds R t T
cross-sectional area microstructural and mechanism dependent constant magnitude of the Burgers vector molar solubility of the crystal in fluid grain size diffusion coefficient grain boundary diffusion coefficient volume diffusion coefficient shear modulus Boltzmann constant temperature and structure dependent parameter initial length of a segment initial gauge length strain rate sensitivity parameter stress sensitivity of the strain rate inverse grain size exponent tensile force imposed hydrostatic pressure activation energy cavity radius critical radius cavity growth rate per unit strain gas constant time absolute temperature
a
a Q
factor depending on the structure of the boundary strain rate total strain rate strain hardening coefficient surface energy intragranular strain diffusion creep strain grain boundary sliding strain grain boundary free energy thickness of the boundary viscosity of grain boundary phase stress atomic volume
DB GBM GBS
diffusion bonding grain boundary migration grain boundary sliding
8
y y Vis 7GBS
r s n
List of Symbols and Abbreviations
NCM SEM SPF TEM TZP
nanocrystalline materials scanning electron microscope superplastic forming transmission electron microscope tetragonal zirconia polycrystals
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9 Superplasticity in Metals, Ceramics and Intermetallics
9.1 Introduction The art and technology of metal forming date back to the bronze age. The technology was born with the ancient's discovery that heating metals and their alloys greatly improved their malleability. This ease of metal deformation at elevated temperatures presents the problem of maintaining dimensional limits in high temperature service applications as in modern power-plants, in an increasingly energyconscious industrial age. The understanding of the related phenomenon of creep deformation is the focus of Chapters 7 and 8 in this Volume. In relatively recent years, another line of investigation, also at elevated temperatures, has opened up new and exciting possibilities. Instead of trying to restrict the creep deformation, this newer approach takes full advantages of the ease of deformation at elevated temperatures; the metal or alloy, and increasingly ceramics, can be made to deform to very large strains in tension. This extremely large ductility associated with such deformation is termed superplasticity. Elongations of 200-500% are quite common, and in aluminum bronze Higashi et al. (1985) have attained 5500% elongation. The work of Jenkins (1928) in England in the late 1920s was probably the forerunner of superplasticity investigation. Pearson (1934) convincingly demonstrated the existence of superplasticity in a 1934 publication. During the ensuing period, the major part of superplastic investigation (Bochvar and Sviderskaya, 1945) was conducted in the USSR. The current emphasis on superplasticity owes much to the review work of Underwood (1962) and the early experimental work of Backofen and co-workers (1964). Superplasticity is being investigated now-a-days both for its scientific merit in
the context of fundamental flow and failure mechanisms as well as for its technological significance in forming operations. The superplastic forming of titanium, aluminum and nickel based commercial alloys is a reality, particularly in the airframe and gasturbine industry. There is considerable saving in materials cost and in laborintensive machining cost in this near-netshape forming process. The forming stresses or pressures are often quite low. Hence, the parts can be formed using methods (e.g., blow molding) that are usually reserved for thermoplastics. The finished product usually has excellent surface finish and quite isotropic mechanical properties. Because of this technological importance, there has been an accelerated growth in publications in both basic and applied studies in superplasticity. There has been a parallel increase in the number of international symposia on superplasticity and superplastic forming (Paton and Hamilton, 1982; Pearce and Kelly, 1985; Baudelet and Suery, 1985; Hamilton and Paton, 1988; Kobayashi and Wakai 1989; Heikkenen and McNelley, 1988; McNelley and Heikkenen, 1990; Merrilea et al., 1990). This presentation is going to concentrate on superplasticity in fine-grained polycrystalline solids, i.e., micrograin superplasticity. This type of superplasticity is by far the most significant one from a technological viewpoint and currently is receiving the most attention in research and development work. In the other type of superplasticity, i.e., the internal stress superplasticity, one observes Newtonian viscous behavior by thermal cycling of pure metals (having anisotropic thermal expansion coefficients), composite materials (with constituents having different thermal expansion coefficients) or thermal cycling through a solid-state phase change. This type of superplasticity will not be discussed
9.2 General Requirements for Superplasticity
here. Interested readers may wish to pursue recent reviews on internal stress superplasticity by Sherby and Wadsworth (1985 and 1989). It is not the intent to present here an exhaustive review of the phenomenon of micrograin superplasticity. Several books and reviews for general reading have appeared over the past several years (Padmanabhan and Davies, 1980; Gifkins, 1982b; Suery and Mukherjee, 1985; Pilling and Ridley, 1989; Mukherjee et al., 1989; Sherby and Wadsworth, 1989; Maehara and Langdon, 1990). Instead the current author wishes to present a short, but critical, review outlining the most salient aspects of superplasticity. This review is being presented in several sections. The general requirements for micrograin superplasticity are discussed first, the relation between the strain rate, stress, grain size and temperature is discussed in the context of the constitutive expression for superplasticity. The microstructural aspects of superplastic deformation e.g., grain boundary sliding, boundary migration, grain rotation, grain growth, etc., are discussed after that. The essence of the various theoretical models for superplasticity is reviewed based on the type of accommodation process and the rate controlling mechanisms. The very important role of the strain rate sensitivity parameter in promoting neck stability is discussed with relation to the microstructural alterations, specifically grain growth. The superplastic cavitation process is discussed in terms of nucleation, growth and interlinkage of voids. The effect of cavitation on post formed service properties and superimposition of hydrostatic gas pressure during deformation that minimizes the extent of cavitation are mentioned. Finally, the recent advances in superplasticity in ceramics and intermetallic compounds are explored.
411
9.2 General Requirements for Superplasticity The three prime requirements for the manifestation of micrograin superplasticity are (a) a fine (less than about 10 jim) and equiaxed grain size that is reasonably stable during deformation, (b) a temperature that is more than about half of the melting point of the matrix in absolute degrees and (c) a strain rate that is typically not too high (less than 10" 2 s" 1 ) or too low (more than 10 ~6 s" 1 ). Most superplastic materials are either dual-phase alloys (often eutectics of eutectoids) or they have quasi single phase microstructure where there are particles at the grain boundaries. Such materials satisfy the condition of some degree of stability of the microstructure (i.e., criterion a) because the strain enhanced grain growth is minimized by the chemical dissimilarity of the two phases in the dualphase microstructure or by pinning of the grain boundary by particles in the quasi single phase alloys. The second criterion essentially refers to the fact that solid state diffusion controlled processes are operative in superplasticity and as such superplasticity is a generic cousin of all other elevated temperature creep mechanisms. The third criterion essentially assures that the strain rate sensitivity parameter has a value large enough to promote the stability of external necks. As will be discussed in Sec. 9.4, the phenomenon of grain boundary sliding constitutes an important topological requirement in superplasticity. In order to facilitate the process of grain boundary sliding, the sliding interfaces should be high angle boundaries. (Low angle boundaries do not slide readily.) For similar reasons, the grain shape should be equiaxed because three dimensional grain sliding cannot take place easily in a matrix with elongated
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9 Superplasticity in Metals, Ceramics and Intermetallics
grain structure. The phenomena of grain boundary sliding and migration are often intimately linked processes. Hence, the grain boundaries of superplastic matrix should also be mobile.
9.3 Mechanical Aspects One of the remarkable characteristics of superplastic material is its large resistance to neck formation in tensile deformation. This resistance to necking is best measured by the value of the strain rate sensitivity parameter, m, given by aces™ where a is the stress and s is the strain rate. (In creep literature the more common relation is eoccr", where n is the stress sensitivity of the strain rate and obviously m = 1/n.) The simplest way to understand the importance of m in promoting large elongations is to consider the change in the specimen crosssectional area A, with time t, during a tensile test: dA/dt = - (P/K)1/m (A){m~1)lm where P is the tensile force and K is the temperature and structure dependent parameter in the simplified constitutive equation o = Ksm. The above discussion shows that the evaluation of the cross-section of the specimen depends on the current crosssection through the parameter m, becoming progressively independent of A, as m approaches 1. It follows that the necking tendency decreases a s m ^ l so that an intermediate value of m = 0.5 will lead only to diffuse necking and thus high macroscopic elongations to failure. A large rnvalue will be an ideal situation for neck stability. The maximum value of ra = l, i.e., w = l, is predicted if the whole polycrystalline material deforms by some form of diffusional creep, i.e., Coble (1963) or Nabarro-Herring (1948) creep. This has been observed in some fine-grained ceram-
ics (Venkatachari and Raj, 1986). In metallic systems and also in some ceramic and intermetallic matrix, however, the significant deformation mechanisms often include components of dislocation creep with n&5, along with other mechanisms which are related to grain boundary sliding, and diffusional creep. This produces an overall w-value that is usually closer to 2. Superplasticity is a close cousin of creep and in fact, it is one of the various elevated temperature micromechanisms of deformation. The general form of constitutive relations for high temperature deformation of crystalline solids was given by Mukherjee et al. (1969). In a subsequent paper, they incorporated additionally (Bird et al. 1969) the aspect of grain size in order to handle diffusion creep and superplasticity. The different theoretical models of high temperature creep and superplasticity can be rephrased in terms of this semiempirical formalism. Since this formalism has received wide acceptance in the literature in correlating the steady-state relationship between stress, strain rate, grain size and temperature, we shall also adopt it in this presentation on superplasticity. The general correlation of both theoretical and experimental constitutive relations for high temperature deformation according to this formalism (Mukherjee et al., 1969) is given by: Q
where Do is the diffusion coefficient, Q the activation energy of diffusion, R the gas constant, T the absolute temperature, G the shear modulus, d the grain size, b the magnitude of the Burgers vector, p the inverse grain size exponent, n the stress dependence of the strain rate, A the mi-
9.3 Mechanical Aspects
9.3.1 Region III
crostructural and mechanism dependent constant, and kT has its usual meaning. The high temperature deformation of micrograined superplastic materials over wide ranges of strain rates reveals three distinct regions in the log stress {a) vs. log strain rate (s) plot as shown in Fig. 9-1. In this plot, the three important regions are labelled I, II and III, respectively. The slope of this curve is the strain rate sensitivity, m. The strain rate sensitivity is maximum in the intermediate strain rate region II. This region is associated with optimum superplasticity and also with a maximum in grain boundary sliding. The region III at higher strain rate is usually associated with dislocation creep mechanism. The origin of the low strain rate region I is not exactly clear. The experimental evidence in this region is often limited and also contradictory. Each of these three regions has characteristic values of the parameter /?, m, and Q of the constitutive relation and distinguishable microstructural and topological features. These are being described next.
REGION 0
The high strain rate region III can be correlated with a modified form of dislocation creep controlled by climb of edge dislocations. The value for m is usually 0.2 or less, the grain size exponent/? is either zero or nearly unity and the activation energy is equal to that for volume diffusion or slightly less. However, even in those experiments (Logan and Mukherjee, 1982) where no dependence of the flow stress on grain size was observed in this region, an activation energy less than that for volume diffusion and a dependence of total elongation on grain size indicate that grain boundary sliding does play a singificant role (Gifkins, 1982 b) in this region. The role of this grain boundary sliding will be discussed in more detail later. 9.3.2 Region II In the intermediate strain rate in region II the m value seems to be equal to 0.5 or higher. The activation energy of the pro-
REGION II
REGION 1
413
REGION III
SUPERPLASTICITY
DIFFUSION CREEP
Maximum Grain Boundary Sliding Minimal Grain Elongation
CO
CO LU
Grain Rotation
Grain Elongation
I
^S^ JS
Multiple Slip Grain Elongation Texture Increase
POWER LAW CREEP ^ s ^
Texture Reduction
CD
g
/
p:2-3 Q:Q gb
m:1 or m:0.2 p:2-3 Q:QV
m:0.5 p:2-3
m:0.2 p:0-1 Q:QV
LOG (STRAIN RATE) Figure 9-1. Schematic illustration of the strain rate dependence of flow stress in a superplastic material (from Pilling and Ridley, 1989).
414
9 Superplasticity in Metals, Ceramics and Intermetallics
cess is usually low and is often equal to that for grain boundary diffusion. However, several exceptions to this general suggestion exist (Arieli and Mukherjee, 1982). In particular Ruano and Sherby (1982) reviewed a very large volume of data in various stress ranges. Their analysis includes the operation of dislocation pipe diffusion and it gives a value of stress dependence of strain rate equal to 2 at lower stresses, becoming equal to 4 at higher stresses. The approach predicted strain rates that are quite satisfactory for a significant number of superplastic materials. The grain size exponent in this region has a value between two and three, including often fractional values. A grain size distribution in the original microstructure as well as strain enhanced alteration in grain size may be responsible for such fractional value for the grain size exponent. The variation in rate parameters in Eq. (9-1) may sometimes arise due to the presence of unstable microstructure. The values for the various rate parameters in the three regions mentioned earlier pertain to steady state. It was shown by Kashyap and Mukherjee (1983), that in a microduplex stainless steel, during the early stage of deformation, the strain rate sensitivity m, grain size exponent p and activation energy Q depend on strain due to microstructural instability. After suitable prestraining, such variations are substantially reduced and then the experimental data conform to the steady state rate parameters as given by theoretical models. It is interesting to note that the value for the activation energy, Q, obtained after suitable prestraining of the specimens in this alloy compared well with the activation energy for grain boundary diffusion in contrast to the activation energy for lattice diffusion as reported in the earlier investigation on this material.
The experimental results in region II are generally consistent with several theoretical models for superplasticity. However, no single model, at present, can account for all of the microstructural and mechanical observations. It is even possible that no such universal theoretical model for superplasticity which is applicable to all polycrystalline solids is attainable. The different theoretical models for superplasticity will be discussed in a later section. It will be worthwhile to state here that this region II is associated with maximum superplastic ductility, maximum extent of grain boundary sliding and in alloys where the grain boundaries are pinned by particles, often with maximum extent of cavitation. 9.3.3 Region I The precise details of the rate controlling mechanism in region I are uncertain at present. The form and even the existence of region I is controversial. The region I has been variously ascribed to: (a) a distinct deformation mechanism, (b) a consequence of grain growth, (c) due to the presence of a threshold stress, (d) Coble creep or a modified form of interface-controlled Coble creep. There is also the suggestion that there is another region (region 0 in Fig. 9-1) at yet lower stress level (Vale et al., 1979). Langdon (1982 a) has reported an activation energy of deformation in region I equal to that for lattice diffusion, a strain rate sensitivity value of 0.2 or 0.3, a grain size exponent equal to 2 or 3. It was suggested that region I and II may result from two distinct deformation processes operating in sequence. The very real possibility that grain growth in region I (where test may run for many hours) may influence the shape of the sigmoidal curve has been shown by Rai and Grant (1975). Differ-
9.3 Mechanical Aspects
ences in test methods also can cast doubt on some of the reported values of m in region I, as demonstrated by Arieli and Rosen (1976). This region has also been attributed to the operation of a threshold stress (Martin and Backofen, 1967). Mohamed (1983) reanalyzed the earlier data on Pb-Sn eutectic and Al-Zn eutectoid. He concluded that the transition in behavior between region II and region I may not necessarily reflect a change in deformation process but can arise from the presence of a threshold stress which decreases strongly with decreasing strain rate. He suggested that such a threshold stress may arise from impurity atom segregation at boundaries and their interaction with boundary dislocations. However, in a careful series of experiments on the Pb-Sn eutectic, Schneibel and Hazzledine (1983) could not find any evidence for a threshold stress even at very low ( ^ 1 0 ~ l o s ~ 1 ) strain rates. Instead they observed an interesting effect of grain size, which was not associated with the phenomenon of grain growth, on deformation mechanisms. In the dilute Sn-2wt.% Pb alloy, they found that region I is controlled by Coble (1963) creep when grain sizes were larger than 50 jim. At smaller grain sizes vacancies may be created at grain boundaries at such a low rate that Coble creep ceases to be diffusion-controlled and instead becomes interface-controlled. The suggestion that region I may be associated with Coble creep has been put forward earlier by Mukherjee and coworkers (Mukherjee, 1979; Misro and Mukherjee, 1975) and also by Vaidya et al. (1973). One should then observe values for the strain rate sensitivity of 1, the grain size exponent of 3 and an activation energy equal to that for grain boundary diffusion. The available experimental results in the literature on Al-Zn eutectoid have been examined by
415
Logan and Mukherjee (1982) who concluded that the nature of the rate controlling mechanism in region I remains unclear. The concept of Coble creep has also been used by Ghosh and Raj (1981) in their treatment of grain size distribution effects in superplasticity. Their approach basically combines diffusional Coble creep with power law creep as has been done in a number of other studies. The result of this distribution in grain size is to increase the strain rate range over which power-law creep switches over to Coble creep. The mix of grain sizes can also produce an apparent approach towards a threshold stress at low strain rates. However, at yet lower strain rates it is suggested that the flow stress changes back toward a Newtonian viscous behavior of Coble creep again. The ambiguity in the rate parameters in region I may then be possibly explained by the precise stress-strain rate range where the experimental results are being evaluated. It has also been suggested (Gifkins, 1982 a) that the mechanism in region I is essentially the same as in region II, but that instead of the rate control being through accommodation of grain boundary sliding, it is by solute drag on grain boundary migration. Perhaps the view of Gifkins (1982 b) summarizes the situation best, i.e., that region I may be due to the operation of a truly different mechanism, but special circumstances may replace it with other. The present author believes that these other mechanisms inlcude Coble creep or interface-controlled Coble creep, i.e., some type of diffusion creep involving the grain boundary and/or the grain mantle region.
416
9 Superplasticity in Metals, Ceramics and Intermetallics
9.4 Microstructural Aspects of Superplasticity The investigations on microstructural aspect involve the role of: grain shape and size, grain growth, grain rotation and grain rearrangement, dislocation activity and the distinction between phase boundaries and grain boundaries in the superplastic deformation process. Despite substantial amount of investigations, several microstructural issues remain unclarified. Significant ones among these issues are the role of grain growth specifically at low strain rates, the precise details of the process of accomodation at triple points due to grain boundary sliding (GBS) and migration, and the role played by discloations in this deformation process in general and in the accommodation mechanism in particular. 9.4.1 Grain Shape and Grain Size
One of the prerequisites for micrograin superplasticity as stated earlier, is a small and equiaxed grain structure. However, many wrought materials do not have equiaxed microstructure initially. The grain aspect ratio (ratio of longitudinal to transverse dimension) decreases during superplastic deformation as the elongated structure evolves towards a near-equiaxed structure, eventually stabilizing at the aspect ratio of approximately 1.2 (Suery and Baudelet, 1975). This nearly equiaxed stable structure is attained after about 30 percent strain. Hence, mechanical data on superplasticity gathered at strains lower than that needed for attainment of stable structure may not produce the stress-strain rate relation that is representative of true steady state condition. Some of the variations and controversy on reported mechanical data on the same material in the literature may arise because of this factor.
Usually, after achieving an equiaxed condition, the structure, as observed in metallographic examinations remains so, even after hundreds or thousand percents of strain (Alden and Schadler, 1968) when deformed in superplastic regime. Some investigators, however, observed slight grain elongation in superplastic deformation both in single phase (Valiev and Kaibyshev, 1976) and two-phase (Ghosh and Hamilton, 1979) alloys. When examined metallographically, the grain and phase boundaries appear as straight lines before deformation. After deformation the interfaces often become curved and have a bulbous aspect (Watts et al., 1976). The usual interpretation for curved boundaries has been that grain boundary migration has occurred. There is also metallographic evidence for a tendency for the grains to cluster together at low (Dunlop and Taplin, 1972) and high (Hatz et al., 1975) strain rates and for transient grain shape change (Valiev and Kaibyshev, 1977) during deformation, which occurs in narrow region near the boundaries. Another important microstructural feature, now firmly established, is that deformation-enhanced grain growth takes place concurrently during superplastic flow (Ghosh and Hamilton, 1979; Prada et al., 1990). This is illustrated in Fig. 9-2 where the grain growth as a function of strain is depicted for both alpha and beta phases in a Ni-modified Ti-Al-V alloy (Prada et al., 1990). Caceres and Wilkinson (1984) analyzed the grain growth vs. strain rate data on several superplastic materials as shown in Fig. 9-3. At intermediate strain rate the rate of grain growth appears to be linearly dependent on strain rate. At lower strain rate the grain growth rate reaches a plateau (the static annealing limit). At higher strain rate the grain growth rate
9.4 Microstructural Aspects of Superplasticity
417
Figure 9-2. Microstructures exhibiting grain growth in a/p microduplex Ni-modified Ti-Al-V alloy after superplastic deformation (T=1088K, e = 2 x l O " 4 s"1) to different levels of strain (from Prada et al., 1990).
shows a tendency to reach another plateau, controlled by inability of the grain boundary to migrate at the higher strain rates. One criticism that may be raised regarding this model is that it considers only the process of grain growth, and ignores the process of dynamic grain refinement that may take place by 'in situ' dynamic recrystallization. In several materials and as specifically demonstrated in Ti-6A1-4V (Gurewitz, 1983; Prada et al., 1990), such grain refinement is not uncommon partic-
ularly at the lower temperature and higher strain rate end of superplastic deformation. Since the superplastic strain rate shows the large sensitivity to grain size, a reliable model for strain enhanced alteration of grain size will be of immense practical help in correctly assessing superplastic strain rate as a function of strain during the forming process. McNelley and coworkers (Hales and McNelley, 1988 a and 1988 b; Hales et al., 1990) have made a detailed and systematic
418
9 Superplasticity in Metals, Ceramics and Intermetallics -2
10
-
'8
• • o o
I I Cu-AI-Si-Co Sn-Bi Zn-AI Ti-AI-V
I
I
i
^
*>— .^
^
——a
-
io5. -
-6 IO
KD"
•
—
^
/
I
I
ion
IO"5
I
I0'4
i
10"
I
ID"1
Figure 9-3. The strain rate dependence of the rate of grain growth normalized by the initial grain size dQ for various materials (from Caceres and Wilkinson, 1984).
1
e(se
study of the refinement in the grain size and increase in the grain boundary misorientation in an Al-Mg alloy by continuous recrystallization. They studied the microstructural alterations both during thermomechanical processing and simultaneous with superplastic deformation. They observed that existing concepts of continuous recrystallization involing subgrain rotation and coalescence mechanisms were inadequate to explain their microstructural data. They proposed a model based on dislocation recovery to preexisting subboundary walls (which were stabilized by a precipitated phase in their alloy). Individual dislocations migrate to sub-boundaries, increasing their misorientation and leading to the evolution of boundaries better able to support superplastic deformation. There was evidence (Crooks et al., 1988) that the population of coincidence-site-lattice boundaries was shifted towards higher I values in the more refined and ductile microstructure. 9.4.2 Grain Boundary Migration and Sliding
Grain boundary migration usually accompanies the superplastic flow. Nichol-
son (1972) observed precipitate free zones near both grain boundaries and interphase boundaries in Al-Zn eutectoid alloy and interpreted his observation in terms of grain boundary migration. Lee (1970) has put forward an explanation where grain boundary sliding is followed by grain boundary migration at triple points to minimize grain boundary energy. As a result, two adjacent grain boundaries will have opposite curvatures which were observed in Al-Cu eutectic (Nicholson, 1972). However, this model cannot fully account for the bulbous aspect of the previously straight boundaries. A possible explanation might be that, in addition to the grain boundary migration (with the boundary moving towards its center of curvature) driven by the necessity to reduce the boundary energy, a second type of boundary migration, this one being straininduced (King and Smith, 1976), will take place concurrently where the boundary moves away from its center of curvature. Grain boundary migration is also important from the point of view of grain boundary sliding. As Ashby (1972) had pointed out "sliding without change of boundary structure is possible only if the boundary also migrates". Extensive inter-
9.4 Microstructural Aspects of Superplasticity
face (either grain or phase boundaries) sliding is commonly observed during superplastic flow (Valiev and Kaibyshev, 1977; Langdon, 1987). Scratch mark experiments are usually used to estimate the contribution of the grain boundary sliding to the total strain. The results of such studies are summarized in Table 9-1. The quantitative evaluations of the contribution of sliding strain can often be contradictory. This fact is probably connected with the uncertainties associated with the choice of geometrical coefficients during the estimations of sliding strain in scratch mark experiments (Valiev and Kaibyshev, 1983; Valiev et al., 1985). Values of the estimated contribution of the grain boundary sliding to the total strain obtained in these investigations using the expression yGBS = 1 - yIS - yDS, where yIS and yDS denote values of the intragranular dislocation strain and diffusion creep contributions respectively, equal to 75-90%, i.e., significantly higher than the ones obtained by scratch mark methods. Gifkins (1991) has very recently reanalyzed the data from the literature with respect to yGBS in the context of possible erTable 9-1. Contribution of grain boundary sliding to the total axial strain (yGBS) (see Arieli and Mukherjee, 1983). Material
TGBS(%)
Region I Zn-22% Al Pb-Tl Mg-Al eutectic Pb-Sn eutectic Pb-Sn eutectic Mg-1.5%, Mn-0.3%C Al-9%, Zn-1.0%Mg Zn-O.4%A1
Region II Region III 30 33 29
21
60 50 64 70 56
33
41-49
30
42
63 48
26 28
30 12
20
419
rors and modifying features. He took into consideration methods of measuring sliding, limitations imposed by total strain and by facet exposure at the surface, contributions from mantle shear, etc. His reanalysis suggested that many of the reported measurements may be in error and more importantly, the values of yGBS may not be as high as previously thought and that the phenomenon of grain rotation should receive more important consideration in such discussion. In particular, according to Gifkins, the revised values of yGBS in region II in both dual or single phase alloys are not higher than about 50%. Thus, it appears that the maximum possible value of GBS in region II is open for further careful study. However, the observation that yGBS is maximum in region II was reconfirmed in reanalysis by Gifkins. This is also confirmed by the data shown in Table 9-1. Although the results shown in the table were calculated by different procedures, which makes direct comparison difficult, the general trend is the same in every investigation: the contribution of grain boundary sliding is maximum in region II (superplasticity) and decreases at both high (region III) and low (region I) strain rate values. There is only very limited amount of information available in the literature regarding the extent of sliding at boundaries between similar and dissimilar phases, or the orientation of the sliding boundary relative to the applied stress. Chandra et al. (1978) observed that in alpha/beta brass, sliding occurred on oc-p boundaries more readily than on oc-oc and (3-(3. Similarly, Vastava and Langdon (1979) observed that in Pb-Sn eutectic sliding took place predominantly at the Sn-Sn boundaries and none at Pb-Pb boundaries, Pb-Sn boundaries having an intermediate value. They also observed considerable migration
9 Superplasticity in Metals, Ceramics and Intermetallics
420
at the Sn-Sn boundaries where most of the sliding occurred. Valiev and Kaibyshev (1977) found a strong dependence of the amount of grain boundary sliding on the angle between the axis of the specimen and the sliding boundary trace on the surface. In region I, sliding occurred mainly at the transverse boundaries, whereas in regions II and III the largest amount of sliding was measured at boundaries lying at 45° to the specimen axis. Also in regions II and III, the sliding anisotropy was less marked than in region I. In all three regions the boundaries which exhibited the largest amount of sliding also showed the most active boundary migration. Figure 9-4 summarizes their results. In such studies for GBS, one sometimes observes striated bands that develop between original grains on the surface. Novikov et al. (1981) have studied this in 6=6% (b) €=
30
I
»
m—
9.4.3 Grain Rotation and Grain Rearrangement
2
(a) •
8.3x1(rV rV
(c)
—m 1
1.6x1(rV
I
ClxloV1
20
S
15 10
o
0° 45° 90°
0° 45° 90°
fair detail and concluded that these are not facets exposed by GBS nor emerging grains. Instead it was suggested that they represent special local diffusional movement of material of one of the two phases that are present, somewhat reminiscent of the suggestion of Spingarn and Nix (1978) and Mayo and Nix (1989). Gifkins (1982 b) has put forward the hypothesis that such striated bands can be viewed as one more of the various ways in which material continuity in a specimen is maintained when GBS occurs. In ending this section it must be remarked that besides sliding and migration, grain rotation and grain rearrangement also take place during superplastic deformation. These are not completely separable effects and the results of grain boundary migration and sliding discussed earlier would obviously be influenced by grain rotation and rearrangement, which will be discussed next.
0° 45° 90°
Figure 9-4. The value of the vertical component of grain boundary sliding, h, versus boundary angle with the tensile axis (from Valiev and Kaybyshev, 1977). (a) fi = 1.6xlO- 4 s" 1 (region I); (b) £ = 8.3xl(T 3 s" 1 (region II); (c) £ = 4.1 x 10" 1 s"1 (region III).
Appreciable amounts of grain rotations have been observed during superplastic flow in both post deformation metallographic study of scratch mark offset records (Watts et al., 1976) and in 4in situ' (Hatz et al., 1975) surface experiments. These studies reveal that grains never rotate more than 45° and that they often change their sense of rotation during the deformation. Matsuki and coworkers (1977) measured the amount of grain rotation during superplastic deformation as a function of strain rate and strain. Part of their results are summarized in Table 9-2. It is interesting to note that rotations higher than 35° were observed only at the lowest strain rate. As the strain rate was increased (but still within region II) the
421
9.4 Microstructural Aspects of Superplasticity
Table 9-2. Relative frequency of grain rotation as a function of strain-rate at s = 60 percent (after Matsuki et al., 1977). Region
eCs-1)
Relative frequency (%) 0-5
II II III
3.33 xlO" 5 1.12 xlO" 4 1.83 x!0~ 3
65 50 75
relative frequency of the intermediate rotation (5°-25°) increased to 53 percent compared to only 30 percent at lower strain rates. In region III, the relative frequency for intermediate rotation (5°-25°) dropped to 23 percent, whereas most (75 percent) of the rotation was limited to less than 5°. The grain rotation will not contribute to total strain, but may provide an additional degree of freedom during superplastic deformation. During deformation, the grain rotation together with grain boundary sliding will lead to an overall reduction in texture. Another possible contribution of the grain rotation to the superplastic flow will be to facilitate grain rearrangement. Grain rearrangement takes place to a large extent during superplastic deformation and two grains which are initially neighbors may end up being many grain diameters apart after deformation. Naziri et al. (1975) in their 'in situ' study of thin foils of Al-Zn eutectoid alloy observed extensive grain rearrangement and their study verified a model for neighbor-switching events put forward by Ashby and Verrall (1973). Subsequently, Kobayashi and coworkers (1977) also noticed such grain rearrangement as well as extensive interface migration and grain boundary sliding in 'in situ' experiments of Al-Cu eutectoid alloy. However, such three-dimensional grain rearrangement is not invariably seen. In
Rotation angle (degrees) 15-25 25-35 5-15
34-40
5 8 3
2 0 0
25 40 20
3 2 3
Supral (Lloyd and Moore, 1982) and in a/P Ti-6A1-4V alloy (Hildalgo-Prada and Mukherjee, 1985) dynamic recrystallization may take the place of the phenomenon of grain rearrangement. 9.4.4 Dislocation Activity The subject of dislocation activity during superplastic deformation had produced some debate. An earlier investigation (Nicholson, 1972) pointed out a lack of dislocation activity. However, subsequent experimental work suggested that lack of dislocations in TEM examination of thin foils prepared from deformed specimens does not necessarily prove that dislocation mechanisms are not operative during superplastic deformation. In fact, it emphasizes the problems associated with the preservation of the dislocation structure after deformation. The earlier evidence of lack of dislocation activity by Nicholson (1972) has been refuted by Edington et al. (1976) who showed that the internal markers used by Nicholson had not been effective barriers for dislocations. Similarly, the lack of dislocation activity in the 'in situ' experiments of Naziri et al. (1975) has been challenged by Bricknell and Edington (1977) who demonstrated that under the experimental condition used by Naziri, an extremely high vacancy flux was created throughout the specimen,
422
9 Superplasticity in Metals, Ceramics and Intermetallics
which is not typical for superplastic deformation. This effect, which facilitates dislocation climb might have masked or eliminated the evidence of dislocation activity. In recent years there is mounting evidence of dislocation activity during superplastic flow. These include observation of slip lines on gage surfaces (Hatz et al., 1975), stabilization of texture and formation of new texture (Melton et al., 1974; Kaibyshev et al., 1976; Matsuki et al., 1977) and direct observations of dislocations in TEM studies (Falk et al., 1986; Valiev and Kaibyshev, 1977; Samuelsson et al., 1974). They all point out that considerable dislocation activity occurs during superplastic deformation. Texture studies invariably show a reduction in the overall texture due to grain boundray sliding and grain rotation. In addition to the overall texture reduction, some texture components are stabilized and new ones are created indicating intergranular slip activity (Melton et al., 1974). The slip behavior is different not only among different alloys, but also among the different phases present in the same alloy. Additional evidence for dislocation activity during superplastic flow is found in carefully conducted TEM studies where the tensile specimens were quenched under load. In the very detailed study of the Alrich phase of a Zn-40 percent Al alloy by Samuelsson et al. (1974), most of the dislocations were observed to climb, the proportion of climbing dislocations increasing with increased strain rate. They found that more than one slip system operated within a grain even in the lower end of region II. They did not find any sudden change in dislocation density or type in the transition region from region II to region III. In the upper strain rate range of region II, dislocations were found in almost every grain containing precipitates, but many grains
free of precipitates were also free of dislocations. In the specimen deformed at a strain rate corresponding to the middle portion of the region II dislocations were observed almost exclusively in grains containing precipitates. This only reemphasizes the difficulty to preserve dislocation structure for TEM observation. The Burgers vector distribution was similar at all but the lowest strain rate. This might indicate that at this low strain rate the dislocations only accommodate the grain boundary sliding, whereas at higher strain rates the dislocations also contribute to intragranular slip. Other investigators (Kaibyshev et al., 1976) have reported dislocations being emitted from grain boundaries. Such dislocations can be generated at ledge and protrusions under large stress concentrations due to grain boundary sliding (Mukherjee, 1971). A method based on a measurement of distances between marker lines or points situated inside grains was used by Valiev and Kaibyshev (1983) and Valiev et al. (1985) in order to study dislocation activity. These experiments gave an evidence of dislocation activity in all three regions, and also they showed some special characteristics of such intragranular dislocation strain. It was shown that the intragranular strain has an oscillatory character and the magnitude of this strain may be considerable, even though its average contribution to the total deformation may not be big, due to mutual cancellation of dislocation accommodation processes in opposite directions, somewhat akin to dislocation motion in fatigue. From the foregoing presentation it is evident that a common microstructural pattern emerges for all the materials and conditions investigated in conjunction with superplastic deformation phenomenon. Specifically, the major microstructural
9.5 Characteristics of Models and Constitutive Relations
characteristics of superplasticity are: (a) The grain structure, if not equiaxed from the onset of the deformation, achieves a nearly equiaxed condition in the first few tens percent of strain. (b) Even after hundreds and thousands percent strain, grains remain essentially equiaxed. (c) Previously straight interfaces (both grain and phase boundaries) become curved, and sometimes the phase boundaries have a bulbous aspect. (d) There is strain-enhanced grain growth, especially at low strain rates. (e) Deformation zones are formed on boundaries approximately perpendicular to the axis of tension. (f) Extensive grain boundary migration and grain boundary sliding occur. (g) There are large relative rotations of individual grains and groups of grains. (h) Three-dimensional grain rearrangements can take place during superplastic flow. (i) Considerable dislocation activity occurs during superplastic flow. (j) 'In situ' continuous recrystallization can take place during superplastic deformation in some systems.
9.5 Characteristics of Models and Constitutive Relations One of the dominant microstructural features in superplasticity is the role played by grain boundary sliding (GBS). The grain compatibility during GBS is maintained by concurrent accommodation processes which may involve grain boundary migration (GBM), grain rotation, diffusion or dislocation motion. Most of the models proposed in the literature for superplasticity, generally consider one or the other accommodation process in conjunc-
423
tion with GBS. The accommodation mechanisms considered can be divided into three groups: (a) diffusional accommodation, (b) accommodation by dislocation motion, and (c) combined models having elements of both dislocation and diffusional accommodation. Thus, more than one mechanism may operate during superplastic deformation. Beere (1977), Padmanabhan (1977) and Bhattacharya and Padmanabhan (1989) had suggested that grain boundary sliding itself could be the rate controlling mechanism. This would require that the accommodation processes can take place at a rate that is faster than that for grain boundary sliding. If, however, the grain boundary sliding is intrinsically a faster process, as suggested by Schneibel and Hazzledine (1983), then the characteristics of flow stress will be determined by the details of the rate-controlling accommodation process. The rate equations (correlating flow stress, strain rate, grain size and test temperature) derived from different models have essentially the same form. This is true in spite of the differences in the detailed concepts associated with the individual models, as discussed below. 9.5.1 Diffusion-Accommodation Model by Ashby and Verrall
Ashby and Verrall (1973) put forward a model which explains superplasticity as a transition region between the diffusionaccommodated flow, operative at low strain rates, and dislocation creep at high strain rates. In order for grains to maintain compatibility during the deformation process, they suffer a transient but complex shape change or accommodation strain. This accommodation is accomplished by
424
9 Superplasticity in Metals, Ceramics and Intermetallics
diffusional transport. Due to transient increase in the grain boundary area during grain rearrangement process, the deformation is limited at very low strain rates by a threshold stress. At high strain rates, where dislocation creep contributes dominantly to the total strain rate, the specimen elongation is achieved by the change of shape of the individual grains. At intermediate strain rates these two mechanisms superimpose on each other. The microstructural and topological features of the plastic flow at these strain rates will be those characteristic of the two mechanisms. Since these two processes are independent and take place simultaneously, the total strain rate, etot, will be the sum of the strain rates contributed by each process, i.e., £
tot
=
^diff. ace + £disloc. creep
where Q
(
0.
d £>, ^disloc. creep
-*1
DvGbU\" i rp
I si J
(9-2b) (9-2 c)
In Eq. (9-2 b), Q is the atomic volume, F is the grain boundary free energy, <5 is the thickness of the boundary, Dv is the volume diffusion coefficient and Dgh is the grain boundary diffusion coefficient. For a discussion of diffusion mechanisms and methods of measurement, please see Chapter 2 in Volume 5 of this series. When Eq. (9-2 a) is expressed in an empirical form as Q o — const • s™ot dp exp (9-2 d) RT the model predicts that m is a strong function of strain rate with a maximum, at a strain rate close to that corresponding to
diffusional creep, approaching unity but never quite reaching it. Furthermore, m is predicted to be dependent on temperature and grain size. The grain size dependence coefficient, p, will vary between 1 and 3 depending upon the strain rate and temperature. The model also predicts that the activation energy for superplastic flow will be dependent upon temperature and stress, varying between the value of the activation energy for grain-boundary diffusion and that for lattice self-diffusion. The Ashby-Verrall model has many attractive features and it integrates the essential topological features in superplasticity. However, as pointed out by Mohamed and Langdon (1975), the predicted transition stress between region I and II and its dependence on grain size are at variance with experimental observations. Also, Smith et al. (1976) observed that the experimentally measured threshold stress was (a) considerably lower, (b) had opposite dependence on temperature and (c) had opposite dependence on grain size compared to that predicted by this model. However, the most serious objection to this model has to do with the unrealistic diffusion path that is proposed. The model requires diffusion to occur in different direction on opposite sides of a given grain boundary under the usual state of tensile stress. Nix (1985) has pointed out that the resultant shape change also violates the symmetry criterion. Spingarn and Nix (1978) and Nix (1985) have suggested that the shape change suggested earlier by Lee (1970) will produce a more realistic transient configuration during the shape change (Fig. 9-5). The process allows each grain in a four-grain cluster to undergo the same (and symmetrical) change in shape and yield the same final configuration.
9.5 Characteristics of Models and Constitutive Relations
425
9.5.2 Dislocation-Accommodation Mechanisms
e=o
The dislocation-accommodation mechanisms can be grouped into a process due to dislocation pile-ups in the grain interior and that due to pile-ups in the grain boundary (Fig. 9-6). 9.5.2.1 Pile-Ups Within the Grains
Figure 9-5. Grain switching through diffusional mass transport as postulated by Ashby and Verrall (1973) (a x -a 2 ) and modified by Spingarn and Nix (1978) (b 1 -b 2 -b 3 ) (from Pilling and Ridley, 1989).
These models (Ball and Hutchinson, 1969; Mukherjee, 1971) are based on the common assumption that dislocations, which are emitted at the grain boundary due to stress concentration, traverse the grain by glide process and are annihilated at the opposite grain boundary. In these sequential processes, the slowest one will control the strain rate. It is usually assumed that the dislocation emission and dislocation glide/climb through the grain are easy processes and that their eventual absorption into opposite boundaries by the process of grain boundary climb is the rate controlling step. Ball and Hutchinson (1969) proposed that groups of grains slide as a unit until unfavorably oriented grains obstruct the process. The resultant stress concentration
climb of lead dislocation
dislocation pile-ups in sliding and blocking grains glide and climb of grain boundary dislocations
Figure 9-6. Dislocation pile-up models of superplastic flow. The rate at which the grains slide past each other can be controlled by (i) the removal of a pile-up of lattice dislocations (Ball and Hutchinson, 1969), (ii) the emission of lattice dislocations from grain boundary ledges (Mukherjee, 1971) and (iii) the removal of pile-ups of grain boundary dislocations (Gifkins, 1982 b) (from Pilling and Ridley, 1989).
426
9 Superplasticity in Metals, Ceramics and Intermetallics
three. This was very likely due to a Mg solute atom related drag effect on the glide motion of the dislocations. In a similar vein, for solid solution superplastic alloys, Sherby and Wadsworth (1989) suggested the n = 2 stress-sensitivity of strain rate is to be expected in finegrained class II solid solution alloys where climb of dislocations (after they traverse the grain interior by glide process) in the grain boundary plane is the rate-controlling step. The activation energy for superplasticity for such alloys is equal to that for grain boundary diffusion. On the other hand, for class I solid solution alloys, where glide is the rate-controlling step, the strain-rate-sensitivity exponent is equal to unity, because there is no pile-up stress. Hence, this approach predicts that fine grained class I solid solution alloys can exhibit high values of ra, because in these 2 2 skT (b\ (o\ —— = const • I — I I — 1 (9-3) alloys the solute-drag controlled viscous gb \ / \ / creep is often the slowest process. Sherby and Wadsworth (1989) obtained quite It should be mentioned that although it good correlation in a wide range of class I is usually assumed that the climb of disloand class II solid solution alloys with the cations in the grain boundary is the rate prediction of this model. controlling step, it need not always be so. Admittedly, if there is activation of a single dislocation source per grain, then the 9.5.2.2 Dislocation Pile-Up strain rate will be controlled by climb of in Grain Boundary Plane dislocations at the grain boundary. However, in the Mukherjee (1971) model, In this model, due to Gifkins (1982 b), where grain boundary ledges act as dislosliding is considered to take place by the cation sources, the density of such sources motion of grain-boundary dislocations can change the predicted strain rate by two that pile up at triple points. The resulting orders of magnitude and the process can stress concentration is relaxed by dissociabe controlled by the dislocation emission tion of the leading grain-boundary dislocarate. Similarly in an alloyed material, the tion capable of moving in the two other boundaries making up the triple point, glide motion of dislocations through the and/or into lattice dislocations which acgrain interior can be controlled by viscous commodate sliding. These new dislocadrag due to solute. In a recent investigations then climb or glide in or near these tion on superplasticity in a mechanitwo boundaries until they meet each other. cally alloyed aluminum alloy, Bieler and Then they may annihilate or combine with Mukherjee (1990 b) observed the stressthem to form different grain-boundary disdependence of the strain rate to be equal to is then relieved by dislocation motion in the blocking grains. These dislocations pile up against the opposite grain-boundary until the back stress prevents further activation of the source and stops sliding. The leading dislocation in the pile-up can thus climb into and along the grain-boundaries to annihilation sites. Mukherjee (1971) has proposed a modification of the above model in which grains slide individually rather than in groups. Dislocations are generated by ledges and protrusions in the grain-boundaries, traverse the grain, and are again held up in the pile-ups at grain-boundaries. The rate of sliding is then controlled by the climb rate of the lead dislocation into annihilation sites located at grain-boundaries. The models lead to the following rate equations:
9.5 Characteristics of Models and Constitutive Relations
locations. This complete sequence of events will reproduce grain rotation and rearrangement in agreement with microstructural studies. The rate equation according to this model is identical to Eq. (9-3) except that the substructure related constant parameter has a different value. The physical basis of this model, however, puts emphasis on the role played by grainboundary dislocations. An important feature of this model is that it does not require that the compatibility between adjacent grains be maintained at all stages of the deformation. Instead, as proposed by Hazzeldine and Newbury (1973), gaps (voids) are opened at interfaces to be filled up by grain sliding from adjacent planes in the specimen. Gifkins' depiction of the grain boundary and grain interior has often been referred to as the core and mantle model. The accommodation of grain boundary sliding is supposed to be taking place within a narrow viscous mantle (the grain boundary region) around a rigid core of the grain interior. In the regular two-dimensional depiction of arrangement of hexagonal grains, the predicted width of the mantle will be 0.07 times the grain diameter (Gifkins, 1982b). In a typical superplastic grain size of few micrometers, the mantle in Gifkin's model would be few tens of nm wide. Recently the concept of core and mantle model of the grain boundary has been invoked by Mayo and Nix (1989). However, they consider a flexible width for the mantle region. They assume that deformation occurs faster in the mantle than in the grain interior. The size of the mantle remains fixed throughout deformation and the size is inversely proportional to the applied stress. The core is assumed to deform by dislocation-climb creep with n = 5 (m = 0.2). The creep deformation of the
427
core is the dominant process in region III. The mantle region is considered to be weak compared to the core region. Mayo and Nix assume that the mantle region increases in volume with a decrease in stress and the deformation of the mantle essentially dominates the deformation in region I. With an increase in stress the transition from essentially mantle-dominated deformation in region I to core-dominated deformation in region III gives rise to region III. Their model allows for obtaining values of m higher than 0.2. However, no specific value of m is predicted for region II. The experimental work (in torsion) of Pb-Sn and Zn-Al alloys by Mayo and Nix (1989) does indicate that the deformation of the mantle region dominates the deformation in the low stress region. Some of the objections that have been raised against the dislocation pile-up models include the following: (a) Dislocations pile-ups are not usually observed and they may not be stable at the relatively high temperature for superplastic deformation. (b) If a threshold stress is really part of the picture of superplasticity, then these models do not predict such a threshold stress. (c) In the absence of any implicit mechanism for grain rotation (i.e., Ball and Hutchinson, 1969) the grains will elongate due to dislocation glide on a limited number of slip system. Pilling and Ridley (1989) have suggested that grain rotation may result from a nonbalanced system of unrelated grain boundary shear stresses. Random variations in the direction and magnitude of rotation will cause slip to switch from one slip system to another. In such case, grain elongations will not be observed. In the Gifkin's (1982 b) model, however, grain rotation is an implicit process. The shear
428
9 Superplasticity in Metals, Ceramics and Intermetallics
stresses acting on the grain boundary, due to the pile-up of grain boundary dislocations would produce the torque needed for grain rotation. As discussed earlier, oscillatory grain rotation by angles as high as 35° to 40° has been observed by Matsuki et al. (1977). The absence of extensive dislocation activity or dislocation pile-ups in transmission electron microscopy do not necessarily invalidate the pile-up models. At the superplastic deformation temperature, climb of dislocations from the head of the pile-up can be rapid and dislocation pile-up in the strict sense may not be sustainable, even though there is a stress concentration at the head of the impinging slip band to cause the dislocation to climb in the grain boundary plane. Furthermore, upon unloading the dislocations in the pile-up can easily run back to the source and subsequent TEM investigation may not reveal too much of their erstwhile activity. Nevertheless, as the investigation of Samuelsson et al. (1974) has shown, analysis of texture from diffraction work does indicate a significant measure of dislocation activity. Post-superplastic deformation studies have revealed clear evidence of dislocation activity in microstructures containing twins that can anchor dislocations (Falk et al., 1986). The recent model (Kaibyshev et al., 1985; Valiev, 1988) considers the motion of two types of grain boundary dislocations an intrinsic one, whose pile-ups in triple points initiate a generation of lattice dislocations and an extrinsic one, formed as a result of dissociation of trapped lattice dislocations in the grain boundaries. The motion of extrinsic dislocations leads to grain boundary sliding and the annihilation of such dislocations in the grain boundary and/or triple points is responsible for the recovery process. The model based on this
concept was shown to describe not only the o — s relationship but also the decrease in the value of strain rate sensitivity in regions I and II. 9.5.3 Accommodation by Both Diffusional and Dislocation Motion Many investigators have pointed out that the activation energies and the stress dependence of the strain rate in region I and III are often very similar. This has led to the suggestion that the deformation process in the low strain rate (hence, also low stress) region I is essentially dislocation-climb controlled recovery creep. One difference relates to the fact that in region I, the stress is so low that the subgrain size is of the order of grain size and hence, there is no subgrain boundary to impede the dislocation motion. The region II of superplasticity then becomes simply a region of transition from the mechanism operating in region I to that operating in region III. Such considerations lead to the description of two such transitional models. (A) Spingarn and Nix Slip Band Model This model (Spingarn and Nix, 1978) considers a stress concentration mechanism through slip bands and sliding boundaries impinging upon grain boundaries. The strain at the boundary is accommodated by diffusional flow in the boundaries. The slip band spacing decreases as the strain rate is increased. At very small stresses the slip band spacing is equal to the grain size, d, and the rate equation is given by:
where we have taken the atomic volume Q = b3 and the grain boundary width
9.5 Characteristics of Models and Constitutive Relations
d = 2b. Equation (9-4) is essentially identical to the pure diffusional creep by the Coble mechanism within a factor of two. At large stresses, the slip band spacing is taken to be equal to the subgrain size and the rate equation is given by skT
(d\(o = const - | - ^—-
(9-5)
The authors suggest that the transition zone (which spans more than one order of magnitude in stress) from n = 1 to n = 5 behavior coincides with the superplastic range. No specific rate equation is given for the superplastic region, although the total strain rate in this region must be different from the simple sum of the rates given by Eqs. (9-4) and (9-5) since the slip band spacing varies with strain rate. According to this model, not only n varies from 1 to 5 but also the strain rate dependence on grain size coefficient will vary between — 3 and + 1 . The activation energy for deformation will be that for grain-boundary diffusion. The model produced quite reasonable agreement with several sets of results. This model in its present form, however, cannot explain the possible presence of a plateau or threshold stress observed in some superplastic materials. The model demands the presence of subgrains and slip bands in region II, contrary to most observations. Furthermore, the model cannot explain some of the topological features in superplasticity, e.g., the neighbor switching events during sliding (if it is significant) or the reason why grains remain equiaxed after extensive strain. (B) Distributed Parameter Model Ghosh and Raj (1981) suggested a model that can also be a plausible explanation for the apparent similarities between deformation parameters in region I and re-
429
gion III. It is based on a distribution of some significant parameters for the deformation process. There may be a distribution in the grain size, i.e., a bi-modal grain size distribution. Alternatively, there may be a large difference in the diffusivity of individual phases in a two-phase microstructure as between the alpha and beta phase in the Ti-6A1-4V alloy. The mechanical properties, e.g., the variation of stress with strain rate will be influenced by the relative proportion of the two phases or by the type of distribution of grain size. In the latter case, the finer grains may deform by Coble mechanisms and the coarser grain may deform by recovery controlled dislocation creep as demonstrated by a recent investigation on Al-Li alloy (Chokshi and Mukherjee, 1988 a). The transitional model of Ghosh and Raj (1981) based on distributed microstructural parameter and isostrain rate rules of mixtures is reasonably successful in predicting the stress-strain rate response in a/p titanium alloy and also in 7000 series aluminum alloy. Depending on the distribution of grain size or the proportion of soft vs. hard phases, the stress dependence of the strain rate can have any value from 1 (corresponding to Coble creep) to 5 (corresponding to dislocation creep). Because of the nature of the transition, i.e., the commencement of Coble creep in finer grains, the flow stress at transition will be a sensitive function of grain size. The mix of grain sizes can also produce an apparent approach to a threshold stress at low strain rates. However, this transitional model, in common with the one presented earlier, cannot explain the topological features, i.e., the equiaxed grain shape and the role of grain boundary sliding during superplastic deformation. The models for superplastic deformation have been developed with an emphasis
430
9 Superplasticity in Metals, Ceramics and Intermetallics
on the operation of grain boundary sliding. The models based on pure GBS do not appear to be promising. Therefore, most models concentrate on formulating some suitable accommodation process for the continued operation of GBS. The AshbyVerrall model introduces grain switching events in a unit of four grains with diffusional accommodation. Some of the concepts of this model were used in a modified manner in several models which were proposed subsequently. The concept for dislocation climb creep and that for Coble creep constitute important elements for formulating rate equations for superplastic deformation in the transitional models. It is commonly believed that grain rotation, grain boundary migration and grain emergence contribute to the accommodation process. However, no systematic attempt has been made to put these elements on a quantitative basis and to integrate them in the rate processes of individual models. 9.5.4 Reflections on Microstructure and Mechanisms
Most of the models attempt to predict some of the changes in microstructural and topological features in superplasticity. Some of the models do not explicitly address this aspect. However, in such instances often indirect inferences can be drawn from the basic analysis of the models. While the qualitative aspect of microstructural and topological characteristics of superplastic deformation are reasonably well established, the quantitative data on such aspects are very scarce. The direct measurements show that the maximum GBS occurs during superplastic deformation and the values of sliding related strain rate sensitivity (mgbs) in region II compares well with the m value calculated
from the mechanical a — s data (Furushiro and Hori, 1979). The values of m and Q for grain boundary sliding, dislocation creep and diffusion creep and their contribution to total elongation in a Zn-Cu-Mn alloy suggest that GBS is the predominant deformation mode but slip and diffusional flow also occur (Matsuki et al., 1983). Similarly, investigation suggests that the contribution of GBS to total strain is maximum while that of the diffusional creep is minimum with an intermediate contribution from intragranular dislocation slip (Kaibyshev, 1981). We believe that the accommodation by diffusional creep continuously decreases while that by dislocation activity continuously increases with increasing strain rate in the superplastic region. However, evidence from several studies (Matsuki et al., 1983; Kaibyshev, 1981) indicates that GBS is accompanied by both diffusional and dislocation modes of deformation and therefore, the models which suggest the operation of only one type of accommodation process may not be adequate to completely describe the deformation behavior (Kashyap and Mukherjee, 1985 b). Rai and Grant (1983) obtained microcreep curves between initially adjacent grains which show that superplastic deformation is cyclical in nature and varies by several orders of magnitude locally from point to point. The highly heterogeneous nature of superplastic deformation at the local level, as observed from this 'in situ' study provides support for the operation of different micromechanisms for accommodation. It is envisaged that they are operative concurrently but in different regions in the microstructure at any one instant. Gifkins (1982 b) has called this the micro-multiplicity effect. The effect provides many closely-spaced shear paths in three dimensions for GBS and its accom-
9.6 Cavitation and Failure in Superplasticity
modation by mechanisms conforming to Eq. (9-1). It means that the obstacles to GBS can be overcome by any number of mechanisms (Fig. 9-7) provided that they conform to n = 2, p = 2, Q = Qgh or Qyol. Some of these mechanisms can be (a) grain emergence, (b) Friedel creep, (c) mantle climb and glide, (c) grain switching, (e) local slip specially in larger grains or in softer phase and (f) grain rotation, etc. Such a concept has the advantage that it can provide the basis to explain the discrepancies reported in the experimental values of strain rate sensitivity, activation energy, grain size dependence of strain rate, etc. However, it is doubtful that a quantitative theory for superplasticity incorporating such concept will evolve in the near future, because of the complexity in formulating it.
9.6 Cavitation and Failure in Superplasticity At elevated testing temperatures, failure in tension may occur by two processes: (a)
431
external necking and (b) internal cavitation. It is now widely known that a high value of m provides superplastic alloys with a resistance towards external flow localization. It is also being recognized increasingly that most superplastic alloys cavitate during deformation (Langdon, 1982 b; Kashyap and Mukherjee, 1986; Pilling and Ridley, 1988), and although these materials are generally capable of withstanding extensive cavitation before failure, recent studies have shown that even small levels of cavitation lead to a deterioration in the subsequent room temperature properties of superplastically formed components (Bampton and Edington, 1983). Clearly, in order to decrease the rate of cavitation damage accumulation during superplastic deformation, it is necessary to develop a good understanding of the processes leading to cavitation failure. This is particularly important for many of the more recently developed quasi-single phase Al-based alloys where there are particles at the grain boundaries. This section reviews briefly the processes involved in the failure of superplastic al-
Figure 9-7. The micro-multiplicity effect provides many closely-spaced shear paths in three dimensions for GBS and its accommodation by mechanisms conforming to Eq. (9-1) - e.g. (a) grain emergence, (b) Friedel creep, (c) mantle climb and glide, (d) grain switching, (e) local slip especially in larger grains and (f) grain rotation. Faster processes may also accommodate GBS and slower ones may be bypassed temporarily (from Gifkins, 1982 b).
432
9 Superplasticity in Metals, Ceramics and Intermetallics
loys. It describes the occurrence of flow localization in superplasticity, and the important role of cavitation in the failure of superplastic alloys. Finally, the section discusses experimental procedures that may inhibit the occurrence of cavitation during superplastic deformation, e.g., superposition of hydrostatic gas pressure during deformation. 9.6.1 Flow Localization During Superplastic Deformation
In general, superplastic alloys do not exhibit much external necking when tested under optimum conditions, so that even in specimens exhibiting elongations to failure of several thousands of percent, there is very little evidence of flow localization (Langdon, 1982b). The importance of strain rate sensitivity in imparting superplastic alloys with resistance towards flow localization has been demonstrated experimentally by measuring with increasing deformation the variation in the local strain along the gauge length. Mohamed and Langdon (1981) conducted experiments to determine the onset and degree of flow localization in a superplastic Zn-22 % Al eutectoid alloy. Tensile specimens were tested at strain rates representative of region I (m = 0.22), region II (m = 0.5) and the onset of region III (m = 0.2), respectively. The specimen gauge lengths were divided into equal segments, with an initial length / 0 , and the local strain along the gauge length, A///o, was measured at different total specimen elongations AL/L 0 , where A/ is the change in the length of a given segment, AL is the change in the total gauge length and Lo is the initial gauge length of a specimen. The variation in the local elongation at different sections along the gauge length is shown in Figs. 9-8 and 9-9 for specimens
4000 Low stress region I - fracture 3500
1500 -
1000
500
2
0 2 SECTION No.
Figure 9-8. Variation in local strain during deformation in region I (m = 0.22) (from Mohamed and Langdon, 1981).
tested in regions I and II, respectively. Inspection of Figs. 9-8 and 9-9 reveals that, in contrast to region I where the deformation becomes non-uniform at elongations of >100%, the deformation in region II is fairly uniform up to elongations of ~ 800 % and non-uniform flow is observed only at large elongations of >1000%. These results demonstrate clearly that an increase in the strain rate sensitivity delays the onset of flow localization to higher strains. Following the general analyses of flow localization during testing, it is anticipated that the occurrence of strain hardening may also contribute to the stability of tensile deformation (Nichols, 1980). The overall stability of tensile deformation is related to a combination of m and y, where y is the strain hardening coefficient. Cac-
9.6 Cavitation and Failure in Superplasticity 2500 Zn -22*/o Al Superplastic region II T=473K Lo=1 27cm €O = 1 67x10"3S-1 2000 •
62 A 550 118 v 770 f 150 o 1130 o 230 a 360 A
433
parameters such as m and y, and they were summarized recently by Hamilton (1986). It suffices to note here that, while most of these expressions predict the correct trend in increasing ductility with increasing m, they do not predict the exact ductility of any given material satisfactorily.
1500
9.6.2 Cavitation Failure in Superplastic Alloys 1000
500
2 0 2 SECTION No.
Figure 9-9. Variation in local strain during deformation in region II (m = 0.5) (from Mohamed and Langdon, 1981).
eres and Wilkinson (1984) examined in detail the influence of strain hardening due to grain growth on fracture in a superplastic quasi-single phase Cu alloy, and they demonstrated that concurrent grain growth at low strain rates in region I may provide a substantial contribution to the stability of tensile deformation. However, concurrent grain growth also tends to decrease the transition strain rate from region II to region III. Therefore, as emphasized recently by Hamilton (1986), concurrent grain growth may also have a destabilizing effect on tensile deformation if the tensile experiments are conducted close to the transition from region II to III. Several attempts have been made to express the elongation to failure in a tensile test analytically (Lian and Suery, 1986) in terms of the specimen dimensions and
Although a high value of m is generally considered to be a pre-requisite for the observation of superplasticity, a close inspection of the available data suggests that cavitation frequently plays an important role in the failure of superplastic alloys to such an extent that in some alloys excessive cavitation may lead to premature failure (Langdon, 1982b; Ridley et al., 1984). It is clear that a fundamental understanding of cavitation in superplasticity may lead to the development of better materials, processing or experimental conditions, so that cavitation damage may be controlled, if not eliminated. Cavitation failure involves the nucleation, growth and interlinkage of cavities. Cavity interlinkage (recently reviewed by Pilling and Ridley, 1988), will not be discussed in the present paper. It is less well understood. We shall note, however, that a recent model study has shown that an increase in the strain rate sensitivity delays the onset of interaction effects and the interlinkage of cavities in a direction perpendicular to the tensile axis (Chokshi and Langdon, 1985). 9.6.3 The Nucleation of Cavities It has been suggested frequently that cavities may pre-exist in many alloys prior to superplastic deformation because of the extensive thermo-mechanical treatment that is used to produce a fine grain size (Stowell, 1983). The indirect evidence for
434
9 Superplasticity in Metals, Ceramics and Intermetallics
pre-existing cavities is obtained from an extrapolation to zero strain of semi-logarithmic plots of the variation in the total level of cavitation with strain. However, since cavitation is plotted on a logarthmic scale, an extrapolation of such plots to zero strain will always yield a positive offset, irrespective of whether cavities pre-existed or not (Chokshi and Mukherjee, 1989 a). Inspection of polished cross-sections of superplastically deformed alloys at different strains generally reveal a wide distribution in cavity sizes at any strain and also an increase in the number of observable cavities with increasing deformation (Chokshi, 1987). Both of these observations suggest strongly that cavities nucleate continuously during superplastic deformation. There have been many theoretical studies of cavity nucleation during high temperature creep deformation. An important conclusion of these studies is that grain boundary sliding is necessary to develop stress concentrations of the magnitude necessary for cavity nucleation (Argon, 1982). Experimental measurements reveal that grain boundary sliding contributes ~ 50-70% of the total strain during superplastic deformation (Chokshi and Langdon, 1985) and that this high contribution of sliding is maintained even at large elongations. In view of the significant contribution of grain boundary sliding, it is not surprising that cavities nucleate during superplastic deformation. It is generally assumed that cavities may nucleate if the stress concentrations caused by grain boundary sliding are not relieved sufficiently rapidly. Kashyap and Mukherjee (1986) tabulated the cavity nucleation sites reported in many superplastic alloys. In quasi-single phase superplastic alloys, where the fine grain size is stabilized by particles pinning the grain boundaries, cavities nucleate predominantly at coarse
grain boundary particles. Frequently, the extrusion and/or rolling processes used in producing a fine grain size leads to a break-up of large agglomerates into coarse particles that are aligned in stringers along the rolling/extrusion direction. The subsequent nucleation of cavities at such particles leads to the observation of cavities aligned in stringers parallel to the tensile axis which is usually parallel to the rolling direction (Ridley and Pilling, 1985; Chokshi, 1987). Cavities have also been observed in microduplex alloys which do not contain coarse grain boundary particles. In such alloys cavity nucleation is reported to occur frequently at the interphase boundaries, such as a/9 boundaries in the Al-Cu eutectic and oc/P boundaries in the Zn-22% Al eutectoid alloy. These results suggest that coarse grain boundary particles are not always necessary to nucleate cavities in superplastic alloys. The above suggestion is supported by observations of cavitation in a high purity laboratory grade Zn-22 % Al eutectoid alloy with only 15 ppm impurities (Miller and Langdon, 1978) and a Ti-6% Al-4% V (Cope and Ridley, 1986) alloy which did not contain any coarse particles. Recently, the nucleation of cavities at ledges was modelled theoretically and conditions favorable for cavity nucleation at grain boundary ledges were identified (Chokshi and Mukherjee, 1989 a). It was demonstrated that the analysis is consistent with the experimental results on the Zn-Al eutectoid alloy. Therefore, in microduplex alloys cavities may nucleate at other potential sites such as ledges and triple point grain junctions, which tend to obstruct grain boundary sliding.
9.6 Cavitation and Failure in Superplasticity
9.6.4 The Growth of Cavities The growth of cavities during high temperature creep deformation has been modelled extensively and this process is reasonably well understood. The mechanisms of cavity growth developed for creep deformation have been used successfully to study cavitation during superplasticity (Miller and Langdon, 1979). The cavity growth models may be classified broadly into two categories: one based on diffusion and the other based on plastic deformation by power-law creep. Diffusion cavity growth occurs by the stress directed diffusion of vacancies into cavities, usually along grain boundaries. The Speight and Beere (1975) model for diffusion growth, as applied to superplastic alloys, leads to the following expression for the diffusion cavity growth rate: dr de
Q5D gb 5kTr2 a —
(9-6)
where r is the cavity radius, dr/ds the cavity growth rate per unit strain, Q the atomic volume, 3 the grain boundary width, Dgh the coefficient for grain boundary diffusion, k Boltzmann's constant, T the absolute temperature and y the surface energy. Power-law cavity growth occurs by the plastic deformation of the matrix surrounding a cavity. The Hancock (1976) model for power-law cavity growth leads to the following expression: dr _ _ 2cr
(9-7)
The diffusion and power-law growth mechanisms operate independently so that cavity growth is dominated by the mechanism giving rise to a higher value of dr/ds. From the cavity size dependencies of the cavity growth rate shown in Eqs. (9-6) and (9-7), it follows that small cavities will tend
435
to grow by a diffusion mechanism whereas larger cavities will grow by a power-law mechanisms. Thus, there is a critical radius, r c , marking the transition from diffusion growth to power-law growth. It has been demonstrated for a number of superplastic alloys including the Zn-22 % Al and Fe-based (Miller and Langdon, 1979), Albased (Chokshi, 1986 a; Pandey et al., 1986) and Cu-based (Chokshi, 1986 b; Livesey and Ridley, 1982) alloys that the change in the cavity morphology with cavity size follows the trend anticipated from combining Eqs. (9-6) and (9-7). The diffusion models were developed for cavity growth in coarse-grained alloys undergoing creep deformation and they assumed that small cavities are situated on grain boundaries perpendicular to the tensile axis. However, in fine-grained superplastic alloys, cavities frequently grow quickly to dimensions greater than the grain size, and it was suggested by Miller and Langdon (1979) that this may lead to an enhancement in the diffusion cavity growth rate. A new diffusional cavity growth model, termed superplastic diffusion growth, was developed recently to account for the difference in geometry between cavities growing under creep and superplastic conditions ( Chokshi and Langdon, 1985). The superplastic diffusional cavity growth model essentially modifies the expression for diffusional cavity growth by allowing vacancy diffusion into cavities to occur along the many grain boundaries intersected by a large cavity. This model predicts the following growth rate: dr
45QdDgha UTdH
(9-8)
where d is the grain size. In general, cavity growth in superplastic alloys may occur by the three mechanisms
436
9 Superplasticity in Metals, Ceramics and Intermetallics
outlined above. Since these three mechanisms operate independently, cavity growth is dominated by the mechanism giving rise to the most rapid growth rate. Figure 9-10 illustrates schematically the variation in the cavity growth rate with cavity radius on a logarithmic scale for the three relevant mechanisms. The upper broken line indicates the overall anticipated growth behavior. Initially, a small cavity grows by a diffusional mechanism until it attains a dimension approximately equal to half the grain size. At this stage, r = rOsp the cavity intersects more than one grain boundary and there is a transition from the diffusion to the superplastic diffusion growth mechanism. Finally, when a cavity grows to a critical radius, r , there is an-
I SUPERPLASTIC DIFFUSION GROWTH
DIFFUSION GROWTH
csp
± Figure 9-10. Schematic illustration of the variation in cavity growth rate with cavity radius for the relevant cavity growth mechanisms (from Chokshi and Langdon, 1985).
:
other transition from the superplastic diffusion to the power-law mechanism. It is important to note that rcsp may be substantially larger than rc under experimental conditions of low strain rates and fine grain sizes. It has been demonstrated that the superplastic diffusion growth mechanism may account for the experimental observation of large rounded cavities in a fine-grained superplastic Cu-based alloy and Zn-22% Al eutectoid alloy (Chokshi and Langdon, 1987) tested at low strain rates. Analysis of experimental data has also shown that the superplastic diffusional cavity growth mechanism is not important during the deformation of relatively coarse-grained Al-Zn-Mg and Al-Li alloys (Chokshi, 1986a and 1986b). It is possible to visualize the influence of grain size on the cavity growth mechanisms from cavity growth maps that were developed to display pictorially the experimental conditions over which the different mechanisms are dominant. A cavity growth map is shown in Fig. 9-11 for a typical experiment condition of Q SDgb a/ (kT s) = 10" 1 9 m 3 , and it depicts the cavity radii between 0.1 and 100 \im and grain sizes between 0.1 and 100 |im. This map shows that the superplastic diffusion growth mechanism is not important for alloys with grain sizes greater than ~ 3 jim. Under such conditions, small cavities will grow initially by a diffusion mechanism and large cavities will grow by a power-law mechanism. Several attempts have been made to quantitatively compare the experimental observations with the theoretical predictions (Langdon, 1982; Kashyap and Mukherjee, 1986; Pilling and Ridley, 1988). In this context it is important to note that the theoretical cavity growth mechanisms do not take into account the experimental observations of either con-
437
9.6 Cavitation and Failure in Superplasticity 10'
10 POWER-LAW
SUPERPLASTIC DIFFUSION 1.0
DIFFUSION 10" 10"1
K)
102
Figure 9-11. A cavity growth map for a superplastic alloy tested under a set of typical experimental conditions (from Chokshi and Mukherjee, 1988 b).
tinuous cavity nucleation or cavity interlinkage. Recently, Franklin et al. (1988) examined cavitation in detail in a commercial quasi-single phase 7475 Al alloy. Tensile specimens were tested to different strains at 748 K and a strain rate of 5 x 10" 4 s" 1 . The size-distribution of cavities at different strains was obtained using a Quantimet, and the experimental cavity growth rates were determined from the slopes of a plot of the variation in the average radii of the largest 1 % population against the true strain. Figure 9-12 shows the variation in cavity growth rate with cavity radius for the theoretical diffusion, superplastic diffusion and power-law mechanism along with the experimental data. Inspection of Fig. 9-12 shows that there is good agreement between theory and experiment at low cavity radii, corresponding to low strains. At radii greater than ~ 10 [am, the experimental cavity growth rates are
higher than the theoretical predictions due to the occurrence of interaction effects and extensive cavity interlinkage. For microduplex alloys, with the two phases having substantially different flow properties, Shang and Suery (1984) developed a power-law cavity growth model, where cavity growth is controlled by deformation in the softer phase. This model leads to a cavity growth rate equation having the same form of Eq. (9-7) as that determined by Hancock (1976) for single phase alloys. Pilling (1985) modeled the influence of cavity interlinkage on cavity growth and cavity size distributions, and obtained good agreement with the experimental results. Wilkinson and Caceres (1986) examined the effect of strain hardening on cavity growth by a power-law mechanism.
103 7475 Al T - 748 K
10*
/
d - 10|im
/
"A
a/e-i.exic^MPi
10 U
-yr*'
Experimental
Data
S
• Power-Uw ~ Growth
y
E
1.0 1
10*
r
—i
-
i 10-2 -
I
Superplastic Diffusion Growth
~
1\ Diffusion Growth
\
10*'3 . ....J
10'' 10"
1.0
10
1 10*
Figure 9-12. Comparison between theoretical cavity growth mechanisms and experimental results in a superplastic 7475 Al alloy (from Chokshi and Mukherjee, 1988 b).
438
9 Superplasticity in Metals, Ceramics and Intermetallics
9.6.5 Influence of Hydrostatic Pressure on Cavitation
The occurrence of significant cavitation during superplastic deformation leads to a substantial deterioration in the mechanical properties of post-formed components. Therefore, it is desirable to reduce the extent of cavitation during superplastic deformation. It is now well known that the superimposition of hydrostatic pressure during superplastic forming may reduce the level of cavitation developed. Figure 9-13 illustrates the influence of hydrostatic pressure on cavitation in a quasi-single phase superplastic 7475 Al alloy tested to a true strain of 1.5 at 748 K and a strain rate o f 5 x l O ~ 4 s ~ 1 . A t atmospheric pressure, the specimens exhibited elongations to failure of ~ 4 2 5 % whereas under a hydrostatic pressure of 4 MPa, the specimen exhibited an elongation to failure of ~1050%. Figure 9-13 a shows cavitation in a specimen tested at atmospheric pressure whereas Fig. 9-13 b shows cavitation in a specimen tested under a hydrostatic pressure of 4 MPa. As shown, the
application of hydrostatic pressure is accompanied by a decrease in the level of cavitation. In addition, a careful inspection of similar micrographs suggests that the superposition of hydrostatic pressure reduces both the number of cavities and the sizes of the largest cavities. This observation is supported by a quantitative study of the size distribution of cavities which suggests that the superposition of hydrostatic pressure reduces cavitation by decreasing both the total number of cavities and the sizes of the largest cavities (Franklin et al., 1988). Pilling and Ridley (1986) examined cavitation in detail in three superplastic Albased alloys. The influence of hydrostatic pressure on the total level of cavitation, as obtained from density measurements, is illustrated in Fig. 9-14 for a superplastic AlLi alloy tested at 793 K and a strain rate of 1.2xlO~ 3 s~ 1 . The arrows along the praxis indicate the strains below which cavitation could not be detected using density measurements. Inspection of Fig. 9-14 indicates that the superposition of hydrostatic pressure reduces the rate of increase
Figure 9-13. Optical micrographs of a superplastic 7475 Al alloy tested to a true strain of 1.5 at 748 K, (a) atmospheric pressure, (b) 4 MPa (7) (from Chokshi et al., 1987). (b)
9.6 Cavitation and Failure in Superplasticity
439
AL-LITHIUM <7-117.1Cf3 s- 1
10
superimposed pressure,N
S o
pressure
21
0-1 y305
05
10
l/
1
15
20
25
Figure 9-14. Variation in the level of cavitation with true strain for an Al-Li alloy tested at 793 K and different levels of pressure (from Pilling and Ridley, 1986).
TRUE STRAIN
in cavitation damage and suppresses the initiation of significant cavitation to higher strains. Bampton and Raj (1982) and Bampton et al. (1983) have examined in detail the effect of hydrostatic pressure on cavitation in a 7475 Al alloy. They concluded that hydrostatic pressures of the order of the flow stress are necessary to substantially decrease the level of cavitation. While the influence of hydrostatic pressure on cavity nucleation and cavity growth is not completely understood, it is clear that the imposition of hydrostatic pressures of the order of the flow stress offers a practical means of decreasing the extent of cavitation during superplastic forming. Since the stress concentrations arising from grain boundary sliding have an important bearing on the nucleation of intergranular cavities, the contribution of grain boundary sliding to the total superplastic strain was measured with and without hydrostatic gas pressure. The details are re-
ported elsewhere by Chokshi and Mukherjee (1990). They observed that the grain boundary sliding contribution to superplastic strain was greater than 75% and that the imposition of hydrostatic pressure did not affect the contribution from grain boundary sliding. Although grain boundary sliding was found to be unaffected by hydrostatic pressures, the cavity nucleation rate appeared to be reduced. To account for this, Chokshi and Mukherjee (1990) suggest that hydrostatic pressure reduces the probability of cavity nucleation by increasing the size of a stable cavity nucleus. The radius of a stable cavity is given by r0 = 2y/a9 where y is the surface energy and a is the normal stress. In the presence of a hydrostatic pressure, r0 = 2y/(a-P) where P is the imposed hydrostatic pressure. Thus, for example, when P = a/2, the value of r 0 is increased by a factor of 2 over that at atmospheric pressure. Since the volume of a critical cavity nucleus is proportional to
440
9 Superplasticity in Metals, Ceramics and Intermetallics
r3, an increase in r0 by a factor of 2 will increase the number of vacancies necessary to form the nucleus by almost an order of magnitude. This process essentially retards cavity nucleation by decreasing the probability for the formation of a stable cavity nucleus. Prior to this investigation, the effect of imposed hydrostatic pressure was thought to affect only the rate of cavity growth, described next. As discussed earlier, cavity growth in superplastic alloys may occur either by a diffusion or plasticity controlled mechanism. These mechanisms indicate that during the initial stages of uniaxial tensile deformation, when cavities are small, cavity growth occurs largely by a diffusional mechanism. At higher elongations, when the cavities are larger, cavity growth is dominated by the plasticity controlled mechanism. For a diffusion controlled mechanism, cavity growth is controlled by the maximum principal stress and the cavity growth rate, dr/dt, is proportional to (cr-P). For a plasticity controlled mechanism, the mean stress is the appropriate parameter and dr/dt is then proportional to (<j/3-P)n, where n( = l/m) is the stress exponent. Both of the cavity growth mechanisms are thus expected to be retarded by the imposition of a hydrostatic pressure, in agreement with previous studies (Bampton and Raj, 1982; Pilling and Ridley, 1986). The significance of the observations of Chokshi and Mukherjee (1990) is that cavity nucleation is reduced by hydrostatic pressure as well. The beneficial effect of hydrostatic pressure has been noted in composites and ceramics as well. Mahoney and Ghosh (1987) tested a PM64 aluminum alloy with up to 20 vol% SiC particles in tension under hydrostatic pressure, and reported that the level of cavitation was reduced. Wang and Raj (1984) investigated the extent of
superplastic deformation in two lithium aluminosilicate glass-ceramics. The maximum obtainable elongation was found to be limited either by intergranular cavitation or by the initiation and growth of surface cracks. Tensile tests were performed under a superimposed hydrostatic pressure to distinguish between the two failure mechanisms. They studied two materials; one with a high flow stress and the other with a lower value. Cavitation was substantially suppressed by hydrostatic pressure in both materials. The strain-to-fracture, however, was increased only in the material with the lower flow stress, where intergranular cavitation was the dominant failure mode. In the higher flow stress material, failure resulted from the initiation and propagation of cracks from the surface, so the tensile fracture strain was not affected by hydrostatic pressure.
9.7 Recent Advances in Superplasticity This section reviews and evaluates new results in superplasticity in a few specific areas where significant progress has been made. These areas include the superplasticity now attainable at comparatively high strain rates in some mechanically alloyed aluminum matrix, the very promising investigation of superplasticity in ceramics and the new area of superplasticity in intermetallic compounds. 9.7.1 High Strain Rate Superplasticity For a given superplastic mechanism and the same stress level, the strain rate increases with decreasing grain size. Currently the fastest superplastic deformation has been found to occur in IN90211, a mechanically alloyed fine grained (0.5 jim)
9.7 Recent Advances in Superplasticity
aluminum alloy (Bieler et al., 1988). The composition of IN90211, (4.4% Cu, 2% Mg) is nominally similar to 2124 aluminum. The Mg is in solution at all temperatures, but the Cu forms precipitates of 6 phase (Al2Cu) in a way that depends upon processing. In addition to these alloying elements the mechanical alloying process introduces 0.8 % oxygen and 1.1% carbon, which exists in the form of extremely fine (10-25 nm) oxide and carbide particles that make up 5 vol% of the alloy. The interparticle spacing is between 30 and 45 nm. Elongation and the stress-strain rate behavior of this alloy are shown in Fig. 9-15. Optimum elongation exceeded 500 % at a strain rate of 2.5/s, and 475 °C. (This elongation is highly subject to the thermal gradients in the specimen, which depend strongly upon the geometry and conductivity path of adiabatically generated heat; Bieler and Mukherjee, 1990 a.) Fracture was predominantly intergranular in nature. Observations on initially polished surfaces with marker lines showed that grain boundary sliding and rotation occurred, with the dominant features being the 2-3 jLim platelets. A remarkable feature
441
of the material near the fracture surface was the near absence of cavities in specimens deformed under the optimum superplastic condition (Bieler et al., 1990). Based upon detailed mechanical and microstructural studies, the following deformation mechanism for superplasticity in IN90211 is proposed (Bieler and Mukherjee, 1990 b): Lattice dislocations are limited in their motion by dragging solute atoms and the necessity of climbing over particles by a mixture of local and general climb (causing the threshold stress). When they reach the boundary, they dissociate into grain boundary dislocations that assist grain boundary sliding. Evidence for grain boundary dislocation activity, and lattice dislocations interacting with particles and grain boundaries were observed. Compatibility problems at triple joints and other obstacles are accommodated by slip in the adjacent grains, thus precluding cavity nucleation. Hence, both grain boundary sliding and bulk slip operated together to permit superplastic elongations in IN90211. The predictions from models for grain boundary sliding, solute drag, and the universal climb theory of Weertman and Weertman (1987) were compared
IN90211 data
Grain Boundary Sliding (n=2) Sherby 0.5 u.m
Universal Climb (n=3) Weertman
.001
.01
.1
1
10
True Strain Rate
(1/sec)
100
1000
Solute Drag
Mg Cu
(n=3) Cottrell-Jaswon
10r12
1 0"'
UTS Effective Stress/G
Figure 9-15. Stress and elongation as a function of strain rate and temperature (Bieler and Mukherjee, 1990 b).
