Structure and Properties of Composites

1 Fibrous Composite Materials Anthony Kelly University of Surrey, Guildford, Surrey, U.K.

List of Symbols and Abbreviations 1.1 Composites in Materials Selection 1.2 Economic Background 1.3 Environmental Concerns 1.3.1 Recycling 1.4 Limits to the Mechanical Properties of Fibres 1.5 Control of Cracking in Brittle Laminates 1.6 Novel Forming Methods 1.6.1 Reconstructive Processing 1.6.2 Reactive Processing 1.6.3 In Situ Processing 1.6.4 Metal Matrix Composites and Spray Methods 1.7 Fibre Geometries 1.8 Failure under Multiaxial Stress 1.9 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

2 3 5 8 8 9 11 12 13 14 15 16 17 18 23

1 Fibrous Composite Materials

List of Symbols and Abbreviations b E F,G,H L9M9N V

thickness of 90 lamellae elastic modulus

y

T

work of fracture transverse ply failure stress angle Poisson ratio shear stress (frictional) shear stress

BMC CFRP CTE gdp GNP grp PPTA SMC LCP

bulk moulding compounds carbon fibre reinforced plastic coefficient of thermal expansion gross domestic product gross national product glass reinforced plastics polyparaphenylene terephthalamide sheet moulding compounds liquid-crystalline polymer

£

e V

o

constants area (volume) fraction

1.1 Composites in Materials Selection

1.1 Composites in Materials Selection Composite materials are combinations of materials put together to achieve a particular function. The combinations may be materials of the same class, e.g., two metals, or of a different class, e.g., a glass and a plastic. The generic types of composite are indicated in Fig. 1-1. The advantages to the engineer fall under three headings. First, composites enable a unique combination of properties to be achieved. For example, tungsten wire is very stiff (405 GPa) but very dense (19.3 Mg m~ 3 ). A combination of graphite fibre in epoxy resin is nearly as stiff (306 GPa) but with a density of only 1.5 Mgm" 3 . The carbon fibre itself is much stiffer than the tungsten; values up to 1000GPa at a density of 2.6 M g m " 3 are possible. Besides the attainment of unique combinations of properties, another advantage is that the property can be made to vary con-

tinuously over a range. If only a small number of simple materials are available the variation of a given property among those materials is digitised to the values the individual materials may show. Figure 1-2 shows the variation of coefficient of expansion with volume fraction for a composite consisting of aluminium containing silicon carbide particles. It is seen that matching the coefficient of thermal expansion with that of other materials is easily obtained. The similar non-digitisation of properties is important in, for example, acoustic wave devices and in many other matching situations, e.g., in prosthetic devices. A third important property is that composites can sometimes attain a value of a given physical property not attainable by either of the two components alone. Figure 1-3 shows thermal conductivity of various materials and it is quite clear there that composites can attain a lower thermal conductivity than that of others. This may be regarded as a rather special example since the combination is with air or vac-

METALS Fe,AI,Ni,Cu,Ti,Pbetc Strong,Stiff,Dense,Tough

POLYMERS

CERAMICS C.AI^.SiC.BA C, Hard, High Melting, Stiff, intermediate density

(CH 2 ) n ,(cONH (CH 2 ) 6 ) n etc

[-NH-0-CO-], easily formed, soft but fibres stiff and strong. Low density

GLASS

RUBBER elastomers high extensions

TOUGHENED. POLYSTYRENE

Figure 1-1. Composites are combinations of two or more materials.

inorganic polymeric metallic

1 Fibrous Composite Materials

Thermal Expansion Matching

20-i Brass 18-

J0%

Bronze

^ 1 2 %

Copper

^ 1 5 %

Stainless Steel ^ ^ 1 7 % 16-

^^ 25%

Monel 14-

\ ^

Nickel

X

Iron, Mild Steel, Beryllium

12-

29% 34% ,38%

Carbon Steel

?,

10-

Figure 1-2. Variation of coefficient of thermal expansion with volume fraction

0

10

20

40

30

of silicon carbide particles in aluminium. Ordinate values x 10 "^K" 1 .

Volume % Silicon Carbide in Aluminium Matrix

PSZ ZrO2

Mullite SiO2(q) ice

Si

TiO2

RBSC BeO

AIN SiC (C)BN Diamond C fibre

CERAMICS SiO 2 GLASS (B/Si/O) p-imide CORK

-•

; p-ester -epoxy-

MICROT^ERM

POLYMERS

Cu/p-imide

FIREBRICK

BALSA

glass / p°xy

e

CFRP ( J _ ) CFRP(II) epoxy epoxy COMPOSITES

STILL AIR

SiC/BeO

Tl Li Fe Pb

Na

Ni

CFRP(II)

Cu Au Ag

METALS THERMAL CONDUCTIVITY (Close to R.T.) I 102

10 1

1 ^ ^ I ^ I J

I 10

I 100

Figure 1-3. Thermal conductivity at room temperature of a variety of materials.

1000

1.2 Economic Background

uum as one of the components. However, it indicates the generally important message, which stimulates the imagination, that often empty space can be an important component of the properties of a solid. The second and third advantages of a composite from the point of view of the variation of thermomechanical properties may stimulate the suggestion of new applications. However, the real technological importance of composites is the putative advantage of the fibrous composite over other materials by producing very high strength, very high stiffness per unit weight. This is important in all modes of transportation and in moving bodies. The advantages of such materials though are only being slowly realised in engineering and this is because of the background economic situation which we now describe. I believe it is important to emphasise that this background economic situation is different today from that which has occurred in the previous history of mankind.

1.2 Economic Background Within all developed countries there is now and there will continue to be consistent and mounting public, professional, statutory and commercial pressure to reduce the negative impact of all mining and manufacturing processes and of consumer products on the quality of life. In a word, do more with less. At the same time there is demand for improved living standards, great comfort, ease of transportation and increased leisure. These pressures produce enormous opportunities for the use of performance materials (of which composite materials are the most important subclass) because these are stiffer, stronger and tougher per unit weight than conventional materials, re-

quire less energy to produce them and can, if appropriate measures are taken, be safely recycled. Their surfaces may be engineered, they can be made available as films or fibres or foams. They produce lighter, smaller, more efficient and comfortable machines and structures of all kinds. The tremendous opportunities for the use of performance materials arise against another background, which is one of decreasing importance of all materials as a proportion of gross domestic product (gdp) in all developed countries when this importance is measured as the weight of material consumed per capita as a proportion of the GNP per capita. This point is illustrated by the graphs in Fig. 1-4 for the U.S.A. The curve for the GNP per capita is expressed in both dollars for the year in question and corrected for inflation to 1985 prices. The trend is clear. Steel is the major constructional material. Tin is a material widely used in the electronics industry (30% of tin consumed in the US is in the form of solder), packaging and in the production of bearings and coatings. Rubber is a material very widely consumed in industry and in domestic situations. Synthetic rubber makes up more than 70% of the US consumption of rubber. The consumption of these three materials per head is decreasing while the GNP or gdp per head is rising. The total material input per unit of gdp for these materials is thus decreasing. On the other hand from the data in Fig. 1-5 for polyurethanes and for nylon and for polypropylene it may be said that the rate of growth of consumption can exceed the increase of gdp per head, which indicates expanding use and expanding market and hence the possibility of very profitable commerce. Data for instance for advanced composites show a greater increase in consumption, very much larger than that of the gdp per capita. On Fig. 1-5

1 Fibrous Composite Materials

GNP/Capita($x103)

Consumption/Capita

25

1500

1200

900

600

10

300

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Year Figure 1-4. U.S.A. consumption of various commodities 1950-1988.

GNP/Capita ($x103)

Consumption/Capita

10

1960

1965

1970

1975

1980 Year

Figure 1-5. U.S.A. consumption of various plastics.

1985

1990

1995

1.2 Economic Background

composite sales by a major composite company for the years 1985 to 1990 are added as the single dark line, scaled to sales in 1985, and with sales in succeeding years deflated using the GNP deflator for the U.S.A. For all materials then, with the exception of selected performance materials the contribution of materials to the gdp/capita is going down. The reasons for the decrease are (in the case of any particular material, e.g., synthetic rubber in the U.S.A.) waste minimisation, substitution, improvement of properties (do more with less) and static or only slowly growing markets for the artifacts in which the material is used. This last point is very important. World motor vehicle production shows a steady increase up to the early 1970s and thereafter a much slower rate of increase no larger than the growth of the world GDP. The collapse of total demand for merchant shipping is well known; contrast this with the continued increased production of domestic refrigerators. The historic demand for aircraft passenger seats, which I take as a crude measure of the total volume of materials to be used in a big added value engineering industry, from 1975 to 1990 and the prediction by Boeing (who holds more than 50% of the world market) up to 2005 shows that the total demand grew rapidly in the 1980s but; from 1990 on, the total demand of some 140 000 new seats per year is approximately static. The conclusions from these data are that, after nearly 50 years without a major destructive war the world market for machines and structures employing engineering materials generally, and which composite materials are penetrating, or hoping to penetrate, such as automobiles, aircraft, transportation equipment and housing, etc., are mature in their concept (virtually all passenger cars have four wheels) and

the numbers produced are growing only as the world gdp. The major exceptions to this trend that I detect are in artefacts concerned with communication of information (and entertainment) of all kinds, household appliances and many types of medical equipment and prosthetic devices. The consequence of these quasi-static markets worldwide provides both an opportunity for composite materials suppliers but also involves heavy risk. The opportunity clearly arises because the users of composite materials, e.g., automobile makers, see two prime advantages in introducing them, (a) doing more with less, and meeting environmental standards, and lower fuel consumption; (b) obtaining competitive advantage through improved performance, appearance, economy and comfort, and in the case of medical devices producing properties hitherto unattainable. The producers (again, for example, car makers) are, however, under intense worldwide competition with one another. Some may not survive. While struggling to do so, most do not perceive themselves as able to make the necessary capital investments to develop the full capability of performance materials. At the same time they are not willing to take the risk of not replacing their heavy and expensive capital equipment which uses (improved) conventional materials. Against this background, if I am right, the opportunities for performance materials are strategic. This is because the material quality of life ultimately depends on the availability of performance materials a very obvious fact but one which tends to get overlooked in the heat of the present debate - witness the added value in an artificial heart valve. Before proceeding further, I must emphasise that there are very few, if any, unique performance materials solutions to

8

1 Fibrous Composite Materials

a given problem. This is because, with important reservations in detail, the ultimate mechanical and some other common physical properties of metals, plastics and ceramics are nowadays shown to be very similar. This fact coupled with the idea of using a composite composed of two or many more materials increases the range of materials available to a designer as we saw in Sec. 1.1. The idea of a composite also carries with it the notion of fashioning the surface of any article and introducing additional materials on the surface so as to reduce wear, dispel electricity, improve colour, lustre, attractiveness, etc. It invites the idea of a multi-material solution to many design problems rather than choosing a single material. It follows that the preferred commercial solution will depend on history, processability and quality much more than on inherent properties of a particular material.

1.3 Environmental Concerns Two very important factors limiting market growth of any material are: (1) The customer's requirement to achieve more in terms of product performance with less material - do more with less. (2) (a) (b) (c) (d)

Waste minimisation by use of cleanest raw materials; recycling within the factory; extracting energy from waste; using waste to produce valuable products;

while at the same time reducing airborne emissions, waterborne emissions, solid residue and implicit environmental impact such as energy, feedstock, transport and aesthetic effects.

1.3.1 Recycling There are many legislative moves, national and local, to establish comprehensive programmes to encourage recycling and waste reduction. The mounting concern to be able to recycle the materials of a product may lead to a form of "leasing" of the material. Recycling occurs at three stages (i) in the factory; (ii) in a prompt industrial form, i.e., when identified by a fabricator who knows its provenance and its constitution, and (iii) when the artifact is junked. The last form of recycling involves collection either of a single product, e.g., newspaper or a platinum crucible, or more generally, where the material has been incorporated into an artifact, it involves identification, separation from other materials and transportation to a factory. In many cases the concept of artifacts of which the component materials may be recycled conflicts with two important design considerations influencing the use of composite materials. These are: (a) Part integration-elimination of joints, fixings, fittings. Large parts of complicated shape are made efficiently but become very difficult to dismantle. (b) Concepts of multimaterial solutions to engineering problems. These produce components composed of very different materials which are highly effective in use but must be separated for effective recycling. However, plastics scrap can be sorted and some impurities removed. The secondary market for clean regrind of thermoplastics is well established and clean regrind can command prices of 70% of that of virgin quality. All chemical companies are investigating the possibility of finding applications of this secondary material. The recy-

1.4 Limits to the Mechanical Properties of Fibres

cling of plastic fuel tanks of cars, for example, leads to little deterioration of property. However, there is, in general, a loss of property and a cascade principle must be followed, i.e., use of the material in progressively less demanding applications, e.g., bottle, wood substitutes, fuel, in that order. Recycling of organic materials cannot continue indefinitely and beyond a certain level there is no point from the ecological standpoint of expending energy in purifying contaminated plastics scrap and the final repose must be the incinerator or the rubbish dump. Cured thermosetting plastics scrap is usually treated as household refuse. Incineration does not raise fresh problems for sheet moulding compounds (SMC), bulk moulding compounds (BMC) or glass reinforced plastics (grp) polyester compounds. The ground material can be recycled as a filler where the glass fibres retain their length and used to reinforce other plastics or cement, for example. Recycling procedures for advanced composites are particularly worthwhile. For advanced carbon fibre reinforced thermosets fully one-third of purchased material may be lost during manufacture. Fabricators can clearly visualise cost savings if this material is recycled and without this are greatly encouraged to look for alternative materials. There are opportunities in my opinion for sophisticated high value added methods for identification of plastics, cleaning, dismantling of structures (e.g., by use of stress corrosion effects), chemical methods of separating fibre and resin in advanced composites and of disassociating multimaterials generally, including separation of metals and non-metals. Where performance materials are used in medical or agricultural applications or in packaging

of expensive items, then the question of disposal may offer opportunity for biodegradable polymers. The automobile industry, particularly in Germany, is taking recycling very seriously in collaboration with the major chemical companies. Studies embrace dismantling techniques, sorting techniques, cleaning and size reduction. The long-term targets are closed cycles of material with recycling quotas of 30% aimed at. This figure is comparable to the percentage achieved with metals.

1.4 Limits to the Mechanical Properties of Fibres The latest theoretical estimates of the possible stiffness and strength of fibres attainable at room temperature can be compared with experiment. Figure 1-6 compares experimental values (top) with theoretical values for the elastic modulus divided by the density at room temperature for a number of materials. The theory for the elastic modulus of a perfect specimen represents the latest values reviewed by Macmillan (1988) and Macmillan and Kelly (1992) and the density has been taken as the defect-free X-ray density at room temperature. The experimental values are the largest known to the present author and the values of the density are experimental ones. The values of density in both cases have been converted to specific gravity, taking the value for water to be exactly 1. It can be seen that there is still room for improvement in the stiffness of carbon fibres. The absence of an accurate theoretical value for thin layer platelets of metals, e.g., Cu:Ni, is noteworthy - in fact this high modulus is to my

10

1 Fibrous Composite Materials Kevlar 149 Cu:Ni 2 nm layers

4-3

-S13N4

SiC Ccore) C T-300

-Nylon

EXPERIMENT (fibrous materials)

FP PET|

Cu

fSiO 2 20

AI2O37 saffimax 40

30

50

60

(Wcore)

Polyethylene

•rSiC Nicalon

PBT

I P120 -fGPa)

150

70 80 90 100

300

200

400

500

THEORY AIN(0001)

Al

(100)

7 Silica

A,f

NaCU

S

T AI 2 O 3

30

')l

' Polyethyjene

7Si 3 ni 4

rB4C

I

Si 7

TiB 2


(111)

20

TiDy

-Nylon rBeO

40

50

60

70

80 90 100

125

jB

r Diamond SiC? ^

j Polyethylene j Graphite

^SiC

(GPa) 200

150

300

400

500

Figure 1-6. Experimental and theoretical values of elastic modulus divided by specific gravity. PBT: polybutyleneterephthalate, PET: polyethylene terephthalate), f: filament - long continuous thin rod of material, diameter variable - up to 200 um, w: whisker - material in the form of a filament of length about 1 mm and diameter up to a few micrometers.

mind not fully confirmed, although the increased strength of Cu:Ni nanolayers is (Cammarata etal., 1990). Figure 1-7 compares experimental (top) and theoretical (bottom) values of strength to density (specific gravity). The room for

improvement in the strength of silicon carbide is very noteworthy: the more certain value (no query mark) for polyethylene of 19 GPa exceeds the highest experimental value (not shown) of 6 GPa by a factor of about 3.

• ^ WJ rod

0-^L-Alvap.quench at 77° w

p. Asbestos crocidolite Si

EXPERIMENT rod

Celion

rKevlar149 0-33 «•

[

rNicalon

AI203w

CcOfef Cu

Cu w

[sic 1-5

2

3

rB 4 C w 4

5

6

7

8

9

10

15

20

30

40

50

60 70

(GPa)

Si din

THEORY

Nylon T Si

Diamond

Polyethylene?

Polyethylene

(100! MgO

Jl

15

rr

Graphite in oiooi >

Silica

T NaCI

dm

5

6

7

8 9 10

15

20

30

40

50

60 70 80

(GPa)

Figure 1-7. Experimental (room temperature) and theoretical values of the tensile strength divided by the experimental specific gravity at room temperature. For symbols see Fig. 1-6.

1.5 Control of Cracking in Brittle Laminates

11

1.5 Control of Cracking in Brittle Laminates

needed so that the following factors are controlled or their effects known to be eliminated:

Cracking in laminates designed to resist biaxial in-plane stress occurs in the plies transverse to the major principal stress when the laminate is strained (in uniaxial straining) to a strain comparable with the transverse failure strain of a single ply, see e.g., Bailey et al. (1979). It is very important in assessing the load bearing capability in the direction of the major principal stress and occurs in

(1) variations of microstructure of prepreg; (2) temperatures of cure and temperature below which the matrix does not flow and is effectively solid; (3) effects due to differences of thermal expansion coefficient between the matrix and the fibres and between plies of different orientation; (4) variation of transverse cracking strain of a single ply.

(1) uniaxial straining, (2) fatigue conditions under cyclic loading, (3) under temperature change and thermal cycling.

When all of these effects are sufficiently well controlled and/or eliminated then there is very good evidence that, all other things being held constant, the transverse ply cracking strain (stress) of a cross ply laminate is increased if the plies are made thinner, and this effect is important for thickness of plies of around 0.5 mm or less. There are two main explanations. The first is that the failure strain (stress) of brittle bodies is governed by stochastic processes so that thinner plies will be found, on average, to be stronger than thicker plies. The second explanation depends on the energetics of cracks in partially cracked media, where one component, longitudinal plies (or fibres, say), can sustain the full applied instantaneous load when the other transverse ply (or matrix) fails. The first explanation is not easily amenable to quantitative test or rather its proponents perhaps prefer not to subject it to such rigorous analysis. The second has been moderately well developed. What experiments there are satisfying conditions (1) to (4) above indubitably favour the second type of explanation (Kelly, 1988). The quantitative evaluation of the energy approach has been most fully developed by McCartney (1992). Subject to the assumption of a sliding interface between

It also occurs under bending and torsion and other more complicated stress states. The phenomenon has been most explored with thermosetting resin matrices, all of which shrink on curing and few of which are cured at the temperature of test. In addition, many experiments are carried out which use prepreg obtained from the manufacturer or other supplier which is used to build up plies of the laminate and cured by the experimenter who later investigates first ply and other failures. There is very good experimental evidence that nominally similar prepregs have very different microstructures, degrees of cure and even may be partially crystallised! Such effects are well recognised with the resins of higher temperature capability such as the various imides. However, in my opinion, their effects are not sufficiently well recognised with epoxies and polyesters. It may also be noted in passing that glass matrix composites show similar effects depending on the method of fabrication. In carrying out rational analysis and, more importantly, when predicting transverse cracking strains, therefore, care is

12

1 Fibrous Composite Materials

plies with that sliding governed by a frictional shear stress T, a lower limit to the transverse ply failure strain is given, on the simplest assumptions, by ZL c T

(1-1)

where £ c is the elastic modulus of the laminates, V1 and V2 the area (volume) fractions of the 90° and 0° lamella respectively, E± and E2 their moduli, a = Ex VJ(E2 V2) and b the thickness of the 90° lamella(e). The unknowns in this equation are % the shear stress, y the work of fracture and, most important, there is, in general, a lack of knowledge of the origin of the value of s, i.e., the effect of interlaminar stresses, due usually to differences of CTE between the various plies and between fibre and matrix in a given ply; sT above. Recently Pryce and Smith (1992) have gone some way to extracting experimentally the value of T and of the origin from which the strain is to be measured. This has not yet been done for a laminate but has been accomplished for a uniaxial fibre reinforced system. Two sets of specimens are prepared and one of these loaded continuously to failure to yield tangent values of modulus and of Poisson's ratio. Another set are loaded, unloaded and reloaded to progressively higher levels. For each loading and unloading cycle the residual strain (permanent set) is determined from the resultant hysteresis loop. As the specimen is stressed the matrix damage is in the form of an array of cracks spanning the width and thickness of the specimen; the density of cracks increases with increasing applied strain. In Pryce and Smith's experiments, in Nicalon fibre (SiC plus additives) in calcium aluminosilicate glass ceramic cracking commences at a strain of 0.08% and the stress-strain

curve becomes linear again at strains greater than 0.4%. Matrix cracking has saturated at this stage, which is confirmed by crack counting. On unloading from relatively low maximum stresses and reloading several times to the same stress a closed hysteresis loop was obtained, i.e., the residual strain reached a constant value. The crack density also remained constant during the loop. These two quantities were measured and also the maximum stress attained during that particular hysteresis loop. From these measurements values of T for sliding between fibres and matrix (not between plies of the laminate) and of the residual stress (strain) can be found, by assuming a constant sliding friction stress between fibres and matrix and calculating the stress-strain curve over the region where the crack density is increasing. An example is shown in Fig. 1-8. The method gives great promise for a quantitative evaluation of the parameter x in Eq. (1-1) by application of the same method to laminates. Perhaps for clarity one should emphasise finally that the value of T and of the thermal strains may be found if the stressstrain curve is analysed quantitatively over the range of strain where crack density is changing. For a given crack density in the transverse ply the longitudinal modulus, Poisson ratio and thermal expansion coefficient can, of course, be calculated (as McCartney and others have shown) for a given crack density without the need to know T (or the thermal strain).

1.6 Novel Forming Methods The recognition of the composite principle as a means of providing very strong, stiff materials of light weight and high tern-

1.6 Novel Forming Methods

13

OOU"

300«T 250CL

w co 200to T5 150 J Shear strength = 8 MPa <

100

j

/ /

50

Thermal stress (fibres) = -50 MPa

/

j

/ 0

0.1

0.2

0.3 Strain (%)

0.4

perature resistance coupled with high toughness has led to a number of quite new forming processes, not previously envisaged for composites, plastics or ceramics. These can be considered under the headings of: • reconstructive processing, where the desired material is reassembled, e.g., inorganic foams; • in situ processing, where a fibre is grown or aligned within a matrix, e.g., directionally solidified metals and some forms of co-polymerised polymers; • reactive processing, in which one material is totally transformed into another of the required shape, e.g., assisted oxidation. 1.6.1 Reconstructive Processing

Reconstructive Processing - a term introduced by Ray (1978) - describes processes whereby a naturally occurring substance, a mineral, a rock or a tree, is dissected by chemical means - perhaps also with some mechanical force - and then reassembled into the required shape. Asbestos has been processed in this way for many years and

0.5

0.6

Figure 1-8. Continuous stress strain curve prediction (solid line) compared with the data (dots) from Pryce and Smith (1992).

paper is made in this way with the introduction of fillers such as chalk or clay into the mat of cellulose fibres produced from trees after the elimination of the lignin. A similar type of process is used to make reproducible sheets of graphite crystals consisting of thin packets of graphite. Graphite crystals are decomposed into thin lamellae by treatment with a strong acid, so as to introduce water between the layer planes, and then heated to boil the water and expand the graphite, which is subsequently sheared into sheets and reassembled. The result is a superb sealing gasket. Similar very clever methods have been used to expand vermiculite by judicious ion exchange to separate the layer planes by chemical swelling. The chemically swollen material is then mechanically delaminated in a high shear mixer in water. Subsequent removal of the water leads to an extremely adherent film. A foam can also be made at this stage by introducing air into a suspension with maximum solids content. In reconstructive processing the dispersing fluid or gas is introduced in order to disperse the material and put it in the required form and is then removed. Closely

14

1 Fibrous Composite Materials

related to this of course are processes whereby the dispersing fluid, which has the property of penetrating between particles of the desired solid and of causing them to flow adequately, is itself used as the matrix of the composite. Synthetic granite, which consists of 80-90 vol.% of granite in epoxy resin, forms a castable composite which may be used for machine tool beds. Similarly very highly filled polymers consisting of 55 vol.% of sand in methylmethacrylate fluid (72% by mass) is very easily cast if attention is paid to the range of particle sizes of the sand and the rheology of the polymer coupling agent. The material called Asterite is a fine substitute for kitchen and sanitary ware ceramics. The important physical questions here are the rheology of the fluid and the surface tension relations between the fluid and solid. The use of a particular range of particle sizes in specific volume fractions is also of enormous utility in assisting rapid flow and some of the "lore" has recently been subject to empirical analysis (Cheng et al, 1990). There are very intriguing hints in the patent literature to easily shearable mixtures of solids of high packing fraction (>80%) provided specific volume fractions of definite size ratios are used, see e.g., Ritter (1971). The use of a polymer to fashion material to shape and its subsequent removal is the basis of much processing of ceramics. A good example is found in a recent paper by Wright etal. (1990), who have made thin springs from ceramics, e.g., A12O3 and YBa 2 Cu 3 O :c . The suspension consists of some 86-88% (by weight) of ceramic with a mixture of various phthalates with stearic acid and polystyrene and sometimes other additives. This is subject to high shear mixing. The softening point may be controlled by the composition as well as the volatility of the diluent so as to effect removal of the

binder. The physics needed concerns not only the particle sizes of the powder of ceramic but the influence of additives on softening point and volatility. Attention to the range of particle sizes is necessary so as to avoid shrinkage (maximum volume fraction) and to maintain the ability to extrude (minimum volume fraction). 1.6.2 Reactive Processing

It is the difficulty of plastically deforming or even of inducing change of shape of the hard high melting point ceramic materials which otherwise have many desirable properties that leads, I believe, to what I call reactive processing, in which one material is totally transformed into another of the required properties and sometimes shaped at the same time. A process which has attracted a good deal of attention is illustrated in Figs. 1-9 a and l-9b - see Urquhart (1991) - which is a novel technique for fabricating ceramics or composites with a ceramic matrix by the directed oxidation of a molten metal either to produce the ceramic alone or through a mass of filler material if a ceramic matrix composite is required. The figure illustrates the outward growth of a ceramic-metal reaction product from a molten metal exposed to a vapour phase oxidant. Additives will be made to the metal to assist oxidation, e.g., Si, Ge, Sn and P plus a little Mg in aluminium. A wide range of composites comprising both particulate and fibrous filler materials have been demonstrated, e.g., A12O3/A1, A1N/A1, TiN/Ti and ZrN/Zr. The putative attractive features of the technology, in comparison with conventional ceramic technology, include the ease of making a ceramic matrix composite directly at low temperature, the elimination of shrinkage, the potential to make large ceramic bodies and projected low

1.6 Novel Forming Methods

15

Filler With Reaction Product Matrix

Refractory Container

(a)

Refractory Container

(b)

cost. An obvious problem is the final elimination of the metal. Of course the reaction need not be with a material extraneous to the composite (in the above case, oxygen). Tani and Wada (1990 a) describe a process which they call internal synthesis for preparing (particulate) ceramic composites such as SiC reinforced with internally synthesized TiB 2 . The process gets over the difficulty that TiB2 is not usually available in small enough particles sizes and furthermore reacts with water. In one method SiC powder mixed with TiN and amorphous boron was hot pressed at above 2000 °C in an argon atmosphere. The molar content of B is more than twice that of TiN. During hot processing the reaction TiN + 2B -> TiB 2 + Y2N2 occurred at between 1100°C and 1700°C. The excess boron aids the densification (sintering) of the SiC so that the process provides hot pressed SiC with small TiB2 particles dispersed within it. The necessity to eliminate a gas is, of course, a disadvantage, if hot pressing has to be used and a proposed process shows promise in which the TiB 2 is synthesised by a pressureless sintering reaction involving SiC, TiO 2 , B4C and car-

Figure 1-9. (a) Schematic of formation of a ceramic-metal reaction product from a molten metal and vapour phase oxidant. (b) Schematic of (a) occurring through a mass of filler material (possibly fibres).

bon. Again excess B4C and C are needed as sintering aids for the SiC (Tani and Wada, 1990b). Of course reaction injection moulding to make reinforced plastics is a very wellknown process. 1.6.3 In Situ Processing Most types of strengthening and stiffening methods used in metal and ceramic processing consist of the introduction of an extraneous phase, for example to block dislocation motion. The same is true of polymers par excellence, where a filler stiffens and, of course, in a composite, reinforces as well. In situ processing to produce the reinforcement directly within the matrix has been accomplished in the metal and ceramic case by directional solidification of eutectics, where the temperature gradient at the solid liquid interface and the rate of growth are both closely controlled so as to produce a rod or lamellar eutectic of the desired orientation. The process is well described by McLean (1983). The disadvantages of this elegant conception is that the volume fraction of reinforcing phase is controlled by the equilibrium phase dia-

16

1 Fibrous Composite Materials

gram and although the addition of other elements may vary this a little, a large change is impossible. Other problems needing to be attacked are that the processing is very sensitive to the presence of trace elements whereas a viable commercial method should not require super purity of the ingredients. The other problem - one where a better understanding of the physical limitation of the process would be useful - is that the current rates of production possible with eutectic alloys is very slow relative to directionally solidified and single crystal superalloys. In situ processing also naturally suggests itself as a means of introducing reinforcement directly into a polymer. Takanyagi etal. (1980) describe the production of a thermoplastic (nylon 6 or 66) containing fibres of polyparaphenylene terephthalamide (PPTA) directly, essentially by polymerising the nylon and then extruding the sulphuric acid solution of the reaction product into a large amount of water and ethanol. The molecular mass and the aspect ratio of the rigid rod component (similar to some form of Kevlar) has of course a marked effect on the mechanical properties. As with metallic eutectics, processing speeds would need to be greatly increased if the method were to show real commercial possibilities. Prevorsek and Mills (1989) discuss other possibilities and point out why melt processing is not possible in this case - see also Paulokowsky et al. (1991). The problems to be solved are related of course to processing of rigid rod polymers which form liquid crystals - lyotropic in this case. The high speed processing of such crystals to produce high strength fibres such as Dupont's or Akzo's aramid fibres is well known. This question also naturally arises of the use of liquid crystals to assist the process-

ing of more conventional polymers. These are the thermotropic liquid crystals which are used to produce moulded parts with a nice surface finish. Tupperware made of Amoco's Xydar is a good example and others are Vectra LCP of Celanese. So-called structural liquid crystal polymers are thermoplastic polyesters or polyamides of aromatic acids, amines, or hydroxyl compounds. They retain some order in the molten state. On solidification they tend to be more highly ordered than other polymers and so have superior properties. Sometimes these can be deliberately increased by incorporation of rigid rod aromatic molecules. The advantage is that they show much lower viscosities than ordinary thermoplastics at similar molecular weights, though with rather high processing temperatures: 340-370 °C. Of course such polymers can be filled with up to 50% of glass or talc or other filler. LCPs compete with the higher temperature resins for use in electrical components, where they must remain stable at soldering temperature. There are many interesting problems in the rheology of such materials. See for example Doraiswamy and Metzner (1986). 1.6.4 Metal Matrix Composites and Spray Methods It is the ease of casting processes that seems to be the key to the commercial viability of metal composites. The advantages in specific strength and stiffness of the material incorporating ceramic particles or fibres over the unreinforced metal are well known. The overriding considerations to achieve commercial success are safety, speed (hence low cost) and subsequent processability of the composite by conventional means. Continuous fibres are too expensive, whiskers are toxic and so the

1.7 Fibre Geometries

reinforcement preferred is a flake form. Consideration must then be given to the size and shape and the method of introduction. Particles about 10-50 jim in diameter and with aspect ratio 1 -10 are considered. The possible methods of introduction are powder blending, mechanical alloying, melt stirring, compocasting and spray codeposition. The third and fifth of these are the ones chosen by the major aluminium producers. In melt stirring, the particles (up to 20% volume fraction) are stirred into the molten material under vacuum by a high shear mixer (at ~ 600 °C!); the problems are clearly avoidance of oxidation and the presence of water. Cospray involves clever design of the atomiser of the aluminium alloys into drops - mainly between solidus and liquidus when the substrate is struck - and 15 jum diameter particles of SiC (say) are introduced and the aluminium droplets are maintained at 100 jim size for safety reasons. A nice set of physical problems arise concerning the process of extracting heat from the aluminium - can one alter the latent heat of melting? - and again the geometry of the relative particle shapes and sizes to obtain the various packings. The material builds up on the collector almost without voids, although a subsequent extrusion is necessary in order to eliminate all porosity. Without the introduction of a ceramic filler or reinforcer, the method has great advantage for conventional powder processing since the powder (spray) is produced and deposited with very little porosity in one operation. Most processes are developments of the so-called Osprey method invented about 1977. The atomisation produces particles with a density in the spray of that of a heavy gas (a few grams per litre) moving very rapidly at 50100ms" 1 , i.e., at gas kinetic velocity. The process physics is fascinating, involving

17

measurement of the particle size, speeds, number per unit volume, state (i.e., solid, liquid, mushy) and the proportions of these and problems of course of the extraction of heat and the effects of impact.

1.7 Fibre Geometries Fibre composites require application in shapes other than thin sheets stressed in their plane so as to overcome poor damage tolerance and delamination resistance. The need to consider three-dimensional arrays of fibres arises and interest is renewed in the physics of, and computer control of weaving and braiding processes. Some surprisingly simple geometrical arrays do not appear to have been fully explored in the literature open to composite specialists. Questions arise as to possible maximum density of packing of fibre bundles of various cross sections. If straight fibres are arranged along three orthogonal directions, the maximum packing fraction attainable is 0.75 (or 3A) for fibres of square cross-section of 3 TI/16 = 0.589 for fibres of circular cross-section. Such a structure yields easily in shear parallel to any of the (orthogonal) axes and in any direction on these planes. It is well known to structural engineers that the rigid frame most economical of material to resist a given set of forces is tetrahedral (Michell, 1904). An array of uniform fibres aligned along the direction normal to the faces of a regular tetrahedron has a surprisingly high packing fraction of 0.5 (or Vi) provided fibres of triangular section are provided. Such an array resists all tensile stresses and most shears except for those in highly specific directions on specific planes. The directions turn out to be the edges of the tetrahedron defining the directions of the fibres. Fibres arranged solely along these directions form

18

1 Fibrous Composite Materials

a dodecahedral arrangement with the fibres running normal to the faces of a rhombic dodecahedron. This dodecahedral arrangement has fibres running in six directions in space and among these six directions no more than three lie in any one plane. An array in these directions is such that any general strain can be resisted by the need to extend at least one set of fibres and there is no easy direction of shear or of extension. This would be called in weaving parlance a 6-D arrangement. Questions then arise such as: (i) What is the densest array of fibres possible in this 6-D arrangement? The answer is 1/3. (ii) What is the cross section of fibre bundle necessary to achieve this packing fraction? Answer: rhombic. (iii) What is the simplest technique of weaving or braiding that can attain this structure? (iv) Can this technique be followed by a machine? Studies of this type are important because it may not be, for instance, necessary to resist a general strain by elongation of one set of fibres. A lack of resistance to shear in a very small number of clearly defined directions may not be of great importance. Then it would follow that perhaps the tetrahedral arrangement of fibres, where a packing fraction of lA in a 4-D arrangement can be attained, would be suitable in a particular application. For all these arrangements the tensile and shear moduli must be estimated without the complication of fibre-fibre interaction. In all of these considerations there is a developing need to be able to display in two dimensions not only the directions in which the fibres are to run but also the cross-sectional form of the fibre. Of course,

a stereogram will represent in two dimensions directions arranged in three dimensions in space, and this representation is what is called angle true. Parkhouse and I find that the stereogram is of course fine for directions but is not good for finding the gaps between the arrays of fibres running in the various directions.

1.8 Failure under Multiaxial Stress As high performance composites go into service in critical situations, there is being recognised an increasing need for soundly based criteria for failure under multiaxial stresses. Physically satisfying theories are of two types only. A maximum stress criterion where it is envisaged that there is a certain upper limit to the force (identified as producing either a shear or a tension) that a solid can withstand. The second is based on the idea of a maximum failure strain, whereby it is envisaged that the relative displacement of the component parts of a solid cannot exceed a certain amount if the cohesion of the solid is to remain. In order to clarify the present situation it may be appropriate to review quickly the classic case of yield criteria in metals. The Tresca yield criterion states that yield will occur (not fracture) when the magnitude of the maximum shearing stress has a value (
(1"2)

where o0 is a constant of the material. This gives equal yield stresses in tension and compression, which is approximately true in metals [but cf. Spitzig et al. (1975)] though it is far from true for fracture. It is

1.8 Failure under Multiaxial Stress

also assumed, clearly, that the mean normal stress does not affect the process, which is approximately true in metals but far from true in plastics (Bowden, 1973). In order to use Eq. (1-2) one must find the principal stresses and subtract the smallest of these from the largest. This can be done but makes the use of Eq. (1-2) unwieldy. Clearly the choice of a criterion should be invariant with respect to change of axes and Tresca's criterion obeys this prescription. Moreover the invariance demanded by physical reasoning suggests that one should formulate a yield criterion in terms of the invariants of the deviatoric stress tensor Stj = atj — lA ou where °n = (ai + G2 + °^ the first invariant. The second most simple invariant is {S2 + Si + Sf). The von Mises yield criterion takes this invariant to be equal to a constant, namely 2<J O /3, which on substituting for Sl9S29S3 and multiplying out gives (1 _ 3) (G2 -
Transverse Tension

c

let

(ji,(72)0*3

19 be the

stresses in given principal directions of the stress tensor. The criterion for yield will be expressed as r \Gi, G2 , G3 , GQ) — U

^1-^+j

which is the equation to a yield surface. Consider the intersection of this surface with the plane G3 = 0. Tresca's criterion requires that the difference between the greatest and least principal stresses be a constant G0. If GX > G2 > 0, this requires GX = G0, giving the line AB in Fig. 1-10 because along this line G1 — G3 = G0 since G3 = 0, irrespective of the value of G2 . Shear would occur on a plane intersecting the G1,G2 plane along the line AB and inclined at 45° to that plane. If G2> G1>0 the criterion requires G2 = G0, giving the line BC in Fig. 1-10. In the quadrant where G1 and G2 are both positive they produce opposing shear stresses on any plane inclined to the GUG2 axes. Where GX > 0 > G2 the condition becomes

where 6 is the angle in the GXG2 plane between the normal to a plane and the GX axis. Since sin 2 6 has its maximum value of 1

Biaxial Tension

B Longitudinal Tension (Shear on 1,3 Plane)

//A

Longitudinal Compression

/ / / ^

»

Pure Shear On 1,2 Plane

F Biaxial Compression

Figure 1-10. Section of yield surface in 1,2 plane with <J3 = 0.

20

1 Fibrous Composite Materials

when 9 = 45°, yield cannot occur for any representative stress point situated closer to the origin than the line AF. This intersection of the yield surface with the ol,a2 plane is then the irregular hexagon ABCDEF The von Mises criterion for yield is, with o3 = 0, 2

I

2

2

(\

f\

and gives in section the ellipse in Fig. 1-10. For what comes later in dealing with fibrous composites, designate direction 1 as that of the fibres - the longitudinal direction - and label the failure modes in Fig. 1-10. In three dimensions all points on the line <jl = a2 = o3 have zero stress deviation. Since the yield criterion involves only the principal stress deviations the yield surface must be a cylinder in stress space with its axis along the line a1 = o2 = o3 equally inclined to the principal directions. The cross section of the cylinder where it intersects the plane cr1 + cr2 + ( j 3 = 0 , which is perpendicular to its axis, is known as the yield locus and is a circle for von Mises criterion and a regular hexagon for Tresca. The cylindrical form of the yield surface is caused by the assumption that the mean normal stress has no effect on the process of yield. In the more general case when this is abandoned, as it must be for polymers, the surface given by Eq. (1-3) is not necessarily a cylinder. If we now apply these ideas to an anisotropic substance and again for the simple case of yield, and still assuming that this is unaffected by the mean normal stress we must now, in the general case, relate for an orthotropic material the yield surface of the material to the imposed stress - also represented by a stress ellipsoid or quadric. In order to account for the onset of yield in an anisotropic material Hill (1948) introduced a theory as follows. He considered an orthotropic material, i.e., one showing three mutually perpendicular planes of

symmetry, the intersection of these planes being the principal axes of anisotropy. Choosing these axes as reference axes Hill proposed the simplest yield criterion that would reduce to von Mises criterion when the anisotropy is vanishingly small. The Huber-Mises assumption of a homogeneous quadratic in the stresses is taken over, which implies that the yield stress in tension and that in compression in the same direction are equal. In view of the assumed symmetry, terms in which any one shear stress occurs linearly must be rejected and a third assumption is that a pure hydrostatic pressure does not affect yielding. Hill thus assumes that yield occurs when F(<J2 - o3)2 + G{o3 - a,)2 2Lx\3 + 2MT231

2NT22

o2)2 =

(1-7)

where the reference axes are those of the principal axes of anisotropy and F9 G, H, L, M, N are constants. If other axes are assumed, the expression changes in form in the same way as the stress components transform. If O01,G02,G03 are the tensile yield stresses in the principal anisotropic directions then by considering the effect of each stress component acting alone it is immediately seen that (G + H) = i/a^; (H + F) = 1/G22; (F + G) = 1/<J203; and if T 0 2 3 , T 0 3 1 and T 0 1 2 are the yield stresses in

pure shear with respect to the principal axes of anisotropy we have 2L=1/T223;

2M = 1/T231;

2N=1/T212

Thus to describe such anisotropic yielding, six independent yield stresses need to be given. If there is rotational symmetry about an axis, say 3, then obviously p = G = H and L = M = N = 3F Equation (1-7) is then identical with Eq. (1-3) with2F = l/(j§. When Hill introduced this anisotropic theory of yielding he was well aware of the

1.8 Failure under Multiaxial Stress

assumptions made and that he was introducing it empirically with little suitable data to check the individual assumptions, in particular the identity of yield stress in tension and in compression and that hydrostatic pressure does not affect yielding. He was well aware that even with these assumptions that six independent yield stresses would need to be measured for an orthotropic material. None of the above theories have been applied to the failure of metals. However, the use of a modified Hill-like criterion has been proposed for the failure of composite materials and has been quite widely quoted. It is often associated with the names of Tsai and Wu (1971). I have developed the (well-known?) derivation of Hill's approach in detail above in order to point out that even in a material with orthotropic symmetry six independent failure stresses would need to be determined. This has certainly not been fully appreciated or acted upon according to the composite literature.

21

By a series of careful experiments involving a critique of the test methods employed, Hart-Smith in a series of papers has succeeded in producing recently a good approximation to a physically satisfying failure model. Carefully prepared and carefully tested unidirectional carbon fibre composites fail both in tension and in compression by shear (Ewins and Potter, 1980). Hart-Smith is able to substantiate in many cases a maximum shear stress condition for failure of (carbon fibre) fibrous composites. He first expresses the (Tresca-type) shear failure criterion in the strain plane for laminated structures under in-plane loads. The failure envelope is then generalised to transversely isotropic materials. If we proceed directly from Fig. 1-10 and draw the shear failure envelope for a transversely isotropic material with fibres say parallel to axis 1 but having equal tensile and compressive strengths, it will appear in strain space as in Fig. 1-11. It is un-

In this author's opinion, results of experiments done so far for simple laminates follow a maximum strain theory more closely than a maximum stress criterion. A careful review of (strength) failure theories up to the early 1980s is given by Rowlands (1985). Because a theory using a formulation which appears mathematically similar to Hill's [Eq. (1-7)] has been widely used in producing computer codes for predicting failure it has been difficult according to Hart-Smith (1991b) to introduce ideas based on the physics of failure and of observed failure modes. This must be, however, the only sound approach and however complicated it is it must be followed through even if eventually a useful numerical approximation can be made.

Shear failures except as noted for transverse tension

Figure 1-11. Shear failure envelope in strain coordinates. Transverse tension cut-off illustrated by line with dotted area on one side.

22

1 Fibrous Composite Materials

symmetric because two Poisson's ratios occur. If fibres are aligned along axis 1 and axis 3 is normal to the plane of the laminates, we have in the usual notation e± = oJE^ - v21 (T2/E22; e2= - v12 aj E1± + o2/E22 and y12 = T12/G12 with v12E22 = v21Ell9 so v12 is the "large"

Poisson ratio. The lettering of the corners of the hexagon is similar in Figs. 1-10 and 1-11 so the point B corresponds to failure in biaxial tension and C to tensile failure transverse to the fibres. Clearly along FA failure occurs by inplane shear; cf. Fig. 1-10. In Fig. 1-11 the line BC corresponds to failure in transverse tension and in Fig. 1-10 such a failure corresponded on the Tresca criterion to shear occurring on a plane parallel to axis 1 and inclined to axes 2 and 3. In a fibre composite consisting of a single lamina, transverse tensile failure may occur within the matrix. HartSmith points out that this should be recognised by inserting a cutoff at a particular strain as indicated in Fig. 1-11. Other failure modes can be dealt with in an analogous fashion, e.g., high strain carbon fibre laminae fail by instability under com-

pressive loads at a lower strain than that which they can withstand in tension. This would be taken account of, for instance, by inserting a cutoff in Fig. 1-11 which is parallel to DE but closer to the origin. Most composite structures contain patterns of fibres based on 0°, 90° and ± 45° directions, with roughly equal percentages of the fibres in each of these directions, in order to prevent matrix dominated failures from occurring. Further, the percentage of + 45° and — 45° is the same. Under these conditions the failure envelope of the structure is best represented by removing the asymmetry in the pattern in Fig. 1-11 by reflecting it in the line EB and further, since v21 is usually very small, by making the lines nearly parallel to the axes of strain precisely so. A further truncation is introduced in the in-plane shear quadrants to produce the failure envelope of the shape shown together with some experimental points in Fig. 1-12. In summary, Hart-Smith's approach is to work in terms of strain rather than stress and to produce a failure envelope for a composite structure which takes into ac-

Solid symbols: first series of tests Open symbols: second series of tests Tests by Swanson et al. on ASA/3501 carbon-epoxy tubes Shear failure of fibers

Failure by microbuckling (?)

Unidirectional load line

Flat laminate compression test result

Figure 1-12. Comparison of test results with theory for the truncated maximum strain failure model for carbonepoxy composites (from Hart-Smith, 1991a).

1.9 References

count realistically the possibilities of a variety of failure modes (matrix, interface) by introducing cutoffs of appropriate form for each different failure mode superimposed on what is essentially a Tresca-type failure criterion for the fibres in the composite.

1.9 References Bailey, J. E., Curtis, P. T., Parvizi, A. (1979), Proc. R. Soc. London A 366, 599-623. Bowden, P. B. (1973), in: The Physics of Glassy Polymers: Haward, R. N. (Ed.). London: Applied Science Ltd., pp. 279-337. Cammarata, R. C , Schlesinger, T. E., Kim, C , Quadri, S. B., Edelstein, A. S. (1990), Appl. Phys. Lett. 56, 1862-1864. Cheng, D. C.-H., Kruszewski, A. P., Senior, M. R., Roberts, T. A. (1990), J. Mater. Sci. 25, 353. Doraiswamy, D., Metzner, A. B. (1986), Rheol. Acta 25, 580. Ewins, P. D., Potter, R. T. (1980), Philos. Trans. R. Soc. A 294, 507. Hart-Smith, L. J. (1991a), in: Failure of Polymeric Composite Structures - Mechanisms and Criteria for the Prediction of Performance. SERC/IMechE Annual Expert Meeting, London: pp. 19-31; and references therein. Hart-Smith, L. J. (1991b), at 9th DOD/NASA/FAA Conference on Fibrous Composites in Structural Design, Lake Tahoe Nevada, November 4-7, 1991. San Diego, CA: SAMPE. Hill, R. (1948), Proc. R. Soc. London A 193, 281. Kelly, A. (1988), in: Mechanical and Physical Behaviour of Metallic and Ceramic Composites: Andersen, S. I. (Ed.). Riso, Denmark: Rise National Laboratories, pp. 75-88. Macmillan, N. H. (1988), in: Mechanical and Physical Behaviour of Metallic and Ceramic Composites: Anderson, S. I. (Ed.). Riso, Denmark: Ris0 National Laboratories. Macmillan, N. H., Kelly, A. (1992), Annu. Rev. Mater. Sci., in press.

23

McCartney, L. N. (1992), /. Mech. Phys. Solids 40, 27, and references therein. McLean, M. (1983), Directionally Solidified Metals for High Temperature Service. London: The Metals Society. Michel, A. G. M. (1904), Philos. Mag. Ser. 68, 589. Paulikowski, G. T., Dutta, D., Weiss, R. A. (1991), Annu. Rev. Mater. Sci. 21, 159-184. Prevorsek, D. C , Mills, P. J. (1989), in: Concise Encyclopedia of Composites Materials: Kelly, A. (Ed.). London: Pergamon. Pryce, A. W, Smith, P. A. (1992), J. Mater. Sci. 27, 2695. Ray, N. H. (1978), Inorganic Polymers. London: Academic. Ritter, J. (1971), App. Polym. Symp. 15. New York: Wiley, p. 238. Rowlands, R. E. (1985), in: Handbook of Composites, Vol. 3: Failure Mechanics of Composites: Sih, G. C , Skudra, A. M. (Eds.). New York: Elsevier, p. 71. Spitzig, W. A., Sober, R. J., Richmond, O. (1975), Metall Trans. A 7, 1703. Takanyagi, M., Takayuki, O., Masanori, M., Takashi, K. (1980), J. Macromol. Sci. Phys. B17(4), 591. Tani, T., Wada, S. (1990a), /. Mater. Sci. 25, 157. Tani, T., Wada, S. (1990 b), /. Mater. Sci. Lett. 9, 22. Tsai, S. W, Wu, E. M. (1971), J. Comp. Materials 5, 58. Urquhart, A. W. (1991), Mater. Sci. Eng. A 144, 7 5 82. Wright, J. K., Thomson, R. M., Evans, J. R. G. (1990), /. Mater. Sci. 25, 149.

General Reading Brunsell, A. R. (Ed.) (1988), Fibre Reinforcements for Composite Materials. Amsterdam: Elsevier. Chou, T.-W. (1992), Microstructural Design of Fibre Composites. Cambridge: Cambridge University Press. Kelly, A. (Ed.) (1989), Concise Encyclopedia of Composite Materials. Oxford: Pergamon Press. Kelly, A., Macmillan, N. H. (Eds.) (1986), Strong Solids, 3rd ed. Oxford: Clarendon Press.

2 Fibers and Whiskers Azar Parvizi-Majidi Department of Mechanical Engineering and Materials Science Program, University of Delaware, Newark, Delaware, U.S.A.

List of Symbols and Abbreviations 2.1 General Introduction 2.2 Evaluation of the Mechanical Properties of Fibers 2.2.1 Introduction 2.2.2 The Strength of Individual Fibers 2.2.3 The Strength of Fiber Bundles 2.2.4 Testing of Fibers 2.3 Glass Fibers 2.3.1 Introduction 2.3.2 E-Glass Fibers 2.3.3 AR-Glass Fibers 2.3.4 Oxynitride Glass Fibers 2.4 Boron Fibers 2.4.1 Introduction 2.4.2 Structure of Boron Fibers 2.4.3 Boron Elongation and Residual Stresses 2.4.4 Structural Defects in Boron Fibers 2.4.5 Surface Coating of Boron Fibers 2.5 Carbon Fibers 2.5.1 Introduction 2.5.2 PAN-Based Carbon Fibers 2.5.3 Rayon-Based Carbon Fibers 2.5.4 Pitch-Based Carbon Fibers 2.5.5 Structure of Carbon Fibers 2.5.6 Surface Treatment of Carbon Fibers 2.6 High Performance Organic Fibers 2.6.1 Introduction 2.6.2 Aramid Fibers 2.6.2.1 Structure of Aramid Fibers 2.6.2.2 Properties of Aramid Fibers 2.6.3 Other High Performance Organic Fibers 2.7 Alumina and Alumina-Based Fibers 2.7.1 Introduction 2.7.2 Sapphire Fibers 2.7.3 Polycrystalline oc-Alumina Fibers Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

27 29 29 29 30 33 33 34 34 36 37 37 38 38 39 41 42 43 44 44 45 48 48 50 52 54 54 55 56 57 59 60 60 63 63

26

2.7.4 2.7.5 2.8 2.8.1 2.8.2 2.8.3 2.8.4 2.9 2.10 2.10.1 2.10.2 2.10.3 2.10.4 2.11 2.12

2 Fibers and Whiskers

Alumina-Based Fibers Polycrystalline Alumina Staples Silicon Carbide and Silicon Carbide-Based Fibers Introduction CVD-Derived SiC Filaments Polymer-Derived SiC-Based Fibers Recent Developments in SiC-Based Fibers Silicon Nitride-Based Fibers Whiskers Introduction Silicon Carbide Whiskers Silicon Nitride Whiskers Oxide Whiskers Closing Remarks References

65 66 66 66 68 72 76 77 78 78 79 82 82 83 83

List of Symbols and Abbreviations

List of Symbols and Abbreviations

5,

C44

E L L

m P P(a)

equilibrium spacing of the atomic planes elastic constants Young's modulus fiber length gage length width of turbostratic crystallites (parallel to fiber axis) height of turbostratic crystallites (perpendicular to fiber axis) shape parameter (Weibull modulus) failure probability density function cumulative failure probability radius of loop (test procedure) fiber volume

<7U,<JL

bundle efficiency surface energy (per unit area) polar, dissipative components of surface energy gamma function stress, mean fiber strength scale parameters fiber strength theoretical strength of a material upper and lower strength limit

BET CG CMC CVD DMPS FT-IRRS HPZ HVR IMC LVR MMC PA PAN PBA PBBI PBBO PBZT PCS PET

Brunauer-Emmett-Teller equation ceramic grade ceramic matrix composite chemical vapor deposition dimethylpolysilane Fourier transform infrared reflection spectroscopy hydridopolysilazane high volume resistivity intermetallic matrix composite low volume resistivity metal matrix composite polyamide polyacrylonitrile poly-p-benzamide polybenzobisimidazole polybenzobisoxazole polybenzobisthiazole polycarbosilane polyester

8

7

r o

cr0, a 01 ,
27

28

PPTA STEM STM TC TEM UV VLS

2 Fibers and Whiskers

poly-p-phenylene terephthalamide scanning transmission electron microscopy scanning tunneling microscopy turbostratic carbon transmission electron microscopy ultraviolet light vapor-liquid-solid process

2.2 Evaluation of the Mechanical Properties of Fibers

2.1 General Introduction The introduction of glass fiber in the 1930s and the development of polyester resins shortly after ushered in a new era in the field of engineering materials, that of man-made composites. As the performance advantages of polymer matrix composites became apparent, demand for higher modulus fibers grew and led to the development of boron, carbon, aramid and other high performance organic reinforcements. Today, the large variety of carbon and high-modulus organic fibers commercially available fulfill virtually all the requirements demanded from the reinforcement for polymer matrix composites. With polymeric composites firmly established, the next stage in the evolution of man-made reinforced materials began almost a decade ago in extending the range of their applications to much higher temperatures such as encountered in aircraft or automobile engines. Towards this end, a large volume of research has been devoted to the development of metal matrix (MMC), ceramic matrix (CMC), and intermetallic matrix (IMC) composites. The reinforcements for these composites must not only possess high strength and high modulus, but must also exhibit superb chemical and thermomechanical stability in order to withstand the processing and use temperatures that may well exceed 1500°C. Based on these criteria, fibers of silicon-based nonoxides and alumina-based oxides have been identified as the most suitable reinforcements for high temperature composites (DiCarlo, 1991). However, despite major developments and the commercial introduction over the last 10-15 years of a dozen or so fibers in the above two systems, there is presently not yet a fiber that can sustain continuous exposure to temperatures much above 1100-1200 °C with-

29

out major loss of properties. The search for a truely high temperature fiber, therefore, continues to be a highly dynamic area of research. This chapter presents the state-of-theart in reinforcements for polymeric and high temperature matrices. It treats both fibers and whiskers. Fibers are defined as filamentary materials with a length larger than about 100 jim and a length to diameter ratio (aspect ratio) much greater than 10 (Katz and Brandmair, 1987). They can be amorphous, polycrystalline, or single crystalline. Whiskers, on the other hand, are elongated single crystalline particles with diameters usually less than 1 jim and aspect ratios that can be as low as 10. Owing to their very small dimensions and single crystalline nature, whiskers possess a high degree of structural and chemical perfection which give them an extremely high strength, elastic modulus and elongation at break. Because of their low aspect ratios, whiskers are not as effective as continuous fibers as reinforcing agents. Nonetheless, they have found many useful applications particularly in MMCs and CMCs, e.g., where improved wear resistance is required. While emphasis is placed primarily on reinforcements that are commercially available, effort has been made to also cover significant developments that have not yet reached a commercial scale.

2.2 Evaluation of the Mechanical Properties of Fibers 2.2.1 Introduction

Testing single fibers or even fiber bundles presents many problems due to the fine diameter of the fibers. Fiber strength determination is further complicated by the fact that stiff fibers are brittle materials,

30

2 Fibers and Whiskers

and as such their strength is strongly dependent on the presence of pre-existing flaws. Since the probability of encountering a life-determining flaw increases with increasing the fiber surface area or volume (depending on whether the surface or volume flaws control the strength), fibers with longer gauge lengths or larger diameters break at lower values of stress. Hence the fiber strength is not a single-valued property but a statistical variable. The statistical distribution of the fiber strength is usually described by the Weibull model (Weibull, 1951). Coleman (1958) showed that the Weibull distribution can be used not only for the strength of the individual fibers but also of fiber bundles. Detailed descriptions of the Weibull statistics as a tool for the experimental strength analysis for single fibers and fiber bundles can be found in Kelly and MacMillan (1986) and in Chou (1992). 2.2.2 The Strength of Individual Fibers

The Weibull distribution is based on the weakest link theory in which failure of the most serious flaw leads to the catastrophic fracture of the entire material. The model assumes that flaws are homogeneously distributed throughout the material, that the strength along a fiber has the same distribution as between the individual fibers, and that fracture is perfectly brittle and hence independent of time and environment (van der Zwaag, 1989). The cumulative failure probability P (o\ i.e., the fraction of fibers in a large population of fibers that fail at or below a stress a, is thus given by = l -exp[-FAT((j)]

empirical form for this function where

N(a) =

^O — G

(2-2)

The cumulative failure probability function is now n ^.

If fibers are assumed to all have the same diameter, V in the above equation can be replaced by the fiber length L thus giving P (a) =

1 - exp f -

(2-4)

In the above equations, au is a location parameter and denotes the stress below which P (a) = 0, a0 is a scale parameter with dimensions of strength x (length)1/m, and m is a shape parameter, usually called the Weibull modulus. For most cases, au is zero, thus Eq. (2-4) yields = 1 — exp

(2-5)

— L\ —

Van der Zwaag (1989) has discussed the relationship between the Weibull parameters and the material properties and pointed out that the Weibull modulus is not a material property but reflects the shape of the defect population in the material. Differentiating Eq. (2-5) yields the failure probability density function

W

exp - L

-

(2-6)

L W J

The mean fiber strength is obtained from p(a) as 1

(2-7)

(2-1)

where V is the fiber volume and N (a) is an unknown function. Weibull assumed an

where F is the gamma function. Manders and Chou (1983) showed that the Weibull modulus can be obtained from the plot of

2.2 Evaluation of the Mechanical Properties of Fibers

log (mean strength) vs. log (fiber length) which has a slope of — 1/m. For the experimental analysis of fiber strength, Eq. (2-5) is usually expressed in the following form

ln{-ln[l -P{a)]}-lnL = = m\n(a)-m\n{a0)

pends on m. A useful feature of Weibull distribution is that strengths obtained at various gauge lengths can be plotted on a single curve. The Weibull moduli of a variety of fibers are given in Table 2-1. Figure 2-3 shows the strength distributions of heat cleaned Nicalon SiC fibers with various gauge lengths on Weibull probability graph. The prediction of the Weibull model of a straight line for log (mean strength) — log (length) does not always correlate with experimental data where a change in slope has been often reported at short fiber lengths (see for example Goda and Fukunaga, 1986; Hitchon and Philips, 1979; Warren and Andersson, 1980). This is troublesome as it leads to erroneous data when extrapolating the Weibull plot to obtain fiber strength at short fiber lengths. Extrapolation of the strength to the critical fiber length (usually about 0.1 to 0.2 mm) is required for the prediction of the composite strength. A change in the slope indicates a change in the type of the flaw that controls the strength. To overcome this limitation, the function N (a) in Eq. (2-2) was modified by Kies (1958) and further modified by Phani (1987, 1988a, 1988b). Phani's modification gives N (a) the form

(2-8)

where 1 — P (a) denotes the probability of survival. Therefore, a plot of In {- In [1 - P (a)]} against In (a) is a straight line with slope m. Figures 2-1 and 2-2 show the plots of Eqs. (2-6) and (2-8) for various values of m. It is seen that the higher the Weibull modulus the narrower the strength distribution, and that the shape of the strength distribution only dem = 100

/

f(a)

I/

m=2

0.0

i m = 10

— •

0.5

1 1.0

V

2.0

1.5

31

Stress

Figure 2-1. Plots of the Weibull probability density function for various values of the Weibull modulus (from van der Zwaag, 1989).

N(a) =

G —

'01

(2-9) '02

X

99% 90% >

babil

n -

' J^.—-^

-2 -

m = 100

-4 / / 0.5

10% £

1% 1

10 i

i

1.0 Stress

1.5

2.0

Figure 2-2. Plots of Eq. (2-8) for various values of the Weibull modulus (from van der Zwaag, 1989).

32

2 Fibers and Whiskers

Table 2-1. Weibull shape parameter for several reinforcing fibers and whiskers. Fiber

Nicalon SiC

SCS-6(CVD-SiC) Carbon Type I Carbon Type II Boron (W core)b Boron (W core)c Kevlar49 d Fiber FP SiC Whisker a

Heat cleaned;

b

Gage length (mm)

Weibull modulus, m

References

2 10 10-220 10-175 3 40 0.5-50 0.5-50 40 40 — — -

8 5.5 2.2-3.6 2.9-3.4 10-16 5.0-8.1 3.8-8.0 ~7 -19 ~9 6.5 1.77

Warren and Andersson, 1980 Andersson and Warren, 1984 Bunsell et al., 1988 Wu and Netravali, 1992 Le Petitcorps et al., 1988 Hitchon and Phillips, 1979 Hitchon and Phillips, 1979 Le Petitcorps et al., 1988 Le Petitcorps et al., 1988 Wagner et al., 1984 Nunes, 1983 Petrovic et al., 1985

140 urn, uncoated;

c

140 urn , thick BdC coating;

d

fiber yarn.

Table 2-2. Values of the parameters of modified Weibull distribution for several fibers (Phani, 1987, 1988 a, b). Fiber

E-Glass S-994 Glass Nicalon SiC Fiber FP (A12O3) Carbon Type I Carbon Type II

Gage length (mm)

(GPa)

(GPa)

(GPa)

(GPa)

15-120 0.25-60 10 254 50 50

6.0 6.07 16.0 3.0 9.9 10.15

0.62 0.55 0 0 1.0 1.0

0.55 0.28 2.0 0.2 2.4 2.3

2.07 4.48 4.5 1.2 5.0 5.0

1.13 1.29 2.88 0.05 0.41 1.41

5.30 5.98 4.66 6.19 12.18 13.07

where au and
(2-10)

0.100 _

= m1 In 0.010

(7-(7 L

—ra2In

-

0.001 100

1000

10000

Stress (MPa)

Figure 2-3. Weibull plots of heat-cleaned Nicalon SiC fibers (from Wu and Netravali, 1992).

The above form of the strength distribution predicts an increase in the slope of the log (mean strength) — log (length) curve with increasing the gauge length. Phani's modified Weibull parameters for glass, carbon, silicon carbide, and alumina fibers are given in Table 2-2.

2.2 Evaluation of the Mechanical Properties of Fibers

2.2.3 The Strength of Fiber Bundles

Coleman (1958) used Weibull statistics to calculate the strength distribution of untwisted fiber bundles containing N fibers (N > 100). Assuming that all fibers in the bundle have the same diameter and length, that the bundle length is equal to the length of the fibers, and that there is no interaction between the individual fibers in the bundle, the maximum stress sustained by the bundle is obtained from the statistical parameters of the individual fibers by the following equation: KuJbundle = ffo

(2-11)

where e is the base of the natural logarithm. The bundle efficiency, e, is defined as the ratio of amax to the mean strength of the fibers of the same length, s is given by (van der Zwaag, 1989) e

= (em)" 1/m /r(l + 1/m)

(2-12)

Equation (2-12) shows that e is only a function of the Weibull modulus and is independent of the bundle length, e is always less than unity (unless fibers have a singlevalued strength), indicating that the bundle strength is always less than the mean strength of the individual fibers.

