Advances in Applied Mechanics Volume 36
Editorial Board Y. C. F U N G AMES DEPARTMENT OF CAI-IFORNIA, Sm D r w o UNIVERSITY LA JOLLA,CALIFORNIA PAUL. GERMAIN ACADEMIE DES SCIENCES PARIS,FRANCE C.-S. Y I H(Editor, 1971-1982) JOHN W. HUTCHINSON (Editor, 1983-1997)
Contributors to Volume 36 ALANc. F. COCKS W. A. CURTIN
S ~ M OP. N A. GILL MICHAEL ORTIZ
JINGZHEPAN ROB
PHII.LIPS
ADVANCES IN
APPLIED MECHANICS Edited by Erik van der Giessen
Theodore Y Wu
DELFT UNIVERSITY O F TECHNOLOGY DELFT. T H E NETHERLANDS
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VOLUME 36
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Contents vii
CONTKIBUTORS
ix
PREFACE
Nanomechanics of Defects in Solids Michael Ortiz and Rob Phillips I. Introduction 11. Atomistic Models of Material Behavior 111. Patched Atomistic/Continuum Models IV. Lattice Statics V. Cauchy-Born Theory of Crystal Elasticity VI. Quasicontinuum Theory VII. Applications of the Quasicontinuum Method VIII. Concluding Remarks Acknowledgments References
7 5 11 29 51 55 61 72 73 73
Modeling Microstructure Evolution in Engineering Materials Alan C. F. Cocks, Simon P. A. Gill, and Jingzhe Pan I. 11. 111. IV. V. VI. VII.
Introduction Microscopic Constitutive Laws Thermodynamic Variational Principle Numerical Models Rayleigh-Ritz Analyses Structure of Constitutive Laws for the Deformation of Engineering Materials Concluding Remarks References
82 84 90 95 150 154 159 160
Stochastic Damage Evolution and Failure in Fiber-Reinforced Composites W A . Curtin I. 11. 111. IV.
Introduction Preliminary Issues Single-Fiber Composite Multifiber Composites: Global Load Sharing V
164 168 173 I US
Contents
vi
V. Multifiher Composites: Local Load Sharing VI. Future Directions Acknowledgments References
212 244 241 248
AUTHORINDEX
255
SUBJECTINDEX
26 1
List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
ALANC. F. COCKS (81) Department of Engineering, Leicester University, Leicester LE1 7RH
W. A. CURTIN (163), Division of Engineering, Brown University, Providence, Rhode Island, 02912 SIMONP. A. GILL(80, Department of Engineering, Leicester University, Leicester LE1 7RH MICHAELORTIZ(l), Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125 JINGZHE PAN(811, School of Mechanical and Material Engineering, University of Surrey, Guildford, Surrey GU2 5XH ROB PHILLIPS (11, Division of Engineering, Brown University, Providence, Rhode Island 02912
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Preface
Since the first issue of Advances in Applied Mechanics in 1948, the field of applied mechanics has witnessed numerous changes in focal points. One of the most noticeable areas of intense activity today is the application of mechanics in materials science. One of the central issues is to explain the relationship between a material’s microstructure and its mechanical properties. The work in the past on the deformation of purely elasticmicrostructured systems is classical by now. In recent years, attention is focussing on more complicated phenomena, which have posed challenging issues on the interface between mechanics and other disciplines. Three articles in the present volume highlight some of the recent advances in these areas. Microstructures exist at various length scales. In metals, atomic defects such as dislocations can control the mechanical behaviour at length scales several orders of magnitude larger than the atomic scale. Structure-property relationships require the bridging of many length scales. One of the most difficult, and least understood scale transitions is that from the atomic scale to the continuum level. The article by Michael Ortiz and Rob Phillips addresses this subject from a novel point of view by which the standard continuum description of the material is imbued with atomistic content. Numerous examples show that this approach promises to be an effective method to link the behavior of individual atomic defects to macroscopic inelastic behavior. A different class of microstructural changes are considered in the article by Alan C. F. Cocks, Simon P. A. Gill and Jingzhe Pan. The kinetic processes addressed here include grain-boundary and surface diffusion, interface reactions and grain-boundary migration, which can be modeled by continuum theory. The authors present a comprehensive variational framework for the description of all these processes in competition with each other and in dependence of their respective thermodynamic forces. The emphasis in this article is on numerical techniques, and in this sense supplements the article by Z . Suo in Advances in Applied Mechanics, Vol. 33, 1997, pp. 193-294. ix
X
Preface
The last article, by W. A. Curtin, addresses failure in fiber-reinforced composite materials. Rather than concentrating on the details of elementary fracture events locally in the microstructure, the main emphasis is on the role of stochastic variations in fiber strengths on macroscopic failure. Starting from the consideration of a single-fiber composite, the author addresses the modeling of damage evolution in multifiber composites, emphasizing the role of accompanying internal stress redistributions. Results of detailed simulations of damage development are used to guide a number of relatively simple analytical models, both are confronted against a wealth of experimental observations. Theodore Y. Wu and E. van der Giessen
.
ADVANCES IN APPLIED MECHANICS VOLUME 76
Nanomechanics of Defects in Solids MICHAEL ORTIZ
and
ROB PHILLIPS
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
I1. Atomistic Models of Material Behavior . . . . . . . . . . . . . . . . . . . . . .
5
111. Patched AtomisticKontinuum Models . . . . . . . . . . . . . . . . . . . . . . .
11 12 A . Peierls-Nabarro Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Cohesive-Zone Theories Applied to Fracture . . . . . . . . . . . . . . . . . 23 C . Other Patched AtomisticKontinuum Models . . . . . . . . . . . . . . . . . 28
IV . Lattice Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 A . Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 B . Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Lattice Statics Solutions via the Discrete Fourier Transform . . . . . . . . . 33 35 D. FCC Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . Diamond Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 39 F . Other Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Mura’s Theory of Eigendistortions . . . . . . . . . . . . . . . . . . . . . . . 40 V . Cauchy-Born Theory of Crystal Elasticity . . . . . . . . . . . . . . . . . . . .
51
VI . Quasicontinuum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Reduced Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . SummationRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . MeshAdaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55 56 57 58 60
VII . Applications of the Quasicontinuum Method . . . . . . . . . . . . . . . . . . . 61 A . Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 65 B . Interfacial Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
.
ADVANCES IN APPLIED MECHANICS VOL. 36 Copyliphi 0 I W hy ~ Acadeinic Press All righrr of reproduction in any torm rrxrved. ISSN 0 0 6 5 2 I6S/Y9 $30.00
Michael Ortiz and Rob Phillips
2
C. Fracture Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Dislocation Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66 71
VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
I. Introduction The mechanics of materials is played out against a background of structural imperfection in which defects mediate the inelastic deformation of materials. The recent development of microscopies that allow for the examination of defects at the atomic scale now permit a more direct connection between the defects and the response they engender. Techniques ranging from high-resolution electron microscopy, which makes possible the determination of the atomic-level structure of dislocation cores and grain boundaries, to atomic force microscopies, which bring new meaning to experiments such as those on nanoindentation, all pose deep challenges in modeling the mechanics of materials. Each of these experiments calls for renewed efforts to establish the connection between defect mechanics and constitutive phenomenology that is relevant to the direct simulation of processes in the mechanics of materials. However, the link between the defects themselves and the observed macroscopic behavior is often a difficult one to forge theoretically and remains an active area of research. The objective of this article is to review methods by which the classical boundary-value problems of continuum mechanics can be imbued with atomistic content. One of the difficulties that has stood in the way of efforts to forge the link alluded to above is the fact that, in some cases of interest, there is no natural separation of scales. Consideration of single defects in isolation is insufficient to yield relevant insights into material behavior, whle the attempt to build up sufficient numbers of such defects to be of macroscopic relevance is computationally unfeasible and conceptually inelegant. In the present article, we first aim to review the strengths and weaknesses of the conventional microscopic and continuum perspectives. Our ambition in this respect is to identify the powerful features of both of these approaches with the aim of extracting those parts that are especially appealing in a way that permits a synthesis. This discussion is followed by an assessment of some of the ideas that seem particularly promising for effecting a linkage between understanding at the single-defect level and higher-Ievel approaches. Microscopic modeling is founded on the fundamental assertion that beneath the details of observed macroscopic phenomenology there is a set of microscopic
Nanomechanics of Defects in Solids
3
processes which, when understood, rationalize the observed macroscopic behavior to the extent of enabling quantitative predictions. A microscopic simulation is one in which the relevant microscopic degrees of freedom and their evolution are treated explicitly. In this context, one may start with a prescription for computing the total energy. Given this prescription, a variety of tools are at hand to compute an energy-minimizing configuration-possibly metastable-for a given set of atoms. For example, if we interest ourselves in the geometry of a particular dislocation core, there are well-understood avenues for determining its structure on the basis of known interatomic interactions. For instance, given a high-resolution micrograph of a dislocation core such as shown in Figure 1, different structural alternatives may be evaluated by minimizing their respective energies and asking for that structure which is simultaneously lowest in energy and accounts for the observations. This process is carried out explicitly in Mills et al. (1994). Continuum mechanics, on the other hand, is founded on the assumption that the spatial variations in a given field variable are sufficiently slow as to make possible the smearing out of the atomistic degrees of freedom upon which they are founded. In particular, there is an implicit mapping from the large set of atomistic degrees of freedom to a single vector field of displacements, namely, {Ti} H Nx>.
(1)
One of the most significant virtues of the continuum approach is the considerable reduction it implies in the number of degrees of freedom that must be accounted
F I G . I . Lonier dislocation core i n aluminum (Mills er ~ d . 1994). , (a) High-resolution image of dislocation core; (b) Atomic positions in same dislocation core as obtained by energy minimization. Reprinted with permission of Elsevier Science.
4
Michael Ortiz and Rob Phillips
for in a given model. By replacing the set of atomic coordinates (r;} by a reckoning in terms of displacements, we pass from the realm of 3N discrete differential equations to a set of three coupled partial differential equations. We may then defer to powerful tools of analysis and approximation theory, such as the finiteelement method, to examine their solutions. One of the important recognitions that stands at the foundation of the approaches that we review in the present article is that, in some circumstances, progress can be made in linking the continuum and microscopic perspectives in a way that results in more theoretical power than either offers alone. One of the areas in which such models have had a significant impact is that of cohesivezone theories, in which a conventional continuum mechanics statement of the boundary-value problem of interest is supplemented by constitutive laws governing interfacial slip and decohesion. The link to atomistics arises from the fact that, in many cases, atomic-level calculations can be used to inform the cohesive constitutive description. Alternative schemes have been devised in which parts of the body being modeled are treated explicitly via the tools of atomistic simulation, with a set of boundary conditions being specified which anchors the atomistic region to a conventional continuum. As a final example, the recently developed quasicontinuum method constructs a seamless connection between the two perspectives by explicitly taking advantage of atomistic constitutive insights throughout the body. The remainder of the article is organized as follows. In Section 11, we briefly review the foundations of atomistic modeling, including a discussion of the presently available approximations to the total energy. These approximations provide an important basis for many of the current efforts in nanomechanics. Section TI1 addresses the development of patched atomistidcontinuum models with special reference to cohesive-zone strategies, in which the constitutive nonlinearity is confined to particular planes. Section IV takes up an analysis of the insights offered into crystal elasticity by the atomistic perspective. This section culminates in a discussion of the way in which a range of microstructures arise by virtue of constitutive nonconvexity. Section V treats mixed atomistic and continuum computational schemes for effecting the atomistidcontinuum linkage described in the preceding sections. The centerpiece of Section VI is the quasicontinuum method. This method makes possible large-scale atomistic analyses by systematically constraining the atomistic degrees of freedom through the application of finite-element discretization and interpolation. Finally, Section VII offers a perspective on the applications which have been made possible by mixed atomistic and continuum models. It is important to note that the primary aim of this article is to recount our own involvement in efforts to imbue continuum boundary-value problems with atom-
Nanomechanics of Defects in Solids
5
istic content. As such, the present article should not be regarded as a systematic review of the literature, but rather as a personal account of modeling in the mechanics of materials.
11. Atomistic Models of Material Behavior As stated above, there are circumstances in which the insights that are gleaned from atomistic analysis are indispensable. In this section, we provide an overview of how atomistics can inform higher-level continuum descriptions. We begin with a few illustrative examples for purposes of motivation, and follow with a description of precisely what it means to carry out an atomistic calculation. Finally, the section closes with more detailed examples of the use of atomistic analysis to link scales in the context of the mechanics of materials. From a fundamental perspective, the microscopic simulation of materials is based on the evolution of degrees of freedom that are governed by the Schrodinger equation. Specifically, given a collection of atoms, each of which carries with it a set of electrons, the problem is to compute the total energy of the ensemble as a function of the relevant microscopic degrees of freedom, e.g., the positions of the nuclei and the electrons. Full-scale calculations such as just described are computationally intensive. The aim of the present discussion is to examine the various approximation strategies that have arisen from the desire to model such systems, with a view to characterizing precisely the compromises that are made in adopting such strategies. We begin by considering an arbitrary collection of N atoms identified by some convenient labeling scheme 1,e.g., simple enumeration in an unstructured gas or Miller indices in a crystal (e.g.. Hammond, 1990). Our objective is the determiOne immediate scheme is nation of the total energy of this collection Etot((r;}). that presented by pair potential descriptions of the total energy in which E'"' is given by
1 E'"' = -
c v"f(r;i), i.
;€I
where r,,; = (r; - ',,I is the distance between atoms i, j E 2.It bears emphasis that in this description only the nuclear coordinates appear explicitly in the energy function and the electronic degrees of freedom have been condensed out, a situation which is hinted at in labeling the potential Veff. Beyond these general restrictions, the choices of the energy function V available to the materials modeler are numerous.
6
Michael Ortiz and Rob Phillips
In many contexts, and especially in cases involving large numbers of atoms within the molecular dynamics setting, it is often useful to resort to multibody expansions of the form
where the potential VX.accounts for k-body interactions. The term V I reduces to an inconsequential constant in the absence of external fields. In writing (3), the hope is that the expansion converges quickly and can be truncated after a few terms to a good approximation. The simplest such truncation scheme consists of keeping two-body interactions only, leading to the pairwise form of the energy:
Here again we emphasize that, either phenomenologically or via explicit calculational strategies, the electronic degrees of freedom are implicitly subsumed in the effective pair potential. Once the pair potential has been identified, it is a straightforward matter to evaluate radial derivatives and the corresponding force fields. These force fields, in turn, provide the basis for lattice statics or molecular dynamics analyses of the problem of interest. As an example of the state of the art in this regard, Figure I1 shows a sequence of temporal snapshots from a molecular dynamics simulation of dynamic fracture in an fcc crystal due to Abraham et al. (1997). One of the outcomes of this series of calculations is the observation of different fundamental mechanisms depending on the underlying crystal orientation. For the orientation shown in the figure, the generation of large amounts of dislocation activity at the crack tip in the form of dislocation loops is particularly noteworthy. It is immediately clear-even from casual inspection of the figure-that the short-range interactions between dislocations are a key part of the physics taking place in the crack tip region. A second key feature of the simulation concerns how to properly meld the boundary conditions of continuum mechanics and those that are a forfirion' used in the atomistic setting. The interatomic interactions used in the example discussed above are of the pair potential variety and, in particular, of the Lennard-Jones form, namely,
where the parameters CI and h can be determined, e.g., by insisting that the crystal have the correct lattice parameter and cohesive energy. Using these highly sim-
Nanornechanics of Defects in Solids 7
8
Michael Ortiz and Rob Phillips
plified interactions, simulations have now been performed involving 10' atoms, and this figure will no doubt continue to rise with each increment in computing performance. As is well described elsewhere (Carlsson, 1990; Pettifor, 1995), without a significant increase in overhead, one may move beyond the pair potential limit to so-called pair functionals which reflect some of the extra physics of bonding. In particular, what such total energy schemes bring to the description of bonding is the idea of environmental dependence. What this means is that, unlike pair potentials, the strength of the bonds of a given atom to its neighbors varies depending on its local environment. For example, near a free surface, the bonds of the surface atoms to their subsurface neighbors are strengthened relative to their bulk strengths. Within the embedded-atom method (EAM) (e.g., Daw, 1990, and references therein), which is one example of a pair functional scheme, the total energy is given by
I
tZ
i.
/ E l
In the first term, known as the embedding energy, p, represents the electron density at site i , while the second term is a conventional pair potential term of the type described above. The local electron density p, is determined by the number and proximity of neighboring atoms. The presence of the embedding energy in ( 6 )renders the interaction between atoms dependent on their environment, e.g., whether the atoms lie in the bulk or near a free surface, an effect which simple pair potentials fail to take into account. More recent versions of the embeddedatom method, e.g., the modified embedded-atom method (MEAM) (Baskes et al., 1989, 1992, 1994), properly account for the angular dependencies characteristic of the covalent bond, and can, therefore, be applied to a broader class of materials. The use of potentials of the pair functional variety has become routine. From the standpoint of modeling extended defects, and particularly dislocations, such potentials suffer from the shortcoming that they often underestimate the stacking fault energy. This has the unfortunate side effect that the predicted dislocation core structures have an unphysically large stacking fault ribbon. Nevertheless, much progress has been made using such potentials and, indeed, they form the basis of much of our later discussion. There are a variety of circumstances in which the central force schemes advocated above are inadequate. In particular, for examining the structures of extended defects such as dislocation cores and grain boundaries in covalently bonded semiconductors and central transition metals, such as tungsten, the introduction of multibody energy terms of order higher than 2 is essential. One avenue for the
Nanomechanics of Defects in Solids
9
introduction of such terms is via angular potentials. In this representation, the multibody expansion (3) is truncated after the three of four-body terms. In the former case, this truncation leads to the energy
where the three-body term has been written in terms of the distances r , , , rJx and the angle e l , k subtended by the vectors rk - r, and r, - r i . The significance of potentials of this variety is that they impose a penalty not only on bond stretching deformations but on bond bending deformations as well. A rich background has been developed on the energetic characteristics of covalent materials, and the bond bending terms have been found to be indispensable in that setting. For silicon, an empirical potential including up to three-body interactions was proposed by Stillinger and Weber (1985). The various terms in the energy potential have the form
otherwise. (9) Based on careful lattice dynamics calculations, Stillinger and Weber (1 985) optimized the constants of their model to match a wide array of thermomechanical properties of silicon. The resulting values are: A = 19.9949 eV, B = 30.5979 eV, C = 273.4452eV, CT = 2.0951 A, p = 4, q = 0, a = 1.80, y = 1.20. Numerous other potentials for silicon and related materials have been proposed in the literature (e.g., Halicioglu et al., 1988; Stoneham et al., 1988; Duesbery et al., 1991). Stillinger and Weber's potential has been successfully employed in a wide range of atomistic simulations (Stillinger and Weber, 1985; Landman et al., 1986; Abraham and Broughton, 1986; Dodson, 1986; Nandedkar and Narayan, 1990; Gallego and Ortiz, 1993). Potentials such as described above allow for the investigation of dislocation core structures in both covalent materials such as silicon and for transition metals such as molybdenum (Bulatov et al., 1995; Xu and Moriarty, 1996). Both of these examples reveal the sometimes counterintuitive atomic rearrangements that occur in the dislocation core itself. For example, in the case of the bcc screw dis-
10
Michael Orriz and Rob Phillips
location (Xu and Moriarty, 1996), there is a symmetry-breaking reconstruction of the core with slip accumulated in increments along three distinct planes. In fact, this reconstruction has been impugned as the origin of the anomalous temperature dependence associated with the flow stress seen in some materials. In particular, with increasing temperature thermally activated constrictions can occur which allow the screw dislocation to cross slip onto other planes with the consequence that the segment of interest is pinned. Equally interesting insights into the complex core behavior of 60" dislocations in silicon have been built up using similar analyses (Bulatov et al., 1995). It has been found that the core supports a variety of defects within the core itself. The various schemes described in the foregoing have made no explicit reference to the electronic degrees of freedom themselves. However, it is well known that in most circumstances the existence of bonds is an inherently electronic phenomenon. We finish this section with a brief overview of the extra work that must be carried out in order to bring the electrons explicitly into the problem. As was noted earlier, the fundamental governing equation (neglecting relativistic effects) for the electronic degrees of freedom is the Schrodinger equation. The disposition of the electronic degrees of freedom is characterized by a complex scalar field known as the wave function, @(r),which satisfies the governing equation
where h is Planck's constant and m is the mass of the electron. It should be carefully noted that (i) the time dependence of the wave function has been removed from the problem and only the time-independent equation is considered; (ii) the energy eigenvalues, E , give the allowed energies of the various quantum states; and (iii) V(r) is the potential experienced by the electron. There are many different levels of complexity that must be faced in solving (lo), and we refer the reader to reviews on the topic for more details (e.g., Tuszynski et al., 1994; Acioli, 1997; Goringe ef al., 1997; Ernzerhofet a!., 1996). In its most general formulation, the problem is a many-body problem in which the behavior of the different electrons is essentially coupled. It is the domain of density functional theory to find ways of managing this complexity. For our purposes, it suffices to note what can be accomplished once solutions to the full electronic problem are available. In simple terms, the calculations provide the total energy as a function of the atomic positions. From this function, properties of interest such as the elastic constants, the dynamical matrix, or the energy minimizing configurations of interest themselves may be computed.
Nanomechanics of Defects in Solids
11
As noted in Section I, each of the atomistic schemes just discussed is used at the price of some compromise. Though pair potentials allow for the investigation of systems comprising millions of atoms, the level of accuracy that can be expected from such potentials is minimal. At the other extreme, state-of-theart density-functional calculations allow for the determination of many material properties to high accuracy. However, even on supercomputers the system sizes that can presently be explored by density-functional methods are on the order of 1000 atoms. One of the central messages of the present article is the belief that, size limitations notwithstanding, there are strategies that allow for atomistic calculations to inform higher-level models. It is precisely this notion of informing higher-level calculations on the basis of atomistic insights that we take up next.
111. Patched AtomistidContinuumModels Cohesive-zone models are of fundamental interest in the mechanics of materials as they effectively balance the sometimes conflicting demands of analytical tractability versus the desire to base descriptions of material behavior directly on atomistic theories. Cohesive-zone models first arose in the context of both plasticity (the Peierls-Nabarro model) and fracture (Barenblatt-Dugdale-type models). In these areas of application, it was noted that, in many cases, the key nonlinear features underlying material behavior can be attributed to processes confined to an interfacial or interatomic plane. In the context of dislocations with a planar core structure, this refers to the fact that the slip, i.e., the displacement jump that is transmitted by the dislocation, is primarily supported on a single plane. Similarly, in the case of a crack, one may imagine the breaking of atomic bonds across a cleavage plane. The fundamental connection between atomistics and the formulation of boundary-value problems within the cohesive-zone framework resides precisely in the constitutive description of the interfaces. From a continuum viewpoint, this description requires that the volume energy be supplemented by an additional term that reflects the energy stored in the cohesive zones themselves. Assuming for simplicity linearized kinematics and elastic behavior in the bulk, the total energy may be written as
E'""u] =
s,
W(VU) d V
+
s,
@(S) dS,
(11)
where R is the domain of the body, 'c is the cohesive surface, u is the displacement field over R, and
12
Michael Ortiz and Rob Phillips
is the displacementjump across Z. A more general treatment of cohesive behavior which accounts for finite kinematics and irreversible behavior may be found in Ortiz and Pandolfi (1998). The success of the strategy just described depends critically on the existence of some cohesive-zone constitutive insight. In formal terms, one may postulate the existence of a reversible cohesive potential q5 (a), the derivatives of which yield the tractions acting on the cohesive zone (Needleman, 1987, 1990a. 1990b). In some cases, one may resort to atomistic analysis for the determination of the cohesive potential. In order to illustrate this link, we begin by considering crystalline slip. In this context, a dislocation line is conventionally defined as a boundary between slipped and unslipped regions of the slip plane. However, the distinction between the slipped and unslipped area is blurred within the dislocation core region, and it is precisely this ambivalence that cohesive-zone theories address.
A. PEIERLS-NABARRO THEORY
Many mechanical and electronic properties of crystals are strongly influenced by the core structure of dislocations (Veyssikre, 1988; Vitek, 1992). For instance, dislocation cores in microelectronics devices may either act as donor- or acceptorlike sites or behave as high conductance regions (Haasen, 1983; Ourmazd, 1984). In many cases, a detailed knowledge of the atomic structure of the core is required to make quantitative predictions possible. A case in point is provided by the nonplanar structure of dislocations in BCC metals, which strongly affects their mobility (Vitek, 1992). A similar situation is encountered in the case of dislocations on nonbasal planes in HCP metals, and dislocations in some intermetallic compounds (Kear and Wilsdorf, 1962; Paidar et al., 1984), ordered alloys, and nonmetallic crystals (Veyssikre, 1988). In those cases in which the dislocation core is ostensibly planar, the venerable Peierls-Nabarro model may reveal useful insights into features such as the core size and the stress required to move the dislocation. The accuracy of the Peierls interplanar potential approximation tends to break down when the slip distribution has structure on the length scale of the lattice parameter (Miller and Phillips, 1996). This breakdown may be delayed by recourse to a nonlocal extension of the interplanar potential, as discussed in Section III.A.4. These limitations notwithstanding, the Peierls interplanar potential approach remains a widely used tool of analysis of lattice defects in crystals.
Nanomechanics of Defects in Solids
13
1. Formulotion of the Theory
Consider an anisotropic linear elastic solid referred to a Cartesian reference frame. Let c i j k ! be the elastic moduli of the solid. Imagine introducing a perfect straight dislocation along the x j axis, resulting in displacements u ; ( X I , x 2 ) and stresses oij ( X I , x.2) in the crystal. Let bi be the Burgers vector of the dislocation. A number of salient features of the displacement and stress field can be established directly (Rice, 1985).Let ( r ,0) be polar coordinates centered at the dislocation, with 0 measured from the X I axis. Let
cos e (=(~i;8)~
- sin0
q=(
;;so)
(13)
be the unit vectors in the radial and circumferential directions, respectively, so that 6, = r , , and q, = rO,a. Assume that the displacement gradients have the separable structure ~ , , , ( r6. ) =
f(wjam
(14)
where here and subsequently in this discussion Greek indices have the range { 1,2] and the summation convention is in effect. Let C be a circle of radius Y centered at the origin. Taking C as a Burgers circuit, it follows that r
which necessitates that f ( r )
-
r - ’ . We therefore have
It also follows from equilibrium (Rice, 1987) that 5,(8)q,(0)
= constant.
(18)
To see this, recall that the relevant equations of equilibrium are, for the case at hand, 4a,(Y
= 0.
(19)
Inserting (1 7) into this equation, we find
o = [ r - 1e , , ~=, -r-26,(y6, ~ +r-’5,,,, = rP2[6,,qk
+ 6;,qa1
= r-’[C,,qa1’.
=r
-2
[--6,,6,
+ e,;q,l (20)
Michael Ortiz and Rob Phillips
14
' denoting the derivative with respect to 19,which demonstrates the veracity of eq. (18). By linearity, iYIwva must be linear in hi. This relation can be expressed in the form
for some tensor K. From (17) and (21), it follows that
~
;
2
= -K;kbk. ~ v r
~
On the slip plane, X? = 0, one has r = 1x11 for x j = 0, and ( q l , 172) = (0, 1) if X I > 0, while (171, q 2 ) = (0. -1) if X I c 0. It therefore follows from (22) that [;(XI)
2
3
a;?_(xl,0) = -K;khk, XI
(23)
which gives the familiar singular distribution of tractions near the core of a dislocation. From (23) and the work-energy identity, it follows that the energy of the crystal per unit dislocation length is EtOt
---
L
ECOI'C
+
I, R
E"re
r i b ; d r = __
L
+ K;kb;bk log -,yoR
(24)
which identifies K as the prelogarithmic energy tensor of anisotropic elasticity (Bacon e f a[., 1979). Barnett and Swanger (1971) derived the particularly convenient formula
where
is the elastic dynamical matrix and
Formula (25) reduces the computation of K to a simple quadrature. For an isotropic solid, K is diagonal and
where is the shear modulus and v is Poisson's ratio. Following Peierls ( 1940) and Nabarro ( 1947). a dislocation is now idealized as a plane of displacement discontinuity in an otherwise perfect crystal. Let the
Nanomechanics of Defects in Solids
15
displacement jump (1 2 ) be supported on the plane x 1 - xj, or slip plane, and be a function of X I only. These displacement jumps may be regarded as a distribution of infinitesimal dislocations of Burgers vectors
Therefore, the crystal lattice offers a resistance to slip in the form of tractions
which is obtained from (23)by superposition. These tractions must be in equilibrium with those due to atomic-level forces acting across the slip plane. Following Needleman (1987, 1990a, 1990b), we may postulate the existence of an interlayer potential 4(S) (cf. eq. ( 1 l ) ) , whence the interlayer tractions follow as
Assume that the crystal additionally deforms under the action of externally applied uniform stresses imparting tractions tK upon the slip plane. Equilibrium of the tractions (30),(31), and t" acting on the slip plane then demands that
These equations, in conjunction with the essential conditions S(-OO) = 0,
S(CO)
= b"',
(33)
determine the slip distributions on the slip plane, which generally will contain varying numbers of dislocations, some positive and some negative. The second of (33) then fixes b"', the net or total Burgers vector of the dislocation ensemble. The problem just defined may be accorded a far-reaching variational interpretation. The total energy of the dislocated crystal is the sum of the elastic energy and the misfit energy and the work done by the applied tractions on the interplanar slip, i.e.,
Michael Ortiz and Rob Phillips
16
In the presence of applied tractions, the potential energy of the crystal may additionally be defined as
The stable equilibrium dislocation structures 6 over the slip plane may now be identified with the minimizers of @, which leads to the variational statement
@[S] = inf @ [ q ] ,
(36)
tJ
where the minimization is subject to the essential boundary conditions (33). Indeed, provided that the slip distribution is sufficiently smooth, the Euler-Lagrange equation corresponding to (36) is (32). Often, the direction of 6 is known on the basis of crystallographic considerations. For instance, for a dislocation that is not dissociated 6 may be presumed to point only along the direction of the Burgers vector b, leading to the constrained displacement hypothesis of Rice (Rice, 1992; Sun et nl., 1993). This hypothesis is born out in some instances by atomistic simulations (Sun et al., 1991; Yamaguchi et a f . , 1981). For a dislocation split into two partials in accordance with the reaction b = bl b2, a simple model is to take XI) to point in the direction of bl up to the midpoint of the stacking fault ribbon, and in the direction b elsewhere. In general, if the direction XI), Is(xl>l = 1, of & X I ) is known a priori, then the full vector displacement jump 6(x-I)may be derived from a scalar field XI 1 through the relation
+
6(Xl)
= S(Xl)S(Xl).
(37)
Under these circumstances, the scalar slip distribution may be obtained by effecting a constrained minimization of (36) over slip distributions of the form (37), which gives the reduced variational problem:
@[a]
= inf @ [ q ] , '1
(38)
where the constrained potential energy is
@[Sl = @(6 = SS]. A simple calculation gives the constrained potential energy in the form
(39)
Nunomechanics of Defects in Solids
17
where
The minimization in (38) is subject to essential boundary conditions
6(-0) = 0,
S ( 0 ) = bj"'s,
(00).
(44)
Differentiation of the constrained interplanar potential (43) with respect to S gives the identity:
which shows that #(S, X I ) acts as a potential for the resolved shear stress r in the slip direction s. As a well-known example of these ideas, assume an interplanar potential of the Frenkel form:
#CS) = A ( 1 - cos
y)
The constant A is determined so as to match the elastic shear modulus in the relevant slip direction, with the result
where d is the interplanar separation. The resolved shear stress follows from (45) as 271 , 2x6 sin -. h b
s(S) = A -
(48)
Assuming the slip direction to be constant and the applied stress to vanish, the slip distribution is given by Peierls celebrated solution (e.g., Cottrell, 1953, p. 61) S(xl) =
b s1 -b2 + -arctan--, n C
where
B h' A 4n
c = --
(49)
18
Michael Ortiz and Rob Phillips
is a measure of the dislocation core width and
B =2K;kSiSk.
(51)
For an edge dislocation in an isotropic solid, the various constants previously introduced reduce to 2
A=(&)
5,
B=
CL
2n(1 - u ) '
d c = - 2(1 - u ) '
(52)
where p is the shear modulus, u is Poisson's ratio and d is the interplanar distance. Simple though the Peierls-Nabarro theory may be, it does establish a first clear link between lack of convexity and the emergence of microstructures. As illustrated by Frenkel's potential (46), the energy (34) of the crystal is nonconvex, in consequence of the lack of convexity of 4.Experience teaches us that nonconvex minimization often leads to the development of microstructure (e.g., Dacorogna, 1989). Here microstructure arises in the guise of dislocation structures. For instance, a crystal subjected to an applied shear ra3 may reduce its energy by nucleating a dislocation dipole. It bears emphasis that the Peierls-Nabarro theory possesses an intrinsic length scale commensurate with the magnitude b of the Burgers vector and may, therefore, be regarded as a regularized theory. The presence of a characteristic length precludes the development of infinitely fine microstructure and accords well-defined dimensions to such microstructural features as dislocation cores and dipole lengths. 2 . Piecewise Quadratic Model In the quest for analytical tractability, the simple piecewise quadratic interplanar potential stands out as a particularly useful basis for further analysis, as it lends itself to the use of the Fourier transform. In this model, the constrained interplanar potential is assumed to be of the form
This potential consists of quadratic wells centered at eigenslip 6" which are integer multiples of b. The concept of eigenslip is closely related to Mura's eigendistortions, to be discussed in Section 1V.G. As in Frenkel's model, the constant A may be determined so as to match the elastic shear modulus in the slip direction, with the result
which reduces to p / d in the isotropic case.
Nanomechanics of Defects in Solids
19
The energy may conveniently be regarded as a function of the two fields 6 and 6 " , which, in the absence of applied forces, then follow jointly from the variational problem
where, in effecting the minimization, S " ( x l ) / b is constrained to take values in 2 for all X I . Evidently, this constraint renders the problem nonlinear and nonconvex. For fixed 6 , minimization with respect to 6" gives
6E (XI) =lh,
I
E
Z,
This rule simply assigns to 6(xl) the nearest multiple of b. The main appeal of the piecewise quadratic model stems from the fact that, if S E is known n priori, then the slip distribution 6 follows from a linear problem which can be solved by recourse to the Fourier transform. Let j ( k ) and j" ( k ) be the Fourier transforms of S ( x 1 ) and 6" ( X I ), respectively. Then, an application of the convolution theorem and Parseval's identity gives the energy in the form
Minimization with respect to s^ then gives $E
6=
+ lklc
p 1
1
where
c=-
X B
A
(59)
has the dimensions of length. For instance, c=-
d
2(1 - u )
(60)
for an edge dislocation in an isotropic solid. The energy corresponding to (58) is computed to be
20
Michael Ortiz and Rob Phillips
As an example, consider the case of an isolated dislocation induced by an eigenslip distribution of the form:
whose Fourier transform is
where S r , is the Dirac delta function. Inserting this result into (58) and taking an inverse Fourier transform gives
We verify that & ( X I ) > h/2 for X I < 0 and & ( X I ) < b/2 for X I > 0. Hence, the constraints (56) are satisfied and the ansatz regarding S"(x1) is proved right. The energy of the dislocation follows from (61) as
where we have introduced a lower cutoff 2 n l R for the wave number in order to avoid divergent integrals. As may be seen, the piecewise quadratic model is analytically solvable in terms of quadratures. Recently, the piecewise quadratic model has been investigated by Movchan et al. (1998), who have extended it to account for kinetics of dislocation motion and three-dimensional effects such as dislocation kinking. 3. Identijcation of the Cohesive Energy
Interplanar potentials have been derived on phenomenological grounds by combining Frenkel's sinusoidal potential (46), or a variant of it (Xu et al., 1995). with the Rose-Ferrante-Smith (Rose er id., 1981) universal bonding relation (Belt2 and Rice, 1991; Rice, 1992). Interplanar potentials have also been derived by recourse to first-principles quantum mechanical calculations (Kaxiras and Duesbery, 1993). By way of contrast, a direct method of experimental identification of the interplanar response does not appear to be available at present. K. S. Kim (private communication) has suggested that the interplanar response can be inferred from observations of the structure of lattice defects such as dislocations. Next we describe a simple method for accomplishing such identification. As shown in Section 1II.A.1, the stable slip distributions over a slip plane in an elastic
Nanornechanics of Defects in Solids
21
crystal are solutions of the variational problem (39). Assuming, for simplicity, that the slip direction s is constant, the corresponding Euler-Lagrange equation is
The general case of variable slip direction may be treated likewise, but this extension will not be pursued here. We proceed to show that, if the displacement jump S ( x - 1 ) on the slip plane of a dislocation is known from direct observation, the integral equation (66) can be used to determine the interplanar law. To see this, assume that the applied stress vanishes and 6(.rl) increases monotonically from 0, at s~ = -00, to h , at X I = 00.Let X I ( 8 ) : (0, 6 ) + (-00, 00) be the inverse mapping. Changing integration variables from X I to 6 in (66) gives
s(6) = -
I”
B SI (6) - X I
(6’)
d6’.
(67)
which is the sought identification formula. As a check on this formula, let the slip distribution be given by Peierls’ solution (49). The inverse mapping is then
Inserting this relation into (67) gives
whence (48) follows by recourse to identity (50). Remarkably, the identification formula correctly returns Frenkel’s interplanar potential from the Peierls slip distribution. The explicit formula (67) furnishes an avenue for a direct identification of the shear-slip relation from high-resolution TEM observations of dislocation cores (Mills et d.,1994; Hemker, 1997).
4. Nonlocal Extension As described in detail above, a pivotal assumption in the formulation of the Peierls-Nabarro model is the existence of an interplanar potential for the atomiclevel forces acting across the slip plane. One of the key features of such a model is the assertion that the local state of stress is strictly a local function of the displacement jump across the interface, as evidenced in eq. (31). In physical terms, the locality assumption is equivalent to the view that the slip distribution 6(x) is slowly varying on the length scale of the underlying crystal lattice. One of the opportunities presented by atomistic analyses is the ability to directly test this locality assumption. Via the device of artificially constructed slip
Michael Ortiz mid Rob Phillips
22
geometries, it is possible to evaluate the energetics of crystalline slip both using the continuum approximation advanced in the Peierls-Nabarro framework and explicitly using atomistics (Miller and Phillips, 1996).What emerges from this study was the conclusion that, in the case of narrow dislocation cores, the locality assertion may be charged with producing inaccurate estimates of the misfit energy. To remedy this problem, a nonlocal extension of the Peierls-Nabarro framework has been undertaken by Miller et al. (1998a) and will be briefly summarized here. The starting point of this analysis is the contention that the conventional formulation of the Peierls-Nabarro model, as stated in eq. ( 3 2 ) ,must be amended to account for nonlocal interactions across the slip plane. For simplicity, throughout this discussion we assume, as in Section 1TI.A.I , that the displacement jump (1 2 ) is supported on the plane X I - x3,or slip plane, and is a function of X I only. In addition, we shall suppose that the slip direction is constant and known a priori. Then, we argue that, in addition to the term that derives from the interplanar potential, there is an additional shear stress of the form
s,
c*3
rNL(x-1) =
K ( X I - xi)&(xi)
dX{
(70)
for some function K . The significance of this term is that it introduces an energy penalty for nonuniform slip distributions. One of the initial questions posed by Miller et al. ( 1 998a) in this setting is whether or not the nonlocal kernel, K(x-1 - xi). can be determined on the basis of atomistic insights. They proceeded to show that the nonlocal interplanar kernel may indeed be so determined by appealing to Fourier methods. The basic idea is to identify the nonlocal kernel, Fourier component by Fourier component. This is accomplished by considering sinusoidal slip distributions of the form
The energy associated with each such distortion may be computed explicitly by recourse to direct atomistics. Furthermore, the contribution due to slip may be extracted by subtracting off the bulk elastic energy. As a result, the exact misfit energy is determined from atomistics. It is then found that the misfit energy may be approximately reckoned by an expression of the form -
~niibfit
s_,
cu
4(S, dxl
+
2
Srn11
K ( x ~- . x ; ) G ( x I ) S ( . X ; ) ~ X ~
d ~ ; .(72)
--oo
By virtue of the fact that the prescribed slip distribution is sinusoidal, this expression depends only on the kth Fourier component of the Fourier transformed
Nanomechanics of Defects in Solids
23
nonlocal kernel. In this manner, as noted above, the nonlocal potential may be computed wavevector by wavevector. This procedure has been systematically followed for a few different slip systems in aluminum (Miller et al., 1998a). One of the immediate outcomes of this analysis is the observation that the energy of nonuniform slip distributions is more accurately captured by the nonlocal continuum approximation than by its local counterpart. An appealing feature of the nonlocal interplanar energy is that it is expressed in convolution form, which, as already noted, lends itself to the application of Fourier methods. In particular, if the local interplanar potential is assumed to be piecewise quadratic, then the same methods of analysis developed in Section III.A.2 carry over to the present setting. For example, the Fourier transform of the slip distribution attendant to a dislocation dipole of spacing r is found to be
with the notation of Section III.A.2. Just as with the solution obtained via the series of quadratic wells discussed earlier, the determination of the slip distribution has been reduced to a problem in Fourier inversion. Evidently, eq. (73) extends the local slip distribution for a dipole, which is obtained by setting the nonlocal kernel K to 0, to the nonlocal range. The nonlocal analysis serves to demonstrate one possible avenue for extending the range of applicability of the cohesive-zone framework within the context of dislocations. This analysis is also instructive in that it calls attention to the role of constitutive nonlocality in the immediate vicinity of defects, a subject that will be taken up again in our discussion of the quasicontinuum method. B. COHESIVE-ZONE THEORIES APPLIEDTO FRACTURE The brittle-to-ductile (B-D) transition furnishes a prime example of the utility of cohesive-zone theories. In the brittle-to-ductile (B-D) transition, a competition is believed to take place between cleavage fracture and plastic shielding, with eventual blunting of the propagating cleavage crack, by either dislocation emission from the crack tip or background plastic relaxation (St John, 1975; Brede and Haasen, 1988; Hirsch et al., 1989; George and Michot, 1993; Brede et al., 1991; Hsia and Argon, 1994). During the emission of a dislocation from the crack tip, the surrounding crystal remains elastic and can be modeled as a continuum. By contrast, it is now well established (Schock and Piischl, 1991; Rice and Beltz, 1994; Xu et al., 1995)that at the saddle point the critical activation configuration,
24
Michael Ortiz and Rob Phillips
or “embryo” of the nucleated dislocation consists only of partially completed core matter. Ultimately, therefore, a full understanding of the nucleation-controlled B-D transition must come from atomistic models of dislocation emission from the crack tip such as discussed in Section VI1.C. However, some progress can be made by recourse to cohesive-zone models such as described above (Schock and Piischl, 1991; Rice and Beltz, 1994; Xu et al., 1995). Following Rice et al. (Rice, 1992; Rice et a/., 1992; Sun el al., 1993). the slip plane on which the dislocation loop nucleates may be viewed as an extension of the crack surfaces with a nonlinear interlayer potential of the type already discussed acting across it. In a recent development of this technique by Xu et al. (1995), the elasticity of the crystal is modeled by recourse to a variational boundary integral method advanced by Xu and Ortiz (1993). They show that the energy of a cracked solid, corresponding to the first term in (1 l), may conveniently be calculated by representing the cracWslip plane system as a distribution of dislocation loops. The associated integral equations, which follow by rendering the potential energy of the solid stationary, are only mildy singular and their discretization by conventional finite-element techniques proceeds without difficulty. The discretized equations are automatically symmetric in consequence of the variational character of the formulation. Atomistic simulations have shown that displacements, and the attendant shear resistance, take place predominantly in the direction of the dominant Burgers vector (Sun e t a l . , 1991; Yamaguchi etal., 1981). These results bear out the constrained displacement hypothesis of Rice (Rice, 1992; Rice et al., 1992) and Sun et al. (1993), whereby the interplanar shear displacement is presumed to be aligned with the Burgers vector direction. Using a reciprocity argument advanced by Needleman (1987) and Rice et al. (Beltz and Rice, 1991; Rice, 1992; Sun et al., 1993), Xu et al. (1995) derived an interlayer potential I$(& by combining the universal binding energy relation of Rose et al. (1981) with a skewed shear resistance profile (Foreman, 1955). The result is
Nanomechanics of Defects in Solids
25
with (78)
and p = -
A* L’
In these expressions, Al is the shear displacement, A2 is the opening displacement, ti is the shear traction, t:! is the normal traction, p is the shear modulus, c is a uniaxial strain elastic modulus, b is the magnitude of the Burgers vector, h is the interplanar spacing, L is the interplanar tensile displacement at r2 = om,,, y;:’ is the unrelaxed unstable stacking energy, yf is the surface energy, A* is the relaxed interplanar tensile displacement at t2 = 0 in the saddle point configuration, and is a skewness parameter in the interplanar shear resistance. Constants representative of a-Fe are collected in Table 1. A traction-displacement relation including the effect of surface production on an inclined slip plane at the crack tip has been proposed by Xu ef al. (1995). Using this approach, Xu et al. (1997) have examined three plausible modes of nucleation: on inclined planes containing the crack front, on oblique planes intersecting the crack front, and on cleavage ledges along the crack front (Figure 3). Their analysis confirms an earlier finding (Xu et al., 1995) that nucleation on inclined planes in a-Fe entails energy barriers that are too high to be overcome, at impending crack advance, at temperatures below the melting point. Contrary to TABLE1 COHESIVE-LAW C O N S T A N T S FOR OI-FE(XU E l 67l.. 1997). REPRINTEDWITH PERMISSION OF TAYLOR A N D FRANCIS
26
Michael Ortiz and Rob Phillips
A
cleavage ledge
(c) FIG. 3 . Alternative modes of dislocation nucleation from crack tips in a-Fe (Xu ~ / d1997). , (a) Inclined plane; (b) Oblique plane; (c) Cleavage ledge. Reprinted with permission of Taylor and Francis.
expectations, the analysis of Xu et al. (1997) has also established that dislocation nucleation on oblique planes in a-Fe requires even higher energies, which translates into transition temperatures well above the melting point. In this nucleation mechanism, the incipient dislocation embryo is kidney shaped and expands in an ostensibly self-similar fasluon from the crack tip (Plate 1). Since in this mode of nucleation no significant free surface is produced, the high activation energy computed is surprising but can be explained by noting that the resolved shear stress driving the dislocation decays as r - ' / * in all directions away from the tip, which tends to stunt the growth of the dislocation. Both the inclined plane modes and the oblique plane modes are instances of homogeneous nucleation, in as much as every segment of the crack front constitutes an equally likely nucleation site. However, numerous experiments (Chiao and Clarke, 1989; Samuels and Roberts, 1989; George and Michot, 1993) have demonstrated that nucleation is a rare event and occurs only at particular sites along the crack front, particularly cleavage ledges. A detailed analysis of dislocation nucleation from cleavage ledges in a-Fe has been reported by Xu et al.
Nanomechanics of Defects in Solids
lo4,
. . . . . . . . . . . . . . .,.
27
. . . . . . . . . ......,
. . . . . . .,
--
Tm
.............................................................
lo3 ............................................ e,
-5a c .-.-e "
. . . . . . .' c
.
;
7 i . $
_
I......'
0 a
.-G-
g
........
""""
( l-v)AUz,/pb3
(b) F I G .4. Dislocation nucleation at a crack tip in a-Fe. (a) Activation energy versus loading parameter GIcd, representing the nominal energy release rate attendant to the remotely applied K field; (b) Inferred B-D transition temperature TBD (Xu rt al., 1997). Reprinted with permission of' Taylor and Francis.
(1997). They take the crack front to coincide with the (110) direction and the ledges to be on (1 12) planes (Figure 3). This mode is favored in two important ways. First, the embryo is of a predominantly screw type and, hence, has a low line energy, and involves no surface production. Indeed, the results of Xu et al. (1997) show that the energetics of t h s mode in a-Fe are so favorable that it borders on being a spontaneous process (Figure 4). The B-D transition temperatures that are estimated for this mode are well within the expected range for low carbon steel, i.e., around 250-300 K (Xu et al., 1997). This mode of initiation of disloca-
28
Michael Ortiz and Rob Phillips
tion activity also furnishes a ready explanation for the observation of George and Michot (1993) that such activity often occurs on planes with low resolved shear stress. The finding of Xu et al. (1997) that dislocation nucleation from a crack tip is aided by heterogeneity is in keeping with most other nucleation-controlled phenomena in nature (see Martin and Doherty, 1976, for a lucid discussion). However, models based on Peierls potentials are approximate at best and, consequently, the results of Xu et al. (1997) must be viewed as qualitative. These limitations notwithstanding, cohesive-zone models are contributing to the understanding of the B-D transition and gradually closing the previously existing gap (Argon, 1987) between theory and observation.
C . OTH E R PATCH ED ATo M 1s T I c/C o N T I N u uM M oD E L s The realization that much of the computation in straightforward atomistic simulations is wasted due to the sufficiency of continuum approximations far from defects is not new. A number of mixed continuum and atomistic models have been proposed in recent years to capitalize on this feature (some were referenced in Section I and others can be found in Tadmor et al. (1996b)). A frequently used method (for reviews, see Vitek, 1988; Stoneham et al., 1988; Daw, 1990) consists of truncating the lattice at some distance away from the defect and holding the atoms on the boundary in their isotropic or anisotropic elastic configuration. This, however, can overconstrain the lattice, e.g., by preventing the change of volume attendant to a discrete dislocation. Flexible boundary methods have been proposed which overcome this difficulty, most notably those of Sinclair and coworkers (Sinclair, 1971; Sinclair et al., 1978; Gehlen et al., 1972). Other models apriori identify both an atomistic and a continuum region and tie them together with some appropriate boundary conditions (Kohlhoff et al., 1991). Yang et al. (1994) have developed a particularly comprehensive mixed model of process zone evolution at crack tips which combines atomistic descriptions near the tip, discrete dislocations at the mesoscale, and continuum plasticity in the far field. However, some features of these approaches contribute to making their implementation onerous. For instance, atoms in the interior and on the boundary of the lattice require different treatment. In addition, cumbersome equilibrium and compatibility conditions need to be enforced between the lattice and the exterior region. These limitations notwithstanding, mixed continuum/atomistic approaches provide a very interesting twist on conventional lattice statics schemes in that they simultaneously reduce the computational overhead and allow for the consideration
Nanomechanics of Defects in Solids
29
of larger length scales than are normally contemplated in the traditional atomistic setting. Such simulations have been especially revealing in consideration of the competition between cleavage and dislocation emission at an atomically sharp crack tip (Kohlhoff et al., 1991; Yang et al., 1994).
IV. Lattice Statics A second means by which atomistic insights can be exploited in the context of the mechanics of materials is through the performance of calculations using lattice statics. Here the idea is the explicit treatment of each and every atomic degree of freedom, although many of the tools associated with these discrete problems have direct analogs in the continuum setting. At the simplest level of modeling, the lattice may be treated in the harmonic approximation where there is a linear relation between applied forces and displacements (Born and Huang, 1954; Maradudin, 1958; Celli, 1961; Boyer and Hardy, 1971; Babiiska et al., 1960; Heinisch and Sines, 1976; Flocken and Hardy, 1970; Holzer and Siems, 1970). The resulting system of equilibrium equations is linear and can be conveniently solved by recourse to Fourier analysis or Green's function (Kanzaki, 1957; Tewary, 1973; Bullough and Tewary, 1979; Thomson et al., 1992). Our emphasis here will be on the analytical headway that can be made in the context of harmonic lattice statics as opposed to the conventional numerical approaches to energy minimization that attend the full nonlinear treatment of such problems. Regardless of whether one exploits the convenient pair potential or pair functional approaches or the computationally demanding but highly accurate density functional approaches, the total energy is ultimately a function E'"' of the atomic positions. The basis of the lattice statics approach to be adopted here is the recognition that, as long as the energy is being computed in the vicinity of some local minimum in the full energy function, it is possible to express this energy in powers of the variables that represent the excursion from local equilibrium, namely, the displacements. Before embarking on a full-fledged treatment of lattice statics, we first consider some geometrical preliminaries. A. BRAVAIS LATTICES
The mathematical abstraction of a continuum as a collection of points and subsets of measurable volume and mass may be constructed using the tools of measure theory and integration. In an entirely similar vein, a mathematical abstraction of a crystal may be built upon the notion of a Bravais lattice. For simplicity, we
30
Michael Ortiz and Rob Phillips
shall consider a simple Bravais lattice of points
x(1) = t a j ,
(83)
where 1 E Z 3 is a multi-index and (ai, 8 2 , 83) is some suitable lattice basis. The dual basis (a', a*, a3) is characterized by the property 8 I . aJ . -8' j .