442
9 Superplasticity in Metals, Ceramics and Intermetallics
with the superplastic data of IN90211. A detailed analysis of the activation energy suggests that slip in the grains with n = 3 is the rate limiting step of deformation here. The fact that the constitutive relation for grain boundary sliding also predicted the correct magnitude for the strain rate, based upon the 3 jim platelet size, suggests that the grain boundary sliding mechanism is not much slower than slip in the grains. However, the bulk of the evidence suggested that the motion of dislocation overcoming Mg atom solute drag was the rate controlling step (giving n = 3) and that mixed climb of the gliding dislocation over the oxide and carbide particles was responsible for the highly temperature sensitive threshold stress that was observed. 9.7.2 "Low-Temperature" Superplasticity
Recent experimental observation has shown evidence of the so called low-temperature superplasticity in alloys with grain sizes about 0.1 jam (Valiev et al., 1988; Valiev and Tsenev, 1990). This effect was observed in Al-based and Mg-based alloys. Table 9-3 presents test conditions and parameters for superplastic deformation of the Al-4%, Cu-0.5% Zr alloy with grain sizes 0.3 and 8 jum and the Mg-1.5 % Mn-0.3 % Ce alloy with grain sizes 0.3 and 10 jam. The data presented indicate that the alloys having sub-micrometer grain microstructure display typical superplastic
behavior at temperatures that are often 200 to 250 K lower than that required for the typical 8 to 10 jam grain diameter microstructure. For example, in the Al-CuZr alloy with a grain size of 0.3 jim, the m-value and large elongation properties at 493 K are similar to the corresponding properties of this alloy with a grain size of 8 jLim at 773 K. In the sub-micrometer grained Mg-Mn-Ce alloy, the superplasticity can be manifested at a comparatively lower temperature as well. Electron microscopic studies indicated that the structural changes occurring in the case of the deformation of sub-micrometer grained alloys are also typical for conventional superplastic alloys. After deformation, grains remain equiaxed, no dislocations are found in the structure, and some extent of grain growth is observed. Thus, after 100% tensile straining, the grain size increased from initially 0.1 to a final value of 0.5 jim. The nature of this low-temperature superplasticity is obviously associated with considerable acceleration of the grain boundary diffusion in alloys when the grain size is about 0.1 jim (Valiev et al., 1990). The physical situation here is similar to the one observed in nanocrystalline materials (see Section 9.8). 9.7.3 Superplasticity in Ceramics
In ceramic materials, crystal bonding is strong and directional, so motion of dislo-
Table 9-3. Conditions and parameters of superplastic deformation of some alloys (from Valiev et al., 1988). Alloy Al-Cu-Zr Mg-Mn-Ce
J(urn)
T(K)
fife"1)
G 4
(MPa)
m
£(%)
8 0.3
773 493
3xl0" 3xl0"4
14 23
0.50 0.48
800 >250 a
10 0.3
673 453
5xlO~ 4 5xlO"4
25 33
0.42 0.38
320 >150 a
Measurements were restricted by test conditions.
9.7 Recent Advances in Superplasticity
cations is a difficult process. The consequent lack of adequate ductility has hampered the introduction of ceramics as structural materials for a long time. Recently, however, superplastic-like ductility has been reported in several ceramic materials at attractively moderate strain rates. Often superplasticity is the only forming mechanism that can be used for ceramics. In line with the earlier part of this presentation, the discussion on superplasticity in ceramics will be restricted to micrograin superplasticity. Interested readers may wish to refer to a recent review (Sherby and Wadsworth, 1988) on transformational superplasticity in ceramics where the enhanced ductility is induced by thermal cycling of materials about their phase transition temperature. Furthermore, this review will focus primarily on tensile ductility. The stress distribution in compression (as well as superimposition of hydrostatic pressure) can minimize cavitation and, hence, improve ductility. However, this behavior does not necessarily indicate the presence of neck stability under tensile loading. A high value of the strain rate sensitivity exponent m is certainly necessary, but it is not a sufficient criterion for superplasticity. The structure of the interface is important for a fundamental understanding of superplasticity. The rate of atom transport along the interface and the rate at which atoms can be attached or removed from an adjoining grain are properties that are expected to be strongly dependent on the structure of the interface. The presence of a thin film of fluid at the grain interface can drastically enhance diffusion through the interface and can have a dramatic effect on the superplastic characteristics as well. The grain (or phase) boundaries of most of the superplastic ceramics that have been
443
reported so far are of high angle. For such high angle boundaries, two types are possible (Raj, 1989) depending on whether or not there is any liquid at the boundary (at the temperature of testing). In the first case, with no liquid in the boundary (type I superplasticity in the classification by Raj), the deformation is limited by the transport properties of grain interfaces, i.e., some form of Nabarro-Herring/Coble creep. The atomistic mechanism is related to the transport of atoms in order to change the shape of the grains by a crystalgrowth-and-dissolution-like process. The interface structure is such that it leads to strain induced grain growth. Hence, such a process should give a stress dependence of strain rate n = 1. To the best of our knowledge, this type of superplasticity has not been observed in tension up until now. However, an w-value equal to one has been observed in superplastic compression in fine-grained alumina (Venkatachari and Raj, 1986; Carry and Mocellin, 1985). According to Raj, the type II superplasticity is exhibited by fine grained polycrystals that often contain a very small amount of a fluid phase in grain interface. Often such a fluid phase consists of a low temperature eutectic, formed due to impurity segregation in graiun boundaries. The liquid phase is expected to enhance creep because it increases the rate of diffusional transport through the interface. The flow equations for this type of superplasticity have been reviewed (Wang and Raj, 1984). The model by Raj depends on (i) the postulate that the atomic structure of the two grain junctions consists of islands, where adjacent crystals meet and (ii) the presence of an interpenetrating liquid phase. The island structure of the interface provides a solidto-solid contact to support traction gradients and the interpenetrating fluid provides a path for fast diffusion. The strain
444
9 Superplasticity in Metals, Ceramics and Intermetallics
rate may be controlled either by the interface reaction between the crystal and fluid or by the diffusion rate of matter through the fluid, whichever provides the slower rate. The strain rate in the diffusion limited case is given by: aQca
(9-9)
where c is the molar soubility of the crystal in the fluid, rj is the viscosity of the grain boundary phase, Q is the atomic volume, d is the grain size and a is a factor that depends on the structure of the boundary. The equation does not depend on the volume fraction of the liquid phase, but it does depend on its viscosity. The linear relation between stress and strain rate and the cubic grain size dependence of Eq. (9-9) has been convincingly demonstrated in tension in p-spodumene glass-ceramics (Wang and Raj, 1984). Their investigation revealed that ultrafine grained ceramics are capable of high rates of deformation (10 ~ 4 to 10~3/s) at quite low stresses (1 to 20 MPa). Equation (9-9) predicts that the viscosity of a ceramic polycrystal is directly proportional to the viscosity of the glass phase even when the glass is present in small amounts. In an interesting correlation Wang and Raj (1984) pointed out that a wide range of strain rates can be obtained in pspodumene glass-ceramics simply by changing the chemistry of the intergranular glass. The presence of dopants can change the viscosity of the glass by several orders of magnitudes and can be an important processing variable. However, there are now reported data on superplasticity in ionically bonded ceramics, e.g., zirconia polycrystals (for a review, see Wakai et al., 1989 a) and on covalently bonded ceramics, e.g., Si3N4/SiC
composite (Wakai et al., 1990) that show n = 2 dependence instead of the n = 1 dependence expected from Eq. (9-9). In both of these ceramics, it was claimed that an intergranular liquid phase was present. It is possible that one may have to distinguish between (a) liquid phase that is present at triple points and at two-grain junctions (i.e., as postulated in the model by Raj) and (b) very thin amorphous film (1-2 nm in thickness) that may be present at two-grain junctions. There has been some discussion in the literature (Clark, 1987; Marion et al., 1987; Ruhle et al., 1986) about the presence of very thin amorphous film at two grain junctions in some oxides and in some non-oxide ceramics. There has been speculation that some kind of structure exists in the film with the thickness of a few nm due to atomic interaction at the interfaces (Clark, 1987). However, in a recent investigation on superplasticity in yittria-stabilized tetragonal zirconia polycrystals (Y-TZP). Nieh and Wadsworth (1990), using high resolution electron microscopy, could not find any evidence of amorphous or liquid phase at the grain boundary. Lattice fringes from adjacent grains, can be followed to their intersections at both the grain interfaces and triple junction, thereby demonstrating the absence of any second phase. Furthermore, Yoshizawa and Sakuma (1990 a) noted that the absence of (deliberately added) glassy phase in the grain boundary of Y-TZP did not alter the manifestation of n = 2 superplastic behavior in their alloys, which they ascribed to grain boundary sliding. It appears that for the grain boundary sliding related n = 2 superplasticity, at least, the presence of a glassy phase at the grain boundary is not a prerequisite in ceramics. The work of Yoshizawa and Sakuma demonstrated that YZP containing 5 wt.% of lithium silicate
9.7 Recent Advances in Superplasticity
glass can be deformed superplastically at a temperature as low as 1100°C. It is clear that the issue of a glassy phase vs. amorphous structure at the grain boundary and their contribution to superplasticity in ceramics is not settled yet. However, these debates on basic understanding should not detract us from the remarkable progress that has been demonstrated in recent years in superplasticity of structural ceramics. Two of the outstanding examples of superplasticity in structural ceramics are provided by the work of Wakai et al. (1986) and Wakai and Kato (1988). In tensile superplastic deformation of fine-grained (~0.3|im diameter) yttriastabilized tetragonal ZrO 2 polycrystals (Y-TZP), an elongation-to-failure of 200% (Fig. 9-16) was reported by Wakai et al. (1986). The temperature and strain rate were 1450°C and 2.8xlO~ 4 /s. This was a landmark experiment which clearly demonstrated for the first time the manifestation of micrograin superplasticity in tension tests in a structural ceramic. The same material tested at higher temperature, i.e., 1550°C and at a strain rate of 8.3xlO~ 5 s" 1 produced a tensile ductility in excess of 800% (Nieh et al., 1990). These are exciting results that have stimu-
Figure 9-16. Undeformed (bottom) and superplastically deformed (at 1450 °C) specimens of 3 mol% Y 2 O 3 stabilized TZP (from Wakai et al., 1986).
445
lated considerable interest among the materials profession. The results of Wakai et al. (1986 and 1989 a, b) and Wakai and Kato (1988), revealed a value for the stress exponent equal to 2, a strain rate that was proportional to the inverse square of grain size and an activation energy of deformation that was much higher than that of grain boundary diffusion or that of lattice diffusion of cation, i.e., the slower diffusing species. The microstructure revealed evidence of cavitation that increased with strain rate and also of strain enhanced grain growth during superplastic deformation. There was no evidence of heavy dislocation activity or subgrain formation in individual grains. Some of these observations are very similar to those noted in superplasticity in metallic systems (Kashyap and Mukherjee, 1985 a). The investigation on superplasticity of TZP/ A12O3 composite (grain size ~0.5 jam) by Wakai and Kato (1988) revealed trends very similar to that for Y-TZP alloy, including a stress sensitivity of 2, strainenhanced grain growth and strain-rate dependent cavitation. In addition, Wakai and Kato estimated that the contribution of grain boundary sliding to total strain was between 60% to 80% - a value remarkably similar to that obtained in studies of micrograin superplasticity in metals. These experimental results on tetragonal zirconia polycrystals (TZP) and also on TZP/20 wt.% A12O3 composite have been plotted in Fig. 9-17 as normalized strain rates vs. normalized stresses. The parameters for the normalization procedure, i.e., temperature dependence of modulus, etc., are described in Mukherjee et al. (1989). The plot of Fig. 9-17 correlates quite well the superplastic stress-strain rate results in TZP and TZP/A12O3 fine-grained ceramics at various temperatures with an rc-value equal to two. The deviation of the data
446 10 1 6
10 1 5 cc
9 Superplasticity in Metals, Ceramics and Intermetallics Y TZP/20w% Alumina n Q • • •
n=2
1250°C 1300°C 1350°C 1400°C 1450°C
YJ2E 10 1 4
• A • 4
1150°C 1250°C 1350°C 1450°C
10 13
10 1 2 10"
10" Stress/E
Figure 9-17. Temperature and modulus compensated stress-strain rate relation in Y-TZP and A12O3/YTZP composite [from Mukherjee et al. (1989), after results from Wakai et al. (1986) and Wakai and Kato (1988)].
for 1350°C for the TZP/A12O3 composite from this correlation was also evident in the primary experimental date (Fig. 5 of Wakai and Kato, 1988). The explanation for this deviation may lie in the original test conditions and circumstances. Wakai et al. (1989 b) have subsequently studied the superplastic properties of ZrO 2 /Al 2 O 3 composites where the A12O3 content was varied from 20 to 80%. The stress dependence of the creep rate n was nearly equal to two as Y-TZP polycrystals. The tensile elongation in the composites decreased with increasing A12O3 content. They correlated their results of the composite with Chen's (1985) rheological model of a composite containing a spherical inclusion that can be either harder or softer than the matrix. It is interesting to note that the degree of strain-enhanced growth in the zirconia/ alumina composite was considerably less
than that observed in Y-TZP (Nieh et al., 1990; Wakai et al., 1989 a). As mentioned earlier, Nieh and Wadsworth (1990) obtained a much higher elongation in Y-TZP by testing at a higher temperature. At these higher temperatures, the strain enhanced grain growth was significant and the measured n-value was closer to three. It is only when they normalized their data to take into account the phenomenon of grain growth, that they observed a value of n « 2. However, in their investigation of the ZrO 2 /Al 2 O 3 (Nieh et al., 1990) no such normalization was deemed necessary, since in the presence of alumina phase the grain growth was not excessive and both they and Wakai and Kato (1988) observed ft = 2 in this composite. The strain enhancement of grain growth has also been studied in Y-TZP by Yoshizawa and Sakuma (1990 b). They observed that the strain induced grain growth rate was proportional to strain rate. But the proportionality constant was found to be two orders of magnitude smaller than in superplastic metals. It was concluded that the smaller value of the constant is caused by smaller deformation zone formed by unit grain boundary sliding in TZP. 9.7.3.1 Reflections There is much in common in the description of micrograin superplasticity between metals and ceramics. The basic structural requirement remains the same, i.e., a fine, equiaxed grain size that is reasonably stable during deformation and high angle grain boundaries. In several metallic systems the alloy is either an eutectic or an eutectoid, where the chemical or structural dissimilarity between the two phases across a phase boundary minimizes the tendency for grain growth. It is interesting to note that in Y-TZP, it is believed (Wakai
9.7 Recent Advances in Superplasticity
et al, 1986 and 1989a; Yoshizawa and Sakuma, 1989) that the coexistence of tetragonal and cubic phases under testing conditions contributes to the stability of grain size. As stated before, the phenomena of grain boundary sliding, generally equiaxed nature of deformed grains, strain-enhanced grain growth and intergranular cavitation are common for both metals and ceramics. Several investigators (Nieh et al., 1990; Nieh and Wadsworth, 1990; Wakai et al., 1989 a) have commented that whereas in metallic systems, superplastic behavior can be observed for grain sizes that are typically ~10 |im, the corresponding grain sizes for superplastic ceramics are significantly finer, usually less than 1 jim. In fact, Wakai et al. (1989 a) could observe superplasticity in yttria or CeO2 stabilized zirconia polycrystals having grain sizes equal to 0.8 jim or less. But the superplastic behavior was lost at a grain size of 1.9 jim. We believe this is associated with the scaling law for high temperature plasticity. As an illustration, let us compare superplasticity in aluminum-based matrix and that for Y-TZP at 0.75 of their respective melting point. We assume that the activation energy is that for grain boundary diffusion. Rewriting Eq. (9-1) and noting that the diffusivity D = Z) o exp[- Q/(RT)] and taking ratios, one can write very approximately: A1 Zr4
at constant strain rate, constant moduluscompensated stress and comparable homologous temperature and on the assumption that the rate parameters remain unchanged. The grain boundary diffusivity (Frost and Ashby, 1982) of aluminum is given by:
447
= 2.0x10" 1 4 • exp [ - 60 (kJ/mol)/RT] m 3 /s and the diffusivity for boundary diffusion of Zr 4 + ion in 16mol% Y2O3-stabilized ZrO 2 is given (Oishi et al., 1983) by: • exp [ - 309 (kJ/mol)/i?7] cm3/s The melting point of cubic ZrO 2 is 2680°C (Wakai etal., 1989a). So 0.75 of the melting point of this matrix is 2214 K. Similarly, 0.75 of the melting point of aluminum is 700 K. One can now evaluate the ratio: J2214K '
:(5.7xlO" 19 m 3 /s)/ (6.08xl
448
9 Superplasticity in Metals, Ceramics and Intermetallics
retical models for superplasticity differ from each other primarily in the details of accommodating the stress concentration that arises due to grain boundary sliding. Due to the intrinsically strong covalent bond strength in ceramics and also because of the directionality of the bonds (and consequently high Peierls stress), it is anticipated that dislocation slip motion related accommodation process will contribute less in ceramics than diffusional stress accommodation. This suggests that extensive dislocation activity and slip-related texture formation should be minimal in superplasticity in ceramics. This conjecture is borne out by experimental results (Wakai et al., 1989 a and b). The precise details of the grain or phase boundary structure are expected to be extremely important in ceramics. Carry and Mocellin (1985) have emphasized the extreme sensitivity of grain boundary diffusivity to the local chemistry. The grain or phase boundary structure is not very well understood and certainly difficult to control. The creation, annihilation and interaction of point defects with impurities, deviation from stoichiometry, etc. will have very important effects on the activation energy of interface diffusion. Under such a situation, conventional experimental measurements of activation energy of deformation do not correlate very well with model predictions. There are already some advancements in translating the ductility of superplastic ceramics to forming processes. Kellet et al. (1988) succeeded in hot extruding 0.23 |im grain size Y-TZP powders to full density at 1500°C using an 8 to 1 reduction in area. Fridez et al. (1984) have successfully hot forged fine grained A12O3 at 1500 to 1600 °C. Kellet and Lange (1988) have also successfully hot forged 20 vol% ZrO 2 / A12O3 composite at 1500 °C to nearly 60 %
reduction in thickness. These results would have been unthinkable a decade ago. 9.7.4 Superplasticity in Ordered Intermetallics
Ordered intermetallic alloys often exhibit excellent high temperature strength properties. However, their ductility (and hence, fabricability) is often very low. Therefore, until very recently they have not been seriously considered for structural applications. Recent studies suggest (for review, see Liu and Stiegler, 1984) that the ductility and fabricability of ordered intermetallic alloys can be substantially improved by alloying and by control of microstructure with thermomechanical processing. One outstanding example of such an approach is the effect of boron addition to nickel aluminides. These aluminides based on Ni3Al (ordered cubic L l 2 structure), are ductile as single crystals but are exceptionally brittle in polycrystalline form. Microalloying studies (Liu and Koch, 1983; Taub et al., 1984) have shown that the ductility at ambient temperature can be improved very significantly by adding ~ 200 ppm boron, which strongly segregates to the grain boundaries and improves the grain boundary cohesion. Recently, another avenue of investigation has opened up the possibility of inducing excellent high temperature ductility in such ordered alloys by making them superplastic. The substoichiometric (with respect to Al) cast and wrought nickel aluminide alloy with 8 wt.% Cr, 0.8 wt.% Zr and 200 ppm boron, was thermomechanically processed to produce a fine ~13 jura grain size. The addition of Cr introduced 5 to 15% of the disordered f.c.c. gamma phase in this otherwise single phase (gamma prime) ordered structure (Ll 2 ). This alloy (IC218) demonstrated (Choudhury
9.7 Recent Advances in Superplasticity Boron Doped Ni3AI Alloy With 8 wt % Cr
1 cm
ef = 6 4 1 %
8 = 8.94 x 10" 4 S" 1
T = 1373 K,
Figure 9-18. Comparison of superplastically deformed specimen with undeformed specimen of Ni3Al (from Mukhopadhyay et al, 1990).
et al., 1989) a superplastic elongation of 638% when tested at a strain rate of 8.3 x l(T 4 /s at 1100°C. It is interesting to reflect on the fact that until recently, in polycrystalline form, this alloy was considered completely brittle and unworkable. Mukhopadhyay et al. (1990) have also investigated the rate parameters and microstructure for superplasticity in cast
449
(IC218) Ni3Al at constant strain rate under argon gas environment. Figure 9-18 depicts a 641% superplastic elongation obtained in this intermetallic alloy. The grain size of this alloy was about 6 jim and it also had the approximately 10% disordered gamma phase in the micro structure. The strain rate vs. stress relation is shown in the double logarithmic plot of Fig. 9-19. At high strain rate (region III) the m-value is 0.32 and preliminary examination suggests that this region may be associated with the microcreep mechanism of viscous glide of dislocations through ordered lattice with an «-value = 3 (i.e., m = l/n = 0.33). At intermediate strain rates (region II), the m-value was high, approximately 0.75 to 0.9. The 638 % ductility depicted in Fig. 9-18 was obtained in this region. SEM studies revealed evidence of grain boundary sliding. The activation energy in region II was estimated to be 290 kJ/mol which compares embarrassingly well with the value of 290 kJ/mol, that was determined for interdiffusivity of Ni in Ni 3 Al by Chou and Chou (1985). Mukhopadhyay et al. (1990), also noted cavitation phenomenon and intergranular fracture in
2.4
2.0
CO Q.
1.6
£
1.2 0.8
0.4
-4.6
-4.2
-3.8
-3.4
-3.0
Log e (s-1)
-2.6
-2.2
-1.8
-1.4
Figure 9-19. Stress-strain rate data at various temperatures for boron-doped Ni3Al (from Mukhopadhyay et al., 1990).
450
9 Superplasticity in Metals, Ceramics and Intermetallics
superplastically deformed specimens. The extent of cavitation increased with increasing grain size or strain rate. The alloys based on Ni3Si have been investigated by Nieh and Oliver (1989), Nieh (1990) and Stoner and Mukherjee (1991). The latter observed a superplastic deformation of 700 % in Ni3Si at 1070 °C and at £ = l x l O ~ 3 s " 1 . The cast and hot-rolled alloy had a grain size of 11 |im in the alpha phase and 4 jim for the ordered P phase. The microstructure consisted of approximately 50:50 ordered: disordered phases at the test temperature. The dynamic grain growth in this alloy was quite sluggish. It is possible that the coexistence of the ordered and disordered phases contributed to the stability of grain size by retarding grain boundary migration. The m-value in this material was 0.5 in the superplastic region II and it decreased to a value of 0.33 in region III (Stoner and Mukherjee, 1991). One of the interesting characteristics of superplasticity in Ni3Si is the fact that superplasticity exists over a limited temperature range, about 1000-1100 °C. Over this temperature range Ni3Si is expected to have a two phase microstructure. This suggests that superplasticity in Ni3Si requires the presence of a two-phase microstructure for the retention of a stable and fine grain size. Significant amounts of research work are going on to ascertain if superplastic properties can be obtained in other ordered alloys as well. Much of these are current investigations and the results are, as yet, unpublished. The titanium aluminides have also drawn considerable attention because of their lower densities. In alloys based on Ti3Al (grain size 3-5 jLim), Bampton (1990) and Yang et al. (1991) has obtained an elongation in excess of 500 % at around 1000 °C temperatures at the strain rate of about 10~4/s. The strain rate sensitivity in region II was noted to be
around 0.5 or 0.6 and the neck stability was fairly good. The m-value decreased to 0.33 in region III, i.e., at higher strain rates (Yang et al., 1991). Some of the Ti3Al based alloy systems use niobium as a (disordered) beta-phase stabilizer. At the superplastic deformation temperature, the microstructure may consist of the betaphase as the predominant (disordered) phase plus the alpha phase, which becomes disordered at elevated temperatures. This raises the semantic problem: Is this particular alloy an example of superplasticity in an intermetallic i.e., an ordered alloy or is this simply a two-phase matrix exhibiting micrograin superplasticity like in alphabeta Ti-6A1-4V alloy? Nevertheless, this alloy has very attractive properties for structural application and can obviously be deformed in superplastic mode. The super alpha-two version of Ti3Al (composition Ti-25Al-10Nb-3V-lMo) is noted for its lack of cavitation in superplastic forming. Bampton (1990) observed dynamic grain growth in this material but no evidence of dynamic recrystallization. Fracture occurred by necking to a point and one can produce complete bond by diffusionbonding super alpha-two. The alloys based on y-TiAl with minor alloying additions like vanadium are also being investigated. They remain ordered at high temperatures. The materials has low density and very attractive high temperature properties. This alloy is not usually superplastic but can be made so at extremely high temperatures. Bamptom (1990) has reported superplastic elongation in excess of 300 % in y-TiAl containing 7vol.% of TiB 2 . A large elongation has also been reported by y-TiAl by Kim (1990). In addition, Bamptom has studied the growth and interlinkage of cavities as a function of strain in both powder metallurgy and ingot metallurgy y-TiAl. He also
9.8 Unconventional Approaches
studied the deformation kinetics in gamma TiAl. In high strain rates (10 ~ 3 to 10 ~2 s" 1 ) gamma material deforms by power law creep with an activation energy of 416 kJ/mol and a strain rate sensitivity m = 0.21. At the high rates and at 1200 °C9 he observed grain refinement by dynamic recrystallization. At lower strain rates the material showed typical superplastic behavior with a low activation energy (188kJ/mol). There was also evidence of grain growth at these lower strain rates. The flow stress increased with increases in grain size, pointing to the increasing influence of diffusional flow mechanism. Although these are very recent investigations, they demonstrate the considerable level of involvement on behalf of the materials community in attempting to develop superplastic intermetallic alloys. The outcome of this effort should become evident in the literature in the near future. It is interesting to note that the strainrate sensitivity (m-value) for region III for superplastic intermetallics discussed so far (i.e., Ni3Al, Ni3Si, Ti3Al and TiAl) is between 0.27 and 0.33. Thus, the stress sensitivity of the strain-rate n (which is equal to l/m) is approximately equal to three. Such w-values have been seen in viscous creep studies of nominally coarse grained Ti3Al, TiNi, Ni3Al and NiAl (see Stoner and Mukherjee, 1991). The premise in these observations is that when disorder is introduced into a crystal by the glide of a dislocation, the steady-state velocity is limited by the rate at which chemical diffusion can reinstate order behind the gliding dislocation. In such cases, creep, and perhaps the high strain-rate deformation behavior of superplastic intermetallics, is controlled by a viscous glide process (for a relevant review of creep studies, see Mukherjee, 1975). In region II for these intermetallics, m is generally 0.5-0.6, thus n is approximately
451
2. At lower strain-rates, the viscous motion of the gliding dislocation can keep up with the externally imposed strain-rate. The rate-controlling mechanism is then the climb motion of these dislocations at their annihilation sites at grain boundaries. As given by the models of Mukherjee (1971) and Langdon (1970) this leads to an ^-value of 2. Stress-drop experiments in creep and observation of the ensuing transients (Biberger et al, 1991) in Ti3Al seem to support the conjecture that the rate-controlling step in region III is viscous motion of gliding dislocations (giving alloy-class transient), whereas the controlling step in region II is the climb of dislocations at or near the grain boundaries (giving metalclass transient).
9.8 Unconventional Approaches The large neck-free elongation associated with superplasticity seems to obey a scaling law. At geologically significant flow rates (10~ 12 s" 1 or below) superplasticity can be expected in significantly coarser grain sizes. The possibility of superplasticity in geological materials, e.g., limestones, mylonites, etc. has been discussed by Mukherjee et al. (1989) and by Schmid et al. (1977). As grain sizes decrease, the optimum superplasticity can be observed at increasingly faster strain rates. This is depicted in Fig. 9-20, taken from the work of Nieh and Wadsworth (1989). It will be natural to ask what the behavior would be if the grain size were smaller than that attained in mechanically alloyed materials (see Sec. 9.7.1) or ceramics (see Sec. 9.7.2). An obvious choice will be nanocrystalline materials. As described by Gleiter (1985) and by Birringer (1989), the crystal sizes of nanocrystalline materials (NCM) are about 1 to 10 nm. They can be single or
452
9 Superplasticity in Metals, Ceramics and Intermetallics
Strain Rate(s"1) 101
10°
10 2
10J
1000
g ••3
c ,o
100
LU
i2124AI20%SICw, 525 °C 1 ^im
10
i
i i i i ml
10
i
i i i i i
10 2
i i i i nil
10 J
i
i i i i n11
i
10*
i i i i ml
10°
i
i i i i nil
10'
Strain Rate (%/min) Figure 9-20. Elongation-to-failure as a function of strain rate for various superplastic Al alloys (from Nieh and Wadsworth, 1989).
multiphase and typically about 50 % of the volume consist of grain or interphase boundaries. NCM seems to exhibit an atomic structure which is different from that for the crystalline state (because it has no long-range order). It is also different from the glassy state because it has no short-range order. It has been suggested that the atoms situated in the grain or interphase boundaries of NCM represent a new solid-state structure having features of gas-like disorder. The diffusivity in NCM can be very large. Birringer et al. (1988) have observed that, in nanocrystalline copper (crystal size 8 nm), the self diffusivity was increased by a factor of 1019 in comparison to lattice diffusion of coarse-grained copper. There was a corresponding increase in boundary diffusivity as well. The very high boundary density presumably provides a connective network of short circuit diffusional path. Karch et al. (1987) demonstrated that nanocrystalline ceramics like CaF 2 and TiO 2 became ductile and could undergo very large plastic deformation at a surpris-
ingly low temperature. Since in superplasticity the strain rate £ ex (diffusivity)/(grain size) 2or3 , one can envisage a very large increase in strain rate due to an increase in diffusivity and a decrease in grain size from 10 (im to 10 nm. Birringer (1989) has suggested that Coble creep rate can be enhanced by a factor of about 10 11 in NCM. We believe the same is true for superplasticity in such nanocrystalline material. Alternately, because of the scaling effect that is implicit in Eq. (9-1), if the strain rate is kept constant, say at quasi static rates, then in NCM superplasticity may be observed at uncommonly lower temperatures. As stated before, Karch et al. (1987) could heavily deform nanocrystalline CaF 2 at 353 K. It is an interesting area for exploration. Such an approach may give us ductile ceramics as a new structure material on its own right. Another interesting area for investigation will be superplasticity at explosive strain rates. The intention will be to explore a totally different phenomenon that may also produce large neck free ductility.
9.9 Superplastic Forming Technology
It is not associated with grain boundary sliding. In principle, it should be observed in single crystals. It is the domain of phonon viscosity where lattice phonons interact with high speed dislocations and produce a drag effect on them. The result is a Newtonian-viscous behavior with n = m = i. The process is observed at strain rates higher than about 1 0 4 s " 1 and the mechanisms have been reviewed by Klahn et al. (1970). In an early work with AgMg single crystals Mukherjee et al. (1966) observed large (in excess of 300%) deformation at strain rates higher than 104 s" 1 using a gas-gun and split Hopkinson's bar arrangement. Unfortunately, this work was done in compression. It will be very interesting ro repeat such work in very high strain rate tension tests that can be conducted now-a-days. The modulus normalized flow stress vs. strain rate is independent of temperature in phonon viscosity (Mukherjee et al., 1966; Studt et al., 1973) and in principle, if this approach is successful, one should be able to observe the much enhanced plasticity even at room temperature, at these high strain rates.
9.9 Superplastic Forming Technology Superplastic forming (SPF) and diffusion bonding (DB) processes have become established technology with special application to forming airframe parts and components of gas turbine engines (Agrawal, 1985; Pilling and Ridley, 1989; Hamilton and Paton, 1988). The advantages for using this technology arise primarily from an increased design flexibility and hence, to reduced cost of fabrication of parts. The ability to fabricate integral structures in a single operation greatly reduces the requirements for a large number of detail
453
parts, each with its associated tooling and fabrication costs, and minimizes subsequent assembly operations. The use of fasteners is reduced, thus eliminating the superfluous weight (Kelley, 1985). The SPF manufacturing capability can be substantially enhanced by forming more than one part of a subassembly by nesting compatible parts in a cage to carry out the forming (Arieli and Vastava, 1985). There is also emphasis in the current practice to optimize the microstructure to get faster forming rate and above all, to automating the operation. Superplastic forming technology is being utilized for titanium alloys and aluminum alloys (primarily in fabrication of airframes) and for nickel-base superalloys (in fabrication of gas turbine engine components). The flow stresses in superplastic forming is quite low (typically less than 10 MN • m~ 2 . Hence, the force for the deformation can be provided for by using gas pressure. Hence, this forming process has more similarity to blow-forming of thermoplastics than conventional metal forming techniques. Pilling and Ridley (1989) have summarized the principles of various forming procedures, i.e.: (a) simple female forming, (b) female drape forming, (c) reverse bulging, (d) plug-assisted forming and (e) snap-back forming. They have also discussed the method for the determination of hydrostatic forming pressure for simple shapes. In recent years, the combined use of superplastic forming and diffusion bonding (SPF/DB) or its reverse DB/SPF, has produced spectacular savings in cost. The process has been found to be highly advantageous for the manufacture of a large number of aerospace parts. Diffusion bonding in the solid state is a joining process in which the two surfaces to be joined are brought into contact at an elevated tern-
454
9 Superplasticity in Metals, Ceramics and Intermetallics
perature. Application of a moderate pressure to each component brings the surfaces to be bonded into intimate contact, creating a planar array of irregular voids. Diffusion and creep flow processes can then transport atoms to the void surfaces from neighboring areas, thus reducing the volume of the interfacial voids. The details of bonding mechanisms and the kinetics of diffusional bonding have been discussed in an excellent recent monograph (Pilling and Ridley, 1989). Titanium based alloys can be easily diffusion bonded because the titanium metal can take into solution surface oxide layer and other contaminants which could otherwise prevent the formation of a good metal to metal bond. Unfortunately, however, the diffusion bonding of aluminum alloys is much more difficult to achieve because of the existence of a tenacious surface oxide. We need more detailed understanding of the bonding mechanism in the presence of stable oxide layers. However, studies have shown that solidstate diffusion bonds in DB/SPF aluminum alloys can yield fracture strengths greater than those attained by using polymeric adhesives (Pilling and Ridley, 1989). Diffusion bonding can be used for selective bonding of sheet material into sandwich constructions. Expanding the sandwich superplastically by gas pressure forms a shell structure. The internal architecture of the shell depends on the number of interlayer sheets and the pattern of bond or non-bond areas. The areas in which bonding is to be prevented are silk screen printed with a stop-off compound (a mixture of yttria and boron nitride in a polymeric binder). This enables the two sheets to separate at these points during the forming of the part in the DB/SPF process (Pillings and Ridley, 1989). An example of three-sheet fabrication in the SPF/DB process is shown in Fig. 9-21. An actual part
[A] CORE SHEET STOP OFF
PACK BONDING
I I I ! FORMING
t
t
t t
Figure 9-21. Schematic illustration of SPF/DB for three-sheet fabrication (from Stephen, 1987).
of a Ti-6A1-4V sandwich structure is fabricated by using the DB/SPF process shown in Fig. 9-22 (Winkler, 1988). The remarkable design flexibility allowed in such SPF/ DB method can result in a cost savings of up to 60 % in comparison to other fabrication methods. Another area where SPF/DB has demonstrated its potential is in the fabrication of a hollow fan blade in turbine engine applications. Current fabrication of a fan blade is usually by envelope forgings of solid pieces which are then machined to final shape. If the desired external configuration can be achieved with a lighter weight, i.e., internally hollow fan blade structure, then the advantages of the improved airfoil designs may be realized to a
.10 Conclusion
455
Figure 9-22. Diffusion bonded and superplastically formed Ti-6A1-4V sandwich structure (after Winkler, 1988).
greater degree. It will also improve the efficiency of the turbine engine by improving the thrust-to-weight ratio. The fabrication of such hollow fan blade calls for complicated tooling configurations, closer toler-
ances and complex quality assurance measure. Furthermore, the twist from blade root to tip can be as much as 60 degrees for larger blades (Weisert, 1985). Despite these difficulties, such hollow fan blades are being fabricated today (Fig. 9-23) where one can attain 30-35% weight saving over the solid blade. Such demonstration is a strong stimulus for further development and application of the SPF/DB technology.
9.10 Conclusion
Figure 9-23. Sections of a hollow (titanium alloy) fan blade of military engine size (from Weisert, 1988).
Superplasticity is now an attractive forming process which is no longer restricted only to metallurgical systems. With control of microstructure, the operative ranges for superplasticity can now be extended to significantly higher (and hence, commercially desirable) strain rates. The typical intergranular cavitation in superplasticity can be controlled by the superposition of hydrostatic gas pressure. The demonstration of superplasticity in several ceramics and intermetallic alloys has substantially enhanced their potential for use as high temperature structural materials. The possible manifestation of superplasticity in nanocrystalline materials
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9 Superplasticity in Metals, Ceramics and Intermetallics
and also superplasticity at very high strain rates are areas worthy of investigation.
9.11 Acknowledgements The author thanks Drs. J. Wadsworth and T. G. Nieh (Lockheed Palo Alto Materials Laboratory); Drs. C. T. Liu and V. Sikka (OakRidge National Laboratory); Dr. C. C. Bampton (Rockwell International Science Center); Professor T. R. McNelley (Naval Post Graduate School); Professor T. Sakuma (Tokyo University); Dr. F. Wakai (GIRIN, Nagoya, Japan); Professor J. Pilling (Michigan Technological University); Dr. N. Ridley (University of Manchester); Professor Valiev (USSR Academy of Science, Ufa) and Professor T. G. Langdon (University of Southern California) for discussion of their ongoing research results and for their permission to quote from these results. The author would also like to thank his former student colleagues: Professors R. Arrowood, A. H. Chokshi, T. R. Bieler, B. P. Kashyap, B. H. Prada and Drs. M. C. Pandey and A. Arieli for their contributions. This work was supported by grants from U. S. National Science Foundation and Air Force Office For Scientific Research. Part of the manuscript was written while the author was on a sabbatical leave in Japan. The hospitality of Professor Y. Ishida of Tokyo University and a Fellowship from the Japan Society for Promotion of Science are acknowledged.
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Kashyap, B. P., Mukherjee, A. K. (1983), Deformation of Multi-Phase and Particle Containing Materials: Bilde-Sorensen, I B., Hansen, N., Horsewell, A., Leffers, T., Lilholt, H. (Eds.). Roskilde, Denmark: Riso National Laboratory, p. 325. Kashyap, B. P., Mukherjee, A. K. (1985a), J. Mat. Sci. 20, 2661. Kashyap, B. P., Mukherjee, A. K. (1985 b), in: Superplasticity. Baudelet, B. and Suery, M. (Eds.). Paris: Editions du CNRS, article 4. Kashyap, B. P., Mukherjee, A. K. (1986), Res. Mechanica 17, 293. Kellet, B. X, Lange, R R (1988), /. Mat. Res. 3, 545. Kellet, B. X, Carry, P., Mocellin, A. (1988), in: Superplasticity and Superplastic Forming: Hamilton, C. H. and Paton, N. E. (Eds.). Warrendale, PA: TMSAIME, pp. 625-630. Kelley, L. G. (1985), in: Superplastic Forming: Agrawal, S. P. (Ed.). Ohio: ASM, Metals Park, p. 61. Kim, Young-Won (1990), private communication, will be published. King, A. M., Smith, D. A. (1976), Met. Sci. 8, 113. Klahn, D., Mukherjee, A. K., Dorn, X E. (1970), Second International Conference on Strength of Metals and Alloys, Proceedings, Vol. Ill, pp. 951 — 982. Kobayashi, M., Wakai, R (Eds.) (1989), Superplasticity. Pittsburg, PA: Materials Research Society. Kobayashi, Y, Ishida, Y, Kato, M. (1977), Scripta Met. 11, 51. Langdon, T. G. (1970), Phil. Mag., 22, p. 689. Langdon, T. G. (1982 a), in: Superplastic Forming of Structural Alloys: Paton, N. E. and Hamilton, C. H. (Eds.). New York: Met. Soc. of AIME, p. 27. Langdon, T. G. (1982b), Metal Sci. 16, 175-183. Langdon, T. G. (1987), in: Fundamentals of Diffusion Bonding: Ishida, Y (Ed.). Amsterdam: Elseviers, pp. 305-328. Lee, D. (1970), Met. Trans 1, 309. Lian, X, Suery, M. (1986), Mat. Sci. Tech. 2, 1093. Liu, C. T., Koch, C. C. (1983), Publication National Bureau of Standards, June, 1983 (Washington, D.C.), Number NBS-IR-83-2679-2. Liu, C. T., Stiegler, X O. (1984), Science 226, 636. Livesey, D. W, Ridley, N. (1982), Metall. Trans. 13A, 1619. Lloyd, D. X, Moore, D. M. (1982), in: Superplastic Forming of Structural Alloys: Paton, N. E. and Hamilton, C. H. (Eds.). Warrendale, PA: TMSAIME, pp. 147-172. Logan, R. W, Mukherjee, A. K. (1982), Mater. Sci. and Engrr., 237. Maehara, Y, Langdon, T. G. (1990), /. Mat. Sci. 25, 2275. Mahoney, M. W, Ghosh, A. K. (1987), Met. Trans. 18, 653. Marion, X E., Hsueh, C. H., Evans, A. G. (1987), /. Amer. Ceram. Soc. 70, 708.
Martin, P. X, Backofen, W. A. (1967), Trans. ASM.60, 352. Matsuki, K., Monita, H., Yamada, M., Murakami, Y (1977), Met. Sci. 6, 156. Matsuki, K., Hariyama, N., Tokizawa, M., Murakami, Y (1983), Metal Sci. 17, 503. Mayo, M. X, Nix, W. D. (1989), Acta Met. 37, 1121. McNelley, T. R., Heikkenen, H. C. (1990), Superplasticity in Aerospace II. Warrendale, PA: TMSAIME, to be published. Melton, K. N., Cutler, C. P., Kallend, X S., Edington, X W. (1974), Acta Met. 22, 165. Merrilea, X M., Wads worth, X, Kobayashi, M., Mukherjee, A. K. (Eds.) (1990), Superplasticity in Metals, Ceramics and Intermetallics, to be published. Miller, D. A., Langdon, T. G. (1978), Metall. Trans. 9A, 1688. Miller, D. A., Langdon, T. G. (1979), Metall. Trans. 10A, 1869-1874. Misro, S. C , Mukherjee, A. K. (1975), in: Rate Processes in Plastic Deformations: Li, X C. M. and Mukherjee, A. K. (Eds.). Metals Park, Ohio: Amer. Soc. of Metals, p. 434. Mohamed, R A. (1983), /. Mat. Sci. 18, 582. Mohamed, F. A., Langdon, T. G. (1975), Phil. Mag. 32, 697. Mohamed, R A., Langdon, T. G. (1981), Acta Met. 29, 922. Mukherjee, A. K. (1971), Mater. Sci. Engg. 8, 83. Mukherjee, A. K. (1975), in: Treatise in Materials Science and Technology, Vol. 6: Arsenault, R. X (Ed.). New York: Academic Press, p. 180. Mukherjee, A. K. (1979), Ann. Rev. Mat. Sci. 9, 191. Mukherjee, A. K., Ferguson, W. G., Barmore, W. L., Dorn, X E. (1966), J. Appl. Phys. 37(10), 3707. Mukherjee, A. K., Bird, X E., Dorn, X E. (1969), Trans. Amer. Soc. Metals 62, 155. Mukherjee, A. K., Bieler, T. R., Chokshi, A. H. (1989), in: Materials Architecture: Bilde-Sorensen, X B. (Ed.). Roskilde, Denmark: Riso National Laboratory, pp. 207. Mukhopadahyay, X, Kaschner, G., Mukherjee, A. K. (1990), Scripta Met. 24, 857. Nabarro, R R. N. (1948), in: Report of a Conference on the Strength of Solids. London: The Physical Society, pp. 75-90. Naziri, H., Pearce, R., Henderson-Brown, M., Hale, K. (1975), Acta Med. 23, 489. Nichols, R A. (1980), Acta Met. 28, 663. Nicholson, R. B. (1972), in: Electron Microscopy and Structure of Materials: Thomas, G. (Ed.). Berkeley: University of California Press, p. 689. Nieh, T. G. (1990), Superplasticity in Ll2 Intermetallic Alloys, to be published in: Proceedings of the MRS Symposium on Superplasticity in Metals, Ceramics and Intermetallics. Pittsburg, PA: Materials Res. Soc. Nieh, T. G., Oliver, W. C. (1989), Scripta Met. 23, 851.
9.12 References
Nieh, T. G., Wadsworth, I (1989), Proceedings of the MRS International Meeting on Advanced Materials, Vol. 7: Kobayashi, Y and Wakai, F. (Eds.). Pittsburg, PA: Materials Research Soc, p. 43. Nieh, T. G., Wadsworth, J. (1990), Superplastic Behavior of a Fine-Grained Yittria-Stabilized Tetragonal Zirconia Polycrystals, to be published in: Ada Metallurgica. Nieh, T. G., Wadsworth, X, Sherby, O. D. (1990), Superplastic Behavior in Ceramics, Ceramic Composites, Metal Matrix Composites and Intermetallics, to be published in: Proceedings of Superplasticity in Aerospace II. Nix, W. D. (1985), in: Superplastic Forming: Agrawal, S. P. (Ed.). Metals Park, Ohio: ASH, pp. 3-12. Novikov, I. I., Portnoy, V. K., Levchenko, V. S. (1981), Acta Met. 29, 1077. Oishi, Y, Ando, K., Sakka, Y (1983), in: Adv. in Ceramics, Amer. Ceram. Soc. 7, 208. Padmanabhan, K. A. (1977), Mater. Set Engg. 29,1. Padmanabhan, K. A., Davies, G. C. (1980), Superplasticity. Berlin and New York: Springer-Verlag. Pandey, M. C , Wadsworth, X, Mukherjee, A. K. (1986), Mater. Sci. Engg. 78, 115. Paton, N. E., Hamilton, C. H. (Eds.) (1982), Superplastic Forming of Structural Alloys. Warrendale, PA: TMS-AIME. Pearce, R., Kelly, L. (Eds.) (1985), Super plasticity in Aerospace Aluminum. Curdridge, England: Ashford Press. Pearson, C. E. (1934), /. Inst. Metals 54, 111. Pilling, X (1985), Mat. Sci. Techn, 461. Pilling, X, Ridley, N. (1988), Res. Mechanica 23, 31. Pilling, X, Ridley, N. (1989), Superplasticity in Crystalline Solids. London: The Inst. Metals. Prada, B. H., Mukhopadhyay, X, Mukherjee, A. K. (1990), Japan Inst. of Metals 31, 200. Rai, G., Grant, N. X (1975), Met. Trans. 6a, 385. Rai, G., Grant, N. X (1983), Metall. Trans. A. 14A, 1451. Raj, R. (1989), in: Superplasticity and Superplastic Forming: Hamilton, C. H. and Paton, N. E. (Eds.). Warrendale, PA: TMS-AIME, p. 583. Ridley, N., Pilling, X (1985), in: Superplasticity: Baudelet, B. and Suery, M. (Eds.). Paris: Editions duCNRS, pp. 8.1-8.17. Ridley, N., Livesey, D. W, Mukherjee, A. K. (1984), Met. Trans. 15A, 1443. Ruano, O. A., Sherby, O. D. (1982), Mat. Sci. Engg. 56, 167. Ruhle, M., McCartney, M. L., Claussen, N. (1986), in: Ceramic Materials and Components for Engines: Bunk, W. and Hausner, H. (Eds.). Deutsche Keramische Gesellschaft, p. 593. Samuelsson, L. C. A., Melton, K. N., Edington, X W (1974), Acta Met. 24, 1017. Schmid, S. M., Boland, X N., Paterson, M. S. (1977), Techtonophys. 43, 257. Schneibel, X H., Hazzledine, P. M. (1983), /. Mat. Sci. 18, 562.
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Shang, H. M., Suery, M. (1984), Metal Sci. 18, 143152. Sherby, O. D., Wadsworth, X (1985), Mater. Sci. and Tech. 1, 925. Sherby, O. D., Wadsworth, X (1988), in: Superplasticity in Aerospace. Heikkenen, H. C. and McNelly, T. R. (Eds.). Warrendale, PA: TMS-AIME, p. 3. Sherby, O. D., Wadsworth, X (1989), Prog. Mat. Sci. 33, 169. Smith, C. I., Norgate, B., Ridley, N. (1976), Met. Sci. May, 182. Speight, M. V., Beere, W (1975), Metal Sci. 9, 190. Spingarn, X R., Nix, W D. (1978), Acta Met. 26, 1389. Stephen, D. (1987), in: Proc. Intl. Conf. on Titanium, San Francisco, CA, Oct. 1986, Vol. 2. Dayton, Ohio: Titanium Development Asoc, p. 603. Stoner, S. L., Mukherjee, A. K. (1991), Superplasticity in Nickel Silicide, to be published in: Proceedings of International Conference on Superplasticity, Osaka, Japan, June, 1991: Furushiro, N. (Ed.), p. 323. Stowell, M. X (1983), Metal. Sci. 17, 1. Studt, P. L., Nidick, E., Mukherjee, A. K. (1973), in: Metallurgical Effects at High Strain Rates: Rhode, R. W. et al. (Eds.). New York: Plenum Press, pp. 379-400. Suery, M., Baudelet, B. (1975), /. Mat. Sci. 10, 22. Suery, M., Mukherjee, A. K. (1985), in: Creep Behavior of Crystalline Solids: Wilshire, B. (Ed.). Swansea, G.B.: Pineridge Press, pp. 137-200. Taub, A. I., Huang, S. C , Chang, K. M. (1984), Met. Trans. 15, 399. Underwood, E. E. (1962), /. Metals 14, 914. Vaidya, M. L., Murty, K. L., Dorn, X E. (1973), Acta Met. 21, 1615. Vale, S. H., Eastgate, D. X, Hazzledine, P. M. (1979), ScriptaMet. 13, 1157. Valiev, R. Z. (1988), in: Proc. Int. Conference on Superplasticity and Superplastic Forming: Hamilton, C. H. and Paton, N. E. (Eds.)- Pittsburg: TMSAIME, pp. 45-50. Valiev, R. Z., Kaibyshev, O. A. (1976), Fiz. Metal, and Metalloved 41(2), 382. Valiev, R. Z., Kaibyshev, O. A. (1977), Phys. Stat. Sol. (9) 44, 65. Valiev, R. Z., Kaibyshev, O. A. (1983), Acta Met. 31, 2121. Valiev, R. Z., Tsenev, N. K. (1990), in: Hot Deformation of Aluminum Alloys: Langdon, T. G. (Ed.). Proc. Fall TMS Meeting, Detroit, October 7-11, in press. Valiev, R. Z., Dudina, C. N., Obraztsova, I. S. (1985), Fiz. Metal and Metalloved 60, 1217. Valiev, R. Z., Kaibyshev, O. A., Kuzhetsov, R. I., Musalimov, R., Sh., Tsenev, N. K. (1988), Doklady Academi Nauk, SSSR 301, 186. Vastava, R. B., Langdon, T. G. (1979), Acta Met. 27, 251. Venkatachari, K. R., Raj, R. (1986), J. Amer. Ceram. Soc. 69, 135.
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Wakai, K, Kato, H. (1988), Adv. Ceram. Mat. 3, 71. Wakai, R, Sakaguchi, S., Matsuno (1986), Adv. Ceram. Mat. 1, 59. Wakai, R, Kodama, Y, Nagano, T. (1989a), Japan J. Appl. Phys. Series 2, Lattice Defects in Ceramics, pp. 57-67. Wakai, R, Kodama, Y, Sakaguchi, S., Murayama, N., Kato, H., Nagano, T. (1989b), in: Materials Research Society, International Meeting on Advanced Materials, Vol. 7. Pittsburg, PA: MRS; pp. 259-265. Wakai, R, Kodama, Y, Sakaguchi, S., Murayama, N., Izaki, K., Nihara, K. (1990), Nature 344, All. Wang, G. X, Raj, R. (1984), J. Amer. Ceram. Soc. 67, 339. Watts, B. M., Stowell, M. X, Baikie, B. L., Owen, D. G. E. (1976), Met. Sci. 5, 198. Weisert, E. D. (1985), in: Superplastic Forming: Agrawal, S. P. (Ed.). Metals Park, Ohio: ASM, p. 84. Weisert, E. D. (1988), in: Superplasticity in Aerospace: Heikkenen, H. C. and McNelley, T. R. (Eds.). Warrendale, PA: TMS-AIME, p. 315. Weertman, X, Weertman, X R. (1987), in: Constitutive Relations and Their Physical Basis: Andersen, S. I., Bilde-Sorensen, X B., Hansen, N., Leffers, T, Lilholt, H., Pedersen, O. B., Ralph, B. (Eds.). Roskilde, Denmark: RISO National Laboratory, p. 191. Wilkinson, D. S., Caceres, C. H. (1986), Mat. Sci. Tech., 1086. Winkler, P. X (1988), in: Superplasticity and Superplastic Forming: Hamilton, C. H. and Paton, N. E. (Eds.). Warrendale, PA: TMS-AIME, p. 491. Yang, H. S., Jin, P., Dalder, E., Mukherjee, A. K. (1991), Scripta Metallurgica et Materialia 25,1223.
Yoshizawa, Y, Sakuma, T. (1989), ISIJ International 29(9), 746. Yoshizawa, Y, Sakuma, T. (1990a), J. Amer. Ceram. Soc. 73, 000. Yoshizawa, Y, Sakuma, T. (1990b), High Temperature Deformation and Grain Growth in Fine Grained Zirconia, to be published in: Proceedings of Utsonomiya Conference, August, 1990, Yokobori, T. (Ed.).
General Reading Baudelet, B., Suery, M. (Eds.) (1985), Superplasticity. Paris: Editions du CNRS, article 15. Hamilton, C. H., Paton, N. E. (Eds.) (1988), Superplasticity and Superplastic Forming. Warrendale, PA: TMS-AIME. Hori, S., Tokizane, M., Purishiro, N. (Eds.) (1991), Superplasticity in Advanced Materials, Japan Soc. for Research on Superplasticity, Osaka, Japan. Mayo, M., Kobayashi, A., Wadsworth, X (Eds.) (1990), Materials Research Society Symposium Proceedings, Vol. 196, Pittsburgh, PA. McNelley, T. R., Heikkenen, H. C. (Eds.) (1990), Superplasticity in Aerospace. Warrendale, PA: TMSAIME. Padmanabhan, K. A., Davies, G. C. (1980), Superplasticity. Berlin and New York: Springer-Verlag. Pilling, X Ridley, N. (1989), Superplasticity in Crystalline Solids. London: The Inst. Metals. Wakai, R (Ed.) (1989), Proceedings, Materials Research Society, Vol. 7, Pittsburgh, PA.
10 Inelastic Deformation and Fracture of Glassy Solids Ali S. Argon Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.
List of Symbols and Abbreviations 10.1 Introduction 10.2 Structure of Glasses 10.2.1 Glass Formation 10.2.2 Atomic Structure of Glasses 10.2.2.1 Relation of Structure to Deformation Mechanisms 10.2.2.2 Radial Distribution Functions 10.2.2.3 Distributions of Atomic Packing or Interstitial Sites 10.2.2.4 Free Volume 10.2.2.5 Packing of Atoms in Network Glasses and Chain Polymeric Solids . . . . 10.2.2.6 Atomic Site Stress Tensor 10.2.2.7 Other Topological Characterizations of the Glassy State 10.3 Overview of Phenomenology, Mechanics and Mechanisms of Inelastic Deformation and Fracture of Glasses 10.3.1 Phenomenological Conditions for Inelastic Deformation 10.3.2 Kinematics of Inelastic Strain 10.3.2.1 Distortional Plasticity 10.3.2.2 Dilatational Plasticity 10.3.3 Plasticity Versus Brittle Behavior 10.3.4 Mechanisms of Fracture in Glasses 10.4 Inelastic Response in Metallic Glasses 10.4.1 Experimental Observations 10.4.1.1 Anelasticity 10.4.1.2 Viscous Flow 10.4.1.3 Plastic Flow 10.4.1.4 Structural Aging and its Effects on Plastic Resistance 10.4.2 Model for Viscoplastic Flow 10.4.3 Simulations of Plastic Flow in Metallic Glasses 10.5 Inelastic Response in Space Network Glasses 10.6 Inelastic Response in Polymeric Glasses 10.6.1 Experimental Observations 10.6.1.1 Anelasticity 10.6.1.2 Plastic Flow and its Mechanism 10.6.1.3 Kinetics of Plastic Flow 10.6.2 Dilatational Plasticity in Glassy Polymers Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.
463 466 466 466 467 467 467 468 468 469 469 470 470 470 471 471 472 473 473 475 475 475 475 477 481 482 484 488 489 489 489 490 493 495
462
10.6.3 10.6.3.1 10.6.3.2 10.6.3.3 10.7 10.7.1 10.7.2 10.7.3 10.7.4 10.8 10.9
10 Inelastic Deformation and Fracture of Glassy Solids
Simulation of Plastic Flow in Glassy Polymers Molecular Structure Models Segment Relaxations in the Polycarbonate of Bisphenol-A (PC) Large Strain Plastic Shear in Polypropylene (PP) Fracture of Glasses The Fracture Instability Fracture in Space Network Glasses Fracture in Metallic Glasses Fracture in Thermoplastic Glassy Polymers Acknowledgements References
496 496 497 499 502 502 503 503 504 506 506
List of Symbols and Abbreviations
463
List of Symbols and Abbreviations a A A, B b B(n) Bt c ClyC2 cs d, dt E Eo AE AF Ai 7 / Gl AG AG0 AG* h J k Kl9 Klc 2 m Mc if, nt p, q p(Qf) Q Q R rtj Rp r0 s stj T Tc 71
molecular radius; half crack length [in Eqs. (10-30) and (10-32)], or curvature of local intermolecular energy in ring rotation [in Eq. (10-28)] total craze area per unit volume material constants interatomic distance a numerical constant in Eq. (10-34) dependent on strain rate sensitivity exponent n diagonal element of back-stress tensor volume fraction material constants in craze initiation model volume fraction at steady state flow shear direction unit vector, component Young's modulus binding energy of the pair potential potential energy barrier for segment rotation Helmholtz free energy Helmholtz free energy barrier for viscous flow mode I energy release rate Gibbs free energy scaling constant for Gibbs free energy Gibbs free energy of activation primordial thickness of craze material polar moment of inertia of atom group Boltzmann constant mode I stress intensity factor, critical value Langevin function stress exponent entanglement molecular weight normal unit vector of shear plane, component numerical constant probability factor in yG material constant in craze initiation model heat flow rate universal gas constant, also reaction (Arrhenius) rate constant [in Eqs. (10-9 c) and (10-29)] magnitude of vector connecting atoms i and j plastic zone size separation between atoms (where the pair potential goes to zero) deviatoric shear stress (a scalar quantity) element of deviatoric stress tensor absolute temperature crystallization temperature glass transition temperature
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10 Inelastic Deformation and Fracture of Glassy Solids
t{ Tm U v0 vch vct At;* w W Y Y
craze initiation time melting point potential energy of the system of sample a n d its tractions pre-exponential craze velocity constant velocity of craze border velocity of craze tip shear activation volume specimen width rate of mechanical work expenditure, overall plastic work rate tensile yield strength athermal tensile plastic resistance of the glassy polymer
a otij /? y y0 yG yT 5 <5 C0 e ec 8tj sT 9 9 Xt kn X\ X'n pi v vG Q Q a °m aic <7tj o^(i) f T0 fis
proportionality constant of order unity elements of the geometrical (Schmid) resolution tensor dilatancy coefficient inelastic shear strain rate phenomenological shear strain rate scaling factor preexponential factor of strain rate unconstrained transformation shear strain increment critical crack opening displacement uniaxial strain craze strain rate element of strain tensor unconstrained uniaxial transformation strain, also transformation dilatation dilatation structural reference temperature principal extension ratio limiting extension ratio, locking stretch hardening-modified extension ratio, X of the craze matter shear modulus Poisson's ratio frequency factor density active craze front length per unit volume applied shear or tensile stress, as clarified in text applied mean normal stress, negative pressure ideal cohesive strength (of a glass) element of applied stress tensor element of stress tensor at atomic site i threshold plastic shear resistance (a scalar quantity) athermal intercept value of the plastic shear resistance ideal shear strength pair potential energy function surface free energy
List of Symbols and Abbreviations
co Qc Qf Qt Qo
molecular segment rotation angle volume of cluster average excess volume per a t o m in a finite cluster volume per atom at site i population average of volume per atom
DC DFO DGEBA DSC EAN H-H IR NMR PC PET PMMA PP PPO (a-) PS PVC RDF rms R-R TEM
deformation calorimetry oxydiphenyl (Kapton) diglycidyl ether of bisphenol-A differential scanning calorimetry epoxy-aromatic amine network hydroquinone infrared (spectroscopy) nuclear magnetic resonance polycarbonate of bisphenol-A polyethylene terephthalate polymethyl methacrylate polypropylene polyphenylene oxide (atactic) polystyrene polyvinyl chloride radial distribution function root mean square resorcinol transmission electron microscopy
465
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10 Inelastic Deformation and Fracture of Glassy Solids
10.1 Introduction Under certain conditions many solids can be obtained in a partially or completely disordered form. We are interested here in the inelastic deformation and fracture of such solids in the range of temperature in which their structure does not change substantially, i.e., below their glass transition temperature. While such solids have often been referred to as amorphous which implies the complete absence of structural regularity, we will prefer to call them glassy to accommodate possibilities of some short range or intermediate range order which is often found to be present. It must be emphasized, however, that our principal assignment will not be the description of the atomic or molecular structure of these solids beyond a certain level that is necessary to understand the mechanisms of deformation. Extensive description of the structure of glasses and the experimental tools available for their measurement can be found in Volume 9 of this Series. The subject of inelastic deformation of solids is a broad one that can not be treated adequately in its entirety in a single chapter. Therefore, here we will concentrate primarily on a discussion of the mechanisms of such deformation in nearly homogeneous solids, to large inelastic strains, under the simplest non-trivial states of stress, emphasizing the physics rather than the operational mechanics aspects of the deformation. Heterogeneities, are often important in modulating the deformation process such as in the case of partially crystalline structures where the crystallites do not undergo much deformation. They are also of great importance in fracture where they act as principal sources not only for fracture initiation but also for toughening of some brittle polymers. These
topics will, however, not be developed in this chapter. While we will pay attention to the important structural differences in the deformation mechanisms between atomic glasses (metallic glasses), network glasses (oxide glasses) and polymeric glasses that govern the local kinematics of the deformation process, we will emphasize as much as possible the common unifying features. We will concentrate our attention only on monotonic deformation and fracture rather than on the cyclic variety which has many important differences requiring extensive additional developments. Readers interested in cyclic response of engineering solids in general and also many glasses in particular are referred to Chapter 11 by Suresh in this Volume and also Suresh (1991). Apart from the necessity to understand the special stress and strain conditions of the crack tip environment in the development of the fracture mechanisms we will not discuss in any detail the mechanics aspects of fracture. An elegant and crisp treatment of this subject has been given by Hutchinson (1979) to which the reader is referred.
10.2 Structure of Glasses 10.2.1 Glass Formation Formation of glasses upon cooling from the melt relies principally on the necessity to suppress nucleation of a crystalline phase. That the nucleation and growth of crystallites from a melt, in the absence of heterogeneities, requires substantial undercooling and that the accompanying sharply reduced kinetics of diffusion or ordering makes the establishment of a crystalline phase even more difficult is common knowl-
10.2 Structure of Glasses
467
edge. The specific conditions required to override crystallization to form a glass in different materials, alloys, or mixtures, such as special features in the phase equilibrium diagrams, the nature of diffusion or ordering in the melt, molecular topological constraints, etc. have been widely discussed in the literature for specific glasses and will not be developed here. More extensive coverage of this subject can be found in Vols. 1, 9, and 12 of this Series for all types of glasses including atomic glasses, network glasses and chain polymeric glasses. (See, in particular, Vol. 1, Chap. 4; and Vol. 9, Chap. 3, 8, and 9.)
and even more importantly, between different forms of relative spatial correlation of the extreme portions of the spectrum of structural disorder (Ziman, 1979; Gaskell, 1983). While these spatial variations of disorder and their correlation has often little influence on the overall volume averaged physical properties, they are of overwhelming importance for the understanding of inelastic deformation (and parenthetically, also of diffusion). It is for this reason that computer simulations of the structure and the associated mechanisms of inelastic deformation of glasses of all types have been very informative as we will discuss below.
10.2.2 Atomic Structure of Glasses
10.2.2.2 Radial Distribution Functions
10.2.2.1 Relation of Structure to Deformation Mechanisms
A characteristic form of structural information obtained from X-ray or neutron diffraction experiments is the radial distribution function (RDF) of atom positions around a typical atomic site (see Ziman, 1979). Such RDF's provide useful information on the distribution of volume averaged coordination of atoms in the glassy state, and give a useful visual diagnostic means of gauging the average degree of disorder from the shapes of these functions. The RDF's, however, are not a sensitive indicator of the local correlations of disordered material of different description. Thus, their utility in understanding inelastic deformation is limited. They serve, nevertheless, as a convenient bridge in connecting results of structural simulations to experimental information, where the information provided by the simulation furnishes a degree of specificity often not reflected in the experimental results. (See also Vol. 9, Chap. 4.)
The mechanisms of inelastic deformation by motion of dislocations, transport of point defects, or by twinning or martensitic shear transformations in crystalline solids are well understood because of the clear connection between the regular crystal structure and the relatively simple kinematics of the atomic motions that are involved. In comparison with this the detailed understanding of the mechanisms of inelastic deformation in glasses and the specific kinematics of atom motions that are involved are far less clear. This is a direct result of the difficulty of accurately describing the structure of glasses, and those regions in them that are most favorably endowed for the local atomic motions that produce inelastic strain. Most, if not all, experimental probes of the structure of glasses only provide information on the volume averaged features of the structure which can not uniquely distinguish between different forms of aggregation of atoms associated with variations of local packing, some forms of short range order,
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10.2.2.3 Distributions of Atomic Packing or Interstitial Sites
Bernal (1964) has pioneered the analysis of dense random packed ball bearing spheres as analogous for the structure of simple liquids, and by extension, that of atomic glasses, demonstrating that the disordered structure of such assemblages of hard spheres can be described by a certain distribution of five different polyhedral figures which characterize interstitial space in these assemblages. The exercise was repeated computationally by others. Frost (1982) has determined that characterization of the disordered space made up from dense random packing of spheres by means of polyhedral figures having atom sites at their apexes requires more than the five canonical figures identified by Bernal. He proceeded further and also furnished distributions of sphere sizes that can just fit into interstitial holes. This latter information is of interest in understanding sorption and diffusion of gases. 500 •
400
JQ
200
An alternative representation of tesselation of space is by Voronoi polyhedra constructed around atoms rather than interstitial sites. This was studied first by Finney (1970 a) and subsequently by others. It provides information on the distribution of the volume per atom. Figure 10-1 gives such a distribution of volume per atom in a dense random packing of hard spheres in units of the average atomic volume in the glass where Vt is the volume of the ith atom, V is the population average volume per atom. The distribution shows that while the average volume per atom is only about 9% larger than that in the reference crystal there is a considerable tail of larger volumes, which as we will see play an important role in inelastic deformation. Simulations using actual interatomic interaction potentials rather than hard spheres have given very similar results (Finney, 1970 b). While the atomic packing studies in 3-D should be clearly preferred to compare with experimentally obtained diffraction information, the visualization of such packing, and the forms of its alteration upon deformation is often difficult to achieve. For this reason several 2-D models have also been studied in considerable detail. One of these is a disordered variant of the well-known Bragg bubble model making use of the capillary interbubble forces acting between small soap bubbles floating on the surface of water (Argon and Kuo, 1979). The behavior of such rafts upon deformation in the plane of the raft have given considerable insight into the nature of plastic flow (Argon and Shi, 1982).
100
10.2.2.4 Free Volume 0.912 0.956 1.000 1.044 1.088 Reduced volume AV*
1.132
Figure 10-1. Distribution of volume per atom in an atomic packing model (Finney, 1970 a).
That excess atomic volume concentrated into local regions in a disordered solid may permit relatively easy rearrangements there has been recognized for a long time
10.2 Structure of Glasses
and has been associated with the notion of free volume. While the notion is often loosely used to advance qualitative arguments, it can be made more precise into a structure characterizing parameter in the computer simulation models. There it has been defined as (Deng et al., 1989 b)
-£
(10-1)
n i=
where Qf is the average excess volume per atom in a cluster of n atoms, over the population average volume Qo per atom. Theoretical considerations (Cohen and Grest, 1979) and computer simulations (Deng et al., 1989 a, b) have indicated that regions in which such excess volume clusters, i.e., where free volume aggregates, show a reduced level of local elastic stiffness or reduced cohesion, and have so-called liquidlike behavior. Topological evidence suggests that in the melt, and even in the subcooled melt regime, above the glass transition temperature (Tg), overall compliant behavior results from the percolation of liquidlike material through the structure. Below Tg this percolation is broken and the material exhibits overall stiff behavior as now the relatively more densely packed stiff material becomes contiguously connected (Deng et al., 1989 a, b). In Sec. 10.4.2 we will furnish evidence that the liquidlike material that is spatially isolated below Tg nevertheless constitutes the fertile material in which plastic rearrangements preferentially occur (Deng et al., 1989d).
469
ization has merit. In space network glasses such as fused SiO2 and its many modifications, as well as in chain polymeric glasses the structures must be simulated by the use of the appropriate interatomic force fields between bonded atoms as well as nonbonded atoms. Such simulations have also been performed for many systems. The results of these simulations are best discussed in the context of the deformation studies for which they have been developed. Therefore, we postpone their introduction to Sees. 10.5 and 10.6. 10.2.2.6 Atomic Site Stress Tensor
The glassy state has been associated with excess enthalpy which has been useful in understanding calorimetric measures of structural relaxation. Such average scalar measures of the excess properties of the glass, however, do not convey much information on where the local excess properties congregate, or more importantly, where and how the mechanical response occurs. For such purposes, Egami and Vitek (1983) have emphasized the utility of the atomic site stress tensor defined much earlier by Born and Huang (1954) on the local application of the virtual work principle at each atomic site based on the level of equilibrated forces acting on individual atoms. For atomic solids, where substantially only central forces act between atoms, the atomic site stress tensor is defined as do-2)
10.2.2.5 Packing of Atoms in Network Glasses and Chain Polymeric Solids
The structures obtained from dense random packed spheres are of great value in elucidating complex topological concepts, and also serve as idealized models of the structure of glassy metals where the ideal-
where 0 is the interatomic pair potential from which all interatomic forces are calculated, rtj is the magnitude of the radius vector connecting atoms i and j , rfj and r\} are the magnitudes of the a- and ^-components of the radius vector connecting
470
10 Inelastic Deformation and Fracture of Glassy Solids
10.2.2.7 Other Topological Characterizations of the Glassy State
atoms i and j and Qt is the volume of the central atom i for which the tensor component is defined. The sum is over all j neighbors with which atom i is effectively interacting. Egami and Vitek have calculated the distribution of the two scalar invariants of the atomic site stress tensors: i.e., the mean normal stress om (negative pressure) and the deviatoric (Mises) shear stress for a specific simulation. Their results, given in Figs. 10-2 a and 10-2b show that in a typical computer generated glass the rms value of the mean normal stress is 6% of the bulk modulus, while that for the deviatoric shear stress is fully 18% of the shear modulus. These magnitudes border on the cohesive resistances of the solid and are very substantial. Egami and Vitek (1983) have proceeded further and have demonstrated that it is possible to account for the excess enthalpy of the glassy state by considering the elastic strain energies associated with the atomic site stresses in the glass - in reference to those in the ordered crystalline phase of the material. We will demonstrate in Sees. 10.4.2 and 10.6.3 the general utility of the atomic site stress tensor in the computational simulations of plastic resistance in metallic and polymeric glasses respectively.
In addition to the forms of structural characterization described above there have been many attempts in the nature of abstract topological transformations to find a unique connection between a characteristic disordered state of the glass and a reference ordered state. While many of these studies stand out as ingenious, they have so far failed to provide much help in better understanding of local inelastic rearrangements. The interested reader will find a good selection of these models presented by Vitek (1983).
10.3 Overview of Phenomenology, Mechanics and Mechanisms of Inelastic Deformation and Fracture of Glasses 10.3.1 Phenomenological Conditions for Inelastic Deformation
Inelastic deformation in solids is a rate process which, in the limit of very low temperatures, requires for its initiation a critical deviatoric shear stress to overcome a
N(p)
10
20 (GPa)
Figure 10-2. Distribution of atomic site stresses: (a) negative pressure, (b) deviatoric shear stress (Egami and Vitek, 1983).
10.3 Overview of Phenomenology, Mechanics and Mechanisms of Inelastic Deformation
threshold shear resistance f which for a given state of the solid, is a material property. Thus as T->0, plasticity is initiated when S
= (s.jSij/2)112
= f (<7m)
(10-3)
where stj = atj — am are the so-called deviatoric stresses, am = crfl/3 is the mean normal stress, and repeated indices imply summation over all indices i = 1,2,3, etc. Equation (10-3) indicates that the threshold plastic resistance f may depend on crm. Under these conditions the plastic strain rate y is arbitrarily high. At higher temperatures, but still well below Tg where the structure does not undergo significant relaxation during the time of deformation, the plastic strain rate should be given by Kocksetal. (1975). (.0-4) where the stress dependence of the activation free energy AG* of the deformation process can be often expressed phenomenologically as (10-5) where p and q are exponents close to unity. The preexponential factor of the rate expression yG combines a fundamental frequency factor vG of the deformation producing unit, the local plastic transformation strain increment yT and the volume fraction c of deformation units, i.e., 7k :
cyTvG
(10-6)
The mechanism inspired form of the strain rate expression given in Eq. (10-4) can often be phenomenologically restated as a power-law form y = v<> [ ^
(10-7)
471
where both y0 and m are fitting constants. Equation (10-7), albeit more approximate, incorporates a reverse deformation component that is left out in Eq. (10-4), which permits the strain rate to go to zero when s->0, at least formally. More specific forms of these expressions will be discussed below. In many mechanistic models only an applied shear stress a is considered, so that S-X7.
10.3.2 Kinematics of Inelastic Strain 10.3.2.1 Distortional Plasticity There has been much controversy associated with the mechanism responsible for plastic strain production and its kinematics in glassy solids. Since a number of glassy metals and polymers exhibit strong shear localization, it was proposed first by Gilman (1968), and then by many others, that plastic deformation in these solids is produced by generalized dislocations having similar mobility characteristics as those in metal crystals. While there is considerable merit in considering dislocations for operational purposes of local stress analysis or accounting for the macroscopic strains, there is no firm evidence that there is any analogous mobile defect in a glassy solid resembling a crystal dislocation (Argon, 1981). Instead, much evidence based on computer simulations indicates that shear is produced in glassy solids by means of individual shear transformations occurring in small atom clusters which do not propagate in a contiguous manner outward from the initial cluster. Thus, mechanistically the rate process of plastic deformation appears to be nucleation controlled rather than controlled by the mobility of the boundaries of the transformation. If such shear transformations can be considered to have shapes of oblate
472
10 Inelastic Deformation and Fracture of Glassy Solids
spheroids with principal axes characterized by unit normal vectors d and n in the plane and out of the plane of the transformation shear increment yT, as shown in Fig. 10-3, the resulting macroscopic plastic strain increment ds^ is (10-8) j
^
j
j
i
)
(10-8a)
In Eq. (10-8) atj are elements of a geometrical (Schmid) shear strain resolution tensor relating the local shear strain increment in the spheroid coordinates to the external shape coordinates, and c is the volume fraction of the local material undergoing the transformation. Since the structure of a glass tends to be isotropic down to rather small volume elements, the shapes of the local shear transformations can be expected to be relatively equiaxed, with the directions of the local shear axes coinciding with those of the macroscopic body. Thus, giving Sy = 5 (c yT)
(10-8 b)
For a longer term average point of view that considers the formation of the transformations as "instantaneous", where yT must be considered as a property of the structure, then S(cyT) = yT be
(10-8 c)
Equations (10-8 a) to (10-8 c) are in incremental form. They can be extended into rate form when they transform into (10-9 a)
= csyrR
(10-9 b)
where cs is now a steady state volume fraction of fertile material available to transform, and that the rate of transformation is governed by a stress and temperature dependent exponential Arrhenius rate con-
Figure 10-3. Principal directions of ellipsoidal shear transformation: n is unit normal vector of invariant plane, d is parallel to shear direction.
stant R such as that appearing in Eq. (10-4), i.e., R = vG exp
-
kT
(10-9 c)
The form of the expressions Eq. (10-9 a) to Eq. (10-9 c) implies that a kinematical steady state exists among volume elements so that the transformable (fertile) fraction of material is never exhausted. 10.3.2.2 Dilatational Plasticity Under certain conditions the plastically deforming solid can undergo stable cavitation homogeneously or in planar zones. This produces additional strain of a dilatational nature. A prominent example of this is the crazing phenomenon in certain flexible chain glassy polymers where a primordial slab of polymer becomes fibrillar or spongy, normal to the principal tensile stress as shown in Fig. 10-4. This effectively results in a dilatational transformation associated with a tensile transformation strain sT in the primordial slab. The overall tensile strain increment then, in the direction of the tensile stress is 5s = 5(csT) = eTdc
(10-10)
10.3 Overview of Phenomenology, Mechanics and Mechanisms of Inelastic Deformation
473
craze
Figure 10-4. Craze as a dilatational transformation.
where 5c is the increment in the volume fraction of the initial primordial material undergoing the crazing expansion. Since inside the craze there are no associated other transverse strains, the uniaxial strain of Eq. (10-10) is also the net dilatation increment 59, in the body, i.e., 56 = 5s
(10-11)
10.3.3 Plasticity Versus Brittle Behavior
Because plastic flow in glassy solids is primarily initiation controlled, it occurs at relatively high stresses in relation to the cohesive properties of the solid. Therefore, fracture is usually a close competitor to plastic flow - particularly when it occurs in a tensile field. The choice in the terminal mechanical behavior of solids in tension, between inherent brittleness and inherent plasticity rests with the nature of the atomic bond. In two complementary fundamental developments Kelly et al. (1967) and Rice and Thomson (1974) have stated that this bifurcation in behavior is governed in the limit at the tip of an atomically sharp crack. There, concentrated tensile stresses ahead of the crack probe the ideal cohesive strength of the solid while the maximum concentrated shear stresses on some inclined planes probe the ideal shear strength. Upon stressing the cracked solid, brittle vs. ductile behavior is governed by whether or not the conditions of decohesion in the plane of the crack are reached before the conditions of ideal shear. In the covalently bonded network glasses the balance appears to be tilted clearly in the di-
rection of brittleness in tension while for metallic and polymeric glasses the tilt is toward plastic behavior, for these glasses in the unaged form (Argon, 1982). As we will discuss somewhat more extensively in Sec. 10.7.1 the bifurcation in the behavior of metallic and polymeric glasses, however, depends strongly on the state of structural aging in these materials which produces changes in plastic resistance. Here it should suffice to state that plastic response in a glass is always attainable in principle, but may in specific instances require suppression of fracture by superposition of a pressure to inactivate cracks or flaws. 10.3.4 Mechanisms of Fracture in Glasses
As in other solids, in one limit, fracture in glasses could be a process of brittle separation with no, or negligible accompanying plastic flow. Oxide glasses and many structurally aged glassy metals fracture in this mode responding locally to a critical decohesion criterion at the tip of a propagating crack that can be characterized by a critical mode I stress intensity criterion. In unaged metallic glasses and in most stiff chain glassy thermoplastic or thermosetting polymers fracture is a process of ductile separation. Here the actual separation occurs in two stages. First the continuous solid is rendered discontinuous by cavity formation at heterogeneities, followed by plastic expansion of these cavities to complete local ligament rupture as illustrated in Fig. 10-5, in a manner closely resembling the ductile fracture process in crystalline metals.
474
10 Inelastic Deformation and Fracture of Glassy Solids
Figure 10-5. Formation and growth of ductile fracture cavities.
Many ductile metallic glasses, free of heterogeneities, however, fracture by a variant of a fluid flow instability where the plastically blunted crack tip acts as a fluid meniscus advancing under the deformation induced negative pressure gradient zone of the blunted crack tip. The concave meniscuslike flow field becomes unstable to flow perturbations of a certain wave length which first penetrate into the crack tip zone in the form of tubular fingers followed by rupture of the ridges separating these tubular zones of penetration as sketched out in Fig. 10-6. The separation advances with the crack tip in a self-similar form of steady ductile separation. In flexible chain thermoplastic glassy polymers the fracture process is preceded by crazing where crazes that spread through the polymer undergo internal break-up due to intrinsic nonuniformities in the fibrillar or spongy craze matter. More often, however, crazes begin to fail
Figure 10-6. Sketch of break-up into fingers of ductile crack tip acting as a nonlinear fluid meniscus.