2.2.4 Testing of Fibers

Most composite applications require information on the strength, failure strain, elastic modulus, Poisson's ratio, coefficient of thermal expansion, thermal and electrical conductivities, and diameter or crosssectional area of reinforcing fibers. In the case of ceramic reinforcements for high temperature applications, a knowledge of the fiber thermal stability and creep properties are also required. Determination of these properties by single fiber testing or characterization is not a trivial task, thus it

33

is not surprising to find that often fiber properties are evaluated indirectly from those of unidirectional composites. However, the need for direct determination of fiber properties has led to the development of various test techniques. A very useful review of testing issues and techniques for stiff fibers has been presented by Hughes (1988). The tensile strength, failure strain, and elastic modulus of fibers are usually determined from single fiber tensile tests. Tensile testing problems include the method of specimen gripping, fiber alignment, and strain measurement. Improper gripping or fiber misalignment lead to premature fiber failures and, hence, lower measured properties. To protect the fiber ends from damage by the grips, they are normally sandwiched between heavy paper end tabs by gluing with epoxy or another suitable adhesive. The testing machine used must be stiff and well-aligned, and must offer small load capacities. Determination of fiber elongation from the crosshead displacement may involve large errors due to fiber slippage in the grips. The best way to measure fiber displacement is through the use of non-contact techniques such as optical or laser extensometry or in-situ optical photomicroscopy. Kent and Vary (1992) used scanning laser acoustic microscopy to measure fiber strain. Fibers with preferential orientation such as carbon fibers have different properties in the axial and transverse directions. This anisotropy adds to the number of properties that must be evaluated as well as it leads to new testing difficulties. Krucinska and Stypka (1991) measured the axial Poisson's ratio of single carbon fibers using a laser diffractographic technique. In these tests, the differential transverse contraction of carbon fibers over their cross sections, which resulted from their anisotropy, ne-

34

2 Fibers and Whiskers

cessitated the measurement of contraction along many directions. Bend tests have often been used for the evaluation of fiber strength and failure strain, due to their simplicity. However, properties obtained from bend tests may be significantly different than those determined from tensile tests, because of the nonuniform stress distribution of the bend test and the effects of surface and volume defects. Among the fiber bend tests are the simple free bending of a loop or knot (Kumar et al., 1986), and the depression of a fiber into a soft elastic pad with mandrels of known diameters (Hillig, 1987; Siemars et al., 1988). The latter is particularly suitable for testing short fibers or whiskers. High temperature testing of ceramic fibers for thermomechanical stability presents significant challenges. A new bend stress relaxation test has been developed for single fiber creep measurement (Morscher and DiCarlo, 1991). In this test, the fiber is formed into a loop of radius Ro and subsequently subjected to a specific combination of temperature, environment, and time. The loop is then cooled to room temperature and is broken at a point. Creepinduced stress relaxation is determined from the radius of the loop before and after thermomechanical treatment and the radius of curvature of the broken loop.

2.3 Glass Fibers 2.3.1 Introduction

The ancient glass workers knew how to draw glass into small diameter rods for decorative uses. The development of glass fibers for engineering applications, however, dates back to around 1930, when Owens-Illinois Glass Company introduced an improved process for producing glass

fibers. Shortly after, Owens-Illinois and Corning Glass Works (presently, Corning, Inc.) embarked on a joint research program which was conducted at Purdue University to develop the technology for mass production of glass fibers. This effort led to the formation of the Owens-Corning Fiberglass Corporation in 1938 and the commercial production of glass fibers. The development of polyester resins in 1935 and the world events during 1940s provided the driving force for the development of reinforced plastics and marked the birth of a technology that has grown since at an amazing rate both in size and scope. Reviews of the historical development of glass fibers and their processing, properties, manufacturers and product forms are given by Gupta (1988) and by Dockum (1987). The extremely high strength, low coefficient of thermal expansion, and low cost of glass fibers make them an excellent reinforcement for polymeric matrices in applications where a moderate stiffness is sufficient. Glass fibers have an amorphous threedimensional network of - S i - O - as their basic structural backbone. They are thus isotropic, having equal elastic moduli in the axial and transverse directions. Silica (SiO2) is the most desirable glass-forming oxide because of its excellent temperature capability; however, its extremely high viscosity and melting temperature hinder its easy formation into fiber form. Therefore, to facilitate processing and achieve desirable properties, glass fiber composites incorporate a variety of other oxides such as A12O3, to improve mechanical and chemical performance; B 2 O 3 , to reduce the liquidus temperature, thus preventing devitrification during processing, and to enhance electrical resistivity; alkali oxides (Na 2 O, K 2 O, or Li2O), to improve fluidity and

2.3 Glass Fibers

lower melting temperature; and ZrO 2 , to improve resistance to alkaline solutions. The fabrication of glass fibers has been detailed by Lowenstein (1973). Fibers are drawn at high speeds ( ^ 2 x l O " 3 c m s " 1 ) from melt through orifices at the bottom of a platinum tank containing the molten glass. Glass fibers are usually manufactured in the form of strands consisting of 204 individual filaments drawn simultaneously. Fiber diameter ranges from 5 jim to 20 jim with most fibers averaging at about 12 jim. Several strands are wound together to form rovings which are finally used as continuous rovings, woven rovings, knitted or braided rovings, chopped rovings (or chopped strands), or chopped mats. Instead of denier (weight in grams of 9000 m of fiber yarn), commonly used to denote the linear density of textile fiber yarns, glass fiber strands are characterized by tex (weight in grams of 1000 m of strand).

35

At the time of drawing, a surface coating called a "size" is applied to the surface of individual fibers to protect them against abrasion and surface degradation during handling. Lowenstein (1973) has detailed the formulations of several of these sizes. In addition, a "coupling agent" such as a silane, titanate or chromium compound is also often applied to the fiber surface to improve wetting and adhesion by polymer matrices. A variety of glass fibers with different compositions and properties are commercially available. A system of letter designation, shown in Table 2-3 together with the composition ranges, has been used by manufacturers to distinguish each type of glass fiber based on its specific characteristics. Table 2-4 lists the properties of some of the commercially produced glass fibers that are used as reinforcements. The properties given in Table 2-4 are for single filaments and may be different from those of

Table 2-3. Letter designation and compositions of commercial glass fibers (Gupta, 1988; Dockum, 1987). Designation

Composition (%) SiO 2

A12O3

B2O3

CaO

Na 2 O

MgO

Other

A (Alkali)

72

0.6-1.5

—

10

14.2

2.5

0.7% SO 3

C (Chemical durability)

65

4

6

14

8

3

D (Low dielectric constant)

74

0.3

22

0.5

1.0

E (Low electrical conductivity)

52-56

12-16

5-13

16-25

0-2

0-6

0-1.5% TiO 2

E-CR

58-63

10-13

1.0-2.5

21-23

0-1.2

-

1-2.5% TiO 2 , 0-3.5% ZnO

53.7

-

-

12.9

-

9.0

2.0% ZrO 2 , 8% BeO 8% TiO 2 , 3% CeO 2

S (High strength)

65

25

-

-

-

10

Z or AR (High zirconia or alkali resistant glass)a

71

1

M (High modulus)

CemFIL.

11

0.5% LiO2

16% ZrO 2 , 2% TiO 2

36

2 Fibers and Whiskers

Table 2-4 Properties of commercial glass fiber reinforcements. Type of Fiber

E AR M S

Diameter

12 12 12 12

Specific gravity

-2.54 2.68 2.89 -2.48

Coef. therm. expansion (xlO-^C"1)

Young's modulus

Tensile strength

(GPa)

(GPa)

-5.0 7.5 5.7 2.9-5.0

72.4-76 70-80 110 -86

3.6 3.6 3.5 4.6

strands. For example, the strand exhibits lower strength than the individual fibers. This difference has been related to the statistical distribution of fiber strength (Sec. 2.2). "A"-glass is a soda-lime glass similar to that used in windows and bottles. It has a poor resistance to attack by water. "C"glass is a soda-lime-borosilicate glass with excellent resistance to acids. "D"-glass is a low-density glass with good electrical properties but poor water resistance. "E"-glass is a calcia-alumina-borosilicate glass with a good balance of properties which makes it suitable for a wide variety of applications. "E-CR"-glass is a modified E-glass for better corrosion resistance in acids and alkalis. "M"-glass has a higher modulus than E-glass but its strength is about the same. "S"-glass is a magnesiumalumino-silicate glass and was developed to provide higher strength and stiffness than E-glass. It also exhibits better strength retention at high temperatures and superior corrosion resistance (except in strong alkali solutions) than E-glass. "Z" or "AR"glass is an alkali-resistant glass developed specifically for use in Portland cement. 2.3.2 E-Glass Fibers E-glass, which was originally developed for making electrical insulating tape, is the

Strain at failure

Poisson's ratio

Softening temp.

-2% -2% — -

0.21 0.22 — -

845 — — 968

most commonly used type of glass fiber for reinforcing polymers. It is a calcia-alumina-borosilicate glass with a total alkali content (Na 2 O + K 2 O + Li2O) below 2%. The low alkali content ensures corrosion resistance and high electrical resistivity. Fiber properties are affected by not only the composition but also the processing conditions and fiber dimensions. Studies by Murgatroyd (1944) and by Weyl and Marboe (1964) indicate that both Young's modulus and shear modulus of glass fibers increase significantly with increasing fiber diameter. The strength of glass fibers decreases sharply with decreases in temperature and/or humidity (Konnert and D'Antonio, 1980). Burgman and Hunia (1970) have reported a linear decrease in strength with increasing humidity. A major drawback of E-glass is its poor chemical resistance in both acidic and alkaline solutions. Susceptibility to alkaline attack has limited the use of E-glass in cement matrices. Otto (1959) and Budd (1961a, b) have attributed the rapid loss of strength and diameter of E-glass fibers in alkaline solutions to the breaking of - S i - O - bonds in the glass network by OH ~ ions which leads to surface damage by pitting: -Si-O-Si+OH

(2-13) -Si-OH + - S i - O "

37

2.3 Glass Fibers 4000.

2.3.3 AR-Glass Fibers The poor alkali resistance of E-glass fiber led to the development of alkali-resistant (AR) glass fibers by Majumdar and co-workers (Majumdar and Ryder, 1968; Majumdar, 1970) for use primarily in cement. A comprehensive account of the development and properties of AR-glass fibers has been given by Majumdar (1985). AR glass fibers contain ~ 16% ZrO 2 (hence the alternative designation "Z") which has a much better resistance to OH~ attack than - S i - O - bonds. The first commercial production of AR-glass fibers took place in 1971 by Fibreglas Ltd., a subsidiary of Pilkington Brothers Ltd. of UK, after extensive research on fiber formation of zirconia-containing glass which is considerably more refractory than regular glasses. This fiber which was specifically designed for reinforcing cement was marketed under the tradename CemFIL glass fiber. A comprehensive review of CemFIL fiber and its production has been given by Proctor and Yale (1980). Since the introduction of CemFIL fibers, other variations of ARglass fibers have been developed and marketed by several glass fiber producers including Owens-Corning Fiberglass Corporation (see Wolf, 1974) and Kanebo, Ltd., of Japan (1978). The superior alkali resistance of ARglass fibers is demonstrated in Fig. 2-4 which shows fiber strengths obtained after varying periods of exposure to Portland cement extract solution. Prolonged aging studies of AR-glass fiber reinforced cement have shown that fiber degradation is much more severe in water-aged composites than in naturally-aged ones (Fig. 2-5). While glass fibers are cheap and easily produced, their low modulus and low temperature capability (below ~ 300 °C) prevent them from being used in advanced,

Alkali-resistant glass fiber

1000

I

l

24

48

72

96

Time (hr)

Figure 2-4. The loss of strength after varying periods of exposure to Portland cement extract solution for AR- and E-glass fibers (from Majundar and Ryder, 1970).

3000

.2

il

Figure 2-5. Fiber degradation after aging of AR-glass fiber reinforced cement in various environments (from West and Majundar, 1982).

high performance composites. Many attempts have been made to combine the economical advantages of glass fibers with a higher Young's modulus and temperature capability. 2.3.4 Oxynitride Glass Fibers Glasses with enhanced bond strength can be obtained by the substitution of nitrogen for oxygen in common oxide glass structures (Jack, 1978; Loehman, 1979,

38

2 Fibers and Whiskers

1980, 1983; Drew, 1981). Nitrogen leads to the formation of more atomic bonds and cross linking, thus providing a tighter glass network. Therefore, oxynitride glasses have higher elastic moduli, tensile strengths, hardnesses, and corrosion resistances than conventional oxide glasses. Recognizing the potential of oxynitride glasses as reinforcements for composites, Messier and co-workers (Messier, 1985; Messier et al., 1989; Gleisner et al, 1989) prepared continuous oxynitride glass fibers of various compositions by melt drawing in a nitrogen atmosphere (Fig. 2-6). Fibers with diameters as small as 10 jim could be drawn with chemical and physical properties the same as those of the bulk oxynitride glass. Fibers in the Y - S i - A l - O - N system offered exceptionally high elastic moduli in the range of 140 to 185 GPa, i.e., more than twice that of oxide glass fibers, and microhardnesses up to 10.4 GPa. These glasses, however, had relatively high densities (3.6 to 3.0 g/cm3). Lower density (2.6 to 3.0 g/cm3) fibers in the M - S i - A l - O - N systems where M stands for Mg, Li, Ca, or Zr were also successfully fabricated.

To Controller Pyrometer Quartz Window

BN Crucible Glass Furnace Assembly

Ceramic Insulation Induction Coils Graphite Susceptor N 2 Atmosphere Fiber

Guide Reel

(*y+— Take-up Reel

Figure 2-6. Schematic of the melt drawing unit for the fabrication of oxynitride glass fiber (from Gleisner, Diguiseppe, and Messier, 1989).

2.4 Boron Fibers 2.4.1 Introduction

Boron exhibits a combination of high strength and stiffness with low density which makes it an attractive material for reinforcement. The strength of boron filament matches that of glass fibers while its Young's modulus competes with those of high-modulus carbon fibers. Boron fibers have been used to reinforce both polymeric and metal, particularly aluminum, matrices for applications ranging from military aircraft components and Space Shuttle to sporting goods where high stiffness and low density are required. During the 1960s and 1970s, boron filament was considered the filament of choice for aerospace applications on account of its high stiffness and low density. Much of that early attention, however, has dissipated in recent years due to the high cost and the reactivity of boron on the one hand and the advent of cheaper, high modulus reinforcements on the other hand. The history of boron fiber development and its processing, structure, and properties are reviewed by Tsirlin (1985), Wawner (1988), and Galasso (1989). The formation of boron filaments by the hydrogen reduction of boron halide on the surface of a hot wire substrate was first demonstrated in 1911 by Weintraub (1911). Boron monofilaments as reinforcing agents were first produced by Talley (1959) at Texaco Experiments Co., using a chemical vapor deposition process. Since then, the production and properties of the fibers have undergone considerable improvements. The strength has increased by at least a factor of two, and the price has dropped by more than a factor of ten. Despite the significant price reduction, the high cost of boron fibers, which is due to both material

cost and processing cost, remains the m a i n stumbling block t o their widespread use. The fabrication m e t h o d currently used for the commercial production of b o r o n fibers (B/W fibers) is basically similar to the original process developed by Talley. Boron is deposited by the gas phase reaction of hydrogen with b o r o n chloride, B C 1 3 , onto a heated substrate of tungsten wire about 12 jim in diameter: BCl 3(g)

3/2H

2(g)

B(s) + 3HCl(g) (2-14)

The tungsten wire is drawn continuously through the reaction chamber which may be configured vertically or horizontally in single stage or multi-stage design (Fig. 2-7). The wire passes through mercury seals placed at both ends of the reactor. The mercury seals act as electrodes to resistively heat the wire. The surface of the tungsten substrate is first cleaned by heating the wire to incandescence in a hydrogen atmosphere. The substrate is maintained at a temperature of about 1300°C during the deposition reaction. A higher deposition temperature reduces the deposition rate but leads to lower strength due to the development of a crystalline microstructure. B/W fibers are produced with diameters in the range of 35-125 jim. Table 2-5 lists the properties of B/W fibers with a diameter of 100 |im.

Hg Seal Wire from Spool

Hg Seal •—\ - Gas Out

Fiber Winding Drum

* Bottom Electrode

Figure 2-7. Schematic of the process for the fabrication of boron filaments (from Galasso, 1989).

Figure 2-8. Boron atoms forming an icosahedron (from Vega-Boggio and Schweitz, 1977).

2.4.2 Structure of Boron Fibers

Boron has many polymorphic modifications (Hoard and Newkirk, 1960). The three stable forms are a-rhombohedral, f$rhombohedral, and tetragonal. The basic element in the structure of the unit cell of all of these polymorphs is believed to be a group of 12 atoms forming an icosahedron as shown in Fig. 2-8 (Tsirlin, 1985; Bhardwaj and Krawitz, 1983; Wawner and Lewis,

Table 2-5. Properties of B/W fibers with a diameter of 100 urn. Property Density (g/cm3) Tensile strength (GPa) Poisson's ratio Young's modulus (GPa) Shear modulus Coef. therm, expansion (°C ~1 )

Value 2.6 3.1-4.1 0.21 420 165-179 4.68-5.0 xlO" 6

40

2 Fibers and Whiskers

1966). The P-rhombohedral form, which is more complex than the other two structures, prevails at temperatures above 1200 °C and has been identified as the predominant crystalline phase found in chemical vapor deposited fibers. Boron fibers produced at temperatures below 1300°C are characterized by a high tensile strength and a microstructure that has been variably identified as amorphous (Tally et al., 1960; Gallaso etal., 1967) or microcrystalline (Gillespie, 1966; Otte and Lipsitt, 1966; Lindquist et al., 1968). Wawner and Lewis (1966) measured a crystallite size of approximately 2 nm from X-ray diffraction peaks. Similar crystallite sizes have been reported by Lindquist et al. (1968) and by Bhardwaj and Krawitz (1983). Fibers deposited at temperatures above 1350-1400 °C are crystalline with a P-rhombohedral structure. Gallaso etal. (1966) suggested that high temperature deposition yields a structure that is amorphous at the early stages of deposition but becomes crystalline at later stages. Crystalline boron fibers are at least 50% weaker than those with an "amorphous" structure produced at lower temperatures. Depending on the processing conditions, crystalline structures other than the three mentioned above may also be formed or the structure may pass from one form to another (Naslain, 1972). Boron fibers deposited on a tungsten core have a characteristic nodular, corn cob-like appearance (Fig. 2-9). The oval nodules are 3 to 7 jam in size and 1 to 3 \im in height and form internodular grooves 0.25 to 0.75 ^im deep (Tsirlin, 1985). The nodules nucleate on the surface of the tungsten substrate and grow outwards in a conical form. This gives the fiber a rough surface with internodular grooves which act as surface flaws to reduce the fiber strength. Galasso (1969), Vega-Boggio and Vingsbo (1977 a), Wawner (1967),

Figure 2-9. The nodular structure of a B/W filament (from Galasso, 1989).

and Carlsson (1979) have attributed the conical growth feature of boron to the preferential nucleation of boron nodules along axial die-mark ridges on the tungsten wire surface. During deposition, the tungsten wire reacts with boron and converts almost entirely to various borides including WB, W 2 B 5 , and WB 4 (Wawner, 1967). While tungsten has been used almost exclusively as the core material in the commercial production of boron fibers, its high cost, high density (Q = 19 g cm" 3 ), and reactivity with boron have been the driving force for the consideration of other materials for use as substrates. Carbon filaments about 33 |im in diameter have been considered as a good candidate for the replacement of

2.4 Boron Fibers

tungsten (Mehalso and Diefendorf, 1972; DeBolt et al., 1972 a, b). Carbon is not only nonreactive with boron, but is also lighter (Q = 1.8 gcm~ 3 ) and cheaper than tungsten. Boron filaments deposited on a carbon core are characterized by a relatively smooth surface and a weak bonding between the core and the boron mantle which is due to the absence of reactions between carbon and boron (DeBolt et al., 1972 a, b). The absence of a nodular surface in fibers deposited on a carbon core provides further support for the role of the tungsten substrate in the nodular growth feature of B/W fibers. It has been noted that boron fibers produced at lower temperatures elongate considerably during deposition (Tally, 1959; Mehalso and Diefendorf, 1972; Wawner et al., 1980; Eason et al., 1980). This, combined with the thermal expansion mismatch between the carbon core and the boron mantle, lead to the development of a tensile residual stress in the core that is large enough to cause the fracture of carbon in a periodic manner and the formation of hot spots where the current has to pass through the boron. This problem has been overcome by precoating the carbon substrate with a 1 - 2 jam thick layer of pyrolytic carbon (Basche et al., 1972,1975).

41

2.4.3 Boron Elongation and Residual Stresses

It has been noted that the boron sheath elongates during deposition at temperatures below 1300°C where it has a microcrystalline or "amorphous" structure. Several mechanisms have been proposed. Mehalso and Diefendorf (1972) presented a model in which boron atoms diffuse from the freshly deposited surface layer to vacancies in the interior layers thus causing a volume expansion and elongation. Wawner, Eason, and co-workers (Wawner et al., 1980; Eason et al., 1980) postulated that the elongation was due to the continuous transformation of the interior boron layers during deposition from a random atomic arrangement to one in which groups of 12 atoms made an icosahedral unit. The combined effects of boron elongation, growth stresses, and thermal expansion mismatch between the core material and the boron sheath lead to the development of significant residual stresses in the fiber which play a key role in the fiber strength (Faughnan, 1974; Behrendt, 1977, 1978, 1980). The magnitude and distribution of residual stresses are dependent on the type of the substrate used. Figure 2-10

Radius, r (jim) 150

20

10

30

40

50

Figure 2-10. Axial residual stresses in B/W and B/C filaments (from Behrendt, 1978). o.o

0.5

1.0

Radius, r (mils)

1.5

2.0

42

2 Fibers and Whiskers

shows the distribution of axial residual stresses in B/W and B/C fibers. In B/W fibers the diffusion of boron into the core and the conversion of tungsten core to borides is accompanied by a considerable volume expansion which puts the core under compression (Vega-Boggio and Vingsbo, 1976 a). The core elongation in this case is larger than that of the boron layer. The radial stress is also compressive in the core but becomes tensile in the sheath (Fig. 2-11). The stress becomes compressive again in the outermost region of the mantle upon quenching of the filament at the end of processing. In B/C fibers, on the other hand, the core experiences a tensile residual stress since carbon does not

( + ) 0.7-0.8

(-) 0.3-0.5

(-) 1.0-1.4

Figure 2-11. Radial residual stresses in a B/W filament (from Vega-Boggio and Vingsbo, 1977).

react with boron. As noted above, in order to prevent the fracture of the carbon core under the residual tensile stress, the carbon substrate is coated with a thin layer of pyrolytic carbon which acts as a buffer between the boron sheath and the carbon core. Both B/W and B/C fibers exhibit compressive stresses in the outer surface due to the structural elongation of the deposited boron. 2.4.4 Structural Defects in Boron Fibers

Boron fibers are very brittle, and their strength is thus strongly controlled by defects (DeBolt, 1982). The strength therefore depends on the length of the filament and increases with decreasing the length following a Weibull distribution (Sec. 2.2). The reported fiber strengths are usually measured on a 25 mm gage length. Among the factors contributing to the scatter in the strength of B/W fibers are surface flaws (such as grooves between the nodules), interfacial voids, radial cracks and residual stresses. The role of these flaws in the strength and fracture behavior of boron fibers has been studied by Vega-Boggio and Vingsbo (1976 a, 1976 b), Vega-Boggio et al. (1976,1977), and Carlsson and Lundstrom (1979). The surface characteristics of boron fibers has been studied by Tsirlin (1985) who identified three types of morphologies: small oval nodules were associated with fiber strengths ranging from 3.3 to 3.7 GPa, large oval nodules with fiber strengths of about 2.8 GPa, and round nodules with a low strength of about 1 GPa. It has been shown that chemical polishing of the fiber in order to remove the surface grooves and flaws results in significant increases in the fiber strength (Wawner, 1966). Once the surface flaws have been etched away, the internal defects become the dominant flaws

2.4 Boron Fibers

controlling the strength. The removal of the outer layers of boron also redistributes the residual stresses and thus further affects the strength. In B/W fibers, the origin of fracture is often traced to voids existing in the boron mantle in the immediate vicinity of the core-mantle interface (Vega-Boggio and Vingsbo, 1977 a). These voids are found to be of the order of microns in transverse direction but considerably longer in the axial direction (Fig. 2-12). Wawner (1967), and Vega-Boggio and Vingsbo (1976 a) attributed these interfacial or "proximate" voids to the rapid diffusion of boron into the tungsten core and the condensation of vacancies in the mantle, i.e., the Kirkendall effect. In a later work, however, Vega-Boggio and Vingsbo (1976 b) proposed that interfacial voids are due to the nodular growth of boron from the surface of tungsten substrate. They maintained that boron nodules grow in tight parallel axial rows which, particularly around deep surface grooves of the substate, may hedge in axially extended voids. Besides the interfacial voids forming during deposition, Eason and Wawner (1986) noted that heating the fibers also results in microvoid formation along the core-sheath interface and is accompanied by considerable fiber contraction. This phenomenon was observed in both B/W and B/C fibers and led to severe strength reduction in the fiber. DiCarlo and Wagner (1981) postulated that fiber contraction and microvoid formation were due to the thermally activated migration of boron atoms that were not located in an icosahedron from the interior layers to the inter-nodular boundaries where they were oxidized and removed. The diffusion of atoms outwards was accompanied by a net flux of vacancies inwards, which led to void formation at the core/boron interface. On the basis of this model, DiCarlo (1979) suc-

43

Figure 2-12. Interfacial voids in a B/W filament (from Vega-Boggio and Vingsbo, 1976).

ceeded in increasing the fiber strength from an average value of approximately 3.5 GPa to about 5.5 GPa by subjecting the fibers to a combined chemical polishing, in which only 5 to 10 jim of the surface was etched away, and thermomechanical treatment below 900 °C. This approach simultaneously removed the surface flaws, controlled the void formation by limiting the diffusion of boron atoms to the surface, and yielded a desirable residual stress distribution where the core region which controlled the strength was placed under large compressive stresses. Because of the state of stress shown in Fig. 2-11, B/W fibers sometimes show radial cracks in the mantle (Vega-Boggio and Vingsbo, 1977 b). These cracks extend from the tungsten core outwards to the transition region between the tensile and compressive states of stress and play a major role in the fiber strength. 2.4.5 Surface Coating of Boron Fibers

Boron fibers are easily wetted by the matrix material due to the high surface energy

44

2 Fibers and Whiskers

of boron. The negative effect of the high surface energy, however, is that boron strongly reacts with most metal matrices, including aluminum, titanium, and magnesium. The reaction layer is usually quite brittle and fractures at a composite strain lower than the failure strain of the fiber. The interphase cracks then serve as surface notches for the fiber and lead to a severe loss of strength in the composite. Ochiari and Murakami (1979) related the fiber strength to the thickness of the reaction zone and concluded that, for boron, once the interphase layer exceeds a critical thickness of 0.2-0.5 |im, the strength will drop. In addition to its reactivity with metals, boron oxidizes readily at temperatures above 500 °C. Several types of coatings have thus been developed for the protection of boron fibers against oxidation and interfacial reactions. These include boron nitride, silicon carbide, and boron carbide. Camahort and co-workers (Camahort, 1968; Vidoz et al., 1969) achieved boron nitride coating by first oxidizing the fibers at a temperature of about 1000 °C and then converting the surface oxide layer to BN by treating the fibers with ammonia at a temperature of approximately 1100 °C. It is important that all the boron oxide be converted to boron nitride, otherwise the composite will exhibit a poor transverse strength due to the weakly bonded oxide layer. BN-coated fibers have been found to have a higher strength than uncoated fibers. The silicon carbide coating is applied by means of chemical vapor deposition in which SiC is formed by the reduction of methyldichlorosilane gas (Basche et al., 1968; Prewo and Kreider, 1972). SiC-coated fibers have received the designation "Borsic" and typically have a coating thickness of about 2.5 [im. The coating has been found to have a (3-SiC structure with the

{111} planes parallel to the fiber axis. During the SiC deposition, boron maintains its "amorphous" nature, and its room temperature strength and stiffness are virtually unaffected by the coating. The boron carbide coating is also applied by means of CVD through the reaction of boron trichloride and methane gases (Thebault et al., 1976; Pailler et al., 1979). The B4C coating has proven to be an effective diffusion barrier particularly in the case of a titanium matrix where SiC and BN coatings do not provide adequate protection.

2.5 Carbon Fibers 2.5.1 Introduction

Carbon fibers are currently the most commonly used type of reinforcement for advanced polymer matrix composites. They are characterized by a high stiffness, high strength, low density, and a negative coefficient of thermal expansion along the fiber direction. In addition, carbon fibers exhibit excellent thermomechanical stability at elevated temperature and are thus used in carbon/carbon composites for high temperature space applications. Carbon fiber is currently the only type of reinforcement available with a continuous temperature capability above about 1300°C. Its susceptibility to oxidation at temperatures above 400 °C, however, requires that the composite processing be carried out in vacuum and that the composite components receive an oxidation resistant coating. A comprehensive review of the processing, structure, properties, and applications of carbon fibers has been published by Donnet and Bansal (1990). Carbon fibers were first made by Thomas Edison in 1879 who carbonized rayon to prepare filaments for use in incandescent

2.5 Carbon Fibers

lamps. The history of modern carbon fibers as reinforcements, however, dates back to the late 1950s, when Bacon and Tang (1964) at Union Carbide developed a process for making high-strength carbon fibers by the stress graphitization of rayon filaments. At about the same time, Watt and co-workers in the United Kingdom (Watt, 1970; Watt etal., 1966) and Shindo (1961) in Japan, working independently, prepared highmodulus polyacrylonitrile (PAN)-based carbon fibers. Shortly afterwards, pitch or asphalt were used as precursors for the preparation of low-cost carbon fibers with elastic moduli in the intermediate range (Otani, 1965, 1967). More recent developments have led to the introduction of mesophase pitch-based carbon fibers with elastic moduli and tensile strengths as high as 900 GPa and 3.5 GPa, respectively. While a large number of precursors have been considered for the fabrication of carbon fibers, PAN, rayon, and pitch (including both isotropic pitch and mesophase pitch) remain the principal precursors for the commercial production of these filaments. The conversion of the organic precursor fiber to carbon fiber involves three principal steps of oxidation stabilization, carbonization, and graphitization. Carbon fibers derived from the organic precursors offer a wide range of elastic modulus and strength. The IUPAC has classified carbon fibers based on their mechanical properties and recommended the following designations: UHM type: carbon fibers with an axial elastic modulus greater than 500 GPa; HM type: carbon fibers with an axial modulus greater than 300 GPa (i.e., greater than 30% of the elastic constant C n of the graphite crystal), but with a ratio of tensile strength to tensile modulus below 1%; HT type: carbon fibers with a tensile strength greater than 3 GPa and a tensile

45

strength to tensile modulus ratio of 1.5 to 2%; IM type: carbon fibers with an axial modulus up to 300 GPa, but with a strength to modulus ratio above 1%; Low-modulus type: carbon fibers with a random, isotropic structure. These fibers have a low strength and a modulus as low as 100 GPa. Another classification often used in the literature categorizes the fibers based on their degree of preferential orientation. Type I has a high degree of crystalline orientation parallel to the fiber axis and exhibits a high modulus. Type II is less highly oriented, but exhibits a high strength. Type III has a random orientation which gives it low strength and low modulus. While the commercially available carbon fibers are prepared exclusively from organic precursors, carbon fibers with diameters up to 10 |im have also been grown recently from a vapor phase. Tibbetts (1983, 1989) has described the growth of fibers up to 30 cm long with a graphitic structure from a mixture of hydrocarbon and hydrogen gases. The gas mixture is passed over a heated substrate (about 1000 °C) which has been seeded with ironcontaining catalytic particles. The fibers thus grown exhibit a concentric annular ring structure consisting of partially aligned turbostratic carbon (defined in Sec. 2.5.5.) with a misalignment angle of 10-15°. Properties reported for vapor-grown carbon fibers with a diameter of 7.5 jim include a relatively high tensile strength of 2.9 GPa and an intermediate modulus of 237 GPa. 2.5.2 PAN-Based Carbon Fibers

High-modulus carbon fibers are prepared primarily from PAN. Table 2-6 lists a variety of commercially available PAN-

46

2 Fibers and Whiskers

Table 2-6. Selected PAN-based carbon fibers and their properties (Donnet and Bansal, 1990; Thorne, 1985). Manufacturer

Designation

Modulus

(g/cm3)

Tensile strength (GPa)

(GPa)

Ultimate strain (%)

Density

Filament count

AVCO, U.S.A.

Avcarb G-160

160 K

1.75

2.8

Celanese, U.S.A.

Celion 3000 Celion 6000 Celion 12000 Celion ST

3K 6K 12 K 3K, 6K, 12 K

1.77 1.77 — -

3.6 3.6 2.5 -4.3

220 — 234 227 -230

— — 1.77 -

Courtaulds, U.K.

Grafil XAHS Grafil XAHP Grafil HM

6 K, 12 K 6K, 12 K 6 K, 10 K

1.79 1.79 1.86

-3.0 3.3-3.6 2.3-2.6

220-240 220-240 330-350

Fortafil 3 Fortafil 5

40 K, 160 K 40 K, 160 K

1.73 1.80 — — — -

2.7 2.4

207 330

-1.3 -1.4 0.7 _ -

3.1 3.6 2.2 2.9

228 235 365 270

3.1 4.5

231 238

-3.0

230

1.3 1.5 0.6 — —

-3.5 5.0 2.2-2.4

230 294 392

1.7 1.7 0.6

Great Lakes Carbon, U.S.A. Hercules, U.S.A.

Magnamite Magnamite Magnamite Magnamite

AS-1 AS-4 HMS HTS

10 K 3 K, 6 K, 12 K 10 K 10 K, 12 K

Union Carbide, U.S.A.

Thornel 300 Thornel 400

1 K, 3 K, 6 K 3K

Nippon Carbon, Japan

Carboton

3 K, 6 K, 12 K

1.75 1.79 —

Torayla 300 Torayla 800 Torayla M40

I K , 3K, 6K, 12 K 12K 1 K, 3 K, 6 K

-1.45 1.80 1.80

Toray, Japan

based carbon fibers and their properties. These fibers have an equivalent diameter of 6-9 jam and a cross-section that could be circular or irregular in shape depending on the shape of the precursor PAN fiber. The chemistry and synthesis of PAN and its conversion to carbon fiber has been described by Watt (1971, 1985) and by Fitzer and Muller (1975). Polyacrylonitrile (-[CH 2 CHCN]-) n contains highly polar nitrile (-C = N) groups. The nitrile groups bond to the neighboring chains thus giving rise to a polymer that has a high melting point and is insoluble in common organic solvents such as alcohols. Only highly polar liquids could dissolve PAN. To produce carbon fiber, PAN is spun into fiber form by melt or solution (dry or wet) spinning.

The spun fiber is stretched at about 100150°C to achieve an oriented structure. The main stages in the conversion of PAN precursor polymer to carbon fiber are shown in Fig. 2-13. The oxidation stabilization is carried out in air or oxygen between 200 to 250 °C. During this process, cyclization and dehydrogenation reactions occur and give rise to a thermally stable, ladder-type structure in which, unlike aromatic polymers, carbon atoms are not in one plane. The exact mechanism of oxidation has been the subject of many investigations and a variety of structural models have been proposed for the oxidized PAN (see for example Watt, 1971; Strandage and Matkowsky, 1971). The oxidative stabilization of the PAN fiber is a very impor-

2.5 Carbon Fibers

47

Tension Control

]-•[ PAN Fiber

c

i

N

r

A N

Oxidation

Carbonization

1500 - 2800°C

Graphitization

ii N

Oxidized PAN

Carbonized PAN

Figure 2-13. The main stages in the conversion of PAN precursor polymer to carbon fiber.

tant step since it determines the final structure of the fiber. It is also the rate controlling step in the manufacture of carbon fiber from PAN, taking several hours in air. Oxidation is carried out under tension in order to prevent shrinkage which is caused by cross linking during the cyclization and dehydrogenation processes. The next step is carbonization during which the stabilized molecular structure of PAN is decomposed to carbon by a sequence of thermal treatments at successively higher temperatures up to 1500 °C in an inert atmosphere. In the early stages of carbonization at lower temperatures, further cyclization of the PAN fiber takes place while at the same time the cyclized sections coalesce and reorganize along the fiber length. At higher temperatures, the cyclized polymer chains undergo dehydrogenation reaction by linking up on the lat-

eral direction to form a graphite-like structure consisting of three hexagons bounded by nitrogen atoms. The lateral growth of the graphite structure occurs at higher temperatures by a condensation reaction which eliminates nitrogen. During the carbonization treatment, almost all of the noncarbon impurities including hydrogen, nitrogen and oxygen which existed in the PAN polymer are removed, giving rise to over 50% weight reduction. The final step in the production of PANbased fibers is graphitization which takes place under tension at temperatures in the range of 1500-2800 °C in an inert atmosphere. This leads to the orientation of the graphitic layer planes along the fiber direction. The Young's modulus of the fiber increases during this step and is directly related to the graphitization temperature and degree of stretching.

48

2 Fibers and Whiskers

2.5.3 Rayon-Based Carbon Fibers

With the introduction of the less expensive and higher yielding PAN route, the importance of rayon as a precursor to carbon fiber has largely decreased. Rayonbased fibers are mostly made from synthetic cellulose fibers due to their higher degree of orientation, lower impurity levels, and lower crystallinity than natural cellulose fibers. Rayon-based carbon fibers and their properties are listed in Table 2-7. During the low-temperature decomposition (< 400 °C) of rayon fibers, some of the oxygen and hydrogen present in the structure of rayon are eliminated in the form of H 2 O, CO 9 , and CO. The carbonization step is usually carried out in an inert atmoTable 2-7. Selected rayon-based carbon fibers and their properties. Manufacturer

Designation

Hitco, U.S.A.

HMG-50 HMG-40 HMG-25

2.1 1.7 1.0

345 276 172

Thornel 40 Thornel 50 Thornel 75

1.7 2.1 2.5

276 345 517

Union Carbide, U.S.A.

Tensile Modulus strength (GPa) (GPa)

sphere at temperatures below 1500°C. In this process, hydrogen is further removed, and an aromatic ring structure is formed. The fiber is then graphitized above 2800 °C under stress to form a high-modulus, oriented graphitic structure. 2.5.4 Pitch-Based Carbon Fibers

Pitch is a low-cost and widely available raw material for the manufacture of carbon fiber. Carbon fibers obtained from isotropic pitches have a randomly oriented crystalline structure which gives a low to intermediate strength and modulus. Mesophase pitches, those which possess liquid crystalline order, on the other hand, yield ultra-high modulus carbon fibers with extremely high degrees of orientation and perfection. Carbon fibers obtained from ordinary and mesophase pitches are listed in Table 2-8 along with their properties. There are many sources of pitch. The most commonly used are polyvinylchloride, coal tar, asphalt, and petroleum. The pitch is first prepared and then spun and drawn into continuous fibers which subsequently undergo oxidization, carbonization and graphitization treatments to obtain carbon fibers. During the oxidation treatment, pitch fiber is first exposed to a temperature

Table 2-8. Selected pitch-based carbon fibers and their properties. Manufacturer

Designation

Density (gem3)

Union Carbide, U.S.A.

Osaka Gas, Japan Kureha, Japan

Thornel P-55 Thornel P-75 Thornel P-100 Thornel P-120 Donacarbo F-140 Donacarbo F-600 — KCF 100

— 2.15 — _ 1.60 1.65

Tensile strength (GPa)

Modulus (GPa)

Ultimate strain (%) .

2.1 2.0 2.2 2.2

380 500 690 820

0.5 0.4 0.3 0.2

1.8 3.0

140 600

1.7 0.9

262 38

1.3 0.5 _ 2.4

49

2.5 Carbon Fibers

below 70 °C in ozonized air and then to a temperature up to 300 °C in air. This produces a cross-linked structure that is infusible and can thus be carbonized without melting. Carbonization is carried out in nitrogen at temperatures up to 1350 °C. It is possible to obtain high modulus pitch-based fibers by stretching of the fiber during high temperature heat treatment (see for example Hawthorne et al., 1970). This approach, however, is very expensive and difficult. Carbon fibers produced from textile precursor fibers or isotropic pitch precursors do not have a real, three-dimensional graphitic crystal structure. Truly graphitic fibers with Young's modulus approaching the theoretical maximum are available from pitch precursors containing a liquid crystalline phase (mesophase). The preparation of carbon fiber from mesophase pitch has been described by Rand (1985). Mesophase pitch is formed by prolonged heating of isotropic pitch at temperatures between 350 and 500 °C. The mesophase has a much higher surface energy than the isotropic liquid pitch. A phase separation therefore takes place with the mesophase forming small spherules which consist of a parallel array of aromatic hydrocarbons. A model for the lamellar structure of hydrocarbon molecules in mesophase pitch is shown in Fig. 2-14. Upon further heating, the spherules grow into anisotropic domains within the two-phase mesophase emulsion. The two-phase emulsion is then spun into fiber yarns using conventional melt spinning, jet spinning, or centrifugal spinning. Spinning of mesophase pitch is made difficult by its heterogeneous structure and high viscosity. The high spinning temperatures required lead to problems of gas evolution and pore formation. During spinning, the anisotropic domains elongate in

CH,

10A Figure 2-14. A model for the lamellar structure of hydrocarbon molecules in mesophase pitch (from Zimmer and White, 1982).

the direction of the fiber axis, thus resulting in a highly oriented structure. Therefore, unlike the PAN and rayon routes, the manufacture of carbon fiber from anisotropic pitch does not require stretching to achieve basal plane orientation. The as-spun mesophase pitch fibers are thermoplastic and require an oxidization treatment to render them infusible. Oxidization is carried out in air during which the mesophase molecules cross link and form a stable structure. Due to the higher glass transition temperature of mesophase pitch compared to conventional isotropic pitch, oxidation is conducted at a high temperature of 250 to 350 °C which cuts down the oxidation time to only a few minutes. Carbonization then follows through heat treatment at successively higher temperatures up to 2000 °C. During carbonization the fiber maintains its highly oriented structure. Subsequent heating to temperatures between 2500 to 3000 °C would further increase the degree of basal plane orientation. The graphitized fiber exhibits the three-dimensional crystal structure of graphite and may have modulus approaching 900 GPa which is close to the

50

2 Fibers and Whiskers

theoretical modulus of single crystal graphite ( C n = 1060 GPa). 2.5.5 Structure of Carbon Fibers

Carbon comes in two crystalline forms: diamond and graphite. In diamond, each carbon atom forms four sp 3 hybrid orbitals with four neighboring atoms which together make a tetrahedron. Graphite, on the other hand, has a layered structure. In the layer plane of graphite, carbon atoms are arranged in a hexagonal pattern in which each atom forms sp 2 hybrid orbitals with three neighboring atoms and has one unhybridized orbital. The unhybridized orbital is responsible for the high thermal and electrical conductivity of graphite parallel to its layer planes. The layers are stacked together to form the three-dimensional crystalline structure of graphite. Graphite with a hexagonal crystal structure has an ABAB stacking sequence with a layer separation of 0.334 nm as shown in Fig. 2-15. The bonding between the layers is by van der Waals forces. This gives a highly anisotropic structure with elastic constants C n - 1060 GPa, C 33 = 36.5 GPa, and C 44 = 4 GPa, where Cxiis for loading parallel to the basal planes, C 33 is for loading perpendicular to the basal planes, and C 4 4 is for shear loading between the basal planes (Ward, 1974). The graphitic planes may be rotated about the c axis in which case the ABAB stacking sequence of the three-dimensional crystalline graphite will be lost. This gives rise to a two-dimensional crystalline structure known as turbostratic carbon (TC). In turbostratic carbon, the separation between the graphitic layers is larger than that in graphite. High modulus carbon fibers prepared from PAN or rayon consist of turbostratic crystallites arranged in such a way that the

0.669 nm

0.142 nm Figure 2-15. The hexagonal structure of graphite (from Johnson, 1985).

graphitic layer planes are preferentially oriented parallel to the fiber axis. The crystallites are identified by two parameters, namely, the width L a (parallel to fiber axis) and the height L c (perpendicular to the fiber axis). Johnson and Watt (1967) studied PAN-based carbon fibers heat treated to 2500 °C and measured a crystallite size with L a of 6-12 nm and L c of at least 12 layer atoms. Watt et al. (1966) showed that the modulus of the fiber is highly dependent on the degree of layer plane orientation (Fig. 2-16). The orientation increases with increasing the heat treatment temperature or the amount of stretching during fiber processing. Another feature of the structure of oriented carbon fibers is the presence of 15 20% porosity between the crystallites. The 500

Width of (002) Line at Half Height (degrees)

Figure 2-16. The variations of the Young's modulus of carbon fiber with the degree of layer plane orientation (from Watt, Phillips, and Johnson, 1966).

2.5 Carbon Fibers

pores have an elongated shape and are preferentially oriented parallel to the fiber axis. Using small-angle X-ray scattering, Ruland and co-workers (Ruland, 1969; Perret and Ruland, 1970) measured pore sizes of the order of l - 2 n m in diameter and at least 20-30 nm in length. The presence of porosity combined with the larger layer separation of turbostratic carbon gives the fibers a lower density than the theoretical density of graphite. Several models have been proposed for the structure of carbon fibers. The experimental techniques used to study the structure have included X-ray and electron diffraction, electron microscopy, optical microscopy, and more recently scanning tunneling microscopy (STM). Watt and Johnson (1969) studied the structure of PANbased carbon fibers using X-ray and electron diffraction and showed that the fibers had a branched microfibrillar structure. Ruland and coworkers (see for example Fourdeux et al., 1968, 1971) in a series of publications also showed a microfibrillar structure for rayon-based and PAN-based fibers. They concluded that the basic structural unit was a ribbon-like layer plane about 6 nm wide but thousands of nanometers long. Several ribbons group together to form a wrinkled microfibril. The microfibrils are highly oriented parallel to the fiber axis and branch to form elongated pores about 1-2 nm in diameter as shown in Fig. 2-17. Further support for a wrinkled ribbon-like model has been offered by Diefendorf and Tokarsky (1975) who examined high modulus fibers by transmission electron microscopy (TEM) and deduced a basket-weave ribbon structure in which the width of ribbons ranged between 4 and 9 nm, and their thickness was of the order of 13-30 layer planes. A somewhat different structure has been proposed by Johnson and Tyson (1969) for

51

20 nm i

1

Figure 2-17. The microfibrillar model of the structure of carbon fiber (from Fourdeaux, Perret, and Ruland, 1971).

PAN-based carbon fibers. Using X-ray diffraction, these workers proposed the block model shown in Fig. 2-18. Each block represents a turbostratic carbon crystallite with L a and L c of about 6.5 nm. The crystallites are stacked together with tilt and twist boundaries to form columns parallel to the fiber axis. Between the columns, there exist voids of about 1 nm diameter and finite length. The block model was later developed into the three-dimensional interlinked structural model illustrated in Fig. 2-19 (Johnson et al., 1971).

Figure 2-18. The block model of the carbon fiber structure (from Johnson and Tyson, 1969).

52

2 Fibers and Whiskers

tron microscopy to the existence of a skin-core structure in the fiber. 2.5.6 Surface Treatment of Carbon Fibers

Figure 2-19. A three-dimensional structural model for carbon fiber (from Johnson, Crawford and Oates, 1971).

Other three-dimensional structural models have been proposed by Bennett and Johnson (1979) and by Oberlin (1984). The structural model of Johnson is depicted in Fig. 2-20. This model presents a skin-core structure in which within the skin region the layer planes are highly parallel to the fiber surface. Using scanning tunneling microscopy, Marshal and Price (1991) examined the surface topography of intermediate-modulus PAN-based carbon fibers and determined a surface crystallite size of 100 x 30 nm in the layer plane and 20 nm perpendicular to the layers. They attributed their considerably larger measurements compared to the earlier size determinations by X-ray diffraction or elec-

Figure 2-20. A three-dimensional structural model for carbon fiber showing a skin-core structure (from Bennett and Johnson, 1979).

The interlaminar shear strength of carbon fiber reinforced composites is directly related to the degree of fiber/matrix interfacial bonding. The bond is comprised of a physical component, which is related to the extent of mechanical interlocking between the fiber and matrix, and a chemical component which is due to the chemical bonds forming between the matrix and the active sites on the fiber surface. Donnet (1982) and Hoffman et al. (1984) have shown that these active sites are located at the edges of the crystallite basal planes emerging at the surface, as well as the structural defects such as vacancies, dislocations, or steps in the basal planes oriented parallel to the fiber axis. The degree of mechanical interlocking is dictated by the surface porosity and surface roughness, hence the total surface area, of the fiber. The total surface area can be measured by the physical adsorption of a gas such as neon, krypton or nitrogen using the Brunauer-Emmett-Teller (BET) equation. As discused earlier, all carbon fibers have porosity due to the undulations in microfibrils. The size and distribution of the pores depend on the heattreatment temperature; the higher the temperature, the fewer, but larger, the pores. The fiber surface morphology is strongly dependent on the precursor fiber material. For example, rayon-based fibers have an irregular cross section and a rough surface which lead to a larger surface area than PAN-based fibers with their smooth surface and virtually circular cross section. Carbon fibers are thus given a surface treatment in order to enhance fiber/matrix bonding by increasing either or both, the surface area and the surface functionality.

2.5 Carbon Fibers

There have been several attempts to determine the degree to which each of the above factors controls the interfacial bonding, however, no consistent results have yet been offered. A factor that complicates the investigation of the separate contributions of the surface area and the active sites is that increasing the surface area by pitting usually also increases the density of active sites since the active groups are located at surface defects including the edges of pores. The pores are sometimes small enough that they become blocked by the active groups at their edges, or are inaccessable to large matrix molecules. Error in the measurement of the surface area by gas adsorption when the pores are very small so that the gas molecules cannot penetrate them is another reason for inconsistencies among results obtained by different investigators (Donnet and Bansal, 1990). While the exact type of the treatments used commercially and their application is generally not available in the open literature due to the proprietary nature of the subject, numerous studies have been conducted to understand the effects of the various types of treatments on the surface properties. Extensive reviews of the surface treatment of carbon fibers have been published by Ehrburger and Donnet (1985) and by Donnet and Bansal (1990). Surface oxidation is the most common type of treatment applied to carbon fibers. It is done in a gas phase or a liquid phase. The gas-phase surface oxidation treatments are usually carried out in air or oxygen diluted with an inert gas, although other oxygen-containing gases such as ozone, carbon dioxide, and nitrogen dioxide have also been used. These processes must use a relatively high temperature (about 400 °C or higher) in order to reduce the oxidation time. The treatment temperatures are therefore very close to the ignition temper-

53

ature of carbon which may lead to extensive weight loss and surface pitting. McKee (1970 a, b) has shown that the oxidation temperature can be lowered significantly by the presence of small amounts of certain metal oxides. In general, gas-phase surface treatments enhance interfacial bonding primarily by increasing the surface rugosity of the fibers which is usually accompanied by a degradation in fiber strength. Liquid-phase oxidative processes are much more efficient than gas-phase processes and are the preferred method of surface treatment by most carbon fiber manufacturers. These are done either chemically or electrochemically. Chemical oxidation in nitric acid is used and studied extensively (see for example McKee and Mimeault, 1973). Nitric acid increases both surface area and surface functionality of carbon fibers, thus considerably enhancing the composite interfacial strength. The concentration of nitric acid and the oxidation time are both very important in determining the extent of surface modification. It has been reported that nitric acid smoothes the fiber surface by removing the existing surface irregularities while at the same time introducing surface microetch pits and polar carboxylic groups (Bahl et al., 1980). In addition to nitric acid, several other oxidizing agents including hypochlorites, sodium dichromate, potassium permanganate, hydrogen peroxide, and potassium persulfate have also been used. Electrochemical oxidation of carbon fibers in acidic or alkaline aqueous solutions has proven quite successful in the modification of surface functional groups and is of great interest because it allows the surface treatment to be carried out in a batch process (Ehrburger et al., 1976). Fitzer et al. (1983) showed that the amount of oxygen that is chemisorbed on the fiber surface increased sharply with increasing

54

2 Fibers and Whiskers

the applied potential. Anodic etching is quite mild so it only slightly affects the surface topography of the fibers. Depending on the nature of the electrolyte used, surface functional groups forming may be strong acidic groups such as carboxylic and phenolic species or weak acidic groups such as carbonyl and hydroxyl species. More recent studies have used plasma treatment to increase fiber/matrix bonding. Donnet and co-workers (Donnet et al., 1987) conducted microwave plasma treatment in air and argon on both high modulus and high strength carbon fibers and found a considerable increase in the polar component, ?£, and a decrease in the dispersive component, y!?, of the surface energy. The enhancement of the polar component is due to the introduction of polar oxygen groups, which in the case of argon plasma treatment has been attributed to the formation of surface radicals which subsequently react with oxygen upon exposure to air. Several non-oxidative approaches have also been employed in the surface treatment of carbon fibers. The most notable among these is surface coating by organic polymers such as organosilanes, polyamide, polyimide, and polydivinyl benzene. The polymer forms an interlayer phase that is capable of chemical bonding to both the fiber and the matrix. By using a flexible interlayer as a stress damper, the energy absorption capability of the composite can be significantly increased (Hancox and Wells, 1977). Reiss et al. (1974) used a polymer coating that served the dual purposes of a flexible interlayer and a chemical bonding agent. The coating comprised a copolymer containing flexible polyisoprene (an elastomer) segments which bonded to the fiber and styrene maleic anhydride copolymer segments which bonded to the matrix. In addition to polymer coatings,

fiber coating by a layer of pyrolytic carbon has also been successfully demonstrated. The pyrolytic carbon layer is applied either by chemical vapor deposition (Schmidt and Hawkins, 1965) or by first coating the fibers with a precursor resin and subsequently pyrolyzing the resin (McKee and Mimeault, 1973). The increase in the interfacial bonding has been attributed to the extensive surface roughness of the pyrolytic coating which results in a considerable increase in the surface area. Another non-oxidative approach is whisker formation at the surface in which mechanical interlocking between fibers and matrix is increased by growing whiskers of silicon carbide, silicon nitride, or titanium dioxide perpendicular to the fiber surface. This approach, however, is costly and difficult and therefore has not been used on a commercial scale.

2.6 High Performance Organic Fibers 2.6.1 Introduction

Since the development of the first synthetic fibers, i.e., polyester (PET) and polyamide (PA 66) fibers, in the 1930s by W. H. Carothers at E.I. DuPont de Nemours & Co. for textile applications, the field of polymer science and technology has experienced a phenomenal expansion particularly over the last two decades. Advances in polymer processing and a better understanding of the microstructure-property relationships in polymers have led to the development of organic fibers with extremely high tensile strength and excellent Young's modulus comparable to those of glass and steel filaments. A common feature of all these fibers is their highly oriented extended chain structure, which for

2.6 High Performance Organic Fibers

polymers with rigid backbones, such as aramids, is easily achieved by wet spinning and drawing processes, particularly when the polymer solution is a liquid crystal. In the case of polymers with random coil chain structures such as polyethylene, chain extension can be obtained by careful selection of molecular weight, temperature and drawing rate. In addition to their high tenacity and stiffness, advanced organic fibers offer low density, high toughness, and good chemical resistance. Their major drawbacks, however, are their low compressive and shear properties, which stem from their highly anisotropic structure, and their limited use temperature which is generally below about 300 °C. Calundann et al. (1988) have classified the high performance organic fibers into four groups based on the chain stiffness and reviewed the chemistry, processing, and microstructure of fibers in each group. A more complete treatment of this subject is also given in Chap. 13 of Vol. 12 in this Series on "High Performance Polymer Fibers". 2.6.2 Aramid Fibers

Aramid fibers are the most established and most extensively used amongst the high performance organic fibers. A detailed account of the processing, structure and properties of aramid fibers is given by Yang (1988). Also, a summary of the product forms and applications is provided by Smith (1987). Aramid is a generic term applied to fully aromatic polyamides. Aramid molecules have an extended chain structure and are very rigid. The rigidity is mainly provided by the aromatic rings as well as the amide (-NH-) and carbonyl (-CO-) bonds which are highly resistant to rotation.

55

Aramid fibers with very high axial tensile strength and good modulus can, therefore, be produced by chain orientation along the fiber axis. Compared to aliphatic polyamides, aramids are more chemically resistant and thermally stable. They exhibit a glass transition temperature above 250 °C and a melting point that can be greater than 550 °C. In some cases the fibers have been reported to decompose without melting. The development of aramid fibers originated in the 1950s. The early aramid fibers, such as Nomax [poly(m-phenylene isophthalamide)] marketed by E. I. Du Pont de Nemours & Co. offered excellent thermal stability but lacked high tensile properties. The new high strength, high modulus aramid fibers were developed in the 1970s based on the pioneering work of Kwolek and co-workers (Hill et al., 1961; Kwolek etal., 1962; Kwolek, 1972, 1974). Two major classes of aramid fibers currently exist: Kevlar fibers which were introduced in 1972 by E.I. Du Pont de Nemours & Co., and Technora (formerly HM-50) fibers which were marketed in 1985 by Teijin in Japan. The properties of these fibers are listed in Table 2-9. Aramid fibers are usually copolymers containing rigid backbones of poly-pphenylene terephthalamide (PPTA)

or poly-p-benzamide (PBA)

Aramid fibers have the highest specific strength of all other fibers and are very light and tough. These attributes make them attractive reinforcements for polymeric composites used in aircraft, sporting goods and electronic packaging applica-

56

2 Fibers and Whiskers

Table 2-9. Properties of aramid fibers (Bentur and Mindess, 1990). Fiber

Kevlar 49 Kevlar 29 Technora (HM-50)

Diameter (jun)

Density (g/cm3)

Tensile strength (GPa)

Young's modulus (GPa)

Elongation (%)

11.9 12 12.4

1.45 1.44 1.39

2.8-3.6 2.9 3.1

125 69 77

2.2-2.8 4.4 4.2

tions. The specific modulus of aramid fibers, however, is not as good as those of inorganic fibers. They also perform poorly under compressive loads; a shortcoming that has limited their use to secondary aircraft structures. Kevlar fibers were the first high modulus, high strength organic fibers to be produced by chain orientation. These fibers have gradually replaced glass fibers in polymeric composites used in the interior of aircrafts. Kevlar fibers are based on PPTA. To form the fibers, PPTA is first dissolved in a highly concentrated solution of sulfuric acid and then wet-spun into fiber form. Kwolek et al. (1962) found that the PPTA/ H 2 SO 4 solution exhibited the anisotropic behavior of a liquid crystal: it flowed like a liquid but had the ordered, anisotropic structure of a crystal. The liquid crystal achieved a higher degree of molecular orientation along the direction of shear during flow. Blades (1973) designed a dry-jet wet spinning process for spinning the liquid crystal solution. The method eliminated the problems of low attenuation and solution freezing associated with a conventional wet spinning process. In the dry-jet wet spinning method, the solution is extruded from spinnerette holes and passed through an air gap before being introduced into a coagulant bath. The spun filaments are then washed, neutralized, and dried to form Kevlar fibers. The molecular chains in the aramid solution undergo a significant reorientation along the direction of

shear during flow, thus resulting in fibers with a high degree of axial orientation and crystallinity. The degree of crystallinity can be further increased by heat treating the as-spun fibers at high temperature while under tension. Three types of Kevlar fibers have been produced by E. I. DuPont de Nemours & Co. through variations in spinning and heat treatment: Kevlar (formerly Fiber B), Kevlar 29, and Kevlar 49. Kevlar 49 possesses a higher degree of crystallinity than the other two grades. These fibers are available in various linear densities measured in denier or tex and in the form of yarns, rovings, and woven fabrics. Kevlar fibers are used as reinforcements for elastomers such as in tires while Kevlar 29 and Kevlar 49 fibers are used in conventional polymeric matrix composites. Technora aramid fibers are manufactured from copolymers of PPTA and 3,4-diphenyl ether which were first synthesized by Ozawa (1978). Fibers are spun from an isotropic solution of the copolymer into an aqueous coagulating bath. Despite their modest crystallinity, Technora fibers offer high strength and modulus beause of their fully extended chain structure. The oriented structure of the fiber results from very high draw rates used during its fabrication. 2.6.2.1 Structure of Aramid Fibers

Structural studies of Kevlar fibers have revealed the existence of a highly crystalline

2.6 High Performance Organic Fibers

state. The apparent crystallite size is less than 52 A for Kevlar 29 fiber and greater than 58 A for Kevlar 49 fiber (Blades, 1973). Northolt (1981) and Haraguchi et al. (1979) proposed two different crystal lattice models for the coagulated PPTA. In both structures the rigid PPTA molecules form planar sheets which are linked together by hydrogen bonds. The sheets are stacked together radially to form the pleated structure shown in Fig. 2-21. Dobb et al. (1977, 1980) suggested that the radial pleats are formed by alternate bands of (200) crystal planes being arranged at equal but opposite angles. The intermolecular forces between the sheets are of Van der Waals nature which explains the low longitudinal shear modulus and poor transverse properties of Kevlar fibers. The orientation angle between the PTA crystallites and the fiber axis is 15-20° for Kevlar and 10° or less of Kevlar 49 fibers (Yang, 1988). A characteristic of Kevlar fibers is their highly fibrillar structure which is superimposed on the crystalline structure. Panar etal. (1983) conducted plasma etching of Kevlar fiber and proposed the oriented fibrillar structure shown in Fig. 2-22. These investigators measured a width of about 600 nm and lengths up to several centimeters for the curved fibrils oriented along the fiber axis. Each fibril exhibits a series of bands with about 30-40 nm spacing. These have been attributed to defect layers separating stacks of crystalline layers. The structure of Kevlar fiber is not uniform across its cross section but exhibits a skin-core behavior in which the skin demonstrates a higher degree of fibrillar orientation and therefore a higher density than the core.

57

(b) Figure 2-21. Schematics of (a) the unit cell of poly-pphenylene terephthalamide and (b) the pleated structural model of Kevlar fiber (from Dobb, Johnson and Saville, 1980).

2.6.2.2 Properties of Aramid Fibers

Kevlar fibers offer an extremely high specific tensile strength along the fiber axis and are very ductile and tough. These attributes make them excellent reinforcements for applications where impact resistance and damage tolerance are key requirements. The tensile strength is controlled by the distribution of flaws as shown by Bunsell (1975) and by Wagner et al. (1984). It has

58

2 Fibers and Whiskers

Cb)

FIBRIL

Figure 2-22. The oriented fibrillar model of the Kevlar fiber structure (from Panar et al., 1983).

been shown that the strength can be fitted to a two-parameter Weibull distribution with a shape parameter of 8.8. The strength is sensitive to the moisture content of the fiber. The equilibrium moisture content of Kevlar fibers varies with the relative humidity and can reach 6 - 7 % at high humidities. For this reason, fiber strength is measured on fibers that have been dried at 140 °C and then conditioned at 20 °C and 55% relative humidity for 16 h (Yang, 1988). For Kevlar yarns, the strength and modulus depend on the degree of yarn twist; the strength reaches a maximum at a twist multiplier level of 1.1 while the modulus shows a continuous drop with increasing the degree of twist (Smith, 1987).