(84)
Explicitly, the dual basis vectors are given by I
2
'
a = -a2 x
V
83,
1
a = -a3 x at,
V
1 a3 = -a1 x a2, V
(85)
where V = a1 . (a2 x 83)
(86)
is the volume of the unit cell of the lattice. The reciprocal basis (b', b2, b3) is conventionally defined as b' = 2xa'
(87)
and will prove useful in connection with the discrete Fourier transform. The dual and reciprocal lattices are those which are spanned by the dual and reciprocal bases, respectively, and have associated cell volumes l / V and ( 2 ~ )V.~ / The choice of lattice basis is clearly not unique. Any triad (a;, a;, a;) related linearly to the original basis as a!1 = p 1? a j
(88)
also defines a lattice basis provided that p! E Z and (Ericksen, 1979) det(p) = f l .
(89)
Perhaps more importantly for the applications that follow is the fact that the class of matrices pi just defined also represents deformations that preserve the crystal lattice. Thus, affine mappings of the form
y = FX
(90)
such that
FT = F j a ' @a,
(91)
map lattice points into lattice points. A particularly important example of relevance to dislocations concerns integer matrices which are rank-one connected to
Nanomechanics of Defects in Solids
31
the identity, i.e., matrices of the form p: = ti/ + m ; l j ,
(92)
with m, 1 E 2'. Requirement (89) is met provided that the orthogonality condition Pm; = O
(93)
is satisfied. The corresponding deformation gradient is
F = I + ( l j a , ) 8 (mja').
(94)
This deformation represents crystallographic slip on the plane normal to miai in the direction l j a j . B. HARMONIC APPROXIMATION Next, we specialize the discussion on atomistic models in Section I1 to crystals. An appropriate atom-labeling scheme is to identify atoms with lattice sites in some perfect reference configuration of the crystal. Assuming that such a reference configuration can be described as a simple Bravais lattice, the atoms of an infinite crystal are labeled by indices 1 E 2'. Extensions to finite crystals are treated in Section VII. For a crystal, the energy function specializes to the form
Etot= V(x(l), 1 E
z'),
(95)
where, here and subsequently, x(1) denotes the position of atom 1. The crystal is in equilibrium with applied forces F(l) if
One of the key provisos concerning such equilibrium conditions is that they must respect the various symmetries of the crystal lattice itself. A symmetry of the lattice is a linear transformation Q: R' += R3 which brings the lattice into coincidence with itself. The collection of all symmetries forms a group S under composition called the symmetry group of the lattice. Clearly, the energy (95) of the crystal must be invariant under its symmetry group. In order to make analytical tools such as Fourier methods possible, we shall resort to the harmonic approximation. Let
x(l) = X(1)
+ u(1)
(97)
32
Michael Ortiz and Rob Phillips
denote the positions of the atoms after deformation. In (97), u(1) is the displacement of atom 1 and X(l) is the position of the same atom in the reference configuration of the crystal. Expanding (96) in Taylor series up to linear terms in u(1) gives
c
@jk(l, l’)ux.(l’) = Fj(l),
(98)
I‘
where
are the stiffness coefficients, or force constants, of the harmonic lattice. The force constants have the following properties:
C
@itn
(1, 1’)xIl(l’) -
I‘
c
@in
(1, 1’)xm (1’) = 0.
(104)
I‘
Property (100) follows directly from definition (99) and corresponds to equality of mixed partial derivatives. Property (101) is a consequence of the translational invariance of the lattice. Equation (103) expresses the restrictions imposed upon the force constants by the symmetry group S of the lattice. Properties (103) and (104) follow as a consequence of the invariance of the energy density of the crystal under rigid translations and rotations of the lattice. A useful consequence of (101) is that the force constants only depend on the relative positions of the atoms, i.e.,
@j,j[l, 1’) = @ij(l - 1’).
(105)
Further restrictions on the force constants, which facilitate their identification, are obtained from macroscopic properties of the lattice such as the elastic moduli, thermal expansion coefficients, and specific heat. In the harmonic approximation, the energy of the crystal is given by the quadratic form 1 E‘”‘ = -
C @ir((l- l ’ ) ~ i ( l ) ~ k ( l ’ ) , I . I’
Nunornecliunics of Defects in Solids
33
Since the lattice is assumed to be of infinite extent, the displacement field u(1) must decay to 0 sufficiently rapidly at 00 for the sum (106) to be finite. Using (103), an equivalent expression for the energy is found to be
Indeed, (106)is recovered simply by expanding (107) term by term and noting that two of the four resulting terms vanish by virtue of (103). The ability to write down the energies and forces prepares us to consider the explicit solution of boundaryvalue problems which we take up now using Fourier transform methods.
c. LATTICESTATICS SOLUTIONS V I A THE DISCRETEFOURIERTRANSFORM The discrete Fourier transform (DFT) furnishes a powerful tool of analysis in the context of harmonic lattice statics (Babiiska et al., 1960). Given a lattice function f (I), its DFT is f ( 1 ) exp[-ik. x(l)],
f(k) = V
k
E
B,
(108)
I
where V is the unit cell volume, eq. (86), and B is the first Brillouin zone of the crystal. The original function is recovered by an application of the inverse DFT, namely,
The properties of the DFT include the Parseval identity
-
where * denotes complex conjugation, and the convolution theorem
(f * g)(k) = ,f(k)i?(k),
(111)
where .f’ and g are lattice functions and the discrete analog of the convolution operator * is
(f * g)(U = v
c
f (1 - l’)g(l’).
(1 12)
I‘
These identities enable a direct solution of (98), the left-hand side of which is in convolution form by virtue of (105). Thus, an application of the DFT to (98)
Michael Ortiz and Rob Phillips
34
gives the system of three linear equations
where 1 D;k(k) = -6)ik(k) V2 is the dynamical matrix of the lattice and
fi (1)
1 V
= - FI (1)
(115)
may be regarded as a lattice body-force field. The dynamical matrix (1 14) may be shown to reduce to (26) when expanded in Taylor series of up to second order in k about the origin, which establishes yet another link between atomistic and continuum descriptions of material behavior. The DFT of the lattice displacements now follows from (1 13) as Gk(k) = D;'(k)j(k),
k E B,
(1 16)
and the lattice displacements are recovered by an application of the inverse DFT (109), with the result
which furnishes the solution of (98) up to quadratures. This result serves as the cornerstone in the solution of boundary-value problems in the discrete setting. We close this section by noting that integrals over the Brillouin zone B , such as involved in the evaluation of the inverse DFT (109), may conveniently be computed by using the representations
x = Pa;,
k = m;a'
(1 18)
in terms of the lattice and dual bases, respectively, whereupon (109) may be expressed in the form 1 1 f ( l ) = - -/ 2 r v (2n)' 0
s,'" s,'"
f(m) exp(im. I) dml dm2 dm3.
(1 19)
The numerical evaluation of these integrals is straightforward for small values of the lattice indices (l' , 12, 1 3 ) . Large values lead to highly oscillatory integrands whose evaluation requires special techniques (Gallego and Ortiz, 1993; Stroud, 1971).
Nanomechanics of Defects in Solids
35
D. FCC LATTICEMODEL As an explicit example of the ideas developed above, we consider the solution of boundary-value problems involving the FCC lattice. Consider an FCC lattice with lattice parameter a . A suitable lattice basis is
a1 = (a/2)(0, 1, I), a3 =
(a/2)(1, 0, I),
a2 =
( ~ / 2 ) ( 1 1, , 01,
(120)
where all components are defined relative to an orthonormal Cartesian frame coincident with the cubic directions of the lattice. The dual basis is
a' = ( I / u ) ( - l , 1, 11, a3 = ( l / u ) ( l , 1, -1).
a2 = ( l / u ) ( l , -1, I),
(121)
The volume of the unit cell is V = a3/4. A simple model of the energetics in this case is obtained by restricting interactions to the 12 nearest neighbors of each atom. The lattice and Cartesian coordinates of the atoms in the stencil so defined are collected in Table 2. Under these assumptions, it can be shown that the most general form of the force constants which is consistent with the cubic symmetry of the lattice is (Musgrave, 1970; Sengupta, 1988)
B @(O, -1, 1) = - ( 0
0
@(l,O, - 1 ) =
-Y
-( : -Y
-ly)
O a!
0 /3
B),
-Y
0
-Y @(-1, 1,O) = - ( ; y
01
O
,
0 0 ) B
Michael Ortiz and Rob Phillips
36
TABLE2 CARTESIAN A N D LATTICE COORDINATES OF INTERACTING ATOMSI N MODELOF FCC LATTICE
0 1 2 3 4 5
0
0
0
aJ2
aJ2
1
1112
0
(112
uJ2
nJ2 -a12
0 0
0 0
0
0
0 -uJ2
-a12 -uJ2
7
0
8 9
-uJ2 aJ2
0
(112
-a12
10
0 (112
-a12 0
0 u/2
II 12
-1112 uJ2
1112
-1112
-uJ2
-a12
1
0
-1
0 0 I
-1
-1
1 0 0
0
-1
0
-1
1
0
1
I 0
0
-I
0
0 0 0
0 0
-1
-u/2 0
6
0 0
1
-1
1 0
-I
for some constants a , B , and y . The remaining force constants follow from the relation
@;,;(-l) = @ i j ( I )
(1 23)
and from (103), which gives @(O, 0, 0) =
-C@(l) = 4(2a+B) I#O
A standard identity (Musgrave, 1970; Sengupta, 1988) gives the three cubic elastic moduli as 2 2 4 c11 = -a, c12 = -(2y - a - B ) , c44 = -(a B ) . (125) a a a
+
These relations permit the identification of the force constants a , B , and y of the lattice from elastic modulus data. A few selected examples are collected in Table 3. The corresponding dynamical matrix is computed to be
where the remaining components follow by permutation of the indices.
Nanomechanics of Defects in Solids
37
TAIXE3 ELASTICMODULI( e V / A 3 ) .LATTICEPAKAMETLR ( A ) , A N D FORCE CONSTANTS (ev/A2)OF SEI ECTCD FCC CRYSTALS Material
CI I
C12
C41
n
CY
B
Y
Ag Al Au Cu Ni Pb Th
0.759 0.737 1.156 1.032 1.559 0.309 0.468
0.564 0.389 0.976 0.745 0.932 0.263 0.304
0.282 0.229 0.26 I 0.470 0.768 0.093 0.297
4.084 4.032 4.079 3.615 3.523 4.948 5.09 1
0.775 0.743 1.179 0.933 1.373 0.382 0.596
-0. I99 -0.281 -0.646 -0.084 -0.021 -0.15 1
0.864 0.623 1.261 1.098 1.497 0.44 I 0.765
0.161
One may also compute the values of a , p, and y , e.g., from an embedded-atom model. In the case of gold, the values obtained in this manner (Oh and Johnson, 1988) are 0.558, -0.174, and 0.673 for a , p , and y . respectively (cf. Table 3), while those of copper obtained from the same potentials are 0.933, -0.103, and 1.012, which are in close agreement with those in Table 3. In cases where the near-neighbor model is not satisfactory, the formulation may be generalized to include higher neighbors. For instance, Akgun (1993) used force constants up to second nearest neighbors for FCC copper and FCC Cu-Zn alloys and obtained phonon dispersion curves that were in good agreement with experiment.
E. DIAMOND STRUCTURE A general treatment of the force constants of cubic crystals has been given by Sengupta (1988). The particular case of the diamond structure may be understood as the result of superposing two FCC lattices displaced by a vector (a/4) ( I , 1, I), where a is the cubic cell size. Restricting attention to nearest-neighbor interactions, the force constants corresponding to each of the constituent sublattices is of the form (122), whereas the most general force constant matrix coupling the constituent sublattices consistent with the crystal symmetry is of the form (Sengupta, 1988):
where p and 0 are constants. The corresponding values of the elastic moduli are (Sengupta, 1988)
38
Michael Ortiz and Rob Phillips
‘Atom 2 F I G . 5 . Displacement coordinates for linearization of Stillinger and Weber’s polential (Gallego and Ortiz, 1993). Reprinted with permission of the Institute of Physics.
Wei and Chou (1992) have reported ub initio calculations of the force constants of silicon up to eighth nearest neighbors, and used the force constants to compute the phonon dispersion curves. For the case of silicon, a simple set of nearest-neighbor force constants may be obtained by a direct linearization of the Stillinger-Weber potential (8>, (9). With the notation of Figure 5, this gives (Gallego and Ortiz, 1993) 1 2
v2 = -@2(u1 - u o y
for two-body interactions, and v3 = - @ 3
2
2
(133)
for three-body interactions. From the constants provided by Stillinger and We2 ber (1989, a simple computation gives @’ = 10.3337 eV/A and @3 = 0.5663 eV/A2 for silicon. A comparison between the unharmonic potentials (8), (9) and their harmonic counterparts (132), (133) reveals that they are nearly indistinguishable for small departures from the equilibrium configurations of the bonds (Gal-
Nanomechrinics ojDqfecrs in Solids
39
lego and Ortiz, 1993). A calculation of the elastic moduli from the StillingerWeber potential furnishes a first check on the accuracy of the model. A straightforward computation gives
Because the harmonic part of Stillinger and Weber’s potential depends on two constants only, the resulting elastic moduli are related by
For silicon, the actual value of this ratio is equal to 0.85 (Simmons and Wang, 1971), or 15% lower than the prediction of these potentials. This sobering realization illustrates the fact that, despite their conceptual and physical appeal, most atomistic models are themselves phenomenological and, therefore, necessarily approximate. The above example also shows that the adoption of an atomistic perspective is not always an improvement over the corresponding continuum theory. For instance, in a situation that is well described by anisotropic linear elasticity, e.g., the large wavelength deformations of a perfect silicon crystal, the continuum theory trivially permits the matching of all experimentally measured elastic moduli of the material. By way of contrast, a lattice statics calculation based on Stillinger and Weber‘s potential would inevitably introduce the spurious constraint (137), and the elastic moduli would not be matched exactly. Of course, within the harmonic approximation this deficiency may be overcome by the adoption of more extensive systems of force constants, e.g., Zielinski (1991).
F. OTHERCRYSTALSTRUCTURES Force constant models have been formulated for a wide array of crystal classes and to varying degrees of approximation. Models of FCC, diamond structure, and rocksalt crystals and their application to acoustics may be found in Musgrave (1970). A general discussion of the force constants of hexagonal and FCC and diamond structure lattices, including applications to the determination of elastic moduli, may be found in Sengupta (1988). Boyer and Hardy (1971) have devel-
40
Michael Orriz and Rob Phillips
oped force constants for BCC crystals accounting for interactions out to the fifth neighbor shell, and have applied their model to the study of the core of screw dislocations. Force constants extending to 15th neighbors have been computed by Saxena et al. (1995), who proceeded to use the constants for the study of the phonon dispersion curves of BCC zirconium and titanium. Trampenau et al. ( 1993) and Guthoff et al. (1994) have experimentally investigated the phonon dispersion of BCC chromium and niobium over a large temperature range and determined force constants up to the fifth and sixth nearest-neighbor shells. Robertson ( 199I ) measured phonon dispersion curves for iron-aluminum alloys with compositions close to Fe3Al in three states of order: BCC,B2,and D03, and analyzed the data by fitting sixth nearest-neighbor force constants. Akgiin (1993) used force constants up to second nearest neighbors for HCP zinc and determined them from the four experimentally measured independent elastic moduli. As yet, we have constructed the relevant formalism for performing lattice statics calculations. We now undertake the application of such methods to the solution of boundary-value problems involving defect structures in solids.
OF EICENDISTORTIONS G. MURA’STHEORY
The total energy of a crystal is a nonconvex function of the atomic displacements. The significance of this observation is that such nonconvexity allows for the emergence of defects such as dislocations. A seeming deficiency of the harmonic approximation, which would appear to disqualify it as a suitable framework for the study of dislocations, is that (106) is a convex function and therefore is not invariant with respect to lattice-preserving deformations such as those described in Section 1V.A. However, this limitation can be overcome by recourse to Mura’s theory of eigendistortions (Mura, 1987; Gallego and Ortiz, 1993). To this end, we define a lattice distortion B(1, 1’) to be a two-point lattice function B: Z3 x Z’ -+ R’. A lattice distortion field is further said to be compatible if there exists a displacement field u(1) such that
It is easily seen that a given lattice distortion field is compatible if and only if
80, 19 = B(1, 1’9
- B(I’,
I”),
vl” E
z’,
( 139)
Nanomechanics of Defects in Solids
41
which is a lattice compatibility equation. The energy associated with a compatible lattice strain field follows from (107) in the form 1 Ef"' = - -
c
@;k(l - l')B;(l, l')BX.(l, 1').
I. I'
Within the framework of harmonic lattice statics, dislocations and certain other lattice defects may be identified with an incompatible lattice distortion field /3" (1, l'), or eigendistortion, with the additional understanding that the energy of the defective crystal is
This expression is analogous to the continuum form of the energy of an elastic-plastic solid undergoing small deformations (Lubliner, 1990). In general, eigendistortions are constrained by crystallography, and are built from latticepreserving deformations such as the crystallographic slip (94). These restrictions endow the energy density with multiple wells and bring about the expected nonconvexity of the energy. The equilibrium lattice displacements corresponding to an eigendistortion field B E ( l , 1') may be obtained by minimizing the energy function (141) with respect to u(l), with the result
c
@,x (1 - 1 h(1') = F,%
(142)
I'
where F,E
(1) =
c
@,A
(1 - l')@(l', 1)
(143)
I'
is the eigenforce field. The energy of the defective crystal follows from (141) in the form
As discussed in Section IV.C, an application of the discrete Fourier transform yields the solution of ( 142) as
Lik(k) = D , ' ( k ) l E ( k ) ,
(145)
42
Michael Ortiz and Rob Phillips
where 1
f i E ( l ) = - FF(1) V
( 146)
and the lattice displacements are recovered by an application of the inverse DFT, with the result Uk(1)
'
= - DL'(k)&E(k)exp[ik.x(l)]d3k. (2n13 B
(147)
Finally, Parseval's identity (110) and the convolution theorem (1 11) enable the energy (144) to be recast as
which provides, up to quadratures, an explicit analytical expression for the energy of the defective crystal. For the solution to be valid, the computed displacements must bear out the assumptions regarding which energy wells are operative. The precise form of this restriction will, in general, depend on the crystallography, the level of applied load, and the nature of the defects studied. 1. Screw Dislocation Dipole in a Square Lattice As a first illustration of Mura's theory, we treat the elementary case of a screw dislocation dipole in a square lattice. In this case, the Brillouin zone is B = [ - n / u , n/ul*, where a is the lattice parameter. Assuming nearest-neighbor interactions only and using identity (103), the force constants are found to necessarily be of the form
=
1
-pa, 4pu,
0,
+
Il'I 11'1 = 1, 1 = 0, otherwise.
(14%
The sole nonzero component of the dynamical matrix is a
+ sin* *). 2
Expanding this expression up to quadratic terms in k and comparing with (26) identifies p as the shear modulus of the lattice.
Nanomechanics of Defects in Solids
43
Let the dislocations in the dipole be spaced N lattice cells apart, where we take N to be odd, N = 2n 1. Placing the dipole center at the origin and talung the plane x2 = 0 as the cut plane, the eigendistortions are
+
where b = (0, 0, a ) is the Burgers vector. This eigendistortion field represents a distribution of crystallographic slip over an interval on the cut plane bounded by the dislocations. The corresponding eigenforces are
where r = N a is the length of the dipole. From (145), the displacement field is found to be
As expected, the continuum limit is realized by formally letting u upon
--f
0, where-
The inverse Fourier transform of the function ( a / k 2 ) ( k 2 / k l )is a 0 / ( 2 n ) ,and the displacement field (154) is that of two linear elastic screw dislocations of Burgers vector a at X I = f r / 2 . The energy per unit length of dislocation follows from (148) as
Carrying out the integral with respect to k2 explicitly gives
where
Michael Ortiz and Rob Phillips
44
Tnntrl 1 NORMALIZED ENER(,ICS
I'tK U N l l LLNGTH
( E ' " ' / L ) / ( p 2 )OF LATTICE
A SCREW D I S L O C A T I O N D I P O L E I N A S Q U A R E
N
CN
Continuum
0 I 3 5 1
0 I I4 0.43028 1 0.5 I2902 0.566760 0.606874
-cxJ
-
0.274056 0.448906 0.530206 0.583757 0.623755
9.62 4.32 3.37 3.00 2.78
9
Error ( S )
The continuum limit of the energy corresponds to the behavior of (156) for large N . A straightforward analysis gives
E'" -
N
L
pa'
r
2n
ro
M
0.178718~
-log -,
where U
ro = -ePY n
(159)
is the dislocation core cutoff radius predicted by the theory. The preceding example illustrates how consideration of the discreteness of the lattice eliminates the divergence at the dislocation core and determines a precise value of the core cutoff radius ro. The normalized energies ( E ' " ' l L ) / ( w a ' ) of small screw dislocation dipoles predicted by harmonic lattice statics and continuum elasticity are compared in Table 4. As expected, as the size N of the dipole becomes large, the lattice energies exhibit the logarithmic variation characteristic of linear elasticity. By contrast, the discrepancies are large for small dipoles.
2. Lomer Dislocation in Silicon Gallego and Ortiz ( 1993) have applied Mura's method of eigendistortions to the analysis of a Lomer dislocation core in silicon. This dislocation has been studied by Nandedkar and Narayan (1990). While Lomers are not the most common dislocations in silicon, they nevertheless are of some interest for their role in the nucleation of stacking faults, the accommodation of interfacial misfit, the formation of subgrain boundaries, and other phenomena (Hornstra, 1958; Tan, 1981; Bourret et al., 1982; Alexander, 1986). A Lomer dislocation in silicon may be obtained by cutting the crystal through a half plane of the (001) type, the edge of the cut, or dislocation line, lying on
Nanomechanics of Defects in Solids
45
W h
4
8
0
0 9
9
0
0 9
.
0
0
0 9
0 0
8
0 8 0 @ o e o e a . 0 . 0
0
o @ o 9 o
0
0
( B e 0
(b) F I G . 6. Calculation of eigenforces: (a)Cutting plane in the perfect lattice; (b) Equivalent eigendistortions (Gallego and Ortiz, 1993).Reprinted with pemiission of the Institute of Physics.
the [ 1101 direction, and subsequently displacing the upper half of the crystal by a distance b = a / & in the direction [ilO], as shown in Figure 6. Using the Stillinger and Weber (Stillinger and Weber, 1985) potential, eqs. (8) and (9), and the corresponding harmonic approximation, Section IV.E, the eigenforces describing the Lomer dislocation are computed to be (Gallego and Ortiz, 1993)
where PI = 55.57 eV/W, Ql = 1.63 eV/& P2 = 18.30 eV/& and Qz = 0.38 eV/A. This system of forces is self-equilibrated. Isolated dislocations follow in the limit of W -+ 00. For convenience, the eigenforces may be grouped into multipoles of zero resultant force and moment. The structure of these multipoles is shown in Figure 7. The forces at C" and D'' are
46
Michael Ortiz and Rob Phillips
D
7 7
FIG. 7 . Structure of multipoles for Loiner dislocation in silicon (Gallego and Ortiz, 1993). Reprinted with permission of the Institute of Physics.
respectively. The complete system of eigenforces consists of a semi-infinite row of multipoles. Finally, the atomic displacements corresponding to the Lomer dislocation are obtained analytically up to quadratures by inserting the eigenforce system just defined into eq. (147). Gallego and Ortiz (1993) have also performed a fully nonlinear analysis of the Lomer dislocation in silicon. They express the displacements as the effect of unknown forces applied near the core to a perfect harmonic lattice of infinite extent. Displacements are related to the unknown applied forces by means of the Green’s function of the perfect harmonic lattice. In this manner, equilibrium at 00, where the behavior of the crystal is asymptotically harmonic, is ensured. The unknown forces, which decay rapidly away from the core, are determined so as to maximize the complementary energy of the crystal, as computed from the full anharmonic potential. Tewary (1973) (see also Bullough and Tewary, 1979) developed a similar Green’ function method for unharmonic lattice statics. The resulting harmonic and unharmonic core structures are shown in Figure 8. In both cases, the core consists of a pentaring and a heptaring (five-atom ring and seven-atom ring, respectively). Indeed, this is one of the core structures suggested by Hornstra (1958) for edge dislocations on the {loo)plane, and is characterized by the fact that it does not contain any dangling bonds. Though Lomer dislocations in silicon often dissociate into Shockley partials, they have also been observed undissociated (Bourret el al., 1982), which suggests that perfect Lomer dislocations are stable. It is probable that Lomer-Cottrell locks have a lower energy than perfect Lomer dislocations, which would account for their abundance. It is interesting that the core structure predicted by the harmonic analysis remains stable when unharmonic effects are taken into account. Indeed, the unharmonic region, conventionally defined as the collection of bonds which are stretched in excess of 1%, only extends six atomic distances away from the dislocation center, The harmonic solution, however, is not without deficiencies. For instance, the harmonic core exhibits an unphysical lack of symmetry (Figure 8a),
Nanomechanics of Defects in Solids
(a)
47
(b)
FIG. 8. Lomer dislocation in silicon computed using Stillinger-Weber potentials. (a) Harmonic approximation; (b) Fully relaxed unhamionic solution (Gallego and Ortiz. 1993).Reprinted with permission of the Institute of Physics.
which stems from the breakdown of the harmonic approximation near the core. Thus, in the harmonic approximation, equilibrium is established with reference to the undeformed lattice, and, because the eigendistortions used to introduce the dislocation are unsymmetrical, so is the resulting harmonic solution. This situation corrects itself when nonlinear effects are fully taken into account. By enforcing equilibrium in the deformed configuration of the lattice using the anharmonic potential, the core solution exhibits the full symmetry of the parent lattice (Figure 8b).
3. Nascent Dislocation Loops in FCC Crystals The tools of analysis developed in the foregoing have been applied to the calculation of the structure and energies of nascent dislocation loops in some FCC crystals. It is necessary to begin by defining the unit eigendistortion at (0, 0, 0), or loopon, as
BE"(l, 1') = b[S(1 - ~ 3 S(1') ) - S(1) S(1' - ~ 3 ) ] b +(I - z3)S(1' - ZI) - S(1 - Z I )6(1' - Z j ) ] 2 b + -[S(Iz j ) S(1' - ~ 2 ) S(1 - ZZ) S(1' - ~ 3 ) ] , 2
+
(162)
where b = (b/&)(Oll) is the Burgers vector, b = a/&. In (162), S(1) represents the discrete delta function, i.e., S(1) = 1, I = 0 and S(1) = 0,1 # 0; and we write z1 = (1, 0, 0), z2 = (0, 1, 0), and z1 = (0, 0, 1). To stabilize this loopon,
48
Michael Ortiz and Rob Phillips
i.e., to ensure that the energy well introduced by the eigendistortions is indeed operative, the homogeneous state of deformation
F=I+yms@m
(163)
needs to be imposed on the crystal. Here, s = (0, 1, 1)/& and m = (1, 1, - 1 ) / 8 in components relative to the cubic axes, and y" is chosen such that the total displacements satisfy the condition that
( ~ ( 00,, 1) - ~ ( 00 , 0 ) ) . b = b / 2 ,
(164)
where b is the unit vector along b. The corresponding resolved shear stress on the slip plane is
tm
where p is the effective shear modulus defined by P=
CII +C44-c12.
3
The displacements due to the eigendistortions may be evaluated using (147) and the corresponding energy of the crystal follows from (148). The potential energy A@, activation energy AU,,,, and the resolved shear stress 7" required to stabilize a loopon have been calculated for a variety of FCC crystals. The potential energy A@ is computed relative to the homogeneously deformed lattice. In order to compute the activation energy AU,,, between two eigenstates, a one-parameter family of eigendistortions joining the eigenstates may be defined. Then nu,,, represents the difference between the maximum and initial potential energies along the path. The results of the calculations are tabulated in Table 5 . The negative value of the potential energies denotes a relaxation of the crystal with respect to the homogeneously strained state, as expected. It can be shown that, for the simple near-neighbor model considered, the strength of the i.e., the maximum possible resolved shear stress on the slip plane, is crystal tmax, given by tmax =g
p
x 0.612p,
(167)
while the resolved shear stress required to stabilize the loop is of the order of 0 . 9 ~At~ the ~ ~resolved . shear stresses listed in Table 5, the activation energy for the nucleation of a loopon is found to vanish; i.e., the loopons form spontaneously.
Nunomechanics of Defects in Solids
49
TABLE 5 RESOI.VEI>S H E A R S T R E S S REQLIIREI)T O S'I'ABll.lZE T H E SMAL.1.EST DISLOCAI'ION LOOP IN St:VEI
Material
A0
Ag
- 1.27768 I
Al
- 1.334087
A ti
-2.561564
cu
- I.097950
Ni Pb
- 1.057933
Th
- 1.535036
T,--IP
0.542755 0.528888 0.570245 0.526683 0.5 13675 0.568959 0.533369
- 1.575 13 I
The loopon eigendistortions (162) can be aggregated to define arbitrary loops. For the present case, consider the specific example of a hexagonal loop corresponding to loopons placed at (0, 0, 0), (1, 0 , O ) , ( 0 , 1, 0), (-1, 1, 0), (- 1, 0 , 0), ( 0 , - 1, 01, and (1, - 1, 0). The calculated potential energy, activation energy, and resolved shear stress for hexagonal loops in selected FCC crystals are tabulated in Table 6. In this case, the nucleation is not spontaneous; i.e., an activation barrier needs to be overcome in order for the hexagonal loop to be formed at the resolved shear stress for which it is stable. An arrow plot of the displacements computed for a hexagonal loop in copper is shown in Figure 9.
TABLE 6
E N E R G YD R O P ( e V ) F R O M T H E HOMOGENEOUSLY DEFOKMED LATTICE.ACTIVATION ENERGY( e v ). A N D EQUILIUI
P H I L L I P S , LINPUULISHED)
Material
A0
AUK,
Ag Al A ti
-7.037581 -7.034 I63 - 13.993365 -5.880701 -8.140570 -6.0468 I7 -9.82 I797
0.128037 0. I492 13 0.0295 I8 0.275247 0.600260 0.0 I9922 0.253 190
Cu Ni Ph Th
~ " I I ~ 0.465264 0.445600 0.495855 0.439 I27 0.430613 0.508019 0.480566
Michael Ortiz and Rob Phillips
50
0
z
0
0
0
0
m
-_ . ;
2
- 1 I
J - 7
X
7 0
FIG.9. Arrow plot of displacements due to a hexagonal dislocation loop in copper. Open circles I 3 = I. The longest arrow in the top figure corresponds to 0.744 A. The X axis corresponds to a (01I ) crystal direction, Y corresponds to (2i I ). and Z to ( 1 1 i). Length is in angstroms. The displacements due 10 denote the atoms on the plane l 3 = 0. while solid circles represent atoms on the plane
the homogeneous strain are not included (Shenoy, Ortiz. Phillips. unpublished).
It is tempting to speculate that, in some materials, the stress required to nucleate a loopon may be attained in the vicinity of a crack tip prior to the extension of the crack by cleavage, especially when thermal activation is operative. Indeed, it is evident from Table 6 that, provided that the resolved shear stress is high enough, the activation energies involved in the early stages of expansion of a dislocation
Nanomechanics of Defects in Solids
51
loop may be quite modest in some materials. Under the right conditions, this might lead to an avalanche of dislocation loop nucleation and growth near the crack tip, with the attendant plastic shielding resulting in ductile behavior. This scenario is consistent with the theory of the brittle-to-ductile transition recently advanced by Khantha et al. (1997). The above examples demonstrate that harmonic lattice statics may be a useful tool for the analysis of dislocation core structures. The chief advantage of harmonic lattice statics is that it lends itself to analytical treatment and leads to simple analytical representations of the displacement fields and the energy of the dislocation. However, the harmonic approximation tends to break down near the core, where unharmonic effects are important. In particular, the core energies predicted by the harmonic theory may be considerably in excess of the fully relaxed energies.
V. Cauchy-Born Theory of Crystal Elasticity Some models of crystal elasticity stand to benefit greatly from the type of atomistic calculations discussed above. The intention of this section is to explain the significance of such models and to offer insights into how such models may be informed by atomistic calculation. Perhaps the simplest paradigmatic example of a continuum theory is furnished by anisotropic linear elasticity. Within the confines of this theory, the deformation of a material has definite calculable consequences. In particular, we note that if a material is forced to suffer a deformation that is parameterized via the small strain tensor E , , , then the strain energy density is given by
in terms of the elastic moduli c, j k l , and the total energy of the solid may be written simply as E'O'
=
J,, W ( r ) d V .
For our purposes, one of the significant and immediate observations that can be made is that, by conduit of the elastic modulus tensor, the energy associated with the possibly complicated atomic rearrangements that may accompany the deformation has been subsumed into a few material parameters. The linear elastic moduli serve as a convenient point of departure in our examination of the ways
52
Michael Ortiz and Rob Phillips
in which atomic-level insights may be brought to bear on the elastic-possibly nonlinear-behavior of a material. As noted in the previous section, once the total energy has been determined as a function of the atomic positions, any number of ancillary questions may be addressed. One such question is that of the energy cost of infinitesimal deformation. Recall that in Section IV we have written the total energy of a crystal in powers of the atomic displacements. Those terms quadratic in these variables are precisely the source of the linear elastic moduli. By way of contrast, in this section we are primarily interested in the emergence of nonlinear effects once the deformation exceeds the regime that may be safely treated using linear elasticity. One route to connecting the energetics of finite elastic deformation to analysis at the atomic scale is the use of the so-called Cauchy-Born rule. In this context, it is imagined that each point in the solid is represented locally by an infinite crystal subjected to homogeneous deformation. A consequence of the homogeneity of the local deformation is the loss of a global origin linking the underlying crystal lattice to the continuum. As a result, the choice of representative atom is immaterial since all atoms are equivalent. We can thus imagine our infinite crystal as surrounding a representative atom located at the origin. Following the Cauchy-Born prescription (Milstein, 1982; Chu and James, 1995; Zanzotto, 1996),this infinite crystal is deformed according to the local continuum deformation gradient. In order to render the kinematics in mathematical terms, let
be the coordinates of the atoms in the undistorted configuration of the crystal. Here {A/, I = 1, 2, 3} is a Bravais basis for the undistorted crystalline lattice. The positions of the atoms in the deformed configuration are then taken to be
where F is the local deformation gradient. To illustrate the fundamental geometric ideas, we begin with an example that may be easily visualized. Consider the transformation of a crystal from a cubic reference state into a deformed orthorhombic crystal; i.e., the angles between the Bravais lattice vectors remain 90°, but the lengths of the three axes are changed. The initial and final Bravais lattices are connected as described above, with the deformation gradient itself being nonzero only in its diagonal elements, which are F I I = a, F22 = f i , and F33 = y . This simple geometric example already contains within it the seeds of the energetic ideas that the present section aims to describe. In particular, from the standpoint of the elastic energy, the transfor-
Nanomechanics ojDefecrs in Solids
53
mation just described brings with it both nonlinearity and nonconvexity (Chu and James, 1995). Within the Cauchy-Born prescription, the strain energy density W(F) follows simply as the energy per unit volume of the uniformly distorted lattice and may be computed using an atomistic potential. The crystal is subsequently regarded as a nonlinear elastic continuum endowed with the resulting strain energy density. In particular, the first Piola-Kirchhoff stress tensor and the Lagrangian tangent stiffness tensor follow as
(173) Thus, the presumption is that the local deformation gradient F(X) is truly the gradient of a continuum deformation mapping ~0 defined over the reference configuration of the crystal, i.e.,
and that the equilibrium deformations obey the classical field equations of continuum elasticity (e.g., Marsden and Hughes, 1983). As an example, one may consider the case in which the atomic-level interactions are described by the embedded-atom method, eq. (6). Then, the strain energy density follows in the form
where Q is the unit cell volume. A tedious calculation then gives (Tadmor et af., 1996b)
and
Michael Ortiz and Rob Phillips
54 where
and a?r(l) ~ F , Ja F L L
-
[ 6 , r ( r ( 1 ) ) 2- - ~ l ( ~ ) x r ( l ) ] x , ( I F) ~- l/ (Fl )p l J ,
L/
.
(180)
(r(1))'
A more complete treatment of the higher-order elasticity of cubic metals in the framework of the embedded-atom method has been given by Chantasiriwan and Milstein ( 1 996). They have shown that, in order for the EAM to model the anharmonic properties accurately, at least the third-nearest-neighbor interactions must be included in the expressions for the cohesive energies of both BCC and FCC metals. The Cauchy-Born theory is consistent with the two leitmotifs of this paper: the exclusive use of atomistic information to construct descriptions of material behavior; the desirability of formulating boundary-value problems which are tractable by functional-analytical means. It also exemplifies the relation between lack of convexity and the emergence of microstructures. Indeed, hyperelastic models of single crystals built upon the Cauchy-Born rule lack quasiconvexity as a direct consequence of the periodicity of the lattice and the existence of lattice-preserving deformations (Fonseca, 1987, 1988; Chipot and Kinderlehrer, 1988; Dacorogna, 1989). Under these conditions, the infimum of the potential energy of the solid is not realized in general; i.e., there may not exist any energy-minimizing deformation mapping. It is often possible, however, to construct minimizing sequences of deformations which attain the minimum energy in the limit. Minimizing sequences often exhibit increasingly fine oscillations (see, e.g., Ball and James, 1987; Chipot and Kinderlehrer, 1988).This unphysical fineness of the microstructures is permitted by the local character of the material description, which lacks an intrinsic length scale. The locality of the Cauchy-Born model also precludes key material properties such as stacking-fault energies and surface energies from being properly accounted for. In other areas of application, regularized theories have been devised by building additional physics into the model, such as surface energy, capillarity, bending, and exchange energy. An effect of these higher-order terms in the energy is to introduce an intrinsic length scale 1 which sets a lower bound for the fineness of the microstructure. Additionally, the higher-order terms may have a direct influence on
PLATE1, Dislocation emission from a crack tip into an oblique plane (Xu el al., 1997).(a) Boundary element mesh; (b) Level contours of slip in the direction normal to the crack front, showing a stable configuration of the emitted dislocation loop. Reprinted with permission of Taylor and Francis.
PLATE2 . Structure of Z 5(210) symmetric tilt boundary in gold as computed using quasicontinuum method and by a full atomistic simulation. The solid circles show the equilibrium positions of the representative atoms predicted by the quasicontinuum method; the open circles show the equilibrium positions of the atoms as predicted by the atomistic calculation (Shenoy et al., 1998b). Reprinted with permission of Elsevier Science Ltd.
PLATE3 . Mesh used in a quasicontinuum analysis of dislocation-grain boundary interaction in aluminum; two snapshots from the deformation history showing absorption of a Shockley partial pair into the boundary and dislocation pile-up after a second nucleation event (Shenoy et al., 1998,). Reprinted with permission of the American Physical Society.
PLATE4. Level contours of displacement in the y direction showing a dislocation junction formed by two conventional edge dislocations in aluminum (courtesy of D. Rodney).
PLATE5. Level contours of displacement in y direction showing the interaction between a conventional edge dislocation and a sessile Lomer dislocation in aluminum (courtesy of D. Rodney).
Nanomechanics of Defects in Solids
55
the microstructure, e.g., by promoting twin branching (Kohn and Miiller, 1992). Yet another simple device to eliminate unphysically fine structures from the solution is to restrict the minimization of the energy to deformations such that the support of their Fourier transform is contained in the ball Ikl < 2n/1 = A, where A is the wavenumber cutoff. This device is widely used in statistical field theories (e.g., Chaikin and Lubensky, 1995) to eliminate “ultraviolet” divergences, and is similar in spirit to the use of a core cutoff radius in the theory of linear elastic dislocations. Despite these limitations, the Cauchy-Born theory has been found useful in certain specialized applications, such as in the investigation of instabilities of cubic crystals associated with the vanishing of shear moduli under hydrostatic pressure (Milstein e f al., 1996). These techniques, especially when coupled to statistical mechanical descriptions of entropic effects, e.g., using a quasiharmonic formulation, provide a promising avenue for characterizing the equation of state of materials at high pressures and temperatures (Nagel er al., 1997a, 1997b, 1997c; Fultz et al., 1997).
VI. Quasicontinuum Theory The preceding sections have demonstrated the key insights that atomic-level analyses bring to the study of defects in materials. It has been seen that often in the vicinity of the defect core, there are large atomic rearrangements that arise by virtue of the nonlinearity and nonconvexity of the total energy as obtained from microscopic analyses. It is the purpose of the present section to shed further light on a computational scheme which has been developed with the aim of respecting the type of atomic-level detail that is clearly needed near the defect core without abandoning the entirely satisfactory description of the far field provided by continuum descriptions. In particular, we review the quasicontinuum (QC) theory of Tadmor et al. (1996a, 1996b). The theory starts from an underlying conventional atomistic model and strives to systematically eliminate redundant degrees of freedom. This is accomplished by constraining possibly large numbers of atoms to move in concert with a piecewise linear displacement field in regions where the fields vary slowly. This operation may be regarded as a projection acting on the full atomistic phase space. We consider throughout a crystal whose N atoms occupy a subset of a Bravais lattice
X(I) = ?a;,
IE
c c z“,
56
Michael Ortiz and Rob Phillips
where d 6 3 is the dimension of the lattice (see Section 1V.A). The coordinates of the atoms in a deformed configuration of the crystal are (x(l), 1 E C).For convenience, we shall collect all atomic coordinates in an array x and regard such an array as an element of the linear space RN" = X , the configuration space of the crystal. The energy of the crystal is then given by a function Etot(x),and its potential energy is
t is the potential energy of the applied forces. We wish to characterwhere W x(x) ize the stable equilibrium configurations of the crystal. The problem is, therefore, min @(x), X€X
i.e., to find the energy-minimizing configurations. The central idea of the quasicontinuum method is to replace this minimization problem with a reduced problem in which only a subset of the full set of atomic positions is considered. In the vicinity of lattice defects such as dislocation cores, full atomic resolution is maintained, while in the far field regions the theory reduces to a continuum description.
A. INTERPOLATION
The essence of the static theory of Tadmor et al. (1996a, 1996b) is to replace (183) by a constrained minimization of @(x) over a suitably chosen subspace X I , of X . In order to define X I , , we begin by selecting a reduced set &, c C of Nl, < N "representative atoms". The selection of the representative atoms is based on the local variation of the fields and is discussed in Tadmor etal. (1996a, 1996b). In addition, we introduce a triangulation 5,of Cl,.It bears emphasis that the triangulation ;% may be unstructured. In particular, &, need not define a Bravais lattice. The positions of the remaining atoms are obtained by piecewise linear interpolation of the representative atom coordinates, much in the same manner as displacement fields are constructed in the finite-element method (e.g., Hughes, 1987; Zienkiewicz and Taylor, 1989). We shall regard the resulting displacement fields {xll(l), 1 E L ) as belonging to a linear space Xll of dimension N1,d. Let $h (X I 11,). 11, E &, be a collection of shape functions for Z,. Thus, $h (X [ 11,) is continuous and piecewise linear, its domain is restricted to the simpleces K E lh connected to X(1/,), and it vanishes at all nodes of the triangulation except at X(lrl), where it
Nanomechanics of Defects in Solids
57
takes the value 1, i.e.,
By construction,
where we write
Evidently, {& (1 1 1/1), 4,E & ) constitutes a basis for X h and the fields Xh (1) are entirely determined by their values xh (lh) at the representative atoms. It is clear that to every configuration x E X there corresponds the unique interpolant xlr E X I , given by
This relation defines a projection
from X onto xh. Indeed, it is readily verified that
as required. We note that the projection PI?is not orthogonal in any obvious sense. PI, decomposes X into the direct sum X = X I , @ AX/,, where X I , is invariant under PI,and P/,AX/, = { O ) . In the limiting case in which every atom is accounted for in the reduced model, the projection PlI trivially reduces to the identity and SXI, = (0).
B. REDUCEDPROBLEM What has been accomplished thus far is a reduction in the number of degrees of freedom contemplated in the minimization problem. The reduced counterpart of problem (183) is now
58
Michael Orfiz and Rob Phillips
The minimizers of the reduced problem follow from the reduced equations of equilibrium kC
Here,
f(x) = - @ , x
(192)
(XI
are the forces corresponding to x and f(l 1 x) is the value of f(x) at site 1. Thus, the reduced problem entails the solution of the Nlld equations (191) in Nhd unknowns.
C. SUMMATION RULES The practicality of the method further hinges on the possibility of avoiding the calculation of the full atomistic force array f, as seemingly required in (191). As noted by Tadmor et al. (1996a, 1996b), this may be accomplished by the introduction of summation rules similar to the conventional quadrature rules of numerical integration. The problem is thus to approximate sums of the general form
s=
c
f (1).
(193)
l€C
We may resort to the triangulation ;rh to devise summation rules. In particular, we seek summation rules of the form
s
n/t(l)f(l)
E
sh
( 194)
for some suitably chosen collection of summation points Sh and weights nl7(1), not necessarily integer. Loosely speaking, the weights nh (1) may be regarded as the number of atoms represented by the site 1. Proceeding as in the development of numerical quadrature rules, the weights n h (I) may be determined by the requirement that the summation rule (194) be exact for a restricted class of lattice functions. The use of a summation rule finally reduces the equilibrium equations (1 91) to the computationally efficient form fh(4,)
= C n h ( l ) f ( l I X h ) $ k ( 1 I lh) = 0.
(195)
Itsh
If the number of sampling sites in &, is of order Nh and the atomic interactions , desired. are short ranged, the calculation of (195) is of complexity O ( N / , ) as
Nanomechanics of Defects in Solids
59
1. First-Order Summation Rule The lowest-order summation rule is obtained by requiring that all lattice functions in xh be summed exactly. This is tantamount to setting S/, = CI, and requiring that the summation rule (194) be exact for all shape functions el7(I 1 lh), lh E CI,.This gives, explicitly, equations nh (hi)=
@/I
(1 I I h ) ,
]/I
E ~ I I .
(196)
l€L
In fine regions of the triangulation, the sums on the right-hand side of (196) may simply be computed explicitly. By contrast, in coarse regions of the mesh approaching the continuum limit this explicit calculation becomes impractical. However, in such regions the lattice sum (194) ostensibly reduces to an integral and the corresponding weights are those of conventional Lobatto quadrature (Hughes, 1987). In this limit, each simplex K E 5, simply contributes ( N / V ) I K l / ( d 1) atoms to each of its d 1 nodes, where N / V is the atom density in the undeformed configuration of the crystal. As a simple illustrative example, consider a monoatomic chain discretized into elements, each containing L bonds. For an internal node, the summation weight follows as
+
+
L-1
,
which coincides with the continuum limit. By contrast, for an end node the summation weight is
We note that nh tends to the continuum Lobatto limit of L / 2 as L -+ reduces to nh = 1 in the full atomistic limit, as required.
00
and
2. Second-Order Summation Rule A second-order summation rule may be obtained by requiring that all products of shape functions be summed exactly. This may be accomplished by choosing S/, to consist of the representative atoms C/,and additional points near the midpoints of all simplex sides. For example, in the case of d = 1, the second-order rule requires the addition of one point 1~ in the interior of each element K . If I/, and 1; are the representative atoms connected to K , the weight corresponding to IK
Michael Ortiz and Rob Phillips
60 follows as
Furthermore, if K and K ’ are the elements connected to a representative atom I;,, the corresponding summation weight is
w,(l/,)=
C#lr(1I 11,)
- W , ( I K ) ~ / ~ ( ~I 11,) K -n
/ f ( l r ) ~ / l (I ~1Iz). K~
(200)
I€C
Higher-order summation rules may be obtained by requiring higher-order products of shape functions to be summed exactly. Other alternative summation rules have been discussed in (Shenoy et al., 1998b).
D. MESHADAPTION A key tenet of the quasicontinuum theory is that it be adaptive; i.e., it should automatically provide adequate resolution of the-a priori unknown-fine structure of the deformation fields. Since the basis of the theory is the elimination of superfluous degrees of freedom, a suitable framework for mesh adaption may be based on an investigation of the errors incurred as a result of the degree-of-freedom reduction. The aim is to identify regions where this error estimator is high, and subsequently add degrees of freedom in these regions. Unfortunately, no systematic approximation theory, including bounds on interpolation errors, appears to be in existence for discrete atomistic models. We therefore adopt a continuum perspective and assume that the Cauchy-Born rule is in force (cf. Section V). Then, we may appeal to finite-element literature, where error estimators and automatic mesh refinement have been subjects of extensive research. Recall that the representative atoms are also nodes on a finite-element mesh of constant strain triangles. A simple heuristic error estimator has been introduced by Zienkiewicz and Zhu (1987) in terms of stresses and later modified by Belytschko and Tabbara (1993) to estimate errors in the strain fields. For convenience, we choose to write the Zienkiewicz-Zhu error estimator in terms of the deformation gradient F. Thus, we define the discretization error in element e as
where FR is the deformation gradient in element e, and FI,is a local L2-projection of FI, obtained by averaging the deformation gradients at the nodes. The integral
Nanomechanics of Defects in Solids
61
in eq. (201) can be computed accurately and efficiently using a Gaussian quadrature rule. Elements for which the error ce is greater than some prescribed error tolerance TOL are targeted for refinement. Refinement then proceeds by adding three new representative atoms at the atomic sites closest to the midsides of the targeted elements. Notice that since representative atoms must fall on actual atomic sites in the reference lattice, there is a natural lower limit to the element size. If the nearest atomic sites to the midsides of the elements are the atoms at the element comers, the region is fully refined and no new representative atoms are added. Actual examples of evolving mesh refinement are given in subsequent sections. In addition to mesh refinement, mesh coarsening is also an important requirement. For example, consider the passage of a dislocation. As the dislocation moves, it leaves a trail of fully refined mesh in its wake corresponding to previous core positions. Far behind the dislocation the solid is undistorted and the high mesh resolution is unnecessary and may be coarsened at no significant loss of accuracy. In order to effect this coarsening, the following algorithm is applied: (1) For each local node/atom, the elements surrounding the node and the polygon defined by their outer sides are identified. (2) If none of these elements satisfy the adaption criterion, remove the current local node and create a new Delaunay triangulation of the outer polygon. (3) If none of the new elements satisfy the adaption criterion, then the local node and all the old elements connected to it are deleted and the new elements are accepted. Essentially, the idea is to examine the necessity of each node. To prevent excessive coarsening of the mesh far from defects, the nodes corresponding to the initial mesh may be protected from deletion.
VII. Applications of the Quasicontinuum Method The use of numerical simulation to examine materials offers promise as a means of uncovering the fundamental mechanisms that give rise to their mechanical properties. Some of the key mechanical properties for which direct numerical simulaton could be hoped to deliver meaningful insights concern plasticity and fracture. The quasicontinuum method has emerged as a viable alternative to traditional lattice statics, and offers the possibility of direct simulation, with atomistic resolution at defect cores, of systems demanding the application of remote boundary conditions of the same type favored in traditional continuum mechanics modeling. The goal of the present section is to showcase selected examples of the uses of the method to date. These examples serve the dual purpose of validating the method and illustrating its possible application to problems in the mechanics of materials.
62
Michael Ortiz and Rob Phillips
We have selected nanoindentation as the first representative of application of the quasicontinuum method. Indeed, recent experimental work has revealed an array of interesting features which call for numerical simulation (Gerberich et al., 1996; Zielinski et al., 1995). This discussion is followed by examples of application of the quasicontinuum method to problems involving internal interfaces. In particular, we demonstrate the scope and versatility of the method in this respect by simulating the complex interactions which ensue when a crack encounters a grain boundary in its path. As a final example, we consider preliminary threedimensional quasicontinuum simulations aimed at uncovering the structure and energetics of dislocation junction formation. As remarked above, the examples collected in this section are primarily intended as an illustration of the ability to formulate conventional boundary-value problems of continuum mechanics in a quasicontinuum setting without essential limitations on the size of the system; and the seamless bridging of the continuum and atomistic realms by the quasicontinuum method. The results that have emerged thus far have shown that, in conjunction with the appropriate contact with experiments, such calculations show promise for yielding mechanistic insights into the nanomechanics of materials.