Figure 10-7. A sequence of fracture of a craze: (a) an advance cavitation event from a dust particle ahead of the main crack in a craze at point P; (b), (c) the fracture spreading from the advance cavitation site joins the main crack (Doyle et al., 1972).
from the interface of a particulate heterogeneity that the growing or widening craze acquires as shown in the sequences of states in Figs. 10-7 a-10-7 c. Since crazes act both as ingredients promoting dilatational plasticity and also as the prominent sites of fracture, they can become an important means of controlling toughness in such glassy polymers.
10.4 Inelastic Response in Metallic Glasses
10.4 Inelastic Response in Metallic Glasses o
10.4.1 Experimental Observations 10.4.1.1 Anelasticity
The distributed nature of atomic disorder in a metallic glass permit different volume elements to undergo shear relaxation under an applied stress with different ease. This can be characterized by assigning different volume elements different terminal plastic shear resistances f or, in an associated manner, consider them to have different activation barriers to shear transformation under a stress o <^ f. This difference can be probed best, and in the least invasive manner, by internal friction experiments (Berry, 1978; Morito and Egami, 1984; Deng and Argon, 1986 a) or by anelastic creep and creep recovery experiments (Berry, 1978; Argon and Kuo, 1980). Figure 10-8 gives a schematic representation of the response of the solid by anelastic creep where the application of a shear stress a mechanically polarizes the structure by permitting the accumulation of shear relaxations that can occur at a given temperature during the period under stress. Increasing the time of observation, or increasing the temperature, increases the total strain by permitting more of the more difficult transformations to occur. In an internal friction experiment such transformations occur in the forward and reverse manner and the associated hysteresis produces energy dissipation. Figure 10-9 gives a typical internal friction spectrum of the cyclic loss (tan 3) in torsional oscillations in a Cu 59 Zr 41 glass at different levels of aging (Deng and Argon, 1986 a) while Fig. 10-10 shows the typical linear, reversible anelastic creep response of a Pd 80 Si 20 glass at 176°C (Berry, 1978). Argon and
475
o
o o °0 o o o o o o o o a (a) (b) Figure 10-8. Sketch depicting development of shear distortion by the accumulation of isolated shear transformations under stress (Argon and Shi, 1982).
Kuo (1980), and Deng and Argon (1986 b) have demonstrated how the distribution of critical activation barriers to the local stress relaxations can be obtained from the analysis of the anelastic creep and the internal friction experiments. Figures 10-11 a and 10-11 b give the distributions of activation energies for transformations in Pd 80 Si 20 and Cu 5 9 Zr 4 1 , deconvoluted from anelastic creep and recovery creep experiments, and from internal friction experiments, respectively. Figures 10-9 and 10-1 l b show that the internal friction experiment that produces nearly no permanent change in the structure during the measurement is a very sensitive method for the study of structural relaxations. This field is also discussed in Vol. 9, Chap. 13. 10.4.1.2 Viscous Flow
When creep experiments at low stress levels (a
476
10 Inelastic Deformation and Fracture of Glassy Solids C° 100
200,
300
400
500
600
40x10" Cu
f = 0.14 Hz 35 -
30 -
5gZr 41
o As received * After 738K/0.5h aging A After 738 K/2.0 h aging * After 738K/8.0h aging 0 After 710K/2.0h aging o After 71 OK/8.Oh aging
25 -
g
20
15
10
Figure 10-9. Change of internal friction with temperature and its dependence on aging below Tg in a Cu 59 Zr 41 alloy (Deng and Argon, 1986 a). 300
400
500
600'
700
800
T(K)
AMORPHOUS Pd-20at. %Si (70 hrs 200°C) CREEP AT I76°C
ELASTIC AFTER-EFFECT (CREEP RECOVERY)
I76°C
100 TIME (SECONDS)
Figure 10-10. Linear anelastic creep and creep recovery in a Pd 80 Si 20 glass at 176 °C (Berry, 1978). The separate symbols on the individual curves identify separate experiments at different stress levels or relaxation of 10000 strain from different initial levels.
10.4 Inelastic Response in Metallic Glasses (eV/atom)
3.0x10^
b
0.5
1.0
I
I
1.5
2.0
2.5 i
2.5x10"2
- 2.0
—
1.5
2.0 -
o 'o CO
- 1.0
- 0.5
i
0
i
0
(a)
i
10
| r r l il 20
i
30
40
i
50
0 60
Activation energy AF (kcal/mol)
o a a
As manufactured, batch 1 As manufactured, batch 2 Aged 2h at 710 K Aged 8h at 710 K V
G
Cu
59Zr41
Q. X <
25
30
35
40
45
(b) AG* (kcal/mol) Figure 10-11. Distribution of activation energies for anelastic response: (a) Pd 80 Si 20 (Argon and Kuo, 1980), (b) Cu 59 Zr 41 (Deng and Argon, 1986 a).
component of irrecoverable viscous strain. Taub and Spaepen (1979, 1980) have used viscous flow experiments as a probe to determine the effects of structural relaxations in a Pd 80 Si 20 glass. The temperature dependence of the viscosity of such a glass after a series of aging treatments at differ-
477
ent temperatures is shown in Fig. 10-12. The remarkable finding is that the activation energy for the flow is unaltered by the different aging treatments but that the viscosity is systematically and dramatically increased with increasing aging temperature or increasing structural compaction. Since viscous flow is always controlled by the highest activation energies of the spectrum, i.e., the most difficult to deform component, and since the internal friction studies show that structural relaxation primarily alters the lower end of the spectrum as shown in Fig. 10-8, these results are not unexpected. The dramatic increase in the viscosity, however, indicates also a key interdependence between different parts of the activation energy spectrum of shearing sites. The elimination of low energy sites with large free volume does not only reduce the anelastic response of the glass, but it also removes the "triggering" mechanism that permits the shear relaxation in the viscous flow of the high energy component of the spectrum. This suggests that the main effect of structural relaxation is the progressive removal of the more readily rearrangeable component of material having large free volume. This can be formally considered as a change occurring in a factor p (Qf) in the preexponential factor yG of the principal strain rate expression, as defined by Eq. (10-4), where p(Gf) = exp(-aG f /Q 0 )
(10-12)
represents the probability of finding a local region with free volume in excess of Q{ (Argon, 1985). The computer simulations of inelastic relaxations in model glassy solids, discussed in Sec. 10.4.2 below are in support of this interpretation. 10.4.1.3 Plastic Flow As the applied shear stress a becomes closer to the mechanical threshold resis-
478
10 Inelastic Deformation and Fracture of Glassy Solids T(K) 580 560 540 520 500
480
460
440
420
10"-
101'
192±17kJ/mol 487 K ] *507K Anneal • 537 KJ 10 1 1.7
1.8
2.1 3
I
L
2.2
2.3
2.4
2.5
Figure 10-12. Dependence of shear viscosity of Pd 82 Si 18 on temperature for different aging treatments (Taub and Spaepen, 1980).
1
10 /T (K' )
tance f the inelastic strain rate increases nonlinearly with stress and tends to become asymptotically unbounded as o-*i. Alternatively upon the imposition of a relatively high strain rate of the level usually applied by laboratory testing machines, a state of plastic flow with negligible hardening behavior is reached after a more or less stretched out elastic-to-plastic transition as is shown in Fig. 10-13 for the case of a Pd 80 Si 20 sample strained in tension (Megusar et al., 1979). Plastic flow in metallic glasses has a number of features that are characteristic of plastic flow in crystalline metals. First, the temperature dependence of the plastic resistance is relatively small in the low tem-
perature region, as shown for the typical case of Pd 80 Si 20 and Pd 77 5 Cu 6 Si 16 5 in Fig. 10-14. Second, the strain rate sensitivity of the plastic resistance is small, or stated alternatively, the strain rate is a very strong function of the applied stress, with a phenomenological stress exponent m of the strain rate, as defined in Eq. (10-7) becoming ever larger as the temperature decreases. This trend is shown in Fig. 10-15 for a Pd 80 Si 20 glass in the relatively high temperature range near Tg where the deformation is homogeneous and conventional strain rate change experiments can be performed relatively easily. The figure shows that as T -> Tg and the resistance to deformation becomes progressively smaller in
10.4 Inelastic Response in Metallic Glasses 1
-
rrrj
I [jtl % :: :l I
lr;T
TJ7; :;!:
!
•
•
•
::;;h:;
:
•
;
•:
^
•
rryj- I;:"
Till
iiii iiii
479
;
!
.
1
-
:
.
•
_
•
; '
I
E
u— „
.
y
/
o o
— -
/
—
•
II
b i
•> «.0.01
•Hin
__i
•
•
•
:
:
These are listed in column 4 of Table 10-1 and apart from some scatter, appear to be constant at a level of about Av* = 9.13 x xlO~ 2 9 m 3 . In the range in which the temperature dependence of the flow stress becomes very small, i.e., in the lower temperature region of Fig. 10-14, the deformation is found to become quite inhomogeneous, with the deformation being almost entirely confined into a set of intense shear bands, as shown in Fig. 10-16. Here the tensile response of
=
~ \ d° JT JT
(10-13)
a
•
3-LJ.
:
comparison with f, the behavior apparently reverts to a viscous type. The data shown in Fig. 10-15 are given also in columns 1 - 3 in Table 10-1, from which the usual shear activation volumes Ar* can be determined according to the well known relations Av
-
Figure 10-13. A stressstrain curve for Pd 80 Si 20 at 433 K for a strain rate of 1.4 x 10~5 s~\ with a sudden excursion to a strain
T (K) 0
100
200
I
I
300
400
500
600
I
I
700
a, R = 2.5a, d =2.67x10"10m
3
1.0
_ Yo=0.125l'Tg
0.8
n
.5
\
D eft
D
_
0.6 D
\
0.4
\
_
O
0.2 -
\
n,
I 0.2
.
I
I
0.4
0.6 T/Tg
0.8
1.0
Figure 10-14. Dependence of the flow stress on temperature for Pd 80 Si 20 (o) (Megusar et al., 1979), and Pd 77 . 5 Cu 6 Si 16>5 (n) (Pampillo and Chen, 1974). Solid curve is the fit obtained with the model of Sec. 10.4.2.
480
10 Inelastic Deformation and Fracture of Glassy Solids
T(K) 0
100
200
300
400
500
600
700
Figure 10-15. Temperature dependence of stress exponent of strain rate for a Pd 80 Si 20 glass (Megusar et al, 1979).
Table 10-1. Stress exponents m, and activation volumes Av*, measured in strain rate change experiments by Megusar et al. (1979) on Pd 80 Si 20 . T (K)
(GPa)
m = d In e/d In o
Au* (m3)
413 433 453 473 493 513 533 553
1.29 1.13 1.02 0.94 0.86 0.65 0.64 0.57
11.74 8.64 7.21 6.15 7.16 5.42 4.51 5.23
8.98 xlO~ 2 9 7.90 7.65 7.40 9.81 10.20 8.98 12.10
Tensile flow stress at a strain rate of e1 = 1.4:
the samples becomes too unstable to perform strain rate change experiments. The specimens undergo early fracture inside the intense shear bands by the "meniscus instability" mechanism discussed in Sec. 10.3.3. In any case, however, the combination of
•isili
tp Figure 10-16. Intense shear bands in a bent ribbon of a metallic glass Fe 29 Ni 19 P 14 B 6 Si 2 (Argon, 1980).
negligible strain hardening and a decreasing strain rate sensitivity of the flow stress should result in hastened rupture by necking even if such fractures did not occur (Argon, 1973 a). In fact the mechanical instability is hastened further since the strong shear localization suggests the presence of a strain softening process. Much evidence (Megusar et al., 1982) indicates that this strain softening is due to the strain induced generation of free volume which does not decay during deformation at low temperatures and high strain rates. The accumulating dilatation inside the sheared regions then results in a drop in plastic resistance which, in turn, rapidly concentrates the deformation further to form intense shear bands (Argon, 1979). Dilatometric measurements and associated kinematic analysis of shear localization in a Pd 77 5 Cu 6 Si 16 5 glass have verified that there are indeed very substantial density reductions inside shear bands which imply a flow dilatancy level of ft = d#/dy of 0.018 and a total local retained dilatation of about 0.5 inside the bands for the total shear strains suffered by the bands of the order of 30 (Argon et al., 1985). Clearly, the level of inferred local dilatation can not have been uniformly dis-
10.4 Inelastic Response in Metallic Glasses
persed but must have produced substantial cavitation. The reported tendency to form etch pits in shear bands (Pampillo and Chen, 1974) is most likely to be related to these supra-atomic scale cavities. Figure 10-17 shows that there is a welldefined combination of strain rate and temperature which characterizes the bifurcation condition for formation of shear bands (Megusar et al, 1979). In the higher temperature and lower strain rate domain the strain induced free volume collapses diffusively as fast as it forms so that the plastic resistance remains relatively unaltered, and deformation remains homogeneous (Spaepen, 1977; Argon, 1979). Operationally, the measurement of the plastic resistance of metallic glasses in the lower temperature region is much more readily accomplished by means of indentation hardness measurements which circumvent most of the difficulties associated with mechanical instabilities in tension. Table 10-2 shows the Vickers microhardnesses of a collection of prominent metallic glasses obtained by Davis (1978). Applying the usual plastic constraint factor of 0.33 to the hardness values, the tensile plastic resistances can be obtained from these measurements, and are also given in Table 10-2. 10.4.1.4 Structural Aging and its Effects on Plastic Resistance
Since glass transitions resulting from higher cooling rates result in higher Tg's and higher levels of frozen-in specific volume, structural compaction, or relaxation, can continue in the glassy state, if the temperature is not too low. Thus, the glass will undergo structural aging during which the density will systematically increase (as the free volume decreases), and all measurable mechanical properties will exhibit
481
-400
Tensile strain rate (s"1)
Figure 10-17. Regimes of homogeneous and inhomogeneous flow in a Pd 8 0 Si 2 0 glass (Megusar et al., 1979).
monotonic changes. Figures 10-9 and 10-12 show the resulting systematic decreases in internal friction and fluidity that accompanies structural aging in metallic glasses. The changes in hardness in a Fe 8 0 B 2 0 glass are shown in Fig. 10-18. Associated TEM studies have indicated that the rise in plastic resistance up to an aging time of 10 4 s at a temperature of 0.96 Tg (0.92 of the crystallization temperature Tc) occurs as a result of structural aging in the glassy state alone without the benefit of crystallization. When crystallization sets in at times exceeding 10 4 s a further rise is recorded in Table 10-2. Young's moduli £, hardnesses H, and plastic resistances in tension Y of a selection of metallic glasses (Davis, 1978). Alloy
E (GPa)
H (GPa)
Y = (H/3) (GPa)
Pd 8 0 Si 2 0 Pd 7 7 5 Cu 6 Si 1 6 5
89.7 89.7 116.0 127.0
4.9 5.0 6.1 7.5
1.63 1.67 2.03 2.50
133 169 85.1 98.6
7.5 11.0 5.8 6.1
2.50 3.67 1.93 2.03
^80^20
Ni 4 0 Fe 4 0 P 1 4 B 6 (Metglass 2826) Fe 8 0 P 2 0 Fe 8 0 B 2 0 Cu 50 Zr 5O Cu 5O Ti 5O
482
10 Inelastic Deformation and Fracture of Glassy Solids I
!
658 K
Fe
80B20
E 1300
-— 0.50
"S •
—
v
'
har dne
CO
Dull side
H
^
—i^_
1200 \ ^"^
\
Shiny side
- 0.10
1100
CO
0.05
Vick
CD
{ f
1000]/ p
I
>
IE
900
/ \
°
wo
o £ n
i
I
i
5
10
15
f
n
- 0.03 " 0.02
laxations under stress. As we will discuss in Sec. 10.4.3 below this, can be viewed either as a system with a distribution of relaxation times, obeying conventional Arrhenius kinetics or a process that obeys Williams-Watts kinetics characterized by so-called "stretched exponential" decays (Dengetal, 1989 c).
i
20x10 3
Aging time (s)
Figure 10-18. Changes in microhardness and strain to fracture of an Fe 80 B 20 glass as a function of aging time at 658 K (Deng and Argon, 1986a).
the hardness which results from the relatively high plastic resistances of the small crystallites that act as a first approximation as rigid filler in the amorphous phase (Argon, 1986). We note from Fig. 10-18 that the aging also results in a sharp reduction in strain to fracture, first in the amorphous range and then in the range where crystallization sets in where the crystallites evidently act as sites for cavitation. Such two stage aging, associated with a stepped increase in hardness and a stepped decrease in strain to fracture is quite common in amorphous metals. While there is general agreement that the first rise in hardness (drop in strain to fracture) is due to loss of free volume, the second step is not always due to crystallization. Thus, Gerling et al. (1988, 1989) have demonstrated that in Fe4ONi4OP2o the second step is due to phase separation, still in the amorphous state. Deng and Argon (1986 b) have studied the kinetics of structural aging of metallic glasses and found evidence from a limited set of measurements that the distributed properties of structural aging resemble closely the distributed nature of shear re-
10.4.2 Model for Viscoplastic Flow Since access to the mechanism of viscoplastic flow in metallic glasses by direct experimental methods on the atomic level has not been possible, the results of various types of simulation of the process have provided considerable insight and guidance. One such fruitful simulation has been the amorphous Bragg soap bubble raft explored by Argon and Kuo (1979) and Argon and Shi (1982) which has indicated that based on this model the inelastic relaxations in atomic glasses are most likely to be by shear transformations, producing deformation in a manner sketched out in Fig. 10-8. In these experiments of simulating the plastic deformation in the sheared amorphous bubble rafts Argon and Kuo noted that the two dimensional forms of the sheared clusters could vary from concentrated forms of translation across nearly planar rows of bubbles to coupled diffuse rearrangements inside relatively equiaxed clusters, as depicted in Fig. 10-19. The kinetics of inelastic straining by the diffuse shear mechanism, which is applicable in the low stress region of behavior could then be given readily for the 3-D variant, by a strain rate expression of the following form (Argon, 1979; Argon and Shi, 1982)
.
T
/
AF\ .
T
h[<jy
Qe\ (10-14)
10.4 Inelastic Response in Metallic Glasses
483
logical stress exponent of ayTQ
(10-16)
kT
Comparing the predictions of the above model with the experimental measurements of Megusar et al. (1979) and making use of the Bragg soap bubble raft simulations, the following values could be established: £ = 1 . 0 ; / = 0.135; QC = 1.S x 10" 2 8 m 3 , for a Pd 80 Si 20 type metallic glass. We note that when the applied stress decreases, the nonlinearity of behavior progressively decreases, the stress exponent m->l and Eq. (10-14) goes smoothly to an expression for Newtonian viscous flow and becomes AFV* (10-17) exp If kT (b) Figure 10-19. Idealization of two limiting forms of shear transformations observed in sheared soap bubble rafts: (a) concentrated shear translation, (b) diffuse shear transformation (Argon and Kuo, 1979).
where AF
J7-Sv •|_30(l-v)
2(1+v) 1 9(l-v) P J
(10-15) is the Helmholtz free energy associated with a shear transformation occurring in a spherical region of volume Qc having a transformation shear strain yT and a transformation dilatation sT at the saddle point (= p yT) (Eshelby, 1957). In Eqs. (10-14) and (10-15), a is the applied shear stress \i and v are the shear modulus and Poisson's ratio respectively, while c is the volume fraction of the potentially transformable material and vG is the fundamental frequency of the clusters. This form of the viscoplastic strain rate would result in a phenomeno-
where AFV* is now the Helmholtz activation energy of viscous flow and c may incorporate a structure given by Eq. (10-12). The smooth change from linear to highly nonlinear behavior as the stress is increased, and supporting the above picture was experimentally established by Taub (1980). The strain rate expression that results for the concentrated shear process is given in turn (Argon, 1979) by T
i
kT
(10-18)
where
~1
(10-19 a) (10-19 b)
with fis being the ideal shear strength of the structure, estimated to be 0.03 \x by comparison with results from 3-D computer simulations of Maeda and Takeuchi (1982). The degree of agreement of the forms of the expressions given by Eqs. (10-14) and
484
10 Inelastic Deformation and Fracture of Glassy Solids
(10-8) when stated as flow stress relations, and compared with experimental results for Pd 80 Si 20 and Pd 77 5 Cu 6 Si 16 5 is shown in Fig. 10-14 and for stress exponent, is shown in Fig. 10-15. The agreement in the lower temperature region must be considered less good since deformation undergoes extensive shear localization where the smaller temperature dependence of the flow stress could be influenced strongly by the strain softening process referred to in Sec. 10.4.1.3 above.
10.4.3 Simulations of Plastic Flow in Metallic Glasses Simulation of plastic shearing in the computer has been considered by many investigators as a fruitful avenue toward a better understanding of the kinematics of the process at the atomic level in a glassy metal. Of those, the 3-D simulations of Maeda and Takeuchi (1982) and that of Srolovitz et al. (1983) have pointed out the difficulties of required size of the simulation cell and those associated with extracting kinematical information out of 3-D space. To avoid these problems, of size and complexities in 3-D space, albeit at the risk of missing some degrees of freedom, Deng et al. (1989 a) have chosen a relatively large 2-D simulation cell with periodic boundary conditions that had previously been melted and quenched in the computer. In this extensive simulation, as in nearly all others, individual atoms interact with each other by means of a Lennard-Jones potential, smoothly truncated between the third and fourth nearest neighbor atom. The details of the interatomic potential, and its application to 2-D simulations are given by Deng et al. (1989 a, d). Here we will discuss some of the more important findings of those investigators.
Figure 10-20. Arrangements of Voronoi polygons of atomic sites of a 2-D model glass quenched to T/Tg = 4 x l O " 3 . Note the 5-7 sided liquidlike material, (b) RDF of atom positions in (a) (Deng et al., 1989 a).
Figure 10-20 a shows the starting configuration of atom environments after a perfect 2-D mat of hexagonal packing was melted and quenched in the computer to a temperature of 4 x 10" 3 Tm(5.56 x 10~ 3 Tg) in a number of steps each incorporating some structural relaxation, followed by holding further at the final temperature. The resulting 2-D radial distribution function of the final structure, is shown in Fig. 10-20b. Several features of the mat in Fig. 10-20 a must be noted. First, while the RDF shows the characteristics of a well-re-
10.4 Inelastic Response in Metallic Glasses
laxed glass, it is far from being homogeneously disordered. There is a substantial fraction of reasonably well-ordered material of hexagonal symmetry with characteristic dimensions of 8-10 atomic spacings. Second, the disordered material, readily recognizable by its chief ingredient of structural dipoles of atoms of 5 and 7 coordination makes up a volume fraction of 0.16 of the total. At the melting point this fraction is as high as 0.40 and percolates through the structure. The 5/7 coordinated structural dipolar material has an excess volume that is on the average 9% larger than a pair of hexagonal atom sites and can be viewed as a structural element of free volume. Between Tm and Tg the 5/7 dipolar material fraction decreases monotonically to about 0.3, remaining, however, contiguous and maintaining the percolation condition. Below Tg the percolation condition is broken. In anticipation of the important properties of the 5/7 sided material resulting from its large free volume it has been identified as the liquidlike material of Cohen and Grest (1979). The plastic deformation simulation has consisted of imposition of successive increments of external shear strain of 5 x 10 ~ 4 applied to the borders of the simulation mat, held at constant volume, followed by reequilibration of the atoms by means of some structure relaxation to remove the most disequilibrium atom configurations. For each step of shear increment, after reequilibration was achieved, the atomic site stress tensor was calculated together with an atomic site strain increment tensor from the changes in the displacement gradients of each atom with respect to its immediate neighbors. Of particular interest were the two scalar invariants of the strain increment: the dilatation increment and the maximum (Mohr circle) shear strain increment. The process of shearing was
485
continued in this manner until a total shear strain in excess of 0.27 was achieved for the simulation mat. Figure 10-21 a shows the shear stress/shear strain curve for the mat. 1 A clear transition from nearly linear behavior to flow behavior occurs at a shear strain of about 0.05. Figure 10-21 b shows that while the mat is plastically sheared at constant volume the external pressure on the mat increases slightly, but monotonically until a strain of 0.15 is reached, after which it remains constant. Figure 10-22 shows that when the shearing process is stopped at total strain levels of 0.025,0.075 and 0.125 and is reversed, there is a prominent Bauschinger effect as reverse yielding is initiated even before complete unloading. The hysteresis is observed even in the apparent linear elastic range. Examination of the distribution of deformations has resulted in the following findings: (a) Inelastic processes occur in the form of local shear transformations incorporating cooperative action of a number of neighboring atoms. (b) These transformations occur preferentially in regions where the liquidlike material (free volume) congregates as is shown in Fig. 10-23 giving the distribution of local maximum shear strain increments at yield for a further step of external shear (observe the resemblance of this distribution to the distribution of liquidlike material in Fig. 10-20 a). (c) Inelastic processes, in the form of local shears in excess of the imposed affine shear, begin in the initial linear range (there is no real elastic limit).
1 The stresses in Fig. 10-21 are given in units of EQ/TQ, where Eo is the binding energy of the pair potential, and r0 is the separation between atoms where the pair potential goes to zero.
486
10 Inelastic Deformation and Fracture of Glassy Solids
2 -
Figure 10-21. Shear response of 2D model glass: (a) shear stress, shear strain curve; (b) increase of evoked pressure due to dilatancy effect (Deng et al, 1989 d). 0.2
0.1
7 (d) The elastic-to-plastic transition occurs approximately when the total accumulated shear transformations become contiguous as was anticipated by Argon and Shi (1983) and as is familiar in crystal plasticity (Kocks, 1966). 1
I
•
i
•
(e) At each stage of deformation, the residual back stresses of the shear transformations of the immediately preceeding history help reverse the deformation and are the cause of the large Bauschinger effect. 1
'
2 o
l .025^1 / ^
i
t
.
L/075
l
-l
1
-0.1
1
1
0.0
0.1
7
1
0.2
Figure 10-22. Simulation of reversal of shear in 2-D model glass showing pronounced Bauschinger effect (Dengetal, 1989d).
10.4 Inelastic Response in Metallic Glasses
'• •.•.VHVG. 13 .". ••°. • " t a ^ X V . ' c i S
a B.'BB,
Figure 10-23. Distribution of atomic site maximum shear strain spikes in the sheared 2-D model glass (Dengetal., 1989d).
(f) The shear transformations are generally dilatant [as observed to be the case by Argon and Shi (1982) in the soap bubble rafts] which results in the buildup of a system pressure when the defor-
487
mation is performed at constant volume. (g) As a result of the deformation dilatancy the volume concentration of liquidlike material can fluctuate, this is shown in Fig. 10-24 where the variation of the liquidlike material concentration with strain is shown. Comparison of this with Fig. 10-21 a shows that the depression in flow stress coincides with the peak in concentration of the liquidlike material. (h) Eventually when the deformation induced liquidlike material becomes contiguous, shear localization can set in as is shown in Fig. 10-25 a, b,c as the total shear strain increases from 0.100.15-0.20. Prominent localization has occurred at a shear strain of 0.15. (i) At any stage of the deformation imposition of an external affine shear strain increment can evoke a substantial fraction of the local shears in directions other than the external shear increment - apparently relieving disorder related preexistent misfit, and always reducing
150
Figure 10-24. Variation with strain of the total number of liquidlike atomic sites in the sheared 2-D model glass (Deng etal, 1989 d).
488
10 Inelastic Deformation and Fracture of Glassy Solids
Figure 10-25. Sequence of development of shear localization in the sheared 2-D model (b) after y = 15%, (c) after y = 20% (Deng et ah, 1989d).
the Gibbs free energy of the entire system. (j) Finally, in all the observed sequences of deformation, production of strain from motion of unambiguous crystal dislocations (displacement of isolated 5/7 dipoles) was quite rare resulting only in a fraction of about 0.11 of the total strain, but demonstrating the overwhelming efficiency of this mode of strain production when it is present. Clearly, the results of the simulation that were presented relate to 2-D material which must quantitatively differ from what occurs in 3-D material. Nonetheless, the reported observations make up a complete set of phenomena which provide important insight not only into the deformation mechanism but also into the glassy state.
10.5 Inelastic Response in Space Network Glasses Observation of anelastic and viscous behavior in space network glasses (oxide
after y = 10%,
glasses) at elevated temperatures, but still under their Tg and at relatively small stresses, has been widely reported (Jones, 1948; Argon, 1968). The distributed nature of relaxation times, in unmodified and modified network glasses measured by internal friction experiments or by creep and recovery creep experiments is very similar to that discussed in Sees. 10.4.1.1 and 10.4.1.2 for metallic glasses (Argon, 1968). Plasticity in network glasses, however, is quite unfamiliar, since they fracture under normal conditions in tension well before any nonlinear behavior characteristic of plastic flow can be initiated. When fracture is suppressed, however, by the superposition of a large pressure or in microhardness experiments where a large pressure exists in the plastic flow field as an integral part, oxide glasses can undergo large strain plastic flow. The diverse, common and uncommon, experiments of many investigators carried out under such favorable conditions have been reviewed by Argon (1980). Of these the most informative are the microhardness indentation experiments of Marsh (1964a,b) on a soda glass
10.6 Inelastic Response in Polymeric Glasses
and type E glass at different temperatures and indentation rates. These results are shown in Figs. 10-26 a and 10-26b, where the athermal intercept value of the plastic shear resistance fn is the ideal shear
V
I.D
1
1
1
1.4
-l
-
1.2
-\
-
1.0
\
\ \ -
0.4
0.2
Theory with Strength-Differential Effect
x \ \ \\ \
0.8
0.6
—
—
(a)
0
1
1 1 ^ \ 0.8 0.6 1.0 T/Tg , Normalized Temperature 0.2
(a) 1.4
I
0.4
I
i
i
i
I
i
489
strength which has been taken as /I/2K on a sinusoidal model of shear interaction of two parallel planes in the solid. The solid curves in these figures are the predictions of a variant of a homogeneous flow model similar to that discussed in Sec. 10.4.2 for metallic glasses. It parallels the results well in the low stress and high temperature region but severely underpredicts the plastic resistance at very low temperature or very high strain rate. This has been accounted for by Argon (1980) as a manifestation of a strong strength differential effect where the plastic resistance of a solid increases sharply with increasing pressure when the accompanying background mean normal elastic strains become very large. The dotted curves take such a strength differential into account based on the anticipated pressure dependence of the plastic resistance. It is worthwhile to emphasize that plasticity of oxide glasses even under the most favorable conditions is very limited and of little technological importance. Thus, oxide network glasses can be considered as intrinsically brittle solids that will fracture before any significant plastic flow can occur under states of stress not containing a very high level of superimposed pressure component.
1 2
3 - I-
A
CD
1.0 " Theory with strength differential effect
10.6 Inelastic Response in Polymeric Glasses
^
0.8 |
0.6
10.6.1 Experimental Observations
b
-a-——
. 0.4 -
-f—^^
-
0.2 I
i
l
I
10' 5 10~4 10' 3 10~2 10"1
I
I
1
10 1
I
10 2
i
10 3
10 4
(b) y. strain rate (s' ) Figure 10-26. Temperature dependence (a) and strain rate dependence (b) of normalized flow stress of two network glasses: (D) E glass, (o) soda glass (Marsh, 1964 a, b; Argon, 1980).
10.6.1.1 Anelasticity
The anelastic response of polymers has been studied extensively over a wide frequency range between 0 K and Tg (see McCrum etal., 1967). In these studies it has been customary to report not only the temperature dependence of the cyclic energy loss but also the so-called dynamic modu-
490
10 Inelastic Deformation and Fracture of Glassy Solids
lus as a function of temperature at a given frequency. As in the case of metallic glasses the spectral distribution of the mechanical relaxation times have been viewed as a sensitive probe of the structure of the polymer. Figure 10-27 shows the temperature dependence of the dynamic modulus and the associated internal friction (tan<5) spectrum of the atactic glassy polymer of polycarbonate of bisphenol-A(PC) which is typical of most such information for other polymers. Examination of Fig. 10-27, shows, however, that there are some discrete relaxations below the glass transition temperature which must have a specific molecular level interpretation. In PC, where the molecule contains a regular arrangement of phenylene rings, their rotation about the molecule axis can be a major component of a local shear relaxation. Similarly, other conformational rearrangements of specific molecular segments can also be responsible for these discrete relaxations superimposed on more diffuse rearrangements of the entire molecule. Such rearrangements have been discussed by McCrum et al. (1967) and have also been partially simulated in the computer as we will discuss in Sec. 10.6.3.
10"" -200
-100 0 100 Temperature (°C)
200
Figure 10-27. Temperature dependence of the shear modulus and cyclic energy loss in PC (Kambour, private communication).
10.6.1.2 Plastic Flow and its Mechanism All glassy polymers exhibit plastic behavior under high stresses in compression where fracture can be suppressed in those that tend to be brittle in tension. In the subgroup of stiff chain thermoplastic glassy polymers which do not craze, or in which crazing can be suppressed, well developed plasticity is observed also in tension. A characteristic feature of this plastic behavior, differentiating it from ductile metals, is the strong molecular orientation hardening that occurs at large strain which results in a sharp upturn in the tensile plastic resistance curve, as is shown in the set of stress strain curves of PMMA in Fig. 10-28 (Hope etal., 1980). The form of these curves is reminiscent of the large strain behavior of cross linked rubbers, albeit with a substantial additive component of strainindependent plastic resistance. That this resemblance is real is established from the nearly complete recoverability of shape of highly distorted glassy polymers when they are heated to above their Tg. The temperature dependence of the yield strength of glassy polymers closely parallels the temperature dependence of the modulus as is shown in Figs. 10-29 a and 10-29b for a set of stiff chain polyimides (Argon and Bessonov, 1977a). This indicates that the plastic resistance is governed primarily by intermolecular interactions. The dependence of the inelastic strain rate on the applied stress is a strongly non-linear one, also reminiscent of metal plasticity, as the results for polyethylene terephthalate (PET) in Fig. 10-30 show. Argon and Bessonov (1977 a) have carried out experiments on the strain rate dependence of the tensile or compressive yield strength of a series of glassy polymers ranging from flexible chain polymers to some stiff chain polymers to probe the shear activation vol-
10.6 Inelastic Response in Polymeric Glasses .
.
.
.
.
'' 1
.
e = 90°c
100
491
fa,
-
CO
CL
pi */' /
50
Exp. e = 1.0 s'1 yr>r|
a
Exp. e Mod. e - 0.1s' 1 Exp. e = 0.01 s"1 —
i
Oi
i
i
.
1
.
•
—
•
—
•
'
Mod. e = 0.01 s"1
i
0.5
1.0
1.5
Figure 10-28. Tensile true stress, true strain curves for PMMA at 90 °C (Hope et al., 1980; Boyce et al., 1988 a).
True strain (m/m)
umes At;* of the plastic flow event. Their results are given in Table 10-3. Calorimetric experiments of Oleynik (1990) and co-workers (Rudnev et al., 1990) have revealed important and striking parallels of the plasticity of glassy polymers to
that of glassy metals discussed in Sees. 10.4.1-10.4.3. Their coordinated measurements of stress strain behavior, concurrent deformation calorimetry (DC) and subsequent differential scanning calorimetry (DSC) have established that heat evolution
zw 180 160
z
X X
140
K
CO CO CD
}—.
_
e = 10x10" s"
S
120 -fc?
Q NNNV
~
~Z. CD
_
CO 3
3
CO
T3
100
.
^V^°
P
O
ton
80
3
60 _ 40
-
20
-
o >-
yo
x
\x
Lji
DFO
R-R°^ H - H N \
o
.
I
I
300
(a)
Ka
ng's
nsi le yii
m
o i
N
100 , 400
6i
S i
200
t
500
Temperature
300 ,°C 600 K
100
(b)
200
300
400
500
Temperature (K)
Figure 10-29. Temperature dependence of tensile yield stress (a), and Young's modulus (b) of a set of polyimides: (R-R) resorcinol, (H-H) hydroquinone, (DFO) oxydiphenyl, (Kapton) pyromelitic acid (Argon and Bessonov, 1977 a).
10 Inelastic Deformation and Fracture of Glassy Solids
492
10 4 1- Glassy PET at 330 K
30
IT(T =
1
1
1
a -PS 10
:
*-> 1 1 —
20 -
'
10"'
Wy
o •^ 1(TL
Plasticity
Qdef^
_ d 10 -
range
Linear viscoelasticity range .*'
9
/
r
AF
^ ^
10"1
1
10
5
10
6
10
7
10
8
10
10
9
2
Shear stress (N/m )
Figure 10-30. Stress dependence of the inelastic strain rate in PET at 330 K. At the mechanical threshold stress the strain rate becomes unbounded (Argon and Bessonov, 1977 b).
20 Strain, %
1 30
40
Figure 10-31. Rate of mechanical work expenditure (W), free energy storage (AF), and evolution of heat (Q) during the plastic deformation of atactic PS (Oleynik, 1990).
Table 10-3. Shear activation volumes Av*, and actual volumes Qc of relaxing clusters of glassy polymers. Polymer PS PMMA PET PC PPO R-R DFO Kapton
At?*a
(m3)
(m3)
284 x l O " 3 0 375 991 1060
0.78 x l 0 ~ 2 6 1.01 2.68 2.86 1.66 2.78 4.62 6.16
613 1030 1710 2280
-100
0 50 Temperature T (°C)
100
150
From experiments of Argon and Bessonov (1977 a); calculated using yT = 0.037 based on simulation of Mott etal. (1992).
Figure 10-32. Strain recovery spectrum with temperature of previously deformed EAN-polymers: (1) T = 150°C, £ = 0.04; (2) T = 60°C, e = 0.11; (3) T = 20°C, £ = 0.09; (4) T = - 8 5 ° C , e = 0.12; (5) T=- 85°C; e = 0.045 (Oleynik, 1990).
begins below yield and monotonically and smoothly increases with increasing strain to a relatively high rate Q, paralleling the overall plastic work rate W at a strain of about 25%, as is shown for the case of atactic polystyrene (a-PS) in Fig. 10-31. At that stage the stored energy of cold work, AF, reaches a steady state (presumably dependent, to some extent on the applied strain rate), while there are still no important changes in infra red (IR) spectroscopy
indicating that significant, large scale changes in molecular conformations have not occurred. The DSC measurements have shown that on samples deformed to plastic strain (compression) levels of 1030%, cooled under stress to low temperatures, followed by stress removal, and upon heating, stored energy release begins at the previous deformation temperature, and has a characteristic bimodal spectrum shown in Fig. 10-32 for an epoxy-aromatic
a
b
10.6 Inelastic Response in Polymeric Glasses
amine network (EAN) polymer. The sharp high temperature peak occurring at Tg ( = 140°C) is related to conformational recovery, while the broad low temperature hump, starting from the previous deformation temperature is of a different, nonconformational, character, and is more akin to the stored elastic strain energies around shear transformations discussed in Sec. 10.4.2 for metallic glasses. Moreover, these investigators have established that in lightly strained samples only the low energy hump is present, and that the appearance of the conformational peak requires strains in excess of those where AF reaches its initial plateau value. This indicates that the development of the molecular orientation, characteristic of large strain behavior akin to cross linked rubbers, requires the establishment of a steady plastic flow state. The validity of the above sequence of processes is verified by the results of computer simulations of plastic flow in vinyl polymers that we will discuss in Sec. 10.6.3.
10.6.1.3 Kinetics of Plastic Flow
While the actual kinematics of molecular segment rearrangements during the plastic deformation of glassy polymers remains elusive, mechanism inspired and quite successful constitutive laws can be stated for deformation in the glassy state. One such formalism is that of Argon (1973 b). In this approach a specific form of strain producing molecular segment rotation is conceived and the saddle point free energy AG* for it is calculated in terms of some molecular scale parameters. The resulting Arrhenius rate expression for plastic shear strain rate is AG* 7=
7G
~kf
(10-20)
493
with AG* = AG0 l - ( - r )
(10-21 a)
Argon and Bessonov (1977 a) have noted that this constitutive form can be stated as
-J
=A~B(T/fi)
(10-22)
where 0.077 T / 6
(10-23 a) (10-23 b)
are material constants. That this is indeed the correct form of the constitutive behavior for the thermoplastic glassy polymers listed in Table 10-3 is shown in Figs. 10-33 a and 10-33 b and for a DGEBA (diglycidyl ether of bisphenol-A) epoxy thermoset polymer in Fig. 10-33 c (Yamini and Young, 1980). More extensive discussions about the possible molecular level meanings of the above form of the constitutive relation were provided in the original reference. When large strains are suffered by the glassy polymer, strain induced molecular orientation produces an important back stress to deformation that can be modeled as a first approximation by the entropic resistance of rubber elasticity (Haward and Thackray, 1968; Argon, 1973 b; Boyce etal., 1988 a) which gives the following principal back stress (resistances) 3MC' (10-24)
where g R T/Mc is the rubbery regime shear modulus, Xi are the principal extension ra-
494
10 Inelastic Deformation and Fracture of Glassy Solids 0.20
74 phr
015 010 0.05 0
9.8 phr
015 010 k
A
*~*^A
0.05 0
12.3 phr
0.15 010 ^
0.05 0
>
^
<
A
^
A
K7phr
015
0
0.2
0.4 0.6 0.8 T / u , m2 °K/N
1.0
I. 2XI0
6
0.10 005
c 01
0. I 6 x
I
I
I
0.4 T/fL,
0.2
0.3
A
OA
A
0.5
A
0.6
T
0. 14 P
0.2
^ ^ " ^ ^ _
0.6 0.8 m 2 o K/N
1.0
l.2x|0'
Figure 10-33. Dependence of normalized plastic shear resistance on temperature for a series of glassy polymers. The straight lines fit the form of the model of Sec. 10.6.1.3: (a) flexible chain polymers, (b) stiff chain polymers (Argon and Bessonov, 1977 a), (c) DGEBA - epoxy polymer (Yamini and Young, 1980).
tios, kn the uniaxial locking stretch of the polymer where full molecular orientation occurs, and fi"1 is the inverse Langevin function (Boyce et al., 1988 a). Figure 10-28 shows the agreement between experimen-
tal results and the theoretical formalisms represented by Eqs. (10-20) to (10-24). This constitutive relation for the plastic resistance incorporating other improvements related to strain softening, aging, and pressure dependence of the plastic resistance has been used extensively by Boyce et al. (1988a,b; 1989) in computational applications of boundary value problems for large strain deformation processing. The above theoretical model is meant to apply for deformation in a stable structure where an increase in temperature only accelerates the kinetics of transfer of deformation units from unflexed to flexed. Near a glass transition where a temperature increase produces important structural alterations that increase the fraction of liquidlike material (see Sec. 10.4.3), different considerations are necessary to change the emphasis from intermolecular to intramolecular resistances to deformation. This was achieved by Robertson (1966, 1968) in a theoretical model in which thermal equilibrium concentrations of strain producing "flexed" molecular conformations under
495
10.6 Inelastic Response in Polymeric Glasses
the applied shear stress are calculated from Boltzmann statistics through which a structural reference temperature 6 (above Tg) is defined that in turn defines the effective deformation resistance through the free volume model of the glass transition of Williams et al. (1955). Argon and Bessonov (1977 a) have compared the Robertson model to theirs and have concluded that it should be the model of choice near Tg. 10.6.2 Dilatational Plasticity in Glassy Polymers As discussed in Sec. 10.3.2.2 certain glassy polymers undergo crazing in tension leading to a mode of dilatational plasticity. While the molecular level requirements for a crazing vs. noncrazing response are not well established, two complementary conditions are known to be important. First, polymers that undergo crazing are predominantly flexible chain material which are known to have unusually high atomic level stresses due to a high degree of molecular structural disorder (Theodorou and Suter, 1986 a, b)> while noncrazable polymers tend to be stiff chain material with considerable evidence for short range order - and possibly having much lower atomic level disorder stresses (Argon and Cohen, 1990). Second, noncrazable polymers include those that have very high levels of entanglement densities, resulting in very small natural draw ratios, not permitting the formation of craze matter fibrils (Kramer, 1983). Initiation of crazes is a rate process that is sensitive to a certain combination of the deviatoric shear component s, and the negative pressure am of the local applied stress. Figure 10-34 shows a set of experimental observations of Argon and Hannoosh (1977) on craze initiation time t{, from the surfaces of polystyrene (PS) samples with a
10' 1
10
10^
10 d
104
10°
tj, craze initiation time (s)
Figure 10-34. Dependence of craze initiation time on different levels of negative pressure and deviatoric shear stress at room temperature in atactic PS (Argon and Hannoosh, 1977).
known level of controlled microroughness, subjected to different combinations of s and am. This combination can be stated as a craze initiation rate as follows: (10-25) The constants C1? C 2 , Q [C1 = 1.66xl0 7 s 1 ; C2 = 0.95GPa; Q = 0.0133 at 295 K (Piorkowska et al., 1990)] are material parameters with the above experimentally determined values being appropriate for
496
10 Inelastic Deformation and Fracture of Glassy Solids
PS at room temperature; and where Y is the tensile plastic resistance. A mechanistic rationale for the above form has been given by Argon and Hannoosh (1977). Crazes viewed as dilatational transformations, as discussed in Sec. 10.3.2.2, grow by the continued conversion of solid polymer into spongy or fibrillar craze matter by displacing the craze borders under a tensile stress acting across the craze plane. This conversion has been established to be by an interface convolution process (Argon and Salama, 1977; Kramer, 1983) that results in the following overall craze strain rate 8c = 8T — (Ah)
(10-26)
dr where A is the total craze area per unit volume, and h is the primordial craze thickness undergoing the tensile transformation strain eT. In the early stages of crazing the craze thickness tends to remain fixed, and e -> sThQVct
(10-26 a)
where Q is the active craze front length per unit volume and vct is the craze tip velocity, transverse to the applied tensile stress. In the latter stages of crazing when the total craze area per unit volume remains relatively fixed a n d the craze widens by translating the craze borders, the craze strain rate becomes cb
l+ fi T"-<">
(10-26 b)
where vch is the craze border velocity, in the direction of the applied tensile stress a and the factor 2 arises from the fact that the craze can thicken by the outward displacement of both of its borders. The craze front velocity and the craze border velocities have a kinetical dependence on the applied stress a given by (Argon and Salama, 1977;
Piorkowska et al., 1990) (10-27) Ct
r
{ AG0
5/6"
In Eq. (10-27) 7[ = 0.133/V(l ~v)] is the athermal plastic resistance of the glassy polymer, k'n is an orientation hardeningmodified extension ratio of the craze matter, v0 is a preexponential rate constant, oc a geometrical constant and AG0 is an effective activation free energy of plastic flow [Eq. (10-26 b)]. For PS at room temperature [a = 0.282; v0 = 1.23 x 106 m/s; AG0/ k T = 44.7; k'n = 1.85 (Piorkowska et al., 1990)]. For additional details of craze plasticity, the effect of polymer type, molecular weight, and related matter, the reader is referred to the two treatises edited by Kausch (1983, 1990). Finally, it should be noted that while crazes do transform the continuous polymer to a discontinuous spongy form it can nevertheless, result in very substantial overall strains - provided that the crazing process is carefully "managed" to occur at stress levels that do not result in craze fracture (Volynskii and Bakeev, 1984). 10.6.3 Simulation of Plastic Flow in Glassy Polymers 10.6.3.1 Molecular Structure Models As in the case of atomic glasses discussed in Sec. 10.4.3, in polymeric glasses the molecular level process of plastic deformation can be very effectively simulated by computer. Considering the interactions between atoms along the backbone of a chain molecule of a polymer glass, it is found that relative separation of atoms as well as flexing of bond angles between atoms along
10.6 Inelastic Response in Polymeric Glasses
the molecule are very strongly resisted. In comparison, the rotation of molecular groups about a bond needs to overcome only a modest torsional restoring moment which tends to make chain molecules conform primarily by rotation along the backbone bonds. Thus, as a first approximation the backbone bonds act as inextensional and bond angles as inflexible. In addition, in a dense arrangement of chain molecules, portions of the molecule will interact with portions of surrounding molecules by van der Waals interaction. Based on these idealizations, utilizing known forms of torsional potentials and van der Waals interactions molecular structures of several glassy polymers have been obtained by static energy minimization techniques. These include polypropylene (PP) (Theodorou and Suter, 1985,1986 a, b) polyvinyl chloride (PVC) (Ludovice and Suter, 1991), polycarbonate of bisphenol-A (PC) (Hutnik etal., 1991a). Figure 10-35 shows a typical fully dense configuration of glassy PC in the form of a single molecule of molecular weight of 4.53 kg/mol together
497
with its many images reflected back into, and filling a cube with periodic boundary conditions. The atomic radii in this figure have been reduced in size to permit viewing into the structure. The details of how these molecular structures are obtained from the best information on interatomic force fields are too extensive to be presented here. The interested reader must consult the above references for this detail. It should suffice here to state that these structures that have been obtained for the appropriate densities, have X-ray scattering factors, cohesive energy densities, and elastic properties that agree very well with corresponding experimental results. They all have rather high atomic level stresses that have root mean square values of the same order as the elastic moduli themselves, and are thus about a factor of 3 higher than the corresponding stresses in atomic glasses on a normalized basis. Below we will discuss simulations of molecular segment relaxations in PC that play a role in internal friction and large strain pure shear deformation in PP. 10.6.3.2 Segment Relaxations in the Polycarbonate of Bisphenol-A (PC)
Figure 10-35. The molecular structure of a typical configuration of glassy PC of bisphenol-A (Hutnik et al, 1991 a).
The so-called /J-relaxation, loss peak shown in Fig. 10-27 in PC has commonly been attributed to a local shear relaxation process involving, at least in part, the rotation of a phenyl ring, or alternatively involving a conformational rearrangement in another specific atomic group along the chain backbone such as perhaps the carbonate group. These possibilities were investigated by Hutnik etal. (1991b) in molecular structure models such as the one shown in Fig. 10-35, by imposing increments of rotation to a selection of phenyl rings, fixing the rotation angle in one or the other bonds on either side of the ring, fol-
498
10 Inelastic Deformation and Fracture of Glassy Solids
lowed by energy reminimization of the entire structure. This has resulted in either the actual rotation of the ring or, in some cases, in the conformational rearrangement in the adjoining carbonate group. For either case the derived potential contour along the reaction path for the imposed change exhibited clear energy maxima, bounding ranges of stable flexing of the local system. Figures 10-36 a and 10-36 b show the cumulative distribution and frequency distributions of the curvature a of the local energy well and the peak energy barriers A£ respectively for the phenyl ring rotations. Considering the ring in its potential well as a rotational simple harmonic oscillator, its frequency factor vG
>o i.o c 3
r
i
i
T
can be obtained as Vn
(10-28)
=
giving an overall rotational transition rate R (reciprocal average waiting time for ring rotation) /
A F \
(10-29)
where J is the effective rotational mass moment of inertia of the phenyl ring. Using appropriate values for the latter and the average value of a = 23 kcal/mol rad2, vG = 2.3 x 1012 Hz was found. The average value ~SE = 10.4 ±6.7 kcal/mol for the energy barrier for ring rotation obtained
0.15
I
a = 23.0 ± 10.1 kcal/mole rad2
cr
o 3 £
0.5
(b)
6 12 18 24 30 AE, Energy Barrier (kcal/mole)
36
0.0
Figure 10-36. Energetic characteristics of phenyl ring rotations in PC: (a) cumulative distribution and frequency distribution of curvatures of energy wells of rings, (b) distribution of peak energy barriers to ring rotations (Hutnik etal., 1991b).
10.6 Inelastic Response in Polymeric Glasses
from the simulation compares very well with the range of experimental values from 9.1 to 12.0kcal/mol measured in NMR experiments. The distribution of energies given in Fig. 10-36 b is extremely broad ranging up to nearly 30 kcal/mol in comparison. The intra-molecular energy barrier for a ring rotation in an isolated molecule in solution is, in comparison, only in the neighborhood of 3 kcal/mol. Thus, clearly the energy barrier to ring rotation is primarily of an intermolecular nature. That this is so has been verified by noting that the segmental displacements in the surroundings of a rotating ring are always far reaching. A complementary simulation of the conformational rearrangement of the carbonate group have given an equally broad distribution of energies with an average of M = 10.1 ±6.5 kcal/mol. These barrier energies for ring rotation and carbonate group rearrangements are considerably higher than the experimental value of about 7 kcal/mol. Clearly, when stimulated by external conditions of imposed cyclic strains the material will respond by selecting the lower energy portion of available processes. There is initial evidence from additional simulations of large strain plastic behavior that local shear relaxations in PC have indeed important components of recognizable ring rotations. Considering that the simulations with their imposed rotations, must be an upper bound, the agreement between them and anelastic damping experiments is encouraging. 10.6.3.3 Large Strain Plastic Shear of Polypropylene (PP)
A molecular structure model of glassy PP of a molecular weight of 2.968 kg/mol (= 76 monomer units) similar to that shown
499
in Fig. 10-35 for PC, and having a density and thermal properties appropriate for a temperature of 233 K (= Tg - 20 K) was subjected to increments of extension of 2 x 10 ~ 3 along one edge of the cube and — 2 x 10" 3 along one of the transverse directions, to achieve conditions of pure shear at constant volume (Mott et al., 1992). In the fully equilibrated structure, after each increment of deformation, the atomic site stress tensors and strain increment tensors were calculated for each atom. Figure 10-37 a shows the resulting deviatoric tensile stress-strain curve obtained from the volume average stress and strain increments for a configuration average of 9 such simulation experiments. Figure 10-37b shows the tensile deviatoric stress-strain curve of just one of these configurations. The latter curve which samples the behavior of only a very small volume element has a high initial atomic level "stress noise" which is much reduced in Fig. 10-37 a in the configuration average of 9 such responses. The configuration average behavior of Fig. 10-37 a shows clear yielding behavior at a strain of about 5%, a broad transition to fully developed flow at about 10% and no important hardening after this. It also shows that the system pressure increases monotonically, up to about 10% strain, due to the fact that the deformation of the glassy material is of a dilatant nature and was enforced at constant volume. While more "noisy", the response of the single configuration shown in Fig. 10-37 b shows important detail. It shows no clear elastic to plastic transition in the early stages but rather a protracted reversible anelastic behavior to about 10%. After this range of straining, which has apparently mechanically polarized all pockets of more compliant material, the behavior becomes a succession of reversible elastic loading steps, with the expected
500
10 Inelastic Deformation and Fracture of Glassy Solids 300 Change in System Pressure Von Mises Equivalent Tensile Stress
250 '_ 200 w w
150
CD
100
(a) 0.00
0.05
0.10
0.15
0.20
Work Equivalent Strain o 600
0.00
o "Forward" imposed extension \ "Backward" extension
0.05
0.10
0.15
Work Equivalent Strain
slope, followed by abrupt and discrete steps of large irreversible plastic relaxations occurring at constant overall strain. Figure 10-38 shows a "spike map" of deviatoric atomic strain increments for each of the 153 backbone carbon atom sites for each of the 100 strain increment steps. Clearly, outside the seven large plastic drops where intense strain activity is registered at nearly all atom sites, the strain increments during the other intervening 93 imposed strain steps are very small and at a background level. Intensive examination of molecular segment motions in the large plastic drops, by several techniques, including incremental stereo-imaging, revealed no readily recognizable recurring
0.20
Figure 10-37. Simulation of stress-strain relation in the plastic deformation of atactic PP at 253 K: (a) stress-strain curve for an average of 9 configurations; the lower curve shows the increase of system pressure under constant volume due to the dilatant nature of the deformation, (b) stressstrain curve for a single configuration (Mott et al., 1992).
segmental motions, but established that the relaxations are far reaching, over the entire simulation cell having a volume of 6.0 x 10" 2 7 m3. Simulations carried out on a volume of 4.7xlO~ 2 6 m 3 gave similar but far less noisy results, suggesting that the inextensional bonds and inflexible bond angles of the molecules can accommodate the transformation shear strains of the plastic events by complex cooperative rotations about bonds only over substantially large volume elements. To verify this, first the distribution of transformation shear strains yT was computed from the plastic stress drops and is shown in Fig. 10-39. The average value of this broad distribution is yT = 0.037 ±0.035. Then, the
10.6 Inelastic Response in Polymeric Glasses
calculated transformation shear strain distribution of the plastic stress drops of the simulation was compared with the experimentally determined activation volumes of Table 10-3. Recognizing that the measured activation volumes are products of the transformation shear strain yT and the actual volumes Qc of the regions undergoing the relaxation, estimates of these volumes were obtained for all the polymers in Table 10-3 on the assumption that the average values of yT are of similar magnitudes as those given in Fig. 10-39 for PP. These computed volumes Qc are given in the last column of Table 10-3, and indicate that the plastic relaxations indeed require cooperative segmental rearrangements over a large volume.
501
The pure shear simulation has also searched for the rate of development of orientational alignment of segments over the range of straining of up to 20% axial extension. No significant development of molecular orientation above the background "noise" level was found. From all of these it must be concluded that the early regions of plastic deformation in glassy polymers bear a remarkable similarity to deformation in atomic glasses. Both undergo deformation by localized, elastic strain-energystoring shear transformations which, however, are less localized in glassy polymers because of the constraints imposed by the chain molecule. Recognizable free energy storage by reduction in configurational entropy that must eventually occur to ac-
c "6
Q.00
o
Figure 10-38. The chronology of the development of plastic strain increments for each backbone carbon atom on the PP molecule for the stress-strain response of Fig. 10-37 b. The extensive "plastic" activity along the entire length of the molecule coincides with the sharp stress drops a to g shown in the figure (Mott et al., 1992).
502
10 Inelastic Deformation and Fracture of Glassy Solids
Total number of events = 35
0.02
0.04
0.06
0.08
0.10
Strain dissipated during event
Figure 10-39. Distribution of "transformation" shear strains calculated from the discrete stress drops in the 9 separate configurations entering into the stressstrain response of Fig. 10-37 a. The figure shows both the cumulative distribution and the frequency distribution (Mott et al., 1992).
count for the complete recoverability of shape change above Tg apparently does not set in until the later stages of deformation, all in very good agreement with the deformation calorimetric measurements and post deformation DSC measurements of Oleynik (1990). In a partial simulation of molecular motions in a glassy polymer undergoing tensile extension, Yannas and Luise (1983) have obtained estimates of the deformation resistance using interatomic pair potentials, based on an assumed uncoiling motion of molecules which they have labelled "Strophon" motion. The above simulations indicate that this view has considerable merit.
10.7 Fracture of Glasses 10.7.1 The Fracture Instability
The role of cracks and flaws in brittle fracture under monotonic loading has been well appreciated since the pioneering work of Griffith (1920, 1924). The modifications that are necessary to Griffith's theory to understand fracture in more or less
plastically deformable solids have been subjects of intense interest to ever-widening groups of researchers. The basic directions to these developments were given early by Orowan (1949) and by Irwin (1948), who in particular founded the branch of study of fracture mechanics that concerns itself with the precise statement of the condition of the fracture instability in structural components. Discussion of this very extensive subject will be outside the scope of this chapter. For a more detailed treatment of fracture, refer to Chapter 12 by Riedel in this Volume. We shall find it sufficient to note that, for relatively low levels, of inelastic deformation that is necessary to propagate a crack in a solid, the condition of the fracture instability can be stated alternatively: as a critical stress intensity factor KY\ a critical crack opening displacement 5CO defined at the root of the crack; or a critical energy release rate GY. These terms, defined for tension in relation to the macroscopic parameters under the control of the experimenter, are given for a plane strain setting as follows: X, = (7(7i;fl)1/2F(fl/iv)
(10-30) (10-31)
G, = - QU/da = [(1 - v)2/E] K?
(10-32)
In Eqs. (10-30) to (10-32), a is the applied tensile stress, a the half crack length, F(a/w) a function of the specimen width w(F-»l for a/w->0), Y the yield strength in tension in a nonstrain-hardening idealization, E Young's modulus, v Poisson's ratio, a a constant of order unity, and U the potential energy of the system of sample and its tractions. The functions F(a/w) have been calculated for a large number of shapes in which the crack length is of finite proportions with respect to the width w and are readily available in the literature (Paris and Sih, 1965; Tada et al., 1973;
10.7 Fracture of Glasses
Rooke and Cartwright, 1976). In glasses below Tg, where the inelastic deformations at the time of fracture are confined to the surroundings of the tip of the crack (smallscale yielding), the three alternative forcing functions given above are of equal utility. At temperatures very near Tg, where the glasses can become more compliant and tough so that the inelastic deformation zone spreads out over a large portion of the sample before the crack begins to propagate, different forcing functions based on nonlinear constitutive behavior become necessary. Since this falls outside our range of interest, however, we shall not expand on this topic further. The interested reader will find an elegant treatment of this subject in Hutchinson (1979). The subject of interest to us will be the mechanisms that govern the critical levels of these "forcing functions" for fracture in the different glasses. The process of fracture needs a crack that can be propagated across the specimen against the resistance of the material when the appropriate forcing function becomes large enough. It is an easy exercise to show (Argon, 1977) that the cracks that are necessary to bridge the gap between the technological strength levels and the cohesive strength cannot form by thermal motion under stress but must result from other processes that differ in complexity and importance between metallic glasses, oxide glasses and glassy polymers. Particularly in thermoplastic glassy polymers the process of crack formation requires crazing, which can in many instances provide significant dilatational strains before turning into unstable cracks. The various processes that lead to the formation of supercritical cracks in glasses to produce fracture under stress have been discussed by Argon (1980). This topic and some related topics are also treated in Vol. 9, Chap. 13.
503
10.7.2 Fracture in Space Network Glasses Griffith (1920) in his classical work on fracture was first to demonstrate that cracks propagate in oxide glasses when the rate of release of elastic energy equals the rate of production of the energy of fresh surfaces or, as Orowan (1934) pointed out somewhat later, when the concentrated stress at the tip of the often atomically sharp crack reaches the ideal cohesive strength aic of the glass. Since oxide glasses are potentially brittle solids according to the basic classification of Kelly et al. (1967), and Rice and Thomson (1974), this propagation is not accompanied by any significant amount of plastic deformation. Hence, the critical stress intensity factor Klc becomes If n F i/^l/2 i v j c — yZ LL X)
=
^- (U ,-.\l/2 ®\z \P *V
(\ A 'X'W \1\J-JD)
where E is Young's modulus, % the surface free energy, and b the interatomic distance. In most inorganic glasses, the surface energy is of order 0.5-1.5J/m 2 (Griffith, 1920) and Young's modulus of order 70GPa. This makes the critical stress intensity factor Klc of order 0.3-1.0 MPam 1 / 2 , which is very close to the value of 0.28MPam 1/2 measured by Griffith (1920) on precracked tubes and spherical bulbs of a conventional soda glass of 0.692 SiO 2 ,0.12 K 2 0,0.009 Na 2 0,0.118 A12O3, 0.045 CaO, and 0.009 MnO. Most experiments since the time of Griffith have confirmed this picture. 10.7.3 Fracture in Metallic Glasses As we have already shown in Fig. 10-17, fracture occurs in metallic glasses by an intrinsic cavitation process involving the meniscus instability. Deep surface offsets at shear bands act as the initiating sites from which cracks propagate inward, usually, but not necessarily always, along the shear
504
10 Inelastic Deformation and Fracture of Glassy Solids
bands where the deformation-induced excess free volume has lowered the plastic resistance. In this fracture process, the basic mechanism of separation is ductile rupture along the steady-state ridges between the finger-shaped protrusions at the convoluted crack tip penetrating almost monolithically into the region ahead of the crack tip - in a manner shown in Fig. 1040, for the classical case of separation in a simple fluid. Figure 10-41 a shows how this happens in Pd80Si20> a s described above. The tell tale ridges of the meniscus fracture are shown in Fig. 10-41 b. The development of Argon and Salama (1976) for the convoluted meniscus interface of a nonlinear fluid permits the determination of the fracture toughness Klc as 1/2
(10-34)
where a ^2.7 is a numerical constant giving the ratio of the critical crack opening displacements to the product of the tensile yield strain and the critical plastic zone size. In metallic glasses at low temperatures where the strain rate sensitivity of the flow stress is nil, B(n)«1.2. In addition most metallic glasses have Young's moduli of order 140 GPa, tensile yield stresses of order 2.5 GPa, and surface energies of order 2J/m 2 , which gives for the fracture toughness Klc about 10 MPam 1 / 2 . Davis (1976) has performed a number of plane strain fracture toughness experiments on samples of metallic glass precracked by fatigue crack propagation. He has found these fracture toughnesses to range from a low of 9.5 M P a m 1 ^ for a glass of Ni 49 Fe 29 P 14 B 6 Si 2 to a high of 12.65 MPam 1 / 2 for the strongest glass of Fe 8 0 B 2 0 . These values are in remarkably good agreement with the prediction of the meniscus convolution model of crack propagation. Embrittlement as a consequence of heat-treatment of metallic glasses is treated in Vol. 9, Chap. 9, Sec. 9.4.3. 10.7.4 Fracture in Thermoplastic Glassy Polymers
In thermoplastic glassy polymers, when crazes are transformed into supercritical cracks, or when other inclusions or largescale surface irregularities act as supercritical cracks, catastrophic fracture follows. In their growth, such cracks will be blunted by inelastic deformation at the crack tip that can be a mixture of plastic flow and additional crazing occurring in a zone having the dimensions Rp given by the smallscale yielding theory as Figure 10-40. Instability in an advancing fluid meniscus between two glass plates producing characteristic surface convolutions: (a) to (c) are stages in the development of the instability (Taylor, 1950).
R=a{o/Y)2
(10-35)
where the symbols have their previously defined meaning. Crack propagation oc-
10.7 Fracture of Glasses
505
TENSION AXIS
(a)
10// Figure 10-41. The development of fracture by the propagation of the convoluted meniscus at the tip of a crack: (a) sketch showing the location of the fracture along a concentrated shear band, (b) fracture surface showing the characteristic rupture ridges (Megusar et al, 1979).
curs when craze matter fracture begins, starting from particulate inclusions entrapped in the craze, as shown in Fig. 10-7 or when a series of ruptures in adjoining crazes are bridged by some plastic flow and tearing. The details of the breakdown of
craze matter under stress from extrinsic or intrinsic imperfections have been discussed by Kramer (1983), and Argon and Cohen (1990). Thus in most cases, the fracture instability occurs when the KY stress intensity reaches a critical value K,n.
506
10 Inelastic Deformation and Fracture of Glassy Solids
most prominent of these have been NSF through Grants: DMR-77-22753; DMR85-17224; DMR-84-18718; DMR-87-19217; DARPA/ONR under contract N00014-86K-0768 for general support, and the Allied Signal Corporation, through a continuing doctoral fellowship. The author is also grateful for many stimulating discussions with numerous colleagues whose researches are acknowledged in the references.
IOXIO
10.9 References
I2XIO
Figure 10-42. Dependence of fracture stress on initial crack length for atactic PS. The slope of the line gives the fracture toughness Klc (Williams, 1977).
Williams (1977) has examined the plane strain fracture condition of a number of glassy polymers and has found them to be governed by a critical stress intensity factor criterion. The case for PS in Fig. 10-42 is typical. Evaluation of the slopes of the lines such as that in Fig. 10-42 at different temperatures has given that the critical stress intensity factor of PS is relatively temperature independent. For a more extensive discussion of the fracture mechanics of polymers the reader is referred to Williams (1984).
10.8 Acknowledgements The author's research on amorphous media has been supported by many agencies throughout more than a decade. The
Argon, A. S. (1968), J. Appl. Phys. 39, 4080. Argon, A. S. (1973 a), in: The Inhomogeneity of Plastic Deformation. Metals Park: ASM, p. 161. Argon, A. S. (1973 b), Phil. Mag. 28, 839. Argon, A. S. (1977), in: Surface Effects in Crystal Plasticity: Latanision, R. M., Fourie, J. F. (Eds.). Leyden: Noordhoff, p. 383. Argon, A. S. (1979), Ada Metall 27, 47. Argon, A. S. (1980), in: Glass: Science and Technology, Vol. 5: Uhlmann, D. R., Kreidl, N. J. (Eds.). New York: Academic Press, p. 79. Argon, A. S. (1981), in: Dislocation Modeling of Physical Systems: Ashby, M. R, Bullough, R., Hartley, C. S., Hirth, J. P. (Eds.). Oxford: Pergamon Press, p. 383. Argon, A. S. (1982), J. Phys. Chem. Solids 43, 945. Argon, A. S. (1985), in: Rapidly Solidified Metals, Vol. 2: Steeb, S., Warlimont, H. (Eds.). Amsterdam: North Holland, p. 1325. Argon, A. S. (1986), in: Strength of Metals and Alloys, Vol. 3: McQueen, H. J., Bailon, J.-P., Dickson, XL, Jonas, X I , Akben, M. G. (Eds.). Oxford: Pergamon Press, p. 2007. Argon, A. S., Bessonov, M. I. (1977a), Phil Mag. 35, 917. Argon, A. S., Bessonov, M. I. (1977b), Polym. Eng. Sci. 17, 114. Argon, A. S., Cohen, R. E. (1990), in: Advances in Polymer Science, Vol. 91/92: Kausch, H. H. (Ed.). Berlin: Springer, p. 301. Argon, A. S., Hannoosh, X G. (1977), Phil. Mag. 36, 1195. Argon, A. S., Kuo, H.-Y. (1979), Mater. Sci. Eng. 39, 110. Argon, A. S., Kuo, H.-Y. (1980), J. Non-Cryst. Solids 37, 241. Argon, A. S., Salama, M. M. (1976), Mater. Sci. Eng. 23, 219. Argon, A. S., Salama, M. M. (1977), Phil. Mag. 36, 1217.
10.9 References
Argon, A. S., Shi, L.-T. (1982), Phil Mag. A 46, 275. Argon, A. S., Shi, L.-T. (1983), Acta Metall. 31, 499. Argon, A. S., Megusar, I , Grant, N. I (1985), Scripta Metall 19, 591. Bernal, X D. (1964), Proc. Roy. Soc. A 280, 299. Berry, B. S. (1978), in: Metallic Glasses. Metals Park: ASM, p. 161. Born, M., Huang, K. (1954), Dynamical Theory of Crystal Lattices. Oxford: Clarendon Press. Boyce, M. C , Parks, D. M., Argon, A. S. (1988 a), Mech. Mater. 7, 15. Boyce, M. C , Parks, D. M., Argon, A. S. (1988 b), Mech. Mater. 7, 35. Boyce, M. C , Parks, D. M., Argon, A. S. (1989), Intern. J. Plast. 5, 593. Cohen, M. H., Grest, G. S. (1979), Phys. Rev. B20, 1077. Davis, L. A. (1976), in: Rapidly Quenched Metals: Grant, N. X, Giessen, B. C. (Eds.). Cambridge: M.I.T. Press, p. 369. Davis, L. A. (1978), in: Metallic Glasses. Metals Park: ASM, p. 190. Deng, D., Argon, A. S. (1986a), Acta Metall. 34, 2011. Deng, D., Argon, A. S. (1986b), Acta Metall. 34, 2025. Deng, D., Argon, A. S., Yip, S. (1989 a), Phil Trans. Roy. Soc. A 329, 549. Deng, D., Argon, A. S., Yip, S. (1989 b), Phil. Trans. Roy. Soc. A 329, 575. Deng, D., Argon, A. S., Yip, S. (1989c), Phil. Trans. Roy. Soc. A 329, 595. Deng, D., Argon, A. S., Yip, S. (1989d), Phil. Trans. Roy. Soc. A329, 613. Doyle, M. J., Maranci, A., Orowan, E., Stork, S. T. (1972), Proc. Roy. Soc. A 329, 137. Egami, T., Vitek, V. (1983), in: Amorphous Materials: Modeling of Structure and Properties: Vitek, V. (Ed.). New York: AIME, p. 127. Eshelby, J. D. (1957), Proc. Roy. Soc. A 241, 376. Finney, J. L. (1970 a), Proc. Roy. Soc. A 319, 479. Finney, J. L. (1970b), Proc. Roy. Soc. A319, 495. Frost, H. J. (1982), Acta Metall 30, 889. Gaskell, P. H. (1983), in: Topics in Applied Physics: Glassy Metals II, Vol. 53: Beck, H., Giintherodt, H.-J. (Eds.). Berlin: Springer, p. 5. Gerling, R., Schimansky, F. P., Wagner, R. (1988), Acta Metall. 36, 575. Gerling, R., Schimansky, F. P., Wagner, R. (1989), Acta Metall 37,2961. Gilman, J. J. (1968), in: Dislocation Dynamics: Rosenfield, A. R., Hahn, G. T., Bement, A. L., Jaffee, R. I. (Eds.). New York: McGraw-Hill, p. 3. Griffith, A. A. (1920), Phil. Trans. Roy. Soc. A 221, 163. Griffith, A. A. (1924), Proc. Intern. Congr. Appl. Mech. 1st. Delft: p. 55. Haward, R. N., Thackray, G. (1968), Proc. Roy. Soc. 302, 453. Hope, P. S., Ward, I. M., Gibson, A. G. (1980), /. Mater. Sci. 15, 2207.
507
Hutchinson, J. W. (1979), Nonlinear Fracture Mechanics. Lyngby, Denmark: Technical U. Denmark. Hutnik, M., Gentile, F. T, Ludovice, P. I, Suter, U. W, Argon, A. S. (1991a), Macromolecules 24, 5962. Hutnik, M., Argon, A. S., Suter, U. W. (1991b), Macromolecules 24, 5970. Irwin, G. R. (1948), in: Fracturing of Metals. Metals Park: ASM, p. 147. Jones, G. O. (1948), Reports Prog. Phys. 9, 136. Kausch, H. H. (Ed.) (1983), Advances in Polymer Science, Vol. 52/53. Berlin: Springer. Kausch, H. H. (Ed.) (1990), Advances in Polymer Science, Vol. 91/92. Berlin: Springer. Kelly, A., Tyson, W. R., Cottrell, A. H. (1967), Phil. Mag. 15, 567. Kocks, U. F. (1966), Phil. Mag. 13, 541. Kocks, U. E, Argon, A. S., Ashby, M. F. (1975), in: Prog. Mater. Sci.: Chalmers, B., Christian, J. W, Massalski, T. B. (Eds.). Oxford: Pergamon Press, Vol. 19. Kramer, E. J. (1983), in: Advances in Polymer Science, Vol. 52/53: Kausch, H. H. (Ed.). Berlin: Springer, p.l. Ludovice, P. X, Suter, U. W (1991), Macromolecules, in press. Maeda, K., Takeuchi, S. (1982), /. Phys. F12, 2767. Marsh, D. M. (1964a), Proc. Roy. Soc. A279, 420. Marsh, D. M. (1964b), Proc. Roy. Soc. A282, 33. McCrum, N. G., Read, B. E., Williams, G. (1967), Anelastic and Dielectric Effects in Polymeric Solids. New York: Wiley. Megusar, X, Argon, A. S., Grant, N. X (1979), Mater. Sci. Eng. 38, 63. Megusar, X, Argon, A.S., Grant, N. X (1982), in: Rapidly Solidified Amorphous and Crystalline Alloys: Kear, B. H., Giessen, B.C., Cohen, M. (Eds.). Amsterdam: Elsevier, p. 283. Morito, N., Egami, T. (1984), Acta Metall 32, 603. Mott, P., Argon, A. S., Suter, U. W (1992), Phil Mag., submitted. Oleynik, E. F. (1990), in: High Performance Polymers: Baer, E., Moet, S. (Eds.). Munich: Hauser, p. 79. Orowan, E. (1934), Z. Kristallog. 89, 327. Orowan, E. (1949), Reports Prog. Phys. 12, 185. Pampillo, C. A., Chen, H. S. (1974), Mater. Sci. Eng. 13, 181. Paris, P. C , Sih, G. C. (1965), in: Fracture Toughness Testing and Its Applications. Philadelphia: ASTM, STP-381, p. 30. Piorkowska, E., Argon, A. S., Cohen, R. E. (1990), Macromolecules 23, 3838. Rice, X R., Thomson, R. (1974), Phil Mag. 29, 73. Robertson, R. E. (1966), /. Chem. Phys. 44, 3950. Robertson, R. E. (1968), Appl. Polym. Symp. 7, 201. Rooke, D. P., Cartwright, D. X (1976), Compendium of Stress Intensity Factors. London: H.M. Stationery Office. Rudnev, S. N., Salamatina, O. B., Voenniy, V V, Oleynik, E. F. (1990), Colloid and Polym. Sci. 268, 1.
508
10 Inelastic Deformation and Fracture of Glassy Solids
Spaepen, F. (1977), Acta Metall 25, 407. Srolovitz, D., Vitek, V., Egami, T. (1983), Acta Metall 31, 335. Suresh, S. (1991), Fatigue of Materials. Cambridge: Harvard University Press. Tada, H., Paris, P. C , Irwin, G. R. (1973), The Stress Analysis of Cracks Handbook. Hellertown: Del Research Corp. Taub, A. I. (1980), Acta Metall. 28, 633. Taub, A. L, Spaepen, F. (1979), Scripta Metall. 13, 195. Taub, A. I., Spaepen, F. (1980), Acta Metall. 28,1781. Taylor, G. I. (1950), Proc. Roy. Soc. 201, 192. Theodorou, D. N., Suter, U. W. (1985), Macromolecules 18, 1467. Theodorou, D. N., Suter, U. W. (1986a), Macromolecules 19, 139. Theodorou, D. N., Suter, U. W. (1986b), Macromolecules 19, 379. Vitek, V. (Ed.) (1983), Amorphous Materials: Modeling of Structure and Properties. New York, AIME. Volynskii, A. L., Bakeev, N. P. (1984), Highly Dispersed Oriented State of Polymers (in Russian). Moscow: Chimia Publ. Williams, J. G. (1977), Polym. Eng. Sci. 17, 144. Williams, J. G. (1984), Fracture Mechanics of Polymers. New York: Halsted-Wiley Press. Williams, M. L., Landel, R. F , Ferry, J. D. (1955), /. Amer. Chem. Soc. 77, 3701. Yamini, S., Young, R. X (1980), /. Mater. Sci. 15, 1814.
Yannas, I. V., Luise, R. R. (1983), in: The Strength and Stiffness of Polymers: Zachariades, A. E., Porter, R. S. (Eds.). New York: Marcel Dekker, p. 255. Ziman, J. M. (1979), Models of Disorder. Cambridge: Harvard University Press.
General Reading Argon, A. S. (1981), in: Glass: Science and Technology, Vol. 5: Uhlmann, D. R., Kreidl, N. J. (Eds.). New York: Academic Press, pp. 79-132. Davis, L. A., Hasegawa, R. (1981), in: Metallurgical Treatises: Tien, J. K., Elliot, X F (Eds.). New York: AIME, pp. 301-320. Deng, D., Argon, A. S., Yip, S. (1989), Phil. Trans. Roy. Soc. 329, pp. 549-640. Egami, T, Vitek, V. (1983), in: Amorphous Materials: Modeling of Structure and Properties: Vitek, V. (Ed.). New York: AIME, pp. 127-145. Kramer, E. J. (1983), in: Crazing in Polymers: Advances in Polymer Science, Vol. 52/53: Kausch, H. H. (Ed.). Berlin: Springer, pp. 1-56. Spaepen, F , Taub, A. I. (1983), in: Amorphous Metallic Alloys: Luborsky, F E. (Ed.). London: Butterworths, pp. 231-256. Williams, X G. (1984), Fracture Mechanics of Polymers. New York: Halsted-Wiley.