Kevlar fibers exhibit three modes of tensile fracture (Yang, 1988). The highly crystalline Kevlar 49 fibers usually show a pointed fracture in which considerable reduction in fiber diameter occurs prior to failure. The less oriented Kevlar fibers, on the other hand, fibrillate at fracture. A kink band type of failure has also been observed and is associated with the premature tensile fracture of fibers in the presence of kink band defects. The highly anisotropic structure of Kevlar fibers, in which planar sheets of PPTA are held together by van der Waals bonds, results in poor compressive strength and wear resistance. Fibers subjected to axial compression or bending tests demon-

59

2.6 High Performance Organic Fibers

strate kink bands forming at 55-60° to the fiber axis. A mechanism for kink band formation has been proposed by Dobb et al. (1981). The poor compressive strength also leads to fiber strength degradation in fatigue where an alternating tensile/compressive state of stress exists. Aramid fibers do not melt but decompose at high temperatures. Both strength and modulus fall gradually as temperature increases. The strength reaches about 50% of its room temperature value at 260 °C and is completely lost at 625 °C. Kevlar fibers are quite resistant to most chemicals including organic solvents, however, they are attacked by certain strong acids at high temperatures. Smith (1987) has listed the strength retention of Kevlar yarns exposed to various environments. Significant attack occurs by hydrochloric acid, concentrated sulfuric and nitric acids, and sodium hydroxide. A characteristic of Kevlar fibers is their strength degradation when exposed to ultraviolet (UV) light. UV light damage is less severe in thick yarns and in composites where fibers in the interior are protected by those at the surface or by the matrix. 2.6.3 Other High Performance Organic Fibers

Following the development of aramid fibers, a variety of high strength, high modulus organic reinforcements have been produced from both rigid-chain and flex-

ible-chain polymers. A review of high-performance organic fibers has been published by Calundann et al. (1988). Among the flexible-chain fibers are ultra-high molecular weight polyethylene fibers with highly oriented, extended chain structure (Keller and Barham, 1981; Barham and Keller, 1985; Capaccio and Ward, 1985). The conversion of conventional polyethylene with its random coil molecular structure to an extended chain structure is achieved by either superdrawing of melt-spun precursor fibers at elevated temperature or a gel spinning process using the dry-jet wet solution spinning method similar to that described for the fabrication of aramid fibers. Table 2-10 lists the high-modulus polyethylene fibers and their properties. In these fibers, the modulus is primarily controlled by the degree of molecular orientation while the strength is highly dependent on the molecular weight. High performance polyimide and polyamide-hydrazide copolymer fibers have been produced by wet spinning and drawing. During drawing, these polymers attain a transient mesophase state, resulting in extensive chain orientation. UBE and Toray/MITI manufacture polyimide fibers with elastic moduli of 140 and 165 GPa, respectively. Another class of organic fibers which like Kevlar aramid fiber have a semi-rigid backbone and are produced from a liquid crystal are thermotropic polyester fibers (Calundann et al., 1988). These fibers have

Table 2-10. Properties of gel-spun crystalline polyethylene fibers (Calundann et al., 1988). Designation

Spectra 900 Spectra 1000 Dyneema SK60 Tekilon

Producer

Density (g/cm3)

Young's modulus (GPa)

Tenacity (GPa)

Elongation

Allied Fibers Allied Fibers DMS/Toyoboa Mitsui Petrochem

0.97 0.97 0.97 0.96

119 175 50-125 60-100

2.6 3.0 2-3.5 1.5-3.5

3.5 2.7 3-6 3-6

60

2 Fibers and Whiskers

an aliphatic-aromatic or wholly aromatic structure and are characterized by excellent mechanical properties. The tensile modulus of wholly aromatic thermotropic polyester fibers increases with degree of aromaticity and can be as high as 90 GPa for wholly aromatic fibers. The highest modulus organic fibers are those produced from mesophase solutions of high molecular weight hetrocyclic polymers with no semi-flexible groups such as - C O N H - or - C O O - in their structural backbones (Calundann et al., 1988). Examples of these rod-like polymers are polybenzobisthiazole (PBZT), polybenzobisoxazole (PBBO), and polybenzobisimidazole (PBBI). These fibers have tensile moduli close to 200 GPa, i.e., comparable to that of steel, and tensile strengths of about 2 GPa.

2.7 Alumina and Alumina-Based Fibers 2.7.1 Introduction

The development of alumina fibers in late 1960s and early 1970s was in response to (a) a need for larger diameter refractory fibers to replace the submicron asbestos fibers as thermal insulating materials, and (b) an increasing need for continuous refractory fibers as reinforcements for high temperature composites. The introductions of "saffil" staple alumina fiber by Imperial Chemical Industries (ICI) and FP alumina fiber by E. I. DuPont de Nemours & Co. were to address the above needs, respectively. Today a variety of alumina fibers with a range of compositions, microstructures, geometries, and properties are available for reinforcement of metal and ceramic matrices. Some of these fibers and their properties are listed in Table 2-11.

The most attractive feature of alumina fibers is their excellent high temperature stability in air. Most alumina fibers can be heat treated in air up to 1000-1100 °C without any significant loss of strength. At higher temperatures, however, the strength and stiffness gradually fall due to grain growth, phase transformations, or the softening of the amorphous phase which is present in most alumina-based fibers. Alumina fibers offer a high elastic modulus comparable to some carbon fibers, however, their strength is only moderately high. In addition, they are electrically insulating which makes them attractive for certain applications. Reviews of the fabrication, microstructures and properties of alumina fibers have been given by Birchall et al. (1985) and by Bunsell et al. (1988). Fabrication routes for alumina fibers vary depending on their composition and microstructure. Continuous monocrystalline a-alumina (sapphire) filaments are grown from the melt. For polycrystalline fibers, the most commonly used preparation method is by solution spinning. The process involves first preparing an aqueous solution of a soluble alumina precursor such as aluminum oxychloride (A12(OH)5C1) or another suitable salt. The solution may also contain precursor to a second phase, such as an acidic oxide which is added for phase stability and grain size control, or fine a-alumina particles which would later serve as seeds for the crystallization and conversion of the salt to a-alumina. Also, the aqueous solution may contain or be replaced by an oxide sol. The solution or the sol is then mixed with an organic polymer to control the rheology and is subsequently extruded through a spinerette to form filaments which are mechanically drawn onto a bobbin. Next, the filaments are heat treated for conversion to dense oxide. Heat treatment

Table 2-11. Alumina-based fibers and their properties. Designation

Producer

Composition

Diameter

Density

(wt.%)

(jam)

(g/cm3)

Tensile strength (GPa)

Young's modulus (GPa)

Coef. therm, expansion

Elongation

Fibers per tow

Fiber FP*

DuPont

>99%a-Al 2 O 3

20

3.95

1.38

379

7.0

0.35

200

PRD-166*

DuPont

80%aAl 2 O 3 -20% ZrO 2

20

4.20

2.07

379

9.0

—

200

Sumitomo

Textron/ Sumitomo

85% Al 2 O 3 -15%SiO 2

-17

3.25

1.50

200

4.0

-

500,1000

Mitsui Mining

99.5% a-Al2O3

10

3.60

-1.80

-280

7.0

—

Nextel 312

3M

62% A1 2 O 3 -14% B 2 O 3 -24% SiO2

11

2.70

-1.72

152

3.0

1.2

130,400,780

Nextel 440

3M

70% Al 2 O 3 -2% B 2 O 3 -28%SiO2

11

3.10

1.4-2.1

193-241

4.5

1.2

130,400,780

Nextel 480

3M

70% Al 2 O 3 -2% B 2 O 3 -28% SiO2

11

3.05

-1.90

220

4.5

-

130,440,780

Saphikon

Saphikon

a-Al2O3 (Sapphire)

150

3.96

2

480

-

-

NA

ICI

95% Al 2 O 3 -5% SiO2

3

2.80

1.00

100

—

—

NA

Almax

* DuPont has recently discontinued the production of this fiber.

ro

Alumima and Aluminai-Based Fibe

Saffil (Short Fiber)

1000

CJ>

62

2 Fibers and Whiskers

is conducted in two steps. First is a low temperature treatment (400-500 °C) to dry, remove the fugitive species, and decompose the salt to aluminum oxide. During this stage, the hydrated species lose water and form polymeric networks of Al-O-Al and ultimately colloidal species. As a result of the removal of water and additives, the fibers become very porous. The second heat treatment is performed at temperatures above 1000° to remove the porosity and induce crystallization. The final structure and hence properties of the fiber are critically determined by the conditions, e.g., time, temperature, and atmosphere, of these post-spinning heat treatments. During the high temperature firing the amorphous alumina crystallizes by going through a series of transition phases until it assumes its most stable form which is oc-alumina. The sequence of transitional phases that may occur in alumina is shown in Fig. 2-23 along with associated changes in some properties. It must be noted, however, that in pure alumina it is extremely difficult to control the conditions such that the above progression through the phases can be observed. Such a controlled progression can only be achieved, and some of the transition phases stabilized, through the addition of a second phase such as silica. From a phase stability point of view it is desirable to produce fibers of pure oc-alumina. It is also desirable for the fiber to have low porosity and small grain size that does not grow significantly with temperature. Unfortunately, in pure alumina the

conditions required for low porosity and small, stable grain size are mutually exclusive. Grain boundaries in pure alumina are extremely mobile which leads to extensive grain growth during the high temperature firing required for the removal of porosity or in high temperature service applications. Grain growth can be inhibited by the addition of a second phase such as acidic oxides of Si, P, B and Zr. The price to pay, however, is the fall in fiber stiffness and maximum use temperature due to reduced crystallinity. Alumina fibers currently produced are mostly of a, y or 8 types, a-alumina fibers incorporate a small amount of MgO as a grain growth inhibitor, a-alumina, which is the densest form of A12O3, has a hexagonal structure with oxygen anions occupying the regular close-packed hexagonal lattice sites and the small aluminum cations positioned at 2/3 of the octahedral interstices. It is thermodynamically very stable and thus has a nonreactive surface. 7-Alumina has the structure of spinel, MgAl 2 O 4 , (Ervin, 1952) and occurs as very small crystallites (about 0.01 jim). Fibers of y-alumina contain silica as the stabilizing second phase. Their structure consists of small y-alumina crystallites embedded homogeneously in a matrix of amorphous silica. y-Alumina has a lower density and lower modulus of elasticity than oc-alumina, and its surface is active. 8-Alumina falls between a and y phases; its crystallite size is larger than that of y-alumina and reaches about 500 A. Compared to a-alumina, both y and 5 phases are less prone to grain

T|/y -

Small

- Crystallite Size -

Large

High

-Pore Volume—

Low

High

-Tensile Strength-

Low

Figure 2-23. The sequence of transitional phases of alumina and associated changes in some properties.

2.7 Alumina and Alumina-Based Fibers

63

2000

1500 J

i

K

1000 _

55

I

o

s

Figure 2-24. The variation of strength versus temperature for sapphire loaded along the C-axis (from Shahinian, 1971). 400

800

1200

1600

2000

Temperature ( C )

growth during high temperature exposure. However, the amorphous silica phase in these fibers softens at temperatures around 1100 °C and leads to a drop in strength and modulus. 2.7.2 Sapphire Fibers

Sapphire filaments are single crystalline a-alumina fibers. They were first produced in 1967 by Tyco. The production method involves drawing the filament from molten alumina contained in a molybdenum crucible. The crystallographic orientation and shape of the fibers is controlled by the geometry and orientation of the sapphire seed used to draw the filament. Sapphire filaments with c-axis perpendicular to the fiber axis have been shown to have excellent tensile creep resistance (Tressler and Barber, 1974). The orientation of the sapphire is critical to its thermomechanical stability. Small misorientations may cause a significant drop in strength at temperatures above ~ 1000 °C due to occurrence of uninterrupted slip along the crystallographic planes. Sapphire fibers have received increasing attention as reinforcements for ceramic

matrix composites because of their inherent oxidation resistance and excellent thermomechanical stability. The early sapphire fibers had a diameter of 225 jim, Young's modulus of 275 GPa and tensile strength of 2.1 GPa. New sapphire fibers with a diameter of 140-150 jim and Young's modulus of ~ 470 GPa are currently commercially available from Saphikon. The strength of sapphire filaments as a function of temperature (Fig. 2-24) exhibits three distinct regions with a minimum occurring around 600 °C (Shahinian, 1971). In the lower temperature range of 25-800 °C, it has been proposed that strength is controlled by stress corrosion at surface flaws (Charles and Shawn, 1962). 2.7.3 Polycrystalline a-Alumina Fibers

The first commercially produced alumina fiber was Fiber FP develop in 1974 by researchers at E. I. DuPont de Nemours & Co. (Dhingra, 1980). Fiber FP is 99% a-alumina with an initial grain size of 0.5 |im and a diameter of 20 jim. The crystallinity of the fiber gives it a rough surface as shown in Fig. 2-25. It is prepared by extruding a slurry of oc-alu-

64

2 Fibers and Whiskers

! 2/

Figure 2-25. The surface morphology of alumina fiber FP (from Dhingra, 1980).

mina particles in an aqueous solution of an aluminum precursor such as aluminum oxychloride. The a-alumina particles are added to seed crystallization during the high temperature firing. Fiber FP is also available with a SiO 2 coating which is applied to facilitate the handling of the fibers and to reduce its strong bonding to metal matrices. SiO2 has been shown to improve the room temperature strength of the fiber (1900 MPa for sized vs. 1380 MPa for unsized) by removing the surface roughness present due to the alumina grain boundaries. However, the softening of SiO2 at high temperatures can be detrimental to the elevated temperature strength of the fiber. Being almost pure oc-alumina the temperature capability of Fiber FP is rather limited as grain growth occurs at temperatures above ~ 1000 °C, leading to considerable fall in the strength. The presence of MgO as grain growth inhibitor in the fiber also leads to plastic deformation and soft-

ening around 1000 °C and thus drop in strength. Another limitation of Fiber FP is its low elongation at break of ~ 0.35%. The strength of Fiber FP is very sensitive to flaws and demonstrates a Weibull distribution with respect to gauge length with a Weibull modulus of 6.5 (Nunes, 1983). In an effort to solve the above problems with the intention to extend the application of the fiber to high temperature ceramic composites, DuPont has introduced a new alumina fiber under the developmental designation PRD-166 (Romine, 1987). PRD-166 contains 15 to 20% yttria partially stabilized zirconia grains in a polycrystalline matrix of a-alumina matrix with a grain size of 0.5 |im. Like Fiber FP, it has a diameter of 20 jim but exhibits an almost 100% improvement in its failure strain and ~ 50% improvement in its strength over Fiber FP. The transformation of zirconia from a tetragonal to a monoclinic phase may be providing this improvement (see also Chaps. 7 and 8 in

2.7 Alumina and Alumina-Based Fibers

Vol. 11). The presence of 0.1 jim zirconia grains in the alumina microstructure also inhibits grain growth thus giving the fiber a better high temperature strength retention capability. PRD-166 fibers may also receive a silica coating to remove the surface roughness and to enhance wettability by metals. It has been noted that the silica coating increases the room temperature strength of the fiber by about 25% but reduces the high temperature strength by 15% at 1000 °C and 25% at 1200 °C (Pysher et al., 1989). A new a-alumina fiber has recently been introduced by Mitsui Mining Company under the tradename Almax (Horiki et al., 1978). Almax is 99.5% a-alumina in polycrystalline form. It has a diameter of only 10 |im which renders it suitable for textile preforming. It is available in untwisted, woven and braided forms as well as chopped fibers of various lengths. Despite its a-alumina structure, a maximum use temperature of 1400 °C has been cited by the manufacturer. Another high temperature alumina fiber has recently been developed by Interatom GmbH in Germany. This fiber has no organic additives and is thus expected to show good thermomechanical stability. 2.7.4 Alumina-Based Fibers

Sumika alumina fiber, which is commercially available from Sumitomo, contains 85% A12O3 and 15% SiO2 and is produced from an aqueous solution of an organo-aluminum precursor (Horikiri et al., 1978; Abe et al., 1982); the processing steps are shown in Fig. 2-26. The presence of 15% SiO2 completely stabilizes alumina in the y-form up to a temperature of 1127 °C in both inert and oxidizing environments. The microstructure of Sumika fiber consists of very small crystallites of y-alumina

65

\ Polymerization

Preparation of spinning mix

t Spinning mix

Dry-spinnig

Precursor fiber (polyaluminoxane and alkyl silicate)

\ Calcination

Alumina fiber

(85wt%AI2Q3-

Figure 2-26. The processing steps for the fabrication of Sumika alumina fiber (from Abe et al., 1982).

(about 0.01 jim) embedded uniformly in a matrix of amorphous silica. The softening of this amorphous silica phase above 800 °C leads to a gradual fall in strength and a more pronounced fall in the modulus up to 1150°C (Lesniewski et al., 1990). Above 1150°C the fiber retains little strength. Transformation of the alumina/ silica structure to mullite (3 A12O3 • 2SiO2) starts above 1127 °C and becomes very vigorous around 1400 °C. The variations of srength and elastic modulus with temperature are shown in Figs. 2-27 and 2-28. A series of alumina-based fibers containing A12O3, B 2 O 3 and SiO2 are also produced by 3 M Company under the tradename Nextel (Snowman and Johnson, 1985; Johnson et al., 1987). Nextel fibers with a diameter of 11 |nm are prepared by

66

2 Fibers and Whiskers 1.2 1.0

Q.

£ 0.6 OAir • Argon

0.2 0.0

200

Each point represents the average of 30 tests.

400

600

800

1000

1200

1400

Temperature (C)

Figure 2-27. Variations of the strength of Sumika alumina fiber with temperature (from Lesniewski, Aubin and Bunsell, 1990). 200

small crystallite size of mullite results in good strength and fiber flexibility. Nextel 480 has been reported to stand ceramic composite processing temperatures of up to 1300°C and service temperatures of up to 1200 °C when used with suitable matrix materials (Snowman and Johnson, 1985). The room temperature strength of Nextel 480 is reportedly higher than those of FP or PRD-166 fibers partly due to its smaller diameter (Pysher et al., 1989). All three fibers, however, exhibited appreciable strength and modulus reductions at temperatures above 800 °C and none retained any significant strength at 1300°C.

150

2.7.5 Polycrystalline Alumina Staples 100 LU O Air • Argon

50

0

200

Each point represents the average of 30 tests.

400

600

800

1000

1200

1400

Temperature (C)

Figure 2-28. Variation of the elastic modulus of Sumika alumina fiber with temperature (from Lesniewski, Aubin and Bunsell, 1990).

a sol-gel process and are available in both unsized condition and with an appropriate sizing for textile handling. Fibers have an oval shape with the major diameter almost twice the minor diameter. Nextel 312 consists of 62% A12O3, 14% B 2 O 3 and 24% SiO 2 . Its structure is composed of crystallites of aluminum borate and some mullite in an amorphous silicate phase. This makes it useful up to a temperature of -1000 °C. In Nextel 440 and 480 the concentration of boria is increasingly reduced and alumina increased. The reduction of boria and increase of alumina promotes the formation of mullite and raises the elastic modulus but also the density. Nextel 480 is virtually all mullite with a 3 to 2 mole ratio of alumina to silica. The

Polycrystalline alumina fibers are also available in short staple form. An example is Saffil fibers produced by ICI Ltd. by aluminum salt decomposition (Birchall et al., 1985). Saffil fibers have a mean diameter of about 3.5 jim and a composition of about 95% alumina and 5% silica. Upon heating the as-spun fibers, r|-alumina appears as the first crystalline phase. Further heating at higher temperatures leads to the conversion of r|- to y- and finally 5-alumina with a grain size of about 500 A. The fibers may also contain some alpha alumina. Saffil fiber was originally produced in the form of random mat nonwoven blanket as a thermal insulator. The staple form, which can be as long as several centimeters, later found use as reinforcement for metals and ceramics.

2.8 Silicon Carbide and Silicon Carbide-Based Fibers 2.8.1 Introduction

Silicon carbide (SiC) is an excellent candidate for reinforcement of ceramics and

67

2.8 Silicon Carbide and Silicon Carbide-Based Fibers 0 -1 -2

Passive

_ ^

-3 -4 . -5 .

Active

Figure 2-29. The active-passive transition in the oxidation behavior of SiC (from Antill and Warburton, 1971).

-6 For reaction SiC (s) + 2SiO ^ tr 3SiO(g) + CO (9) -7

i

1100

i

1150

i

i

1200

1250

1300

1350

1400

Temperature (°C)

some metals because of its high strength and stiffness, good thermomechanical stability, and lower density and coefficient of thermal expansion than refractory oxides. Like some metals and ceramics, SiC has an interesting oxidation behavior (Costello and Tressler, 1986; Narushima et al, 1989). In the presence of sufficient levels of oxygen, as in air, and below a temperature of ~ 1200°C, SiC undergoes "passive" oxidation in which a stable SiO2 film forms according to the following reaction: 3 O 9 - + 2 S i O 7 + 2 C O (g)

Virgin SiC Fiber Heat Treated for 54Hrs at 1200°C Passive Oxidation, POa = 10~ 3 atm

(2-15)

The SiO2 film acts as a diffusion barrier and slows down further oxidation of SiC markedly. At very low oxygen levels or high temperatures, however, SiC exhibits "active" oxidation by the following reaction: SiC + O 2 -> SiO(g) + CO

fibers exposed to passive and active heat treatments. The SiO2 surface layer formation under conditions of passive oxidation gives fibers a smooth appearance while active oxidation results in a rough, pitted surface.

(a) Virgin SiC Fiber Heat Treated for 180 Hrs at 1200°C Active Oxidation, PQ2 = 10 ~ 8 atm

(2-16) (g)

In this case a stable oxide layer does not develop, and the oxidation rate increases with increasing partial pressure of oxygen until it finally drops when the oxygen partial pressure becomes large enough so that an active/passive transition takes place. Conditions for the active/passive oxidation of SiC are shown in Fig. 2-29. Figure 2-30 shows the surface characteristics of SiC

(b) Figure 2-30. The surface appearance of Nicalon SiC fiber oxidized under (a) passive and (b) active conditions (from Sands and Parvizi-Majidi, 1992).

68

2 Fibers and Whiskers

scale and their properties. The processing, microstructure and thermomechanical stability of some of these fibers are discussed below.

SiC is not easily sintered. Therefore, its production in fiber form through the conventional inorganic processes is very difficult. SiC fibers are at present produced either by chemical vapor deposition (CVD) or from a polymer precursor. CVD-derived SiC monifilaments grown onto a tungsten or carbon substrate have been in existence since the mid 1960s. These fibers exhibit good thermomechanical stability due to their stoichiometry p-SiC structure, but have a large diameter which limits their effectiveness for toughness enhancement in ceramic matrix composites and precludes textile preforming and net-shape manufacturing of complex geometries. The polymer precursor route is not only potentially cheaper than CVD method but is currently the only method for the production of small diameter SiC fibers. Table 2-12 lists SiC fibers currently available on a commercial or experimental

2.8.2 CVD-Derived SiC Filaments Chemical vapor deposited SiC monofilaments were first fabricated in the mid 1960s by General Technologies Corporation using the processing technology developed for boron fibers. The production of SiC fibers, however, was less costly than that of boron fibers due to the cheaper starting materials and greater deposition rates. In 1972, AVCO Corporation started the commercial production of a CVD-SiC fiber with a tensile strength over 3 GPa (DeBolt and Krukonis, 1973; DeBolt et al, 1973). These early fibers used a tungsten wire, about 12.5 jim in diameter, as the substrate material. It was found, however,

Table 2-12. SiC-based fibers and their properties. Designation

Producer

Composition

(wt.%) p-SiC Textron p-SiC Textron Nippon Si-C-O Carbon a Nicalon (HVR) Nippon Si-C-O Carbon a Nicalon (LVR) Nippon Si-C-O Carbon a Carburandum Carburandum oc-SiC SiC p-SiC Dow Corning Dow Corning SiC X9-6371 HPZ Dow Corning 57% Si-28% N 10% C-4% O UBE Ind. Tyranno Si-Ti-C-O Lox-M UBE Ind.b 38% Si-2% Ti50%C-10%O SCS-6 SCS-9 Nicalon (CG)

Marketed in U.S.A by Dow Corning;

b

Diameter Density Tensile Young's strength modulus

Coef. Elontherm. gation expan. ( x K T ^ C T 1 ) (%)

(g/cm3)

(GPa)

(GPa)

143 78 12-15

3.00 2.80 2.55

-3.8 -2.9 2.97

-400 -330 193

4.2 — 4.0

_ — 1.5

12-15

2.32

2.93

186

—

1.4

15

-2.50

2.93

186

—

—

30-50

3.1

1.38

415

—

—

8-10

2.9

2.28

393

-3.5

—

11

2.4

2.24

206

3.0

1.0

8.10 8-10

2.35 -2.50

3.38 —

193 —

3.10 —

— —

(urn)

Marketed in U.S.A. by Textron Speciality Materials.

2.8 Silicon Carbide and Silicon Carbide-Based Fibers

that the tungsten core limited the thermal stability of the fiber as tungsten reacted with SiC to form W 2 C and W 5 Si 3 . The reaction layer developed during filament production thickened considerably when the fiber was thermally treated at temperatures above 1000 °C thus leading to significant strength reductions (DeBolt et al., 1974; Crane and Krukonis, 1975; Lindley and Jones, 1975). As a result, the tungsten substrate was later replaced by carbon which not only provides better thermodynamic stability with SiC but is also cheaper and lighter (Q = 19 g/cm3 for tungsten and 1.8 g/cm3 for carbon) than tungsten. CVD-derived SiC fibers are deposited from a mixture of silane gasses and hydrogen. The process involves drawing the resistively heated W or C wire substrate through a cylindrical reaction chamber. The gas mixture is then introduced into the chamber and allowed to react and deposit SiC onto the surface of the substrate which is maintained at a temperature above 1200 °C. SiC fibers are currently produced at a temperature of ~ 1350 °C. The rate of reaction and the composition and structure of the resulting fiber are critically dependent on the composition, pressure and supply rate of the precursor gases (Martineau et al., 1984; Weiss and Diefendorf, 1973) as well as the deposition temperature. Higher deposition temperatures increase the rate of production but lead to a coarse, and hence weak, crystalline structure. Low temperatures are associated with slow deposition rates and an amorphous fiber structure which has a high initial strength but loses the strength due to crystallization at high temperatures. A variety of silane gases including methyltrichlorosilane (CH3SiCl3), methyldichlorosilane (Cl2CH3SiH), ethyltrichlorosilane (C2H3SiCl3), or tetrachlorosilane (SiCl4) have been used as reactant gases for the deposition of SiC. Methyl-

69

dichlorosilane provides the fastest production rate and requires a lower deposition temperature; the fibers produced, however, have a low tensile strength due to the rough fiber surface (DeBolt and Krukonis, 1973). Methyltrichlorosilane or the more widely available ethyltrichlorosilane, on the other hand, yield fibers with high strength and smooth surface but at prohibitively low growth rates. A compromise between good mechanical properties and high deposition rates is achieved by using a mixture of silane gases. Present CVD-SiC fibers utilize a 3 3 37 ^m carbon substrate which has been given a pyrolytic carbon coating about 1.5 jim thick. The pyrolytic carbon layer provides good thermal conductivity and serves as a buffer to reduce processing stresses which result from the thermal expansion mismatch between the carbon core and the SiC mantle (Nutt and Wawner, 1985). At the end of SiC deposition, the fiber surface is given a carbonaceous coating by varying the processing conditions and composition of the reactant gases. Carbon coating results in a significant increase in fiber strength due to the healing of surface flaws (Cornie et al., 1981). It also provides the weakly bonded interface critical to the toughness of ceramic matrix composites. However, carbon suffers from rapid oxidation. Furthermore, it is not suitable for many metal matrices where chemical reactions may take place at the fiber/metal interface. For these reasons, the carbon coating often incorporates, or is overcoated with, SiC. A family of SiC monofilaments are currently produced by Textron Corporation under the designation "SCS". These filaments have a C-SiC coating. Several SCS fibers with different coating thicknesses and C/Si ratios in the coating are available. SCS-2 and SCS-8 fibers were developed primarily

70

2 Fibers and Whiskers

for use in aluminum matrices while SCS-6 fibers with a thicker coating were offered for the more reactive titanium matrices (DeBolt et al., 1982). These fibers have a diameter of about 140jnm which is too large for toughening of ceramic matrices but provides a convenient model system for the interfacial studies in SiC reinforced CMCs. Textron Corporation has recently introduced a new fiber, SCS-9, with a diameter of only about 70 jam for use primarily in ceramic matrix composites. Depending on the processing conditions, fibers with a wide range of compositions from stoichiometric SiC to SiC with free C or Si can be obtained. Free carbon improves resistance to self-abrasion. SiC has two polymorphs: a-SiC with a hexagonal wurtzite structure and P-SiC with a cubic zinc blende structure. SiC fibers produced by CVD are primarily of P-SiC type with a high density of stacking faults and microtwins. Wawner and Nutt (Nutt and Wawner, 1985; Wawner et al., 1983) studied the microstructure of CVD-derived SiC fibers by transmission electron microscopy and found that columnar grains of p-SiC grow in the direction of deposition with their closed-packed {111} planes perpendicular to the direction of deposition. An interesting feature of the structure of these fibers is a distinct transition in the grain structure occurring at a diameter of ~80 |im as shown in Fig. 2-31. Columnar P-SiC crystallites in the inner SiC ring are about 40-50 nm in diameter while those in the outer sheath are ~90-100 nm. This change in the microstructure has been attributed to the design of the reaction chamber and the method of processing. The most widely used member of the SCS series of fibers is SCS-6 fiber. A detailed study of the microstructure and composition of SCS-6 fiber has been carried out by Ning, Pirouz and co-workers (1990,

y^-^M:

Figure 2-31. The transition in the grain structure of CVD-fabricated SiC fiber (from Wawner, Teng, and Nutt, 1983).

1991) using transmission electron microscopy and scanning Auger microscopy. They found the fiber to consist of many concentric layers as shown in Fig. 2-32. Starting from the fiber center, the core (carbon substrate) structure consists of randomly oriented blocks of turbostatic carbon (TC) about 1-5 nm in size. The next layer is the pyrolytic carbon coating (~1.5 j^m thick) which also has a turbostratic structure. However, the TC blocks in this layer are in the range of 30-50 nm and are preferentially oriented with their c-axis parallel to the radial direction. The SiC mantle with a total thickness of ~ 50 |im consists of four distinct layers. The transition from one layer to the next is marked by a sharp change in the grain size and degree of preferential orientation. The sharpest transition occurs between SiC-3 and SiC-4 layers. The first three layers, i.e.,

2.8 Silicon Carbide and Silicon Carbide-Based Fibers

71

outermost coating total thickness 3 jam

inner coating 1.5 (im

Figure 2-32. The ring structure of SCS-6 SiC fiber (from Ning and Pirous, 1991).

SiC-1, SiC-2, and SiC-3 layers, consist of elongated (3-SiC crystallites with the long axis parallel to the <111> crystal directions. Moving away from the center, the crystallites become increasingly larger and aligned with the radial direction. The crystallite length is — 5-15nm immediately adjacent to the pyrolytic carbon layer and of the order of microns within the SiC-3 layer. The enlargening of the grains is accompanied by increased faulting on the {111} planes. The transition from the SiC-3 region to the SiC-4 region is marked by an almost doubling in the size of the elongated p-SiC grains, which are now heavily faulted, and a sharp increase in the alignment of the <1 1 1 > directions of SiC parallel to the radial direction. A significant finding is that the SiC-4 layer is almost perfectly stoichiometric silicon carbide while the SiC-1, SiC-2 and SiC-3 regions have 10-20% excess carbon. The presence of boundaries between the first three layers of SiC is attributed to a step-wise decrease in the Si/C ratio outwards from the fiber center. The outermost carbonaceous coating on the SCS-6 fibers exhibits three distinct sublayers (Ning and Pirouz, 1991; Ning et al, 1990; Pirouz et al, 1989): the outermost sublayer is carbon-rich, containing SiC crystallites that grow in density and decrease in size in a radial direction inwards

from the fiber surface; the thin middle sublayer is silicon-free carbon; and the third, innermost sublayer is very similar in structure to the first layer. In all sublayers, carbon has a turbostratic structure with the weakly bonded graphitic planes parallel to the fiber surface. The debonding of the turbostratic graphite layers in the coating provides the weak interface desirable in ceramic matrix composites. Debonding occurs in the middle, silicon-free sublayer or at its interface with the innermost sublayer since SiC crystallites in the other two sublayer strengthen the carbon (Ning et al, 1990). The occurrence of debonding in or near the middle sublayer is desirable since debonding between the SiC sheath and the innermost coating sublayer would result in the complete removal of the coating and, therefore, a loss in fiber strength. During exposure to oxygen at high temperatures, SiC crystallites in the outer coating sublayer oxidize to SiO 2 , thus providing some protection against further oxidation of the carbon coating. DiCarlo (1989) noted that for short times of at least 30 min, the carbon content is not significantly affected by exposure to 100% oxygen up to 1200 °C. He further noted that the carbon layer is virtually unaffected by argon treatment up to 1400 °C. However, a pretreatment with argon above 900 °C followed by a 100% oxygen treatment results in accel-

72

2 Fibers and Whiskers

erated carbon attack compared to direct oxygen treatment. It has been suggested that argon pretreatment above 900 °C causes the outer coating sublayer to become enriched in carbon, thus eliminating the protective effect of SiC particles.

Dichlorodimethvlsilane Dechlorination with Na (to NaCI)

2.8.3 Polymer-Derived SiC-Based Fibers In 1975, Yajima and co-workers introduced a new process for making SiC-based fibers from a spun polymer precursor (Yajima et al., 1976 a, b, 1978; Hasegawa et al., 1980; Hasegawa and Okamura, 1983; Yajima, 1985). Following some improvement of the initial process, polymer-derived SiC multifilaments were produced on a pilot scale in 1978. In 1981, Nippon Carbon Company initiated the commercial production of these fibers in continuous form of 500 filaments/yarn under the tradename "Nicalon" The main stages in the preparation of Nicalon SiC fiber are shown in Fig. 2-33. Dimethylpolysilane (DMPS) is obtained by dechlorination of the starting material, dichloromethylsilane, with molten sodium at about 130°C under a nitrogen atmosphere. DMPS is polymerized to polycarbosilane (PCS) by heating at 450-470 °C in an autoclave under argon atmosphere. The molecular weight of PCS rises markedly with treatment temperature within the range of 450 to 470 °C. The molecular weight of PCS has a significant effect on the properties of the resultant SiC fibers. At present, an average molecular weight of about 1500 is used. Following a vacuum distillation treatment at up to 280 °C to remove the undesirable low-molecularweight PCS components, PCS is melt spun in the presence of N 2 gas at about 350 °C to form continuous polycarbosilane fibers. To improve the melt-spinnability of the PCS polymer, thus reducing the fiber di-

Dimethylpolysilane Polymerization at 470°C in autoclave (in argon)

•j-Si—(p4 Polycarbosilane Melt spinning at 350°C (in nitrogen) Polyca rbosilane fibers

Curing at 190°C in air or RT in ozone Polycarbo silane fibers made infu sible through cross-linki ng by oxygen Pyrolysis-heating to 1300°C or less in hydrogen or vacuum (100°C/hr) Si C fiber with arnorphous or microc rystalline p-SiC structure

Figure 2-33. The steps in the processing of Nicalon SiC (from Andersson and Warren, 1984).

ameter, a few percent of phenylchlorosilane is added to the starting material. This promotes a chain-like molecular structure in polycarbosilane which results in almost 30% reduction in fiber diameter. The spun PCS fibers are then cured in air for about 30 min at a temperature up to 200 °C. This induces cross-linking of PCS polymer chains by oxygen which is necessary to prevent melting of fibers during the subsequent high temperature heat treat-

73

2.8 Silicon Carbide and Silicon Carbide-Based Fibers

ment. The cured fibers are finally converted to SiC by heating at 1200-1300 °C in N 2 gas or vacuum. The pyrolysis temperature has a profound effect on the microstructure and properties of the SiC fiber (Yajima, 1985). Maximum tensile strength is achieved around 1200-1250 °C, while the Young's modulus remains virtually independent of the heat treatment temperature as shown in Figs. 2-34 and 2-35, respectively. X-ray powder diffraction patterns of SiC fibers pyrolyzed at temperatures from 900 to 1500°C indicate that fibers have an amorphous structure up to 1000 °C (Fig. 2-36). Beyond 1000 °C, crystallization of SiC to P-SiC takes place. Crystallization and grain growth become

900° 1000°

a "E & fi

1100°

5

1200°

I

1300°

1500° 20

30

40

50

60

70

80

26

Figure 2-36. X-ray diffraction patterns of Nicalon SiC fibers pyrolyzed at different temperatures (from Yajima, 1985).

400

600

800

1000

1200

1400

1600

1800

Temperature (°C)

Figure 2-34. The variation of the strength of Nicalon SiC fiber with the pyrolysis temperature (from Yajima, 1985). 250

400

600

800

1000

1200

1400

1600

1800

Temperature (°C)

Figure 2-35. The variation of the Young's modulus of Nicalon SiC fiber with the pyrolysis temperature (from Yajima, 1985).

significant beyond 1300 °C and are accompanied by a drop in fiber strength. Nicalon fibers thus produced have a non-stoichiometric composition consisting of SiC, SiO2 and free C. The presence of SiO2 accounts for the lower Young's modulus of Nicalon compared to the CVDderived SiC fibers. The oxygen is introduced mainly during the curing treatment. The initial laboratory grade fibers had a composition of 64% (by weight) SiC, 15% C and 21% SiO2 (Yajima et al., 1979). Nippon Carbon Company has since produced several generations of Nicalon fibers with progressively lower oxygen levels and slightly varying microstructures. The family of Nicalon fibers currently produced commercially includes the ceramic grade (CG) for high temperature mechanical applications, the high volume resistivity (HVR) grade (low dielectric constant fiber), and the low volume resistivity (LVR) grade (higher conductivity fiber). A low-oxygen-

74

2 Fibers and Whiskers

content Nicalon fiber with improved thermal stability has been developed recently but is not yet commercially available (Okamura et al., 1988, 1989); this fiber is discussed in Sec. 2.8.4. The nominal compositions of several generations of the ceramic grade Nicalon fiber and the experimental low oxygen fiber are shown in Table 2-13. The standard grade Nicalon fibers produced earlier had a virtually amorphous structure while the ceramic grade fibers consist of very fine crystallites of f$-SiC, about 1.7-2.5 nm in size, in an amorphous matrix (Simon and Bunsell, 1983, 1984; Okamura, 1987; Sawyer et al., 1985, 1986). The continuous amorphous phase is a silicon oxycarbide, SiCaOb (a + b = 4) and controls the modulus and fracture properties of the fiber (Lipowitz, 1991; Sawyer etal., 1987). The P-SiC crystallite size is dependent not only on the heat treatment temperature but also on the average molecular weight of the polycarbosilane Precursor (Hasegawa and Okamura, 1983; Hasegawa, 1989,1990). Stronger and stiffer fibers are obtained by using larger molecular weight PCS. Laffon etal. (1989), and Maniette and Oberlin (1989) conducted TEM studies of the fiber microstructure which have indicated that the excess carbon is in the form of nanocrystallites with a turbostratic structure. Using X-ray scattering, Lipowitz et al. (1990) found fine porosity in the range of ~ 5-20 vol.% in polymer-derived S i - C - O (Nicalon) and S i - C - N - O fibers. The pores were on the Table 2-13. Compositions of Nicalon fibers3 (Takeda etal., 1991). Fiber Ceramic grade Low oxygen a

Atomic ratio.

Si

C

O

H

1.00 1.00

1.29 1.31

0.38 0.02

0.10 0.05

order of a few nanometers in diameter. The average pore size increased and volume fraction of porosity decreased with increasing pyrolysis temperature. Porosity leads to reduced density and elastic modulus compared to a fully dense fiber. Scanning Auger depth profiling of the fiber surface has revealed that fibers possess a carbonrich surface layer of ~ 60 nm thickness (Lipowitz, 1991). The diameter of Nicalon fibers varies in the range of 10 to 20 |im with most fibers having a diameter of ~ 15 jam. Yajima (1985) showed that both fiber strength and Young's modulus decrease rapidly with increasing fiber diameter. The strength averages around 2.9 GPa, the Young's modulus around 190 GPa and density around 2.55 g/cm3. The density is lower than that of stoichiometric (3-SiC (3.16 g/cm3) due to the presence of SiO2 and C in the structure. Fiber strength varies markedly with the gage length due to flaw distribution. Testing both pilot and commercial fibers with a gage length in the range of 2-300 jim, Andersson and Warren (1984) found that fiber strength could, to a good approximation, be given by the following relationship fu

= 8.12-(lnL/7.8)

(2-17)

where L is the gage length. A similar relationship was also found by Simon and Bunsell (1982) for commercial fibers but at somewhat lower strength levels. Nicalon fibers exhibit significant thermal degradation at temperatures above ~1200°C. The thermal stability and strength loss of Nicalon fiber in various environments have been investigated by many researchers (Mah et al., 1984; Clark et al., 1985, 1986, 1987; Pysher et al., 1989; Bibbo et al., 1991). The loss of strength is more severe in non-oxidizing environments such as argon, nitrogen or vacuum where crystallization and grain growth of

2.8 Silicon Carbide and Silicon Carbide-Based Fibers

p-SiC as well as the development of a porous structure have been reported (Clark etal, 1985, 1986, 1987). Thermodynamic instability of S i - C - O system in nonoxidizing environments at temperatures above 1200 °C is due to the following carbothermic reduction reactions (Lipowitz, 1991; Luthra, 1986). 3C + SiO2 C + SiO2

SiO + CO

2C + SiO

SiC

(2-18)

The oxygen in the fiber reacts with excess carbon causing the evolution of CO and some SiO gases. This intrinsic oxidation process is accompanied by a weight loss and the development of porosity in the fiber. It has been suggested that free carbon particles in the fiber promote its thermomechanical stability by hindering creep and grain growth (Catoire etal., 1987). Once the carbon is removed, grain growth of the P-SiC crystallites occurs as observed in elevated temperature aging of fibers in argon 1200°C (Bibbo etal, 1991). Thermodynamic calculations by Luthra (1986) and experimental studies by Mah et al. (1984) have indicated that CO gas is the predominant decomposition product of the intrinsic oxidation of Nicalon fiber at temperatures above 1200°C. Bibbo etal. (1991) studied heat treated ceramic grade Nicalon fibers at 1300°C under various partial pressures of CO gas and noted that fibers retained 75% of their strength when heat treated in CO as compared to only 25% strength retention in argon. The loss in strength is much less severe and fibers exhibit a weight gain when aging takes place in an oxidizing environment such as air. This has been attributed to the formation of a stable SiO2 skin which serves as a diffusion barrier for SiO and CO gases,

75

thus slowing down the oxidation and consumption of free carbon (Mah et al, 1984; Clark et al, 1985, 1986, 1987). Mah et al. (1984) noted that strength retention was higher for a 1300-1400 °C aging than a 1200 °C aging in air for 2 h. This was linked to a better formation of the SiO2 shell at the higher temperature range. Clark et al. (1987) studied the oxidation behavior of Nicalon fibers after 12 h exposure to 1400 °C under a O 2 /He atmosphere with varying oxygen levels. They recorded a weight loss accompanied by P-SiC grain growth and a significant drop in fiber strength when heat treatment was carried out at a low oxygen level of 8 ppm. This was indicative of active oxidation of the fiber. At oxygen levels greater than 100 ppm, however, fibers exhibited a weight gain, due to the protective SiO2 layer formation, and a less severe strength reduction. The thickness of the silica layer increased with the oxygen level. The results indicated that there exists an optimum SiO2 thickness below which diffusion control is insufficient and above which residual stresses caused by the thermal expansion mismatch between the fiber and silica become large enough to bring about cracking of the SiO2 shell. The benefit of higher temperatures in producing a thicker, more protective silica coating is somewhat offset by the crystallization of silica to cristobalite at temperatures above 1400°C (Clark etal, 1985, 1986, 1987). Upong cooling to room temperature cristobalite undergoes a phase transformation to ot-cristobalite at about 250 °C. This phase change is accompanied by a volume change which leads to the cracking of SiO2 shell. The need for a fiber with better thermal stability than Nicalon resulted in the introduction of a S i - T i - C - O fiber by Ube Industries, Ltd. under the tradename "Tyranno" (Yamamura etal, 1988). The

76

2 Fibers and Whiskers

synthesis and pyrolysis of polytitanocarbosilane) have been described by Yajima et al. (1981) and by Okamura (1987). Tyranno fibers are produced by the melt spinning and pyrolysis of polytitanocarbosilane in a process analogous to the production of Nicalon fiber. Polytitanocarbosilane is formed by the cross-linking of polycarbosilane with a titanium compound such as titanium tetraisopropoxide and is easily spun into fiber form. The addition of 1.5 to 4.0 wt.% Ti was made to improve the thermal stability of the fiber. Tyranno fiber is reported to have a predominantly amorphous structure up to 1300°C (Yamamura et al., 1988). It has a strain to failure of about 1.4 to 1.7% which is considerably greater than that of Nicalon fiber. Thermal stability studies of the Tyranno fiber in a glass-ceramic matrix, however, have shown it to be less stable and more reactive than Nicalon fiber (Brennan, 1991). The reason for this was determined to be the significantly higher oxygen level of the Tyranno fiber (~20at.%). Fischbach et al. (1988) investigated the microstructure and mechanical properties of Tyranno fibers at room temperature in the as-fabricated condition and after heat treatment at 1300°C and 1350 °C under a nitrogen atmosphere. They reported significant reductions in both strength and modulus of fibers after heat treatment, accompanied by an almost doubling of the mean crystallite size. 2.8.4 Recent Developments in SiC-Based Fibers

The inability of both Nicalon and CVDderived SiC fibers to withstand continuous exposure to the high temperatures required of ceramic composites applications has sent off a wave of research activities in pursuit of a high temperature SiC fiber. None of these efforts, however, have yet

produced a new commercial fiber. Among these efforts is the recent development of a low oxygen content Nicalon fiber (the HiNicalon fiber). It was described above that the thermal instability of the commercial Nicalon fiber is due to its high oxygen concentration of about 10 wt.% which is introduced during the oxidative cross linking step of the fiber production (Clark, 1985). By replacing the oxidation curing with electron beam radiation curing, Okamura etal. (1988, 1989) have succeeded in preparing fibers with a low oxygen content of about 0.4%. The low oxygen fiber reportedly has a high strength of 2.8 GPa and an elastic modulus of 270 GPa which is considerably higher than that of the ceramic grade fiber (Takeda etal., 1992). Takeda etal. (1991, 1992) compared the thermal stability of the low oxygen content fibers with the oxidation-cured ceramic grade fibers. Fibers were exposed to temperatures in the range of 1500 to 2000 °C in an argon atmosphere for periods of 1 h and 10 h. In contrast to the dramatic compositional changes and severe crystallization of the ceramic grade fiber, the low oxygen fiber virtually retained its chemical composition and exhibited only a ten-fold increase in its crystallite size up to 2000 °C. The beneficial effect of a low oxygen content has also been demonstrated by Toreki et al. (1992) who prepared continuous silicon carbide fibers with an oxygen content of less than 2 wt.% by dry spinning of polycarbosilane solution and subsequent pyrolysis of the polymer fiber. Ube Industries Ltd. has also developed a new S i - T i - C - O fiber with lower oxygen content (~13 wt.%) and therefore more refractoriness than Tyranno fiber. The new fiber is marketed under the tradename Lox-M. It has a diameter in the range of 8.5-11 jam and is available in various tow sizes from 400 to 1600 filaments per tow.

2.9 Silicon Nitride-Based Fibers

The microstructure of the fiber borders on the amorphous, consisting of extremely fine-grained (3-SiC (<0.04nm) with dispersed TiC particles of up to 0.2 nm in size (Brennan, 1991). A major research effort at Dow Corning has led to the development of a nearly stoichiometric crystalline SiC fiber by high temperature pyrolysis (>1600°C) of meltspun organosilicon polymers. The fiber is over 95% (3-SiC with a crystallite size of 30 to 40 nm. Also among the new developments is the fabrication of a sintered oc-SiC fiber by Frechette and co-workers (1991) at Carborundum Company. The fiber is produced in a continuous process by sintering spun filaments of submicron SiC power in a polymeric binder. Fibers with diameter in the range of 25-100 jim and an average strength of about 1 GPa have been produced. oc-SiC fibers have shown almost no loss of strength after 4 h of exposure to a temperature of 1550°C in air.

2.9 Silicon Nitride-Based Fibers The polymer precursor route offers considerable flexibility in terms of the fiber compositions and microstructures that can be produced (Lipowitz, 1991; Okamura, 1987; Okamura et al., 1988). In addition to silicon carbide fibers described above, continuous silicon nitride (Okamura et al., 1987), silicon carbonitride (Legrow et al., 1987; Penn et al., 1982), and silicon oxynitride (Okamura, 1987) fibers have been prepared by the synthesis and pyrolysis of organosilicon polymers. However, at the time of this writing, none of these fibers have been commercialized. Silicon carbonitride fibers with a typical composition of 60.0 wt.% Si, 32.6 wt.% N, 2.3 wt.% C, and 2.2 wt.% O are produced

71

by Dow Corning Corp. from hydridopolysilazane (HPZ) (Lipowitz, 1991; Legrow et al., 1987). HPZ polymer is synthesized by the reaction of trichlorosilane with hexamethyldisilazane and is melt-spun into fiber form. The preceramic fiber is cured in an oxygen-free argon gas saturated with trichlorosilane vapor. The cured fibers with chlorosilane cross linking are then pyrolyzed at temperatures up to 1200 °C under flowing high purity nitrogen to silicon carbonitride. Silicon carbonitride fibers thus produced have a diameter ranging from 10-12 Jim, a tensile strength of ~ 2.8 GPa, and an elastic modulus of ~ 180 GPa. The fiber microstructure consists of amorphous silicon carbonitride with about 5 wt.% nanocrystalline carbon with a turbostratic structure. The fiber is not fully dense, but contains about 20 vol.% fine porosity. Fibers show a surface layer of ~0.5 nm thickness that is richer in oxygen and poorer in nitrogen compared to the uniform S i - C - N - O composition of the interior. It has been reported that HPZ fibers tested in air at temperatures up to 1400 °C behaved similarly to Nicalon and Tyranno fibers in that strength and modulus dropped sharply above about 1200 °C (Lipowitz, 1991). Okamura et al. (1987) have prepared continuous silicon carbide (Si3N4) and silicon oxynitride (Si-N-O) fibers from polycarbosilanes. The melt-spun polycarbosilane fibers were given an electron beam radiation curing treatment for the preparation of silicon nitride fibers and an oxygen curing treatment for silicon oxynitride fibers. The cross-linked preceramic fibers were then heat treated in NH 3 gas at temperatures up to 1400 °C to obtain silicon nitride and silicon oxynitride fibers with a diameter of 11-13 jam. X-ray diffraction patterns of fibers nitrided at different temperatures revealed that both fibers were

78

2 Fibers and Whiskers

amorphous up to 1300°C, but crystallized to oc-Si3N4 at higher temperatures. For the silicon nitride fiber, the strength and modulus reached their maxima, ~ 1.0-1.5 GPa for strength and -100-130 GPa for modulus, at 1300°C, beyond which temperature they both dropped sharply with strength reaching almost zero at 1400 °C due to the crystallization of oc-Si3N4. The strength and modulus of the silicon oxynitride fiber depend on the level of oxygen introduced during the curing process. Fibers cured at 160 °C (O/Si = 0.36) showed the highest strength, modulus, and thermal stability. These fibers had a higher strength and modulus than the silicon nitride fibers.

2.10 Whiskers 2.10.1 Introduction

Whiskers are single crystal reinforcements with an aspect ratio of usually over 20. The high degree of structural perfection of whiskers which stems from their single crystalline morphology and small diameter (usually submicron) bestows on them a very high Young's modulus and a strength that approaches the theoretical strength of the material calculated by Orowan (1949) to be (2-19) where E is the Young's modulus, y is the surface energy per unit area, and a0 is the equilibrium spacing of the atomic planes. Carbon whiskers have the highest specific strength and specific Young's modulus among all existing reinforcements. The absence of crystalline defects is also responsible for the better creep resistance and, therefore, higher temperature capability of the whiskers compared to their fiber counterparts. A variety of ceramic whiskers are

currently produced for use primarily in ceramic and metal matrices. Some of these whiskers and their properties are shown in Table 2-14. The addition of whiskers to a monolithic ceramic raises the toughness through a variety of mechanisms which include crack deflection, whiskers bridging, and whisker pullout. Three to four-fold increases in the toughness of alumina have been reported by the incorporation of SiC whiskers (Wei and Becher, 1985; Tiegs and Becher, 1987). Although whiskers are usually quite short, they possess large aspect ratios, usually over 20, due to their small diameters and are thus efficient reinforcements. Whisker diameter is also a critical parameter in terms of its strength; increasing the diameter raises the probability of the presence of defects, hence lowering the strength. Being single crystals, most whiskers have a non-circular cross section. The shape of the cross section depends on the crystal structure and the growth orientation of the whisker. Thus, it can be triangular, hexagonal, rhombohedral, or another shape. For a non-circular cross section, whisker diameter is defined as the square root of the cross-sectional surface area. The shape of the whisker affects the total whisker/matrix interfacial area and hence the composite properties. Whiskers with submicron diameters may pose a potential health hazard similar to that of asbestos (Bogoroch and Luck, 1988). Asbestos fibers with a diameter less than 1 jim and aspect ratio greater than five are known to cause irritation of the eyes, skin, and pulmonary system and may even lead to pulmonary fibrosis and cancer. Precautions must, therefore, be taken during handling of dry whiskers or machining of whisker reinforced composites.

79

2.10 Whiskers

Table 2-14. Whiskers and their properties. Material

Designation

Phase

Manuf.

Diameter Density

(mm) SiC SiC

Silar SC-9

sew

a 95% p-5% a

SiC

Tokama

P

SiC SiC

VLS Versite

P

Si 3 N 4

SNW

>97%a

Si3N4 C

Versite —

— —

SiO2

XPVI

10-6oC-l

Young's Tensile Length modulus strength )

(GPa)

(GPa)

(mm)

3.2 3.21

4.7 4.4-4.7

689 483

6.89 20.68

10-80 10-40

3.19

—

400-700

3-14

50-200

— —

— —

578 -560

8.40 -11

0.2-0.5

—

—

—

—

-5,000 up to 3,000 50-300

— <1.0

— —

— —

-245 —

-5.6 —

2.25

—

-

-

ACMC 0.6 0.05-0.2 Tateho Chemical Tokai 0.1-0.5 Carbon Los Alamos 5-6 Versar 1-5 Tateho Chemical Versar NiKKiso

(g/cm3) (x

Coef. therm. expan.

Amorphous J. M. Huber up to 0.02 Corp.

2.10.2 Silicon Carbide Whiskers

At present, SiC whiskers are the most widely used reinforcement in whisker reinforced composites. They have been incorporated into ceramic matrices, including alumina (Shalek et al, 1986) and silicon nitride (Buljan et al., 1987), as well as metal matrices (Arsenault and Fisher, 1983; Webster, 1982). SiC whiskers come in two crystallographic variations; a and p. Some whiskers contain both phases. The cubic crystal structure of SiC is known as p and the hexagonal and rhombohedral structures are referred to as a. The mechanical properties of commercially produced P-SiC whiskers are generally superior to those of oc-SiC whiskers. This is probably due to less structural variation and better consistency of the P-type whiskers. A variety of manufacturing processes exist for SiC whiskers. The physical charac-

— up to 2,000 10-20 (aspect ratio)

teristics of the whiskers produced depend largely on the fabrication method as shown by the variation in the diameter, length, crystal structure, and therefore properties of SiC whiskers given in Table 2-14. One method of manufacture is chemical vapor deposition in which P-silicon carbide whiskers are deposited onto a carbon substrate by the hydrogen reduction of methyltrichlorosilane gas at a high temperature. Advanced Composite Materials Corporation (formerly ARCO Metals) manufactures a predominantly oc-silicon carbide whisker under the tradename Silar SC-9 by the heat decomposition of rice hulls (Lee and Cutler, 1975). Rice hulls contain about 1520% silica and are a cheap and abundant source for silicon carbide whiskers. Submicron whiskers are produced by heating the rice hulls initially at 700 to 900 °C and subsequently at 1300 to 1500°C in a flowing inert gas. The maximum yield of SiC is on the order of 60%, of which only 10 to 15%

80

2 Fibers and Whiskers

is in the form of whiskers, and the remainder is in the form of SiC particulates (Bowman and El-Rahiby, 1991). Higher yields have recently been achieved by Krishnarao et al. (1992) who produced SiC whiskers by direct pyrolysis of rice hulls in a single stage vacuum pyrolysis process. These investigators have identified four competitive processes during direct pyrolysis of rice hulls: (i) formation of SiC whiskers, (ii) formation of SiC polycrystals, (iii) crystallization of amorphous silica, and (iv) formation of graphite from amorphous carbon. The rice hull whiskers are characterized by a hexagonal cross section as shown in Fig. 2-37. The structural and chemical characteristics of these whiskers have been studied in detail by Sharma et al. (1984) and by Nutt (1984). They are found to contain a high density of planar faults lying on

(b) Figure 2-38. The defect structure of Silar SC-9 SiC whisker on cross sections (a) perpendicular and (b) parallel to the whisker axis (from Nutt, 1984).

Figure 2-37. A Silar SC-9 SiC whisker with its hexagonal cross-section (from Wereszczak and ParviziMajidi, 1991).

closed-packed planes perpendicular to the whisker axis. Faulting resulted in the formation of thin lamellae of a and (3 polytypes normal to the whisker axis. Core regions of whiskers are often filled with small cavities ranging in size from 1 to 20 nm, accompanied by partial dislocations. Figure 2-38 shows the defect structure of a Silar SiC whisker. The thermal stability of Silar SC-9 whiskers under oxidizing conditions has been studied by LaTorre et al. (1987) by

2.10 Whiskers

means of the FT-IRRS method. They reported minimal oxidation of whiskers heat treated for 5 hours below 1000 °C. Heat treatment at 1000 °C for up to 96 hours resulted in partial oxidation of the whiskers and the formation of amorphous silica. Whiskers exhibited extensive oxidation after exposure at 1300 °C for 24 hours which was accompanied by the emergence of a non-stoichiometric crystalline silica phase. The same silica phase was observed at 1500 °C where whiskers oxidized severely after only 1 h of exposure. Another method for growing SiC whiskers is the vapor-liquid-solid (VLS) process which was first reported in the early 1960's by Merz (1960) and Wagner and Ellis (1965) and further developed by Milewski and co-workers (Shyne et al., 1986; Milewski etal., 1985). The VLS process has been adopted by Los Alamos National Laboratory and, more recently, by Carborundum Company for the manufacture of p-SiC whiskers. A summary of advances made in the VLS process over the twenty five years of its development is given by Hollar and Kim (1991). Fig. 2-39 illustrates a schematic of the VLS process for SiC whisker growth. A layer of stainless steel catalyst particles are placed on a graphite substrate. Upon heating to temperatures above 1400 °C, the stainless steel particles melt and form liquid catalyst balls on the graphite plate. A mixture of SiO (generated by the reduction of SiO2 with carbon), methane, and hydrogen gases are passed

Liquid Catalyst Solid Catalyst

1

Figure 2-39. Schematic of the VLS process for SiC whisker growth (from Milewski and Katz, 1987).

81

over the liquid metal balls whereupon Si and C are extracted from the gases by the liquid catalyst. Liquid iron becomes supersaturated in Si and C, and SiC precipitates and grows in the form of a single crystal rod between the catalyst ball and the substrate. As the whisker grows, it lifts the ball off the substrate. Whiskers thus grown are almost completely cubic P-phase with their growth axis along the <111> direction. They have a rounded triangular cross section with an effective diameter of 0.5 to 1.0 jim and lengths ranging from 1 mm to several centimeters. The VLS-grown SiC whiskers contain a lower concentration of defects, and therefore a higher strength and thermal conductivity, than whiskers from rice hulls. Petrovic et al. (1985) measured the tensile strength and Young's modulus of VLSgrown SiC whiskers through direct experiments and found quite a large scatter in both properties. The strength decreased with increasing whisker length, following a Weibull distribution with a Weibull modulus of 1.77 (see Sec. 2.2). Whiskers exhibited a considerable variation in their diameter which contributed to the scatter in strength. The strength averaged at 8.40 GPa but reached as high a value as 23.74 GPa which was almost half of the calculated theoretical strength of 46.3 GPa for SiC. The authors predicted a Young's modulus of 511 GPa for cubic SiC along the <111> direction. The experimentally determined modulus was about 15% higher at an average value of 581 GPa. Whisker surface chemistry and morphology affect the fabrication and properties of the resulting composites. Karasek and co-workers (Karasek, 1989; Karasek etal., 1989, 1991) examined the chemistry and morphology of commercial SiC whiskers from various sources, using photoelectron spectroscopy (XPS) and scanning

82

2 Fibers and Whiskers

transmission electron microscopy (STEM). They found that all whiskers contained some carbon in their bulk composition. These investigators identified four categories of whisker surface chemistries: (1) a high oxygen (> 20 at.%) surface layer resembling SiO 2 , prevalent in the earlier versions of whiskers when a HF acid wash was not employed to remove the silica surface layer, (2) a low oxygen (<20%, >10%) layer resembling S i - O - C glass (black glass), (3) a high oxygen layer resembling S i - O - C glass, and (4) a predominantly hydrocarbon layer in those cases where a carbon layer had been deposited on the surface of the whiskers. All commercial SiC whiskers contain particulate matter which is defined as particles with aspect ratio less than 5. Particle sizes typically range from submicron to > 50 \im. Lee and Sheargold (1987) measured the particulate levels of SiC whiskers using an optical microscopy method coupled with a wet sieve analysis. They showed that the total particulate level ranged from 19% to 46% among whiskers produced by different manufacturers. The authors further examined the effect of the particulate content on the flexural strength and fracture toughness of SiCw/borosilicate glass composites and concluded that particulate levels found in commercial whiskers did not significantly degrade the mechanical properties of the composite. Because of the difficulty in homogeneous mixing of whiskers with ceramic powders and the potential health hazards of handling small whiskers, several attempts have been made to form SiC whiskers in situ in a ceramic matrix. Wang and Fischman (1991) prepared (3-SiC whiskers in situ in a Si 3 N 4 matrix by the reaction of silicon nitride with graphite in an argon atmosphere. (3-SiC formation initiated at 1400 to 1450 °C. The whiskers were needle-like

with lengths and diameters changing with temperature. The product phases were dependent upon the Si 3 N 4 : C ratio of the raw materials and the processing temperature. This "chemical mixing process" route was also followed by Yamada et al. (1988) for in situ preparation of SiCw/Si3N4 composites. Jagota and Parvizi-Majidi (1990) used sol-gel processing to produce in situ composites of SiC whiskers in a mullite matrix. This was achieved by mixing mullite precursor gel with carbon-doped silica gel and crystallization above 1400 °C. 2.10.3 Silicon Nitride Whiskers

Like silicon carbide, silicon nitride (Si3N4) whiskers have two crystallographic modifications. oc-Si3N4 has a trigonal structure while (3-Si3N4 is hexagonal. Whiskers are available in a wool product form with average diameter of 10 (im and in small-diameter needle form. a-Si 3 N 4 whiskers are produced by the VLS technique in essentially the same way SiC whiskers are manufactured with the exception that methane is replaced by ammonia gas. 2.10.4 Oxide Whiskers

Sapphire (a-Al2O3) whiskers have been produced in the form of a wool product with a whisker diameter of 1 to 3 |im as well as ribbon and needle products. The strength of sapphire whiskers is found to decrease with increasing the whisker diameter (Milewski and Katz, 1987). The strength improves considerably when the surface flaws are removed by chemically polishing the whiskers in, for example, orthophosphoric acid (Bayer and Cooper, 1967). A cobweb whisker of amorphous SiO 2 with a silicon core has been recently commercialized by J. M. Huber Corporation under the designation XPV1 (Milewski

2.12 References

and Katz, 1987). Cobweb whiskers are defined as single crystal fibrous materials with diameters below 0.1 |im. The XPV1 whisker has a diameter of about 10 nm. These whisker-like fibers are expected to have extremely high strengths due to their small size and have demonstrated great efficiency in reinforcing polymer matrices where they can be used on their own or together with large-diameter fibers.

2.11 Closing Remarks A large number of fibers and whiskers have been developed for the reinforcement of composites with polymeric, metallic, intermetallic, or ceramic matrix materials. This Chapter has provided an introduction to the synthesis and manufacturing, structure, and properties of fiber and whisker reinforcements available on both commercial and developmental scales. Attempt has been made to give as comprehensive and accurate a description as possible. However, the field of fiber development, particularly for high-temperature composites, is a highly active and competitive one which inhibits the presentation of an in-depth and exhaustive review of the subject.

2.12 References Abe, Y, Horikiri, S., Fujimura, K., Ichiki, E. (1982), in: Proc. 4th Int. Conf. on Composite Materials, ICCM-IV: Hayashi, T, Kawabata, K., Unekawa, S. (Eds.), Tokyo, Japan, p. 1427. Andersson, C. H., Warren, R. (1984), Composites 15, 16. Antill, J. E., Warburton, J. B. (1971), Corrosion Science 11, 337. Arsenault, R. X, Fisher, R. M. (1983), Scr. MetalL 17, 67. Bacon, R., Tang, M. (1964), Carbon 2, 221. Bahl, O. P., Mathur, R. B., Seth, R. L., Dhami, T. L. (1980), at: Carbon '80, 3rd Int. Carbon Conf., Baden-Baden, F.R.G.: Preprints, p. 602. Barham, P. X, Keller, A. (1985), /. Mater. Sci. 20, 2281.

83

Basche, M., Fanti, R., Galasso, F. (1968), Fibre Sci. Technol.

1,1.