A. NANOINDENTATION
Nanoindentation constitutes the earliest application of the quasicontinuum method, and serves as a strjngent validation test of the approach (Tadmor et al., 1996a, 1996b).The calculations were motivated by a host of recent experiments in which load-displacement curves and subsurface dislocations have been measured (Gerberich et al., 1996). One of the critical questions that arise in this setting concerns the conditions attendant to dislocation nucleation. Upon indentation, and after a preliminary elastic stage, the onset of permanent deformation is mediated by the nucleation and propagation of dislocations. As a first investigation of this problem, a two-dimensional quasicontinuum analysis was performed for aluminum. The embedded-atom potential, as fitted by Ercolessi and Adams (1994) to the results of their first-principles calculations, was used in the analysis. Again it bears emphasis that the only input to the model as regards the description of the material is the interatomic potential, and that the transition from the atomistic to the continuum description is strictly based on kinematics (see Section V1.A). ' h o different indentation geometries are considered as are two different indenter shapes. In Figure 10, we show schematics of the indentation geometries used in these calculations. For simplicity, the indenter
Nanomechanics of Defects in Solids
63
FIG. 10. Example of indentation geometry considered in quasicontinuum simulations (Tadmor. unpublished).
is assumed to be rigid and the loading is imparted by prescribing the distance of travel. One of the compelling features of the calculations pertaining to the underlying crystallography and for which interatomic interactions preside over the deformation history is the natural way in which the discreteness of the slip systems reveal themselves. With increasing displacement, the shearing deformations near the indenter edges increase monotonically. Eventually, dislocations are released from the indenter surface and propagate toward the crystal’s interior. The dislocation activity predicted by the calculations is shown in Figure 12. Another key feature that is captured in these simulations is the load-displacement curve (Figure 11). The steps in this cullre are associated with dislocation nucleation events. One of the immediate conclusions afforded by this analysis is that crystal plasticity emerges as a natural consequence of the underlying crystallography without further ado. In particular, the preferred slip systems make themselves known crystallographically, and changes in crystal orientation result in different deformation modes. For the simplest orientation shown above, the dislocations are nucleated onto the awaiting (1 11) planes aligned parallel to the indentation direction. Alternatively, when the crystal orientation is altered, it is again the crystallographically preferred (1 11) planes which see action despite no longer being parallel to the indentation direction. One of the most difficult questions to arise in the context of numerical simulation is how calculations can be used to instruct intuition. One point of contact
64
Michael Ortiz and Rob Phillips
FIG. 1 I . Load-displacement curve determined from simulation of indentation (Tadmor. unpublishcd).
F I G . 12. Dislocation nucleation beneath indcnter. The dissociated cores of two injected dislocations are revealed by the contours of vertical displacement (Tadmor. unpublished).
Nanomechanics of Defects in Solids
65
between the results presented above and more traditional thinking is afforded by linear elasticity. For example, for a given indentation depth, the equilibrium distance of the injected dislocations beneath the crystal surface may be estimated analytically. If one idealizes the Shockley partial pair as a perfect dislocation and considers the balance between the Peach-Koehler force on the dislocation and image forces, the linear elastic equilibrium distance is in reasonable accord with the calculated value. Similarly, analyses of the maximum resolved shear stresses on various planes beneath the indenter can rationalize the different deformation modes: nucleation of Shockley partials or the creation of twin needles. As already noted, the application of the quasicontinuum method to nanoindentation served as the jumping-off point for the method. The results presented above illustrate some of the key features of the calculations. In particular, mesh adaption provides the system with the freedom to follow whatever deformation pathway it wants. In subsequent sections, we consider other fruitful avenues that have been pursued using these ideas.
B. INTERFACIALDEFORMATION Internal interfaces form the backdrop for many of the key microstructural features which materials science aims to control (for detailed reviews of the role played by interfaces, see Sutton and Balluffi, 1995; Wolf and Yip, 1992). The treatment of internal interfaces requires the extension of the simple version of the quasicontinuum method presented in Section VI. This extension is nontrivial as it requires a revision of the conceptual foundations of the method, which is built around the use of a crystalline reference configuration as a means of obviating the need for large lists of neighbors. In the presence of multiple grains, there are correspondingly multiple sets of Bravais lattice vectors to be considered, and the crystalline reference state is composed of a series of disjoint grains. It is the accountancy of these multiple grains that poses the central challenge. Here we content ourselves with demonstrating the application of the quasicontinuum method to problems with multiple grains without recapitulating the method itself. For further details, the reader is referred to Shenoy et al. (1998a). As a first validation test on the method, we have investigated bicrystalline interfaces which have previously been simulated using conventional atomistic analysis. For example, there is an extensive literature on atomistic studies of tilt grain boundaries. A quasicontinuum simulation of a C5(2 10) symmetric tilt boundary in gold is shown in Plate 2. Lattice statics simulations carried out by Ackland et al. ( 1987) produced the low-energy B structure. The quasicontinuum simulations also predict a B structure. It is evident from Plate 2 that the agreement between the
66
Michael Ortiz and Rob Phillips
quasicontinuum and lattice statics atomic positions is satisfactory. In addition, the quasicontinuum method predicts a grain boundary energy of 670 mJ/m2, which is in good agreement with the value of 676 mJ/m2 predicted by lattice statics. This test demonstrates that grain boundary structures are well within reach of the quasicontinuum method. Other quasicontinuum calculations have been performed in which interfacial deformation assumes a central role. In particular, problems involving step motion, dislocation-grain boundary interaction, and fracture in the presence of a grain boundary have been considered. For present purposes, we make particular reference to calculations concerned with the interaction between dislocations and grain boundaries. This application is of special interest to a range of phenomena which have been extensively documented in the experimental literature (Clark et al., 1989, 1992; Lee et al., 1990). The central question being posed concerns the nature of the interaction between grain boundaries and dislocations, in particular, the conditions under which dislocations are absorbed at a grain boundary; and the mechanisms by which dislocations are transmitted or re-emitted at boundaries. Our approach is to resort to nanoindentation as a device for injecting dislocations into a subsurface grain boundary. The finite-element mesh used in this simulation is shown in Plate 3. As in the indentation simulation already described, at a certain critical load dislocations are nucleated at the surface of the aluminum crystal and subsequently propagate toward a subsurface C7(24i) tilt boundary awaiting below. Two snapshots from the deformation history are shown in Plate 3. The initial snapshot shows the crystal after the initial nucleation event. As may be seen from the figure, a Shockley partial pair is produced at the crystal surface and propagates all the way to the grain boundary where it is absorbed. The kinematics of dislocation absorption at the boundary requires that the step which is produced be consistent with the Burgers vector of the incident dislocation. The final load step shown in this sequence shows the crystal after a second nucleation event, this time with the outcome that the dislocations are piled up against the grain boundary. Evidently, the geometry considered in this example is not favorable for slip transmission. The determination of such conditions as result in slip transmission is the thrust of work in progress. C. FRACTURE PHENOMENA
Rice’s recent elucidation of the unstable stackmg energy concept (Rice, 1992) has rekindled the debate surrounding atomic-level criteria for the distinction between brittle and ductile behavior in materials. While it may be unrealistic to ex-
Nanomechanics of Defects in Solids
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o O O D O o O O O
0 0 0 0 0 0 0 0 0
/[1-10] ~
j
[ill]
I
(c)
I
L
(4
FIG. 13. Two different crack tip geometries resulting in cleavage (a, b) and dislocation emission (c, d) (Miller et nl., 1998b). Reprinted with permission of the Institute of Physics.
pect highly idealized models to settle the complex question of brittle versus ductile behavior (see Knap, 1997, for a discussion on the validity of Rice's model), atomic-level insights may nevertheless help to ascertain the more specific question of the competition between the cleavage and dislocation emission from atomically sharp crack tips. The purpose of the present section is to describe the contributions made by the quasicontinuum method to these questions. As a first exercise in considering the competing processes at an atomically sharp crack tip, Miller et al. (1998b) carried out a series of calculations on cracks in single-crystal nickel. These calculations were performed in two spatial dimensions using embedded-atom potentials. Figure 13 shows the near crack tip region for two different geometries: a crack in the (001) plane propagating in the [ 1101
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FIG. 14. Mesh used to study interaction between cracks and grain boundaries (Shenoy e/ ol.. 1998a). Reprinted with permission of The American Physical Society.
direction and a crack in the (1 i0) plane propagating in the [ 1 1 11 direction. The significance of these two geometries is that they reveal different failure mechanisms; in the first configuration, the crack propagates by cleavage, while, in the second, the onset of permanent deformation is accompanied by the emission of dislocations. The calculations provide a venue for evaluating Griffith’s and Rice’s criteria for crack extension. By virtue of the simplicity of the response in Figure 13a, it may be expected that the simple idea of an energy release rate tied completely to the creation of new free surface is satisfied, and this is indeed what is found. In the second orientation, the emission of dislocations occurs at a load that is approximately 50% lower than that implied by Rice’s Peierls analysis. These observations raise the question of whether the simple conceptual schemes provided by Griffith
Nanornechanics of Defects in Solids
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FIG. 15. Crack-grain houndary interaction for C21(421) grain boundary. Different snapshots correspond to different load levels with increasing loads in latter snapshots (Miller et d ,1998b). Reprinted with permission of the Institute of Physics.
and Rice do indeed suffice to rationalize the fracture behavior of the crystal in the two different orientations. In addition to the insights provided into the brittle-ductile dichotomy, the calculations also allow for contact to be made between atomic-level simulation and the more traditional approaches to fracture founded upon linear elasticity. Just as many salient features in the nanoindentation simulations can be understood on the basis of linear elasticity, the quasicontinuum calculations of Miller el al. (1998b) can be rationalized similarly. In particular, one question of interest concerns the equilibrium distance attained by a dislocation once it has been emitted from the crack tip. Unlike in the nanoindentation example, the image forces acting on the nucleated dislocation do not tend to attract the dislocation back to the tip and, hence, only the lattice friction ultimately leads to an equilibrium position for the dislocation.
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F I G . 16. Crack-grain boundary interaction for C5Ci20) grain boundary (Miller el NI., 1998b). Reprinted with permission of the Institute of Physics.
It is well known that many deformation processes involve the simultaneous operation of more than one type of defect. As a first step aimed at incorporating additional microstructural features into our analysis of fracture, calculations have been carried out in which a grain boundary awaits in the path of the incoming crack. The finite-element mesh used in these calculations is shown in Figure 14. As a concrete example of the degree-of-freedom reduction implied by the use of the quasicontinuum method, it is interesting to note that if the same geometry were considered within the context of strict lattice statics it would demand on the order of eight million atoms. Two different tilt boundaries, C21(421) and C5(i20), are considered. The latter geometry is distinguished by the absence of any available slip planes which could support the nucleation of dislocations. A series of snapshots from each deformation history is contrasted in Figures 15 and 16. In the first orientation, a series of dislocations is emitted from the grain boundary and, in addition, the grain boundary itself migrates toward the crack tip. By way of contrast, the second
Nanornechanics of Defects in Solids
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geometry is relatively inactive save for the motion of the crack tip itself, which impinges on the awaiting grain boundary and ultimately branches along it. The case in which the grain boundary bows out has been considered from the perspective of both the continuum theory of energetic forces on interfaces and dislocation theory, with the result that the bowed out geometry can be rationalized as a natural outcome of the large crack tip stresses.
D. DISLOCATION JUNCTIONS Dislocation-based models of hardening must confront the failure of linear elasticity to properly account for the short-range interactions that occur once dislocations are in sufficient proximity that their cores overlap (see, e.g., Baskes et a/., 1997, for an atomistic study of Lomer-Cottrell locks in FCC nickel). One of the avenues being pursued at present for realizing the goal of dislocation-based theories of hardening is the construction of mesoscopic dislocation dynamics codes in which the fundamental degrees of freedom are not atoms, but are rather the dislocation segments themselves (Five1 er nl., 1997; Zbib et al., 1998). The dynamical evolution of complex dislocation rearrangements is effected by computing the forces on the various dislocation segments and allowing them to move according to a simple relation between force and velocity. However, if two segments are found to be too close, the elastic reckoning of this interaction is superseded by simple rules of the cellular automaton variety which dictate whether a juncton will form or not. The quasicontinuum method suggests itself as an effective means of computing the structure, energetics, and strength of dislocation junctions. These calculations offer the possibility of using junction strengths based on atomistic insights as the basis of the mesoscopic dislocation dynamics cellular automaton. In this section, we show two examples of application concerning the interaction of a conventional FCC edge dislocation with a Lomer dislocation, and the interaction between two conventional FCC edge dislocations. One of the interesting features of these calculations, which is in sharp contrast with our earlier examples, is the unequivocal three dimensionality which they demand. The three-dimensional quasicontinuum method is practiced in an essentially similar way to its two-dimensional counterpart with the distinction that the two-dimensional triangular elements are replaced by three-dimensional tetrahedral elements. The present formulation does not as yet exploit mesh adaption, though the development of three-dimensional mesh adaption capability is in progress.
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An example of the quasicontinuum calculations that have been performed to date is shown in Plate 4. This figure shows the interaction between two conventional FCC edge dislocations. It is remarkable that the initially perfect dislocations split into Shockley partials at equilibrium. The key point to be gleaned from this figure is that the interaction between dislocations at close range leads to complex structures that are beyond the purview of linear elastic models of junctions. In particular, we note the presence of a triangular region which constitutes one face of a stachng fault tetrahedron formed at the node of the intersecting dislocations. One question that has also been addressed is how a glissile dislocation circumvents an obstacle in its path. As a model for the interaction of a glissile dislocation with a sessile forest dislocation in an FCC crystal, the Lomer dislocation may be used as an obstacle for a conventional edge dislocation. In Plate 5 , we show one snapshot of the resulting interaction history as increasing strain is applied to the computational cell. The net effect of the applied strain is a force tending to push the conventional dislocation into the awaiting Lomer dislocation. The calculation reveals that a complex interaction takes place when the two dislocations are in immediate proximity. This interaction leads to the creation of additional stacking fault triangles on (111) planes other than that of the incoming dislocation. The conventional dislocation then bows out on both sides of the Lomer, a pattern that is reminiscent of the Orowan mechanism by which dislocations bypass point particles. Once a critical strain is reached, the conventional dislocation breaks away from the Lomer and continues on its way.
VIII. Concluding Remarks This article has had as its central objective the discussion of ways in which boundary-value problems in the mechanics of materials may be imbued with atomistic content. As has been shown in the different sections of this article, a number of different strategies have been developed for effecting the coupling between the solution of boundary-value problems, on the one hand, and atomistic analysis, on the other. The key strategies outlined here include the use of cohesivezone models, lattice statics in conjunction with the method of eigendistortions, and the quasicontinuum method in which a systematic thinning of the full set of atomistic degrees of freedom is effected. The examples presented throughout the paper demonstrate how the consideration of atomic-level nonlinearities and length scales has the beneficial effect of resolving many of the pathologies that plague conventional continuum theories. For simple two-dimensional geometries, analytic progress can be made in the study of defect geometries, such as those of
Nanomechnnics of Defects in Solids
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dislocation cores. If one is ready to adopt less inspiring but more flexible numerical approaches, then three-dimensional problems such as dislocation nucleation from crack tips with cleavage ledges and dislocation junctions may be considered. Now more than ever it appears that the emerging area of nanomechanics may yield to the efforts of modelers.
Acknowledgments It is our special pleasure to acknowledge fruitful collaborations with Ron Miller, David Rodney, Vijay Shenoy, and Ellad Tadmor. Tadmor's Ph.D. thesis laid the groundwork for all subsequent efforts on the quasicontinuum method and included the first applications to the problem of nanoindentation. Shenoy developed the version of the quasicontinuum method in which multiple grains could be accounted for simultaneously, whereas Rodney created the first three-dimensional version of the code. We are also grateful to the AFOSR which funded much of this work through Grant F49620-951-0264 and the NSF which supported much of this work under Grants CMS-94 14648 and DMR-9632524.
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ADVANCbS IN APPLIED MECHANICS, VOLUME 76
Modeling Microstructure Evolution in Engineering Materials ALAN C. F. COCKS and SIMON P. A. GILL
and
JINGZHE PAN
I. Introduction.. . . . . . . . . .
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11. Microscopic Constitutive Laws . . . A. Diffusional Processes . . . . . . B. Interface Reactions . . . . . . . . C. Migration of Interfaces . . . . . .
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IV. Numerical Models . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . A. Grain Boundary Diffusion Controlled Processes . . . . . . . . . . . . . . B. Coupled Grain Boundary and Surface Diffusion . . . . . . . , . . . . . . C. Interface Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Coupled Grain Boundary Diffusion and Self-Diffusion . . . . . . . . . . E. Grain Boundary Migration. . . . . . . . . . . . . . . . . . . . . . . . . . F. Effect of Changes in Elastic Stored Energy on Microstructure Evolution V. Rayleigh-Ritz Analyses . . .
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VI. Structure of Constitutive Laws for the Deformation of Engineering Materials . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . I54 A. Stage 1 Compaction of Ceramic Components . . . . . . . . . . . . . . . . . I56 VII. Concluding Remarks . . . References . .
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I. Introduction In recent years there has been a growing interest in modeling the microscopic processes that occur within a material when it is processed or suffers inelastic deformation and damage growth under mechanical loading. During processing, it is often important to control certain features of the microstructure. A simple example is the grain size, which can have a significant influence on a range of mechanical and electrical properties (Honeycombe and Bhadeshia, 1995; Spriggs and Dutta, 1974; Heywang, 1971).It is therefore important to develop appropriate kinetic relationships that can predict the evolution of the grain structure. In components processed by a powder route the size and shape of the evolving porosity must also be considered and the influence of this on the grain structure and properties of the finished component evaluated (Du and Cocks, 1992). In alloy systems careful control of the processing conditions are required to ensure that the desired precipitate structure, composition, and distribution are obtained Lo acliicve optimum mechanical or electrical performance (Matan el al., 1998). Similarly, in thin-film devices the exact composition and geometry of the film or islands needs to be controlled to obtain the desired optical or electrical properties (Howes and Morgan, 1985). The microstructure of a material can continue to evolve in the design environment. Dislocation and precipitate structures can change; voids and cracks can nucleate and grow, leading eventually to failure of the component (Cocks and Ashby, 1982). Similarly, the structure of thin films can evolve further: For example, they can break up into islands or fail by the growth and migration of voids (Amerasekera and Nam, 1997; Suo, 1996, and references therein). Before developing modeling strategies to deal with the above situations, it is important to ask what the purpose of the modeling exercise is and to determine what information is available about the fundamental properties of the system being evaluated. For example: Is the aim to exactly model the way in which a given system evolves under a prescribed loading history and in a given environment; or are only the major features of the evolution process required, together with an identification of the dominant kinetic processes; or is the aim to develop a macroscopic constitutive law for the material response in terms of a limited number of state variables? It is also important to determine what is considered to be the fundamental starting point for the development of the models. The terms microscopic andfundamental properties mean different things to different research communities. In ab initio simulations the fundamental properties are the quantum states; in atomic simulations they are the interatomic potentials; in microplasticity models they are the properties of individual dislocations; while in continuum plasticity
Modeling Microstructure Evolution
83
and lunetic models we think in terms of local continuum properties such as yield strength and diffusivities. The more microscopic the starting point the smaller the scale of problem which can be sensibly considered. We also need to consider the accuracy within which the fundamental properties are known. It is not sensible to undertake a large-scale simulation if an accurate measure of the fundamental properties is not available. Here we start from a description at the continuum micromechanical level and describe a general strategy for modeling microstructure evolution which can be adapted to deal with any of the situations identified above. The approach is based on a fundamental variational principle which considers the competition between all possible dissipative processes that are driven by a number of thermodynamic driving forces. Basic microscopic constitutive laws for the different kinetic processes are presented in Section I1 and the variational principle is derived in Section 111. In Section IV we show how numerical schemes can be developed from the variational principle. A number of examples are given which involve a range of kinetic and dissipative processes, such as grain boundary, lattice, and surface diffusion, and interface migration, driven by changes in the total potential energy of the system and changes in surface, grain boundary, and interface energy. By examining the form of the variational principle, it is possible to identify the dissipative processes that will dominate in a given situation. Also the approach allows for a range of different modeling and numerical strategies. The utility of different approaches is discussed for a range of practical physical situations. Even by adopting a reasonably large intrinsic length scale for our starting point, i.e., starting from a micromechanical continuum description, the computational requirement of problems of practical interest can rapidly grow beyond the capabilities of currently available computers. Two different strategies can be adopted to overcome this problem. In the first of these a number of representative fundamental problems can be identified and detailed simulations conducted. Thermodynamically consistent constitutive evolution laws can then be obtained by identifying major features of the evolution process (e.g., in grain growth studies one might examine features of the distribution of grain sizes within the analyzed region) and feeding these characteristics back into the variational principle. This provides evolution laws in terms of a reduced number of degrees of freedom. Large-scale problems can then be analyzed by treating these new relationships as the fundamental relationships for this new situation. The second approach differs in detail but not in philosophy. From the detailed analysis of a fundamental problem, major geometric features of the evolving microstructure can be identified. It is then possible to identify a very coarse description of the microstructure which retains these essential global features. Again, use of this coarse description
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for the local problem readily allows much larger scale problems to be analyzed. Both these approaches naturally provide a mechanism of smoothing information up the hierarchy of length scales. They also prompt the investigator to think about the physical situation under examination, to consider the purpose of the analysis being conducted and to identify the overall objectives. These procedures are fully described in Sections IV and V. The variational principle employed in this paper also allows general features of the structure of constitutive laws for the macroscopic response to be identified which consider the competition between the different mechanisms. By necessity, these constitutive laws need to be expressed in terms of a limited number of state variables and these procedures can be thought of as an extension of those described above for analyzing large-scale problems. We examine these general features in Section VI and describe a range of constitutive laws which have been developed for creeping and sintering materials. 11. Microscopic Constitutive Laws
In this section we develop a strategy for analyzing a wide range of physical processes. We seek to describe the micromechanical physical processes in terms of continuum-type properties. Typical examples of the range of kinetic and dissipative processes we are interested in are shown diagrammatically in Figure I and listed in Table I . These principally involve the diffusional rearrangement of material within a body and/or the motion of internal boundaries and interfaces. The thermodynamic driving forces for each of these processes are identified below. There is no unique starting point for the analysis of physical situations in which these mechanisms operate and which fully take into account the interactions between the different mechanisms and driving forces, but any procedure must be thermodynamically consistent and allow the relative importance of all the competing mechanisms to be identified. Our objective is to determine the macroscopic response of a body. We consider an element of material within the body as a thermodynamic open system, which can exchange material with surrounding elements and whose state can change as a result of these exchanges. The macroscopic response can then by obtained by integrating over all the elements. Before doing this we need more information about the constitutive behavior at the microlevel. At this stage we wish to express the material response in as general a way as possible. To clarify our exposition of the method of analysis proposed here, however, it proves useful to limit the range of kinetic processes, number of diffusing species, and different types of physical problems in which we are interested. In particular, we assume that there
Modeling Microstructure Evolution
Ti
t t t t t
85
r-----------
1 1 1 1 1 FIG.I . General situation considered iii this paper. The microstructure can evolve by a number of kinetic/dissipative processes: grain boundary. lattice. or surface diffusion: the migration of interfaces: evaporation and condensation of material: or the rate of operation of any interface reactions.
is always an equilibrium concentration of vacancies in the matrix, so that the diffusivities are not a function of state, and we limit our attention to situations in which diffusion of a single element determines the response. We do, however, allow for phase changes, which can be accompanied by a change in volume of the material. It is possible to extend the approach to a wider range of situations and this is discussed more fully in subsequent sections.
TABLE1 F U L LR A N G EOF KINETIC/DISSIPATIVE PROCESSES AN11 THERMODYNAMIC DRIVING FORCESC O N S I D E R E I ) IN THIS PAPER KineticlDissipative Mechanisms ~~
Origin of Thermodynamic Driving Forces
~
Lattice diffusion Grain boundary dilfusion Surface diffusion Interface reactions Grain boundary migration
Potential energy of applied load Stored strain energy Surface energy Grain boundary energy
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It proves convenient in this section, and in the derivation of the variational principle in Section 111, to express the material response in terms of the transport of atoms within the body. When considering practical problems in Sections IV to VI, it is generally more convenient to express the behavior in terms of continuum quantities. In particular, in the absence of phase changes, it proves more convenient to consider the transport of volumes of rnaterial from one location to another. The normalizations and groupings of material parameters adopted in this section are dictated by this practical utilization of the constitutive laws.
A. DIFFUSIONAL PROCESSES The constitutive behavior of all the diffusional processes identified in Table 1, namely lattice, grain boundary, and surface diffusion, can be represented in the same general form. For lattice diffusion we consider the flux of atoms across unit area within a body, J l , where i has a value in the range I to 3 and refers to a right-handed coordinate system. According to Fick’s first law (Shewmon, 19631,
where w is the excess chemical potential of an atom at position x;, Dl = D / Q / k T is an effective diffusivity, with D/ representing the lattice diffusivity, which, in general, is a function of the state of the material, Q is the atomic volume, k is Boltzmann’s constant, and T is the absolute temperature. The reference state for definition of the chemical potential is a flat stress-free surface. The flux is driven by the potential gradient s,.We will see later that it is instructive to express this relationship in potential form, such that
where 1 Dl
$1
= 5 $SI.Sl.
The rate of energy dissipation per unit volume for material rearrangement due to lattice diffusion 21 can be expressed in terms of the potential &:
(2.3)
87
Modeling Microstructure Evolution is the dual potential to q5! and si
= --.a @I
a J;
When analyzing situations where material rearrangement occurs by diffusion in an interface, such as along a grain boundary, along a free surface, or along the interface between two different solid phases, we consider the material to flow through a thin interface layer of thickness &in.It is then convenient to express the diffusional constitutive laws in terms of the atomic flux across unit length of interface. Then (2.4) with
The effective diffusivity DInis now related to the interface diffusivity D,, through the relationship V,,= D1nSlnR/kT. In each of these expressions we have represented the response in terms of a local coordinate system, x;, in the plane of the interface, such that the subscript a has a value in the range 1 to 2. Throughout this paper we use Greek subscripts to refer to a coordinate system within the plane of an interface. A repeating suffix then implies summation over two components. Arabic symbols refer to a general three-dimensional coordinate system and in this instance a repeating suffix implies summation over all three components. If n is the outward normal to the material on a given side of an interface, the chemical potential of an atom of that phase in the interface is
( - a , J n l n / - ? 4 n ( K l + K 2 ) +UP)" = (- fin(K1 + K 2 ) -k "c)",
P =
(2.5)
air
where a,, is the stress normal to the boundary, vIl is the free energy per unit area of interface, K I and K? are the principal curvatures of the interface, which are positive for concave surfaces with respect to the outward normal, and u p is the strain energy density. For a free surface, a,]is simply - p , where p is the pressure exerted by the vapor phase on the surface, which is constant for a given continuous surface. For a grain boundary, the phases are the same on each side of the boundary, with the interface curvatures equal in magnitude but opposite in sign. The net flux of material in the boundary is then only dependent on the stress normal to the boundary and the strain energy density, i.e., p = (-a,l u,)".
+
Alan C. F: Cocks et al.
88
When considering energy dissipation it is convenient to consider the energy dissipated per unit area of interface d i n . Now d i n = @in
+ +in,
where
is the dual potential to @in and s, =
a +in a J,
-*
B. INTERFACE REACTIONS In the above expressions for diffusion in an interface, we have implicitly assumed that these interfaces are perfect sources and sinks for the diffusing species; i.e., all the work done during the diffusion process is available to drive the diffusive flux of material. In practice, some of this work is used to drive the sources and sinks. Following Ashby (1969), Cocks (1992) took this into account by modifying eq. (2.4). He identified part of the total chemical potential with the diffusion process and part with the deposition process, so that, for diffusion in an interface,
Interface reactions can also be important when material diffuses through the grains. The effect is then to change the chemical potential boundary conditions at the interfaces where material is either removed or deposited. We will refer to F, as an interface reaction potential. If atoms are added to an interface of unit area at a rate N,,, the rate of energy dissipation is dr = -N,,wr. Cocks (1992) proposed that p r depends on the rate of plating:
where 4,. is a scalar positive function of p r . This ensures that the rate of energy dissipation is positive. Here we will assume a simple power law function of the form
a,.
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are suitable reference quantities. Interface reactions determine where fi" and the creep response of fine-grained materials at low stresses. Chen and Xue (1990) obtained creep exponents of approximately 2 when fitting a Norton law to creep data for a number of fine-grained engineering ceramics. This observation implies that the interface reaction index m is 2. A value of m = 2 is also consistent with the mechanistic models of Burton (1972) and Arzt et al. (1983). The rate of energy dissipation per unit area is now given by dr = @in
+ + + +in
@r
@rq
where
and
OF INTERFACES c . MIGRATION
The microstructure of a material can change by material jumping across an interface, resulting in migration of the interface. Consider the situation where an interface exists between two phases, which we designate as A and B . We define the normal n to the interface such that it points away from phase A and into phase B . If Mi,, is the mobility of the interface, then the velocity in the direction of n is
(2.10) where f is the thermodynamic driving force, which is simply the change in free energy that results when an atom is transported across the interface from B to A . Changes in interface energy, elastic stored energy and chemical energy, all contribute to the thermodynamic driving force. Then
where AM, is the change in elastic stored energy experienced by the element of material and AwCis the change in free energy associated with the phase change at constant temperature and pressure, 'i2 is the atomic volume of phase A , and A f i is the difference in atomic volume between phases B and A (the elastic deformation required to accommodate this volume change contributes to the change in elastic stored energy A u e ) .
Alan C. E Cocks et al.
90
For situations where the two phases are the same, but of different crystallographic orientation separated by a grain boundary, and there are no internal stresses,
f = (KI
fK2)YbQr
where y ~ is , the grain boundary energy per unit area. When modeling phase transformations in materials, all three contributions to the driving force must be considered. Equations (2.10) and (2.11) are also valid for the analysis of material rearrangement by evaporation and condensation. Material rearrangement in the vapor phase, which we take as phase B , is then much quicker than diffusional processes in the solid phase, so that a constant chemical composition and free energy can be essentially maintained throughout the vapor. For a deposition process at constant pressure and temperature, Au, is simply the elastic strain energy in the solid at the surface and, provided there are no compositional or phase variations, Auc is constant. We can express the governing equation for interface migration employing the same potential structure as for diffusional rearrangement, i.e., (2.12) and the rate of energy dissipation per unit area is d1n
where the dual potential
$,n
= 4m
+
$,,I
,
is now given by (2.13)
111. Thermodynamic Variational Principle
In the previous section we identified a number of kinetic and dissipative processes for the evolution of the microstructure. For a given mechanism and driving force, it is possible to develop a number of approaches for analyzing the response of a system, which will all produce essentially the same results. Suitable approximate procedures can also be readily identified which capture the major features of the evolution process and which are thermodynamically consistent. It is then a matter of personal choice as to which procedure should be adopted for a given problem. In situations where a range of mechanisms and driving forces are involved, greater care is required in the development of analysis procedures and the
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status and appropriateness of any approximate procedures need to be carefully assessed. Sun et al. (1996) have demonstrated that a common assumption that the microstructure evolves in a direction which maximizes the change in Gibbs free energy is generally incorrect. Both the thermodynamics and the kinetics of the process determine the direction of microstructure evolution. We illustrate this result in Section 1V.A where we examine material behavior when grain boundary diffusion is the dominant mechanism of material transport. It is evident from the microscopic constitutive laws presented in the previous section that a given change in microstructure, e.g., a change in surface profile, can result from a number of kinetic processes. It is obviously important to identify the appropriate kinetic process with this change and to determine under what conditions a given mechanism dominates. In this section we derive a general variational principle which naturally takes into account the competition between the different kinetic processes and which can be used to aid the development of computational simulation procedures and approximate methods of analysis. The full utility of this variational principle is explored in subsequent sections. Consider an isothermal body of volume V which is subjected to constant tractions T, over part of its surface rr and displacement rates over the remainder rl,. We assume that the microstructure has evolved to a particular state and we wish to determine the rate of change of state and the inelastic deformation rate of the system. We isolate an elemental volume of material, d V , which consists of a single phase and which we consider to be a thermodynamic open system. For simplicity, we only consider the situation where there is one diffusing species, but it can adopt a number of phases. We can identify two types of elements: elements that lie wholly within a grain and elements that have part of their boundary coincident with a free surface or interface. The specific internal energy of an elemental volume is e = e(ei,, s ) ,
(3.1)
where e;,j is the total strain, measured with respect to a suitable reference configuration, and s is the specific entropy. The Gibbs free energy of the entire body is then given by
where ui are the surface displacements, p is the density of the material, and T is the temperature of the body. By considering the different contributions to the free energy, we can relate the rate of evolution of the microstructure to the rate
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of change of the Gibbs free energy. This is the purpose of the remainder of this section. For any arbitrary change of state accompanied by a change in the number of atoms d N of an element, the internal energy changes by an amount
dEdV =depdV+pdN= = (gij dE;j
+ T ds)pdV +p d N .
In the above equation we identify r ~ ; ,and p with equilibrium stress and chemical potential fields, oi?;.and /A*, in the body which satisfy all the imposed boundary conditions and are continuous functions which are differentiable to second order, d t , Y d t , and N " d t where d t is an arbitrary inand d e ; j , d s , and d N with crement of time and the subscript c indicates that the fields k:), ?', and N" are compatible but not necessarily the exact fields. Material can enter an element by diffusing through the grain, along an interface layer or by jumping across the interface. We divide N into three components N i ' d V , N/il d A , and &; d A , representing these three respective contributions. Dividing by dt and integrating over the volume and total interfacial area Ail, gives
where E is the rate of change of internal energy associated with the assumed fields and p; is that portion of p* required to drive any interface reactions. The atomic flow rate into an element of material is related to the flux in the grain
a J, -+N, ax;
=o,
and the rate of plating on the interface due to the diffusional transport of material in the interface is related to the flux through the interface layer
a
JLY
-
ax:,
+ $'il
= 0.
(3.3)
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Modeling Microstructure Evolution
Substituting these continuity conditions into (3.2)and making use of the principle of virtual work and the divergence theorem gives
s,
T,*ui d r
EdV =
+
J:
+T
s,
p i ' dV
% dV + S,,n(p* p : ) N i d A dx, -
- ip*J:n,dT.
(3.4)
where r is the total surface area with outward normal n and J:nj is the flux of material out of the body. In (3.4) Ain is the total interfacial area for all phases. A grain boundary is an interface to two phases and it contributes twice to the above equation. If we now identify the stress and chemical potential fields with the exact fields, q,,p. and p,., and the assumed compatible field with arbitrary increments d u f , d i e , d J f , dN;,, d J i , d N h , which satisfy any prescribed boundary conditions and which result in a small change in the rate of change of specific internal energy 2,rearranging (3.4) after noting the constitutive laws of Section I1 gives
=-
JI.
d$/ dV -
S,,.
d@md A -
S,,,
d$m d A -
S,,,
d$r d A = -d'JJ,
where G is the rate of change of Gibbs free energy of the system and \I, =
$1
dV
+ S,
,n
$rn
dA
+ S,, @indA+
S,). $r
dA.
(3.5)
Thus the exact flux and velocity field provide a stationary value of the functional
n,. = w + G .
(3.6)
It can further be shown that for any class of assumed compatible flux and velocity fields there is only one stationary value and this represents a minimum value
Alan C. E Cocks et al.
94
of n,. In situations where our range of assumed fields is limited, we assume that the most appropriate field, i.e., that which is closest to the exact field, is that which provides the minimum value of n, within the assumed set of fields. Suo (1996) has demonstrated how this variational principle can be employed to analyze a wide range of problems, in which the behavior is expressed in terms of a limited number of geometric parameters. In this paper we mainly concentrate on the development of numerical procedures for large-scale simulations, but in the process also examine how simple geometric descriptions can be employed to provide information about the way in which the microstructure evolves. In order to use this variational principle, we do not need to specify the exact form of the constitutive laws described in Section I1 nor determine all the contributions to the chemical potential. These are provided by the variational principle and the exact solution also satisfies all the necessary internal equilibrium requirements. We discuss these conditions more fully in Section IV when we consider the macroscopic response in more detail. An alternative stresdchemical potential-based variational principle can be obtained by identifying u f , iC, J:, N;, J i , and NFn with the exact field u i , i,J j , N m , J a , and &in and o;, w*, and with arbitrary increments doij, d w , and d w r . We now find that the exact stress and chemical potential field is that which minimizes the functional
n,=a+G, where =
S,
4, d v +
(3.7)
lin + S,'" 4m
d~
4in d~
+
Lin
4r
d ~ .
(3.8)
We can further show that the rate of energy dissipation b is given by
D
= -G = @
+ 9.
(3.9)
In practice, it is much easier to base numerical schemes on the flux-based variational principle represented by (3.5) and (3.6) than the stredchemical potentialbased variational principle of (3.7) and (3.8). In the following section we therefore base our numerical schemes on this first variational principle. In many practical situations it proves more appropriate to express the variational principle in dimensionless form. Rather than provide a general normalization here, it proves more convenient to identify appropriate dimensionless quantities when analyzing a given class of practical problems. We return to the variational principle of (3.7) and (3.8) in Section VI where we examine the structure of constitutive laws for the inelastic deformation of engineering materials.
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95
In this section we have considered a reasonably wide range of mechanisms which encompass the full range of situations considered in this paper. It would have been possible to consider a much wider range of situations, involving the diffusion of a number of elements, which interact to form a number of different phases, and other dissipative processes, such as viscous flow or creep of the grains. It can readily be demonstrated that the general forms of the variational principles represented by (3.5) to (3.8) are still valid for this more general problem and, provided information about the effect of changes of composition on the Gibbs free energy is available, it is possible to extend the procedures described here to this more general class of problem.
IV. Numerical Models In this section we describe how the kinematic variational principle presented in the previous section can be used to aid the development of numerical procedures for modeling microstructure evolution in engineering materials. Throughout this section we limit our attention to the diffusion of a single component and do not consider phase transformations. It then proves convenient to consider the volumetric flux rate of material across unit area, ji = QJi, or unit length of interface, j , = nJ,, rather than atomic fluxes, and the rate of plating of material onto interfaces, u,, = n&,,,rather than the rate of addition of atoms. We start by considering the response of a network of rigid grains in which material rearrangement occurs by grain boundary diffusion, driven by changes in the potential energy of the applied loads. We then gradually increase the number of kinetic processes and thermodynamic driving forces and examine the interaction between these for different classes of problems. The examples described in the following subsections are not exhaustive. They have been chosen to illustrate the proposed methods, to demonstrate that they produce accurate results by comparing them with either available analytical results or physical observations, and to show how the simulations can be evaluated using the variational principle to provide simple macroscopic models of the way in wluch the microstructure evolves. Also, equal emphasis is not given to all mechanisms and the reader is referred to the cited references for more detailed descriptions of the techniques and a more comprehensive evaluation of the results of the simulations. The variational principle described in Section I11 allows the rate of evolution of the microstructure to be determined. In the simulations described in this section we update the microstructure incrementally using a simple explicit Euler scheme by multiplying the velocities by a suitable time step.
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Alan C. E Cocks et al.
FIG. 2 . A two-dimensional network of grains taken from a micrograph of a ceramic component. Reprinted from Pan and Cocks (1993a). with permission of Elsevier Science.
A. GRAINBOUNDARY DIFFUSION CONTROLLED PROCESSES For situations in which grain boundary diffusion is the dominant kinetic process and the thermodynamic driving force arises from changes in potential energy of the applied load, the variational functional of (3.6) becomes
where Ab is the total area of the grain boundary, L b is the total length of the boundary that intersects a free surface, and m is the unit normal along the the line of the boundary where it meets a free surface, such that it points away from the body. In this section we consider the class of problem represented by Figure 2 in which each grain is bounded by straight boundary facets and all boundaries to the element are axes of symmetry. The last term of eq. (4.1) is then 0. It proves convenient to express eq. (4.1) in nondimensionalized form. If d is a characteristic physical length scale for the problem under consideration, such as the grain size, and a0 is a suitable reference stress, we can identify a characteristic strain rate
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FIG.3. The degrees of freedom and constraints associated with a grain boundary element.
All lengths I , stresses CT, velocities u , grain boundary fluxes j , and time t , can then be normalized such that -
I=-
1 d’
-
0
(T=-
07,
,
u u=hod ’
; J = -
J
iod’ ’
i = tho.
(4.3)
Then
and the exact solution occurs when
an,. = 0.
(4.5)
Cocks (19901, Cocks and Searle (1991), and Pan and Cocks (1993a) have described how numerical schemes can be developed from (4.4) and (4.5)when the grains are hexagonal, and Pan and Cocks (1993b) have generalized these schemes to deal with arbitrary two-dimensional shapes of grains. The flux at any point of the boundary can be expressed in terms of the velocities of the centers of the grains on either side of a grain boundary (two translational velocities, i f and , one rotational velocity, w , for each grain in two dimensions) and the flux at the center of the boundary, as shown in Figure 3, by using the continuity condition of eq. (3.3). The contribution to the variational functional for a given grain boundary facet can therefore be presented in terms of seven discrete quantities, and the first term on the right of eq. (4.4) can be determined by summing the contributions
Alan C. E Cocks et al.
98 from each individual facet:
where [u]is the matrix of grain velocities and fluxes at the center of each facet, [ B b ] is a matrix relating the flux to the degrees of freedom, and [ K b ] is the assembled viscosity matrix for grain boundary diffusion. In the derivation of (4.6) we solved a series of local problems and did not examine how these local problems interact with each other. Matter conservation is only guaranteed along a boundary and not at its ends where it is connected to other boundaries. We must therefore further constrain the range of admissible fields by requiring that the flux of material into a triple-point be 0; i.e., at each triple-point, boundaries
where the summation is over all intersecting boundaries and now m, are the components of the unit normal along the line of the intersecting boundary pointing away from the triple-point. We account for these constraints by introducing a series of Lagrange multipliers, one per triple-point, so that the variational functional of (4.4) becomes
points
where the outer summation is over all triple-points and h are the Lagrange multipliers, which physically represent the values of -ii/fi at the triple-point. Thus the chemical potential is continuous throughout the network of grains. The advantage of enforcing conservation of matter using Lagrange multipliers in this way is that the resulting viscositykonstraint matrix can be organized into a narrowly banded matrix, for which there are many efficient solvers. An alternative approach would be to directly relate the fluxes at the triple-points to each other. This would be much more difficult to formulate and would result in a dense viscosity matrix, although there would be fewer variables for which to solve. We can express the functional of (4.7) in discretized form: fic
1 = j[UI'[Kbl[u]
- [UlT[FI
+ [AlT[Cbl[UI,
(4.8)
where IF] is the matrix of applied forces, [ c b ] is a constraint matrix relating the fluxes at the triple-points to the velocity and flux degrees of freedom [u] through matter conservation, and [A] is the matrix of Lagrange multipliers. Differentiating
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Modeling Microstructure Evolution
FIG.4. Superplastic deformation of an m a y of grains. Reprinted from Pan and Cocks (1993a). with permission of Elsevier Science.
(4.8) with respect to [u]and [A]to find the stationary value of I?,
gives
This equation represents a set of linear simultaneous equations wluch can be solved using standard matrix methods, providing the exact values of [u]and [A] for the posed problem. Care needs to be taken, however, in the choice of method, because the leading matrix is not positive definite. The geometry can be updated by multiplying the calculated velocities by an appropriate time step and adjusting the positions and relative orientations of the grains. In the process, the lengths of the boundary facets will change. As the microstructure is allowed to evolve, grains can change neighbors and facets, and eventually grains can disappear. Methods of dealing with this are fully described by Pan and Cocks (1993b). These topological changes are similar to those that occur during normal grain growth by grain boundary migration. These are described more fully in Section IV.E.l. Pan and Cocks (1993b) have used this method to model superplastic deformation in engineering ceramics. A simulation starting from a grain structure taken from a micrograph for alumina is shown in Figure 4. All boundaries were assumed to be axes of symmetry and the simulation was continued until the structure experienced 100% strain. Near the boundaries the grains experience significant elongation as a result of the constraint imposed by the boundary conditions, but away from the boundaries, where the grains are able
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Alan C. E Cocks et a1
to slide and rotate over each other, an equiaxed grain structure is maintained. As the microstructure evolves, 8% of the grains are lost and there are many grainswitching events. A small cluster of grains is highlighted in Figure 4. These have changed neighbors and become strung out in the axial direction as the body has been deformed. The results of this simulation are consistent with the major observations of the way in which the grain structure evolves during superplastic deformation of ceramic materials. The grain-switching events do not, however, have a significant effect on the overall deformation process, contrary to the model proposed by Ashby and Verrall (1973), and the strain rate is well represented by a simple Coble creep analysis for an array of regular hexagonal grains with the same mean grain size (Cocks and Searle, 1990). Simulations of this type provide valuable information about the details of the way in which the microstructure evolves and the geometric and material features that most strongly influence the material response. If the major objective, however, is to determine the overall, or average, response, then analyses based on simple geometric models, which can often be performed analytically, generally provide an adequate description of the material response. In the above description we limited our attention to situations in which the grains are formed by a small number of straight facets. In practice, the boundaries might be curved. The above procedures can be readily extended to this situation by discretizing a curved boundary by a number of straight elements. We follow exactly the same procedure as used above, relating the flux along each element to the velocities of the adjacent grains and the flux at the center of the element and introducing Lagrange multipliers at each nodal point formed where two elements meet to ensure continuity of flux (see Pan et al., 1997, for further details). The set of simultaneous equations represented by (4.9) was obtained by considering both the dominant kinetic process and the thermodynamic driving force. We obtain similar relationships for all the situations considered in this paper. In Section I11 it was noted that a common assumption employed when modeling microstructure evolution is that the microstructure evolves in the direction that maximizes the rate of change of Gibbs free energy. It is instructive to examine how the governing equations obtained from such a model would differ from those obtained from the variational principle. Consider the situation where the evolving geometry can be represented in terms of a number of state variables, x l , such that G = G(x,) and
where Fj are the thermodynamic forces associated with the state variables x i . The rate of change of Gibbs free energy is maximized if Xi is normal to the surface of
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101
constant G in state variable space, i.e., if .ii
=Me,
(4.10)
where M is a constant, which is determined by the lunetic processes. Equation (4.10) implies that the viscosity matrix is simply the identity matrix multiplied by a kinetic constant, allowing the rate of evolution of each state variable to be determined independently of each other. This is obviously an attractive feature of this approach, as it permits simple and efficient numerical schemes to be developed. Problems can, however, be encountered with the implementation of internal compatibility requirements, and often ad hoc adjustments are required to ensure continuity of the microstructure. Numerical schemes based on (4.10) only provide an approximate model of microstructure evolution. Schemes based on the variational principle of (3.6) provide the most appropriate model for a given discretization of the real problem. So far we have only considered situations in which there is a single kinetic process. In the following sections we examine situations in which more than one dissipative process contributes to the way in which the microstructure evolves. It is not immediately evident what the single most appropriate kinetic constant is in eq. (4.10) when assuming that the microstructure evolves in the direction of maximum G.The variational principle of (3.6),however, naturally takes into account the full interaction between the different lunetic processes. We therefore employ the variational approach when modeling these more complex situations.
B.
COUPLED
GRAINBOUNDARY AND
SURFACE
DIFFUSION
In this section we examine situations in which both surface and grain bound-
ary diffusion contribute to the evolution of the microstructure. We also include the influence of changes of surface and grain boundary energy on the evolution process. Employing the normalizations of (4.3), the variational functional of (3.6) becomes
(4.11)
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Alan C. F Cocks et al.
FIG. 5. The classical model of Hull-Rimmer void growth. A represenlative cylindrical unit ccll of radius I centered on a grain boundary pore of radius r/, is isolated from the body and assumed lo experience the same stress cr that is applied to the body.
where
- the total normalized grain boundary and surface areas, The areas A h and A,y are which evolve at rates A and A , F . The associated changes in total grain boundary and surface energy provide additional contributions to the change in Gibbs free energy of the body.
1. Fast Su$ace Diffusion
It proves instructive to first consider some simple geometric situations and extreme material parameters using (4.11) before examining the full interaction between surface and grain boundary diffusion. If fi,. is large, i.e., if the surface diffusivity is much faster than the grain boundary diffusivity, the second term of eq. (4.11) only makes a significant contribution to the functional if the surface flux is much faster than the flux along the grain boundaries. If they are comparable, then the influence of surface diffusion on the global evolution process is small. It proves useful to discuss this point further by examining a simple problem. Figure 5 represents the classical model of Hull-Rimmer void growth (Raj and Ashby, 1975; Cocks and Ashby, 1982). A void of radius I-/, grows by material flowing along the surface of the void to the point of intersection with the grain boundary and then along the grain boundary, as illustrated in the general picture
Modeling Microstructure Evolution
103
of Figure 1, where it plates out uniformly over an area ~ ( - 1r;),~allowing the applied load to do work. As well as providing material for the growth process, surface diffusion also provides a mechanism for internal rearrangement of material toward a local equilibrium configuration. If 2>, is large, this local rearrangement will occur rapidly and only a slight perturbation from the equilibrium shape, which is sufficient to provide the flux for the growth process, will quickly evolve. In our analysis we can therefore assume an equilibrium profile for the pore, as in the classical model (Raj and Ashby, 1975), and ignore the contribution from the second term in the variational functional of (4.1 1). To simplify the analysis further, we assume that the grain boundary energy is 0, so that the equilibrium shape of the pore is spherical. This is a one-degree-of-freedom problem: The volumetric growth rate of the pore can be directly related to the rate of elongation, Li, of the unit cell of Figure 5 ,
v = 41rr;i,
= xi2ci.
(4.12)
The continuity condition for flux in the boundary is
where r is the radial distance from the center of the pore and j,. is the radial flux. Solving this equation, noting that j,. = 0 at r = I , gives l i ,
j,. = -2r (I-
-
r’).
Substituting this expression for the flux into (4.1 I), noting that
and differentiating the resulting expression with respect to ci to obtain the stationary value, we obtain
where t i , = rill’ is the area fraction of voids on the grain boundary. This expression agrees with the result of Raj and Ashby (1975) as corrected by Raj et al. (1977), although it has been derived in a much more straightforward way. For situations where surface diffusion is rapid, the variational functional only requires a consideration of the flux along the grain boundaries. The above type of
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104
FIG.6. A closed contour r h around a grain boundary pore used to compute the volumetric growth rate.
analysis can therefore be readily extended to the class of two-dimensional problems examined in Section IV.A, where now the body can contain a distribution of near-equilibrium-shaped pores. Consider the situation where the pores are situated at the triple-points, as is commonly observed in engineering ceramics. We again assume for simplicity that the grain boundary energy is 0, so that the pores adopt a circular profile. The rate of increase of surface area for the pore can then be related to the volumetric growth rate:
There are two contributions to the volumetric growth rate: due to the flux of material away from the pore and as a result of the grain boundary plating process. These contributions can readily be expressed in terms of the degrees of freedom identified above by constructing a closed contour rl, around each pore, as shown in Figure 6 . Then V =
fr
i;nidr+ h
c
j,m,,
boundaries
where n; is the outward normal to the chosen contour, the summation is over all boundaries cutting the contour, and m, is a unit normal along the line of the boundary pointing away from the contour. The velocities u i and fluxes j , can be related to the degrees of freedom [ u ] and thus -
A,&.
= -[U]"F,],
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105
FIG. 7. ( a ) A macroscopic crack in a polycrystalline material which defomis by Cohle creep. Triple-point voids cxisl along the line of boundaries directly ahead of the crack tip. ( h ) The geometry after an extensive ainount of crack growth.
Alan C. F: Cocks et al.
106 56
crack tip position
54-
52
------
strain rate
-
.-55 50a" .a 48k Y
46-
0 44
-
42-
1 - 1
-/' ____------I
I
where [FYIis a force matrix associated with the change in total surface energy. Substituting this into (4.1 1) and following the derivation of Section IV.A, we obtain
This system of equations can be solved using the same procedures as described in Section 1V.A. Cocks and Searle (1991) have used this method to evaluate the effect of competing length scales on the process of void growth when the spacing of the voids is much greater than the grain size. More recently, Pan and Cocks (1998) have evaluated the process of void growth ahead of a dominant crack. Clearly, the crack tip in this situation does not adopt a near-equilibrium profile (we discuss this more fully in Section 1V.El), but by determining the shape of the crack tip region using the profiles predicted by Chuang et al. (1979) for the same volumetric growth rate they were able to compute the propagation of a crack through an array of triple-point voids lying directly ahead of the crack tip. Computations were performed for regular square arrays of cracks, so that the boundaries shown
Modeling Microstructure Evolution
107
in Figure 7a are axes of symmetry, and for situations where a stress field corresponding to a K I field of prescribed magnitude was imposed on the boundary. A typical profile of the crack and voids after a large extent of crack growth for the former case is shown in Figure 7b and the extent of crack growth as a function of normalized time, as defined in (4.3). is plotted in Figure 8, together with the variation of strain rate for the entire cracked body, normalized with respect to i o , the strain rate of the uncracked body at the same stress. The position of the crack is measured in terms of the number of grain boundary facets that lie along its length. Two processes contribute to crack growth: the diffusional transport of material away from the crack and the growth of triple-point voids. The second contribution results in instantaneous increments of crack growth as the pores link with the crack, indicated by the vertical portions of the curve in Figure 8. It is evident from Figure 7a that toward the end of the life of the component there is significant damage in the uncracked ligament in the plane of the crack, and these voids are all approximately of the same size. Final failure is effectively due to the coalescence of these voids across this minimum section. A full evaluation of these results is provided by Pan and Cocks ( 1998).