11 Cyclic Deformation and Fatigue Subra Suresh Division of Engineering, Brown University, Providence, RI, U.S.A.
List of Symbols and Abbreviations 11.1 Introduction 11.2 Mechanisms and Mechanics of Cyclic Deformation 11.2.1 Cyclic Hardening in Ductile Single Crystals 11.2.2 Cyclic Stress-Strain Curves for F.C.C. Single Crystals 11.2.3 Some Characteristic Features of Cyclic Deformation in Single Crystals . 11.2.4 Cyclic Deformation of Polycrystalline Metals and Alloys 11.2.5 The Bauschinger Effect 11.2.6 Continuum Aspects of Cyclic Deformation 11.3 Mechanisms of Fatigue Crack Initiation 11.4 Characterization of Fatigue Life Based on Cyclic Stress and Cyclic Strain 11.5 Fatigue Design Philosophies 11.6 Fracture Mechanics Approach to Fatigue 11.6.1 The Stress Intensity Factor 11.6.2 X-Dominance 11.6.3 Fatigue Crack Growth Characterization 11.6.4 Plastic Zones in Tension and Fatigue 11.6.5 Nonlinear Fracture Mechanics 11.7 Characteristics of Fatigue Crack Growth 11.7.1 Different Regimes of Fatigue Crack Growth 11.7.2 Microscopic Stages of Fatigue Crack Growth 11.7.3 Micromechanical Considerations 11.8 Fatigue Crack Closure and Other Retardation Mechanisms 11.8.1 Fatigue Crack Closure 11.8.1.1 Plasticity-Induced Crack Closure 11.8.1.2 Oxide-Induced Crack Closure 11.8.1.3 Roughness-Induced Crack Closure 11.8.1.4 Other Closure Mechanisms 11.8.2 Other Retardation Mechanisms 11.9 The Growth of Short Fatigue Cracks 11.10 Crack Growth Retardation During Variable Amplitude Fatigue 11.11 Fatigue at Notches 11.12 Compression Fatigue of Metals and Nonmetals 11.13 Fatigue of Ceramic Materials 11.14 Cyclic Deformation and Fatigue of Polymers 11.15 Concluding Remarks 11.16 Acknowledgements 11.17 References Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
510 513 514 514 515 518 518 520 522 523 525 527 528 528 529 530 530 532 534 534 534 537 538 538 538 540 542 543 544 545 548 552 552 553 556 559 561 561
510
11 Cyclic Deformation and Fatigue
List of Symbols and Abbreviations a a* a0 Af Am b b c C dtrap da/dN da/dt D E G & hT H J K Kl9KU9Km Klc Kc Keff Kmax, Kmin Kf AK0 AX th m n N AT* Nf nm nf P q r rc rp R S T
distance of crack growth retardation crack length critical crack size for transition from short crack to long crack cyclic strength coefficient monotonic strength coefficient Burgers vector fatigue strength exponent fatigue ductility exponent material constant in the Paris equation trapping distance of edge dislocations crack increment per fatigue stress cycle crack velocity deflected segment length of the periodically deflected crack Young's modulus a function of the stress components atj and atj strain energy release rate height of the transformation zone scalar function of X characterizing expansion of yield surface integral of Rice stress intensity factor stress intensity factors in modes I, II, and III, respectively plane strain fracture toughness in mode I fracture toughness effective value of K maximum and minimum values of K, respectively fatigue notch factor nominal value of threshold stress intensity factor range intrinsic threshold stress intensity factor range material constant in the Paris equation inverse of nm n u m b e r of fatigue cycles n u m b e r of cycles of transient crack g r o w t h n u m b e r of cycles t o failure monotonic strain hardening exponent cyclic strain hardening exponent load notch sensitivity index radial distance cyclic plastic zone size monotonic plastic zone size load ratio ( = KmJKmax) straight segment length of the periodically deflected crack traction vector in the Rice integral
List of Symbols and Abbreviations
511
T u Feff Y
nonsingular stress displacement vector in the Rice integral volume of the transformation zone geometrical correction factor
octj ypl <5 CZ 5fj 8 8e £f £p £T Ae b k ju v vc X a
stress components characterizing translation of yield surface resolved plastic shear strain maximum opening of the craze at crack tip Kronecker delta strain elastic strain fatigue ductility exponent plastic strain transformation strain Bauschinger strain scalar quantity related to plastic strain shear modulus Poisson's ratio cyclic frequency mismatch ratio stress stress amplitude back stress stress at which crazing occurs endurance limit fatigue strength coefficient net yield stress during forward deformation contribution to overall strength from intersection of mobile dislocations with forest dislocations threshold stress amplitude universal functions of the polar angle 6 net yield stress during reversed deformation ultimate tensile strength monotonic yield strength cyclic yield strength contribution t o overall strength from solid-solution hardening Bauschinger stress polar angle resolved shear stress at saturation peak value of TR in the plateau regime of cyclic stress-strain curve for f.c.c. single crystals
ath Stj <jR
c.s.s. b.c.c. CTOD
cyclic stress-strain body-centered cubic crack-tip opening displacement
512
11 Cyclic Deformation and Fatigue
DGB f.c.c. h.c.p. HCF LCF PMMA PSB PSZ PTFE PVC S-N curve TRIP
discontinuous growth band face-centered cubic hexagonal close-packed high-cycle fatigue low-cycle fatigue polymethylmethacrylate persistent slip band partially stabilized zirconia polytetrafluoroethylene polyvinylchloride stress-life curve, Wohler curve transformation-induced plasticity
11.1 Introduction
11.1 Introduction The susceptibility of engineering materials to cyclic deformation and to failure by the nucleation and growth of flaws under fluctuating loads is a topic of much importance in many structural applications. The phenomenon by which changes are induced in the properties of materials due to the repeated application of stresses or strains is commonly referred to by the term fatigue, especially when these changes lead to cracking or fracture. Research on the fatigue of engineering materials dates back approximately 150 years. Although intimately linked to the development of design methodologies and engineering approaches in industrial practice, the study of fatigue of materials has also become a major discipline for scientific research in its own right. With the growing interest in advanced nonmetals and composites, topics of fatigue research also continue to expand at a rapid rate. There are different processes by which failure of materials is instigated under cyclically varying loads. Each of these fatigue processes can be further subdivided into different stages in which cyclic damage and crack formation progressively occur. When the origin of material degradation is attributable solely to externally imposed pulsating loads, the ensuing failure process is referred to as mechanical fatigue. When small vibratory relative displacement and rolling contact between surfaces takes place in conjunction with alternating stresses, the attendant failure processes are termed fretting fatigue and rolling contact fatigue, respectively. Fluctuating loads applied to a material along with aggressive environments and elevated temperatures cause corrosion fatigue and creep fatigue, respectively.
513
In a nominally defect-free, high purity material subjected to alternating loads, the progression of fatigue damage generally occurs in the following stages: (i) Creation of irreversible changes in the microstructural and substructural constitution of the material which are accompanied by cyclic hardening or softening, (ii) The nucleation of microscopic defects, (iii) The formation of a dominant macroscopic flaw by the growth and coalescence of the microscopic defects, (iv) The subcritical growth of the macrocrack. (v) Final failure of the material or instability. From the standpoint of fatigue life prediction in engineering design, clearly identifying the initiation stage of a fatigue crack is a very challenging task. This task is compounded by the fact that even small changes in the mechanical loading, and in the microstructural and environmental conditions to which a fatigued material is exposed can have a strong bearing on the rate of inception and growth of fatigue cracks. Therefore, it is not surprising that the manner in which the crack initiation and crack growth stages of fatigue are quantitatively accounted for mostly determines the fundamental differences in the fatigue design philosophies currently employed in industry. This article seeks to provide an overview of the micromechanisms and mechanics of cyclic deformation, fatigue crack initiation and fatigue crack growth in materials. Because of space restrictions, it is not feasible to provide full details of all the issues discussed in this chapter and to cite all the sources from which an understanding of the subject has emerged. However, a sufficiently broad perspective on major issues in this subject area is provided, and a comprehensive list of references is included so that the interested reader can easily gain access to the details of the topics and original sources. An attempt is made to present
514
11 Cyclic Deformation and Fatigue
a summary of various approaches to fatigue along with a discussion of the salient features and limitations of each approach. Although the vast majority of available results on fatigue pertains to ductile metals and alloys, this review also presents an extension of fatigue concepts developed for metals to nonmetallic brittle solids, such as ceramics, and to partly crystalline and noncrystalline solids, such as polymers.
11.2 Mechanisms and Mechanics of Cyclic Deformation Experimental investigations of the micromechanistic origins of fatigue can be traced back to the work of Ewing and Humfrey (1903). These workers examined fatigue failure along slip planes in Swedish iron. Since that time, advances in electron microscopy have greatly contributed to our current understanding of cyclic deformation mechanisms in materials. This section focuses attention on the phenomena of cyclic deformation from the viewpoints of microstructural and substructural changes as well as of continuum formulations. 11.2.1 Cyclic Hardening in Ductile Single Crystals The most conclusive concepts of cyclic deformation have evolved from research work on high purity materials, in particular single crystals of face-centered cubic (f.c.c.) metals. A common method of performing cyclic load tests on ductile single crystals involves fixed amplitudes of resolved plastic shear strain. The application of fully reversed cyclic plastic strains to a well-annealed f.c.c. single crystal causes rapid hardening even in the initial few cycles. On a typical stress-strain plot of the variation of the resolved shear stress
against the resolved shear strain, this hardening behavior is manifested as a precipitous increase in flow stress (for a fixed amplitude of resolved plastic shear strain) with increasing number of fatigue cycles. The rate of rapid hardening progressively diminishes with continued cycling and a state of 'saturation' is reached. Beyond the onset of saturation, the stress-strain hysteresis loop reaches a stable configuration and remains unaltered with further fatigue cycling. Transmission electron microscopy of fatigued copper single crystals (e.g., Basinski et al., 1969; Hancock and Grosskreutz, 1969) has shown that the rapid hardening in the initial stages of cyclic loading is due to the creation of dislocations which accumulate on the primary slip plane. With an increase in the number of fatigue cycles, edge dislocations of opposite signs on closely-spaced, parallel slip planes trap each other forming a network of dipoles known as bundles, veins or loop patches. Fully reversed (equal tension-compression) fatigue loading does not cause any rotation of the slip system with respect to the loading axis. Since plastic deformation generates approximately equal numbers of positive and negative edge dislocations, the veins also do not produce long-range internal stresses at low amplitudes of cyclic strains. Positive and negative screw dislocations are not accumulated to such an extent because of the possibility of cross slip. Most screw dislocations annihilate by stress-induced cross slip (i.e., that occurring between closely-spaced screw dislocations of opposite sign). The vein structure has a dislocation density of approximately 10 1 5 m~ 2 and a mean spacing of 30 nm for Cu at room temperature. The veins have an elongated shape, with the long axis being parallel to the primary dislocation lines and the cross
11.2 Mechanisms and Mechanics of Cyclic Deformation
section (about 1.5 \im wide in Cu at room temperature) perpendicular to the long axis being equi-axed. The veins are separated by channels where the dislocation density is some three orders of magnitude smaller than that within the veins. The channels also contain screw dislocations which move back and forth under the influence of cyclic strains. The impediment to dislocation motion on the primary slip system due to the increase in the network of veins (which become up to 50% by volume with the progression of cyclic deformation) contributes to rapid hardening. Figure 11-1 shows an example of matrix vein structures in a Cu monocrystal cycled to saturation at 77.4 K. The Burgers vector is along [101] and the plane of the figure is the primary slip plane. The basic characteristic length corresponding to the dislocation vein structures is the average spacing of the edge dislocations in the veins. This dimension can be related to the trapping distance, dtrap, of edge dislocations: <
fib
(11-1)
8TT(1-VK
where \i is the shear modulus, b is the magnitude of the Burgers vector, v is the Poisson's ratio and TR is the resolved shear stress at saturation. Taking typical values for Cu at room temperature, i.e., /nb = 11J -m~ 2 , v = 0.33 and TR = 28 MPa, it is seen that dtrap < 24 nm (Neumann, 1983). This estimate is in fair proximity to the value of 30 nm, which is estimated from the measurements of dislocation densities. Similarly, it can be shown that the channel width (^1.5 jim) is of the order of the diameter of the dislocation loops generated by the Orowan-FrankRead mechanism under the influence of the shear stress T R .
515
Figure 11-1. An example of the dislocation loop patches in monocrystalline Cu fatigued at 77.4 K. The primary slip plane is the plane of the figure and the Burgers vector is along [101] (from Basinski et al., 1980).
11.2.2 Cyclic Stress-Strain Curves for F.C.C. Single Crystals
As noted in the preceding section, the application of fatigue loads causes a rapid increase in flow stress (under a fixed amplitude of plastic strain) which is followed by saturation leading to stable stress-strain hysteresis loops. Different values of imposed plastic shear strain amplitude lead to different saturation hysteresis loops. Figure 11-2 a shows a plot of the resolved shear stress at saturation TR against the resolved plastic shear strain, ypl. If one connects the tips of stable hysteresis loops, as shown by the dashed curve in Fig. 112 (a), the cyclic stress-strain curve for a single crystal is obtained. The cyclic stressstrain curve (see Fig. 11-2 b) exhibits three distinct regimes (Mughrabi, 1978, 1980): (i) Regime A in which the resolved shear stress at saturation TR increases rapidly with increasing values of resolved plastic shear strain amplitude y pl . (ii) A plateau region, termed regime B, in which TR is insensitive to variations in ypl. This con-
516
11 Cyclic Deformation and Fatigue
(a)
(b)
T,
\J
I
y
I
-r
L
s
/I ' I
A
8
I I 1
i p\,AB
Figure 11-2. (a) A plot of the resolved shear stress at saturation against the resolved plastic shear strain showing stabilized hysteresis loops. The cyclic stressstrain curve at saturation for the ductile single crystal is the dashed line connecting the corners of the loops, (b) Different regimes of the cyclic stress-strain curve (after Mughrabi, 1978).
stant value of TR = zf extends from ?Pi = Vpi, AB t 0 7Pi = 7pi, BC- (iii) Regime C in which the saturation stress-strain curve exhibits a hardening behavior similar to regime A. Table 11-1 lists some typical stress-strain characteristics of f.c.c. single crystals of different compositions subjected to cyclic deformation over a wide range of test temperature. As noted in the previous section, the accumulation of primary dislocations is mainly responsible for the rapid hardening seen in regime A of the saturation stressstrain curve. In this regime of plastic strain amplitude, i.e., for yv\
Table 11-1. Effects of test temperature and composition on the (saturation) cyclic stress-strain characteristics of f.c.c. crystals. Material
yPi,A,B
Reference
y P i.iic
(MPa) Cu (4.2 K) Cu (77.4 K) Cu (295 K) Cu (523 K) Cu-2at.%Al (295 K) Cu-5at.%Al (295 K) C u - 2 a t . % C o (295 K) Ni (295 K) Ag (295 K) Al-1.6at.%Cu (295 K)
_ -
6.0xl0~5 l.OxKT4 l.OxlO"4 -
3.0xl0~ 4
l.oxicr4
6.0-xHT5 1.5 xl(T 5
_
8.0xl0~ 3 7.5 x l O " 3 l.OxHT 3 3.0xl0~ 3 5.0xl0~ 3 7.5 xlO~ 3 7.5 x l O " 3 1.5xlO~ 3
73.0 48.0 27.5 14.0 33.0 32.0 27.5 52.0 17.5 95.0
Basinski et al. (1980) Basinski et al. (1980) Mughrabi (1978) Lisiecki and Weertman (1990) Wilhelm and Everwin (1980) Woods (1973) Wilhelm and Everwin (1980) Mughrabi (1978) Mughrabi et al. (1979) Lee and Laird (1983)
11.2 Mechanisms and Mechanics of Cyclic Deformation
where they originally appeared. Many studies have shown that the persistent slip bands (PSBs) form through the thickness of the fatigue specimen in f.c.c. single crystals; the slip markings seen on the surface merely correspond to the locations where the PSBs terminate at the surface. A PSB in Cu (fatigued at room temperature) is made up of about 5000 slip planes. The PSB, oriented along the primary Burgers vector 6, is composed of parallel arrays of dislocation ladders or walls. For Cu, oriented for single slip and fatigued at room temperature, the dislocation ladders are about 0.1 |im in thickness and tens of micrometers in depth, and they have a mean spacing of about 1.3 jim. The walls are divided by channels. The dislocation density in the PSB walls is of the order of l o i 5 _ l o i 6 m - 2 a n d i s a b o u t 2__3 orders of magnitude smaller in the PSB channels. Figure 11-3 a is a transmission electron micrograph of matrix vein structures (marked 'M') and PSB wall structures in a Cu single crystal fatigued at room temperature. The volume fraction of PSBs in the crystal increases linearly from zero at 7pi ~ yPi, AB t o n e a r l y u n i t y a t vPi ~ W e Since the PSBs are much softer than the surrounding matrix material, plastic deformation in regime B is almost fully accommodated by the PSBs. It is also known (Mughrabi et al, 1979) that, for Cu, Ni and Ag at room temperature, the ratio of the threshold saturation stress for PSB formation rf to the shear modulus \i is approximately the same (^6.5 x 10" 4 ). While the slip steps on the surface of the crystal form in proportion to ypl, the displacements within the persistent slip bands are not fully reversed during fatigue loading. This kinematic irreversibility of cyclic slip, whose origins are discussed in later sections, is a principal mechanism for the nucleation of fatigue cracks.
517
Although there exists a vast amount of experimental information on the dislocation structure and deformation within the PSBs, the mechanisms of the very formation of these slip bands have not been fully understood. Some progress, however, has been made recently on theoretical grounds where the formation of dipolar walls within PSBs has been demonstrated by recourse to the stability considerations of finite sections of dislocation arrays. Neumann (1983, 1986) has shown that finite sections of Taylor-Nabarro lattices (madeup of diamond-shaped planar arrays of positive and negative edge dislocations) can be transformed into the dipolar wall structure of PSBs under certain loading conditions. Furthermore, Dickson et al. (1986) have shown that the stacking of twins of dislocation loops into regular networks can result in stable dislocation configurations characteristic of PSBs. Towards the end of regime B and in regime C of the saturation cyclic stressstrain curve (i.e., for ypl > yphBc)> a n *n" creased contribution of secondary slip produces a gradual evolution of labyrinth and cell structures instead of the PSB structure seen in the early stages of regime B. (The labyrinth structure consists of walls of two sets of orthogonal dislocations; in Cu, the labyrinths are oriented parallel to the <100> direction and have a mean spacing of 0.75 jam.) Secondary slip, which originates at the PSB-matrix interface, gradually spreads by the formation of cell structures. Ackermann et al. (1984) have documented the following sequence of substructural evolution in the later stages of regime B and into regime C of f.c.c. crystals: matrix phase with labyrinth structures -> PSBs and labyrinth structures -* cell structures. Figure 11-3 b shows the cell structure formed in annealed polycrystalline Cu which was cycled to saturation at ypl =
518
11 Cyclic Deformation and Fatigue M
PSB
(b)
Figure 11-3. (a) A transmission electron micrograph showing the matrix vein structure (M), the PSB ladder structure and screw dislocations in the channels between the dislocation walls of the PSB in a Cu crystal fatigued at ypl = 0.001. b denotes the primary Burgers vector. The section investigated is perpendicular to the primary glide plane and contains the primary Burgers vector (from Mughrabi et al, 1979). (b) Dislocation cells in polycrystalline Cu fatigued at ypl = 0.0033 (courtesy of C. Laird, University of Pennsylvania).
3.3 x 10 3. The cell structures seen here are also characteristic of the ones that form in regime C of f.c.c. single crystals. The mechanisms of substructural evolution discussed thus far are also known to hold, under certain conditions, to the fatigue deformation of hexagonal closepacked (h.c.p.) materials undergoing basal slip (Kwadjo and Brown, 1978), germanium (Scoble and Weissman, 1973) and Cu-Zn (Lukas and Klesnil, 1973).
11.2.3 Some Characteristic Features of Cyclic Deformation in Single Crystals The creation of dislocation networks, the localization of deformation along slip bands and the formation of cell structures during the cyclic deformation of ductile single crystals have an apparent analogy to those seen during monotonic tensile deformation. However, the imposition of cyclic strains causes several unique features to develop. (i) The most striking difference between monotonic tension and cyclic (tensioncompression) deformation is found in the manner in which slip steps develop at the free surface. Monotonic loading produces slip traces on the surface of the specimen which have a 'staircase-like' geometry. However, cyclic deformation leads to surface roughness composed of'hills' and 'valleys', which are known as 'extrusions' and 'intrusions', respectively. These depressions and protrusions are sites at which the persistent slip bands egress at the free surface. (ii) The creation of 'persistent' slip bands (with their characteristic wall structures) is unique to cyclic deformation. (iii) The existence of a region in the saturation stress-strain curve (regime B in Fig. 11-2) where the peak resolved shear stress is independent of the plastic shear strain amplitude is also specific to cyclic deformation. (iv) The density of dislocations and of point defects produced during cyclic deformation is significantly higher than that observed in monotonic tension. 11.2.4 Cyclic Deformation of Polycrystalline Metals and Alloys The micromechanisms of cyclic deformation described earlier for ductile single crystals also have some applicability to
11.2 Mechanisms and Mechanics of Cyclic Deformation
polycrystalline ensembles. However, the presence of grain boundaries, inclusions, pores or other microstructural complexities in commercial alloys alter the microscopic mechanisms by which cyclic deformation occurs in polycrystals. Furthermore, considerable uncertainties arise in the interpretation of fatigue mechanisms in commercial systems because of the presence of a variety of microstructural inhomogeneities. Transmission electron microscopy observations in fatigued polycrystalline Cu reveal that within grains located near the surface as well as in the bulk, persistent slip bands restricted to a single slip system do exist (e.g., Winter et al., 1981). These persistent slip bands contain dislocation arrangements which are similar to those described earlier for monocrystals. On the other hand, while a single PSB can span the whole cross-section of a monocrystal, PSBs in polycrystalline metals are confined to individual grains because of slip incompatibility between neighboring grains. (The PSBs may, however, cut across low angle grain boundaries.) At intermediate to high values of plastic strain amplitudes, polycrystals also exhibit labyrinth and cell structures. The procedure for obtaining the cyclic stress-strain (c.s.s.) curves for polycrystals is similar to that discussed earlier for single crystals. Polycrystalline metals generally exhibit cyclic hardening or softening during fatigue loading; the hysteresis loops stabilize after an initial "shakedown" period of transient deformation. Common methods for obtaining c.s.s. curves involve (i) cyclic tests conducted under constant values of imposed plastic strains, (ii) multiple step tests where the plastic strain range limit is incremented after saturation at the previous lower strain level, or (iii) incremental step tests where strain amplitudes
519
Cyclic hardening / ^ - — Monotonic f
Cyclic softening
/
Strain Figure 11-4. Schematic representation of cyclic hardening and softening.
consisting of linearly increasing and decreasing total strain limits are repeatedly imposed. The cyclic stress-strain curve for the polycrystalline metal is drawn by connecting the loci of the tips of stable axial stress-strain hysteresis loops, in much the same way as it is determined for the monocrystal (see Fig. 11-2). Figure 11-4 schematically illustrates the stress-strain response for materials which cyclically harden and soften (with respect to the monotonic response). The following general trends are observed during the cyclic deformation of most engineering metals: well-annealed metals usually exhibit cyclic hardening as a consequence of dislocation multiplication, whereas cold-worked materials exhibit cyclic softening. An illustration of the role of slip behavior in influencing cyclic deformation characteristics was provided by Feltner and Laird (1967). They showed that the saturated cyclic state of pure metals with a strong tendency for wavy slip is independent of the initial state in that annealed metals (which cyclically harden) and severely cold-worked metals (which cyclically soften) exhibit the same saturated stress-strain curves.1 On the 1
Feltner and Laird (1967) suggested that increasing the propensity for cross slip with a high stacking fault energy metal/alloy gave rise to the same saturated stress-strain curves for annealed and cold-worked metals. However, more recent studies by Gerold et al. (1989) indicate that the slip mode in solid-solution alloys is governed probably to a greater degree by short-range order (which favors planar slip) than by the stacking fault energy.
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11 Cyclic Deformation and Fatigue
other hand, when there is a strong propensity for planar slip, the annealed and coldworked conditions do not attain a common saturated state; they exhibit different c.s.s. curves. In precipitation-hardened alloys, cyclic softening can also be caused by fatigue-induced overaging, re-solution or disordering of metastable precipitates, or Ostwald ripening of unsheared precipitates (see, for a review, Brett and Doherty, 1978). The stress-strain response of metallic materials is usually represented by an elastic-plastic constitutive equation of the type
-HIT where Am is a material constant known as the monotonic strength coefficient, nm is the monotonic strain hardening exponent, E is Young's modulus, a is the axial stress and s is the axial strain. nm varies typically in the range 0-0.5. (In the mechanics literature, the strain hardening response is usually characterized by the exponent, n = l/nm. n = oo for an elastic-perfectly plastic material and n = 1 for a linear elastic material.) The cyclic stress-strain curve for a material is also characterized by a similar expression where As
T
Ad
/ A(j V /nf
\2A~J
(11-3)
Here ACT and As denote the range of cyclic stress and strain, respectively, Af is the cyclic strength coefficient and nf is the cyclic strain-hardening exponent. Metallic materials with very different microstructures, composition and deformation characteristics show only a narrow variation in nf, which falls typically between 0.1 and 0.2.
11.2.5 The Bauschinger Effect
When a ductile alloy, which is subjected to a certain amount plastic deformation in tension, undergoes reversed loading into compression, it often exhibits a lower value of initial yield stress with load reversal than for continued forward deformation in tension. This phenomenon is known as the Bauschinger effect (Bauschinger, 1886). An understanding of the physical processes causing the Bauschinger effect is of importance for quantitative modelling of cyclic plasticity. Figure 11-5 a shows the stress-strain response of a material which is subjected to reverse loading in compression following plastic straining in tension. The segment AB in the stress-strain curve represents initial elastic deformation and B C denotes plastic flow. If the material is now unloaded and subjected to compression, the existence of the Bauschinger effect is manifested as a lower magnitude of stress at the initial yield point in compression (i.e., at loading point E) than at B. It is a common practice to denote the Bauschinger effect on a plot of the magnitude of the stress versus the magnitude of the accumulated strain, as in Fig. 11-5 b which readily reveals the extent of the Bauschinger effect (e.g., Wilson, 1965; Atkinson et al, 1974). If the forward loading line BC is extrapolated until the point C and compared with the reverse loading line EF, the stress difference between these two (nearly parallel) lines represents the Bauschinger stress Aah. Similarly, the Bauschinger strain, Aeb, may be defined as the difference in strain magnitudes between point C and the corresponding point C" (located at the same magnitude of applied stress) on the reverse loading curve. Pure metals generally exhibit a much smaller Bauschinger effect than metals
11.2 Mechanisms and Mechanics of Cyclic Deformation
521
Figure 11-5. (a) Forward tensile deformation and reverse compressive deformation for a ductile solid subjected to uniaxial loading, (b) A plot of the stress magnitude against the accumulated strain magnitude and the nomenclature associated with the Bauschinger effect.
strengthened by incoherent precipitates or dispersoids. The dissolution by reverse loading of cell walls or subgrain boundaries which were formed during forward loading and the development of long range internal stresses due to the strain mismatch between PSB walls and channels can contribute to some Bauschinger stress in pure metals (e.g., Mughrabi, 1983; Hasegawa et al., 1986). However, a significantly greater Bauschinger effect can be induced in precipitation-hardened alloys with nonshearable and incoherent particles and in dispersion-strenghtened alloys on the basis of the following factors: (i) An increment in strength, cr0, which arises from solid solution strengthening and from the stress necessary to bend and move the dislocations beyond the impenetrable obstacles, (ii) a contribution, afor, which arises from the interaction of the forest dislocations with mobile dislocations, and (iii) a back stress or internal stress, oB, exerted on the matrix by the particles (which is a consequence of inhomogeneous plastic flow on a microscopic level). The net yield stress developed during forward deformation is (e.g., Wilson, 1965) <* = *b + <*or + <%
(H-4)
Upon reversal of the direction of loading, (7o aids deformation rather than counter-
acting it, with the result that the reverse yield stress is given by
The corresponding Bauschinger stress is Aa b = a¥ — aR = 2 aB
(11-6)
The Bauschinger effect has been studied extensively in many alloys. In particular, studies of aluminum alloys containing 9\ S' and Y] precipitates have shown the existence of a large Bauschinger effect (e.g., Wilson, 1965; Stoltz and Pelloux, 1976). Investigations of plastic deformation in Cu-SiO 2 have suggested two contributions to strengthening at low temperatures and low strains: (i) A mean stress which develops in the matrix as a consequence of the formation of Orowan loops around the impenetrable particles and which opposes forward deformation, and (ii) An increase in the internal stress (in addition to the initial Orowan stress), known as the "source-shortening stress", which arises because the presence of the Orowan loops around the particles repels subsequent dislocations reaching the particle (Brown and Stobbs, 1971; Atkinson et al., 1974). The Bauschinger effect is induced in such dispersion-strengthened systems as a result of shrinkage of Orowan loops (Gould et al., 1974) or removal of Orowan loops by the
522
11 Cyclic Deformation and Fatigue
formation of prismatic loops and secondary dislocations (Atkinson et al., 1974). Similar mechanisms have also been advanced for interpreting the Bauschinger effect in particle- and fiber-reinforced metalmatrix composites (Brown and Clarke, 1977; Lilholt, 1977; Pederson, 1990). 11.2.6 Continuum Aspects of Cyclic Deformation
The foregoing discussions pertain to cyclic deformation at the microscopic level. However, the design of cyclically loaded engineering components inevitably requires a knowledge of continuum formulations for cyclic plasticity. The following features are incorporated into any general continuum plasticity model (e.g., Hill, 1950; Malvern, 1969): (a) A yield condition (i.e., the von Mises or Tresca yield condition) which provides the criterion for the initiation of plastic flow under general multiaxial loading; (b) A hardening rule (such as the isotropic or kinematic hardening rule) which accounts for the changes in yield condition during plastic deformation in order to incorporate strain hardening of the material; (c) A flow rule (such as the associated flow rule) linking the stress components to the plastic rate of deformation. Isotropic hardening is characterized by the uniform expansion of the yield surface (which represents the yield condition in stress space), with no change in its shape or location. If the yield surface does not change its size and shape, but merely translates the stress space in the direction of the outward normal to it, kinematic hardening is obtained. In mathematical terms, the hardening rules are described by the relation (11-7)
where G is a function of the stress components
11.3 Mechanisms of Fatigue Crack Initiation
(ii) Mroz (1967) and Iwan (1967) introduced the concept of nested yield surfaces in stress space (termed 'the field of work hardening moduli') in an attempt to apply the known rules of kinematic hardening to fatigue. This field involves a configuration of nested yield surfaces of constant work hardening moduli in stress space. In this approach, an overall nonlinear cyclic deformation response is derived although the individual yield surfaces follow a linear strain hardening law. Several criteria are imposed on the translation of the yield surfaces for both uniaxial and multiaxial deformation such that fatigue characteristics, such as the Bauschinger effect, can be accounted for. (iii) Two-surface models of cyclic plasticity, where a bounding surface encloses a loading surface in stress space, were introduced by Dafalias and Popov (1975) and Krieg (1975). These models were developed to account for cyclic hardening or softening, the Bauschinger effect and cyclic creep (i.e., the phenomenon by which the deformation progressively shifts to increased mean strain levels in a stress-controlled fatigue with a nonzero mean stress). Two parameters are used to characterize the cyclic deformation: (a) the location of the present loading point with respect to the bounding surface in stress space and (b) the amount of accumulated plastic work. Modifications of the two-surface models of cyclic plasticity have been proposed in conjunction with various isotropic and kinematic hardening formulations (e.g., Moosbrugger and McDowell, 1989). Such studies have indicated that the two-surface models for rate-independent plasticity offer good predictions of experimentally observed cyclic deformation in stainless steels subjected to nonproportional fatigue loading.
523
(iv) Several types of internal variable concepts (e.g., Bodner and Partom, 1975; Chaboche, 1986) and combined isotropic kinematic hardening models (e.g., White et al., 1990) have been developed to quantitatively describe cyclic plasticity on a continuum level. (v) Finite element analyses invoke von Mises yield criterion and isotropic or kinematic hardening rules for the analyses of cyclic deformation in monolithic alloys and metal-matrix composites (e.g., Llorca et al., 1990; White et al, 1990). The aforementioned continuum approaches have provided differing levels of accuracy when evaluated in the context of experimental observations. A recent survey and quantitative description of these approaches is provided in Suresh (1991). It should, however, be noted that most of the aforementioned models lack a clear micromechanical basis which captures the mechanistic aspects of cyclic deformation.
11.3 Mechanisms of Fatigue Crack Initiation Under cyclic straining, the initially smooth surface of a crystal becomes progressively rougher if different proportions of net glide displacements develop on different glide planes. This roughening of the surfaces leading to the formation of 'hills' and 'valleys' (also known as 'extrusions' and 'intrusions', respectively) is an outcome of the kinematic irreversibility of cyclic slip (Forsyth, 1953; Wood, 1958; May, 1960). The valleys function as micronotches which further accelerate the roughening process. Figure 11-6 a shows an example of a large extrusion created in the direction of the primary Burgers vector b on the surface of a Cu single crystal which was fatigued for 35 000 cycles at a
524
11 Cyclic Deformation and Fatigue
Figure 11-7. An incipient fatigue flaw situated along the wall structure of a PS B in poly crystalline Cu. The inset shows the location of the fatigue crack with respect to the surface of the material (from Katagiri et al., 1977).
Figure 11-6. (a) A large extrusion oriented along the primary Burgers vector b and protruding from the top surface of a Cu single crystal strained at ypl = 2 x 10 ~3 for 35 000 cycles at 77 K. (b) A direct view of the end of the extrusion with the electron beam parallel to b (from Differt et al., 1986).
plastic shear strain amplitude of 0.002 at 77 K. The microscopic hills and valleys at the end of the protrusion are seen in Fig. 11-6 b, which is a scanning electron micrograph taken with the electron beam parallel to b. The interface between the PSB and the adjoining matrix can function as a site for fatigue crack nucleation because of the strong gradients in the density of dislocations and the attendant strain mismatch across the interface. Many experimental studies (e.g., Hunsche and Neumann, 1986; Ma and Laird, 1989) employing scanning electron microscopy have shown that microscopic flaws nucleate along the interface between the PSBs and the matrix. Figure 11-7 shows a transmission electron micrograph of an incipient fatigue crack
along the wall structure of a PSB in polycrystalline Cu. Kinematic irreversibility of cyclic slip, which is the principal cause of surface roughening and the attendant nucleation of fatigue cracks in well-annealed high purity metals, arises from a variety of sources: (i) Slip on different glide planes during the loading and unloading portions of the fatigue cycle as a consequence of cross slip or formation of dislocation nodes, locks and jogs, (ii) Slip asymmetry2 arising from changes in the shape of the crystal or from different amounts of back stress due to slip on different planes, (hi) Production of point defects, (iv) Oxidation of slip steps or adsorption of foreign atoms on the freshly formed slip steps. In polycrystalline metals and alloys, many investigations have revealed the somewhat puzzling result that, among a 2 If slip occurs on different glide planes during the tension and compression portions of fatigue, a crystal must undergo shape changes due to this slip asymmetry. Nine (1973), Neumann (1975), Mughrabi and Wuthrich (1976), and Guiu and Anglada (1980) have shown that this geometrical effect (which is observed in b.c.c. crystals, but not in f.c.c. crystals) transforms an initially circular cross section of cylindrical crystal into an ellipse. Kinematic irreversibility of cyclic slip can be correlated with the shape changes occurring as a result of slip asymmetry during fatigue in b.c.c. crystals.
11.4 Characterization of Fatigue Life Based on Cyclic Stress and Cyclic Strain
stack of lamellar twins, fatigue cracks initiate at every other twin boundary (e.g., Thompson et al., 1956; Boettner et al., 1964; Neumann and Tonnessen, 1988). This observation is also surprising because twin boundaries are of the lowest energy among all grain boundaries. Neumann and Tonnessen (1988) and Heinz and Neumann (1990) have rationalized this observation on the basis of two mechanisms: (i) In f.c.c. metals, the elastic anisotropy on either side of the boundary results in different levels of local strain concentration. Consequently, every other twin boundary experiences the largest levels of local strains and becomes a preferential site for the nucleation of fatigue cracks, (ii) Since the twin boundary is also a glide plane for an f.c.c. metal, the large travel distances for dislocations along the boundary greatly influence this strain concentration effect. Using electron channeling techniques for measuring grain orientations and laser interferometry to monitor local plastic strain across twin boundaries, Heinz and Neumann have provided justification for the validity of these two mechanisms. In polycrystalline metals, high angle grain boundaries also provide sites for the nucleation of fatigue cracks. The propensity for crack nucleation at the boundary is dictated by many factors which include the relative orientations of adjoining grains as well as the orientation of the grain boundary with respect to the free surface and to the tensile axis (e.g., Kim and Laird, 1978). In commercial materials, additional sites for fatigue crack initiation are provided by inclusions, pores, surface roughness (induced by the machining and fabrication of the engineering component), microscopic regions of chemical segregation and scratches. Furthermore, in the presence of a chemically aggressive environment, pitting corrosion and fretting
525
corrosion can provide nuclei from which dominant fatigue cracks grow.
11.4 Characterization of Fatigue Life Based on Cyclic Stress and Cyclic Strain The discussions up to this point were concerned with the nucleation of permanent damage due to the repeated application of stresses and strains and with the attendant initiation of fatigue flaws in nominally defect-free materials. While these descriptions provide insights into the mechanistic processes underlying fatigue, it is necessary to develop quantitative estimates of the useful fatigue life of a material for implementation into design codes. Although such fatigue life estimates are invariably empirical in nature, they have long been an integral part of fatigue analyses in the design of engineering components. The classical stress-life approach to fatigue life dates back to the work of Wohler (1860) who was concerned with the cyclic life of railroad axles. In this method, cyclic stress-controlled tests are conducted on specimens in uniaxial tension, plane flexure or rotating bending and the number of stress cycles to failure N{ is determined for different values of stress amplitude aa (or one-half the stress range in fully reversed loading). A plot of aa versus logiVf then provides the so-called stress-life (or S-N) curve, which is schematically shown in Fig. 11-8. Most mild steels and strain-age-hardening materials exhibit a well defined plateau (known as the endurance limit, ae) in the S-N curve. For imposed stress amplitudes equal to or smaller than
526
11 Cyclic Deformation and Fatigue
Iog/Vf
Figure 11-8. A schematic of a stress-strain plot (Wohler curve, S-N plot) for a ductile solid exhibiting a well-defined fatigue endurance limit.
cru. Aluminum alloys and many high strength steels do not exhibit a well-defined endurance limit. In such cases, the operational definition of an apparent endurance limit is the stress amplitude which provides a life of at least 1 million to 10 million fatigue cycles. It was shown by Basquin (1910) that the following expression describes the stress-life curve for many materials which is valid for low stress amplitudes/long fatigue life, i.e., for high-cycle fatigue (HCF): (11-8) where b is the fatigue strength exponent or Basequin exponent (which is generally in the range -0.05 to -0.12) and af is the fatigue strength coefficient (which is generally of the order of the true fracture strength in a tension test). Although the above relationship (commonly known as the Basquin relationship) strictly pertains to fatigue fracture in a smooth specimen subjected to fully reversed tension-compression loads, many empirical approaches, such as the Gerber, Soderberg or Goodman relationships, are available to account for nonzero mean stress levels (see Mitchell, 1979, for a comprehensive review). Similarly, the fatigue notch factor, Kf (or equivalently, the notch sensitivity index, q) is used to define fatigue life in notched specimens subjected to stress-con-
trolled fatigue [see Peterson (1959), and Section 11.11 of this review]. In this method, high cycle fatigue life for notched members is obtained by empirically determining a modified S-N curve where GJK{ is plotted against JVf. The stress-based approach to total fatigue life becomes less relevant in situations where significant plastic deformation occurs. Under these conditions, characterization methods based on cyclic strains are widely used for fatigue life estimation. The role of plastic strains in influencing fatigue lives was popularized by Coffin (1954) and Manson (1954) who, working independently on thermal fatigue problems, suggested the following empirical relationship between the plastic strain amplitude Asp/2 and the number of load reversals to failure 2Nf: (11-9) where e'f is the fatigue ductility coefficient (which is of the order of the true fracture ductility in tension) and c is the fatigue ductility exponent (^ — 0.5 to —0.7 for many metallic materials). Equation (11-9) is generally known as the Coffin-Manson relationship. Since the elastic strain amplitude Aee/2 is given by 2E~ E
(11-10)
where E is Young's modulus, the total strain amplitude can be written as As
As,. Asn c
(11-11) Note that the total strain amplitude is composed of elastic and plastic parts [see Eq. (11-11) and Fig. 11-9]. At long fatigue lives, the elastic part of the strain amplitude dominates and the fracture strength
11.5 Fatigue Design Philosophies
527
Figure 11-9. Strain-based characterization of fatigue life (after Mitchell, 1979).
Total = elastic and plastic
Reversals to failure,2A/f log scale
primarily determines fatigue life. At short fatigue lives, i.e., for low cycle fatigue (LCF), the plastic part dominates and the fatigue ductility usually determines fatigue life (e.g., Morrow, 1965; Landgraf, 1970; Mitchell, 1979). Strain-based approaches to fatigue have found widespread use in the design of many components used in ground-vehicle industries.
11.5 Fatigue Design Philosophies The stress-life and strain-life approaches to fatigue estimate the total life of a cyclically loaded material. As noted earlier, the total fatigue life or number of cycles to failure is composed of the number of cycles to initiate a fatigue crack in a nominally defect-free specimen and the number of cycles to propagate the dominant crack to final failure. Total life estimates derived from smooth laboratory specimens primarily denote the life to initiate a fatigue crack because the crack initiation component of total life in such cases can be as much as 80%. In this sense, total life approaches may be construed as focusing primarily on design against fatigue crack ini-
tiation in smooth specimens. The cyclic stress-based approach is mainly valid for unconstrained deformation involving primarily elastic deformation (high cycle fatigue). On the other hand, the strain-based approach applies to situations where considerable plastic deformation occurs (e.g., the region ahead of a stress concentration) and where the fatigue lives are small (low cycle fatigue). By contrast to the stress-based and strain-based approaches, the defect-tolerant method invokes the notion that all engineering components intrinsically contain defects. In this approach, which is embedded in the principles of fracture mechanics (to be discussed in the next several sections), the useful fatigue life is merely that expended in propagating a flaw of a certain initial size to some critical size; the final crack size depends on the particular design philosophy and may be determined from the fracture toughness, limit load, allowable strain or allowable compliance change. The fracture mechanics or defecttolerant approach to fatigue is, therefore, a conservative design philosophy and its most widespread use in engineering applications is found in situations where structural failure will result in the loss of human
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11 Cyclic Deformation and Fatigue
lives (e.g., nuclear and aerospace industries). The different approaches to fatigue also provide seemingly different guidelines for the microstructural design for improved fatigue properties. A careful examination of these approaches, however, reveals that these apparent differences are merely an outcome of how each approach handles crack initiation and crack growth in a quantitative manner in the estimation of useful fatigue life. This point is discussed further in Sec. 11.15.
11.6 Fracture Mechanics Approach to Fatigue The subcritical growth of fatigue cracks is characterized in terms of continuum formulations by recourse to the principles of fracture mechanics (see Chap. 13 of this volume for detailed discussions of fracture mechanics). In particular, linear elastic fracture mechanics has found a major practical application in the characterization of fatigue crack growth. 11.6.1 The Stress Intensity Factor Linear elastic fracture mechanics originated from the pioneering work of Irwin (1957). He demonstrated that the amplitude of the stress, strain and displacement
singularity fields ahead of a crack can be determined in terms of a single scalar quantity known as the 'stress intensity factor'. In this approach, the stress, strain and deformation fields in the immediate vicinity of the crack-tip are determined by asymptotic continuum analyses. For plane strain and generalized plane stress, it can be shown that, for a stationary crack subjected to fracture in a tensile opening mode (known as mode I), the variation of the stresses u{j as a function of the radial distance r and the polar angle 9 (see the coordinate system shown in Fig. 11-10) are of the form J \/2nr J + higher order terms ...
(11-12)
where Xx is the scalar amplitude of the asymptotic singular fields in mode I, d\j are universal functions of the polar angle 9 for mode I, Otj (r°) is a nonsingular (but nonvanishing) stress term, and Oij(r1/2) is a nonsingular stress term which vanishes at the crack tip. Kx is known as the mode I stress intensity factor which is a function of the far-field loads, the crack dimensions and the geometry of the specimen containing the crack. The appealing feature here is that KY uniquely characterizes the near-tip fields and the conditions for the progression of fracture; the use of this approach
<*xy mrr-
Figure 11-10. The coordinate axes, geometrical parameters and stress components for an element located in the vicinity of the crack-tip.
11.6 Fracture Mechanics Approach to Fatigue
does not require a detailed understanding of the mechanisms by which fracture takes place. The strain and displacement fields ahead of a tensile crack can also be expressed in a form similar to that of Eq. (11-12). Furthermore, similar expressions can be derived for other modes of fracture, viz., in-plane sliding (mode II) and antiplane shear (mode III). Equation (11-12) can also be rewritten in the following manner which is useful for later discussion: lJ
Six8jx + (11-13) \/2nr lJ + terms which vanish at crack tip ...
The first term on the right hand side is the leading term of the asymptotic singular field for mode I crack problems in linear elastic fracture. The second term is a nonsingular term, commonly known as the T-stress (e.g., Larsson and Carlsson, 1973). (There is no summation over x for the second term, where S^ is the Kronecker delta with the property that Smn = 0 for m # n and 3mn = 1 for m = n) The higher order terms all vanish at the crack-tip. The magnitude of KY can be computed from the far-field loads and the geometry of the cracked specimen; standard procedures for the estimation of KY for a wide range of specimen geometries and loading configurations can be found in fracture handbooks (e.g., Tada et al., 1973) and standard fracture textbooks (e.g., Kanninen and Popelar, 1985). As an example, consider a plate containing a central through-thickness crack of length 2 a and subjected to a far-field tensile stress, o^. For this case, KY is given by the expression, (11-14) na Y is a geometrical correction factor which is equal to unity for an infinitely large plate.
529
11.6.2 ^-Dominance The proper use of linear elastic fracture mechanics to crack problems inevitably requires a thorough understanding of the conditions of '^-dominance'. This involves identifying the regions of dominance of the leading terms of the asymptotic singular fields (in the case of mode I loading, Kx) on the basis of a knowledge of the accuracy of asymptotic solutions and the microscopic deformation processes at the crack-tip. The region of dominance of K field is an annular zone. Its outer radius is estimated as the radial distance at which the singular solutions based on the leading term deviate by more than a prescribed error level (say 10%) from the full-field solutions which include all the higher order terms in Eqs. (11-12) and (11-13). The K solutions would also lose validity in the immediate vicinity of the crack-tip where finite deformation processes take place. [Equations (11-12) and (11-13) imply that the stresses become infinitely large as the crack-tip is approached. This situation is not realized in practice because of plastic deformation or other damage processes occurring at the crack-tip.] The inner radius of the annular zone of K dominance is controlled by the distance over which nonlinear deformation occurs. Provided that the size of the process zone (i.e., the zone of permanent, nonlinear deformation at the crack-tip) is small compared to the characteristic dimensions of the cracked specimen, including the crack size and the size of the uncracked ligament, 'small-scale yielding' or 'smallscale damage' conditions prevail; then the stress intensity factor K can be regarded as a unique measure of the intensity of the singular field. Under small-scale yielding conditions, the imposition of quasi-static loads to a cracked body causes the onset of fracture
530
11 Cyclic Deformation and Fatigue
when the stress intensity factor reaches a critical value, Kc. If tensile (mode I) fracture occurs in plane strain, the critical value of K for fracture initiation is denoted as the fracture toughness Klc. Similarly, fracture toughness values can also be defined for modes II and III. In addition to the mode of deformation, the fracture toughness is sensitive to such factors as stress state, environment, strain rate, temperature and microstructural constitution. However, when these variables are held fixed, the fracture toughness is a material property, i.e., it is independent of the geometry of the test specimen. Standardized procedures for the evaluation of Klc are described in Standard E-399 of the American Society for Testing and Materials (Philadelphia). 11.6.3 Fatigue Crack Growth Characterization
When a cracked body is subjected to cyclically varying loads of a constant amplitude, the stress intensity factor also fluctuates between a maximum value Kmax and a minimum value Kmin during a fatigue cycle. These extremum values of K are computed from the maximum and minimum loads of the fatigue cycle and from the geometry of the cracked body, using Eq. (11-14). It was suggested by Paris et al. (1961) that the range of stress intensity factor can be used to characterize the growth of a fatigue crack subjected to a fixed amplitude of cyclic stresses under conditions of small-scale yielding. Paris et al. (1961) showed that the increment of crack advance per stress cycle, da/diV, is a function of the stress intensity factor range, AX, such that m
dN
=C(AK)
(11-15) "'max
^min
where C and m are material constants which are influenced by the microstructure, yield strength and environment, m is commonly referred to as the Paris exponent, which varies typically between 2 and 4 for most ductile metallic materials. In the regime where this power law characterization is applicable, Eq. (11-15) implies that a plot of log(da/dAT) versus log(AK) is a straight line with a positive slope equal to m. In addition to the microstructure, test temperature, material strength and environment, the rate of fatigue crack growth is strongly influenced by the mechanical load variables such as the mean stress or load ratio R (which is defined as the ratio of the maximum load of the fatigue cycle, i.e., R = KBdJKnax)9 stress state (i.e., plane stress or plane strain), as well as cyclic frequency and waveform (usually in the presence of an aggressive environment). Although the Paris equation is empirical, this method is the most widely used for characterizing the subcritical growth of fatigue cracks in engineering practice. 11.6.4 Plastic Zones in Tension and Fatigue
In order to ensure that small-scale yielding conditions prevail, it is important to derive quantitative estimates of the extent of plastic deformation at the crack-tip in ductile solids. A rough measure of the size of the plastic zone can be obtained by taking the boundary of the plastically deformed region to be that within which the von Mises equivalent stress exceeds the tensile yield strength, oy. The distance of the elastic-plastic boundary directly ahead of the crack-tip (i.e., along the crack plane, with 9 = 0) is given by l (K V rp w - ( — I ,
i /*Y
for plane stress
rv « — I — I , for plane strain 371\
{1M6)
11.6 Fracture Mechanics Approach to Fatigue
(a)
\ \ \ P
(b) M l
-AP ay(x,O)
i 111
m (c)
P-AP
Cyclic plastic zone Monotonic plastic zone
Figure 11-11. A schematic representation of the development of the cyclic plastic zone and the near-tip stress field (after Rice, 1967). See text for details.
As an operational definition, small-scale yielding conditions are expected to be satisfied when the characteristic dimensions of the cracked body (i.e., the crack size and the size of the uncracked ligament) are typically greater than 15 times the plastic zone size. Under cyclic loading conditions, a zone of reversed plastic flow develops within the monotonic plastic zone which is created at the peak far-field tensile stress. An estimate of the size of the reverse yield zone or cyclic plastic zone was provided by Rice (1967). Consider the idealized case of an elasticperfectly plastic solid subjected to a tensile load P. A plastic zone of dimension rp, Eq. (11-16), develops at this tensile load. Consider now that the load is reduced by a small amount to a lower value, P — AP. If
531
crack-tip blunting and the possibility of any premature contact (closure) between the faces of the fatigue crack are excluded, the drop in far-field tensile load would lead to the creation of a zone of reversed plastic flow within the monotonic yield zone as a consequence of the infinitely large stress concentration at the crack-tip. For proportional loading (i.e., the loading situation in which different components of the stress tensor vary in constant proportion to one another), the solutions given in Eq. (11-16) can also be used to derive the cyclic plastic zone dimensions. However, in the cyclic case, the stress intensity factor range AK corresponding to the load range AP is used in Eq. (11-16) rather than the stress intensity factor K. Similarly, the yield strength ay in Eq. (11-16) is replaced by twice its value (i.e., the range from tensile yield to compressive yield for an elastic-perfectly plastic solid). This process is schematically illustrated in Fig. 11-11. The cyclic plastic zone size in plane stress is thus given by (11-17) n Note that, for fully tensile cyclic loads, a zone of residual compressive stresses develops within the cyclic plastic zone ahead of the fatigue crack. For a cyclic variation in the stress intensity factor from 0 to a maximum value of Xj, the cyclic plastic zone size is one-fourth the size of the monotonic plastic zone. For materials which cyclically harden or soften, the monotonic yield strength in Eq. (11-17) should be replaced by the cyclic yield strength, o'y. The existence of the reversed yield zone and of a region of residual compressive stresses at the tip of a tensile fatigue crack has many important implications for crack growth, especially in the context of vari-
532
11 Cyclic Deformation and Fatigue
able amplitude fatigue (see Sec. 11.10). Furthermore, a nonclosing notch or other stress concentration, when subjected to cyclic compressive loads, can develop a zone of residual tensile stresses in the neartip region because of reversed plastic flow. This process is the principal cause for the nucleation and growth of stable fatigue cracks in notched plates of metallic materials subjected to fully compressive load fluctuations (see Sec. 11.12). While the cyclic plastic zone for a stationary fatigue crack in an elastic-perfectly plastic solid is onefourth the size of the monotonic plastic zone, a smaller extent of cyclic plasticity is developed for a fatigue crack propagating steadily in plane stress. Budiansky and Hutchinson (1978) showed that for steady crack growth, the reverse yield zone is onetenth the size of the plane stress monotonic plastic zone. 11.6.5 Nonlinear Fracture Mechanics While the stress intensity factor, K, uniquely characterizes the near-tip fields under small-scale yielding conditions, the scalar amplitude of the asymptotic singular fields for elastic-plastic fracture is the J-integral of Rice (1968). This path-independent integral is defined as j= Uwdy-T-^ds) A\
(11-18)
OX J
where s is the arc length along any contour completely surrounding the crack-tip, y is the distance along the normal to the crack plane, u is the displacement vector, T is the traction vector, and w is the strain energy density defined by the relationship otj = dw/dstj. For a nonlinear elastic material, J is also the rate of change of potential energy with respect to crack length, and for a linear elastic material, J equals the strain energy release rate 0. Y§ = K2/E in plane
stress and ^ = K2 (1 — v2)/E in plane strain, where v is Poisson's ratio.] Take the case of an elastic-power law plastic material with the stress-strain relationship of the form s/ey = a (cr/cry)n, where <7y = 8yE and a is a material constant. For small-strain deformation of such a solid, it was shown by Hutchinson (1968) and by Rice and Rosengren (1968) that the stress and strain fields ahead of the crack-tip take the form: ai} = a, n/(n+l)
(11-19)
oc(iyeylnr
where a and s are universal functions of the polar angle 9 and the strain hardening exponent n, In is weakly dependent on the strain hardening exponent, and r is the radial distance from the crack-tip. The singular fields given in Eq. (11-19) are the so-called Hutchinson-Rice-Rosengren (HRR) fields. Analogous to Klc, the initiation of quasi-static fracture under elasticplastic conditions can be defined when J approaches a critical value J lc in plane strain. Procedures for determining this elastic-plastic fracture toughness of ductile materials are described in the Standard E-813 which has been published by the American Society for Testing and Materials (Philadelphia). The use of J to characterize fracture is generally valid if (i) the deformation theory of plasticity holds (i.e., when proportional loading occurs everywhere) and (ii) the zone within which finite deformation occurs at the crack-tip (which is of the order of the crack-tip opening displacement) is small compared to the region of dominance of the asymptotic singular fields of the small-strain analysis. Dowling and Begley (1976), Dowling (1977) and Wiithrich (1982) suggested that
11.6 Fracture Mechanics Approach to Fatigue
the ./-integral concept could be extended to fatigue situations where the rate of crack growth under large-scale yielding da/dN could be characterized by the cyclic J-integral, AJ: da/dN cc(AJ)m\ where m1 is an exponent similar to m in Eq. (11-15). Standard procedures are available for evaluating AJ under load-controlled as well as displacement-controlled fatigue loading (see Dowling and Begley, 1976; Sadananda and Shahinian, 1979). Experimental results in some alloy systems show that this elastic-plastic approach for fatigue fracture shows a better characterization of crack growth data at high stress amplitudes, short crack lengths and for crack growth from notches (see Sec. 11.8 for an example). Such a correlation, however, is somewhat surprising from a fundamental standpoint because proportional loading conditions would not be expected to hold, considering the rapid advance of the fatigue crack and elastic unloading (especially at shorter crack lengths and larger imposed stress range values). Experimental observations revealing a reasonable characterization appear to be the primary justification thus far for the adaptation of cyclic J-concepts to fatigue loading conditions. Another equivalent method of characterizing nonlinear fracture involves use of the crack-tip opening displacement. Although the opening between the crack faces varies continuously with distance along the wake of the crack-tip, an operational definition of crack-tip opening displacement is often given by the distance between two points on the upper and lower surfaces of the crack at which two lines drawn at +45° and —45° from the cracktip intersect. On the basis of this geometrical convention, the crack-tip opening displacement is found to be J
(11-20)
533
where dn (ranging in value from 0.3-0.8 for n = 3-13) is a function of a, the yield strain and the strain hardening exponent (Shih, 1981). From earlier discussions of this section, it is readily seen that 3tocK2/ {Eay).
The crack-tip opening displacement is useful in the study of nonlinear fracture on several accounts: (i) It provides a useful size scale for the characterization of intense nonlinear deformation at the crack-tip. This length scale is also useful in relating the microscopic failure processes to macroscopic fracture behavior (see Sees. 11.7.2 and 11.7.3). (ii) The crack-tip opening displacement is directly related to the J-integral by the functional form given in Eq. (11-20). (iii) Under highly nonlinear fatigue conditions, the crack growth rates can be characterized by the relationship: da/dN oc (A(5t), where A<5t is the cyclic variation in the crack-tip opening displacement (also commonly denoted by the symbol ACTOD). (iv) The characterization based on ACTOD is also convenient for comparing the fatigue crack growth behavior of materials under multiaxial loading conditions on a common scale because a crack opening displacement can be defined for tension as well as for in-plane shear and torsional loading conditions. The limitations in using the crack-tip opening displacement to characterize crack growth are: (i) The crack-tip opening displacement varies continuously along the crack wake and it is often experimentally difficult to quantify the extent of crack opening in the vicinity of the crack tip, and (ii) The very definition of a crack opening displacement becomes uncertain as a consequence of fatigue crack closure and frictional sliding of crack faces in mode I as well as in multiaxial fatigue.
534
11 Cyclic Deformation and Fatigue
11.7 Characteristics of Fatigue Crack Growth 11.7.1 Different Regimes of Fatigue Crack Growth When the rate of fatigue crack growth da/dN under small-scale yielding conditions is characterized in terms of the stress intensity factor range AK, a plot schematically shown in Fig. 11-12 is obtained. On a log-log scale, the plot of da/dN versus AK exhibits a sigmoidal variation with three distinct regimes of fatigue fracture. Regime A is the near-threshold fatigue region in which the crack growth rates are typically lower than 10" 6 mm/cycle. The growth rates vary precipitously with the stress intensity factor range. The threshold stress intensity factor range for the onset of fatigue crack growth, AK0, is usually defined as that below which fatigue cracks are dormant or propagate at an undetectably slow growth rate («10 " 8 mm/cycle). It is important to note that the maximum stress intensity factor of the fatigue cycle at threshold can be as low as one-hundredth of the fracture toughness (i.e., the stress intensity factor for the initiation of quasistatic fracture) in very ductile solids. This
observation serves to underscore the significance and ease of occurrence of fatigue fractures in structural engineering applications. Regime B in Fig. 11-12 is the region in which the crack growth rates obey the Paris power law relationship, Eq. (11-15), where log da/dN exhibits a linear relationship with logAK. As noted earlier, the slope of this linear region, m, varies typically between 2 and 4 for most ductile metals and alloys. As the maximum stress intensity factor of the fatigue cycle Kmax = AK/(1—R)-+Klc, the crack growth rates increase rapidly and catastrophic fracture ensues. This region is denoted as C in Fig. 11-12. The effects of various mechanical, microstructural and environmental factors on the characteristics of crack growth in the three regimes of fatigue are summarized in Table 11-2. 11.7.2 Microscopic Stages of Fatigue Crack Growth Depending on the size of the crack-tip plastic zone in relation to the characteristic microstructural dimension and on the microstructural constitution of the metallic material, the microscopic mechanisms of
AK
_ I
E
10"4
Nearthreshold regime
One lattice - spacing — per cycle
O)
o
10-8
AK0 logAK
Figure 11-12. The macroscopic regimes of fatigue crack growth characterized in terms of linear elastic fracture mechanics.
535
11.7 Characteristics of Fatigue Crack Growth
Table 11-2. The effects of various mechanical, microstructural and environmental factors on fatigue crack growth in the three regimes shown in Fig. 11-11. (From Suresh, 1991. Courtesy of Cambridge University Press.) A
B
C
Slow-growth rate (near-threshold)
Mid-growth rate (Paris regime)
High-growth rate
Microscopic failure mode
Stage I, single shear
Stage II, (striations) and duplex slip
Additional static modes
Fracture surface features
Faceted or serrated
Planar, with ripples
High
Low
Additional cleavage or microvoid coalescence -
Regime Terminology
Crack closure levels Microstructural effects
Large
Small
Large
Load ratio effects
Large
Small
Large
Environmental effects
Large
a
Small
-
Large
Large
r c >^g
r
Stress state effects Near-tip plasticity a b
b
rc
c>dg
Large influence on crack growth for certain combinations of environment, load ratio and frequency, rc and dg refer to the cyclic plastic zone size and the grain size, respectively.
fatigue crack growth also undergo drastic variations. When the zone of plastic flow at the crack-tip is typically smaller than several grain diameters, deformation restricted to a single slip system is observed in many ductile solids. The fatigue crack advances along the most dominant slip plane with the attendant crystallographic fracture resulting in a highly serrated or faceted fracture morphology on a microscopic scale. This crack growth process has been termed stage I by Forsyth (1962). This type of failure is commonly observed in the near-threshold fatigue regime of metallic materials (where the stress intensity factor range is not large enough to induce appreciable plastic deformation at the crack-tip and where the plastic zone size is typically smaller than the grain size) or in the case of short flaws emanating from free surfaces. Figure 11-13 shows an example of crystallographic stage I fatigue fracture along {111} slip planes in a Mar M-200 nickelbase superalloy. Here crack growth occurs
in the same slip system over thousands of fatigue cycles before the crack tip is deflected to another {111} plane. When the plastic zone size spans many grain diameters, a different mode of fatigue fracture emerges as a consequence of simultaneous or alternating slip in two dif-
Figure 11-13. Crystallographic (stage I) fatigue fracture in a Mar M200 nickel-base superalloy single crystal. The [100] direction is vertical (Courtesy of P. B. Aswath, Brown University).
536
11 Cyclic Deformation and Fatigue Crack opening (tension)
7\ Crack closing (compression)
2nd tension
Figure 11-14. The extension of a fatigue crack by duplex slip (after Neumann, 1969).
ferent slip systems (Neumann, 1969). The macroscopic path of the fatigue crack is, on the average, normal to the tensile loading axis. Figure 11-14 schematically illustrates the process of duplex slip ahead of a fatigue crack (termed stage II fatigue by Forsyth, 1962), and the attendant fracture by "unzipping" along crystallographic planes. On a microscopic scale, this cyclic slip process manifests itself in the form of distinct ripple-like markings on the fracture surface. These surface features, apparently first observed by Zappfe and Worden (1951), are commonly known as ductile fatigue striations. Figure 11-15 shows an example of fatigue striations formed in an Al-3.5 wt.% Cu alloy matrix reinforced with 6vol.% SiC particles. The stage II fatigue crack growth process has also been conceptually modelled in terms of the continual blunting and resharpening of the crack-tip (Laird, 1967). Here, the crack effectively extends by an amount comparable to the crack-tip opening displacement during the tensile portion of the fatigue cycle. The reverse loading resharpens the crack-tip only to be blunted again in the next tensile loading phase (Fig. 11-16). It is important to note that this process, at least in principle, can occur
even in the absence of cyclic (dislocation) plasticity; striation formation in semi-crystalline and noncrystalline solids can also be rationalized on the basis of crack-tip blunting and resharpening. Striation formation is strongly influenced by the environment. If slip steps freshly formed at the crack-tip during the tensile loading phase are oxidized or chemically attacked, kinematic reversibility of slip upon load reversal will be inhibited. Consequently, the spacing between striations as well as the rate of crack advance
Figure 11-15. Fatigue striations formed in the ductile matrix of an Al-Cu alloy reinforced with 6 vol.% of SiC particles. Crack growth is from left to right (courtesy of Y. Sugimura, Brown University).
11.7 Characteristics of Fatigue Crack Growth
\ \
Figure 11-16. An idealization of the fatigue crack growth process by the blunting and resharpening of the crack-tip. The arrows denote slip directions. Left column (from top to bottom): zero load, tensile load, and maximum tensile load. Right column (from top to bottom): unloading, compressive load, and tensile load in the next fatigue cycle (after Laird, 1967).
per cycle is affected by the environment (e.g., Pelloux, 1969; Lynch, 1988). These studies have also shown that many alloys exhibit no striations in vacuo, and that the crack growth rates are significantly less in vacuo than in moist air. 11.7.3 Micromechanical Considerations
Several hypotheses have been put forth to rationalize the existence of a threshold for the growth of fatigue cracks. McClintock (1963) suggested that the fatigue threshold occurs when a critical local strain or accumulated damage over a characteristic microstructural size scale falls below a certain critical value. This argument can be rephrased such that the threshold is determined by the attainment of a critical size of the cyclic plastic zone. Paris et al. (1972) associated the threshold with a critical value of the stress intensity factor range, AK0. Other arguments for accounting for the existence of threshold include the critical crack-tip opening displacement criterion or the critical shear stress required to generate and move dislocations
537
at the crack-tip. (These hypotheses do not fully account for the effects of various mechanical and microstructural factors on fatigue thresholds.) Research work in the recent past has also shown that the premature contact between the crack faces due to corrosion deposits or the mismatch between fracture surface asperities can lead to apparent thresholds in many materials (Suresh and Ritchie, 1984 a and 1984 b). Such crack closure effects (to be discussed later) have also been suggested as a reason for the very existence of fatigue thresholds. Quantitatively predicting the fatigue crack growth rates solely on the basis of theoretical considerations is a formidable task in view of the synergistic effects of environment, mechanical load variables, and material microstructure on fatigue fracture. Nevertheless, there have been numerous attempts aimed at providing quantitative estimates of fatigue fracture in various materials. One of the physically appealing models in this regard involves the geometrical arguments (McClintock, 1967; Pelloux, 1969), whereby the cyclic crack-tip opening displacement ACTOD is related to the distance of crack advance per fatigue cycle, da/dN: ACTOD ACTOD
(11-21)
where o'y is the cyclic yield strength and E' = E/(l — v2) is Young's modulus in plane strain. The constant of proportionality in Eq. (11-21) is a fraction of unity (-0.1). Note that the geometrical model based on crack-tip opening displacement predicts that da/dN is proportional to AK raised to the power 2, which is reasonable for many metallic materials. However, the quantitative predictions of growth rates often deviate significantly from experiments. Equation (11-21) is merely a mathematical
538
11 Cyclic Deformation and Fatigue
statement of the crack-tip blunting process schematically sketched in Fig. 11-16. Another popular approach to modelling fatigue crack growth involves damage accumulation concepts in which the plastic strain or hysteresis energy accumulated over a certain distance ahead of the cracktip reaches a critical value (McClintock, 1963; Weertman, 1966; Rice, 1967). The damage accumulation models predict that da/dN is proportional to AK4 (or to the square of the plastic zone size), which is also in reasonable agreement with experimental data for many metals. However, these models often ignore the details of microscopic separation processes (which have a significant effect on fatigue crack growth) and the gradients in the stress and deformation fields ahead of the crack-tip (which cannot be accounted for by the uniaxial data gathered on smooth specimens which are used to estimate the accumulated damage by recourse to Coffin-Manson-type plots). At high AK levels approaching the final failure regime (regime C in Fig. 11-12 and Table 11-2), the mechanism of deformation changes from purely striated fracture. As Kmax = AK/(1 -R)-> Klc9 additional static modes involving transgranular cleavage, ductile void growth or intergranular failure are initiated. In this regime of crack growth, an increase in JR ratio results in catastrophic fatigue fracture results at lower AK levels.
11.8 Fatigue Crack Closure and Other Retardation Mechanisms Designing for improved damage tolerance in engineering materials involves microstructural modifications whereby the (apparent or intrisic) resistance to fatigue crack growth can be enhanced. In the past
several decades, many mechanisms have been identified for both monolithic and composite materials which offer the possibility of improving the fatigue crack growth properties. These mechanisms include different types of crack closure processes (involving plastic stretch in crack wake, fracture surface corrosion layers, roughness of asperities on crack faces, viscous fluids trapped within the crack walls, or stress- or strain-induced phase transformations), periodic deflections in the patch of the crack, mitigating the influence of farfield loads by shielding the crack-tip from external loads, or bridging of the crack faces by fibers and particles reinforcing the matrix. Figure 11-17 is a schematic representation of different mechanisms by which the effective stress intensity factor range responsible for the advance of fatigue cracks can be reduced. 11.8.1 Fatigue Crack Closure 11.8.1.1 Plasticity-Induced Crack Closure The characterization of fatigue crack growth discussed thus far in terms of fracture mechanics concepts is predicated upon the premise that the crack remains fully open during a tensile loading cycle. However, it was first shown by Elber (1970, 1971) that even a tensile crack can be fully or partially closed during part of the fatigue loading cycle. Elber measured the changes in the compliance of cracked aluminum alloys during the loading and unloading phases of a tensile fatigue cycle to demonstrate the premature contact between fracture surfaces. He argued that, as the crack advances through the near-tip plastic zone, a residual stretch of material from prior plastic zones is left appended to the upper and lower faces of the crack. Upon unloading from the peak far-field tensile load, the crack faces close prema-
11.8 Fatigue Crack Closure and Other Retardation Mechanisms
539
(0 Plastic wake
Fatigue crack
Current plastic zone
Crack
Obstacle
(9)
(b) Oxide film
(c)
Crack
Fibers
(h) Fracture surface asperity
\
X
Crack
Particles
Microcracks
(j)
Transformed zone
Dislocation array
turely at a 'closure load' even before being unloaded completely. If the nominal stress intensity factor range imposed on a fatigue crack is AK( = K m a x -K m i n ), the crack-tip experiences only a lower effective value at its tip: = Kmax-Xcl
(11-22)
as a consequence of crack closure due to the plastic stretch in its wake. Kcl is the stress intensity factor of the fatigue loading cycle at which the fracture surfaces first contact upon unloading. (In practice, the fatigue crack closes gradually during unloading such that a precise value of Kcl cannot be defined. An average value, equal to the mean of the stress intensity factor at
Figure 11-17. A schematic of the various mechanisms by which the effective driving force for fatigue crack growth can be reduced, (a) Plasticity induced crack closure, (b) Oxide-induced crack closure, (c) Roughness-induced crack closure, (d) Fluid-induced crack closure, (e) Transformation-induced crack closure. (f) Fatigue crack deflection. (g) Crack bridging by continuous fibers, (h) Crack trapping by particles, (i) Crack-tip shielding by microcracks. (j) Crack-tip shielding by dislocations (from Suresh, 1991; courtesy of Cambridge University Press).
which the crack faces begin to close and that at which they are completely closed, is used to define Kcl. Furthermore, there is usually a hysteresis in the load-crack opening displacement plot between the unloading phase and the reloading phase. Consequently, the opening stress intensity factor upon reloading, Kop, is different from Kcl.) Elber's closure mechanism, which is now widely known as 'plasticity-induced crack closure', has several important implications for the characterization of fatigue crack growth, (i) The effective driving force for crack growth is influenced by the conditions of contact in the wake of the advancing crack-tip and by the very history
540
11 Cyclic Deformation and Fatigue
of crack growth, (ii) The nominal stress intensity factor does not uniquely characterize the rate of fatigue fracture. That is, cracks of different dimensions and different loading histories, but with the same nominal value of AK, can exhibit significantly different growth response, (iii) Apparent retardations can be induced in the rate of advance of a fatigue crack subjected to tensile overloads because of the possibility of creating larger plastic stretch (see Sec. 11.10 for further details). Following the work of Elber, the phenomenon of plasticity-induced crack closure was applied to explain many fatigue crack growth effects, including the influence of load ratio on fatigue thresholds and transient crack advance following the application of tensile overloads (for a review, see Suresh and Ritchie, 1984 a and Suresh, 1991). However, certain experimental observations were apparently counter to the predictions of plasticity-induced crack closure. For example, both experiments and analyses showed that crack closure due to prior plastic stretch is more dominant under plane stress than under plane strain. However, in the near-threshold fatigue
regime where predominantly plane strain conditions prevail, a significantly greater amount of crack closure is seen than at higher growth rates. This apparent contradiction was resolved in the late 1970s and in the 1980s where other mechanisms of fatigue crack closure, which take place under both plane stress and plane strain conditions, were identified. The following sections provide a description of these additional closure processes. 11.8.1.2 Oxide-Induced Crack Closure
During the near-threshold fatigue fracture of many metallic materials at low load ratios, the presence of a moist environment causes considerable oxide formation on freshly formed fracture surfaces (Fig. 11-18). Given the slow crack growth rates in the threshold regime as well as the repeated contact between the crack faces (due to some plasticity-induced closure) and the combined mode I-mode II displacements (due to microscopically tortuous crystallographic fracture), the oxide layers build up by a "fretting" process to thicknesses of the order of the crack-tip opening displace-
0=
AK 0 = 7.7
3.1 MPa./m
• notch
notch -
- 005
R - 075
Figure 11-18. Appearance of fracture surfaces of 2JCr-lMo steel (bainitic microstructure, yield strength = 500 MPa) subjected to fatigue at low and high R ratios. Note the formation of thick corrosion deposits on the fracture surface for the R = 0.05 test (after Suresh et al., 1981).
11.8 Fatigue Crack Closure and Other Retardation Mechanisms 2 1/4Cr-1 Mo Steel
8CD C
0.2 - SA 542 Class 3 Ambient >' Temperature (23 °C) / Frequency = 50 Hz y 0.1
Max. excess oxide *< thickness (cf0) L - A i r ( f l =0.05)
/
CD
\
Airffl = 0.75)V / a ~^=«
JS n ©
541
~
^
.
^
-JQ-4
E 10"5 E •E
106
5
10"7
Threshold /l
108 0
1
2
3
4
5 6 7 8 9 Crack length in mm
I
i
l
Figure 11-19. Scanning Auger spectroscopy results of fracture surface oxide thickness at low and high R ratios on the fracture surface of the bainitic 2JCr-lMo steel as a function of crack length (measured from the notch-tip) and crack propagation rates (after Suresh et al., 1981).
10 11
merit at threshold (Suresh et al., 1981). The premature contact of the crack faces resulting from the built-up of corrosion layers on crack faces is known as 'oxide-induced crack closure' (Stewart, 1980; Ritchie et al., 1980). As an example, consider the case of a bainitic, lower strength 2iCr-lMo steel. At near-threshold growth rates, the excess oxide thickness formed within the fatigue crack in this material is sufficient to completely wedge the crack closed at low load ratios. Scanning Auger spectroscopy analyses of the fracture surface oxide layers reveal that the excess oxide thickness wedging the crack faces is as much as 0.2 |jm (Fig. 11-19). This is orders of magnitude greater than the oxide thickness (~50 A) formed on a freshly polished surface of the same material exposed to the same environment for the same amount of time as a fatigue threshold test. The oxide thickness at threshold and at low load ratios is also substantially greater than that seen at higher growth rates or at higher load ratios. (At higher load ratios, even the minimum crack opening displacement of the fatigue cycle is too large to permit the re-
peated contact between the crack faces leading to oxide build-up.) If the threshold test is conducted in dry, moisture-free environments, oxide-induced crack closure is essentially suppressed. In such cases, the near-threshold crack growth rates in dry environments can be higher than in moist media where corrosion layers form at low load ratios. Such differences between dry and moist environments are virtually eliminated at high load ratios where oxide-induced crack closure is not an important consideration. These observations are illustrated in Fig. 11-20 with the aid of experimental data for the 2iCr-lMo steel which is fatigue-tested in dry hydrogen and moist laboratory air at low and high load ratios. It is important to note here that oxideinduced crack closure results in crack growth characteristics which are counter to the expectations based on traditional corrosion fatigue theories. While conventional interpretations of corrosion fatigue commonly focus on the deleterious effects of moist environments (as a result of anodic dissolution or hydrogen embrittle-
542
11 Cyclic Deformation and Fatigue
10'
: 10" r : 4
1
z
iim i J _
10 r c 10 5 g '"S
AK in ksi /In" 5 6 7 8 9 10 — 1 — 1 — 1 — I l l l
—1
20 - 1 —
30
40
1
50
II_
2 1 / 4 O - 1 Mo Steel H 2 (fl = 0.05)
^
SA 542 Class 3 Frequency = 50 Hz, Ambient Temperature **Je R H2 (R = 0.3) /** • » 0.05 1 moist H2(f? = 0.5) ^ <^**!fi<>hr*' 0.75 J air • 0.05 ] > 0.3 dry ' 0.5 V H2 t 0.75 0.05 (0.5 Hz) J n2 j> r •
= 105 -
JO) o
io-7i
i
Q.
2 10 6
IS io
Air
=
Q.
\ a • 9
.
t
P
-
• •
• •
1 lattice — " -~i 108 spacing per cycle :
-
7
10-*
r :
" o °
fl=0.75
H2^J
•
1
l
3
^Air
H = 0.05# * Threshold, AK 0 I * I I * I I I
-1 10" 9 i
4 5 6 7 8 9 1 0 20 Alternating stress intensity, AK in MPa /m
i 30
II
40
=
50 60
Figure 11-20. Fatigue crack growth rates of 2JCr-lMo steel as a function of the stress intensity factor range for tests conducted in dry hydrogen and moist laboratory air at several different load ratios and test frequencies. See text for details (after Suresh and Ritchie, 1982).
ment mechanisms), oxide-induced crack closure can lead to apparently beneficial crack growth resistance in certain materials in certain regimes of fatigue crack growth. One is cautioned that materials which benefit from enhanced closure due to corrosion layer formation in one regime of fatigue crack growth can, in fact, exhibit a deterioration in fracture resistance in a different regime of fatigue. This point is evident from the results of Fig. 11-20 where the presence of dry hydrogen (or more precisely, the paucity of moisture and the attendant absence of oxide-induced crack closure in the dry hydrogen environment) causes apparently higher growth rates in the near-threshold regime. However, hydrogen promotes a significant acceleration in fatigue crack growth rates in the Paris regime of fatigue for a variety of load ratios and test frequencies as compared to moist
air. This detrimental effect of hydrogen has been ascribed to hydrogen embrittlement phenomena in the steel. 11.8.1.3 Roughness-Induced Crack Closure
While oxide-induced crack closure rationalizes the effects of environment and load ratio in the near-threshold regime, the effects of microstructure on apparent fatigue crack growth resistance in many engineering alloys can be accounted for on the basis of 'roughness-induced crack closure' (Walker and Beever, 1979; Minakawa and McEvily, 1981; Suresh, 1985 a). At near-threshold growth rates, where the size of the plastic zone at the crack-tip is generally smaller than the microstructural sizescale (such as the grain size), crystallographic (stage I) fracture is induced in many alloys. The attendant crack path is
11.8 Fatigue Crack Closure and Other Retardation Mechanisms
highly tortuous on a microscopic scale, especially in coarse-grained materials. Given this microscopic roughness of the crack faces and the possibility of mismatch between the mating fracture surface asperities (as a consequence of such factors as inelastic deformation at the crack-tip, kinematicirreversibility of cyclic slip, etc.), one can envision the possibility of premature crack closure due to the very roughness of the fracture surfaces. Figure 11-21 contains in situ SEM observations of the closing of the crack faces, illustrating the process of roughness-induced crack closure in an underaged condition of a 7xxx series aluminum alloy. The imposition of cyclic stresses in the Paris regime of fatigue crack growth promotes a straight crack profile (the left half of Fig. 11-21 a). However, in the near-threshold regime, a rough fracture surface emerges as a result of crystallographic fracture (the right half of Fig. 11-21 a). Figure 11-21 a is a scanning electron micrograph showing the crack profile at about 35% of the peak load of the fatigue cycle. Note that the mismatch between the mating asperities on the fracture surfaces leads to crack wake contact
Time
Figure 11-21. In situ SEM observations of the development of crack closure at two different tensile load levels of a fatigue cycle in the underaged temper of a high purity 7 x 75 aluminum alloy (after Schulte et al., 1980).