Basche, M., Fanti, R., Galasso, F , Kuntz, U., Schile, D. (1972), U.S. Patent No. 3 679 475. Basche, M., Fanti, R., Galasso, F , Kuntz, U., Schile, D. (1975), U.S. Patent No. 3 861 953. Bayer, P. D., Cooper, R. E. (1967), J. Mater. Sci. 2, 347. Behrendt, D. (1977), Longitudinal Residual Stresses in Boron Filaments, Special Technical Publication 617, American Society for Testing and Materials, Philadelphia, PA, USA, pp. 215-276. Behrendt, D. (1978), Axial Residual Stresses in Boron Fibers, NASA TM-73894. Behrendt, D. (1980), Calculation of Residual Principal Stresses in CVD Boron on Carbon Filaments, NASA TM-81456. Bennett, S. C , Johnson, D. J. (1978), in: Proc. 5th London Carbon and Graphite Conf, Vol. 1, London: Soc. Chem. Ind., p. 377. Bennet, S. G., Johnson, D. J. (1979), Carbon 17, 25. Bentur, A., Mindess, S. (1990), in: Fiber Reinforced Cementitious Composites: Amsterdam: Elsevier Applied Science, p. 360. Bhardwaj, X, Krawitz, A. (1983), J. Mater. Sci. 18, 2639. Bibbo, G. S., Benson, P. M., Pantano, C. G. (1991), /. Mater. Sci. 26, 5075. Birchall, I D . , Bradbury, X A. A., Dinwoodie, X (1985), in: Strong Fibers: Watt, W, Perov, B. V. (Eds.). Amsterdam: Elsevier Science Publishers, pp. 115-154. Blades, H. (1973), U.S. Pat. 3 767 756. Blades, H. (1975), U.S. Pat. 3 869 429. Bogoroch, R., Luck, S. R. (1988), in: Whisker and Fiber Toughened Ceramics: Bradley, R. A., Clark, D. E., Larsen, D. C , Stiegler, X O. (Eds.). Metals Park, OH: ASM International, pp. 81-89. Bowman, K. X, El-Rahaiby, S. K. (1991), CIAC Newsletter 2(1), 1. Brennan, X (1991), Interfacial Studies of Refractory Glass-Ceramic Matrix/Advanced SiC Fiber Reinforced Composites, R91-918246-2. East Hartford, CT: United Technologies Research Center. Budd, S. M. (1961 a), Phys. Chem. Glasses 2, 111. Budd, S. M. (1961b), Phys. Chem. Glasses 2, 114. Buljan, S. T, Baldoni, X G., Huckabee, M. L. (1987), Am. Ceram. Soc. Bull. 66, 347. Bunsell, A. R. (1975), /. Mater. Sci. 10, 1300. Bunsell, A. R., Simon, G., Abe, Y, Akiyama, M. (1988), in: Fiber Reinforcements for Composite Materials: Bunsell, A. R. (Ed.). Amsterdam: Elsevier Science Publishers, pp. 427-478. Burgman, X A., Hunia, E. M. (1970), Glass Technol. 11, 147. Calundann, G., Jaffe, M., Jones, R. S., Yoon, H. (1988), in: Fiber Reinforcements for Composite Materials: Bunsell, A. R. (Ed.). Amsterdam: Elsevier Science Publishers, pp. 211-248. Camahort, X (1968), /. Compos. Mater. 2, 104.

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Capaccio, G., Ward, I. M. (1985), Nature 243, 2281. Carlsson, J. (1979), J. Mater. Sci. 14, 255. Carlsson, I, Lundstrom, T. (1979), /. Mater. Sci. 14, 966. Catoire, B., Sotton, M., Simon, G., Bunsell, A. R. (1987), Polymer 28, 751. Charles, R. X, Shawn, R. R. (1962), Delayed Failure of Poly crystalline and Single Crystal Alumina, General Electric Technical Report, No. 62-RL-3081M. Chou, T. W. (1992), Microstructural Design of Fiber Composites. Cambridge: Cambridge University Press. Clark, T. X, Arons, R. M., Stamaloff, X B., Rabe, X (1985), Ceram. Eng. Sci. Proc. 6, 576. Clark, T. X, Jaffe, M., Rabe, X, Langley, R. (1986), Ceram. Eng. Sci. Proc. 7, 901. Clark, T. X, Prack, E. R., Haider, M. L, Sawyer, L. C. (1987), Ceram. Eng. Sci. Proc. 8, 111. Coleman, B. D. (1958), J. Mech. Phys. Solids 7, 60. Cornie, X, Suplinskas, R., DeBolt, H. (1981), Surface Enhancement of Silicon Carbide Filament for Metal Matrix Composites, ONR Contract N0001-4-79-C0691, Summary Report. Costello, X A., Tressler, R. E. (1986), J. Am. Ceram. Soc. 69, 674. Crane, R., Krukonis, X (1975), Am. Ceram. Soc. Bull. 54, 184. DeBolt, H. (1982), in: Handbook of Composites: Lubin, G. (Ed.). New York: Van Nostrand Reinhold, p. 171. DeBolt, H., Krukonis, V. (1973), Improvement of Manufacturing Methods for the Production of Low Cost Silicon Carbide Filament, AFML-TR-73-140. DeBolt, H., Krukonis, V., McKee, X, Prescott, R., Wawner, F. (1972 a), Development and Demonstration of a Low Cost Boron Filament Formation Process, Tech. Rept. AFML-TR-72-271. DeBolt, H., Krukonis, V., Neff, R., Wawner, F. (1972b), in: 17th National SAMPE Symp., April 11-13, Los Angeles, CA. The Society for the Advancement of Material and Process Engineering. Covina, CA, USA. DeBolt, H., Krukonis, V., Wawner, F. (1973), in: Silicon Carbide-1973, Proc. 3rd Int. Conf. on Silicon Carbide. Miami, FL: University of South Carolina Press. DeBolt, H., Krukonis, X, Wawner, F. (1974), 19th National SAMPE Symp. The Society for the Advancement of Material and Process Engineering. Covina, CA, USA. DeBolt, H., Suplinskas, R., Cornie, X, Henze, T, Hanze, A. (1982), U.S. Pat. 4 340 636. Dhingra, A. K. (1980), Phil. Trans. R. Soc. London A294, AW. DiCarlo, X (1979), Mechanisms of Boron Fiber Strengthening by Thermal Treatment, NASA TM79077. DiCarlo, X, Wagner, T. (1981), Ceram. Eng. Sci. Proc. 2, 872.

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General Reading Advanced Fibers and Composites, by F. S. Galasso (1989). Gordon and Breach Science Publishers. Carbon Fibers, by J. B. Donnet and R. C. Bansal (1990). Marcel Dekker, Inc., New York. Fibre Reinforced Cementitions Composites, by A. Bentur and S. Mindess (1990). Elsevier Applied Science. Fibre Reinforcements for Composite Materials, ed., A. R. Bunsell (1988). Elsevier Science Publishers. Handbook of Reinforcements for Plastics, eds., J. V. Milewski and H. S. Katz (1987). Van Nostrand Reinhold Company Inc. Strong Fibers, eds., W Watt and B. V. Perov (1985). Elsevier Science Publishers. Strong Solids, Chapter 2: Strong Fibrous Solids, by A. Kelly and N. H. Macmillan (1986). Oxford University Press. Fiber Reinforced Ceramic Composites, by K. S. Mazdiyasni (1990). Chapters 3, 4, 5 and 6, Noyes Publications.

3 Polymer Matrix Composites Golam M. Newaz Mechanics Section, Battelle Memorial Institute, Columbus, OH, U.S.A.

List of 3.1 3.2 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.2 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.1.1 3.5.1.2 3.5.1.3 3.5.1.4 3.5.2 3.5.2.1 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.6.5 3.7 3.7.1 3.7.2 3.7.3 3.8

Symbols and Abbreviations Introduction Major Types of Polymer Composites Fabrication of Advanced Composites Thermoset Composites Hand Lay-Up Technique Semiautomatic Lay-Up Automatic Lay-Up Thermoplastic Composites Micromechanics Stiffness Properties Thermal Expansion and Transport Properties Quality Assurance Characterization Techniques - Overview High Performance Liquid Chromatography Infrared Spectroscopy Thermal Analysis Rheological Analysis Calculation of Statistically Based Material Properties Estimation of B-Basis Values Test Methods Tensile Test Method Compression Test Methods Shear Test Method Flexure Test Methods Mode I Interlaminar Fracture Test Damage Tolerance Damage Characterization and Detection Compressive Loading and Laminate Buckling Residual Properties References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

90 92 94 95 95 95 96 96 96 97 98 98 99 100 100 101 102 103 104 106 107 108 109 Ill 112 113 114 117 118 119 119

90

3 Polymer Matrix Composites

List of Symbols and Abbreviations a a\ b' b C d E' E" EB Ef Em Ex Ey El Gx Gf Gm Gxy H Kx Km K{ L P r rx S Sm t T Tg Tm Ff Vm w Z

crack length fiber cross-section dimension beam width compliance thickness storage modulus loss modulus modulus of elasticity in bending fiber modulus matrix modulus longitudinal modulus transverse modulus flexural rigidity strain energy release rate fiber shear modulus matrix shear modulus in-plane shear modulus heat longitudinal thermal conductivity matrix thermal conductivity fiber thermal conductivity span length load radial distance slope of initial tangent line bending stress m a x i m u m tensile stress time temperature glass transition temperature melting temperature fiber volume fraction matrix volume fraction width of specimen out-of-plane direction

<xx af am y12 712 sx zv

longitudinal composite linear thermal expansion coefficient linear thermal expansion coefficient of fiber linear thermal expansion coefficient of matrix in-plane shear strain in material coordinate system ultimate shear strain longitudinal normal strain transverse normal strain

List of Symbols and Abbreviations

r\ rj* vf vm vxy € ax rxy T12

factor in Halpin-Tsai equation dynamic complex viscosity Poisson's ratio of fiber Poisson's ratio of matrix Poisson's ratio in the L-T plane measure of fiber reinforcement normal stress along loading axis in-plane shear stress in-plane shear stress in material coordinate system

AN OVA ATR CTE DCB DDA DMA DSC DTA FTIR HPLC IFT IITRI IR NDT TBA TGA TMA

analysis of variances method attenuated total reflection coefficient of thermal expansion double contilever beam dynamic dielectric analysis dynamic mechanical analysis differential scanning calorimetry differential thermal analysis Fourier transform infrared spectroscopy high performance liquid chromatography interlaminar fracture toughness Illinois Institute of Technology Research Institute infrared nondestructive testing torsional braid analysis thermal gravimetric analysis thermal mechanical analysis

91

92

3 Polymer Matrix Composites

3.1 Introduction Composite materials are made from two or more different materials which include a compatible matrix or resin material and a reinforcing phase e.g. fillers and fibers. The basic purpose of making composites is to tailor properties of materials geared toward specific applications. Composite materials provide the designer with considerable flexibility in designing the structure. One of the most prominent composite materials is the class of composites known as polymeric composites which is also often referred to as reinforced plastics. Polymeric composites were introduced in the early 1940s. The need for specific structural capabilities during the second World War led to limited applications of composite materials. Cotton fiber reinforced phenolic resin composites were used for semistructural applications as early as 1941. By the end of the war, glass fiber-reinforced composite materials were beginning to find a way into structural applications. Gradually, applications of polymeric composites continued to grow over the years finding use in aircraft and naval ships and in commercial engineered structures such as boats, cars and appliance housings. And, it appears that polymeric composites will continue to have the special status for high volume applications and in dominating the composites industry as one of the most practical and useful light-weight structural materials. There are two major types of polymeric composites, which include short-fiber and continuous-fiber composites. The shortfiber composites have found major applications in body panels of automotive vehicles as secondary structural components. Short-fiber composites are primarily reinforced with glass fibers. They are also extensively used in housings for many types

of appliances. Advanced polymeric composites are material systems which contain aligned strong and stiff fibers in a resin matrix and are used for secondary as well as primary structural applications. A generic example is carbon fiber reinforced epoxy composite. Application of composite materials are most favorable for structures where gross weight to payload ratio is most unfavorable. Table 3-1 lists the ratio for many structures to illustrate the point. Compared with conventional metallic materials such as steel and aluminum, commonly used polymeric composites offer several advantages in relation to properties, namely, in terms of specific strength, specific tensile modulus, and coefficient of thermal expansion as shown in Figs. 3-1 and 3-2. Given the consideration of gross weight to payload ratio, it is clear that composite materials offer excellent opportunities. Tailoring properties by orienting fibers to provide directional strength and stiffness are other advantages only offered by composite materials. Typical material properties of unidirectional carbon fiber composites are shown in Table 3-2. In this chapter, we will discuss several important aspects of continuous fiber polymeric composites primarily used for Table 3-1. Gross weight to payload ratio for common structures. Bicycle 1st Benz Car Suitcase Truck Passenger Car Fokker Aircraft Boeing 747 Metro Train Concorde Aircraft Space Shuttle Saturn V Rocket

1.1 1.6 1.7 2.0 4.0 4.0 6.0 10.0 12.0 70.0 500.0

93

3.1 Introduction

Table 3-2. Typical properties of unidirectional-fiber-reinforced epoxy resins (Agarwal and Broutman, 1980). Fiber type Property

E-Glass

Kevlar-49

Graphite (Thornel 300)

Fiber volume fraction Specific gravity Tensile strength, 0° (MPa) Tensile modulus, 0° (GPa) Tensile strength, 90° (MPa) Tensile modulus, 90° (GPa) Compression strength, 0° (MPa) Compression modulus, 0° (GPa) Compression strength, 90° (MPa) Compression modulus, 90° (GPa) In-plane shear strength (MPa) In-plane shear modulus (GPa) Longitudinal Poisson's ratio (vLT) Interlaminar shear strength (MPa) Longitudinal coefficient of thermal expansion (10~6/°C) Transverse coefficient of thermal expansion (10~6/°C)

46 1.80 1104 39 36 10 600 32 138 8

60-65 1.38 1310 83 39 5.6 286 73 138 5.6 60 2.1 0.34 69 -2.3 to - 4 . 0 a 35 b

63 1.61 1725 159 42 10.9 1366 138 230 11 95 6.4 0.38 113 0.045 20.2

- 7 9 ° C t o +100°C;

b

0.25 31 5.4 36

- 1 9 5 ° C t o +120°C.

monly used thermoset and thermoplastic composites reinforced with carbon fibers. There is detailed discussion of commonly used processing methods followed by information on quality assurance methodologies and test methods to produce reliable mechanical properties. There is a

aerospace applications with special emphasis on properties of these composites. Information on application and use of short fiber composites and their properties can be found in Chap. 7 in this Volume. This chapter is organized into several subareas which include a description of com-

Boron/epoxy l01

Graphite / aluminum (GY70/AI) Graphite / metal composiles Graphite / magnesium Preferred range of CTE Graphite / glass (GY70/GU

12

16

Specific tensile modulus, m

20x10*

Figure 3-1. Composite materials: specific tensile modulus and CTE. HMS, GY70: carbon fibers; Ep: epoxy (abbr.); Gl: glass (abbr.).

94

3 Polymer Matrix Composites

T300Gr/Ep Specific tensile strength (cmx2.54*10«)

2P100Gr/mg HMS Gr/Ep P 100 Gr/Al

Beryllium

Aluminum — *

1

2

3

4

5

6

Specific tensile modulus (cmx 2.54x10* )

separate discussion on critical properties of polymer matrix composites which includes the issue of damage tolerance. It may be noted that detailed micromechanics of continuous fiber composites is not recounted in this presentation. This type of analysis can be easily found in any text book on composites. The primary goal of this chapter is to present information to the readers on properties and advantages of polymeric composites for aerospace applications and to serve as a reference for composite materials scientists and engineers.

3.2 Major Types of Polymer Composites Historically, the advanced polymer composites made from thermoset matrices, such as epoxies, polyesters, and polyimides (addition polyimides) have received most

Figure 3-2. Composite materials: specific tensile modulus and strength. HMS, GY70, Ep, Gl: see Fig. 3-1; T 300, P100: carbon fibers; P 1700: epoxy resin; Gr: graphite (abbr.).

attention. The processing experience and strong fundamental understanding of these materials have led to significant progresses over the years. As a result, these materials have found widespread application in aerospace, automotive, electronic, and other applications. Although thermoset composites are very popular, they have some inherent deficiencies. These include limited shelf life, low toughness (low strain-to-failure) and, sensitivity to moisture. Also, these materials require multistep processing, and cannot be reprocessed or reformed. Improved processability and increased toughness for greater damage tolerance of the structure are major needs for aerospace applications. Another factor that is important is improved supportability. This includes reduced repair frequency, ability to reprocess the material, and lower cost repairs. A new class of materials, namely, advanced thermoplastic composites, are receiving considerable attention to meet these needs.

3.3 Fabrication of Advanced Composites

Thermoplastic composites are unique materials and are significantly different from their thermoset counterparts. Thermoset composites will undergo various degrees of cross-linking during "cure". In contrast, a thermoplastic composite will not undergo chemical changes during "consolidation". Generally, the thermoplastic composites are melt fusible and can be consolidated by application of heat and pressure only. The two major technical driving forces for aerospace composites are processability and damage tolerance. Of course, cost is also an important issue. The current interest in thermoplastic composites stems from their improved processability and damage tolerance characteristics. The most important advantage offered by thermoplastics is the potential for fast and easy part fabrication. Multistep processing is not required for thermoplastic composites (Fig. 3-3). Although material costs are currently high, it is anticipated that total structure life cycle costs will be lower for thermoplastic composites sometime in the future.

3.3 Fabrication of Advanced Composites

95

tors in the successful application and use of advanced composites. The key steps in fabricating a part made from advanced composites are (1) lay-up of prepreg material in a mold, (2) application of heat and pressure, and cure followed by consolidation of the prepreg material into a structural shape, (3) joining and assembly of subparts to form the final component. In making the structure, attention needs to be paid to the following factors: (1) handling and the storage of materials, (2) quality of the prepreg material, (3) process control to achieve superior processing of the part, (4) potential for process-induced flaws, (5) complexity in joining composite structural parts. 3.3.1 Thermoset Composites

In this section, the fabrication of thermoset composites is discussed. For polymeric matrix composites based on resin matrix, the most common method of layup is the hand lay-up technique which is followed by the semiautomatic lay-up and automatic lay-up. 3.3.1.1 Hand Lay-Up Technique

Fabrication of parts with consistency and quality is one of the most critical fac-

Fabrication of advanced composite structure requires the placement of the prepreg material in a specified direction to Thermoplastic

Thermoset Temp

Temp Heat + Pressure

Time (hr)

tr

Heat + Pressure

Time (hr)

Figure 3-3. Processing temperature and time profile of thermoplastic and thermoset resin composites.

96

3 Polymer Matrix Composites

maximize the strength and stiffness of the fibers consistent with properties required by engineering. Plies of the prepreg tape are placed in a desired stacking sequence. When hand lay-up is used, it is common practice to pre-assemble the prepregs on a special plastic template. This technique is performed in a controlled atmosphere. In the hand lay-up technique, separate plies are stacked up to build the desired thickness of the part with desired orientation of plies. During ply lay-up, it is important to insure that the prepreg quality is good. Each ply lay-up should be inspected for filament breaks, cross-overs, gaps between tapes and foreign matter contamination. Any poor material must be discarded. The curing set up is illustrated in Fig. 3-4. During cure, each ply reduces from its original thickness by voiding itself of entrapped air and excess resin. Therefore, each advanced composite material and part requires a different boundary support (dam). 3.3.1.2 Semiautomatic Lay-Up Semiautomatic lay-up techniques for composite material fabrication can be extremely beneficial in terms of cost-reducCaul plate Perforated Teflon fabric Bleeder (120 Glass fabric or paper) <

\ \ \

x

/Vacuum bag vent cloth '(181 Glass fabric) ' / /Vacuum bag / / - - I k - Controlled gap Ll ' * Vacuum bag sealant

Release fabric

Boundary support or dam (cork or metal member with cork caps)

Cork Tool plate

Figure 3-4. Controlled-flow bleed system for curing polymeric composite laminate.

tions and lay-up uniformities. In most cases, the method used must be tailored to the part or shape being made to realize maximum benefits. Multilayer composite materials can be easily made from which simple shapes may be performed, rather than laying up each ply to the final shape. 3.3.1.3 Automatic Lay-Up Automatic lay-up of aerospace composites is not very common. This is the general case, because the manufacturing efficiency of aerospace structures does not call for automatic lay-ups. Aerospace advanced composite part production rate is manageable using hand lay-up and semiautomatic lay-up techniques. Although automatic tape laying machines are quite popular, the number of parts produced using such tools is only a small fraction of the total. In automatic lay-ups the ultimate goal is to make lay-ups, either flat or contoured, by numerically controlled machines with tape laying heads capable of uniformly and consistently laying up ply after ply as programmed without manual assistance. 3.3.2 Thermoplastic Composites Thermoplastic composites for aerospace applications can be found in two forms, namely, those with continuous fibers randomly oriented in a mat and those with continuous fibers in a unidirectional or woven fabric form. Short fiber reinforced thermoplastics are common but are primarily used in automotive and appliance applications due to lower requirements on strength and stiffness. The major thrust in processing of advanced thermoplastic composite parts to date has involved conversion of prepreg material to articles by thermoforming and stamping (high speed compression mold-

3.4 Micromechanics

97

damp Frame

1

Oven

Fa Laminate

Heating

Forming

ing). The stamping process or laminating is very common and is widely used primarily for discontinuous fiber reinforced systems. The stamping process is also suitable for continuous fiber reinforced systems. This technique is currently being used by many in the composite industry. Thermoforming offers a good potential for processing advanced thermoplastic composites (Fig. 3-5). The key elements of the thermoforming process are: • laminate support frame to carry the laminate into the heat source and rapid transfer of the formable laminate to a forming tool, • a heat source for heating during consolidation, and • a thermoformer with rapid closing speeds and proper pressure capabilities to form the laminate. For continuous fiber reinforced composites, the laminate does not thin and stretch in the mold. Before the tool closes, the laminate is released from the clamp frame and lies in the lower half of the tool. During consolidation, the laminate slips into the tool from the edges to cover the contours. The thickness of the laminate does not change during the thermoforming operation. There is much interest in thermoplastic pultrusion and filament winding processes. The key requirement for either of these processes is a means of applying sufficient

Figure 3-5. Thermoforming of advanced thermoplastic composite.

heat and pressure to consolidate the already impregnated rovings as they are wound or formed by the pultrusion die. A major advantage of thermoplastic winding and pultrusion is the ability to postform the part after consolidation. These processing techniques will have to be further improved before they are widely accepted for thermoplastic composite part manufacturing. When thermoset composites are processed, the scrap material is thrown away. Thermoplastic composite scraps, on the other hand, have the potential to be used either as ground-up material for injection molding compounds or chopped-fiber-reinforced sheet for stamping. Considering the higher cost of thermoplastic composite raw materials especially where carbon fibers are used, utilization of the scrap material will be very cost-effective.

3.4 Micromechanics The properties of a composite material depend on the properties of its constituents and their distribution and the quality of fiber and matrix bonds. Prediction of composite properties from fiber and matrix properties is known as micromechanics. Micromechanics analyses procedures have been extensively used to predict the properties of polymeric composites. (Prediction

98

3 Polymer Matrix Composites

of strength has been less of a success and is not presented in this discussion.) 3.4.1 Stiffness Properties Mechanics of materials approach has been used to estimate the stiffness and strength of polymeric composites. Elasticity techniques have also been used extensively. However, we will not discuss these detailed approaches here. Prediction of stiffness using simple equations has been quite successful. The longitudinal and transverse stiffness can be predicted using the following equations: (3-1)

As mentioned earlier, several elasticity based micromechanics approaches have received considerable attention. However, they are somewhat complicated. One result that requires special attention is the Halpin-Tsai equations (Tsai and Hahn, 1980). Halpin and Tsai developed an interpolation procedure that is an approximate representation of more complicated micromechanics result. The results obtained from Halpin-Tsai equations are quite correct for fiber volume fractions less than 0.8. The approximate form offered by Halpin-Tsai equations is M _

(3-5)

and where

1

(3-2)

where £ f , Em are the fiber and matrix moduli, Vi, Vm are the fiber and matrix volume fractions, and EX9 Ey are the longitudinal and transverse composite moduli. The x, y directions represent longitudinal (L) and transverse (T) directions with respect to the loading axis and the 1, 2 directions discussed later represent material coordinate system; 1 represents direction along the fiber and 2 represents direction transverse to the fiber. Also, the Poisson's ratio and shear modulus are given by: v

=v V

xy

m

r

vfFf

(3-3)

m

and G

(3-4) xy

where v f , vm are the fiber and matrix Poisson's ratios, G f , G m are the fiber and matrix shear moduli, vxy is Poisson's ratio in the L - T plane, and Gxy is the shear modulus in the L - T plane.

(M f /MJ-l

(3-6)

Also, M is the composite modulus E2,G12 or v 2 3 and M f is the corresponding fiber modulus E f , Gf or v f . M m is the corresponding matrix modulus Em, Gm or v m , and £ is a measure of fiber reinforcement of the composite that depends on the fiber geometry, packing geometry and loading conditions. For circular fibers, £ = 2 is suggested. The longitudinal modulus and Poisson's ratios are given by Eqs. (3-1) and (3-3). 3.4.2 Thermal Expansion and Transport Properties Thermal expansion coefficients of unidirectional composites can be obtained by the following equations as proposed by Schapery (1968): (3-7) (3-8)

3.5 Quality Assurance

where a f , am are the coefficients of thermal expansion for fibers and matrix. The longitudinal thermal conductivity can be calculated using the rule of mixtures:

a' and V represent fiber cross-section (width and thickness). For a circular fiber a'/b' = l.

3.5 Quality Assurance Quality assurance for polymeric composites for aerospace applications require special considerations since reliable materials are essential for developing structures which are durable and can meet stringent safety requirements. Several steps of screening, qualification tests and quality conformance criteria are needed to ensure that the composite material meets requirements and offer design properties for the application under consideration. The various aspects of quality assurance are best shown in a generic form, considering the steps required to obtain design allowables as illustrated in Fig. 3-6.

where Kf, Km are the thermal conductivities of fiber and matrix. The transverse thermal conductivity is given by Halpin-Tsai equations:

\+ifl ? 1-rjV;

Km

(3-10)

where

JKf/KJ-l

99

(3-11) (3-12)

Material systems requirement

i Screening

i

Stop

Material systems preliminary evaluation

I • Qualification test plan

Stop

i First batch evaluation

1 Completion of qualification test program

I Data reduction and normalization

I

Statistical evaluation of data package

i "A" & "B" basis values Establishment of quality conformance values for specification Component testing for applications Composite material design allowables

i Documentation of test data and publication of allowables Figure 3-6. Generic procedure to obtain composite material design allowables.

100

3 Polymer Matrix Composites

The processability and properties of high-performance composites depend upon the composition of the fiber/resin preimpregnated-materials (prepregs) from which they are manufactured. In general, prepregs consist of "modified" or surfacetreated glass, graphite, or aramid fibers impregnated with 28-60 wt.% of a reactive and chemically-complex thermoset resin formulation or a semi-crystalline thermoplastic resin. A typical thermoset resin formulation may contain, for example, several different types of epoxy resins, curing agents, diluents, rubber modifiers, thermoplastic additives, accelerators or catalysts, residual solvents, and inorganic materials, plus various impurities and synthetic byproducts. Furthermore, such resins are often "staged" or partially reacted during the prepregging process and may undergo compositional changes during transport, handling, and storage. Although less likely to undergo compositional changes, polymer molecular weight, molecular weight distribution and crystalline morphology have major effects on the processability and properties of thermoplastic prepregs and composites. Inadvertent or minor changes in resin composition may cause problems in processing and have deleterious effects on the performance and longterm properties of composites. Modern analytical techniques and detailed knowledge relating to fibers, fiber surface treatments, and resin types and formulations are needed to characterize prepregs and composite materials. Further information can be obtained in chapters on thermal analysis, spectroscopy, and dynamic mechanical properties in Vol. 2 of this Series.

3.5.1 Characterization Techniques Overview

The most widely utilized techniques for the characterization and quality assurance of composite material precursors are (1) high performance liquid chromatography (HPLC), (2) infrared (IR) spectroscopy, (3) thermal analysis, (4) rheological analysis. HPLC and IR spectroscopy provide the capability for rapid screening and quality control fingerprinting of individual resin constituents as well as of the prepreg resin and, therefore, may be used advantageously by both the prepregger and composite manufacturer. Thermal analytical techniques, such as thermal gravimetrical analysis (TGA), differential thermal analysis (DTA), differential scanning calorimetry (DSC), thermal mechanical analysis (TMA), dynamic mechanical analysis (DMA), and torsional braid analysis (TBA) are not strictly chemical analysis techniques; however, they provide useful information relating to the composition and processability of resins. Similarly, rheological and dielectric techniques are used frequently to evaluate the chemoviscosity properties of thermoset resins during cure, and there is increasing interest in applying such techniques for process monitoring and process control of both thermoset and thermoplastic resins. 3.5.1.1 High Performance Liquid Chromatography

High performance liquid chromatography (HPLC) is a versatile and economically viable quality assurance technique for soluble resin materials (Hagnauer, 1980, 1981; Hagnauer and Dunn, 1982; Mestan and Morris, 1984). HPLC involves the

3.5 Quality Assurance

101

Benzaldehyde DDS

MY 720

Reaction product

Figure 3-7. Chromatogram of Narmco resin matrix showing major peak assignment. DDS: diaminodiphenyl sulfone (curing agent for epoxy resin); MY 720: epoxy resin.

I 5

10

15

20

Elution time, min.

liquid-phase separation and monitoring of separated resin components. Dilute solutions of resin samples are prepared and injected into a liquid mobile phase which is pumped through column(s) packed with a stationary phase to facilitate separation and then into a detector. The detector monitors concentrations or the separated components; its signal response, recorded as a function of time after injection, provides a "fingerprint" of the sample's chemical composition. A typical example of a chromatogram is shown in Fig. 3-7.

3.5.1.2 Infrared Spectroscopy

Infrared (IR) spectroscopy is sensitive to changes in the dipole moments of vibrating groups in molecules and, accordingly, yields useful information for the identification of resin components. IR spectroscopy provides a fingerprint of the resin composition and is not limited by the solubility of resin components (May et al., 1979; Koenig, 1983; Garton, 1984). Indeed, gases, liquids and solids may be analyzed by IR spectroscopy. Advances in technology

102

3 Polymer Matrix Composites

have led to the development of Fourier transform infrared spectroscopy (FTIR) a computer-supported IR technique for rapidly scanning and storing infrared spectra. Multiple scans and Fourier transformation of the infrared spectra enhance the signal-to-noise ratio and provide improved spectra for interpretation. In addition, the FTIR attenuated total reflection (ATR) technique may be applied for quality assurance of thermoset composite materials to assess their state of cure; i.e., residual epoxide concentration. 3.5.1.3 Thermal Analysis

Thermal gravimetric analysis (TGA) monitors the weight changes in a sample as a function of temperature. Although primarily used for studying the degradation processes, TGA can also be applied as a quality assurance technique to provide information about the volatiles, resin, fiber, and inorganic residue content of prepreg materials (Bellenger et al., 1984). Since dissimilar materials often degrade and volatilize at different temperature and rates, compositional differences may be reflected by differences in their TGA thermograms. Thermal oxidative degradation rates determined by TGA are useful for estimating the life cycles of resin materials (Flynn and Wall, 1966). Differential scanning calorimetry (DSC) and differential thermal analysis (DTA) techniques are frequently employed for characterizing resins and composite materials (May etal., 1979; Carpenter, 1977). Both DSC and DTA monitor enthalpy changes in materials as a function of temperature (DSC) directly and DTA indirectly and thereby provide similar information useful for quality assurance of prepreg materials. An example of DSC thermogram is shown in Fig. 3-8. The bottom

solid line represents the baseline to allow calculation of exotherm energy under the peak. The first intersection (Tl) represents the start of chemical reaction and the second intersection (T4) represents the end of reaction. DTA measures the temperature difference (AT) between the epoxy resin specimen and a reference material; whereas DSC measures the rate of heat evolution (dH/dt) or enthalpy absorption of the specimen relative to a reference. DTA and DSC measure thermal changes (1) as a function of time with both the specimen and reference material held at the same temperature (isothermal), or (2) as a function of temperature with both the specimen and reference material heated at the same heating rate (dynamic). For quality assurance applications, DTA and DSC are usually run in the dynamic mode with the weighed specimen in an aluminum specimen holder and an empty holder used as the reference. Dynamic DTA and DSC measure the glass transition temperature Tg and heat of reaction AH of the prepreg resin but do not directly provide information about chemical composition. By monitoring the fraction of heat evolved as a function of temperature or time, information relating to the extent of cure and curing kinetics can be obtained. DSC and DTA may also be applied to evaluate the melting temperature Tm and to estimate the degree of crystallinity of thermoplastic resins and composites. Since the average specimen size used in DSC is only about 10 mg, special care must be taken in obtaining representative materials. Multiple specimens runs are advisable. Thermal mechanical analysis (TMA) is used in conjunction with DTA and DSC to study the thermal transition behavior (e.g., Tg) of prepreg resins and cured laminates. TMA simulates a linear dilatometer to measure the thermal expansion and con-

3.5 Quality Assurance

103

213 °C {415.4 °F) (T3)

5 ° C ( 4 1 °F)/min Heat-up rate

100 °C (212°F) (T1)

0 (32)

50 (122)

100 (212)

I 150 200 (302) (392) Temperature, °C (°F)

traction of specimens under dynamic or isothermal heating conditions. Adjustable loads are applied via a specially designed probe resting upon the specimen surface. Sensitive displacement devices are employed to monitor the "nominal" thermal response of a material. Since thermal transition behavior is related to the chemical composition and extent of cure of a prepreg resin, TMA can be applied as a quality assurance technique.

253 °C (487.4 °F) (T4)

I 250 (482)

I 300 (572)

Figure 3-8. DSC thermogram for 3501-6 resin.

3.5.1.4 Rheological Analysis

Dynamic mechanical analysis (DMA), torsional braid analysis (TBA), and various mechanical spectrometers measure the rheological response of resins and composites as a function of frequency, temperature and/or state of cure. Both DMA and TBA provide information relating to the storage modulus (£") and the loss modulus (E") of the resin during cure and to the Tg

104

3 Polymer Matrix Composites

of the cured resin (Gillham, 1976; Enns and Gillham, 1983; Zukas et al., 1983). The dynamic complex viscosity (rj*) and tan delta parameters can also be determined from rheological measurements of thermoset resins during cure. Rheological techniques are most often used to optimize processing parameters and evaluate the onset of gelation and vitrification during cure. However, since rheological properties are related to resin composition and morphology, rheological techniques may also be applied for the quality assurance of prepregs. Rheological data are often less reproducible when taken from fiber reinforced samples. Whenever possible, neat resin or extruded film samples of the resin lot should be procured along with the prepreg for rheological testing. Dynamic dielectric analysis (DDA) techniques involve the use of electrical measurements to monitor changes in the dielectric constant, the dissipation factor, capacitance and/or conductance of the prepreg resin during processing as a function of frequency, time, and temperature. Measured electrical parameters are highly responsive to changes in resin viscosity and, similar to the rheological techniques, are often employed to investigate and optimize prepreg processing parameters such as resin flow and gelation time/temperature. Since chemical composition affects the electrical properties and curing behavior of prepregs, DDA techniques may also be applied for their quality assurance. Whenever possible, complementary techniques should be used for the chemical quality assurance of resins and composites. Techniques, such as HPLC and IR spectroscopy, are fundamentally different from one another and provide direct, but different, information about a resin's composition. An example of IR comparison of resin is shown in Fig. 3-9. The top IR spectrum

represents an uncured resin and the bottom spectrum is that of the same resin after curing. Wave numbers 2800-3400 represent hydroxyl functionality. Wave numbers 1600 or less represent the fingerprint region (this is the region for organic materials' functional groups). In the cured resin IR spectra, changes from uncured condition are depicted. If appropriate test methods are applied, HPLC and IR spectroscopy are usually powerful enough to detect difference or changes in the chemical compositions of resins. DTA and DSC complement HPLC and IR spectroscopy by providing information relating to the handleability (i.e., the Tg and extent of reaction of the resin) and the processability of the prepreg. TGA and GC head-space analysis techniques for volatile components are secondary, but important, techniques. Special techniques for analyzing specific components or elements should be used if knowledge of the concentrations of the components is critical for processing the resin or if their presence could adversely effect the performance and durability of the cured composite. The information provided by mechanical, rheological, and dielectric analysis techniques is related to the chemical composition of the prepreg resin and thereby complements the more direct chemical techniques. However, caution is recommended in applying non-chemical techniques since the information obtained is complex and frequently ambiguous when attempts are made to relate measured parameters to chemical composition. 3.5.2 Calculation of Statistically Based Material Properties

Statistical considerations for mechanical properties of polymer based composites can be quite extensive. Given the scope of this chapter, we will only confine our atten-

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106

3 Polymer Matrix Composites

tion to estimation of B-basis values which is often chosen for a wide variety of aerospace composites as design allowables. B-basis value is a statistically based material property value above which at least 90% of the population of values is expected to fall with a confidence of 95%. It should be noted that the magnitude of a B-basis value is a function of the amount of data obtained, the number of batches represented and the uniformity of the batches produced. In general, B-basis values will tend to increase if the batch-to-batch variability or the within-batch variability is reduced. 3.5.2.1 Estimation of B-Basis Values

The procedure used to determine a Bbasis value depends on the characteristics of the data. The step-by-step procedure for selecting the appropriate computational method is illustrated by the flowchart in Fig. 3-10. If the sample is made up of data from several different batches, each of the batches should be screened for outliers by the maximum named residual outlier test. One should then determine whether or not the data from the different batches should be analyzed as one sample by employing the k-sample Anderson-Darling test. If the k-sample test does not reject the hypothesis that the batches are from the same population, the batches should be combined and the data should be analyzed as a single sample. If the hypothesis that the batches are from the same population is rejected, indicating significant batch-to-batch variability, then the B-basis value should be computed by the analysis of variances (ANOVA) method, provided that it can be assumed that the data in each batch are normally distributed and that the within-batch vari-

ances are equal. If either of these assumptions appears to be invalid, then there is currently no approved method for computing a B-basis value. Discussion of these techniques are detailed in MIL-HDBK-17 (1987). If the sample represents a single batch of material, or several batches which are to be analyzed as a single sample, the single sample of data should be screened for outliers as discussed earlier. Note that if only a single batch is represented, then the B-basis value should only be considered a "preliminary" value. One should then determine which, if any, distributional form will be assumed in calculating the B-basis value. (The goodness-of-fit test establishes the degree to which the underlying population may be fitted by the two-parameter Weibull, normal or lognormal distribution.) These tests establish the degree to which the underlying population may be fitted by one of these distributions. The two-parameter Weibull distribution is tested first. If it does not adequately fit the data then the normal distribution is tested. If neither of these provide an adequate fit, then the lognormal distribution is tested. If one of the distributions mentioned above provides a good fit to the data, the B-basis value is computed by the corresponding two-parameter Weibull, normal or lognormal method. If none of the distributions adequately fit the data, and the sample contains 29 or more observations, the nonparametric method is used to compute the B-basis value. If none of the distributions fit the data adequately and the sample size is less than 29, the HansonKoopman method, may be employed (MIL-HDBK-17, 1987). This method, however, does not produce an approved B-basis value due to the small sample size. Before a B-basis value is computed, the data must be screened for erroneous values

3.6 Test Methods

107

Start

Is the data from a single batch?

No

Test batch samples for outliers

Yes Batch-to-batch variation?

Yes

No Test single sample for outliers

Equality of variance?

Yes

)

ANOVA method

No Investigate departures from standard models and/or sources of variability

No Exploratory data analysis

Yes Test for Weibullness

>

No

Yes Weibull method

v

Test for normality Yes Normal method

No

Test for lognormality

No

Nonparametric method

Yes Lognormal method

Figure 3-10. Flow chart illustrating computational procedures for B-basis material property values.

and the structure of the population from which the data were sampled must be investigated.

3,6 Test Methods To obtain reliable properties of composite materials, appropriate test methods are to be used. A good understanding of the test methods is essential to use them to characterize the mechanical properties of the composites. The basic goal is to achieve useful properties by conducting meaningful tests. To obtain reliable properties, it is often recommended to conduct tests which ensure a uniform stress field within the

specimen. Because of this requirement, it is easy to see that flexural tests are not very desirable although they can be quite faster and can be used for qualification purposes. Whitney et al. (1982) provide an excellent description of major test methods for polymeric composites. Laminates made from composite materials can have many types of lay-ups. Their characterization is very similar to the lamina (unidirectional composite) evaluation. In this section, we will confine our attention to lamina tests. And, in the following section, a discussion of damage tolerance tests for laminates is presented. Extensive coverage of test methods of composites can also be found in Chaps. 11 and 12 of this Volume.

108

3 Polymer Matrix Composites

3.6.1 Tensile Test Method

The tensile test is performed in order to determine uniaxial tensile strength, Young's modulus and Poisson's ratio. ASTM Standard D-3039 describes the general requirements and test procedures. The tensile specimen is straight-sided and of constant cross-section with adhesively bonded and beveled tabs for load introduction. The lamina 0-deg test specimen is 12.7 mm in width, and typically six piles in thickness while the lamina 90-deg test specimen is 25.4 mm in width and eight plies in thickness. The overall length of the specimen is 22.9 cm and the test section is 15.2 cm. The test specimen geometry is illustrated in Fig. 3-11. Test specimen thickness variation should not exceed ± 2 % . Specimens should be precision machined from plates

with tabs bonded in place. Test specimen edges should be undamaged and parallel to within 0.127 mm. Tab surfaces should be parallel to within 0.127 mm. Tab surfaces should be parallel to a reference surface within 0.0508 mm. The tensile test is performed utilizing wedge-action friction grips. The specimen is first aligned in the grips and tightened in place. The specimen is then loaded monotonically to failure at a recommended rate of 0.02 cm/min. In order to determine specimen strains, electrical resistance strain gages are mounted on the specimen and monitored during the test. Foil gages with a resistance of 350 Q and a gage length of 3.175 or 6.35 mm have been found to minimize gage heating during the test. Typically, three gages are applied to each specimen, two longitudinal and one

(228.7mm) — £=9" — Edges parallel to each ofher within 0.005" | (0.127 mm)

(152.4mm)

(38.1mm)

(25.4) Transition edges to be perpendicular to reference edge "A"within 0.003"(0.0762mm)

Ref. edge"A"

(0.102 mm) 0.004" allowable offset between transition edges Measured (7"T)

Ref. surface "B"

w

L

Upper tab surface to be parallel to reference surface "B" within 0.002" (0.051 mm)

Measured (7"T) ±0.002"

r

^G

h

±0.0156 ±0.1250 ±0.0625 ±0.0625 ±0.0150 ±0.0050 0.396

3.175

1.5875

1.5875

0.361

0.127

—

inches

—

mm

Figure 3-11. Geometry of tensile coupon.

3.6 Test Methods

109

2500 -^2

2000-

£

1500Graphite/Epoxy.IOle T-l

0.4

0.6

£1= 126 GPa v12= 0.29 X j= 1980 MPa €{= 1.4 %

1.2

0:8 1.0 Strain, %

1.4

1.6

Figure 3-12. Tensile stress-strain response for a [O]8 graphite/epoxy specimen.

£

1200-

2

X

x 1000-

y^

/

800-

/ /

600400-

£

d*

i-

^Kevlar f1= v 12= ] X{=

I

200-

,T=1.75% i

0.2

0.4

0.6

0.8

1.0

49/SP328,[01 6 68.7 GPa 0.36 1140 MPa

1.2

Strain,

i

i

i

1.4

1.6

1.8

2.0

Figure 3-13. Tensile stress-strain response for a [O]6 Kevlar/epoxy specimen.

%

transverse to the load direction. The gages are located at the geometric center of the test section with the longitudinal gages mounted on opposite faces of the specimen. The gages may be monitored continuously or at discrete load intervals. Typical test results for the 0-deg tensile test are shown in Figs. 3-12 to 3-14 for unidirectional composites. Extensometers can also be used to monitor strains. 3.6.2 Compression Test Methods

Perhaps the most difficult of the intrinsic composite material properties to measure

are the compressive strength properties. This is the case due to the fact that slight specimen geometric variations result in eccentricity of the applied load, thereby enhancing the opportunity for failure to occur due to geometric instability. Thus, in order to achieve an accurate measure of the compressive strength of a given composite material, rather complex loading fixtures and specimen configurations have been developed. In this section a generic compression test method will be discussed. It is characterized by a specimen geometry having a test section length that is relatively short and

110

3 Polymer Matrix Composites

1200

1000-

800-

600-

-glass/epoxy, ( 0 ] 8

400-

£ 1= 42.7 GPa vn= 0.27 Xj= 1040 MPa £ 1= 2.6 %

200-

1.0

3.0

2.0 Strain, %

Figure 3-14. Tensile stress-strain response for a [O]8 E-glass/epoxy specimen.

completely unsupported. The test specimen is generally loaded through friction by means of wedge action friction grips. Associated test fixtures which insure colinearity of the applied load and specimen centerline are required. To meet this need, several test-fixture configurations have been developed. One such fixture shown in Fig. 3-15 is the Celanese test fixture (ASTM D-341075). The Celanese fixture employs conical friction grips contained in matching cyclindrical end fittings. Colinearity of the cylindrical end fittings is insured by a hollow cylinder which contains the fittings. The test specimen geometry is also illustrated in Fig. 3-15 where the specimen is shown to be 141 mm in length and 6.4 mm in width with a test section length of 12.8 mm. Beveled end tabs are bonded to the specimen

Fixture

Specimen

Specimen

L

U

1 Steel or Ola

fiberglass tab Dimension A B C

mm 39.9 6.35 57.2

inches

Dimension

0.157 0.250 2.250

D E F

mm

inches

12.7 3.16 63.5

0.500 0.125 2.500

Figure 3-15. Celanese compression specimen and test fixture, ASTM D 3410-75.

3.6 Test Methods

111

w—

v Strain

gage

Composite specimen Specimen dimensions Lj,mm

Z.2,mm

w* mm

12.7 + 1

127 + 1.5

12.7 ±0.1 or 6.4 ± 0.1

for load introduction. Strain gages can be used with this specimen to measure modulus. The IITRI test method developed by Illinois Institute of Technology Research Institute employs a test sample identical in geometry to that of the Celanese test method with a different loading fixture. Shown in Fig. 3-16, the IITRI test method employs linear bearings and hardened shafts to insure colinearity of the load path. Load is applied to the specimen through serrated wedges which are contained in solid steel bases. The IITRI fixture may also be modified to accept specimens of widths greater than 0.64 cm. In addition, support rollers to restrain column instability of the specimen may be added to the wedge-action friction grips. 3.6.3 Shear Test Method

Shear tests are performed in order to establish the ultimate shear strengths, ulti-

Figure 3-16. Modified IITRI compression fixture and specimen dimensions.

mate shear strains, and shear moduli of the composite material system. Proper characterization of the shear properties of the composite lamina or laminates requires evaluation of three distinct moduli and strengths. The lamina properties in the plane of lamination (1-2) are known as the inplane shear properties, while the properties in the 1-3 and 2 - 3 plane are known as interlaminar shear properties. To date, test methods have been developed for evaluation of the lamina inplane properties. Shear tests for the composite laminate have generally been restricted to evaluation of properties in the plane of the laminate. No test method has been developed to date which yields intrinsic interlaminar properties for composite laminates. There are four generally accepted test methods for evaluation of lamina inplane shear properties. They are the [±45] 2S coupon test, the off-axis coupon, the rail shear test method, and the torsion test. The [±45] 2S coupon test method consists of laminate

112

3 Polymer Matrix Composites

having a typical geometry specimen length, 229 mm, gage section, 152 mm and width 38 mm. For {±45} 2S laminates, it can be shown from lamination theory that the state of shear stress and strain in the lamina coordinate system can be expressed in terms of the laminate axial stress
T19 = '

(3-14) where x and y are the longitudinal and transverse directions of the laminate. Hence the lamina shear modulus is readily determined by plotting oJ2 vs. \sx\ + \zy\ and establishing the slope of the curve. The ultimate shear stress is defined as the maximum value of oJ2 and the ultimate shear strain, y"", is the corresponding strain

( I S J + le>>l) value.

,

0

0.4

0.8

1.2 1.6 2.0 2.4 Shear s t r a i n , %

2.8

3.2

Figure 3-17. Shear stress-strain curve obtained from a tension test of a [±45] 2S graphite/epoxy specimen. p.

2

Load span - L

Support span

Note that as an alternative, the lamina shear modulus may be directly determined from the relation G12 = ^4Ex

100-

Figure 3-18. Four-point loading.

(3-15)

where Ex and vxy are the longitudinal modulus and major Poisson's ratio, respectively, of the {±45} 2S laminate. An example of shear response is shown in Fig. 3-17. 3.6.4

Flexure Test Methods Support span

The four-point flexure test is shown in Fig. 3-18 and the three-point flexure test is shown in Fig. 3-19. The quarter-point loading has been utilized primarily for high modulus materials such as graphite/epoxy and boron/epoxy composites. These test methods are used for determining flexural strength and modulus. The test is not recommended for generating design data. However, it does provide a simple test for

Figure 3-19. Three-point loading.

quality control. Flexure tests for high modulus, continuous filament composites are generally limited to unidirectional materials with fibers oriented at either 0° or 90° to the beam axis. Specific requirements for radius of load noses and supports are given in ASTM

3.6 Test Methods

D790-71. Span-to-depth ratios, L/d, depend on the ratio of tensile strength parallel to the beam axis to interlaminar shear strength. For strength ratios less than 8 to 1, an L/d ratio of 16 is recommended. This is typical of fiberglass composites. For high modulus materials such as graphite/epoxy and boron/epoxy a value of L/d = 32 is recommended for 0° unidirectional composites. For 90° unidirectional properties of these materials, L/d = 16 is appropriate. Recommended specimen dimensions can also be found in ASTM D790-71. For tensile strength determination the specimen is loaded until failure occurs. The maximum tensile stress, Sm, is determined for the three-point loading from the relationship

3PL 2bd2

3PL

four-point loading with a load-span of one-half of support span, 0.17L3

~~b¥

(3-17)

It should be noted that Eqs. (3-16) and (3-17) are valid only for material in which the stress-strain curve is linear to failure. If some nonlinear stress-strain behavior occurs, an error will be introduced into the relationships for Sm. An adjustment in the calculation for Sm must also be made if the deflections become large (ASTM D790-71). Modulus is determined for the three point flexure test from the relationship

3,6.5 Mode I Interlaminar Fracture Test Mode I interlaminar tests are conducted with a double cantilever beam (DCB) specimen, Fig. 3-20. The compliance, C, of the DCB specimen may be obtained from elastic beam theory as C =

2a3 3E/

(3-20)

where a is the crack length and El is the flexural rigidity of the beam. The strain energy release rate, Gl9 is given by: 2w da

P2a2 wEI

(3-18)

where: EB is the modulus of elasticity in bending in N/m 2 , and r1 is the slope of the tangent to the initial straight-line portion of the load-deflection curve in N/m. For

(3-21)

where P is the applied load and w is the width of the specimen. Equations (3-20) and (3-21) give

r3

E * B ~4bd3

(3-19)

(3-16)

where b is the beam width. For four-point loading at quarter points

Td2'

113

Figure 3-20. DCB specimen.

(3-22)

114

3 Polymer Matrix Composites

Critical conditions occur when GY reaches its critical value GIC for P = Pc; i.e., Pc2a2 G ir = wEI

(3-23)

There are several ways to reduce the data from interlaminar fracture tests (Carlsson and Pipes, 1987).

3,7 Damage Tolerance Damage tolerance for aerospace composites is defined as the capability of the composite structure to sustain a level of low-velocity impact with barely visible damage and retain appropriate residual strength. The low-velocity impact has become a major issue due to a concern or damage resulting from drop of tools during maintenance of composite material structures. The main problem is the effect of accidental lateral impact loading on the extent of damage, which is often located in the internal part of the laminate, nonvisible on its external surfaces. The main goal for damage tolerance evaluation and design is to develop empirical and analytical methods aiming to predict the deterioration in structural performance due to such damage. The tests on basic material properties discussed in the preceding section are essential for generating data on design allowables. However, it is also needed to determine the damage tolerance capability of the composite material. Because of the weak interlaminar fracture resistance of polymeric composites, damage tolerance evaluation or tests primarily concern delamination and low-velocity impact resistance of the composite. Graphite/epoxy composites have been reliably used in many secondary aerospace structures. In military aircraft, this class of

materials also has been used in primary structures such as wings and tails. Conventional graphite/epoxy composites which are brittle have the limitation that maximum weight savings cannot be achieved due to their ease of susceptibility to damage. The primary impediment to full utilization of continuous fiber polymeric composites is their inherent tendency to delaminate. Delamination (Fig. 3-21) is the most prevalent type of life-limiting failure in advanced composites. Technologically, it is also one of the most significant problems in advanced composites. There are various sources of delamination. Delamination may develop during manufacturing with improper consolidation of plies. They may result from impact damage or from three-dimensional interlaminar stresses that develop at stress-free edges or discontinuities such as the free surface of a hole. Both interlaminar normal (oz) and shear stresses (TXZ and xyz) as shown in Fig. 3-22 can be detrimental. The interlaminar stresses developing at discontinuities in typical composite structures which may promote delamination are presented in Fig. 3-23. Cut fibers

Matrix cracks

Delamination

Figure 3-21. Delamination and other local damage mechanisms in a composite laminate.

3.7 Damage Tolerance

115

Interface Free-edge

Figure 3-22. Interlaminar normal and shear stresses (circled) in a composite laminate.

Free edge

Notch (hole)

Bond joint

Bolted joint

Figure 3-23. Interlaminar stresses arise from typical discontinuities in composite structures.

Three major failure modes may occur during loading of composite materials namely: fiber fracture, interfiber transverse matrix cracking, and interlaminar fracture or delamination. Fiber fracture rarely occurs in multidirectional laminates under working loads. Presence of a hole may reduce residual strength. Interfiber matrix cracking, which occurs at a low tensile stress transverse to fiber direction, does not significantly reduce composite performance.

Interlaminar stresses can also develop due to compressive loading in laminates. A consequence of compressive loading can be local and global buckling of plies in a laminate. This situation is enhanced due to the propensity of delamination crack growth and separation of plies. An area of considerable concern for conventional graphite/epoxy composites is their response under low velocity impact. The damage may not be visible. However, internal delamination can be quite large

116

3 Polymer Matrix Composites

resulting in the loss of compression strength and structural integrity. The compressive residual strength of the composite structure may be controlled by the size and location of the delamination. Hence, the design of the composite structure should be made resistant to both formation and growth of delaminations. Since it is impossible to control the low energy impacts that can initiate delaminations, composite structural designer must assume that such delaminations exist in the laminate a priori and control their growth through intelligent design. The current damage tolerance philosophy for composite structures quite justifiably leans heavily on impact damage. It is anticipated that composite structures should sustain lifetime design loads in a damaged condition up to the condition where a barely visible impact damage is detectable. Due to these requirements, the design allowable ultimate strain is limited to ~0.4%. However, for further weight savings and for greater damage tolerance performance, design strains of 0.6% are considered highly desirable. Traditional graphite/epoxy composites are brittle and show poor resistance against delamination growth and interlaminar separation resulting from low velocity impact. Although toughened graphite/epoxy composites show good properties, particularly delamination resistance, their performance under hot-wet conditions is of concern. Interleaving of graphite/epoxy composites have shown much promise. In recent years, thermoplastic composites, particularly graphite/PEEK composite, have shown considerable promise in delamination fracture resistance. Thermoplastic matrices give an order of magnitude increase in interlaminar toughness compared to existing epoxy resin composites (GIc of 1500 J/m2 vs. 150 J/m2). Dis-

cussion on toughened polymeric composites including thermoplastic composites can be found in two ASTM publications edited by Johnston (1987) and Newaz (1989), respectively. Fatigue loading is common in aerospace structures. Delamination initiation and growth due to fatigue is also of concern. Delamination growth under fatigue loading in both thermoset and thermoplastic composites has been studied well except that the knowledge base for thermoplastic composites is not as extensive. It is now clear that delamination crack growth rate under fatigue loading is relatively fast in both graphite/epoxy and thermoplastic composites as exhibited by high n values 8-20 in the Paris-type equation. An implication is that fatigue propagation life is short. Thus, an approach to proper damage tolerance design calls for design on the basis of threshold stress-intensity factor; i.e., to avoid any possible growth of the small delamination crack due to fatigue loading. For anisotropic layered material systems, the complexity in actual structures arises from the inherent coupling of Mode I, II and III fracture and possible geometric and material discontinuities. Furthermore, one can envision possible nonlinearities - making the problem of delamination quite complex. The source of nonlinearity may be either due to large deflection or due to ductility of the material as in thermoplastic composites. Thus, the challenge rests on both materials and structural design to avoid this failure mode in aircraft structures. The most severe and frequent event during service and maintenance is the local transverse impact due to foreign objects. The evaluation of their distribution of stress waves due to impact at function of time and space is highly complex due to the

3.7 Damage Tolerance

heterogeneous and anisotropic nature of the composite material and is complicated by loading rate. Hence, the loading rate is an important variable to consider. Another important factor which affects impact loading consequences is the boundary conditions of the loaded part. In real situations such conditions may vary from rigidly fixed to free simply supported and their exact translation to experimental and analytical models are difficult to determine. Hence one way to tackle such variable is by comparing the extreme conditions and establish some empirical factors to adjust the laboratory finding to the real life structural behavior. On the other hand, delamination is considered to be highly critical to the durability of the composite structure under subsequent long term (or cyclic) environmentloading service conditions. Even transverse cracking may be crucial under certain loading conditions as a source for additional delaminations growth initiating from the transverse crack root. Various structural threats vs. local damage are illustrated in Fig. 3-24. Under flexural loading two types of delaminations may develop, namely:

117

(a) Edge delaminations at the tensile zone close to the outer surface, where tensile stresses attain their maximum value. (b) Shear delamination, close to the neutral plane where interlaminar shear stresses are predominant under flexure. The chance of predominance of one failure mode above the other depends on span to thickness ratio of the beam or panel considered and on boundary conditions. It is also dependent on loading rate as edge delamination propagates slowly as a function of tensile axial strain, while shear delamination propagates both fast and abruptly when loading energy attains its critical level. Hence shear delamination mode is expected to be predominant under impact loading whereas both modes may develop under quasistatic loading. 3.7.1 Damage Characterization and Detection During fabrication, interlaminar flaws may be formed. During service and maintenance period, damage may be developed. In most cases these flaws or damages may be described geometrically approximately as two-dimensional circular

Resulting local damage Cut fibers Material flaw

i iT

Matrix cracks

i i

Nature of threat

Few MFG error Impact Puncture Hydrodynamic ram

Many

i T Very large number

i

Delamination

Minimal

Characteristic spacing assumed

t

Moderate

i

i i

Extensive

Figure 3-24. Various structural threats cause different amounts of local damage.

118

3 Polymer Matrix Composites

interlaminar separations. In thin sections, or laboratory specimens, such cracks may propagate through the element width and form a "one-dimensional" delamination. The damage (flaw) dimension and its location through the laminate thickness are functions of loading mode and direction, and laminate lay-up. Thus, damage due to processing and due to in-service mechanical loads is a crucial factor which determines the subsequent performance of the laminate under working loads. Hence the importance of reliable and accurate detection methodology for damage (flaw) as well as its geometrical characterization, and correct interpretation cannot be overstated. To date the best method for detection of interlaminar separation pattern is the ultrasonic C-scan technique. With a modern ultrasonic system it is possible to obtain both planar dimensions and level locations of any delamination as long as it is not hidden by another separation located closer to the exposed surface. The full pattern of flaws and damages may be found by C-scan detection applied from the two surfaces together with other NDT methods (radiography etc.). Checking of NDT results may be accomplished by microscopical analysis of sections cut from the damaged laminate (fractography). In the case of narrow laboratory specimens in which one dimensional damages have been formed, the dimension, location and growth of interlaminar damages, under loading may be detected and followed-up by direct on-line video inspection. The chances of existing interlaminar damage (flaw) to propagate under loading depends on its geometry, loading mode and direction, and mainly on the interlaminar fracture toughness (IFT) characteristics of the composite laminate. IFT characterization and its dependence of composite material composition was dealt

widely in numerous publications. Conclusions from these findings point out that by proper design of laminate lay-up and composition (hybridization), a significant improvement of IFT may be achieved, which will in turn, have a positive effect on damage tolerance capacity of the material. Propagation of in-plane cracks in a multidirectional laminate is arrested by the existence of fibers which are misaligned to the crack direction. Interlaminar damage, on the other hand, may propagate almost uninterrupted through the quasi-homogeneous and isotropic interlaminar plane (similar to crack propagation in isotropic homogeneous media). Hence, delamination is a major concern in structural composite materials. Detailed discussions on damage tolerance and interlaminar damage are forwarded by Ishai et al. (1988). 3.7.2 Compressive Loading and Laminate Buckling

The most severe loading condition for a laminate with interlaminar damage (or flaw) is the compressive mode. In such a mode, the damaged laminate is inclined to fail due to sublaminate local buckling mechanism. Such a mechanism was widely studied analytically and investigated experimentally. It was found that the buckled sublaminate provide the driving energy for additional delamination propagation, which in turn enhances premature laminate compressive failure due to the weakening effect and eccentricity caused by the partial separation of the sub-laminate. In standard testing specifications proposed by NASA and some aircraft industries, uniaxial compressive loading after impact procedure is recommended. These loading procedures are carried out on relatively large scale specimens, and can be defined as structural tests.

3.8 References

These tests are expensive especially when considering the number of such large specimens which are required to obtain reliable data. In the future, a cyclic compressive loading test will be probably required in order to investigate the residual fatigue life of impacted specimens. Such a program using panel testing will necessitate substantial funds and hence will be limited to exclusive research centers. A new approach to simplify the post-impact residual characterization procedure is suggested here, based on preliminary tests. In this method, impact and post-impact loading are conducted on small beams, where damage formation and propagation mechanism and its effect on ultimate failure can be detected directly during loading procedure. Residual compressive fatigue test may also be accomplished using "beam specimens". The limitation and advantages of "beam test" vs. "panel test" for damage tolerance evaluation, in view of the different requirements for structural design, material engineering and quality assurance disciplines should be assessed. 3.7.3 Residual Properties

Engineering design considerations must take into account the fact that the composite structure may consist of preformed fabrication flaws and damages due to accidental lateral loads, which will effect the design allowables. Interlaminar flaws are not sensitive to tensile inplane loading. On the other hand under compressive inplane loading such damages (flaws) are likely to propagate and consequently residual compressive strength of damaged laminate is expected to be significantly lower than its virgin reference. Similarly, existing delaminations are expected to propagate under cyclic loading and hence fatigue life is liable to be reduced as compared with that of reference laminate. In most damage tol-

119

erance investigations, residual compressive properties of laminated panel after impact is plotted vs. initial impact energy. Damage tolerance methodology for advanced composites is not fully mature primarily due to the complexity in progressive failure modes encountered in service. Except for the case of dominant cracks such as delamination, damage tolerance methodology for interacting failure modes are still continuing to evolve. Current approaches are generally semi-empirical in nature with considerable bias toward experimental data. One current approach is to assess residual properties and relate them to the damage size and propagation modes, which in turn can be related to the state and distribution of stresses within the laminate.

3.8 References Agarwal, B. D., Broutman, L. I (1980), Analysis and Performance of Fiber Composites. New York: John Wiley & Sons. Bellenger, V., Fontaine, E., Fleishmann, A., Saporito, X, Verdu, J. (1984), Polym. Degrad. Stab. 9(4), 195. Carlsson, L. A., Pipes, R. B. (1987), Experimental Characterization of Advanced Composite Materials. Englewood Cliffs, NJ: Prentice-Hall, Inc. Carpenter, J. F. (1977), Assessment of Composite Starting Materials: Physicochemical Quality Control of Prepregs, AIAA/ASME Symp. Aircraft Composites, San Diego, CA. Enns, J. B., Gillham, J. K. (1983), /. Appl. Polym. Sci. 28, 2567. Flynn, J. H., Wall, L. A. (1966), Res. Nat. Bur. Stds. 70A (6), 487. Garton, A. (1984), Polym. Prepr. (Am. Chem. Soc, Div. Polym. Chem.) 25 (2), 163. Gillham, J. K. (1976), Polymer Eng. Sci. 16, 353. Hagnauer, G. L. (1980), Polymer Composites 1, 81. Hagnauer, G. L. (1981), It Res. Dev. 23 (4), 123. Hagnauer, G. L., Dunn, D. A. (1980), in: Materials 1980,12 (12th Natl. SAMPE Technical Conf.), pp. 648-655. Hagnauer, G. L., Dunn, D. A. (1982), Ind. Ed. Chem. Prod. Res. Dev. 21, 68. Hagnauer, G. L., Dunn, D. A. (1984), in: Plastics in a World Economy, ANTEC '84, Society Plastics

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Eng. 42nd Annu. Technical Conf. and Exhibition, pp. 330-333. Ishai, O., Rotem, A., Lifshitz, J. (1988), Damage Tolerance Evaluation of Structural Composite Materials. Haifa: Technion R & D Foundation. Johnston, N. J. (Ed.) (1987), ASTM STP937. Philadelphia, PA: Am. Soc. Test. Mat. Koenig, J. L. (1983), Quality Control and Nondestructive Evaluation Techniques for Composites, Part II: Physicochemical Characterization Techniques. A State-of-the-Art Review, U.S. Army Aviation R & D Command, AVRADCOM TR 83-F-6. May, C. A., Hadad, D. W, Browning, C. E. (1979), Polymer Eng. Sci. 19, 545. Mestan, S. A., Morris, C. E. M. (1984), Macromol. Sci., Rev. Macromol. Chem. Phys. C24(l), 117. MIL-HDBK-17 (1987), Plastics for Aerospace Vehicles: Part 1 - Composite Materials for Aircraft and

Aerospace Applications, Vol. 1 - Guidelines. Watertown, MA: U.S. Army Mater. Lab. Newaz, G. M. (Ed.) (1989), ASTM STP1044. Philadelphia, PA: Am. Soc. Test. Mat. Schapery, R. A. (1968), /. Composite Materials 2 (3), 280. Tsai, S. W, Hahn, H. T. (1980), Introduction to Composite Materials. Lancaster, PA: Techmonic Publishing Co. Whitney, J. M., Daniel, I. M., Pipes, R. B. (1982), Experimental Mechanics of Fiber Reinforced Composite Materials. Brookfield Center, CT: Soc. Experimental Mechanics. Zukas, W X., MacKnight, W. J. Schneider, N. S. (1983), in: ACS Symp. Ser. 227 (Chemorheology of Thermosetting Polymers). Washington, D.C.: Am. Chem. Soc, pp. 223-250.