2. Fullv Coupled Analysis In this subsection we examine situations in which we cannot assume a nearequilibrium profile for a free surface and the evolving shape is determined by a coupling between surface and grain boundary diffusion. We limit our discussion initially to plane two-dimensional problems, as before. We have already dealt with the boundary diffusion part of the analysis process in Section 1V.A. Here we present an equivalent analysis for surface diffusion. Material rearrangement by surface diffusion results in a change of surface profile. If material is deposited on a section of boundary, it moves in the direction of the outward normal and the continuity equation (3.3) provides a relationship between the flux in the plane of the boundary and the velocity of the surface u,! in the direction of the outward normal n. To model a particular problem, we discretize a section of free surface by a series of straight elements (Figure 9). Having described the initial geometry, care needs to be taken in identifying the degrees of freedom that are used to model the evolution process. Consider the situation represented in Figure 9a in which we associate two velocities with a node at the point of intersection of two elements. In the limit where the normals to the two elements are in the same direction, it is possible for the node to move along the line of the elements without any diffusional rearrangement of material. This is equivalent to a zero-energy mode in a static finite-element analysis and results in a singular surface diffusion viscosity matrix, as defined below. If there is a small angle between the elements, then a much smaller volumetric flux of material is required to translate the node by a
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Alan C. E Cocks et al.
FIG.9, A section of surface can he discretized by a number ofelements. (a) Two elements with nodal velocities at each node. (h) The degrees of lreedoni and position ofthe Lagrange multipliers for the surface elements.
IWO
given amount in the transverse direction compared with that required to translate the node in the near-normal direction. This results in a low transverse viscosity. This, coupled with the fact that our description of the geometry is approximate (in practice, there are no discontinuities in slope, apart from where a surface is intersected by a grain boundary), results in the transverse velocities dominating the time step and providing only slight modifications to an already idealized situation. It is therefore necessary to constrain the transverse motion of the nodes; i.e., the only degree of freedom associated with a node is in the near-normal direction. The problem now is: How do we define this near-normal direction? There are a number of possibilities, which all produce equivalent results. Pan et al. (1997) choose the velocity to be in a direction normal to a circular arc constructed through the node of interest and its two adjacent nodes. Another approach is to associate it with the direction that bisects the normals of the two adjoining elements. This latter description was used in the simulations described in Section 1V.F. This type of construction is only required for the discretization of a continuous surface. At the point of intersection of a grain boundary with a free surface, there is a distinct discontinuity in slope and two mutually orthogonal velocities are required to model the response. Both types of nodal degrees of freedom are shown in Figure 9b. In practice, the surface and grain boundary tensions at a point where a grain boundary meets the surface must be in equilibrium. This condition is satisfied by
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the variational formalism; i.e., the exact solution to the problem satisfies all the equilibrium requirements. If our initial specified geometry does not satisfy this requirement, then rapid local diffusion will occur to ensure equilibrium of the surface and grain boundary tensions. We need not, therefore, enforce this condition. As with conventional finite-element analyses, when using a discretized description of the problem the optimum solution does not completely satisfy the equilibrium conditions. Similarly, here, the microstructure that evolves may not exactly satisfy the dihedral angle requirement, but by not enforcing this condition we can readily evaluate the appropriateness of our geometric description in the vicinity of a surface/grain boundary junction by comparing the local angle between the intersecting interfaces with that required by equilibrium. We can follow a similar procedure to that described in Section 1V.A for boundary diffusion when examining the contribution of surface diffusion to the variational functional. For a given element, the velocity normal to the boundary at any position can be related to the nodal velocities at its ends. Integration of the continuity condition (3.3) then allows the surface flux to be determined in terms of these velocities and the flux at the center of the element, which represents an additional degree of freedom, giving three in total. Our local flux field does not ensure global conservation of mass and, as in our analysis of grain boundary diffusioncontrolled processes, we introduce a series of Lagrange multipliers, one per node, to enforce the condition that there is zero net flux of material into a node. Also, the rate of change of surface and grain boundary area can be related to the velocities at the nodes. Following the analysis presented for grain boundary diffusion in Section IV.A, we can express the variational principle of (4.1 1) in the following form:
and [ F b ] are the where 3 , ; ' [ K s ] is the surface diffusion viscosity matrix, [Fs] thermodynamic forces determined from the surface and grain boundary tensions, and [C,] is the constraint matrix arising from the conservation of mass condition at the surface nodes. The matrices [ u ] and [A] contain the degrees of freedom and Lagrange multipliers associated with both the surface and the grain boundary diffusion processes. The only common quantities are the Lagrange multipliers associated with the points where the grain boundaries meet a free surface; i.e., the coupling arises from the requirement that the flux into a boundary is equal to the sum of the fluxes away from the adjoining surfaces. These Lagrange multipliers
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Alan C. I;: Cocks et al.
are effectively the capillarity stresses at these points. All the above matrices are fully defined by Pan et al. (1997). Taking the stationary value of (4.13) gives
This set of equations can be solved using the same standard matrix procedures as for grain boundary diffusion alone and the geometry can be updated by multiplying the velocities by a suitable time step. For large-scale problems a large viscosity/constraint matrix must be inverted at each time step, which can lead to long computer run times. It is evident from the geometric description of Figure 9 that the surface diffusion elements have limited connectivity to other elements, only interacting with two adjoining elements in general. The matrix [&I is therefore a narrow-banded matrix. Under these conditions finite-difference methods can be much more efficient than matrix procedures developed from a variational formulation. Pan and Cocks (1995) describe a combined finite-difference/matrix method for the coupled problem in which the above Euler explicit integration scheme is employed. Recently, Kucherenko and Pan (1998) have described a procedure based on an implicit integration scheme and provided detailed comparisons of the different methods and integration schemes. For problems in which only surface and grain boundary diffusion contribute to the deformation process and in which the grains are assumed to be rigid, these combined schemes are much more efficient than the above matrix method when using the same number of degrees of freedom. When using the numerical schemes developed from the variational principle, very coarse descriptions of the microstructure can be adopted. Even with a small number of degrees of freedom, the computations remain stable, unldce conventional finite-differencemethods. In Section V we illustrate this by considering the cosintering of a row of initially spherical particles by coupled grain boundary and surface diffusion and use the variational principle to model this process in terms of only two degrees of freedom and compare the overall features of the evolution process with that obtained from a finite-difference/matrix method. Numerical procedures based on the variational principle have a distinct advantage over finite-difference methods when other processes or driving forces contribute to the component response. These can produce a stronger interaction between the surface diffusion elements. We consider one such example in Section IV.F, where we take into account the contribution of the elastic deformation of the material to the driving force. It is not immediately evident how finite-difference methods can be adapted to deal with this situation.
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c. I N T E R F A C E REACTIONS In Sections 1V.A and 1V.B we assumed that the surfaces and grain boundaries are perfect sources and sinks for the diffusing species. As noted in Section II.C, some of the work that is available to drive the overall process is required to drive the sources and sinks. In fine-grained materials the kinetics associated with these interfacial processes can control the overall rate of microstructure evolution. The incorporation of interface reactions into the variational principle has been described by Cocks (1992) and Cocks and Pan (1994). No additional geometric information or degrees of freedom are required to model these processes. For a given element, be it a grain boundary or a surface element, we can express the velocity at any point of the element in terms of the velocity degrees of freedom associated with that element. For the purpose of discussion, we concentrate on the situation where interface reactions only operate on grain boundaries; i.e., we consider the surfaces to act as perfect sources and sinks for the diffusing species. The variational functional eq. (3.6) can then be written in the following dimensionless form:
where a! = (iod/vo)'/''' measures the relative importance of the diffusive transport of material and the interface reactions to the rate of dissipation of energy, u, = I U , ~ 1, the magnitude of the rate of plating of material on the grain boundary, and the rate constant uo is related to the constant & defined in (2.8) through the relationship uo = NOR. As before, d is a suitable physical length scale for the problem of interest, which is generally chosen to be the grain diameter. For small a! the kinetics associated with the interface reactions are fast compared to those associated with the diffusional process and only a small proportion of the available work is required to drive the interface reactions. As a result, we can ignore the contribution of the interface reactions to the variational functional and to the overall microstructure evolution process. Conversely, when a! is large the interface reaction kinetics are sluggish, these now determine the overall rate of the process. We can then ignore the contribution of the diffusional flux of ma-
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FIG. 10. A simple rheological model representing the creep behavior of a polycrystalline material. Poc/ and Por are the uniaxial strain rates at a stress 00 in the limits of grain boundary diffusion and interface reaction controlled creep.
terial to the variational functional. Diffusional processes of the type considered here consist of removing material from a section of the grain boundary and transporting it to another part of the body where it is deposited. These are all necessary sequential parts of the same process which must operate at a compatible rate; thus the slowest process determines the overall rate of microstructure evolution. Cocks and Pan (1994) have described this type of response using simple rheological models. For example, the uniaxial strain rate of a polycrystalline material can be represented in terms of the rheological model of Figure 10, where the two parallel dashpots represent the material response in the limits of grain boundary diffusion controlled and interface reaction controlled creep, respectively. The strain rate at intermediate values of a! is given by i. for m = 2, where B = 1 5 . 5 ~ 1= ~ &/&I;, & is the uniaxial strain rate in the limit of grain boundary diffusion controlled (Coble) creep, and Boi is the uniaxial strain rate in the limit of interface reaction controlled creep.
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Following the analyses of Sections 1V.A and IV.B, we can express the variational functional in terms of the velocity and flux degrees of freedom and the nodal constraints, represented by the Lagrange multipliers. The values of these quantities, which provide a stationary value of the functional, satisfy the following set of nonlinear simultaneous equations:
where 1K , 1 is the interface reaction viscosity matrix, which is a function of the degrees of freedom [ u ] . Cocks (1992) describes a Newton-Raphson method for solving this system of equations. Cocks and Pan ( I 994) have used this method to evaluate the process of void growth in fine-grained materials and to determine the nature of the stress and velocity fields that develop ahead of a crack tip in a creeping material, and how these depend on the parameter a. Consider the situation where a body is subjected to a single force F and u is the velocity in the direction of F . For the purpose of discussion, we only consider the dissipative processes of grain boundary diffusion and interface reactions and ignore all other contributions to \I, and G. In the limit of grain boundary diffusion controlled microstructure evolution, i.e., when a is small, the variational functional of (3.6)becomes
fl,.=
L,,5
1;
7
ju ja d
A -FC.
This functional can be minimized by minimizing the distance that the material diffuses, subject to the constraint that the resulting velocity field is compatible. In polycrystalline materials, a compatible mechanism can be obtained by the material diffusing a distance of the order of the grain diameter. In all the problems that have been analyzed, we observe that material never diffuses distances greater than this, even when the body contains large defects. For example, in the crack problem of Figure 7 the high plating rates in the vicinity of the crack tip are accommodated by Coble creep of the surrounding cage of grains and material that flows into this region from the growing crack is plated onto the first boundary facet. In the limit in which the interface reactions control the rate of microstructure evolution, i.e., for large a,the variational functional of (3.6)becomes
For the situation where the body contains defects, in the form of either voids or cracks, the functional is now minimized by material flowing away from the defects and plating uniformly over all available facets that allow the applied load
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to do work. For the crack problem of Figure 7, we now find that material from the growing voids and crack plates out uniformly on all the grain boundary facets which make an angle of 30" with respect to the line of the crack; i.e., material flows from the surface of the crack to the extremes of the repeating cell. For a macroscopic crack this limit will never be achieved in practice. In fine-grained materials the velocity and flux patterns are determined by both dissipative processes. Cocks and Pan (1994) demonstrate that the distance that material diffuses depends on the value of a. They demonstrate that, on average, material diffuses a distance that scales with ad if a > 0.5, while the diffusion distance is of the order of d / 2 for smaller values of a. Thus, if a fine-grained material contains a macroscopic crack extending over hundreds of grains, a diffusion zone of mean dimension ad forms ahead of the crack tip. Material that diffuses away from the growing crack remains within this zone and the resulting deformation is accommodated by interface reaction controlled creep of the surrounding network of grains.
D. COUPLED G R A I NBOUNDARY D~FFUSION A N D SELF-DIFFUSION In the previous subsection we considered the situation where two sequential processes contribute to the material response. Here we consider a simple situation where two parallel/competing processes occur. We simply examine the form of the variational principle and do not attempt to develop numerical procedures from the variational formalism. Consider, for simplicity, the situation where only lattice and grain boundary diffusion contribute to the component response, so that, by default, there are no free surfaces in the body. The variational principle then becomes, in dimensionless form,
(4.17)
where L' $ = Vld/'D/,. The lattice and grain boundary fluxes are, in general, independent of each other and combine to produce the displacement rate l i i , particularly at extreme values of 81.In situations where V / is of the order of unity, flux
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patterns can develop where the lattice and grain boundary flux fields are incompatible when taken in isolation. The actual rate of energy dissipation is then faster than that determined by simply summing the contributions from the two mechanisms operating in isolation. This can be readily illustrated by considering the situation where the applied surface tractions T, are specified. The rate of energy dissipation associated with the exact field is then
In the limit where grain boundary diffusion dominates,
where the superscript b indicates rates determined by only considering grain boundary diffusion. Only considering material rearrangement due to lattice diffusion. we find
where the superscript I indicates the fields determined in this limit. According to the variational principle, the value of I?,. determined from the full coupled analysis is less than or equal to the value obtained by adding the two extreme mechanisms. Mahng use of the above relationships, we find
If the body is subjected to a single load on its boundary and the rate of change of grain boundary energy is small compared to the external work rate, then the displacement rate in the direction of the applied load is greater than the sum of the displacement rates for each mechanism operating in isolation. For small 31it is evident that the minimum value of fi,. occurs when most of the displacement rate is due to grain boundary diffusion, so that the second term of the functional is comparable to the first. Similarly, if 2)/is large, lattice diffusion dominates the response. This is an obvious example, but it illustrates the way in which the variational principle can be used to determine the relative importance of different mechanisms in a given situation. We consider other examples of this in subsequent sections. Numerical and approximate procedures for situations where lattice diffusion is the dominant mechanism have been described by Cocks (1996).
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E. G R A I NB O U N D A RMIGRATION Y In this section we examine how the variational principle of Section 111 can be used to model situations in which grain boundary migration contributes to the way in which a microstructure evolves. We start by considering the process of normal grain growth in Section IV.E.1, in which grain boundaries move under the action of grain boundary tension. In the process of normal grain growth described here, the only factors that determine whether a grain grows or shrinks are the local topology of the grain network and the grain boundary geometry, which wants to satisfy surface tension equilibrium. Abnormal (or secondary) grain growth, which occurs when some grains have a preferential growth rate due to some energetic advantage over the others, is considered in Section IV.E.2. In Section IV.E.3 we examine situations in which grain boundary migration combines with other diffusional processes to determine the way in which the microstructure evolves.
1. Normal Grain GI-owrh We consider the case here of two-dimensional normal grain growth (Cocks and Gill, 1996; Gill and Cocks, 1996; Du et al., 1998). This removes some of the geometric complexity of working in three dimensions and the results have the benefit of being easier to visualize and interpret. Pseudo two-dimensional grain structures do exist in thin films in which the mean grain diameter of the grains is larger than the film thickness. Such grain structures are said to be columnar and can be represented by a two-dimensional plan view of the microstructure. To represent such a microstructure mathematically, we need a geometric description for the grain boundaries and a topological description of how these boundaries are organized to form the grain network. In this subsection we assume that the only dissipative mechanism is grain boundary migration. Then, from (2.13), for a thin film of thickness h ,
where Lr, = A/,/ h is the total grain boundary length, and the driving force for microstructural evolution is the reduction in the total grain boundary energy
6 =h
IL
Y/,K U,,dS,
h
where K is the curvature of a boundary in the plane of the body. In addition, equilibrium of the grain boundary tensions must be considered. Assuming that the grain boundaries are chosen to be smooth and continuous, the
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FIG. I I , For uniform grain boundary energy per unit area y/,. ( a ) an equilibrated grain boundary junction and (b) a grain boundary junction in which the grain boundary tensions are not in equilibrium leading to a net force F.
only location at whch equilibrium must be enforced is where the grain boundaries meet. In two dimensions, grain boundary junctions of more than three boundaries are assumed to be unstable and, consequently, grain boundaries are said to meet at a triple-point. To satisfy equilibrium, the angle between any two boundaries meeting at a triple-point must by 120" as shown in Figure I la. The microstructural description can be chosen so that this condition is automatically satisfied. Alternatively, this condition can be imposed in a weak form through the variational formulation. Consider the situation shown in Figure 1l b of a triple-point that is not necessarily in equilibrium. There is a net force acting at the triple-point and, therefore, for uniform grain boundary energy yt,,
i=l
where si are the unit tangents to the grain boundaries at the triple-point. If the triple-point is in equilibrium, then F = 0. If the triple-point has a velocity v, then the rate of work done is F v and hence the variational functional can be written in dimensionless form as
-
all triple points
(4.18) potnis
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FIG. 12. Thc cubic polynomial grain boundary description. The boundary is completely defined by the position ( - 4 - 1 , y1, .r2, y2) and orientation ( H I , 8 2 ) of its two end points. assuming a profile of the forin uI(.y)= rrs' /xs2 C S t / .
+
+ +
where n, = n,/cq)uohRo, F = F/oohRg, V,, = v,,/uo, S = s/Ro, and k = K R O with , 00 = yh/Ro, u g = Ml,oo, and Ro representing the initial mean grain size. We describe a given polycrystalline microstructure by the position of its triplepoints and the orientation of the grain boundaries at these triple-points. Hence we wish to choose a grain boundary description using a smooth function that is completely defined by the position and orientation of its end points but also one that allows their position and orientation to vary independently of one another. The simplest function of this type is a cubic polynomial. A typical boundary is shown in Figure 12. The rate of evolution of this boundary is completely described by the four translational and two rotational velocities of its two end points, which combine to provide the elemental contribution to the degree-of-freedom matrix [ u ] . By considering an infinitesimal temporal displacement of this boundary subject to these end point velocities, it is possible to find a linear relationship between the normal velocity of the boundary u,, at any point and the end point velocities, i.e., u,, = [B,,,l[u],where [B,,] relates the velocity at a material point to the nodal velocities. The variational functional can then be discretized in terms of the triplepoint velocities given the current geometry of the microstructure. The consequent minimization of the variational functional therefore gives the actual triple-point velocities in a similar way to that described in Sections 1V.A and 1V.B for situations in which grain boundary or surface diffusion are the dominant mechanisms. This allows the microstructure to be updated over a suitable time increment. The construction of the initial grain structure is described by Gill and Cocks (1996). They model a large periodic array by identifying a repeating cell and connnecting the boundaries on one side of the cell to those on the opposite side. However, this periodicity means that the viscosity matrix is sparse (cannot be diagonalized) and hence it is computationally more demanding.
Evolution ofthe microstructure Inspection of (4.18) shows that the last term, introduced to ensure equilibrium at the triple-points, acts as a driving force for
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(b)
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(c)
F I G . 13. Dcgrees offreedom o i a triple-point for grain growth models: (a) Case 1, (b) Case 2, and (c) Case 3.
grain growth even when all the boundaries are straight (i.e., G = 0). Therefore there are two terms that contribute to the driving force. This suggests three different models by which the microstructure could be allowed to evolve:
Case 1 : Straight boundaries and hence not in equilibrium at triple-points ( G = 0 but F # 0). Case 2: Cubic boundaries fixed to be in equilibrium at triple-points (G # 0 but F = 0). Case 3: Cubic boundaries not geometrically constrained to be in equilibrium at triple-points (G # 0 and F # 0). As shown in Figure 13, the first case is the most restrictive model as each triplepoint only has two degrees of freedom (both translational), followed by Case 2 which has three (one rotational and two translational), and, finally, Case 3 which has five (three rotational and two translational). As grains are growing and shrinking, it is necessary to incorporate the topological changes that allow this to happen into the updating procedure. These topological processes can be modeled by the combination of the two types of events shown in Figure 14 (Ashby and Verrall, 1973). The first of these, the grain boundary switching event, occurs when a boundary connecting two grains A and B becomes very small and the grains A and B come into contact. This is modeled by reorienting a boundary when it shrinks below a critical length. The second, three-sided cell removal, is simply the contraction of a three-sided cell to a single point at a grain boundary junction. This is modeled by removing three-sided cells when their area falls below a critical size. Combining these processes allows grains with any number of sides to be removed. In the numerical simulations it was assumed that the grain boundary energy y/, is constant. The solution is found to be independent of the film thickness h and the time is normalized with respect to the only material parameter MhaoRo =
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(b) FIG. 14. Schematic diagram of the two types of topological changes: (a) neighbor switching and (b) disappearance of a three-sided grain.
Snapshots of a typical Case 2 simulation starting with 1024 grains are shown in Figure 15 (Gill and Cocks, 1996). Simulations of the simpler Case 1 model conducted by Du et al. (1998) possess the same general features of the Case 2 model. For a given grain structure the Case 1 model requires fewer degrees of freedom to describe the microstructure, resulting in a reduction in the total time required for a simulation. In each case, however, as in all the numerical simulations considered in this paper, a system of N, (where Nt is the number of degrees of freedom used to describe the system) linear simultaneous equations is solved at each time step. Du et al. (1998) have used the variational principle to develop an approximate nodal model for an array of straight-sided grains. The force matrix, with each component representing the force on the associated triplepoint, is determined as for the Case 1 model. A viscosity matrix is determined for each node by arbitrarily associating half of each grain boundary facet with its two end nodes and assuming that the normal velocity along the length associated with a given node is constant and equal to that at the node. The velocity of a given node is then determined by solving two local linear simultaneous equations; thus the rate of change of microstructure is determined by solving Nt / 2 pairs of equations. Although approximate, this procedure is much quicker than inverting the full viscosity matrix at each time step. Simulations using this method differ in detail from those produced using the full variational method, but when the results are examined in terms of various macroscopic parameters there is good agreement between the two sets of simulated results. YbMh.
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T i m =OOO
Numer of cell 4 0 2 4
T i m =I95
Number of cel?, -707
Time =a54
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Nmber of cdk =269
f)
I
Time =4 21
Nmber of cels -464
Flc;. I S . Thc grain growth evolution of a network of p i n s as a function or noimalized time T = t/y/,M/,:(a) starting with 1024 grains, (b)-(rS some grains grow at the expense of others causing sonic of ihern to disappeai-. Reprinted front Gill and C o c k (1996) with permission of Elsevier Science.
When a similar simulation to that shown in Figure 15 was conducted for the more general Case 3 (Gill and Cocks, 1997b), it was found that the grain growth occurred roughly 5% quicker but the general characteristics of the evolution were exactly the same. It is thought that the growth was slightly faster as the additional
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degrees of freedom allow a slightly more optimal energy reduction path to be followed. However, the principal result from this study was the fact that the computational time required was 10 times that for the simpler Case 2 study as the more complex model proved to be less stable. This clearly demonstrates that one has to be careful in choosing the degrees of freedom for a given problem. If the problem description is not given enough degrees of freedom to encompass the physical behavior of the system, the results will be misleading. However, if the problem is given a lot of freedom, it may become less stable, i.e., the functional minimum becomes shallow, and it may not become possible to obtain a solution at all. To get the full value out of these simulations, it is necessary to investigate the evolution in more detail and extract some simple laws or parameters that more readily encapsulate the characteristics of the simulation. There are many statistical methods and theories to monitor and predict the kinetic and topological evolution of cellular networks (Atkinson, 1988; Gill and Cocks, 1996). The most widely used model for grain growth kinetics is the parabolic grain growth equation proposed by Burke (1 949): R” - R; = k t ,
(4.19)
where n takes the value of 2 and is known as the grain growth exponent, R is the mean grain radius at time t , and k is a rate constant. As the mean grain area (taken to be R 2 ) is inversely proportional to the number of grains N , the grain growth exponent obtained from the Case 2 simulations can be readily calculated. In this case it was found to be n = 1.92 in reasonable agreement with (4.19). In real material systems, however, n is found to be between 2 and 4. This discrepancy will be discussed further in the section on abnormal grain growth. Rhines and Craig (1974) introduced the concept of a sweep constant in an attempt to bridge the gap between the topological and kinetic models of grain growth. Their definition was later revised by Doherty (1975) to define the sweep constant O* as “the number of grains lost when the grain boundaries sweep through a volume equal to the mean grain volume.” The numerical results were found to justify Doherty’s definition with a constant value of O* % 1.32. This definition is important for the development of a simplified variational model. Simplified variational approach The numerical results demonstrated that there is significant evidence that a number of parameters exist that can accurately represent the characteristics of normal grain growth evolution. These are fully described by Gill and Cocks (1996), who demonstrate that it is possible to represent the observed self-similar evolution of the system by a single variable. One choice for this degree of freedom is N ( r ) , the number of grains in the network as a function of time r .
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Let the constant Ao be the total cross-sectional area of the unit cell and let A be the average area swept per grain boundary per unit time. Then the number of grain boundaries in the unit cell is 3 N (as the average number of sides in a trivalent cellular network is necessarily 6), the number of grains lost per unit time is -N, and the mean grain size at time t is A o / N . Thus the sweep constant is given by
If the average grain boundary length is L and its average absolute normal drift velocity is u , , ~ ?then . A = u,,,,? L . The total grain boundary length L, = 3 N L is empirically found to be Lt = c2N1/’, where c i = 2 f i a ’ A o and (Y = 0.96. Combining these expressions yields
A
utlln
N
=---cI- N I / Z ’
L
where c: = A 0 / 2 f i ( a e * ) ’ . The variational functional (4.18) can be expressed in terms of these average quantities as
where u,~,.is the root mean square of the normal drift velocity, which has been written as u:,. = q(u,,,,,)’ and q > 1 is an unknown constant. This functional is a minimum when d & / d ~= 0, which gives, upon rearrangement and integration,
,/m,
As the mean grain size R = it can be seen that the above is similar in form to (4.19) with k = f i ( ( ~ O * ) ’ y [ , M b / q .Hence we have shown that the variational principle predicts a parabolic grain growth law without making any assumptions about the grain boundary dynamics except that 8* is a constant. We have also derived an expression for the time constant k in terms of 8* and an unknown parameter q. From the numerical results we find that q = 2.76 and 8* = 1.32, which predicts that k = 1.01, in good agreement with the actual value of k = 0.98. The important point to note about the above analysis is that as well as providing the theoretical background for the development of the numerical procedures the variational principle also allows the results of the simulations to be evaluated and
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a simple macroscopic law to be developed. We extend this concept further in the following subsection. 2. Abnormal Gmin Growth The evolution of polycrystalline thin films can be strongly influenced by the substrate on which the film is deposited. In general, the interfacial energy between the film and the substrate is lower if a grain has a similar crystallographic orientation to the substrate than if it does not. This variation in the interfacial energy between grains means that the energetic conditions for the growth of some grains are more favorable than those for the growth of others. This leads to abnormal (or secondary) grain growth (Thompson, 1990). In this case, unlike in normal grain growth, no time-invariant grain size distribution exists. Abnormal grain growth is characterized by the development of a bimodal grain size distribution in which abnormal grains (those with a low film-substrate interfacial energy) generally grow at the expense of so-called normal grains (those not preferentially favored for enhanced growth). The variational approach is flexible enough to incorporate many driving forces and dissipative mechanisms. The functional described here is similar to that for normal grain growth (4.18) but with an additional driving force. For the purposes of clarity and illustration, it is assumed here that the grain boundary energy per unit area, p,,is constant and that there are only two different interfacial energies. The difference between these energies, Ax,,, is the driving force for abnormal grain growth. The variational functional is therefore (Gill and Cocks, 1998)
h
u i ds
+ hy17
L,,
-
+
K U ds ~ ~
l),
F v f Ayln grain boundary junctions
u,, ds
or, in dimensionless form, using the same normalizations as for normal grain growth, (4.20) all triple points
where the final term is the rate of change of Gibbs free energy due to the change in the total energy of the film-substrate interface and the parameter (4.21)
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FIG. 16. The coalescence of two abnormal (light) grains in a matrix of normal (dark) grains.
where Ro is a reference length taken to be the initial mean grain radius. This parameter has been proposed elsewhere (Floro and Thompson, 1993) and represents the propensity of a system toward abnormality.
Evolution of the microstructure The initial microstructure is generated as for normal grain growth. A certain percentage of the grains are randomly chosen to be abnormal, i.e., to have a film-substrate energy that is A x n less than the others. The boundaries between abnormal grains are assumed to have a very low energy as they have a similar crystallographic orientation. Consequently, such boundaries are simply removed from the simulation. The process of two abnormal grains coming into contact is therefore one of coalescence, as shown in Figure 16. Once a boundary is removed, the two boundaries that remain at a junction are no longer in equilibrium. The microstructure is therefore allowed to restore itself to equilibrium under the action of the grain boundary tension by assigning such boundaries independent rotational freedom. Otherwise, the triple-points are chosen to be of the type Case 2 (see Section IV.E.l) to facilitate the computational process. Typical values for the relevant material parameters are approximately A x n = 0.05 J/m2, yb = 1 J/m', h = 50-500 A, and Ro = 500 A (Floro and Thompson, 1993). This gives a range for realistic values of 20of 0 5 Zo 5 0.5, where
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20 = 0 corresponds to normal grain growth. The results of two typical simulations for the same initial microstructure and ZO = 0.047 and Zo = 0.47 are illustrated in Figures 17 and 18, respectively. Gill and Cocks (1998) have fully evaluated the results of these simulations. Here we examine some of their major observations. It is immediately evident from Figures 17 and 18 that for small 20 there is significant normal grain growth before the growth of the abnormal grains takes over and consumes the entire film, while for the larger value of ZO there is limited normal grain growth and that preferential growth of the favorably orientated grains occurs early in the evolution process, with the final size of the normal grains only slightly larger than at the start of the simulation. Also, when ZO is small a significant number of the preferentially orientated grains are lost during the early stages of grain growth, while the majority of these grains remain when Zo is large and contribute to the abnormal grain growth process. After a time t, the abnormal grains consume all the normal grains to produce a single crystal. Over the range of conditions considered in the simulations, this time is well represented by the following empirical relationship:
t- =
.
g ~
(Zo+C.)’
where c and g are dimensionless material constants, which depend on the initial abnormal grain area fraction f . For ,f = 4%, c = 6.3 x lo-’ and g = 6.4. The area fraction of abnormal grains at a given time, , f ( t ) , is of practical interest to experimentalists. Figure 19 illustrates the temporal evolution of this quantity for the two numerical simulations mentioned above. As we shall see later, it is of great interest to know the length of the grain boundary interface between the abnormal and normal grains, L(,. Von Siclen (1996) predicts that L,, = ( 1 - f’)L,‘,X, where L:,X is called the “extended” length of the abnormal-normal interface and represents the length of the boundary in the absence of grain impingement. Using the Kolmogorov-Johnson-Avrarn-Mehl equation (see Von Siclen, 1996), f = 1 - exp(-fex), where f“ is the extended abnormal grain area fraction, and, assuming that grains can be represented as circular in the absence of impingement, we find L,, = 2(1 - flJ-arN,,Aolog(l
-
f),
(4.22)
where the total area under consideration, Ao, the initial number of abnormal grains, N,, and a , the fraction of abnormal grains that continue to grow and do not shrink, are all constants. The parameter a can be estimated from Hillert’s ( 1965)
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F I G . 18. Snapshots of the abnormal grain growth evolution of a polycrystalline thin film microstructure for Zo = 0.47 at normalized time: (a) t = 0.0 s, (b) t = 1.4 s. (c) r = 2.8 s, (d) r = 4.8 s. (e)r = 6.7 s. and (f) I = 8.4 s.
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a
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grain growth law with an additional driving force term due to the interfacial energy difference. A grain of radius R, is assumed to grow initially if (4.23)
+
i.e., when R,/Ro > 1/(1 2Zo). Assuming the initial grain size distribution can be represented by the steady-state grain size distribution for normal grain growth proposed by Hillert (1965), we can find the fraction of grains that satisfy this condition 3 a!
+4zo
= (iTG)exp(&).
This predicts that a! = 0.50 for ZO = 0.047 and a! = 0.85 for ZO = 0.47. For the simulation shown in Figure 17 for Zo = 0.047, it was found that only 30% of the abnormal grains survived (given the initial random microstructure shown in Figure 17a) so a value of a! = 0.30 is used in subsequent calculations. The model is much better for larger ZO = 0.47, when a much larger fraction of the favorably orientated grains survive and contribute to the abnormal grain growth process. Simplijied variational approach To develop a macroscopic model of abnormal grain growth, we return to the variational principle of (4.20) and, following the analysis of normal grain growth, express the material response in terms of a limited number of state variables. There is one driving force for normal grain growth and the state of the microstructure at any point can be represented by one variable, N i l ,the number of normal grains. For abnormal grain growth there are two driving forces and so an additional variable is required. We choose the abnormal grain area fraction f . Gill and Cocks (1998) have expressed the variational functional of (4.20) in terms of these two state variables and by determining the values of f and N,,that minimize the functional obtained evolution laws for both J' and N i l . Here we concentrate on the evolution of the area fraction of normal grains: (4.24) where the parameter K",, is the increased curvature of the abnormal grain boundaries over that of the normal grain boundaries due to changes in the grain boundary geometries from geometrical constraints. This is the net mean curvature of the abnormal grain boundaries, i.e., a directional mean curvature taking the curvature to be positive if its center lies on the normal grain side of the abnormal-normal grain interface. It has been found from the numerical results that K,,, quickly attains a reasonably constant value, once any abnormal grains that are not going to
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0.9
0.8 0.7 f 0.6 0.5 0.k
-ZO = 0 047 - - - Z0=0.47
0.3
01
0
I
I
I
I
0.1
0.2
0.3
0.4
I
0.5
I
I
I
I
0.6
0.7
0.8
0.9
F I G .i 9. The abnormal area fraction f as a functioii of time I normalized with respect to the total evolution time r for 20 = 0.047 and Zo = 0.47.
survive have disappeared, until the very final stages of growth, during which the net curvature can increase. Consequently, using (4.22), assuming a constant value of K(!,, , and integrating (4.24) after separation of variables, we obtain f = 1 - exp(-h2(t - t i ) 2 ) ,
(4.25)
where h=
M O Y O ~ ( z f0 ROKori)
4%
,
tl
=
J-
- fo)
h
and fo is the abnormal grain area fraction at time t = 0. Equation (4.25) is the classical expression commonly used to obtain a fit to experimental data (Thompson, 1985).However, the asymptotic behavior of (4.25) as f -+ 1 is not correct, i.e., f + 1 only as t + 00, although the majority of the evolution is faithfully represented. The quality of the fit is determined by the accuracy of the model for the abnormal grain boundary length, eq. (4.22). This equation provides a good representation for 20 = 0.47 and hence eq. (4.25) provides a good fit to the numerical results of Figure 19 except for the final stages of abnormal grain growth, i.e., f > 0.9. The fit is not as good for 20 = 0.047 when there is a much stronger interaction between the normal and abnormal grain growth processes.
Modeling Microstructure Evolution
135
We have investigated here the case of a thin-film microstructure in which there are two different film-substrate interfacial energies. A more general formulation for multiple interfacial energy systems could be obtained using a variational approach with multiple state variables. This is discussed more fully by Gill and Cocks (1998).
3 . Coupled Grain Boundap Migration and Surjiace Diflusion In the above analysis we considered a two-dimensional plan view of the surface of the thin film when modeling the process of abnormal grain growth. In practice, the surface profile is not flat. Grooves can develop where the grain boundaries meet the free surface, with the depth of these grooves and the surface profile depending on the velocity of the migrating grain boundary. A two-dimensional analysis of this situation has been presented by Suo (1996) and Mullins (1958). Recently, Pan et al. (1997) modeled this problem by combining the surface diffusion relationships of Section 1V.B with the variational functional of Section IV.E.2. Using the same normalizations as in Section IV.E.2, the variational functional becomes
poiII t s
where 8f,,,s = D , s , / M ~ Rand i Ps = yr/y/,, which we assume to be constant for is small, then the surface diffusion hnetics are slugthe entire surface. If fi,,,,, gish compared with the rate of grain boundary migration. The surface therefore remains flat as the boundaries migrate and the analysis of Section IV.E.2 is valid for the full three-dimensional situation. If Bfii,, is large, significant grooving of the grain boundaries will occur before the boundaries have had the chance to move, which will effectively pin the boundaries, preventing their migration. When is of the order of unity, both surface diffusion and grain boundary migration determine the way in which the microstructure evolves. Pan et al. (1997) have used the variational principle to analyze the twodimensional situation considered by Mullins (1958) and Suo (1996), which is shown in Figure 20a, and consists of a single migrating grain boundary. The only length scale in this problem is the film thickness h , which we take as our normalizing length, rather than the grain size Ro. The steady-state profile is shown in Figure 20b for fi,,,,= 0.096, 20 = 0.5, and y, = 1.0 and compared with the predictions of Suo and Mullins. These values were chosen to violate the small-
136
Alan C. R Cocks et al.
---_-Finite element solution. Steady state solution by Mullins and Suo.
(b) FIG.20. Thermal grooving at a migrating grain boundary. (a) The initial geometry. (b) The steadystate solution compared with lhe analytical solution of Mullins (1958) and Suo (1996). Reproduced from Pan e/ a/. (1997) with permission of The Royal Society.
slope assumption of the analytical solution (in the analysis of Suo and Mullins, it was assumed that the curvature of the surface is approximately given by the second derivative of height with respect to distance along the film). Despite this, there is good agreement between the two sets of results. A fuller evaluation of the predictive capability of this variational principle is given by Pan et al. (1997). Full three-dimensional calculations of the evolution of the grain structure in a thin film are yet to be undertaken.
F. EFFECTOF CHANGES IN ELASTICSTORED ENERGY ON MICROSTRUCTURE EVOLUTION In this section we add the contribution of changes of elastic stored energy to the variational principle. Two methods of analysis are described, which make use of boundary element and finite-element methods to determine how the Gibbs free energy changes as the microstructure evolves. We use the first of these approaches to evaluate the diffusive growth of a grain boundary crack. This is presented in Section 1V.F. 1. The second method is described in Section IV.F.2 and is employed to evaluate the growth of instabilities in thin films. 1. Difusive Growth of an Intergranular Crack in an Elastic Bicrystal
In Section 1V.B we examined the classical model of Hull-Rimer void growth in which it is assumed that surface diffusion is sufficiently rapid for the pores to
Modeling Microstructure Evolution
137
F I ~ ;2. I , Hall geometry of the intergranulai- crach problem for a linite crack of halt' width ( I . The stress licld due to the prewnce of thc crack o<.(.v)is singular at the crack tip. Material elastically wcdgcd into the grain boundary causes ;in opening displacement 6( Y ) and alters the stress distribution along the boundary. Thc displacemeni utlopta a confguratioti such that the stress singulaiity at thc crack tip disappears and the resulting stress normal to the houndary O , ~ ( . I) is finite.
maintain an equilibrium profile as they grow. In this section we examine the situation where both surface and grain boundary diffusion processes determine the rate of void growth (Chuang ef ul., 1979; Thouless et al., 1983).In a rigid material, any matter removed from the crack must be uniformly plated over the grain boundary. This situation has been investigated by Pharr and Nix (1979). In an elastic material, however, this constraint does not apply as nonuniform plating of material over the grain boundary can be accommodated by elastic defoimation of the matrix. The elastic stored energy associated with this wedging of material into the boundary depends on the current plating distribution (or opening displacement) over the entire length of the boundary, and, consequently, it depends on the entire history of plating on the grain interface. Chuang ( 1982) analyzed the steady-state propagation of a semi-infinite crack in an elastic bicrystal. This necessarily assumes that a steady-state plating distribution exists and is reached within a finite time. The method presented here is a numerical model for the growth of a finite crack which is dependent on the loading history, and hence the plating history, of the specimen. Consider an intergranular cavity of finite length 2u at the center of a grain boundary of length 2w,as shown i n Figure 21. The height of the cavity is assumed to be small with respect to its length, i.e., it is a cracklike cavity, and plane strain conditions are assumed. The x axis is taken to coincide with the grain boundary with the origin at the center of the crack so that the ends of the crack are at
138
Alan C. I? Cocks et al.
x = f a . The remotely applied stress
acts normal to the grain boundary and is assumed to be constant. The dissipation rate in the variational functional contains terms due to grain boundary and surface fluxes 0 ,
\Ir=\k,+\Irb=
and is accompanied by suitable Lagrangian terms to ensure matter conservation as described in Sections 1V.A and 1V.B. The driving force arises from the rate of change of the surface area of the grain boundary A b and crack surface A, and the rate of change of total potential energy of the system A so that
G = y S A &+ Y,,Ah
+ A.
The variation of elastic stored energy in diffusive cavity growth has been considered by Chuang and h c e (1981). Assuming that the strain energy density generated is low, even near the crack tip, energy variations due to the flux of strained material can be considered negligible. The rate of change of potential energy is then simply the difference between the rate of change of elastic strain energy due to removal of material to extend the crack and work done rearranging material on the grain boundary. If the stress normal to the boundary at any instant is a,,(x) and the opening displacement of the boundary is 6(x), then Chuang and Rice (1981) demonstrated that
(4.26) Chuang (1982) proposed that the boundary opening displacement could be represented by a continuous array of infinitesimal dislocations. This approach will be adopted here and is outlined below for the case of a crack of finite length. Consider an edge dislocation with Burgers vector b a distance u in front of the crack. The misfit stress at a position x on the grain boundary associated with this dislocation is (Luo, 1978)
a;n ( u , x-> =
bE
4 ~ ( -l U * ) ( . X - U )
for a material of Young’s modulus E and Poisson’s ratio u. The formation of a dislocation is due to a change in the opening displacement of the boundary and, following Chuang (1982), we write b = -d8/dx = -S’(x). Consequently, the stress normal to the grain boundary is the stress due to the presence of the crack
Modeling Microstructure Evolution
139
F I G . 2 2 . Discretization of the crack problem. Thc evolution of the geometry is described by the nornial velocities to the crack surface 1 1 ; and the opening displacement ratcs 8, at given points on the grain boundary.
a,.(x) and the sum of the stresses due to the array of dislocations
where a,, is the mean stress acting normal to the boundary. The stress distribution can be determined at any instant given a knowledge of the opening displacement distribution 6(x). Therefore the current state of this system can be represented by a discretization of the crack surface profile and the grain boundary opening displacement, as shown in Figure 22. Modeling the motion of a free surface by surface diffusion has been discussed previously in Section 1V.B. The opening displacement distribution is described by a number of linear elements which are defined by their end points 6,. However, the element at the crack tip requires further consideration. The crack tip opening displacement adopts a configuration so that the stress field associated with it interacts with the stress field due to the presence of the crack to remove the singularity in the stress field at x = a. This suggests that the stress field due to the opening displacement in the locality of the crack tip
Alan C. I? Cocks et al.
140
must also be singular. Inverting the Cauchy integral in (4.27) gives
Assuming that the stress at the crack tip is finite, one can evaluate this integral for x in the locality of the crack tip to give S’(X)
=
I+
4(1 - u2) X - U ITE all( a )In I T
finite terms
for (x - u ) / a << 1. Thus the opening displacement at the crack tip is described by
where a is an unknown constant dependent on the stress at the crack tip, a,( a ) , and the opening displacement at the crack tip, 60, is chosen to ensure continuity between elements. The stress distribution along the grain boundary can now be evaluated in terms of the discretized opening displacements 6; from (4.27). This yields an expression of the form
where A and B are constants and fi,, (1)is singular to the order ln(x). When evaluating the rate of change of elastic stored energy (4.26), the integral of the firstorder singular stress component O;,(x) is finite. The first term represents the stress singularity at the crack tip and can be removed by prescribing a = - B / A . This determines the value of stress at the crack tip. The constant B depends on an integral over the entire opening displacement distribution and A depends on the size of the opening displacement tip element. It is interesting to note that the piecewise linear opening displacement distribution yields a finite elastic stored energy, but not a finite stress. Also, a piecewise step function description for the opening displacement, such as that utilized by Thouless et al. (1983), does not yield a finite stress or finite elastic stored energy. The stress presented in the numerical results is the average integral of the stress over a single element. The variational functional can be formulated in terms of the degrees of freedom of this system, the normal velocities to the crack surface u; , and the opening displacement rates & at the discretization nodes, with appropriate Lagrangian terms to ensure mass conservation between the elements; see Figure 22. The geometry can then be updated using a suitable time step. Implementation and validation
Modeling Microstructure Evolution
141
of the method is presented by Gill et al. (1998). Here we present the results of some simulations which allow a direct comparison to be made with the results of Chuang (1982). Rather than write the variational principle in dimensionless form, it proves more convenient here to retain the full expression for the variational principle and employ material parameters for a representative high-temperature ceramic. Following Chuang ( 1982), we use the following material parameters, which represent the response of Salon at 1400°C: y5 = 0.75 J/m’, yl, = 0.375 J/m2, E = 200 GPa, w = 0.27, D, = 2.52 x m4kg-’s, and DI,= 2.75 x m4kg-’s. The initial crack was propagated from an equilibrium cavity of half length a0 = 1 p m on a grain boundary of total half length w = 100 pm. This is not a cracklike cavity, but a cracklike profile develops almost immediately. The remote applied stress was held constant at a, = 100 MPa and so the mean boundary stress a,, increases with the crack length. Physical quantities are normalized with respect to a reference length ao, the initial radius of the void, a reference stress ,,a and a reference time to = ai/Dha,. In this case ro = 420 days. This is large because the applied stress is quite low. Chuang (1982) predicts that the minimum stress intensity factor for steady-state crack growth is Kmjn = 0.83 M P a 6 which is roughly 1.7 times the critical intensity factor predicted by the Griffith theory. This stress intensity factor corresponds to a crack with a length of roughly 20 p m in this problem. It is of interest to study the crack evolution in this relatively low stress situation, because it is not something that can be inferred from Chuang’s steady-state results. Chuang (1982) found that the crack velocity is extremely sensitive to the stress intensity factor, and even when the stress intensity is only 20% above the minimum value, the crack is predicted to propagate the distance of 100 p m in just 30 minutes. The results presented here consider how a defect develops over a considerable period during the lifetime of a component. Figure 23a shows how the crack length increases over time. It can be seen that the crack grows at a stress intensity much less than that predicted by Chuang or the Griffith theory, albeit very slowly. The complete lifetime of the component in real time is approximately 16 years. From the initial development of the opening displacement in front of the crack in Figure 24a, one can see that matter is distributed over a large area for a small increase in crack length. Chuang’s analysis predicts that a steady-state crack traveling at its minimum velocity due to Kmin will plate material over a length of about 3 pm. In addition, such a crack will have a maximum opening displacement of roughly 5 nm at its tip and the crack width will decrease as the crack velocity increases. At first the crack travels very slowly and has a substantial width of about 100 nm. Therefore the increase in crack surface area is relatively small for the amount of material redistribution required
Alan C. I;: Cocks et a1
142
0.045
0 04 0 035
d(al 0.03 a0
0 025 0 02
0 015 0 01 0 005 0
1
1
5
10
t/to (b) FIG.2 3 . The temporal evolution of(a) the crack length and (b) the crack tip opening displacement under a remotely applied load of 100 MPa.
Modeling Microstructure Evolution
143
to extend it. There is ample time for the material to redistribute over a significant fraction of the grain boundary and from Figure 24c it can be seen that this reduces the stress at the head of the crack very effectively. Figure 23b shows that, as the crack slowly grows, the opening displacement at the crack tip monotonically increases; i.e., the amount of material wedged into the boundary, and hence the total elastic stored energy, increases. By the time the crack reaches the critical length of 20 b m predicted for steady-state growth (at a speed that would cause the crack to completely propagate the length of the boundary in 20-30 days), substantially more material has been wedged in front of the crack (35 nm at the tip) than would be expected if it had time to adopt a steady-state distribution (5 nm at the tip). It can be seen that this effect due to the plating history of the boundary is quite significant. The large mass of material wedged in front of the crack presents a substantial hindrance to its forward motion. In fact, it can be seen from Figure 23b that material continues to accumulate in front of it for a while longer. Eventually, this reservoir of elastic stored energy loses it battle with the applied load and the crack finds a mechanism by which it can release this energy. Relatively suddenly, the crack becomes very thin and moves forward through the bulk of the material wedged in front of it, leaving i t behind as indicated by the sharp drop in the opening displacement in Figure 23b which is associated with the rapid increase in the crack growth rate seen in Figure 23a when the crack is about 30 p m in length. Figure 24d shows that a large peak in the stress develops at the crack tip and it can be seen from Figure 24c that a substantial amount of material is removed from the grain boundary to the crack tip in an effort to try to reduce this (this was also found by Chuang ( 1982) for high-velocity cracks). As the stress increases the opening displacement and stress become more and more localized in the vicinity of the crack tip until the component fails. Similar behavior is observed for greater applied loads and the length at which the crack growth rate rapidly increases is found to be approximately inversely proportional to this load. Here we have examined a relatively simple loading situation and cracked geometry. More complex loading situations and geometries of the cracked body are examined more fully by Gill ef ul. (1998). The analysis described in this subsection takes into account the influence of the elastic deformation on the grain boundary deposition process, but the effect of changes of elastic stored energy on the shape of the growing crack has not been taken into account. In the next subsection we describe how this can be modeled and examine a situation where the elastic stored energy has a significant effect on the way in which the microstructure evolves.
Alan C. E Cocks et al.
144
a=ao
- - - a=4ao
0.035 2:
-. -.-
a = llao ............. a=23ao
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L
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20
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-a=40ao
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- - - a=49ao
6(x) -
-._._ a=64ao
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............ a = 8 8 a o
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.
"..(--"'................................................................................................ , -.- - -.-.-.-.-.- - -.- - - - - -.- - - - - - - - . __---*
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FIG, 24. Illustration of (a)-(b) the opening displacement and (c)-(d) the slress field. ahead of a growing crack of finite length u under a remotely applied load of 100 MPa.
2.0
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a=ao
1.8
- - - a = 4ao
- _ _ _a =_Ilao
1.6
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............. a = 23ao
........................ ... ........... .... ..... .... -.-.-.-..... ............. -.-.....---._. -. .....-..... ...........
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20
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a = 49ao
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a = 64a0 ............ a = 88ao
-
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146
Alan C. E Cocks et al. 2. Su f a c e Evolution of an Elastically Strained Epitaxial Thin Film
Thin films are widely used in high-technology applications, such as microelectronic and optoelectronic devices. They are generally manufactured by the deposition of a material from the liquid or vapor phase onto a supporting substrate. Under certain deposition conditions, epitaxial films are formed in which the atomic lattices of the film and substrate materials are commensurate. If no defects are formed, large elastic mismatch strains can result (roughly 4% for Si on Ge and up to nearly 7% for GaAs on TnAs) to ensure that the interface is coherent. The major contributions to the Gibbs free energy of the system are therefore the surface energy and the elastic stored energy. The elastic stored energy of the system can be reduced by the surface becoming wavy (Gao, 1991; Freund and Jonsdottir, 1993) but this results in an increase of surface energy. The driving force for the growth of these waves (which generally occurs by surface diffusion) and the ultimate breakup of the film (and hence failure of the device) is determined by the competition between these two contributions. We assume that the surface profile of the film evolves by surface diffusion, which is the only kinetic process. Therefore
The surface profile evolves in order to reduce its surface energy y 5 A , and the elastic strain energy due to the film-substrate mismatch strain U,, so that
G = y,A,
f
u,.
Assuming that the elastic strains are small, the elastic strain energy in the body can be determined through a transformation strain analysis in which the film is given a transformation strain E,; (the negative of the strain required to make the film coherent with the substrate). There are two contributions to A , : the surface distortion resulting from the elastic deformation of the film and substrate and changes of the surface profile arising from the diffusional redistribution of material. For the situations considered in this section, the contribution arising from the elastic deformation of the film is small. We are then able to describe the evolution of the profile of the film in terms of a single dimensionless parameter Eo, defined below. An evaluation of the effect of the elastic deformation of the body is given by Cocks and Gill ( I 998). For a film of volume V f on a substrate of volume V,, the elastic stored energy is given by
Modeling Microstructure Evolution
I47
where D;,kl is the elastic stiffness tensor and EL, is the total strain at a material point. Discretization of the film into a number of finite elements allows the elastic strain energy to be expressed as
(4.28) where all displacements and lengths are normalized with respect to the film thickness h , [ K,] is the dimensionless stiffness matrix for a given configuration, [ R ] is a normalized transformation strain matrix, [S]is the matrix of elastic displacements of the elemental nodes, and E , , ~ is the magnitude of the mismatch strain. Since we are only considering rearrangement due to surface diffusion, the last term of eq. (4.28) is constant throughout the evolution process. The nodal displacements are those that minimize the elastic strain energy:
I S 1 = ~ , , l K , l - ' l R l = &,,[GI. The rate of change of strain energy is then given by
where l u ] is the velocity matrix of the nodes on the surface of the film with normalized coordinates [x ], and driving force matrix [ F(,] determined from changes in the elastic strain energy is the assembly of
where F,, represents the contribution to the force matrix from the change in elastic stored energy associated with an increase of the ith degree of freedom. To evaluate this contribution, one must determine the differentials of the stiffness and transformation matrices. Analytical expressions for these differential matrices can be readily evaluated for isoparametric finite elements (Cocks and Gill, 1998)in terms of the mapping from the parent (reference) element to the mapped (physical) element. We normalize the variational functional in terms of the reference quantities
so that
I48
Alan C. F: Cocks et al.
where the dimensionless parameter Eo = Eh&:,/ y, represents the relative significance of the elastic mismatch strain energy and surface energy as driving forces. As in the analysis of surface diffusion in Section IV.B, we must modify this functional by introducing a series of Lagrange multipliers associated with the surface nodes to ensure that matter is conserved as the surface evolves. The nodal velocities of the film surface are given by the stationary solution of this modified functional:
and hence the surface profile of the film can be updated by a suitable time increment. Once this has been done, the finite-element mesh must also be reconstructed. Freund ( 1995) analytically investigated the evolution of small surface perturbations in an elastically strained thin film using a Fourier series representation for the surface profile, Results were obtained for a number of axisymmetric imperfections by the numerical integration of an infinite series of Bessel functions. The results presented here are for two-dimensional plane strain problems. Therefore, using the same notation as Freund. the perturbation from the uniform flat surface profile is chosen to be
A h ( x ) = -Aexp)Z: ; (
= -Aexp
)?I;(
__
,
where A and u define the shape of the imperfection and the asterisk represents the normalization x * = x / l for lengths and I*= r / t for time in which 1 = y , / U o = 2 h / E o is a reference length, Uo is the strain energy density in an unperturbed film, and t = l 3 / D 0 , yis , a reference time. Freund estimated that T x 10 s and 1 FZ 1 F m for silicon under a biaxial strain of 1% at 800 K. An initial imperfection of this type of amplitude 0.01h was introduced into a film of thickness h and width 2h and a suitable finite-element mesh was chosen. The results of the numerical simulation are shown in Figures 25 and 26 for two different imperfection sizes. Although these results in two dimensions are not directly comparable with those of Freund in three dimensions, their general form and behavior are very similar. The strain energy acts to increase the amplitude of the perturbation. However, when a* = 0.5 the imperfection is quite sharp, and the driving force to reduce the curvature at the center of the imperfection dominates initially, causing the amplitude to decrcase momentarily. This is accompanied by a spreading out of the imperfection along the length of the film. After this transient period the imperfection propagates toward the substrate. Using the method
Modeling Microstructure Evolution
149
I -1.5
t*=O ........... t*=0.05
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-
/
I
-.-.-
/
-2
---
4
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1
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t"z0.22 t*=0.43 I
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-
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,'..'