543
(roughness-induced crack closure) at this tensile load. This point is evident more clearly in Fig. 11-21 b which is a micrograph taken at about 60% of the peak load. The contact between the crack faces can be seen here even at the large tensile load. Roughness-induced crack closure is more dominant for lower load ratios (where the minimum opening displacement along the crack wake is significantly smaller than the height of the fracture surface asperities), coarser-grained materials (where crystallographic failure mechanisms result in a higher degree of fracture surface roughness) and in certain microstructural conditions (such as the underaged microstructures of aluminum alloys or composites where periodic deflections are induced in the crack path when the crack-tip encounters the grain boundaries or the reinforcement phase). 11.8.1.4 Other Closure Mechanisms
In addition to the above mechanisms, certain special circumstances can also promote additional crack closure effects in both metallic and nonmetallic systems. These closure processes involve the presence of viscous fluids trapped within the walls of a fatigue crack (fluid-induced crack closure) and phase transformations which take place at the highly stressed region of the crack-tip (transformation-induced crack closure). The former closure process involving viscous fluids is governed by several concurrent and mutually competitive effects: the role of viscosity in exerting pressure on the crack walls (which increases with increasing viscosity), the extent of penetration of the viscous fluid within the crack (which decreases with increasing viscosity), and the suppression of oxide-induced
544
11 Cyclic Deformation and Fatigue
crack closure and environmental embrittlement by precluding the access of the environment to the crack-tip. A survey of available literature indicates that closure effects arising from viscous fluids are not as pronounced as those of the earlier-mentioned mechanisms. Phase-transformation-induced crack closure arises from the process whereby stress- or strain-induced microstructural changes at the crack-tip mitigate the effect of far-field loads by either shielding the crack-tip from external loads or by enhancing crack closure in the wake of the crack-tip. Martensitic transformation in TRIP (TRansformation-Induced Plasticity) steels and partially stabilized zirconia (PSZ) ceramics as well as phase changes in polytetrafluoroethylene (PTFE) are examples of situations in which transformationinduced crack closure could be effected. In many systems, phase transformations at the crack-tip are accompanied by both dilatational and shear strains. When the enlarged volume of material is left in the wake of the crack as the crack-tip advances through the transformation zone, there results a net reduction in the crack opening displacement. Under cyclic loading conditions, this process can be envisioned as a crack closure phenomena. For monotonic loading involving purely dilatational phase changes, McMeeking and Evans (1982) calculated the reduction in the apparent stress intensity factor range due to transformation to be 0.22 (1-v)
(11-23)
where hT and V{ are the height and volume of the transformation zone, sT is the transformation strain, and E and v are the Young's modulus and Poisson's ratio. Constitutive modelling of cyclic deformation using a combined dilatational-shear
transformation model has been reported by Suresh and Brockenbrough (1988). A quantitative description of fluid-induced and transformation-induced closure phenomena is provided in Suresh (1991). 11.8.2 Other Retardation Mechanisms
In addition to crack closure, several other mechanisms have also been identified based on which the macroscopic resistance of a material to fatigue fracture can be significantly altered. These processes offer limitless possibilities for microstructural modification with a view toward improving the fatigue properties. Prominent among these mechanisms is fatigue crack deflection by which the neartip driving force as well as the extent of crack closure can be altered by means of periodic deflections in the path of the crack. Crack deflection processes have received considerable attention in recent years in the context of fatigue crack growth in metals and metal-matrix composites (as well as in the context of resistance-curve (incurve) behavior in ceramic-matrix composites subject to monotonic tension). Changes in fatigue crack growth resistance are effected by crack deflection as a consequence of the following factors: (i) A cracktip oriented away from the nominal mode I growth plane has a lower effective driving force than a straight crack of the same projected length, (ii) At identical values of effective driving force, a deflected crack propagates at an apparently lower rate than a straight crack, if the average crack length is measured along the mode I growth plane, (iii) Periodic deflections in crack path enhance roughness-induced crack closure by increasing the extent of fracture surface roughness. Idealized geometrical models providing quantitative predictions of the effects of
11.9 The Growth of Short Fatigue Cracks
each of these factors have been developed by Suresh (1985 a). These calculations, incorporating the effects of the above three factors associated with crack deflection, show that the ratio of the apparent stress intensity factor for a periodically deflected crack, AKD, to that of a straight crack, AKS, of the same projected length is AKD AX,
(11-24)
"T 1 Here 9 is the angle of periodic deflection in the path of the crack, D and S are the deflected and straight segment lengths of the periodically deflected crack, and x *s the mismatch ratio (defined as the ratio of the mode II displacement to the mode I displacement of the crack faces upon first contact). In other words, the effective driving force to propagate a periodically deflected (two-dimensional, linear elastic) fatigue crack at the same rate as a straight fatigue crack is apparently larger by the factor equal to the right hand side of Eq. (11-24). At the same effective values of the driving force, the deflected crack propagates at a rate which is apparently slower than that of a straight crack by a factor equal to {[D/{D + S)]cos6] + [S/(D + S)l if the crack length is measured solely along the mode I crack plane. With the increasing demand for composite materials in many fatigue-critical structural applications, there is growing interest in tailor-making microstructures which offer improved damage-tolerance. To this end, mechanisms such as crack deflection, bridging of the crack by continuous fibers or discontinuously reinforced particles, or shielding of the crack-tip by microcracks, transformation zones or dislocations (see Fig. 11-17) have become top-
545
ics of considerable interest in fatigue research. All these phenomena promote an apparently lower effective stress intensity factor range under cyclic loading conditions, thereby leading to a slower macroscopic rate of crack extension. (Under certain loading situations, such mixed-mode far-field loading, crack-tip dislocations can result in an amplification of the stress intensity factor range at the crack-tip due to the possibility of fracture during both the loading and unloading phases of the fatigue cycle.) It is important to note here that all of the retardation phenomena illustrated in Fig. 11-17 do not give rise to any intrinsic improvements in the resistance of the material to fatigue fracture. They merely enhance the apparent or extrinsic resistance to failure. Furthermore, in many situations, especially in polycrystals and composites with highly anisotropic grains or reinforcement distribution, the apparent improvements in fatigue properties may be highly directional. Microstructural modifications which enhance crack deflection, closure or bridging in one direction of loading may, in fact, lead to a severe deterioration in the resistance to fatigue fracture in a transverse orientation. Furthermore, the improvements in fatigue crack growth resistance may be accompanied by a lowering of the resistance to fatigue crack initiation. Therefore, caution should be exercised in applying such retardation effects to the fatigue design of damage-tolerant microstructures.
11.9 The Growth of Short Fatigue Cracks The application of fracture mechanics principles to characterize the growth of fatigue cracks is implicitly based on the no-
546
11 Cyclic Deformation and Fatigue
tion of similitude. Similitude merely reflects the feature inherent in the asymptotic solutions that the scalar amplitude of the singular fields (i.e., K for linear-elastic fracture mechanics and J for nonlinear fracture mechanics) uniquely characterizes the near-tip stress, strain and deformation, irrespective of the geometrical conditions of the cracked body. Therefore, different cracked bodies with different far-field loads, crack sizes and crack geometries would be expected to undergo the same crack growth characteristics provided that the value of K at the crack-tip (for linearelastic fracture) is identical. Although such similitude concepts are reasonably valid for the growth of long' fatigue cracks (typically 10 mm or longer) in most engineering materials, there arises a serious problem in the use of fracture mechanics for characterizing the growth of small cracks (typically smaller than 1-2 mm). This socalled short crack problem (seemingly first identified by Pearson, 1975) is particularly ominous from a practical standpoint in that cracks which are much longer than the size of the near-tip plastic zone or the grain size, but are merely physically small, can propagate at rates that are orders of magnitude faster than the corresponding longer flaws when subjected to identical values of nominal AK in air or in a chemically aggressive medium. The work of a number of researchers (e.g., Kitagawa and Takahashi, 1976; Gangloff, 1981; Lankford, 1982) has shown the role of microstructure, environment and crack size in influencing the apparent fatigue response of a wide variety of engineering alloys. There are several different definitions of what constitutes a small fatigue crack (Suresh and Ritchie, 1984 b). The different types of small fatigue cracks and the origins of their apparently anomalous
growth behavior are described in the following. (i) When the size of the fatigue crack is small in relation to the characteristic microstructural dimension, continuum descriptions of crack advance become invalid. These microstructurally small fatigue flaws often exhibit crystallographic crack growth, with crack extension periodically retarded or even arrested as the crack-tip encounters a grain boundary or other microstructural inhomogeneity (e.g., Lankford, 1982). Obviously, the use of linear elastic fracture mechanics to characterize the propagation of such cracks is questionable. (ii) Even if the crack size is large compared to the microstructural dimension, linear elastic fracture mechanics is not applicable when the size of the near-tip plastic zone is comparable to the crack size. In such cases involving nonlinear fatigue fracture or mechanically small fatigue flaws, it has been suggested that crack growth can be characterized in terms of AJ (Dowling, 1977). (See Sec. 11.6.7 for a discussion of this approach.) Figure 11-22 shows an example of AJ characterization for an AISI A533B steel where the growth rates for cracks typically larger than 25 mm (filled symbols) and smaller than 0.18 mm (the remaining symbols) exhibit good correlation with AJ for different values of imposed strain amplitude in the range 0.005-0.04. As noted earlier, the use of AJ to characterize fatigue crack growth under elasticplastic conditions is by no means unambiguous. The occurrence of non-proportional loading, elastic unloading and the rapid growth of the short crack would seriously violate the conditions on which the development of the J-integral concept is predicated. Despite such basic limitations, some correlations, such as Fig. 11-22, appear to provide a reasonable characterization of
11.9 The Growth of Short Fatigue Cracks
10"
|10"
Scatter band from long crack data (LEFM)
5
6
E c
i 10-7
10"
8
10- 3
10"2
10"1
A J in MPam
Figure 11-22. Characterization of elastic-plastic fatigue crack growth in terms of AJ for an AISI A533B steel. The closed symbols refer to long fatigue cracks (crack size longer than 25 mm) and the remaining symbols refer to small cracks (crack size smaller than 0.18 mm). All flaws were subjected to far-field strain amplitudes in the range 0.005-0.04 (after Dowling, 1977).
nonlinear fatigue fracture in certain cases. The issue of elastic-plastic fracture mechanics also arises in situations when a fatigue crack emerges from the tip of a stress concentration. In this instance, the entire crack may be fully submerged in the plastic strain field of the notch. Although no reliable method for characterizing the growth of such mechanically small (nonlinear) flaws is presently available, the cyclic J concepts have also been applied to the notch-fatigue problem involving short cracks. (iii) Even when the crack size is large compared to the scale of the microstructure or plastic zone (i.e., when the crack is
547
suitable for characterization on the basis of linear elastic fracture mechanics), the growth of the crack may exhibit apparently anomalous behavior because of the physical smallness of the crack (typically smaller than several millimeters). The origin of this physically small crack problem lies in several issues: (a) A physically small crack with a small wake undergoes a relatively lower level of crack closure than a longer flaw with a fully developed wake. Therefore, the growth of small flaws can be apparently faster than the corresponding longer flaws as a result of the differences in the level of crack closure, (b) The region of X-dominance (see Sec. 11.6.2) spans a distance of about 10% of the crack size. Since this zone is much smaller for a short flaw than for a long crack, the higher order terms in the asymptotic expansion, such as the nonsingular T-term and the terms which vanish at the crack-tip [Eqs. (11-12) and (11-13)], cannot be ignored in the determination of K. It is often found that the apparent differences between the growth rates of physically small and long fatigue cracks are merely an outcome of the test techniques employed to measure crack growth rates. In the standard methods for the measurement of fatigue thresholds, AK0, involving progressive 'load-shedding' from high AK levels (Standard E-647-86a of the American Society for Testing and Materials, Philadelphia), an apparently high value of AK0 can result because of the development of crack closure. However, if the AK0 for tension fatigue is determined under increasing stress intensity conditions using small through-thickness cracks introduced previously in cyclic compression (see Sec. 11.12), a more conservative (lower) threshold results (Suresh, 1985 b). Similarly, fatigue threshold tests conducted at high load ratios and constant Kmax test
548
11 Cyclic Deformation and Fatigue
conditions also produce lower AK0 even for long fatigue cracks because of the preclusion of crack closure effects (e.g., Herman et al, 1988). (iv) For some metallic systems and loading conditions, the growth of small cracks (of the order of a mm or so in size) is essentially the same as those of longer cracks subjected to the same value of AK in inert environments. However, the smaller flaws exhibit a significantly faster growth in aggressive media such as distilled water or sodium chloride solution than the longer flaws. This (chemically short fatigue crack) effect arises from the dependence of environmental interaction on the crack size as well as the effect of crack size on the transport of the embrittling species to the cracktip and on surface reactions (Gangloff, 1981). Whatever its origin, the existence of the short crack anomaly poses considerable challenge in the characterization of fatigue crack growth. Since the use of conventional fracture mechanics methodology for short flaws may result in dangerously nonconservative estimates of useful fatigue life, much further research is needed to develop reliable characterization methods. It is perhaps worth concluding this section by noting that the study of short cracks offers one possible way in which the seemingly contradictory philosophies of total-life approaches and defect-tolerant approach can be merged. In the stress-life approach, the fatigue threshold (in smooth specimens) is characterized by the endurance limit, Acre. In cracked components, the threshold for the onset of crack growth under linear elastic loading conditions is characterized in terms of the threshold stress intensity factor range, AK0. Accordingly, as a fatigue crack nucleates in a smooth specimen and becomes long enough to be characterized in terms of
fracture mechanics, the threshold conditions would be expected to change, with respect to the crack size, from a threshold stress range Acrth = Aae to the nominal value of threshold stress intensity factor range, AK0. The transition crack size, a0, for a change from the cyclic stress-based analysis to the fracture mechanics analysis can be written as (Kitagawa and Takahashi, 1976) 1 AK( Ac; AKth = AK0 for
a> a0
Aath = Acre
a < a0
a0
=
for
(11-25)
(On the basis of the similitude concept of fracture mechanics, the intrinsic threshold stress intensity factor range, AKth, for a long crack should be independent of crack size such that AKth = AK0.) Figure 11-23 shows experimental data for a wide range of ferrous and nonferrous alloys of yield strength values 30-770 MPa where stressbased characterization at crack lengths below a0 and stress intensity characterization for crack length above a0 appear to hold.
11.10 Crack Growth Retardation During Variable Amplitude Fatigue Although scientific research on fatigue of materials primarily deals with constant amplitudes of stresses or strains, engineering components are invariably subjected to varying amplitude fatigue involving periodic overloads, block overloads or random loads. The classical approach to variable amplitude fatigue design involves use of the Palmgren-Miner linear damage rule in which the amount of 'damage' introduced in each stress cycle varies with the amplitude of stresses in that cycle. In this approach, if a cracked component is sub-
11.10 Crack Growth Retardation During Variable Amplitude Fatigue
549
0.1 0.01
1.0 -
0.5
0.2
0.1 / 0.01
1 0.1
i
i
i
1 ala0
10
100
jected to n} stress cycles of different amplitudes (Ao-jJ = l,p) the linear summation of fractional damage over the entire number of stress cycles must equal unity: Z ^ = l j=i
(H-26)
Nfj
where Nfj is the number of cycles to failure if all the stress amplitudes had been equal to AGJ (with N^ determined from S-N curves). This approach predicts that the larger the stress amplitude, the larger is the amount of damage introduced. Although this expectation based on total-life ap-
Figure 11-23. The approach due to Kitagawa and Takahashi (1976) to characterize the growth of short flaws (below the transition size a0) in terms of the endurance limit and the growth of long flaws (above a0) in terms of the threshold stress intensity factor range. See text for details. The data points, taken from Tanaka et al., 1981, are for a wide variety of ferrous and nonferrous alloys of yield strengths in the range 30-770 MPa.
proaches may hold in some cases pertaining to crack initiation or to random loading involving a narrow band of load spectra, the predictions of the Palmgren Miner rule generally run counter to a wide variety of experimental data. The clearest evidence against the validity of this method is found in crack growth data for variable amplitude fatigue involving tensile overloads where it is known that the application of a tensile overload to a fatigue crack in a ductile solid can result in a severe retardation (or even complete arrest) of crack growth.
550
11 Cyclic Deformation and Fatigue
In industrial practice, one often needs to relate the accumulated fatigue damage in a structural component subjected to variable amplitude fatigue with the fatigue life of the same material determined in the laboratory (using smooth specimens) under constant amplitude loading conditions. Several 'cycle counting' techniques have been developed over the years in an attempt to reduce complex loading histories to a series of discrete events so that the appropriate amount of 'cyclic damage' can be properly accounted for by recourse to laboratory data on smooth fatigue specimens. These techniques include: rainflow counting, range pair, level crossing, and peak
counting. Among these, the rainflow counting algorithms have found particularly widespread use, in conjunction with digital computers, for fatigue life prediction in the context of both local strain approach and defect-tolerant approach (e.g., Mitchell, 1979; Fuchs and Stephens, 1980). Figure 11-24 is a schematic of the transient crack growth characteristics following a single tensile overload applied to a metallic material. During the application of the tensile overload, one generally observes a burst of crack extension (which creates a stretch zone on the fracture surface). If constant amplitude fatigue loading is resumed in the post-overload region at
Nominal load ratio, fl = K^ mm / K1 max Baseline stress intensity range, A K B = K 1 m a x - K 1 m j n Overload stress intensity range, A K o L = K 2 m a x - Kimm Overload factor, rOL= K2max/ K-imax (a)
o Ka
8 0) CO
Time, t
o Number of cycles, N
Figure 11-24. A schematic of the transient crack growth effects following the application of a single tensile overload to a metallic material. The nomenclature associated with various overload parameters is defined (after Suresh, 1983).
11.10 Crack Growth Retardation During Variable Amplitude Fatigue
the same nominal AK level as in the preoverload regime, the rate of crack growth progressively slows down, reaching a minimum value after a delay distance ad. (The extent of crack deceleration or even the possibility of crack arrest is determined by the magnitude of the overload in relation to the baseline cyclic load, the stress state, environment, load ratio, and crack closure effects.) Beyond this point, crack growth accelerates until the rate of crack propagation (da/dN)K reaches a value representative of the constant amplitude AK level (da/diV)B. Transient crack growth occurs over a total distance a* and a total number of cycles AT*. Various arguments have been advanced to rationalize the retardation of fatigue cracks due to the application of overloads. Although a single mechanism is unlikely to govern completely all the transient effects associated with variable amplitude fatigue, a combination of the following mechanisms provides realistic descriptions of the prominent features of post-overload retardation. (i) Residual Compressive Stresses at Crack-tip: It was shown in Sec. 11.6.4 that there exists a zone of residual compressive stresses within the cyclic plastic zone of a tensile fatigue crack. The application of a tensile overload leads to an enlargement in the extent of the residual compressive field. It has been argued by many researchers that this enlarged compressive zone "clamps down" the crack-tip subjected to a lower magnitude of post-overload AK and causes a retardation in crack growth. This retardation effect is expected to persist until the current crack-tip traverses through the greatest prior elastic-plastic boundary. These yield zone-type models are also widely used by the aerospace industry for quantitative predictions of variable amplitude fatigue crack growth. However, such
551
models cannot rationalize the existence of delayed retardation. Furthermore, the predictions of total delay distance on the basis of plastic zone dimensions often deviate significantly from experimental observations. (ii) Plasticity-Induced Crack Closure: It was suggested by Elber (1970) that enhanced residual plastic stretch created by the overload causes higher levels of plasticity-induced fatigue crack closure as the post-overload crack advances through the overload plastic zone. This hypothesis has been substantiated by experimental data which show enhanced levels of crack closure following overloads. However, microstructural and environmental effects on transient crack growth cannot be accounted for by this mechanism alone. (iii) Changes in Crack Geometry Involving Crack-tip Blunting or Branching: Certain materials, such as aluminum alloys, exhibit a propensity for blunting or branching/deflection of the crack-tip upon application of a spike overload. The changes in the near-tip stress intensity factor range resulting from such modifications of crack-tip geometry can play some role in aiding post-overload retardation. (iv) Activation of Near-Threshold Mechanisms in the Post-Overload Regime: If the effective AK in the post-overload regime is reduced by one of the above mechanisms to a value typical of the near-threshold regime (even though the nominal AK is typical of the Paris regime), near-threshold mechanisms, such as stage I fracture and oxide-induced crack closure, may be activated following the application of overloads (Suresh, 1983). These additional changes induced by the overloads prolong the delay in crack growth. In addition to the above mechanisms, crack-tip strain hardening is also considered a contributing factor to transient ef-
552
11 Cyclic Deformation and Fatigue
fects. All of the aforementioned effects also play a role in influencing the large transient effects commonly seen during the application of block overloads. Furthermore, the application of compressive overloads is sometimes detrimental to fatigue fracture resistance because flattening of the fracture surface asperities, resharpening of the blunted crack-tip or the generation of residual tensile stresses by the compressive overload can accelerate crack growth.
11.11 Fatigue at Notches In the stress-life approach to fatigue, the fatigue notch factor provides an estimate of the severity of stress concentrations, analogous to the theoretical elastic stress concentration, Kt, in monotonic loading. The fatigue notch factor, K f , is defined as the ratio of the unnotched bar endurance limit to the notched bar endurance limit. Kf is generally smaller than Kt. For high strength materials and large notch-tip radii, the two factors are comparable. Sometimes, a parameter known as the fatigue notch sensitivity index, q = (Kf — 1)/ (Kt — 1), is used to gauge the acuteness of a stress concentration, q varies from zero for no notch effect to unity for the full effect predicted by the elasticity theory. K{ and q are generally determined from experimental measurements or empirical engineering approximations and are listed in many fatigue handbooks and data sources (e.g., Peterson, 1959). The stress-life approach is often unsuitable for notch fatigue problems because of appreciable plastic deformation at the root of the notch. Consequently, a more popular total-life approach to handling notch fatigue involves the so-called local strain approach'. This method relates the stresses and strains at the tip of the notch to far-
field loading via constitutive relations determined from simple laboratory test specimens, often subjected to uniaxial loading. This process usually involves two steps: (i) The determination of local stresses and strains in the vicinity of the notch-tip for given specimen geometry and far-field loading, (ii) The prediction of fatigue life that is appropriate for the deformation conditions prevailing in the region immediately ahead of the notch-tip (often using the low cycle fatigue data obtained on smooth laboratory specimens). Fracture mechanics concepts are also applicable to notch fatigue problems provided that the strain field associated with the notch-tip deformation is properly accounted for and that small-cycle yielding conditions exist (for linear elastic fracture mechanics). A review of various approaches to notch fatigue problems can be found in Dowling etal. (1977).
11.12 Compression Fatigue of Metals and Nonmetals The vast majority of scientific and applied research has focused on the initiation and growth of fatigue cracks in cyclic tension loading. However, it has become increasingly more evident in the past decade that fully compressive cyclic loads can lead to both initiation and subcritical propagation of cracks. Notched plates of metallic materials, when subjected to cyclic compression loads, undergo model fracture along the plane of the notch in a direction macroscopically perpendicular to the farfield compression axis (e.g., Hubbard, 1969; Reid etal., 1979; Suresh, 1985b). The cracks decelerate progressively before arresting. This crack growth process is promoted by the inducement of residual tensile stresses locally in the vicinity of the
11.13 Fatigue of Ceramic Materials
notch-tip upon unloading from the farfield peak compressive stress. For an elastic-perfectly plastic solid containing a sharp nonclosing notch, the residual tensile stresses are of the order of the flow stress in compression spanning a distance of the order of the cyclic plastic zone. This zone of reversed flow is about one-fourth the size of the monotonic plastic zone directly ahead of the notch-tip (see Fig. 11-25). The compression fatigue crack arrests completely once the residual tensile stresses are progressively exhausted and the fraction of the loading cycle during which the crack remains open gradually decreases with an increase in crack length (as a consequence of crack closure). Although the occurrence of compression fatigue fracture has traditionally been associated with cyclic plasticity, recent work has established that brittle solids such as ceramics and ceramic composites (Ewart and Suresh, 1987; Suresh, 1990) and semicrystalline and amorphous polymers (Suresh and Pruitt, 1991) also exhibit mode I fracture under cyclic compression. A particularly noteworthy feature of this phenomenon is that both metallic and nonmetallic materials exhibit a macroscopically similar crack growth behavior in cyclic compression despite vast differences in their microscopic deformation processes. The reason for this universality is Oyy
x
0
Time
w
Figure 11-25. The generation of a zone of residual tensile stresses at the tip of a sharp, nonclosing notch in an elastic-perfectly plastic solid subjected to one cycle of zero-compression-zero loads.
553
the inducement of a zone of residual tensile stresses (confined to the notch-tip region) upon unloading, irrespective of whether the deformation mechanism leading to this permanent damage is microcracking, shear banding, dislocation plasticity, craze formation, creep or phase transformation. Figures ll-26a-c show examples of mode I fatigue fracture under fully compressive cyclic loads in an Al-3.5 wt.% Cu alloy reinforced with 20 vol. % SiC particles, a polycrystalline alumina (grain size = 18 jum), and an amorphous polystyrene polymer (weight average molecular weight = 240000 and number average molecular weight « 86000), respectively. Note in Fig. 11-26 c the presence of an intense deformation zone at the crack-tip containing a craze, with the craze oriented parallel to the crack plane and perpendicular to the far-field compression axis. The formation of the craze in a direction normal to the far-field compressive axis is indicative of the generation of residual tensile stresses on the crack plane, since crazes always grow perpendicular to the local tensile direction.
11.13 Fatigue of Ceramic Materials The majority of fatigue research over the past 150 years has been devoted to the study of metallic materials. However, with the advances in the synthesis of modern ceramics intended for potential applications such as high temperature propulsion systems, there is a growing interest in the study of fatigue in these brittle materials. Fatigue research on ceramic materials has long been hampered by the presumed absence of a mechanical cyclic effect on account of the paucity of appreciable plastic deformation. This lack of effort is also an outcome of the implicit assumption that
554
:-- ^ *y>w.
11 Cyclic Deformation and Fatigue
-5.
Figure 11-26. Examples of mode I fatigue crack growth normal to the compression axis (vertical direction) in (a) Al-3.5Cu alloy reinforced with 20 vol.% of SiC particulates, (b) an alumina ceramic and (c) polystyrene (from Suresh and Pruitt, 1991).
cyclic (dislocation) plasticity is a requirement for fatigue and that materials which do not exhibit cyclic plasticity may not undergo true mechanical fatigue effects. Although there has long been sporadic, and often controversial, evidence pointing to possible mechanical fatigue effects in ceramic materials (e.g., Williams, 1956; Guiu, 1978; Evans, 1980), more conclusive results
on mechanical fatigue in ceramics have emerged in the last several years. The first unequivocal documentation of a purely mechanical fatigue failure in a wide variety of brittle ceramics was reported in connection with crack initiation and growth in cyclic compression (Ewart and Suresh, 1987; Suresh and Brockenbrough, 1988). This process, described in the preceding section, is specific to cyclic fatigue loading conditions and is distinctly different from the monotonic compression failure of brittle solids where axial splitting rather than mode I crack growth is observed. The mechanics of cyclic compression fracture in ceramics is also similar to that seen in ductile metallic materials and is influenced by the effects of load ratio, crack closure and debris particles trapped within the crack faces. There has also been recent experimental data which indicate that the rates of fatigue crack growth in cyclic tension or tensioncompression loading at room temperature can be significantly higher than those seen under quasi-static tension (Reece et al., 1989; Dauskardt et al, 1990; Suresh, 1990). Figure 11-27 shows the variation of fatigue crack growth rates da/dN as a function of the maximum stress intensity factor Kmax for fully-reversed (R = - 1 ) cyclic loading (frequency, vc = 5 Hz) in polycrystalline alumina (grain size ^lOjum) at room temperature. Also shown in this figure are static crack velocities da/dt plotted on the same scale by using the conversion da/dN = vc (da/dt). It is seen that the cyclic crack growth rates are up to two orders of magnitude faster than the static load crack velocities. It is interesting to note that the exponent m in the Paris equation, da/dN = C(AK)m for the alumina is as high as 14, while it is generally in the range 2 - 4 for most metallic materials. The exponent takes even higher values (as high as 42) for
11.13 Fatigue of Ceramic Materials 1 0 6 r—
§> 10" 7 E c •D
10£
10£ I
2
3
4
5
Kma* in MPaVrff
Figure 11-27. Variation of fatigue crack growth rates (at R = — 1 and vc = 5 Hz) in polycrystalline alumina (filled circles) as a function of the maximum stress intensity factor. Also shown are the crack velocities under static loads (filled triangles) plotted on a similar scale using the procedure described in the text (after Reece et al, 1989).
transformation-toughened zirconia fatiguetested at room temperature. Table 11-3, taken from a recent literature survey by the author, provides the tension and tensioncompression fatigue crack growth characteristics of a variety of ceramics and ceramic composites. As categorized by Suresh (1990), mechanical fatigue effects in ceramic materials (often termed cyclic fatigue in the ceramics literature) have been documented on the basis of the following definitions: (i) A mechanical phenomenon whereby the mode of fracture under cyclic loading occurs distinctly differently under cyclic loading conditions than under monotonic tension or compression. (The compression fatigue effects described above are examples of this mechanical fatigue process.) (ii) A micromechanical phenomenon whereby cyclic damage is induced by kinematic irreversibility of cyclic deformation (due to microcracking, martensitic trans-
555
formation, creep, frictional sliding along grain boundaries and interfaces), analogous to slip irreversibility in metal fatigue. [This phenomenon has been documented in room temperature compression fatigue (Suresh and Brockenbrough, 1988) as well in high temperature tension fatigue (Han and Suresh, 1989) of ceramics and ceramic composites.] (iii) A mechanical phenomenon or a combined mechanical-environmental effect whereby the growth rates in fatigue are vastly different from those seen under static loads even when the basic mechanisms of failure are the same in the two cases (e.g., Reece et al., 1989; Dauskardt et al, 1990). Although there have been a number of studies which show large differences between static and cyclic crack growth rates in ceramics (e.g., Fig. 11-27), the underlying failure mechanics appear very similar. It has been suggested that such apparent differences arise in ceramics primarily as a consequence of contact between the faces of the crack under cyclic loading conditions (e.g., Evans, 1980; Reece et al, 1989) and not as a result of any differences in the deformation processes ahead of the cracktip. Such wake effects at room temperature include wedging of the crack faces by debris particles or uncracked grain facets. However, detailed transmission electron microscopy of crack-tip damage at elevated temperature in ceramics and ceramic composites indicate that the mechanisms of near-tip deformation can be affected by whether static or cyclic loads are imposed (Han and Suresh, 1989; Suresh, 1990). Furthermore, the presence of any pre-existing glass phase or the in situ formation of amorphous phases in the elevated temperature environment is known to influence significantly the rates of both static and cyclic fracture. Since the viscous flow of the
556
11 Cyclic Deformation and Fatigue
Table 11-3. A survey of available data on tension or tension-compression fatigue of ceramic materials. Listed are experimentally determined constants in the Paris relationshipjor fatigue crack growth, da/dN = C (AK)m. C and AK are listed in units of m/cycle • (MPa j/m)~ m and MPa j/m, respectively. (From Suresh, 1991. Courtesy of Cambridge University Press.) Test conditions
C
m
AX range
Al 2 O 3 a (99% pure)
vc = 5 Hz, R = - 1.0 room temp.
1.1 xlO" 1 1
14 ±5
2.7-4.0
Al 2 O 3 b (90% pure)
vc = 0.13Hz, JR = 0.15 T=1050°C vc = 2Hz, # = 0.15 T=1050°C
2.8 xlO" 1 0
10
1.0-3.0
6.3 xlO" 1 1
8
2.0-3.5
vc = 50 Hz, R = 0.1 room temp. vc = 50 Hz, R = 0.1 room temp. vc = 50 Hz, R = 0.1 room temp. vc = 50Hz, # = 0.1 room temp.
2.0xl0~ 1 4
21
1.5-2.1
5.7xlO~ 28
24
5.2-7.2
1.7 x l 0 ~ 4 8
42
7.7-10.0
4.9 xlO" 2 2
24
3.0-4.2
4.5xl0~ 1 0
7
3.5-6.0
4.0xl0~ 1 0
4
3.5-6.0
Material
Mg PSZC (over-aged) (peak strength) (peak toughness) (low toughness) Al 2 O 3 -33vol.% SiC whiskers d
vc = 0.1Hz, # = 0.15 T=1400°C vc = 2Hz, # = 0.15 T=1400°C
a
After Reece et al. (1989); direct push-pull tests on wedge opening load specimens with AX = X max . After Ewart (1990) and Suresh (1990); four-point flexure specimens. c After Dauskardt et al. (1990); compact tension specimens. The peak strength, peak toughness and over-aged conditions of Mg-PSZ were obtained by heat treating at 1100°C for 3, 7 and 24 h, respectively. The low toughness condition pertains to the as-fabricated material. d After Han and Suresh (1989) and Suresh (1990); four-point flexure specimens. b
glass phase in the elevated temperature environment is sensitive to loading rates (and test frequency), one would expect differences between the fracture characteristics seen under static and cyclic loads. Such differences, including frequency effects on fatigue fracture, have recently been published for polycrystalline alumina (Ewart, 1990) and alumina reinforced with SiC whiskers (Han and Suresh, 1989; Suresh, 1990). It should be noted that very little information is presently available on the origin of fatigue effects in ceramic materials because of the limited number of investiga-
tions conducted in this field. Although experimental data on fatigue crack growth in ceramics are beginning to emerge, further work is necessary before a clearer understanding of mechanical fatigue processes can be obtained.
11.14 Cyclic Deformation and Fatigue of Polymers A vast amount of literature exists on the fatigue behavior of polymeric materials. Although many aspects of cyclic deforma-
11.14 Cyclic Deformation and Fatigue of Polymers
tion and fracture in semi-crystalline and amorphous polymers remain poorly understood, this area of research is becoming a topic of considerable practical significance owing to the increased use of plastics in structural applications. A number of reviews and a research monograph are available on the fatigue of polymers (e.g., Hertzberg and Manson, 1980, 1986; Takemori, 1990). Further details on the plastic deformation and fracture of polymers can also be found in Chaps. 10 and 15 of Vol. 12 in this Encyclopedia series. In this section, a brief review of the similarities and differences in the cyclic deformation characteristics and fatigue crack growth behavior of polymers and metals is provided. Under monotonic tension, the most prominent deformation processes in polymers involve crazing and shear flow. While shear flow in polymers has the connotation of a ductile deformation process (analogous to that found in metals), crazing is a pseudo-brittle process. The craze zone ahead of a crack in a polymer finds an analogy in the plastic zone ahead of a crack in a metal. A craze contains fibrils of highly aligned molecules; the fibrils are separated by porous regions. The density of the craze itself is 20-40% of the bulk density of the polymer. Crazes generally form in a direction normal to a tensile principal stress. Under a monotonic tensile stress, the craze extends by taking more fibrils from the surrounding matrix. Comprehensive reviews of crazing in polymers can be found in Kramer (1982) and Kramer and Berger (1990). (See also Chapters 10 and 15 of this Volume for further details.) Under cyclic loading conditions, the deformation characteristics of polymers are seemingly different from those of metals. These differences merely reflect the distinctions in their homologous temperature (vis-a-vis the fatigue test temperature),
557
thermal diffusivity and the basic structural unit, (i) As noted in Sees. 11.2 and 11.3, metallic materials exhibit cyclic hardening or softening depending on prior loading history, heat treatment conditions and slip mode. However, polymers always exhibit only cyclic softening. Even cyclic stability is very seldom seen in polymers (Rabinowitz and Beardmore, 1974). (ii) Since room temperature can be a significant fraction of the homologous temperature, and since viscous damping effects are common in polymers, cyclic loading at high frequencies (typically in excess of 30 Hz) can lead to hysteretic heating and thermal softening in plastics. On the other hand, extremely slow strain rates and frequencies can lead to creep. This strain rate sensitivity at room temperature is therefore markedly more pronounced in polymers than in metals, (iii) While cyclic plasticity is the predominant deformation mode in the fatigue of metals, polymers exhibit a variety of homogeneous deformation processes (such as molecular chain disentanglement, reorientation or slip, or crystallization) or heterogeneous deformation processes such as craze formation and shear banding. In fact, crazes play essentially the same role in initiating a fatigue crack in polymers as persistent slip bands do in metals. The effects of molecular variables on crazing and shear flow during fatigue have been the subject of many investigations (e.g., Sauer and Hara, 1990). The stress-life and strain-life approaches discussed earlier for metallic materials are also widely adapted for polymeric materials. (In the latter case, however, special precaution should be taken to ensure that the mechanical fatigue effects are isolated from possible thermal effects arising from hysteretic heating.) Similarly, the fatigue crack growth characteristics of polymers are also characterized in terms of
558
11 Cyclic Deformation and Fatigue
linear elastic fracture mechanics using the Paris-type correlation. A comprehensive survey of experimental data on fatigue crack growth can be found in Hertzberg and Manson (1980, 1986). Several distinct fracture surface features and failure modes also occur in polymers. (i) At high AK levels characteristic of crack growth rates in excess of 5 x 10 ~ 4 mm/cycle, polymers exhibit striations on the fracture surface in much the same way as that seen in metals (Fig. 11-15). Figure 11-28 a shows an example of fatigue striations in polymethylmethacrylate (PMMA). The spacing between adjacent striations correlates very well with the
macroscopic fatigue crack propagation rates. Similar correlations have been reported by Hertzberg and Manson (1980) for several other polymers. (ii) At lower AK levels, many polymers exhibit discontinuous growth bands (DGBs) which have apparent resemblance to fatigue striations. However, DGB spacing is often many times greater than the crack growth rate per cycle indicating that the crack does not advance in each stress cycle, but rather jumps in bursts after tens or hundreds of cycles. Figure 11-28 b shows an example of discontinuous growth bands in polyvinylchloride (PVC). The burst of discontinuous crack growth in polymers is often accompanied by craze formation coplanar with the crack. Yield zone models similar to the Dugdale strip yield model for metals have been employed to rationalize the discontinuous crack growth by craze formation (Skibo et al., 1977). Here the maximum opening of the craze at the crack tip, Scz, is estimated from the Dugdale expression for crack opening displacement where , oc-
Figure 11-28. (a) Striation formation in PMMA. (b) Discontinuous growth bands in polyvinyl chloride. Arrows in both figures indicate the crack growth direction. (Photos courtesy of R. W. Hertzberg, Lehigh University.)
(11-27)
acz is the stress at which crazing occurs. It has been proposed by Skibo et al. that during cyclic loading, the accumulation of fatigue damage results in a gradual increase in 3CZ even though the crack tip remains stationary. The crack front jumps when 8CZ reaches a certain critical value. Figure 11-29 shows a sequence of micrographs which illustrate the process of discontinuous crack growth and crack-tip crazing in PVC. Here the positions of the crack tip and of the tip of the craze are indicated. More recent studies by Konczol et al. (1984) and Doll and Konczol (1990), employing optical interferometry, have also provided detailed observations of craze
11.15 Concluding Remarks
Stretch zone $3
New crack tip
Craze
(a)
A/=0
N=100
559
formation ahead of fatigue crack tips. Although there is at present no complete understanding of fatigue crack advance through crazes, small-angle X-ray scattering studies (e.g., Mills et al, 1985; Brown et al., 1987) have shown evidence for fibril buckling within the crazes during fatigue. (iii) In some polymers, short surface cracks emanating from free surfaces advance by a combination of crazing (coincident with the crack plane) and shear banding (oriented asymptotically at 45° to the crack plane). The resulting crack profile resembles the Greek letter epsilon (e). This fracture process, observed in polycarbonate, polysulfone, polyester carbonate copolymers and polyacrylate block copolymers, has been termed the epsilon discontinuous crack growth (Takemori, 1984). (iv) At high AK regimes or high test temperatures, the epsilon discontinuous growth process gives way to a pure shear fracture where failure takes place along one of the shear bands.
N=200
11.15 Concluding Remarks
A/=300
A/=450 (b)
Figure 11-29. (a) A schematic representation of craze formation at the tip of a fatigue crack, (b) Micrographs showing discontinuous fatigue crack growth and crack-tip crazing in PVC. The arrows on the left indicate positions of the fatigue crack tip and the arrows on the right indicate the tip of the craze. JV denotes the number of stress cycles. (After Hertzberg and Manson, 1973. Photo courtesy of R. W. Hertzberg, Lehigh University.)
This review has been aimed at providing a perspective of cyclic deformation and fatigue fracture in materials. Although the discussion has been necessarily brief, major advances in the field have been addressed and areas in which there is a lack of quantitative understanding have been pinpointed. Particular attention has been devoted to providing a survey of both the mechanics and micromechanistic aspects, as well as of the fatigue behavior of metals, ceramics and polymers. A comprehensive list of references is provided. The discussions presented here readily reveal the complexity of the microscopic and macroscopic processes by which fatigue failures occur. Although further re-
560
11 Cyclic Deformation and Fatigue
search work is needed in several areas, significant advances have also been made in many aspects of fatigue. Comprehensive experimental information is available, thanks to modern developments in electron microscopy, on the microstructural and substructural changes by which cyclic deformation leading to 'permanent damage' occurs. Similarly, research work in the last several decades has provided a wealth of information on the effects of various microstructural, environmental and mechanical factors on the growth of fatigue cracks. Considerable success has been attained in developing reliable characterization methodology for estimating the growth life of long fatigue cracks, particularly in terms of linear elastic fracture mechanics. Such methods are presently in widespread use in a variety of engineering industries. However, further work is necessary in order to develop quantitative models and design methodology for fatigue crack initiation, environmentally-assisted fracture, multiaxial fatigue, and the complex area of small fatigue cracks (which often falls beyond the bounds of continuum analyses). These topics inevitably require collaborative work among researchers in a number of diverse disciplines. It is perhaps worth highlighting here some ostensible differences in the guidelines for microstructural design offered by the various approaches to fatigue. For example, improvements to fatigue behavior on the basis of total-life approaches would generally call for higher strength, finergrained and wavy-slip microstructures. On the other hand, enhancing the fatigue crack growth resistance on the basis of defect-tolerant characterization methodology, especially in the near-threshold regime where a large fraction of the crack growth life is expended, usually requires lower-strength, coarse-grained and planar-slip microstruc-
tures. These apparently conflicting views of what leads to improved fatigue resistance are merely a reflection of the distinctions in how the crack initiation and crack growth stages of fatigue are included in the quantitative analysis of useful fatigue life. In the total-life approaches to fatigue, crack initiation constitutes a significant fraction of fatigue life. However, the defect-tolerant approach deals with only the crack growth life. This point serves to illustrate as to how a balance must be struck between modelling of crack initiation and crack growth for a complete understanding of the fatigue behavior. The discussions presented in this review on the topic of short fatigue cracks (e.g., Fig. 11-23) also provide some helpful ways in which the seemingly diverse viewpoints of the total-life approaches and the defect-tolerant approach can be interpreted in a unified manner. While fatigue of metals and alloys has long been the subject of fundamental and applied research, recent developments in materials science call for the extension of existing knowledge and the generation of new information in connection with the fatigue of advanced nonmetallic systems and composites. Research on metallic materials over the past several decades has highlighted the role of kinematic irreversibility of cyclic slip in initiating and advancing a fatigue crack. However, there is a growing body of evidence which points to the existence of mechanical fatigue effects in crystalline materials which do not exhibit appreciable dislocation motion, such as ceramics at room temperature. In these cases, kinematic irreversibility of microscopic deformation under cyclic loads can promote fatigue effects which apparently resemble metal fatigue. The microscopic mechanisms of deformation in these nonmetallic materials exhibiting true fatigue effects can be as diverse as microcracking, martensitic
11.17 References
transformation, frictional sliding along grain boundaries and interfaces, crazing or creep cavitation. A study of cyclic deformation and fatigue crack growth in ceramics, polymers, inorganic composites and organic composites necessarily requires a priori knowledge of their constitutive response under both monotonic and cyclic loading conditions and under complex stress states. These new avenues provide challenging prospects for researchers pursuing studies of fatigue in advanced engineering materials.
11.16 Acknowledgements The author gratefully acknowledges the support of the U.S. Department of Energy (Contract No. DE-FG02-84-ER-45167) and the Office of Naval Research (Contract No. N00014-89-J-3099) during the preparation of this manuscript. Thanks are due to Profs. H. Mughrabi and E. J. Kramer for many helpful comments on the manuscript.
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Manson, S. S. (1954), National Advisory Commission on Aeronautics, Report No. 1170. Cleveland: Lewis Flight Propulsion Laboratory. Masing, G. (1926), Proc. Second Intl. Conf Appl. Mech. Zurich, pp. 332-335. May, A. N. (1960), Nature 185, 303. McClintock, F. A. (1963), Fracture of Solids: Drucker, D. C , Gilman, X X (Eds.). New York: Wiley, pp. 65-102. McClintock, F. A. (1967), M.I.T Report, Cambridge, MA; cited in Pelloux (1969). McMeeking, R. M., Evans, A. G. (1982), /. Amer. Ceram. Soc. 65, 242. Mills, P. X, Brown, H. R., Kramer, E. X (1985), J. Mater. Sci. 20, 4413. Minakawa, K., McEvily, A. X (1981), Scripta Met. 15, 633. Mitchell, M. (1979), Fatigue and Microstructure: Meshii, M. (Ed.). Metals Park: American Society for Metals, pp. 385-437. Moosbrugger, X C , McDowell, D. (1989), /. Eng. Mater. Tech. Ill, 87. Morrow, X D. (1965), Internal Friction, Damping and Cyclic Plasticity, Special Technical Publication 378. Philadelphia: American Society for Testing and Materials, pp. 45-60. Mroz, Z. (1967), J. Mech. Phys. Sol. 15, 163. Mughrabi, H. (1978), Mater. Sci. Eng. 33, 207. Mughrabi, H. (1980), The Strength of Metals and Alloys: Haasen, P., Gerold, V., Kostorz, G. (Eds.). Oxford: Pergamon Press, Vol. 3, pp. 1615-1639. Mughrabi, H. (1983), Acta Met. 31, 1367. Mughrabi, H., Ackermann, F, Herz, K. (1979), Fatigue Mechanisms, Special Technical Publication 675. Philadelphia: American Society for Testing and Materials, pp. 69-105. Mughrabi, H., Wuthrich, Ch. (1976), Phil. Mag. A, 33, 963. Neumann, P. (1969), Acta Metallurgica 17, 1219. Neumann, P. (1983), Physical Metallurgy: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: Elsevier Science, pp. 1554-1593. Neumann, P. (1986), Mater. Sci. Eng. 81, 465. Neumann, P., Tonnessen, A. (1988), Strength of Metals and Alloys: Kettunen, P. O., Lepisto, T. K., Lehtonen, M. E. (Eds.). Oxford: Pergamon Press, Vol. 1, pp. 743-748. Neumann, R. (1975), Zeit. Metallkunde 66, 26. Nine, H. D. (1973), /. App. Phys. 44, 4875. Paris, P. C , Gomez, M. P., Anderson, W P. (1961), The Trend in Eng. 13, 9. Paris, P. C , Bucci, R. X, Wessel, E. T, Clark, W G., Mager, T. R. (1972), Stress Analysis and Growth of Cracks, Special Technical Publication 513. Philadelphia: American Society for Testing and Materials, pp. 141-176. Pearson, S. (1975), Eng. Fract. Mech. 7, 235. Pederson, O. B. (1990), Acta Met. Mater. 38, 1201. Pelloux, R. M. N. (1969), Trans. Amer. Soc. Met. 62, 281.
11.17 References
Peterson, R. E. (1959), Metal Fatigue: Sines, G., Waisman, X L. (Eds.). New York: McGraw-Hill, pp. 293-306. Rabinowitz, S., Beardmore, P. (1974), 1 Mater. Sci. 9, 81. Reece, M. X, Guiu, R, Sammur, M. E R. (1989), J. Amer. Ceram. Soc. 72, 348. Reid, C. N., Williams, K., Hermann, R. (1979), Fat. Eng. Mater. Struct. 1, 267. Rice, I R. (1967), Fatigue Crack Propagation, Special Technical Publication 415. Philadelphia: American Society for Testing and Materials, pp. 247-309. Rice, J. R. (1968), J. Appl. Mech. 35, 379. Rice, J. R., Rosengren, G. F. (1968), J. Mech. Phys. Solids 16, 1. Ritchie, R. O., Suresh, S., Moss, C. M. (1980), J. Eng. Mater. Tech. 102, 293. Sadananda, K., Shahinian, P. (1979), Eng. Fract. Mech. 11, 73. Sauer, X, Hara, M. (1900), Adv. Polymer Sci. 91/92, 69. Schulte, K., Nowack, H., Trautmann, K. H. (1980), Z. Werkstofftech. 11, 287. Scoble, W. R., Weissman, S. (1973), Cryst. Latt. Def. 4, 123. Shih, C. F. (1981), J. Mech. Phys. Solids 29, 305. Skibo, M. D., Hertzberg, R. W, Manson, J. A., Kim, 5. (1977), /. Mater. Sci. 12, 531. Stewart, A. T. (1980), Eng. Fract. Mech. 13, 463. Stoltz, R. E., Pelloux, R. M. N. (1976), Metall. Trans. 7A, 1295. Suresh, S. (1983), Eng. Fract. Mech. 18, 577. Suresh, S. (1985 a), Metall. Trans. 16A, 249. Suresh, S. (1985b), Eng. Fract. Mech. 21, 453. Suresh, S. (1990), Intl. J. Fract. 42, 41. Suresh, S. (1991), Fatigue of Materials. Cambridge: Cambridge University Press. Suresh, S., Brockenbrough, X R. (1988), Ada Metall. 36, 1455. Suresh, S., Pruitt, L. (1991), in: Proc. 8th Intl. Conf. on Deformation, Yield and Fracture of Polymers, Churchill College, Cambridge, April 1991. London: Plastics and Rubber Institute, pp. 32-1-32-4. Suresh, S., Ritchie, R. O. (1982), Met. Sci. 16, 529. Suresh, S., Ritchie, R. O. (1984a), Fatigue Crack Growth Threshold Concepts: Davidson, D. L., Suresh, S. (Eds.). Warrendale: The Metallurgical Society, pp. 227-261. Suresh, S., Ritchie, R. O. (1984b), Intl. Met. Rev. 29, 445. Suresh, S., Zamiski, G. E, Ritchie, R. O. (1981), Metall. Trans. 12A, 1435. Tada, H., Paris, P. C , Irwin, G. R. (1973), Stress Analysis of Cracks Handbook. Hellertown, PA: Del Research Corporation. Takemori, M. T. (1984), Ann. Rev. Mater. Sci. 14, 171. Takemori, M. T. (1990), Adv. Polymer Sci. 91/92, 263-300.
563
Tanaka, K., Nakai, Y, Yamashita, M. (1981), Intl. J. Fract. 17, 519. Thompson, N., Wadsworth, N. X, Louat, N. (1956), Phil. Mag. 1, 113. Walker, N., Beever, C. X (1979), Fat. Eng. Mater. Struct. 1, 135. Weertman, X (1966), Intl. J. Fract. 2, 460. White, C. S., Bronkhorst, C , Anand, L. (1990), Mech. Mater., in press. Wilhelm, M., Everwin, P. (1980), The Strength of Metals and Alloys: Haasen, P., Gerold, V., Kostorz, G. (Eds.). Oxford: Pergamon Press, Vol. 2, pp. 1089-1099. Williams, L. S. (1956), Trans. Brit. Ceram. Soc. 55, 287. Wilson, D. V. (1965), Ada Met. 13, 807. Winter, A. T, Pederson, O. B., Rasmussen, K. V. (1981), Acta Met. 29, 735. Wohler, A. (1860), Zeitschrift fur Bauwesen 10, 160. Wood, W. A. (1958), Phil. Mag. 3, 692. Woods, P. X (1973), Phil. Mag. 28, 155. Wuthrich, Ch. (1982), Int. J. Fract. 20, R35. Zappfe, C. A., Worden, C. O. (1951), Trans. Amer. Soc. Met. 43, 958.
General Reading Bannantine, X A., Comer, X X, Handrock, X L. (1990), Fundamentals of Metal Fatigue Analysis. Englewood Cliffs: Prentice-Hall. Fong, X T. (Ed.) (1979), Fatigue and Micro structure, Special Technical Publication 675. Philadelphia: American Society for Testing and Materials. Hertzberg, R. W. (1989), Deformation and Fracture Mechanics of Engineering Materials, 3rd ed. New York: Wiley. Hertzberg, R. W, Manson, X A. (1980), Fatigue of Engineering Plastics. New York: Academic Press. Klesnil, M., Lukas, P. (1980), Fatigue of Metallic Materials. Prague: Academia. Kitagawa, H., Tanaka, T. (Eds.) (1990), Fatigue 90, Proceedings of an International Conference, Honolulu, July 1990. Birmingham: Materials and Components Engineering Publications. Laird, C. (1983), "The Application of Dislocation Concepts in Fatigue", in: Dislocations in Solids: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, Chapter 26. Mughrabi, H. (1985), "Dislocations in Fatigue", in: Dislocations and Properties of Real Materials, Book No. 323. London: The Institute of Metals, pp. 244-262. Neumann, P. (1983), "Fatigue", in: Physical Metallurgy, 3rd edn, Cahn, R. W, Haasen, P. (Eds.). Lausanne: Elsevier, Chapter 24. Suresh, S. (1991), Fatigue of Materials. Cambridge University Press.
12 Fracture Mechanisms Hermann Riedel
Fraunhofer-Institut fur Werkstoffmechanik, Freiburg, Federal Republic of Germany
List of Symbols and Abbreviations 12.1 Introduction 12.1.1 Distinction of Mechanisms 12.1.2 Fracture-Mechanism Maps 12.1.3 Ductility Versus Brittleness 12.1.4 Material Laws 12.1.5 Elements of Linear Elastic Fracture Mechanics 12.1.5.1 Conventions 12.1.5.2 Linear Elastic Crack Tip Fields 12.1.5.3 Relation to the Remote Loadings 12.1.5.4 The Fracture Toughness 12.2 Cleavage Fracture 12.2.1 Introduction 12.2.2 Cleavage and Plasticity 12.2.3 The Ideal Strength 12.2.4 Brittle Fracture from Cracks 12.2.4.1 Strength as a Function of Crack Size 12.2.4.2 Statistics of Brittle Fracture 12.2.5 Cleavage Versus Dislocation Emission from Cracks 12.2.5.1 The Model of Kelly, Tyson, and Cottrell 12.2.5.2 The Model of Rice and Thomson 12.2.5.3 Atomic Simulation of Cleavage and Dislocation Emission 12.2.6 Initiation of Cleavage Fracture by Plastic Slip 12.3 Fracture by Plastic Void Growth 12.3.1 Introduction 12.3.1.1 The Role of Second-Phase Particles and Inclusions 12.3.1.2 Complete Necking or Superplasticity 12.3.1.3 The Orientation of the Fracture Surface 12.3.2 Void Nucleation 12.3.2.1 Observations 12.3.2.2 Nucleation Theory Based on Dislocations 12.3.2.3 Nucleation Theory Based on Continuum Plasticity 12.3.3 Independent Growth of Widely Spaced Voids 12.3.3.1 Spherical Void in Ideally Plastic Material 12.3.3.2 Spheroidal Voids in Power-Law Viscous Materials Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.
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12.3.3.3 12.3.4 12.3.5 12.3.5.1 12.3.5.2 12.3.5.3 12.3.6 12.3.6.1 12.3.6.2 12.3.6.3 12.3.7 12.3.7.1 12.3.7.2 12.3.7.3 12.4 12.4.1 12.4.2 12.4.2.1 12.4.2.2 12.4.2.3 12.4.2.4 12.4.3 12.4.3.1 12.4.3.2 12.4.3.3 12.4.3.4 12.4.4 12.4.4.1 12.4.4.2 12.4.4.3 12.4.4.4 12.4.5 12.4.5.1 12.4.5.2 12.4.5.3 12.4.5.4 12.5 12.5.1 12.5.1.1 12.5.1.2 12.5.1.3 12.5.1.4 12.5.2
12 Fracture Mechanisms
Other Models for the Growth of Isolated Voids Interacting Voids Strain to Failure Coalescence of Independently Growing Voids Thomason's Model Comparison of Ductilities Predicted by Different Models Constitutive Equations for Porous Plastic Solids The Gurson Model The Modified Gurson Model Performance of the Modified Gurson Model Compared to Micromechanical Models Comparison with Experiments Ductility Void Growth Constitutive Behavior Creep Fracture Introduction Cavity Nucleation Nucleation Sites The Role of Grain Boundary Sliding The Observed Nucleation Kinetics Theory of Nucleation Models for Cavity Growth Overview Constrained Diffusive Cavity Growth Comparison with Measured Growth Rates Rupture Lifetimes Cavity Growth and Continuous Nucleation The Cavity Size Distribution Function Similarity Solutions of the Continuity Equation Lifetimes for Continuous Nucleation Comparison with Experiments Constitutive Behavior of Cavitating Solids The Kachanov Model Hutchinson's Model Constitutive Equations for Higher Densities of Cavitating Boundary Facets Remanent Life The Chemistry of Fracture Weakening or Strengthening of Interfaces by Segregants Segregation Equilibria Segregation Kinetics Grain Boundary Energy and Cohesive Strength Atomic Simulations Hydrogen Embrittlement
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12 Fracture Mechanisms
12.5.2.1 12.5.2.2 12.5.2.3 12.5.3 12.5.3.1 12.5.3.2 12.5.3.3 12.5.3.4 12.6 12.6.1 12.6.2 12.6.2.1 12.6.2.2 12.6.2.3 12.6.3 12.6.3.1 12.6.3.2 12.6.3.3 12.6.4 12.6.5 12.6.5.1 12.6.5.2 12.6.5.3 12.6.6 12.6.6.1 12.6.6.2 12.6.6.3 12.6.6.4 12.6.7 12.6.8 12.6.8.1 12.6.8.2 12.6.8.3 12.7
General Observations Possible Micromechanisms Mechanism-Oriented Observations Stress Corrosion Cracking (SCC) General Observations Stress Corrosion Cracking by Hydrogen Embrittlement The Slip/Dissolution Mechanism (with Possible Participation of Hydrogen Embrittlement) Experimental Evidence for the Importance of Slip Nonlinear Fracture Mechanics Introduction The J-Integral and the Hutchinson, Rice, and Rosengren (HRR) Crack-Tip Field The J-Integral in Nonlinear Elastic Materials Power-Law Elastic and Power-Law Plastic Materials Crack-Tip Fields in Power-Law Materials Stationary Crack in Elastic-Plastic Material Small-Scale Yielding Transition from Small-Scale Yielding to General Yield Range of Validity of the HRR Field Limitation to J by Crack Blunting Models for Crack Growth Initiation Slip-Induced Cleavage Fracture from Cracks Crack Growth Initiation by Ductile Void Growth The Cleavage/Ductile Transition Stable Crack Growth The R-Curve and the Stability of Crack Growth The Source of the R-Curve Effect in Ductile Metals Limitations to J in Fully Yielded Specimens During Stable Crack Growth Damage Mechanics Models for Stable Crack Growth A Note on Fast Crack Growth Creep Crack Growth The C*-Integral A Model for C*-Controlled Crack Growth Limitations to C* and Other Load Parameters References
567
607 607 608 610 610 612 612 614 614 614 615 615 616 616 617 617 618 618 619 620 620 620 621 621 621 622 623 624 625 625 626 626 627 628
568
12 Fracture Mechanisms
List of Symbols and Abbreviations a a A Ao b c c d 5D b E / /* / c ,/f /0 G AG AGb, AGS h (x//) J J* JR k Kx KR Klc Kld X Ie Kss ^issc m n N N (R, t) P quq2 r R R rpl r0 t T
crack length lattice spacing material parameter of Norton's creep law material parameter of power-hardening material law Burgers vector carbide thickness shear wave speed grain size or grain facet diameter grain boundary diffusion coefficient Young's modulus void volume fraction extended void volume fraction to consider the enhancement of the effect of porosity adjustable parameters in modified Gurson model initial void volume fraction shear modulus free enthalpy per mole free enthalpy per mole for segregation to the grain boundary, the free surface dimensionless function which accounts for the lenticular cavity shape contour integral number of cavities nucleated per unit time and grain boundary area crack resistance in terms of J Boltzmann constant stress intensity factor crack resistance in terms of KY fracture toughness dynamic fracture toughness of a fast crack critical value of KY for spontaneous dislocation emission steady-state value of the crack resistance threshold for stress corrosion cracking Weibull modulus stress exponent of Norton's creep law hardening exponent distribution function for cavity sizes load adjustable parameters in modified Gurson model polar coordinate, distance from the crack tip cavity radius gas constant plastic zone size dislocation core width time absolute temperature
List of Symbols and Abbreviations
u ti
u u{ Au V V
W
w X
*c Y
Y
a' 7b 7s
A A <S« £
% «*;
^nuc
X X v (T ^b
(J I? (T T <7M (7 y
Y
time to cavity coalescence time to fracture tearing modulus characteristic time displacement relative to equilibrium lattice spacing displacement field separation distance of the crack faces volume normalized growth rate of a spherical void (dimensionless) strain energy density specimen width concentration critical distance Cartesian coordinates parameters proportionality factor specific grain boundary energy specific surface energy load-point displacement load point displacement rate critical crack-tip opening displacement crack-tip opening displacement strain in uniaxial tension equivalent tensile strain equivalent tensile strain rate fracture strain strain tensor strain rate macroscopic strain at which nucleation of a void occurs angle (polar coordinate) relative coverage of available grain boundary sites distance between two cavities on a grain facet strain ratio Poisson's ratio of lateral contraction creep damage parameter tensile stress stress on cavitating boundary critical tensile stress for decohesion von Mises equivalent stress fracture stress normal, transverse tensile stress mean (or hydrostatic) stress equivalent flow stress of the matrix material plastic yield stress
569
570
12 Fracture Mechanisms
aid atj a'ij of T Tj Ty
ideal strength stress tensor deviator cleavage fracture stress resolved shear stress frictional shear stress plastic yield stress in shear flow potential dihedral angle at cavity tip cavitated area fraction of the grain facet damage parameter in Kachanov model atomic volume fracture value of co
CCP CT HRR p/m SCC SENB SENT
center-cracked tension specimens compact-tension Hutchinson, Rice, and Rosengren (field) powder-metallurgical stress corrosion cracking single-edge cracked bend single-edge cracked tension
12.1 Introduction
12.1 Introduction 12.1.1 Distinction of Mechanisms
Solid materials can fracture under constant or increasing load by one of several fundamental mechanisms: - Rupturing of atomic bonds by tensile stresses (Sec. 12.2: Cleavage Fracture), - Void growth and coalescence, or complete necking by plastic flow (Sec. 12.3: Fracture by Plastic Void Growth), - Cavity growth and coalescence by stress-directed diffusion of atoms or vacancies (Sec. 12.4: Creep Fracture), - Strain-rate-assisted localized chemical attack at crack tips (Sec. 12.5.3: Stress Corrosion Cracking). This list covers the most important fracture mechanisms of metallic materials and applies to some aspects of the fracture of ceramics and glasses. Not included in this chapter are specific features of polymers (Kausch, 1990), composite materials (Sih and Skudra, 1985), concrete (Sih and DiTommaso, 1985), geomaterials, and bimaterial interfaces (Rice, 1988).
12.1.2 Fracture-Mechanism Maps
Some materials strongly prefer one of the fundamental mechanisms over the others. The covalently bonded ceramics, for example, fail by cleavage or by brittle intergranular fracture under a wide range of conditions, whereas most of them never fail by plastic flow. In ferritic steels, on the other hand, all mechanisms play a role under technologically important conditions. For a given material, it depends on the temperature, on the stress, on the loading rate, on the stress triaxiality, and on the chemical environment which mechanism predominates.
571
The dependence on temperature and stress is illustrated by a diagram called fracture-mechanism map (Wray, 1969; Ashby, 1977). Figure 12-1 schematically shows a simplified version of such a map. At low temperatures, cleavage fracture occurs, once a critical stress is reached. Above a certain brittle-to-ductile transition temperature the fracture mode changes to ductile void growth. The regime of ductile void growth broadens towards higher temperatures, since plastic flow becomes more and more time-dependent. This means that void growth continues at stresses below the instantaneous fracture stress, and fracture ensues after a hold time. At even lower stresses, and correspondingly longer times, the fracture mode changes to diffusive cavitation of grain boundaries. In this stresstemperature regime, metallic materials usually deform by dislocation-climb-controlled creep flow or by diffusion creep. Fracture-mechanism maps for actual materials were established by Ashby et al. (1979) and Gandhi and Ashby (1979 a) for face-centered cubic metals, by Gandhi and Ashby (1979 b) for cleavable materials, i.e., body-centered cubic and hexagonal metals, ceramics and ionically bonded materials like rock salt and ice, by Fields et al. (1980)
LU 10- 2
^Cleavage
Transgranular Dimpled Fr acture , Rupture by Dynamic ^Recrys \zation
I 10-3
CO N
TO 10- 4 o
No Fracture 10-5 0
0.5
1
Homologous Temperature, T/Tm Figure 12-1. Fracture mechanism map (schematic). Tm is the melting temperature, E is Young's modulus.
572
12 Fracture Mechanisms
for iron and steels, and by Krishnamohanrao et al. (1986) for titanium alloys. 12.1.3 Ductility Versus Brittleness
Ductility, as opposed to brittleness, is the ability of a material to undergo plastic deformations before it breaks. A measure for the ductility is the strain to fracture, sf. Ductility is a desirable material property, since plastic flow tends to reduce stress peaks at notches and cracks and to smoothen the stress distribution. This makes a ductile material tolerant against defects, whereas a brittle material fractures, once the stress reaches a critical value in any small portion of a component. It is usually difficult to achieve high strength and high ductility at the same time, and it is a goal of alloy design to provide materials with an optimum combination of these properties for a given application. The ductility depends on the prevailing fracture mechanism. Cleavage fracture is usually associated with low or zero ductility, e.g., in ceramic materials, but it may also occur after some plastic deformation, as in ferritic steels at cryogenic temperatures. Dimpled fracture is a ductile fracture mode with typically 20 to 100 percent strain to fracture, but in high-strength steels or nickel-base alloys, the strain to dimpled fracture may be as low as a few percent. Intergranular cavitation under creep conditions leads to relatively low ductilities between typically 3 and 20 percent strain. The triaxiality of the stress is another factor that affects the ductility. Generally, compressive stress components enhance the ductility as, for example, during the rolling of steel blanks, whereas multiaxial tension reduces the ductility, a situation encountered by the material near crack tips.
An alternative measure for the flaw tolerance of materials is the fracture toughness Klc. It describes the resistance of the material against crack extension in a way defined within the framework of linear elastic fracture mechanics (see Sees. 12.1.5 and 12.6). 12.1.4 Material Laws
All fracture processes are preceded by, and are intimately related to the deformation of the material. Deformation processes are described here in the framework of (static) continuum mechanics. This theory comprises equations for the equilibrium of the stress field and for the compatibility of the strain field, and a material law. Several types of materials will be considered. The simplest case is a linear elastic material characterized by Hooke's law 1+v
l-2v
(12-1)
which reduces to s = ojE in uniaxial tension. Here, etj is the strain tensor, otj is the stress tensor, E is Young's modulus, v is Poisson's ratio of lateral contraction, the prime denotes the deviator
(r'ij = <Tij-
om
is
the
mean (or hydrostatic) stress defined by G = m (°"n + Gn + °"33)A an( * (>ij is the Kronecker symbol (or unit tensor). Second, power-law elastic materials are considered, which are described by 3 2J
T d/JV)-i
(12-2)
which reduces to 8 = Aoa1/N in uniaxial tension. Here, N is called the hardening exponent, Ao is another material parameter and ae = (3 aftj a^j/2)112 is the von Mises equivalent stress. (Throughout this chapter, the summation convention for repeated indices is applied.) Hardening expo-
12.1 Introduction
573
nents range from typically N = 0.05 for high-strength materials to 0.3 for lowstrength alloys; N = 0 is the nonhardening, or ideally plastic, limit. The parameter Ao can be expressed by the plastic yield stress,
different fields, if nonproportional stressing and unloading are involved. Finally, power-law viscous materials are considered, which are described by
ay9 as
fi0. = -,4 (7TV 0 . (12-3)
A
°
where a is a dimensionless factor. If <7y is defined as the 0.2% flow stress, a = 0.002 £/<7y. In the following, we set a = 1. Equation (12-2) describes only incompressible behavior, since it is intended to serve as an approximation for describing metal plasticity, which is incompressible. In this sense, nonlinear elasticity is also called deformation plasticity. Plasticity of metals is modeled more realistically by an incremental theory. If power-law hardening is assumed, the plastic strain rate is given by ij
~2
N
e
ij
(12-4)
if the equivalent stress is on the current yield surface and de > 0; otherwise is etj = 0. The superposed dot denotes the time derivative. For uniaxial tension with no unloading, Eq. (12-4) reduces to s = Aoa1/N just as in the nonlinear elastic case, and Ao has the same meaning as in Eq. (12-3). The power-law elastic description, Eq. (12-2) often provides a good approximation to the stress fields in incrementally plastic materials. In fact, the two material laws lead to exactly the same stress and strain fields for monotonically increasing proportional loading. In this case, the stress field is a proportional field, i.e., the stress components at each point increase in a fixed proportion, so that Eq. (12-4) can be integrated to give Eq. (12-2). On the other hand, the models lead to significantly
(12-5)
which reduces to 8 = A an in uniaxial tension. This is called Norton's creep law, with a stress exponent n and a coefficient A. It describes the stress dependence of the creep strain rate of metals and ceramics approximately. Nonlinear viscous and nonlinear elastic materials are analogous (Hoff, 1954). Under the same loads, they develop the same stress field, while the elastic strain field corresponds to the viscous strain rate field. 12.1.5 Elements of Linear Elastic Fracture Mechanics
The analysis of cracks is an important branch of fracture research, firstly, since some fracture mechanisms (cleavage fracture from microcracks, stress corrosion cracking) directly involve cracks. Hence, for the discussion of these mechanisms a brief introduction into fracture mechanics is needed here. Second, fracture mechanics is the discipline which allows one to assess the importance of cracks in components, irrespective of the micromechanism by which the cracks grow. Fracture mechanics in this sense will be described in Sec. 12.6. 12.1.5.1 Conventions
Figure 12-2 shows the three modes by which cracks can generally be loaded. The tensile opening of the crack (Mode I) is the most important one in engineering metallic materials and will primarily be considered in the following. Mode III is mathematically the simplest of the three modes and
574
12 Fracture Mechanisms
i
universal angular functions independent of the specimen geometry. The Cartesian components of ftj{9) are (see, e.g., Rice, 1968 a):
X
2 /
a Figure 12-2. Modes of crack loading (from Riedel, 1987).
/n(0) = cos-( 1 - sin-sin — . 9 . 39
has therefore often been used to explore basic fracture mechanical issues. In reality, mixed-mode loadings are frequent, e.g., if a surface crack in a rail grows under the loadings of rolling wheels. Another important distinction is that between plane strain and plane stress. If x3 is the coordinate along the crack front, a plane-strain deformation field is defined by u3 = 0 where ut is the displacement field. Plane strain is approximated in specimens that are thick in the direction of the crack front compared to the distances over which the in-plane stress components vary significantly. Plane stress is defined by a3i = 0. This limiting case is approached in thin specimens. In anti-plane strain the only nonzero components are u3 (for the displacement),
fiffl
(12-6)
Here, r is the distance from the crack tip, r and 9 are polar coordinates (Fig. 12-3), Kt is the coefficient of the leading term and is called stress intensity factor; and fij(O) are
9
9
(12"7)
39
/ 33 (0) = 2vcos12.1.5.3 Relation to the Remote Loadings The stress intensity factor must be proportional to the applied stress and to the square-root of the crack length. For a crack of length 2 a in an infinite body subjected to a tensile stress a, the stress intensity factor is (12-8)
= -sjn a cr
For a circular (penny-shaped) crack with radius a is (12-9) For other specimen geometries, Eq. (12-8) is augmented by a dimensionless factor, which is usually determined by a numerical solution of the continuum-mechanical equations. Handbook tabulations for X2
}a22
Figure 12-3. Crack tip coordinate systems.
12.2 Cleavage Fracture
many specimen geometries are available (Tada etal., 1973; Sih, 1973; Rooke and Cartwright, 1974). From the foregoing it is clear that the crack tip fields are universal for all cracks in elastic bodies, The outer specimen geometry and the loading can only affect the stress intensity factor but not the radial and angular distribution of the stress and strain fields. Hence one concludes that all cracks in a given elastic material at a given temperature and chemical environment must respond identically when exposed to the same stress intensity factor, irrespective of the specimen size and shape. 12.1.5.4 The Fracture Toughness
The simplest response of a crack to an applied stress intensity factor is to remain static as long as KY is smaller than a critical value Klc and to propagate once KY = Klc; Klc is called the fracture toughness of a material. Its magnitude depends on the fracture mechanism by which the crack propagates. A lower-bound estimate for Klc can be obtained from an energy argument (Griffith, 1920). If the elastic energy release rate (per unit area of crack advance) equals the energy of the newly created surfaces, 2y s , the crack propagates. This leads to 1-v2
(12-10)
For ys = 2 J/m2, E = 200 GPa and v = 0.3, which are typical values for metals, Klc is found to be of the order 1 MPa-y/m. Only the most brittle materials exhibit fracture toughnesses as low as predicted by Eq. (12-10) (e.g., silicon: St. John, 1975). The fracture toughness of glasses is typically three times the theoretical value (0.8 vs. 0.3 MPa^/m: Kerkhof, 1970). Structural ceramics have values between 3
575
and 12 MPa^/m. Steels range from 30 MPa^/m upwards. More detailed models for estimating Klc follow in later sections.
12.2 Cleavage Fracture 12.2.1 Introduction
Cleavage fracture is the separation of crystalline solids along certain crystallographic planes. This separation occurs by forces which are large enough to rupture the atomic bonds. In most cases the applied stresses must be magnified by stress concentrators such as cracks in order to reach the atomic bond strength. Cleavage is the predominant fracture mode in covalently and ionically bonded materials. It is frequent also in body-centered cubic and hexagonal metals and alloys, but nearly never occurs in face-centered cubic metals. A collection of materials which are capable of cleavage is given by Gandhi and Ashby (1979 b). Figures 12-4 and 12-5 show cleavage fracture surfaces of a ferritic steel and of an alumina ceramic, respectively. The fracture surface of alumina exhibits a feature which is frequent in ceramics, namely that the fracture mode changes easily from transgranular cleavage to brittle intergranular fracture. Also in metals such transitions occur, especially when the grain boundaries are weakened by the segregation of impurities such as sulfur or phosphorus. Noncrystalline solids can fracture in a similar way by the breaking of atomic bonds, but, of course, there are no preferred cleavage planes. Hence much of the following can be applied to glassy materials except when special crystal properties are involved. For a review of the mechanical properties of glassy materials see Chap. 10 of this Volume by Argon. Specific
576
12 Fracture Mechanisms
shown in Figs. 12-4 and 12-5 result from the compromise between a macroscopically flat fracture surface and fracture along cleavage planes of individual grains. 12.2.2 Cleavage and Plasticity
Figure 12-4. Cleavage fracture surface of a coarsegrained mild steel fractured at — 100 °C.
Figure 12-5. Transgranular cleavage and intergranular fracture of alumina ceramic.
aspects of the fracture of ceramic materials are treated by Cook and Pharr and by Becher in Chaps. 7 and 8 of Vol. 11 of this series. Polymer fracture is described by Yee and Narisawa in Chap. 15 of Vol. 12. Cleavage fracture, as well as brittle intergranular fracture, lead to fracture surfaces that are macroscopically oriented normal to the applied tensile stress. On a microscopic scale the fracture path follows the possible cleavage planes of the grains or the grain boundaries. The jogs and the "river patterns" on the cleavage facets
The relation between cleavage and plasticity is a complicated one. Depending on material, temperature and strain rate, the two processes can preclude each other, or they can coexist, or plastic slip can initiate cleavage fracture. Covalently bonded materials, such as diamond, silicon or ceramics, can fracture by cleavage in a completely brittle manner, at least at low temperatures. Even on a microscopic scale, there is no sign of dislocation activity (Thomson, 1986). At high temperatures and slow strain rates, however, dislocations can be formed and can move in some of these materials, which leads to a brittle-to-ductile transition. The nature of the transition, is not necessarily the same in different materials. In silicon, the sudden increase in toughness at temperatures above 700 °C is associated with the occurrence of plasticity at crack tips (St. John, 1975; Brede and Haasen, 1988). The fracture mode, however, remains cleavage. Similarly, Vehoff and Neumann (1979), who investigate crack growth in Fe-3%Si single crystals, vary the amount of plasticity that accompanies crack growth within wide margins by varying the temperature and the loading rate, but the fracture mode remains cleavage. On the other hand, the brittle-to-ductile transition in body-centered cubic and hexagonal metals, e.g., in ferritic steels, is often associated with a change in fracture mode from cleavage (already accompanied by some plasticity) to a ductile fibrous mode. The decrease in flow stress with increasing temperature, which is typical for these
12.2 Cleavage Fracture
metals, favors fracture by ductile void growth and makes cleavage less and less likely. In many applications of ferritic steels the brittle-to-ductile transition temperature is a major design consideration in order to prevent catastrophic brittle failures at low temperatures. The transition temperature is usually measured by conducting a series of Charpy impact tests at different temperatures. In a certain temperature range, the energy absorbed during fracture rises sharply. Figure 12-6 shows an example for a fine-grained structural steel typically used in large offshore structures.
577
o Mo • K A CU
OBa 2+ + Sm vSm 3 +
-(1+x) exp(-x) -exp(-0.23x2 1/(1+0.37x2) -1.0 - 1 0 1 2 3 4 5 6 7 8 Normalized Separation, x=u/u 0
Figure 12-7. Universal potential describing the cohesive energy of numerous solids.
12.2.3 The Ideal Strength
The ideal strength of solids is determined by the maximum atomic forces between neighboring planes of atoms. Pertinent calculations based on pair potentials or on more advanced quantum-mechanical methods were reviewed by Pettifor (1983), Rose etal. (1983) and Carlsson (1990). If the energy, U, of a crystal (per atom) is calculated as a function of the atomic spacing, a, it is found that many different metals with quite different electronic structures
300 =7 250 -
can be described by a universal potential (Rose etal., 1983): U
/
E U
a (u) = — exp -120
-80
-40 Temperature | C]
Figure 12-6. Charpy impact energy vs. temperature. (Burger, 1990).
(12-11)
Here AU is the amount of the binding energy per atom for the equilibrium atomic spacing, a 0 , u = a — a0 is the displacement relative to the equilibrium spacing, and u0 is a normalizing length. Figure 12-7 shows the results quoted by Rose et al. (1983). Recent density-functional calculations of Paxton etal. (1990) for the first-row transition metals lead to the same universal behavior. The mechanical stress a, which results if neighboring atoms are pulled apart by the displacement u, is obtained by differentiating Eq. (12-11): t ^
O
u\
— = - 1 + - exp AU \ uoj \
(12-12)
Here AU and u0 were replaced by Young's modulus, E, such that the behavior for small u/u0 reproduces Hooke's law. [It is assumed here that for uniaxial tension an analogous universal potential exists as
578
12 Fracture Mechanisms
Eq. (12-11).] The maximum of the stress represents the ideal strength (12-13) Since u0 is some fraction of the lattice spacing a0 (typical is u0 = ao/4) one obtains the ideal strength to be of the order E/10. Alternately, u0 has been eliminated by the argument that the integral under the stress/displacement curve should be of the order of the specific energy of the two fracture surfaces, 2 ys (Riedel, 1987). This yields = 0.52(^^
(12-14)
a0 j
For typical values of the parameters, this leads to the same conclusion that oid should be of the order E/10. Most real materials, except defect-free whiskers, fail at applied stresses one or two orders below the ideal strength, since preexisting defects concentrate the applied stress or since plastic yielding intervenes. 12.2.4 Brittle Fracture from Cracks 12.2.4.1 Strength as a Function of Crack Size
As already mentioned, cleavage fracture of real materials occurs at stresses one or two orders below the ideal strength. In glasses and ceramics this is due to pores and crack-like defects resulting from the manufacturing process. According to linear elastic fracture mechanics, the strength, i.e., the fracture stress crf, is then determined by the diameter of the crack, 2 a, and by the fracture toughness of the material, Klc. For an embedded penny-shaped crack one obtains from Eq. (12-4)
Figure 12-8. Fracture origin in a sintered silicon-nitride ceramic.
(12-15)
Equation (12-15) indeed describes the strength of ceramic materials approximately correctly. Figure 12-8 shows a fracture origin in sintered silicon-nitride (Gehrke and Riedel, 1990). Fracture started from a flat subsurface pore having an average diameter of 100 juim. The fracture stress is 442 MPa, while Eq. (12-15) gives 627 MPa, if a value of Klc = 5 MPa^/m is chosen, which is appropriate for macroscopic cracks in this material. As is typical in the fractography of ceramics, the order of magnitude of the fracture stress is obtained correctly, but the numerical value cannot be expected to be accurate for several reasons. First, Eq. (12-15) ignores the proximity of the pore to the free surface; second, the fracture origin is not a sharp-tipped crack; and thirdly KIc may be smaller for microcracks due to an R-curve effect influencing the X lc value of macrocracks (Sec. 12.6.6). 12.2.4.2 Statistics of Brittle Fracture
Ideally brittle solids fracture, once the conditions at the most dangerous defect become critical ("weakest-link" hypothesis). Assuming that there are sufficiently many defects in a specimen, the strength
579
12.2 Cleavage Fracture
P(a) = 1 - expf- f — V ]
(12-16)
I
95
I
I
I
I
Bi
-
,im ^50 o_
bil
should obey one of the asymptotic distributions of extremes. Among these the Weibull distribution is commonly used to describe the strength distribution of brittle materials
A
-
/•• m/
f
m/
L \ao/ J where P is the probability that a specimen has a strength less than cr, a0 is the characteristic strength of the distribution and m is called the Weibull modulus. Figure 12-9 shows a Weibull plot for a test series of 20 specimens of sintered silicon-nitride (Berweiler and Riedel, 1990). The solid line represents the maximumlikelihood fit, which gives a0 = 746 MPa and m = 9.8 with 90% confidence intervals from 714 to 779 MPa and m = 6.7 to 12.3.
££
12.2.5 Cleavage Versus Dislocation Emission from Cracks
other hand, materials like rock salt and diamond are correctly predicted to cleave long before the ideal shear strength is reached on a slip plane through the crack front. The body-centered cubic metals Fe and W are found to be border-line cases slightly in the regime of cleavage.
The question which material cleaves and which fails in a ductile manner has been examined by Kelly et al. (1967) and by Rice and Thomson (1974). They consider a crack and ask whether it is easier to expand the crack by cleavage or to initiate plastic slip at the crack tip in the absence of pre-existing dislocations (Fig. 12-10). They assume that a material is intrinsically ductile, if shear occurs first, and intrinsically brittle, if cleavage occurs first.
o
•
Q_
/ • /'
o 10 Jj 5
•
•/
1
Sintered silicon-nitride i
500
i
i
i
i
600 700 800 Strength, a f [MPa]
i
900
Figure 12-9. Weibull plot of the strength distribution of sintered silicon-nitride.