4 Metal Matrix Composites Krishan K. Chawla Department of Materials and Metallurgical Engineering, New Mexico Institute of Mining and Technology, Socorro, NM, U.S.A.

List of 4.1 4.1.1 4.1.2 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3 4.3.3.1 4.3.3.2 4.3.4 4.3.5 4.3.5.1 4.3.5.2 4.3.5.3 4.3.5.4 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7 4.4.8 4.4.9 4.5 4.6

Symbols and Abbreviations Introduction Types of Metal Matrix Composites Important Metallic Matrix Materials Processing Liquid State Processes Solid State Processes Deposition Techniques In Situ Processes Interfaces in Metal Matrix Composites Importance of the Interface in Composites Crystallographic Nature of the Fiber/Matrix Interfaces Interfacial Bonding in Metal Matrix Composites Mechanical Bonding Chemical Bonding Thermal Stresses in Metal Matrix Composites Microstructure of the Interface Region Alumina/Metal Composites Silicon Carbide/Metal Composites Carbon Fiber/Metal Matrix Composites Interaction Between Interfacial Phenomena Properties Importance of the Metallic Matrix Modulus Strength Matrix Aging Response in Composites Toughness Thermal Properties Fatigue Creep Tribological Properties Applications References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

122 125 125 126 128 128 129 134 135 137 138 138 139 139 140 141 144 145 147 150 150 151 151 154 156 157 160 162 163 173 174 176 179

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4 Metal Matrix Composites

List of Symbols and Abbreviations a a A b d D E E Eo Et

f g G

h IA Jc k K

AK AK th

Klc (1)

I m m n N Nc P P

P P r R R R{ Ro (s) s S S,Sf

t

T

crack length radius of fiber fatigue crack propagation parameter radius of matrix shell diameter of fiber, density diffusivity elastic constants (general) Young's modulus modulus of uncracked material transverse Young's modulus subscript: fiber acceleration due to gravity shear modulus height of composite area of fiber/matrix interface critical current density (of superconductors) Boltzmann constant bulk modulus cyclic stress intensity factor threshold stress intensity plain strain fracture toughness liquid length of composite, length of shaft fatigue crack propagation exponent subscript: matrix creep stress exponent number (of cycles) critical speed (rpm) pressure between interface and matrix property subscript: particle load geometric parameter solidification rate stress ratio inner radius outer radius solid aspect ratio of discontinuous reinforcement stress amplitude precipitates in A l - C u - M g system time temperature

List of Symbols and Abbreviations

AT Tc Tm V w w x Y

temperature interval critical temperature (of superconductors) melting temperature volume fraction subscript: whisker width of composite thickness of the reaction zone a constant, a geometric factor

a Aa s e 0'90" X v a o Aa G0 cra
linear expansion coefficient thermal mismatch creep strain strain precipitates in A l - C u system interfiber spacing Poisson's ratio strength stress cyclic stress range friction stress cyclic stress amplitude mean stress stress in radial direction stress in tangential direction stress in direction of fiber

ARALL C-PFZ CMC CTE CVD CVI DC FP GP HIP HREM LEFM MMC MRI NASP OPIT PMC PRD-166 PVD

aramid aluminum laminate carbon-precipitation free zone ceramic matrix composite coefficient of thermal expansion chemical vapor deposition chemical vapor infiltration direct chill polycrystalline a-alumina fiber (trade name) Guinier-Preston zones hot isotopic pressing high resolution electron microscopy linear elastic fracture mechanics metal matrix composite magnetic resonance imaging National Aerospace Plane (U.S.) oxide-powder-in-tube method polymer matrix composite Alumina + Zirconia fiber (trade name) physical vapor deposition

123

124

SCS-6 SEM SHS TEM XD

4 Metal Matrix Composites

monofilament silicon carbide fiber (trade name) scanning electron microscopy self propagating high temperature synthesis transmission electron microscopy proprietary process to produce a composite

4.1 Introduction

4.1 Introduction A composite material can be defined (Chawla, 1987a) as a material consisting of two or more physically and/or chemically distinct, suitably arranged or distributed phases. It has characteristics that are not depicted by any of the components in isolation. Generally, one component acts as a matrix in which the reinforcing phase is distributed. The matrix component is, thus, the continuous phase. When the matrix component is a metal, we call such a composite a metal matrix composite (MMC). The reinforcement can be in the form of particles, whiskers, short fibers, or continuous fibers. There are three entities that determine the characteristics of a composite, viz., reinforcement, matrix, and interface. Earlier on, the role of matrix was considered to be that of a medium or binder to hold the strong and stiff fibers or other types of reinforcement. This view was particularly prevalent in the polymer matrix composite area. Over the years, however, it has been realized that the matrix microstructure and consequently its mechanical properties have a considerable influence on the overall performance of a composite. This is particularly true of the metal matrix composites because the very act of incorporating a reinforcement - be that fiber, whisker, or particle - can result in change(s) in the microstructure of the metallic matrix and, consequently, in their structure-sensitive properties such as strength and toughness. In this article we first briefly describe some of the important metal matrix materials and some commonly used reinforcements. This is followed by a description of the various processing techniques for making metal matrix composites, the structural characteristics of the interface region in a

125

variety of MMCs, and properties of different metal matrix composites. Finally, we indicate various applications of metal matrix composites. 4.1.1 Types of Metal Matrix Composites There are three kinds of metal matrix composites (MMCs): particle reinforced MMCs, short fiber or whisker reinforced MMCs, and continuous fiber or sheet reinforced MMCs. Figure 4-1 (a and b) shows typical microstructures of particle (SiC particle/Al) and continuous fiber reinforced (Al2O3/Al-Li) composites. Particle or discontinuously reinforced MMCs (one uses the term discontinuously reinforced MMCs to indicate metal matrix composites having reinforcements in the form of short fibers, whiskers, or particles) have assumed special importance for the following reasons: - Particle reinforced composites are inexpensive vis a vis continuous fiber reinforced composites. Cost is an important and essential item for large volume usage. - Conventional metallurgical processing techniques such as casting or powder metallurgy, followed by conventional secondary processing by rolling, forging, and extrusion can be used. - Higher use temperatures than the unreinforced metal. - Enhanced modulus and strength. - Increased thermal stability. - Better wear resistance. - Relatively isotropic properties compared to the fiber reinforced composites. Within the broad category of discontinuously reinforced composites, the cast metal matrix composites are cheaper to produce than the powder metallurgy composites. There two types of cast metal matrix composites:

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4 Metal Matrix Composites

(a) cast composite having local reinforcement. (b) cast composite in the form of a billet having general reinforcement with a wrought alloy matrix. Such composite billets are forged and/or extruded, followed by rolling or other forming operations.

(a)

4.1.2 Important Metallic Matrix Materials A variety of metals and their alloys can used as matrix materials. We describe the important characteristics of some of the more common ones: Aluminum alloys Aluminum alloys, because of their low density and excellent strength, toughness, and resistance to corrosion, find important applications in aerospace field. Of special mention in this regard are the Al-Cu-Mg and Al-Zn-Mg-Cu alloys, very important precipitation hardenable alloys. The latest and by far the most important development in the precipitation hardenable aluminum alloys is a series of aluminum-lithium alloys. Lithium when added to aluminum as a primary alloying element has the unique characteristics of increasing the elastic modulus and decreasing the density of the alloy. Understandably, the aerospace industry has been the major target of this development. Al-Li alloys are precipitation hardenable much like the Al-Cu-Mg and Al-Zn-Mg-Cu alloys mentioned above. The precipitation hardening sequence in Al-Li alloys is, however, much more complex than that observed in conventional precipitation hardenable aluminum alloys. Generally, these alloys contain, besides lithium, some copper, zirconium, and magnesium. Vasudevan and Doherty (1989) provide an account of the various aluminum alloys. Rapidly Solidified Aluminum Transition Metal Alloys

(b) Figure 4-1. Typical microstructures of some metal matrix composites: (a) particle (SiC particle/Al) and (b) continuous fiber reinforced (Al2O3/Al-Li) composites.

Aluminum alloys capable of competing with titanium alloys on a specific strength basis up to 375 °C can be obtained by strengthening with transition metal intermetallic compounds. Allied-Signal Corpo-

4.1 Introduction

ration has developed a series of Al-Fe-V-Si alloys using rapid solidification (cooling rates > 1 0 6 K s ~ 1 ) having an excellent combination of mechanical properties. The high cooling rate results in an extremely fine grain size which, in turn, results in excellent property combinations unobtainable by conventional metallurgy route. Allied-Signal uses what it calls a planar flow casting process to produce rapidly solidified aluminum-based quaternary alloys having enhanced solubility of iron, vanadium, and silicon with good room temperature strength and ductility. The silicon addition results in a thermally stable, 40-50 nm spherical silicides (Al12(Fe,V)3Si). Titanium alloys Titanium is one of the important aerospace materials. It has a density of 4.5 gem""3 and a Young's modulus of 115 GPa. For titanium alloys, the density can vary between 4.3 and 5.1 g cm" 3 while the modulus can have a range of 80130 GPa. High strength/weight and modulus/weight ratios are important. Titanium has a relatively high melting point (1672°C) and retains strength to high temperatures with good oxidation and corrosion resistance. All these factors make it an ideal material for aerospace applications. Titanium alloys are used in jet engine (turbine and compressor blades), fuselage parts, etc. It is, however, an expensive material. At very high speeds, the skin of an airplane heats up so much that aluminum alloys become weak or even melt. Titanium alloys must be used at such high temperatures. In the U.S. National Aero Space Plane (NASP) program, the plane will fly at speeds over Mach 2 and the temperatures will be even higher than what titani-

127

um alloys can withstand. Titanium aluminides are one of the candidate materials in this case. Titanium has two polymorphs; alpha (a) titanium has an h.c.p. structure and is stable below 885 °C, and beta (p) titanium which has a b.c.c. structure and is stable above 885 °C. Aluminum raises the a -• p transformation temperature, i.e., aluminum is an a stabilizer. Most other alloying elements (Fe, Mn, Cr, Mo, V, Nb, Ta) lower the a -> P transformation temperature, i.e., they stabilize the phase. Thus, three general alloy types can be produced, viz., a, a+(3, and P titanium alloys. The Ti-6%-Al-4%-V, called the work horse Ti alloy of the aerospace industry, belongs to the (oc + P) group. Most titanium alloys are not used in a quenched and tempered condition. Generally, hot working in the (a + p)-region is carried out to break up the structure and distribute the a phase in an extremely fine form. Titanium has a great affinity for oxygen, nitrogen, and hydrogen. Parts per million of such interstitials in titanium can change mechanical properties drastically, producing embrittlement. That is why welding of titanium by any technique requires protection from the atmosphere. Electron beam techniques, in vacuum, are frequently used. Magnesium alloys Magnesium and its alloys form another group of very light materials. Magnesium is one of the lightest metals, its density being 1.74 g cm" 3 . Magnesium alloys, especially castings, are used in aircraft gearbox housings, chain saw housing, electronic equipment, etc. Magnesium, being a hexagonal close packed metal, is difficult to cold work.

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4 Metal Matrix Composites

Copper Copper has a face centered cubic structure. Its use as an electrical conductor is quite ubiquitous. It has good thermal conductivity. It can be cast and worked easily. One of the major applications of copper in a composite as a matrix material is in niobium based superconductors. Intermetallic Compounds There are many other intermetallics, not necessarily ordered, available. Ordered intermetallic alloys possess structures characterized by long range ordering, i.e., different atoms occupy specific positions in the lattice. Because of their ordered structure, dislocations in intermetallics are much more restricted than in disordered alloys. This results in retention (and, in some case, even an increase) of strength at elevated temperatures, a very desirable feature. For example, nickel aluminide exhibits a marked increase in strength with temperature up to 800 °C. An undesirable feature of intermetallics is their extremely low ductility. Among the explanations put forth to explain this are: (a) Availability of an insufficient number of slip systems because of the low symmetry of the structure; (b) Weak grain boundaries. Attempts at ductility enhancement in intermetallics have involved a number of metallurgical techniques. Rapid solidification is one method. Another technique that has met great success is the addition of boron to Ni 3 Al. Boron segregates to the grain boundaries, strengthening them. With extremely small amounts of boron (0.06 wt.%), the ductility increases from about 2% to about 50%. Long range order also has significant effects on diffusion controlled phenomena such as recovery, recrystallization, and grain growth. The

activation energy for these processes is increased, and these processes are slowed down. Thus, ordered intermetallic compounds tend to exhibit a high creep resistance. Enhanced toughness by making composites with intermetallic matrix materials is a potential possibility. Chief among the disordered intermetallics is molybdenum disilicide (MoSi2) (Vasudevan and Petrovic, 1992). It has a high melting point and shows good stability at temperatures greater than 1200 °C, in oxidizing atmospheres. It is commonly used as a heating element in furnaces. The high oxidation resistance comes from a protective SiO2 film that it tends to form at high temperatures.

4.2 Processing Many processes for fabricating metal matrix composites are available. For the most part, these processes involve processing in the liquid and solid state. Some processes may involve a variety of deposition techniques or an in situ process of incorporating a reinforcement phase. We provide a summary of these fabrication processes. 4.2.1 Liquid State Processes Casting or liquid infiltration involves infiltration of a fiber bundle by liquid metal. Alternatively tows of fibers can be passed through a liquid metal bath, where the individual fibers are wet by the molten metal, wiped of excess metal, and a composite wire is produced. A bundle of such wires can be consolidated by extrusion to make a composite. Relatively few MMCs have been made by simple liquid phase infiltration, mainly because of difficulties with fiber wetting by the molten metal. When the infiltration of a fiber preform occurs read-

129

ily, reactions between the fiber and the molten metal can significantly degrade fiber properties. Fiber coatings applied prior to infiltration, which improve wetting and control reactions, have been developed and can produce impressive results. In this case, however, the disadvantage is that the fiber coatings must not be exposed to air prior to infiltration since surface oxidation will alter the positive effects of coating (Katzman, 1987). Squeeze casting or pressure infiltration involves forcing the liquid metal into a fibrous preform. Pressure is applied until the solidification is complete. Figure 4-2 shows a schematic of this process. By forcing the molten metal through small pores of a fibrous preform, this method obviated the requirement of good wettability of the reinforcement by the molten metal. Composites fabricated with this method have minimal reaction between the reinforcement and molten metal, and are free from common casting defects such as porosity and shrinkage cavities. Infiltration of a fibrous preform by means of a pressurized inert gas is another variant of liquid metal infiltration technique. Net-shaped components can be produced inexpensively (Cook and Warner, 1991). The process is conducted in the controlled environment of a pressure vessel and rather high fiber volume fractions and complex shaped structures are obtainable (Cook and Warner, 1991). Alumina fiber reinforced intermetallic matrix composites (e.g., TiAl, Ni 3 Al, and Fe3Al matrix materials) composites have been prepared by pressure casting or squeeze casting (Nourbakhsh et al. 1990). The matrix alloy is melted in a crucible in vacuum while the fibrous preform is heated separately. The molten matrix material (at about 100 °C above the Tm) is poured onto the fibers and argon gas is introduced simultaneously. Argon gas

Molten Metal Heater

mmk

MBS*

WHO*

Fiber preform

mum

Mold

Figure 4-2. Squeeze casting or pressure infiltration process. The liquid metal is forced into a fibrous preform. Pressure is applied until the solidification is complete.

pressure forces the melt to infiltrate the preform. The melt generally contains additives to aid wetting of the fibers. 4.2.2 Solid State Processes Diffusion Bonding Diffusion bonding is a common solid state welding technique for joining similar or dissimilar metals. Interdiffusion of atoms from clean metal surfaces in contact at an elevated temperature leads to welding. The major advantages of this technique are: (i) ability to process a wide variety of matrix metals, (ii) control of fiber orientation and volume fraction. Among the disadvantages are: (i) processing times of several hours, (ii) cost of high processing temperatures and pressures, (iii) only objects of limited size can be produced.

130

4 Metal Matrix Composites

There are many variants of the basic diffusion bonding process; however, all of them involve a simultaneous application of pressure and high temperature. Matrix alloy foil and fiber arrays, or composite wire, or monolayer laminae are stacked in a predetermined order. Figure 4-3 shows a schematic of one such diffusion bonding process, while Fig. 4-4 shows the microstructure of a transverse section of silicon carbide fiber/aluminum composite made by diffusion bonding. Vacuum hot pressing is a most important step in the diffusion bonding processes for metal matrix com-

posites. Hot isotopic pressing (HIP), instead of uniaxial pressing, can also be used. Hipping involves gas pressure pushing against a can which consolidates the composite piece contained inside the can. With HIP it is relatively easy to apply high pressures at elevated temperatures over variable geometries. Hot roll diffusion bonding is another variant of diffusion bonding technique which can be used to produce a composite consisting of different metals in the sheet form. Such composites are called sheet laminated metal matrix composites.

(1) Apply metal foil

(5) Heat to fabrication temperature (2) Cut to shape

ULL :T H K H

n n n n.

(3) Lay up desired plies

(6)Apply pressure and hold for consolidtion cycle

M M U M )l il Li~i

(4) Vacuum encapsulate

OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO (7) Cool, remove, and clean part

Figure 4-3. Schematic of a diffusion bonding process.

4.2 Processing

Figure 4-4. Microstructure of a transverse section of silicon carbide fiber/aluminum composite made by diffusion bonding.

Deformation processing Deformation processed metal/metal composites are formed during mechanical processing (by swaging, extrusion, drawing, or rolling) of a ductile two-phase material. The two phases co-deform, causing the minor phase to elongate and become fibrous in nature within the matrix. These materials are sometimes referred to as in situ composites. The properties of a deformation processed composite depend largely on the characteristics of the starting material which is usually a billet of two-phase alloy that has been prepared by casting or powder metallurgy methods. Deformation processed composites are limited to starting materials in which both phases are ductile and have similar flow stresses so that co-deformation occurs and large total deformation strains may be employed. Niobium based conventional filamentary superconductors in a copper matrix and the oxide based high-Tc superconductors in a silver matrix constitute a special

131

class of deformation processed metal matrix composites. Conventional niobium based superconductors, niobium/titanium filaments in a copper matrix and Nb 3 Sn filaments in a copper matrix, constitute perhaps the largest commercial market for continuous fiber MMCs. High-Tc superconductors were hailed in 1987 as a revolutionary discovery. It was pointed out at that time (Chawla, 1987 a, p. 164) that eventually some kind of filamentary route will have to be used for making useful components out of these materials. There are two main problems with high-Tc superconductors that any process technique must contend with. These are: (i) High-Tc superconductors are difficult to make into desirable forms such as wires or films, especially wires. (ii) They have a very low current carrying capacity. This has been attributed to grain boundaries as weak links and flux creep. We shall come back to the high-r c superconductors shortly. We first describe the processing of niobium based copper matrix composite superconductors (Foner and Schwarz, 1981; Hillmann, 1981; Hillmann et al., 1989). There are two main types of the conventional niobium based superconductors: Nb-Ti/Cu and Nb 3 Sn/ Cu. Niobium-titanium (about 50-50) form a ductile system. Rods of Nb-Ti are inserted in holes drilled in a block of copper, evacuated, sealed, and subjected to a series of drawing operations interspersed with appropriate annealing treatments to give the final composite superconducting wire. This process is fairly straight forward. In the case of Nb 3 Sn/Cu, an ingenious process called the bronze route is used to make this composite. Nb 3 Sn, being an A-15 type, brittle intermetallic, cannot be processed like Nb-Ti. Instead, one starts with a bronze (Cu-13 % Sn) matrix.

132

4 Metal Matrix Composites

Pure niobium rods are inserted in holes drilled in bronze, evacuated, sealed, and subjected to wire drawing operations as in the case of Nb-Ti/Cu. The critical step is the heat treatment at about 700 °C that drives out the tin from the bronze matrix to combine with niobium to form stoichiometric, superconducting Nb 3 Sn, leaving behind copper matrix. In view of the highly important and publicized discovery of high temperature superconductivity, made in 1987, we digress briefly to provide some background on the subject and then come to the processing problem of these superconductors. Oxide superconductors, (Y, Ba, Cu)-oxide or the so-called 1-2-3 compound ceramic superconductors, with a transition temperature around 90 K were discovered in February, 1987. Such a high transition temperature enables the use of a cheap and easily available coolant, viz., liquid nitrogen (boiling point 77 K). Since the discovery of these high-Tc superconductors, a lot of hyperbole has been made of the potentialities and possibilities of these superconductors. There is no gainsaying the fact that the discovery of the high temperature superconductivity does represent an extremely important discovery. In all fairness, however, it should be recognized that these high-Tc superconductors have a number of problems to be ironed out before they come anywhere close to the conventional superconductors in terms of applications. One of the major problems is that these superconductors carry very low current densities. The electric resistance goes to zero at Tc« 90 K, but the supercurrents seem to have a great difficulty in going from one grain to another. This may be due to impurities or misshaped grains. The troublesome fact remains that these materials lose their superconductivity at very modest current densities of about a few 100 A/

cm2. Also, their performance deteriorates as the applied magnetic field increases. At a magnetic field of 1 tesla, the Jc«1 -10 A/ cm2, whereas one would like to have a / c ^ 105 A/cm2. The inherent brittleness of ceramics is another problem in these oxide superconductors. Due to the phenomenon of Lorentz force, extremely high tensile and shear forces develop in dipole magnets that would shatter the brittle ceramic oxides. This is particularly true for large magnets which are like pressure vessels wherein the pressure on the superconductor due to the Lorentz force goes as square of the magnetic field. It is easy to see that in such a system the mechanical characteristics of the composite superconductor will dictate the design. Critical current density, / c , of a superconductor is a function of material processing. The standard process for making (Y, Ba, Cu)-based superconductors is to take oxide powders of these elements, cold compact them and then sinter. All that such a material can be used for is to demonstrate the phenomenon of high temperature superconductivity. The main reasons are the ones given above, viz., low current carrying capacity and brittleness. Jin (1991) used a technique called melt textured growth technique which gave impressive current density values. This technique involves melting at 1300°C and cool back to room temperature via a proprietary regimen, which results in needle-like crystals (about 100 \im long). These needle-like grains are aligned (locally) in the grain conduction direction. 1-2-3 superconductors are highly anisotropic with their ability to superconduct. Their current density results are as follows: / c « 7400 A/cm2

at

/ c « 1000 A/cm2

at

£ =0 B=\T

4.2 Processing

In a manner similar to the so-called bronze route of making conventional superconductors described above, where the superconductor is a very brittle intermetallic, Nb 3 Sn, one must avoid the synthesis of the brittle ceramic until the very end. The sintering step should be carried after the formation of the desired shape. This technique involves loading the superconducting powder in the elemental form into a metal tube, draw into a fine wire, and wind into a coil before sintering. For example, in one technique, the metallic elements (a rare earth, Ba, Cu) are melted and solidified as a ternary metal alloy by melt spinning. Melt spinning gives a rapidly solidified alloy in a ribbon form. The ribbon is then shaped and thereupon heated in oxygen to convert into an oxide superconductor. (Jin, 1991; Singh and Narayan, 1991; Sandhage e t a l , 1991). A process that is increasingly gaining importance is called the Oxide-Powder-InTube (OPIT) method (Sandhage etal., 1991). In this process, the oxide powder of appropriate composition (stoichiometry, phase content, impurities, etc.) is packed inside a metal tube (generally silver). This assembly is first deformation processed into the desired geometry (round wire or flat tape) and then given an annealing treatment to obtain the superconductor composite. Intermediate anneals are given during deformation processing. This OPIT process is very similar to the bronze route of making Nb 3 Sn/Cu composite superconductors. Silver is the most common metal tubing material for the following reasons: (i) relatively high oxygen diffusivity (ii) good thermal conductivity (iii) relatively low reactivity with bismuth cuprate-based superconducting oxides.

133

The oxide powder is packed in silver tube (4-7 mm inside diameter and 611 mm outside diameter). The silver tube is sealed after packing and degassed. The deformation processing techniques commonly used are swaging and drawing for wires, and rolling is used for tapes. Heat treatments, intermediate and/or subsequent to deformation are given to form the correct phase, promote grain interconnectivity, crystallographic alignment of the oxide, and to obtain proper oxygenation (Sandhage, etal., 1991). Powder Processing Powder metallurgy methods involving cold pressing and sintering, or hot pressing can be used to fabricate metal matrix composites. A number of powder processing techniques are used for making particle or whisker reinforced MMCs. The matrix and the reinforcement powders are blended to produce a homogeneous distribution. Most of the information on this critical step is considered to be proprietary. The blending stage is followed by cold pressing to produce what is called a green body, which is about 80% dense and can be easily handled. The cold pressed, green body is canned in a sealed container and degassed to remove any absorbed moisture from the particle surfaces. The final step is hot pressing, uniaxial or isostatic, to produce a fully dense composite. The hot pressing temperature can be either below or above the matrix alloy solidus. Continuous filament reinforced composites can also be made by this technique. Fiber tows are infiltrated by dry matrix powder, followed by hot isostatic pressing. Alternatively, matrix alloy powder that has been dry blended with short whiskers, chopped filament, or particles is hot pressed to produce a composite material

134

4 Metal Matrix Composites

with a random fiber orientation. This material can become partially oriented by hot mechanical deformation, usually extrusion or hot forming. Cold pressing and sintering is a less desirable process since the high pressure needed to press the powders to the required density can break the fibers, or the sintering process can degrade the fibers. 4.2.3 Deposition Techniques

Deposition techniques for metal matrix composite fabrication involve coating individual fibers in a tow with the matrix material needed to form the composite, followed by diffusion bonding to form a consolidated composite plate or structural shape. Since the composite is composed of identical units, the microstructure will be more homogeneous than cast composites. The main disadvantage of using deposition techniques is that they are time consuming. However, there are several advantages: - The degree of interfacial bonding is easily controllable; interfacial diffusion barriers and compliant coatings can be formed on the fiber prior to matrix deposition, or graded interfaces can be formed. - Filament wound thin, monolayer tapes can be produced which are easier to handle and mold into structural shapes than other precursor forms; unidirectional or angle-plied composites can be easily fabricated in this way. Several deposition techniques are available: immersion plating, electroplating, spray deposition, chemical vapor deposition (CVD), and physical vapor deposition (PVD). Dipping or immersion plating is similar to infiltration casting except that fiber tows are continuously passed through baths of molten metal, slurry, sol, or organometallic precursors. Electrode-

position or electroplating produces a coating from a solution containing the ion of the desired material in the presence of an electric current. Fibers are wound on a mandrel, which serves as the cathode, and placed into the plating bath with an anode of the desired matrix material. The advantage of this method is that the temperatures involved are moderate and no damage is done to the fibers. Problems with electroplating involve void formation between fibers and between fiber layers, adhesion of the deposit to the fibers may be poor, and there are limited numbers of alloy matrices available for this processing. A spray deposition operation typically consists of winding fibers onto a foil coated drum and spraying molten metal onto them to form a monotape. The source of molten metal may be powder or wire feedstock which is melted in a flame, arc, or plasma torch. The advantages of spray deposition are easy control of fiber alignment and rapid solidification of the molten matrix. In a CVD process, a vaporized component decomposes or reacts with another vaporized chemical on the substrate to form a coating on that substrate. The processing is generally carried out at elevated temperatures. The product phase can be homogeneously nucleated if it occurs in the vapor phase or heterogeneously nucleated if it occurs on a substrate surface. Metallic, oxide, carbide, or nitride coatings or films can be produced with CVD. These films can be grown amorphous, single crystal, or polycrystalline form. When the matrix material is developed on a reinforced preform, such as a woven mat, it is called chemical vapor infiltration (CVI). The three basic PVD processes for creating metal matrix composites differ mainly in how the vapor is generated: evaporation, ion plating, and sputtering. Evaporation methods for matrix materials under high

vacuum, generally contained in a crucible, include resistive heating, electron beam evaporation, arc evaporation, radiation heating, or laser ablation. Ion plating involves passing the vaporized component through an argon gas glow discharge around the substrate which ionizes some of the vapor molecules. These ionized atoms subsequently deposit onto the substrate. Sputtering techniques involve vaporization of the coating material (the target or cathode) via a momentum transfer from an ionized argon gas molecule in a glow discharge around the cathode. The primary advantage of the PVD is versatility in the compositions and micro structure of the coating produced. These processes can produce coatings with superior bonding to the substrate and excellent surface finishes. In addition, no chemical reactions by products are produced by these processes. The disadvantage is that the processing equipment is complex and expensive. Spray-forming of Particulate MMCs A relatively new and promising development for making particle reinforced MMCs deserves a separate description. It involves the use of spray techniques that have been used for some time to produce monolithic alloys (Willis, 1988). One particular example of this, a cospray process, is shown in Fig. 4-5. This process uses a spray gun to atomize a molten aluminum alloy matrix, into which heated (to dry them) silicon carbide particles are injected. An optimum particle size is required for transfer. Whiskers, for example, are too fine to be transferred. The preform produced in this way can be up to about 97% dense. The metal matrix composite is subjected to scalping, consolidation, and secondary finishing processes. The product is a wrought material. The process is totally

Manufactyre of Billets 1 2 3 4 5 6

Tundish Gas atomi/

.

""""" ' 6 '

Figure 4-5. Cospray process of making an MMC. A spray gun atomizes a molten alloy matrix, into which heated ceramic particles are injected.

computer controlled and is quite fast. It should be noted that the process is essentially a liquid metallurgy process. One avoids the formation of deleterious reaction products because the time of flight is extremely short. Silicon carbide particles of an aspect ratio (length/diamter) between 3-4 and volume fractions up to 20% have been incorporated into aluminum alloys. Alcan can cast 200 kg ingots at a rate of 6-10kg/min (Lloyd, 1992). A great advantage of the process is the flexibility that it affords in making different types of composites. For example, one can make in situ laminates using two sprayers. One can have selective reinforcement. This process, however, is more expensive because of the expensive capital equipment. 4.2.4 In Situ Processes In these techniques, as the name suggests, one forms the reinforcement phase

136

4 Metal Matrix Composites

in situ. The composite material is produced in one step from an appropriate starting alloy, e.g., Co-NbC, Ni-TiC, etc., thus avoiding the difficulties inherent in combining the separate components as done in a typical composite processing. Controlled unidirectional solidification of a eutectic alloy is a classic example of in situ processing. Unidirectional solidification of a eutectic alloy can result in one phase being distributed in the form of fibers or ribbon in the other. One can control the fineness of distribution of the reinforcement phase by simply controlling the solidification rate. The solidification rate in practice, however, is limited to a range a of 1 - 5 cm/h because of the need to maintain a stable growth front. The stable growth front requires a high temperature gradient. Some of the newer processes for producing metal matrix composites include Lanxide's PRIMEX, pressureless metal infiltration processes, and Martin Marrietta's XD self propagating reaction process. A brief description of these follows. The Lanxide Process, called PRIMEX, involves a pressureless infiltration of metal into a ceramic preform. The ceramic preform can be prepared by tape casting, molding, or machining of a green body. The metal or alloy ingot is placed on the ceramic preform in a graphite mold and the assembly is heated in a nitrogen atmosphere, so that the molten metal spontaneously infiltrates the ceramic preform. It appears that some specific aluminum-magnesium alloy compositions must be used in a nitrogen atmosphere in order to get wetting and infiltration of silicon carbide preform by the molten alloy. The XD process is another in situ process which uses an exothermic reaction between two components to produce a third component. Sometimes such processing

techniques are referred to as Self Propagating High Temperature Synthesis (SHS) processes. Specifically, the XD Process is a proprietary process developed by Martin Marietta Corp. to produce ceramic particle reinforced metallic alloy. Generally, a master alloy containing rather high volume fraction of reinforcement is produced by the reaction synthesis. This is mixed and remelted with the base alloy to produce a desirable amount of particle reinforcement. Typical reinforcements are SiC, TiC, TiB 2 , etc. in an aluminum, nickel, or an intermetallic matrix. Figure 4-6 gives a schematic of this process, while Fig. 4-7 shows the microstructure of a 15%-volume fraction TiC/Al composite made by this process, followed by extrusion.

Matrix and reinforcement precursor materials A+B+C+X+Y+Z Mixing

Heating

(^Exothermic reaction"^

Metallic or intermetallic matrix with intermetallic or ceramic reinforcements, as particles or whiskers A + B + A B + C + BC + XY + Y Z + . . .

Reinforcements

Figure 4-6. Schematic of an SHS process of making an MMC.

4.3 Interfaces in Metal Matrix Composites

Figure 4-7. Microstructure of a 15% volume fraction TiC/Al composite made by the XD process, followed by extrusion.

4.3 Interfaces in Metal Matrix Composites One of the objectives of this article is to highlight the importance of the interface in metal/ceramic composites, discuss the different types of interfacial bonding, microstructural characterization of the interface region, and the effects of interface on composites properties. We shall do this by giving some specific examples from representative composites. The interface region in a given composite has a great deal of importance in determining the ultimate properties of the composite. An interface is, by definition, a bidimensional region through which there occurs a discontinuity in one or more material parameters. In practice, there is always some volume associated with the interface region over which a gradual transition in material parameter(s) occurs. The importance of the interface region in composites stems from two main reasons: (i) the interface occupies a very large area in composites, and (ii) in general, the reinforcement and the matrix will form a sys-

137

tem that is not in thermodynamic equilibrium. In this seciton, I give examples of interface microstructure in different metal matrix composite systems (particle and fiber reinforced) and discuss the implications of various interface characteristics on the resultant properties of the composite. We can define an interface as a bounding surface between two phases where a discontinuity in some material parameters occurs. Among the important discontinuities are: elastic moduli, thermodynamic parameters such as chemical potential, and the coefficient of thermal expansion. The discontinuity in chemical potential is likely to cause chemical interaction, leading to an interdiffusion zone or a chemical compound formation. The discontinuity in the thermal expansion coefficient means that the interface will be in equilibrium only at the temperature which the reinforcement and the matrix were brought into contact. At any other temperature, there will be present biaxial or triaxial stress fields because of the thermal mismatch between the components of a composite. Thermal stresses due to a thermal mismatch will generally have an expression of the form

<j=f(E,a,b,r)A(xAT

(4-1)

where/(ii, a, b, r) is a function of the elastic constants, E, and geometric parameters, a, b, and r to be defined later, Aa is the difference in the expansion coefficient of the components, and AT is the temperature change. The term (Aa AT") is, of course, the thermal strain. It what follows, I describe the importance of the interface, its crystallographic nature, bonding type and microstructure of the interface in some important composites.

138

4 Metal Matrix Composites

j A = particle/matrix interface area = = Nnd2 (4-7)

4.3,1 Importance of the Interface in Composites The importance of the interface in composites is recognized on all sides. The following simple exercise will help bring home this importance. For a given volume fraction, the total area of fiber/matrix interface increases as the fiber diameter decreases. We can visualize this in a very simple manner. Consider a composite bar containing N continuous, unidirectionally aligned fibers. Let the length, width, and height of the composite be /, w, and /z, respectively, and let the fiber diameter be d. Then, we can write Vf =

volume of fibers volume of composite

Nnd2 /wh

(4-2)

and area of the fiber/matrix interface IA = Nndl

(4-3)

From Eqs. (4-2) and (4-3), we get u-A-^y,

(4-4)

Let the volume of composite be 1 m3. Let us take the fiber volume fraction to be 0.25. Then, Eq. (4-4) reduces to

/.-i(m')

(4-5)

Equation (4-5) says that the fiber/matrix interfacial area, for a given fiber volume fraction, varies inversely with the fiber diameter. We can do a similar calculation for a particle reinforced composite. Assume N spherical ceramic particles embedded in a spherical shell of the matrix. Let the total volume of the composite be 1 m3. Then, we can write p

volume of particles volume of composite N(n/6)d3

i

(4-6)

J

(4-8)

d Taking Vp = 0.25, we have (4-9) Comparing Eqs. (4-5) and (4-9), we find that for a given diameter and volume fraction of reinforcement, a particulate composite will have a larger interfacial area than a fibrous composite. However, the more important conclusion to be drawn from Eqs. (4-5) and (4-9) is that the interfacial area increases with decreasing reinforcement diameter. It is easy to visualize that the interfacial area can indeed become very large for reinforcements less than 10-20 ^im in diameter. Since chemical and/or mechanical interactions between reinforcement and matrix occur at interfaces, an extremely large area of interface has a tremendous importance in determining the final properties and performance of a composite. 4.3.2 Crystallographic Nature of the Fiber/Matrix Interface In crystallographic terms, ceramic/metal interfaces in composites are, generally, incoherent and high-energy interfaces. Accordingly, they can act as very efficient vacancy sinks, provide rapid diffusion paths, segregation sites, sites of heterogeneous precipitation, as well as sites for precipitate free zones. Among the possible exceptions are the eutectic composites (Chawla, 1987 b) and the newer XD type particulate composites (Wang and Arsenault, 1991; Mitra et al., 1991). The in situ or eutectic composites do show semi-coherent interfaces, i.e., the lattice mismatch between the

4.3 Interfaces in Metal Matrix Composites

matrix phase and the reinforcement phase is accommodated by creating a network of dislocations. Figure 4-8 shows an example of a network of interface dislocations between NiAl-matrix and Cr-rod in a directionally solidified NiAl-Cr eutectic (Cline etal., 1971). The XD process gives particulate composites with stable and clean interfaces (Lewis, 1991). Wang and Arsenault (1991) used HREM to study interfaces in some XD composites. They observed semicoherent interfaces between OC-A1203 and NiAl, but incoherent interface between NiAl and TiB2.

139

10011 WPHji .._„-.-~IO1O)

0.4 fim

4.3.3 Interfacial Bonding in Metal Matrix Composites

Some bonding must exist between the ceramic reinforcement and the metal matrix for load transfer from matrix to fiber to occur. Neglecting any direct loading of the reinforcement, the applied load is transferred from the matrix to the reinforcement via a well-bonded interface. Here, it is important to distinguish between wettability and bonding. Wettability refers to ability of a liquid to spread on a solid. Frequently, the contact angle is taken as a measure of wettability, a contact angle of 0° indicating perfect wetting, and a contact angle of 180° indicating no wetting. Wettability just tells about the possibility of attaining an intimate contact between a liquid or solid. Good wetting is a necessary but not sufficient condition for strong bonding. One needs a good wetting even for purely mechanical bonding or weak van der Waals bonding, otherwise voids may form at the interface. Besides wettability, there are other important factors such as chemical, mechanical, thermal, and structural that will affect the nature of bonding between reinforcement and ma-

Figure 4-8. Network of interface dislocations between NiAl-matrix (light area) and Cr-rod in a directionally solidified NiAl-Cr eutectic (Cline et al., 1971).

trix. As it happens, these factors frequently overlap, and it may not always be possible to isolate these effects. 4.3.3.1 Mechanical Bonding

Mechanical keying effect between two surfaces can lead to bonding. Hill e t a l . (1969) confirmed this experimentally for tungsten filaments in an aluminum matrix while Chawla and Metzger (1978) observed mechanical gripping effects at A1 2 O 3 /A1 interfaces. Hill et al. etched tungsten wires along a portion of their length to produce a rough interface. The tungsten filaments were incorporated into an aluminum matrix by the liquid metal infiltration technique in vacuum. They evaluated three interface conditions by longitudinal tensile tests of composites. In the case of a smooth surface, a chemical reaction between aluminum matrix and tungsten led to a high strength. In the case of a smooth surface with a graphite layer,

140

4 Metal Matrix Composites

the graphite barrier layer prevented the reaction from taking place, that is, there was no chemical bonding, and because the interface was smooth (no roughness), there was also very little mechanical bonding. The resultant strength was, therefore, very low. In the case of the etched wires with a graphite layer, there was no reaction bonding, but there was a mechanical keying effect because of the rough surface produced by etching. This mechanical bonding restored the strength to the level achieved with chemical reaction at the interface. Chawla and Metzger (1978) artificially produced interfacial roughness by etchpitting and microhardness indentation. Figure 4-9 shows plots of number of cracks per mm (crack density) vs. percent strain. Cracks in the alumina film first appeared at about the same strain and initially the crack density rose at about the same rate for a rough and a smooth surface. However, for a smooth interface, the crack density remained constant above 8% while for a rough interface, the crack density continued to increase beyond this value, i.e., in the case of a rough interface the load transfer continued to higher strain values than in the case of a smooth surface. Similar mechanical bonding effects due to interface roughness have been observed in babbitt-bronze composites (Liaw et al., 1990). It should be mentioned that in fiber reinforced composites mechanical bonding will be effective mostly in the longitudinal or fiber direction. Transverse to the fiber, it will provide only a minor effect. 4.3.3.2 Chemical Bonding Most metal matrix composite systems are nonequilibrium systems in the thermodynamic sense; that is, there exists a chemical potential gradient across the fiber/matrix interface. This means that given favor-

Polished substrate

Steep sided pits 106 per cm2

70 pm

Gently sloping pits 103 per cm2

100

90

5

10 15 Elongation (%)

20

Figure 4-9. Number of cracks per mm (crack density) vs. percent strain in an alumina/aluminum composite for different degrees of roughness (Chawla and Metzger, 1978).

able kinetic conditions (which in practice means a high enough temperature or long enough time), diffusion and/or chemical reactions will occur between the components. The interface layer(s) formed because of such a reaction will generally have characteristics different from those of either one of the components. It should be pointed out, however, that at times, some controlled amount of reaction at the interface may be desirable for obtaining strong

4.3 Interfaces in Metal Matrix Composites

bonding between the fiber and the matrix, but, too thick an interaction zone will adversely affect the composite properties. Ceramic/metal interfaces are generally formed at high temperatures. Diffusion and chemical reaction kinetics are faster at elevated temperatures. One needs to have knowledge of the chemical reaction products and, if possible, their properties. It is thus imperative to understand the thermodynamics and kinetics of reactions such that processing can be controlled and optimum properties can be obtained. As pointed out above, interfacial bond degradation can occur by an adverse chemical reaction between the reinforcement and the matrix. Therefore, it is important to characterize the fiber/matrix interfacial reaction zone. One generally does this by studying a diffusion couple made of the components of the composite with a planar interface. Diffusion controlled growth in an infinite diffusion couple with a planar interface follows the relationship x2 = Dt

(4-10)

where x is the thickness of the reaction zone, D is the diffusivity, and t is the time. For a composite containing cylindrical fibers of small diameter, the diffusion distance is very small. However, to a first approximation, we can write

x2 = kt

141

inforcement and the matrix. This thermal mismatch can lead to thermal stresses large enough to cause plastic deformation in the matrix. Plastic deformation in the matrix means introduction of defects such as dislocations, vacancies, etc. in the matrix, especially in the interface region. The introduction of such defects can and do affect the phenomena responsible for chemical reactions at the interface as well as characteristics of the matrix such as the precipitation kinetics. A brief review of the thermal stress effects is therefore presented in the next section. 4.3.4 Thermal Stresses in Metal Matrix Composites In general, ceramic reinforcements (fibers, whiskers, or particles) have a coefficient of thermal expansion smaller than that of most metallic matrices. This means that when the composite is subjected to a temperature change, thermal stresses will be generated in both the components. A simple two-element model shown in Fig. 4-10 can be used to make a thermoelastic analysis. It can be shown that for

(4-11)

where k is a material constant and has the dimensions of diffusivity (m 2 s~ 1 ). Such parabolic relationships between the reaction zone thickness and time have been observed by many researchers (see for example DiCarlo, 1983; Schulyer et al., 1990). A factor that ends up complicating the simple chemical incompatibility problem at the interface is the ever present thermal expansion incompatibility between the re-

Figure 4-10. Simple two-element model for thermoelastic analysis, a and b are the radii of the fiber and matrix, respectively.

142

4 Metal Matrix Composites

this unit fiber composite, using cylindrical coordinates, we have the following principal stresses in the matrix (Poritsky, 1934; Hull and Burger, 1934; Chawla, 1987 c): b2

of the cylinder, b is the radius of the matrix shell, and the subscripts m and f denote the matrix and the fiber, respectively. Figure 4-11 shows the thermoelastic stress distribution for an alumina fiber/magnesium composite with 35% volume fraction of fibers as a function of a normalized distance, r/a, with r/a = 1 representing the fiber/matrix interface. Note that or and ae fall off rapidly with the distance, r. It turns

(4-12)

-AU+Z

(4-13) (4-14)

= B 2

A

Em(am-al)AT(a/b) • - [ • l+(a/b)2(i-2v)[(b/a)2-l]EJE

(4-15)

1 + (a/b)2 (1 - 2 v) + (g/b)2 (1 - 2 v) [(b/a)2 - 1 ] g j ^ l l+[(ft/a) 2 -l]£ m /£ f J

D

where a is the coefficient of linear expansion, v is the Poisson ratio (assuming vf = vm = v), E is the Young's modulus, AT is the temperature interval, a is the radius 1600

1

1

1

1 1

(4-16)

out that or and oe are relatively insensitive to fiber volume fraction while az depends on the fiber volume fraction.

I

'

I

• J

,/Mg

1400

AT= -

1200 1000 800 600 400 200

Figure 4-11. Stress distribution for an alumina fiber/magnesium composite with 35% volume fraction of fibers as a function of a normalized distance, r/a, with r/a = 1 representing the fiber/matrix interface. Note that or and oe fall off rapidly with the distance, r.

0 -200 -400 -600 -800 0

I.I

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2 . 0

4.3 Interfaces in Metal Matrix Composites

If we regard a ceramic particle in a metal matrix as more or less an elastic sphere embedded in an infinite elastic continuum, then it can be shown from theory of elasticity (e.g., Brooksbank and Andrews, 1972) that an axially symmetrical stress distribution around the particle will result. The pressure at the interface between the particle and the matrix due to the thermal mismatch is given by Eq. (4-17), where a is the coefficient of linear expansion, v is the Poisson ratio, E is the Young's modulus, AT is the temperature interval, a is the radius of the particle, b is the radius of the matrix, Vp is the particle volume fraction, and the subscripts m and p denote the matrix and the particle, respectively. Figure 4-12 shows, schematically, the stress distribution in a particulate composite.

P = 0.5(l+v ) m

(am-ap)AT (l-2v m )F p V)

(l-2v p ) E

143

matrix and a higher dislocation density. Chawla and Metzger (1972) showed, in a definitive matter, the importance of thermal stresses in metal matrix composites. They used a tungsten fiber/copper single crystal matrix and by employing a dislocation etch-pitting technique showed that near the fiber the dislocation density was much higher in the matrix than the dislocation density far away from the fiber. The enhanced dislocation density in the copper matrix near the fiber arises because of the plastic deformation in response to the thermal stresses generated by the thermal mismatch between the fiber and the matrix. The situation in the as-cast composite can be depicted as shown schematically in Fig. 4-13 where a section along a primary plane of the single crystal copper matrix of the composite is shown (Chawla, 1975 a). (4-17)

The principal elastic stresses in the matrix, the radial and tangential components, in terms of p are: 1- K -P

l-VpUr3

(4-18) (4-19)

The stresses in the particle are: GrV = P (constant)

(4-20)

°e? ~ P (constant)

(4-21)

It can be readily shown from these expressions that the thermal stresses generated in a composite, fibrous or particulate, for rather small temperature changes will exceed the yield stress of the metal matrix and, thus, cause plastic deformation of the

Figure 4-12. Schematic of the stress distribution in a particulate composite.

144

4 Metal Matrix Composites TUNGSTEN FIBER COPPER MATRIX

DISLOCATION ETCH PITS

Figure 4-13. Schematic of a section along the primary plane of the single crystal copper matrix containing tungsten wires. There exists a hard zone (high dislocation density) around each fiber and a soft zone (low dislocation density) away from the fiber.

^PRIMARY SLIP PLANE

There exists a hard zone (high dislocation density) around each fiber and a soft zone (low dislocation density) away from the fiber. Later on, the existence of a plastically deformed zone containing high dislocation density in the metallic matrix in the vicinity of the reinforcement was confirmed by transmission electron microscopy by a number of researchers, both in fibrous and particulate metal matrix composites (Arsenault and Fisher, 1983; Suresh et al., 1989; Christman and Suresh, 1988a; Rack, 1987; Chawla et al., 1991). Figure 4-14, a TEM micrograph, shows dislocations in aluminum matrix near an SiC particle (Chawla etal., 1991). Such high densities of defects in the metal matrix will, of course, lead to a different set of properties of the matrix. In nonprecipitation hardening metals, this will simply cause a strengthening due to a higher dislocation density. In precipitation hardenable matrix alloys such as aluminum-copper, one would expect faster aging kinetics (Arsenault and Fisher, 1983; Suresh et al., 1989; Christman and Suresh, 1988 a; Rack, 1987; Chawla etal., 1991). Preferential precipitation at the reinforcement/matrix interface in an age-hardenable matrix has

been observed by many researchers (see for example Mahon etal., 1990). We discuss this subject of aging behavior of an alloy matrix in the presence of a reinforcement in Sec. 4.4.5. 4.3.5 Microstructure of the Interface Region

In this section, we describe the microstructural features in some important composites.

Figure 4-14. TEM micrograph showing dislocations in aluminum in the region near a silicon carbide particle (SiCp) (Chawla et al., 1991).

4.3 Interfaces in Metal Matrix Composites

145

4.3.5.1 Alumina/Metal Composites Wetting is rather poor between different forms of alumina and molten aluminum, with contact angles ranging from 180° (no wetting) at the melting point to greater than 60° at 1800 K. In particular, a-Al 2 O 3 (FP) based systems may involve the formation of spinels at the fiber/matrix interface, and they enhance the chemical bond at the interface (Levi et al., 1978). Alloying Al matrix by Li to promote wetting of polycrystalline A12O3 (FP) fibers is a very good and successful example of the matrix modification approach to obtain enhanced wettability and bonding. Thus, Al/2-3%Li matrix would wet the A12O3 fibers in vacuum and the technique of metal infiltration of a fiber bundle can be used. X-ray diffraction studies of A12O3 fibers extracted from matrix showed the presence of LiA102 on the fiber surface (Champion et al., 1978). The following chemical reaction has been proposed for the metallurgical bonding in FP fiber/Al-Li matrix (Dhingra, 1980):

Figure 4-15. Interface in a continuous fiber a-Al2O3 (FP)/Mg alloy (ZE41A) matrix showing the reaction zone (RZ) (Pfeifer et al., 1990). BF denotes bright field TEM micrograph

6 Li + A12O3 -» 3LiO 2 + 2A1 LiO 2 + A12O3 -> 2LiAlO 2 Electron diffraction analysis showed the presence of Li2O • 5A12O3 spinel at the fiber surface (Prewo, 1977). Interface in a continuous fiber a-Al 2 O 3 (FP)/Mg alloy (ZE41A) matrix made by casting is shown in Fig. 4-15 (Pfeifer et al., 1990). The reaction zone consists of MgO and is about 100 nm thick. The chemical reaction occurring at the interface can be represented by the reaction: Al2O3(s) + 3 Mg(l) -> 3 MgO(s) + 2 Al(l) Figure 4-16 shows deformation twins in the magnesium matrix in the alumina/ magnesium composite. These deformation twins form because of thermal stresses in

Figure 4-16. Dark field (DF) TEM micrograph showing deformation twins in the matrix in the same system as in Fig. 4-13. These deformation twins form because of thermal stresses in the matrix during cool down. M and F indicate matrix and fiber regions, respectively (Chawla, 1987).

the matrix during cool down. Not unexpectedly, the thermal mismatch gives a zone of the matrix around each fiber that suffers plastic deformation during solidification. Figure 4-17 shows the microhardness profiles of a-Al 2 O 3 (FP)/Al-Li and a-Al 2 O 3 (FP)/Mg alloy composites from the fiber through the reaction zone to the

4 Metal Matrix Composites

146

-FP/Al Fiber ^Fiber surface o FP/Mg 1300 -^&^ 1000'-Interface reaction zone ~500^ x Matrix
c x 100 5025 0

1

2

3 4 Distance (r/a)

5

Figure 4-17. Microhardness profiles of a-Al2O3 (FP)/Al-Li and a-Al2O3 (FP)/Mg alloy composites from the fiber through the reaction zone to the matrix. Note the high hardness of the matrix in the vicinity of the interface, indicating a workhardened matrix.

matrix. Note the high hardness of the matrix in the vicinity of the interface, indicating a workhardened matrix. PRD-166 fiber is a continuous fiber with the composition of a-Al 2 O 3 with about 15-20 wt.% of ZrO 2 . The interface in a composite of PRD-166/Ti-48.4%-Al1%-Mn made by high temperature pressure casting showed ZrAl 3 at the fiber periphery. There was a zone just below the surface free of ZrO 2 , and the fibers contained some entrapped ZrAl 3 (Nourbakhsh etal., 1990). Discrete facetted particles of ZrAl 3 were observed at the interface in a PRD-166/ high purity aluminum made by liquid metal infiltration, Fig. 4-18 (Isaacs etal., 1991). These particles formed and grew into the matrix when the processing was done above the melting point of the matrix for thirteen minutes or more. This is as expected from thermodynamic calculations. However, when the fibers were infil-

(b) Figure 4-18. (a) Discrete facetted particles of ZrAl3 were observed at the interface in a PRD-166/high purity aluminum made by liquid metal infiltration, (b) Zr mapping of the region shown in (a) (Isaacs et al., 1991).

4.3 Interfaces in Metal Matrix Composites

trated with the preform temperature below the melting point of aluminum and with a hold time of less than one minute, no reaction products were visible in an SEM, indicating thereby that by controlling the reaction kinetic conditions, one can control the extent of interfacial reaction. 4.3.5.2 Silicon Carbide/Metal Composites Silicon carbide in a particulate or fibrous form has been incorporated in a number of metallic matrix materials. Silicon carbide is unstable in titanium alloys. Depending on the specific titanium alloy composition, a series of compounds can form in the interface zone. Generally, a very brittle interface region forms in titanium-based composites containing SiC, fiber or composite (Lorreto and Konitzer, 1990; Warren and Andersson, 1984). Das (1990 a, b) has characterized the microstructure of silicon carbide fiber (SCS-

-2 Fiber

0 2 > Reaction Zone j f y t a t r i x Distance (nut)

147

6) reinforced a-Ti, (5-Tx, Ti3Al, and TiAl matrix composites. He examined the microstructure of these composites in the asfabricated state and after thermal exposure. SCS-6 silicon carbide fiber has an outer diameter of 142 |im and a core of about 37 }^m diameter of a carbon filament. The SiC is produced by chemical vapor deposition onto this core (Chawla, 1987 a). A surface coating rich in carbon is applied to heal the "intrinsic" surface flaws due to the chemical vapor deposition technique and to promote compatibility with the matrix. This surface layer consists of carbon-rich nonstoichiometric amorphous SiC. There occurs a local change in the chemical composition of the matrix in the vicinity of the fiber/matrix reaction zone. Figure 4-19 shows a TEM micrograph of the reaction zone in SCS-6/ near a-Ti composite after exposure to 975°C for 4 hours (Das, 1990a). Recrystallized grains and voids can be seen in the

Figure 4-19. TEM micrograph of the reaction zone in SCS-6/near a-Ti composite after exposure to 975 °C for four hours (Das, 1990).

4 Metal Matrix Composites

148

reaction zone. The bottom part of this figure shows the concentration profiles of various elements as determined by energy dispersive analysis. Knoop indentations on the matrix in the vicinity of the reaction zone of the same composite after exposure

MATRIX

FIBER COATING

\ 25jim

CRACK

REACTION ZONE

Figure 4-20. Knoop indentations on the matrix in the vicinity of the reaction zone of the same composite after exposure to 975 °C for 16 hours (Das, 1990). A crack around the indentation near the fiber can be seen in the brittle matrix.

to 975 °C for 16 hours are shown in Fig. 4-20 (Das, 1990 b). A crack around the indentation near the fiber can be seen. This is because the matrix around the fiber is more brittle than far away. Chou et al. (1991) studied interfacial reactions between some refractory metals (Co, Ni, and Pt) and silicon carbide at temperatures between 800 and 1200 °C. They observed a series of silicides and free unreacted carbon precipitates in the reaction zones. A cross-section of the reaction zone in SiC/Ni after 6 hours at 1100°C is shown in Fig. 4-21 (Chou et al., 1991). Adjacent to Ni, there is a carbon precipitation free zone (C-PFZ). Between C-PFZ and SiC, random and modulated carbon precipitates appear. Quantitative electron microprobe profiles of nickel, silicon, and carbon across the reaction zone after 6 hours at 1100°C are shown in Fig. 4-22 (Chou etal., 1991). The arrows indicate the relative nickel concentrations for Ni3Si and Ni 5 Si 2 . The layered regions of the unreacted nickel, carbon precipitate free zone, and carbon precipitate zone are also indicated. These authors observed that the thickness of the reaction zone on the metal side was always greater than on the SiC side.

Figure 4-21. Cross-section of the reaction zone in SiC/Ni after six hours at 1100 °C. C-PFZ next to nickel is the carbon precipitate free zone (Chou et al., 1991).

4.3 Interfaces in Metal Matrix Composites

149

100 90

*—.•....

Ni

.

- ^ 80 §:

Ni

Ni5Si2

70

^T 60 o •"S

8

50 40 30 20

S Si

Ni 3 Si

10 0

'Ni 5 Si 2

Figure 4-22. Quantitative electron microprobe profiles of nickel, silicon, and carbon across the reaction zone after six hours at 1100°C. The arrows indicate the relative nickel concentrations for Ni3Si and Ni 5 Si 2 . The layered regions of the unreacted nickel, Ni, carbon precipitate free zone, C-PFZ, and carbon precipitate zone, CPZ, are also indicated (Chou et al., 1991).

Distance (20jum/Division)

They also did not observe a parabolic reaction zone between the thickness of the SiC reaction zone and the annealing time. This was attributed to the reaction being controlled by the decomposition of SiC, rather than by interdiffusion of Si or metal. Figure 4-23 shows a series of cross-sectional views of the reaction zone in SiC/Ni after 4 hours at 1170°C (Chou etal., 1991). Modulated carbon precipitation, random carbon precipitation, as well as voids are seen in this figure. Porosity results from melting and solidification. The formation of low melting point metal silicides as well as a large enthalpy of mixing between the metals and silicon at the interface lead to melting. Krishnamurthy (1990) studied interfacial reactions in TiAl-based alloys containing SCS-6 fibers and single crystal alumina fibers. The former showed significant reaction at the fiber/matrix interface while the latter did not show any noticeable reaction. Many researchers have examined the interface region in silicon carbide particle or whisker/aluminum alloy composites made by powder metallurgy or casting routes (see for example Arsenault and Fisher, 1983; Suresh etal., 1989; Christman and

Suresh, 1988 a; Rack, 1987; Chawla et al., 1991; Mahon etal., 1990; Nutt and Carpenter, 1990; Lagace and Lloyd, 1989). In magnesium containing aluminum alloys, MgO is frequently observed at the interface (Nutt and Carpenter, 1990). In the case of liquid metal infiltration route, molten aluminum can react with SiC to form aluminum carbide at the interface as per the reaction given below: 4Al + 3SiC -> A14C3 + 3 Si Besides giving a brittle constituent, A14C3 at the interface, such a reaction will inject silicon into the matrix, i.e., a changed alloy composition of the matrix will result. The casting route of making ceramic particle/metal matrix composites involves higher temperatures than those in the powder metallurgy route. Severe interfacial reactions between ceramic particle and metal matrix can occur. Extensive reaction between the ceramic particle and the molten metal matrix can result in degradation of the reinforcement, formation of brittle constituents at the interface, and/or alteration in the matrix microstructure. Wang etal. (1992) used electron microscopy techniques to study the interface in a cast SiCp/Al alloy (A356-7 wt.% Si, 0.3 wt.%

150

4 Metal Matrix Composites

>.;" r ? ;**T.;.'*~ '•

a

"

' b

Figure 4-23. Series of cross-sectional views of the reaction zone in SiC/Ni after four hours at 1170 °C. Modulated carbon precipitation, random carbon precipitation, as well as voids are seen (Chou et al., 1991).

Mg, and the rest Al). They observed magnesium aluminate, MgAl 2 O 4 , a spinel, at the interface. 4.3.5.3 Carbon Fiber/Metal Matrix Composites Carbon fiber reinforced metals have been investigated by a number of researchers (Fuet al., 1986; Katzman, 1987). Fu et al. did a detailed analysis of the interface region in carbon fiber/aluminum composites by TEM and observed fine-grained y-Al 2 O 3 and some A14C3 at the interface. An interesting example of interface tailoring comes from the C/Mg system. Molten Mg does not wet and thus does not bond C fibers. Codeposition of Ti-B on carbon fiber before incorporating it into an alu-

minum matrix is commonly resorted to. The Ti-B coating, however, oxidizes in air, and molten Mg does not wet the oxidized coating. The option of not exposing the coated fibers to air is not very feasible. Katzman (1987) used a silicon dioxide coating from an organometallic precursor solution. SiO2 coating facilitates melting and bonding of C fibers by molten Mg. Such innovative barrier coatings between reinforcement and matrix can be a powerful means of designing a composite with appropriate properties. 4.3.5.4 Interaction Between Interfacial Phenomena Various processing parameters such as time, temperature, pressure combined with

4.4 Properties

the thermodynamic, kinetic, and thermal data can be used to get an optimum set of interface characteristics in a given metal matrix composite. Such a control of interface characteristics can result in an increase in a given mechanical property of the composite. Figure 4-24 shows this idea in a schematic manner. This figure is somewhat simplistic inasmuch as many of the thermal, mechanical, chemical, and structural factors frequently overlap and it is not always easy, in practice, to separate the effects of these factors. For example, consider the interaction between chemical interaction and thermal mismatch. A chemical reaction will produce one or more chemical compounds at the interface. It is of great importance to take into account the presence of any new species at the interface when computing the thermal stresses. In other words, if it is known that a compound forms at the fiber/matrix interface, then it is pointless to use a two-element cylindrical model to compute the thermal stress distribution because the new species is very likely to have a different coefficient of thermal expansion than either one of the original components. Even

li Processing Parameter — • (temperature, time, pressure)

Figure 4-24. Schematic of correlation between processing parameter and property of a composite.

151

a thermally matched composite can have thermal stresses if an interfacial product forms. The extent of interfacial region in composites can be very large. Mechanical as well as chemical bonding can contribute to the bond strength. A controlled interfacial reaction between a metal and a ceramic reinforcement can provide good adhesion. A diffusion barrier coating between the reinforcement and the matrix is frequently employed to control the extent of interfacial reaction. It is therefore important to understand the thermodynamic, kinetic, mechanical, and microstructural aspects of a metal/ceramic interface in order to use that information to optimize the processing to obtain a desirable set of characteristics in the composite. In particular, such an understanding is important if the composite is to have an ability to sustain high temperatures during processing as well as long time exposures at operating temperatures.