/r;
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- 1.5
........4'
.' I
-t * =o
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-2
............ t* =0.15 -.-.- t*=o.2
- - - t*=0.25 -3.5
0
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1
1.5
2
2.5
3
3.5
4
4.5
X*
(b) F I G . 25. Surface profile evolution of a small perturbation for (a) (I* = 0.5 and (b) (I* = I .0
Alan C. E Cocks et al.
150
L bhlojl A -1.2
t ..
t
\
\ \
FIG.2 6 . Temporal evolution of the amplitude of a small surface perturbation.
presented here, the development of these perturbations into sizeable defects can be followed through to the breakup of the film or other geometrical configuration (Cocks and Gill, 1998). The procedures presented here can also be readily extended to consider the formation of continuous and discontinuous thin films by including the process of condensation from the vapor phase and accounting for the accompanying increase in volume of the film.
V. Rayleigh-Ritz Analyses In the previous section we described how numerical simulation procedures can be developed from the variational principle of Section 111. An accurate description of the geometry is often required to obtain a simulation that fully describes the way in which the microstructure evolves. We should bear in mind, however, that the geometric situations we consider in these simulations are necessarily idealized and when we compare the simulations with experiments we are often interested in global features of the evolution process. We recognized this in our modeling of grain growth in Section IV.F, where we focused on a limited number of features of the evolving microstructure and used the variational principle to develop constitutive laws in terms of a small number of state variables and global macroscopic parameters. Also, diffusion coefficients are often not known with any degree of
Modeling Microstructure Evolution
(a)
151
(b)
FIG.27. The cosintering of two spheres showing the detailed shape of the particles determined using the techniques described in Section 1V.B. (a) Early in the sintering process, showing the formation of a neck. (h) Late i n the process. when diffusional rearrangement of material over the entire surtace of the particles has occurred.
accuracy; surface diffusion coefficients can only be taken as order-of-magnitude estimates. In any modeling exercise we should determine the overall objectives of the modeling and determine the simplest description that retains the major features of the evolving microstructure. Simulations can then be undertaken in an efficient manner and larger-scale simulations can readily be undertaken which allows information to be passed up the full hierarchy of length scales. Here we demonstrate how relatively crude descriptions of the evolving microstructure can be employed to provide insight into the dominant physical processes. In Section 1V.B we examined the cosintering of a row of identical spherical particles. The evolving shape of two contacting spheres computed using the methods described in Section I1 is shown in Figure 27. Early in the sintering process a neck region develops (Figure 27a). Sintering occurs by material flowing away from the grain boundary formed between two adjoining particles and plating out on the surface of the particles in the vicinity of the neck, with the detailed shape of the neck region determined by the surface and boundary diffusion processes. Away
Alan C. E Cocks et al.
I52
Frc;. 2 8 . An idealized representation of Figure 27. with the geonietry represented by truncated spheres of radius R separated by cylindrical disks of thickness t and radius .r.
from the neck region, the particles retain their initial equilibrium spherical shape. As sintering progresses the neck region grows and material rearrangement occurs over the entire particle surface (Figure 27b). Eventually, an equilibrium configuration develops in which the surface has a constant curvature. The early stage of sintering is similar to that experienced by a row of contacting spheres. Parhami er al. (1998) modeled this situation by adopting a simple two-degree-of-freedom geometric description (Figure 28). At a given instant the geometry is represented by a row of truncated spheres of radius R separated by disks of radius x and thickness d , with a grain boundary located at the center of the disk. The condition that volume must be conserved provides a relationship between x , d , R , and the initial radius Ro. The disk represents the development of the neck region and changes in the radius and thickness of this region indicate the extent of the local diffusion process. Changes in the value of R reflect the relative importance of the global diffusion process. Parhami et al. (1998) computed the evolution of the simple profile using the variational principle of (4.2) and compared the resulting evolution of the neck radius, x, and center-to-center spacing with more detailed in the range 0.1 to 10 for both free and forced sintering. These calculations for general features of the evolving microstructure are in very good agreement with the more detailed calculations over the full evolution process. Here we consider in more detail the early stages of the sintering process under the action of an axial force F . During the early stages of sintering, material only diffuses local to the neck region. As a result, as observed by Parhami et al. (1998), R does not change and remains equal to its initial value Ro. Thus we can treat the problem as a singledegree-of-freedom problem. Volume conservation gives, for x and d much less
es
Modeling Microstructure Evolution
153
than Ro,
We express the rate of change of the microstructure in terms of the height h of the unit cell shown in Figure 28, where 7
7
h=2[R,j-x-]
112
+d=2Ro-d
(5.2)
and = -2.The flux along the circular grain boundary can be obtained from the continuity condition of (3.3) and the flux along the surface of the cylinder from the continuity condition
Substituting the resulting flux fields into the variational principle of (3.6), noting that
1 . A/, = - - A , = 2 2
~
~
i
for small x and d and optimizing with respect to 6, gives
or
In general, 3,is greater than 1 and for the conditions considered here x' << Ro. The term involving 3,can therefore be ignored, and integration of (5.3) from x' = 0 to its current value gives
(5.4) For F = 0 and y/, = 0, eq. (5.4)reduces to Coble's (1958) free sintering result. We can therefore regard (5.4) as a generalization of Coble's analysis. Parhami et al. (1998) demonstrate that this expression is in good agreement with their fuller simulations, in which they also allowed R to vary, and more complex numerical models up to values of x / Ro of the order of 0.1. For larger values of x / R o , diffusion over the entire particle surface becomes significant and the rate of neck growth decreases as the geometry approaches the equilibrium configuration.
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The real power of this approach is not that we are able to model this problem with a reasonable degree of accuracy using only a small number of state variables, but it allows us to model problems over a much wider range of length scales, i.e., to analyze large arrays of particles. This does not mean that there is no role for detailed computations of the type described in Section IV. It is only by performing simulations of that type that we are able to identify the major features of the way in which the microstructure evolves. Thus simple models of the type described here are guided by the physical insight developed from the more detailed simulations. Parhami (1 996) has used the simple model described here as a major building block in the development of a discrete element model for the sintering of random arrays of particles. In the following section we use the result developed here to obtain an analytical constitutive model of the sintering of ceramic compacts.
VI. Structure of Constitutive Laws for the Deformation of Engineering Materials In Section IV we used the variational principle of Section 111 to develop numerical procedures for the detailed modeling of microstructure evolution in engineering materials. In many practical situations we wish to develop constitutive laws for the behavior of a material in terms of a limited number of state variables. In this section we identify a general structure for constitutive laws where inelastic deformation is due to the class of diffusional mechanisms considered in Section 11. In the process we simplify the structure of constitutive laws proposed by Cocks (1994). Consider a body of unit volume which is subjected to macroscopic stresses Xi,, and experiences inelastic straining at a rate E i j . We limit our discussion to situations where the free energy is dominated by the potential energy of the applied load and the energy associated with any interfaces in the body. For this situation the rate of change of Gibbs free energy is given by
For situations where the interfaces consist of grain boundaries and surfaces and the specific energies are independent of state, we can simplify this relationship:
Modeling Microstructure Evolution
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Substituting for G in eq. (3.5) and differentiating the functional to obtain the exact solution gives Cij dEij
- YbdAb - y.5 d A s
=d q .
(6.2)
In general, the interface areas are a function of the inelastic strain. Then
where Xi’/is a capillarity, or sintering, stress, which arises from the change in interface area that accompanies an increment of strain, i.e.,
+
where A = ybAb ysA,. Similarly, substituting (6.1) into (3.7) and considering chemical potential and stress fields that satisfy the surface and boundary energy requirements, we obtain
E,, dC,, =d @ ,
(6.5)
i.e.,
a@ 8..= IJ axij
’
The rate of energy dissipation per unit volume is given by
D
= -G = @ + P ‘
and the macroscopic work rate per unit volume is lQ = X,j E;, = 0
+ \v + A .
These results simplify the expressions originally proposed by Cocks (1994) by relating all contributions to the capillarity stress to changes in internal interface areas and combining all these contributions into the scalar potential A. Following Cocks (1994), we can obtain bounds on the macroscopic potentials. We do not repeat the complete details here and further limit our attention to the kinematic bounds derived from the variational principle of (3.6). Bounds on 0 and Q A can be obtained by identifying suitable compatible flux and velocity fields, which results in a macroscopic strain rate E c . For a body subjected to a prescribed stress C i j , the strain rate potential is bounded from below by
+
@
>_ & j E Y .‘ J - Q c - A‘,
(6.7)
Alan C. F: Cocks et al.
I56
where the superscript c indicates quantities determined from the assumed compatible field. If instead the strain rate k,, i5 prescribed and the assuined field is scaled such that hi, = E , , , then
W
+ A 5 W' + A ' .
(6.8)
These bounds and their extensions allow the nature of the interactions between different mechanism$ to be determined and suitable simplified structures for constitutive laws to be identified. Full details are given by Cocks (1994). In this section we demonstrate the utility of the above bounds by developing an anisotropic constitutive law for stage 1 sintering controlled by grain boundary diffusion employing the bound of (6.8).
A. STAGE1 COMPACTION OF CERAMIC COMPONENTS Consider the random array of spherical particles of initial radius Ro shown in Figure 29a, which has been strained to a state E,, and is subjected to a strain rate E , , . We limit our attention to the early stages of compaction, when the strains and contact patches formed between the particles are small. We further restrict our consideration to conditions where all the principal components of strain, but not necessarily strain rate, are compressive. It was demonstrated in Section V that, if the necks formed between the contacting particles are small, diffusional processes only occur local to the neck region. We can therefore assume that there are limited interactions between the necks. Following the analysis of Section V, we idealize
(a)
(b)
FIG. 29. (a) A random array of spherical particles subjected to an applied strain rate conlacring particles from [he array of (a) with outward normal n.
b;,.(h) Two
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the geometry at a given instant as an array of truncated spheres of radius Ro separated by disks of material of radius x and thickness d . We assume, as in the analysis of Fleck et al. (1997),that we can relate the center-to-center spacing of two particles whose contact patch has an outward normal n; (Figure 29b) to the strain: Nn,)=2Rdl
+ E,,n,n,).
(6.9)
During the early stages of compaction, the average number of contacts per particle, Z, will remain essentially constant. Geometric considerations allow us to relate the contact radius x ( n , ) to h ( n , ) (see Section V) and hence E l , : x ( n , ) = 2Ro(-E,JnlnJ)'/'.
(6.10)
We now assume a compatible velocity field such that h(n,')= 2R&,,nln,
(6.1 1)
and, from the analysis of Section V, the contribution of the boundary diffusion process on the contact patch to the potential W is (6.12)
Consider an element of the compact which contains a single particle and has a volume :rr R i / D , where D is the relative density of the compact. Since the array of particles is random, the number of contacts per unit area is
z=- Z
4rr R,'
and
(6.13)
W becomes (6.14)
where the factor of arises because a given boundary is shared between two particles and S is the total surface area of a particle. Substituting for z and @'(ni) using (6.12) and (6.13) gives (6.15)
To determine the capillarity stress contribution, we need to relate the rate of change of the area of the contacts to the applied strain rate. Employing the above
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TABLE2 M A T R I XDEFINIIIII N ( 6 .18): cj,,i\=
670K +c,,ji[
compatible field and making use of the analysis of Section V, we find 1 . 7 ‘ A b ( n , )= - - A s h f ) = -4rrR6El,n,n, 2
(6.16)
and
where EL,is the volumetric strain rate. Choosing a coordinate system whose axes lie in the principal strain directions, combining (6.15) and (6.17), and evaluating the integrals of (6.15), we find
where
The viscosity matrix CfJx/can be expressed in terms of the three principal strain components and four constant parameters to give six independent components of the viscosity matrix. All the nonzero components are given in Table 2. It is interesting to note that the sintering potential defined in (6.18) is only a function of the volumetric strain (i.e., relative density of the component). This does not, however, mean that the material will densify uniformly after experiencing a prior loading history which results in an anisotropic structure. The vis-
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cosity in the direction of the maximum compressive stress is greater than in the transverse directions. As a result, the material will strain at a slower rate in this direction and the material will deform toward a more isotropic state. If the material is compressed isostatically, the three strain components are equal and the material response is isotropic. The macroscopic response can then be expressed in terms of a single state variable, the relative density D . The general form of the model then reduces to that developed by McMeeking and Kuhn (1992). The approach presented here can be readily extended to other mechanisms of material redistribution. For isolated contacts, once the solution for the deformation between two contacting particles is known, the strategy described here can be employed to determine the macroscopic response.
VII. Concluding Remarks In this paper we have described a general variational principle for the analysis of microstructure evolution in engineering materials. We have demonstrated how self-consistent numerical schemes can be developed from it which naturally take into account the competition and synergy between the different mechanisms. The form of the variational principle allows the relative importance of different mechanisms to be readily identified and strategies to be developed in which the full range of possible situations can be systematically evaluated. In Section IV we limited our attention to situations in which material rearrangement occurs by direct diffusional processes or through the migration of interfaces. In the process we have considered a wide range of thermodynamic driving forces. The full range of kinetic processes and the origin of the thermodynamic driving force are summarized in Table 1. This is not an exclusive list. The techniques described here can be extended to a wide range of mechanisms. For example, power law viscous flow of the matrix and multicomponent diffusion problems can be considered within the current framework as well as other contributions to the driving force, such as chemical contributions related to composition and phase changes. Similarly, the range of physical problems we have examined is not intended to be exhaustive. These have been chosen to illustrate the techniques and to demonstrate how macroscopic constitutive laws can be developed in a systematic manner from the results of the simulations. We have also shown how simplified methods of analysis of the Rayleigh-Ritz type can be developed. These provide a good indication of the general way in which the microstructure evolves. Care, however, needs to be taken in the use of these methods. Enough degrees of freedom need to be chosen to allow for the different major ways in which the microstructure can evolve. Use of this method
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therefore generally requires a deeper physical understanding of the way in which a given material system is likely to behave than use of the full computational methods. It has also been shown how bounding theorems can be developed from the variational principle of Section 111. A major use of these bounds has been in the development of constitutive laws for the inelastic behavior of engineering materials, as described in Section VI. A fuller evaluation of the use of these bounding theorems is provided by Cocks (1994).
References Amerasekera. E. A,, and Nam, F. N. (1997). Failure Mechanisms in Semiconductor Devices. Wiley, Chichester. Arzt. E., Ashby. M. F., and Verrall, R. A. (1983). Interface controlled diffusional creep. Acra Mefall. 31. 1977. Ashby, M. F. (1969). On interface reaction control of Narharro-Herring creep and sintering. Scripra Meftrll. 3, 837. Ashby, M. F., and Verrall, R. A. (1973). Diffusion accommodated flow and superplasticity. Acra Metall. 21, 149. Atkinson. H. V. (1988). Theories of normal grain growth in pure single phase systems. Acra Mefall. 36, 469. Burke, J. E. (1949). Some factors affecting the rate of grain growth in metals. Trans AIME 180.73. Burton, B. (1972). Interface reaction controlled diffusional creep: a consideration of grain-boundary dislocation climb sources. Marer: Sci. Engrg. 10, 9. Chen, I. W., and Xue, L. A. (1990). Development of superplastic structural ceramics. J. Amer: Ceram. Soc. 73. 2585. Chuang, T.-J. (1982). A diffusive crack growth model for creep fracture. J. Amer: Cerani. Soc. 65. 93-103. Chuang, T.-J., Kagawa, K. I., Rice, J. R., and Bank-Sills, L. (1979). Non-equilibrium models for diffusive cavitation of grain interfaces. Acra Mefull. 27, 265-284. Chuang, T.-J., and Rice, J. R. (1981). Energy variations in diffusive cavity growth. J. Amer: Ceram. Soc. 64, 46-53. Coble, R. L. (1958). Initial sintering of alumina and hematite. J. Amer: Ceram. Soc. 41, 55. Cocks, A. C. F. (1990). A finite element description of grain-boundary diffusion controlled processes in ceramic materials. In Applied Solid Mechu,iics--3 (I. M . Allison and C. Ruiz, eds.). Elsevier, North-Holland, Amsterdam. Cocks, A. C. F. (1992). Interface reaction controlled creep. Mech. Mater: 13, 165. Cocks. A. C. F. (1994). The stmcture of constitutive laws for the sintering of fine-grained materials. Actcr Metall. Mate,: 42, 2191. Cocks, A. C. F. (1996). Variational principles, numerical schemes and bounding theorems for deforMech. . Phys. Solids 44, 1429. mation by Nabarro-Herring creep. .I Cocks, A. C. F., and Ashby. M. F. (1982). On creep fracture by void growth. Prog. Mater: Sci. 27, 189. Cocks. A. C. F., and Gill. S. P. A. (1996). A variational approach to two dimensional grain growth. 1. Theory. Acta Murer: 44. 47654775. Cocks, A. C. F., and Gill, S. P. A. (1998). A numerical model for stress-driven diffusion in elastically strained epitaxial thin films. Leicester University Engineering Department Report 98-3.
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Cocks. A. C. F.. and Pan. J. i 1994). Thc influelice o f an interface reaction on the creep response daniaged materials. M t 4 Mrrwr: 18. 269. Cocks, A C . F.. and Searlc. A. A. 11990). Cavity growth in ceramic materials tinder multiaxial stress states. A C I ~MJ r ~ l l38. . 2493. Cocks. A. C. F.. and Searle, A. A. 119911. Void growth by grain-boundary diffusion i n tine grained materials. Meelr. M ~ J I 12, ~ J 279. : Doheny. R. D. 11975). Discusion of Mechanism of steady-stair grain growth i n Aluminium. Metrrl/JlJ:qfCrf/ Trtrtrv. 6A. 588-590. Du. Z.-2.. and Cocks. A. C. 1;. (1992). Constitutive models for the sintering ofcet-amic components. 1. MLiterial models. Acm Mmrll. 40. 1969. Du. 2.-2..McMceking. R. M.. and Cocks. A. C. F. i1998). A numerical model of grain growth. Philos. Mtrg. To appear. Fleck. N. A,. Storikers. B.. and McMeeking. R. M. ( 1997). The viscoplastic compaction of powders. 111Mrchtrr~icsof Grtrrirrkrr t r ~ i r Po,nir.\ l Mrrtt~ricrls(N. A . Fleck and A. C. E Cocks, eds.). Kluwer Academic, Dordrecht. Floro. J. A,. and Thompson. C. V. i 1993).Numerical analysis o f interface energy-driven coarsening in thin films and its connection to grain growth. Arm Metnll. Mtrtrr: 41. 1137-1 147. Freund. L. B. i 1995). Evolution of waviness on the suilace of a strained elastic solid due to stressdriven surface diffusion. I r m w i r r / . J. Solid.\ Srrrrcnrres 32, 9 1 1-923. Freund, L. B., and Jonsdottir. F. i1993).Instability o f a hiaxially stressed thin tilm on a substrate due to materials diffusion over its free surface. J. Mecli. Phw. Solir/.s 41, 1245-1264. Gao. H. ( 1991). A boundary perttirhation analysis for elastic inclusion\ and interfaces. Irrternot. J. Solirl.~Strrrcrrrres 28. 703-725. Gluier. J. A.. and Weaire. D. (1992). The kinetics of cellular patterns. J. P l y Co!it/ert.st,d Mrrftrr 4, 1867. Gill. S . P. A,. and Cocks. A. C. E i1996). A variational approach to two dimensional grain growth. 11. Numerical results. Ac/tr Mrrwc 4 4 . 1 7 7 7 1 7 8 9 . Gill. S. P. A,. and Cocks. A. C. F. (1997a). An investigation of mean-field theories for normal grain growth. Plrilos. Mtrg. A 75. 30 1-3 13. Gill, S. P. A,, and Cocks. A. C. F. (1997b). A short note on a variational method for two-dimensional normal grain growth. Scrip~trMLJWJ: 35. 9. Gill. S. P. A,. and Cocks. A. C . F. ( 1998). A variational approach to modelling abnormal grain growth i n thin tilnis. Leicester University Engineering Report 98- I. P. A.. Cocks. A. C. F.. and Gao, H. (1998). A variational model of the diffusive growth of an k of finite length. Leicester University Engineering Department Report 9X-2 Heywang. W. (197 I). Semiconducting barium titanate. J. Mtrtrt: S r i . 6, 12 14. Hillert. M. 11965). A mean field theory foi- nornial grain growth. Amr Mrtrill. 13. 227. mid Proprrrirs. Honeyconihe. R. W. K., and Bhadeshia. H. K. D. H. 1 1995). Sfrrls~Mi~.ro.strrrc/rrrr Edward Arnold. London. Howes. M. J.. and Morgan. D. V. ( 19851. Gtrllirtt~rAtwrrielr: Mrr/t,riu/s. Drvicrc rirttl Cinwits. Wiley. Chichester. Kucherenko. S.. and Pan. J. 1 1998).A coupled linite element and finitc diffcrcncc formulation with an implicit time integration scheme for analysih of inicrwtructure evolution. LJniversity of Surrey Report. Luo, K. K . (1978). Analysis of branched cracks. J. Appl. Mrch. 45. 797-802. Matan. N.. Winand. H. M. A,, Bogdanufl. P. D.. and Reed. R. C . (1998). Coupled thermodynaniickinctic modelling of diffusion reactions i n superalloys, Actrr Mnrrr. To appear. McMeeking. R. M.. and Kuhn. L. T. i1997).A diffusional creep law for powder compacts. Acrrr M e ~ l l . Mrrtc.~:40. 96 I.
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Mullins, W. W. (1958). The effect of thcrmal grooving on grain-boundary migration. Acfa M ~ t u l l6, . 414. Pan, J., and Cocks, A. C. F. (1993a). The effect of grain-size on the stress field ahead of a crack in a material which deforms by Coble creep. Internut. J. Fracture 60, 121. Pan. J., and Cocks, A. C. F. (1993b).Computer simulation of superplastic deformation. Compuf.Mare): sci. 1,95. Pan, J . , and Cocks, A. C. F. (1995). A numerical technique for the analysis of coupled surface and grain-boundary diffusion. Act0 Metall. Muter: 43, 1395. Pan, J . . and Cocks, A. C. F. (1998). Modeling crack growth in engineering ceramics. To appear. Pan, J . , Cocks, A. C. F., and Kucherenko, S. (1997). Finite element formulation of coupled grainboundary and surface diffusion with grain-boundary migration. Proc. Roy. Soc. London Se): A 453,2161. Parhami, Z. (1996). Ph.D. thesis. University of California at Santa Barbara. Parhami, F., McMeeking, R. M., Cocks, A. C. F.. and SUO,Z. (1998). A model for the sintering and coarsening of rows of spherical particles. Submitted for publication. Perduijn, D. J., and Peloschek, H. P. (1968). Mn-Zn ferrites with very high permeabilities. Proc. British Ceram. Soc. 10, 263. Pharr, G . M., and Nix, W. D. (1979). A numerical study of cavity growth controlled by surface diffusion. Actu Mefall. 27, 1615-1631. Raj. R., and Ashby, M. F. (1975). Intergranular fracture at elevated temperature. A c f a Mefall. 23,653. Raj. R., Shih, H. M., and Johnson, H. H. (1977). Correction to: Intergranular fracture at elevated temperature. Scripta Merull. 11, 839. Rhines, F. N., and Craig, K. R. (1974). Mechanism of steady-state grain growth in Aluminium. Merullurgical Trans. 5.413425. Shewmon, P. G . (1963). Difusion in Solids. McGraw-Hill, New York. Spriggs, R. M., and Dutta, S. K. (1974). Pressure sintering-recent advances in mechanisms and technology. Sci. Sintering 6 , 1. Sun, B., SUO,Z . , and Cocks, A. C. E (1996). A global analysis of structural evolution in a row of grains. J. Mech. Phys. Solids. Suo, Z. (1996). Motions of microscopic surfaces in materials. Adv. Appl. Mech. 33. Thompson, C. V. (1985). Secondary grain growth in thin films of semiconductors: theoretical aspects. J. Appl. Phys. 58,763-772. Thompson, C. V. (1990). Grain growth in thin films. Ann. Rev. Mates Sci. 20, 245-268. Thouless. M. D., Hsueh, C. H., and Evans, A. G . (1983). A damage model of creep crack growth in polycrystals. Aclu Memll. 31, 1675-1 687. Van Siclen, C. Dew. (1996). Random nucleation and growth kinetics. Phys. Rev. B 54, 11845-1 1848.
ADVANCES IN APPLIED MECHANICS. VOLUME 36
Stochastic Damage Evolution and Failure in Fiber-Reinforced Composites W. A . CURTIN* Engineering Science and Mechunics Materials Science und Engineering Virginia folyrechnic Institute cind Stcite University Blackshurg Virginia
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I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I1. Preliminary Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 168 A . Fibers, Matrices. and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . B . Critical Strength and Critical Length Sc . . . . . . . . . . . . . . . . . . . 172
I11. Single-Fiber Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Solutions to the s.f.c. Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Comparison to Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
173 173 174 179 182
IV . Multifiber Composites: Global Load Sharing . . . . . . . . . . . . . . . . . . . . 185 A . Global Load Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 B. Fiber Pullout and Work of Fracture . . . . . . . . . . . . . . . . . . . . . . . 187 C . Stress-Strain Behavior: Exact and Approximate Results . . . . . . . . . . . . 189 D . In situ Fiber Strength and Fracture Mirrors . . . . . . . . . . . . . . . . . . . 197 198 E . Localization and Numerical Simulations . . . . . . . . . . . . . . . . . . . . F . Initial Fiber Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 G . Comparison to Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 H . Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 I . S u m mary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 V . Multifiber Composites: Local Load Sharing . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Local Load Sharing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Statistical Models of Composite Failure under Local Load Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Local Load Sharing Model of Zhou and Curtin (1995) . . . . . . . . . . . . E . Analytic Models and Weak-Link Scaling . . . . . . . . . . . . . . . . . . . . F . Comparison to Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
212 212 213 217
. 220 232 237 243
.
*Present address: Division of Engineering Brown University Providence. RI 029 I2.
163 ISBN 0- I?-(M?O3O-X
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ADVANCES IN APPLIED MECHANICS VOL. 36 Coplright 0 1999 by Academic Press. All n p h b of rrpruduclion in any form reserved. ISSN 006S-21hS/99 $30.00
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VI. Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,248
I. Introduction A major theme in the field of materials for structural applications is the development of the relationship between the material microstructure and its performance under various thermomechanical loading conditions. Such a relationship allows for the tailoring of the microstructure to obtain a desired behavior or for optimization of a material for multifunctional use. In the engineering design of structural components, the “performance” includes, among others, material stiffness, inelastic deformation, ultimate failure, reliability, and lifetime. Heterogeneity, or nonuniformity, can, however, severely affect these material properties and complicates the development of microstructure/performance relationships. Heterogeneity is used here in the broadest sense, and includes spatial variations in the elastic, thermal, inelastic, and toughness properties of the constituent materials. Heterogeneity is particularly important in understanding material failure and reliability because failure is controlled by the development of a “critical” amount of damage somewhere in the material, damage that becomes unstable to growth at the failure stress. The damage may be a single crack (preexisting or induced) or a more diffuse damage zone. In a structural component, there are several sources of heterogeneity that influence the formation and propagation of damage. First, there can be a “distributed’ or spatially varying stress field, imposed externally and/or caused by heterogeneous elastic and thermal properties in the material. Second, in real materials with flaws and heterogeneities, there are spatially distributed strength and/or toughness fields. The stress and toughness fields are coupled after damage occurs: Damage at areas of low strength or low toughness redistributes stress over length scales comparable to the damaged region and so drives further damage in the vicinity of the existing damage. Ultimately, the damage becomes unstable, requiring no further increase in the externally applied field to grow indefinitely across the entire material. While conceptually clear, the formal coupling of evolving stress fields in the presence of damage caused by heterogeneous strength or toughness is a very difficult problem. One major difficulty stems from the fact that the final failure requires only one critical damage region somewhere in the entire material so that any type of averaging over the heterogeneity or stress fields is precluded, since the averaging process can eliminate precisely those fluctuations that drive the failure. Since the formation of the “critical” crack or damage zone can occur
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anywhere in the material, failure is controlled by such “weak links.” Weak-link failure is accompanied by a volume scaling of the strength (larger systems have lower strengths). Hence, models must determine the strength probability distributions at volumes that are large enough to be in the “weak-link’ scaling regime if volume effects are to be described properly. In heterogeneous materials with intrinsically brittle constituents, or materials where the brittle constituents are the main load-bearing phase, the problem of accurately describing the damage evolution and failure is most acute. Damage is in the form of cracks with sharp crack tips, and a model cannot eliminate the crack tips a priori without eliminating the possible source of initiation of the macroscopic failure. There are thus at least three (widely differing) scales of importance: the atomic crack tip scale, the microstructural scale, and the macroscopic scale. The atomic scale might be replaceable by a larger “near-tip’’ zone but such a scale is still expected to be much smaller than the microstructural scale. In a homogeneous material or weakly heterogeneous material, failure is entirely controlled by the atomic crack tip and the larger scales are not important. However, failure in a strongly heterogeneous material can require proper treatment of phenomena over all the length scales, which span orders of magnitude in physical size. Traditional analyses of material failure typically involve either (i) direct continuum finite-element models (FEMs) of stress fields, damage growth, and crack growth at the microstructural scale; or (ii) continuum damage mechanics (CDM) wherein a stable damage state is assumed and associated with, or measured by, a stiffness reduction. In FEM modeling, it is computationally difficult to study large “representative” regions of a heterogeneous material, large numbers of cracks, or substantial crack growth. In CDM, there is usually no micromechanics of actual damage mechanisms and the material is assumed homogeneous on some largehepresentative length scale; this immediately removes the underlying true heterogeneous toughness distribution and local stress fluctuations that may be critical to failure. The material becomes a deterministic nonlinear continuum medium. The point of failure is then also deterministic and not specifically connected to the microscale damage. Stochastic CDM models can accommodate statistical variations in the properties of the various “representative elements” but usually such variations are not derived from the study of known damage modes at the microstructural scale. In one class of materials, fiber-reinforced composites (an important class of engineered structural materials), some of the atomickrack tip issues can be eliminated in favor of the cumulative damage at a larger scale. The brittle reinforcing fibers have intrinsic cracklike flaws and known toughness so that the strengths of these flaws can be characterized. More importantly, once a flaw becomes unstable and fails a single fiber, the crack formed is generally arrested
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at the fibedmatrix interface by matrix plasticity/yielding, interface debonding, or fibedmatrix sliding at a previously debonded interface. All of these mechanisms prevent the sharp crack from extending up to the surface of the neighboring fibers, which would cause them to fail spontaneously. The neighboring fibers thus experience a higher load, or stress concentration factor, but not a singular stress field or stress intensity factor. The fibers can thus be considered as stochastic load-bearing entities that interact through load transfer from broken to unbroken fibers. The details of the stress concentration factors or load sharing among fibers, and its dependence on constituent mechanical properties and interface properties, is a formidable problem but at a length scale of the microstructure (fiber diameter or spacing) that is far more tractable than the crack tip problem. The fiber composite does contain heterogeneity in the fiber strengths and in the local stress fields at the microstructural scale once some fiber damage has occurred. So, the connection between the microstructural scale and the macroscopic scale is still nontrivial and involves mechanics, stochastics, and volume scaling. Coupling mechanics and stochastics exactly is impossible, which has driven the development of approximate models wherein the mechanics of the load transfer is approximately described, although in a manner consistent with the physical requirements such as macroscopic mechanical equilibrium. The purpose of this article is to review the accomplishments over the last 10 years in the area of modeling of the mechanical properties of fiber-reinforced composites, with an emphasis on accurately predicting ultimate tensile strength, stress-strain behavior, and reliability. A key aspect of the problem is the effect of heterogeneous or stochastic (nondeterministic) damage evolution that eventually leads to overall material failure. In particular, we discuss recent efforts to develop coupled mechanics/stochastic models for fiber composites in a manner that yields predictive, accurate results for the deformation and strength of real fiber-reinforced composite systems. These models, although using simple approximations to the complex three-dimensional stress fields, provide the microstructure/performance relationships in these materials, demonstrating the basic dependencies of macroscopic material performance on underlying constituent properties. The general problem addressed here can be posed as follows:
Given a statistical distribution of fiber strengths, a matrix strength, prescribed interfacial fibedmatrix interface behavior, and material thermoelastic constants, then Predict the macroscopic stress-strain behavior, tensile strength and failure strain, work of fracture, reliability (probability of failure versus stress), and any dependence of these properties on the physical composite volume.
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The success of the resulting predictions is assessed by comparing them to as many detailed experimental results as possible. This review will attempt to unify much of the recent work in this field into a common framework and common notation, with derivations of key concepts and models, presentation of major results, and extensive comparisons of the theories to experimental data. This review can thus serve both as a compendium of results for workers in the field and as a primer for scientists and engineers interested in the subject. The specific work described below is confined to continuous, unidirectionally reinforced composites. However, extensions to cross-ply and woven fiber geometries will be discussed. Particulate-reinforced composites are not considered although aligned short-fiber composites are a natural extension of the present work. We will also consider only fiberlmatrix interfaces having a constant shear yield strength or interfacial sliding resistance T at the interface; purely elastic interfaces, Coulomb friction, and rough asperityxontact interfaces will not be discussed although work in these areas shows them to be relevant in some materials (Parthasarathy and Kerans, 1997, and references therein). Lastly, the detailed micromechanics of the stress concentration factors or load transfer will not be discussed extensively. Rather, we will focus on two extreme cases that circumscribe the range of behaviors physically expected. These limitations are not fundamental ones, however. The major concepts, methods, and general results are anticipated to survive the relaxation of these restrictions, although not without significant additional effort to obtain accurate results. The remainder of this article is organized as follows. In Section 11, we discuss the basic model of the unidirectional composite, including fiber strength statistics, matrix behavior, and interfacial mechanics. In Section 111, we study the singlefiber composite (s.f.c.), which is the fundamental problem in understanding the interplay of fiber strength statistics and interface mechanics. The s.f.c. problem highlights the key aspects relevant for subsequent models for ceramic, metal, and polymer composites (CMCs, MMCs, and PMCs, respectively). In Section IV, we introduce the concept of global load sharing and demonstrate how composite failure properties follow from the s.f.c. problem. Applications to CMCs are made to demonstrate the capabilities of this model. In Section V, we investigate local load sharing, which brings many of the important and subtle features of brittle, weaklink-driven failure into the problem of composite failure. Models for combining mechanics, stochastics, and size scaling are presented and applied to various MMCs and PMCs. Finally, in Section VI, we discuss the limitations of the existing models, and aspects of the existing theories that require improvement or refinement, possible extensions of these models to a host of other problems, and the coupling of these models to macroscale models for predicting structural performance.
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11. Preliminary Issues
A. FIBERS,MATRICES, AND INTERFACES
The model composite studied throughout this work consists of a volume fraction f of continuous cylindrical fibers of radius r embedded in a matrix material in a unidirectional (aligned) arrangement. The axial (along the direction of fiber alignment) Young’s moduli of the fibers, matrix, and composite are E f , E , , and E,, respectively, with the rule of mixtures estimate E, = f E f (1 - f)Em being highly accurate. Under an applied axial stress cr, and with no interface debonding, fiber breakage, or inelastic behavior, mechanical equilibrium and strain compatibility require that
+
=f0.f
+ (1 - f)%,
(14
where
are the average axial stresses on the fibers and matrix, respectively. In formulating simple approximations below, we neglect many complicating features of the true stress fields, such as the radial variations in axial stresses (Weitsman and Zhu, 1993),but retain the dominant features so that the analysis is tractable and yet the predictions are accurate. Fiber composites are designed to take advantage of the strong fibers, which are thus generally much stronger than the matrix material. We consider three possible types of matrix behavior: (i) elastic but with Em << E f , as for PMCs; (ii) elasticbrittle where the matrix failure strain is much lower than that of the fibers, as for CMCs; and (iii) elastic/perfectly plastic where the tensile yield stress of the matrix, av,is lower than the fiber bundle strength, as for MMCs. In all of these cases, the matrix then plays only a secondary role in determining the ultimate composite strength. In case (i), the matrix remains elastic but carries negligible stress. In case (ii), after cracking the matrix carries essentially zero stress at higher applied stresses. In case (iii), after yielding the matrix carries a fixed stress uY.Therefore, after cracking/yielding, all of the additional applied load is carried by the fibers. Mechanical equilibrium can then be written as cr = fcrf
+ (1 - f>q,
(above yield),
(2)
where cry M 0 for CMCs and PMCs. The ultimate tensile strength of the composite, out,,is then controlled by the maximum load-carrying capability of the fibers,
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169
a;, so that
The major task below is to determine the deformation behavior at versus (T and the maximum a;. Our subsequent discussion for obtaining a; and hence outsis not strictly limited to the three matrix cases above. For work-hardening materials or a fully elastic matrix, the matrix contribution to the stress can be included approximately by retaining the constitutive dependence of a,,, versus strain with the composite strain determined self-consistently. Also, in a CMC the matrix cracking itself is a stochastic process of multiple matrix crack evolution with a range of “strengths” but typically all cracking is completed well prior to the onset of fiber breakage so that ( T ~ 0 remains accurate (Ahn and Curtin, 1997; He et nl., 1994a). The reinforcing fibers are generally brittle materials (ceramic, glass, graphite) and so are linearly elastic up to the point of failure. The point of failure in any individual length of fiber is determined by the largest flaw or crack in that particular fiber. Different pieces of nominally identical fiber have different largest flaws and hence different strengths. The “strength” of a fiber is thus a stochastic variable with some probability distribution. The stochastic nature of monolithic ceramics and glasses is widely recognized and the situation is identical for most reinforcing fibers. To formalize the discussion, consider a fiber of length L to have a distribution of flaw sizes, with each flaw having a corresponding strength. The total number of flaws in this fiber that are weaker than a stress (T is @(o,L ) where
Equation (4) assumes a power law distribution of flaw strengths and is a “Weibull model” for the flaw distribution with a “Weibull modulus” rn (Weibull, 1952); typical values of m for structural fibers are in the range of 2 5 ni 5 20. The number of flaws is proportional to L since the flaws are assumed to occur randomly along the length. 00 and Lo are reference parameters for the fiber strength, with typically one flaw weaker than strength a()in a length Lo of fiber. In testing of fibers with a flaw distribution given by eq. (4), failure of a fiber occurs at a stress (T if there is a flaw of strength (T and no flaws weaker than a. The cumulative probability of fiber failure in L at stress (T is therefore given by
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which is the well-known Weibull cumulative failure probability distribution. From eq. (9,it is evident that not only is the fiber strength at fixed length a statistically distributed quantity but also that the “typical” fiber strength depends on the length L tested. For a Weibull distribution, the “typical” strength is the characteristic strength such that Pf(a,L ) = 1 - e-’ = 0.632, or @(u, L ) = 1 in eq. (4).So, 00 is the characteristic strength at length Lo and
is the characteristic strength at any other length L. The strengtMength relationship embodied in eqs. (4)-(6) is very important to understanding composite behavior. Lastly, the underlying assumption that the fibers are governed by eqs. (4) and (5) is usually assessed by measuring the fiber strength distribution at a length Lo and fitting the measured distribution to eq. (5j to obtain the parameters no and rn. The accuracy of eqs. (4)-(6) in describing fiber strengths at other lengths L # Lo depends critically on the number of tests performed at length Lo. For typical numbers of 25-50 tests, the extrapolation of strength using eq. (6) to much smaller or larger lengths L < Lo/lO or L > lOL0 begins to be inaccurate. So, predictions on composite properties based on such extrapolations must be regarded cautiously. The interface between the fibers and matrix occupies a vanishing fraction of the total composite volume but plays a critical role in determining many composite properties related to damage and strength. In polymer and metal matrices, the interface becomes important when fibers break. In ceramic matrices, the interface is critical first when the matrix cracks and then again when the fibers break. So, consider a fiber in the matrix that is broken due to failure at some preexisting flaw. At the fiber cracWmatrix intersection, the stress state is very complex (He et al., 1994b),but can drive crack extension into the matrix, yielding of the matrix, crack deflection along the fibedmatrix interface, or yielding along the interface. The latter two modes are driven by the high shear stresses acting along the interface. In effective composites, either (i) the interface is engineered to be weak enough (low toughness) to promote cracking and debonding or (ii) the matrix undergoes shear yielding along the interface. After interfacial debonding, there can remain a residual shear sliding resistance across the fibedmatrix interface due to friction. The precise nature of such “friction” is the subject of considerable study, and the roles of thermal clamping stresses, interface roughness, Coulomb friction, and Poisson effects are all important to some degree. For tractability, however, a common assumption is that there exists some constant interfacial sliding stress r across the debonded interface.
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FIG. 1 . Schematic of stresses around a broken tiber. and axial fiber stress of vs position :around a break.
Many, but not all, experimental results on the sliding interface can be interpreted fairly well with the constant 7 assumption. For the shear-yielding interface in the absence of work-hardening and multiaxial stress states, the constant t is also an adequate approximation. So, the constant t model allows for a commonality among ceramic, metal, and polymer matrix composites. Many of the important results below will seem to depend explicitly on the constant 7 assumption. But, in fact, the general concepts developed below can be applied to many more complex interface models; doing so is simply unwieldy and obscures the main physical features that are captured clearly within the context of the constant 7 model. Returning to the broken fiber, we now have a “sliding” or “debond’ or “slip” zone along the fibedmatrix interface (Figure 1). Neglecting the radial variations of the axial stress along the fiber, equilibrium between the fiber axial stress and the interface shear stress is given by, with the break at z = 0 and fiber radius r ,
or, upon integrating,
2tz =
7
in the slip zone. The axial fiber stress increases linearly in the slip zone. Neglecting the elastic behavior at the end of the slip zone, the slip length 6 is the distance at which the fiber axial stress attains the far-field value T (which will, due to damage and matrix yielding, differ from the value a E f / E , ) . Setting of ( z = 6) = T leads to rT 6 = -. 2s
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Numerous works have shown that the above “shear lag” type of approximation is quite accurate for the slip length and average axial fiber stresses, particularly in systems with E f E,,, and “low” t values (He et al., 1994a). B. CRITICAL STRENGTH a,.AND CRITICAL LENGTH6,. The fiber strength depends on the gauge length tested. In a single-fiber tension test, the length can be selected arbitrarily. In a composite, the fiber strengths control the composite tensile strength. So, is there a particular fiber strength, as well as an associated length, that is related to the composite strength and deformation? The answer is yes. To determine the critical strength ac and critical length 6, requires consideration of both the fiber strength statistics and the fiber slip in the composite. Imagine applying a stress T to the fibers, causing them to break into fragments of average length (x) = Lo(ao/T)”’.If the spacing (x) is larger than twice the slip length B = r T / 2 r around each break, then there remain some regions of fiber which experience the far-field load T ; further load could be applied to the fibers and they could break into smaller pieces. So, the typical maximum stress that the fibers in the composite can be subjected to is a stress a, for which the average spacing is exactly twice the slip length at this stress, 6,. = ra,/r. Since the average fragment length is also related to the applied stress as indicated above, we have 6, = L o ( o ~ ) / a ~as) ~well. ~ ’ Solving these two relationships simultaneously, we obtain (Curtin, 1991b; Henstenburg and Phoenix, 1989)
Physically, there is typically one flaw of strength a,. in a length 6, of fiber and 6, is twice the fiber slip length at an applied stress of cr,. Equations (9) are the generalization of the Rosen (1964) and Kelly (1965) critical length and stress to the case of stochastic fibers. These quantities control several major composite properties, as we will see below. In particular, we shall find that, within the “global load sharing” approximation (Section IV), Tensile strength
c(
Fiber pullout
a 6,,
Work of fracture
0:
oC,
a,6,,.
All of the dependencies on the constituent properties no, r, Lo, and t enter only through a,.and 6,., with only slight additional dependence on the Weibull modulus m. Even in the more general “local load sharing” approximations (Section V), the
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173
dominant dependencies remain as indicated above. It is of some interest to note that the composite tensile strength depends on the interfacial sliding resistance and fiber radius in a strongly nonlinear manner, dependencies that probably would never be directly anticipated otherwise.
111. Single-FiberComposite A. INTRODUCTION
Consider a single fiber embedded in a matrix material (rigid or plastic) and subjected to uniaxial tension. Under increasing load. the fiber will break at its weaker flaws and form slip zones around those breaks. Ultimately, at higher loads, there will be enough fiber breaks so that the entire length of fiber is slipping within the matrix. The fiber can then be loaded no further and the damage ceases. This single-fiber composite (s.f.c.) test, though conceptually simple and seemingly far from the reality of the multifiber composites of practical interest, holds the key to understanding some fundamental aspects of all multifiber composites (CMCs, MMCs, and PMCs). In addition, the s.f.c. test can be used to derive information about the in situ fiber critical length 8, , critical stress ac,and the Weibull modulus m appropriate at these gauge lengths. These in turn then imply a value for the interfacial sliding resistance, T = r o c / & , for the particular fibedmatrix interface studied. The s.f.c. test has a long history of application in the PMC field, primarily to study the effects of fiber surface treatments on adhesion (a few references are Fraser et al., 1983; Netravali et al., 1989; Wagner and Eitan, 1990; Rao and Drzal, 1991; Gulino and Phoenix, 1991). More recently, the s.f.c. test has been used to study the interfacial debonding in Ti-MMCs as well (Majumdar, 1996; Majumdar et al., 1996a, b contain a number of examples and references). The questions we wish to answer about the s.f.c. test are as follows. Given the fiber strength distribution (00, m ) at gauge length Lo and an interfacial T, what are (i) the number of fiber breaks versus applied stress, (ii) the spatial distribution of the breaks or, equivalently, the fiber fragment lengths created by the breaks versus stress, and (iii) the average fragment length distribution at the end of the test? If these questions can be answered quantitatively, then experimental data on the fragment distributions and break evolution versus stress can be inverted to derive the values of a,, &., m , and t. Also, the damage evolution will be used in the multifiber composite to predict tensile strength, fiber pullout lengths, and work of fracture, as discussed in Section IV. The s.f.c. problem is difficult to solve because fiber breaks can only occur where the stress on the fiber is at the far-field applied value. Flaws in regions of the fiber
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174
within the slip zone around a previous break experience a lower stress and hence are strictly excluded from causing another fiber break. Therefore, not all flaws in the fiber, and not even all of the weakest flaws in the fiber, can actually cause breaks. To demonstrate the strict existence of the exclusion zone, consider a section of fiber that has survived intact up to some stress o ,with a flaw at position z = 0 and of strength (T just about to fail. When this flaw fails, a slip zone is formed in the region -6 < z c 6, within which the stress is, according to eq. (7b), 2 t z f r (see Figure 1). The stress everywhere in the slip zone is now lower than it was just before the break occurred, and will remain lower for all further increases in the applied stress o. Since the region -6 < z < 6 survived the original stress (T everywhere except at the one break point, this region will also survive with no further breaks for all future applied stresses. Regions within a slip length of an existing break are thus excluded from subsequent failure. The existence of the excluded regions accounts for the cessation of fiber damage when all regions of the fiber are within a slip length of some break. The existence of an exclusion zone does not depend on the constant t assumption (Curtin, 1991a; Hui et al., 1996). A range of behaviors for the interface shear can be shown to create a rigorous exclusion zone. A notable exception, however, is the perfectly elastic interface where all stresses are always proportional to the applied stress. Figure 2 shows a schematic example of the s.f.c. test for a small section of fiber with the 12 weakest flaws along the fiber shown explicitly (Curtin, 1991a). Increasing stress causes flaws to fail, slip/exclusion zones to form or increase in length, and exclusion zones to increasingly overlap until the entire fiber is subsumed within the exclusion zones and the test “saturates.” In the case shown, the flaws having strengths os,o7,08, (TI I , ( ~ 1 2 and , all other stronger flaws not shown never fail because they become part of an exclusion zone before the applied stress reaches the value needed to fail those flaws.
B.
SOLUTIONS TO THE S.F.C.
1. nz -+
00
TEST
Limit
In this limit, all flaws have the same strength (TO and when a break occurs the slip length is always 60 = roof2r. So, no break can occur within a distance 60 of any other break and breaks continue to appear until there are no break spacings larger than 260. The s.f.c. problem in this case is identical to the “car-parking” problem where cars of length 60 are randomly “parked” along a road until there are no spaces large enough to accommodate another car. The average spacing of
I75
Fiber-Reinforced Composites
4. q-
I
0.
a,,
f 04
t tt 4 all %
t
4
t
t
09
a,
-FIBER
AXIS
t t t t
t
%%a,
olo
-
-
Flci. 2 . Schematic of the evolution of fiber damage and fiber slip (exclusion zones) with iiicreasing applied stress (bottoni to top) in an s.f’.c. test. The 12 weakest defects are shown explicitly. and are at strengths rr,, = C J ~ , ! ’ / ’ . At stress “10 the s.f.c. test “saturates” because all regions of the fiber are within a dip length of an existing break. From Curtin (1991a). Reproduced with permission of‘ Chapman & Hall.