12.2.5.2 The Model of Rice and Thomson
Rice and Thomson (1974) pose the problem of brittle-versus-ductile behavior slightly differently than Kelly et al. (1967). Shear of the lattice
12.2.5.1 The Model of Kelly, Tyson, and Cottrell
Kelly etal. (1967) compare the ideal strength of crystal lattices under tensile loads with that under shear loads. From atomistic pair-potential calculations they conclude that the face-centered cubic lattice is usually much easier to shear by the stress field of a crack than to cleave. On the
Dislocation emission
Figure 12-10. Cleavage fracture vs. ductile crack-tip blunting.
580
12 Fracture Mechanisms
They argue that, rather than considering the ideal shear strength, it would be more appropriate to compare the ease of cleavage with that of the spontaneous formation of a dislocation at the crack tip. An intrinsically brittle material cleaves before a dislocation is emitted from the crack tip, and an intrinsically ductile material emits a dislocation before it cleaves. The essence of the Rice and Thomson model is that a continuum dislocation with Burgers vector b nucleated at the crack tip experiences a repulsive force KY b/y/r by the stress field of the crack and an attractive image force, which varies as b2 G/r. (G is the shear modulus.) As Fig. 12-11 shows, the attractive force dominates at small distances, while at distances larger than a critical distance, rc « ( G fe/Kj)2, the repulsive force dominates. Rice and Thomson assume that spontaneous dislocation emission takes place when the critical length rc, which decreases with increasing Kl9 becomes as small as the dislocation core width, r 0 , an assumption which was later supported by a more detailed analysis of the emission process by Schoeck (1991). Thus, equating rc with r0 gives a critical value of KY for
0
5 Norm. Distance, r/rc
10
Figure 12-11. Force on a dislocation emitted by a crack tip.
spontaneous dislocation emission K
>
Gb
(12-17)
Comparing this critical value with the critical value Klc for cleavage, Eq. (12-10), gives a criterion for brittle versus ductile behavior. If the inequality (A^ A o. (lz-15)
Kle > Klc or equivalently b2 G > 10 ys r0 is fulfilled, brittle cleavage is predicted. The numerical factor 10 is somewhat arbitrary, and the reader is referred to Rice and Thomson (1974) for a more detailed analysis. They find that LiF, NaCl, MgO, A12O3, the covalent materials C, Si, Ge and the hexagonal metals Be and Zn are far in the brittle regime. Their results for f.c.c. and b.c.c. metals are reproduced in Table 12-1. The cleavage plane was assumed to be (001) in all cases. The crack front lies along [110] in f.c.c. metals and [100] in b.c.c. metals. The slip system is (Til) [101] in f.c.c. and (011)[111] in b.c.c. metals. The numerical values for rc are calculated including the detailed cleavage and slip geometry and the surface energy of the ledge formed at the crack tip when a dislocation is emitted. The values for the dislocation core width are based on general arguments without detailed considerations of the individual materials. Rice and Thomson (1974) suspect that the prediction for Na would have been different, had a more realistic core width been used. Contrary to the fact that sodium is a ductile metal, the numbers in Table 12-1 suggest that it would cleave. As in the Kelly, Tyson and Cottrell model all f.c.c. metals are predicted to be ductile, whereas the b.c.c. metals are predicted to cleave. In the Rice and Thomson model the reason for the ductility of f.c.c. metals is the large width of the dislocation cores, as compared to the low ideal shear strength in
12.2 Cleavage Fracture
Table 12-1. Cleavage is predicted for r0 < rc. Material
Normalized core width
Normalized critical distance rjb
2 2 2 2 2 2
1.1 0.85 1.0 1.1 1.4 1.7
2/3 2/3 2/3
1.2 1.9 4.0
f.c.c. Pb Au Cu Ag Al Ni b.c.c. Na Fe W
the Kelly, Tyson and Cottrell model. Of course, the two quantities are related to one another. Their common root is the stacking fault energy. Rice and Thomson (1974) as well as Kelly et al. (1967) consider the possibility that dislocation emission is aided by thermal activation. They calculate the activation energy necessary to produce a stable semicircular dislocation loop under an applied stress intensity which would just cause cleavage crack propagation. They find activation energies of 0.02 eV for Na, 2.2 eV for Fe and values between 58 eV (for LiF) and 852 eV (for A12O3) for a number of other materials. Hence thermal activation at usual temperatures and loading rates occurs easily in Na and is possible in Fe, but is quite unlikely in the other materials. For a more recent investigation of the thermally activated emission process, see Anderson and Rice (1986). 12.2.5.3 Atomic Simulation of Cleavage and Dislocation Emission
The Rice and Thomson (1974) model, although it deals with processes on an
581
atomic scale, uses continuum mechanical quantities and methods. Its predictions should be compared with atomic simulations available in the literature (e.g., deCellis etal., 1983; Mullins, 1984a,b; Kohlhoff etal., 1991; Baskes and Daw, 1990). All results of the atomic calculations agree in that iron (modeled with Johnson pair potentials or with the embedded atom method) and tungsten cleave along {100} planes, whereas cracks in copper (modeled with Morse potentials) blunt by the emission of dislocations. Nickel is found to be ductile. Hence the atomic calculations qualitatively confirm the conclusions of the phenomenological theories and are consistent with observations. 12.2.6 Initiation of Cleavage Fracture by Plastic Slip
Plastic slip is not only a competing process to the propagation of a crack, but it can also initiate cleavage fracture. This possibility was discussed mainly in connection with cleavage of metals, especially with that of ferritic steels. The subject attracted particular attention in the 1940's in connection with the liberty ships; some 200 of them developed large cracks, while about a dozen broke completely in two (Stroh, 1957). Much of the research work in the decades following World War II has been summarized by Knott (1971, 1973). Theories of cleavage fracture of ferritic steels generally assume that no pre-existing cracks are present in the material. Rather, there is evidence that microcracks are nucleated by dislocation pile-ups or deformation twins, since cleavage fracture of steel's is always preceded by more or less plastic yielding. Even the most brittle fractures do not occur before the yield stress is reached (Low, 1956; Cottrell, 1958). Figure 12-12 shows a micrograph from Riedel and
582
12 Fracture Mechanisms
their values at fracture, af and Tf, are related by 4 7T
Figure 12-12. Cracking of a pearlite colony by deformation twins in mild steel pulled in vertical direction at 57 K (Riedel and Kochendorfer, 1979).
Kochendorfer (1979), in which a twin initiates a microcrack in a pearlite colony in a mild steel at T = 57 K. Below about 100 K twins predominate compared to slip bands. Following work of Stroh (1957) and Cottrell (1958), Smith (1966 a, b) proposed a model in which a slip band initiates a stable microcrack in a carbide. A higher stress is needed to propagate the crack into the ferrite grain (Fig. 12-13). The problem is one of linear elasticity with traction-free cracks and a slip plane with a frictional shear stress x{. Crack propagation into the ferrite occurs once the elastic energy release rate equals twice the surface energy for cleavage in ferrite, 2y s . If T is the resolved shear stress on the slip plane and a is the normal stress on the microcracks,
crack nucleus in carbide particle
7
'
\
Figure 12-13. Smith's model.
Id
71(1 - V 2 )
(12-19) Qualitatively, Smith's (1966 a) model predicts the dependencies of the fracture stress on grain size, d, carbide thickness, c, and shear-to-tension ratio correctly. A special case of the model is discussed in the following. Since a notch concentrates the applied stress, plastic yielding occurs first at the notch root in a plastic zone which grows with increasing load. In the plastic zone, the shear stress is limited to the plastic yield stress in shear, ry, which depends on the grain size according to the Hall-Petch relation 1
(12-20)
Here ky is the Hall-Petch constant, which has values of typically around 1 MPa^/m. Because of rf = ry and of Eq. (12-20), it has become customary to replace i f — x{ in Eq. (12-19) by 0.5 ky/^d (Knott, 1971). The resulting tensile fracture stress is called the cleavage fracture stress of: (12-21) k 4 A 2 ~|i Remarkably, the so calculated af is independent of the grain size, depends strongly on the carbide thickness, and only slightly on the temperature (through T{). Further, the fracture criterion reduces to a criterion for the tensile stress (which is not surprising since the shear stress is fixed and equal to the yield stress). Contrary to these predictions, Riedel and Kochendorfer (1979) show by reference to published and their own experi-
12.3 Fracture by Plastic Void Growth
583
ments that the cleavage fracture stress depends markedly on the grain size (for a fixed carbide thickness) and on the degree of stress triaxiality. The mild dependence of of on temperature, however, is confirmed. They also analyze a modified model, which qualitatively describes the observed trends, but a quantitative description is not possible. «
12.3 Fracture by Plastic Void Growth
Figure 12-14. Dimpled fracture surface of a reactor pressure vessel steel (22NiMoCr37) (Honig, 1990).
12.3.1 Introduction 12.3.1.1 The Role of Second-Phase Particles and Inclusions
Engineering alloys usually contain second-phase particles, which contribute substantially to their strength (see Chap. 5 of this Volume). In order to impede the motion of dislocations effectively, these particles have to be finely dispersed and are usually in the submicrometer range. Examples are carbides in steels, ordered intermetallic phases in aluminum alloys and oxide particles in oxide-dispersion strengthened nickel-base alloys. Besides the small second-phase particles, commercial materials also contain larger inclusions, often as undesirable relics from the production process or from the raw materials. Examples are aluminum-oxide and manganese-sulfide inclusions in steels. When the material is deformed plastically, these inclusions and second-phase particles fracture or decohere from the matrix. The resulting voids grow and their eventual coalescence leads to fracture. This type of fracture is called fibrous, or dimpled fracture, since the fracture surface is covered by dimples resulting from the voids (Fig. 12-14).
Figure 12-14 reveals an additional feature which is typical for many steels. There are two size classes of dimples, one around large inclusions (like MnS or A12O3), the other around small carbides. Large inclusions are often weakly bonded and initiate voids easily. As these voids grow, the intervoid material is strained intensely. This eventually leads to breakage or decohesion of the strong and well bonded small particles. There is evidence that the growth of the large voids is terminated and fracture intervenes, as soon as the small particles break. The processes of void nucleation, growth and coalescence are described in the following sections. For details, the reader is referred to the original literature or to more comprehensive reviews on ductile fracture (Thomason, 1990; Tvergaard, 1990). 12.3.1.2 Complete Necking or Superplasticity
In the absence of void-nucleating particles, plastic deformation can go on nearly indefinitely to very large strains. Depending on the strain-hardening properties and the
584
12 Fracture Mechanisms
strain-rate sensitivity of the flow stress, the material may either be superplastic, or it may fail by necking. Superplastic behavior means that the whole specimen extends uniformly to very large strains, whereas necking implies localization of large strains in the neck. Ideally, the specimen can neck down to a point (Fig. 12-15). 12.3.1.3 The Orientation of the Fracture Surface
The macroscopic orientation of a dimpled fracture surface to the applied stresses does not obey simple rules. Whereas cleavage fracture occurs by the propagation of one dominant crack, which prefers the orientation normal to the tensile stress, dimpled fracture is brought about by the growth of many voids distributed in the specimen according to the strain and stress history experienced by the material at different locations. Depending on the stress and the distribution of voids prior to final failure the fracture path can be normal to the applied tensile stress, but inclined fracture surfaces are at least as frequent. Round tensile bars of ductile materials develop a so-called cup-and-cone fracture
1 1 a) complete I necking \
j /
b) cup-cone / fracture /
\ \
T
Figure 12-15. Ductile fracture: (a) Material developing no voids, (b) material with voids.
surface (Fig. 12-15). This is explained as follows. Plastic elongation of the specimen leads to the formation of a neck. In the neck region, strains are concentrated, and the conditions for void growth are particularly favorable in the center of the neck. Once these voids are large enough to reduce the load-carrying capacity of the central region of the specimen markedly, this region starts to act like a circular crack. Such a crack concentrates strains on inclined slip bands emanating from the crack tip. At some point, it may become easier for the crack to traverse the damaged material along the slip bands, rather than to follow its original direction normal to the tensile axis. A finite-element calculation by Tvergaard and Needleman (1984) based on the Gurson model (Sec. 12.3.6) quantifies the interplay between the tendency of the central crack to follow the highly strained conical surfaces and the axisymmetric constraint, which makes conical fracture unfavorable at first. 12.3.2 Void Nucleation 12.3.2.1 Observations
Experiments on void nucleation have been carried out in several alloy systems. By an appropriate heat treatment, carbides in steels can be made to assume a rounded form, and most experiments have been carried out on such spheroidized steels as relatively simple model materials (Argon and Im, 1975; LeRoy et al, 1981; Fisher and Gurland, 1981). Void nucleation in a commercial 9Cr-lMo steel was investigated by Senior et al. (1988). For reviews of the literature, see Goods and Brown (1979) and Thomason (1990). These investigations show that the number of voids increases with increasing strain, sometimes according to a linear relation (Gurland, 1972), sometimes in a
12.3 Fracture by Plastic Void Growth
nonlinear fashion (Cox and Low, 1974; Fisher and Gurland, 1981; Senior et al., 1988). Further it is found that large particles usually nucleate voids earlier than small particles (Fisher and Gurland, 1981; Thomason, 1990). 12.3.2.2 Nucleation Theory Based on Dislocations
For small particles (say, below 1 (im), it is appropriate to describe the plasticity of the surrounding matrix by the motion of dislocations, rather than by continuum plasticity. Decohesion is assumed to occur when the tensile stress across the particle/ matrix interface reaches a critical value ac. With these assumptions, Brown and Stobbs (1976) derive the following expression for the macroscopic strain at which nucleation of a void occurs (12-22) where rp is the particle radius, b is the magnitude of the Burgers vector, G is the shear modulus, and crm is the applied mean (or hydrostatic) stress. Goods and Brown (1979) show some evidence that the nucleation strain indeed increases with the particle size, a dependence which is opposite to the trend observed in other investigations. For a further discussion of this issue, see Thomason (1990). 12.3.2.3 Nucleation Theory Based on Continuum Plasticity
The stresses exerted by a plastically deforming matrix on a brittle inclusion have been calculated in the framework of continuum mechanics by Argon et al. (1975) and by Needleman (1987), who uses the finite-element method. If oc is the critical stress for decohesion of the inclusion, and
585
t7e and am are the applied equivalent and hydrostatic stress, respectively, Needleman gives the criterion for decohesion 1.4 (je + 0.5 am = ac
(12-23)
This modifies the result of the approximate analysis of Argon et al. (1975), who obtain all factors in Eq. (12-23) equal to unity. If
If the spacing of the voids is large compared to their diameter, they can be considered to grow independently, each in an effectively infinite block of material with a remotely uniform stress and strain field. As a consequence of plastic flow, the voids grow (or shrink) in volume, they change their shape to prolate (or oblate) forms, and the principal axes of the voids may rotate depending on the stressing system and history.
586
12 Fracture Mechanisms
scribed by the closed-form solution
12.3.3.1 Spherical Void in Ideally Plastic Material
A sufficiently general, yet not too complicated starting point is the model of Rice and Tracey (1969). They consider the growth and incipient shape change of spherical voids in ideally plastic materials under arbitrary stress states. They generate approximate solutions to the continuummechanical equations using the RayleighRitz method with two types of trial functions, one of them describing the sphericalsymmetric void growth, the other describing a volume-preserving shape change. Rice and Tracey's results for the volume growth rate can be summarized by the formula (1225) V
3
= 1.674 s i n h — - + 0.024 vL cosh ^ 2(7. 2o' Equation (12-25) shows that the void growth rate is proportional to the remotely applied strain rate; ee is the equivalent tensile strain rate defined by se = (2 &'tj e'f y3) 1/2 . Further the void growth rate depends on the ratio of hydrostatic to equivalent tensile stress, where in a nonhardening plastic material ae is constant and equal to the tensile yield stress. A minor additional dependence exists on the Lode variable vL defined by 3 fin
vT = —
(12-26)
— 8 III
where £r > % > £m are the principal components of the remotely applied strain rates; vL can vary from — 1 to + 1 for an incompressible material. Equation (12-25) was obtained by an interpolation between the high-triaxiality limits in tension and compression. The interpolation formula reproduces the numerical results of the Rayleigh-Ritz method closely. The high-triaxiality limit is de-
T^T = 0.849 e x p ( ^ = )
(12-27)
if vL is set equal to 1 as appropriate for axisymmetric loading. This showns that, for a given strain rate, the void growth rate increases exponentially with the triaxiality ratio (7m/cre. To illustrate this strong dependence, one compares the void growth rates for uniaxial tension ((7m/(7e = 1/3) and for the stress field ahead of a crack {oj (7e = 2.49 for nonhardening material). Using Eq. (12-25) one finds normalized growth rates V/(seV) of 0.9 for uniaxial tension and 35 for the crack-tip field. 12.3.3.2 Spheroidal Voids in Power-Law Viscous Materials
Budiansky et al. (1982) consider spheroidal voids including the limiting cases of needles and flat cracks in power-law viscous material under axisymmetric loading. The material law is given by Eq. (12-5), which reduces to s = A an in uniaxial tension. Besides various other cases, Budiansky et al. (1982) analyze the growth of a spherical void. In the high-triaxiality limit, crm/(7e ^> 1, a closed-form solution is possible:
(12-28) 3|_2n<7e
T
Figure 12-16 shows this approximation (dashed lines) together with the full numerical solutions obtained with the RayleighRitz method (solid lines). For n = 1, Eq. (12-28) is exact for all values of ojoc. 12.3.3.3 Other Models for the Growth of Isolated Voids
A few further aspects of isolated void growth are treated by the following au-
587
12.3 Fracture by Plastic Void Growth
0
1 2 3 Triaxiality, a m / a e
4
Figure 12-16. Normalized void growth rate v = F/(ee V) after Budiansky et al. (1982).
thors: McClintock et al. (1966) and McClintock (1968 a) on two-dimensional (cylindrical) voids, Tracey (1971) on voids in strain-hardening materials, Fleck and Hutchinson (1986) on voids in nonlinear viscous materials under combined shear and hydrostatic stresses (as opposed to the axisymmetric loading case treated by Budiansky et al., 1982), and Huang et al. (1991) on the effects of elastic strains on void growth. Huang et al. also improve the numerical accuracy of the Rice and Tracey model and show that the growth rate in Eq. (12-27) should be multiplied by a factor of about 1.5.
of the periodic void array by a circularcylindrical cell. This reduces the three-dimensional problem to an axisymmetric problem (provided that the loading is axisymmetric), which is computationally much simpler, with little loss in accuracy (Worswick and Pick, 1990). Most of these calculations are based on large-strain theory, i.e., the shape change of the voids is taken into account and is part of the result. Figure 12-17 shows a numerical result of Koplik and Needleman (1988) for the increase of the void volume fraction, /, as a function of the equivalent strain ee. The initial void volume fraction is / 0 = 0.0013, stress triaxialities of ojoe = 1,2 and 3 are considered and the hardening exponent is N = 0.1. For comparison, the predictions of Budiansky et al. (1982) for isolated voids are shown as dashed lines. Here it was assumed that the voids remain spherical, so that the void growth law, Eq. (12-28), can be integrated readily to give / = /oexp(i5fic)
where v = V/(se V) is the dimensionless growth rate of a spherical void, which is given by Eq. (12-28) in the high-triaxiality limit and numerically otherwise (Fig. 12-16). Interestingly, the isolated-void model does 0.08
12.3.4 Interacting Voids
Early finite-element calculations of void interactions were done for two-dimensional arrays of (cylindrical) voids (Needleman, 1972; Tvergaard, 1981). Later, threedimensional void arrays were analyzed either directly (McMeeking and Horn, 1990; Worswick and Pick, 1990) or by somewhat simplified cell models (Tvergaard, 1982; Bourcier et al., 1986; Koplik and Needleman, 1988; Becker et al., 1988). The cell models replace the actual unit cell
(12-29)
0.06 0.04o Q.
0.020.00,
u 0.5 Equivalent Strain, e e
I 1
/
/
/
1
1 1 1
_
1.0
Figure 12-17. Evolution of the porosity. Solid lines: Finite element result of Koplik and Needleman (1988). Dashed lines: Noninteracting voids.
588
12 Fracture Mechanisms
not deviate much from the numerical solution for interacting voids, although at the circles on the curves strains start to localize rather abruptly to the ligaments between the voids and fracture occurs after a small additional strain increment. Figure 12-18 shows the evolution of this strain localization between 14.6% and 18% strain for ajae = 2 and f0 = 0.0104. Interestingly, the figure also confirms the prediction of Budiansky et al. (1982) that at higher triaxialities the voids assume an oblate shape perpendicular to the maximum straining direction. The three-dimensional calculations of McMeeking and Horn (1990) include the cases of uniaxial tension (crm/cre = 1/3) and pure shear, which were not considered by Koplik and Needleman (1988). In uniaxial tension, the voids become elongated and
the material between the voids experiences nearly uniform strains. For pure shear, there is a trend to localize strains along 45°-lines to the principal tensile axis. 12.3.5 Strain to Failure
Although numerical analyses like that of Koplik and Needleman (1988) provide values for the strain to fracture under various conditions, it appears worthwhile to recall previous attempts to achieve the same goal by simpler means. 12.3.5.1 Coalescence of Independently Growing Voids
The simplest approach is to assume that the void growth rates calculated for isolated voids remain valid until the voids touch each other. Assuming that the voids retain a spherical shape leads to a strain to fracture T l n ^
(12-30)
where snuc is the strain at which voids nucleate, f0 is their initial volume fraction, (TT/6) is their volume fraction at mutual contact (assuming that they form a simple cubic lattice), and v = V/(ec V) is the normalized void growth rate. In uniaxial tension, v is of the order 1. If enuc = 30% and f0 = 1 %, the strain to fracture is calculated to be £f = 426%. Since real materials usually exhibit a far smaller ductility, this estimate is probably too simpie. 12.3.5.2 Thomason's Model (c)
(d)
Figure 12-18. Contours of constant equivalent plastic strain for IV = 0.1, f0 = 0.0104, ajae = 2, and macroscopic strains se = (a) 2.93%, (b) 9.45%, (c) 14.6%, and (d) 18% (Koplik and Needleman, 1988).
In a series of papers, Thomason (1968, 1990) pointed out that the independent growth of voids is terminated when the load on the ligaments between the voids reaches the plastic limit load. Then the
12.3 Fracture by Plastic Void Growth
strain localizes to the ligament and the material fails with little additional strain outside the ligament. This general idea is supported by the finite-element models described in the previous section, and it was similarly and independently suggested by McClintock (1968 a, b). Instead of attempting a numerical solution of the plastic field equations, Thomason (1990) introduces several approximations. First he assumes that up to the point of strain localization, the voids grow independently, an assumption which appears to be reasonable in view of Fig. 12-17. To calculate the volume and shape change of the voids, Thomason employs the Rice and Tracey (1969) model. Then he performs a plastic limit-load analysis for the remaining ligaments between the voids, both for two- and three-dimensional void configurations. For the plane-strain case, slip-line field theory is used, whereas in the three-dimensional case the void shape is idealized to be square-prismatic, and kinematically admissible velocity fields are constructed to obtain upper-bound estimates for the limit load. Results for the strain to fracture are shown in Fig. 12-19 below in comparison with other models. 1.5 — Thomason — Rice and Tracey ••• Budiansky et al. Koplik and Needleman
1.0
Om 1
<5Q
-
N=0.2 N=0.1 N=0
B
1
2
A O
• •
•
•
Becker et al.:
i 0.5 c GO
Vl.6
0
£.67
0.05 0.1 Initial Porosity, fo
0.15
Figure 12-19. Strain to fracture vs. initial porosity calculated from various models.
589
12.3.5.3 Comparison of Ductilities Predicted by Different Models
Figure 12-19 shows a comparison of the strain to fracture predicted by the models described above. A possible strain to void nucleation is neglected. The results are shown as a function of the initial void volume fraction / 0 with the stress triaxiality <7m/cre as a parameter. Axisymmetric loading conditions are considered. Uniaxial tension corresponds to ojoc = 1/3, but no complete set of results is available for that special case. In Fig. 12-19, the solid lines represent the results of Thomason's (1990) three-dimensional analysis. As the figure shows, the model predicts a rapid decrease of the ductility with increasing initial void volume fraction and with increasing triaxiality. Above a certain / 0 , the ductility is zero (apart from a possible strain to void nucleation), since the limit load of the ligaments is reached, before plastic flow of the matrix commences. The data points in Fig. 12-19 represent the critical strain at the onset of strain localization as calculated by Koplik and Needleman (1988) and by Becker etal. (1988) using the same finite-element cell model. Only a small additional strain increment (typically 5% of the strain to localization) is needed to achieve the strain to fracture. The open symbols represent a case of mild triaxiality, Gm/ae = 1. In this case, the Thomason model agrees well with the numerical solution. For the case of higher triaxiality {ojoe = 2, full symbols), Thomason provides no comparable results, but extrapolation of his results leads to too small ductilities compared to the numerical model. Among other possibilities, this might be a consequence of the fact that he uses the Rice and Tracey (1969) model, which is inaccurate as far as the
590
12 Fracture Mechanisms
evolution of the void shape at high triaxialities is concerned. At higher triaxialities the voids become oblate, whereas the Rice and Tracey model predicts prolate shapes. Prolate shapes yield too small limit loads and ductilities. The dashed and dashed-dotted lines in Fig. 12-19 represent Eq. (12-30). This is the ductility which is calculated if the voids are assumed to grow independently, until they touch each other. Obviously, this overestimates the ductility, especially at higher void volume fractions, irrespective of whether the model of Rice and Tracey (1969) or that of Budiansky et al. (1982) is used.
sponds to the void spacing, so that void interactions are modeled in an approximate way. Then he assumes simple displacement rate fields, which are compatible with the macroscopic strain rate field on the outer shell surface, and calculates the pertinent macroscopic stresses. This leads to a macroscopic yield condition and a flow rule in terms of the following flow potential (12-31) 2(J
M
with $ = 0: yield condition
(12-32)
and 12.3.6 Constitutive Equations for Porous Plastic Solids
flow rule
(12-33)
}
ij
For practical reasons, the stress analysis of a macroscopic body containing millions of voids cannot be based on the modeling of the individual voids. Rather, macroscopic constitutive equations are needed for the response of a porous solid on a size scale large compared to the void spacing, so that each volume element of the continuum contains sufficiently many voids. These constitutive equations are usually implemented into finite element codes and applied, for example, to the failure of tensile bars after necking or to the failure of notched or cracked specimens. This way of describing the behavior of porous solids is called damage mechanics. 12.3.6.1 The Gurson Model
The most widely used constitutive equations for porous elastic-plastic solids are based on work of Gurson (1977). Besides several other, less important cases, Gurson considers a spherical void centered in a spherical shell of nonhardening plastic material. The outer shell diameter corre-
Here,
Weaknesses of the Gurson model were revealed when the model was applied to predict the onset of strain localization as a consequence of a constitutive instability (Yamamoto, 1978; Needleman and Rice, 1978). It turned out that the strain at which the Gurson model develops such an instability is much higher than the ductilities which are obtained from both experiments and cell-model calculations. Therefore Tvergaard (1981, 1982) suggested to
591
12.3 Fracture by Plastic Void Growth
introduce two adjustable parameters, q± and q2 into the Gurson model as shown below. This modification still did not describe the rapid loss of strength near final failure, when the ligaments between the voids start to collapse. Therefore Tvergaard and Needleman (1984) proposed to enhance the effect of porosity on the constitutive response by using a variable / * instead of the void volume fraction / and by letting / * grow faster than / near final fracture. Then Eq. (12-31) is replaced by the modified flow potential (I 2g 1 /*cosh( (12-34)
123.6.3 Performance of the Modified Gurson Model Compared to Micromechanical Models The three-dimensional finite-element models of interacting voids referred to in Sec. 12.3.4 have been used to check the accuracy of the Gurson model. Among these comparisons an example from Koplik and Needleman (1988) is shown here. They obtain an only moderately good agreement for the stress-strain response, if they use the unmodified porosity / in the flow potential of the Gurson model. With the modified porosity / * , however, the agreement is very good for both, the stressstrain response and for the evolution of the porosity (Fig. 12-20).
Values of qx and q2 are obtained by comparisons with cell-model calculations and with experiments. Typically one finds qt = 1.25 to 2.4 and q2 = 0.75 to 1. For / * , a bilinear dependence on / is suggested for / < / c (12-35) The adjustable parameters fc and ff correspond to the values of / at the onset of enhanced void growth and final fracture, respectively. By comparison with cellmodel calculations it turns out that fc should depend on the initial porosity, but not on the hardening exponent nor on the stress triaxiality (Koplik and Needleman, 1988; Becker et al, 1988). For f0 = 0.0013 to 0.07 they find fc = 0.03 to 0.12. The model is insensitive to the choice of / f , and a value of f{ = 0.25 has been suggested. Other constitutive models for porous plastic solids have been proposed. Among them the approach of Duva (1986), which is based on a Rayleigh-Ritz analysis of a cell model, appears promising.
0.4
0.6
Strain, 0.08 -.
3 1/
M
0.06
W
2 o
1
0.04
Q_
0.02
oon
UL——-^^
0.0 (b)
0.2
0.4
0.6
0.8
1.0
Strain, e
Figure 12-20. Comparison of modified Gurson model (dashed lines) with numerical cell-model calculations (solid lines), (a) Stress-strain curve, (b) evolution of porosity; N = 0.1, f0 = 0.0013.
592
12 Fracture Mechanisms
12.3.7 Comparison with Experiments 12.3.7.1 Ductility
Figure 12-21 is a comparison between calculated and measured ductilities. The ductility is measured in uniaxial tensile tests as the average natural strain in the neck, ef = ln(A0/Af% where Ao is the initial cross-sectional area of the specimen and Ai is the area of the neck at fracture. The experimental scatterband comprises data from the following sources: Edelson and Baldwin (1962) on copper containing various types and amounts of inclusions; LeRoy et al. (1981) on spheroidized steel; Magnusen etal. (1985) on powder-metallurgical (p/m) nickel and titanium; Bourcier et al. (1986) on p/m titanium; and Spitzig et al. (1988) on p/m iron. In the case of LeRoy et al.'s data, it is not the strain to fracture itself which is plotted, but the strain to nucleation was subtracted, which was found to be 0.5 to 0.6 (Thomason, 1990). The p/m materials were partially sintered and contained various degrees of porosity. Within the scatterband, materials with a high strain hardening exponent tend to lie near the upper edge (iron with N = 0.32), while those with a lower expo-
1.5
ar
A Becker et al. cm/oe = 0.67 om Becker a m /tfe = 0.65 ® Becker (nonprop. loading) Isolated void growth 0.8
Thomason
0.05
0.1
0.15
Initial Porosity, f0
Figure 12-21. Comparison of experimental and theoretical ductilities in the uniaxial tensile test.
nent lie near the lower edge (Ti with AT = 0.19). The nearly universal behavior shown in Fig. 12-21 should not be generalized to all types of ductile metals. Counterexamples are many steels, the ductility of which may be substantially reduced by secondary void nucleation at particles between the primary voids. Figure 12-21 also shows various theoretical results. First one notes that the assumption of independent void growth overestimates the ductility: the dasheddotted line represents Eq. (12-30) with 8nuc = 0 and with v taken from Budiansky et al. (1982). This is so, although a triaxiality of <JJOQ = 0.8 was assumed, which is about the highest value that can reasonably be assumed for the neck region. (Uniaxial tension without necking implies oj <7e = 1/3.) All other theoretical predictions included in Fig. 12-21 take void interactions into account and are compatible with the experimental scatterband. The dashed line represents Thomason's (1990) model with the void volume and void shape integrated for the nonproportional stressing history in the neck. For the shape evolution of the neck, Thomason uses a relation found empirically by LeRoy etal. (1981). The triangular data point in Fig. 12-21 is from Becker et al. (1988) for ojoe = 0.67, a triaxiality which appears realistic for the neck. This result is based on a finite-element cell-model calculation, and the strong strain hardening of iron was assumed. Therefore, the prediction should lie near the upper edge of the scatterband. This is actually the case. The circular symbols are results of Becker (1987) based on the modified Gurson model (with ql = 2.38, q2 = 0.748 and fc = 0.15). The symbol
12.3 Fracture by Plastic Void Growth
final triaxiality in the neck). The open circle is valid for a homogeneous void distribution, while the full circle and the crossed circle refer to an inhomogeneous distribution. It should be mentioned that Becker's results cannot be regarded as truly theoretical predictions, since the parameters q± and q2 were fitted to experiments of Spitzig etal. (1988). Ductilities as a function of triaxiality have been measured by Hancock and Mackenzie (1976) and Hancock and Brown (1983) using notched specimens of various steels. They show that the ductility drops rapidly with increasing triaxiality, and that this drop can approximately be described by the exponential relation following from the Rice and Tracey (1969) model, Eq. (12-27). A much better theoretical description of the data is obtained by Needleman and Tvergaard (1984), who simulate the tests using the modified Gurson model (with qt = 1.5) including continuous void nucleation.
593
Spitzig et al. (1988) measure the increase in porosity with strain in powder-metallurgical iron and find
£
(12-36)
/o
in the uniform-elongation regime of their tests (e < 0.3). Bourcier et al. (1986) find the same result for titanium. It agrees nearly exactly with Eq. (12-29) for small e, no matter whether v is taken from Rice and Tracey (1969) or from Budiansky et al. (1982). For the factor in front of e in Eq. (12-36) (which is unity experimentally) Rice and Tracey's model yields 0.9, whereas Budiansky et al. give 1.0 for N = 0.33 as appropriate for the powder-metallurgical iron. The three-dimensional finite-element model of Worswick and Pick (1990) gives the same result. The (modified) Gurson model predicts an initial increase of the porosity for uniaxial tension according to ^ = 1+0.78^(1-/0)8
(12-37)
Jo
12.3.7.2 Void Growth
By interrupting tensile tests at various stages, Marini et al. (1985) investigated the growth of voids in a powder-metallurgical steel containing A12O3 particles. They find that the Rice and Tracey (1969) model underestimates their measured void growth rates by a factor 3 to 5, but that the dependence on the triaxiality is predicted correctly. LeRoy et al. (1981) evaluate their experiments on spheroidized steels in relation to the Rice and Tracey (1969) model and, contrary to Marini etal. (1985), find reasonable agreement. Worswick and Pick (1990) arrive at the same conclusion, when they evaluate their own experiments as well as results from the literature.
(for small / 0 ). This agrees with the experimental result, Eq. (12-36), if one chooses qt = 1.35, a value which is close to that suggested by cell-model calculations. 12.3.7.3 Constitutive Behavior
The presence of porosity reduces the yield and flow stress of metals. Empirically, the reduction of the yield stress ay is described by (12-38) 'yO
where oy0 is the yield stress of the fully dense material. The factor a is found to be 3 by Spitzig etal. (1988) for iron and by Magnusen et al. (1985) for nickel and tita-
594
12 Fracture Mechanisms
nium, while Bourcier et al. (1986) find oc = 4 to 5 for titanium and Ti-6A1-4V. The micromechanical finite-element model of McMeeking and Horn (1990) gives a value of a = 3 to 5, which is consistent with the observations. On the other hand, the modified Gurson model gives a = 1.13q1 for uniaxial tension and small f0. Thus the experimental value a = 3 is recovered, if q1 = 2.66. This result illustrates the problem the modified Gurson model has in fitting the observed growth rate of the porosity and the constitutive response simultaneously. Fitting the observed void growth rates requires q1 = 1.35, whereas the constitutive response is better fitted by choosing q1 = 2.66.
12.4 Creep Fracture 12.4.1 Introduction
At temperatures exceeding 30 to 50% of the absolute melting temperature, metallic materials undergo continuing plastic deformation ("creep") under low sustained stresses. In the same stress and temperature range, many metals and alloys fracture along their grain boundaries with relatively low ductilities when loaded for prolonged times. This type of fracture is brought about by the nucleation, growth and coalescence of cavities on grain boundaries. (In accord with the common usage, small voids developing on grain boundaries at high temperatures are called "cavities" as distinct from the "voids" developing at inclusions at room temperature.) Figure 12-22 shows such cavities in a CrMoV steel tested at 540 °C. The resulting intercrystalline fracture surface is covered by shallow dimples corresponding to the prior cavities. Besides
Figure 12-22. Creep cavities in a l/2Cr-l/2Mo-l/4V steel tested at 540 °C. Polished and etched section.
creep, fatigue, corrosion and a degradation of the microstructure, grain boundary cavitation is one of the predominant factors that bound the useable temperature range and the lifetime of materials. The common cause for most of these problems is the increasing mobility of atoms by diffusion at higher temperatures. In some materials, creep damage has the form of grain boundary cracks often emanating from triple grain junctions. These crack-like defects have been called wedge cracks because of their shape (Fig. 12-23). Wedge cracks tend to predominate at relatively high stresses (McLean, 1956/7). They may result from the coalescence of ordinary rounded cavities. In this case, the (a)
Figure 12-23. Wedge cracks by (a) brittle decohesion or (b) cavity coalescence (from Riedel, 1987).
12.4 Creep Fracture
cracks have serrated edges, and small cavities can be detected in front of the crack tip (Chen and Argon, 1981). There may be other cases, however, in which wedge cracking occurs by a brittle decohesion of the grains probability assisted by segregated impurity atoms. This possibility was discussed especially in connection with stress relief cracking (McMahon, 1984). It will not be pursued here. Grain boundary cavitation is of concern to all industries in which structural materials are exposed to high temperatures. A few examples will be mentioned here. For a more comprehensive survey of published work on cavitation in various metals, alloys and ceramics, the reader is referred to Riedel (1987). Cavitation plays an important role in the creep-resistant ferritic steels that are widely used in fossil-fired power plants in the temperature range up to 550 °C. While numerous investigations on the lCr-l/2Mo, 2iCr-lMo and l/2Cr-l/2Mo-l/4V steels are available, publications on cavitation in the 12% and 9% chromium steels are less frequent (Eggeler et al., 1989; Biirgel et al., 1990; Wachter et al., 1991). Since the safety of the high-pressure piping systems in fossil-fired power plants depends on the control of grain boundary cavitation, critical locations such as weldments and pipe bends are regularly inspected by the replication technique, and recommendations on continued operation, reduced inspection intervals or immediate exchange are based on the detected degree of creep damage (VdTUV-Merkblatt Dampfkessel, 1983; Neubauer, 1981; Bendick et al., 1985). Austenitic steels can be employed at higher temperatures (up to, say, 750 °C), since the diffusion coefficient in the facecentered cubic (austenitic) lattice is about two orders lower than in the body centered (ferritic) lattice. Many but not all austenitic
595
steels develop grain boundary cavities depending on the heat treatment and the impurity content. For higher temperatures (up to about 1000 °C), for example in gas turbines, nickel base superalloys are used. Some of them, but again not all, fracture by intergranular cavitation at high temperatures. Small additions of Zr or B tend to impede or even suppress intergranular cavitation, so that failure becomes transgranular as at room temperature. Creep cavitation was further investigated in alloys of magnesium, copper, titanium, tungsten, and zirconium. Also ceramic materials fracture by intergranular cavitation at high temperatures (above 1000 °C) and long times of loading. 12.4.2 Cavity Nucleation 12.4.2.1 Nucleation Sites In relatively pure metals, cavities may nucleate at intersections of slip bands with grain boundaries, probably because of the stress concentration at these locations. Similarly, grain boundary sliding focusses stresses on ledges in grain boundaries and on triple grain junctions (Fig. 12-24). Cavity nucleation has been observed to occur at each of these locations. In materials containing second-phase particles, such as most structural alloys, these particles are often preferred cavity
Figure 12-24. Cavity nucleation sites.
596
12 Fracture Mechanisms
nucleation sites. For an evaluation of experimental investigations, see Riedel (1987). In ferritic steels, cavities are nucleated at carbides, oxides and sulfides. Also in austenitic steels cavities form at carbides (Argon et al., 1981; Swindeman et al., 1981). If sigma-phase particles are present, their edges are prone to cavitation. In nickel-base alloys, coarse carbides have been identified as cavity nucleation sites, often in conjunction with a slip band impinging on the particle (Shiozawa and Weertman, 1983). A low impurity level generally reduces the susceptibility to cavitation and therefore improves the creep ductility (Tipler and Hopkins, 1976). 12.4.2.2 The Role of Grain Boundary Sliding
As a consequence of the increasing diffusivity of atoms, grain boundary sliding becomes possible at elevated temperatures. This concentrates stresses on triple grain junctions and on ledges and asperities in the grain boundaries. Hence cavity nucleation at these stress concentrations has often been postulated. Indeed, the importance of grain boundary sliding for cavity nucleation was convincingly demonstrated by Chen and Machlin (1957) and Intrater and Machlin (1959). They found that copper bicrystals develop no cavities under pure tension, but exhibit profuse cavitation, if the grain boundary is sheared prior to tensile loading. On the other hand, Raj (1975) and Gandhi and Raj (1982) observe cavitation at particles on bicrystal boundaries under pure tension. Contrary to the observations on copper bicrystals, grain boundary sliding can hardly be essential for cavity nucleation in polycrystalline structural materials, since cavities are preferentially found on boundaries that are normal to the tensile stress
and therefore slide the least (Chen and Argon, 1981; Chen and Weertman, 1984). There are counter-examples, where inclined boundaries cavitate preferentially, but even in those cases the preference is shifted to normal boundaries at low stress. 12.4.2.3 The Observed Nucleation Kinetics
Numerous investigations arrive at the common conclusion that cavity nucleation starts early and continues over the entire creep life. Roughly speaking the number of cavities often increases approximately in proportion to the creep strain (Dyson, 1983), i.e., the nucleation rate is proportional to the strain rate 7* — a' F
(12-19^
Here J* is the number of cavities nucleated per unit time and grain boundary area. The factor of proportionality a' varies greatly from material to material. It ranges from a' = 109/m2 or less for high-purity steels over 4 x 1010/m2 for a commercial steel quality to 4 x 10 12 /m 2 for a coarse-grained heat affected zone material which contains finely dispersed sulfides on its grain boundaries. The corresponding creep ductilities vary between 20% and 2% strain to rupture. Dyson (1983) and Riedel (1987) compile experimental evidence for the approximate validity of Eq. (12-39) and also quote results which deviate from that simple relation. Especially, the possibility of a critical stress required for cavity nucleation is discussed. For ferritic steels such a critical stress certainly lies below the usual design stresses (50 to 100 MPa), since many of them develop cavities during service. In pure metals, cavity nucleation is still observed at stresses below 10 MPa.
12.4 Creep Fracture
12.4.2.4 Theory of Nucleation
12.4.3 Models for Cavity Growth
Cavities can be nucleated by a thermally activated condensation of atomic vacancies. It is likely that at high temperatures this process becomes more favorable than the rupturing of atomic bonds by high local stresses. Within the framework of classical nucleation theory, it has been shown (Raj and Ashby, 1975; Riedel, 1987), that the cavity nucleation rate is small below, and becomes very large, above a certain nucleation stress, which is given by
12.4.3.1 Overview
y'fAi 10kT
(12-40)
Here the dimensionless factor / v accounts for the cavity nucleus shape; it is defined as cavity volume divided by radius of surface curvature cubed. Taking the surface energy as ys = 1.5 J/m2 and T = 850 K, Eq. (12-40) simplifies to cjnuc = 5360 v //^MPa. This numerical example shows that the predicted nucleation stress is too high compared to observed values, unless the volumetric function / v is of the order 10 ~5 to 10 ~4. It is not likely, however, that cavity nuclei generally have the extreme shapes corresponding to such small values of / v . It has therefore been discussed (e.g., Argon, 1982; Riedel, 1984, 1987) whether the stress could be raised locally to the required levels by stress concentrations at slip bands or due to grain boundary sliding. As a result of the stress analyses it appeares that the stress concentrations cannot be exceedingly large, and are hardly sufficient to reach the theoretical nucleation stresses. Thus the problem of cavity nucleation cannot be regarded as being quantitatively understood.
597
Cavities can grow by several mechanisms. The basic mechanism at elevated temperatures is probably the diffusion of atoms away from the cavity into the grain boundary (Hull and Rimmer, 1959). At high stresses, diffusive cavity growth may be enhanced by plastic flow of the matrix. At low stresses, diffusive cavity growth on an isolated cavitating boundary can be constrained by the rigidity of the surrounding material (Dyson, 1976, 1979). In the case of diffusive growth, the cases of rapid and slow surface diffusion are distinguished (Chuang et al., 1979). All these and other models were summarized by Riedel (1987). Comparisons with experiments suggest that for structural alloys tested under low-stress, long-time conditions, the constrained diffusive mechanism is the most important one. 12.4.3.2 Constrained Diffusive Cavity Growth
Over a substantial fraction of the lifetime, cavitation is confined to isolated grain boundary facets that are surrounded by undamaged material as shown in Fig. 12-25. To understand the idea of the constrained growth model (Dyson, 1976, 1979) suppose that the cavities grow easily under the influence of the applied stress by diffusion of atoms away from the cavity into the grain boundary and that the surrounding material is rigid. Then the excess volume of the growing cavities cannot be accommodated, or, in other words, the rigid material exerts a back stress on the cavitating boundary. Thus the stress oh which effectively acts on the grain boundary and drives diffusive cavity growth is reduced such that the rate of diffusive cav-
598
12 Fracture Mechanisms
The cavity radius R, the spacing X and the facet size d were defined in Fig. 12-25. Further Q is atomic volume, and 5Db is the grain boundary diffusion coefficient (in m3/s), co = (2 R/X)2 is the cavitated area fraction of the grain facet, q' = n2 (1 + 3/n)1/2 is an abbreviation, a e = | oY —
ity growth is compatible with the rate at which the surrounding material can accommodate the excess volume by creep flow. Therefore, in a relatively creep-resistant material, the rate of creep is rate controlling for the cavity growth process. This is called constrained cavity growth. On the other hand, if the material creeps readily, the excess volume is accommodated easily, and the rate of cavity growth is controlled by the diffusive growth process itself. This is the unconstrained limit. Rice (1981) cast this idea into a quantitative model, which was subsequently improved by several authors (see, e.g., Riedel, 1987). Rice idealizes the cavitating facet as a circular crack in a creeping matrix described by Norton's power law, Eq. (12-5). A normal tensile stress, oY, and a transverse stress, aT, are applied remote from the facet, and the facet transmits a stress <7b, which is determined from the compatibility of the diffusive void growth rate and the volume growth rate of the facet by creep. Under these conditions the cavity growth rate is obtained as R=
01-(1
-CO)
(12-41) X2d
(12-42)
the dimensionless function 2(1+oos,fr)-'-co,* 2smi/f
accounts for the lenticular cavity shape, and q((o) = - 21nco - (3 - ©)(1 - co) (12-44) Equation (12-41) comprises the limiting cases of constrained and unconstrained cavity growth. If the strain rate is high, the second term in the denominator vanishes. Then the stress on the boundary is equal to the applied tensile stress, and the cavity growth rate is that of unconstrained diffusive growth. If the strain rate is low, however, the first term in the denominator becomes negligible, and the cavity growth rate is proportional to the strain rate. In this case the stress on the boundary is reduced to the sintering stress, i.e., to a value that is generally small compared to the applied stress. This means that in the constrained limit the boundary facet acts mechanically like a microcrack which transmits no appreciable tractions. 12.4.3.3 Comparison with Measured Growth Rates
Needham (1983) and Needham and Gladman (1984) report measured cavity
12.4 Creep Fracture
599
12.4.3.4 Rupture Lifetimes
By integration of Eq. (12-41) from an initial cavity radius R0toR = A/2, the time to cavity coalescence, tc, is obtained to be: c
315QdDbaI
' (12-45)
100
150 200 250 300 O in MPa - * -
Figure 12-26. Cavity growth rates in steels (Riedel, 1985 a, 1987). Data points from Needham (1983). Solid lines: constrained growth; dashed lines: unconstrained growth.
growth rates in ferritic steels. Figure 12-26 shows a comparison with the predictions of the constrained and unconstrained models (after Riedel, 1985 a). The solid lines represent Eq. (12-41) with the measured creep rates inserted and with the values given by Needham for the grain size (d = 18|im) and for the cavity spacing (A = 3.2 jim for the lCr-l/2Mo and 4.5 |im for the 2|Cr-lMo steel). The cavity diameter was chosen as 2 R = 1 jam, and the material parameters were taken from tabulations of Frost and Ashby (1977). The agreement with the data is very good for the 2^Cr-lMo steel and fair for the lCr-l/2Mo steel. On the other hand, the unconstrained growth rate represented by the dashed lines predicts a wrong stress dependence and too high growth rates. From this comparison, it is concluded that cavity growth at low to moderate stresses is constrained in these steels and the growth rates are predicted well by the model for constrained growth.
The first term results from the unconstrained growth rate with the sintering stress and higher-order terms in Ro/A neglected. The second term, which is due to the constraint, is negligible at high strain rates, but leads to greatly prolonged times to coalescence for small strain rates. While the unconstrained term predicts a linearly inverse dependence on stress, tc oc I/a, the constrained term predicts a stronger stress dependence, tc oc \jon. Experimental rupture times tend to obey the latter relation. However, the time to coalescence, tc9 was deliberately not identified with the time to fracture, tf. A grain boundary undergoing constrained cavitation acts mechanically like a microcrack no matter whether the cavities are still small or have already linked up to form a real microcrack. Cavitating facets with relaxed stresses reduce the strength of the material, until a sufficient number of them has formed to cause rapid fracture. Hence the fracture process is not controlled by the coalescence of individual cavities on boundary facets, but rather by cavitation on an increasing number of facets. Despite this objection, tc was found to describe the time to fracture approximately correctly (Dyson, 1979; Riedel, 1987). Detailed comparisons are shown in the next section.
600
12 Fracture Mechanisms
obey power-law relations
12.4.4 Cavity Growth and Continuous Nucleation
R
12.4.4.1 The Cavity Size Distribution Function
Contrary to the models considered so far, cavities in real materials have unequal sizes, which can be described by a size distribution function N(R,t); NdR is the number of cavities per unit grain boundary area having radii between R and R + dR. Unequal cavity sizes may result from statistical variations of local stress and material properties, but also from continuous cavity nucleation. Older cavities tend to be larger than younger ones. Statistical effects will be ignored here, so that the sole cause for unequal cavity sizes is continuous nucleation. Under these circumstances the size distribution function obeys a continuity equation in size space (Riedel, 1985 a, 1987): =0
(12-46)
u/v
where the growth rate of the individual cavities, R (R, t\ depends on their size and on the time. This time dependence arises from the fact that R usually depends on the cavity spacing, which decreases as a function of time due to continuous nucleation. The nucleation rate enters into the problem as a boundary condition to Eq. (12-46): The flux in size space at R = 0 must be equal to a prescribed nucleation rate, NR = J* at 11 = 0. 12.4.4.2 Similarity Solutions of the Continuity Equation
The partial differential Eq. (12-46) together with the boundary condition can be solved by similarity solutions, if both, the cavity growth rate and the nucleation rate,
— A
T!~Pt~a
T* — A
ty
(12-47)
with the parameters a, fl,y, At, A2. Then the cavity size distribution function is found to have the form (Riedel, 1985 a, 1987):
N(R,t) =
(12-48) a+ y 1 -a
l+PA±t
1-a
Examples for the evolution of the distribution function according to Eq. (12-48) are shown in Fig. 12-27. Depending on the sign of 1 — a, the maximum of the distribution moves to larger or smaller cavity sizes, and the distribution is cut off at a maximum size or it extends to infinity. 12.4.4.3 Lifetimes for Continuous Nucleation
From the distribution function, the cavitated area fraction of grain boundary can be calculated as <x> = J n R2 N {R, t) dR. At a critical value, co{9 fracture is assumed to occur. The time to reach the critical cavitated area fraction was evaluated by Riedel (1985 a, 1987) for various growth mechanisms taking the nucleation rate in the empirical form of Eq. (12-39). For unconstrained diffusive growth, the growth rate does not exactly have the power-law form of Eq. (12-47) required for the similarity solutions to be valid. However, the sintering stress can often be neglected, and q (co) can be approximated by a power-law function which is always greater than q(co). Then the similarity solutions are applicable, and a lower-bound estimate for tf is obtained: tf = 0.25
\3/5
(12-49)
601
12.4 Creep Fracture i
i
a =0
a = -5/4
30
p
P =2 T =0
/
J ML 0
/ -
/
i
J
A\
/
/ /
-
J A\ y
1
\
0
10 R —-
1
\ l\ \ \ 2
3
in
0
-
—
r\
^—x3 \ \
i 5
P7 =
/ l\
/
=2
a= 2
= 2 - T = 1
/
—
I
t=9'
P
I
10
i
i
-
^
^^
I /1 1
R — -
R
Figure 12-27. Evolution of the cavity size distribution function. (From Riedel, 1985 a.)
If Norton's creep law is valid, the nucleation rate, which was assumed to be J* oc s, depends on stress as J* oc a", so that the stress dependence of the lifetime is tf oc 1/(7(3M + 2 ) / 5 . Thus the strain-controlled nucleation rate and the stress-controlled growth rate lead to an intermediate stress dependence of the lifetime. In the limit of constrained cavity growth, the growth rate is given by Eq. (12-41) with the first term in the denominator deleted. If the sintering stress is neglected, the growth rate has a form permitting the use of the similarity solutions. The time to reach a certain cavitated area fraction on a grain boundary, cof, is found to be
12.4.4.4 Comparison with Experiments
Despite the objections against an interpretation of tc as the rupture lifetime, Eq. (12-50) is compared now with Eq. (1249) and with experiments. Figure 12-28 shows rupture lifetimes of the 2iCr-lMo steel tested by Needham (1983), for which cavity growth rates were already given in Fig. 12-26. A comparison with the theoretical lines shows that the constrained growth model leads to a better
constrained 10'
tn = 0.78
\
(12-50)
As in Eq. (12-45), this time is denoted by tc (rather than by £f), since cavity coalescence on a grain boundary facet is not directly related to the rupture process, if the cavities grow by the constrained diffusive mechanism. For strain-rate-controlled nucleation, Eq. (12-50) gives a constant Monkman-Grant product stc with values of typically 2 to 10%.
•s w3
w2
strained 2l/4Cr-lMo steel 550°C 100
ISO 200 250 CfinMPa -»-
Figure 12-28. Rupture lifetimes measured by Needham (1983). Solid line: constrained growth; dashed line: unconstrained growth. (From Riedel, 1985 a.)
602
12 Fracture Mechanisms
lution law is written as
•constrained 10000 continuous nude at ion 1000
co =
(1 - cof
and the effect on the stress/strain-rate relation is assumed to be
unconstrained
c 100
instantaneous nucleation
a',
v
2kCr-1Mo steel 565°C 10 100
(12-51)
150 200
oinMPa - • • Figure 12-29. Rupture lifetimes measured by Cane (1979). Theoretical curves (bottom to top): Eq. (12-45) first term only, Eq. (12-49), Eq. (12-45) all terms, Eq. (12-50). (From Riedel, 1985 a.)
agreement with the data than does the unconstrained-growth model. Figure 12-29 shows a comparison with data by Cane (1979). It also illustrates the effect of continuous vs. instantaneous nucleation of all cavities at the beginning of the test. For unconstrained growth the difference is substantial, while it is nearly negligible for constrained growth. In this case the agreement of the data with the predictions of the constrained-growth model is excellent.
12.4.5 Constitutive Behavior of Cavitating Solids 12.4.5.1 The Kachanov Model
A phenomenological constitutive model for tertiary creep was proposed by Kachanov (1960) and generalized to multiaxial states of stress by Hayhurst and Leckie (1984). Here, a damage parameter co is introduced which varies from 0 in the undamaged material to 1 at fracture. The evo-
2
(l-co)n
(12-52)
Here, A, n, D, #, <> / and x are material parameters. The parameter x serves to describe the relative importance of the maximum principal stress and the von Mises equivalent stress. The damage parameter co is sometimes interpreted as the cavitated area fraction of grain boundary, but such an interpretation has no micromechanical basis. The features of the Kachanov model are similar to those of a model-based description (Sec. 12.4.5.3), and will not be discussed here. 12.4.5.2 Hutchinson's Model
In the limiting case of constrained cavity growth, cavitating grain boundary facets can be regarded as nearly traction-free microcracks. Hence the constitutive response of a creeping solid undergoing constrained cavitation was analyzed by Hutchinson (1983 a) by considering a power-law creeping matrix containing circular cracks. For widely spaced cracks he obtains a three-dimensional constitutive equation, which, for uniaxial tension, reduces to e = Aan(l + e )
(12-53)
where s = A on is the response of the matrix material and ^ d3N (13-54) 1/2 mc 2(l+3/n) is a measure of the number density of cracks. Here Nmc is the number of cracks per volume, and d is their diameter.
603
12.4 Creep Fracture
12,4.5.3 Constitutive Equations for Higher Densities of Cavitating Boundary Facets
Hutchinson's (1983 a) constitutive Eq. (12-53) which is valid for small Q, was extended to greater densities of cavitating facets using the differential self-consistent scheme by Rodin and Parks (1986, 1988). A simplified approximate approach proposed by Riedel (1987) using the self-consistent method leads to the uniaxial form of the stress/strain-rate relation 8 =
Aan
0
(12-55)
0.01
0.02
normalized
0.03 0.04 0.05 time is t
•
Figure 12-30. Creep curves according to Eq. (12-57).
with the damage parameter Q as defined in Eq. (12-54). The accuracy of Eq. (12-55) can be checked by comparison with numerical cell-model calculations by Tvergaard (1984). For n = 5 and Q = 0.38 the strain rate predicted by Eq. (12-55) agrees with the cell-model result to within 1%. A complete constitutive description involves an evolution equation for the damage parameter Q. Since cavity nucleation was found empirically to be related to the strain, the evolution equation is assumed to have the form Q=
-
graphic sections or replicas were made by Riedel (1989 a). Integrating the constitutive Eqs. (12-55) and (12-56), for constant stress gives the creep curve in the inverse form t = t (e): y+l
L t = £—
y+
(12-57)
where ss = A on is the steady-state creep rate of the matrix material. Equation (12-57) above describes secondary and tertiary creep as shown in Fig. 12-30.
(12-56) 12.4.5.4 Remanent Life
The exponent y is an adjustable parameter and 8f is the strain to fracture, which may be stress dependent, but is assumed to be constant in the following. Equations (12-55) and (12-56) form a constitutive model to describe the material behavior under constant- or variable-load conditions. It is similar to the phenomenological model of Kachanov (1960), but, contrary to the formal damage parameter used by Kachanov, the damage parameter Q has a well-defined meaning. Suggestions for the measurement of Q using metallo-
By replacing strain in Eq. (12-57) by the damage parameter Q through Eq. (12-56) one obtains a relation between the consumed lifetime fraction and the state of damage, t
(12-58)
where k = 1 + 1/y; X can also be interpreted as the strain ratio X = ef/(es t{), where es t{ is the Monkman-Grant product. Equation (12-58) is shown graphically in Fig. 12-31.
604
12 Fracture Mechanisms 1.0
12.5.1 Weakening or Strengthening of Interfaces by Segregants
0.2
0A
0.6
0.8
1.0
Figure 12-31. Damage parameter Q VS. lifetime fraction for various strain ratios X.
It was suggested to use Eq. (12-58) for remaining lifetime estimates by determining Q from metallographic sections or replicas and by taking the strain ratio X from creep tests carried out at stresses not too far from the service stress (Riedel, 1989 a). Attempts to apply the proposed procedure to material from the heat affected zone of a weldment were successful, whereas applications to the base material of ViCrV^Mo-ViV steel revealed shortcomings of the theory. Alternative formulations, which appear to be more appropriate for the base metal, were suggested by Riedel (1992).
12.5 The Chemistry of Fracture Among the effects, which the chemical environment and trace elements have on fracture processes, temper embrittlement, hydrogen embrittlement, and stress corrosion cracking are especially important and are described in the following.
Low-alloy ferritic steels can experience a severe embrittlement by a long-time exposure to temperatures between 350 °C and 500 °C. This so-called temper embrittlement is of concern to the electric-power generating industry in connection with steam turbine components operating in this temperature range. Its cause is a segregation of impurity atoms, such as phosphorus, tin, antimony or arsenic, to grain boundaries. As a result the fracture mode changes from ductile dimpled fracture before tempering to brittle intergranular fracture afterwards, with a substantial loss in Charpy impact energy. Previous reviews of the subject were given by McMahon (1968, 1984), Yoo et al. (1985) and Riedel (1987). Segregated atoms can also strengthen grain boundaries. A prominent example is the effect of boron in the ordered intermetallic compound Ni3Al. Intermetallic compounds have been investigated intensively in the past decade, since among them there are promising candidates for hightemperature materials (Sauthoff, 1989). Some of them are intrinsically brittle, while others (e.g., Ni3Al) are ductile in singlecrystal form, but exhibit brittle intergranular fracture in polycrystalline form. However, it was discovered that this intergranular brittleness can be removed by adding boron in the range of 200 ppm, besides some additional measures (Aoki and Izumi, 1979; Liu et al., 1985). 12.5.1.1 Segregation Equilibria
For many of the substitutional impurity atoms and interstitials, it is energetically favorable to occupy a site in a grain boundary rather than one in the lattice. The balance between the energetic attraction to grain boundaries and the entropy,
12.5 The Chemistry of Fracture
which favors a homogeneous distribution over the grains, results in a relative coverage of available grain boundary sites, 0, (McLean, 1957; Guttmann and McLean, 1979): 0
AG
605
increase of the coverage, 9 — 90 oc y/t9 and for long times, 9 saturates to 0^. 12.5.1.3 Grain Boundary Energy and Cohesive Strength
(12-59)
This relation is called the LangmuirMcLean isotherm, x is the concentration (more precisely, the activity) of the foreign atoms in the grains; the denominator, 1 — 0, accounts for the limited number of sites (0 cannot be larger than unity); and AG is the free enthalpy per mole of foreign atoms segregated. For phosphorus in iron, Erhart and Grabke (1981) give AG = - [34.3 + 0.021 T/(K)] kJ/mol. A similar value is obtained for a commercial steel (Moller et al., 1984). 12.5.1.2 Segregation Kinetics
After the usual heat treatments of steels, impurity atoms are distributed more or less homogeneously over the grains with no or little enrichment on grain boundaries. At elevated temperatures, the atoms start to become mobile and diffuse to grain boundaries. The diffusion problem can be solved (McLean, 1957) with the result that the grain boundary coverage increases with time according to - eT erfc
(12-60)
where 0O is the coverage at the beginning of the test, and 0^ is its value in the final equilibrium, which is given by Eq. (12-59), "erfc" denotes the complementary error function, and T is the time normalized as T = 4Dt [xftOn d)]2 where D is the diffusion coefficient of the foreign atom in the lattice, and S is the grain boundary thickness. For short times, Eq. (12-60) gives a parabolic
Segregated atoms change the energy of a grain boundary, and it can intuitively be expected that strong segregants reduce the grain boundary energy strongly. The relation is made quantitative by the Gibbs adsorption equation (see, e.g., Hirth and Rice, 1980; Riedel, 1987). A reduction of the grain boundary energy by impurity atoms does not yet mean a reduction of the cohesive strength. In fact, some species of segregating atoms, like carbon in steels or boron in Ni3Al, strengthen the grain boundaries, whereas others, like sulfur and phosphorus, weaken them. Strengthening or weakening is determined not only by the reduction of the grain boundary energy but also by the reduction of the free surface energy. Loosely speaking, an impurity atom that prefers to reside on a free surface rather than on a grain boundary reduces the cohesive strength, since separation of the grains leads to a favorable location of the impurity atom. Hirth and Rice (1980) analyze the process of grain separation in the presence of foreign atoms in greater detail. They distinguish the limiting case of slow separation, in which the newly created fracture surfaces exhibit their equilibrium coverage, from that of fast separation, in which each fracture surface inherits half of the foreign atoms of the grain boundary. Generally, the loss in cohesive strength is greater in the slow-separation limit, and only this case will be described in the following. The work of separation of a grain boundary is 2y = 2ys — yb9 and its reduction due to
606
12 Fracture Mechanisms
12.5.1.4 Atomic Simulations
segregation, 2 Ay, is found to be: 2 Ay = -2JVsln kT
+
+ Nb
(12-61)
(-
Here AGS and AGb are the free enthalpies for segregation to the free surface and the grain boundary, respectively, and Ns and JVb are the number of sites available for the foreign atoms per unit surface and grain boundary area. Figure 12-32 shows an evaluation of Eq. (12-61) for sulfur and phosphorus in iron (Riedel, 1987). Additions in the range 0.1 ppm S or 0.1% P reduce the work of separation by typically 1 J/m2, which corresponds to about one third of the value of the clean material. Apparently, sulfur is a much more effective embrittler than is phosphorus. Therefore, the sulfur activity in commercial steels is reduced by the addition of manganese, which binds sulfur in the form of MnS particles. The dashed line in Fig. 12-32 a describes the sulfur activity in the presence of 1 % Mn. In this case, the loss in work of separation amounts to at most 0.7 J/m2.
Clean grain boundaries have been modeled on the atomic level by several authors using pair potentials or more advanced methods (e.g., Wang et al. 1984, for nickel and aluminum; Wang and Vitek, 1986, for copper). Chen et al. (1989) model grain boundaries in the ordered intermetallic compound Ni3Al containing various foreign atoms. They find that grain boundaries are strengthened substantially if boron atoms are inserted into low-density grain boundary sites, and the beneficial effect of boron increases with increasing nickel concentration in the planes adjacent to the boundary. In fact the fracture path in the numerical simulations sometimes runs through the material away from the strengthened grain boundaries. Aside from its direct strengthening effect, boron seems to attract nickel (rather than aluminum) to the boundary, which additionally toughens the grain boundary. Sulfur has a pronounced deleterious effect by reducing the strength by a factor of about three. Of course, also the orientation of the grain boundary plays a role. Figure 12-33 shows
a) Sulfur
b) Phosphorus
-0.5
WOOK .
-W
0.05 s
x j at-ppm
0.1
0J5
0.1
0.5
S
Figure 12-32. Loss of cohesive strength by the segregation of (a) sulfur and (b) phosphorus in iron (from Riedel, 1987).
12.5 The Chemistry of Fracture
30
CD
I
I
20 -
|
I
/
10 -
/ //
1 \ If Bulk
,GB+B\GB+Ni
in
CO
I
GB
GB+B+Ni
\GB+S i
10
20 Strain [%]
i
30
40
Figure 12-33. Stress-strain response of a grain boundary in Ni3Al with various segregants. (After Chen et al, 1989).
the stress-strain response of a special (210) boundary in the presence of sulfur, boron and excess nickel (Chen et al., 1989). For this orientation, nickel alone suffices to strengthen the grain boundary. 12.5.2 Hydrogen Embrittlement 12.5.2.1 General Observations
Many metals and alloys are embrittled by hydrogen. In slow strain rate tests the presence of hydrogen reduces the ductility, hydrogen dissolved in the metal causes delayed fractures under constant loads, and cracks continue to grow at low, constant stress intensity factors. For examples see Tyson (1979). Usually the susceptibility to embrittlement increases with strength. Maraging steels, for example, are particularly sensitive, but the phenomenon occurs also in low-strength materials. At low temperatures, the cracking rate increases with increasing temperature, while above a certain temperature the phenomenon disappears. Sometimes this transition is rather sharp (Gangloff and Wei, 1977), whereas in other cases it is more gradual.
607
In many practical cases the deleterious hydrogen stems from electrochemical reactions of the metal with an aqueous solution or with humid air. Operations like pickling or electroplating are associated with hydrogen formation and can lead to hydrogen blisters in the metal. The pick-up of hydrogen from humid air during steelmaking or welding causes flakes to form during cooling or leads to hydrogen-induced cold cracking. Another source of hydrogen in metals are corrosive processes during service. Hydrogen may be produced by a reaction directly at the crack tip, or it may be provided to a crack by a corrosion reaction taking place far removed from the crack. Ratke and Gruhl (1980), for example, expose the inside of an AlZnMg tube to a corrosive environment and find that a brittle intergranular crack develops at an outside notch, very much like if the notch itself had been exposed to the corrosive environment. This is explained convincingly by hydrogen diffusion from the inside of the tube to the outside notch. Embrittlement can also be caused by dry hydrogen gas, but this is less frequent, since gas pressures are usually much lower than the fugacities developed by chemical reactions and since metal surfaces are usually covered with oxygen or oxide, so that hydrogen gas has no access to the metal. 12.5.2.2 Possible Micromechanics
Hydrogen embrittlement may be caused by several micromechanisms. It appears that hydrogen can simply accelerate ductile fracture by nucleating a greater number of voids or by generating gas pressure in the voids, but often embrittlement involves a transition to brittle intergranular, or transgranular cleavage fracture modes. Possible mechanisms are:
608
12 Fracture Mechanisms
(1) Hydrogen bubbles are formed at internal interfaces or at pre-existing defects. The hydrogen pressure, if in equilibrium with an electrochemical reaction at the surface, can be very high (several hundred MPa) and may cause a plastic expansion of the bubbles or internal cracking or blistering (Zapffe and Sims, 1941; Tetelman and Robertson, 1963). (2) Hydrogen may cause fracture by reducing the cohesive strength of metals. This may happen at the first atomic bonds at the tip of the crack or within the material, especially in the stress and strain field ahead of a crack. Hydrogen is attracted to the crack-tip region by the hydrostatic tensile stresses, which increase the solubility of hydrogen (Troiano, 1960; Oriani and Josephic, 1977), and by the high density of hydrogen traps at dislocations in the plastic zone (Sofronis and McMeeking, 1989). Although there seems to be no unambiguous experimental proof for the bond-weakening mechanism, there is much indirect evidence for its importance (see Sec. 12.5.2.3). The idea is further supported by atomic calculations, which indicate that hydrogen weakens the bonds of transition metals with more than half-filled d bands (e.g., Mullins, 1984 b, for iron). (3) Some metals (Ni, Ti, Zr, Nb, V) can form hydrides. It was suggested that these hydrides are preferentially formed in the tensile stress field of a crack. Once present, the brittle hydrides offer a path for rapid crack propagation (Westlake, 1969). A review is given by Birnbaum (1983). (4) At least in some cases, hydrogen reduces the yield stress of metals. It was suggested that this might lead to enhanced plastic void growth near the crack tip and to a corresponding reduc-
tion in toughness (Beachem, 1972; Lynch, 1979). However, a great enhancement of void growth by a local softening cannot be anticipated, since the material near the crack tip is still constrained by the surrounding material. However, a hydrogen-induced strain localization instability near the crack tip cannot be ruled out (Robertson and Birnbaum, 1986). (5) Hydrogen attack is a phenomenon which is usually not included under the heading of hydrogen embrittlement. Whereas ordinary embrittlement is confined to temperatures below and not much above room temperature, hydrogen attack occurs in the temperature range 300 °C to 500 °C (in carbon steels). If exposed to high-pressure hydrogen, e.g., in petrochemical plants or (formerly) in ammonia synthesis reactors, these steels develop methane bubbles on grain boundaries resulting from a reaction of hydrogen with the carbon contained in the steel. Equilibrium methane pressures can be high enough to expand the bubbles and to cause internal cracking (Shewmon, 1985; Riedel, 1987). 12.5.2.3 Mechanism-Oriented Observations
Among the mechanism-oriented experiments in the literature, those of Vehoff and Rothe (1983) will be briefly described here. These authors investigated crack growth in Fe-2.6%Si single crystals exposed to small to moderate hydrogen pressures, in the temperature range 300 to 400 K and at loading rates at which the material is ductile in the absence of hydrogen. (Tests on Ni lead to similar conclusions but in a different range of testing conditions.) First it is noteworthy that a crack growing in a ductile manner along a noncleav-
609
12.5 The Chemistry of Fracture
age plane in vacuum immediately develops a facetted fracture surface, the fracture facets being {100} cleavage planes, once hydrogen is admitted to the specimen. When hydrogen is removed, the crack returns to a ductile mode. This pronounced preference of cleavage in the presence of hydrogen suggests that the effect of hydrogen in this case is to weaken the atomic bonds. In order to locate where exactly hydrogen develops its embrittling action, Vehoff and Rothe (1983) suddenly admitted oxygen to the test specimen in which a brittle hydrogen crack was growing. Oxygen is known to displace hydrogen from the metal surface and to prevent hydrogen from penetrating into the metal. Upon oxygen admission the crack immediately became ductile (within the temporal resolution of the method). The instantaneous response of the cracking mode could not be explained, if the hydrogen deep inside the material were responsible for the embrittlement. Vehoff and Rothe conclude that hydrogen must act either directly at the first bonds at the crack tip or at a distance of at most 150 nm ahead of the crack tip. The crystal orientation shown in Fig. 12-34 lends itself to a quantitative evaluation of the hydrogen effect. In this orientation, both ductile and brittle crack propagation in the (100) plane are possible. In the absence of hydrogen, the crack is opened by plastic slip on {112} slip planes and develops the relatively large crack-tip angle corresponding to the slip geometry (Fig. 12-34 a). In the presence of hydrogen, plastic crack opening and hydrogen-induced cleavage occur at the same time (or possibly in small, nonresolvable alternate steps), so that the crack tip angle becomes more acute. This is described quantitatively by the normalized crack growth rate
a) Ductile Crack Opening (No Hydrogen) [111]
b) Simultaneous Crack Opening and Hydrogen Cracking
5/2 Figure 12-34. (a) Ductile crack opening vs. (b) hydrogen-induced cleavage in Fe-2.6% Si single crystal (after Vehoff and Rothe, 1983).
due to hydrogen, which is defined as
= a~ at
(12-62)
where an is the total normalized crack growth rate defined by an = 2 a/5, from which the merely ductile contribution a* is subtracted (for the definition of the quantities see Fig. 12-34). Vehoff and Rothe (1983) hypothesize that the hydrogen-induced crack growth rate a^ is proportional to the coverage of the crack surface with hydrogen, 6. The surface soverage is calculated from the Langmuir-McLean isotherm: (12-63) Here k0 is a factor of proportionality, p is the hydrogen pressure in the gas phase, and AG is the free enthalpy for atomic hydrogen adsorption from the molecular gas phase. The hypothesis is checked by plot-
610
12 Fracture Mechanisms
ting Eq. (12-63) with a" oc 9 inserted. Figure 12-35 shows that the measured crack growth rates can be described well by the theoretical curves except for the lowest pressures. The saturation behavior at high pressures and the transition to low growth rates are predicted correctly. In summary, all observations of Vehoff and Rothe are compatible with the assumption that hydrogen embrittlement is caused by a weakening of atomic bonds at or very near the crack tip. 12.5.3 Stress Corrosion Cracking (SCC) 12.5.3.1 General Observations
Stress corrosion cracking (SCC) is a brittle form of crack nucleation and growth, which is caused by localized corrosive processes at the crack tip. It is a safety-relevant problem for various industries. Design against SCC may be the primary criterion for the selection of materials in the chemical industries, where ferritic and austenitic steels and nickel base alloys are common materials (Spaehn, 1984). Most highstrength aluminum and titanium alloys 15
n T = 313K A T = 343K 10 1
10 3
105
Hydrogen Pressure, p [Pa] Figure 12-35. Normalized hydrogen-induced cracking rate, a^, vs. hydrogen pressure. Comparison of theory (solid lines) and experiments (symbols). (After Vehoff and Rothe, 1983.)
used in the aviation industry are susceptible to SCC (Speidel and Hyatt, 1972). The phenomenon of SCC is of concern also to the electric-power generating industry, since several components of steam turbines and generators are endangered. Welded austenitic steel pipes of nuclear reactors have developed cracks by SCC (Ford, 1984, 1990). Slow cracking in ferritic pressure vessel and piping steels of light-water reactors cannot entirely by excluded (Speidel and Magdowski, 1988). Various materials exhibit SCC in different solutions. Rather mild chemical conditions may suffice to cause stress corrosion cracking. High-strength aluminum alloys, for example, crack in pure water and in most air. The presence of water is essential, as these alloys do not exhibit SCC in any of several dry gases including hydrogen (Speidel and Hyatt, 1972). Specific anions greatly enhance SCC. For aluminum alloys, Cl", Br~ and I" ions are deleterious, and crack growth rates can then be very high, whereas F~ and SO4" are innocuous (or even beneficial). Further, SCC may depend on the pH of the solution, and an electrical polarization has a pronounced effect. Depending on the applied electric potential, the metal may be stable against any chemical attack (under cathodic polarization), it may be passive due to the formation of a protective surface layer, general active dissolution may occur (in the anodic regime), or, often in the vicinity of the active-passive transition, one observes pitting corrosion or SCC. The mechanical stress is another essential factor for stress corrosion cracking. The lifetime of an initially uncracked specimen decreases as a function of the stress, sometimes according to an exponential relation, tf oc exp (— G/O0), sometimes with a threshold stress below which the lifetime becomes very long. If the crack growth rate
12.5 The Chemistry of Fracture
is measured as a function of the stress intensity factor, one often obtains a relationship as shown schematically in Fig. 12-36. At small stress intensity factors the crack growth rate drops sharply, so that it appears justified to speak of a threshold for stress corrosion cracking KISCC, which is typically 5 to lOMPay/m for steels. At somewhat higher Kl9 the crack growth rate increases until it reaches a plateau, on which a further increase of KY does not lead to higher crack growth rates. Growth rates in the plateau depend strongly on the material and the environment and range from typically 10 ~9 m/s in humid air to 10 ~5 m/s in concentrated KI solution (for an AlZnMg alloy). Not all SCC systems exhibit such a plateau. Another factor which affects cracking rates is the temperature. Usually higher temperatures lead to faster cracking, with apparent activation energies of about 40 kJ/mol in the plateau region. Among the metallurgical aspects of SCC, the most obvious one is the increasing susceptibility to SCC with increasing yield stress (Pedrazzoli, 1989). Of course, there are exceptions to that general rule. An example is the cold-worked manganese-bearing austenitic steel (with nitrogen instead of carbon), which is used for retaining rings in turbo-generators (Speidel, 1981; Pedrazzoli and Speidel, 1990). Another well known metallurgical effect is the sensitization of austenitic stainless steels. During welding, the material is exposed to a heating cycle which results in the precipitation of chromium carbides on grain boundaries. As a consequence the vicinity of the grain boundaries is depleted of Cr. In the absence of enough Cr, however, no stable passivation layer can be formed, so that corrosive attack can penetrate along grain boundaries. This intergranular corrosion occurs even without an applied stress. Cor-
611
• 05
O
logK, Figure 12-36. Stress corrosion cracking rate vs. stress intensity factor (schematic).
rosion cracks may grow along grain boundaries (as in sensitized steels or in AlZnMg alloys) or through the grains. Steels exhibit both modes of cracking. The basic mechanisms for SCC that have been proposed are hydrogen embrittlement and the slip/dissolution mechanism. The latter assumes that plastic slip at the crack tip is necessary to break the protective oxide layer, so that localized dissolution can continue along the crack path. Of course, the dissolution reaction can produce hydrogen aside from propagating the crack tip, and the two basic mechanisms can thus cooperate rather than being mutually exlusive. For other proposed mechanisms, see Newman and Sieradzki (1987) and Kaufmann and Fink (1988). In addition to the references given above, the following papers cover a wide range of experimental data or discuss the underlying mechanisms of SCC: Engell and Speidel (1969), Gruhl (1984), Parkins (1984), Magdowski and Speidel (1988), Wei (1989).