4.4 Properties 4.4.1 Importance of the Metallic Matrix The differential thermal expansion between the reinforcement and the metal matrix introduces a high dislocation density in the matrix, especially in the near interface region of the matrix. This high matrix dislocation density as well as the reinforcement/matrix interfaces can provide high diffusivity paths in a composite, an important topic that we discuss in Sec. 4.4.5 on the aging behavior of particle reinforced MMCs. Most reinforcements for use in metals are essentially elastic right up to the fracture point. Once we know their physical and elastic properties (elastic constants, expansion coefficient, etc.), we have pretty well defined their behavior. The interface

152

4 Metal Matrix Composites

region, however, can suffer changes just as the matrix does during processing and/or service. The importance of the interface region, however, has been recognized on all hands almost from the very beginning, and we have described some of these in Sec. 4.3 above. The recognition of the importance of the matrix is more recent. The metallic matrix is much more than a mere medium or glue to hold the reinforcement, be that fibers, whiskers, or particles (Chawla, 1985, McKimpson and Scott, 1989). It should be chosen after due consideration of its chemical compatibility and thermal mismatch with the reinforcement. Processing induced chemical reaction and/or thermal stresses can cause changes in the matrix microstructure of the matrix. These microstructural changes in the matrix, in turn, can and do affect the mechanical and physical behavior of the composite. Typically, commercial producers have taken off-the-shelf wrought or casting alloy compositions for the matrix. Let us consider an aligned fiber reinforced metal matrix composite under a load Pc in the direction of the fibers. This load is distributed between the fiber and the matrix. Thus, Pe = PmVm + P( V{ where P m and P f are the loads on the matrix and the fiber, respectively. This equation can be converted to the following ruleof-mixtures relationship under conditions of isostrain: <7f V(

(4-22)

where a is the stress, V is the volume fraction, and the subscripts c, f, and m denote the composite, the fiber, and the matrix, respectively. Equation (4-22) is commonly referred to as the Rule of Mixtures. This equation says that the strength of the composite is a

volume weighted average of the strength of the fiber and the matrix. On the face of it, there is nothing wrong with this statement. The problem, however, is that one does not have a very good idea of the <7m and af values, i.e., the in situ values of the stress on the fiber and matrix. If the fiber remains essentially elastic up to the point of fracture, then o{9 the fiber strength in the composite, is the same as that determined in an isolated test of the fiber. This is particularly true of ceramic fibers. The same, however, cannot be said for the metallic matrix. The matrix strength in the composite (the in situ strength) will not be the same as that determined from a test of an unreinforced matrix sample in isolation. The metal matrix is likely to suffer several microstructural alterations during processing and consequently changes in its mechanical properties. Let us consider the process of liquid metal infiltration of a bundle of ceramic fibers. After the fiber bundle is impregnated by the liquid matrix, the composite must cool down to room temperature. In view of the fact that, in general, ceramic fibers have a coefficient of thermal expansion less than that of most metallic matrices, thermal stresses will be generated in both the components, fiber and matrix. A series of events can take place in response to the thermal stresses: (i) plastic deformation of the ductile metal matrix (slip, twinning, cavitation, grain boundary sliding and/or migration), (ii) cracking and failure of the brittle fiber, (iii) failure of the fiber/matrix interface. Some very fundamental work has been done on a simple system, tungsten filament/copper matrix, to elucidate the role of matrix in MMCs. The tungsten/copper system is an ideal system for studying pure fiber reinforcement. The two components

4.4 Properties

are mutually insoluble in the liquid as well as solid states. Thus, there is no interfacial reaction between the fiber and the matrix. In the case of a single crystal matrix, microstructural changes induced in the matrix by the mere act of incorporating fibers into it can be studied. Polycrystalline but pure matrix is the next step in complication while an alloy matrix will be even more complex. We provide a summary of these results (Chawla and Metzger, 1972; Chawla, 1973 a, b, 1975 a, b, 1976 a, b). Single crystal copper matrix is amenable to etch-pitting techniques for dislocation density and distribution studies (Livingston, 1962). Low volume fraction (0 to 15%) and large diameter fibers (228 |im) were used to ensure the availability of large matrix areas for direct structural observations. For the etch-pitting technique, the copper surface has to be within 3° of (1 1 1) plane. Dislocation density profiles in the copper matrix for different fiber volume fractions of tungsten filament are shown in Fig. 4-25. Close to the fiber, the curves tend to merge. The differences in the dislocation density were greater away from the fiber (the plateau region). Near the fiber

153

the dislocation cell structure was better defined. The dislocation densities can be thought of as a combination (not necessarily a linear one) of two contributions: (i) a fiber volume fraction independent contribution (near the interface) (ii) a fiber volume fraction dependent contribution (throughout the matrix). In order to understand the origin of this dislocation distribution, Chawla and Metzger (1972) used the simple two-element model shown in Figure 4-10. For the W/Cu system, a was 114 jam and the radius b of the equivalent cylinder of matrix varied from 1590 urn at 0.5 % Vf to 290 (im at 15% Vf. The enhanced dislocation densities in the copper matrix arise owing to the plastic deformation in response to the thermal stresses generated because of the thermal mismatch between the fiber and the matrix. According to the Tresca yield criterion [or — ae] = Y, the matrix yielding will occur first at the interface and then spread out with continued cycling. The dislocation density gradient will tend to even out with continued thermal cycling. In the as-fabricated state, the copper matrix had a cell structure with a diameter

7^7.551 10 9 crn" 2

/ i o1°

density o

/ ° ° ! n0.5 1 15% 10 % ,

CL

107

7.5%" 2.5% ~~i

'•

! 0.5 %

0% 106

i L0

80 120 160 jim 200 Normal distance from interface

Figure 4-25. Dislocation density profiles in the copper matrix for different fiber volume fractions of tungsten filament. Close to the fiber, the curves tend to merge. The differences in the dislocation density are greater away from the fiber (the plateau region).

154

4 Metal Matrix Composites

10 Jim. The cell structure develops due to the high temperature, low (thermal) stress creep that the matrix sufffers during cooling from 1200 °C to room temperature. On thermal cycling to low peak temperatures, the subgrain size and the cell diameter did not change, and the cell diameter did not decrease significantly. The cell walls, however, thickened and the cell structure became better defined. It would appear that the dislocation cell walls formed during the composite fabrication served as obstacles to dislocations generated on subjecting the composite to either mechanical or thermal cycling. Mortensen and coworkers (1991, 1992) as well as others have focussed their attention on matrix microstructural control in cast MMCs. The final matrix microstructure is a function of fiber type, fiber diameter, fiber distribution as well as conventional solidification parameters. For example, in an SCS-2 silicon carbide fiber/ Al-4.5 %-Cu matrix, they observed that in the unreinforced region of the matrix, normal dendritic structure could be seen while in the reinforced region the dendritic morphology was controlled by the fiber distribution. Second phase appeared preferentially at the fiber/matrix interface or in the narrow interfiber spaces. Porosity is a critical defect that is likely to be present in the matrix, especially in cast materials. Porosity can be very deleterious to the overall performance of cast MMCs. The main sources of porosity are any gas evolution and shrinkage occurring on solidification. A high volume fraction of reinforcement may impede the flow of the interdendritic liquid and inhibit any "bulk movement" of the semisolid metal. In short, the micro structure of the metallic matrix in the fiber composite is likely to differ significantly from that of

the unreinforced metal processed in an identical manner. In what we have discussed in this section so far, we have implicitly assumed that the fibers are uniformly distributed in the matrix and that the fiber cross sectional geometry does not have any effect on the strength and plastic deformation of the composite. In practice, there invariably does occur some fiber clustering and the fiber cross sectional shape may also vary. Christman et al. (1989) used a plane strain model to study the effect of fiber clustering on the plastic deformation of silicon carbide whisker reinforced aluminum. They found that when the whiskers were clustered, either horizontally or vertically, the flow stress and strain hardening of the composite decreased significantly compared to the situation where the whiskers were arranged in a perfectly periodic manner. Brockenbrough and Suresh (1990) analyzed continuous boron fiber reinforced aluminum 606 l-O T using the ABAQUS finite element code. The fiber packing arrangement as well as the cross sectional geometry influenced the plastic deformation behavior when the composite was loaded in the transverse direction. The effect of fiber packing arrangement on strain hardening response, however, was more pronounced than that of fiber geometry. Both these variables did not have much effect when the composite was loaded parallel to the fibers. Suresh and Brockenbrough (1993) provide a review of the various continuum models for deformation of continuous fiber reinforced metal matrix composites. 4.4.2 Modulus Unidirectionally reinforced continuous fiber reinforced metal matrix composites show a linear increase in the longitudinal

155

4.4 Properties

Figure 4-26. Example of modulus increase as a function of fiber volume fraction for alumina fiber reinforced aluminum-lithium alloy matrix (Champion et al., 1978). 0.2

(U

0.6

0.8

Young's modulus of the composite as a function of the fiber volume fraction. Figure 4-26 shows an example of modulus increase as a function of fiber volume fraction for alumina fiber reinforced aluminum-lithium alloy matrix (Champion et al., 1978). The increase in the longitudinal Young's modulus is in agreement with the rule-of-mixtures value while the modulus increase in a direction transverse to the fibers is very low. Particle reinforcement also results in an increase in the modulus of the composite; the increase, however, is much less than that predicted by the rule of mixtures. This is understandable inasmuch as the rule of mixtures is valid only for continuous fiber reinforcement. Figure 4-27 shows modulus increase as function of short fiber volume fraction (Girot et al., 1987). Note the rather large variation from the rule-of-mixtures type linear variation with fiber volume fraction. A schematic of the modulus increase in a composite with reinforcement volume fraction for different reinforcement type, continuous fiber, whisker, or particle, is shown in Fig. 4-28. This schematic shows the loss of reinforcement efficiency as one goes from continuous fiber to particle. For discontinuous reinforcement, one may use a Halpin-Tsai-

175 n

tf)

lr

1125; O

1100 ; L

10 20 30 Fiber volume fraction (%)

40

Figure 4-27. Modulus increase as function of short fiber volume fraction (Girot et al., 1987).

Continuous fibres • o_ o in D

Whiskers

/

120-

_ D -o 100o E o 80-

/

^ ' Particle

sti

0

o

LU

600

5

10 15 20 25 Volume fraction (%)

30

35

Figure 4-28. Schematic of modulus increase in a composite with reinforcement volume fraction for different reinforcement type, continuous fiber, whisker, or particle. Note the loss of reinforcement efficiency as one goes from continuous fiber to particle.

156

4 Metal Matrix Composites

Kardos type expression to obtain the modulus (Chawla, 1987 a):

l+2sqVp Pc = Pm

1

_

14

12

y

where p stands for the property of interest, for example, the transverse Young's modulus, Ev s is the aspect ratio of the discontinuous reinforcement, and

^+2s Pm

It would appear that the increase in elastic modulus of a metal matrix composite as a function of the reinforcement volume fraction is fairly straight forward. The modulus of the composite is fairly independent of the reinforcement packing arrangement as long as all the fibers are parallel. For discontinuous reinforcement also the modulus is quite independent of the particle clustering, etc. The modulus does show a dependence on temperature, which essentially comes in from the matrix modulus dependence on temperature, the ceramic modulus being quite independent of temperature for moderate temperatures involved with MMCs. 4.4.3 Strength Commonly, a metal matrix composite has a ceramic reinforcement, however, metal reinforcement in a metal matrix or metal/metal laminated composites are not uncommon. Stress-strain curves of the coppermatrix, the Nb-Ti filaments, and the resultant composite consisting of Nb-Ti filaments in a copper matrix are shown in Fig. 4-29 (Reed et al., 1977). This is an unusual composite in that both the components, matrix and fiber, are ductile. An interesting aspect of the stress-strain curve

298 K 0.02

0.03

0.04

0.05

STRAIN Figure 4-29. Stress-strain curves of the copper matrix, the Nb-Ti filaments, and the resultant composite consisting of Nb-Ti filaments in a copper matrix (Reed et al., 1977).

shown in Fig. 4-29 is the rather high fracture strain of the composite. The strength of a metal matrix composite is difficult to predict by a rule of mixtures type relationship because the strength is a very structure sensitive property. Processing induced thermal stresses and/or chemical reaction at the interface can make it difficult to predict the strength by a simple rule of mixtures type relationship. In Sec. 4.2.1, we pointed out the importance of taking into account the microstructural changes occurring in the metal matrix during processing. If we take due account of these processing induced microstructural changes in the metallic matrix, then it would appear that shear-lag, which involves load transfer from the matrix to the fiber via shear on planes parallel to the fiber, would explain the fiber strengthening phenomenon. For a description of this fiber strengthening model, the reader is referred to one of the

4.4 Properties

references given in the list of suggested reading at the end of the chapter. The shear-lag model would also appear to work for whisker reinforced metal matrix composites. It becomes of rather doubtful validity when one considers particulate composites. Clearly, the matrix alloy and the processing induced micro structural changes in the matrix will influence the final properties of the composite much more in this case. The strengthening of a precipitation hardenable aluminum alloy by incorporating particles of, say, SiC (aspect ratio about unity) is not as significant as in the case of a fiber or whisker of the same material. It appears that dislocation density increase, and the concomitant microstructural alterations in the matrix are an important source of strengthening in particulate composites. This would appear to explain why the composite shows a strength higher than that predicted by continuum mechanics theories. At least in the case of aluminum matrix composites, the increase in strength would appear to be mainly due to the much higher dislocation density and smaller subgrain size in the matrix than in the unreinforced aluminum alloy. The increase in elastic modulus, as discussed above, can be appreciable. McDanels (1985) evaluated the effects of reinforcement type SiC-whisker, -nodule, or -particle reinforcement in various matrix alloys, 6061, 2024/2124, 7075, and 5083 Al matrices. His results showed that in these cases, the elastic modulus of the composites was independent of both reinforcement type and the matrix alloy. The reinforcement content was the dominant factor in the improvement of elastic modulus in these composites. However, the yield and ultimate tensile strength of the composites were primarily controlled by the intrinsic yield/tensile strengths of the matrix alloys and by reinforcement content.

157

The yield strength and ultimate tensile strength of aluminum alloys can be increased to as much as 60% more than those of the unreinforced matrix. However, it is important to emphasize that for a given reinforcement content, yield strength and ultimate tensile strength are controlled by the type and temper of the matrix alloy. The precipitate-hardenable aluminum alloy matrix has a much more important role to play in these composites than a nonhardenable matrix material. A considerable strength increment results due to the age hardening treatments. The particulate or whisker type reinforcements such as SiC, B4C, A12O3, etc. are unaffected by the aging process. These particles, however, can affect the precipitation behavior of the matrix quite significantly, as we discuss below. The strength,
where a0 is a friction stress term, k is a material constant and X is the spacing between rods or lamellae. It turns out that one can vary the interfiber spacing, X, rather easily by controlling the solidification rate, R, because X2 R = constant. Thus, one can easily control the strength of the resultant composite. 4.4.4 Matrix Aging Response in Composites We have pointed above that the microstructure of a metallic matrix is modified by the presence of ceramic reinforcement (particle, whisker, or fiber). In particular, a higher dislocation density in the

158

4 Metal Matrix Composites

matrix metal or alloy than that in the unreinforced metal or alloy has been observed. The higher dislocation density in the matrix has its origin in the thermal mismatch, Aa, between the reinforcement and the metal matrix. For example, the thermal mismatch in the case of SiC/Al has a high value of 21 x 10" 6 K" 1 , which will lead to thermal stresses high enough to deform the matrix plastically and thus leave the matrix workhardened. One would expect that the high dislocation density will also affect the precipitation kinetics in a precipitation hardenable matrix such as 2XXX series aluminum alloys. Aging Process in Al-Cu-Mg

Alloys

The general sequence of precipitation from a supersaturated Al-Cu-Mg alloy when aged at a temperature above room temperature is: supersaturated alloy -> GP zones -• This aging behavior is commonly analyzed in terms of a hardness or strength vs. aging time curve as shown in Fig. 4-30. The shape of the aging curve can be explained as follows: Immediately after quenching, only solid solution hardening is present. As GP zones form, hardness or strength increases because extra stress is needed to

Equilibrium precipitates

Aging time (particle size)

Figure 4-30. Aging curve, hardness vs. aging time, of a precipitation hardenable aluminum alloy.

have dislocations shear the coherent zones. The hardness increases as the GP zone size increase with time, making it ever more difficult for the dislocations to shear them. The peak hardness or strength is associated with a critical dispersion of coherent or semicoherent precipitates. With still further increase in time, equilibrium precipitates with incoherent interface start appearing, and the mechanism of Orowan bowing of dislocations around the particles becomes operative. The stress required for dislocation bowing varies inversely as the interparticle spacing, which decreases with increasing aging time. Continued aging results in an increase in precipitate size and interparticle distance, and a lower strength results as dislocation bowing becomes easier. It is well known that mechanical deformation prior to aging can accelerate the formation of S' at the expense of GP zones. The S' precipitate has the same chemical composition as the equilibrium phase S, orthorhombic structure with [10 0], [010], and [001] directions of S' being parallel to the [100], [0 2 1], and [0 1 2] directions of the matrix. The S' precipitates nucleate on dislocations and grow as laths on {2 1 0} planes along [1 0 0] directions. One expects that the quenching from the solutionizing temperature to room temperature, a AT of about 450°C, will result in a large zone of matrix plastically deformed around each ceramic particle in which the dislocation density will be very high. One would expect that this high dislocation density will tend to accelerate the aging kinetics. Nieh and Karlak (1984) showed that the presence of B4C particles in 6061 Al made the matrix reach the peak hardness in 3 hours at 450 K while a control powder metallurgy processed alloy took 10 hours. This accelerated aging behavior was attributed to the high diffusivity character-

4.4 Properties

istics of the composite. A considerable strength increment can be due to the age hardening treatments in a precipitation hardenable aluminum alloy composite. The particle or whisker type reinforcements such as SiC, B4C, A12O3, etc. are unaffected by the aging process. These particles, however, can affect the precipitation behavior of the matrix quite significantly. Arsenault and Fisher (1983) used transmission electron microscopy to observe a high density of dislocations (4x 109 cm" 2 ) clustered around the ends of SiC whiskers (SiCw) and forming low angle cell boundaries in a 20%-SiCw/Al 6061 composite. They also observed very fine precipitates along the dislocations. Mahon et al. (1990) observed decoration of dislocations by precipitates in SiC particle reinforced aluminum composites. In situ observation of dislocations generated due to thermal mismatch have been made by using a hot stage in a high voltage transmission electron microscopy (Vogelsang etal., 1986). These authors also observed that the presence of SiC particles hindered the subgrain growth in the aluminum matrix. It was also observed that the dislocation density in the matrix was smaller around small, equiaxial SiC particles. As the particle size increased from 1 |j,m to 5 jim, the dislocation density in the matrix around a particle increased. Such an effect of the matrix particle size on the size of the matrix plastic zone around the particle was verified by Kim et al. (1990). These dislocations form heterogeneous nucleation sites for the precipitates in the matrix during subsequent aging treatments. This in turn results in the alteration of the precipitation kinetics in the matrix of the composite as compared to those of the unreinforced material (Nieh and Karlak, 1984; Arsenault and Shi, 1986; Rack, 1987; Christman and Suresh, 1988 a; Papazian,

159

1988; Dutta and Bourell, 1988; Christman etal., 1989; Suresh etal., 1989; Chawla et al., 1991; Dutta et al., 1991). Dutta and Bourell (1989) considered the effect of elastic strain field induced in the matrix by the ceramic particles to cause an enhanced matrix diffusivity and accelerated aging kinetics. Dutta et al. (1991) made a theoretical analysis and experimental observations leading them to conclude that precipitate growth rate was accelerated because of short circuit, pipe-diffusion, through dislocations. An interesting aspect of the effect of ceramic particles on aging kinetics of a matrix is that the aging kinetics are a very sensitive function of the aging temperature. Specifically, the aging behavior of the matrix in SiCp/Al 2014 (Chawla et al., 1991; Esmaeili et al., 1991) and in Al 2 O 3p / Al 2014 (Chawla et al., unpublished) composites made by casting route, was examined at four different aging temperatures and compared with the unreinforced Al 2014. The time to peakage for the Al 2014 matrix and the unreinforced Al 2014 as a function of aging temperature is shown in Fig. 4-31. As the aging temperature increased, the difference in peak aging time between Al 2014 matrix and the unreinforced Al 2014 became larger. Note at 150°C the difference in peak aging time between the reinforced and unreinforced material was rather negligible. At 195 °C this difference was about 7 hours. The relatively large difference between thermal expansion coefficient of SiC and Al resulted in substantial stresses which generated dislocations at and near the interface on quenching. During aging at 195 °C, high density of dislocations would provide a short-circuit path for heterogenous nucleation and fast growth of S' precipitates resulting in accelerated aging of the matrix. As the aging temperature was lowered, the

160

4 Metal Matrix Composites

165 180 Temperature (°C)

195

Figure 4-31. Time to peakage for Al 2014 matrix and Al 2014 as a function of aging temperature. As the aging temperature increases, the difference in time to peakage between Al 2014 matrix and the unreinforced Al 2014 increases.

difference in aging kinetics diminished. This was attributed to the homogeneous nucleation of GP zones and a decrease in growth rate of precipitates at lower aging temperature. It is well known (Lorimer, 1978; Porter and Easterling, 1981) that the formation of GP zone and its transition to Q" and & in Al-Cu system is only possible when the alloy is aged at a temperature below the GP zone solvus. If, for example, the aging temperature is above 0" but below 9' solvus, then there will be no GP zones and the first precipitate will be 0', heterogeneously nucleated at dislocations. At high aging temperatures, the S' precipitates nucleate heterogeneously on dislocations and grow as laths on (2 1 0)a in [0 01] directions (Porter and Easterling, 1981). At lower temperatures the first precipitates to nucleate are the coherent Cu-rich and GP zones. It is important to mention that the nucleation of GP zones at low temperature is mostly homogeneous, and excess vacancies introduced by quenching are thought to play an important role in their formation. Upon quenching the reinforced

composite, the soft matrix undergoes a plastic deformation due to the substantial thermal stresses resulting from the CTE difference between Al matrix and SiC particles. The plastic deformation of the matrix will result in a large number of dislocations and vacancies. These defects would appear to accelerate the aging kinetics of the matrix at high aging temperatures. Such effects do not seem to be effective at low aging temperatures. Differential scanning calorimetry confirmed the presence of GP zones and S' precipitates at 150°C, and only S' precipitates at 195 °C in the 2014 matrix (Chawla et al., 1991). Most of the metal matrix composite work has involved off-the-shelf metallic alloys, especially in the case of particle reinforced metal matrix composites. It is important to bear in mind that in such cases, using the standard heat treatment practices given in the manuals and handbooks for the unreinforced alloys can lead to drastically different results. For a thorough review of the matrix aging behavior in particle reinforced metal matrix composites, see the review by Suresh and Chawla (1993). 4.4.5 Toughness

Toughness of a given metal matrix composite depends on - the matrix alloy composition and microstructure, - the reinforcement type, size, and orientation, and - processing insofar as it affects microstructural variables (e.g., distribution of reinforcement, porosity, segregation, etc.). The general range of the plane strain fracture toughness, Klc, values for particle reinforced aluminum type MMCs is between 15-30 MPa m 1/2 , while short fiber

161

4.4 Properties

or whisker reinforced MMCs show a range of about 5-10 MPa m 1/2 . The reader is cautioned that a lot of data in the literature has been obtained from invalid fracture toughness tests! A summary of fracture toughness results for some particle and whisker reinforced composites with aluminum 6061 matrix and unreinforced aluminum 6061 are given in Tables 4-1 (Friend, 1989) and 4-2 (Hasson and Crowe, 1985). Table 4-2 shows that by substituting silicon carbide whiskers with particles, one can increase the toughness of the composite. Continuous fiber reinforced metal matrix composites show anisotropy in toughness just as in other properties. Table 4-3 shows the results of impact energy for the B/Al system in the longitudinal and transverse directions. Note that, for a given matrix alloy, the longitudinal fracture energy is higher than the transverse energy. Table 4-4 gives the results of 0°, 90°, and 0°/90° reinforcement of aluminum by boron fibers. The 0° and 90° arrangements of fibers result in two extremes of toughness while the 0°/90° arrangement, a sort of pseudo-random arrangement, reduces the degree of anisotropy (Sun and Prewo, 1977). Using fibers in the form of a braid can make the crack propagation toughness increase tremendously due to extensive matrix deformation, fiber bundle debonding, and pullout. Table 4-5 shows this for alumina/Al-2.5 % -Li composite (Majidi and Chou, 1987). Note the increase in the transverse toughness on three-dimensional braiding of the fibers. Matrix alloy composition can also have significant effect on toughness of a composite. Table 4-6 shows the effect of different aluminum matrix alloys on Charpy energy of boron/aluminum alloy composites (McDanels and Signorelli, 1976). The tougher the matrix, the tougher will be the

Table 4-1. Toughness of particle reinforced MMCs vis-a-vis unreinforced alloy (Friend, 1989). Klc (MPa m1/2)

Material

Al 6061 (T6) A12O3/A1 6061 SiC/Al 6061

37 15 7-18

Table 4-2. Toughness of Al 6061 and SiC whisker and particle reinforced composites (Hasson and Crowe, 1985). Klc (MPa m1/2)

Material 20 vol. % SiCw/A16061 T6 25 vol. % SiCp/A16061 T6 Al 6061 T6

7.1 15.8 37.0

Table 4-3. Anisotropy in impact fracture energy (kJ m~2) of B/Al (McDanels and Signorelli, 1976). System

B/Al 1100 B/Al 5052 B/Al6061 B/Al2024

Longitudinal

Transverse

200-300 90 80 40

0.53 0.85 0.27 0.31

Table 4-4. Toughness as function of ply orientation in a B/Al composite (Sun and Prewo, 1977). K IC (MPam 1/2 )

Ply angle 0° 90° 0°/90°

100 34 61-63

Table 4-5. Anisotropy in toughness reduced by 3 D braiding in an alumina FP (36 vol.%)/Al-2.5 Li composite (Majidi and Chou, 1987). iC lc (MPam 0 - 5 ) Longitudinal Unidirectional 3 D braided

32 32.3

Transverse

15-17 21.1

162

4 Metal Matrix Composites

Table 4-6. Effect of matrix alloy on Charpy energy of B/Al alloy composites (McDanels and Signorelli, 1976). Composite

Charpy impact energy (kJ m 2)

B/1100 B/5052 B/6061 B/2024

Thin sheet

Standard

20-300 90 80 40

200-300 170 126 120-170

Table 4-7. Effect of fiber diameter on impact fracture energy (kJm~ 2 ) of B/A11100 (McDanels and Signorelli, 1976). Fiber Diameter (um)

Energy (kJ m 2)

100 140 200

90 150 200-300

composite. The effect of fiber diameter on the toughness of a metal matrix can be seen in Table 4-7. For a given Vf9 the larger the diameter of the fiber, the tougher is the composite. This is because the larger the fiber diameter, for a given fiber volume fraction, the larger is the amount of tough, metallic matrix in the inter/fiber region that can undergo plastic deformation and thus contribute to the toughness.

ficients of thermal expansion of composites (Kerner, 1956; Turner, 1946; Schapery, 1968), determined experimentally these coefficients and analyzed the general thermal expansion characteristics of metal matrix composites (Vaidya and Chawla, 1991; Premkumar etal., 1992; Zweben, 1992). Figure 4-32 shows the variation of thermal expansion coefficient of SiCp/Al composite as a function of particle volume fraction. The solid line represents the Turner model prediction of the expansion coefficient as a =

VmK

VpKp

where a is the coefficient of thermal expansion, Fis the volume fraction, Kis the bulk modulus, and the subscripts p and m represent the particle and the matrix. In Fig. 4-32, we have taken the coefficient of expansion of SiCp to be 4.5xlO~ 6 K~ 1 while that of aluminum i s 2 3 x l O ~ 6 K ~ 1 . Values of the expansion coefficients of some common electronic substrate materials are also included. Turner equation seems to predict the values very well in these particulate composites. In terms of 25 Turner model prediction

4.4.6 Thermal Properties We discussed in the interface section the importance of thermal expansion mismatch between the reinforcement and the matrix. Thermal mismatch is indeed something that is difficult to avoid in any composite. By the same token, however, one can control the overall thermal expansion characteristics of a composite by controlling the proportion of reinforcement and matrix and the distribution of the reinforcement in the matrix. Many researchers have proposed models to predict the coef-

10 — AI 2 O 3 GaAsAlN.

0 0

0.2 (U 0.6 0.8 Volume fraction of SiC

1.0

Figure 4-32. Variation of thermal expansion coefficient of SiCp/Al composite as a function of particle volume fraction. The solid line represents the Turner model prediction.

4.4 Properties

matching the expansion coefficient of the composite with that of a given substrate, all one needs to do is to use the appropriate particle volume fraction. Varying the aluminum alloy composition can allow some additional control of thermal expansion coefficient of the composite. Figure 4-33 shows the predicted influence of the silicon content in the matrix on the coefficient of thermal expansion of the composite as a function of silicon carbide volume fraction (Premkumar etal., 1992). Understandably, the influence of SiCp becomes more 12

1

11

i:

N

-

'

AI-5%S1 matrix " A M 0%SI matrix ...... AI-20%SI matrix AI-25%SI matrix

X

10

1

^

VS.

-

S4 6 5 1

0.3

0.4

1

i

i

0.5 0.6 0.7 Volume fraction of SIC

i

0.8

0.9

Figure 4-33. Predicted influence of silicon content in the matrix on the coefficient of thermal expansion of the composite as a function of silicon carbide volume fraction (Premkumar et al., 1992). The influence of SiCp becomes more dominant at higher particle volume fractions.

Composite Pure copper /^3

163

dominant at higher particle volume fractions. Thermal conductivity is another thermal property of great significance. Unidirectionally aligned carbon fibers in an aluminum matrix can have very high thermal conductivity along the fiber direction. The conductivity transverse to the fibers is about two-thirds that of aluminum. In fact, in the temperature range of-20°C to 140 °C, the thermal conductivity of the composite along the fiber direction is greater than that of copper. On a per unit weight basis, the composite is about four times as conductive as pure aluminum and 2.6 times as conductive a Al 6061. Figure 4-34 compares the thermal conductivity of copper and carbon fiber/aluminum (Sharp and Loftin, 1992). Such a C/Al composite can find applications in heat transfer applications where weight reduction is an important consideration. For example, in high-density, high-speed integrated-circuit packages for computers and in base plates for electronic equipment. Another possibility is to use this composite to dissipate heat from the leading edges of wings in high-speed airplanes. 4.4.7 Fatigue We can define fatigue as the phenomenon of mechanical property degrada-

5.000 2.000 2.000

C

o o

I

1.000

2

1.000 -200

-100

0 100 Temperature (°C)

200

Figure 4-34. Comparison of thermal conductivity of copper and carbon fiber/aluminum as a function of temperature.

164

4 Metal Matrix Composites

tion leading to failure of a material or a component under cyclic loading. Understanding of the fatigue behavior of composites has lagged that of other aspects such as the elastic stiffness or strength. An understanding of the fatigue behavior of composites of all kinds (polymer matrix composites - PMCs, metal matrix composites - MMCs, and ceramic matrix composites - CMCs) is of vital importance, because without such an understanding it would be virtually impossible to gain acceptance of the design engineers. It is well known that a high proportion of failures in metals can be traced to the phenomenon of fatigue. Many high volume applications of composite materials involve cycling loading situations, e.g., automobile components. In general, it would appear that the fatigue resistance of a given material can be enhanced by reinforcing it by short discontinuous fibers (or whiskers) or continuous fibers. Not unexpectedly, the improvement is the greatest when the fibers are aligned parallel to the stress direction. While conventional approaches such as stress vs. cycles (S-N) curves or fatigue crack propagation tests based on the fracture mechanics concepts can be used for comparative purposes and for obtaining information on the operative failure mechanisms, they do not provide information useful to designers. Novel approaches as that epitomized by the measurement of stiffness reduction as a function of cycles in polymer matrix composites are desirable. Since many applications of composites involve temperature changes, it is important that thermal fatigue characteristics of composites be evaluated in addition to their mechanical fatigue characteristics. A major difficulty in this regard is that the application of conventional approaches to fatigue of composites, for example,

the stress vs. cycles (S-N) curves or the application of linear elastic fracture mechanics (LEFM), is not straightforward. The main reasons for this being the inherent heterogeneity and anisotropic nature of the composites. This results in damage mechanisms in composites being very different from those encountered in conventional, homogeneous, or monolithic materials. Despite these limitations, conventional approaches have been used, and we describe them briefly before describing some novel approaches to the problem of fatigue in composites. In particular, we shall describe some of the techniques that have been used to analyze the fatigue behavior of PMCs, and are beginning to be applied to MMCs as well. Following this, we briefly touch upon the important problem of thermal fatigue in composites.

S-N Curves A very popular conventinal fatigue testing technique used with metals involves determination of the so-called S-N curves, where S is the stress amplitude and N is the number of cycles to failure (Fig. 4-35). In general, for ferrous metals, one obtains a

Cycles to failure A/f

Figure 4-35. S-N curve obtained in conventional fatigue testing. S is the stress amplitude, and N is the number of cycles to failure. Curve A shows an endurance or fatigue limit, SL, while curve B does not.

4.4 Properties

fatigue limit or endurance limit, curve A in Fig. 4-35. For stress levels below this endurance limit, theoretically, the material can be cycled infinitely. If, however, such an endurance limit does not exist, for example for nonferrous metals (curve B in Fig. 4-35), then one can arbitrarily define a certain number of cycles, say 106, as the cutoff value. It would be useful at this point to define some important parameters which are used in the discussion of the fatigue phenomenon. Based on Fig. 4-36, we can define the following parameters: cyclic stress range, cyclic stress amplitude,

mean stress,

stress ratio, r>

Fatigue behavior of metals can be conveniently divided into two stages: crack initiation and propagation. In high cyclelow stress fatigue of metals, most of the property changes occur in what has been called the small process zone at the crack tip. The rest of the material away from the crack does not seem to be affected seriously by the phenomenon of fatigue. In the low cycle-high stress fatigue of metals, the phenomena of workhardening and worksoftening are observed. Quite frequently, however, a rule-of-thumb approach in metal fatigue is to increase its monotonic strength which concomitantly results in an increase in its cyclic strength. This rule-ofthumb assumes that the ratio fatigue

in CD

\ \ /

/ /

'V/ \ / \v /

165

Time

Figure 4-36. Various fatigue parameters.

strength/tensile strength is about constant. Maximum efficiency in terms of stiffness and strength gains in fiber reinforced composites occurs when the fibers are continuous, uniaxially aligned and the properties are measured parallel to the fiber direction. Incorporation of fibers certainly improves the fatigue resistance of fiber reinforced composites in the fiber direction. High strength, brittle fibers such as carbon or boron would not be expected to show fatigue behavior similar to that of metals. Not surprisingly, therefore, composites containing these fibers, aligned along the stress axis and in large volume fractions, will show high monotonic strength values which are translated into high fatigue strength values as well. Typically, the S-N curves of such composites are very flat. Figure 4-37 shows the S~N curves in tension-tension for unidirectionally reinforced boron (40 vol.%)/Al 6061, alumina-FP (50vol.%)/Al, and alumina-FP (50vol.%)/Mg composites (Champion et al., 1978). The cyclic stress is normalized with respect to the monotonic ultimate tensile stress. Note the rather flat S-N curves in all the cases and the fact that the unidirectional MMCs show better fatigue properties than the matrix when loaded parallel to the fibers. For example, at 107 cycles, the fatigue to tensile strength ratio of the composite is about 0.77, almost double that of the matrix.

166

4 Metal Matrix Composites I.O

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I04 I0 5 CYCLES TO FAILURE

McGuire and Harris (1974) studied the fatigue behavior of tungsten fiber reinforced aluminum-4%-copper alloy under tension-compression cycling (R= — 1). They found that increasing the fiber volume fraction from 0 to 24% resulted in increased fatigue resistance. This was a direct result of increased monotonic strength of the composite as a function of the fiber volume fraction. It should be mentioned that due to the highly anisotropic nature of the fiber reinforced composites in general, the fatigue strength of off-axis MMCs, just like that of any fibrous composite, decreases with increasing angle between the fiber axis and the stress axis. Fatigue crack propagation studies have been done on aligned eutectic or in situ composites as well. Since many of these in situ composites are meant for high temperature applications in turbines, their fatigue behavior has been studied at temperatures ranging from room temperature to 1100°C. The general consensus is that the mechanical behavior of in situ composites, static and cyclic strengths, is superior to that of the conventional cast superalloys (Stoloff, 1978). A major drawback of studies of fatigue behavior of a material using this S-N approach is that no distinction can be made

Figure 4-37. S-N curves in tension-tension for unidirectionally reinforced boron (40 vol.%)/Al 6061, alumina-FP (50 vol.%)/Al, and alumina-FP (50 vol.%)/Mg composites (Champion etal., 1978).

I0 7

between the crack initiation phase and the crack propagation phase. Fatigue Crack Propagation Tests Such tests are generally conducted in an electrohydraulic closed-loop testing machine on notched samples. The results are presented as log (da/dN) (crack growth per cycle) vs. log AK (the alternating stress intensity factor). Crack growth rate, da/dN, is related to the cyclic stress intensity factor range, AK, according to the power low relationship first formulated by Paris and Erdogan (1963):

where A and m depend on the material and test conditions. The applied cyclic stress intensity range is given by AK=

YAa(a)1/2

where Y is a geometric factor, ACT is the cyclic stress range, and a is the crack length. The major problem in this kind of tests is to make sure that there is one and only one dominant crack that is propagating. It was mentioned above that the complexities in composites lead to the presence of many modes of damage, such as matrix

167

4.4 Properties

cracking, fiber fracture, delamination, debonding, void growth, multidirectional cracking, etc. And these modes appear rather early in the fatigue life of composites. Progressive loss of stiffness during fatigue of composites is a very important characteristic, and it is very different from the behavior of metals in fatigue. The different types of damage zones formed in an isotropic material such as a metal, or ceramic, or polymer and anisotropic material such as a fiber reinforced composite are shown in Figs. 4-38 and 4-39, respectively. In the case of an isotropic material, a single crack propagates in a direction perpendicular to the cyclic loading axis (mode I loading), Fig. 4-38. In the fiber reinforced composite, on the other hand, a variety of subcritical damage mechanisms, Fig. 4-39 (a, b, c), lead to a highly diffused damage zone. Figure 4-40 shows schematically a comparison of damage accumulation in a composite and in monolithic, homogeneous materials under constant stress amplitude fatigue. We plot damage ratio against cycle ratio. The damage ratio is the current damage normalized with respect to the damage at final failure. The cycle ratio, similarly, is the number of cycles at a given instant divided by the number of cycles to failure. In a homogeneous material, the term damage simply represents the crack length, and not surprisingly it increases monotonically with cycling (Fig. 4-40). In the case of a composite, we do not have a simple and unambiguous manifestation of damage, such as a crack length. Instead, damage means the crack density. Note that, unlike in homogeneous materials, the damage in composites accelerates at first and then decelerates with cycling (Fig. 4-40). This distinctive behavior is very important.

Figure 4-38. Damage zone in a conventional, homogeneous, monolithic material (isotropic).

Figure 4-39. Diffuse damage zone in a fiber reinforced composite (anisotropic): (a) fiber break and local debonding, (b) matrix cracking, deflection of the principal crack along a weak fiber/matrix interface.

Composites

Life (cycles or time)

^

Figure 4-40. Comparison of damage accumulation in a fiber reinforced composite and a monolithic material.

Fatigue Behavior of Whisker and Particle Reinforced Composites Williams and Fine (1985 a) studied the microcrack initiation and growth in fatigue in silicon carbide whisker (20 vol-%) reinforced 2124 aluminum alloy under fatigue. Fatigue crack initiation sites were

168

4 Metal Matrix Composites

the unbonded silicon carbide whiskers as well as non-SiC intermetallies. Not unexpectedly, a reduction in the clustering of SiC particles and the number and size of the intermetallics led to an increased fatigue life. The composite showed a slower crack growth rate than the unreinforced matrix. However, the unreinforced 2124T6 matrix was able to support cracks as long as 500 \xm at the peak cycle stress close to its yield stress while the composite could support only a crack 30 \xm long at 60% of its yield strength. Silicon carbide particle reinforced aluminum alloy composites have been studied by researchers (Crowe and Hasson, 1982; Williams and Fine, 1985 a, b; Logsdon and Liaw, 1986; Shang etal., 1988; Davidson 1989; Kumai etal., 1990; Bonnen etal., 1990; Williams and Fine, 1987; Christman and Suresh, 1988 a, b). In general, in terms of S-N curve behavior, the composite shows an improved fatigue behavior vis a vis the reinforced alloy. Such an improvement in stress controlled cyclic loading or high cyclic fatigue is attributed to the higher stiffness of the composite. However, the fatigue behavior of the composite, evaluated in terms of strain amplitude vs. cycles or low cycle fatigue, was observed to be inferior to that of the unreinforced alloy (Bonnen et al., 1990). This was attributed to the generally lower ductility of the composite compared to unreinforced alloy. Fatigue Crack Initiation Particles on short fibers can provide easy crack initiation sites. The detailed behavior can vary depending on the volume fraction of the reinforcement, shape, size, and most importantly on the reinforcement/matrix bond strength (Williams and Fine, 1987). For example, they observed fatigue crack initiation at the poles of SiC

whiskers in 2124 aluminum. They also observed arrest of short cracks at the whisker/Al interfaces. Frequently in aluminum matrix composites, especially those made by casting route, there are present particles other than SiC, such as CuAl2, (Fe,Mn) 3 SiAl 12 , and Cu 2 Mg 5 Si 6 Al 5 (Kumai and Knott, 1991). The phenomenon of particle pushing ahead of the solidification front results in SiC particles and the so-called constituent particles decorating the cell boundaries in the aluminum alloy matrix. The slow cooling rate in conventional castings process leads to rather coarse constituent particles and SiC segregating to the interdendritic region. Some possible reinforcement and crack tip interactions are shown in Fig. 4-41 while Fig. 4-42 shows schematic representations of crack growth vs. cyclic intensity factor (Kumai et al., 1990). It would appear that choosing an optimum particle size and volume fraction together with a clean matrix alloy will result in a composite with improved fatigue characteristics. Shiang et al. (1988) examined the effect of particle size on fatigue crack propagation as a function of cyclic stress intensity in a silicon carbide particle reinforced aluminum. They observed that for fine particle size, the threshold stress intensity, AKth, for the composite was less than that for the unreinforced alloy, i.e., initial fatigue crack growth resistance of the composite was less than that of the unreinforced alloy. For coarse particles, the threshold intensity values were about the same for the two, while at very high values of the cyclic stress intensity, the fatigue crack growth rate of the composite was less than that of the unreinforced alloy. As was pointed out above, laminated fibrous composites can sustain a variety of subcritical damage (cracking of matrix, fiber/matrix decohesion, fiber fracture, ply

4.4 Properties Promoted by coarse particles, high strength matrix

Promoted by strong interfaces, high modulus particles

Promoted by weak interfaces (precipitation, oxidation, reaction)

Figure 4-41. Some possible interactions between particulate reinforcement and a crack tip.

Monolithic alloy

Figure 4-42. Schematic representations of crack growth vs. cyclic intensity factor. (Kumai et al., 1990.)

169

cracking, delamination, etc.). For example, the cracking of a ply will result in a relaxation of stress in that ply, and with continued cycling no further cracking occurs in that ply. Ply cracking generally involves cracking in the matrix and along the fiber/matrix interface but rarely any fiber fracture. Other damage accumulating mechanisms include the growth of existing cracks into interfaces leading to ply delamination. The delamination of a ply results in a reduction of stress concentration on the neighboring plies. As can be seen, the subcritical damage can accumulate rather rapidly on cycling. The various types of subcritical damage mentioned above result in a reduction of the load-carrying capacity of the laminate composite, which in turn manifests itself as a reduction of laminate stiffness and strength (Hahn and Kim, 1976; Highsmith and Reifsnider, 1982; Talreja, 1985; O'Brien and Reifsnider, 1981; Ogin et al., 1985). Many researchers have experimentally related the stiffness changes in the laminated composites to the accumulated damage under fatigue (Hahn and Kim, 1976; Highsmith and Reifsnider, 1982; Talreja, 1985; O'Brien and Reifsnider, 1981). It can safely be said that the change in stiffness values is a good indicator of the damage in composites. The modulus reduction rate — 1/EO dE/dN, at a given value of E/Eo, is the tangent to curve Ev at that value of E/Eo where E is the current modulus, Eo is the initial modulus of the uncracked material, and N is the number of cycles. This rate of modulus reduction can be described over a certain range of peak fatigue stress by the expression (Ogin etal., 1985)

i d£ _ £diV £odiV

170

4 Metal Matrix Composites

where A and n are constants and amax is the peak fatigue stress. Stiffness loss in continous fiber reinforced MMCs and be a useful parameter for detecting fatigue damage initiation and growth (Johnson, 1989). Laminated MMCs There are two type of laminated MMCs: (i) Metallic matrix containing fibers oriented at different angles in different layers. This is similar to that of polymeric laminates, (ii) Two or more different metallic sheets bonded to each other. In the case of the fibrous laminate, Johnson (1982) observed a significant loss of stiffness in boron/aluminum laminate when cycled in tension-tension. Also, in manner similar to that in the laminated PMCs, the ply stacking sequence affected the fatigue behavior. For example, the 90° surface plies in a 90°/0° sequence would develop damage more rapidly than the 0° plies. In the case of laminated composites made out of metallic sheets, e.g., stainless steel and aluminum, enhanced resistance against fatigue crack propagation was observed than in either one of the components in isolation (Chawla and Liaw, 1979).

tional aramid fibers in an epoxy matrix. It is called aramid aluminum laminates (ARALL). Improved fatigue resistance of ARALL over that of monolithic aluminum structures is the main attractive feature. Cracks can grow only a short distance before being blocked by the aramid fibers spanning the crack tip. Figure 4-43 shows the slow fatigue crack growth characteristics of ARALL laminates compared to two monolithic aluminum alloys (Mueller and Gregory, 1988). Applications of ARALL are envisaged in tension dominated fatigue structures such as aircraft fuselage, lower wing, and tail skins. It is expected that the use of ARALL will result in 15-30% weight savings over conventional construction. Structural Observations on MMCs Subjected to Fatigue Structural observations on metal matrix composites after subjecting them to fatigue can reveal a lot of important information on the operating deformation mechanisms, in particular such observations can

2024

Hybrid Composites Composites containing more than one type of fibers are called hybrid composites. Such composites, by using two or more type of fibers, extend the idea of tailormaking a composite material to meet specific property requirements. Partial replacement of expensive fibers by cheaper but adequate fiber types is another attractive feature of hybrid composites. An interesting type of hybrid composite is made of alternating layers of high strength aluminum alloy sheets and layers of unidirec-

l

1OU

10b

10b

Cycles

Figure 4-43. ARALL composite shows slow fatigue crack growth characteristics compared to the monolithic aluminum alloys (after Mueller and Gregory, 1990).

4.4 Properties

tell us how the metal matrix composites behave under cyclic loading vis a vis the monolithic metallic materials. Chawla (1975 a) examined the dislocation structures produced in tungsten fiber reinforced single crystal copper matrix composites. As pointed out above, in single crystal copper one can use etch-pitting technique to study the dislocation structure in the bulk material unlike the TEM techniques which give information about an extremely small region. The fiber/matrix thermal coefficient mismatch results, in cooling through a temperature range of more than 1000 °C, in a very heterogeneous dislocation distribution in the matrix. Recall that the dislocation density in the matrix rose to very high value near the fiber/matrix interface and decreased with distance from the interface (see Figs. 4-13 and 4-25). There was a high dislocation density, hard zone (marked H) immediately next to the fiber and a low dislocation density, soft zone (marked S) away from the fiber. All structural observations on fatigue cycling were made in the plateau or soft region, i.e., at distances greater than 100 |im from the fiber. At distances less than that the cell structure was well developed to start with and the only difference observed on cycling was a reduction in cell size from the original 10 jim diameter (approximate) to a final 5 to 6 |im. The dislocation structure produced after the first cycle up to a stress of 15 MPa, and which give a permanent strain of 8.6xlO~ 6 , showed dislocation pileups in the copper matrix against well-defined sub-boundary obstacles and less well-defined cell wall obstacles. Near the fiber the dislocation distribution, after the first cycle, was indistinguishable from that existing in the as-prepared state. It would thus appear that plastic deformation started in this soft region away from the fiber while

171

the high dislocation region immediately next to the fiber stayed relatively elastic at this stage. The preexisting sub-boundaries and, to a lesser extent some cell walls, acted as obstacles to dislocation motion. The dislocation structure produced on further cycling to the same stress amplitude showed that more plastic deformation took place in this soft matrix region, leading to a greater number of pileups against the above-mentioned obstacles and also thickening of the sub-boundaries by collection of dislocations in their vicinity. Williams and Fine (1985 a, b, 1987) used TEM technique to study the structural changes occurring in SiC whisker reinforced aluminum matrix composites. The unfatigued foil showed a nonuniform dislocation distribution. High initial dislocation density near the pole of a whisker promoted fatigue crack initiation. Thermal Fatigue As pointed out above, thermal stresses in composite materials are very important because, in general, there is a large difference in the thermal expansion coefficient of the ceramic and the metallic components. Thus, even a small change in temperature can cause rather large thermal stresses. The presence of these thermal stresses of large magnitude makes it imperative that we understand the microstructural changes and the mechanical behavior of composites under thermal cycling conditions. In particular, structural damage can occur on thermal cycling, which should be taken into account in any application involving cyclic temperature changes. At the risk of repetition, we wish to reiterate that thermal stresses arise in composite materials due to the large differences in the thermal expansion coefficient, a, of the fiber and the matrix. It should be

172

4 Metal Matrix Composites

emphasized that thermal stresses in composites will arise even if the temperature change is uniform throughout the volume of the composite. Such thermal stresses can be introduced in composites during cooling from high fabrication or annealing or curing temperatures or during any temperature excursions (inadvertent or by design) during service. Turbine blades, for example, are very much susceptible to thermal fatigue. The magnitude of thermal stresses in composites is proportional to the thermal strain, Aa • AT", where Aa is the difference in the expansion coefficient of the two components and AT is the amplitude of the thermal cycle (see Sec. 4.3.4). In MMCs, the matrix generally has a much higher coefficient of thermal expansion than the fiber. Therefore, rather large internal stresses can result when fiber reinforced composites are heated or cooled through a temperature range. When this happens in a repeated manner, we have what is called the phenomenon of thermal fatigue, because the cyclic stress is thermal in origin. Thermal fatigue can lead to cracking in a brittle matrix, e.g., a polymeric or a ceramic matrix. In an MMC, plastic deformation of a ductile metallic matrix is more likely (Chawla, 1973 a, b, 1976 a, b; Chawla, 1974, 1976; Garmong, 1975; Koss and Copley, 1971; Kwei and Chawla, 1992; Li et al., 1989; Yoda et al., 1978). Some examples of microstructural damage that can occur on thermal cycling of metal matrix composites are given in Figs. 4-44 and 4-45. Figure 4-44 shows the formation of slip lines (multiple slip) in a tungsten/copper single crystal matrix thermally cycled in the chamber of an SEM. The polished, undeformed copper matrix shows slip lines due to the plastic deformation. In a polycrystalline copper matrix, on cycling to high temperatures, void formation occurred at grain boundaries in cop-

Figure 4-44. Multiple slip in copper single crystal matrix in a W/Cu composite thermally cycled in the chamber of an SEM. The polished, undeformed copper matrix shows slip lines due to the plastic deformation (Chawla, 1973).

per matrix in a W/Cu composite (Fig. 4-45 a), while grain boundary sliding and migration occurred on cycling to very high temperatures (Fig. 4-45 b). Besides cavitation in the matrix, fiber/matrix debonding can also occur due to thermal fatigue in composites. Li et al. (1989) observed cavitation at the interface in an alumina fiber reinforced magnesium alloy matrix that was subjected to 2500 thermal cycles between room temperature and 300 °C. In general, one can reduce the damage in the matrix by choosing a matrix material that has a high yield strength and a large strain to failure (i.e., ductility). The eventual fiber/matrix debonding can only be avoided by choosing the components such that the difference in the thermal expansion characteristics of the fiber and the matrix is low.

173

4.4 Properties

ulus, AQ is the activation energy for creep, k is the Boltzmann's constant, and Tis the temperature in kelvin. The power law explains the behavior at low to intermediate stress levels and high temperatures while the exponential relationship is generally applicable at high stress levels and relatively low temperatures. A typical creep curve showing the three stages, primary, secondary, and tertiary is shown in Fig. 4-46. Kelly and Tyson (1966) showed that incorporation of tungsten wires in silver can reduce the steady state creep rate, Fig. 4-47. 0.05 0.04

tertiary . 0.03 i ' 0.02

steady state stage

:

-primary stage

0.01

10

Figure 4-45. Microstructural changes from thermal cycling, (a) Void formation occurred at grain boundaries in a polycrystalline copper matrix in a W/Cu composite, (b) Grain boundary sliding occurred on further cycling.

n

e = A (a/G) exp ( - AQ/kT) where e is the strain rate, A is a constant, a is the applied stress, G is the shear mod-

30

Figure 4-46. Typical creep curve showing the primary, secondary, and tertiary stages.

io-

4.4.8 Creep Creep behavior of metals and alloys can be conveniently described by combining two types of expressions, a power and an exponential relationship (Meyers and Chawla, 1984). The basic governing equation of creep can be written in the following form

20 Time (h)

3

• matrix A composite

8 io- 5 0.5

1

10

Stress, MPa Figure 4-47. Incorporation of tungsten wires in silver can reduce the steady state creep rate (Kelly and Tyson, 1966).

174

4 Metal Matrix Composites

/ o

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range of 1100-1400 °C. They observed that the overall creep rate of the composite was lower than that of unreinforced molybdenum silicide. Some efforts have been made at modeling the creep behavior of MMCs. The main problem is that in the regime where the metal matrix shows a steady state creep, the fibers will generally be deforming elastically.

/

log (stress/MPa)

Figure 4-48. Minimum creep rate as a function of applied stress for Saffil short fiber reinforced aluminum alloys and unreinforced aluminum alloys (Dlouhy et al., 1993). Note that the creep rate for the composite is lower than that of the control alloy, but the slope of the composite line is higher than that of the unreinforced alloy.

Figure 4-48 shows the minimum creep rate as a function of applied stress for Saffil short fiber reinforced aluminum alloys and unreinforced aluminum alloys (Dlouhy etal., 1993). Note that the creep rate for the composite is lower than that of the control alloy, but the slope of the composite line is characteristically much higher than that of the unreinforced alloy. Many researchers have verified the high stress exponent, n around 20, for a discontinuously reinforced aluminum vis a vis a stress exponent of n around 4 for the unreinforced aluminum (Nieh, 1984; Nardone and Strife, 1987; Morimoto etal., 1988; Dragone and Nix, 1992). Also, the activation energy for these composite has generally been observed to be very high, in the range of 225-400 kJ/mol while the activation energy for self diffusion in aluminum is about 150kJ/mol. Vasudevan et al. (1991) and Sadananda etal. (1991) examined the creep behavior of monolithic and SiC reinforced MoSi2, in compression, and in the temperature

4.4.9 Tribological Properties

Wear or tribological characteristics are system properties rather than material properties, i.e., operating wear mechanism^) depends on the contact type (rolling, sliding, etc.), the operating conditions, environment, the material characteristics of the test material and the mating material. In the case of metal matrix composites, the hard ceramic constituents such as silicon carbide will be more wear resistant than the metal matrix, the soft matrix keeps the reinforcement in place and transfers the applied load to the reinforcement. Many of the conventional materials, generally not considered to be composites, do have the characteristics of particulate composites because they have hard, brittle intermetallics or carbides distributed in a soft metallic matrix (Fine, 1991). There is a variety of tests available for wear resistance testing. Lubricated rolling and sliding wear are two common ones. For rolling contact tests, or roller-on-cylinder is used, the roller is made of the MMC test material and the cylinder is made of cast iron. The sliding test, called block-on-ring test, involves a stationary test block (MMC) and a rotating steel ring. The wear test is done at specific applied normal load and rotational speed while a jet of lubricant is sprayed on the steel ring. The wear of the test material (MMC) is

175

4.4 Properties

measured by measuring weight loss as a function of sliding distance. As in other areas, one would expect that continuous fiber reinforced composites will show anisotropy in their wear behavior as well (Eliezer et al., 1979; Greenfield and Vignaud, 1985). The wear rate is high when the fibers are parallel to the wear surface but perpendicular to the sliding direction. Ceramic particle (alumina, silicon carbide, graphite, etc.) reinforced aluminum alloys show excellent abrasive resistance with a variety of mating surfaces (Pan et al., 1990; Hosking et al., 1982; Rohatgi, etal., 1990, 1992). Figure 4-49a shows the weight loss vs. sliding distance curves for different orientations of silicon carbide whisker (20% by volume)/aluminum 2124 composites and unreinforced aluminum 2124 under lubricated contact, 168 N load and 2 m s " 1 speed while Fig. 4-49 b shows the behavior under dry contact, 52 N load and l m s " 1 speed (Pan etal., 1992). Analyses of the wear tracks and subsurface micro structure by SEM and TEM showed that the adhesive wear occurred through plastic deformation under dry sliding condition and abrasive wear through plowing and polishing occurred under lubricated conditions. An interesting aspect of precipitation hardenable aluminum matrix particulate composites is the finding that the aging characteristics of the matrix can affect the wear behavior of the composite. Pan et al. (1992) observed a decrease in the wear rate of powder metallurgy alloy 2124 containing silicon carbide particles in the overaged condition because of an easier crack propagation along incoherent precipitate/matrix interface than along SiC/matrix interface. This resulted in less pullout of SiC particles due to the change of the fracture path from the SiC/matrix interface in the underaged and peakaged to the grain

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40 60 80 100 Sliding distance (km)

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(b)

10 20 30 Sliding distance (102m)

40

Figure 4-49. (a) Weight loss vs. sliding distance curves for different orientations of silicon carbide whisker (20 vol.%)/aluminum 2124 composites and unreinforced aluminum 2124 under lubricated contact, 168 N load and 2 ms" 1 speed, (b) Behavior under dry contact, 52 N load and l m s " 1 speed (Pan et al., 1992).

boundaries in the overaged condition. In short, proper heat treatment can reduce the extent of SiC particle pullout and thus reduce the wear damage.

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4 Metal Matrix Composites

4.5 Applications It is convenient to divide the applications of metal matrix composites into aerospace and non-aerospace categories. In the category of aerospace applications, low density coupled with other desirable features such as a tailored thermal expansion and conductivity, high stiffness and strength, etc. are the main drivers. Performance rather than cost is an important item. Inasmuch as continuous fiber reinforced MMCs deliver superior performance than particle reinforced composites, the former are frequently used in aerospace applications. In the nonaerospace applications, cost and performance are important, i.e., an optimum combination of these items is required. It is thus understandable that particle reinforced MMCs are increasingly finding applications in non-aerospace applications. We provide below a brief description of various applications of MMCs. Aerospace Applications Reduction in the weight of a component is a major driving force for any application in the aerospace field. Also, it should be recognized that space environment is not benign. Among the hazardous items that may be encountered in space are orbital debris, meteorites, atomic oxygen, etc. It appears that metal matrix composites can withstand the space environment better than polymer matrix composites. In the Hubble telescope, pitch-based continuous carbon fiber reinforced aluminum was used for waveguide booms because this composite is very light, has a high elastic modulus, E, and a low coefficient of thermal expansion, a. Other aerospace applications of MMCs involve replacement of light but toxic

beryllium. For example, in the US Trident missile, beryllium has been replaced by SiCp/Al composite. Silicon carbide particle/aluminum composite is replacing carbon/epoxy in aircraft electronic equipment racks. Under the US National Aerospace Plane (NASP) program, various US government agencies have funded perhaps the world's first plant devoted solely to the production of SCS type fiber reinforced titanium matrix composites. The NASP will experience very high skin temperatures, especially at the leading edges. Non-Aerospace Applications One of the important applications of MMCs in the automotive area is in the Diesel piston crowns (Donomoto et al., 1983). This application involves incorporation of short fibers of alumina or alumina + silica in the crown of the piston. The conventional Diesel engine piston has an Al-Si casting alloy with crown made of a nickel cast iron. The replacement of the nickel cast iron by aluminum matrix composite resulted in a lighter, more abrasion resistant, and cheaper product. Particulate metal matrix composites, especially with light metal matrix composites such as aluminum, magnesium, also find applications in automotive and sporting goods. In this regard, the price per kg becomes the driving force for application. If we take the price of direct chill (DC) aluminum ingot as unity, then the price of cast SiCp/Al or A12O3/A1 composite will be about 1.5 while the powder metallurgy processed composite will be between 3035, and a composite made by co-spray will be about 10. It is worth pointing out some of the cost drivers. In the production of an extrusion billet, the cost factors are only material and sawing costs. The billets are generally subjected to extrusion to pro-

4.5 Applications

duce a wrought alloy product. The cost factors in the extrusion process are the speed of extrusion, die cost, die wear, etc. Cutting, machining, forming, welding, surface finishing, and transportation are the other cost drivers before a finished product is available. Further items that should be included in this list are recycling and reclamation, particularly so in aluminum matrix composites because recycling has been extremely successful in the aluminum industry. The two terms, recycling and reclamation, are used advisedly for composites. Recycling means reuse of a worn out product as a composite while reclamation involves a recovery of the individual components, i.e., aluminum and ceramic particles separately (Lloyd, 1993). Cast MMCs can be remelted and reused as a composite. Although some of the issues such as particle/molten metal reaction are the same in remelting, it should be appreciated that scrap sorting and melt cleanliness and degassing can be rather tricky problems in MMCs. Special fluxing and degassing techniques have been developed at Duralcan (Lloyd, 1993). It would appear the powder processed MMCs will also involve remelting. The reclamation route also involves remelting. According to Lloyd (1993), using combined argon and salt fluxing, silicon carbide particles can be removed from the melt, and 85-90% of the aluminum can be reclaimed. Hoover (1992) provides a good example of a potential commercial application of particle reinforced aluminum composite to make automotive driveshafts. In the design of a driveshaft, one needs to consider the speed at which it becomes dynamically unstable. This critical speed (in rpm) is given by N =

0.5

15K

L

2

\d

g(R2o-R?)

177

where E is the Young's modulus, L is the length of shaft, g is the acceleration due to gravity, d is the density, Ro and R{ are the outer and inner radii, respectively. This expression says that in terms of geometric parameters, a shorter shaft length and larger diameter will give a higher critical speed, Nc, while in terms of material parameters, the higher the specific stiffness (stiffness/density), the higher will be Nc. The specific stiffness of two common materials used for shafts, steel and aluminum, are about equal. Thus, the driveshafts made out of these materials will have quite similar dynamic properties. Hoover analyzed the use of a 6061 aluminum alloy reinforced with 20 vol. % alumina particles. This particulate MMC had a specific modulus about 40% higher than that of steel or aluminum. The measured critical speed, Nc was over 6000 rpm. Driveshafts of identical dimensions but made of unreinforced aluminum have a critical speed of 5200 rpm. One can also make changes in the driveshaft geometry (increase in length or a reduction in the diameter) while maintaining a constant critical speed. A decrease in the driveshaft diameter can be important because of under-chassis space limitations. Copper based composites having Nb, Ta, or Cr as the second phase in a discontinuous form are of interest for certain applications requiring high thermal conductivity and high strength. Sometimes we refer to these as Cu-X composites, where X is insoluble in copper at room-temperature and forms the second phase. One specific example is high heat flux application in rocket engine thrust chambers. Cu-X systems are very useful for processing such composites. The second phase appears in a dendritic form at room temperature, which can be converted into a filamentary or ribbon form by mechanical

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4 Metal Matrix Composites

working. It should be pointed out that the ribbon morphology is thermodynamically unstable at high temperatures because the ribbons tend to spherodize as a function of time and temperature. Carbon fiber reinforced copper can find applications in the aerospace as a very high thermal conductivity material. The problem with carbon fibers and metallic matrices is that the surface energy considerations preclude wetting of the carbon fibers by the molten metals. The surface energy of carbon fiber is about 100 mJ/m 2 while most molten metals have surface energy of about 1000 mJ/m 2 . Wetting occurs when the surface energy of the substrate (fiber) is higher than that of the molten metal. Thus, one does not expect the wetting of carbon fibers by molten copper. Two possible routes around this difficulty are: fiber surfaces modification or matrix modification. Filamentary Superconductors Conventional commercial superconductors are referred to as niobium-based superconductors because Nb-Ti and Nb 3 Sn are the superconducting materials. It should be recognized that these conventional superconductors became commercially viable after techniques were developed to put the superconducting material in the form of extremely fine filaments in a copper matrix (Chawla, 1987 a, p. 165), i.e., the final product is a copper matrix composite in the real sense of the word. There are two reasons for using the superconducting material in a filamentary composite form. Both of them have to do with stabilization of the superconductor. Pinning of magnetic flux lines is an absolute necessity because a flux jump due to any reason (magnetic, thermal, or mechanical) can lead to quenching of the superconduc-

tor. Dividing the superconductor material into smaller volume units in the form of ultrathin (diameter in the jum range) filaments is helpful because in this way the heat capacity of the material is high enough to limit the temperature increase. This is called adiabatic stabilization. Putting a high conductivity, normal material such as high purity copper can dampen the rate of flux change and can serve as an efficient heat sink. This is called dynamic stabilization (Foner and Schwarz, 1981; Luhman and Dew-Hughes, 1979; Krauth, 1991). Krauth (1991) shows that for a magnetic field perpendicular to the filament axis in the case of Nb-Ti superconductors at 4.2 K, a filament diameter of less than 100 jim is required. Note that this requirement of fine diameter filaments will hold for the high temperature superconductors as well. If we can use them at 77 K, then larger diameter filaments may be permissible because at 77 K, the specific heat capacity values will be higher than at 4.2 K. Because the current carrying capacity of ultrathin filaments will be rather low, a large number of them are incorporated in a normal conductor matrix such as copper. Copper matrix, of course, provides the dynamic stabilization. Niobium-titanium superconductors are used in the magnetic resonance imaging (MRI) techniques for medical diagnostics. The superconducting solenoid made from Nb-Ti/Cu composite wire, is immersed in a liquid helium cryogenic Dewar. The liquid helium is consumed at about 4 mL per hour, and each refill of the Dewar lasts about three months. Nb-Ti/Cu superconducting composites are also used in various high-energy physics applications, such as particle accelerators. They are also likely to find applications in magnetically levitated train project of the future. They are being used in experimental test beds of

4.6 References

magnetically levitated trains. The Nb 3 Sn/ Cu superconducting composites are used for magnetic fields greater than 12 T. Such high fields are encountered in a thermonuclear fusion reactor, representing a sizeable fraction of the capital cost of such a fusion power plant. Electronic Grade MMCs Metal matrix composites can be tailored to have optimized thermal and physical properties to meet the requirements of electronic packaging systems (e.g., cores, substrates, carriers, and housings). We can list the following as the desirable attributes for electronic grade applications: - Dimensional stability. This can be obtained by controlling the following characteristics of a composite: - Coefficient of thermal expansion. For example, the very high coefficient of thermal expansion of aluminum, (24 x 10 ~6 K~ x ) can be easily lowered to a desirable value by adding an appropriate amount of ceramic particle such as SiC or alumina (for isotropy) or carbon fibers (for anisotropy). - high thermal conductivity for efficient heat transfer from the substrate and to even out any hot spots generated on components. - Lightness or low density is an important consideration for airborne and space applications. - Surface quality (as smooth as possible). Use of MMCs for electronic packaging must be regarded as a major application that has come about in the last quarter of the twentieth century. Many MMCs have been produced that meet these requirements. Examples include: SiCp/Al, SiCw/Al, carbon fiber (pitch-based)/Al, etc. A controlled thermal expansion space truss, i.e., one having a high precision dimensional

179

tolerance in space environment was developed from carbon fiber (pitch-based)/Al composite. Continuous boron fiber reinforced aluminum composites made by diffusion bonding have been used as heat sinks in chip carrier multilayer boards. Lanxide Electronic Components makes silicon carbide reinforced aluminum by their pressureless infiltration process for high power, high density electronic circuit enclosures, with a specific gravity about lA of Kovar. Many other companies make such components by different methods, for example, DWA makes SiC particle or carbon fiber reinforced powder metallurgy aluminum alloy composites, Advanced Composite Materials Corporation makes SiC whisker reinforced powder metallurgy aluminum alloy composites, while PCast Equipment Corp. makes SiC/Al, A1N/A1, C/Al, C/Cu, TiB2/Al composites by pressure infiltration casting using an inert gas to force liquid metal into a fibrous preform.