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176
the breaks at the end of the test has been derived many times in the literature in different contexts and is ( x r ) = 1.33760 (Widom, 1966). The spacing distribution, or distribution of fragment lengths x, as a function of the number of breaks N in a length L , was found by Widom (1966). We denote this family of distributions as Pw(x;q ) where q = N & / L is the fraction of space taken up by the slip zones (or “cars”). Widom found this distribution to divide naturally into two components, a distribution for 60 < x < 260 and 260 < x , as
60
< x < 280,
+ ’ e( vx) p [ - ( i - 2 ) + ( q ) ] , Pw(x;77) = m
260<x,
(llb)
with the function + ( q ) defined by
Widom’s result says nothing, however, about the cases of practical interest, corresponding to 2 < m i20. 2. Curtin ‘s Solution The present author developed an approach to calculating the fragment length distribution P ( x , a) for the s.f.c. problem (Curtin, 1991a) that is quite accurate but is not exact, as thought by many prior to the work of Hui et al. (1995) discussed below. The gist of the solution is as follows. Consider a fragmented fiber of some length LT at some stress a with NT breaks in this fiber and a slip length 6 at this stress. There are some fragments, formed earlier in the test, that are smaller than 6 and the distribution of these will never change since they are too small to break again and too small to be formed by the breaking of larger fragments. Let this portion of the distribution be denoted P R ( x , a) (nonzero only for x 5 6) and contain N R of the breaks occupying a length L R . The remaining portion of the fragmented fiber then consists of N = NT - N R breaks distributed in a length L = Lr - L R , with all of the fragment lengths being x > 6 by construction. The key assumption in (Curtin, 1991a) is then that this latter distribution is identical to Widom’s distribution Pw(x;q ) with q = N 6 / L (for x > S only). Note that the stress dependence in PW is implicit in N , L , and 6. From this starting point, now consider how the two parts PR and PW of the overall distribution P ( x , a ) change as the stress is increased. Increasing the stress has two effects: an increase in slip length 6 and an increase in the number of breaks
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in the fiber. Fragments just larger than 6 become smaller than the new 6 and so move from the x > 6 distribution PW to the x < 6 distribution P R , which can be written as
while the remaining length of fiber containing x > 6 fragments is reduced as
The number of breaks N in the remaining portion x > 6 is decreased by the fragments lost to the x < 6 distribution but is increased by the occurrence of new breaks. New breaks can only occur in regions of fiber experiencing the applied stress, which is thus only in the central x - 26 region of those fragments larger than 26. So, we have
d6 d-N- - N P w ( ~ ;q ) da do
L*) + dQ,(a, da ’
where the length L* available for new breaks can be written as
and for a Weibull distribution we have from eq. (4) dQ,(a, L * )
da
L*
CP-’
- _ ni -. L a(;”
(16)
Equations ( 12)-( 15) provide a set of differential equations for determining the overall fragment distribution as the normalized sum of the two parts PR and PW versus applied stress:
as a function of the underlying flaw distribution Q, and the slip length S(a).There is no specijc requirement that be a Weibull distribution or that the i n t e ~ a c e have a constant r as long as 6 is a proper exclusion zone length. For the case of a Weibull Q, and constant t such that 6 = r a / 2 t , the above equations can be simplified by normalizing all lengths by a reference lengths 6~ and all stresses by a reference stress O R , with 6~ = rCfR/2T. The appropriate choice for these reference quantities comes from simplifying eq. (16) which, upon
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W A. Curtin
substituting eq. (15), becomes
To make the term in brackets in eq. (1 8) equal to unity then requires the reference stress and length to be identically OR = o~.and 8~ = &, respectively (see eqs. (9)). So, all features of the s.8~.test are properly normalized by the critical stress and length introduced in eqs. (9). When normalized by a(.and &, the solution to eqs. (1 2)-( 16) leads to a fragment distribution as a function of x E x / 6 , at a normalized stress s = o/a,.. The normalized distributions, the number of breaks versus stress, and the average fragment lengths then depend only on the Weibull modulus m . Specific results will be presented below in tandem with the exact results of Hui et al. (1 995). 3. Exact Solution of Hui et al. (I 995) Recognizing the inadequacy of Curtin's solution, as presaged by some results on multifiber composites by Neumeister (1993) (see Section V), Hui et al. (1995) recently formulated an exact set of differential equations for the fragment distribution evolution for Weibull fibers and a constant t interface. They were then able to obtain analytic closed-form solutions for the fragment distributions in this case. Hui et al. introduced the same reference parameters n,. and 6,. as above, and so worked in the normalized length x and stress s, with a normalized slip length 6/6, = s also. For Weibull fibers, the number of breaks per unit length per increment of stress (the "hazard rate" h ) is 1 d@ h=-L do and can be written in dimensionless units as h ( s ) = ms"'-', following from eq. (16). Hui et al. then considered three different length regimes for the overall fragment distribution P ( x , s ) : x < s / 2 ( x < 6 in dimensional units), s/2 < x < s (6 < x < 26), and s < x (26 < x ) . These are the same regimes of length that arise in Curtin's solution since the Widom solution divides naturally into two parts: 6 < x < 26 and 26 < x . For the short fragments x < s / 2 , no new fragments can occur, so that d P ( x , s) ds
=o,
S
x<-
2'
Fiber-Reinforced Composites
I79
For fragments in the range s / 2 < x < s , fragments can only form by the breaking of a larger fragment of length x’ > x s / 2 > s, so that
+
For the largest fragments, new fragments can form by division of larger fragments but fragments of size x disappear by breaking into smaller fragments. Hence,
dP(x-, s) d.S
= 2h(s)
d x ’ P ( x ’ ,s ) - (x - s ) h ( s ) P ( x , s).
s < x.
(22)
Continuity at x’ = s / 2 is required, so that eq. ( 2 0 ) is essentially the same as eq. (12). The differential equations (20)-(22)are quite simple and, in retrospect, it is surprising that this general form was not found earlier. However, simplicity is also the hallmark of a seminal result. Note also that the normalized distribution P ( x . s ) is entirely characterized by the Weibull modulus 111 contained in h ( s ) .Hui et al. proceeded to demonstrate that eqs. (20)-(22)can be solved analytically, but the analytic forms still involve definite integrals and are unwieldy to reproduce here. C. PREDICTIONS Of primary experimental interest are the final fragment size distribution (break spacings), the associated average fragment length ( x f ) at the end of the test, and the number of breaks versus stress during the test. These major results are easy to suinmarize because they depend only on m .In other words, the distributions and averages shown below are “master curves” that are directly usable for comparison to experimental data. Experimental data are normalized by selecting test values for a, and 6, and then m is used to best-fit to the master curves. Adjustments of a( and 6, along with m to obtain an overall best fit to all of the data are then obtained by iteration. It is important to recognize that all of the measured data, including the distribution functions, are characterized by only three scalar parameters ( m C , 6,. and m ) . Figure 3 shows the predicted average normalized fragment length (sf /6, ) at the end of the test versus the Weibull modulus m . Also shown are results of a Monte Car10 simulation study of the fragmentation test by Henstenburg and Phoenix (1989). All three results are nearly identical. Note also that at small m the average value significantly exceeds the m = 00 value of 1.337/2 = 0.668 (the factor of 2 arising from the normalization by 6, = 260) and that the convergence to the n7 = 00 value is quite slow.
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180
1.1 1
1 D
0
2
4
6
8
10 12 14 16
Weibull modulus m Frc;. 3 . Predicted final normalized average fragnicnt Icngth (.Y/ )is,.i n the s.1.c. test vs Weibull modulus i n . Dianionds: Hui ut ti/. (1995) (exact); .sqtt~ircs:Curtin (I99la)result; open triangles: simulations of Hcnstenburg and Phoenix (1989). The result for 111 = cc is shown as a solid circle at 111
= 16.
Figure 4 shows the predicted cumulative final fragment length distribution, defined as dx’ P ( x ’ , ,s = co),which is the fraction of fragments smaller than x, for various values of m. The fragment distributions for small m are substantially broader than for very large m .The approximate solution agrees fairly well with the exact results although there are some differences (a few percent in probability) at larger fragment sizes and for larger Weibull moduli, and the approximate model does not obtain the correct saturation (maximum) fragment spacing. The approximate results were, however, within the statistical error of the Monte Carlo simulation studies of Henstenburg and Phoenix ( I 989), and are exact, by construction, for the m = 00 limit as well. Figure 5 shows the evolution of the cumulative fragment length distribution dx’ P ( x ’ , s) for m = 5 and for various stresses corresponding to the break fractions, as predicted by the Curtin (1991a) theory. Of some interest is that the smallest fragments must occur early in the test, when the slip lengths are small and there are few overall breaks, and so this part of the distribution does not change as more fragments occur at higher stresses. Hence, when reformulated as a probability distribution, the probability of finding small fragments decreases with increasing stress.
1;
1;
Fiber-Reinforced Composites
0.0
0.5
1 .o 1.5 2.0 Normalized Fragment Length
181
2.5
3.0
FIG.4. Final normalized fragment length distribution for various Weibull moduli m . Solid lines: Hui rial. (1995) (exact); dashed lines: Cunin (1991a).
2.0 1. 1.o01
-2
0.0 -1 .o
? -2.0 C
7 -3.0 -4.0
-5.0 -6.0
-1 .o -1.v
-0.5 0.0 0.5 I n (n (normalized fragment length)
1.o
FIG. 5 . Evolution of normalized fragment length distribution vs applied fiber stress for Weibull modulus ni = 5 as predicted by Curtin (1991a). Reproduced with the permission of Chapman & Hall.
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W A. Curtin D. COMPARISON TO EXPERIMENT
There have been surprisingly few direct comparisons of these new theories for the s.f.c. test, in light of the widespread use of the fragmentation test over the last two decades. Here we note three works illustrating the predictive capabilities of the theory in application to PMCs. Van der Heuvel et al. (1997) recently studied graphite fibers (Apollo IM 43-750 PAN fibers) in an epoxy matrix (bisphenol A (DGEBA)-type) where the fibers had varying oxygen plasma surface treatments. Their goal was to assess the interface properties as a function of this surface treatment. They believed that, for high levels of treatment, the adhesion is excellent and the deformation is controlled by shear yielding of the matrix. To test this hypothesis, and indirectly the theory of Curtin (199 la), Van der Heuvel et al. first performed single-fiber tension tests over a range of gauge lengths to assess the fiber strength statistics ((TO, m ) and verify the length scaling of eq. (6). They also determined the shear yield stress of the matrix as 37 MPa at a moderate test rate. Single-fiber composite tests were then performed and the final fragment lengths were measured, along with various interesting observations of yield propagation and acoustic emission. The measured and predicted fragment probability distributions (not cumulative distributions) are shown in Figures 6a and b for the highest surface treatments (200% and 100%). The agreement is generally very good. The theory predicts slightly longer fragments than observed experimentally but the range and shape of the distributions are very similar. Tests at a surface treatment of 50% showed poorer agreement, but this might stem from a fiber strength distribution that was estimated to be rather different from the strengths obtained at both higher and lower surface treatments ( m = 10.3 at 5096, m = 5.9-6.7 for 0%, lo%, loo%, and 200%). The slight disagreement in typical fragment sizes shown in Figure 6 could also be due to the use of a bulk matrix t value; larger t would slowly decrease the fragment lengths but maintain the same distribution. Overall, the results of Van der Heuvel et al. confirm the general accuracy of the s.f.c. models that employ constant interfacial shear stress. Fukuda and Miyazawa (1994) performed similar s.f.c. tests on a T300 graphite fiber in Epicote epoxy, and showed results for the evolution of the fragment distribution at several applied stresses. Converting applied strain to fiber stress via c f = E f t , their data are replotted in Figure 7a with the length normalized by the projected final average fragment length of (x,) = 298 wm. Also shown are the predictions of Curtin’s model where 0, and m have been adjusted to fit all of the data simultaneously (Curtin and Takeda, 1998b). The agreement is quite reasonable, although not excellent, which can stem from having only one test
v)
5
m
...-.... Experimental
.
Experimental t *'
4
:A
I
4
I
--
1-1 d
a
Q
1 .o
v!
0.5
c
0.0
9
Fragment length [mm]
1.5
c
1 .o
0
0.5
Y
0 0.0
I
1
1
1
2
-
N
2
1.5
Fragment length [mml
FIG. 6. Final fragment length distribution as measured experimentally and as predicted (simulation) by the Cunin theory: (a) 200% 0-plasma treament: (b) 100% 0-plasma treatment. From Van der Heuvel er 01. ( 1997).Reproduced with permission of Elsevier Science.
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3
3
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184 1 0.8
0
E 0.6
3 n
0.4
o. 0.2
0 0.5
1.5
1
2
2.5
Normalized Fragment Length
(a)
.-a
0.8
2e
0.4 -
0.6
~
-
a 0.2 04
0.5
1
1.5
2
Normalized Fragment Length
(b) FIG. 7. (a) Normalized fragment length distribution vs applied fiber strains of 2.5%, 3.0%. and 3.510 as measured experimentally (Fukuda and Miyazawa, 1994) and predicted by the Curtin theory (Curtin and Takeda, I998b). (b) Normalized final fragment distribution as measured by Winiolkiatisak and Bell (1989) and predicted by the Curtin theory (Curtin and Takeda, 1998b).
specimen against which to compare. The derived values of nc = 5800 MPa and m = 7 were obtained, along with a critical length 6, = 500 pm. For these fibers of radius r = 3.5 pm, this suggests r = 40 MPa, which is quite close to a Von Mises estimate r % IT,/& = 45 MPa of the shear yield stress based on the tensile yield stress CT,, for this epoxy matrix. Fukuda and Miyazawa separately measured the fiber strength at Lo = 1 in. and found no = 3000 MPa, m = 5. Using eq. (4) to extrapolate the s.f.c. results up to 1 in. yields 00 = 3100 MPa, m = 7. The characteristic strengths are quite close. The difference in Weibull modulus may stem from variations in the single-fiber testing that broaden the measured distribution (m = 5) relative to the true one (m = 7). Using m = 5 in the size scaling with the measured strength at 1 in. and extrapolating to 6, would yield a much larger, and erroneous, value for the critical strength. This
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highlights the importance of an accurate measure of the fiber strength at the relevant gauge length 6, , as opposed to the extrapolation from a much larger length scale. The latter can be appropriate, but must be carried out with care and accuracy. Finally, Figure 7b shows the normalized final fragment distribution for AS-4 graphite fibers (Y = 3.5 p m ) in a PDA matrix, as measured (Wimolkiatisak and Bell, 1989) and as predicted by the Curtin theory (Curtin and Takeda, 199%). Single-fiber tension tests were performed at various gauge lengths by Wimolkiatisak and Bell to yield DO = 4275 MPa, in = 10.7 at Lo = 12.7 mm, and were used in making the theoretical predictions. Agreement between theory and experiment is quite good, with no adjustable parameters. These results will be used for prediction of the strength of an AS-4Iepoxy PMC in Section V. Having demonstrated in at least three cases the accuracy and predictive capability of the theories for the s.f.c. test, we can now return to the multifiber composites that are our main interest.
IV. Multifiber Composites: Global Load Sharing A. GLOBAI. LOADS H A R I N G In the real multifiber composites of practical interest, many features of the s.f.c. are preserved. Namely, the fibers have the same statistical strength distribution and, once broken, slip with sliding resistance r over a slip length 6 = r a / 2 r (eq. (7)).The evolution of the fiber fragmentation during loading is, in principle, different because each individual fiber experiences a nonuniform stress due to the uniform applied stress plus stresses transferred from other broken fibers in the composite. The evolution of fiber damage thus depends crucially on the nature of the load transfer from broken or slipping fibers to unbroken (elastic) fibers. In this section, we make the assumption of global load sharing (GLS) to deal with the load transfer (Curtin, 1991b). Global load sharing assumes that the load lost by a fiber at some axial position z , due to breakage and slippage, is transferred equally to all unbroken (elastic) fibers in the cross-sectional plane perpendicular to z . For a load drop of Aa(7) along a single broken fiber in a composite of n j fibers, the remaining r z / - 1 each experience an increased stress of Aa(z)/(n f - 1). For n broken/slipping fibers at position z with stress drops AD,( z ) ,i = 1, . . . , i i , the remaining n - n fibers experience an increased stress [ l / ( n / - n ) ] C:’, Aa, ( z ) .In GLS there are no local stress concentrations; damage in any part of the composite at z influences the stress state everywhere else in the composite at z to an equal extent.
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Is the GLS assumption reasonable? The extent of stress concentrations around broken fibers depends on many factors: the interface t, fiber and matrix elastic moduli, fiber arrangement, crackeayielded state of the matrix, among others. However, if t is sufficiently low, then the local stress concentrations cannot be too large and the stress must be redistributed over some fairly large number of fibers, approaching the GLS limit. In the limit t += 0, the slip length becomes infinite and so a broken fiber is decoupled from the remainder of the composite except through the gripping system and hence GLS becomes exact. In any case, Section V is devoted to analyzing the effects of local load sharing in considerable detail. The assumption of GLS leads to several important simplifications to the multifiber composite problem that then allow for analytic calculations of the damage and failure. First, since all unbroken (elastic) fibers experience equal stress increases from all of the brokedslipping fibers, the stress on each of these fibers is the same, and is given by
where of is the remote applied stress on the fibers. Furthermore, since (i) the fiber strengths are all selected from the same probability distribution and (ii) these strengths are statistically uncorrelated along the length of the fiber, every crosssectional plane is statistically identical to every other cross-sectional plane in the composite. In other words, if one analyzed the strength distribution of all the fibers in arbitrary planes ZI and z 2 , one would find these distributions to be identical, and also identical to the strength distribution along the length of any one single fiber in the composite. Therefore, the average amounts of damage n and stress ( z ) are independent of location z so that the stresses on the transfer C:, ACT, unbrokenfibers are independent of z as well, T ( z ) = T . These statements hold in a statistical sense, or for large numbers of fibers in the cross section and, moreover, for any state of damage at any applied fiber stress. The stress T , being constant along the unbroken portions of every fiber, can therefore be interpreted as an effective remote applied fiber stress on each fiber. The value of T is related to both the true applied fiber stress and the damage (fiber breakage) that is caused by the stress T itself acting on the fibers. Now, as the true applied fiber stress of is increased monotonically, T also increases monotonically (increasing damage). Therefore, eachfiber in the mult$ber composite is, during the loading history of the composite, subjected to an effective stress T along its length and so undergoes fragmentation that is identical to the fragmentation of thefiber in a single-fiber composite. To reiterate, every fiber
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behaves as if it were in an s.f.c. under stress T . The relationship between T and of is as yet unknown but is uniquely determined, as we shall see below. So, at any applied stress of the fibers are fragmented into pieces having precisely the statistical distribution P ( x , T(of))where P is the s.f.c. distribution derived in Section 111.
B. FIBER PULLOUT A N D WORK
OF
FRACTURE
Putting aside the T (of) dependence temporarily, imagine that the composite has been tested to infinite strain or stress so that of + 00, T + 00. Then, each fiber is fully fragmented with the final fragment spacing distribution P ( x , 00) (see Figure 4 ) . The composite will separate into two pieces along some arbitrary plane (recall all cross sections are statistically identical). The fibers will pull out of the matrix around this fracture plane, with the shorter of the two embedded lengths (one on each side of the plane) being pulled out because the force to pull out an embedded length h against the interfacial shear is ( 2 n r ) s h . Knowing the distribution of fragment lengths P ( x , oo),the distribution of pullout lengths Pp,(h) is obtained as follows (Curtin, 1991b). The probability of a plane intersecting a fiber fragment of length x is simply x P ( x , o o ) / ( x f )where ( x f )is the final average fragment length as before. The pullout length distribution p t ( h )caused by such a size-x fragment has equal probability between 0 and x / 2 , i.e., 2 p , ( h ) = -,
X
0 < h < -. (24) 2 X The probability of obtaining a pullout length h , Pp,(h), considering all possible fragment lengths x is then
versus h/6, is shown in Figure 8 for various The normalized distribution Ppo(h)6, rn values. Since most fragments are longer than 6,/2 (see Figure 4 ) , the pullout length distribution is nearly constant for lengths h < 6,./4. The average pullout length is ( h ) = 1; d h h Pp,(h), but is easier to understand as follows. For an intersected fragment of length x , the average resulting pullout is x / 4 since it is equally likely to intersect the fragment anywhere along the length and the shorter piece pulls out. So,
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1.o
Pullout Length h/Sc F I G .8. Normalized liber pullout length distribution lor various Weibull moduli. Reproduced from Curtin ( I99 I b) with permission of the ./o~trucr/q f r h Anrericm Cerrrmic Sociefv.
where (x; ) = dx x ” P ( x , 00) is the nth moment of the final fragment length distribution. Introducing the reference length S,, we thus find
1 ( h ) = - h(m)S,. 4
(271
Here, h ( m ) = ((xf /S02)/(xf /ac) is a pure number that vanes slowly with rn and is reasonably approximated (within 5%) by
0.664 W) = ,0.6 +0.716,
m
1.
Fiber pullout thus scales directly with the characteristic length 6,. . Measurement on ( h )can thus be used to estimate 6,. even if in is not known. Measuring the full distribution P,,(h) can be used to determine the value of m as well. The work to pull out the broken fibers is the major contribution to the work of fracture per unit area in many composites. The work to pull out an embedded length h is ( 2 x r ) r ( h 2 / 2 )Therefore, . the average work to pull out a fragment of
Fiber-Reinforced Composites
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length x is nrr d h ( n r r h z ) p , y ( h= ) -x . 12 The work per unit area of composite to pull out all of the fibers, W,, is then (Curtin, 1991b)
upon introduction of a,.and 8,. Here, y ( m ) = ( ( x j /S,.)')/(xj number that is well approximated by 1.87 v ( m > = ,0.75 +0.50,
/ac)
is another pure
m 2 1.
The work of fracture per unit area is determined only by the material parameters through the product of the critical stress and critical length, 0~8,. It should be noted that several workers preceding Curtin (199 1b) derived results that were subsequently shown to reduce to the same forms as eqs. (27) and (30) but with different statistical prefactors. Specifically, Thouless and Evans ( 1988) studied the case of a ceramic composite with a single matrix crack, an important limiting case, and found the pullout and work of fracture to be somewhat smaller (smaller coefficients h and y ) than for the multiple matrix cracking case in eqs. (27) and (30). Sutcu (1989) considered multiple matrix craclung with an approximate statistical analysis and obtained results fairly close to those of eqs. (27) and (30) (see Curtin, 1991b).
c. STRESS-STRAIN
BEHAVIOR: EXACTAND APPROXIMATE RESULTS
To obtain the fiber portion of the composite stress-strain relation, we first recognize that the composite strain is controlled, at all cross sections, by the elastic stretching of the unbroken fibers. Since these fibers have a stress T , the composite strain is precisely E = T / E f . Therefore, the T (a/ ) relationship is the straidfiber stress relationship for the composite. The relationship between T and af is now found as follows. Static mechanical equilibrium requires that the average stress borne by all the fibers (unbroken, broken, sliding) in any cross section must equal the remote applied fiber stress af. Considering an arbitrary cross section, it is clear that there is some load-bearing capacity on all fibers, including the slipping fibers, except for those fibers broken exactly at this cross section. Because the fibers are statistically identical and under
W A. Curtin
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an identical effective applied stress T , the average of the fiber stresses across the cross section is identical to the average stress along the length of any one fiber. Therefore, we must have
where L , is the length of the composite and a(z) is the stress distribution along the length of an arbitrary fiber. Since the fiber is broken into a set of fragment lengths x according to P ( x , T ) and since the stress in a fragment of length x is known, the integral in eq. (32) can be rewritten as a sum over all fragment lengths of the integrated stress over a fragment of length x:
where o,( z ) is the stress along a fragment of length x and N is the total number of fragments (breaks) in the length L ; . With an “applied” stress T and with reference to Figures 1 and 2, it is trivial to find the integrated stress over a fragment length x as
Substituting these results into eq. (33) yields = pT
lh
X2
dx P ( x , T ) -
46
+ pT
r
dx P ( x , T ) ( x -a),
(35)
with p = N / L ; the number of breaks per unit length. Using the identities dx ~ ( x T, ) = I - Jo2h d x ~ ( x T, I , J20$5 d x x ~ ( x T , ) = (x) d x x P ( x , T ) , and ( x ) = L , / N = p - ’ allows eq. (35) to be rewritten in the convenient and physical form
Jg
Jib
The first term on the r.h.s. of eq. (36) is the linear relationship that exists in the absence of damage. The second term on the r.h.s. is primarily due to the reduced load-carrying capability in the slip regions at the ends of the longer fragments (x > 26) while the third term on the r.h.s. accounts for the reduced load-carrying capability in regions of overlapping slip lengths between two nearby breaks. The damage parameter p includes the total number of breaks, independent of relative
Fiber-Reinforced Composites
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positions or fragment lengths, and thus makes the second and third terms on the r.h.s. ofeq. (36) not completely due to long and short fragments, respectively. Since P ( x , T ) is the s.f.c. fragment distribution at stress T and p is the fraction of breaks per unit length in the s.f.c. problem, eq. (36) is a unique relationship between the applied fiber stress a, and the effective stress T , which is also proportional to the composite strain. Both P ( x , T ) and p are really only functions of the normalized length and stress variables x’ = x / S , and F = T / o ( .so that, upon introduction of ac and 6,. and the normalized quantities iif= a f /ac,5 = p6, (the number of breaks in length a(.), and 6 / & = T / o C= T , eq. (36) becomes
i
i
Equation (37)is the nonnalized stress-strain curve for the j b e r portion of the composite, and in this form depends only on the Weibull modulus in. It is the major resultfor the GLS theory of comnposite defonnation and failure. The leading nonlinear term is explicitly linear in the damage parameter and quadratic in the strain, as expected physically. The factor of in the second term on the r.h.s. of eq. (37) expresses the fact that, on average, sections of slipping fiber that intersect the cross section of interest carry - of the stress camed by an elastic fiber. The broken but slipping fibers can thus make a substantial contribution to the loadbearing capacity of the composite. The ultimate tensile strength (u.t.s.) a,,,for , the composite corresponds to the maximum of a/.versus T or, in dimensionless form, 6, versus f. Exact and various approximate solutions to eq. (371, discussed below, all lead to the general form for the maximum fiber stress as 6; = cp(rn) or 01 = cp(m)o,, where cp(m) depends rather weakly on m . Substitution into eq. (3) then yields the composite strength. The present author obtained a simple analytic form for af versus T by assuming that the damage at the maximum stress was small, in some sense (Curtin, 1991b). Then, with few breaks, it is unlikely to find small fragments x < 26 so that the third term on the r.h.s. of eq. (37) due to such short fragments can be neglected. Furthermore, the occurrence of fiber breaks is not substantially affected by the small fraction of fiber length excluded by slip, so that
4
4
5
@(6(., T ) = f”’.
With these assumptions, eq. (37) becomes simply
(38)
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0'9 0.8
c
I
I
I .
0
1
i
2
3
4
5
6
7
8
9 10
Weibull modulus m
6* vs Weibull modulus m as predicted by various analyt ses. Squares: eq. (40a) (Curtin, 1991b); triangles: eq. (48) (Curtin and Zhou, 1995); crosses: following from eq. ( 5 2 ) (Neumeister, 1993); diamonds: Hui rr al. (1995) (exact); circles: eq. (54) (Curtin e t a / . , 1998) (identical to Neumeister for larger m ) . FIG. 9. Normalized fiber bundle strength
The maximum of eq. (39) is at
so that the tensile strength of the fibers is
These results for 6; and f* versus m are shown in Figures 9 and 10. Returning to dimensional units and using eq. (3), the composite u.t.s. is
the failure strain (uncorrected for thermal residual strains) is
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1.6
1.4
'5C
1.2
5
e .--m
U
1
0.8
0.6
0
1
2
3
4
5
6
7
8
9
10
Weibull modulus m
FIG. 10. Normalized tiher bundle failure strain f* vs Weihull modulus ~n as predicted by various analyses. Squares: eq. (JOb) (Curtin, I991 h): triangles: eq. (49) (Curtin and Zhou. 1995): crosses: following from eq. (52) (Neumeister. 1993): diamonds: Hui er ril. ( 1995) (exact); circles: eq. ( 5 5 ) (Curtin rt d.. 1998).
the stress-strain curve after yielding or matrix cracking is
and the damage per length 8,. at failure is
These results were anticipated to be accurate for ,Z << 1, which is valid for m ? 3 or so. An indication that eqs. (39)-(44) become inaccurate at small m is that the predicted 6;drops below the average fiber pullout stress upo,which is not physically possible. Recalling that the force to pull out the average embedded fiber of length ( x l ) is 2 r r r s ( x f ) ,the stress up" to pull the composite apart against the sliding resistance after all of the fibers have broken is, using eq. ( 2 6 ) , op0=
51 h(m)a,
or
1 ap0= - k ( m ) .
2
(45)
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For m < 2, 8ppo exceeds the 8; of eq. (40); eq. (45) must be a lower bound for the true 5;. To improve the tensile strength and failure strain predictions at lower m and to include the possibility of initial fiber damage due to processing, Curtin and Zhou (1995) accounted approximately for the random occurrence of close fiber breaks (small fragments) as follows. If N breaks are placed randomly in a fiber of length L , with no regard for the exclusion zones, the fragment distribution is exponential P ( x , T ) x pePPX,
(46)
with p = N / L z as before. Some break spacing will be smaller than 26 merely due to the random placement. Using eq. (46) in eq. (37), the integrations are easily performed to yield, after a bit of algebra, -
fff
1 = z ( 1 - e-Pf), P
(47)
as the dimensionless stress-strain curve. Assuming ,ij = Fnl again, an approximate analytic form for the maximum of eq. (47) was derived as
with the damage parameter at failure fi* = (2/m)m/(mf')and the failure strain given by
The predictions of eqs. (48) and (49) are shown in Figures 9 and 10, and the predictions are improved significantly at lower m values. Neumeister (1993) followed the work of the present author by using the Curtin (199 1a) solution for P (x , T ) to obtain the stress-strain relationship more accurately. Neumeister (1993) nicely showed that the integrals required, such as in eqs. (36) and (37), could be performed analytically so that the resulting equations involved only simple functions of the break density q that enters the Widom fragment distribution (see Section 111). Moreover, Neumeister then showed that the Curtin solution provides no actual maximum in the stress-strain curve for m 5 1.6, although down to m = 1.6 the results looked very reasonable. Thus, there was something amiss with the Curtin solution, although the pullout stress a,,. (eq. (45)) does exist below m = 1.6 and so provides the maximum tensile stress that would be obtained asymptotically at infinite strain.
Fiber-Reinforced Composites
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In light of the limitation of the Curtin theory and its unwieldy implementation, Neumeister (1993) then developed an excellent analytic model for the stressstrain behavior by better accounting for the exclusion zones formed by slip and for the stresses carried by the short fragments x < 26. His result cannot be obtained directly from eq. (37), however, and so we merely quote the analytic result of
The damage parameter, including the effect of the exclusion zones in preventing some breaks, could be written as
leading to
The ultimate strengths and strains predicted by maximizing eq. (52) are shown in Figures 9 and 10, respectively. The predictions for m > 1.6 are essentially identical to the values obtained from using Curtin’s theory and so the latter result is not shown separately. The Neumeister expression of eq. (52) does not have a maximum for m < 1.2, which has not been previously recognized in the literature; quotes of Neumeister’s result for smaller m do appear in the literature but may have been obtained through an assumption that the maximum strain is given by eq. (49). Jansson and Kedward (1996) found an alternative analytic form that reproduces the tensile strength results of Neumeister (1993) quite closely but this result does not have an associated stress-strain curve with a maximum at the analytic strength. Hui et al. (1995) used their exact solution for P ( x , T ) to numerically calculate 3f versus T , using the concepts leading to eq. (37), and numerically evaluated the ultimate stress and failure strain. Exact analytic results were not possible. The exact stress-strain curves for various m are shown in Figure 11 , and the predicted 3; and F* are shown in Figures 9 and 10. Hui et al. proposed approximate analytic relationships for the stress-strain curve and failure stress for various regimes of m, but these results are no more accurate than the Neumeister results form > 1 and so are not reproduced here. Hui et al. did demonstrate, however, that the failure strain estimate of eq. (49), also derived from separate analyses of Ibnabdeljalil and Phoenix (1995b), was quite accurate for m > 1.
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W A. Curtin
: 4
0.8 -
a
2 *
2-
0.6-
0 3
0.4 -
0 LL
0.2-
0.0 0.0
0.5
1.o
1.5
2.0
2.5
Normalized Strain T 1 I . Exact normalized fiber bundle stress-strain curve ( f i t vs f*)for various Weihull moduli Reproduced from Hui ei ol. (1995) with permission of Elsevier Science. Fici. 117.
Finally, Curtin et al. (1998) recently derived an approximate result by alternative means that is surprisingly simple and accurate. The analysis is based on considering more completely the presence of discrete matrix cracks in CMCs, but in the limit of zero matrix crack spacing such an analysis leads to a new result for the present problem. Here, we quote the results only. The stress-strain curve was found to be
with a tensile strength of
and a failure strain of
These predictions are numerically almost identical to those of Neumeister (shown in Figures 9 andlo), but retain a maximum for all m. Figures 9 and 10 show the predictions of eqs. (54) and (55) at rn = 1, where the Neumeister results do not exist. The failure strain of eq. (55) is smaller by a factor of ((in l)/(m 2))”(’”+’) than eq. (49), which is fairly negligible for m 3 3. In CMCs with finite matrix crack spacings, Curtin et al. (1998) showed that the tensile strength
+
+
Fiber-Reinforced Composites
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(eq. (54)) is only modestly affected whereas the failure strain (eq. (55)) can be substantially affected by the degree of matrix cracking. Several other works are worthy of note. Sutcu (1989) and Schwietert and Steif (1990) derived models for tensile strength that, when analyzed carefully, embody the same mechanics and approximations as found in Curtin (1991b). Thouless and Evans (1988) also analyzed the composite strength in the presence of only one matrix crack to obtain an upper bound to the strength with multiple cracks. Phoenix and Raj (1992) developed strict, tight, upper and lower bounds on the tensile strength but their results were superseded by the more precise results of Neumeister (1993) and of Hui et al. (1995). In summary, a number of results have been derived for the stress-strain curve, tensile strength, and failure strain. It is evident that all of these results are nearly identical for m 2 3. This similarity arises from the fact that, at higher m , the amount of damage at the peak applied stress is fairly small when measured as the fraction of damage per length 6,. Hence, the amount of overlap in slip zones and exclusion of regions from breaking is small enough to be almost neglected entirely. While the deviations for m < 3 are important in some systems, the deviations in tensile strength are still relatively small. We shall compare these predictions to detailed experiments in Section 1V.F.
D.
INSlTU
FIBERSTRENGTH AND FRACTURE MIRRORS
The strengths of the individual fibers in situ, i.e., after full processing of the composite, are often different from the values measured on pristine fibers. The observed degradation is not surprising since the brittle ceramic fibers are exposed to high temperatures and possible abrasion during composite fabrication. Two options for assessing the in situ strength have been used. The first method is dissolution of the matrix, using reagents that do not affect the fibers, followed by direct single-fiber tension tests on the extracted fibers. The second method involves interrogating the fracture surfaces of the pulled-out fibers on the fracture surface of a tested composite. Many ceramic fibers, particularly the widely used Nicalon CG fibers, show “fracture mirrors” on the fracture surfaces. A mirror is indicative of an underlying critical flaw that caused the fiber failure. Empirical relationships between the mirror radius a , and the fiber strength S due to failure at the flaw indicate that
W A. Curtin
198 1.2
f
+.L
-
I
1 -
!i
0.9 -
5
0.8
-
0.7
-
0.6
0
10
5
15
Weibull rn F I G . 12. Fracture mirror parameters S* and in* vs true Weibull modulus. Squares: S*/o;.; diamonds: m * / m . Reproduced from Curtin (1991b) with permission of the Journol o f t h e Americrrri Cerortiic. Society.
where K is a fracture-toughness-like parameter. The value of K can be obtained by comparing measured S and values for single fibers tested in tension ex siru. The measurement of many mirror radii ( a w zon } the composite fracture surface allows for the creation of an in situ strength distribution ( S ] . Often, the distribution can be adequately characterized by a Weibull distribution with Weibull modulus m* and characteristic strength S*. However, S* and m* are not identical to the needed quantities 0;.and m. Nonetheless, since within the GLS approximation the fibers break as if in an s.f.c. test, there is a unique relationship between ( S * , m*) and (ac,m ) (Curtin, 1991b). The distribution ( S } is precisely the number of breaks versus stress in the s.f.c. test and does indeed follow an approximate Weibull form over the major portion of the probability range. The relationship between ( S * , m*) and (ac,m ) is expressed as ratios of S*/uCand m * / m versus the true Weibull modulus m , and is shown in Figure 12. The differences are small (less than 10%) for m 3 3 but become significant for smaller m. Even for larger m , quantitative agreement with experiments requires proper conversion of (S*, m*) to (cC, m). E. LOCALIZATION A N D NUMERICAL SIMULATIONS Prior to the availability of the exact results of Hui el al. (1993, several workers devised numerical simulation methods to check the accuracy of approximate
Fiber-Reinforced Composites
199
analytic results within the GLS theory. In addition, the issue of strain and damage localization is important, in principle. Since the macroscopic stress-strain behavior has a peak at 67 followed by a softening regime (see Figure 1l), localization of damage is expected to occur somewhere in the material once 6 ; is reached. The localization then interrupts the overall progression of fiber damage envisioned in the analytic models: i.e., each fiber is no longer undergoing an s.f.c. test. The local stress T in one region of width % 26, along the length becomes larger while damage elsewhere ceases since the applied stress is not increasing. Since the total damage 5" at the maximum stress (e.g., eq. (44) or prior to eq. (49)) is small, much of the observed fiber damage on the fracture surface occurs after the localization. The localization thus formally invalidates the predictions of pullout, work of fracture, and fracture mirrors using the s.f.c. theory. Although the stress and length can continue to be referenced by cr( and 6, , the values of the coefficients h ( m ) ,y ( m ) ,and in Figure 12 are not necessarily correct. The simulation methods are thus valuable to obtain accurate coefficients for these important quantities. The simulation methods have been adequately described (Curtin, 1993a-c; Ibnabdeljalil and Phoenix, 1995a; Zhou and Curtin, 1995; Iyengar and Curtin, 1997a, b), but various small errors, such as incorrect pullout planes and possible effects due to the small lengths of composite simulated, raise some doubts as to the validity of some of the results for pullout properties. We have thus used the simulation models most recently described by Iyengar and Curtin (1997a, b) to revisit the issue of pullout and to analyze the fracture mirror statistics, which have not previously been studied numerically. We performed five simulated tests at various Weibull moduli, on composites composed of 1000 fibers and a length 46,. The longer length was used to avoid possible boundary effects in the determination of the pullout. The coefficients h ( m ) ,y(m), S*/a,, and m " / m so derived are shown in Table 1, labeled as Sim., along with the predictions of the analytic models from eqs. (28) and (31) and Figure 12, labeled as Anal. The agreement is remarkably
TAIjLk I COMPARISON O F ANALYTIC A N D SIMULATION-DERIVED VALllES FOR PULLOLIT, WORK O F FRACTURE, A N D F R A C T I I RM E I K K O KPARAMETERS, FOR SEVERAL WE1BUI.L MODULIm
m 1
2
s 10
Pullout
h(,rr)
Work
y(m)
Minors
S*/rr,
Minor5
~JI*/JII
Anal. 1.36 1.18 0.97 0.86
Sirn. 1.38
Anal. 2.24 1.62 1 .05 0.81
Sim. 2.62 1.95 1.16 0.96
Anal. 0.65 0.89 I .o I I .03
Sim. 0.7 1 0.86 1.01 I .01
Anal.
Sim. 1.21 I .08 0.98 0.98
1.19
1.01 0.94
1.1s I .07 0.98 0.94
W A. Curtin
200
31
I /’
I
I
l
l
,
-6 -5 -4 -3 -2 -1 0
I
/
1
2
3
In (stress) FIG. 13. Simulated fracture mirror strength distribution for various Weibull moduli, plotted in Weibull form. Note the rollover in the distributions at higher stresses. Derived parameters S* and m * are shown in Table 1.
good. The fiber fracture strength probability distributions, or mirror distributions, IS} obtained in these simulations are shown in Figure 13. These distributions show a characteristic rollover at higher probabilities such that the m* is best determined by only considering the probability range below about 0.7 or so (below In(- ln(0.3)) = 0.2 on a Weibull plot, as in Figure 13). Otherwise, the simulated results are quite consistent with the analytic models. F. INITIAL FIBERDAMAGE Another important issue in practical composites is that of actual fiber breakage during composite processing. Preexisting fiber damage modifies the overall stress-strain behavior and reduces the composite tensile strength and failure strain. Experimental work on Ti-MMCs by Groves et al. (1994) has clearly identified various causes of fiber damage during processing. Since early analytic models were based on the assumption of limited fiber damage, such concepts were anticipated to fail in applications to predamaged composites at moderate damage levels. Fiber damage is, however, easy to incorporate into the exact and approximate GLS models derived in Section 1V.C. The total possible fiber damage (preexisting plus stress-induced) is simply represented by a new “flaw” population which is not of the Weibull form. Various workers have investigated the case where the number of flaws per unit length weaker than the stress CT is described by
Fiber- Reinforced Composites 0.8
.--
-
20 1
~
0.75
5
0.7 0.65
C 2
0.6
3 Q
0.55
7 I-
0.5 0.45 0.4 0.35 0.3
I
0
1
2
1 3
Damage parameter
FIG. 11. Predicted normalized fiher hundle strength Ci* vs diinen\ionless initial damage parameter t J,.) for several Weibull moduli as obtained in the model of
pg& (number of initial break5 per length
Curtin and Zhou ( 199s). Note the convergence at high darnage.
rather than by eq. (4), such that there is a certain preexisting density of breaks po that adds to the additional damage from the “usual” fiber flaws. This assumption is probably the most severe case, since it assumes that the preexisting damage does not deplete the remaining available flaws in the fiber. Curtin and Zhou (1995) and Duva et al. (1995) used eq. (47) along the dimensionless damage parameter 6 modified to accommodate eq. (57) as
6 = p0Sc + f’”,
(58)
which shows that the relevant scale for preexisting damage relative to induced damage is the amount of damage per length &.. Curtin and Zhou ( 1995) and Ibnabedeljalil and Phoenix (1995a) used the general result of eq. (57) in simulation models, and they, Duva ef al. ( 1995), and Hui et al. ( 1995) used eq. (57) in analytic models based on Curtin’s or the exact s.f.c. solution. Figure 14 shows typical results for the tensile strength versus initial damage po& for various Weibull moduli, from the model of Curtin and Zhou (1995); other workers show essentially identical results. Figure 14 shows that the decrease in tensile strength is nearly linear with increasing damage, with significant decreases occurring only for ,006, > 0.25 or so. Since 6, is typically on the order of a few millimeters or smaller, as we shall see below, the amount of damage required to degrade the tensile strength is on the order of a few breaks per inch in every fiber. This is a fairly large amount of damage; the ability of the composite to maintain strength at such damage levels is due to the fact that it is the damage over the crit-
202
W A. Curtin
ical length 6, that is important. These results also suggest that aligned, short-fiber composites with lengths only somewhat longer than 6, should perform nearly as well as continuous-fiber composites. Elzey et al. (1994) used these numerical studies in tandem with models for initial damage formation in Ti-MMCs to develop comprehensive time-temperature-pressure-damage-strength maps relating the detailed processing to the predicted final composite tensile performance. G. COMPARISON TO EXPERIMENTS The GLS theory has been used to predict and derive properties of a large number of CMCs and Ti-MMCs. Here we describe applications to CMCs because it is anticipated that local load sharing provides an even better description of MMCs and so we defer most of the discussion of MMCs until Section V. The interested reader should see Curtin (1993~)for the initial applications of the GLS model to MMCs. Majumdar (1996) and the references therein show limitations of the GLS model in application to Ti-MMCs. In CMCs, the fibers used to date have been almost exclusively the Nicalon CG fibers. However, fiber degradation during processing makes composites with different interface coatings and matrices somewhat different. Here we present all of the complete applications of the theory known at present. Inputs to the theory are the in situ fiber strength parameters ao, rn at length Lo, fiber radius r , interfacial sliding resistance t,fiber volume fraction f , and fiber elastic modulus E f . For CMCs, a, = 0 is assumed after matrix cracking. Predictions of the theory are then outs,the stress-strain curve, the pullout length distribution and ( h ) ,and the work of fracture W,?.When fracture mirrors are used for the in situ strengths, the inputs are S* and rn*, which lead directly to a, and rn via Figure 12, f , and E f . Values of t can then be derived from the measured pullout length ( h )using eq. (26). Below, we use the results of eqs. (48) and (49) from the model of Curtin and Zhou (1995) because of their reasonable accuracy for both strength and strain and their relative simplicity. Differences with the exact results are generally negligible. Prewo (1986) extracted Nicalon fibers ( r = 7.5 pm, E f = 200 GPa) from a unidirectional LAS-I1 glass matrix CMC and measured a 0 and rn at Lo = 25.4 mm. Pushout tests showed r = 2-3 MPa so we use 2.5 MPa here. Interface debonding is accomplished due to a very thin carbon layer formed in situ in this material. The strength and t values lead to a, and 6, as shown in Table 2 along with f . The predicted aUt,and E / compare very well with the measured values obtained by Prewo, also shown in Table 2. The predicted pullout length is ( h ) = 1.7 111111, which is in the range estimated by Prewo for these systems.
Fiber-Reinforced Composites
203
TABLE 2 PROPERTY DATAA N D M E A S I J R LAI )N D PREDICTED STRENGTH A N D FAILURE STRAIN FOR LAS-II/NICAI.ON (PKEWO. 1986), LAS-V/SCS-6 (JAKMONA N D P R E W O , 1986). A N D PYKEX/NICALON (TsuD.4 ('t d . . 1996) COMPOSITES CONS'I'ITUTIVE
LAS-I1 LAS-I1 LAS-11 LAS-I1 LAS-V Pyrex Pyrex'
0.46 0.46 0.44 0.44 0.20 0.48
1730 3.8 1740 2.7 1615 3.9 1632 3.1 3500 8 2900 3.0
2.5 2.5 2.5 2.5
2411 2657 2257 2129 10 3986 4.5 4069 1.0 2993
7.23 7.97 6.77 7.39 2X.3
6.78 22.5
758 664 670 680 557 71 1
756 796 678 697 605 I360 98 I
0.97 0.86 0.90 I .03 -
I.03
I.05 I.22 0.95
I .09 0.85 I .75 I.29
Stresses in MPa, lengths in mm. strains in (2: Lo = 25.3 iiiiii in all cases. A sample calculation for the NicalonPyrex system for a reduced r is indicated by the asterisk and is for illustration purposes.
Table 2 contains four slightly separate systems with marginally different matrix structure, fiber strengths, f,tensile strengths, and failure strains. The theory predictions agree exceptionally well with three of the four and are within 10% for the fourth system, and so seem to capture even subtle changes in material properties. Jarmon and Prewo (1986) investigated the properties of LAS-V glass reinforced with Textron SCS-6 S i c fibers ( r = 71 pm, E f = 400 GPa). The fiber strength statistics at 1 in., the estimated r . and derived CT~ and 6, are shown in Table 2. The ultimate strength and failure strain, as predicted and measured, are also shown in Table 2 and very good agreement is again obtained. Tsuda et al. (1996) have used a similar approach for the system of Nicalon fibers coated with 140 nm of pyrolitic carbon and embedded in a Pyrex matrix. The in situ fiber strengths measured after matrix dissolution were quite high, showing almost no degradation relative to the pristine fibers, as shown in Table 2. The interface r = 4.5 MPa was quoted based on pushout tests, leading to a, and 6, as shown in Table 2. The predicted tensile strength and failure strain are also shown in Table 2 along with the experimental values. The theory greatly overpredicts the strength and failure strain in this material. The origin of this discrepancy is unknown at present. Corroborating data on fiber pullout and fiber fracture mirrors for in situ strengths were not presented. It is possible that the interface r is lower than quoted; Table 2 shows predicted results using a much lower hypothetical r = 1 MPa for illustration. Analysis of pullout would help confirm the value of r for these materials. Also, stress concentrations at the fiber surface may play a role as well, although in all other applications such an effect has been neglected and good results have been obtained. This system clearly requires further study
W A. Curtin
204
TARLC 3 CONSTITUTIVE PKOI’EKTY DATAANI) DliRlVED QUANTITIES I:OR VARIOUS NICALONFIBER C M C SYSTEMS Matrix
Arch.
f
s*
Carbon Alumina CAS Blackglas in-Cord. in-Cord. LAS-111 Soda-lime
Woven Woven Uni x-ply Uni x-ply Uni Uni
0.22 0.20 0.37 0.20 0.37 0.19 0.45 0.43
2200 1875 2000 1704 1618 2409 2470 1380
)?I*
4.5 5.0 I .8 2.1 7.0 1.8 2. I 3.1
(!I)
410 305 300 256 1620 482 -
uc
rti
2200 1875 2250 1982 1585 2890 2800 1440
4.5 5.0 1.7 2.0 7.3 1.6 2.0 3.0
6,.
1
I640 1258 I016 853 7200 1517
I0 I I (25) 17(17-20) 17.5 I .7 14
-
Stresses are in MPa, lengths in pin. strains in ‘%. Other values for r are shown in parentheses. See text for references.
to elucidate any new mechanisms that may drive composite failure much earlier than predicted by the present GLS models. In a large number of other CMCs, fracture mirrors have been used to assess a,.and m for Nicalon fibers. These systems include a carbon matrix with no fiber coating (Heredia et al., 1992), an alumina matrix with dual BN/SiC fiber coating (Heredia et al., 1995), a CAS glass matrix with no explicit fiber coating (Beyerle er al., 1992), a Blackglas matrix with pyrolitic carbon fiber coating (Stawovy e f al., 1997),a modified-cordierite matrix with pyrolitic carbon fiber coating (Stawovy et al., 1997), an LAS-I11 glass matrix with no explicit coating (Jansson and Leckie, 1992), and a soda-lime glass matrix also with no explicit coating (Cao et al., 1990). Table 3 shows the measured constitutive properties of these various systems and the derived values of oc,m , a,., and r . Values of r measured by other methods are noted for comparison. Note that these are not all unidirectional materials, and in applications to cross-ply and woven fiber geometries f refers to the fiber volume fraction in the loading direction only (generally $ of the total). Table 4 shows the predicted tensile strength and failure strain for all of the systems mentioned above. Agreement between theory and experiment is excellent in almost all cases, particularly for the tensile strength. The failure strain can be affected by three factors. First, the stress-strain curve approaches zero tangent modulus at failure so that even small differences in tensile strength can have much larger effects on the failure strain. Second, it has recently been shown that the matrix crack spacing, usually neglected, can have a marked effect on the failure strain but not the tensile strength (Curtin et al., 1998); this effect is neglected here.
205
Fiber-Reinforced Composites TAHLE 4 PREDICTED AN11 M E A S U R E D TENSILE STKENGTH A N D FAILURE STRAIN FOR VARIOUS N I C A L O N FIBERCMC SYSTEMS (SEE T ~ 8 i . e3 ) Ten5ile strength qII\ Matrix Carbon Alumina CAS
Blackglas in-Cord uni m-Cord x-ply LAS-111 Soda-lime
Expt 300 270 460 244 454 348 790 348
Theory 330 265 520 26 I 474 334 857 417
Failure strain E Expt. 0.62 0.83 1.01 0.78 0.64 0.98 -
,
Theory 0.95 0.79 1.19 0.99 0.60 I .49 1.40 0.65
Third, there is a relief of residual thermal strains upon matrix cracking so that an additional strain of (59)
must be added, where Aa is the thermal expansion mismatch and AT is the cooling range during processing. This correction has been added to the modifiedcordierite materials and is not needed for the Blackglas materials, which are cracked extensively upon cooling, or the LAS materials, for which the residual stresses are negligible. The predicted ultimate tensile strength is generally within 5-10% of the measured value, with the one exception being the sodalime system where the difference is 20%. This system is also one in which the fibers experience the most severe degradation during processing. Overall, note that there are no adjustable parameters in this theory-all quantities input to the theory are measured-and hence the general level of agreement found here is impressive. The Blackglas material cracks extensively upon cooling. Therefore, the entire stress-strain curve is dominated by the response of the fibers and, as noted above, thermal strains are relieved at the outset. The predicted and measured stress-strain behaviors are shown in Figure 15a, and excellent agreement is evident. The entire nonlinearity in the deformation is controlled by the statistical evolution of the fiber damage and provides strong validation of the general phenomenon of cumulative fiber damage in CMCs. Figure 15b shows the stress-strain curve for an modifiedcordierite cross-ply material and, above the matrix cracking regime, the predicted behavior again agrees well with experiment.
W A. Curtin
206
400
-/
500
400
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Strain FIG. 1 5 . Measured and predicted stress-strain curves for (a) Nicalon/Blackglas CMC and (b) Nicalon/modified-cordierite glass cross-ply CMC. The dashed portion in (b) shows the fiber contribution prior to saturation of the matrix cracking. Reproduced from Stawovy et cil. (1997) with permission of Elsevier Science.
The modified-cordierite unidirectional and cross-ply materials are an interesting pair of materials. Although processed similarly, and with fairly small residual stresses, there is a factor of 10 difference in the interface t.This is also accompanied by a large difference in critical fiber strength. The origin of the t difference is not understood. But the fiber strength difference is, at least in part, due to the strengtMength scaling (eq. (6)) and the very different critical lengths 6, due to the different t values (demonstrated by the very different pullout lengths). The theoretical analysis thus helps show how macroscopic measured properties are connected to underlying constituent properties in each specific system under study. Note that the theory works equally well for unidirectional, cross-ply, and woven fiber geometries. The effects of local fiber volume fraction and weave geometry apparently play a very secondary role in these materials. Although weave geometry is often cited as a source of material variation in PMCs, in CMCs it appears that, since the critical length 6,. is much smaller than the weave length scale, the weaving has a negligible effect once in situ strength differences (if any) are accounted for properly.
Fiber-Reirforced Composites
207
40
35 30
10
5 0 0
100 200 300 400 500 600 700 800 900 100 0
Pullout length (microns)
FIG. 16. Measured (squares) and predicted (solid line) pullout length distributions for Nicalonl CAS. Predictions use !)I = 2 and 6,. = 1000 pni.