612
12 Fracture Mechanisms
12.5.3.2 Stress Corrosion Cracking by Hydrogen Embrittlement
In aluminum alloys of the AlZnMg type it is obvious that hydrogen, which is produced by corrosion reactions can lead to hydrogen-induced crack growth. This is demonstrated by the experiments of Ratke and Gruhl (1980), which were described in Sec. 12.5.2.1. Since hydrogen is produced far from the crack in these experiments, it is not necessary that the oxide film at the crack tip be ruptured by plastic slip, and dissolution can play no role, since there is no liquid in the crack. Similarly, SCC of aluminum alloys in humid air cannot be brought about by dissolution, since even the presence of small pockets of liquid due to capillary condensation can be excluded. Regarding SCC of aluminum alloys in aqueous solutions the evidence for the preponderance of hydrogen embrittlement compared to slip/dissolution is less direct than in the above-mentioned cases. Nevertheless, there seems to be a general agreement that indeed hydrogen embrittlement is responsible for SCC under these conditions (Speidel, 1973). In a recent review, Kaesche (1989) arrives at the same conclusion for ferritic steels in dilute aqueous solutions. His conclusion is mainly based on the occurrence of cleavage fracture facets during SCC, which would be difficult to explain by a dissolution mechanism. 12.5.3.3 The Slip/Dissolution Mechanism (with Possible Participation of Hydrogen Embrittlement)
Contrary to the noble metals, the usual structural metals and alloys are thermodynamically unstable against oxidation even at very low oxygen partial pressures or against dissolution in aqueous environments. Under most operating conditions, these reactions either proceed very slowly
or are impeded by the formation of protective oxide or other passivation layers on the metal surface. At the tip of a slowly growing crack, however, the passivation layers may be unstable. Plastic slip at the crack tip continually exposes fresh metal to the environment, so that the passivation layer near the crack tip is young and therefore thinner and less protective than elsewhere. In aqueous environments this means that the anodic partial reaction of metal dissolution is focussed to a small surface area near the crack tip, whereas the balancing cathodic reaction (e.g., hydrogen formation by the reduction of H + or reduction of dissolved oxygen to O H ) may take place anywhere on the specimen surface. These coupled processes of exposure of new surface by plastic slip and passivation are often viewed as a sequence of discrete steps: The passivation film is ruptured by a slip event at the crack tip, then dissolution takes places at a decreasing rate, as a new passive layer is built up, until the next slip event ruptures the new film (Vehoff et al., 1987). Thus the occurrence of this type of SCC is tied to an intermediate stability of the protective surface film. The film must be stable enough to protect the specimen surface including the side walls of the crack. Otherwise, a crack nucleus would be blunted by wide-spread corrosion and would be transformed into a flat corrosion pit. On the other hand, the film must not be so stable as to suppress the dissolution process completely. From this it appears plausible that an alloy/solution combination which is usually inert to SCC can become susceptible, if the stability of the passivation layer is shifted into the critical range by small changes in the anion content of the solution, in the electric potential or in the
12.5 The Chemistry of Fracture
metallurgical condition of the alloy. For example, the drastic increase of the cracking rate in aluminum alloys and stainless steels by halide ions is ascribed to a destabilization of the otherwise rather stable passivation layers. The slip/dissolution model, in its strict sense, assumes that it is the anodic dissolution reaction itself which propagates the crack by the removal of material from its tip. However, it is also possible that the hydrogen, which is produced by the corrosion reaction, is the intrinsic cause for crack propagation. Then the slip/dissolution event is a necessary prerequisite, and may be rate controlling, but hydrogen-induced cracking is the actual micromechanism. As Ford (1984) remarks, a distinction between crack growth directly by dissolution or indirectly via hydrogen is experimentally difficult and has no great practical importance in such a case. Ford (1990) attempts to quantify the slip/dissolution model with the special goal to describe SCC in austenitic steels in highpurity water under conditions representative for U.S. light water reactors. Applications to ferritic steels have also been proposed (Ford, 1988). As a working hypothesis, Ford starts from the slip/dissolution model and writes the crack growth rate as the product of material layer thickness dissolved during one film-rupture/repassivation event, times the frequency of these events: MQ e a = ZQF ef
(12-64)
Here, the thickness of dissolved material is expressed by electrochemical quantities (F is the Faraday constant, z the number of electrons involved in the oxidation of a metal atom, Q the charge density passed through the crack tip between two film rupture events, and M, Q are the atomic
613
weight and density of the solid). The frequency of film rupture events is given by the strain rate at the crack tip, a, and the strain to fracture of the film, ef. The charge per rupture/passivation event, Q, depends sensitively on the alloy and solution chemistry near the crack tip as distinct from the conditions far away. A difference between local and remote conditions must be anticipated to exist, since the dissolution process at the crack tip adds ions to the solution and consumes others. The exchange of ions with the free solution outside the crack may well be rate controlling for the whole process. Moreover, the local solution chemistry may be essential for the continuation of stress corrosion cracking, when the free solution, were it present at the crack tip, would not support SCC. Therefore, Ford's model involves theoretical and experimental input about the anion content, the pH and the potential near the cack tip as a function of the conditions in the remote solution. Knowing the local chemistry, one can make measurements of the oxidation currents at freshly exposed surfaces in the simulated crack-tip environments. In this sense, the first factor in Eq. (12-64) can be considered to be known. The details of the model cannot be reproduced here. It should only be mentioned that the calculation of the crack-tip strain rate might be a weak point of the theory. Nevertheless, the model is able to describe the dependence of measured crack growth rates on parameters like electric potential and solution conductivity (as a measure of the anion content). This indicates that the model takes the controlling steps adequately into account, especially the local solution chemistry and its effect on the dissolution rate.
614
12 Fracture Mechanisms
12.5.3.4 Experimental Evidence for the Importance of Slip
While evidence for hydrogen-induced cracking as a mechanism for SCC was already presented, it remains to show that plastic slip at the crack tip can also play an important role. The experiments of Vehoff et al. (1987) on Fe-Si bicrystal specimens in (NH 4 ) 2 CO 3 solution lend strong support to this idea. The authors interpret their results in terms of the slip/dissolution model, but a participitation of hydrogen embrittlement cannot entirely be ruled out. Among several observations showing the importance of plastic slip, the most striking one is shown in Fig. 12-37. Two series of tests on symmetric tilt boundaries were conducted. The series differ only by the direction in which the crack grows. The normalized cracking rate an as defined in the text following Eq. (12-62) was measured as a function of the tilt angle. It turns out that the growth rate is high in the growth direction for which the {211} slip planes experience a high resolved shear stress and therefore undergo substantial
30° 60° Tilt Angle, 6
90°
Figure 12-37. Normalized stress corrosion cracking rate vs. tilt angle of bicrystal. Crack growth directions are shown in the inserts.
plastic slip. For unfavorably oriented slip planes and correspondingly little slip, the crack grows more slowly. The only obvious explanation for the difference in cracking rate is the different slip geometry.
12.6 Nonlinear Fracture Mechanics 12.6.1 Introduction
Fracture mechanics is the discipline which quantifies the deleterious effects that a crack has on the load-bearing capacity or the lifetime of an engineering structure. Cracks may result from the fabrication process or they may develop during service by fatigue loading, by stress corrosion cracking or by some other problem. The importance of cracks increases with the strength and brittleness of materials. In ductile low-strength steels a crack must be large (several centimeters) in order to reduce the load-bearing capacity significantly. In high-strength steels or nickel base alloys, cracks of a few millimeters length are dangerous, and glasses and ceramics fracture from defects having sizes between a few and a few hundred micrometers. It is convenient to distinguish two aspects in the analysis of cracks, namely (1) the continuum-mechanical deformation fields in the cracked body, and (2) the fracture micromechanisms that operate near the crack tip. The aim of the continuum-mechanical analysis is to identify universal crack-tip fields, and load parameters which characterize them under various conditions (e.g., the stress intensity factor, the J-integral, Sec. 12.6.2, or the C*-integral, Sec. 12.6.8.1). By means of the appropriate load parameter one can transfer the crack growth behavior measured in a laboratory test speci-
12.6 Nonlinear Fracture Mechanics
men to cracks in engineering structures of different sizes and shapes. Thus the continuum-mechanical analysis provides a basis for the transfer of measured data to other specimen geometries. Micromechanical models, on the other hand, describe the fracture processes near crack tips, and thus make predictions on the crack growth behavior. The following examples will be considered: Crack extension in steels by cleavage; crack growth initiation and slow stable cracking by ductile void growth; fast crack growth; and creep crack growth at elevated temperatures. Figure 12-38 shows the standard test specimen configuration used in fracture mechanics tests. (Some features of the figure are referred to later.) A crack is generated at a premachined notch by fatigue loading. Subsequently its behavior under monotonically increasing load is investigated. Detailed rules for measuring the fracture toughness of relatively brittle materials are specified in the ASTM-E399 Standard (1978). Rules for tougher materials are given in ASTM-E813 Standard (1981). Numerous books and reviews on fracture mechanics are available covering both fundamental and practical aspects (e.g.,
Rice, 1968 a; McClintock, 1971; Hutchinson, 1983 b; Knott, 1973; Broek, 1982; Kanninen and Popelar, 1985). 12.6.2 The /-Integral and the Hutchinson, Rice, and Rosengren (HRR) Crack-Tip Field
Path-independent contour integrals such as the J-integral play an important role in fracture mechanics, since they allow to correlate the crack-tip field to the remote loading. 12.6.2.1 The /-Integral in Nonlinear Elastic Materials
Nonlinear elastic materials are defined as materials in which the strain is a function only of the current stress, independent of the prior loading history. As Rice (1968 a, b) has shown, the J contour integral defined by
/ = J [Wdx2 -
(12-65) r \ is path independent in nonlinear elastic materials. In Eq. (12-65) W = Jo-^de^ is the strain energy density, F is the integration path around the crack tip as shown in Fig. 12-38, n( are the components of the outward normal unit vector on the path, ds is the differential arc length, a(j, 8tj and Uj are the stress, strain and displacement fields, and x1 and x2 are Cartesian coordinates. The J-integral can be determined by measuring the mechanical work done on a specimen (or a pair of specimens) at two incrementally different crack lengths: J= - — f^dzl
Figure 12-38. Standard compact specimen for toughness testing.
615
(12-66)
Here, Pt is the load per unit specimen thickness and A is the load-point displacement.
616
12 Fracture Mechanisms
12.6.2.2 Power-Law Elastic and Power-Law Plastic Materials Power-law elastic materials represent a convenient special case for the further discussion. They are described by Eq. (12-2), i.e., by s = Ao a1/N for uniaxial tension. Recall that for proportional monotonically increasing loading, power-law elastic and incrementally plastic materials are equivalent, so that the following equations are valid for both material classes. In a power-law material the stress field scales with the load, P, the strain field with P 1/iV , and the J-integral must have the form j —
g1\
a crn
(12-70)
J=
for plane strain. For plane stress the factor 1 — v2 is deleted.
(12-67)
where crnet is the net section stress (i.e., the applied load per uncracked-ligament area), a and W are defined in Fig. 12-38 above, and g± is a dimensionless function of the specimen geometry and of the hardening exponent N. Approximate, but accurate, analytical expressions for g± are available for plane-strain and penny-shaped cracks in infinite bodies (He and Hutchinson, 1981). Further, Kumar et al. (1981) provide finite-element solutions for a large number of specimen geometries. Introducing the load-line deflection, A9 which must vary in proportion to Ao a
able, and is still used frequently. For deeply cracked bend geometries, it has the advantage to be very simple, viz. rj « 2 (Rice et al., 1973). For compact specimens, the relation rj = 2 + 0.522(1 - a/W) approximates numerical results well (ASTM-E813 Standard, 1981). In the special case of linear elastic material, J is related to the stress intensity factor by
(12-68) (12-69)
where g2 and rj are dimensionless functions, which can be extracted from Kumar et al.'s results (Riedel, 1987), and which are related by g2 = rj/(l + N). Equation (12-69), has been used widely, before systematic finite-element calculations became avail-
12.6.2.3 Crack-Tip Fields in Power-Law Materials
Asymptotic fields near crack tips in power-law materials were derived by Hutchinson (1968) and by Rice and Rosengren (1968). Again the results are valid for both power-law elastic and power-law incrementally plastic materials. The "HRR fields" have the form:
A0INr
(12-71) (12-72)
The dimensionless angular functions &ij(89N) and eo(0,iV) are generally calculated numerically; JN is a dimensionless factor normalizing the maximum of
12.6 Nonlinear Fracture Mechanics
identically. Limitations to the validity of this conclusion will be discussed later. 12.6.3 Stationary Crack in Elastic-Plastic Material An elastic-plastic material is described by the material law 3^o
2 77
(12-73)
where the elastic strain rate is given by Eq. (12-1), and Ao is set equal to zero if the stress is inside the yield surface or if the material is unloaded from the yield surface. For uniaxial tension, Eq. (12-73) reduces to s = o/E + a((7y/£)(cr/(jy)1/iv, when Ao is replaced by ay through Eq. (12-3). 12.6.3.1 Small-Scale Yielding Under small loads, an elastic-plastic material responds elastically except in a small plastic zone near the crack tip, where the stress exceeds the plastic yield stress. As long as the plastic zone is small enough ("small-scale yielding"), the problem can be analyzed under the remote boundary condition that the stress field at large distances from the crack tip must approach the elastic singular field given in Eq. (12-6). This so-called boundary layer approach is an important concept in fracture mechanics (Rice, 1968 a). Its practical consequence is that the evolution of the plastic zone is governed by the linear elastic stress intensity factor alone, while details of the specimen geometry play no role apart from that they affect Kv Hence under small-scale yielding conditions, KY can serve to unify the behavior of cracks in differently shaped specimens (Irwin, 1957). Since the only length scale in the boundary layer problem is the ratio (Kjoy)2, the plastic zone size must be proportional to
617
this quantity. The factor of proportionality and the shape of the plastic zone can be obtained in closed analytic form for Modelll (Hult and McClintock, 1957; Rice, 1967). For Mode-I loading, the boundary-layer problem was solved numerically by the finite-element method (e.g., Larsson and Carlsson, 1973). The maximum extent of the plastic zone is approximately (12-74) for plane strain and about twice as large for plane stress. Figure 12-39 shows the upper half of the plastic zone for plane strain tension (Larsson and Carlsson, 1973). The range of validity of the small-scale yielding approximation ends, when the plastic zone size reaches a certain fraction of the crack length, a. Experimental experience, primarily with compact-type specimens, showed that the condition for smallscale yielding should be (12-75)
0 0.05 Coordinate, x/(K ( /a y ) 2 Figure 12-39. Shape of the plastic zone.
0.10
618
12 Fracture Mechanisms
This condition has become a standard requirement for valid Klc tests (ASTM-E399 Standard, 1978). If it is fulfilled at the onset of fracture, i.e., at KY = KIC, one expects Klc values that are independent of the specimen size and shape. 12.6.3.2 Transition from Small-Scale Yielding to General Yield
Outside the small-scale yielding limit, the evolution of the plastic zone and of the deformation fields becomes specimenshape dependent and can in general be described only numerically. With increasing load, the plastic zone spreads over the whole ligament of the specimen. Eventually plastic strains dominate nearly everywhere in the specimen compared to elastic strains, i.e., the material response approaches that of a power-law plastic material. In this fully plastic limit, the J-integral is given by Eqs. (12-67) to (12-69), while the crack-tip fields are still the HRR fields. In the whole range from small-scale yielding to general yield, the J-integral can be approximately described by J = J el + J pl , where the elastic (low-load) term is given by Eq. (12-70) and the plastic (high-load) term is given by Eq. (12-67) (Shih et al., 1981; Kumar et al, 1981).
fields are disturbed by crack-tip blunting and by the fracture process zone. At greater distances from the crack tip the nonsingular terms of the fields become comparable in magnitude to the HRR field. The question is whether there remains an annular zone, in which the HRR field determines the fields to a desired degree of accuracy. The situation is illustrated schematically in Fig. 12-40. Under fully yielded conditions the outer range of validity of the HRR field scales with the crack length or ligament width. The factor of proportionality depends strongly on the specimen shape and on the hardening exponent N. Slip-line field considerations for nonhardening material indicate that the range of the HRR field should be very restricted in specimens like the center-cracked plate under tension, in which the strain tends to localize in slip bands inclined 45° to the crack plane, a strain distribution which is not compatible with the HRR strain distribution. Strain hardening counteracts the tendency towards slip localization. In accord with these arguments finite-element calculations show that the HRR field dominates over 1% of the ligament in a centercracked plate under tension if N = lA, but only over an unresolvably small distance if
12.6.3.3 Range of Validity of the HRR Field
Cracks in differently shaped specimens can be expected to behave identically, if their tips are surrounded by the same stress and strain fields; in power-law materials, for example, by the HRR fields. In this case a single parameter, e.g., the J-integral, can serve as a universal fracture parameter as proposed by Broberg (1971) and by Begley and Landes (1972). However, the applicability of J in that sense is limited by the following arguments. Very near the crack tip, the HRR
disturbance by nonsingular terms
disturbance by blunting and fracture processes
Figure 12-40. Range of validity of the HRR field.
12.6 Nonlinear Fracture Mechanics
619
N = 0.1 (e.g., Shih and German, 1981; Shih, 1985). On the other hand, the fields are approximated by the HRR field to within 10% in a zone of about 8% of the ligament in bend and compact specimen geometries, nearly independently of the hardening exponent (Shih, 1985). Distance from Crack, ro y / J
12.6.4 Limitation to / by Crack Blunting In the framework of the theory mentioned so far, which neglects crack geometry changes by blunting, the crack profile near the crack tip is determined by the HRR field: Auoc
(12-76)
Here, Aw is the separation distance of the crack faces. Figure 12-41 shows such a profile along with the usual definition of the crack-tip opening displacement, St, based on 45°-lines through the crack-tip. Apart from small dependencies on N and cry/E, the crack-tip opening displacement is obtained from Eq. (12-76) as 1.2 A$
(12-77)
where the numerical factors are valid for N = 0.1 and E/ay = 500. Similar crack profiles are obtained by theories, which take geometry changes into account consistently. Rice and Johnson (1970) carry out an approximate analysis of blunting based on slip-line theory, whereas McMeeking (1977) and McMeeking and Parks (1979) perform finite-element calculations in the framework of large-strain plasticity theory. These analyses show how the HRR field is disturbed by the blunting of the crack (Fig. 12-41). Whereas the HRR stress field diverges, the actual stress field near a blunting crack is limited to a few times the yield stress. Typ-
Figure 12-41. Disturbance of the HRR field by blunting. Comparison of HRR field with finite element calculation.
ically the disturbance extends over a distance of three times the crack opening displacement. Thus there remains a zone of dominance of the HRR field between the disturbance by blunting and the zone affected by the outer specimen geometry, if (a and W
a)>25J-
(12-78)
for bend and compact geometries. For center-cracked tension specimens, the numerical factor must be 200 instead of 25 for strong strain hardening and even higher for low hardening. Equation (12-78) is a necessary prerequisite for a one-parameter characterization of fracture, e.g., by the Jintegral. This criterion has been established also experimentally (e.g., Landes and Begley, 1974), and it is part of the ASTME813 Standard (1981). Considerations of the three-dimensional crack-tip fields show that a universal plane-strain HRR field dominates near a blunting crack tip, as long as the specimen thickness B is large enough compared to the crack-tip opening displacement (see, e.g., Riedel, 1987). Numerically the condition (12-79)
620
12 Fracture Mechanisms
should be satisified, if the limiting case of plane strain is to be investigated. Otherwise the triaxial stressing conditions at the crack are less severe than in plane strain, and too optimistic fracture parameters are measured.
8 C
"I
12.6.5 Models for Crack Growth Initiation 12.6.5.1 Slip-Induced Cleavage Fracture from Cracks
Ritchie et al. (1973) consider cleavage fracture initiation in the plastic zone of a crack. Their model refers to plastic-slip induced cleavage fracture in materials like ferritic steels. Following the model of Smith (1966 a, b; Sec. 12.2.6), they assume that fracture occurs once the temperatureindependent cleavage fracture stress of, is attained over a critical distance, x c , which should be of the order of the grain size. For the stress distribution in the plastic zone near the crack tip, they use the HRR field, Eq. (12-71), with the small-scale yielding expression for J, Eq. (12-70), inserted. Setting r = xc and o = of and resolving for KY gives the critical stress intensity factor Klc for crack propagation: Klc oc of
l-N 2N
(12-80)
Since the plastic flow stress oy decreases with temperature, Eq. (12-80) predicts an increase of the fracture toughness. In a limited range at low temperatures, Eq. (12-80) indeed describes the observed behavior correctly, but at higher temperatures the observed toughness rises more sharply than predicted (see Fig. 12-42 below). This can be explained by recalling that the stress ahead of a blunting crack is limited to a few times the yield stress, in contrast to the HRR field, which was used in deriving Eq. (12-80) and which increases indefinitely.
Temperature, T
Figure 12-42. Fracture toughness of steels vs. temperature. RKR: theory of Ritchie et al. (1974), RJ: theory of Rice and Johnson (1970).
Hence above a certain temperature, the cleavage fracture stress of is no longer reached ahead of a blunting crack. The fracture mechanism must then change to a ductile fibrous mode. 12.6.5.2 Crack Growth Initiation by Ductile Void Growth
Fracture by ductile void growth is controlled by a critical-strain criterion, where the critical strain, sf, depends strongly on the triaxiality of the stress field (Sec. 12.3). Combining the HRR strain field, Eq. (12-72), with a critical-strain criterion at a distance xc from the crack tip gives a critical value J lc for crack growth initiation
-y
1+N
(12-81)
where xc should now be of the order of the spacing between void-nucleating inclusions. The angular function £e depends strongly on the direction in which the crack grows. Directly ahead of the crack, se is 0.018 (for N = 0.1) and increases to 1 at an angle 6 = 98°. Thus the crack should start growing out of its original plane. However, since parts of the crack front start growing along an angle + 0, while
12.6 Nonlinear Fracture Mechanics
others grow along the equally favorable direction — 0, it is well possible that the crack extends macroscopically in its original plane as a compromise between the two inclined directions. Rice and Johnson (1970) point out that it is important to take blunting of the crack into account, since it focuses large strains in the region ahead of the tip. They calculate the critical crack-tip opening displacement, (5C, at which the crack just coalesces with a void having an initial radius Ro and an initial distance xc from the tip. Clearly, (5C should be of the order of the void spacing. The detailed analysis, after expressing 3C by J Ic through Eq. (12-77), gives R<
(12-82)
where / is a slowly varying function of its argument and has values between about 1 and 2.5 for xJR0 = 3 to 30. Equation (12-82) is very similar to Eq. (12-81) except that the strain to fracture is replaced by a function of microstructural quantities. Numerous investigations (e.g., Hahn etal, 1972; Hahn and Rosenfield, 1975; Green and Knott, 1976; McMeeking, 1989) have shown that the <SC- or Jlc-values calculated by Rice and Johnson (1970) are realistic for various steels and aluminum alloys. However, the actual toughness values can be considerably lower than predicted, if a shear localization occurs between the crack tip and the void, for instance by the formation of secondary voids. 12.6.5.3 The Cleavage/Ductile Transition
Figure 12-42 summarizes the temperature dependence of the fracture toughness of ferritic steels as calculated in the preceding two subsections. At very low temperatures the fracture mechanism is cleavage, and the increase of the toughness is de-
621
scribed by the Ritchie, Knott and Rice model. The increase is enhanced by the effect of blunting, which makes cleavage fracture impossible above a certain temperature. Then crack extension by ductile void growth takes over, which is described by Eqs. (12-81) or (12-82). Ductile crack growth exhibits a negative temperature dependence, since ay decreases with increasing temperature. 12.6.6 Stable Crack Growth 12.6.6.1 The R-Curve and the Stability of Crack Growth
In an ideally brittle material a crack starts to propagate and continues to propagate if the stress intensity factor reaches or exceeds a critical value, the fracture toughness Klc. Ductile materials, however, exhibit a more complicated crack growth behavior: The stress intensity factor needed to propagate a crack increases, while the crack grows. Figure 12-43 illustrates this so-called Rcurve effect, where "R" denotes the resistance of the material against crack propagation. The solid curve represents the crack resistance of the material KR, which increases as a function of Aa = a — ai?
i
K
ss
instability
/
^
^
^
applied Kj for 3 loads initiation
Crack Length, a
Figure 12-43. Crack resistance, KR, and applied stress intensity factor, Kl9 vs. crack length. Schematic of R-curve effect.
622
12 Fracture Mechanisms
where a{ is the initial crack length produced, for example, by fatigue crack propagation. The crack resistance rises from an initiation value K{ to a steady-state value Kss after some growth. Also shown is the dependence of the applied stress intensity factor on the crack length, a relation which depends on the specimen shape and on the loading conditions. While the sketched behavior is typical for constant-load conditions, fixed-grip conditions often lead to decreasing curves. Once the applied stress intensity factor reaches the initiation value, the crack starts growing stably the current length of the crack being given by the intersection of the KY (applied)- and the KRcurve. However, when the K r curve becomes tangent to the resistance curve, the crack starts to extend unstably, since a small disturbance of the crack length renders KY (applied) greater than the resistance KR. Obviously, the point of instability is not a material constant, but depends on the specimen shape and on the loading conditions. Increasing R-curves are observed in practically all ductile metallic materials and in certain ceramics. The effect is more pronounced in low- and medium-strength steels than in high-strength steels. Often the more ductile materials cannot be tested in the small-scale yielding regime, where KY is applicable, since the specimens would have to be impractically large. Then one must resort to an analogous description in terms of the J-integral, which has, however, severe limitations (Sec. 12.6.6.3).
model for Mode III. For their explanation it is essential that the material is realistically described as incrementally plastic rather than as nonlinear elastic. In incrementally plastic material the nature of the strain singularity changes from the strong HRR-type singularity, eocr1/(N+1\ at the stationary crack tip to a weak logarithmic singularity at the growing crack tip (McClintock and Irwin, 1965; Rice, 1968 a; Gao, 1985). Assuming that crack growth is controlled by a critical-strain criterion at a distance xc ahead of the crack tip, which is appropriate for ductile failure modes, it is plausible that the applied loading must be increased in order to achieve the same critical strain by the weak singularity, as was achieved by the strong singularity at the stationary crack. A quantitative model for stable crack growth under plane-strain conditions was developed by Rice and Sorensen (1978), and by Rice et al. (1980). Instead of imposing a critical-strain criterion, these authors prefer to formulate the crack growth criterion in terms of the crack profile. This has a practical advantage, since the strain directly ahead of the crack is difficult to evaluate in plane-strain finite element calculations. The two types of criteria are similar, though not quite equivalent. When the conditions at a blunted, stationary crack tip become critical, the crack starts growing with a profile that is much sharper than that of the stationary crack before growth initiation. For elastic-plastic material the profile of the growing crack is obtained approximately as /.^ o^n
12.6.6.2 The Source of the R-Curve Effect in Ductile Metals
5
An explanation for the fact that the crack resistance increases while the crack grows, was given by McClintock and Irwin (1965), who also provide a quantitative
This result is valid for small-scale yielding, but it applies approximately to large-scale yielding within limits to be specified in the following section.
0.13 EdJl +—2"- —
623
12.6 Nonlinear Fracture Mechanics
Now the criterion is imposed on Eq. (12-83) that Au should have the critical value 5C at a distance xc behind the crack tip. This leads to a differential equation for the incurve, J = J R (Aa). The value for crack growth initiation, Ji9 serves as an initial condition and can be taken from Eq. (12-81) or (12-82). The differential equation can be solved in terms of the exponential integral. The initial slope of the R-curve, in dimensionless form, is found to be
1.5£<5_ da
- 7 . 7 In
(12-84) 0.5 EJ^
which is called the tearing modulus TR. Eventually the R-curve saturates to the steady-state value /.~ on ^
5ayx{
While the initiation value, Ji? depends linearly on the ductility, %, the steady-state value, Jss, increases exponentially with the ductility measure SJxc. This means that materials with a high ductility (and with a low oy/E ratio) exhibit high JJJ{ ratios, i.e., strongly rising R-curves, whereas materials with low ductility and high ay/E exhibit nearly no R-curve effect. Hermann and Rice (1980) compare Rcurves of a high-strength steel with theoretical curves as shown in Fig. 12-44. They obtain good agreement, if the initiation value, J i? and a second parameter, e.g., Jss or the tearing modulus are used as adjustable parameters. 12.6.6.3 Limitations to / in Fully Yielded Specimens During Stable Crack Growth Material elements that are passed by a growing crack tip experience strongly nonproportional stressing and are unloaded when they are shielded from the applied
Theory: J ^ = 80 kN/m
4140 Steel
2 4 6 8 Crack Growth, Aa[mm]
10
Figure 12-44. Calculated and measured R-curve after Herrmann and Rice (1980).
stress by the growing crack. In the region, in which these effects are not negligible, the J-integral is not path independent and in fact goes to zero on paths very close to the moving crack tip. However, crack growth is still controlled by J, if the nonproportional stressing zone is well contained within the J-controlled HRR field. Hutchinson and Paris (1979) estimate the size of the disturbed zone to be of the order JR • (dJR/da)~*. In analogy to the disturbance by blunting, the nonproportional stressing zone is contained within the HRR field if -l
(a and W - a)> 10
JR(^ (12-86) da where the numerical factor was tentatively established empirically. Further, Hutchinson and Paris (1979) argue that crack growth itself, due to the associated unloading zone, represents a disturbance of the HRR field. Therefore the amount of crack growth should be limited to some fraction of the crack length or the ligament width: Aa<0.07(W-a)
(12-87)
624
12 Fracture Mechanisms
Again the numerical factor is tentative. If Eqs. (12-86) and (12-87) are satisfied in addition to the requirements derived already for the stationary crack, Eqs. (12-78) and (12-79), one can expect to obtain a JR-curve which is independent of the specimen shape and size. Incidentally, the present author surmises that it suffices to satisfy Eq. (12-86) or (12-87), since small amounts of crack growth should be admissible even for materials with a flat R-curve, and since a very steep R-curve enforces a nearly proportional field close to the crack tip even for large amounts of crack growth. For ductile materials, like low- and medium-strength steels, the conditions for valid / testing are very restrictive and it is often only possible to measure the initial parts of the R-curve within these limits with specimens of a practical size. Several proposals were made for testing the crack resistance outside the range of validity of J. Since a loss in triaxial constraint compared to that of the plane-strain HRR field seems to be one of the most important aspects, it appears promising to introduce the triaxiality ratio ojat at some structural distance ahead of the crack tip as an additional fracture characterizing parameter besides J (Kordisch et al, 1989). A more fundamental, but computationally expensive approach is to model crack growth by damage mechanics equations, e.g., by the Gurson model, and to abandon any direct transferability of fracture mechanics parameters from one specimen to another.
Sun et al. (1989, 1990, 1991) determine the parameters of the Gurson model from smooth tensile tests on ASTM A 710 steel with yield stress 612 MPa. They verify the usefulness of the model by predicting results of notched bar tests, and finally apply the model to crack growth in various types of specimens. The calculations are done for plane strain. Figure 12-45 shows a calculated R-curve based on the material parameters taken from the tensile test in comparison with a curve measured in a side-grooved CT specimen. The agreement is good. Figure 12-46 shows R-curves calculated for compact-tension (CT), single-edge cracked bend (SENB), single-edge cracked tension (SENT) and center-cracked tension (CCP) specimens (Sun et al., 1990, 1991). Apparently no geometry-independent R-curve is obtained. The tensile ge800
simulation
0.5 1.00 1.50 Crack Extension, Aa[mm]
Figure 12-45. Measured R-curve compared with calculation based on the Gurson model by Sun et al. (1990). 1500
12.6.6.4 Damage Mechanics Models for Stable Crack Growth Several attempts were made to simulate stable crack growth by finite-element calculations based on the (modified) Gurson model, which was described in Sec. 12.3.6.
0.5 1.00 1.50 Crack Extension, Aa [mm]
2.00
Figure 12-46. R-curves calculated for various specimen geometries based on the Gurson model. After Sun et al. (1990).
625
12.6 Nonlinear Fracture Mechanics
ometries lead to higher initiation values and greater slopes of the R-curve. This sort of non-uniqueness is generally observed in experiments, and it is not unexpected, since it occurs outside the range of validity of J. The horizontal lines represent the limitation expressed by Eq. (12-78) with a numerical factor 25 for the bend geometries and 200 for the tensile geometries. The limit is calculated with a ligament width of 19 mm for all specimens. All other criteria for J dominance are less stringent in this case. In this (calculated) example it appears that the criterion for the tensile geometries could be a little less restrictive, while the criterion for the bend geometries appears to be reasonably chosen. 12.6.7 A Note on Fast Crack Growth
The subject area of high loading rates, fast crack growth and possible crack arrest has several theoretical aspects. Effects of inertia and waves in the elastic field and in the plastic zone, the strain-rate sensitivity of the deformation response, and possible transitions from a ductile fracture mode at low rates to cleavage at high rates play a role (e.g., Freund, 1989). Here, only the influence of inertia in the plastic zone on the dynamic fracture toughness of a fast crack, Kld, will be considered. A strain singularity at a growing crack tip is associated with high accelerations of material elements and with a high kinetic energy density. It is therefore plausible that, if inertial forces are taken into account, they will tend to reduce the strain singularity in order to reduce the kinetic energy. In fact, Freund and Douglas (1982) have shown for Mode III that the strain singularity of the quasistatically growing crack, e oc In2 (r/rpl% is reduced by inertial forces to s oc In (r/rpl). Hence, if a criticalstrain criterion is required for crack growth,
6
1
1
i
i
5 c /x c = 32.5 4-
>
/
c
/
30
^27.5
-
O H o "E 2 CO
1-
• 4340 Steel: c = 3200 m/s
-
30 MPaVm i
i
0.1
0.2
i
0.3
0.4
Crack Speed, a / c
Figure 12-47. Dynamic fracture toughness vs. crack speed after Lam and Freund (1985).
the applied stress intensity factor must increase in order to achieve the critical strain in the weaker singularity, if the crack speed grows. This idea was worked out numerically for Mode I (plane strain) by Lam and Freund (1985). Figure 12-47 shows their result for the dynamic fracture toughness, KId, of a crack running at a constant speed a. The fracture toughness is normalized by the value Ki5 which is necessary for static crack growth initiation. Different theoretical curves are shown for different values of the ductility measure SJxc, which has the same meaning as in Sec. 12.6.6.2. As Fig. 12-47 shows, the dynamic fracture toughness starts rising at a relatively small fraction of the shear wave speed, c9 solely by inertia effects in the plastic zone. Also shown are data on AISI4340 steel by Rosakis et al. (1984), which agree with the predicted trend. 12.6.8 Creep Crack Growth
Creep crack growth is the slow extension of a macroscopic crack under more or less
626
12 Fracture Mechanisms
constant load at elevated temperatures. It usually, but not necessarily, occurs along grain boundaries by the formation of grain boundary cavities ahead of the crack tip. High-temperature corrosion at the crack tip plays an additional role in some alloys. In creep-brittle materials, creep crack growth occurs under "small-scale creep" conditions. This means that creep strains are confined to a small creep zone near the crack tip. In creep-ductile materials, crack growth is accompanied by large creep strains in the whole specimen. Figure 12-48 shows an example for each of these limiting cases. In brittle materials, the stress intensity factor will be the appropriate load parameter to describe the crack growth behavior, while in ductile materials the so-called C*-integral is applicable under conditions to be specified more precisely later. Creep crack growth rates have been measured extensively in the past two decades, primarily in ferritic and austenitic steels, but also in nickel-base alloys and, to a lesser extent, in cobalt and aluminum alloys and in ceramic materials. The subject has been reviewed repeatedly with extensive reference lists of the experimental literature (e.g., Riedel, 1985b, 1987, 1989b, c).
12.6.8.1 The C*-Integral
In the secondary creep range, materials can approximately be described as nonlinear viscous, which means that the strain rate is a function of the current stress only, but not of the prior loading history. According to Hoff s (1954) analogy, there is a path-independent integral, which is analogous to the ./-integral and which is denoted by C*. In the definition of J, strain and displacement are replaced by their rates to yield C*. Most frequently experimentalists use a formula of the type a
w'1
(12-88)
where A is the load point displacement rate and g2 is the same dimensionless function as in Eq. (12-68) with N being replaced by l/n. Several authors independently suggested to use C* as a load parameter to describe creep crack growth (Ohji et al., 1974; Landes and Begley, 1976). As in the case of the J-integral, the argument for its use is that it can be measured at the loading points of a specimen and that, owing to its path independence, it characterizes the crack-tip fields, and hence controls the fracture processes.
12.6.8.2 A Model for C*-Controlled Crack Growth
tCr-1/2 Mo steel
Nimonic 80A
Figure 12-48. Compact specimens after creep crack growth tests in creep-ductile and creep-brittle materials.
Creep crack growth by grain boundary cavitation ahead of the crack tip is modeled by requiring a critical-strain criterion (Riedel, 1981, 1987). For power-law viscous material (s = A an) the crack-tip stress field is an HRR field, which, in conjunction with the critical-strain criterion, leads to
627
12.6 Nonlinear Fracture Mechanics
the crack growth rate
1 ' lncoloy800H, T = 800°C
E, 10"8 •ctf
a oc-
(12-89)
where ef is the local strain to fracture. The predicted dependence on (C*) n/(n+1) has been confirmed in numerous experiments. Figure 12-49 shows an example for the austenitic ferrous Alloy 800 H from Hollstein and Kienzler (1988). At the same time, the figure demonstrates that C* unifies the crack growth rates in many different specimen geometries including semi-elliptical part-through cracks in plates, and cracks in pipes.
?
B
o
6
I 6
- r*
f
i
1
\
10-3
10-2
CT25/50 CT25/80 CT13/40 CCT PTC Tube i
10' 1
C* [W/m2]
Figure 12-49. Crack growth rate vs. C*-integral for various specimen geometries. Dashed line: theoretical slope.
creep zone spreads out over the ligament. After a characteristic time j
12.6.8.3 Limitations to C* and Other Load Parameters If the deformation behavior of the material deviates significantly from nonlinear viscous in a substantial portion of the cracked specimen, the range of validity of C* and other load parameters can be conveniently displayed on a load parameter map (Riedel, 1985 b). As Fig. 12-50 shows, this is a diagram with the logarithm of a load measure (i.e., the net section stress or a reference stress) on the horizontal and the logarithm of time on the vertical axis. In this plane, the ranges of validity of various load parameters are separated by lines, which represent characteristic transition times. At short times after load application, the specimen response is elastic or elastic/plastic depending on the load level. Creep strains are still confined to a small creep zone that grows around the crack tip (small-scale creep). Therefore, crack growth occurring in the short-time regime is governed by Kx or J. This is the case in creepbrittle materials. If a specimen of ductile material is held at a constant load level, the
(12-90)
l)C*
(Riedel and Rice, 1980) creep strains start to dominate in the whole specimen, and one enters the regime of decontrolled crack growth. The other characteristic times shown in Fig. 12-50 are calculated similarly. Depending on the importance of primary creep in the considered material, there may be a regime in which the deformation fields are controlled by primary creep, while for very long times at low loads, diffusion creep dominates. The line denoted by th in Fig. 12-50 represents the limitation to C* due to crack blunting. ^7777
K
° O
power- law>\ \yiscous 0 \
N
/primary \ a 2
V
2 /ASTM E 813
elastic- j plastic log CTref
^
Figure 12-50. Load parameter map (schematic).
628
12 Fracture Mechanisms
Finally, tf denotes the time to fracture of the cracked specimen. In Fig. 12-50 the hatched band adjacent to the line for ti represents the limitation to C* due to tertiary creep. Before the specimen finally fractures, it reaches the tertiary creep stage, in which the remaining ligament is severely damaged by creep cavities. Wide-spread damage, as opposed to small-scale damage only near the crack tip, offsets the validity of C* (Riedel, 1985 b, c). In ductile materials, this is often a severe restriction on the applicability of C*, especially if short cracks are considered. It may be more appropriate then to estimate the lifetime of a cracked component by a traditional reference stress concept rather than by a fracture mechanics concept based on C* (Webster et al., 1986; Hsia et al., 1990). Of course, the transition from one regime on the load parameter map to another occurs gradually rather than abruptly. In many practical cases, it is therefore necessary to use methods which interpolate between different regimes. The so-called Ct-parameter is an example, which attempts to interpolate between the regimes of elastic behavior, primary creep and secondary creep (Saxena, 1986; Bassani et al., 1989).
12.7 References Anderson, P. M., Rice, J. R. (1986), Scripta Met. 20, 1467-1472. Aoki, K., Izumi, O. (1979), Nippon Kinzaku Gakkaishi43, 1190. Argon, A. S. (1982), in: Recent Advances in Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Owen, D. R. J. (Eds.). Swansea: Pineridge Press, pp. 1-52. Argon, A. S., Chen, I.-W, Lau, C. W. (1981), in: Three-Dimensional Constitutive Relations and Ductile Fracture: Nemat-Nasser, S. (Ed.). Amsterdam: North-Holland Publishers, pp. 23-49. Argon, A. S., Im, J. (1975), Metall. Trans. 6 A, 839851.
Argon, A. S., Im, X, Safoglu, R. (1975), Metall. Trans. 6A, 825-837. Ashby, M. F. (1977), in: Fracture 1977, Vol. 1: Taplin, D. M. R. (Ed.). Waterloo: University of Waterloo Press, pp. 1-14. Ashby, M. R, Gandhi, C , Taplin, D. M. R. (1979), Acta Metall. 27, 699-729. ASTM-E399 (1978), Standard Method of Test for Plane-Strain Fracture Toughness of Metallic Materials, in: Annual Book of ASTM Standards. Philadelphia: American Society for Testing and Materials, pp. 512-533. ASTM-E 813 (1981), Standard Test for JIc, a Measure of Fracture Toughness, in: Annual Book of ASTM Standards, Part 10. Philadelphia: American Society for Testing and Materials, pp. 810-828. Baskes, M. L, Daw, M. S. (1990), in: Proceedings of the Fourth International Conference on Hydrogen Effects on Material Behaviour. Jackson Lodge, New York, Sept. 1990. Bassani, J. L., Hawk, D. E., Saxena, A. (1989), in: Nonlinear Fracture Mechanics, ASTM STP 995: Saxena, A., Landes, X D., Bassani, X L. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 7-26. Beachem, C. D. (1972), Metall. Trans. 3, 437-451. Becker, R. (1987), /. Mech. Phys. Solids 35, 577-599. Becker, R., Needleman, A., Richmond, O., Tvergaard, V. (1988), J. Mech. Phys. Solids 36, 317-351. Begley, X A., Landes, X D. (1972), in: Fracture Toughness, ASTM STP 514. Philadelphia: American Society for Testing and Materials, pp. 1-23. Bendick, W, Briining, B., Weber, H. (1985), in: Sonderbdnde der Praktischen Metallographie, Band 16. Stuttgart: Dr. Riederer-Verlag, pp. 439-450. Berweiler, W, Riedel, H. (1990), Reliability of Bulk Ceramics, Fatigue Behaviour in Corrosive Media, IWM Report Z 2190. Birnbaum, H. K. (1983), in: Atomistics of Fracture: Latanision, R. M., Pickens, X R. (Eds.). New York: Plenum Press, pp. 733-769. Bourcier, R. X, Koss, D. A., Smelser, R. E., Richmond, O. (1986), Acta Metall. 34, 2442-2453. Brede, M., Haasen, P. (1988), Acta Metall. 36, 20032016. Broberg, K. B. (1971), /. Mech. Phys. Solids 19, 407418. Broek, D. (1982), Elementary Engineering Fracture Mechanics. Dordrecht: Martinus Nijhoff Publishers. Brown, L. M., Stobbs, W. M. (1976), Phil. Mag. 34, 351. Budiansky, B., Hutchinson, X W, Slutsky, S. (1982), in: Mechanics of Solids, The Rodney Hill 60th Anniversary Volume: Hopkins, H. G., Sewell, M. X (Eds.). Oxford: Pergamon Press, pp. 13-45. Burgel, R., Truck, B., Schneider, K. (1990), in: Langzeitverhalten warm)ester Sta'hle und Hochtemperaturwerkstoffe. Diisseldorf: Verein Deutscher Eisenhuttenleute, pp. 94-107.
12.7 References
Burget, W. (1990), private communication. Cane, B. J. (1979), Metal Sci. 13, 287-294. Carlsson, A. E. (1990), Solid State Physics 43, 1-91. Chen, C. W, Machlin, E. S. (1957), Trans. AIME209, J. Metals, 829-835. Chen, I.-W, Argon, A. S. (1981), Ada Metall 29, 1321-1333. Chen, R. T., Weertman, J. R. (1984), Mater. Sci. Eng. 64, 15-25. Chen, S. P., Voter, A. E, Albers, R. C , Boring, A. M., Hay, P. J. (1989), Scripta Met. 23, 217-222. Chuang, T.-X, Kagawa, K. I., Rice, J. R., Sills, L. B. (1979), Acta Metall 27, 265-284. Cottrell, A. H. (1958), Trans. Am. Inst. Min. Metall. Engrs. 212, 192-203. Cox, T. B., Low, X R. (1974), Metall Trans. 5, 1457. deCellis, B., Argon, A., Yip, S. (1983), J. Appl Phys. 54, 4864-4878. Duva, J. M. (1986), Mech. Mater. 5, 137-144. Dyson, B. F. (1976), Metal Sci. 10, 349-353. Dyson, B. F. (1979), Canad. Metall. Qu. 18, 31-38. Dyson, B. F. (1983), Scripta Metall 17, 31-37. Edelson, B. I., Baldwin, W. M. (1962), Trans. ASM 55, 230. Eggeler, G., Earthman, J. C , Nilsvang, N., Ilschner, B. (1989), Acta Metall 37, 49-60. Engell, H. X, Speidel, M. O. (1969), in: Spannungsrifikorrosion (Korrosion 22): Kaesche, H. (Ed.)- Weinheim: VCH Publishers, pp. 1-20. Erhart, H., Grabke, H. X (1981), Metal Sci. 15, 401 408. Fields, R. X, Weerasooriya, T., Ashby, M. R (1980), Metall Trans. 11 A, 333-347. Fisher, X R., Gurland, X (1981), Metal Sci. 15, 185202. Fleck, N. A., Hutchinson, X W. (1986), Proc. R. Soc. London A 407, 435-458. Ford, F. P. (1984), in: Environment-Sensitive Fracture, ASTMSTP821: Dean, S. W, Pugh, E. N., Ugiansky, G. M. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 32-51. Ford, R P. (1988), J. Pres. Ves. Technology 1, 113. Ford, F. P. (1990), in: Environment-Induced Cracking of Metals, NACE-10: Gangloff, R. P., Ives, M. B. (Eds.). Houston: National Association of Corrosion Engineers, pp. 139-166. Freund, L. B. (1989), in: Fracture Mechanics, Perspectives and Directions (Twentieth Symposium), ASTMSTP1020: Wei, R. P., Gangloff, R. P. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 84-97. Freund, L. B., Douglas, A. S. (1982), J. Mech. Phys. Solids 30, 59-1 A. Frost, H. X, Ashby, M. R (1977), Deformation-Mechanism Maps. The Plasticity and Creep of Metals and Ceramics. New York, London: Pergamon Press. Gandhi, C , Ashby, M. F (1979 a), Scripta Metall 13, 371-376. Gandhi, C , Ashby, M. F. (1979b), Acta Metall. 27, 1565-1602.
629
Gandhi, C, Raj, R. (1982), Acta Metall 30, 505-511. Gangloff, R. P., Wei, R. P. (1977), Metall Trans. 8 A, 1043-1053. Gao, Y. C. (1985), Int. J. Fracture 29, 171-180. Gehrke, E., Riedel, H. (1990), Reliability of Bulk Ceramics, Fatigue Behaviour in Corrosive Media, IWM Report Z 14/90. Goods, S. H., Brown, L. M. (1979), Acta Metall. 27, 1-15. Green, G., Knott, X R (1976), /. Eng. Mater. Technology 98, 37-46. Griffith, A. A. (1920), Phil Trans. Roy. Soc. Lond. A 221, 163-198. Gruhl, W (1984), Z. Metallkunde 74, 819-826. Gurland, X (1972), Acta Metall 20, 735. Gurson, A. L. (1977), J. Eng. Mater. Technology 99, 2-15. Guttman, M., McLean, D. (1979), in: Interfacial Segregation: Johnson, W. C , Blakely, X M. (Eds.). Metals Park, OH: American Society for Metals, pp. 261-348. Hahn, G. T, Rosenfield, A. R. (1975), Metall Trans. 6A, 653-668. Hahn, G. T, Kanninen, M. R, Rosenfield, A. R. (1972), Annual Rev. Mater. Sci. 2, 381-404. Hancock, X W, Brown, D. K. (1983), J. Mech. Phys. Solids 31, 1-24. Hancock, X W, Mackenzie, A. C. (1976), J. Mech. Phys. Solids 24, 147-169. Hayhurst, D. R., Leckie, R A. (1984), in: Mechanical Behaviour of Materials, Proceedings of ICM4, Vol. 2: Carlsson, X, Ohlson, N. G. (Eds.). Oxford: Pergamon Press, pp. 1195-1212. He, M. Y, Hutchinson, X W. (1981), ASME J. Appl Mech. 48, 830-840. Hermann, L., Rice, X R. (1980), Metal Sci. 14, 285291. Hirth, X P., Rice, X R. (1980), Metall Trans. 11 A, 1501-1511. Hoff, N. X (1954), Quart. Appl Math. 12, 49-55. Hollstein, T, Kienzler, R. (1988), /. Strain Analysis 23, 87-96. Honig, A. (1990), private communication. Hsia, K. X, Argon, A. S., Parks, D. M. (1990), in: Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Evans, R. W. (Eds.). London: The Institute of Metals, pp. 691-701. Huang, Y, Hutchinson, X W, Tvergaard, V (1991), J. Mech. Phys. Solids 39, 223-241. Hull, D., Rimmer, D. E. (1959), Phil Mag. 4, 673687. Hult, X D., McClintock, R A. (1957), in: Ninth International Congress on Applied Mechanics, Vol. 8. Brussels: pp. 51-58. Hutchinson, X W. (1968), J. Mech. Phys. Solids 16, 13-31. Hutchinson, X W. (1983a), Acta Metall. 31, 10791088. Hutchinson, X W. (1983 b), /. Appl Mech. 105, 10421051.
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Hutchinson, J. W, Paris, P. C. (1979), in: ElasticPlastic Fracture, ASTM STP668: Landes, J. D., Begley, J. A., Clarke, G. A. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 37-64. Intrater, I , Machlin, E. S. (1959), Ada Metall 7, 140-143. Irwin, G. R. (1957), J. Appl. Mech. 24, 361-364. Kachanov, L. M. (1960), The Theory of Creep: Kennedy, A. J. (Ed.). Boston, U.K.: Wetherby. Kaesche, H. (1989), in: Ergebnisse der WerkstoffForschung, Band 2: Speidel, M. O., Magdowski, R. M., Uggowitzer, P. J. (Eds.). Zurich: Yerlag Thubal-Kain, pp. 27-45. Kanninen, M. R, Popelar, C. H. (1985), Advanced Fracture Mechanics. New York: Oxford University Press. Kaufmann, M. X, Fink, J. L. (1988), Ada Metall. 36, 2213-2228. Kausch, H.-H. (Ed.) (1990), Crazing in Polymers, Vol. 2. Berlin: Springer-Verlag. Kelly, A., Tyson, W. R., Cottrell, A. H. (1967), Phil. Mag. 15, 567-586. Kerkhof, F. (1970), Bruchvorgdnge in Gldsern. Frankfurt: Verlag der Deutschen Glastechnischen Gesellschaft. Knott, J. F. (1971), in: Effect of Second-Phase Particles on the Mechanical Properties of Steel. London: The Iron and Steel Institute, pp. 44-53. Knott, J. F. (1973), Fundamentals of Fracture Mechanics. London: Butterworths. Kohlhoff, S., Gumbsch, P., Fischmeister, H. F. (1991), Phil Mag. A64, 851-878. Koplik, J., Needleman, A. (1988), Int. J. Solids Structures 24, 835-853. Kordisch, H., Sommer, E., Schmitt, W. (1989), Nuclear Eng. Design 112, 27-35. Krishnamohanrao, Y, Kutumbarao, V. V, Rama Rao, P. (1986), Ada Metall. 34, 1783-1806. Kumar, V, German, M. D., Shih, C. F. (1981), An Engineering Approach for Elastic-Plastic Fracture Analysis, Report NP-1931 on project 1237-1 for Electric Power Research Institute, Palo Alto. Lam, P. S., Freund, L. B. (1985), J. Mech. Phys. Solids 33, 153-167. Landes, J. D., Begley, J. A. (1974), in: Fracture Analysis, ASTM STP 560. Philadelphia: American Society for Testing and Materials, pp. 170-186. Landes, I D . , Begley, J. A. (1976), in: Mechanics of Crack Growth, ASTM STP 590. Philadelphia: American Society for Testing and Materials, pp. 128-148. Larsson, S. G., Carlsson, A. J. (1973), /. Mech. Phys. Solids 21, 263-277. LeRoy, G., Embury, J. D., Edwards, G., Ashby, M. F. (1981), Ada Metall. 29, 1509. Liu, C. T, White, C. L., Horton, J. A. (1985), Ada Metall. 33, 213-229. Low, J. R. (1956), in: Deformation and Flow of Solids. Berlin: Springer-Verlag, p. 60.
Lynch, S. P. (1979), Metals Forum 2, 189. Magdowski, R. M., Speidel, M. O. (1988), Metall. Trans. 19 A, 1583-1596. Magnusen, P. E., Follansbee, P. S., Koss, D. A. (1985), Metall Trans. 16A, 2273-2281. Marini, B., Mudry, F , Pineau, A. (1985), Eng. Fracture Mech. 22, 989-996. McClintock, F. A. (1968 a), J. Appl Mech. 35, 363371. McClintock, F A . (1968b), in: Ductility. Metals Park, OH: American Society for Metals, pp. 255277. McClintock, F. A. (1971), in: Fracture - An Advanced Treatise, Vol. 3: Liebowitz, H. (Ed.). New York: Academic Press, pp. 47-225. McClintock, F. A., Irwin, G. R. (1965), in: Fracture Toughness Testing and Its Applications, ASTM STP 381. Philadelphia: American Society for Testing and Materials, pp. 84-113. McClintock, F. A., Kaplan, S. M., Berg, C. A. (1966), Int. J. Fracture Mech. 2, 614-627. McLean, D. (1956/7), J. Inst. Metals 85, 468-472. McLean, D. (1957), Grain Boundaries in Metals. London: Oxford University Press. McMahon, C. J. (1968), Temper Embrittlement in Steel, ASTM STP 407, 127-167. McMahon, C. J. (1984), Z. Metallkde. 75, 496-509. McMeeking, R. M. (1977), /. Mech. Phys. Solids 25, 357-381. McMeeking, R. M. (1989), in: Advances in Fracture Research (ICF7), Vol. 3: Salama, K., Ravi-Chandar, K., Taplin, D. M. R., Rama Rao, P. (Eds.). Oxford: Pergamon Press, pp. 1971-1996. McMeeking, R. M., Horn, C. L. (1990), Int. J. Fracture 42, 1-19. McMeeking, R. M., Parks, D. (1979), in: ElasticPlastic Fracture, ASTM STP 668: Landes, J. D., Begley, J. A., Clarke, G. A. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 175-194. Moller, R., Erhart, H., Grabke, H. J. (1984), Arch. Eisenhuttenwes. 55, 543-548. Mullins, M. (1984a), Int. J. Fracture 24, 189-196. Mullins, M. (1984b), Ada Metall 32, 381-388. Needham, N. G. (1983), Cavitation and Fracture in Creep Resisting Steels, in: Report EUR 8121 EN, Commission of the European Communities, Brussels, Contract No. 7210MA/802. Needham, N. G., Gladmann, T. (1984), in: Creep and Fracture of Engineering Materials and Structures, Part II: Wilshire, D., Owen, D. R. J. (Eds.). Swansea: Pineridge Press, pp. 1263-1276. Needleman, A. (1972), /. Appl Mech. 39, 964-970. Needleman, A. (1987), /. Appl. Mech. 54, 525-531. Needleman, A., Rice, J. R. (1978), in: Mechanics of Sheet Metal Forming: Koistinen, D. P., Wang, N.-M. (Eds.). New York: Plenum Publishing Corporation, pp. 237-267. Needleman, A., Tvergaard, V. (1984), J. Mech. Phys. Solids 32, 461-490.
12.7 References
Neubauer, B. (1981), in: Recent Advances in Creep and Fracture of Engineering Materials and Structures: Wilshire, B., Owen, D. R. J. (Eds.). Swansea: Pineridge Press, pp. 617-619. Newman, R. C , Sieradzki, K. (1987), in: Chemistry and Physics of Fracture: Latanision, R. M., Jones, R. H. (Eds.). Dordrecht: Martinus Nijhoff Publishers, p. 597. Ohji, K., Ogura, K., Kubo, S. (1974), Preprint of Japan. Soc. Mech. Engrs. No. 640-11, 207 (in Japanese). Oriani, R. A., Josephic, P. H. (1977), Ada Metall 25, 979. Parkins, R. N. (1984), in: Environment-Sensitive Fracture, ASTMSTP821: Dean, S. W, Pugh, E. N., Ugiansky, G. M. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 5-31. Paxton, A. T., Methfessel, M., Polatoglou, H. M. (1990), Phys. Rev. B41, 8127-8138. Pedrazzoli, R. M. (1989), in: Advances in Fracture Research (ICF7), Vol. 6: Salama, K., Ravi-Chandar, K., Taplin, D. M. R., Rama Rao, P. (Eds.). Oxford: Pergamon Press, pp. 3711-3722. Pedrazzoli, R. M., Speidel, M. O. (1990), in: Proceedings of the International Conference on High Nitrogen Steels (HNS 90). Diisseldorf: Stahleisen-Verlag. Pettifor, D. G. (1983), in: Physical Metallurgy: Cahn, R. W., Haasen, P. (Eds.)- Amsterdam: Elsevier, pp. 73-152. Raj, R., (1975), Metall Trans. 6A, 1499-1509. Raj, R., Ashby, M. F. (1975), Acta Metall. 23, 653666. Ratke, L., Gruhl, W. (1980), Werkstoffe und Korrosion 31, 768-773. Rice, J. R. (1967), /. Appl. Mech. 34, 287-298. Rice, J. R. (1968 a), in: Fracture. An Advanced Treatise, Vol. 2: Liebowitz, H. (Ed.). New York: Academic Press, pp. 191-311. Rice, J. R. (1968b), J. Appl. Mech. 35, 379-386. Rice, J. R. (1981), Acta Metall. 29, 675-681. Rice, J. R. (1988), /. Appl. Mech. 55, 98-103. Rice, J. R., Johnson, M. A. (1970), in: Inelastic Behavior of Solids: Kanninen, M. R, Adler, W. R, Rosenfield, A. R., Jaffee, R. I. (Eds.). New York: McGraw-Hill, pp. 641-671. Rice, J. R., Rosengren, G. R (1968), /. Mech. Phys. Solids 16, 1-12. Rice, J. R., Sorensen, E. P. (1978), /. Mech. Phys. Solids 26, 163-186. Rice, J. R., Thomson, R. (1974), Phil. Mag. 29, 7 3 97. Rice, J. R., Tracey, D. M. (1969), J. Mech. Phys. Solids 17, 201-217. Rice, J. R., Paris, P. C , Merkle, J. G. (1973), in: Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536. Philadelphia: American Society for Testing and Materials, pp. 231-245. Rice, J. R., Drugan, W. X, Sham, T.-L. (1980), in: Fracture Mechanics: Twelfth Conference, ASTM
631
STP 700: Paris, P. C. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 189— 221. Riedel, H. (1981), in: Creep in Structures: Ponter, A. R. S., Hayhurst, D. R. (Eds.). Berlin: SpringerVerlag, pp. 504-519. Riedel, H. (1984), Acta Metall. 32, 313-321. Riedel, H. (1985a), Z. Metallkde. 76, 669-675. Riedel, H. (1985b), in: Flow and Fracture at Elevated Temperatures: Raj, R. (Ed.). Metals Park, OH: American Society for Metals, pp. 149-177. Riedel, H. (1985 c), in: Fundamentals of Deformation and Fracture: Bilby, B. A., Miller, K. X, Willis, J. R. (Eds.). Cambridge: Cambridge University Press, pp. 293-309. Riedel, H. (1987), Fracture at High Temperatures. Berlin: Springer-Verlag. Riedel, H. (1989 a), Int. J. Pres. Ves. Piping 39,119134. Riedel, H. (1989 b), in: Advances in Fracture Research (ICF7), Vol.2: Salama, K., Ravi-Chandar, K., Taplin, D. M. R., Rama Rao, P. (Eds.). Oxford: Pergamon Press, pp. 1495-1524. Riedel, H. (1989 c), in: Fracture Mechanics: Perspectives and Directions (Twentieth Symposium), ASTM STP 1020: Wei, R. P., Gangloff, R. P. (Eds.). Philadelphia: American Society for Testing and Materials, p. 101-126. Riedel, H. (1992), in: VGB-Konferenz ,,Restlebensdauer". Essen: Technische Vereinigung der GroBkraftwerksbetreiber, pp. 1-15. Riedel, H., Kochendorfer, A. (1979), Arch. Eisenhuttenwes. 50, 173-178. Riedel, H., Rice, X R. (1980), in: Fracture Mechanics: Twelfth Conference, ASTM STP 700: Paris, P. C. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 112-130. Ritchie, R. O., Knott, X R, Rice, X R. (1973), /. Mech. Phys. Solids 21, 395-410. Robertson, I. M., Birnbaum, H. K. (1986), Acta Metall. 34, 353. Rodin, G. X, Parks, D. M. (1986), Mech. Mater. 5, 221-228. Rodin, G. X, Parks, D. M. (1988), /. Mech. Phys. Solids 36, 237-249. Rooke, D. P., Cartwright, D. X (1974), Stress Intensity Factors. London: Her Majesty's Stationary Office. Rosakis, A. X, Duffy, X, Freund, L. B. (1984), /. Mech. Phys. Solids 32, 443-460. Rose, X H., Smith, X R., Perrante, X (1983), Phys. Rev. B28, 1835-1845. Sauthoff, G. (1989), Z. Metallkde. 80, 337-344. Saxena, A. (1986), in: Fracture Mechanics: Seventeenth Symposium, ASTM STP 905: Underwood, X H., Chait, R., Smith, C. W, Wilhelm, D. P., Andrews, W A., Newman, X C. (Eds.). Philadelphia: American Society for Testing and Materials, pp. 185-201. Schoeck, G. (1991), Phil. Mag. 63, 111-120.
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Senior, B. A., Noble, F. W., Eyre, B. L. (1988), Acta Metall. 36, 1855-1862. Shewmon, P. G. (1985), Metall. Trans. 7A, 279-286. Shih, C. F. (1983), Tables of the Hutchinson-RiceRosengren Singular Field Quantities, Brown Universtiy Report MRL 3-147. Providence. Shih, C. F. (1985), Int. J. Fracture 29, 73-84. Shih, C. F , German, M. D. (1981), Int. J. Fracture 17, 27-43. Shirh, C. F , German, M. D., Kumar, V. (1981), Int. J. Pres. Ves. Piping 9, 159-196. Shiozawa, K., Weertman, J. R. (1983), Acta Metall. 3/, 993-1004. Sih, G. C. (1973), Handbook of Stress Intensity Factors. Bethlehem: Institute of Fracture and Solid Mechanics, Lehigh University. Sih, G. C , DiTommaso, A. (Eds.) (1985), Fracture Mechanics of Concrete. Dordrecht: Martinus Nijhoff Publishers. Sih, G. C , Skudra, A. M. (Eds.) (1985), Handbook of Composites, Vol. 3: Failure Mechanisms of Composites. Amsterdam: Elsevier. Smith, E. (1966 a), in: Physical Basis of Yield and Fracture. Oxford: Institute of Physics and Physical Society, pp. 36-46. Smith, E. (1966b), Acta Metall 14, 991-996. Sofronis, P., McMeeking, R. M. (1989), /. Mech. Phys. Solids 37, 317-350. Spaehn, H. (1984), in: Subcritical Crack Growth Due to Fatigue, Stress Corrosion and Creep: Larsson, L. H. (Ed.). London and New York: Elsevier, pp. 55-84. Speidel, M. O. (1973), in: Hydrogen in Metals. Metals Park, OH: American Society for Metals, pp. 249276. Speidel, M. O. (1981), VGB Kraftwerkstechnik 61, Ail-All. Speidel, M. O., Hyatt, M. V. (1972), in: Advances in Corrosion Science and Technology, Vol. 2. New York: Plenum, pp. 115-334. Speidel, M. O., Magdowski, R. M. (1988), Int. J. Pres. Ves. & Piping 34, 119-142. Spitzig, W. A., Smelser, R. E., Richmond, O. (1988), Acta Metall. 36, 1201-1211. St. John, C. (1975), Phil. Mag. 32, 1193-1212. Stroh, A. N. (1957), Phil. Mag. Suppl. 6, 418-465. Sun, D.-Z., Siegele, D., Voss, B., Schmitt, W. (1989), Fatigue Fract. Eng. Mater. Struct. 12, 201-212. Sun, D.-Z., Schmitt, W. (1990), in: Numerical Methods in Fractue Mechanics: Luxmoore, A. R., Owen, D. R. J. (Eds.). Swansea: Pineridge Press, pp. 275286. Sun, D.-Z., Kienzler, R., Voss, B., Schmitt, W. (1991), in: 22nd National Symposium on Fracture Mechanics, Atlanta, Georgia, USA. June 26-28, 1990, to appear in an STM STP.
Swindeman, R. W, Farrell, K., Yoo, M. H. (1981), Res. Mechanica Letters 1, 61-11. Tada, H., Paris, P. C , Irwin, G. R. (1973), Stress Analysis of Cracks Handbook. Hellertown, PA: Del Research Corporation. Tetelman, A. S., Robertson, W. D. (1963), Acta Metall. 11, 415-426. Thomason, P. F (1968), /. Inst. Metals. 96, 360. Thomason, P. F. (1990), Ductile Fracture of Metals. Oxford: Pergamon Press. Thomson, R. (1986), Solid State Physics 39, 1-129. Tipler, H. R., Hopkins, B. E. (1976), Metal Sci. 10, 47-56. Tracey, D. M. (1971), Engg. Fracture Mech. 3, 301315. Troiano, A. R. (1960), Trans. ASM 52, 54. Tvergaard, V (1981), Int. J. Fracture 17, 389-407. Tvergaard, V. (1982), Int. J. Fracture 18, 237-252. Tvergaard, V. (1984), Acta Metall. 32, 1977-1990. Tvergaard, V. (1990), Advances in Applied Mechanics 27, 83-151. Tvergaard, V, Needleman, A. (1984), J. Mech. Phys. Solids 32, 157-169. Tyson, W. (1979), Canad. Metall. Quart. 18, 1-11. VdTUV-Merkblatt Dampfkessel 451-83/6 (1983). Essen: Verein der Technischen Uberwachungsvereine. Vehoff, H., Neumann, P. (1979), Acta Metall. 27, 915. Vehoff, H., Rothe, W. (1983), Acta Metall. 31, 1781 1793. Vehoff, H., Stenzel, H., Neumann, P. (1987), Z. Metallkde. 78, 550-556. Wachter, O., Miisch, H., Bendick, W. (1991), in: Werkstoffe und Schweifitechnik im Kraftwerk. Essen: VGB, pp. 3.1-3.39. Wang, J. S., Vitek, V (1986), Acta Metall. 34, 951. Wang, J. S., Sutton, A. P., Vitek, V. (1984), Acta Metall. 32, 1093. Webster, G. A., Smith, D. X, Nikbin, K. M. (1986), in: International Conference on Creep. Tokyo: Japanese Society of Mechanical Engineers, pp. 303 308. Wei, R. P. (1989), in: Advances in Fracture Research, ICF7, Vol. 2: Salama, K., Ravi-Chandar, K., Taplin, D. M. R., Rama Rao, P. (Eds.). Oxford: Pergamon Press, pp. 1525-1544. Westlake, D. G. (1969), Trans. ASM 62, 1000. Williams, M. L. (1957), J. Appl. Mech. 24, 109. Worswick, M. J., Pick, R. J. (1990), J. Mech. Phys. Solids 38, 601-625. Wray, P. J. (1969), /. Appl. Phys. 40, 4018-4029. Yamamoto, H. (1978), Int. J. Fracture 14, 347-365. Yoo, M. H., White, C. L., Trinkaus, H. (1985), in Flow and Fracture at Elevated Temperatures: Raj, R. (Ed.). Metals Park, OH: American Society for Metals, pp. 349-382. Zapffe, C. A., Sims, C. E. (1941), Trans. AIME 145, 225.
12.7 References
General Reading Ashby, M. F. (1977), in: Fracture 1977, Vol. 1: Taplin, D. M. R. (Ed.). Waterloo: University of Waterloo Press, pp. 1-14. Ford, F. P. (1990), in: Environment-Induced Cracking of Metals, NACE-10: Gangloff, R. P., Ives, M. B. (Eds.). Houston: National Association of Corrosion Engineers, pp. 139-166. Hutchinson, I W. (1983), /. Appl. Mech. 105, 10421051. Kanninen, M. R, Popelar, C. H. (1985), Advanced Fracture Mechanics. New York: Oxford University Press.
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Rice, J. R. (1968), in: Fracture. An Advanced Treatise, Vol. 2: Liebowitz, H. (Ed.). New York: Academic Press, pp. 191-311. Riedel, H. (1987), Fracture at High Temperatures. Berlin: Springer-Verlag. Speidel, M. O., Hyatt, M. V (1972), in: Advances in Corrosion Science and Technology, Vol. 2. New York: Plenum, pp. 115-334. Thomason, P. F. (1990), Ductile Fracture of Metals. Oxford: Pergamon Press. Thomson, R. (1986), Solid State Phys. 39, 1-129. Tvergaard, V. (1990), Adv. in Appl. Mech. 27, 83-151. Vehoff, H., Rothe, W (1983), Acta Metall. 31, 17811793.
13 Friction and Wear Koji Kato Mechanical Engineering Department, Tohoku University, Sendai, Japan
List of 13.1 13.2 13.3 13.3.1 13.3.2 13.3.3 13.4 13.4.1 13.4.2
Symbols and Abbreviations Introduction Solid Surface Mechanism of Contact Between Solid Surfaces Elastic Contact Elasto-Plastic Contact Plastic Contact Friction Microscopic Model of Friction The Change of Friction Coefficient due to the Change of Adhesion at the Contact Interface 13.4.2.1 The Effect of Adsorbed Gases and Oxides on Adhesion and Friction . .. 13.4.2.2 The Effect of Adsorbed Organic Molecular Layers on Adhesion and Friction 13.5 Wear 13.5.1 Wear Under Elastic Contact: Fatigue Wear 13.5.2 Wear Under Elasto-Plastic Contact: Fatigue Wear 13.5.3 Wear Under Plastic Contact: Adhesive Wear, Abrasive Wear, Flow Wear . 13.5.4 Wear Mode Transition due to Repeated Sliding on the Same Track 13.5.5 Wear Models and Wear Coefficients 13.5.5.1 Abrasive Wear 13.5.5.2 Adhesive Wear 13.5.5.3 Fatigue Wear 13.5.5.4 Corrosive Wear 13.5.6 The Wear Mechanism Map for a Steel 13.5.7 Microstructure of Subsurface Layers 13.5.7.1 Plastic Strain Distribution in the Subsurface After Sliding 13.5.7.2 Dislocations and Cells in the Subsurface After Sliding 13.6 Outlook 13.7 Acknowledgements 13.8 References
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
636 639 641 641 643 644 645 646 646 649 650 653 654 655 656 658 660 663 663 667 667 669 670 671 672 673 678 679 679
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13 Friction and Wear
List of Symbols and Abbreviations a d Aa At Ar at Aw Aox Ao Al9A2 b b b' c C Cs d d D Dp E1,E2 £* £ad / F ft fm /ab /ad G h Hv ifdef Ho k K Kf K'{ Kab Kad / L Lc
radius of a circular contact area half the unit sliding distance for fatigue wear apparent contact area microscopic contact area for the zth contact point total real contact area thermal diffusivity of metal cross-sectional area of wear groove Arrhenius constant for oxidation microscopic contact area for the case of st = 0 cross-sectional area of two side ridges half contact width between a cylinder and a flat surface Burgers vector magnitude of Burgers vector constant related to the fracture ductility specific heat section through the center of a deformed elliptic grain film thickness mean, transverse, linear, intercept cell size section through the center of an original spherical grain degree of penetration Young's modulus for the asperity and the surface in contact respectively equivalent Young's modulus {(£*)" * = [(1 - vfj/E^ + [(1 - v22)/E2]} adsorption energy dimensionless shearing strength at the interface total friction force r a t i o of sttok volume fraction of molten material removed during sliding volume fraction of wear particle in the groove for abrasive wear volume fraction of wear particle in the contact zone shear modulus depth of the yield point from the contact interface hardness on wear track hardness of the wearing material in the deformed state after abrasive action room-temperature hardness of metal critical shear strength of the material constant wear coefficient for fatigue wear probability of wear of material in contact zone for fatigue wear wear coefficient for abrasive wear wear coefficient for adhesive wear the length of a cylinder sliding distance critical distance to generate the fatigue wear particle
List of Symbols and Abbreviations
637
Lm m, m\ m" m* m** n JV A N{ nm p pt pi0 pm p0 q0 Qo r R Re rg r0 s sf st sQ T* Tb T{ Tm T™ To v v V V W W V z zc z*
latent heat of fusion per unit volume for metal constants experimental constant including the effect of work hardening exerimental constant total number of microscopic contact points Avogadro constant critical number of cycles for the failure of materials number of molecules per unit area contact stress contact stress of the ith contact point contact stress for the case of st = 0 mean contact stress maximum contact stress tangential stress at the interface activation energy for oxidation curvatue of contact asperity molar gas constant electrical contact resistance tip radius of the abrasive particle radius of pin shear stress bonding strength of a film shear stress at the ith contact point c o n s t a n t value of st a n equivalent temperature for metal bulk temperature flash temperature melting temperature of metal melting temperature of oxide sink temperature for bulk heating sliding velocity normalized velocity wear volume normalized wear rate normal load normalized pressure on sliding interface yield stress depth from the surface critical thickness of oxide film thickness of strongly sheared layer
a ah p
constant heat distribution coefficient factor which depends mainly on the work-hardening behavior of the wearing material
638
y ys A
13 Friction and Wear
cre
shear strain shear strain at the interface percentage by which the curved area of the identation exceeds the original plane surface area effective deformation (effective plastic strain) angle of slip line plastic strain at failure in a tensile test m a x i m u m plastic strain a t failure after o n e cycle of loading longitudinal plastic strain range dimensionless parameter for bulk heating adhesion coefficient slope of the plastic surface half wedge angle of groove d u e to the abrasive action contact angle at the ith contact point thermal conductivity of oxide electric resistivity friction coefficient friction coefficient at the ith contact point Poisson ratios of asperity a n d surface, respectively deformation due to a penetrating abrasive particle at the surface level deformation due t o a penetrating abrasive particle at the level below which ploughing dominates effective stress per cycle compressive stress in the direction of x-axis compressive stress in the direction of y-axis principal shear stress flow stress shear stress friction stress o n the slide plane film resistance
AES FIM LEED SEM STM TEM
Auger electron spectroscopy field ion microscopy low energy electron diffraction scanning electron microscope scanning tunneling microscope transmission electron microscope
3 s ef 8 max Ag p C rj 9 9g 9t x ox X H fii v 1? v 2 £s £ lim
13.1 Introduction
13.1 Introduction Friction and wear of materials are the main topics of tribology, which is the science and technology of phenomena occurring at the contact interface between solids. Friction is the resistance to the relative movement between two solids in contact. Wear is the volume loss from solid surfaces in contact which occurs as a result of friction. The following functions have to be taken into consideration in order to understand the mechanisms of friction and wear according to these definitions. (a) Environment: vacuum, gases, liquids. (b) Materials: metals, ceramics, polymers. (c) Surfaces: smooth, rough, curved, flat, clean, contaminated, lubricated, coated. (d) Types of contact or loading: stationary, oscillating, sliding, rolling, impact. (e) Electronic and atomic interactions: attractive forces, repulsive forces, adhesion, diffusion. (f) Deformation: elastic, elasto-plastic, plastic, local, general. (g) Heat generation and transfer: adsorption, desorption, chemical reaction, evaporation, melting, softening, phase transition. (h) Fracture: ductile, brittle, fatigue, creep, microscopic, macroscopic. (i) Movement of wear particles: transfer, re-transfer, mixturing, indenting, abrasion, removal. In these functions mentioned above, (a), (b), (c) and (d) are given conditions for friction and wear, and (e), (f), (g), (h) and (i) are resultant phenomena. Roughly speaking, the conditions of (a), (b), (c) and (d) can be assumed to be steady. But to be exact, they all change during friction and wear.
639
The air humidity generally changes during friction and wear tests. Lubricating oils and metal surfaces are oxidized. Gases absorbed in the bulk of a material are desorbed under vacuum. Wear changes the original shape and roughness of a surface. Such changes in (a), (b), and (c) induce changes in the friction and microgeometry of contact, which cause a change in the dynamic response of the driving system and, finally, the real dynamic load at the interface. The resultant phenomena of (e), (f), (g), (h) and (i) appear at the same time and interact with each other. Heat generation, for example, accelerates the oxidation at the surface and reduces the adhesion at the interface. But the softening of a material near the interface helps plastic deformation around the interface. Brittle fracture can be avoided by the softening of a material, but a phase transition might occur above a certain temperature. This could result in hard and brittle structure after cooling. All these interactions change the nature of the friction and wear. Therefore, friction and wear are characterized by an initial state and by a steady state. In the initial state, the original conditions of (a), (b), (c) and (d) change gradually through the combined effects of (e), (f), (g), (h) and (i). In the steady state, all the conditions of (a), (b), (c) and (d) reach their saturated states, and the friction and wear are controlled by a balance of the combined effects of (e), (f), (g), (h), and (i). It is obvious from this consideration that friction and wear are phenomena of system sensitivity. In addition to the understanding of each function's effect on friction and wear, the total interaction between them has to be understood (Czichos, 1978).
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13 Friction and Wear
Among the functions mentioned above, the environment has the strongest effect on friction and wear. The friction coefficient (frictional resistance/normal load) of a metal can be about 0.001 in liquids, 0.4 in air and 10 in high vacuum. The low friction in liquids is caused by the formation of a thin liquid film at the interface. The mechanism of formation of a thin film has been well-analyzed experimentally and theoretically by hydrodynamics (Cameron, 1966; Dowson and Higginson, 1966). The middle friction in air, and high friction in vacuum, are caused by high contact pressure and strong adhesion at the interface. The high contact pressure, which is much higher than the expected value for the apparent contact area, is generated at small contact points of asperities on rough surfaces. This pressure helps the adhesion at the interface. Adsorbed gases on a surface change this adhesion drastically and give rise to the different friction coefficients in air and under vacuum. This understanding was reached by means of a large number of fundamental and microscopic experiments in air and under vacuum (Bowden and Tabor, 1950,1964). The high contact pressure on the real contact area generates various states of deformation in the contact region. These include elastic, elasto-plastic and plastic deformation corresponding to the geometry, material and applied force. The theory of contact mechanisms is the basis for the analysis (Johnson, 1985). Wear occurs as a result of this contact pressure and adhesion at the interface, and wear coefficient [(wear volume x hardness)/ (load x sliding distance)] changes from about 10~ 9 — 10~ 2 depending on the frictional conditions (Peterson and Winer, 1980).
The environment can cause this wide variation in the wear coefficient of one material due to a change in the friction coefficient. However, a change in the material combination, contact geometry or loading type can also cause a change in the order of magnitude of the wear coefficients (Chiou e t a l , 1985; Chiou and Kato, 1987). This is because wear is a phenomenon which has various different modes decided by the types of deformation (Childs, 1988) and fracture of a material (Rigney, 1990). In other words, wear is microscopic fracture and is controlled by the microscopic stress distribution and microstructure in the contact region (Zum Gahr, 1987). In the case of relatively large scale wear under a large contact area, the model of crack initiation and propagation by repeated friction in the contact region gives a good theoretical estimation of the wear for use in analysis by linear fracture dynamics. This is the case when materials can be assumed to be continuous (Cheng and Keer, 1980; Keer and Bryant, 1983). In the case of small scale wear, which generates wear particles smaller than the contact size, the continuous theory can no longer be applied in the analysis. A more microscopical and microstructural approach is necessary in order to understand the generation mechanism of wear particles smaller than about 1 |4,m. Techniques such as transmission electron microscopy (TEM), scanning electron microscopy (SEM), scanning tunneling microscope (STM), Auger emission spectroscopy (AES), low energy electron diffraction (LEED), field ion microscopy (FIM) and X-ray analysis can give information on small scale wear (for details on analytic techniques see Vol. 2 of this Series). For example, FIM analysis of the surface of tungsten, after contact with gold, reveals
13.3 Mechanism of Contact Between Solid Surfaces
that there are three gold atoms bonded to each tungsten atom on the tungsten (110) surface. This means that wear occurs on the scale of one atom (Buckley, 1981), which suggests that the unit size of wear (a wear particle) is of the order of 0.1 nm to the order of 1 mm. Therefore we have to assume that there are many wear particles at the interface with a large size distribution. These particles behave in a complicated manner at the interface and change the nature of the friction and wear. The space for their movements is the clearance at the interface, which is smaller than the surface roughness. Dynamic response, such as vibration, of the system changes this space easily, which changes friction and wear as a result. It is clear from the above considerations that there are no constant friction coefficients and wear coefficients of materials. Based on this viewpoint, the fundamental mechanisms of friction and wear are explained in the following sections. Efforts were made to show the importance of deformation and fracture in the contact region. The large variety of experimental data or special data obtained under extreme conditions, was not shown.
641
micrometer scale. Each ridge has many peaks of different height. Figure 13-1 b shows the similar grooves and ridges on a steel surface which was polished with an abrasive paper. Figure 13-1 c shows the gold surface lathed with a diamond tool and observed by an STM. This surface is optically flat and is a good mirror, but it still has asperities and waves on a nanometer scale. Figure 13-1 d shows the cleaved surface of graphite, observed by an STM, where atoms are arranged in order. Each atom is an irregularity of the surface in this case. Surface defects such as kinks, voids, jogs and ad-atoms are also irregularities on the surface. In order to measure the roughness values for such solid surfaces, a surface profilometer is popularly used, and the crosssectional profile of the surface is measured as shown in Fig. 13-2. From such a profile, roughness values are given as a maximum roughness, an average roughness or a tip radius of one asperity. A height distribution function is also measured for treating contact as a phenomenon of probability.
13.2 Solid Surface
13.3 Mechanism of Contact Between Solid Surfaces
Practical solid surfaces are generally formed by cutting, grinding and polishing. These surfaces look smooth and are sometimes called mirrors. A so-called mirror surface acts as an ideally smooth surface for light. But even a mirror surface is not perfectly smooth microscopically. Figure 13-1 a shows a steel surface finished by grinding with a grinding wheel and observed by an SEM. There are parallel grooves and ridges on the surface on a
Since a solid surface generally has microasperities, the contact between solid surfaces is the contact between asperities microscopically. Figure 13-3 shows the schematic features of contact between rough surfaces. The unit models of contact between asperities in such a case is given by the contact between a flat surface and a sphere or a cylinder, as shown in Fig. 13-4, where a is the radius of a circular contact area and 2 b is the contact width.