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McKimpson, M.G., Scott, T.E. (1989), Mater. Sci. Eng. A107, 93-106. Meyers, M.A., Chawla, K.K. (1984), Mechanical Metallurgy: Principles and Applications. Englewood Cliffs, NJ: Prentice-Hall. Mitra, R., Chiou, W.A., Weertman, J.R., Fine, M.E., Aikin, R.M. (1991), Scr. Metall. Mater. 25, 26892694. Morimoto, T, Yamaoka, T., Lilholt, H., Taya, M. (1988), /. Eng. Mater. Technol. 110, 71. Mueller, L.R., Gregory, M. (1990), Paper presented at I Annual Metals and Metals Processing Conf. of SAMPE Aug., Cherry Hill. NJ. Nardone, V.C., Strife, J.R., Metall. Trans. 18A, 109114. Nieh, T.G. (1984), Metall. Trans. 15A, 139-146. Nieh, T.G., Karlak, R.F (1984), Scripta Metall. 18, 25. Nourbakhsh, S., Liang, R.L., Margolin, H. (1990), Metall. Trans. 21 A, 213-228. Nutt, S.R., Carpenter, R.W (1990), Mater. Sci. Eng. 76, 169-177. Pan, Y.M., Cheng, H.S., Fine, M.E. (1990), in: Fundamental Relationship between Microstructure and Mechanical Properties of Metal Matrix Composites. Warrendale, PA: The Mineral, Metals and Materials Soc, p. 637-653. O'Brien, T.K., Reifsnider, K.L. (1981), /. Composite Mater. 15, 55. Ogin, S.L., Smith, P.A., Beaumont, P.WR. (1985), Compos. Sci. Technol. 22, 23-31. Pan, Y.M., Fine, M.E., Cheng, H.S. (1992), Tribol. Trans. 35, 482-490. Papazian, J.M. (1988), Metall. Trans. 19A, 29452953. Paris, P.C., Erdogan, F. (1963), J. Basic Eng. (Trans. ASME, Ser. D) 85, 528-534. Pfeifer, M., Rigsbee, J.M., Chawla, K.K. (1990), /. Mater. Sci. 25, 1563-1567. Poritsky, H. (1934), Physics 5, 406-411. Porter, D.A., Easterling, K.E. (1981), Phase Transformation in Metals and Alloys, New York: Van Nostrand Reinhold, p. 308. Premkumar, M.K., Hunt, W.H., Sawtell, R.R. (1992), /. Miner. Met. Mater. Soc. July 24-28. Prewo, K.W. (1977), United Technologies Res. Center. Rep. R77-912245-3. Rack, H.J. (1987), in: Proc. 6th Int. Conf. on Composite Materials. New York: Elsevier, p. 382. Reed, R.P., Mikesell, R.P., Clark, A.F. (1977), Adv. Cryog. Eng. 22, 463. Rohatgi, P.K., Blau, P.J., Yust, C.S. (Eds.) (1990), Tribology of Composite Materials. Materials Park, OH: ASM International. Rohatgi, P.K., Ray, S., Liu, Y. (1992), Int. Mater. Rev. 37, 129. Sadananda, K., Jones, H., Feng, C.R., Petrovic, I I , Vasudevan, A.K. (1991), in: Ceram. Eng. Sci. Proc. Westerville, OH: Am. Ceram. Soc

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Sandhage, K.H., Riley, Jr., G.N., Carter, W.L. (1991), J. Miner. Met. Mater. Soc. March, 21-25. Schapery, R.A. (1968) J. Compos. Mater. 2, 380-404. Schulyer, D.R., Sohi, Mahapatra, R. (1990), in: Interfaces in Metal-Ceramics Composites. Warrendale, PA: The Minerals, Metals and Materials Soc, p. 5. Shang, J.K., Yu, W., Ritchie, R.O. (1988), Mater. Sci. Eng. A102, 181-192. Sharp, G.R., Loftin, T.A. (1992), Personal communication from DWA Speciality Composites Co., CA. Singh, R.K., Narayan, J. (1991), /. Miner. Met. Mater. Soc, March, 13-20. Stoloff, N.S., in: Advances in Composite Materials. London: Applied Sci. Publ., p. 247. Sun, C.T., Prewo, K.M. (1977), J. Compos. Mater. 11, 164. Suresh, S., Brockenbrough, J.R. (1993), in: Metal Matrix Composites, Suresh, S., Needleman, A., Mortensen, A. (Eds.) Stoneham, MA: Butterworth-Heinemann. Suresh, S., Chawla, K.K. (1993), in: Fundamentals of Metal Matrix Composites: Suresh, A., Needleman, A., Mortensen, A. (Eds.). Stoneham, MA: Butterworth-Heinemann. Suresh, S., Christman, T., Sugimura, Y. (1989), Scr. Metall 23, 1599-1602. Talreja, R. (1985), Fatigue of Composite Materials, Lyngby: Technical Univ. of Denmark. Turner, P.S. (1946), J. Res. Natl. Bur. Stand. 37, 239250. Vaidya, R.U., Chawla, K.K. (1991), in: Developments in Metal and Ceramic Matrix Composites: Upadhya, K. (Ed.). Warrendale, PA: The Minerals, Metals, and Materials Society, pp. 253-272. Vasudevan, A.K., Doherty, R.D. (Eds.) (1989), Aluminum Alloys-Contemporary Research Applications. Boston: Academic. Vasudevan, A.K., Petrovic, J.J. (Eds.) (1992), High Temperature Structural Silicides, Amsterdam: North-Holland.

Vasudevan, A.K., Petrovic, J.J., Sadananda, K. (1991), in: Metal Matrix Composites-Processing, Microstructure and Properties. Roskilde, Denmark; Ris0 Natl. Lab., p. 707-712. Vogelsang, M., Arsenault, R.J., Fisher, R.M. (1986), Metall. Trans., 17A, 379-389. Wang, L., Arsenault, RJ. (1991), Metall. Trans. 22A, 3013-3021. Wang, N., Wang, Z., Weatherly, G.C. (1992), Metall. Trans., 23A, 1423-1430. Warren, R., Andersson, C.-H. (1984), Composites 15, 101-111. Williams, D.R., Fine, M.E. (1987), Proc. 6th Int. Conf on Composite Materials, Vol. 2. London: Elsevier, pp. 113-118. Williams, D.R., Fine, M.E. (1985), Proc. 5th Int. Conf Composite Materials., pp. 275-290. Williams, D.R., Fine, M.E. (1985), in: Proc. 5th Int. Conf. Composite Materials (ICCM/V), Warrendale, PA: TMS-AIME, p. 639. Willis, T.C. (1988), Met. Mater., Aug., 485. Yoda, S., Kurihara, N., Wakashima, K., Umekawa, S. (1978), Metall. Trans. 9A, 1229-1236. Zweben, C. (1992), /. Miner. Met. Mater. Soc, July, 15-23.

General Reading Arsenault, R.J., Everett, R. (Eds.) (1990), Metal Matrix Composites. Orlando, FL: Academic. Chawla, K.K. (1987), Composite Materials: Science and Engineering. New York: Springer. Suresh, S., Mortensen, A., Needleman, A. (Eds.) (1993), Fundamentals of Metal Matrix Composites. Stoneham, MA: Butterworth-Heinemann. Taya, M., Arsenault, RJ. (1989), Metal Matrix Composites. Oxford: Pergamon.

5 Ceramic Matrix Composites Gilbert Fantozzi and Christian Olagnon Groupe d'Etudes de Metallurgie et de Physique de Materiaux, INSA, Lyon, Villeurbanne, France

List of 5.1 5.1.1 5.1.1.1 5.1.1.2 5.1.2 5.1.2.1 5.1.2.2 5.1.3 5.1.4 5.1.5 5.2 5.2.1 5.2.1.1 5.2.1.2 5.2.2 5.2.2.1 5.2.2.2 5.2.2.3 5.2.3 5.2.3.1 5.2.3.2 5.2.3.3 5.2.3.4 5.2.3.5 5.3 5.3.1 5.3.2 5.3.2.1 5.3.2.2 5.3.2.3 5.3.2.4 5.3.2.5 5.3.3

Symbols and Abbreviations Reinforcement Mechanisms Transformation Toughening Stress Intensity Approach Trends in Transformation Toughening Microcracking Frontal Process Zone Process Zone Wake Crack Deflection Crack Bridging Fiber Reinforcement Fiber Composites Processing Techniques Liquid Phase Routes Gas Phase Routes: Chemical Vapor Deposition (CVD) Chemical Vapor Infiltration (CVI) Glass Matrix Composites Carbon Fiber Composites Oxide Fiber Composites SiC Fiber Composites Refractory Ceramic Composites Carbon-Carbon Composites C - C - T i C Composites SiC Matrix Composites Other Matrix Materials Functionally Gradient Materials (FGM) Particle, Whisker and Platelet Composites Processing Transformation-Toughened Ceramics Effect of Size and Amount of Zirconia Particles Effect of Solute Content Effect of Temperature Strength-Toughness Relationship Mullite-Zirconia Whisker Composites

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

185 187 187 188 190 191 192 192 194 195 197 202 202 202 204 205 205 206 207 209 209 211 211 216 217 218 218 220 220 221 222 224 224 225

184

5.3.3.1 5.3.3.2 5.3.3.3 5.3.4 5.3.5 5.3.6 5.3.7 5.4 5.5

5 Ceramic Matrix Composites

SiC-Alumina Composites SiC(W)-Silicon Nitride Composites Multiple Mechanisms Particle-Reinforced Composites Platelet-Reinforced Composites Nanocomposites Complex Structure Materials Conclusions References

226 228 229 230 231 232 233 233 234

List of Symbols and Abbreviations

185

List of Symbols and Abbreviations a Aa B

L mf Ms N,n P q r,R R R s AS T T uc U V

crack length crack increment slope of the stress-strain curve associated with nonlinear microcracked or transformed zone particle diameter bridging zone size unconstrained dilational strain associated with phase transformation Young's modulus microcrack density shear modulus strain energy release rate process zone width stress intensity factor fracture toughness toughening length of fiber or whiskers gauge length Weibull modulus of fibers transformation temperature cycle number pressure broken fiber fraction particle or whisker radius length to diameter aspect ratio crack growth resistance crack spacing entropy variation temperature peak stress maximum crack opening of the bridging zone residual energy density volume fraction

a j8 ys 20 eu ejj 9 ft v A@

thermal expansion coefficient ratio of shear rate to densification rate surface energy deformation at which matrix microcracking takes place rupture deformation stress-free strain dilational strain associated with microcracking coefficient of friction Poisson's ratio time dependent densification

d D eT E

f

G G h K Kc AK

I

186

5 Ceramic Matrix Composites

a of crfl <°f,u) T ij

stress normal residual compression due to thermal stress fatigue limit average fiber strength time constant related to the densification of the matrix shear stress at fiber matrix interface

c-ZrO2 CTE CVD CVI FGM HIP HP m-ZrO2 PS PSZ SEM SEP TEM t-ZrO2 TZP ZDC ZTA ZTC

cubic phase zirconia thermal expansion coefficient chemical vapor deposition chemical vapor infiltration functionally gradient materials hot isostatic pressing hot pressed monoclinic phase zirconia pressureless sintered partially stabilized zirconia scanning electron microscope Societe Europeene de Propulsion transmission electron microscope tetragonal phase zirconia tetragonal zirconia polycrystals zirconia dispersed ceramics zirconia toughened alumina zirconia toughened ceramics

5.1 Reinforcement Mechanisms

5,1 Reinforcement Mechanisms The main goal of ceramic composites is to increase the toughness. For this purpose, two types of reinforcement can be used (Fantozzi, 1988): - by particles, either with or without phase transformation - by fibers or whiskers. Several mechanics have been identified and analyzed and classified according to - phase transformation, - microcracking, - crack deflection, - crack bridging, - fiber reinforcement. In the case of particle or whisker reinforcement, the second phase introduces a perturbation in the matrix by introducing two different stress fields. First during cooling, owing to the processing, the thermal expansion coefficient mismatch induces a hydrostatic stress in the particle and at the interface proportional to the temperature difference between a temperature at which any plastic flow is negligible and the ambient temperature (Seising, 1961). The particle is subjected to a pressure P and the matrix to radial and tangential stresses of -PR3/x3 and PR3/x3 where R is the particle radius and x the distance from a point in the matrix to the centre of the particle. The pressure is equal to P=

AaAT (1+vJ (l-2v 2 ) 2E E

(5-1)

where Aa = (ax — a2) is the thermal expansion coefficient mismatch, AT, cooling range on which plasticity is negligible, and vl9 v2 Poisson's ratios of the matrix and the particles. The second stress field is due to the Young's modulus mismatch between the

187

particle and the matrix because of the remote applied stress (Goodier, 1933). Again, a multiaxial stress arises. The presence of those stress fields plays a major role in the behavior of cracks around such fillers and mostly control the interaction. The crack may experience a different stress intensity factor when approaching the stress field resulting from the particles (Khaund, 1977). Furthermore, the crack will propagate in a direction perpendicular to the compressive stress. 5.1.1 Transformation Toughening

Increase in strength and toughness can be obtained in ceramics that undergoes a stress-induced transformation. The case of ceramics containing zirconia particles which present a tetragonal to monoclinic phase transformation is the most spectacular (Garvie et al, 1975). Indeed, zirconia presents three crystalline forms: cubic (c) from 2700°C to 2350°C, tetragonal (t) from 2350°C to 1170°C and monoclinic (m) below 1170°C. The tetragonal -• monoclinic transformation takes place with a volume increase of about 3% which corresponds to a large dilatometric anomaly. For pure zirconia, this transformation leads to the crumpling of the material so that, zirconia cannot be used in the pure state. Cubic zirconia can be stabilized by adding CaO, MgO, Y2O3, CeO 2 ... The ZrO 2 -Y 2 O 3 phase diagram is shown in Fig. 5-1 (Scott, 1975). A decrease in temperature of the t ->m transformation can be observed with an increase of yttria content. A metastable tetragonal solid solution can therefore be obtained at room temperature. The volume change and the shear strain developed during the martensitic t-*m transformation impede the crack opening and therefore lead to an increase of the crack propagation resistance of the ceramic. Different

188

5 Ceramic Matrix Composites

2500

"m

A 11 I!'

B

C m+c

U

W

Figure 5-1. Phase diagram for ZrO 2 -Y 2 O 3 system. (After Scott, 1975).

Once the transformation has occurred, loading and unloading follow NR with the same slope than OP. The strain of the transformed material under zero stress is VeT(R\ where eT is the unconstrained transformation strain and V the volume fraction of transformed particles. The remote wake is subjected to a residual compression stress crr(S). The stress reduction is linked to magnitude of the irreversible transformation strain VeT. Near the tip, the stress can be described by a local stress intensity factor Kl lower than the applied stress intensity factor Ka (Fig. 5-2): the transformation has shielded the crack tip from the applied load (McMeeking and Evans, 1982). Near the crack tip, stresses depend on Kl: (5-2)

approaches can be considered to describe the transformation toughening mechanism: a thermodynamic approach or a stress intensity approach with some equivalence between these approaches. A review of the developed models was made by Evans and Cannon (1986). An extensive treatment of transformation toughening can be found in Chaps. 7 (Cook and Pharr) and 8 (Becher and Rose) in Volume 11 of this Series.

Or

p

(a/^/Q

5.1.1.1 Stress Intensity Approach

The stresses near a crack tip are reduced by the phase transformation involving shear and dilational strains. The stress change can be regarded as deriving from the stress-strain curve shown in Fig. 5-2. The loading OP is elastic until to the critical stress oc required to transform the tetragonal particles. Some unloading occurs during the transformation (PQ) because the material is under a residual compressive stress oT in the remote crack wake.

wV in/ <7 r

s

Figure 5-2. Nonlinear stress-strain curve (dilational component) of an element near the crack tip of a body experiencing a) microcracking or b) phase transformation, under the assumption of supercritical behavior. Note that the modulus after microcracking (Bl) is lower than the initial modulus (B). Area I = OPQ'R, area II = PQ'Q, area III = RST. After Evans (1984b); Budiansky et al. (1983).

5.1 Reinforcement Mechanisms

189

0.22-£-fT-l/F • Vh 1 (1 - v) / /^ /

/

-

Extended zone

" Partial zone

-Mjji 7)\

/

I CrackW j \j

Transformation zone

Frontal zone

Figure 5-3. Crack resistance curve as a function of transformation zone extent. From Evans (1984).

I

Outside the transformation zone, stresses depend on X a . The phase transformation leads to a change in stress intensity factor AX = X 1 - X a < 0

(5-3) 1

Crack growth occurs when X is equal to Xj?, the fracture toughness of the transformed material. The measured toughness is the critical value of the applied stress intensity factor Kl: K*C = K°C+AKC

(5-4)

where AXC is the value of —AX corresponding to the critical condition. In order to determine AXC, only the effect of a purely dilational transformation will be considered, the shear stresses induced by the phase transformation having a negligible contribution. The value of AXC is dependent on the zone shape. The change in the stress-intensity factor at a crack tip is the integral over the zone 1

AK= {JdX

(5-5)

For regions at angles smaller than TT/3, dX 1 is positive whereas dX 1 is negative for

9 > 7i/3. Thus, for the frontal zone (Fig. 5-2) determined by the condition o = oc, the net effect is zero and there is no toughening (AX = 0). The calculation of AX for the extended zone (Fig. 5-3) can be performed using the double integral and determining the half height h of the zone. As in the case of plastic deformation zones at crack tip, the size of the transformation zone is related to the applied stress intensity factor X and the critical stress oc by (Evans, 1984 a) h=

1271

(5-6)

For a dilational and supercritical transformation (all particles transform in the zone), the toughening is given by (Evans and Cannon, 1986) 0.22 EVeT^ fh

(5-7)

For other conditions, similar results are obtained, with only the toughness constant 0.22 being modified. The AX levels are different for the two zones, resulting in a R-curve effect (Mar-

190

5 Ceramic Matrix Composites

shall etal., 1983). AXC increases with Aa following the relation (Evans, 1984 a) (Fig. 5 3

")

(5-8) 0.44 =

n(l-v)Ee

T

rF

_x f Aa

v / l t a n [~h

and asymptotically tends to the limit given by Eq. (5-7). Substituting h given by Eq. (5-6) into Eq. (5-7) yields

= AEVeTKJac

(5-9)

where E is Young's modulus and A is a constant. Up to that point, only applied tensile stress is considered in order to trigger the transformation. However, spherical particles imbedded in a matrix subjected to differential thermal contraction are subjected to residual stress aR (Becher, 1986): (7R = P [1 + 3 F/{(4TI) 1 / 3 - ( 3 K)1/3}3] (5-10) with P given by Eq. (5-1). When oc2>ai> m transformation is modified by crR: <7 a ><7 c =GC — &R

P-H)

<7°, the effective critical transformation stress, must replace oc in Eq. (5-9). oc is given by AG t . m = a c e T

(5-12)

where AGt^m is the variation of free energy by unit volume during the t ->m transformation of particles. For spherical particles, GR is not a function of particle size but polyhedral particles exhibit stress concentration at edges and corners and oR increases with increasing size for these polyhedral particles. When oR > oc spontaneous transformation occurs. Thus polyhedral particles lead to the size dependence.

5.1.1.2 Trends in Transformation Toughening

From the preceding analysis of the stress induced phase transformation on fracture toughness, it can be said that the amount of toughening increases as the transformation zone size increases or as the critical stress ac decreases. Therefore the trends in transformation toughening are linked to the trends in the critical stress. Effect of Temperature and Solute Content The critical stress oc is determined by the nucleation condition of the t ->m transformation given by Eq. (5-12). From thermodynamic considerations AGt_>m can be expressed as (Anderson and Gupta, 1981; Becher and Rose, Vol. 11, Chap. 8 of this Series). AG t ^ m = (T—MS)AS

(5-13)

T is the test temperature, M s is the temperature at which the tetragonal particles transform without requiring any extra energy, AS is the entropy change per unit volume between the two phases m and t. From Eq. (5-12), (5-9) and (5-11), one can see that AKC is proportional to (AGt_m)~1 and the toughness will increase as T decreases towards Ms until a maximum is reached at T « M s (
5.1 Reinforcement

For small particle sizes compared to the critical size dc at the test temperature T, the critical stress is high and thus h and toughness are small. As the particle size increases, the critical stress decreases (Ms increases) and toughness increases. The maximum toughness occurs when all the particles have a size just below dc (Ms « T). For larger particles size, the transformation becomes spontaneous (Ms > T) and the amount of tetragonal particles available for a stress induced transformation decreases. Increasing the amount of tetragonal particles increases the toughness increment but decreases the critical size. This effect could result from interaction effects between particles leading to an autocatalytic nucleation or from the decrease of the elastic modulus as zirconia content increases. The particle shape plays a role in nucleating the transformation (Riihle and Heuer, 1984): angular particles exhibit larger M s temperatures and lead to higher toughness. Up to that point, it has been assumed that an increase of h leads to a toughness improvement but there is a limit above which Kc decreases. If the reduction of ac is such that the stress induced phase transformation occurs throughout all the specimen, the transformation zone is no longer constrained and the crack closure forces are reduced. Thus an optimum toughness occurs when h is large but still small relative to the specimen size. 5.1.2 Microcracking The fundamental point of this analysis is that if microcracking can be induced when loading the material owing to the stress field present around particles, it can absorb elastic energy and thus contribute to a fracture energy increase. The analysis is conducted on a particle embedded in an in-

Mechanisms

191

finite matrix without bonding at the interface. Although Eq. (5-1) shows that the magnitude of the stresses are independent of the particle size, it has been observed experimentally that cracking occurs only around particles greater than a critical size, showing that the elastic energy (due to the thermal contraction mismatch) stored in the matrix and the particle must be greater than the surface energy needed to create the crack. This critical size is given by (Davidge and Greene, 1968) 8 ys

(5-14)

-+2where ys is the surface energy of the matrix. Therefore, when the size of the particles is greater than the critical size r c , microcracking arises, which leads to a decrease in the strength of the material. However, if the particle size is smaller than rc9 microcracks need additional stress to be induced which can be provided by the external applied load. For an efficient reinforcement, the microcracking must be confined in a so-called 'process zone' in front of the crack tip. When the crack further advances, a process zone wake will develop around the crack, which modifies the crack opening, resulting in an increase of crack growth resistance with crack length. An analysis similar to that conducted for transformation toughening has therefore been conducted (Evans and Faber, 1983). The development of an i^-curve is predicted by the existence of a nonlinear process zone as the crack advances, that can be evaluated from the nonlinear stressstrain curve caused by the microcracking (Fig. 5-2). It is assumed that there is a saturation of microcracking that corresponds to the linear regions of the stress-strain curve at large strains. The calculation of

192

5 Ceramic Matrix Composites

both the frontal and wake process zone can therefore be evaluated by means of stress intensity. 5.1.2.1 Frontal Process Zone

When the process zone is contained ahead of the crack, each element is only subjected to continuous loading, the energy (J integral) is nearly path independent and therefore unchanged after microcrack ing. The crack shielding due to modulus variation can therefore be written as Klil Vl)

~

=

Ka(1

~V)

(5-15)

where Ka is the stress intensity associated with the external applied field, E the elastic modulus for the crack free material, E± the elastic modulus for the microcracked material and v, vx the corresponding Poisson's ratios. A reduction of K is therefore expected. However, this reduction is limited because the toughness K™ of the microcracked zone is reduced. The toughness of the microcracked body can be approximately calculated under the assumption of the similarity to a crack extending through a porous body K? = K C ( 1 - / )

(5-16)

where / is the microcrack density. The measured toughness is subsequently reduced to (5-17) The ratio EJE is only a function of the volume fraction of microcracks present and is given by

Finally, the ratio K™/Kc is solely dependent on the volume fraction of microcracks. The toughening obtained by the frontal process zone is rather modest and is shown to give a negligible contribution to the overall toughness. In fact, the toughness increase imparted by crack shielding is counteracted by the damage produced by the process zone. 5.1.2.2 Process Zone Wake When the crack advances, a process zone wake is formed which provides an additional contribution to crack shielding because elements in the wake experience unloading with a nonlinear stress-strain curve. For a fully developed wake of uniform size h the shielding can be expressed aS

where U(y) is the residual strain energy density which can be evaluated from the stress-strain curve. U(y) is due to the hysteresis of the curve. The residual energy density is twofold: the permanent strain and the other related to the change in elastic modulus that however gives a less pronounced contribution. This relation shows that an i^-curve analogous to that evaluated from transformation toughening can be determined. The stress-strain curve exhibits a critical value oc for the onset of microcracking followed by a stress decrease associated to the microcrack strain development of slope B related to shear modulus G: S=-|G

(5 18b)

'

(5-19)

(5-20)

Microcracks present in the process zone result in a permanent dilatation (Of) due to localized residual tension schematically

5.1 Reinforcement Mechanisms

shown in Fig. 5-2, that contributes to the toughening. Another contribution is due to the change in shear modulus induced by the microcracks. The crack shielding component due to the dilatation is caused by the unloading of the microcrack zone that can occur in the wake and is therefore calculated directly from the stress-strain curve. The residual strain energy density, U(y\ can be evaluated from the stressstrain curve OPQRS (Fig. 5-2), which includes the energy stored in the wake, U{y) = area(I) - area (II) + area (III) (5-21) This area can therefore be evaluated, since it is assumed that the microcracked zone subjected to dilatation does not perturb the stress field outside the zone. The zone can be expressed by a relation similar to that established for transformation toughening (Eq. (5-6)). The residual energy associated with the deviatoric component can be obtained from the area under the stress-strain curve of the deviatoric components. Finally the overall toughening is represented in Fig. 5-4. An analytical fit has been proposed as = ( 1 + 0 . 4 / ) + 0.25

(5-22)

The analysis defines the steady state toughening corresponding to the fully developed process zone. However, process zone morphology intermediate between the frontal zone and the steady state zone will lead to intermediate values of toughness. The steady state value is considered to be obtained when the crack advance approaches 5k The magnitude of the #-curve is of the form Kr = K* + [0.025 Vf 92 E3 (1 + v) <7C~x] •Aa1/2 (5-23)

1

193

10 EQ\[R/KZ

Figure 5-4. Crack growth resistance for process zone of width h as a function of dilational parameters. Note that the modulus toughening imparts only a negligible contribution whereas dilational toughening can lead to significant toughening. After Evans (1984 b).

where K* depends on the microcrack conditions when the crack begins to advance, Aa = 0. An alternative solution for the process zone size was given by Evans and Faber (1981). A similar analysis was conducted but only the formation of the frontal zone was considered. Microcracking occurs when the total elastic energy exceeds the energy needed to create the crack surface. The process zone width can be calculated from the knowledge of the stress state at the crack tip: I

(5-24)

-i where 6 = — 3 ATAa is the dilational transformation strain, rc represents a critical particle size above which spontaneous microcracking occurs, r is the particle size. This relation brings the benefit of taking into account the particle size on toughening (Fig. 5-5). It shows the particle must be as close as possible to a critical size to induce significant toughening. Therefore, the introduction of a particle size distribution

194

5 Ceramic Matrix Composites

1.2 Relative particle size —

Figure 5-5. Prediction of toughening by microcracking as a function of particle size and by taking into account a distribution of particle sizes characterized by the factor k. Two fractions of particles with sizes larger than the minimum value necessary to induce a process zone are considered. After Evans and Faber (1981).

results in a degradation of the maximum toughening anticipated by the uniform particle distribution analysis, which is however a more realistic description of a real composite. 5.1.3 Crack Deflection Cracks can be deviated either by stress fields present around particles or by hard particles of higher resistance (Fig. 5-6). This mechanism therefore leads to a non-

Figure 5-6. Schematic representation of crack deflection by particles.

planar crack, subjected to a mixed mode stress intensity lower than that experienced by the corresponding planar crack. This mechanism has been modeled by evaluating (Faber and Evans, 1983) the strain energy release rate for a planar (Go) and a deviated (G) crack. The basic idea is that when a crack previously in mode I approaches a particle, it will tilt an angle 9 out of its original plane to be subjected to mixed I + II mode, and it will also twist by an angle (f) to be subjected to a mixed I + III mode. The increase in fracture toughness imparted by deflections of the cracks is evaluated from the local stress intensity factor (Kl9 X n , Km) as a function of the deflection angles. Crack advance is then assumed to be governed by the strain energy release rate G pertinent to each segment of the crack front along its deflected trajectory, that can be related to the stress intensity factors by (5-25)

where E and v are the Young's modulus and Poisson's ratio of the matrix. The average strain energy release rate is then considered to represent the net crack driving force that can be compared to that of a planar crack Go. The toughening increment is therefore obtained by Go/Go =G 0 /

(5-26)

The analysis has first been carried out for spherical, rod and disc shaped particles for isotropic composite materials. In any case the average energy release rate has been obtained by taking into account all the possible position angles between the several particles. In the case of discs and rods, the influence of the aspect ratio has been investigated. High deflection and therefore high toughening are obtained with discs and rods, but the most effective morphology is

195

5.1 Reinforcement Mechanisms

5.1.4 Crack Bridging

Such a reinforcement occurs when a crack goes around a reinforcement particle without breaking it. The second phase exerts a bridging stress on the crack surfaces that counteracts the crack opening and thus reduces the stress intensity factor (Fig. 5-8) (Evans and Meeking, 1986). If the bridging zone D is small compared to the crack length, the stress intensity factor variation due to the stress of the ligaments can be expressed by (5-27)

£

o

7

•

—

7

\

ive to ugh ning,

6 QJ

k

that of the rod with a high aspect ratio in a concentration of about 10 to 20%. This analysis has also been extended to the case of rods randomly oriented in planes in order to simulate transverse isotropic whisker like composites (Liu et al, 1989). The result evaluated for a crack perpendicular to the rod plane shows that a more significant reinforcement can be obtained by crack deflection for this more specific case (Fig. 5-7). The toughening obtained by such a mechanism must be tempered when applied to a composite material because it has been evaluated from the comparison of a deflected and a planar crack. This result describes relatively well the toughness increment from a single crystal to a polycrystal experiencing a partly intergranular fracture where the crack is no longer planar as it is in the case for a composite matrix. The real toughening that can be practically achieved will be therefore lower than that given by the model. The typical benefit that can be obtained by such a mechanism can be appreciated by consideration of the silicon nitride containing rod-like particles leading to toughness values between 7-lOMPa^/m.

a:

5

•

/ /

2D

y'

3

/

12

if

3D

3

——

0.1

0.2

0.3

0.4

Rod volume fraction, Vf

Figure 5-7. Toughening predicted by crack deflection for rods of different aspect ratios, randomly and planar oriented. After Faber and Evans (1983) and Liu etal. (1989).

Since, it may also be obtained by the expression of the strain energy release rate as AG= \o{u)du (5-28) o where u is the crack opening displacement, the well known relation between K and G leads to 2GAGC 1-v

(5-29)

The analysis has been developed for a Dugdale zone, i.e., under the assumption that the tensile stress exerted by whiskers is uniform along the bridging zone. In such a case, when the stress intensity of the matrix is small compared to the closing stress ex-

Figure 5-8. Schematic representation of whisker bridging. Bridging zone D and debond length ld.

196

5 Ceramic Matrix Composites

erted by the bridging zone relations, Eqs. (5-27) and (5-29) lead to (5-30) 1-v where V{ is the volume fraction of ligaments, a the closing stress applied to the ligaments, G the shear modulus, v the Poisson's ratio, and uc the crack opening at the end of the bridging zone. Although this relation is general and can be applied to different reinforcements, it shows that the reinforcement depends on ligament density, the stress exerted by the ligaments as well as the crack opening in a square root factor. It means that the reinforcement must exhibit a high strength in order to exert the highest possible stress and must be of large dimension in order to bridge over a large crack opening. This might be the case of whiskers and this analysis has been worked out in detail for this specific case (Becher et al, 1988). The whiskers are considered as uniform square array of aligned fibers of diameter 2r and fracture strength aj. The closure stress is then defined as

where Ew is the Young's modulus of the whiskers. Under these assumptions, the reinforcement is therefore equal to

AKC=2

<7 0 =
A K =
VtrEGGm a/2 6(l-v)£ w G i

(5-34)

This result shows that, in order to obtain a high reinforcement, one must maximize the whisker strength and whisker radius and minimize the interface bonding strength. An example of the calculation evaluated for silicon nitride can be seen in Fig. 5-9. According to the analysis, the expected toughness might be quite high. However, it must be stressed that this analysis has been conducted with the hypothesis of aligned whiskers. In an isotropic material the fraction of whiskers perpendicular to the crack is reduced. It has been shown (Evans et al., 1989) that the debonding is enhanced by inclination of the whiskers to the stress axis but in that case, due to a change of fracture mode, the crack tends to deviate from the interface to the whiskers (kinking), leading to whisker failure. As a result, the toughness rather than the strength of the fiber is the key parameter. If the debonding is sig-

(5-31)

The whiskers are considered to be debonded over a length /d from the crack surface in which the tensile stress is uniform. It is therefore required that the ratio of the fracture energy at interface (G{) and in the matrix (Gm) to be low. The debonded length is then defined as (5-32) 6 G The maximum crack opening at the end of the bridging zone is imposed by the maximum displacement held by the whisker before fracture, <=~

(5-33)

10 15 20 25 Whisker content (%)

30

Figure 5-9. Theoretical toughening imparted by crack bridging as a function of volume fraction of SiC whiskers in Si 3 N 4 matrix from the relation of Becher et al. (1988). Influence of the whisker debonded length

5.1 Reinforcement Mechanisms

nificant, the bridging zone can increase with crack length over a long distance. The contribution of the reinforcement therefore depends on the crack length as (5-35) A crack growth resistance increase with crack length can therefore be expected for such a reinforcement. 5.1.5 Fiber Reinforcement The fundamental reinforcement mechanisms of fiber composites are dominated by the interface between the fibers and the matrix (Evans, 1990). These basic mechanisms are governed mainly by the relative fiber-matrix interface debond toughness, the residual stresses which are mostly of thermal origin, the friction coefficient, the fiber strength distribution, the matrix properties (such as toughness and statistical nature of matrix cracking) and the fiber volume fraction. Fibers introduced in a matrix lead to two effects: thermal stresses due to the thermal expansion mismatch and load transfer involving the fiber-matrix interface (Rouby and Navarre, 1990). Thermal stresses depend on the mismatch of the thermal expansion coefficients of fiber (af) and matrix (am). If af < am, the interface is in compression and there is a fretting of the fiber but large mismatch is undesirable. If af > am, the matrix is compressed but the interface is in tension, leading to very poor load transfer. Control of the interface properties (thermal stress, friction conditions) can be achieved by using coatings or interphases between the fibers and the matrix (Hsueh etal., 1988). Further discussions on interfacial behavior can be found in Chap. 6 of this volume. In the case of ceramic composites, both fibers and matrices show a brittle behavior. For ceramic

197

matrix, generally the rupture deformation of the matrix sm u is lower than that of the fiber ef u . Thus the microcracking starts in the matrix without fracture of the composite for a fiber fraction higher than Ff c . When the matrix microcracks at s m u , there is a load transfer from the matrix to the fibers through a shear stress T{ at the fiber matrix interface. The stress evolution during matrix microcracking is shown in Fig. 5-10. The load transfer takes place on a distance lc given by h=^~

(5-36)

with r being the fiber radius and afu the fiber fracture stress. Outside the lc zone, the

Matrix

U//////7/T//7T//m

Fiber

Figure 5-10. Evolution of stresses in fiber and matrix after matrix microcracking for a constant interfacial shear stress.

198

5 Ceramic Matrix Composites

matrix supports c7mu, the fracture stress of the matrix, and can again develop microcracks. Subsequently a multiple parallel microcracking of the matrix occurs. The matrix multicracking becomes saturated when the applied load is raised and the saturation crack spacing s is governed by the sliding stress x{ (Evans, 1990), , „ . T^[(l-Vf)2GmcEfEmr2/(EcVfs^2 where Gmc is the critical strain energy release rate of matrix. The stress profiles after matrix cracking depend on the interface behavior. For a constant interfacial shear stress the stress profiles are triangles. x-x should be due to friction and may be written (5-38)

T{ = aJ jLL

where \i is the friction coefficient, and aj the residual compression normal to the interface due to thermal stresses. Thus the mechanical behavior of the composite can be summarized as follows (Fig. 5-11). First, there is an elastic loading (modulus Ec = VmEm + VfEf\ then multiple microcracking of the matrix, lengthening of fibers with friction and finally fracture. The fracture can be catastrophic or controlled according to the characteristics of fibers and the fiber matrix interface. The friction of fiber leads to a permanent strain after loading and to an open loading-unloading cycle. The damage of the ceramic composite begins by matrix microcracking at s0. Aveston et al. (1971) have proposed an analysis based on an energy approach which leads to the following relationship for a frictional interface: VmEcr)]^

(5-39)

This relation shows that the matrix fracture is not due to the critical flaw itself but is controlled by the interface (s0 increases

with Vf and T{). SO, S0 can exceed the maximum strain of the monolithic ceramic. A more accurate analysis must take into account the residual thermal stresses, debonding and the statistical nature of matrix and fiber fracture (Korczynskyj et al., 1981; Budiansky et al., 1986). In the case of matrix microcracking and bonded interface, the interfacial stresses can lead to a debonding phenomenon. Initial debonding along the interface at the crack front requires that fracture energy of fiber Gfc must be higher than 4G ic , the fracture energy of the interface (Evans, 1990). Due to crack opening, the debonding extends along the fibers located in the wake. The debonding stress can be determined by using an energy approach. The critical fiber stress for the onset of debonding was calculated by Outwater and Murphy (1969)
(5-40)

More precise analyses were made by many authors (Evans and Marshall, 1989). A more accurate analysis based on finite elements methods was made by Peres (1988), taking into account the effect of defect distribution on the matrix multicracking process and the statistical nature of matrix and fiber fracture. A similar ap-

Fiber unloading and friction

at

f f
Elastic unloading of fibers Unloading of fibers with friction

Figure 5-11. Composite behavior with multiple microcracking of the matrix.

5.1 Reinforcement Mechanisms

proach based on failure probability concept was made by Lamon and Guillaumat (1992) which shows that the weak interfaces (high debond length) limit the matrix cracking process whereas with strong interfaces, a larger matrix crack density is obtained at saturation. The energy released in front of the crack and the energy dissipated by friction during matrix crack propagation for a multifiber composite can also be calculated by finite element caculation (Peres, 1988). These energies present, during the fiber crossing, large variations, which depend on the volume fraction of fibers and the interface shear stress x{. Matrix multicracking which depends on the interface properties is significant because the pull-out length is influenced by the crack spacing. After matrix microcracking, fibers undergo overloading near the crack and exhibit a distribution of strengths described by Weibull statistics (Fig. 5-12). Fiber can therefore break at a given distance /p from the matrix crack which is the pull-out length. This distance increases with both decreasing Weibull modulus and interfacial shear stress value (T{) (Thouless and Evans, 1988; Sutcu, 1989). In the case of constant Ti5 the average pull-out length can be given by the following relationship (Navarre, 1990):

Fiber

199

0

mans Matrix crack 2

1

'

^^r x

Figure 5-12. Fiber fracture near a matrix crack: a) fiber stress profile after matrix cracking; b) local fiber strength. From Navarre (1990).

pull-out length. On the other hand, small values of x{ result in large pull-out lengths. In order to encourage large pull-out lengths, high mean fiber strengths and small mf values are needed also. So, when a main crack propagates in a ID composite, fibers bridge between the crack surfaces and different zones can be observed (Fig. 5-13). The zone a is the notch. In the zone b, no interaction between the crack surfaces occurs (all the fibers are pulled out). In the zone c, the crack is bridged by broken fibers whereas unbroken fibers intervene in zone d. The

(5-41) with <<7f5U> and mf respectively average fiber strength and Weibull modulus of fiber measured on fibers with gauge length L. Sutcu (1989) showed that the distribution of pull-out lengths depends on the nature of interface: bonded, frictional, Poisson contraction effect. Large values of T{ cause the fiber stress to decrease rapidly with the distance from the matrix crack and fiber failures occur near the crack with a small

"max

Figure 5-13. Schematic of a fracture crack in a ID composite. From Navarre (1990).

200

5 Ceramic Matrix Composites

zone e is undamaged. The toughness increment in fiber reinforced ceramics is due to the following processes (Evans, 1990): debonding which creates new surfaces, frictional dissipation during pull-out, loss of residual strain energy and dissipation of elastic energy stored in fiber through acoustic waves during fiber failure. The increase in toughness due to matrix crack bridging can be calculated from the following relation (Thouless and Evans, 1988): (5-42) = Vf J a(u)du o where u is the crack opening, umax the opening at the edge of the bridging zone and a (u) the traction on the crack surfaces exerted by the bridging fibers, a (u) is given by a (u) = (1 — q) T -f q af

p

(5-43)

where q is the broken fiber fraction, T the peak stress (Fig. 5-12) and a{ p the pull-out stress. The toughness increment is given by (Evans, 1990) AG

u

f *d

2Gin I

ld is the debond length and £?• the stressfree strain. The first term is the bridging contribution due to elastic energy dissipated as acoustic waves, the second term is linked to the loss of residual strain energy, the third term is due to debonding and the last term corresponds to pull-out dissipation by frictional sliding of the interfaces. Different cases of composite fracture can occur (Fig. 5-14). Strong interface: If the stress crf d given by Eq. (5-40) is much greater than the fiber strength <jfu, the fibers will break in the

g)

Figure 5-14. Fracture mechanisms: a) ID composite submitted to a tensile stress; b) initial fracture occurs in the matrix; c) a{ d > a{ u : brittle behavior; d) debonding between fiber and matrix; e) high Weibull modulus of fiber; f) small Weibull modulus; g) pullout. After Osmani et al. (1990).

matrix cracking plane without debonding of the interface. So, the matrix cracking leads to the fiber failure and the composite shows a brittle behavior (Fig. 5-14 c). The fracture energy is then directly linked to the created surfaces: = F f G fc

+ (l-K f )G m c

(5 _ 45)

This energy is of the order of the fracture energy of monolithic ceramic (10-100 J/ m2). High Weibull modulus of fibers and weak interface: If the adhesion between the fiber and matrix is low (crf d < <7f>u), debonding occurs between fiber and matrix, before fiber fracture. As the loading continues, the debonding extends and the fibers elongate while dissipating energy by friction with the matrix (Marshall and Cox, 1985). If the fiber strength distribution is very narrow (the Weibull modulus is high), the fiber will break in the matrix crack plane

201

5.1 Reinforcement Mechanisms

(Fig. 5-14e) and the fracture energy due to the fiber frictional dissipation in the sliding zone (debonding) is then

pull-out must focus our attention on fiber strength and sliding resistance which govern the pull-out length and the frictional dissipation. The sliding resistance can be AG2 = AGX + V( r {a,, u ) 3 /(3 E{ x-x) (5-46) optimized by using fiber coatings. 2 AG2 is of the order of 1000 J/m . It is evident that the mechanisms of crack propagation are complex and a sinSmall Weibull modulus of fibers: In this gle parameter Kc or Gc is not sufficient to case, the fibers have a significant probabilcharacterize the fracture behavior of fiber ity of breaking at one side or other of the composite materials. The notion of crack matrix cracking plane (Fig. 5-14f). The propagation resistance curves (.R-curves) separation then requires the pull-out of must be used, because each contribution of fibers with an important energy dissipation the reinforced mechanisms is only fully redue to the friction (Fig. 5-14 g): alized when fibers break and pull-out. AG3 = AG2 + V{ T{ / 2 /3 r (5-47) From the Dugdale model, Sakai et al. (1988) have determined the R-curve due to AG3 is of the order of 10000 J/m 2 . bridging of crack lips: The pull-out length is observable on the (5-49) fracture surface, which resembles a brush. Aa < Aa* As shown by Eq. (5-41) the pull-out length ,bTS/Aa*/2n Aa>Aa* increases with <jf u and r and with decreasing mf and T{ or Gic. where Kl is the initiation toughness, Aa* c

Short fibers: For aligned short fibers, with a length lower than the critical length /c, all the fibers are extracted and the maximum fracture energy is obtained for a fiber length of the order of /c and we have AG4 = V{ r (fff,J2/12 ii

(5-48)

The fracture energy is very high (> 10000 J/m 2 ). For disoriented short fibers, the fracture probability in the matrix crack plane is significant because of the occurrence of bending. So, pull-out is suppressed and the reinforcement decreases. Therefore low fiber-matrix adhesion and high fiber strength are needed in order to obtain matrix microcracking and fiber pull-out with friction and energy absorption giving controlled rupture. If the interface is too strong, there is no extraction and the failure is straight (Fig. 5-14). However, the interface should not be too weak, in order to retain a high composite strength. The dominance of toughness by

the size of the steady state bridging zone, and ohr the bridging stress. A plot of XR as a function of y/Ka gives the bridging stress abr and an estimation of the crack opening at the limit of the bridging zone (5.50) 'Aa*\ 1 / 2 f

/Aa*V / 2

+

K[c

—'—) IM—J 3 . E Llorca and Elices (1990) have developed a cohesive crack model that allows the crack growth resistance curves of the fiber-reinforced ceramics to be computed. R-curves can be obtained by assuming the cohesive stresses in the process zone are due to fiber bridging and pull-out (the debonding effect is neglected) and from the knowledge of the stress-crack opening displacement law (Thouless and Evans, 1988). So the influence of fiber strength
202

5 Ceramic Matrix Composites

tance is improved by increasing a{, mf, or by decreasing T{ . ^-curves exhibit three regions: the first stage with a fast increase of R, a second region with almost a plateau and again a fast increase. The bridging stresses are responsible for the first region, fiber pull-out occurs in the second stage, the third region being linked to a singularity.

5.2 Fiber Composites 5.2.1 Processing Techniques Fiber-reinforced ceramic matrix composites can be processed essentially according to liquid or gas phase routes (Naslain, 1986). 5.2.1.1 Liquid Phase Routes Slurry Infiltration This technique is the one most commonly used to produce ceramic matrix composites. A fiber preform (e.g. a tow) is impregnated by passage through a slurry mixture usually containing at least three components: the carrier liquid, the matrix powder, and an organic binder. In fact these kinds of composites are fabricated in a manner totally analogous to that used for resin matrix composites whose dramatic success is well established. This technique has to date been used more effectively with glass, and glass-ceramic matrix composites which are hotpressed at temperatures near that of glass softening. In refractory crystalline matrix systems, slurry processing has been less effective because of the absence of viscous flow and the higher processing temperatures necessary to densify the matrices, which in turn degrade the fibers.

Composites consisting of carbon or silicon carbide fibers with different varieties of glasss such as LAS (lithium alumino silicate) with fracture strength reaching 830 MPa and toughness of 18.9 MPa jm have been recorded. More recently, a technique for infiltration of ultrafine powders (0.2/0.4 jum), leading to efficient and fast densification directly applicable to a thick multidirectional substrate has been investigated (Jamet et al., 1987). In this method a suspension of these powders is poured into the substratum laid on an ultrafine filter. The sedimentation of the particles is promoted both by a vacuum under the filter and pressure on the top of the slip. This processing route has been used to infiltrate a Nicalon SiC fiber network with alumina; the infiltrated network was subsequently sintered at 1250°C. Despite the occurrence of microcracking in the matrix, interesting mechanical properties were obtained, moreover this kind of composite can tolerate very high strains and even after deterioration of the internal components a high stress can be withstood. With a thick 3D SiC substrate, this technique results in an almost fully dense composite with desirable characteristics, e.g. <jf - 250 MPa and Klc = 18 MPa

Sol-Gel and Polymer Pyrolysis Techniques In principle, the sol-gel technique consists of preparation of homogeneous noncrystalline xerogels from metal alkoxides and inorganic salts and subsequent heat treatment to transform these gels into solid glass and ceramics. The gel is formed from a sol, i.e. a colloidal suspension of hydrous oxide in a liquid which is used to impregnate the reinforcing phase with a liquid phase.

5.2 Fiber Composites

Sol-gel or polymer pyrolysis processing routes offer several advantages over other methods, these are (Cornie et al, 1986): - higher compositional homogeneity in matrices - greater ease of infiltration and forming, with a possible adaptation to pressure forming and injection moulding techniques - processing temperatures lower than the ones of conventional processing techniques. In the case of covalent matrices, the pyrolysis of polymeric precursors allows a significant reduction of processing temperatures (<1400°C). Some experiments have been carried out by Fitzer (1988) using SiC fibers in SiO2 matrix and also with ceramic matrices such as alumina and mullite. The processing route was modified, from the simple solgel processing, into two steps which were preparation of fiber gel prepregs followed by a thermal treatment and final densification by hot-pressing. The processing temperature was somewhat higher compared with the simple sol-gel technique. But with a SiC coated C fiber/alumina composite a strengthening was obtained when the amount of fibers was greater than 10%. Values of strength and toughness were equal to 500 MPa and 10 MPa y m respectively. There is much current interest in preceramic polymers which yield, on pyrolysis, various useful ceramic materials. Such a technique has been used for the fabrication of carbon-carbon composites from organic resins or pitches. It has been very recently extended to other composites such as SiC, Si 3 N 4 or oxides, the organometallic precursors being respectively polycarbosilanes, polysilanes, polysilazanes and metalalkoxides (Seyfert and Wiseman, 1984; Seyferth et al, 1983;

203

Naslain, 1986). The choice of the precursor and pyrolysis conditions must be made in order to achieve high pyrolysis yields with limited shrinkage. The shrinkage can indeed lead to severe cracking or poor control over the final dimensions of the part. As a result, several impregnation firing sequences must be performed or inert fillers must be added to the polymer precursors as Jamet et al. (1984) have done with BN, SiO2, SiC and Si 3 N 4 fine particle fillers in a polymer-derived SiC matrix. The important application of preceramic polymers involves the preparation of ceramic fibers, coatings, composites and the infiltration of porous bodies. Melt Infiltration This technique is little used but presents some advantages such as: achievement of fully dense and flaw free matrices in a single step, minimal dimensional change and stability for any geometry. But, some obstacles are encountered such as fiber damage by chemical reaction at high temperatures required for melting and low rates of infiltration (Cornie et al, 1986). The importance of wetting will be great in melt processing since, at the higher viscosities encountered, the exploitation of capillary pressure for infiltration is necessary if reasonable infiltration rates are to be achieved. Reaction Bonding Reaction bonding can also be utilized to prepare Si 3 N 4 reinforced with refractory fiber through a simple modification of the processing normally used for producing RBSN (reaction bonded silicon nitride). Cornie et al. (1986) demonstrate a composite made with whiskers or fibers dispersed into the silicon carbide powder before nitridation, for which a very high room temperature strength has been re-

204

5 Ceramic Matrix Composites

ported (900 MPa) using SiC filaments as the reinforcing phase. BN-composites (Lin et al., 1976) have also been prepared following this manner, by blending the BN fibers with a partially nitrided intermediate, followed by a hot pressing operation. This offers two advantages: - ability to achieve high density without going through multiple reimpregnation cycles. - retention of BN fiber integrity resulting in good reinforcement. 5.2.1.2 Gas Phase Routes: Chemical Vapor Deposition (CVD) Chemical Vapor Infiltration (CVI)

Chemical vapor deposition coatings, chemical vapor infiltration technique and in situ reaction methods for forming ceramic composites have been developed as extensions of the well-established CVD, reaction bonding, and carbon-composite technologies. The primary advantage of CVD methods is the uniform coating of tailored composition, including multiple layers of different compositions, which can be achieved at moderate temperature. The basis of CVD-CVI is that the composite is fabricated by infiltrating low density fibrous structures with vapors that deposit as solid phases on and between the fibers to form the matrix of the composites. As shown by Naslain et al. (1985), the CVI involves several steps: diffusion of the gaseous source species through a boundary layer and within the pores, a surface reaction between adsorbed species and diffusion of the gaseous reaction products in the pores and through the boundary layer. In order to favor deposition in the pore network of the substrate, the process should be controlled by surface reaction kinetics, supersaturation of gas near pore

entrance must be high enough and deposition rate must be sufficiently low. Such conditions imply that the deposition temperature, total pressure and gas flow rate should be kept as low as possible. Three types of infiltration processings have already been used in the fabrication of several composites with SiC, Si 3 N 4 , B4C, BN, and TiB2 matrices: - An isothermal process in which CVD gases diffuse into the preform. - A thermal gradient process in which CVD gases diffuse into the preform. - An isothermal process in which CVD gases are forced into the preform by a pressure gradient. All these processes require a very long process time, typically weeks. More recently, Stinton et al. (1986) have improved the process in order to reduce the time by combining the thermal gradient and pressure gradient approach. Their preform had a hot surface and the opposite one was cool. The coating gas infiltrates through the preform from the cool part where there is no reaction because of the low temperature and arrives at the hot surface where there is a reaction. Deposition of matrix material within the hot region increases the thermal conductivity, so that the hot region moves progressively from the top of the preform toward the bottom. Different kinds of preforms made with SiC fibers have been tested. With unidirectional preforms, containing 45 vol.% fibers, which appeared to produce optimum infiltration, processing to maximum density of 70% - 8 0 % has been reached after 24-36 h. These gave a flexural strength as high as 85% of the value of fully dense sintered a SiC. The fiber-matrix bonds in unidirectional composites allow considerable fiber pull-out which was thought to result in exceptional strain tolerance.

5.2 Fiber Composites

Cloth preforms have also been used and in this case the infiltration time needed to reach the maximum densities of 85% to 90% of theoretical density was 20-30 h, leading to a flexure strength equal to 450 MPa. In this last case the strain tolerance is not as good as for unidirectional composites. In fact it is believed that the limitations of mechanical properties were due to the degradation of the SiC fibers at this high temperature of 1200°C. As a conclusion it is suggested that the most important points for the future will be to reduce the deposition temperature and to determine some method for the fabrication of fibrous preform with improved properties (uniform permeability and porosity) in order to simplify the infiltration process, reduce the critical flaw size and improve the mechanical properties of the finished composite. The CVI process has several advantages: it can be applied to complex shape preforms; it is performed at rather low temperatures (800-1000 °C) thus limiting fiber degradation; in situ fiber surface treatment can be made before densification.

5.2.2 Glass Matrix Composites Prewo (1987) considers that from a fabrication point of view, the glass matrix composites, as compared to other ceramic candidates, probably offer the greatest commercial potential for ease of densification, low cost and also achievement of high performance aspects. He outlined the following important attributes: - Glass can be created with a broad range of chemistries to control fiber-matrix chemical interaction. - Glasses can be created with a wide range of thermal expansion coefficients to tai-

205

lor them to nearly match those of the reinforcing fibers. - The low elastic modulus of glasses (5090 GPa) permits high elastic modulus fibers to provide significant reinforcement. - The composite densification process can be rapid since glass matrix flow is all that is required. Moreover with the low viscosity a high density can be achieved without damage to the fibers. Fiber reinforced glass ceramics with strengths above 700 MPa were demonstrated in the 1970s (Sambell et al., 1972) through the use of carbon fibers. The key to the successful development of glass matrix composites lies in the fact that it has been carried out as a direct extension of metal and resin matrix composite efforts. Indeed, the elastic moduli of glass and glass ceramic allow reinforcement by load transfer like in fiber-polymer composites. The general types of matrices used to date are indicated in Table 5-1. 5.2.2.1 Carbon Fiber Composites Graphite fiber-reinforced glass matrix composites have demonstrated a wide range of attributes, including high strength, high stiffness, excellent toughness and low density. These composites exhibit strengths between 600 and 700 MPa and moduli between 200 and 300 GPa for about 50 vol.% fiber content. These values are competitive with fiber-reinforced resins and can be maintained over a much wider temperature range than those of the resin matrix systems. In inert atmospheres the strengths of the composites are retained up to temperatures at which the matrix becomes softened. However, in air, oxidation of the fibers begins at around 400- 500 °C leading to a loss of strength. Substantial increases

206

5 Ceramic Matrix Composites

Table 5-1. Glass-ceramic matrices. (From Prewo et al., 1986). Matrix

Glasses 7740 1723 7933

Type

Borosilicate B 2 O 3 , SiO2 Aluminosilicate A12O3, MgO, CaO, SiO2 Silica SiO2

Glass-Ceramics LAS-I LAS-II LAS-III MAS BMAS CAS MLAS

Major constituents

Li2O, A12O3, MgO, SiO2 Li 2 O, A12O3, MgO SiO2, Nb 2 O 5 Li2O, A12O3, MgO SiO2, Nb 2 O 5 MgO, A12O3, SiO2 BaO, MgO, A12O3, SiO2 CaO, A12O3, SiO2 MgO, Li2O A12O3, SiO2

in toughness were obtained from about 3 Jm~ 2 for unreinforced glass to about 400 Jm~ 2 for composites containing 20-30 vol.% of short fibers (Sambell et al., 1972) and to about 3000 J m " 2 for unidirectional composites (Davidge, 1985). Because carbon fibers have a higher strain capability than the matrix, the composites exhibit matrix microcracking during loading at stress lower than the fracture stress of the composite. Since carbon fibers have a highly negative axial thermal expansion coefficient (CTE), when combined with glass matrix it is possible to achieve a resultant composite CTE of very close to zero. Another interesting characteristic of carbon fiber is that when combined with glass, the resultant material exhibits a low coefficient of friction since carbon fibers possess a certain degree of lubricity and also it has high wear resistance due to the presence of the glass matrix. However, the major disadvantage of this type of composite is that graphite fibers are readily oxidized at elevated temperature in air.

Minor constituents

Major crystalline phase

Na 2 O, A12O3 B 2 O 3 , BaO B2O3

max

°c

E GPa

600 700 1150

65 90 65

ZnO, ZrO 2 , BaO ZnO, ZrO 2 , BaO

(3-Spodumene (3-Spodumene

1000 1100

90 90

ZrO 2

P-Spodumene

1200

90

BaO

Cordierite Barium osumilite Anorthite p-Spodumene a-Cordierite

1200 1250 1250 1250

105 90

5.2.2.2 Oxide Fiber Composites Oxide fibers have also been used to fabricate composites; however, in these cases the fiber to matrix bond is significantly greater than that of either carbon or silicon carbide fibers composites. The fracture strength is generally lower but it has been possible to achieve composite materials which prevent unstable crack growth. For instance, successful composites having excellent oxidation stability and electrically insulating properties have been achieved with the use of alumina fibers in both aluminosilicate glass and LAS (lithium alumino silicate) glass-matrices. Alumina fiber-reinforced glass was shown to be unaffected by exposure to temperature up to 1000°C (Prewo et al, 1986). The mechanical behavior of Nextel and SiC fiber-reinforced silica has been studied by Osmani et al. (1990). incurve behavior of Nextel and SiC fiber-reinforced silica is shown in Fig. 5-15. The methods of incurve determination in the context of nonlinear and nonelastic behavior are de-

5.2 Fiber Composites i

1

C-C2CU~N C-C3D 6 -

\ \

>^

NEX-SiO2 3D

\

N

'If 1/

4 -

0.5

SiC-SiO23D I

7

1

1.5

—

i

i

2.5

207

length is small and the absorbed fracture energy decreases strongly. At high temperatures (above 700 °C), the SiO2 matrix crystallizes. This leads to a decrease of extracted length and thus of absorbed energy. The fracture behavior becomes similar to that of bulk ceramic. For short random fiber (Nextel or SiC) composites, the absorbed energy is much lower (^100 Jm" 2 ), the pull-out length is about 100 jim.

La(mm)

Figure 5-15. Crack growth resistance curve for 3D Nextel-SiO 2 (7f = 40%), SiC-SiO 2 (Ff = 36%), 3D (Ff = 45%) and 2D C - C (7f = 60%) composites. From Osmani et al. (1990).

scribed by many authors (R'Mili et al., 1990; Bouquet et al., 1990; Gomina, 1987). The 3D Nextel-SiO 2 presents a good example of pull-out mechanisms owing to weak interface parameters (
Figure 5-16. Fracture surface of 3D Nextel-SiO 2 composite tested at room temperature. From Osmani et al. (1990).

5.2.2.3 SiC Fiber Composites Silicon carbide fiber-reinforced glasses and glass-ceramic matrices have been found to combine strength, toughness and the potential for high temperature oxidation resistance. The Nicalon SiC fiber has received the greatest attention because of its ease of handling in fabrication of complex shapes, as well as the experimental determination of its superior ability to provide high strength, and high fracture toughness composites. One of the more important characteristics of these fiber-reinforced glass composites is the softening of the glass with temperature. It is considered to be the cause of the loss of strength at high temperature and also, at least in part, the cause of the higher strength observed at intermediate temperatures. The use of high silica glasses to achieve high temperature composites however, has been difficult due to the higher temperatures required to densify the composite during hot-pressing. Glass-ceramics have been investigated in order to overvome the softening problems. They provide the unique capability to densify a composite in a low viscosity glassy state and subsequently to crystallize the matrix to achieve high temperature mechanical properties. Thus, it is possible to

208

5 Ceramic Matrix Composites

get 90% of the matrix crystallized, the rest remaining glassy. An example of mechanical behavior of long fiber composites is given by Fig. 5-17, which shows clearly the reinforcement effect due to fibers. Figure 5-18 shows the dependence of flexural strength and toughness with temperature for LAS glass-ceramics reinforced by SiC 300 Unidirectional

250 500 Displacement ( pm)

Figure 5-17. Load-displacement flexural curve for a LAS glass-ceramic matrix reinforced by SiC fibers. After Brennan and Prewo (1982).

800-

400-

0 0 100

600

800

1000

12 000

n°o

unidirectional

0 100

800

1000

1200

Figure 5-18. Dependence of flexural strength (a{) and toughness (Klc) on temperature for LAS glassceramics reinforced by SiC fibers. After Brennan and Prewo (1982).

fibers. High strength and toughness is retained up to about 1100°C in an inert environment. As in the case of the glass matrix composites, the eventual loss of strength at elevated temperature is due to the softening of the matrix, and thus is highly dependent on the percentage of matrix still left in the glassy state. The formation of a very thin carbon-rich interfacial layer is responsible for the excellent fracture toughness and crack deflecting ability of these composites. The effect of the interface strength on the mechanical behavior of a ID SiC-pyrex composite formed at 950°C by hot-pressing has been studied by Jamet et al. (1987). When the fibers are not treated, the interfacial bonding strength z{ is high and the composite has a brittle behavior. On the other hand, when a carbon coating of 50 nm is deposited on fibers by CVD, the material behavior is modified: the shear stress of the interface x{ is lowered {x{ = 9 MPa), and matrix microcracking occurs around 800 MPa and fracture takes place above 1000 MPa. The interfacial shear stress can be measured by using methods based on mechanical testing or indentation or fiber pull-out (Rouby and Navarre, 1990). In the first method, the distance between matrix cracks (Eq. (5-47)), the opening of matrix cracks (Marshall and Evans, 1985) and the fracture surface analysis by fitting the pullout length distribution with Eq. (5-51) are used. For the methods based on indentation, fiber indentation by a Vickers indentor (Peres, 1988; Benoit et al., 1992), pushout test or slice compression test (Shafry et al., 1989) are involved. Unfortunately, when tested in air (Fig. 5-19), the composite exhibits a greater loss of flexure strength with temperature and the tensile side of the sample exhibits rather brittle behavior while the compres-

5.2 Fiber Composites 1200

1 1 0 0 0-

_

ih

~

"I

~

'•

-

r•

-'

r-

T

1

Tested in argon A—,,N

A \

\ \ \

800-

(V

-

a.