There is one system, the S i c matrix deposited by chemical vapor infiltration onto Nicalon fibers, for which the GLS theory has not apparently been successful and strongly overpredicts the strength and failure strain. Unfortunately, this is one of the more important materials for current applications. One difficulty in applying the theory is that the in situ strengths have been more difficult to measure accurately. In addition, the interfacial t in these materials may be rather higher than in the CMCs discussed here so that local load sharing may be an important additional factor. Finally, the influence of incomplete matrix cracking may also play some role in reducing the failure strain. Recent work based on the GLS concept has, however, taken into account matrix crackmg and has been quite successful in predicting the stress-strain behavior, strength, and failure strain of single-tow “minicomposites” of CVI-SiC/Nicalon material (Curtin et al., 1998). Future work on full-scale composite coupons using similar approaches may thus rectify the theory and experiment in this particular system. To complete our comparisons of theory and experiment, we consider the fiber pullout length distributions. The measured distribution for CAS glass/Nicalon materials (Beyerle et ul., 1992) is shown in Figure 16 along with the predicted value using the Curtin s.f.c. theory with m = 2 and 6, = 1000 pm. There are some differences at small lengths, which may be associated with the difficulty of seeing small pullout lengths in a forest of much longer lengths or due to factors not yet understood and missing in the theory. However, the overall shape of the distribution is generally well predicted by the theory. Such a measurement helps confirm the value of the Weibull modulus determined through fracture mirrors.
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208
H. OTHERAPPLICATIONS The GLS model shows that the composite strength and deformation are controlled almost entirely by the characteristic strength a,..Changes in the material parameters DO, m , and t induced by fatigue, temperature, corrosion, and creep can thus be directly translated into changes in composite tensile strength. Fatigue, fiber strength degradation, and creep have all been studied carefully to date. ' ) eq. (49) The GLS model shows that strength is proportional to t ' / ( ' ' +(see coupled with the definition in eq. (9)). The physical reason for this is that t establishes the critical gauge length and the critical fiber strength is the strength at this gauge length. Under cyclic fatigue loading above matrix cracking, the relative fibedmatrix interface sliding leads to wear of the interface, among other possible effects. The wearing phenomenon decreases the interfacial t with increasing cycles N , which can be measured directly during testing through the unloadreload hysteresis. Assuming no other material changes with cyclic loading, strength decreases as t (N)l/'"+l'as the critical gauge length increases. When the strength decreases to the maximum applied stress S in the fatigue test, the material fails. The fatigue life "S-N" curve can thus be mapped out knowing only t ( N ) and m. Specifically, the peak stress S at which failure occurs in N cycles satisfies, with nutsthe unfatigued strength,
If the wear leads to some nonzero asymptotic value t ( N + oo),then the fatigue threshold stress Sth (life is infinite for stresses below the threshold) is
+
Because of the (m l)-I power in eq. (61), even a reasonably large decrease in t can yield a high fatigue threshold. The above fatigue model has been applied to several CMC systems (Rouby and Reynaud, 1993; Evans et al., 1995). Figure 17 shows t versus N for CAS/Nicalon (Evans er al., 1995), which was found to be largely independent of the maximum applied load S, and t decreases by about a factor of 4 in less than 100 cycles. Using this t ( N ) in eq. (60) with rn = 3, Evans el al. predicted the S-N curve shown in Figure 18, along with the measured behavior. The agreement is quite good, with a fatigue threshold at Stl, = 325 MPa based on outs= 460 MPa. For m = 2 (see Table 3), the threshold would decrease slightly to 290 MPa, which
Fiber-Reinforced Coinposites 25
2 I
20
-J 15 d QI
0
f
10
) .
C
-
5 0 0
1
2
3
4
5
loglO(Cycles)
PI
FIG. 17. Interfacial sliding resistance T vs fatigue cycles N for N i c a l d C A S glass. After Ev;m c d . (19%) with permission of Elsevier Science.
still agrees reasonably well with the data and shows that the sensitivity to m is not too strong for moderate m values. Interfacial shear stresses can also decrease under constant load by creep. Du and McMeeking (1993, Fabeny and Curtin (1996), and Ohno et al. (1997,1998) have developed creep models for t ( t ) that are driven by the creep relaxation of the matrix in shear, after the tensile stresses have been relaxed by cracking or tensile
450
250 200
I_._--
I 0
1
2
3
,A
4
5
logl O(Cycles to failure) FIG. 18. Measured and predicted S-N curve foi- NicaloidCAS glass. Solid symbols denote experimental failure; open symbols denote test slopped (“run-out”). Predictions are shown for Weibull moduli rzi = 3 (solid line) and m = 2 (dashed line). After Evans cr t i / . (1995) with permission of Elsevier Science.
210
W A. Curtin
creep. Inserting t ( t ) into a,.then leads to a time-dependent composite tensile Similar models for PMCs were developed earlier by Mason et al. strength auts(t). (1992). For a power law shear creep relation given by
where f J n is the shear strain rate, t is the shear stress, and n and B are the matrix power law creep exponent and prefactor for tensile creep, Ohno et af. (1997,1998) have derived a form for t ( t ) that is particularly convenient
Here,
is a characteristic relaxation time for the shear creep in the composite, and w is the fiber spacing. Combining eqs. (60) and (62b) and solving for the failure time yields
Predictions and data for time to failure in a Ti-MMC using this approach are shown in Figure 19 and the theory clearly shows the trend exhibited by the data. The absolute time scale is in poor agreement, however. While this may be due to uncertainty in some of the constitutive property data used, other physical features may be missing from the simple model. In any case, the failure time is sufficiently long at these low stresses that the tensile creep has effectively driven the matrix contribution 0)to 0 well before t f is reached; hence, the analogy with fatigue in the CMCs is quite appropriate. Du and McMeeking (1993, Ohno et al. (1997, 1998), Fabeny and Curtin (1996), and Weber et al. (1996) have all addressed the issue of failure at higher loads and shorter times during the period of effectively decreasing 0).Predictions of another Ti-MMC by Fabeny and Curtin (1996) in this regime are depicted in Figure 20, and show generally good agreement with the experimental failure times. Ibnabdeljalil and Phoenix ( I995b) and lyengar and Curtin (1997b) have also investigated composite strength versus time due to time-dependent fiber strength loss. However, experimental data on such phenomena is limited at present and so these theories will not be discussed here. Iyengar and Curtin (1997~)have also re-
21 1
Fiber-Reinforced Composites 1400
800
-1
I
-1
0
1
2
3
4
5
6
7
loglO(fai1ure time, hrs) F I G . 19, Applied stress vs hilure time for a Ti-MMC. Solid symbols: measured failure; open symbols: test stopped ("run-out"): solid line: prediction due to shear creep relaxation. After Ohno PI d. (1998).
cently theoretically investigated the strength versus time when both creep/fatigue and fiber strength loss operate simultaneously; their results show a marked nonlinear coupling of these two mechanisms which sharply reduces the composite lifetime relative to either mechanism acting separately. This may have strong implications in the interpretation of real experimental data.
1 1,100 ,100
-a" I
ao=l.47GPa 1,000 5
Y
v)
$
-oO=1.29G
(13
U Q,
a.
8
800 -
700
'
-4
0 I
1
0 2 4 Base10 Logarithm of Time-lo-Failure in Hours -2
6
FIG. 20. Applied stress vs failure time for a Ti-MMC. Symbols: measured failure: solid lines: predicted due to matrix tensile creep lor two different initial fiher strength values. Reproduced from Fabeny and Curtin ( 1996) with permission of Elsevier Science.
212
M? A. Curtin I. S U M M A R Y
The GLS model of composite failure is based on the assumption that fiber breaks do not cause local stress concentrations. From that one assumption flows a complete theory for the stress-strain behavior, tensile strength and failure strain, fiber pullout, and work of fracture as functions of the constituent material properties. The deformation and failure are controlled by the stochastic evolution of the fiber damage, with many key composite properties determined by the critical stress rr, and critical length 6, . The success of this theory in applications to CMCs appears quite good, and with no adjustable parameters. This success has motivated extensions to time-dependent and cycle-dependent phenomena, and these models appear to capture the major factors controlling lifetime.
V. Multifiber Composites: Local Load Sharing A. INTRODUCTION
When a single fiber fails in a composite, its stress is certainly not transferred to fibers infinitely far away except in the limit r -+ 0. Some sort of localized stress transfer must exist in the real composite, a situation we generally refer to as “local load sharing” (LLS). Although GLS is a good approximation, as seen in Section IV, the existence of LLS to any degree has some important consequences for the nature of the composite failure. First, fiber damage in some local region increases the stresses around that region and drives further damage locally. Composite failure is then caused by the formation, at some place in the material, of a “critical cluster” of damaged fibers that grows/propagates in an unstable manner across the remainder of the composite. This situation is thus similar to failure of a monolithic ceramic or the individual fibers themselves, where one critical flaw causes failure. As a result, the composite failure becomes statistical, with some probability distribution and some limited reliability. Furthermore, the composite strength depends on the sample volume, decreasing as the volume increases because the larger volume provides more possible locations for the “critical cluster” to develop. Finally, the composite is sensitive to local stress concentrations, stress gradients, or localized induced damage due to notches, holes, or impact events. None of these features is present within the GLS approximation, yet all are critically important to the engineering application of composites. There is considerable experimental evidence in PMC materials that the failure strength is both statistically distributed and volume dependent. Although difficult
Fiber-Reinforced Composites
213
to determine and quantitatively assess because of competing failure modes (e.g., compression and tension in bend tests), the volume scaling has been presumed to be due to the localized stress transfer in PMCs. We reference some recent work along these lines by Wisnom (Wisnom, 1991a, b) and, for ceramics, the work of McNulty and Zok (1997). To understand how LLS influences composite behavior, one must address (i) how to determine the nature of the load sharing in any particular system for arbitrary spatial locations of breaks and (ii) how to use such information to assess the damage evolution and failure in large composites. In this section, we review progress on both of these issues and present in detail one recent model for LLS that appears to capture features observed in experiments on MMCs and possibly PMCs.
B. LOCALLOADSHARING MODELS The existence of localized stress transfer has long been recognized in work on predicting the strength of PMCs. The earliest model was a fully elastic model developed by Hedgepeth and Van Dyke (1967). Hedgepeth and Van Dyke modeled the fibers as one-dimensional elastic elements aligned in a regular array and surrounded by a matrix capable of carrying only shear stresses. They considered shear coupling between near-neighbor fibers only and were able to derive a second-order differentialldifference equation for the coupled axial displacements { u ( z ) } of the elastic fibers, with the axial stress following from equilibrium as p ( z ) a duldz. For fibers in a regular array, they showed that (i) the differential/difference equations in the presence of a single broken fiber could be solved exactly to obtain the stress transfer, or “break influence function,” and (ii) the induced stress transfer in the presence of multiple breaks could be obtained by solving a matrix equation involving only the fiber displacements at the break locations followed by simple matrix multiplication. Specifically, for a square lattice of fibers denoted, Hedgepeth and Van Dyke showed that the in-plane stress on fibers (tr,l , ) due to a single break at the origin (0, 0) under stress c could be expressed as K c , , e,u with K c , , y , = -ky,, e , lko, 0 and
kL6 -
-1. 7r2
ln 1’
x (1
+ sin2
do
dg2cos(t,,g2)cos(tr8)
(v)) 1 /2
-
sin2
,
W A. Curtin
214
When multiple breaks exist at locations (t.r,t ,,} in the plane, one then solves
It:. :0 broken)
for the displacements uy,. y , of all the ( t r ,t , }interacting broken fibers under unit compressive loads (the source of the -1 on the r.h.s. of eq. (66)) and then the stresses Kp, , p, (r transferred to the remaining unbroken fibers (l:, ti 1 are obtained by solving
( V , , Y , broken)
The matrix size in the set of eqs. (66) is equal to the number of actual breaks, not the total number of fibers, and is therefore optimally efficient. Similar results are obtained for other lattices, and the generalization to out-of-plane stresses straightforward. Hedgepeth and Van Dyke (1967) utilized the above model to analyze stress concentration factors for various compact clusters of in-plane broken fibers for linear (two-dimensional), square, and hexagonal fiber arrays. The in-plane stress concentrations for single breaks in the two types of arrays are shown in Figure 21; clearly, the load transfer is quite localized, with about 60% of the stress from a single fiber transferred to the nearest neighbor fibers. In spite of the great power of this approach, in which only the interactions between the broken fibers need be determined, it was not utilized to study composite failure. A number of subsequent workers analyzed the same model problem and presented further results, both numerical and analytical. Suemasu (1984) included some analysis of the probability of damage around preexisting clusters of breaks and concluded that the enhanced stresses did not translate into significant increases in failure probability. However, the extension to full predictions of composite failure with stochastic fibers was not accomplished. Sastry and Phoenix (1993, 1995) and Beyerlein and Phoenix (1996~1, b, 1997a, b) have revived the Hedgepeth-Van Dyke (HVD) model. Sastry and Phoenix (1993, 1995) investigated interactions between out-of-plane breaks to demonstrate the full capability of the model but did not pursue problems in three dimensions. Beyerlein and Phoenix (1996b) showed that, for a large linear array of aligned breaks, the stresses predicted by the HVD shear lag model agree very well with linear elastic fracture mechanics for orthotropic materials down to the scale of one fiber diameter. The stresses in this case were also investigated previously by Hikami and Chou (1989), who obtained an analytic form for the stress concentration at the crack tip. Beyerlein and Phoenix (1996a) also devel-
215
Fiber-Reinforced Composites
(3 u (3 @@@
/I@@@@
\ 0.004
@Be@@@
@@@a@ (@@@a@@ @a@@@ (b)
FIG 2 1 In-plane load transfer for a single broken fiber under unit apphed load. the Hedgepeth and Van Dyke ( 1967) method (a, square lattice, (b) hexagonal lattlce
a5
calculated by
oped a method for incorporating matrix yielding, beyond what was done in the HVD model, and showed the strong decrease in stress concentration factors near fiber breaks when the yielding is included. This detail is accompanied, however, by a marked increase in the computational requirements since the matrix regions must also be treated as non-linear elements. Beyerlein and Phoenix (1997a, b) also analyzed the failure of a two-dimensional composite with an initial crack and stochastic fiber strengths, for in-plane breaking only, and showed how the statistical variability in fiber strength could give rise to toughening or “resistance-curve’’ behavior even in the absence of fiber pullout toughening (whlch dominates in real
216
W A. Curtin
composites). Lastly, Beyerlein and Phoenix ( I 998) have extended this approach to consider a viscoelastic matrix, and hence the evolution of stress concentrations as a function of time. All of these efforts put the HVD technique on the verge of exploring full composite failure but the final step has not yet been taken. Several other workers have recently analyzed the load transfer around broken fibers using finite-element modeling aimed at CMCs, MMCs, or PMCs. He et al. (1992) investigated the stresses around an elastic fiber with a break and interface debonding embedded in an elastic axisymmetric geometry. A ring of elastic fiber material represented the neighboring fibers, and the stresses on the neighboring fibers were studied as a function of the applied fiber stress C T ~ interfacial , t,elastic moduli, and volume fraction. He ef al. found nonconstant stress concentrations that were very roughly linear in t/of for small t/af < 0.1 and attained values approaching the HVD result for hexagonal geometry for large E f I E , , , and t/a/% 0.2. Since the matrix remained elastic throughout and was uncracked, the He et al. results showed significant reductions in stress concentration for Em M E f . Du and McMeeking (1993) analyzed stress concentrations in a similar axisymmetric model but with an elastic/plastic matrix and a thin elastic/plastic fiber coating. They found the stress concentration on the neighboring fibers was a function of the applied fiber stress af divided by the matrix yield stress no, in a manner similar to Beyerlein and Phoenix (1996a), and a function of the coating shear yield stress to. For equal matrix and coating shear yield stresses (to = no/&), the stress concentration typically increased rapidly beyond full matrix yielding, reached a peak of 543% at around af/oo = 5-10, and then decreased rapidly with further increases in applied fiber stress. For coatings with a fairly reduced shear yield stress (TO 5 O.loo), the peak stress concentration factor typically decreased slightly, occurred at small applied fiber stress, and decreased rapidly at higher applied stresses. An example of this behavior is shown in Figure 22 for f = 0.35, E I / E , ~= 3, and varying ro. Load transfer to the level found in the HVD model was never achieved. Caliskan (1996) used a different numerical method for the same axisymmetric geometry and elastic fiber, matrix, and interface to investigate the very small E,/Ef regime. He found that the average stress concentration on the neighboring fibers was almost identical to that predicted by the HVD model for both 1 broken fiber, 7 broken fibers (central fiber and the near-neighbor ring of fibers), and 19 broken fibers (central fiber and two nearest-neighbor rings of fibers). The in-plane load transfer was also independent of the matrix modulus when small, as in the HVD model. This work confirmed the accuracy of the HVD shear lag result in the limit of elastic interfaces and very low matrix modulus, cases of possible practical applicability in PMCs. Finally, Nedele and Wisnom (1994) performed full three-dimensional finite-element models (al-
217
Fiber-Reinforced Composites 1.1
W
;1.02 1 0
2
4
6
8
10
Fiber stress normalized by matrix yield strength. o,/q
FIG.22. Load transferred to near-neighbor fibers (stress concentration factor) for a single broken fiber in an elastic-plastic matrix (yield s i r e s U O ) with an elastic-plastic fiber coating vs fiber stress 0, and for various normalized coating shear yield stresses rv(,,/cr0as indicated. Reproduced from Du and McMeeking (1993) with permission of Kluwer Academic. Dordrecht.
though using a cylinder of effective composite material beyond the neighboring fibers) for one specific case of a PMC (low E , , , / E f , elastic behavior throughout). They found a stress concentration of only about 6% in the plane of the break, which is rather smaller than that obtained by Hedgepeth and Van Dyke. Such results should be revisited again, and in more depth, to fully map out the stress concentrations predicted in three-dimensional models. None of the above works included matrix cracking, as relevant to CMCs, and only Du and McMeeking (1993) permitted full matrix yielding, as expected in MMCs prior to failure. In total, the above results show that some local load transfer does exist and that the HVD results appear to be an upper bound. However, a full correlation of the load transfer and the material properties E f , E,,, f ,r , and matrix yielding does not yet exist in the literature although the Du and McMeeking results are a fairly comprehensive set of data. And, most unfortunately, none of these detailed numerical techniques can easily accommodate multiple-fiber breaks in a realistic manner including out-of-plane breaks.
c. STATISTICAL MODELSOF COMPOSITE FAILURE UNDER
LOCALLOADSHARING
With the difficulty of obtaining approximate load transfer values for single broken fibers in composites, workers interested in the general effects of LLS on composite failure have developed models using the HVD values or other approximate
218
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A. Curtin
rules for the load transfer which are tractable for use in composite theories. An early model is that of Zweben (1968) and Zweben and Rosen (1970), who considered the development of successively larger compact clusters of broken fibers when there is LLS and the fibers have a statistical strength distribution. Exact enumeration of the statistics became difficult, however, beyond clusters consisting of more than just a few fiber breaks so that Zweben and Rosen could only provide some general insight and approximate guidelines for composite strength. Such models do contain the important feature of size scaling, but could not consider all of the likely “critical” damage clusters causing failure. Around the same time, Scop and Argon (1967a, b) developed similar models for two-dimensional structures. They recognized that a composite of length L; could be broken into a series arrangement of L;/S smaller composites, where the slip length 6 was also stress dependent, and then weak-link scaling could be used to relate the lengthS composite strength to the length-l, composite strength. The use of weak-link ideas, discussed further in Section V.E, was also introduced earlier by Gucer and Gurland (1962). The full composite problem then reduces to finding the strength statistics of a length4 composite. Like Zweben and Rosen, Scop and Argon presented a conceptual approach to obtaining the strength statistics but could only operationally carry out the calculations for fairly small numbers of broken fibers in a two-dimensional linear arrangement. They did propose asymptotic forms for the strength for large numbers of fibers in the cross section but the accuracy of these forms remained untested. A decade later, Phoenix and co-workers revisited the “chain of bundles” (chain of length4 fiber bundles) and performed very impressive analyses of failure. Harlow and Phoenix (1981) studied linear arrays of fibers under the assumption that all of the stress from a broken fiber is transferred onto the two nearest-neighbor fibers (one on each side). With such a simple load transfer, Harlow and Phoenix found exact recursion relations for the composite failure probability for increasing composite size. They then showed that, since failure in very large systems is controlled by small amounts of local damage at low overall applied stresses, asymptotic methods could be used to accurately relate failure in a small system to larger systems. These works clearly demonstrated the effects of local damage-driven failure and stochastic/size-dependent composite strengths that are the hallmark of LLS. However, the simplified load sharing and geometry prevented direct predictions of strength in realistic three-dimensional fiber composites. Smith et al. (1983) and Pitt and Phoenix (1983) followed this effort with work on three-dimensional composites and with “tapered” local load sharing, respectively, but did not make any applications to specific systems. Various other models for composite failure have been developed since the work of Harlow and Phoenix (1981), including numerical simulation models. Most re-
Fiber-Reiiqorced Composites
219
cently, Leath and Duxbury ( 1994) have revisited the problem studied by Harlow and Phoenix and obtained new exact recursion relations. None of these models has demonstrated broad prediction capabilities and so they have not obtained widespread use to date. One model, developed by Batdorf ( 1982), captures much of the dominant features of the Harlow-Phoenix, Zweben-Rosen, and ScopArgon models, however, and is fairly easy to apply. Batdorf (1982) considered the probability of forming Q I isolated breaks in a composite, and then determined the probability that the neighbors of these breaks would also break under the stress concentration (‘1 to form Q2 pairs of breaks. The number of triplets (three breaks) emanating from the pairs was then determined, and so on. Batdorf thus obtained a recursive relationship between the number Qi of i-clusters and Qi+l in terms of the number of neighbors ni around an i-cluster and the stress concentration ci on those neighbors. An underlying assumption, when applied to three-dimensional fiber arrangements, was the neglect of specific different shapes and neighbor stress concentrations of the i-clusters. The n ; and c; were considered averages over the different types of i-clusters. The Batdorf model can be rewritten to read as (Foster et ul., 1998)
for a composite of n / fibers in length L ; . The composite strength in this model is defined as the stress at which the largest cluster (size i * for which Qi* = 1) is unstable to growth because Q;*+l 1. That is, once there exists one cluster of size i*, it will grow to larger sizes with no further increase in the applied stress. Batdorf showed that this model produces results very close to those of Harlow and Phoenix (1981) for the linear array. Batdorf and Ghaffarian (1982) applied the model to polymer composites with some success using compact clusters and the HVD results to obtain the requisite ti; and c i . In principle, the Batdorf model can also incorporate load transfer rules that are a function of both local damage cluster size and applied stress, although these aspects have not been included to date. Two limitations of many of the above models are as follows. First, there is the lack of spatial variability of the fiber breaks within the length 6 ; this eliminates the load carried by slipping fibers, which was shown to be one-half the full load within the GLS model. Second, there is the assumption that the clusters of fiber breaks are compact and near neighbors; more-diffuse clusters of breaks and long-ranged interactions among damage are not included in these models. As remarked earlier, however, the general three-dimensional version of the HVD model, as presented
220
W A. Curtin
by Sastry and Phoenix (1993, 1995) for elastic matrices and by Beyerlein and Phoenix (1996a) for elastic-plastic matrices, is ideally suited toward pursuing numerical studies of composite failure. Such studies remain to be carried out, however. As a final note, several workers attempted to use the GLS models to approximately investigate the effects of LLS. Specifically, Curtin (1993b), Phoenix and Raj (1992), and Phoenix et al. (1997) envisioned that failure due to some critical cluster occurring in LLS could be approximated by failure in GLS in a bundle of some effective number of fibers n. Since a finite-size bundle of fibers in GLS shows statistical variations, as known since the early work of Daniels (1945), the statistical variations in GLS at size n were thought to somehow reflect those of LLS. The strengths of large systems could then be obtained by weak-link scaling in a manner similar to that shown much earlier by Gucer and Gurland (1962). However, none of these workers could show a definite connection between LLS and GLS at some particular size n so that while the concept was interesting, specific predictions could not be developed for specific systems. This work presaged some very recent work (Ibnabdeljalil and Curtin, 1997a) in which a specific connection between LLS and GLS has been shown to exist (see below).
D. LOCALLOADSHARING MODELOF ZHOU A N D CURTIN (1995) 1. Model Development Zhou and Curtin (1995) developed a model for studying composite failure that retained all of the features of the GLS model but included local load sharing within any plane z of the composite. The introduction of LLS was accomplished through a discrete model very similar to the continuum model of Hedgepeth and Van Dyke. However, because of the discretization, Zhou and Curtin (ZC) were able to induce load sharing that varied from GLS to that of Hedgepeth and Van Dyke with the tuning of one single parameter in the model. With a tunable load sharing, this method also is suitable for incorporating results from much more detailed determinations of stress transfer (Section V.B) with relative ease. The restriction to in-plane load transfer also allows the full three-dimensional composite to be considered as a coupled set of two-dimensional (in-plane) problems, thereby yielding considerable computational efficiency. In the ZC model, each individual fiber is modeled as a linear set of springs in series, with each spring representing a small section of the linearly elastic fiber. These springs are then arranged in a regular array (e.g., square lattice) and are coupled through the introduction of pure shear (leaf) springs between the nodes
Fiber-Reinforced Composites
22 1
of nearest-neighbor fiber springs. A schematic of this setup of springs and nodes is shown for the two-dimensional case in Figure 23. The fiber springs have a spring constant k, and the “matrix” shear springs have spring constant k, related to the matrix shear modulus G,fl.As in the HVD model, the “matrix” carries no tensile load. The entire composite consists of n f fibers labeled by index n (1 5 n 5 n t ) of length L ; , each of which is divided into N z elements of length 8 = L , / N , labeled by index m (1 5 m 5 N 2 ) . A remote applied stress o f per fiber is applied to the composite which, in the absence of any fiber damage, induces a stress (force) o,],= o f in the nth fiber at axial location m . The key issue is the treatment of fiber breaks within the above spring model for the composite. Suppose spring element ( n , m ) is broken (i.e., k, for this element is set to 0). The induced slip along this nth fiber must be put in “by hand,” since the matrix is purely elastic, as follows. According to eq. (7b), the stresdforce is linear with distance from the break and takes on the values
(69) This slip occurs over those fiber elements m’ for which the slip-controlled stress is less than the total “far-field” applied stress o,]. which includes both the applied fiber stress and stress transferred to element ( n , m’) from other broken fibers. The slipping fiber elements carry the stress p l l , independent of the displacements of the nodes connected to spring ( n , m’) and so are “perfectly plastic” springs with zero spring constant k, = 0. Operationally, the slipping fiber elements ( n , m’) are then treated as breaks, k, = 0, but with an effective applied stress of - p I I . The broken and slipping regions along the discretized fiber n must also transfer the loads af - p n , to the neighboring fibers n’ # n . In this model, we assume that the transferred load at - p N nl . in plane m is transferred only to otherfibers in the same axial plane m, as if there were no other damage in any other plane. Equilibrium requires that this be true on average; i.e., the axial stress carried across any plane m must be n f o f , independent of any details of the load transfer. However, the spatial distribution of the loads is, in principle, affected by the damage or slipping that occurs in other planes (as evident in the three-dimensional HVD results of Sastry and Phoenix (1993), for example). With this one approximation, the full three-dimensional problem of determining the stress fields arl, on every unbroken, nonslipping fiber element, given the applied load of plus the transferred loads (of - P , ~,,? , ) from the brokedslipping fibers, is reduced to a sequence of N z highly coupled planar (two-dimensional) problems. The coupling in the two-dimensional problems occurs since an actual break in element ( n , m ) affects, through the slip pI1, the state of stress in planes m’ # m for those m’ within a slip length of plane m . Thus, the induced stress state around a broken nlj,
,,lj,
W A. Curtin
222
n.m+l
n.m+ I
nm
n.m- I
F I G . 23. Schematic of spring/node model for unidirectional composites with load applied in the vertical ( z ) direction. (a) Labeling of fibers and nodes by spatial position. (b) Schematic after some fiber damage (denoted by -x-) with slip represented by “plastic” springs (denoted by (c) Representation of an individual layer with plastic springs replaced by broken springs and applied closing forces acting against the remote applied forces.
-m-).
Fiber-Reinforced Composites
223
fiber is three-dimensional but not the exact solution of the full three-dimensional problem. Consider now the planar problem of a set of broken fibers ( k , = 0 ) in some plane in with applied stresses a/ - p,, ,,! on each broken fiber and applied stresses o f on each unbroken fiber. We wish to obtain the stresses (forces) on all of the unbroken fiber elements (springs) in this plane m ,assuming no other damage in the material. This problem is essentially identical to the HVD problem for inplane breaks, but for a discrete lattice rather than a continuum medium. We solve this general problem using a Green’s function technique that is also the underlying basis for the HVD “break influence function” approach. 2 . Lattice Green’s Function Method
Green’s function methods are widely used in applied mechanics and in physics. The discrete models used here are carried over from their applications in atomic deformations in solids and are well developed in the solid-state physics literature (Tewary, 1973; Hsieh and Thomson, 1973; Thomson et al., 1987). Because of its origins outside of applied mechanics, we discuss the method in some detail here to provide a clear reference for future applications. Consider the lattice of springs and nodes shown in Figure 23. Denote the position of the node located at physical position t , ; t!, $ t!,? by t! and the axial displacement of that node by ug. To hold the nodes at specified positions u = ( u t ) requires the application of forces F = (Ft ] of
+
F = $u
or
+
Ft = @ ~ , p u ~ ~ ,
6‘0)
where @ = ( @ i , ~ is ~ ) the force constant matrix with entry @[,!’ equal to the spring constant connecting sites f! and t’.The repeated index t!’ in eq. (70) implies summation over that index, as usual. The form of $ for the perfect lattice (no broken springs) is straightforward to determine by writing out eq. (70) for some particular site t!. For a square lattice, this becomes Fu,.e,. PI = k t [ u L , t , . / ; + l
+
U I , . t v ,(:-I
+ k s [ ~ ( . , + I . t , . i+, u t , - l , t , . f :
- 2ut.,,i,.,t,] +uf,,t,+l,t:
+ u t , . t , - l ~ t :-4Uf,,t,,e:]?
(71)
so that $ t . y = -2k, - 4k,y,$(, = k , if e, t’ are in-plane neighbors, IlrU, y~ = k , if t!, e’ are axial neighbors, and @,, = 0 otherwise. To obtain the forces in the axial springs, we must solve for the displacements u = ( u t } given the applied nodal forces F = (Fp].Then, the spring force is 1c
W A. Curtin
224
simply kt(ue,.e , , rz - uy,, (J, !.-I) for the spring with upper axial node l . The displacements are obtained from the inversion of eq. (70):
u
= GF,
=$-IF
(72)
where G = is the Green’s function matrix. The quantity Gel, is the displacement induced at site [’ for a unit force applied at site l . The matrix G is not sparse like its inverse $ but is the quantity of interest for calculating the spring forces. In the perfect lattice (no broken springs), the lattice retains its translational symmetry and hence both $ and G are functions only of the relative displacement l - [’ between two nodes. The perfect lattice Green’s function Go can be obtained from eq. (70) by using Fourier transforms. Defining the Fourier transform relationship between u y and up in the infinite discrete system through $I-’
and similarly for Fr and $[-if, substituting these into eq. (70), and solving for the Fourier coefficients u q , F,, and GI: = u q / F y , we obtain the perfect lattice Green’s function as
1
0 -
G, - 2 ( k , (cos(q1)
+ cos(q, ) - 2 ) + k , (cos(q,) - 1)).
(74)
G: in real space is then obtained using the analog of eq. (73), and is the Green’s function connecting the node at the origin to the node at C in the perfect, undamaged lattice. Now consider the damaged system in which some fibers are broken or, within the spring model, some springs are missing ( k , = 0). The force constant matrix can then be written as $ = $(I - S$,
(75)
where S$ is the portion of @ due to the missing springs, or “defects,” in the lattice. For a defect between neighboring vertical nodes [ and l - ? , the perturbation in S$ has just four entries, S @ t . e = 8$?-:- f - f = -k, and 6@e,c-2 = 6$c-t, = k,. Hence, in general, S$ has a number of nonzero entries equal to only twice the number of nodes associated with the defects in the system. The Green’s function G = @-’ for the lattice with defects can then be written as the solution of G = G”
+ G”S$G.
(76)
The node displacements ( m e } under the forces { Fp} are then 0
u t = G,_,,
F ~ I
+ G ~ - , , , s ~elup. ~,
(77)
Fiber-Reinforced Composites
225
Writing a similar expression for node u f -:, i.e., the node vertically below node C , and subtracting from eq. (77) leads to the displacement Aut across the spring with upper node e as Aut = u i - ~ t - i = ( G ~ - Y , - G ~ - 2 - ~ , ) F t ~ + ( G ~ p i , , - S+tii, G~-~ e’uv‘. - ~ , ,(78) )
Now since 81) only connects pairs of vertical nodes e, e-2, the product S $ p l , I ‘ u p depends only on Auttl, the displacement across the broken spring having top node l” (a node in the defect space). Applying equal and opposite forces to all pairs of nodes, F,-; = - F t , and noting that, by symmetry, G:,(-: = GP-:.. (, and GY = G:-:, y - 2 , we obtain our main result kl
A u ~= gFpv,Fti
+
@-i,,kl
Au,~!
(79a)
e, l” E (defect space), t ‘ E (all space),
+
kf A U Y= G0~ p t ~ F y ~0
Gf-ltjkt
AUC“
t’’ E (defect space), t $ (defect space),
(79b)
e‘
E (all space),
where the sums are only over the top nodes of each spring. In eqs. (79) we have introduced the dimensionless dipole Green’s function
G:-(,
= 2k,(G:-Yr
-
(80)
G:pz-Y,).
Equation (79a) is a matrix equation for the unknown displacements Auy of the broken springs only (in the defect space), in the presence of arbitrary applied forces {Fe} on all nodes in [he system. Equation (79b) then determines the displacements of all of the unbroken springs (outside the defect space) in terms of the displacements of the broken springs obtained in eq. (79a). The main problem is thus the solution of eq. (79a), which involves matrices of size only equal to the number of broken springs in the entire system. Equation (79b) involves matrix multiplication only, and is straightforward. In this formulation of the problem, as in the HVD approach, the size ofthe matrix inversion is only the size of the number of defects in the system and so this is the minimum possible size and an optimally efJicient method. Solving eq. (79a) formally and substituting into eq. (79b), we find that the forces of interest, kl Aue on the unbroken springs, are given by
k, Au, = G:-rrFti
+
-
G:,,pI,,,)-
1G 0l l l l p u I F ~ t .
(81)
where e”, e”’ E (defect space), e $ (defect space), and e’ E (all space), and repeated indices imply summation within the restricted space. A few simple results stemming from eqs. (79) and (81) are in order. First, consider a single broken spring at site e’’ with applied force F at this site only. Then
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W A. Curtin
the forces on all other sites follow from eq. (81) as
or
The quantity G:-t,,/(l - G): is the fractional force transferred to site l from site l", and so is the load transfer function. Summing up the load transferred to all of the sites shows that =1 Y
is required to satisfy equilibrium. Next consider a single break at site l" but with a uniform applied force F on all sites in the system. Equation (81) then yields, using the sum rule of eq. (84),
so that the load transfer is the same as if only the broken site were loaded, as
required by superposition. The above analysis shows that only the dimensionless dipole Green's functions are needed to determine the spring forces in the presence of arbitrary applied forces. In applying this approach to the ZC model for composites, we only consider individual layers of springs (fixed layer rn, or fixed axial top node coordinate l : ) .So, the dipole Green's functions of interest are those which couple the pair of nodes at lri + l , j l,i and l,x^ l , j ( l , - 1 ) i to another (top) node at l',; + l : j + l,i, i.e., to nodes having the same axial coordinate l , . Since the perfect lattice Green's functions only depend on the difference in node positions, we therefore need the Green's function connecting the nodes at the origin 0 and -? to the node l , i + lv.$for all integers l , , l , . The dimensionless dipole Green's functions needed are then
+
+
+
Fiber-Reinforced Composites
227
Hsieh and Thomson ( 1973) showed that the q: integral in eq. (86) could be performed analytically to yield
+ sin’(qy/2) exp(iq,l.,) exp(iq,.e,.) J I + ~:(sin’(q.,/2) + sin’(qy/2))
Jsin’(q,/2) X
.
.
(87)
m.
For a finite system with periwhere we have introduced the quantity R = odic boundary conditions in A- and y, the integrals over the wavevectors q.y,qy in eq. (87) are replaced by discrete sums over the Brillouin zone, i.e., q,, = n,,r / N , for - N , /2 5 n , 5 N , /2 for a system of N., nodes in the x direction, and similarly for the y direction (Thomson et al., 1987). Note that the dimensionless dipole Green’s function depends only on the quantity 52, which is the anisotropy ratio of the shear and tensile springs, and the lattice symmetry. The parameter R thus controls the load transfer function (e.g., eq. (83)) and is an adjustable parameter in the model. There are two limits of particular interest. For R +, 00, the square-root integrands in eq. (87) cancel out and the stress transfer defined by eq. (83) becomes independent of position and vanishingly small but such that the sum rule of eq. (84) is still satisfied. This case thus corresponds precisely to GLS conditions. For R + 0, the stress transfer defined by eq. (83) remains well defined and the resulting integrals become identical to the HVD results, after some algebra. To relate the discrete problem to the continuum problem, Zhou and Curtin showed that, for fibers separated by a 2r, a single spring representing a physical length 8 of fiber has
+
Elxr’
k, = -,
s
while the shear springs have
Thus, k, and k , depend on continuum properties and on the axial discretization 8. In the continuum limit length 8. From eqs. (88) and (89),it is clear that R 6 + 0, we have Q + 0 and hence the expected correspondence with the HVD results. In the ZC model, however, only in-plane load transfer is considered and hence the axial and transverse stress variations are explicitly decoupled. This allows any value for R to be selected to control the in-plane load transfer and, independently, maintain any value of interfacial r for the axial slip. Although the in-plane load transfer and interface r are coupled in reality, as shown by the vari-
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ous finite-element models discussed in Section V.B, it is nevertheless convenient to have a general method which allows both features to be controlled separately rather than fixed at the values required by the (elastic) HVD problem. Figure 2 1 has already shown examples of in-plane stress concentrations around a single break in the HVD limit. Figure 24 shows the in-plane ( z = 0 plane) stress redistributed to surrounding fibers from a cluster of four broken fibers in the z = 0 plane for the square lattice, where each of the broken fibers is loaded with a unit load ( F = 1). Three different cases of 52 are shown: The C2 = 0.001 case is essentially identical to HVD; the 52 = 10.0 case is clearly approaching the GLS limit of equal loads on all remaining fibers; the case !2 = 1.0 is an intermediate result, showing moderate stress concentrations around the broken cluster that are not as large as in the HVD limit. 3. Algorithm and Qualitative Results To employ the overall method described above, the algorithm is as follows. First, select the number of fibers n 1 , the overall composite length L , > 2&, and the desired fiber geometry. Second, select the load-sharing parameter R and calculate the dimensionless dipole Green’s functions G: from eq. (87). Third, discretize the axial length into units 8 << 6,. Fourth, assign each spring ( n , m ) a strength . T , ~ , ~=~ ~q l n ( l / ( l - R ) ) where C T ~= a[6,/6]1/”1 is the Weibull characteristic strength at length 6 and R is a random number in the interval (0, 1). These steps complete the initial setup of the composite to be simulated. A tensile test is then performed by the following algorithm:
1. Apply a uniform stress (force) at to all the springs. 2. Break the springs that are overloaded, otr, 2 s j j ,
3. Introduce an axial slip along the broken fiber(s) by setting kt = 0 and assigning the residual plastic loads p j j ,m . 4.
Use the Green’s functions and eq. (8 1) to calculate the new stresses (forces) on all of the remaining elastic springs in each layer m .
5.
Under the new local stresses, determine if any fibers are overloaded by the transferred stress. If yes, go to step 2. If no, go to step 6.
6. Increase the applied stress by a small increment and recalculate the local stresses on the elastic springs in each layer by solving eq. (81). Return to step 2.
Mechanical equilibrium at any applied stress is obtained when all unbroken springs carry stresses smaller than their strengths, which occurs on proceeding to
Fiber-Reinforced Composites
229
W
(c)
(2@@@@@ 4
FIG. 24. Load transfer around four in-plane breaks to surrounding in-plane fibers (broken fibers have unit applied load) for various load transl‘er parameters R. (a) R = 0.001 (HVD lirnir): (b) R = 1.0; (c) R = 10.0 (approaching global load sharing). After Zhou and Curtin (1995) with permission of Elsevier Science.
a
00000000 000000000
00000000000000000000 0000000000000 000000000000 (b) F I G . 25. Critical clusters of damage at onset of macroscopic failure instability and in the eventual
plane of composite separation. as simulated in the LLS model of Zhou and Curtin. Fihers broken in the plane are shown in black, fibers broken in a different plane hut slipping in the plane of failure are shown in gray. and unbroken tibers are shown in white. (a) Square lattice, m = 10: (b) square lattice, 111 = 5 ; (c) hexagonal lattice. rn = 10: (d) hexagonal lattice, In = 5 . The critical clusters are schematically identified hy the outlined regions.
Fiber-Reinforced Composites
23 1
3 0
step 6 above. The simulation is complete when, after step 3 , all springs in any one or more layers are broken or sliding. At this point, the material cannot support the applied load and must pull apart. Failure under LLS occurs by the formation of some localized cluster of fiber damage that causes stress concentrations that, in turn, drive continued fiber damage with no increase in applied stress. The stress at which failure occurs, and the configuration of fiber breaks, are statistically distributed. Hence, the material shows features similar to the failure of monolithic ceramics. Here we show some of these basic features and defer detailed analysis to the next section. For the remainder of this section, the load transfer studied is identical to the HVD limit (52 + 0 ) ;this is the most-local load sharing obtainable within the present model and so represents the case that differs most from the GLS model of Section IV.
Critical clusters Figure 25 shows the fiber damage in the eventual plane of failure for several different composites just at the point of failure, i.e., where a tiny increase in stress will cause unstable breaking of the remaining fibers. Most
232
W A. Curtin
of the damage is due to fibers failing above or below the plane of separation, and hence the “critical cluster” is a complex three-dimensional object. The critical cluster size increases with decreasing fiber Weibull modulus, and, in general, a fairly large number of fibers must be damaged locally to precipitate failure. There are several features that show a distinct contrast with earlier approximate models of composite failure, such as those of Zweben and Rosen (1970), Scop and Argon (1967a, b), and Batdorf (1982). First, the critical cluster sizes are much larger than a few broken fibers, particularly for smaller m . Second, the critical clusters are not necessarily compact, near-neighbor clusters of breaks but are more diffuse. Third, the breaks are not all coplanar so that the broken fibers retain some load-carrying capability in any one plane. Strength variations Performing many tensile tests on nominally identical materials (only differing in the specific strength values of (sn,m ) in the fibers) leads to a distribution of composite strengths. The cumulative probability of failure for , fiber stress af is shown in Figure 26 for volume n f L , , H n f ~ z ( ~ atj )applied various-sized composites. The data are plotted such that a Weibull distribution would appear as a straight line. Clearly, there is variability in the strength at fixed size. Size scaling Figure 26 also shows that the strength distribution depends on the physical volume of the composite (nf fibers, length L z ) . The characteristic strength (where In(- ln(1 - H ) ) = 0) decreases steadily with increasing size n f andor L,.
E. ANALYTIC MODELSAND WEAK-LINK SCALING Materials that fail by unstable propagation of localized damage must, for sufficiently large sizes, have strength distributions that obey weak-link scaling. That is, a large system can be formally considered as composed of a collection of independent subsystems coupled in series so that failure in the weakest subsystem causes failure across the entire system. So, the strength distributions at different ) the cumulative failsizes must be related as follows. Let &(of) and H r l f ( o fbe ure probability distributions (f.p.d.) at fiber stress C T ~ for systems of size n and n’, respectively. Considering the size n system as composed of n/n’ subsystems of size n’ n, the probability of survival at size n , 1 - & ( o f ) , is then simply the product of the probabilities of survival 1 - H , l j ( a f )for the n/n’ subsystems. The f.p.d. Hn/(aj) at size n can thus be related to that of size n’ by HfI(Uf= )
1 - (1 - Hflr(cTf))fl’n’.
233
2
0 h
-6
-8
FIG. 26. Cumulative probability of tiber bundle failure vs stress as obtained from the LLS simulation model. for ni = 5, length Li = 2&, and n = 100, 196,400, 576. and 900 fibers. Distribution is plotted such that a Weibull distribution appears as a straight line. Reproduced from Ibnabdeljalil and Curtin (1997a) with perniission of the Interncrtionnl Joitnial of Structures and Solids.
Ibnabdeljalil and Curtin (1997a) showed that the data in Figure 26 at different sizes are related by the above weak-link scaling relationship. Moreover, Ibnabdeljalil and Curtin (1997a) postulated that, at some small critical “link” size, the failure probability under LLS is identical to that obtained under GLS for the same size. In other words, failure in a large system following LLS is controlled by failure in a small subregion whose failure probability is the same as the failure probability in GLS. Thus, LLS and GLS are postulated to give identical statistics at some small size. Phoenix er al. (1997) had previously derived the strength probability distributions for finite-sized GLS systems and found that the link length was 0.46, in the axial direction. They also found that the failure probability distribution for a bundle of n fibers of length 0.46,. is a Gaussian distribution with cumulant @((of - p:)/y,:*), having mean pL,*and standard deviation y:*. Ibnabdeljalil and Curtin (1997a) thus argued that there exists a critical link of size n1 fibers of length 0.46, for which the LLS strength = @((of - p.f,)/y,*,*). This correspondence was ardistribution is Hn,0.46,.(~f)
W A. Curtin
234
gued physically by noting that the large amount of spatially distributed damage in the critical cluster just prior to failure tends to homogenize the local stress fields and so approaches the spatially constant value prevailing in GLS. With this correspondence at some (as-yet unknown) size n / , Ibnabedeljalil and Curtin claimed that, by use of eq. (90). the strength distribution in LLS at any size n f and length L z could be expressed as &,L,(q)
= 1 - (1 - @ ( ( O f - PL,T/)/Y,TI*))
,I/
L-/O4b,ll,
(91)
To show that their postulated correspondence between LLS and GLS held, and to find the critical size n / , Ibnabdeljalil and Curtin proceeded as follows. They used the simulation data, as in Figure 26, and the known Gaussian distribution for GLS to determine what size nl would make eq. (91) true for all of the simulation data at any fixed fiber Weibull modulus. Inverting eq. (91) to read as @((Of
- &)/Vn:*)
= 1 - (1
-fL/l
0 4&11//11/ L (Of>)
(92)
shows that one must find a value for n / such that the r.h.s. of eq. (92) becomes a cumulative Gaussian with mean j ~ ; ,and standard deviation y,?. Such a correspondence is highly nontrivial. However, the correspondence does exist: Figure 27 shows how the simulated data from Figure 26 for rn = 5 at n f = 576 and 900 and L , = 26, transform under eq. (92) to a Gaussian for n/ = 54, length 0.46,, with mean and standard deviation identical to those for the GLS case for nr = 54 fibers. Similar correspondences were obtained form = 10 with n / = 21 and m = 2 with n [ = 165. These results show that the failure of the large bundle of fibers in LLS is controlled entirely by the failure of a GLS bundle of size n l , length 0.46,, via weak-link scaling. Observation of the critical damage clusters formed (Figure 25) shows that these clusters (composed of broken, sliding, and unbroken fibers and roughly shown as the outlined regions) are very close in size to the n/ values obtained purely numerically through the GLSLLS. Based on the analysis of their simulation data, Ibnabdeljalil and Curtin proposed the empirical relationship
n/ = 403mF'
'*
(93)
to relate the fiber Weibull modulus m to the critical link size n / . The associated values of the mean j~r*;,and standard deviation y,T,* for each rn were then tabulated, and are shown in Table 5. The detailed origins of the very close relationship between LLS and GLS at size n / remain unknown. However, the result appears to be quite general. Recent work has shown that the strength distribution for hexagonal fiber arrangements is nearly identical to that for square lattices (Curtin and Takeda, 1998a), and the
Fiber-Reinforced Composites
235
i -1
.--:s\
f -2 tj
W
-3 4
-5
' ' ' I 0.62 0.64
"
"
0.66
"
"
0.68
"
'
1
1
0.70
,
,
1
,
0.72
CT F I G . 27. Small points: cumulative probahility of libcr bundle failure vs stress as ohtained from the LLS simulation model after weak-link scaling to critical siie n / = 53. length 0.48,. (parameters )ti = 5. L: = 28,. and ? I / = 576. and 900). Solid line: GLS strength distribution for i f / = 54. length 0.4&. Data are plotted so that a cuniulative Gaussian distribution appears as a straight line. Reproduced from Ihnabdeljalil and Cui-tin ( 1997a) with permis5ion of' the /ntenw/rouci/ J o w t i d of Stnrctirws ofid sol;t/.v.
critical sizes are quite similar (compare Figures 25). Simulations in the absence of pullout (dry bundles, or r + 0) show the same correspondence between GLS and LLS but the critical size nl is different from the case studied here (Curtin, 1998). The relationship between LLS and GLS embodied in eqs. (91) and (92) also leads to powerful analytic results. Specifically, using the analytic asymptotic behavior of the cumulative Gaussian @, Smith and Phoenix (1981) and Phoenix et al. (1997) have found accurate approximations to the weak-link scaling form of eq. (9 1) for large sizes. Of interest here is the result from Phoenix et al. ( 1 997), which shows that the strength distribution for a composite size n and length L , can be written as
W A. Curtin
236 with
+
ln(ln(n)) ln(4n) 4 ln(n) ii =
1.
(94b)
a* +5GGj, Y nI
and where n is the dimensionless composite size n = -.nfLz 0.48,nl
(944
In other words, the strength distribution for nf fibers of length Lz under LLS is accurately described by a Weibull distribution with characteristic strength 6; and Weibull modulus ii. Equations (94) also show that the scaling of strength depends predominantly on Along with the tabulated data in Table 5 for the nl, p:, , and y:* versus m , eqs. (94) provide an analytic formula for the tensile strength distribution for arbitrary fibers and composite sizes. Comparisons of this result with experiment are discussed in the next section.
a.
TABLE5 GLS MEANS T R E N G T H @,; A N D S T A N D A R D DEVIATION y;, , NORMALIZED B Y n,, FOR THE CRITICAL SIZEn / I N T H E L L s FAILURE PROBLEM, FOR V A R I O U S FIBER W E I B U L L MODULIm . THESENUMBERS AREUSED IN THE ANALYTIC THEORY OF EQS. (94)
2.0 3.0 4.0 4.5 5.0 5.5
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
I66 99 68 59 51 45 41 37 33 31 28 26 24 23 21
0.6869 0.6996 0.7256 0.7415 0.7442 0.7516 0.7595 0.7676 0.7760 0.7832 0.7909 0.7979 0.8047 0.8106 0.8173
0.023I 0.0257 0.0278 0.0285 0.0293 0.0300 0.0303 0.0307 0.03 10 0.0314 0.0319 0.0321 0.0322 0.0324 0.0327
Fiber-Reinforced Composites
237
F. COMPARISON TO EXPERIMENT The LLS model summarized by eqs. (94) and Table 5 and/or the numerical simulation results themselves are specifically for the most-local load-sharing case possible, corresponding to the HVD case. Hence, the predictions of these models are expected to best apply to materials with higher sliding resistance or shear yield strengths T . Comparisons to date have been made on Ti-MMCs, Al-MMCs, and graphite/epoxy PMCs. Inputs to the theory are few: As in the GLS case, we require fiber parameters 00, m , LO,r , and f ;interfacial r ; matrix yield stress o v ;and now for LLS the composite size (number of fibers n f and length L z ) .The reference parameters orand 6 , follow directly from eqs. (9). The critical link size nl is obtained from eq. (93), the dimensionless fiber bundle strength 5; is then calculated from eqs. (94) using the requisite data in Table 5, and the composite strength is calculated using eq. (3). Ibnabdeljalil and Curtin (1997a) applied the analytic model to a Ti-24Al11Nb MMC reinforced with SCS-6 S i c fibers carefully characterized by McKay et al. (1994). McKay et al. quoted values of a0 = 4577 MPa, m = 8.6 at LO = 12.7 mm, with r = 56 MPa, crl = 546 MPa, and f = 0.26. The coupons had dimensions of Lz = 25.4 111111,and cross-sectional area A = 8.26 mm2.The measured tensile strength was found to be 1251 f 93 MPa. Application of the LLS analytic model, with 6, = 6.29 mm, n, = 26, and nfL,/0.46,n1 = 53, leads to the predicted strength of 1338 f 71 MPa. The agreement is within 7%. The standard deviations are also very comparable, with the experiments broader in part due to an uncertainty of f0.028 in the fiber volume fraction. This indicates that the LLS model accounts reasonably well for the reliability of the strength as well as the average strength. Foster et al. (1998) have used the simulation model to predict the strengths of Ti-1 100 MMCs reinforced with Textron SCS-6 S i c fibers, a system studied in detail over a range of fiber volume fractions by Gundel and Wawner (1997). Gundel and Wawner determined the postprocessed fiber strengths 00,m at Lo = 12.7 mm by matrix dissolution and single-fiber tension tests. Fiber pushout tests determined the T values for the debonded, sliding interface. The matrix yield strength was a, = 950 MPa. The constitutive information and calculated 0, and 6, are shown in Table 6. Because of the small size of the composites (tape geometry of 4 x 26 fibers), Foster et al. (1998) performed direct numerical simulations of the entire %in. composite gauge length rather than using the analytic model of eqs. (94) valid for large composites. Foster et al. also developed methods to include free edges along the tape but found results nearly identical to those obtained
W A. Curtin
238
TABLF, 6 MATEKIAI. PARAMETERS FOR Ti- 1 100 REINFORCEI) W I T H
Sic F I H E R (CUNDEL S ANI) Sample
f'
00
B
0.15
3930
C
0.18 0.20 0.26 0.28 0.30 0.35
4310 2890 4270 4640 3330 4410
D F G H I
SCS-6
WAWNER, 1997)
111
10.I 13.9
5.8 12.3 12.6 6.8
r
a,.