642
13 Friction and Wear
60 (jm
60 |jm (b)
(a)
0.1 nm
(C)
(d)
Figure 13-1. Surface morphologies: (a) SEM micrograph of a steel surface ground with grinding wheel; (b) SEM micrograph of a steel surface ground with emery paper; (c) STM image of a gold surface lathed with a diamond bit; (d) STM image of a cleaved graphite surface.
0.05 mm Figure 13-2. The cross-sectional profile of a surface measured with a surface profilometer.
Figure 13-3. The schematic features of contact between rough surfaces.
13.3 Mechanism of Contact Between Solid Surfaces
643
w
w
2b
(a)
(b)
Figure 13-4. The unit models of contact between asperities, where W is the load and r is the radius of a sphere or cylinder: (a) Contact between a spherical asperity and a flat surface where a is the radius of a circular contact area; (b) contact between a cylindrical asperity and a flat surface where 2 b is the contact width and / is the length of the cylinder.
13.3.1 Elastic Contact If the deformation at the contact area under the load W stays within the elastic limit and the semi-circular distribution of the normal contact stress p is as shown in Fig. 13-5. a, b, p and the mean contact stress pm are given theoretically as follows (Hertz, 1882) in the contact of a sphere with a flat surface, (13-1) ) (13-2)
fl6E*W\113 A 9n3r2
2a or 2b
(13-3)
in the contact of a cylinder with a flat surface, 1/2
(13-4) (13-5) Pm =
16rJ
(13-6)
where p0 is the maximum contact stress and £* is the equivalent Young's modulus defined by (£*)- 1 = [(l-vf)/£ 1 ] + [(1 — vl)/E2]. Under the cylindrical contact of Eqs. (13-4) to (13-6), the stresses ax, ay and Txy, defined in Fig. 13-5, are distributed in the subsurface below the interface
Figure 13-5. The semi-circular distribution of the normal contact stress p and definition of the stresses o-x, <7y and xxy in the subsurface below the interface.
644
13 Friction and Wear 0
-5
0.2p 0
0.4p 0
0.6p 0
°x
T 1
—I /
L
7
0.8p 0
/
7 /
p0
T
1-
/
V
/i /
r/r/
as shown in Figs. 13-6 a and b (Merwin and Johnson, 1963). ax is the compressive stress in the direction of the x-axis along the plane at x = 0 in Fig. 13-6 a. oy is the compressive stress in the direction of the y-axis along the plane at x = 0 in Fig. 13-6 a and y = — b in Fig. 13-6b. rxy is the shear stress along the plane at y = —bin Fig. 13-6b. x in the figure is the principal shear stress and is given by ? = W K - < 7 y ) 2 + 4Txy
-^2
(13-7)
If a tangential force is applied to the contacting pair, the position of the maximum value of x moves to the direction of the contact interface and to the contact periphery (Poritsky and Schnectaday, 1950).
y
(a) T
0.2p0
13.3.2 Elasto-Plastic Contact
-0.2p 0 -0.4p 0 -
r>7^ / V \
-0.6p 0 1
-2b
1
y
2b
-b (b)
po//c=5.56
-3d (c)
If the load for cylindrical contact exceeds a certain limit, the plastic yield starts locally at the point where the yield criterion is satisfied. Following the yield criterion of Tresca (1864), local yield starts at (0, — 0.786 ft), as shown in Fig. 13-6 a on x-y diagram, where x is the maximum value for yield. This state is generated when p0 = 3.3 k (k is the critical shear strength) and the yield region expands as p0 is increased further. When p0 becomes larger than 4/c, xxy reaches k at (+ 0.876, — 0.5 b) and plastic flow starts in the direction of x. Figure 13-6c shows the circular plastic deformation region and the shadowed region where plastic strain is accu-
Figure 13-6. Stress distributions in the contact region between a cylinder and a flat surface (Merwin and Johnson, 1963). (a) Variation of elastic stresses with depth along the axis of symmetry, x = 0; (b) variation of elastic stresses in a cross-section at y = — b; (c) the circular plastic deformation region in static contact and the shadowed region where plastic strain is accumulated by the stress cycle.
13.3 Mechanism of Contact Between Solid Surfaces
mulated by the stress cycle (Merwin and Johnson, 1963). If a tangential force is applied in addition to the normal load, the local yield region expands further depending on the magnitude of the force and it moves forward and upward as shown in Fig. 13-7 (Jahanmir and Suh, 1977), where the friction coefficient JLL is changed and the tangential stress at the interface is given by q0 = JLL - p0 under a constant contact stress °f Po (= 4 k). When JJL = 0.5 the plastic deformation region appears at the surface and covers the whole contact area. The surroundings of the plastic deformation region are in a state of elastic deformation. Therefore the contact situation is elastoplastic. 13.3.3 Plastic Contact When the plastically deformed region is well developed under a large load or frictional force and it covers the whole contact area, the state of contact is fully plastic and is called plastic contact with general yield. The slip-line field theory for a rigid, perfectly plastic solid can be applied for the theoretical analysis of the contact stress in this case. For the two-dimensional model
Figure 13-8. The two-dimensional contact model between asperities whose plastic deformation region is obtained by using the slip line field theory (Green, 1954).
of plastic contact given in Fig. 13-8, the normal contact stress p and the shear stress s are given theoretically as follows (Green, 1954); p = k[l + § + 2(8 - 6) + sin2a]
(13-8)
s=kcos2s
(13-9)
where k is the critical shear strength (flow stress) of the material, e is the angle of slipline and 9 is the slope of the plastic surface. If the contact is between a rigid punch and a rigid, perfectly plastic flat surface, as shown in Fig. 13-9, the normal contact pressure p is given as an extreme situation of Fig. 13-8 (Prandtl, 1920);
= 2fc(l+i)
(13-10)
Equation (13-10) explains well the experimental relationship between hardness value Hv and the yield stress Y, as follows (Tabor, 1951); H ~3Y
Figure 13-7. Plastic deformation regions for different friction coefficients. In the case of /i = 0.5, the plastic deformation region appears at the surface and covers the whole contact area (Jahanmir and Suh, 1977).
645
(13-11)
Figure 13-9. The plastic deformation region of contact between a rigid punch and a rigid, perfectly plastic flat surface (Prandtl, 1920).
646
13 Friction and Wear W
where Y = 2 k by the yield criterion of Tresca and Y =
13.4 Friction 13.4.1 Microscopic Model of Friction Even in the case of a mirror surface, which gives us an accurate reflection of an image, the surface is rough microscopically, as shown in Fig. 13-1. A schematic model of the contact between two flat surfaces is shown in Fig. 13-10, where the number of microscopic contact points is n for the apparent contact area Aa under a normal load W. If we introduce the microscopic contact area At for the ith contact point between asperities, which is called a junction, the total real contact area Ar is given by the following equation, (13-12)
= Z i= 0
From the two-dimensional junction model shown in Fig. 13-11, the total load W and the total friction force F are given as follows, (13-13)
W= £ Wt= Z {AiPiCosOi-Ai i= O
i=0
i= 0
By supposing all contact points have the same values of At and p{, and by introducing the local friction coefficient \i{, defined by nt = s^, Eq. (13-15) is simplified as follows,
(13-16) i= 0
For the quantitatively exact estimation of the friction coefficient, the distributions of all the possible values of Ai9 pt, st and 9t have to be known. The distribution of Qi is first determined by the initial microgeometry of the surface before contact. This largely depends on the method of surface finishing. It is secondarily
;
=° (13-14) where pt and st are a normal stress and a shear stress, respectively, at the interface of a junction. The friction coefficient ju (= F/W) for the apparent contact area Aa is given by Eqs. (13-13) and (13-14) as follows,
(13-15) i=0
Figure 13-10. The schematic model of contact between two rough surfaces, where Aa is an apparent contact area and At is a microscopic contact area for the ith contact point between asperities.
wj
Figure 13-11. The two-dimensional junction model, where p{ and st are a normal stress and a shear stress respectively at the interface and 6t is an asperity angle.
647
13.4 Friction
determined by the mechanism of deformation of the asperities in the sliding process. Figure 13-12 shows schematic images of the sliding processes of asperities where various types of deformations can occur. In Figs. 1 3 4 2 ^ - a 4 ) , the deformation in the contact region stays elastic and the relative slip at the contact interface forms the larger part of the relative displacement. In Figs. 1342(bi-b 6 ) and {c^c^ the deformation in the contact region is fully plastic. Figures 13-12 (bi-b 3 ) and 13-12 (c1 -c 3 ) show the deformation process for a junction without relative slip at the contact interface, while Figs. 13-12(b 4 -b 6 ) and 1312(c 4 -c 6 ) illustrate the processes where the relative slip at the contact interface forms the larger part of the relative displacement. It is obvious that 6t changes in all these deformation processes and fit at the contact interface also varies depending on the amount of shear strain at the interface. Equation (13-15) or (13-16) should be calculated considering these possibilities for an exact analysis (Kayaba and Kato, 1981). If we consider the elastic contact in Figs. 13-12 (a 1 -a 4 ), pt is given by pm of Eq. (13-3) for spherical asperities and by Eq. (13-6) for cylindrical asperities. By assuming a constant value s0 for st for the whole sliding process, the friction coefficient \ix under elastic contact for spherical asperities is given by the following equation, 3 2
°\16£* W
v
V J \ JJ \ JJ \\J J (^)
(a 2 )
(a 3 )
(b2)
(a 4 )
(b3)
(b6)
(c2)
(c3)
(C5)
(c6)
Figure 13-12. Schematic images of sliding processes of asperities where various types of deformation can occur: (a 1 )-(a 4 ) sliding in elastic contact; (bJ-O^) sliding in plastic contact where a spherical asperity does not deform; ( ^ - ( c j sliding in plastic contact where both asperities deform.
\x is about 0.12 at a load of 5 g and falls to about 0.04 at a load of 70 g, which follows the relation given by Eq. (13-17) (Bowden and Tabor, 1964). When we consider the plastic contact shown in Figs. 13-12 (b!-b 6 ) and 13-12(Ci-Cg), pt and st are given by p of
A diamond pin on a diamond flat
l/3
(13-17)
Equation (13-17) means that the friction coefficient is a function of the elastic modulus, the shape of the asperity and the normal load. Figure 13-13 shows the friction coefficient for a diamond pin on a flat diamond as a function of load. It is seen that
Load in g
Figure 13-13. Friction of a diamond stylus (radius 0.05 cm) on the {110} face of diamond in air as a function of load W (Bowden and Tabor, 1964).
648
13 Friction and Wear
Eq.(13-8) and s of Eq.(13-9). But these equations do not totally define the processes of deformation in these figures. They only give the values of pt and st in static balance. In order to simulate the process of Figs. 13-12 (b 1 -b 3 ) or Figs. 13-12(c 1 -c 3 ), a criterion must be introduced under which the deformation of a junction is continued. Since it is a rather complicated procedure of calculation (Kayaba and Kato, 1981), an empirical, simple equation for the model of Fig. 13-14 (McFarlane and Tabor, 1950) is introduced here as follows,
pf + a sf = a k2
(13-18)
where a is an experimental constant and varies from 3 to 25 (Kayaba and Kato, 1978 b). It should be noted that Eq. (13-18) only applies before the introduction of slip at the contact interface of a junction. By introducing the parameter ft (= sjk) into Eq. (13-18) the following relationship is obtained:
If the contact interface is well lubricated with oil or grease, st is much smaller than k, and ft is close to zero. If the contact surfaces are well cleaned and the contact interface has good adhesion, st is close to k of the softer material in contact and ft is close to unity. It is clear from Eq. (13-19) that ^ becomes infinite when ft becomes unity. This change in the friction coefficient from zero to infinite is caused by the plastic deformation of a junction and the resultant increase in contact area. If we introduce a contact area Ao and contact stress pi0 for the case st = 0 in Eq. (13-18), we obtain the following: Pjo
(13-20)
Pi
Since pi0- A0 = pi- At
and ni = si/pi.
Eq. (13-20) gives the following equation,
Pi
=r 3 -
2 -
1 -
Figure 13-14. Friction coefficient at the interface plotted against area of contact At expressed as a ratio of initial static area Ao. The full line is the theoretical curve calculated from Eq. (13-21) for a = 3.3. It shows the process of junctiongrowth (McFarlane and Tabor, 1950).
13.4 Friction
Figure 13-14 shows the relationship between fii and AJA0 for the case of a = 3.3. It is clearly seen from the schematic model in the figure that fit increases with an increase in the contact area at a junction. This is the fundamental mechanism which explains the large change in the friction coefficient. The process of increasing the contact area is called a "junction-growth process". By equating Eq. (13-21) to Eq. (13-19), AJA0 is given as follows, 1
1 — \ix tan 6t
given by Eq. (13-19). As the definition of ft = sjk shows, its value is determined by the shearing strength st of the interface, and st can be varied from zero to k depending on the microstructure of the interface. Figure 13-15 shows schematically the various types of microstructure possible at the interface. In Fig. 13-15 a the surfaces
Y
(13-22)
Equation (13-22) shows that the extent of junction growth is directly related to the dimensionless shearing strength ft. Junction growth due to this mechanism can take place so long as the plastic deformation continues without fracture. The fracture of a junction is controlled by the ductility of a material and the geometry of a junction in addition to the shearing strength of the interface. In practical contact between two large surfaces of multiple junctions, those functions mentioned above can be different at each junction. Only when we can assume the representative values of ft and 9t to apply at all junctions, or when the apparent contact geometry is very close to one asperity contact, can the friction coefficient of Eq. (13-16) be simplified as follows, tan 0t
649
(13-23)
13.4.2 The Change of Friction Coefficient due to the Change of Adhesion at the Contact Interface
In Eq. (13-23), the main factor which changes the value of fi drastically in ft is \i{
(e) Figure 13-15. Schematic representation of the various types of microstructure at an interface; (a) surfaces are clean without any contamination; (b) surfaces are in contact through layers containing adsorbed gases which commonly exist in air; (c) surfaces are in contact through oxide films in air; (d) surfaces are in contact through physically adsorbed layers of organic molecules such as hydrocarbons, alcohols, or fatty acids; (e) liquid film is thick enough to behave as bulk liquid. As a result the shearing strength of the film is determined by the viscous flow resistance at the interface.
650
13 Friction and Wear
are clean without any contamination, the interface is similar to the grain boundary. The shearing stength of such interface should be almost the same as that of the grain boundary in the bulk. In Fig. 13-15 b the surfaces are in contact through adsorbed gas layers which commonly exist in air. It is well known that inert gases such as argon and nitrogen are weakly bonded to the surface by physical adsorption. Oxygen, on the other hand, is strongly bonded by chemical adsorption and forms an oxide on the surface. So contact through oxide films is common in air, as shown in Fig. 13-15 c. Adsorbed gases can also exist at the interface between oxide films. In Fig. 13-15d the surfaces are in contact through physically adsorbed layers of organic molecules such as hydrocarbon, alcohol or fatty acid. The shearing strength of such molecular layers should be different from that of a liquid since they are under high contact pressure. The plane of shearing is sometimes the interface between the layer and the metal surface. In Fig. 13-15e the liquid film is thick enough to behave as a liquid. So the shearing strength of the film is determined by the viscous flow resistance at the interface, which is a function of temperature, shearing velocity and the microscopic geometry of the mating surfaces. This is a field of hydrodynamics, and material aspects are not included. Out of these 5 types of microstructure of the interface, those of Fig. 13-15 b, c and d will be discussed further. 13.4.2.1 The Effect of Adsorbed Gases and Oxides on Adhesion and Friction
The shearing strength for the type of microstructures shown in Fig. 13-15 b and c are especially hard to measure. This is be-
cause adsorbed gas layers or oxides are so thin that the shearing strength cannot be measured independently from the junction growth and their resultant rupture in the sliding process. Measurement of the adhesive force in the direction normal to the interface gives more sensitive and reliable values for the estimation of the shearing strength affected by adsorbed gases and oxides. In Fig. 13-16 surfaces were newly generated by tensile fracture under high vacuum and they were subject to repeated contact in order to measure the normal adhesive force (pull-off force) at the interface after loading. The adhesion coefficient r\ in the figure is defined as follows;
n
normal adhesive force constant compressive force
(13-24)
The horizontal axis represents the time of exposure of fractured surfaces under vacuum to a certain pressure. The amount of gas molecules which reach the surface is proportional to the value of exposure in Pa • s units, and the brief calculation indicates that a monolayer of gas molecules can be formed on the surface after an exposure of about l(T 4 Pa-s(Gilbreath, 1967). Therefore Fig. 13-16 shows that strong adhesion can take place at the contact interface when the surface has no adsorbed gas molecules, and the adhesive strength is almost the same as the fracture strength of the bulk material. It is evident from the figure, on the other hand, that inert gases such as argon and nitrogen do not reduce adhesion, even if the surface is covered with several layers of molecules. Oxygen, on the other hand, has a strong effect on the reduction of adhesion. A monolayer seems to be enough to reduce the adhesion coefficient from about 0.8 to
13.4 Friction
651
Figure 13-16. The effect of exposure of clean surfaces to various gaseous environments on the adhesion coefficient of OFHC copper (Gilbreath, 1967). The temperature during the exposure was T = 25 °C, the symbols are: o vacuum, o air, n O 2 , (D C 2 H 4 , s H 2 , v N 2 , and A Ar. Exposure in P a s
a value below 0.1. This suggests that an oxide film reduces the adhesive strength effectively, and the adhesive coefficient becomes much smaller than 0.1 and can be assumed to be about zero. Although Fig. 13-16 shows the change in the adhesive strength at the interface caused by the adsorption of gases on surfaces, the shearing strength at the interface is also expected to behave similarly under the effect of gas adsorption on the surface. It is commonly understood that the shearing strength of a material is proportional to the fracture strength and about half as large.
In Fig. 13-17 (Bowden and Hughes, 1939), frictional specimens of copper were heated to bright red heat in a vacuum of 10~ 3 -40~ 4 Pa and adsorbed gases and organic chemicals were removed. After this heating process, the friction coefficient increased from 0.5 to about 5. Then oxygen was introduced into the vacuum chamber and this caused an immediate decrease in the friction coefficient which decreased below 0.5 again after 3 minutes exposure. Pure hydrogen or pure nitrogen had little effect on friction in similar experiments. These observations correspond well with those in Fig. 13-16.
6 5
14 o
1o 3
s
2
t
1
Admit oxgen
Degas cylinder and foil
Stand IG hours
Form a visible film s foil
0 Time
Figure 13-17. The effect of removing the adsorbed film of oxygen and other contaminations from metal surfaces on fi and of deliberately adding a trace of oxygen to clean outgassed metal (Bowden and Hughes, 1939).
652
13 Friction and Wear
This means that adsorbed oxygen effectively reduces the shearing strength at the interface, and as a result the friction coefficient changes drastically. In Fig. 13-18 (Kayaba and Kato, 1978 a), the frictional change observed under vacuum is related to the change in the electrical contact resistance Re. Re is related to the contact radius a and the film resistance (p by the following equation (Holm, 1946; Bowden and Hughes, 1939),
V vV 15 -
o 10 --
n
i
IT
Cu pin on Cu cylinder in 10"7 Pa
it 10 -
V —01
1
1
2
3
D
Au-Au Cu-Cu
o n
where I is the electrical resistivity. The contact radius a is changed by Eq. (13-1) and the film resistance cp is changed by the thickness of adsorbed gas film. The plot in Fig. 13-18 shows observed experimental values and the solid line shows the theoretical relations between \i and Re calculated from Eqs. (13-21) and (13-25) for the case of cp = 0 and a = 3.0. Therefore, the deviation in the experimental values of Re from the solid line at the same friction coefficient corresponds to the film resistance at the interface.
15
• D
(13-25)
Ya na2
V
a
5 i- n
X
V
|
4
i
5
i
6
i
7
i
8
9
10
4
Electrical contact resistance Re in 10~ Q
Figure 13-18. Plot of the friction coefficient of copper against the electrical contact resistance (Kayaba and Kato, 1978 a).
A
A
o o
n
A
A I
I
5
10
Friction coefficient [i
Figure 13-19. Plot of the film resistance of metals against the coefficient of friction (Kayaba and Kato, 1978 a). The solid lines are the average values of film resistance and the broken lines are the values obtained by Holm (1946). The symbols are: o gold, A copper, • nickel, and v iron.
Figure 13-19 shows the relationship between the film resistance and the friction coefficient obtained in this way between similar metals (Kayaba and Kato, 1978 a). Broken lines are values obtained by Holm (1946) under static contacts of gold and copper. It is seen that the values of film resistance for these four metals are in a reasonable order and the film resistance of gold or copper in sliding is about half of that observed in static contact. The film resistance observed can be related to the film thickness at the interface by the theoretical equation of Holm (1946) for the electrical contact of tunneling effect. Table 13-1 shows the average values of film thickness calculated in this way from Fig. 13-19 (Kayaba and Kato, 1978a).
653
13.4 Friction
This indicates that there are still films of about 0.43-0.53 nm thickness when ^ = 1 — 15 and the change of friction is caused by a change in the film thickness of the order of 0.1 nm, which is beyond the exact resolution of the electrical contact method. It is evident from the results of Figs. 13-16, 13-17, 13-18 and Table 13-1 that a few layers of adsorbed gases, especially oxygen, change the shearing strength of the interface and as a result change the friction coefficient drastically from about 0.5 to over 15. 13.4.2.2 The Effect of Adsorbed Organic Molecular Layers on Adhesion and Friction
For the reason discussed in the last section, the adhesion coefficient gives clear insight into the effect of adsorbed organic molecular layers on the shearing strength st at the interface. Figure 13-20 shows the adhesion coefficient of a clean steel ball on an indium surface covered with a monolayer of lauric acid (Bowden and Tabor, 1950). A is the percentage by which the curved area of the indentation exceeds the original plane surface area. If the monolayer is stretched by less than about 2 percent the adhesion is negligible. If its area is increased by more than this there is appreciable breakdown of the monolayer and marked adhesion is ob-
Table 13-1. Film resistance and film thickness by Kayaba and Kato (1978 a).
Au Cu Ni Fe
Work function ineV
Film resistance cp in Q cm2
Film thickness in nm
4.31 4.56 4.54 4.50
2.41 x l ( T 9 1.97 x l ( T 9 7.26 x l O " 9 1.53 x l O " 8
0.46 0.43 0.50 0.53
1.5 Clean surfaces
25
Figure 13-20. Adhesion of a clean steel ball on an indium surface covered with a monolayer of lauric acid. The coefficient of adhesion is plotted against A (Bowden and Tabor, 1950).
served. In contrast to this result, the clean surface of a steel ball and an indium surface give an adhesion coefficient of about 1.25. This means that a monolayer of lauric acid is enough to prevent adhesion at the metallic contact interface, but a small breakdown in the monolayer causes a rapid increase in adhesion. Such a sensitive relationship between the adhesion coefficient and the breakdown of a monolayer is important, as was apparent from the phenomenon of junction growth, described by Eq. (13-22), where a large expansion of the original contact area is clearly shown. Figure 13-21 shows the effect of breakdown and wear of molecular layers (films) on the friction coefficient after repeated sliding cycles on stainless steel surfaces covered with stearic acid molecular layers (Bowden and Tabor, 1950). The figure shows that the friction coefficient increases rapidly with the breakdown of films, except in the case of 53 films. The increase in shearing strength at the interface and the resultant junction growth is the cause of this breakdown. The local
654
13 Friction and Wear
10
20 30 40 Number of sliding cycles
Figure 13-21. Wear of stearic acid films deposited on the lower surface of stainless steel. A single molecular layer gives the same reduction in friction as a thick film, but it is worn away much more rapidly (Bowden and Tabor, 1950).
metallic adhesion occurs at the place where the films have broken down. On the other hand, the friction coefficient for the first cycle of friction is almost constant (/i ~ 0.1) for any number of films. This means that a film is not broken down during the first cycle of friction and the friction coefficient of about 0.1 corresponds to the shearing strength of the film itself at the interface. The shearing strength of a film is determined by the bonding energy for physical or chemical adsorption of a molecule, the film thickness and the number of molecules per unit area. The theoretical analysis gives the following equation for the bonding strength sf of a film (Sakurai, 1958) -ad
sf = m N -d A
(13-26)
where Ead is the adsorption energy, nm the number of molecules per unit area, d the film thickness, NA the Avogadro constant, and m" is an experimental constant. The observation of Fig. 13-21 and Eq. (13-26) means that an adsorbed film behaves as a solid on a metal surface and
results in low friction. This state of lubrication is called boundary lubrication. The temperature range is especially important if the stable solid film is to be kept in boundary lubrication. The combination of reactive metal surfaces and reactive lubricants which form stable solid films is important from the viewpoint of lubrication. When the film is thick enough to behave as a liquid at the interface, as shown in Fig. 13-15 e, it causes viscous hydrodynamic flow resistance which is mainly decided by the shape of surfaces in contact. If the shape is well designed for the smooth flow of a fluid and gives good load support then the friction coefficient is between 0.01 and 0.001, depending on the viscosity of the fluid (Cameron, 1966; Dowson and Higginson, 1966). This state of lubrication is called hydrodynamic lubrication. Although the conditions of Fig. 13-15d and e are usual in practical friction, the mechanism of friction will not be described in detail since it depends mainly on the hydrodynamics of the fluid at the interface for the situation of Fig. 13-15e, and on the chemistry of lubricants under high contact pressure for Fig. 13-15d. This lies beyond the scope of this Volume.
13.5 Wear Wear is the successive removal of surface material by repeated friction and is mainly caused by microscopic mechanical fracture. Even when the surface has some chemical reaction products, such as oxides, the volume loss from a surface occurs mechanically in many cases. The microscopic mechanical wear mechanism is described below from this viewpoint.
13.5 Wear
13.5,1 Wear Under Elastic Contact: Fatigue Wear
If the material is ideally homogeneous, there is no reason for the region of plastic yield under elastic contact, as described before in Sec. 13.3. But a real material has various kinds of inhomogeneity. A single crystal has slip planes for preferential sliding under shear stress. A polycrystal has grain boundaries and inclusions, and all crystalline materials have vacancies. Because of these inhomogeneities, the local stress in the contact region exceeds the yield stress of the material even when the
655
theoretical stress for the homogeneous material does not exceed the yield stress. Figure 13-22 shows the cross-section across the wear track of an MgO single crystal after rolling contact with an AISI 52100 steel ball (Dufrane and Glaeser, 1976). The maximum Hertzian pressure is 148 MPa for Fig. 13-22 a which is small enough to keep elastic contact if the material is homogeneous. Nevertheless, dislocations are introduced along slip planes by rolling in the [100] direction and are observed by etch pits of dislocations. In Figs. 13-22b and c, the maximum Hertzian pressure is 111.2 MPa, and
Track width
45° slip plane: parallel to the ball track
45° slip planes normal to the ball track
Figure 13-22. Dislocation etch-pit pattern on (100) plane of MgO in the subsurface after rolling contact of an AISI 52100 steel ball (Dufrane and Glaeser, 1976). (a) After one rolling-contact cycle (570 g load; [100] rolling direction); (b) after one rolling-contact cycle (244 g load; [110] rolling direction); (c) after 10 rolling-contact cycles (244 g load; [110] rolling direction).
656
13 Friction and Wear
Fig. 13-22b shows that dislocations are again introduced by one roll in the [110] direction, and the dislocation density increases with repeated rolling on the same track, as shown in Fig. 13-22c, after 106 passes. At this stage, cracks are being introduced on the surface and the surface layer becomes delaminated. Figure 13-23 shows inclusions of A12O3 and MnS in a steel and cracks propagating from them after repeated rolling-sliding contacts (Fegredo and Pritchard, 1978). The maximum Hertzian pressure is 870 MPa, and cracks are generated at about the maximum principal shear stress in theoretical elastic contact. The observations in Figs. 13-22 and 13-23 show that dislocations and cracks can be generated in the subsurface even when the contact load is small enough to theoretically keep elastic contact for a homogeneous material. Pitting or delamination is finally observed on the surface after repeated friction cycles. Therefore the mechanism of wear is fatigue and this type of wear is called "fatigue wear".
13.5.2 Wear Under Elasto-Plastic ContactFatigue Wear Under a certain amount of contact load, there is a state where a plastically deformed region appears beneath the interface without reaching the surface, in Figs. 13-24 and 13-25. Figure 13-24 a shows the cross-section of a copper disk along the contact track, where the plastically deformed region is observed as a dark area around 0.24 mm beneath the surface (Hamilton, 1963). The direction of the plastic flow is seen clearly in the figure and the surface layer
U.D
f
0.5 0.4 /
0.3 /
0.2 0.1
/ / 5000
(b)
Figure 13-23. Typical subsurface cracks after rolling contact (slippage 3%, p0 = 870 MPa, fi varies between 0.52 and 0.56). Cracks are started from the inclusions of A12O3 and MnS (Fegredo and Pritchard, 1978).
10000
Number of cycles
Figure 13-24. Plastic flow beneath the contact surface of a copper disk after rolling contact with a steel disk (Hamilton, 1963). (a) Photograph of plastic flow layer (po = 678.4 MPa); (b) forward flow shown against number of stress cycles.
13.5 Wear
400
657
Slippage 0% On driving surface Cycles:
350
2.0x106 Initiation of pit
CO CO
Figure 13-25. The change in the hardness distribution beneath the surface during the repeated rolling of steel (0.45 C, 0.27 Si, 0.85 Mn) (Kayaba and Suzuki, 1976). 500
1000
Distance from surface in urn
remains without plastic deformation. Figure 13-24b shows the displacement of the surface layer observed by the optical microscope in the direction of friction. It was caused by such plastic flow in the subsurface layer as shown in Fig. 13-24 a. The displacement increases linearly with the number of rotational friction cycles. This plastic flow in the subsurface layer and the displacement of the surface layer of copper was caused by repeated rolling contact under a maximum contact pressure of 678.4 MPa. In this process of plastic flow, work hardening occurs in the flow region. This is shown in Fig. 13-25, where the hardness peak is located about 130 jim beneath the surface, and the value of maximum hard-
ness increases with an increase in the number of rolling cycles (Kayaba and Suzuki, 1976). The maximum hardness value of about 400 is reached after about 2 x 106 cycles, at which point pits starts to appear on the surface. As shown in Figs. 13-22, 13-23, 13-24 and 13-25, repeated friction under elastic contact or elasto-plastic contact causes the accumulation of local plastic deformation around some stress concentration points, and cracks are generated and propagated after a certain number of frictional repetitions. The mechanism of crack generation and propagation in such a situation is that of fatigue fracture. Therefore wear caused by such a mechanism is called "fatigue wear", since fatigue is a kind of rate process
658
13 Friction and Wear
controlled by the inhomogeneity of the microstructure of a material. Fatigue wear is treated as a process of probability, and the relationship between the life by wear and the contact pressure is similar to the S-N curve for fatigue fracture. 13.5.3 Wear Under Plastic Contact: Adhesive Wear, Abrasive Wear, Flow Wear
When the contact pressure is large enough to introduce general yield at the contact interface, where the plastically deformed region covers the whole area of contact, the shape of the contact surfaces changes gradually during the process of sliding. Depending on the initial contact conditions, such as the shape of the surfaces, the hardness of the two surfaces and the friction coefficient at the interface, six representative types of wear process could occur, as shown in Fig. 13-26.
S
(b,)
(a2)
(a 3 )
(b 2 )
(b 3 )
(c 2 )
(c 3 )
(f2)
(f3)
S
s H
^
Figure 13-26. Wear types possible in plastic contact and sliding.
In Fig. 13-26, the letter H is given to the harder side and the letter S is given to the softer side. When both surfaces deform as shown in Fig. 13-26(a1), the contact interfaces start to wave, as shown in Fig. 13-26 (a2), and a crack propagates from the edge of the contact interface into the subsurface, and part of the surface is removed by the mating surface (Cocks, 1962), as shown in Fig. 13-26 (a3). Figure 13-27 a shows the experimental model corresponding to Fig. 13-26(a3). This can happen when the adhesion at the contact interface is strong enough to support the shearing force for crack initiation and propagation. Cracks sometimes propagate on both sides of the contact interface and a wear particle rolls up between the two sliding surfaces (Greenwood and Tabor, 1957). When a soft asperity is in contact with a flat surface of the same hardness or harder, as shown in Fig. 13-26^), and the two surfaces adhere strongly, a crack is introduced from the front edge of the contact interface into the soft asperity by relative movement of the mating surfaces, as shown in Fig. 13-26 (b2), and a thin, flaky wear particle is generated by adhesive transfer from the asperity to the flat surface, as shown in Fig. 13-26 (b3). Figure 13-27 b shows an SEM observation corresponding to this. A wear particle of this type is called a shear tongue (Kayaba and Kato, 1981). If there is a small difference in the geometry of a soft asperity, a crack is generated at the trailing edge of the contact interface and propagates into the asperity, as shown in Figs. 13-26^) and (c2). In Fig. 13-26(c3), the crack reaches the front surface of the asperity, and a thick, large wear particle transfers from the asperity to the flat surface. Figure 13-27c shows an SEM observation corresponding to this. The adhesion at the contact interface must be strong enough to support the shearing
13.5 Wear
3 mm
50 pm
lupin
659
IUU prrl
50 pm
Figure 13-27. Typical SEM images of plastic contact and sliding, (a) A crack is generated at the ends of the junction between lead pairs (Greenwood and Tabor, 1957; optical photo); (b) flaky two wear particles are transferred from the lower surface to the upper surface between two specimens of austenitic stainless steel (Kayaba and Kato, 1981); (c) block wear particle is transferred from the lower surface to the upper surface between two specimens of austenitic stainless steel (Kayaba and Kato, 1981); (d) cutting wear particle is generated from brass surface continuously (Hokkirigawa and Kato, 1988); (e) wedge is formed out of brass surface and stays at the head of asperity without growing (Hokkirigawa and Kato, 1988); (f) surface flow layer expands in the direction of sliding and forms a protrusion of thin film which is going to be detached from the surface of low carbon steel (0.45% C) (Kato, private).
force for crack initiation and propagation (Kayaba and Kato, 1981). Clearly adhesion at the interface is essential for the wear modes shown in Figs. 13-26 (a3), (b3) and (c3), and these wear modes are called "adhesive wear". In Figs. 13-26 (di), ( e j and (f^, on the other hand, a hard spherical asperity is in contact with a soft, flat surface which can be easily deformed plastically. Under such a contact situation, the soft flat surface is grooved by the hard, spherical asperity during sliding, and a ribbonlike, long cutting chip is generated as a wear particle by the mechanism of micro-cutting, as shown in Figs. 13-26 (d±)9
(d2) and (d3). Figure 13-27 (d) shows an SEM observation corresponding to Fig. 13-26(d3). Easy flow of the chip against the surface of the spherical, hard asperity is necessary for this wear mode, and adhesion at the contact interface is not necessary. Deep penetration of the hard asperity into the flat surface is necessary for micro-cutting to occur. When the indentation of the asperity is not deep enough and the friction at the contact interface is rather high, micro-cutting cannot take place and a wedge is formed at the head of the asperity, as shown in Fig. 13-26 (ej, (e2) and (e3) where a crack is generated at the bottom of the
660
13 Friction and Wear
contact interface and is propagated until it reaches the surface in front of the plastically deformed region. Figure 13-27e shows an SEM observation corresponding to Fig. 13-26(e3), which is called a wedge (Hokkirigawa and Kato, 1988). Sliding takes place at the fracture surface of the wedge after its formation until the wedge is removed from the asperity head by some interruptions, such as the irregularities of microstructure or geometry. This wear mode of wedge formation is somewhat similar to the mode of Fig. 13-26(c3). The adhesion at the contact interface helps the formation of a wedge, but it is not essentially necessary. The inclining angle of the contact interface and the depth of indentation are important for the onset of wedge formation. In this way, the geometry of the indentation is essentially important for the wear modes of Figs. 13-26(d3), (e3) and the wear due to these wear modes is called "abrasive wear", where adhesion at the contact interface plays a minor role. When both the indentation depth of the hard asperity and the friction coefficient are small, the plastically deformed surface layer is burnished and is not transferred or cut, as shown in Figs. 13-26(f1) and (f2). In Fig. 13-26(f3), the burnished surface layer is elongated horizontally by rubbing and the thin protrusion grows horizontally on the surface due to the plastic flow. Figure 13-27f shows such a thin protrusion observed by SEM. The thin protrusion detaches from the surface after one slide in some cases, but in many cases it grows large and long due to repeated sliding on the same track. Large amounts of plastic flow under high isostatic compressive pressure are possible in the contact region. The wear mode of Fig. 13-26(f3) is called "flow wear" (Akagaki and Kato, 1989).
In principle, all the wear modes shown in Fig. 13-27 can appear after a single slide under plastic contact. 13.5.4 Wear Mode Transition due to Repeated Sliding on the Same Track
If the contact is elastic or elasto-plastic, the accumulation of plastic strain at a local region in the subsurface is necessary for wear. Wear appears when a crack is generated and propagated to the surface through the subsurface. Under such conditions of contact, wear is suddenly observed after a certain amount of repeated sliding as fatigue wear. A wear mode transition is not a major concern in this case. However, in the case of plastic contact, the wear mode transition is a major concern in the repeated sliding process since the contact situation changes from plastic to elasto-plastic, and then to elastic, for the following reasons: (1) Firstly, an accumulation of plastic strain in the contact region induces work hardening of a material, as shown in Fig. 13-28 (Kitsunai et al., 1990) where the hardness is measured on the same wear track during the process of repeated sliding. It is obvious from the figure that the hardness can be increased to twice the value of the initial hardness. Since the hardness is proportional to the yield stress, the yield stress of the surface layer of the track should be almost twice the initial value. This means that the contact situation under the same load tends to change from plastic to elasto-plastic or elastic. (2) Secondly, wear under plastic contact changes the shape of the asperity tip and the wear groove. It generally decreases the contact pressure as a result. This a continuous decrease in the con-
661
13.5 Wear
900 800 700 600 500 -
1
2
3
4
5
6
7
8
10
Number of sliding cycles
plastic deformation during abrasive contact. The wear mode shown in Fig. 13-26(f3), which is called flow wear, is the result of ploughing. This flow wear is one of the mildest wear states obtained by plastic contact, for which repeated sliding is an important controlling factor. Figure 13-30 shows the effect of repeated sliding on the growth of a flow wear particle of steel under lubrication. It is seen that a thin surface layer protrudes in the direction of sliding, and grows as the number of sliding cycles increases from Fig. 13-30 a to d (Akagaki and Kato, 1985). Figure 13-31 shows the amount of flow of the surface layer as a function of the sliding cycles under the same sliding Condi-
Figure 13-28. The increase in hardness on the wear track due to repeated sliding (Kitsunai et al., 1990). The symbols are: o 0.2 N load, n 0.5 N load, and A 1.0 N load.
tact pressure during repeated sliding causes a transition in the contact situation from plastic to elasto-plastic or elastic. (3) Thirdly, repeated sliding on the same track accelerates the chemical reaction on the wear track. The chemical products, such as oxides, generally reduce the friction at the contact interface. This reduction in friction can also help the transition from plastic contact to elasto-plastic or elastic contact. Figure 13-29 shows an example of the wear mode transition for the stainless steel within the region of plastic contact. As a result of sliding a WC spherical pin against a stainless steel disk, the wear mode changes from cutting to wedge forming, then to shear tongue forming and finally to ploughing (Kitsunai et al., 1990). Ploughing is not exactly a description of the wear mode but it describes the state of
WC pin on SUS304 disk
Cutting
A A A
•
A A A A AX) C € € €
O O O O O O O O O O 1
2
3
4
5
6
7
8
9
10
Number of sliding cycles
Figure 13-29. Wear mode transition within the region of plastic contact for a WC spherical pin against stainless steel disk (Kitsunai et al., 1990). The wear mode changes from cutting to wedge forming, then to shear tongue forming and finally to ploughing.
662
13 Friction and Wear
^••:.y-M
^H|;U.::r;.<
:
'--•'••/'A::'?..:
^
-iff
'•&s
Figure 13-30. Flow wear processes for steel during repeated sliding (Akagaki and Kato, 1985).
•'-£.:•
7.5 pm
tions (Akagaki and Kato, 1988). The flow rate observed is a few micrometers per 104 cycles of sliding and it changes depending on the sliding velocity and the contact pressure. Figure 13-32 shows the
£
20 -
2
4
6
8
Number of sliding cycles in multiples of 10 4
Figure 13-31. Amount of flow of the surface layer during repeated sliding (Akagaki and Kato, 1988). The observed flow rates are: 1.0 urn per 104 cycles (upper curve) and 0.5 urn per 104 cycles (lower curve). The contact pressures are 9.8 MPa (o) and 26.5 MPa (A).
resultant filmy wear particles which look transparent in a SEM (Akagaki and Kato, 1987). Even when the friction is high, as in dry contact, similar wear process can occur, although the filmy wear particle cannot grow as large as in lubrication. In such a state of contact, the elastoplastic part of the contact can be on a larger scale than that of the plastic contact, and a crack propagates beneath the surface by the fatigue mechanism. The wear particle should be larger than in plastic contact from this viewpoint. Figure 13-33 shows a delaminating, flaky wear particle of steel which was observed under relatively high friction Gu~0.5) (Jahanmir etal, 1974). The microgeometry of the surface of the delaminating particle suggests that microscopic adhesive and abrasive wear took place and
13.5 Wear
663
that better partial conformity was attained by the delamination of a flake on a much larger scale than in plastic contact. 13.5.5 Wear Models and Wear Coefficients 13.5.5.1 Abrasive Wear
In common macroscopic sliding situations between flat or curved surfaces, parallel grooves or scratches are always observed on the wear surface in the direction of sliding, as in the case of Figs. 13-1 a and b. This is typically the case when a rough, hard surface slides on a soft, flat surface, but it is even observed in the case of conFigure 13-33. A delaminating flaky wear particle of steel after a certain amount of repeated sliding (Jahanmir et al., 1974).
10 pm
Figure 13-32. Filmy flow wear particles of steel (Akagaki and Kato, 1987).
tact between smooth surfaces of similar materials. Figure 13-34 shows the cross-sectional profiles of two mating surfaces of 0.45% carbon steel after a sliding wear test (Chiou and Kato, 1987). It is obvious from the figure that the peak and the valley coincide well for the mating surfaces. This was caused by the microscopic hardness difference on the wear surface, which is shown in Fig. 13-35. The hardness distribution in the figure shows that the maximum hardness is almost twice the minimum hardness. As a result, the contact geometry in the steady state becomes abrasive in many cases. Thus abrasive wear is common and therefore important. Various modes of wear can occur under such abrasive sliding, as shown in Fig. 13-26, depending on the microscopic shape, hardness of the mating surfaces and the shearing strength at the interface.
664
13 Friction and Wear
Figure 13-36 shows how the modes of cutting, wedge forming and ploughing are related to the degree of penetration Dp and the dimensionless shearing strength / at the interface, where / is defined by shearing strength at the interface shearing strength of grooved material (13-27)
/=
and D p is defined by the following equation for the abrasive model with a spherical pin, as shown in Fig. 13-37, L)
=
depth of penetration h — radius of contact area a
,1Q~«x (lJ-Zo)
= 1/2
Figure 13-34. The cross-sectional profiles of two mating surfaces of 0.45% carbon steel after a sliding wear test (Chiou and Kato, 1987). The peak and the valley coincide well with the mating surfaces.
Yw
r2-l
The possible wear mode can be estimated from Fig. 13-36 if values of W, r, Hv
Ploughing
Wedge
Cutting
o A
3 A
• A
•
a
•
0.4
0.6
0.8
Brass S45C SUS 304 1.0
0.8
0.6
Cutting
20
30
40
50
60
Sliding distance in m
Figure 13-35. The relationship between sliding distance and microscopic hardness (Chiou and Kato, 1987). Microscopic hardness increases with sliding distance. The contact load is p = 21.09 N, the sliding velocity is v = 0.20 m/s, and the symbols are: o pin, A ring.
0.2
Shearing strength at the interface f Figure 13-36. Wear mode diagram for abrasive wear (Hokkirigawa and Kato, 1988). Abrasive wear mode is a function of the degree of penetration Dp and dimensionless shearing strength at the interface /
13.5 Wear
Figure 13-37. Model of contact between a hemisphere and a flat surface during sliding (Hokkirigawa and Kato, 1988). Degree of penetration Dp = h/a where a is the radius of contact and h is the depth of the groove.
and / are given (Hokkirigawa and Kato, 1988). The exact boundaries between these three wear modes can be calculated theoretically using slip-line field theory if the contact geometry is assumed to be twodimensional (Challen and Oxley, 1979). Therefore, if the wear volume from the groove is known for the modes of cutting, wedge forming and ploughing, the total wear volume can be estimated as the sum of the wear volumes at all contact points. But this mechanism is rather complicated. In general only a portion of the groove volume is removed as a wear particle (Stroud and Wilman, 1962) and the re-
665
mainder of the groove volume is plastically displaced to the sides of the grooves, as shown schematically in Fig. 13-38, where Av is the cross-sectional area of the wear groove and (A1-\- A2) are the cross-sectional areas of the side ridges formed by plastic deformation and displacement of the material (Zum Gahr, 1987). In this case, the volume fraction / a b of wear in the groove is given by Ab "~
Ay-(Ai
(13-29)
/ a b defined in this way is expressed by the following equation after certain assumptions.
/ab = 1 - expT- ^ n ( ^ - ) l
(13"3°)
where £s is the deformation due to a penetrating abrasive particle at the surface level, £lim is the deformation due to a penetrating abrasive particle at the level below which ploughing dominates, and /? a factor which depends mainly on the work-hardening behavior of the wearing material. The wear volume V after sliding a distance L is given by =
fab-Av-L
(13-31)
Figure 13-38. Theoretical model for calculating the ratio of micro-cutting to micro-ploughing by / a b values and the cross-section through the groove (Zum Gahr, 1987).
666
13 Friction and Wear
When we consider the case of rg = 0 and jd = 0.2-0.5, which is the typical value for unlubricated abrasive friction, Eq. (13-31) can be approximated as follows,
If Av is calculated from the geometrical consideration in Fig. 13-38, Eq. (13-32) is given as follows (Zum Gahr, 1987), V L
w
'1
;v
(13-32)
where fx is the friction coefficient between the abrasive particle and the groove surface, Hdef is the hardness value of the wearing material in the deformed state after the abrasive action, rg is the tip radius of the abrasive particle and 9g the half tip angle of the abrasive particle. Equation (13-32) shows the effects of representative functions such as load, hardness, geometry, friction coefficient and work hardening on abrasive wear, and the theoretical estimations obtained using Eq. (13-31) agree well with experimental values for common metals and alloys, as shown in Fig. 13-39 (Zum Gahr, 1987). The inverse value of the wear rate V/L in the figure represents the volumetric wear resistance.
(13-33)
Equation (13-31) indicates that the abrasive wear volume V is the proportional to the normal load and the sliding distance. The proportional constant Kah is called the wear coefficient for abrasive wear and it is a kind of material constant, as shown in Eqs. (13-30) and (13-31). Table 13-2 shows some representative values of Kah (Wear Control Handbook, 1980). Table 13-2. Representative wear coefficient Kah for abrasive wear, from Wear Control Handbook (1980). File
Dry surface Lubricated
5xlO~ 2 10"
1
Abrasive Loose Coarse paper, abrasive polishing new grain 1(T2 2xl(T2
1(T 3
1(T 4
2xl(T3 2xl0"4
O Cu
1.0 -
/
• Ni A Fe Zn A Cu-10Ni
/ /
•
/
0.8 El •^
w K
0.6 -
® D y D
y/^
yx
+
ta
a
o
3 X
CD
X
A+
0.4 -
-f
y
0.2 -
• X
y
I
0.2
0.4
0.6
0.8
L/V(exp.) in 104mm"2
1.0
Cu-30Ni Cu-30Zn Cu-40Zn Cu-40Zn-2Pb Fe-38Ni-6AI Ti-8.5AI Fe-1Cu(A) Fe-1Cu(C) Fe-1Cu(D) Fe-1Cu(E) C45(A) C45(B) C45(C) C45(D) X10CrAM8 X5CrNM88 AI-1Mg-1Si(a)
Figure 13-39. Theoretical volumetric wear resistance L/V (theor.) versus experimental volumetric wear resistance L/V (exp.) measured on different materials by using a scratch diamond of attack angle 6% = 90°, rg = 8 urn and load W = 2N (Zum Gahr, 1987).
13.5 Wear
13.5.5.2 Adhesive Wear As shown schematically in Figs. 13-26 (a3), (b3) and (c3), adhesive wear can occur through various different modes and the size of the wear particle does not simply correspond to the size of the contact area. Under such a situation, the first point to consider for the wear mechanism is the possible volume from which wear particles can be generated. If a circular contact area, as shown in Fig. 13-40, has the radius a, the possible volume for wear particle generation is assumed to be 27ia 3 /3. The unit sliding distance to produce wear particles from this volume should be 2 a. w 2a
\ Figure 13-40. Schematic model for the generation of an adhesion wear particle. The possible volume for wear particle generation is assumed to be 2na3/3 through a sliding distance of 2 a.
So the wear volume AV after sliding a distance 2 a is given by AF = / a d -f7ia 3
(13-34)
where / a d represents the volume fraction of wear particle in the contact zone. In the case of n multiple contact points (junctions) between large, rough surfaces, one junction disappears by wear after sliding a certain distance. But if a new junction appears in series after the disappearance of the former one, n junction points can be assumed, as though they were always in contact during sliding. Therefore, the total wear volume Fat n junctions after the sliding of L is given as follows, V-hfn-^c?--
(13-35)
667
Since the normal contact pressure can be assumed to be the same as the hardness value Hy of the wearing material, the normal load W is given by W = nna2Hw
(13-36)
By introducing Eq. (13-35) and Kad = / ad /3 into Eq.(13-36), the wear volume V is given by W
(13-37)
Equation (13-37) shows that the adhesive wear volume is proportional to the normal load and the sliding distance. The proportional contact Kad is called the wear coefficient for adhesive wear. It is the principal value for a frictional pair in order to estimate the wear rate. The physical meaning of Kad is the wear volume fraction at the plastic contact zone, and it is strongly affected by the material properties and the geometry of a junction. Table 13-3 a, b shows the representative values of the adhesive wear coefficient observed by Archard (1953) and Hirst (1957) respectively. 13.5.5.3 Fatigue Wear In elastic or elasto-plastic contact under a light load, or after the transition from plastic contact by running-in, wear particles are generated by the initiation and propagation of cracks by the fatigue mechanism. The stress cycle for fatigue at the region of contact is generated in the sliding process by various mechanisms. (1) The microscopic irregularity of a surface profile causes a change in the normal and tangential resistances during sliding, and a cyclic stress is induced. (2) The local difference in surface contamination causes a change in the tangen-
668
13 Friction and Wear
Table 13-3 a. Representative wear coefficient Kad for adhesive wear, by Archard (1953). Unlubricated surface Combination
Cadmium on cadmium Zinc on zinc Silver on silver Copper on copper Platinum on platinum Mild steel on mild steel Cadmium on mild steel Copper on mild steel Platinum on mild steel Mild steel on copper Platinum on silver
Kad 57 530 40 110 130 150 0.3 5 5 1.7 0.3
tial resistance during sliding and a cyclic stress is induced. (3) The negative velocity dependency of the friction coefficient causes vibration and a dynamic load in the frictional system, which induces a cyclic stress. Based on these possibilities, we can assume that the unit sliding distance which is necessary to cause one stress cycle during sliding for the contact of radius a under a load W, is as shown in Fig. 13-41. By introducing an unit sliding distance 2 a! for one stress cycle, the critical distance L c to generate a fatigue wear particle and the critical number of stress cycles Nf are related as follows,
Table 13-3 b. Representative wear coefficient Kad for adhesive wear, by Hirst (1957). Combination
Wear coefficient xlCT 4
Low carbon steel on low carbon steel 60/40 brass on tool steel Teflon on tool steel 70/30 brass on tool steel Lucite on tool steel Molded Bakelite on tool steel Silver steel on tool steel Beryllium copper on tool steel Tool steel on tool steel Stellite # 1 on tool steel Ferrite stainless steel on tool steel Laminated Bakelite on tool steel Tungsten carbide on low carbon steel Polyethylene on tool steel Tungsten carbide on tungsten carbide
70 6 0.25 1.7 0.07 0.024 0.6 0.37 1.3 0.55 0.17 0.0067 0.04 0.0013 0.01
(13-38) If we introduce a probability K'f for fatigue wear as in the case of Eq. (13-29), the wear volume AF generated from the contact zone shown in Fig. 13-41 after sliding a distance L c is given by (13-39) By introducing a similar idea to that for Eq. (13-32) in adhesive wear, the total wear volume V at n junctions after sliding a distance L is given as follows, V = n-K'(-^a*-±-
(13-40)
Introducing Eq. (13-38) into Eq. (13-40) and taking the mean contact pressure as pm
Figure 13-41. Schematic model for the generation of fatigue wear particles. They are generated through sliding a distance L c .
13.5 Wear
for all contact points under a load W gives,
669
(a)
nna2 K't
a W
(b)
(13-41)
As in the cases of Eqs. (13-33) and (13-37), Eq. (13-41) is written as follows, = K{
W
(13-42)
L
Figure 13-42. (a) Schematic images of repeated rollingsliding contact, (b) schematic images of sliding contact on a rough surface.
Pm
where K{ is the wear coefficient for fatigue wear. It should be noted that Kf for fatigue wear includes the terms of N{ and a/a! in addition to K'f which is similar to Kah and In the standard fatigue test for metals, the empirical formula for the number N of cycles to failure is given by Tavernelli and Coffin (1959) as follows, =c,
m=
(13-43)
where Aep is the longitudinal plastic strain range and c is a constant related to the fracture ductility. Based on this relationship and a model for asperity contact at the interface, JVf is given by Hailing (1975) as follows, Nt =
(13-44)
where ef is the maximum plastic strain at failure after one cycle of loading, emax is the maximum strain during contact with an asperity, and m' = 2 for metals. It is obvious, on the other hand, that ajd can vary over a wide distribution depending on the geometry of the friction surfaces. In the schematic images of the rollingsliding contact, as shown in Fig. 13-42 a,
and that of rough contact, as shown in Fig. 13-42b, it is possible for ajd to have a value in the range o f l O ~ 2 - 1 0 ~ 3 i n a general case. If a flat surface is relatively smooth against a wearing asperity, aja! may have a value close to unity. In this way, the order of the value of a/a' may be estimated by knowing the exact geometrical conditions of a frictional pair. It is obvious from these considerations that the physical meaning of Kf is different from that of Kah and Kad. 13.5.5.4 Corrosive Wear When sliding takes place in a corrosive gas or liquid, chemical reaction products are formed on the surface. If these reaction products adhere strongly to the surface and behave like the bulk material, the wear mechanism should be almost the same as that for a metal. The wear coefficient of a chemical product would be introduced for its abrasive or adhesive wear. However, in many cases, chemical reaction products behave very differently from metals. The hydroxide of silicon nitride in water is very weakly bonded on the surface and it is easily removed from the surface by rubbing and it dissolves in water (Fischer
670
13 Friction and Wear
and Tomizawa, 1985; Tsunai and Enomoto, 1989). If the reaction rate is high, a lot of silicon nitride hydroxide can be formed and worn away quickly. The reaction rate determines the wear rate in this case. In the case of an oxide film of steel, it spalls off the surface at a critical thickness of about 10 j^m (Quinn, 1962). If the reaction rate is not high enough to grow an oxide film to the critical thickness within a cycle of sliding, oxidational wear of the steel may not occur. The reaction rate also determines the wear rate in this case. Therefore the real temperature at the real contact interface is important in the determination of the wear rate, because the reaction rate required to form a chemical product is strongly affected by the temperature. The oxidational wear of steel is explained in the next section from this viewpoint as a representative case of corrosive wear in metals.
(13-45)
For melt wear, the whole volume of a layer of molten metal at the interface is assumed to be ejected from the interface and the normalized wear rate, V is given by »=
Tm-To\Ho 1 T* jLmCv - T*£
-1
(13-46)
Oxidation-dominated wear, which belongs in the corrosive wear category, is split into two regimes: mild-oxidational wear and severe-oxidational wear. Figure 13-43 shows the model for mildoxidational wear, where an oxide film is thought to spall off when its thickness reaches the critical thickness. The normalized wear rate V of this model is
13.5.6 The Wear Mechanism Map for a Steel
V=
A large amount of wear data has been published in the past for unlubricated sliding wear between a steel pin and a steel disk. Lim and Ashby (1987) formed the empirical wear mechanism map using data obtained from the following four main areas:
Figure 13-44 shows the model for severe-oxidational wear, where a thick oxide film is formed on the surface and the molten part of the film (molten layers) is lost as wear particles. The normalized wear rate V of this model is given by
(a) (b) (c) (d)
1
seizure; melt-dominated wear; oxidation-dominated wear; plasticity-dominated wear.
In seizure, the whole nominal area of a pin contacts the disk (the real contact area is equal to the nominal contact area), and the normalized pressure W for seizure is given by
ox'0
exp
- RT
(14-47)
(13-48)
xox(T°*
Th)\n
For plasticity-dominated wear, adhesive wear and fatigue wear are included, and the normalized wear rate V is given by V = KW
(13-49)
13.5 Wear
\w
Asperity size = 2rn
When adhesive wear dominates at higher contact pressure,
Pin
K = 5xl(T3
(a)
mfm
D s ik /
\
/
\
Asperity
=
Pin
\ \
/
T\ T '
Thinner oxide films
5
5xl0 ~
(13-50b)
Figure 13-45 shows the wear mechanism map obtained by calculation using Eqs. (13-45), (13-46), (13-47), (13-48), (13-49), (13-50 a) and (13-50b). Contours show the constant normalized wear rate V and field boundaries of thick solid lines show the regimes of the dominating wear modes. This wear mechanism map agrees well with past experimental data for steel.
(b)
T,
(13-50 a)
and when fatigue wear (including delamination) dominates at lower contact pressure,
/
Parabolic growth of oxide due to high flash temperature 7"f
Disk /
671
Oxide film reaches a critical thickness zc and breaks off
\W
Pin (c) Disk
13.5.7 Microstructure of Subsurface Layers
Oxide film reaches a Broken off oxide layer critical thickness z c becomes wear debris and breaks off A new layer of oxide Thinner oxide begins to form on newly film exposed metallic surface
For a given pair in friction, the change in the friction coefficient is caused by the change of a and ft in Eq. (13-19), where a mainly depends on the property of work hardening of a material (Kayaba and Kato, 1981) and ft depends on the shearing strength of the interface and the bulk material. It has already been shown in Sec. 13.4.2 that adsorbed gases or organic molecules change the shearing strength sf at the interface. Another parameter which changes the value of ft is the shearing
Figure 13-43. An idealized mild-oxidational wear model, (a) The oxide films formed at asperity contacts (with a total area of Ar) grow according to a parabolic rate, (b) The critical oxide thickness zc is reached at one of the asperity contacts and the oxide layer there breaks off. (c) The broken-off oxide layer becomes a wear fragment and an oxide layer is again formed on the newly exposed metallic surface; meanwhile the oxide film on another asperity contact has reached the critical thickness zc and has broken off (Lim and Ashby, 1987). \W
Heat input: aq J/m s
Sink temperature = Tb
Oxide
The asperity is completely oxidized
Molten layers of oxides
Asperity size = 2rn
Figure 13-44. An idealized severe-oxidational wear model. The asperities are assumed to be completely oxidized and thin layers of molten oxide are formed at asperity contacts (Lim and Ashby, 1987).
672
13 Friction and Wear 10
I
I
I
Steel wear-mechanism map pin-on-disk configuration
Seizure
1(T -
1 5/ io- 3 ( io- 5 ) ; \S;JV^^^^Twear ° 10-4(10-6)^
1 0 -6
Jj\
10<^*^
Mild^ ^ y \ oxidational ] | i \ \ \ 1 > Martensite { wear ftU\JL^ formation 10 -
Delamination wear
1
°"6<^V 10-7>
nl\\i
l\\\\l 1n-e M 10 n - 8 , _ Mild to severe . l \ \ \ \ l 10 ( ) wear transition ^
U l t r s ^ W ^ 10-9 mild ^ W . wear , " ^ ^
10''
1
Severe
-
'
oxidational 99 wear 10
10 8
,
" Q fAWWV 109~|\\\\\X 1CT
|
10" V
104
Normalized velocity v
Figure 13-45. The wear mechanism map for a steel sliding pair using pin-on-disk configuration. Contours of constant normalized wear rates are superimposed on fields showing the regimes of dominance by different wear mechanisms (Lim and Ashby, 1987).
But observations such as are shown in Figs. 13-28 and 13-35 show that wear surfaces are extremely work hardened by repeated friction. This means that the hardness of a wear track increases with each frictional cycle, and that the wear mode is changed as a result. Therefore, it is important to know the microstructure of the subsurface layers and to modify theoretical solutions on the plastic deformation in subsurface layers by introducing a variable k. Therefore, the micro-mechanism of deformation in the subsurface is considered from these viewpoints on work hardening and the shearing strength. 13.5.7.1 Plastic Strain Distribution in the Subsurface After Sliding
Figure 13-46 shows the cross-section of a worn copper pin after dry sliding, where strength k of the bulk material at the interan aluminum foil is sandwiched between face. The value of k generally increases due two halves of the pin to make the displaceto work hardening with an increase in the ment field visible (Dautzenberg, 1980). plastic strain. As a result of such deformation under On the other hand, the wear mode under the wear track, the hardness distribution is repeated cycles of sliding depends on the as shown in Fig. 13-47 (Hirst and Lanamount of plastic deformation and the caster, 1961). The observed thickness of the state of the microstructure in the subsursevere deformation layer is about 120 j^m face, where the distribution of plastic strain in both Figs. 13-46 and 13-47, which is and the size of the plastic region can be considered to be comparable to the diameestimated theoretically using elastic and ter of the average contact area at a juncplastic theories, as shown in Sec. 13.5.2 and tion. in Sec. 13.5.3. The assumption of constant Figure 13-48 shows the plastic strain 3 k is generally introduced for such an analyas a function of distance from the wear sis. surface (Dautzenberg and Zaat, 1973),
Iff liilillii
mMmmmmmmm
fmm
mmsm
Figure 13-46. The displacement field on a worn copper surface (Dautzenberg, 1980).
673
13.5 Wear Hardness 160 0 I
180
200
220
I
240 I
^
26 • V
V" 0.01
I
• °°o °o
o
0.02 (
10u
B
° ^9
.22 Q
0.03
10"
(a)
25
50
75
100
Distance from surface z in urn Hardness 16C) 0 I
180 I
200
220
240 1
26(
0.01
i
Figure 13-48. Effective deformation (effective plastic strain) of copper as a function of the distance from the wear surface (Van Dijck, 1978): o light-microscope observations, • electron-microscope observations on replicas, x electron-microscope observations on thin foils.
0.02
— (b)
Figure 13-47. Hardness distribution from the surface to grain depth for 60/40 brass pairs for two different sliding speeds (Hirst and Lancaster, 1961): (a) 0.02 cm/s, (b) 65 cm/s.
where S is defined for a grain by (13-51) where D is the section through the center of an original spherical grain and Cs is the section through the center of a deformed elliptic grain. This shows that the effective plastic strain reaches high values of 3 = 100 in the worn surface. However, it has been shown that it can reach 5"- 200 (Van Dijck, 1978), as indicated on Fig. 13-48 by crosses. [A
general definition of 6 is given in Van Dijck (1978), for example.] The result of Fig. 13-48 gives the following empirical relationship between the shear strain y and the depth z (KuhlmanWilsdorf, 1981), nr 1 + z*jz
(13-52)
where y is the shear strain, ys is the shear strain at the surface, z* is the thickness of strongly sheared layer, z is the depth from the surface, and m* is an experimental constant including the effect of work hardening. 13.5.7.2 Dislocations and Cells in the Subsurface After Sliding
As well as the stress distributions of Eq. (13-51) and Eq. (13-52), which were observed through the deformation of grains, the dislocation structure is also changed significantly.
674
13 Friction and Wear
Figure 13-49 shows the dislocations observed on horizontal cross-sections of the subsurface of a mild steel after sliding 4000 m in dry conditions (Garbar and Skorinin, 1978). In the sublayer at 15 |im below the surface, the dislocations are distributed at random, as shown in Fig. 13-49 a, but they start to form cell boundaries in Fig. 13-49b at 8-10 jam below the surface. In Fig. 13-49c at 5 jim below the surface, the cells are now well formed, and they become finer and elliptic at the surface, as shown in Fig. 13-49d. Figure 13-50 shows detailed longitudinal sections through a wear surface of copper after sliding 12 m in an argon atmosphere (Rigney et al., 1986). In Fig. 13-50 a, the structure consists of equiaxed cells bounded by relatively wide walls. These walls become somewhat sharper and the cells are elongated as they come closer to the surface, as shown in Fig. 13-50b. Figure 13-50c shows the part just below the surface including the edge of the surface. There is an ultrafine-grained, mechanically mixed layer on top of the bulk, which appears as a narrow, dark band. It contains small crystallites 3-30 nm in diameter which are a mixture of small pieces of copper and the mating material (440C steel). This layer is newly formed on the bulk surface as a result of the adhesive transfer and retransfer of small pieces from both surfaces during sliding (Heilman et al., 1983). The transfer layer and the bulk material below it are separated by a sharply defined interface. There are well-defined cells or subgrains 0.2-0.4 jim wide just below the interface. These cells are elongated in the sliding direction and the substructure boundaries are sharp. It is obvious from Figs. 13-48,13-49 and 13-50 that dislocation networks are formed at a depth of more than 15 \±m below the
wear surface where effective plastic strain is relatively small, and that cells are formed closer to the surface. Near to the surface the cell is elongated and it becomes smaller and its wall becomes thinner. The mechanism of formation of an elongated cell along the direction of shear stress in the subsurface is very similar to that observed in wire drawing (Langford and Cohen, 1969). In a wire drawing experiment on 0.007% cabon steel, the mean, transverse, linear, intercept cell size d is related to the strength (flow stress) i f by nr
(13-53)
where T 0 is the friction stress on the slide plane and m** is an experimental constant. The mechanism of cell size strengthening given by Eq. (13-53) can be accounted for by assuming that dislocation sources in the cell walls are readily activated at low applied stresses with the aid of high stress concentrations, and that the observed strain hardening arises from the stress required to expand each dislocation loop across the glide plane of the cell until the loop is incorporated in, or annihilated at, the perimeter of the glide plane. Based on this assumption and a model on the work of deformation per unit cell volume to the Burgers vector b and the shear modulus G by m** = 3.0 Gb\ and Eq. (13-53) is written as follows, GV Tf = T o + 3>0
(13-54) a
where b' is the magnitude of b. In the model used for Eq. (13-54) the friction stress T0 is very small compared to the second term of d'1. This second term shows that most of the work of deformation goes into the generation of the total dislocation length necessary to produce the imposed plastic elongation of the cells. These dislocations have three possible
13.5 Wear
675
Hulk
Figure 13-49. Dislocations, cells and subgrains in horizontal sections from below the wear surface of a mild steel (Garbar and Skorinin, 1978): (a) 15 urn below the surface: (b) 8-10 jam below the surface, (c) 5 um below the surface; (d) on the surface.
(a)
(b)
(c)
Figure 13-50. Dislocations, cells and subgrains in longitudinal sections through the wear surface of copper (Rigney et al, 1986): (a) about 15 um below the wear surface; (b) about 8 um below the wear surface; (c) 0-5 um below the wear surface.
676
13 Friction and Wear
fates: (a) loss by annihilation at or in the cell walls, (b) loss by cellular growth, and (c) incorporation into the remaining cell walls (stored energy). On the other hand, it should be noted that the usual grain-size models for strain hardening give the Hall-Petch equation, (Tf = T 0 + m**/y/3) where the strength is a linear function of Jj. Dislocation initiation or source activation is considered as the critical stage in determining the strength. This difference between Eq. (13-54) and the Hall-Petch equation shows clearly the unique strain hardening mechanism for the subsurface layers deformed by friction. The cell size distribution at the section transverse to the sliding direction is primarily important for determining the strength of the subsurface and its behaviour under wear. It is important to know the effects of various frictional conditions on this cell size distribution as the next step in understanding the mechanism of microstructural change in the subsurface. Published observations on the cell size distribution at a transverse section across a wear track are very few at present. However, the cell size distribution at different levels below the surface and the effects of sliding distance, load and velocity on the cell size at a level of 0.2 pm below the surface of aluminum are shown in Fig. 13-51 (Kato et al., 1985). The friction is due to a pin on a ring sliding under the lubrication of turbine oil. The friction coefficient is around 0.1 which is much smaller than in the unlubricated cases of Figs. 13-46 and 13-50. The cell size was calculated as a mean value of the long and the short axes of an elongated elliptic cell. Figure 13-51 a shows the cell size distribution below the wear surface. The
mean value of the cell size at the surface is about 1.4 jim and the minimum is about 0.4 |im. It is not surprising that the mean cell size is much larger than those observed in Figs. 13-49 and 13-50 if we recall the difference in lubricating conditions for Figs. 13-49, 13-50 and 13-51. It is more important to notice that the minimum cell size of about 0.4 jim is very close to that oberved in Fig. 13-50, because the minimum cell size is generated at the contact point where the lubricant film is broken and metal/metal contact is formed. The cell sizes in Figs. 13-51 b, c and d were observed at about 0.2 jim below the surface. Figure 13-51 b shows that the cell size decreases almost linearly with an increase in the sliding distance. The larger the cell size, the more quickly it decreases. Once the cell size reaches a value around 0.4 |im, it almost stops decreasing, which suggests the existence of a minimum size for a cell under a given frictional condition. The effect of load on the cell size is quite remarkable, as shown in Fig. 13-51 c. An increase of the load by a factor of about four reduces the cell size by a factor of two. The minimum cell size observed is about 0.16 (im. The sliding velocity also affects the cell size significantly, as shown in Fig. 13-51 d. Increasing the sliding velocity decreases the cell size. The mechanism is rather complicated since a change in the sliding velocity reduces the friction and generates heat at the interface. Figure 13-52 shows this effect, as reflected by the dislocations and cells in the subsurface of 0.45 percent carbon steel after unlubricated sliding for a distance of 66.6 m. Although only dislocation networks are formed at a sliding velocity of 0.07 m/s (see Fig. 13-52 a), cells of thick walls are formed at 0.48 m/s (see Fig. 13-52b), and elongated cells of thin walls are formed at 4.56 m/s (Fig. 13-52c).
677
13.5 Wear
Specimen: Al Lub: Turbine oil Sliding distance: 1000 m 0.2 um below the surface
4.0
| I73
3.0
I73
= CD
2.0
o
o
1.0
Specimen: Al Lub: Turbine oil Load: 2.6 N Velocity: 0.26 m/s Sliding distance: 1000m 5
(a)
10
15
20
Distance from surface z in um
5.0
Specimen: Al Lub: Turbine oil Load: 2.6 N Velocity: 0.26 m/s 0.2 um below the surface
5.0
10.0
Load W in N
(c)
Specimen: Al Lub: Turbine oil Load: 2.6 N Sliding distance: 1000m 0.2 um below the surface
4.0
4.0
3.0 3.0 173 CD
173
0
o
8
2.0
1.0
1.0
100 (b)
2.0
1000
10000
Sliding distance L in m
Figure 13-51. Cell size distribution in a horizontal section 0.2 um below the wear surface. The maximum values are symbolized by a triangle, the mean values by a circle, and the minimum values by a square.
0.5 (d)
1.0
Sliding velocity v in m/s
(a) The cell structure in the subsurface layer; (b) the change in cell size during sliding; (c) the effect of load on cell size; (d) the effect of sliding velocity on cell size (Kato et al., 1985).
678
13 Friction and Wear
mated using Eq. (13-54) for the given frictional conditions using the quantitative relationships shown in Fig. 13-51.
13.6 Outlook
1
•,'*.*';
'_*.?"„ .
'-'j,',
-
..A*A-\,''*.'-.
•/
:.-.•{•
\
.:
••.'•
Figure 13-52. The influence of sliding velocity on dislocation and cell structures in the subsurface of 0.45% carbon steel after unlubricated sliding over a distance of 66.6 m (Nakajima and Mizutani, 1969): (a) 0.07 m/s, (b) 0.48 m/s, (c) 4.56 m/s.
This observation qualitatively supports the result of Fig. 13-51 d. If a method for translation from d in Fig. 13-51 to J i n Eq. (13-54) is introduced, the strength in the subsurface can be esti-
There is an extremely large number of observations on the friction and wear of materials, especially of metals. But the theories of friction and wear are not yet well enough established for practical usage. One reason is that the frictional condition is not carefully enough controlled in many cases in spite of the sensitivity of phenomena at the interface. People do not generally care about the microscopic viewpoints and observe only the apparent phenomena. Hence the opportunity to observe the real nature of friction and wear has been lost repeatedly. Based on this viewpoint, this chapter was mainly concentrated on providing a bridge between microscopic and macroscopic mechanisms of friction and wear. But it still goes only half-way. The behavior of atoms and molecules at the interface is not well described, and the microscopic chemical reactions are not well explained. It is well known that there may be exceptional types of chemical reaction at the frictional interface and special types of electron emission. A wear particle from the interface can be as small as a cluster of a few atoms. It is clear from the modern trend of engineering that an ultra-microscopic view of friction and wear will become more necessary in the near future. Micro-tribology will be developed in the future for investigating this state of friction and wear.
13.8 References
13.7 Acknowledgements The author greatly appreciates the kind help of Dr. N. Umehara and Mr. K. Adachi in finishing the manuscript for this chapter. Both are research associates at the Department of Mechanical Engineering of Tohoku University.
13.8 References Akagaki, T., Kato, K. (1985), Proc. JSLE Int. Trib. Conf., Tokyo, 891. Akagaki, X, Kato, K. (1987), Wear 117, 179. Akagaki, X, Kato, K. (1988), STLE, Trib. Trans. 31, 311. Archard, I F. (1953), /. Appl. Phys. 24, 981. Bowden, F. P., Hughes, X P. (1939), Proc. Roy. Soc. A172, 263. Bowden, F. P., Tabor, D. (1950), The Friction and Lubrication of Solids, Part 1. Oxford: Clarendon. Bowden, F. P., Tabor, D. (1964), The Friction and Lubrication of Solids, Part 2. Oxford: Clarendon. Buckley, D. (1981), Surface Effects in Adhesion, Friction, Wear and Lubrication. Amsterdam: Elsevier. Cameron, A. (1966), Principles of Lubrication. London: Longmans. Challen, I M., Oxley, P. L. P. (1979), Wear 53, 229. Cheng, H. S., Keer, L. M. (1980), ASME, AMD, 39. Childs, T. H. C. (1988), Proc. Instn. Mech. Engr. 202, c6, p. 379. Chiou, Y. C , Kato, K. (1987), /. JSLE, International Edition 9, 11. Chiou, Y. C , Kato, K., Kayaba, T. (1985), Trans. ASMEJ. Trib. 107,491. Cocks, M. (1962), /. Appl. Phys. 33, 2152. Czichos, H. (1978), Tribology - a System Approach to Friction, Lubrication and Wear. Amsterdam: Elsevier. Dautzenberg, J. H. (1980), Wear 59, 401. Dautzenberg, J. H., Zaat, J. H. (1973), Wear 23, 9. Dowson, D., Higginson, G. R. (1966), Elasto-Hydrodynamic Lubrication. Oxford: Pergamon Press. Dufrane, K. F , Glaeser, W. A. (1976), Wear 37, 21. Fegredo, D. M., Pritchard, C. (1978), Wear 49, 67. Fischer, T. E., Tomizawa, H. (1985), Wear 105, 21. Garber, 1.1., Skorinin, J. V. (1978), Wear 51, 327. Gilbreath, W. P. (1967), in: Adhesion or Cold Welding of Materials in Space Environments, ASTMSTP431. Philadelphia, PA: American Society of Testing and Materials. Green, A. P. (1954), J. Mech. Phys. Solids 2, 197. Greenwood, J. A., Tabor, D. (1957), Proc. Conf. on Lub. and Wear, Inst. Mech. Engrs., p. 314.
679
Hailing, J. (1975), Wear 34, 239. Hamilton, G. M. (1963), Proc. Instn. Mech. Engrs. 177, p. 667. Heilman, P., Don, I, Sun, T. C , Glaeser, W. A., Rigney, D. A. (1983), Wear of Materials. New York: ASME, p. 414. Hertz, H. (1882), J. Reine und Angewandte Mathematik92, 156. Hirst, W. (1957), Proceedings of the Conference on Lubrication and Wear. London: Institution of Mechanical Engineers, p. 674. Hirst, W, Lancaster, J. K. (1961), Proc. Roy. Soc. A 259, p. 228. Hokkirigawa, K., Kato, K. (1988), Tribology International 21, 51. Holm, R. (1946), Electric Contacts. Uppsala: Almquist Wiksells. Jahanmir, S., Suh, N. P. (1977), Wear 44, 17. Jahanmir, S., Suh, N. P., Abrahamson, E. P. (1974), Wear 28, 235. Johnson, K. L. (1985), Contact Mechanics. Cambridge: Harvard University Press. Kato, K., Kayaba, X, Ono, Y. (1985), Wear of Materials. New York: ASME, p. 463. Kayaba, X, Kato, K. (1978 a), Wear 47, 93. Kayaba, X, Kato, K. (1978 b), Wear 51, 105. Kayaba, X, Kato, K. (1981), ASLE Trans. 24, 164. Kayaba, X, Suzuki, S. (1976), Technical Report Tohoku University 41, 21. Keer, L. M., Bryant, M. D. (1983), Trans. ASME 105, 198. Kitsunai, H., Kato, K., Hokkirigawa, K., Inoue, H. (1990), Wear 135, 237. Kuhlman-Wilsdorf, D. (1980), in: Fundamentals of Friction and Wear of Materials. Metals Park, OH: Am. Soc. Met. Langford, G., Cohen, M. (1969), Trans. ASM 62, 623. Lim, S. C , Ashby, M. F. (1987), Ada Metall. 35.1.1. McFarlane, T. S., Tabor, D. (1950), Proc. Roy. Soc. London, Series A, 202, p. 244. Merwin, J. E., Johnson, K. L. (1963), Proc. Instn. Mech. Engrs. 177, p. 676. Nakajima, K., Mitzutani, Y. (1969), Wear 13, 283. Peterson, M. B., Winer, W. O. (1980), Wear Control Handbook. New York: ASME. Poritsky, H., Schnectaday, N. Y. (1950), J Appl. Mech., June, 191. Prandtl, L. (1920), Gottingen Nachr. Math. - Phys. Nachr., 74. Quinn, T. F. J. (1962), British J Appl. Phys. 13,33. Rigney, D. A., Naylor, M. G., Divakar, R. (1986), Mat. Sci. Engr. 81, 409. Rigney, D. (1990), Scripta Metallurgica Materialia 24, 798. Sakurai, X, Baba, X, Ohara, S. (1958), /. JSLE 3, 293. Stroud, M. F., Wilman, H. (1962), British J. Appl. Phys. 13, 173.
680
13 Friction and Wear
Tabor, D. (1951), The Hardness of Metals. Oxford: Clarendon. Travernelli, J. R, Coffin, L. R (1959), Trans. ASM, 438. Tresca, H. (1864), Comptes Rendus Acad. Sci. Paris 59, 754. Tsunai, Y, Enomoto, Y. (1989), Wear of Materials. New York: ASME, p. 369. Van Dijck, J. A. B. (1978), Wear 51, 327. Wear Control Handbook (1980). New York: ASME. Zum Gahr, K. H. (1987), Microstructure and Wear of Materials. New York: Elsevier.
General Reading Bowden, R P., Tabor, D. (1954), The Friction and Lubrication of Solids, Part I. Oxford: Clarendon Press. Bowden, R P., Tabor, D. (1964), The Friction and Lubrication of Solids, Part II. Oxford: Clarendon Press.
Buckley, D. H. (1981), Surface Effects in Adhesion, Friction, Wear and Lubrication. Amsterdam: Elsevier. Cameron, A. (1966), Principles of Lubrication. London: Longmans Green and Co. Ltd. Dowson, D. (1979), History of Tribology. London: Longman Group. Dowson, D., Higginson, G. R. (1977), Elasto-Hydrodynamic Lubrication. Oxford: Pergamon Press. George, G. M. (Ed.) (1982), Microscopic Aspects of Adhesion and Lubrication. Amsterdam: Elsevier. Hailing, J. (1975), Principles of Tribology. New York: Macmillan Press. Kragelsky, I. V., Dobychin, M. N., Kombalov, V. S. (1977), Friction and Wear. Oxford: Pergamon Press. Rigney, D. A. (Ed.) (1980), Fundamentals of Friction and Wear of Materials. Metals Park, OH: Amer. Soc. for Metals. Zum Gahr, K. H. (1987), Microstructure and Wear of Materials. Amsterdam: Elsevier.