4
188 190 190
5082 5191 4608 4856 5229 4280 5126
1.89 1.91 1.70 5.23 5.63 4.61 4.43
65 65 65 81
11.6
-
Stresscs in MPa. lengths in mni
using periodic boundary conditions. The measured and simulated LLS strengths are shown in Table 7, and very good agreement is found, typically within less than 10% over the full range of samples. The GLS results are also shown in Table 7 and are consistently higher in strength, but not by very much because the overall composite size (n f L,/0.46,nl FZ 30) is fairly small. Predictions of the Batdorf model described earlier (eq. (68)) are also shown, and fare slightly better than the LLS model. However, Foster et al. showed that the Batdorf model is actually an approximation to the LLS model so that improved agreement is due to somewhat fortuitous cancellations of various effects. They also showed that the Batdorf model predictions differ significantly from the LLS model for smaller Weibull moduli m 5 5 and do not show the correct trends for notched composites.
TABLE7 100 MMCS AS M E A S ~ J K E(CUNDEL D L L s MODEL(PERIODIC A N D FREE-EDGE). THE GLS MODEL,THE R U L E OF MIXTUKES, A N D THE BATDORF MODEL.ALSO S H O W N I S THE Fll3EK PULLOUT LENG'I'H( I N M M ) O B T A I N E D F K O M THE L L s SIMULATIONS. COMPOSITE TENSILE STRENGTHS ( I N MPA) FOK Ti- I
A N D WAWNEK, 1997) AN11 A S PREDICTEII H Y THE
Sample ~~
Measured ~
B C D F G H I
LLS (per.) ~
I252 1300 1230 1496 I724 I317 1716
1341 1470 I353 1630 1768
I535 I929
LLS (free) ~
1348 I474 I367 1643 1789 IS46 I938
GLS ~~
I392 1531 1414 I708 I856 1605 204 I
ROM
Baldorf
Pullout
1334 1464 1319
0.2 I 0.22 0.19
1631 1766
0.60
~
1384 1543 1326 1802 1972 1654 2151
1502 1900
0.65 0.55 0.5 I
Fiber-Reinforced Composites
239
2500 1 I
-
'E
2000
i
1500
u)
s! 1000
Q
500
0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
% Strain FIG. 28. Measured (symbols) and predicted (solid lines). by GLS and LLS models, stress-strain curves for Ti-I 100 MMC (Sample I). including thermal residual stress corrections. Adapted from Gundel and Wawner (1997) with permission of Elsevier Science.
Gundel and Wawner used the GLS model, and included thermal residual stresses, to predict the stress-strain behavior in this Ti-MMC and found excellent agreement up to failure, aside from the over prediction of the failure point. Their results for two particular samples are shown in Figure 28 along with the GLS and LLS predictions of Foster et al. The LLS model reduces both the tensile strength and the strain but otherwise the deformation follows the GLS prediction, as shown previously by Zhou and Curtin (1995). Foster er ul. also analyzed the predicted fiber pullout for these materials, and found values of about 200 p m for materials B-D (high r ) and about 550 p m for materials F-I (low T). These values are about one-half of those estimated from the GLS model. The measured pullout was about 200 p m for all the samples studied by Gundel and Wawner. The predicted pullout is thus in the right range for samples B-D, but is too large for samples F-I. The latter disagreement may be due to effects of dynamic fracture during final composite failure, which are not in the LLS model, or due to incorrect assumptions about the axial load transfer in the LLS model itself. This issue requires further study. Foster (1998) has recently simulated the behavior of Ti-6-4 MMCs reinforced with BP Sigma fibers, a system studied by Weber et al. (1996) and Ramamurty et al. (unpublished). These fibers have a radius r = 50 p m and a postprocessed strength of 1470 MPa with m = 5.3 at to = 1 m. The composite consisted of a 6 x 40 array of fibers with gauge length 3 in. and .f = 0.326, and other properties are c,,= 820 MPa and T = 130 MPa. The measured tensile strength was found
W A. Curtin
240
*
to be 1482 14 MPa. Using the LLS simulation model, with a,.= 4820 MPa and 6, = 1.85 mm from eqs. (9), the predicted tensile strength is 1589f 14 MPa. The predicted strength is again in good agreement with the measured value (7%)and the reliability (standard deviation) is predicted very well in this system. Ramamurty et al. (1997) have also used the LLS theory to predict the strengths of Al-MMCs reinforced with Nextel alumina fibers tested in tension, 4-point bending, and 3-point bending, and have compared these predictions to their experimental results. This work provides the first explicit test of the predicted size scaling of tensile strength. It is well known that, for a linearly elastic brittle material with a Weibull strength distribution characterized by Weibull modulus &, the tension, 4-point, and 3-point, strengths are related due to the volume scaling of the strength and the different effective volumes sampled in these test configurations. The effective sample volumes are in the ratio of
v , : ( 4(rn m ++21)’
v4) :
(
+
2(rn 1 1)’
v3)
3
(95)
when the overall nominal tested specimen volumes experiencing any amount of tension are V,, V4, and V3, for tension, 4-point, and 3-point bending, respectively. The tensile strengths are then in a ratio of the effective volumes of eq. (95) raised to the ( - l / g ) power, according to the Weibull scaling of strength with size. The , so that AI-MMCs are essentially elastic to failure, due to a very low matrix (T, eq. (95) and the associated strength ratios are expected to be valid. Ramamurty et al. tested the Al-MMC materials with physical volumes of V, = 1.08 x m3, V4 = 4.74 x m3, and V3 = 2.42 x lop7 m3 and obtained characteristic strengths of 1821 MPa, 2051 MPa, and 2171 MPa, respectively, clearly showing the volume scaling of strength in the experiments. Comparing the measured tension and 4-point bending strengths, the strength ratio following from eq. (95) indicates 6 = 5 I z!c 10, while comparing the tension and 3-point bending strengths indicates 61= 59 f 6. To apply the LLS theory to this system, Ramamurty et al. considered the 4-point and 3-point bend tests to be identical to tension tests at the effective volumes given in eq. (95) with & = 55 and calcuIated the tensile strengths at these effective volumes using the LLS theory. Such an approach is valid as long as the effective volumes contain a large number of fibers in both area and through-thickness of the test specimens, which is the case here. The Nextel 610 fiber properties are (TO = 2060 MPa, m = 9 at Lo = 1 m, r = 5.5 vm, E j = 380 GPa, and f = 0.652 0.022. The matrix 0.2% offset yield stress is = 100 MPa with E,,, = 60 GPa, and so with no debonding at the interface the appropriate r is estimated to be the matrix shear stress t, = CT, = 60 MPa. According to the results of Du and McMeeking (1993)
*
/a
5
2000
b v)
1800 1600
~
tI---r ~
1
(Figure 22), such materials should exhibit LLS approaching the HVD results. The reference parameters are then a,.= 4856 MPa, 6,. = 445 pm,with t t / = 24. The predicted composite characteristic strengths versus dimensionless composite size follow directly from eqs. (94) and are shown in Figure 29 along with the experimental results and the GLS results, which are volume independent. The scaling of the strength is predicted fairly well by the LLS model, and the predicted strength is only about 10%too large. In contrast, the GLS result is 35% larger at the largest volume (tension test). The LLS-predicted trend of strength with length is not perfect; the experimental strengths appear to decrease slightly faster than predicted. However, the general level of agreement does demonstrate the capability of the LLS model to capture the major size-scaling effects quantitatively with no adjustable parameters. Ramamurty et 41. (1997) also measured the strength distributions at fixed coinposite size and estimated much lower composite Weibull moduli of 15-20. This was attributable primarily to the variations in fiber volume fraction from sample to sample, variations that do not depend on sample volume and hence do not affect the volume scaling of the strength. This observation highlights the care that must be taken in using the strength distributions at fixed size to predict the strength versus volume via eq. ( 5 ) . As a final application of the LLS theory, we consider a graphitekpoxy PMC system studied by Madhukar and Drzal (1991). The fibers are surface-treated
242
W A. Curtin
AS-4 ( E f = 234 GPa) and analysis of s.f.c. tests on these same fibers (Figure 7b) leads directly to a, = 5783 MPa and 6, = 501 p m (Curtin and Takeda, 1998b). These fibers were incorporated into an Epon 828 matrix cured with mPDA that is very similar to the matrix used in the s.f.c. tests. Uniaxial tension tests were performed on specimens of length 152 mm, width 12.5 mm, and thickness 1.8 mm. The fiber volume fraction was measured as f = 0.677 but the composite Young’s modulus E , is much more consistent with an effective volume fraction of 0.59. To avoid this issue (see Curtin and Takeda, 1998b, for discussion), we consider the tensile strength divided by the composite modulus, nuts/ E, = auts/ f E 1 , which is independent of the fiber volume fraction. Using n[ = 20, p:, = 0.825, and y,T,* = 0.0328 for rn = 10.7, eqs. (94) and eq. (3) (with n) = 0 for the matrix) yields autS/E,= 0.0161,
(96)
~uts/E<= 0.0137.
(97)
while the experimental value is
The predicted strength is 18% larger than experiment. For reference, the GLS prediction is 0.0195, which is 43% larger than experiment and again demonstrates the importance of the volume scaling of the strength for larger specimen sizes when LLS is expected to be important. The difference between LLS theory and experiment is rather larger than found in all of the previous applications to MMC materials but, as a first application to PMCs, it is reasonable. Similar levels of agreement have been obtained in other graphite/epoxy systems (Curtin and Takeda, 1998b). One issue in PMCs that might contribute to the larger discrepancy between theory and experiment is the role of the loads carried by broken fibers. A constant t model may not be appropriate here or the deformations around the broken fiber may be more complex than envisioned by the simple models. The effect of the load-carrying capability of the broken fibers can be assessed using recent work (Curtin, 1998) in which, effectively, 5 = 0 over the length 0.46, with the axial stress abruptly increasing to the far-field value. In such a situation, the composite strength does decrease below the LLS model of eqs. (94) because the broken fibers do not carry load near the fiber breaks. For the graphite/epoxy system of Madhukar and Drzal(1991), the predicted strength becomes auts/E, = 0.0134,
(98)
which is slightly below the measured value. Therefore, the precise nature of the deformation around the fibedmatrix interface and ‘‘pullout” load of the broken fibers may become more important to obtain quantitative agreement in PMC systems.
Fiber-Reinforced Composites
243
A second issue involves the fiber strength distribution for carbon and graphite fibers. Tensile strength data on these fibers often show a deviation from the strength-length relationship of eqs. (4)-(6). Specifically, the strength distribution at fixed size and the scaling of average strength with changing length are not related simply through the Weibull modulus rn. Thus, some additional work is necessary to determine how to incorporate the different strength variations into an appropriate model.
G. SUMMARY A N D DISCUSSION We have reviewed the development of tensile strength models incorporating local load-sharing and fiber strength statistics. The model of Zhou and Curtin ( 1995) and Ibnabdeljalil and Curtin (1997a) has been reviewed in the greatest detail due to its computational tractability, inclusion of many physical features, flexible loadsharing rule, and accurate predictions in several different MMC and PMC systems. However, most of the composite strength models mentioned in Section V.C contain the same common physics and mechanics. Hence, even the most-recent models could, in principle, have been developed and brought to fruition almost 30 years ago except for the limitations of computational power. The reasonably good success of the recent models in applications to MMCs and, to a lesser extent, PMCs suggests that the major features of the failure have been captured in these models. More sophisticated and complete models are being developed at present. The work of Sastry and Phoenix (1993,1995) and Beyerlein and Phoenix ( 1996a) does not make assumptions regarding the three-dimensional nature of the load transfer and is the faithful extension of the HVD approach to three-dimensions and yielding matrices. McGlockton and McMeeking (unpublished) have extended a finiteelement model due to Cox et al. ( 1994) and Xu e f al. ( I 995), which bears some resemblance to the HVD model, to include interface slip. They find strengths that are closer to GLS in simulations on small composites because the load sharing depends on both r and the applied stress (see Figure 22). The detailed results that emerge from these models may show some important differences with the LLS model of Zhou and Curtin and Ibnabdeljalil and Curtin. However, these morecomplete models are also computationally demanding. Since in any numerical model many simulations must be performed on moderate-sized systems to capture the necessary statistics for performing weak-link scaling, comprehensive new results from these models may not be forthcoming in the near future. Another possibility for future directions exists if models such as those of McGlockton and McMeeking (unpublished) or Beyerlein and Phoenix (1 996a) can demonstrate that the planar decoupling approach used by Curtin and co-workers
244
W A. Currin
is reasonably accurate. Then, there exists the prospect of combining the very flexible LLS model described in detail here with load-sharing rules (possibly stressdependent) obtained from detailed finite-element models of load transfer around various arrays of fiber breaks, as in the works of Du and McMeeking (1993), Nedele and Wisnom (1994), Caliskan ( 1996), and others.
VI. Future Directions After 10 years of active research, considerable progress has been made in predicting the strength and deformation in tiber composites, including the very important aspect of stochastic fiber damage. The GLS model, stemming from the underlying s.f.c. problem, provides the basis for understanding the relevant strength and length scales in the composite and shows that many features of composite tensile failure (strength, stress-strain behavior, fiber pullout, fracture mirrors, work of fracture) are related and depend almost entirely on the key parameters mc, 6,., and m . Predictions of CMC properties using the GLS model are in very good agreement with available experimental data, and provide a first approximation to MMCs as well. The GLS model also provides a framework for trying to understand what physical features may exist in some materials that cause them not to agree with the theory. Furthermore, the theory provides a guide for material optimization and for the analysis of the tradeoffs expected in performanc- as constituent properties are varied. The success of this basic model has generated a host of extensions to related time-dependent failure problems, such as fatigue, creep, and fiber degradation. The extended GLS models then provide guidelines for quantitative prediction of composite lifetime under various modes of degradation. The LLS models recently developed have built upon the GLS model but incorporate the nonglobal nature of the load sharing expected to be prevalent in many real materials. The LLS models then give rise to finite reliability, size scaling of strength, and sensitivity to local stresses and/or damage. All of these issues are of considerable importance in taking composite materials from the laboratory coupon scale into realistic engineering applications. Preliminary applications of one LLS model to several MMC systems show the predictive capabilities of these models and provide some new concepts for dealing with the problem. However, the field of “LLS models” for quasistatic tensile strength is still evolving due to the complexity of dealing with both heterogeneous stresses and fiber strengths in large systems. Some important questions still to be resolved are as follows. What is the precise dependence of “load sharing” on the underlying material parameters 5 , fly,E,,,, E t , and j ’ , and on the applied stress and state of damage (multiple
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fiber breaks in three-dimensional, matrix cracking in CMCs)? What are the appropriate models for the interface shear stress, and, in pai-ticular, how do features of Coulomb friction and asperity-controlled friction enter into the problem? How effective are broken fibers in carrying load in the “slip regions” in the presence of multiple breaks, and does this vary from material to material? Do specific deformations around fiber breaks, such as the shear banding in Ti-MMCs observed by Majumdar et al. (1996a, b), drive local fiber failure in a manner different from that envisioned in all current LLS models’! Is it accurate to neglect the radial variations in the fiber axial stress by using the shear-lag models? Recent work by Weitsman and Zhu (1993) shows a stress concentration at the tibedmatrix interface that could drive fiber failure to occur at lower applied stresses than in models that neglect these variations. Will further careful applications of the existing LLS models to PMC systems lead to satisfactory agreement or will new features such those noted above be necessary for obtaining quantitative agreement? Can these models predict notch strengths and strengths in the presence of stress gradients in accord with experimental data? These and other issues are all important in bringing any LLS model to a state of widespread acceptance within the composites community. While questions remain about the LLS models in general, there are also many new problems to address in extending the existing LLS models (with their inherit assumptions) to other problems of importance in composite applications. For instance, what is the dependence of the tensile strength on initial fiber damage that might occur upon processing (Groves rt a/., 1994)? How does damage progress around a preexisting crack in the composite (see Ibnabdeljalil and Curtin, 1997b. for initial progress on this issue)? What is the predicted time- or cycledependent strength degradation due to creep and/or fatigue (see Beyerlein and Phoenix, 1998, for an extension of the three-dimensional HVD models to matrix creep)? What is the stress rupture life under conditions where the fiber strengths degrade in time? How do material strengths degrade when multiple degradation phenomena (fatigue, fiber degradation) are occurring simultaneously? These are issues that have been addressed within the GLS model but may show some significant changes under LLS; in particular, the composite lifetime will show some size dependence. There are also some basic issues within the LLS quasistatic models that warrant further study. In particular, how does the connection between LLS and GLS at some critical size F I ~arise? Can such a relationship be demonstrated fundamentally rather than simply as a correspondence found through analysis of simulation data? Why is the predicted tensile strength so weakly dependent on the fiber arrangement (Curtin and Takeda, 1998a)? How does the tensile strength prediction
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change for “random” arrangements of fibers (see Foster, 1998, for some preliminary results)? For engineering design of composite structures, the micromechanical results obtained within GLS and LLS models must be useable in larger-scale design codes based on continuum methods such as finite-element models. It is computationally impossible to directly simulate large composite structures at the scale of the individual fibers. Thus, coupled micromechanics/macromechanics methods must also be developed. One strategy is simply to use the nonlinear GLS constitutive relation for elements within an finite-element model (although the constitutive relations for shear deformations and transverse tension must also be incorporated). The failure strength of such elements must be dealt with very carefully, however. First, the element strengths must be assigned stochastically based on the actual element volume according to the LLS size scaling of strength. Second, the element sizes cannot be made too small. Elements smaller than the critical size nl and length 0.46, are not appropriate since damage in the fibers is closely correlated over this size scale. Mesh refinement around macroscopic stress concentrators such as holes and notches must therefore be done carefully. Third, after “failure” of an element, the propagation of the localized damage must also be incorporated; this may involve the introduction of interface elements to represent a crack bridged by broken fibers. In any case, it is simply inappropriate to assume either (i) component failure or (ii) element failure, which is insufficient in the treatment of the very localized stress transfer at the real crack tip in the material. Another promising strategy is to intimately combine component simulations at both micromechanical and macromechanical levels. Specifically, each individual finite element in a macro model could obtain state information from an associated fiber-level LLS model, although the LLS model size must be tractable. The LLS model could be invoked when stresses attain a preset minimum to avoid excessive computations for largely undamaged material. Around stress concentrators or induced damage, the LLS models could faithfully represent the entire element. Failure of an element would be obtained naturally in the underlying LLS simulation, along with the formation of a bridged crack or fiber damage zone. The propagation of this damage zone could then be followed by remeshing to larger sizes or by appropriate modification of the “applied stress” in the neighboring element LLS simulations to include the effect of the localized damage zones. Such a “direct” combined simulation approach might also be useful for the study of various time-dependent degradation problems as well. Finally, we return to the very general issue of failure in heterogeneous materials discussed in Section I. The fundamental progress made in fiber-reinforced
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composites is due to the fact that stresses from fiber breaks are transferred over the scale of the fibers themselves. In polycrystalline materials or particulatereinforced composites, such a “blunting” of the microscopic crack tip does not exist generally. Therefore, the direct transfer of results from the fiber-reinforced systems to other materials is not possible, even if the appropriate strength statistics for the polycrystalline grain boundaries or particle/matrix interfaces, etc. are supplied by some means. Careful analysis of mechanisms to inhibit crack propagation is required in these other systems, and such mechanisms are probably very material specific. Heterogeneity alone also may not increase toughness. Recent work on the propagation of a large crack through a material with discrete heterogeneous toughnesses, using the Green’s function technique, indicates little toughness improvement over a homogeneous material (Curtin, 1997). Therefore, crack bridging and microcracking, features that may be an indirect consequence of the heterogeneity, appear needed to provide enhanced toughness. In systems for which crack tip stress intensities can be neglected, such as heterogeneous materials with very weak interfaces failing in shear (Lawn et ul., 1994). materials with a very broad distribution of residual stress (Padture et ul., 1991), or systems with intrinsic mechanical decoupling at some length scale (possibly earthquake fault systems, see Ben-Zion and Rice, 1993), the damage evolution may be approachable using models similar to the LLS models. Interpretation of results must be approached carefully, though, since Jagota and Bennison ( 1995) have shown that unphysical or anomalous results, such as stress distributions dependent on the mesh shape, can arise in discrete models, especially when treating fracture. Nonetheless, several important concepts developed within the LLS model (see also Harlow and Phoenix, 1981) for fiber composites will be important in other heterogeneous systems. Specifically, the concept of weak-link scaling coupled with the existence of a critical damage size that controls the macroscopic failure may be a universal feature of failure in many heterogeneous materials. The resulting size scaling of the strength predicted by eqs. (94) may also be a general form for the failure strength scaling in heterogeneous materials. Thus, although considerable effort must be expended to make significant and practical progress in this broader area of failure in heterogeneous materials, the seeds of some key concepts exist within the work reported here on fiber composites.
Acknowledgments I gratefully thank the Air Force Office of Scientific Research (Grant F4962095-1-0158) and the National Science Foundation (Grant DMR-942083 1) for financial support since 1994, and BP Research for support in prior years. I also
M? A. Curtin thank my colleagues S. J. Zhou, M. Ibnabdeljalil, H. Scher, and S. L. Phoenix for their very valuable explicit and implicit contributions to much of the work reported herein; H. D. Wagner for introducing me to the s.f.c. problem; R. Thomson for guidance in the Green's function method; and A. G . Evans for particularly stimulating suggestions and conversations over the last 10 years.
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253
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Author Index
A Abraham. F. F.. 6, 9. 73 Acioli. P. H.. 10, 7.3 Ackland. G. J . . 6.5. 73 A d a m . J. B.. 62, 75 Agrawal. R. M., 40. 77 Ahn. B. K.. 169. 192. 193. 196. 204. 207. 2f8 2fY
Ahn, C. C.. 55. 75 Akgiin. I.. 37. 40. 73 Alexander. H.. 44. 73 Alp. E. E.. 55, 75 Anierasekera. E. A,. 82. 160 Ancker. F. H.. 173. 2 f Y Anderson. P.M.. 62. 75 Andersson. C. A.. 204. 250 Angelo. J . E.. 8. 7.1 Argon. A. S.. 9. 10. 20. 23-28. 73. 7.1. 76. 78, 21 8.232.252 Arnold, E., 9. 76 A m . E.. 89. 160 Ashby, M. F.. 82. 88. 89. 100. 102. 103, 119. 160,162 Athinson. H. V.. 122. 160
R Babtlska. I.. 29. 33. 72 Bacon. D. J.. 14, 7.? Ball. J . M.. 54, 73 Balluffi. R . W.. 65. 78 Bank-Sills. I-.. 106. 137. 160 Barnett. D. M.. 14. 72 Barnett. R. N., 9. 76
Baskes, M. I.. 8, 71. 73, 74 Bntdorf. S . B.. 219. 232. 248 Baunrgai-t. H.. 9. 76 Bell. J . P., 184. 185, 253 Beltr. G. E.. 16. 20. 22-24. 7.1. 76-78 Belytschko, T.. 60. 71 Ben-Zion. Y.. 247. 248 Bennison, S. J.. 247. 250. 251 Beyerle. D. S.. 204, 207, 248 Beyerlein. I. J.. 214-216. 220. 243. 245. 24K Bliadeshia. H. K. D. H.. 82. I61 Birnbauni. H. K.. 66. 7 f . 76 Bisson. C. L.. 8. 7-1 Bogdanoff. P.D.. 82, I61 Born. M . . 29, 7f Bourret. A,. 44, 46. 7 f Bowler. D. R.. 10. 75 Boyer. L. L.. 29. 39. 74 Brede, M.. 23. 7 f Brennan. J. J.. 204. 248 Broughton, J . 0.. 9, 73 Bulatov. V. V., 9. 10. 7.1 Bullough. R.. 20.29.46. 7.1. 76 Burke, J. E.. 122, 160 Burke, K.. 10. 75 Burton. B.. 89. 160
c Cai. H.. 247. 251 Caliskan. A. G., 216, 244. 2 f 8 Canova. G., 71. 75 Cao, H. C.. 203. 2fX Carlsson. A. E.. 8. 29. 7 f . 78
256
Author Index
Carter, W. C., 243,248,253 Celli, V., 29, 74 Chaikin, P. M., 55, 74 Chan, H., 247,251 Chantasiriwan, S . , 54, 74 Chen. I. W., 89, 160 Chiao, Y.-H., 26, 74 Chipot, M., 54, 74 Chou, M. Y., 38, 78 Chou, T.-W., 214,250 Chu, C., 52, 53, 74 Chuang, T.-J., 106, 137, 138, 141, 143, 160 Clark, W. A. T., 66, 74 Clarke, D. R., 26, 74 Cleveland, C. L., 9, 76 Coble. R. L., 153, 160 Cocks, A. C. F., 82, 88,96,97,99, 100, 102. 106-108, 110-116, 118, 120-122, 124, 129, 133, 135, 136, 141, 143, 146, 147, 1.50, 152-156, 160-162 Cottrell, A. H., 17, 74 Cox, B. N., 243.248.253 Craig, K. R., 122, 162 Curtin, W. A., 163, 169, 172, 174-176, 18&182, 184, 185, 187-189, 191-194, 196--199,201,202,204,206,207, 209-211,216,219,220,229,233-235, 237,239,242,243,245,247-250, 252, 253 D Dacorogna, B., 18.54, 74 Daniels, H. E., 249 Daw, M. S., 3,8,21,28. 75, 76 Desseaux. J., 44,46, 74 Deve, H. E., 240, 241,252 DiBenedetto, A. T., 173, 249 Dixon, J. M., 10, 78 Dodson, B. W., 9, 75 Doherty. R. D.. 28, 76, 122, 161 Draper, S. L., 237, 251 Drzal, L. T., 173, 241, 242, 251, 252 Du,Z.-Z.. 82, 116, 120, 161, 209, 210, 216, 217,239,240,244,249,252 Duesbery, M. S., 9, 20, 75, 76 Dutta, S. K., 82, 162 Duva, J. M., 201,202,249 Duxbury, P. M., 219, 251
E Eitan, A,, 173,252 Elbirli, B.. 173, 249 Elzey. D., 200, 202, 245, 249 Ercolessi. F., 62, 75 Ericksen, J. L., 30, 75 Erkoc, S., 9, 75 Emzerhof, M., 10, 75 Evans, A. G., 137, 140,162, 169, 170, 172, 189, 197,204.207-209, 216,248-250, 252
F Fabeny, B., 209-21 I, 249 Fang, H. E., 55, 76 Ferrante, J., 20, 24, 77 Finnis, M. W., 65, 73 Fischmeister, H. F., 28, 29, 76 Fivel, M., 71, 75 Fleck, N. A,, 157, 161.243,248 Flocken, J. W., 29, 75 Floro, J. A,, 125, 161 Foiles, S . M., 3, 21, 76 Fonseca, I., 54, 75 Foreman, A. J., 24, 75 Foster, G. C., 219.237, 239, 246,249 Fraser, W. A,, 173,249 Freund, L. B., 146, 148, 161 Fukuda, H., 182, 184.249 Fultz, B., 55. 75, 77
G Gallego, R., 9, 34, 3 8 4 0 , 44-17, 75 Gao, H., 141, 143, 146,161 Gehlen, P. C., 28, 75, 78 George, A,, 23, 26,28, 75 Gerberich, W. W., 62, 75, 79 Gemer, I., 6, 73 Ghaffarian, R., 219,248 Gill, S . P. A,, 116, 118, 120-122, 124, 129, 133, 135, 141, 143, 146, 147, 150, 160, 161 Glazier, J. A,, 161 Gong, X. Y., 55, 76 Goringe, C. M., 10, 75 Greenfield, M. R., 218,252 Groves, J. F.. 200,245,249 Grundland. A. M., 10, 78 Gucer, D. E., 218,220,249 Guiberteau, F., 247, 251 Gulino, R., 173,249
257
Author Index Gumbsch. P.. 28. 29, 76 Gundel, D. B.. 237-239.24') Guo. T., 28, 29. 79 Gurland. J.. 218, 220. 249 Giithoff. F.. 40. 75
H Haascn. P.. 12. 23, 74. 75 Halicioglu, T., 9. 75 Hanimond. C.. 5 , 75 Hardy. J. R.. 29. 39. 74. 75 Harlow. D. G.. 2 18, 2 19. 247. 250 Hayashi. R.. 203. 252 He. M. Y.. 169. 170, 172. 250 Hedgepeth. J. M.. 213-215.250 Hcinisch. Jr.. H. L., 29. 75 Henikei-, K. J.. 21. 75 Hennion, B.. 40. 75 Henstenburg, R. B.. 172, 173. 179, 180, 218. 250-252
Heredia. E E.. 204. 2 16. 250 Hernandez. E., 10, 75 Herzig. C., 40, 75. 7X Heywang. W., 82. 161 Hikaiiii. F.. 2 14. 250 Hillcrt, M.. 129. 133. 161 Hirsch, P. B.. 23, 75 Hirth. J. P.. 28. 71. 75. 78. 7Y Hoagland. R. G.. 8. 28. 71. 74, 75. 78 Hogeweg. B.. 182, 183. 252 Holzer, A.. 29. 76 Honeycombe. R. W. K.. 82. 161 Hornstra, J., 44, 46, 76 Howcs, M . J.. 82. 161 Hsia. K. J.. 23. 74. 76 Hsieh. C.. 223. 221, 250 Hsueh. C. H.. 137, 140.162 Huang, H., 62. 7Y Huang, K.. 29. 74 Hughes. T. J. R., 53. 56, 59. 76 Hui. C.-Y.. 174. 176. 178. 180, 181. 192. 193. 195-198. 201. 210, 220. 233. 235, 250. 251 Hutchinson. J. W., 169. 170. 172, 250
I Ibnabdeljalil, M., 176. 178, 180. 181, 192, 193. 195-199. 201, 210,219, 220, 233, 235,237.243.245.249-251
Iyengar. N.. 199. 210, 250
J
Jagota. A., 247, 2S0 James. R. D.. 52-54, 7.3, 74 Jansson. S . . 195, 204,251 .Jarmow D. C.. 203. 251 Johnson, H. H.. 103, 162 Johnson, R.. 73 Johnson. R. A,, 37. 77 Jonsdottir, F.. 146, 16/ Joos. B.. 9. 75
K Kagawa, K. 1.. 106, 137. 160 Kampe. S. L., 204,206.252 Kanninen, M. F,, 28, 75 Kanzaki. H.. 29. 76 Karpur, P., 251 Kawabe, H.. 209-21 I , 251 Kaxiras. E.. 9, 20, 75, 76 Kear, B. H.. 12. 76 Kcdward. K.. 195. 251 Kelly, A., 172. 251 Keinmochi. K., 203,252 b r a n s . R. J., 167. 25/ Khan, B.. 9. 76 Khantha, M.. 51. 76 Kindcrlehrer. D.. 54, 74 Knap, J.. 67. 76 Kogan, L.. 174.250 Kohlhoff, S., 28, 29, 76 Kohn, R. V.. 55. 76 Kroupa. F.. 29, 33. 73 Kuchercnko. S.. 100. 108, 110. 135. 136. 161, 162
Kuhn, L. T., 159, 161
I, Land, B. D. L.. 6. 73 Landman. U.. 9. 76 Lawn, B. R., 247, 251 Lcath. P. L., 219. 251 Lcckie. F. A., 204. 240, 241, 251, 252 Lee, T. C.. 66. 74, 76 Lilleodden. E. T., 62, 75 Lubcnsky, T. C., 55. 74 Lubliner. J., 41, 76 Luedtke, W. D.. 9, 76 Luo. K. K., 138, 161
25 8
Author Index M
MacKay, R., 237,251 Madhukar, M. S., 241,242,251 Majumdar, B. S., 173, 202,245, 251 Maradudin, A. A., 29, 76 Marsden, J. E., 53, 76 Marshall, D. B., 204,248 Martin, J. W., 28, 76 Martinez, A,, 9, 76 Mason, D. D., 210,251 Masri, P. M., 9, 28, 78 Masuda-Jindo, K., 223, 227, 252 Matan, N., 82, 161 Matikas, T., 251 McGlockton, M. A,, 243, 253 McMeeking, R. M., 116. 120, 152, 153, 157, 159,161, 162,208-210,216,217,240, 244.249 McNulty, J., 213, 251 Michel, D. J., 9, 75 Michot, G., 23,26,28, 75 Miller, R., 12, 22,23,60, 65. 67-70, 76-78 Mills, M. J., 3, 21, 76 Milstein, F., 52, 54, 5 5 , 74, 76 Miracle, D. B., 173, 245, 251 Miyake, T., 209-21 I, 251 Miyazawa, T., 182, 184, 249 Morgan, D. V., 82, 161 Moriarty, J. A., 9, 10, 79 Mosher, P., 204,216,250 Movchan, A. B., 20, 76 Miiller, S., 55. 76 Mullins, W. W., 135, 162 Mura, T., 40, 76 Musgrave, M. J. P., 35, 36, 39, 76
N Nabarro, F. R. N., 14,77 Nagel, L. J . , 5 5 , 77 Nam, E N., 82,160 Nandedkar, A. S.. 9.44. 77 Narayan, J., 9. 44, 77 Nedele, M. R., 216,244,251 Needleman, A., 8. 12. 15, 24, 74, 77 Nelson, I., 8.62, 74, 75 Netravali. A. N.. 173,251 Neumeister, J. M., 178. 192-195, 197, 251 Nix, W. D., 137, 162
0 Ochiai, S., 251 Oh, D. J., 37, 77 Ohno, N., 209-21 1,251 Ortiz. M., 9, 12, 20, 22-28, 34, 3 8 4 0 , 44-47, 53,55, 56, 58,60,62.65,67-70, 75-79 Ourmazd, A,, 12, 77
P Padture, N. P., 247,251 Paidar, V., 12, 77 Pamuk, H. 0.. 9.75 Pan, J., 96, 97, 99, 100, 106108, 110-1 14, 135, 136,161,162 Pandolfi, A,, 12, 77 Parhami. F., 152, 153, 162 Parhami, Z . , 154,162 Parthasarathy, T. A,, 167,251 Peierls, R. E., 14, 77 Peijs, T., 182, 183, 252 Peloschek, H. P., 162 Perdew, J. P., 10, 75 Perduijn, D. J., 162 Peters, P. W. M., 251 Petty, W., 40, 75 Pettifor, D. G., 8, 77 Petty, W., 40, 78 Pharr, G. M., 137.162 Phillips, R., 12, 22. 23, 28, 53, 55, 56, 58, 60, 62,65, 67-70, 76-78 Phoenix, S. L., 172-174, 176, 178-181. 192, 193, 195-199,201,210, 214-216, 218-221,233,235,243, 245,247, 248-252 Pitt, R. E., 218, 252 Pope, D., 16, 24, 79 Pope, D. P., 12.51, 76. 77 Prewo, K., 203, 251 Prewo, K. M., 202,203,252 Piischl, W., 23, 24, 77
R Raj. R., 102, 103, 162, 197, 216, 220, 245, 248.251 Ramamurty, U., 240,241,252 Ramesh, S., 9, 76 Rao, V., 173.252 Rasky, D. J.. 55. 76 Rathore, R. P. S., 40, 77 Reed, R. C., 82, 161 Renault, A,, 44, 46, 74
Author Index Reynaud, P.. 208,252 Rhee. M., 7 I , 79 Rhines, E N.. 122. 162 Riharsky. M. V.. 9, 76 Rice, J. R., 13, 16, 20, 23, 24, 66, 74, 77. 78, 106. 137. 138. 160,247.248 Ritter, A. M.. 237,251 Roberts, S. G., 23.26, 75. 77 Robertson, 1. M.. 40. 66, 74. 76. 77 Robertson. J. L.. 55, 77 Rodel. J.. 247. 251 Rodney. D.. 60, 78 Rose. J. H.. 20. 24, 77 Rosen. B. W., 172. 2 18,232.252.253 Rosenkrantz. M., 6, 73 Rouhy, D., 208.252 Ruhle. M., 204. 248 Runyan. J. L.. 247. 251
259
St John.C.. 23. 78 Stawovy, R. H., 204, 206,252 Steif. P. S., 197. 252 Stillinger. E H., 9, 38. 45. 78 Stoneham. A. M., 9, 28. 78 Storbkers. B., 157, I61 Stroud, A. H., 34, 78 Sturhahn, W.. 55, 75 Sucniasu, H.. 214. 252 Sun, B., 91. 162 Sun, Y..16, 24. 77, 78 Sun. Y. M.. 16. 24. 78 Su0.Z.. 82.94. 135. 152. 153. /62 Sutcu. M.. 189. 197, 252 Sutton, A. P.. 65. 78 Swanger. L. A,, 14, 73
T S Samuels. J., 23. 26, 75, 77 Sastry. A. M.. 214, 220, 221. 243. 252 Saxena. S. K., 40, 77 Sbaizero, 0.. 204. 248, 252 Scattergood. R. 0.. 14, 73 Schneider, D.. 6. 73 Schober. H. R.. 9. 28,40. 75, 78 Schock. C.. 23.24. 77 Schulte, K., 251 Schwartz. P.. 173, 251 Schwietert. H. R., 197.252 Scop, P. M., 2 18,232.252 Searle, A. A,. 97. 100, 106, 161 Sengupta, S., 35-37, 39, 77 Shen. Z.. 66. 74 Shenoy. V. B., 60, 65.68. 77. 78 Shewmon. P. C., 86.162 Shih. H. M., 103, 162 Siemers, P. A.. 237, 251 Siems. R.. 29. 76 Sigl, L. S., 252 Simmons. G., 39. 78 Sinclair, J. E., 28, 78 Sines. C.. 29. 75 Skovira, J.. 6, 73 Smith, I. R.. 20. 24, 77 Smith. R. L.. 176. 178. 180. 1x1. 192, 193. 195-198,201,218,235,250,252 Spearing. S. M., 204, 207. 216, 248. 250 Spooner. S., 55. 77 Spriggs, R. M.. 82, 162
Tabbara, M., 60, 74 Tadmor, E. B., 28. 53. 55. 56, 58. 60, 62. 65, 67-70. 76-78 Takahashi, J.. 203. 252 Takeda. N.. 182. 184. 185. 192, 193, 196, 204. 207,234,242.245.249 Tan, H., 2X,29. 79 Tan, T. Y., 44. 78 Taylor. R. L.. 56, 79 Tew'ary, V. K.. 29.46, 74, 78, 223. 227, 252 Thompson, C. V., 124. 125, 134, 161, 162 Thomson. R..29. 78,223,227,250,252 Thouless, M. D.. 137, 140, 162, 189, 197,252 Tichy. C., 65. 73 Toellnel; T. S., 55, 7.5 Torres, V. T. B., 9. 28. 78 Trampenau. J., 40. 75, 78 Truskinovsky, L., 16. 24, 78 Tsuda. H.. 203.252 Tsuji. T.. 71, 74 Tuszynski. J. A.. 10, 78 Tyson. W. R.. 172,251 V Van der Heuvel. P. W. J., 182. 183.252 van Dyke. P. J., 2 13-2 IS. 250 Van Siclen, C. Dew., 129, 162 Venakataraman, S., 62, 79 Verdier, M., 7 I , 75 Verrall. R. A,, 89. 100, 119. 160 Veyssibe, P.. 12. 78
Author Index
260
Vitasck, E.. 29.33. 73 Vitek. V.. 12, 16. 24. 28. 51. 65, 73. 76-7Y
W Wadley. H. N. G.. 200-202, 245.24') Wagner. H. D.. 173, 252 Wagoner. R. H.. 66. 74 Wang, H.. 39. 78 Wawncr. E E., 237-239, 24Y Weaire. D.. 161 Weher. C. H.. 210,239.252 Weher. T. A., 9. 38, 45. 78 Wei. S.. 38, 78 Weibull, W.. 169. 253 Wcitstnan. Y.. 168. 245. 253 Widom, B., 176.253 Willis. J . R., 20. 76 Wilsdorf, G. E, 12, 76 Wimolkiatisak. A. S.. 184. 185. 253 Winand. H . M. A., 82. 1151 Wise. C. E.. 66. 71 Wisnom. M. R.. 213,216.244.251. 253 Wolf, D.. 65, 78 Wright, A.E. 8, 74 Wu. B. X.. 169. 172.250 Wyrobek. J . T., 62. 75
X
xu, c.. 20. 23-28.
78. 7Y
Xu. J., 243. 253 xu. w., 9. 10. 741 Xue. L. A,. 89. 160
Y Yainaguchi. M., 16, 24. 7Y Y:ung, w..28, 29, 79 Yip. S.. 9, 10.65. 74. 78
Z Zanzotto, G., 52. 7Y Zhib. H. M.. 7 I. 7Y Zhen. Z. Y.. 66. 74 Zhou, S. J.. 29, 78. 163. 192-194. 199. 201, 202, 220. 229. 239. 243. 249, 253 Zhu. H.. 168. 245. 252 Zhu. J. Z., 60. 79 Zieliriski. P., 39, 79 Zielinski, W., 62, 7Y Zicnkiewicz. 0.C.. S6. 60. 7Y Zok. F. W.. 204,207-2 10. 2 13.239-24 1.248. 249,251,252 Zweben. C.. 218. 232. 253
Subject Index
A
Cohesive potential, 12 Cohesive surface, I 1 Cohesive zone. 1 I . 12 models, 1 I , 23 Compaction of ceramic components. 156 boundary diffusion, 157 contact radius, 157 number of contacts per unit area, 1.57 velocity field. 157 Composite properties tiber pulloul, 172 tensile strength. 172 work of fracture. 172 Composite strain. I89 Composite strength variations. 232 Continuity condition, 153 Continuity equation. 93. 107 Continuum deformation gradient, 53 Convolution theorem, 19.33,42 Coupled grain boundary diffusion and self-diffusion, I 14 migration and surface diffusion, 135 Crack evolution, 141 Crack growth, 107 Critical clusters of fiber damage. 231 Crystal cubic, 52 orthorhomhic. 52 Cumulative failure probability distribution. 232
Abnormal grain growth. 124 Aluminum crystal, 62. 66 Anisotropic linear elasticity, 39 Atomic coordinates, 4 Atomic How rate, 92 Axial stress on tiber. 168 on matrix. 168 Axisymmetric imperfections, 138
B Bessel functions. 148 Body-force field of lattice. 34 Bravais lattice, 29, 3 I , 55, 65 basis, 30, 52 dual basis vectors. 30 harmonic approximation, 3 I reciprocal basis, 30 Brillouin zone, 227 Brittle-to-ductile (B-D) transition, 23, 28 temperatures. 27 transition theory, 5 1 Burgers vector. 15. 16, 18, 24. 25,43. 47. 66, I38
C Capillarity stress. 155 Cauchy integral, 139 Cauchy-Born theory, 5 I. 54.55, 60 Characteristic relaxation time for shear creep in composite, 2 I0 Characteristic strain rate. 96 Chemical potential. 86, 87. 92, 93 Chromium crystal, 40 Coble creep. 100. I13
D Delaunay triangulation. 61 Diamond structure. 39 Diffusion coupled grain boundary and surface, 101 grain boundary, 96. 102, 1 15 26 I
262
Subject Index
interface, 87, 88 surface, 102, 107 Diffusive growth of intergranular crack, I36 Difusivity effective. 86, 87 grain boundary, 102 interface, 87 lattice. 86 surface, 102 Dirac delta function, 20 Discrete Fourier transform (DFT), 33,41 Discretization of crack surface profile. 139 Dislocation Burgers vector, 13 core, 9. 12, 22 cutoff radius%44 width, 18 dipole, 23 edge, 19,46,71,72, 138 energy, 20 glissile, 72 junctions, 71 kinking, 20 line, 12 Lomer, 7 I , 72 loop nucleation, 24 motion, 20 nucleation, 24-26, 28. 62, 63, 70, 138 screw, 40 Shockley partial pair. 65, 66, 72 straight, 13 Displacement field, 1 1, 33 Displacement hypothesis of Rice, 16, 24 Divergence theorem, 93 Dynamical matrix, 14, 36 lattice, 34
E Effective shear modulus, 48 Eigendistortions, 18 Eigenslips, I8 Elastic deformation of film, 146 Elastic mismatch strain, 146 Elastic moduli, 13, 32, 36, 37,39, 5 1 Elastic shear modulus. 17. 18 Elastic stiffness tensor, 147 Elastic stored energy, 146 Elastic strain energy, 146. 147 Elastically strained epitaxial thin film, 146
Embedded-atom method (EAM), 8,37, 53, 54 modified, 8 Energy constrained potential, 16 Energy dissipation rate, 94. I 1 1, 115, 155 per unit area, 88. 89 per unit volume, 86 Euler-Lagrange equation, 21 Evolution small surface perturbations, 148 microstructure, I 18, I25 Experimental identification of interplanar potential, 20
F Fatigue model for composites, 208 Fiber average spacing of breaks. I76 characteristic strength, 170 critical length. 172. 173, 178 critical strength, 172 critical stress, 173, 178 damage localization, 199 damage parameter, 201 failure probability, 169 failure strain, 192 flaw distribution, 169 fracture mirrors, 197 initial damage, 200, 201 maximum stress, 191 number of breaks, 190 number of flaws. 169, 200 pullout stress, 193 strength, 169 in siru, 197 strenglh/length relationship, 170 stress, 189 stress distribution. 190 stress-strain equation. 196 tensile strength, 20 I tensile strength equation, 196 tests, 170 Fibedmatrix interface. 170, 171 interfacial sliding stress, I70 Fick’s law, 86 Flux in grain, 92 through interface layer, 92 Flux-based variational principle, 94. 109. I I I . I I3 dimensionless form, 1 1 1
Subject Index Force constants, 32, 35, 36, 39 matrix, 37 models, 39 of silicon. 38 Fourier methods, 22. 23, 29. 31, 33 series, 148 transfomi, 19. 20, 30
G Gaussian distribution, 233, 234 Gaussian quadrature. 6 I Gibbs free energy, 91. 93. 95. 100. 102, 124, 136, 146. 154 Global load sharing (GLS) theory, 172. 185. 191. 198 comparison to experiment. 202-207 numerical simulation methods. 199 Gold crystal, 65 Grain boundary energy, 90 migration, I16 Grain structure of alumina. 99 Green’s function. 29. 46, 223, 224 dimensionless dipole. 225-228 method, 46 Griffith theory, 141 Growth of triple-point voids, 107
H Hamionic approximation, 39,30,45 energy of crystal in quadratic form. 32 Hillert’s grain growth law, 129
I Interactions k-body. 6 short-range, 6, 7 I three-body, 9 lwo-body, 6 Interface migration dual potential, 90 energy dissipation rate, 90 equation, 90 mobility. 89 velocity. 89
263
Interface reaction, 1 1 I index, 89 Interfacial deformation, 65 Interfacial sliding resistance, I73 Interlayer potential, 24 Interlayer tractions. 15 Interplanar shear displacement. 24 Inverse Fourier transform, 43 Iron-aluminum alloys, 40
K Kolmogorov-Johnson-Avrami-Mehl equation. 129
L Lagrange multipliers, 98 Lagrangian tangent stiffness tensor. 53 Lattice basis. 35 Green’s function method, 223 model face-centered cubic (FCC), 35 symmetry, 3 I Lobatto quadrature. 59 Local deformation gradient, 52, 53 Local load sharing (LLS) theory, 172 Batdorf model, 2 19, 238 comparison to experiment, 237-243 dimensionless composite size characteristic strength, 236 Hedgepeth-Van Dyke (HVD) model, 214-217 statistical models, 21 7-220 strength distribution. 234, 235 stress on fibers due to multiple breaks. 213 stress on fibers due to single break, 2 13 tensile strength distribution, 236 Zhou and Curtin (ZC) model. 220 Lomer dislocation, 44 eigenforces, 45 in silicon, 44 Lomer-Cottrell locks, 46. 71 Loopon activation energy, 48 eigendistortion. 49 FCC crystals, 48 hexagonal, 48,49 nucleation, 49
Subject Index potential energy, 48 shear stress. 48
N
M Macroscopic model of abnormal grain growth, I33 Macroscopic strain rate, 155 Matrix applied forces. 98 constraint, 98, I09 dimensionless stiffness, 147 driving force. 147 elastic displacements. 147 force. 106 grain velocities and fluxes, 98 normalized transfomiation strain, 147 velocity. 147 viscosity. 158 for grain boundary diffusion, 98 for singular surface diffusion, I07 of surface diffusion, 109 Miller indices. 5 Misfit energy, 15 stress. 138 Models Barenblatt-Dugdale, I I Hull-Rimmer void growth, 102. 136 Mixed continuum/atomistic, 28 Cohesive-zone, 1 1 Peierls-Nabarro, I I Mu1t i fi ber composites average pullout length, 187 distribution of pullout lengths, I87 failure strain, I94 fiber pullout, 187 stress on fibers, I86 work of fracture, 188 Mura’s theory, 40,44 displacement field. 40 eigendistortion field, 41 eigenforce field, 4 I energy of defective crystal, 4 I screw dislocation dipole, 42 displacement field, 43 dynamical matrix. 42 eigendistortion field, 43 eigenforces, 43 energy per unit length of dislocation, 43 force constants, 42
Nanoindentation. 62 simulations, 69 Nascent dislocation loops, 47 Newton-Raphson method, I 1 3 Nickel crystal, 67, 7 I Niobium crystal. 40 Normal grain growth. 116 Norton law. 89
0
Opening displacement at crack tip. 140 Orowan mechanism, 72
P Parabolic grain growth equation, 122 Parseval’s identity, 19, 33,42 Peach-Koehler force, 65 Peierls interplanar potential approximation, 12 Peierls potential, 28 Peierls solution, 17, 21 Peierls-Nabmo theory. I2 Phonon dispersion, 38, 40 Piola-Kirchhoff stress tensor, 53 Poisson’s ratio. 14, 18, 138 Potential energy of crystal, 16 Frenkel form, 17, 20 Frenkel interplanar, 2 I interface reaction, 88 interlayer, I5 interplanar, 17, 20 Lennard-Jones. 6 piecewise quadratic interplanar. 18 sinrering. 158 Principle of virtual work, 93
Q Quasicontinuum (QC) theory, 4. 55, 57, 65. 7 1 applications. 61 discretization error, 60 displacement field, 56 mesh adaption, 60 piecewise linear interpolation, 56 suniination rules, 58 triangulation, 56, 58
Subject Index R Rate of platins of material onto interfaces, 95 Rayleigh-Ritz analyses. 150 Rheological model, I 12 Rocksalt crystals. 39
265
Titanium crystal. 40 Total energy o f N atonis, 5 Total energy of solid. 5 I Total grain boundary energy. I 16
U S Schrfidinger equation. 5 , 10 Screw dislocation. 9. 10 dipoles, 41 Shear modulus. 14. 18 Single-tiher composite (s.1.c.) comparison to experiment. 182-1 85 Curtin‘s solution. 176 distribution of breaks. I76 i~d~lllellt length distrihution, I80 Monte Carlo simulation. I79 nonnaliied fragment distribution. I78 test. 173. 174, 185 Single migrating grain boundary. 135 Sintering process. IS 1 Slip distribution. 17 Specific entropy. 91 Stillinger-Weber potential. 9. 38. 45 three-body. 38 two-body. 38 Strain energy change rate. 147 density. S I , 53 Stress distribution, 139 along the grain boundary. 140 Stredchemical potential-based variational principle. 94 Surface diffusion coefficients. IS I Surface energy. 146 Sweep coiistant. 113 concept. 122
Ultimate tensile strength 01 composite, I68
V Variational boundary integral method. 24 Variational Functional. I 18. 123. 124. 135. 138, 140. 147 dimensionless form, I I7 Variational principle. 114. 150. 1.52, 153 Volumetric Ilux rate of material. 95 Volumetric growth rate. 103. 104 Volumetric strain rate. IS8
w Weibull cumulative failure probability distribution. 170 Weibull distribution. 177. 198. 232. 236 Weibull fibers. 178 number of breaks per unit length, 178 Weibull model. 169 Weibull modulus. 169. 173. 179. 184, 191. 198. 199.201,232.234,236, 238.240 relationship to critical link size. 234
Y Young’s modulus, 138 composite, I68 fibers, 168 matrix, I68
1’ Thermodynamic driving force, 89 Thei~niodynaniicforces. 100. I09
2
Zinc crystal, 40 Zirconium crystal. 10
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