1 600 -

\ Tested in air

en

| 400 to

N

sO//

!

200

-

SiC yarn/LAS -III composite 0

II "

i

600

i

i

i

i

800

1000

1200

1400

T e m p e r a t u r e (°C)

Figure 5-19. Flexural strength in Ar and in air vs. temperature for unidirectional LAS matrix-SiC yarn composites. After Prewo et al. (1986).

sive side retains the very fibrous appearance that is normal for the entire fracture surface composites when tested in an inert environment. The evolution of the microstructure of the matrices and interfaces allows a better understanding of the evolution with temperature of the mechanical behavior of the composites (Larnac et al., 1992). The matrix can transform at high temperatures and the fiber/matrix interface evolves during annealing. This evolution can be followed by measuring the Young's modulus as a function of temperature (Cutard, 1992) or by using pull-out tests (Thouless et al., 1989). For a SiC fiber-reinforced LAS, the original carbon layer present at the fiber matrix interface is responsible for the low sliding resistance of the interface (T{ « 2MPa) and the composite exhibits a nonbrittle behavior. Upon heat treatment in air at 800°C, the carbon layer is removed and replaced by silica and T{ increases up to 200 MPa and the composite shows a more brittle failure. Similar results were obtained by Bleay and Scott (1992) with a calcium aluminosilicate matrix. Some TEM observations made by

209

Ponthieu (1990) show that the presence of carbon layer is not sufficient to obtain small value of x>x but NbC grains provoke the brittleness of the carbon-matrix interface and therefore small T{. The fatigue behavior of these composites has been little studied to date (Kervadec and Chermant, 1991). Zawada et al. (1991) and Harris et al. (1992) have measured the change in stiffness or Young modulus to follow the damage during cycles due to matrix cracking. Prewo (1987) has measured the stressstrain curves for an unidirectional SiC fiber-reinforced LAS glass-ceramic and observed an increasing nonlinearity of the unloading part of the stress-strain curves with increasing the number of cycles. This nonlinearity is due to the mismatch of the debond surfaces, which come in piecewise contact during unloading (Talreja, 1990). Talreja presents a model allowing the construction of a fatigue life diagram. The diagram is divided in three regions. In the first one, the mechanism intervening is the catastrophic fiber failure. In region 2, a progressive damage is observed due to fiberbridged matrix cracking with unstable fiber-failure. In region 3 (under fatigue limit), matrix cracking and stable fiber failure occur (see also Chap. 13). 5.2.3 Refractory Ceramic Composites

In this section the composites made with a very refractory matrix, which cannot be processed in a viscous state, are considered. 5.2.3.1 Carbon-Carbon Composites

Historically, C - C composites were the first materials to be made with highly refractory matrix. A detailed review has been presented by Lamicq and Thebault (1985). Highly fibrous C - C composites were

5 Ceramic Matrix Composites

Table 5-2. Typical mechanical properties of flat laminates. After Lamicq and Thebault (1985). Room temperature Tensile strength (MPa) Compressive strength (MPa) Shear strength (MPa) Young's modulus (GPa)

2000°C

50-100 60-200 60-100 80-150 10-12(||) 30-35 (_L) 15-3O(||) 20-50(||) 3-10 (J.) 6-15

(a)

E

H6

%

+

+

••*

+

CO

c

QJ

(b)

"^4 01

+

+

-•-

ener

made by impregnation of the carbon prepreg by an organic material and subsequent carbonization. A very tough material resulted exhibiting nonbrittle behavior and notch insensitivity, that can be used at very high temperature in inert atmospheres. The typical properties are given in Table 5-2. However, above 600 °C oxidation becomes very significant limiting the application of this material. But protection against oxidation can be obtained by a multilayer coating (Rousseau et al., 1992). A twofold coating constituted of pack-cemented SiC and silica glass presents a good protection up to 1600°C, due to microcrack healing. Beyond 1600°C, chemical reaction takes place betwen SiC and SiO2 coating and a new protection concept with aluminum nitride and alumina successive coatings over SiC was designed for providing oxidation resistance at higher temperature. The fracture behavior of fiber-reinforced ceramics is often nonbrittle and the loaddisplacement curves are not linear. This behavior can be described by a crack growth resistance curve. Some results of fracture behavior of a 2D carbon fiber-reinforced carbon laminate (made by SEP) are presented for two orientations. We can observe the effect of reinforcement orientation on the fracture toughness of the mate-

f.

ro

Sur

210

(a) _ — ____________ ______ — — ——»— — — — —— ——

0 0

2 3 A a (mm)

Figure 5-20. Reinforcement orientation effect on the crack growth resistance curve for a) (3,2) orientation and b) (1,2) orientation for a 2D carbon-carbon composite. After R'Mili et al. (1990).

rials. Following the orientation (3,1) interlaminar cracking is observed and the material has a similar behavior to a densified carbon product without reinforcement. The crack growth resistance reaches a plateau near 100 J/m 2 (Fig. 5-20). In the orientation (1,2) test, an edgewise cracking occurs (the crack plane is perpendicular to the cloth plane). The fracture behavior of the material is nonlinear and the fracture is controlled by fiber processes. For this orientation, the crack growth resistance is quite high (R = 5000 J/m 2 ) due to energy absorbing mechanisms occurring in the process zone in front of the crack (Fig. 5-20). Carbon-carbon (Aerospatiale) (Vf = 60%) 2D and 3D materials present a good example of pull-out mechanisms, due to the weak interface parameters (T{ and the critical stress for the onset of debonding at the interface) leading to pull-out of bundles

5.2 Fiber Composites

which leads to large energy dissipation. The extracted lengths are of the orders of 1 to 1.5 mm; the R-curves for these materials are given in Fig. 5-15. Damage meso-modeling of 3D carboncarbon (Aerospatiale) was developed by Cluzel et al. (1992). The mesoscale (an intermediate scale) is used for the modeling of the deterioration. The degradation mechanisms, complex at macroscale are simple to describe at the mesoscale. The identification of the mesoconstituent depends on the composite type. A 2D composite is modeled as a stacking of damageable single layers. For a 3D composite material the meso-constituents are the fiber yarns, the matrix blocks and their interfaces. The relation between the mesoscale and the macroscale is made by using homogenization techniques. This enables the mechanical behavior to be predicted satisfactory. 5.2.3.2 C-C-TiC Composites A new family of ceramic fiber-reinforced ceramic composite materials referred to as 2D C - C - T i C has recently been developed by Rossignol et al. (1987) to improve the properties of the already known two dimensional carbon-carbon composite. The stack of carbon fiber fabrics remains the same but part of the carbon matrix is replaced by titanium carbide. Compared with 2D C - C - S i C composite, the 2D C - C - T i C material appears to have a higher thermal stability and wear resistance but a somewhat lower oxidation resistance. The processing technique consists of two CVI steps. The first one is to make a 2D C - C composite by infiltration of carbon in a preform made of a stack of carbon fiber fabrics. In the second step, titanium carbide is deposited within the remaining pores at 950°C. This material has been re-

211

ported to have a brittle behavior (although the composite is tougher than the related monolithic ceramic). 5.2.3.3 SiC Matrix Composites In order to overcome the oxidation problems of C - C composites, SiC, which is also very refractory, stiff and has a high strength, has been used to fabricate C-SiC and SiC-SiC composites. Different processing methods that lead to different results and final properties have been investigated: - CVI of a fibrous substrate - Reaction bonding of composites with a carbon matrix - Liquid impregnation with an organosilicon compound and subsequent thermal decomposition to SiC - Processing of fiber-reinforced polycarbosilane-based SiC composites. Unidirectionally and two dimensionally reinforced composites with high flexural strengths can be obtained by chemical vapor impregnation of fiber skeletons with SiC (Fitzer and Gadow, 1986). The C V D SiC fiber-reinforced composite shows the greatest absolute strength compared with all SiC composites, but still exhibits brittle fracture behavior, probably owing to the very strong adhesion between the CVDbased SiC fibers and matrix. In such a case no crack energy can be absorbed by delamination. In the case where high modulus carbon fibers are used as reinforcement, dense CVI-SiC composites show at maximum stress a small region of inelastic deformation that varies with the adhesion between the carbon fiber and the CVI-SiC matrix. This can be influenced by variation of the deposition parameters. Multidirectional^ reinforced CVI-SiC composites have been

212

5 Ceramic Matrix Composites

introduced by Naslain et al. (1980). Carbon-carbon composites with porosity of about 30% were impregnated with CVISiC. A flexural strength of 385 MPa has been recorded. More recently, Societe Europeenne de Propulsion (SEP) has introduced a SiC-fibre-reinforced CVD-SC with two dimensional fiber reinforcement made by CVI of polycarbosilane-derived SiC fiber skeleton by Lamicq et al. (1986) which has a density of about 2400 Kg/m 3 and a residual open porosity near 10%. The bending strength remains constant and sometimes shows a slight increase up to 1300°C (Fig. 5-21) followed by a progressive decrease with some strength remaining at temperature of 2000°C. This retention of some strength at high temperature implies that this material is capable of sustaining temporary overheating without catastrophic failure. The most unusual results for ceramics have been obtained in fracture and damage evaluation. Fracture toughness has been measured on notched samples tested in bending (Fig. 5-21). Values as high as 25 MPa^/m have been reported by Lamicq etal. (1986) but these values must be interpreted with care, as the theory from

which they are derived is only valid for isotropic materials, which SiC-SiC composites are not. Damage evaluation can be made by measuring the variation of the propagation speed of ultrasonic waves (Baste and Gerard, 1987). From the variation of the elastic constants, the damage tensor components can be evaluated, damage being anisotropic. Such evaluation was used for SiC-SiC composites. Damage modeling of SiC-SiC composite was developed by Lebon et al. (1990) using a micro-macro technique based on elementary cells. Crack growth resistance curves of C SiC and SiC-SiC composites were measured by many authors (Bouquet et al., 1990; Fourvel etal., 1990; Navarre etal., 1989). Figure 5-22 shows typical R-curves for 2D SiC-SiC and C-SiC composites. First of all, especially for SiC-SiC composites, the curves are serrated and reveal large peaks. This feature is linked to the reinforcement geometry: in the SiC-SiC composite, half of the bundles are perpendicular to the crack path and nearly arranged in rows. The fracture of each row needs an increase of energy, responsible of each peak. For the C-SiC materials, the

400 30

-300

It Si C - S i C

20 10

0

500

1000

1500

2000

Temperature (°C)

Figure 5-21. Temperature dependence in inert atmosphere of bending strength and fracture toughness for SiC-SiC composites. After Lamicq et al. (1986).

0 0

I

\ c- S i C 1

3 (mm)

Figure 5-22. Crack growth resistance curve for 2D SiC-SiC (K f ^40%) and C-SiC (F f ^30%) composites (relative notch length = 0.4). From Navarre etal. (1989).

5.2 Fiber Composites

l?-curve is smoother, because bundles are oriented obliquely to the crack plane and the crack path is more homogeneous (Navarre et al., 1989). However, it is worth mentioning that mean fracture energy is very high (5-10 KJ/m 2 ) as compared to classical bulk ceramics (100 J/m 2 ). As shown in Sec. 5.1.5, micromechanisms such as matrix microcracking, crack bridging, fiber failure, and fiber pull-out are responsible for this incurve behavior. From the first stage (fast increase of R) controlled by the crack bridging, the bridging stress ahr can be obtained using Eq.(5-57): for the SiC-SiC composite, a value of 490 MPa is obtained (Navarre, 1990), compatible with the values computed by the method of Thouless and Evans (1988). Most of the crack growth energy absorbed for the plateau is due to fiber pullout (Sec. 5.1.5) and it is interesting to analyze quantitatively the fracture surface. The distribution of pull-out lengths can be deduced from theoretical analysis and an estimation of the interfacial shear stress %-x can be obtained by comparison with experimental pull-out length distribution. Such a distribution is shown in Fig. 5-23 for a SiC-SiC composite with a theoretical curve corresponding to the bonded interface model. The agreement is quite good, confirming that Sutcu's model (1989) is an interesting tool to analyze the fracture surface of composites. The fatigue behavior of 2D SiC-SiC composite was studied recently (Jouin, 1990; Reynaud, 1992). Repeated loading may fracture materials at maximum load lower than the monotonic fracture load. Figure 5-24 shows the life-time of samples as a function of the maximum stress of fatigue cycles. Tests were carried out under tension-tension cycling on a cross weave composite of SiC fiber bundles and a SiC

213

Figure 5-23. Nondimensional pull-out length distribution (probability density function times

versus relative pull-out length / p / = 1.6 um). From Navarre (1990).

r

• •S**<S1I^ • • • •^ s < s v s ^^ •

-

^

• • • \

• ^

^

.

T

Fatigue limit • •

1 1

1

10

102

103 104 Cycle number

• 105

106

Figure 5-24. Life-time diagram of 2D SiC - SiC composite (F f ~40%) under tension-tension cycling. From Reynaud (1992).

matrix obtained by CVI. Depending on the maximum applied stress, three stages are observed: (i) samples broke on first loading, (ii) fractures occurred after a limited number of cycles, (iii) no fracture after a conventional number of cycles (typically 106). As illustrated in Fig. 5-25, the first stress-strain curve has a typical shape because of a response under loading similar to monotonic test. Further hysteresis loops are characterized by increasing area and decreasing slopes as shown in Fig. 5-26. The analysis of fatigue damage will refer to the stiffness of the loop: £ sl and En are the

214

5 Ceramic Matrix Composites

cycles JV*, where T{ = Tioo. For n greater than JV*, x{ remains constant (= rioo) and no further fatigue effect occurs. A decrease of t increases the fraction of broken fibers q. Fracture occurs when a critical fraction qc is obtained 0.1

0.2 Deformation (%)

0.3

Figure 5-25. Stress-strain a-e loops during fatigue of 2D SiC-SiC composite. From Reynaud (1992).

4c = l - e x p [ - l / ( m f

with mf as the fiber Weibull modulus. From Eqs. (5-51) and (5-52) and from the evolution of q with x{ and the stress a, the variation of q with the number of fatigue cycles can be deduced. For low stresses (
10

10

10 10 No of cycles

10

10

Figure 5-26. Evolution of the loop slopes versus number of fatigue cycles. From Reynaud (1992).

Ti(n) = Wt

(5-51)

where n is the number of fatigue cycles and t the parameter characterizing the decrease of shear stress with cycles until a number of

(5-53)

where oc u is the fracture strength of the composite. For stresses higher than on, fracture occurs after a number of fatigue cycles JV given by Ar(a) = (a c?u /a) (mf+1)/t

tangential elastic moduli at the beginning and the end of loading, Esu and Efu at the beginning and the end of unloading. A model describing the damage evolution under cyclic fatigue loading was proposed by Reynaud et al. (1992). Matrix cracks and fiber failures appear mainly during the first loading with an average spacing s0 and an average fraction of broken fibers. Then, relative movements between fibers and matrix under further repeated loadings wear away the interface, leading to a decrease of the friction shear stress from r i0 up to rioo. This decrease can be described by

(5-52)

(5-54)

The theoretical life-time diagram obtained with the following parameters mf = 4, t = 0.1 and Tj = 0.4 r i0 is in good agreement with the experimental results (Fig. 5-24). The mechanical properties (particularly toughness) of the ceramic-ceramic fiber composites depend on the degree of bonding between fiber and matrix. If the bonding is too strong, composites exhibit brittle failure while with weaker interfaces fibers pull-out is significant leading to a significant increase of toughness. The degree of bonding depends on the processing of the composites and particularly on the nature and texture of the interphases present between matrix and fiber. One way to control the adhesion is to coat the fibers with a thin layer of pyrolytic carbon. An example of the presence of this carbon layer is given by Fig. 5-27.

215

5.2 Fiber Composites

Fiber

mierphase

Figure 5-27. Transmission electron micrograph of the interfacial zone fiber-matrix in a SiC - SiC composite. From Mozdzierz and Backhaus-Ricoult (1991).

Structural and microtextural investigations of the interfacial zones through TEM observations are needed to understand the mechanical behavior. For SiC-SiC composites the presence of silica is undesirable because either it lowers the fiber-pyrocarbon adhesion too much or it gives rise to too strong interface zones due to chemical reactions (Monthioux and Cojean, 1992). For carbon fiber-SiC composites, a correlation betwen the strain to failure and the textural features of the pyrolitic carbon interphase was established (Despres and Monthioux, 1992). With respect to oxidation resistance, Filipuzzi et al. (1990a,b) have studied the microstructural and chemical evolution of ID SiC-SiC composites with a pyrocarbon interphase. Results have shown that several phenomena occur: (i) oxidation of the carbon interphase which leaves interstitial cavities between fibers and matrix, (ii) gaseous diffusion of oxygen in cavities, (iii) growth of silica layers through the matrix and oxidation of fibers which fill up the cavities. If connections between the silica films occur, oxidation along the fibers is strongly re-

duced. Use of a thin carbon interphase leads to higher oxidation resistance. Oxidation behavior of carbon fiber-SiC composites have also been studied (Berton et al., 1990). The oxidation behavior can be studied by measuring the Young's modulus using ultrasonic waves (Gault and Huger, 1990). The oxidation of the carbon interphase above about 600 °C leads to a decrease in the modulus while the formation of silica layers above about 750 °C increases the elastic modulus. A good correlation with thermogravimetric experiments is observed. The microcracking of the matrix due to internal stresses created by the silica formation can also be detected by measurement of the elastic modulus. The effect of high temperature oxidizing and inert aging treatments on the mechanical behavior has been studied by several authors (Frety and Boussuge, 1990; Filipuzzi et al., 1990 a, b; Navarre, 1990). Results show that aging temperature, duration, environment and initial interphase thickness have a strong influence on mechanical behavior. Figure 5-28 shows 400

300

a. ~ 200 100

0

250

500

750

1000

Displacement (\in\)

Figure 5-28. Three point bending tests (at room temperature) of 2D SiC-SiC composites after aging in air at different temperatures for 10 h: (1): 20°C, (2): 600 °C, (3): 800 °C, (4): 1000°C, (5): 1200°C. From' Navarre (1990).

216

5 Ceramic Matrix Composites

the evolution of the load-displacement bending curves after aging in air at different temperatures during 10 h for a SiC-SiC composite. Between 20 and 800°C, the mechanical behavior is always similar but around 1000°C, a "ductile" rupture occurs whereas the behavior becomes more brittle at 1200°C. The measurement of the interfacial shear stress T{ by indentation shows that ij decreases significantly after an annealing at 1000°C and increases again at 1200°C. These results are limited to the modification of the fiber-matrix bonding due to the oxidation of the carbon interphase and the subsequent growth of silica layers which form a strong interface. When the tests are made as a function of temperature in air, the flexural strength and the pull-out lengths decrease by aging above 600 °C. This evolution is probably linked to the formation of silica layers. The K-curves (Fig. 5-29) show a decrease of the fracture energy above 600°C due to the evolution of the interface bond which becomes stronger. An evolution is also observed during treatment in an inert atmosphere, probably due to the degradation of fibers inducing a modification of the fiber-matrix

Figure 5-29. Crack growth resistance (evaluated from the /-integral) curves for a 2D SiC-SiC composite measured in air at different temperatures: (1): 20°C, (2): 600°C, (3): 800 °C, (4): 1000°C, (5): 1200°C. From Navarre (1990).

bond. So, the oxidation of the carbon interphase constitutes a weak point for the material but protection against oxidation can be obtained by a SiC and oxide glass coating allowing microcracks healing. 5.2.3.4 Other Matrix Materials Fiber-reinforced ceramics with matrices other than glass, glassceramic, C or SiC were studied (Schioler and Stiglich, 1986; Fitzer, 1988). Carbon fiber reinforced alumina and mullite were prepared by the sol-gel technique with high-pressure molding. A toughening effect is observed with SiC coated carbon fibers (Klc increases from 4 to 10 MPaVm with 50 vol.% C fibers). SiC fiber-reinforced alumina has also been investigated. There has been much effort to incorporate SiC or C fibers in a Si 3 N 4 matrix. Nakano and Kamiya (1992) have studied unidirectional carbon fiber-reinforced Si 3 N 4 and Sialon matrices composites fabricated by using a polysilazane slurry process and hot pressing. The composites showed flexural strengths at room temperature and 1400°C in vacuum around 500 MPa and a dynamic fracture energy between 2 and 12 k Jm~ 2 . Hot pressed SiC fiber-reinforced ceramics with Si 3 N 4 matrix were fabricated by Miyoshi et al. (1988). SiC monofilaments obtained by CVD of (3-SiC on a carbon fiber core and coated with a carbon-rich layer were used. The effect of filament-matrix interfacial frictional stress, on the fracture toughness was investigated. The fracture toughness for 15 vol.% SiC filament with flexural strength of 600 MPa increases up to 17 MPa^/m when the interfacial frictional stress is that of the coated filament. In such a case the composite exhibits filament pull-

5.2 Fiber Composites

out. On the contrary, without coating, the frictional stress is too large and no pull-out is observed and the toughness is not significantly improved. Mechanical properties of SiC fiber monofilament reinforced reaction bonded silicon nitride have been studied by Bhatt (1988). The ID and 2D composites show improved properties: the room temperature tensile strength and fracture toughness are 680 MPa and 13 MPa^/m respectively for the ID composites with 30 vol.% of fibers. The strength measured as a function of temperature in air shows that it is maintained up to 1200°C. Measurement of room temperature tensile strength after 100 h exposure at temperatures up to 1400°C in air shows nearly 60% strength loss between 400-1200 °C. Oxidation of the fiber-matrix interface as well as internal oxidation of the porous Si 3 N 4 matrix are likely responsible for strength degradation. The tensile fatigue behavior of unidirectional SiC-fiber - Si 3 N 4 -matrix composite was investigated at 1200°C in air by Holmes (1991). For peak fatigue stresses below the proportional limit (about 196 MPa) fatigue run-out at 5 x 106 cycles was observed. For peak fatigue stresses above the proportional limit, fatigue life decreased as the stress amplitude was increased. Creep appears to be the predominant damage mechanism which occurs during elevated temperature fatigue below the proportional limit. Above the proportional limit, both mechanical fatigue damage and creep contribute to specimen failure. Three promising routes to obtain silicon nitride reinforced with continuous fibers have been demonstrated by Lundberg et al. (1990), Lundberg and Pejrud (1992), Swan etal. (1992). Si 3 N 4 powder slurry infiltration of carbon fibers with subsequent HIPing yields dense composites with

217

a weak fiber-matrix bond leading to a tough behavior. Similar slurry infiltration but with a SiSi 3 N 4 or Si-Si 3 N 4 -ZrO 2 slurry and SiC fibers and nitridation at low temperature (1300°C) lead to a tough composite with substantial pull-out. In order to protect the fiber-matrix interfacial carbon film, fibers are coated with a TiC-TiN layer by CVD. Si 3 N 4 obtained by pyrolysis of polysilazanes at temperatures below 1000°C was also a promising method. Infiltration is easy and the low processing temperature prevents chemical reaction and thermal degradation. The reaction bonded fiber reinforced silicon nitride ceramic composites and the materials obtained by infiltration of a polymer precursor and pyrolysis can be considered as a potential for producing high strength low cost ceramic matrix composites compared to the CVI or the HIP processes. Because of their good strength and toughness and good thermal stability, the Si 3 N 4 composites are potential candidates for high temperature applications. 5.2.3.5 Functionally Gradient Materials (FGMs) FGMs are composite materials that consists of a gradual compositional variation from ceramic to metal from one surface to the other (Sheppard, 1992). This continuous change results in property gradients. For space plane applications, materials must support temperatures up to 1800°C and gradients up to 1300°C. Materials must be oxidation resistant at the surface, tough at the cool surface and must withstand thermal gradient. Different processing methods are developed to produce FGMs. Current research in FGM is reviewed by Sheppard (1992) but some ex-

218

5 Ceramic Matrix Composites

amples are briefly given. A 3D C - C composite incorporating a SiC-C FGM as a coating (with a compositional gradient from SiC to C) shows a better thermal shock resistance than the uncoated material and an excellent performance in a rocket combustor. In order to join metals to ceramics, FGM of PSZ-stainless steel, Ni-MgO, A1-A1N, Ni-Si 3 N 4 can be obtained by using powder metallurgy or plasma spraying methods. Joining applications using combustion synthesis are also developed for TiB 2 -Cu or TiC-NiAl systems.

5.3 Particle, Whisker and Platelet Composites 5.3.1 Processing The main characteristic of this group of material is their ability to be processed by a way similar to that carried out for monolithic ceramics, i.e. powder preparation, compaction and sintering. This is possible because on the one hand of the small size of the reinforcement and on the other hand, of its high stability, even at the high temperature required for sintering. However, specific problems due to the addition of a second phase arise: (i) decrease in the densification, (ii) dispersion of the second phase, (iii) chemical reaction between the two phases. The processing methods of ceramic composites have therefore been more or less modified to take into account these difficulties. (i) The introduction of a hard particle that does not sinter in a compact of powder can reduce the sintering rate and the

final density and also lead to microcrack formation (Bordia and Raj, 1986 a,b). The constraint imposed by rigid particles when the matrix shrinks has been evaluated by Raj and Bordia (1984). It generates a shear stress in the matrix and a tensile hoop stress at the particle-matrix interface. Unless the shear stress can be relaxed by shear flow in the matrix, the hoop stress will become locked in causing the sintering rate to fall. Thus, the magnitude of the tensile hoop stress and the effect of particles on the rate of sintering will depend on the relative magnitude of the intrinsic rate of densification and the rate of shear in the matrix. Practically, it means that for a given temperature at which the sintering rate of the matrix is high, the addition of a second phase prevents the total densification. The magnitude of the maximum tensile hoop stress at the particle-matrix interface depends on the densification rate, the matrix, the particle size and the viscosity of the matrix at the sintering temperature. A general phenomenologic viscoelastic model has been proposed by Raj and Bordia (1984, 1986) and Bordia and Raj (1986 a, b). It is based on the initiation of stresses in biphase materials, showing two different shrinkages during sintering, that are progressively relaxed. The results show that interfacial stresses rapidly increase up to a limit after which they are gradually relaxed over a time corresponding to the sintering duration. Two important values are given by the model: the maximum hoop stress and the densification that are given by 2-9/?

(5-55 a)

5.3 Particle, Whisker and Platelet Composites

where PQ is the intrinsic sintering pressure, A^ is the densification defined as the time dependent matrix density, V{ is the volume fraction of inclusions, T is a time constant related to the densification of the sole matrix, and /? is defined as the ratio of shear rate to the densification rate. This strong stress-induced fall-off of densification can be overcome by (i) increasing the sintering driving force, i.e. the sintering time and temperature and add a pressure that counteracts the induced stress, (ii) modifying the intergranular phase. Pressure sintering: hot pressing (HP), hot isostatic pressing (HIP) or gas pressure sintering have therefore advantageously been used for the processing of dense composites. HIPing has been especially used for fibrous reinforcement such as whiskers where the densification can be very critical. No dense whisker composite has been successfully densified by pressureless sintering. Bordia's equation shows that much benefit is to be gained by reducing the viscosity of the matrix. This has been achieved by liquid phase sintering which consists of adding sintering aids that form a low melting point phase with the matrix. This, plus a fine dispersion has permitted the elaboration of nearly dense SiC whisker silicon nitride composites (Fig. 5-30) (Olagnon, 1990; Yonezawa 1989, Hoffmann et al., 1988) by gas pressure lower than the intrinsic sintering pressure. This pressure has been estimated to be of about 0.5-2 MPa (Bordia and Raj, 1988) for particulate composite constituted of an alumina matrix. (ii) The homogeneity of the second phase dispersion is also a major problem that induces two types of difficulties. First of all, badly dispersed second phase particles lead to heterogeneities in the sintering powder, and thus to local variations of densities that can induce stresses and cracks similar to those described above. Numerous

219

850

5

10 15 20 Whisker content ( v o l . % )

Figure 5-30. Literature compilation of sintered density as a function of whisker content for SiC(W)Si3N4 composites. After (1), (2): Hoffmann et al. (1988); (3): Yonezawa et al. (1989) and (4): Olagnon (1990).

works have shown this problem to be crucial when the densification is limited. Badly dispersed particles also play a major role in the mechanical characteristics of the sintered body, either by inducing local internal stresses or by forming depleted zone of unreinforced matrix or even by making clusters that can be the source of rupture of the component. It should be noted that the existence of clusters is particularly critical in whisker-based composites, where they can be of great size. Hence, numerous negative results of strengths published in the literature have been attributed to heterogeneous dispersions. Studies on the whisker packing have shown that the aspect ratio should not exceed 10 to 30 in order to avoid any clumping (Milewski, 1986) which is always difficult to eliminate. The improvement of the dispersion can be obtained by liquid mixing and forming such as slip casting, pressure casting and to a less extent by plastic forming processes. Methods based on the casting of a slurry brings the advantage of keeping the dispersion up to the compact formation leading to composites of high density with a good dispersion. This has been well illustrated by the fact that sintered whisker com-

220

5 Ceramic Matrix Composites

posites of high density have only been obtained by slip casting. (iii) The need of very high temperature to overcome the densification decrease of two-phase materials promotes chemical reactions. Those reactions may create strong bonding at the interface betwen the matrix and the particle, which can prevent any reinforcement. This has been demonstrated in whisker composites where the resulting toughness strongly depends on the interface strength. The chemical reactions can also degrade the reinforcement such as for instance in silicon carbide whisker-silicon nitride composites where following the nitrogen pressure and temperature during sintering, either nitridation of the SiC or decomposition of Si 3 N 4 can occur (Olagnon et al., 1991 a, b). 5.3.2 Transformation-Toughened Ceramics

Many experimental evidences exist showing zirconia transformation toughening into ceramics. As shown in Sec. 5.1.1, the fracture toughness increase is due to t -»m transformation of zirconia and needs the retention of t-ZrO 2 . This retention can be obtained by using several types of microstructures (Claussen, 1984). Consider composition A in Fig. 5-1: it can be densified in the t range and if the grain size is lower than the critical size required for the t-nn, a single-phase (t-ZrO2) ceramics is obtained: this microstructure is referred to as tetragonal zirconia polycrystals (TZP). For composition B, the ceramic is sintered in the t + c phase field and a microstructure consisting of t-ZrO 2 and c-ZrO 2 can be obtained. For composition C, sintering takes place in the cubic range and with a specific treatment in the t + c region, tZrO 2 precipitate in a cubic matrix: these

materials are referred to as partially stabilized zirconia (PSZ). Zirconia particles can be dispersed in a ceramic matrix other than ZrO 2 : these materials are called ZrO 2 dispersed ceramics (ZDC). An example of such a composite is the ZrO 2 toughened alumina (ZTA) of which a microstructure is shown in Fig. 5-31. Many other microstructures of zirconia-toughened ceramics (ZTC) can be obtained (Claussen, 1984). 5.3.2.1 Effect of Size and Amount of Zirconia Particles

As seen in Sec. 5.1.1 toughness improvement is directly related to the tetragonal particle size. In the case of PSZ, the precipitate size depends on the aging conditions after sintering and solution treatment. A dramatic drop off in toughness occurs when the precipitate size is just above the critical size dc: the t->m transformation becomes spontaneous and the amount of tetragonal precipitates available for transformation decreases. Eq. (5-7) predicts that the toughness increment due to the t->m transformation depends on the volume fraction of transformed particles and the transformation zone width h. The transformation zone size is determined by using X-ray, optical,

' ¥ ' • ,5* " '

Figure 5-31. Microstructure of a zirconia (30vol.%) toughened alumina (ZTA). After Jorand (1991).

5.3 Particle, Whisker and Platelet Composites

TEM or Raman techniques: to be between about 1 and 15 jim. Figure 5-32, shows that experimental results are in good agreement with Eq. (5-7). As foreseen by the transformation toughening models (Sec. 5.1.1), a i^-curve behavior is observed (Fig. 5-33). The most significant R-curve behavior for Mg-PSZ occurs for a high content of t-ZrO 2 precipitates (peak aged), with a large initial rise occurring during the first 50 jim of crack extension as predicted by the models (Fig. 5-2). For overaged PSZ materials with large monoclinic precipitates, #-curve behavior is smaller and extends over large crack extension: this effect can be attributed to crack branching and crack microcracking (Swain and Hannink, 1984; Swain et al., 1983; Saadaoui, 1991). The effect of particle size on the toughness of ZTA was studied by Claussen et al. (1977). Fracture toughness presents a maximum with increasing volume fraction of zirconia particles. The maximum occurs at higher volume fraction with smaller particle size. This behavior corresponds to the theorical predictions of transformation toughening of microcracking models described in Sec. 5.1. Fracture energy (Gc) values for different pressureless sintered (PS) ot hot-pressed (HP) ZTA compositions have been measured (Orange, 1992; Orange et al., 1992) (Table 5-3). For PS materials, Gc is nearly constant up to 15 vol.% ZrO 2 whereas, for HP materials, Gc shows a maximum at 10 vol.% ZrO2. The observed increase of Gc in PS materials and to 10% ZrO 2 in HP materials can be attributed mainly to the stress induced transformation toughening mechanism as indicated by the relative amount of tetragonal phase. Results are consistent with theoretical models presented in Sec. 5.1.1. Above 15 vol.% pure ZrO 2 , the relative

221

Figure 5-32. Fracture toughness increment AKC for Ca and Mg-PSZ with transformation zone size and volume fraction of t-ZrO 2 . After Swain et al. (1983).

0

200

400 600 800 Crack extension, Aa [\im)

1000

Figure 5-33. i?-curves for peak-aged and overaged Mg-PSZ. After Swain and Hannink (1984).

amount of tetragonal phase is considerably reduced and microcracking toughening becomes probably the effective mechanism. The microcracking toughening is not so energy dissipative than transformation toughening. When the transformation toughening mechanism is predominant, the toughness increment follows well Eq. (5-7) as for PSZ materials (Green et al., 1989). 5.3.2.2 Effect of Solute Content

The effect of Y2O3 addition for 20 vol.% ZrO 2 has been studied for ZTA materials (Table 5-3). With stabilizing agent addi-

222

5 Ceramic Matrix Composites

Table 5-3. Fracture Energy of ZTA materials. A20Z

A20Z1Y A20Z2Y A20Z3Y A45Z3Y

A

A5Z

Vol.% ZrO 2 Mol.% Y2O3

0 0

5 0

10 0

15 0

20 0

20 1

20 2

20 3

45 3

41 0

86 98

80 48

95 20

98 15

88 33

120 95

72 98

390 96

50 0

160 96

250 82

115 23

102 7

117 30

270 97

194 100

576 99

PS Mater. G c (Jm" 2 ) fir(%) HP Mater. G c (Jm- 2 ) Gr(%)

A10Z

A15Z

Composition

QT = relative tetragonal zirconia ratio.

tion, the metastable tetragonal phase content increases, which is reflected by a decrease in M s or an increase of critical diameter dc. Consequently, the stress-induced phase transformation toughening effect becomes preponderant on microcracking toughening and fracture energy values increase up to 2 mol.% Y2O3 [Ms reaches the test temperature and AGt^m tends to zero, Eq. (5-13)]. For higher Y2O3 content, AGt_^m increases and therefore the critical stress required to initiate the transformation oc decreases [Eqs. (5-13), (5-10) and (5-12)], the tetragonal phase being too stable. As noted in Eq. (5-10), internal stresses are generated due to mismatch of thermal contraction of particles and matrix. Thus, the effective critical transformation stress a°c [Eq. (5-11)] must replace VC, autotransformation occurs (
volume fraction of t-ZrO 2 and toughness. Thoughness exhibits a maximum as a function of V. This maximum increases with yttria content as expected. By using Eq. (5-10) the value of ac can be obtained as a function of Y2O3 content. For yttria content higher than 2 mol.%, zirconia particles are not transformed in the stress field of the crack and so, Klc decreases. For a given volume fraction of zirconia a® = crc —
The temperature dependence of mechanical properties of TZP and ZTA composites is shown in Fig. 5-34. For all ZTA composites, Klc (or fracture strength af) decreases as temperature increases. However, Klc decreasing rate depends on the material composition and on the toughening mechanism of the material. Y-TZP materials exhibit very high values of toughness at room temperature but there is a drastic decrease in these values at higher temperatures up to about 600°C. This temperature corresponds to the critical temperature where transformation toughening is no

5.3 Particle, Whisker and Platelet Composites 14 •TZP3Y • A12O3 \ (HP) • A45Z3YJ AA20Z0Y (PS)

nj 10

Q_

0

200

400

600 800 1000 Temperature (°C)

1200

Figure 5-34. Temperature dependence of fracture toughness for hot pressed (HP) and pressureless sintered (PS) alumina, TZP and ZTA composites. After Orange et al. (1988).

more effective: the chemical free energy for transformation AGC is equal to zero. For ZTA composites (A45Z3Y) with 45 vol.% ZrO 2 (3mol.% Y2O3), where transformation toughening is the main toughening mechanism, a similar temperature dependence as in TZP is observed: properties are much improved at room temperature but decrease drastically as temperature is increased. Compositions with 20 vol.% ZrO 2 without stabilizing addition (A20Z), where microcracking toughening is the preponderant toughening mechanism, present not so good properties at room temperature but are quite temperature independent up to 800°C. This result supports the fact that microcracking toughening is temperature independent. By addition of 2 mol.% Y2O3, stabilization of t-ZrO 2 is achieved and again the stress-induced phase transformation toughening becomes preponderant and Klc decreases with temperature. Similar observations on the temperature dependence of toughness were made in other zirconia toughened ceramics. The temperature ef-

223

fect on toughness is well described by Eq. (5-11): AKC is proportional to (T-MJ1 and K will increase as T decreases until a maximum is reached at T&MS. For T<MS, autotransformation occurs, leading to a decrease in toughness. Y-TZP materials exhibit degradation of properties under moist environments in the temperature range 200-300 °C, associated with an increase of m-ZrO 2 content. This destabilization depends on the grain size, materials with small grain size showing smaller degradation. The mechanism for this degradation with aging time in the presence of water is still the subject of discussion (Green et al., 1989). The t ->m ZrO 2 transformation involves a volume increase and when the phase transformation occurs only near the surface, it gives rise to surface compression. This compressive stress develops during cooling after sintering due to the absence of the hydrostatic constraint near the free surface. The thickness of the compressive surface layer can be increased by abrasion (Green et al., 1989), leading to a considerable increase in the fracture strength of ZTC. (The compressive stresses up to « 1 GPa can be generated at depth between 10-100 |im below the surface.) In order to obtain the maximum benefit, the thickness of the transformed zone must be larger than the critical flaw size. Many other properties can be improved by zirconia reinforcement. Becher (1981) has shown that thermal shock resistance is greatly increased for alumina zirconia composites. For zirconia amount lower than 15 vol.%, about 80% of ZrO 2 is tetragonal. The critical ATC resulting in the resistance degradation is maximum for the peak strength material (11.5 vol.% ZrO2) in agreement with the models of thermal shock. Thermal fatigue behavior is also improved for TZP or ZTA materials (Orange

224

5 Ceramic Matrix Composites

et al., 1992). ZTC also shows an increase of the slow crack growth resistance. In comparison with other ceramics, zirconia engineering ceramics offer a combination of properties which lead to improvement of the wear behavior of ceramics (Stevens, 1986).

tremely high strength with low fracture toughness due to the small flaw size. The results shown in this section are essentially based on TZP, PSZ and ZTA materials but similar results can be obtained in many ZTC materials or zirconia dispersed ceramics (ZDC) (Green et al., 1989).

5.3.2.4 Strength-Toughness Relationship

5.3.2.5 Mullite-Zirconia

The relationship between strength and toughness for brittle ceramics is

Zirconia has also been added to reinforce mullite matrices. Two different processing routes have been investigated either by classical dispersion of mullite and zirconia powder or by reaction sintering of a mixture of zircon and alumina that is transformed in zirconia plus mullite (Moya and Osendi, 1983; Anseau et al., 1983; Leriche et al., 1988). In the first route, the addition of zirconia promotes the sintering and prevents abnormal mullite grain growth. Zirconia has been considered as promoting the purification of the grain boundaries, especially by eliminating the glassy phase from the mullite-mullite interface. The second route brings the benefit of using a low cost base material, zircon, but leads to a specific microstructure where two different types of zirconia particles are present. Intragranular particles of round shape and of small size, and polyhedric particles of larger size. The processing route is of major importance for the final properties. However, the phase transformation that operates in mullite is far different to what happens in alumina since the Young's modulus of the matrix and the thermal expansion coefficient are lower. Moreover, the existence of solid solution between zirconia and mullite has been shown. Finally, the measured critical particle transformation (1.2 jam) is superior in the case of reaction sintered materials. Toughness between 4 - 5 M P a x / m have

a = Constant

(5-56)

where a is the critical flaw length. From this relation, it follows that for a constant flaw size, af will increase as Kc increases due to decreasing oc (Eq. (5-9)). But a peak will be reached when af = ac: another mechanism intervened with the appearance of inelastic deformation in shear bands (Swain, 1985). Thus strength of the material is "yield" limited (a = oc). Decreasing flaw size allows a higher peak strength. An extensive set of results is reproduced in Fig. 5-35. Mg-PSZ and C e TZP exhibit high value of fracture toughness but low strength due to large flaw size. In contrast, Y-TZP and ZTA show ex-

3000

A ' / 1 2S °Mm // \V / y

2000

• Y-TZP-A1 2 O 3 AY-TZP OCe TZP

1 /* / ^

100(jm

1000 /

0

//

>^N^ % ^ ^

10 Kk (MPa-\/m)

400jjm

15

Figure 5-35. Relation between fracture toughness and strength in ZTC. After Tsukuma et al. (1988) and Swain (1985).

5.3 Particle, Whisker and Platelet Composites

been measured (Wallace et al., 1984) while fracture strengths of 300-500 MPa were obtained. The toughening mechanism is complicated but different works show that mechanisms independent of the phase transformation play a significant role (Orange et al., 1985; Torrecillas et al., 1990). Important reinforcements have hence been obtained by microcracking in composites containing significant amounts of monoclinic phases. Transformation mechanisms do not seem to play a significant role. 5.3.3 Whisker Composites

Whiskers are single crystal fibers with aspect ratio of about 10 and of a small diameter in the range of 10-0.1 jim. The absence of grain boundaries and their high purity brings up several attractive characteristics. They exhibit a good chemical stability, even at high temperature. Owing to their small size, they can contain only small defects and therefore they show a high tensile strength. Owing to the experimental difficulties values are sparse, however, average strength values of about 8 GPa have been measured on relatively large whiskers ( 0 = 6|im) (Petrovic et al., 1985). Note that is one order of magnitude higher than that observed for dense ceramics and four times that of continuous fibers. Moreover the chemical stability allows them to keep those strength values above sintering temperatures. Several types of whiskers have been used for the reinforcement of ceramics, including silicon carbide, silicon nitride and alumina but only the former have been widely developed and tested. Many different sizes have been available with relatively high aspect ratios up to average values of 30-50. The distributions of diameters and lengths are however generally quite wide. Small

225

whiskers of diameters of about 0.3 \xm have however been mostly used. Many microstructural studies have shown the presence of defects such as planar defects, inclusion or cavities. Moreover they often show an irregular convoluted surface that can act as a mechanical joint by clamping the whisker in the matrix which prevents any reinforcement by pull-out mechanism. Whisker being strongly anisotropic, different composite textures that depend on the processing method have been achieved. The reduction of densification induced by the whiskers has mostly led to the use of hot pressing (HP) for the preparation of such composites. Therefore most of the results are given for composite materials where the whiskers are randomly oriented in planes. This is also the case of the materials formed by slip casting technique that make the whiskers randomly oriented in planes parallel to the slip mold interface. In this case, since the structure is built at the green forming stage the final anisotropy is conserved whatever the sintering technique is used. The use of extrusion for the preparation of the green body causes the whiskers to be aligned along the extrusion axis, making a so-called ID composite (Fig. 5-36). Although, it is more difficult to obtain, isotropic bodies on a macroscale can be obtained by powder preparation and pressureless or hot isostatic pressing (HIP). The structure anisotropy can be observed on the microstructure and also by toughness and strength measurements. For instance, increase of both toughness and strength of about 50% have been measured in SiC(W)-Al 2 O 3 composites when the crack plane was perpendicular rather than parallel to the whisker plane (Becher et al., 1984). One of the highest toughness values obtained in SiC(W)-Si 3 N 4 was measured with a crack perpendicular to

226

5 Ceramic Matrix Composites

5.3.3.1 S i C - Alumina Composites

In the case of alumina, spectacular reinforcements have been obtained. At ambient temperature, concomitant increase of toughness and strength have been nearly systematically observed (Fig. 5-37). Toughness values of more than 9 MPa^/m, which implies a two- or three-fold toughness value increase for 20-30 vol. % whisker content, have been recorded. Such

20%

SiC(W)

50pm

Figure 5-36. Optical micrograph of 20% SiC(W)Si3N4 composite made by slip casting and 1 MPa N 2 overpressure sintering. Note that whiskers are randomly oriented in planes.

the direction of aligned whiskers (Kandori etal, 1987). Whiskers have been incorporated in alumina and silicon nitride and to a less extent in mullite, zirconia, glass or other matrices. Silicon nitride and alumina based composites show different behaviors. Since the thermal expansion coefficients of A12O3, SiC and Si 3 N 4 are about 9xlO~ 6 , 4.5 x 10" 6 and 3.1 x 1CT6 respectively, silicon carbide whiskers are in tension in silicon nitride and in compression in alumina (Pradel et al., 1990). This has been experimentally confirmed by measurement of the stress magnitude in the composite by X-ray diffraction technique.

10 20 30 Whisker content (vol.%)

10 20 30 40 Whisker content (vol.%)

Figure 5-37. Literature compilation of toughness and strength data obtained in various SiC(W)-Al 2 O 3 composite materials. After (1): Becher and Wei (1984), (2): Wei and Becher (1985), (3): Tiegs and Becher (1987), (4): Homeny etal. (1987), (5): Porter etal. (1987), (6): Becher etal. (1986), (7): Govila (1988), (8): Lio etal. (1989).

5.3 Particle, Whisker and Platelet Composites

composite materials have been made with small diameter whiskers in the range of 0.3-0.6 (im. The achievement of a similar strengthening is shown to strongly depend on the whisker dispersion in the matrix and on the total densification of the composite material. For this purpose, the different materials have generally been hot pressed in order to reach density close to theoretical values. The importance of the whisker length and diameter has not been definitely clarified and only few studies have been conducted. Baek et al. (1989) have demonstrated the advantage of longer whiskers in the range of 9 -18 |im in an alumina matrix. Both the strength and the toughness were increased but the values were only modest: from 380-560 MPa for the strength and from 4.7-5.3 MPa Jm for the toughness. Fracture surface observations lead the authors to the conclusion that longer whiskers experience more extended pull-out. However, other studies (Tiegs and Becher, 1987) concluded on the benefit of milling the whisker which, by reduction of the length, contributes to a higher green and final density and to a better whisker dispersion. Moreover, theoretical considerations of reinforcement by crack bridging and crack deflection show that a length of 15 (am can produce optimum toughening. The influence of the diameter has been better clarified. Theoretical consideration of crack bridging indicates that large diameters are preferred for significant toughening. Hence debonding is promoted by large whiskers. However, owing to the significant thermal expansion mismatch of the silicon carbide-alumina system, too large diameter whiskers induce microcracking of the matrix around the reinforcement (Tuffe, 1991). The strength is

227

therefore reduced and optimum results for this system has been obtained with whiskers of diameter lower than 1 jim. The main toughening mechanisms have been attributed to crack deflection and to crack bridging. The magnitude of the toughening by crack bridging is controlled by the crack opening at the end of the bridging zone. Experimental evidence of this mechanism has been shown by transmission and scanning electron microscopy analysis. However only small bridging zone lengths of about ten microns, for a maximum debonded length of 1 jam, i.e. about the whisker diameter, have been found (Rodel et al., 1991). Although this can impart significant toughening, no extended K-curve, i.e. over large crack extension, can be expected based solely on whisker bridging. However, in rare occurrences slight incurve effect has been recorded in SiC(W)-Al 2 O 3 composites (Lio et al., 1989; Jenkins et al., 1987). Stable crack growths were recorded on chevron notch specimens and were shown to strongly depend on the hot pressing temperature. Too high a temperature led to a more brittle material where cracks appeared to be undeviated by whiskers. High-temperature strength can be slightly improved up to about 1000°C, where the composite material retains its superior fracture strength even in an oxidizing environment such as air. But above this temperature, owing to the presence of viscous flow that induces a different fracture mechanism, the fall-off of strength is similar to that observed with pure alumina. Furthermore, since the introduction of whiskers impedes the densification, sintering aids are added. Also the SiO2 layer mostly present at the whisker surface increases the amount of intergranular glassy phase in the composite compared to that of the matrix (Becher et al., 1988; Govila,

228

5 Ceramic Matrix Composites 1000

1988). Extended slow crack growth is therefore observed at high temperature. _

800 •

5.3.3.2 SiC(W)-Silicon Nitride Composites Although the final toughness of the best materials is similar to that obtained in alumina, the reinforcement of silicon nitride by silicon carbide whiskers has led to more contradictory results. The toughness which is less sensitive to processing defects has given different results as can be seen in Fig. 5-38, made of a compilation of recent literature data. It is clear that the reinforcement is less significant than in the alumina case and the highest final values have been obtained with matrices exhibiting a high starting toughness. The broad range of matrix toughness is to be noted and is specific of the silicon nitride that shows a strongly processing dependent microstructure. Some results have even led to a toughness decrease with the addition of whiskers. The strength data (Fig. 5-38) are even more sparse. The strengthening does not seem to be related to toughening and many occurrences have shown strong strength decreases. The understanding of this behavior requires us to recall the microstructure of this matrix. Silicon nitride as a highly covalent material that shows very little self-diffusion is difficult to densify without sintering aids even at high temperature and pressure. Sintering aids such as Y2O3, A12O3, CeO 2 , MgO ... are added to silicon nitride powder and by reaction with the SiO2 layer always present around Si 3 N 4 particles form a low melting point ternary phase that promotes densification by liquid phase sintering. Since the sintering aids are generally added in an amount of 5-10 wt.%, an important intergranular phase (often glassy) is present around silicon ni-

10

20

30

Whisker content ( v o l . % )

10

20

30

40

Whisker content ( v o l . % )

Figure 5-38. Literature compilation of toughness and strength data obtained in various SiC(W)-Si 3 N 4 composite materials. After (1): Buljan et al. (1987), (2): Lundberg et al. (1987), (3): Shalek et al. (1986), (4): Kandori et al. (1987), (5): Takemura et al. (1987), (6): Singh et al. (1988), (7): Hayami et al. (1986), (8): Akimune et al. (1989), (9): Braue et al. (1989).

tride grains. Silicon nitride exists under two polymorphic phases a (stable up to 1400°C) and p. The final microstructure is directly linked to the (3 content of the starting powder. Sintering of a powder containing mostly (3 phase leads to a final material consisting of equiaxed P grains. When the starting powder contains a high ratio of unstable a phase, solution is promoted and the recrystallization leads to acicular p grains of aspect ratio reciprocal to the ini-

5.3 Particle, Whisker and Platelet Composites

tial (3 content. Lange (1979) has proposed the following relation between the aspect ratio, R, and starting powder content: R = 1 + a/P

(5-57)

The diameter of such grains can also vary over a wide range that depends on the sintering conditions, time, temperature and liquid phase characteristics. Diameters between 0.3-6 jim for an aspect ratio of more than 10 have been reported. These different microstructures lead to a large variation of toughness from 3MPa v / m for equiaxed grains to ^ M P a ^ / m for large grains of high aspect ratio that reinforced by the same mechanisms used for whisker composites. The term of self-reinforcement is even sometimes used. If one does not consider the results of Shalek et al. (1986) that have been obtained with large diameter whiskers (6 jim) as well as results obtained with aligned whiskers (Kandori et al., 1987), most systematic reinforcements have been obtained with low toughness matrices. Indeed the introduction of whiskers to silicon nitride strongly inhibits grain growth and composites cannot be tailored with a matrix containing large grains. The toughness of the composite is therefore the result of the competition between a toughness decrease of the matrix and toughening due to the whiskers (Olagnon, 1990, 1991b; Becher, 1991). Another difficulty that arises with the introduction of silicon carbide to silicon nitride is the possible chemical reaction mentioned in Sec. 5.3.1 which degrades the whiskers or increases the interface strength. Such a reaction has been shown to be strongly dependent on the impurities present on the whisker and the different whisker sources have therefore led to major differences in composite formation and whisker degradation (Bradley et al., 1989). However, other authors have ob-

229

served similar toughening with various SiC whisker sources, which demonstrates again the high dependence of the silicon nitride matrix structure on the final reinforcement. Fracture surface analyses show that the pull-out in SiC(W)-Si 3 N 4 is less extended than in alumina matrices and has often been attributed to a stronger interface. Although it is detrimental for toughening, owing to significant differences of Young modulus between SiC and Si 3 N 4 , some strength increase by load transfer can be obtained (Kandori et al., 1987; Olagnon et al., 1991 a, b). The analysis of the effect of whisker size on the reinforcement has not been definitely investigated. However, owing to the sign of the thermal coefficient mismatch, no microcracking of the matrix as observed in alumina can occur. Therefore, theoretical analyses recommending a large diameter can be considered and compared to the highest toughness obtained with large 6 (im diameter whiskers. The high temperature strength of SiC(W)-Si 3 N 4 composites is similar to that of the matrix, i.e., the addition of sintering aids that both promotes densification and high toughness is detrimental for high temperature mechanical properties. Pure silicon nitride, that exhibits temperature independent strength can be hardly processed to high density by HI Ping, but is unfortunately made of low toughness structure (3MPaym). Toughening of about 70% can be obtained by introducing whiskers to pure silicon nitride (Takemura et al., 1987), without reduction in strength up to 1400°C (Pezzotti et al., 1989). 5.3.3.3 Multiple Mechanisms In order to further improve the toughness, whisker composites invoking deliberately different toughening mechanisms have been developed. The basics of this

230

5 Ceramic Matrix Composites

idea is that the overall toughness can be decomposed into the matrix toughness plus the whisker toughness plus perhaps another reinforcement Ts

fin

i

A z^w

I

A jy

XVJ^AJ +AAj -fAAj

/C CQ\

[D-Do)

At the edge of this category, similarly to what has been observed with SiC(W)Si 3 N 4 composites, the matrix can be tailored to show a high toughness, which for a similar toughening leads to a higher final toughness. This has been shown by Becher et al. (1991) in alumina where an increase of matrix toughness obtained by increasing the grain size leads to high overall toughness according to Eq. (5-64). Such a microstructure can induce a duplex i^-curve, i.e. of high initial slope but on small crack length owing to the whisker, and a lower slope but over longer crack length due to the large matrix grains that can exert bridging stresses up to relatively large crack opening. This result shows that two different toughening mechanisms can be coupled. More complex structures have been tailored by combining both whiskers and other particles. For instance, the addition of large SiC particles (40 jim) to a SiC(W)-Si 3 N 4 composite leads to reinforcement higher than the relative composite material reinforced either solely by SiC particles or SiC whiskers (Kodama et al., 1990). The combination of ZrO 2 and whiskers to alumina has also been investigated. The primary goal was to increase the high temperature range at which the ZrO 2 transformation is not operative (Becher and Tiegs, 1987, 1988; Mathieu and Cales, 1987; Claussen and Petzow, 1986). Different situations have been observed. Either the toughening behavior is simply the sum of whisker and microcrack contribution, showing that the two mechanisms act independently, or whisker reinforce-

ment and transformation toughening are coupled and the composite toughness is greater than a simple sum of toughness achieved by separate processes. Values of about lOMPaym have been observed for mullite reinforced by both 20 vol. % of SiC(W) and 20 vol. % of t-ZrO 2 . In alumina, toughness and strength of 13.5 M P a y m and 700 MPa, respectively, have been achieved. 5.3.4 Particle-Reinforced Composites

Under this classification, composite materials of particle experiencing no transformation and of micron size are grouped. The concept is similar to that of whisker composites with however, lower potential toughening. But this type of composite material has the possible advantage of being processed by conventional method more readily than whisker-reinforced materials. The development of such composite materials was primarily focussed on tribological applications. The target was therefore to improve both toughness and hardness. Many compositions have been investigated among which the principals are: - reinforcement of alumina by SiC, TiC, BN and TiN; - reinforcement of silicon carbide by TiB 2 , TiC and A1N; and - reinforcement of silicon nitride by SiC and TiC. Since the residual stress due to thermal mismatch is an important parameter that controls the toughening, it is convenient to classify those composite materials into three different groups: (i) a p > a m :

TiC-SiC, SiC-Si 3 N 4 , TiC-Si 3 N 4 , (ii) a p ^ a m : TiB 2 -SiC, TiC-Al 2 O 3 , TiN-Al 2 O 3 , AIN-SiC, (iii)a p
5.3 Particle, Whisker and Platelet Composite

Materials from the last category (ap < am) are more susceptible to experience microcrack toughening where as in the cases of other materials, the probability of microcracking is low and toughening will be mostly due to crack-particle interaction. The most definite example of microcracking toughening has been observed in SiC-Al 2 O 3 composites (Niihara et al., 1986). Two reinforcement sizes (8 ^m and 2 jim) have been incorporated in alumina. Although the highest toughening has been obtained with the smallest particles, both materials exhibit a maximum of toughness as a function of the particle content (Fig. 5-39). The strength shows a similar trend in the case of the small particles, but a strength degradation is observed for the larger size particle composites. The calculation of the critical size from Eq. (5-14) gives rc = 4 jim, showing that microcrack toughening can be imparted by 2 fim diameter reinforcement composite but particles of 8 jum induce the formation of large cracks that are detrimental for strength. However, no general law between the toughness and the particle size can be stated since it strongly depends on the toughening mechanism. For instance in the case of Si 3 N 4 material reinforced with

231

SiC, where ap > am, and where particles of sizes 32, 9 and 5 ^m were investigated, the highest fracture energy has been obtained with the coarse particles. However, in such a case, particle-crack interactions such as crack deflection were responsible for toughening and no microcracking is to be expected or observed. Nevertheless, introduction of large particles still lead to strength reduction by increasing the flaw size. Other results of ap > am composite materials show that under the condition of good processing, strengthening proportional to toughening can be expected since no degradation by microcracking is noted. For instance, in 25% TiC-SiC increases of strength from 500 to 700 MPa and toughness from 4 to 6 MPa^/m have been reported (Wei and Becher, 1984). Similarly, 15% TiB 2 -SiC exhibits increases of toughness and strength of 3.1 to 4.5MPa v /m and 380 to 485 MPa respectively (Janney, 1987; McMurtry and Boecker, 1987). This last mechanism is less critical since no maximum of strength with particle content, such as that observed with microcracking, occurs. A fair example of material showing no thermal mismatch is TiC-Al 2 O 3 . Although some authors (Wahi and Ilschner, 1980) have reported a strengthening due to grain reduction caused by TiC addition, no particle-crack interaction was observed in such composites (Baldoni et al., 1985). Finally, fine particles are generally to be preferred since large sizes can lead to strength reduction. 5.3.5 Platelet-Reinforced Composites

10 20 30 Particle content (vol.%)

Figure 5-39. Strength and toughness of SiC-Al 2 O 3 composites for two different particle sizes: solid line: 8 ^im, dashed line: 2 jam. After Niihara et al. (1986).

Platelets have been proposed as reinforcement candidates for ceramics after the possibility of the toxicity of whiskers was raised.

232

5 Ceramic Matrix Composites

Different types of platelets have been developed but most work has been done on SiC and A12O3 platelets. A broad range of sizes has been proposed and investigated for aspect ratios (length/diameter) between 70 jim/6 jim to 15 jim/1 \xm. They consist of single crystals and can therefore sustain the high temperature needed for densification. They show some theoretical advantages compared to whiskers since they exhibit a smoother surface and a lower defect density (Claussen, 1990; Nischik et al., 1991). Theoretical analysis of toughening shows a high propensity to crack deflection and crack bridging, possibly associated to frictional forces resulting from pull-out of the platelets. Platelets have been incorporated in Si 3 N 4 , alumina, mullite and TZP materials. In the case of silicon nitride, spectacular toughening has been observed, with 40 jam SiC platelets and a value of 14 MPa^/m for 30% loading having been recorded. However, a strength reduction caused by the increase of flaw size due to large platelet size has been noticed. The strength reduction is even more pronounced in the case of mullite and alumina matrices. The development of smaller platelets should lead to a simultaneous increase of both strength and toughness. Owing to their morphology, a strong texture is obtained in the final composite (Warner et al., 1992). This leads to toughness anisotropy. For instance, pressure casting, or hot pressing and, to a higher extent, tape casting leads to alignment of the platelets. 5.3.6 Nanocomposites Although ultrafine microstructures have been recognized for a long time to exhibit remarkable properties in physics applications, interest in nanocomposites is rela-

tively recent. In the case of ceramics, the first desirable property was the large plasticity even at relatively low temperature that could be obtained with nano size grains. The nanocomposites can be classified into different categories according to the microstructure (Niihara, 1991). The reinforcing particle can be either inter- or intragranular, i.e. localized within the matrix grain. The different microstructures can be obtained by a fine control of the processing parameters. Different preparation techniques have been investigated but no significant difference in microstructure has resulted from the techniques of powder preparation, chemical preparation and CVD. Several combinations have been investigated: Al 2 O 3 -SiC, Al 2 O 3 -Si 3 N 4 , Al 2 O 3 -TiC, Mullite-SiC, B 4 C-SiC, B 4 C-TiB 2 , amorphous SiC-SiC and Si 3 N 4 -SiC. The toughening results are relatively modest: from 4-6.5 M P a y m for SiCSi 3 N 4 or 3.5-4.7 MPa^/m for A1 2 O 3 Si 3 N 4 or Al 2 O 3 -SiC. They are therefore comparable to the results obtained with micron-sized particle-reinforced composites. More remarkable are the results obtained in terms of strength at high temperature (Fig. 5-40) and hardness. Room temperature strengths higher than 1000 MPa are reported. Moreover, these values are retained to high temperature. Most of these materials are composed of micronsize matrix grains with nano-sized reinforcing particles. Surprisingly, by comparison to other more classical composite materials, an improvement of properties is obtained even for low percentage of reinforcement (3 to 5%). Reinforcement mechanisms have not been definitely investigated and clarified. They however seem to depend on the material type. In the case of silicon nitride, for

5.4 Conclusions

1500 Si 3 N 4 -SiC Al 2 O 3 -SiC n

1000 MgO-SiC

ro

500

500

1000

1500

Temperature (°C)

Figure 5-40. Temperature dependence of the fracture strength of Al 2 O 3 -SiC, MgO-SiC and Si 3 N 4 -SiC nanocomposites. After Niihara (1991).

instance, nanoparticles promote the development of elongated grains which induce the reinforcement by a more classical mechanism. It should be noted, however, that the processing of nanocomposites from nanopowders leads to final materials where critical defects can be reduced from 23 to 6 jam in size, which in turn means improvement of processing reliability. 5.3.7 Complex Structure Materials

A last group of composite materials made of toughening agents at different scales can be considered. They consist of a dispersoid composite made of a complex structure possessing a macroscopic texture over a large scale. This is the case of so-called duplex ceramics composed of large spherical zones (1050 jim) containing variable fractions of ZrO 2 particles dispersed in a ceramic matrix. Two different toughening mechanisms can operate concomitantly, i.e., phase transformation and microcracking as well as crack bridging. Large toughening (Wang and Stevens, 1988) and tremendous

233

.R-curve behavior over a large crack increment (Lutz et al, 1991) have been observed. However, owing to processing defects caused by differential sintering between TZP agglomerates and matrix, the resulting strength is seriously reduced. The thermal shock behavior is therefore that of resistant ceramics as refractories, i.e. it exhibits stable rather than unstable crack growth during thermal shock. This is also the case of layered composites consisting of stacking of different compositions, such as alternate layers of alumina and zirconia-alumina where interactions between the transformation zone and the interface layer leads to large toughening and crack growth resistance from 5 to ll.SMPa^/m (Marshall et al., 1991). Residual stresses owing to thermal expansion mismatch and difference of shrinkage during sintering have also been shown to contribute to the reinforcement that is therefore anisotropic. This type of material can be prepared either by tape casting (Boch et al., 1986) or by slip casting (Requena et al., 1989) and can be used to produce substrates of high mechanical properties.

5.4 Conclusions Monolithic ceramics are reinforced to modify mechanical properties, primarily to - increase toughness (and thus reliability) - increase strength. The achievable toughness values of monolithic ceramic (typically 2-4MPa N /m) conferred by different types of composite reinforcement were reported (a little optimistically for real applications) by Evans (1990) as in Table 5-4.

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5 Ceramic Matrix Composites

Table 5-4. Toughness values of monolithic ceramics [after Evans (1990)]. Mechanism

Transformation

Highest toughness (M Pa y/m)

Reinforcing phase/ matrix

Limitation

T<900K

20

ZrO2(MgO)

Microcracking

10

Whisker/ platelets

15

Al 2 O 3 /ZrO 2 T<1300K Si 3 N 4 /SiC strength SiC/TiB2 Si 3 N 4 /SiC Si 3 N 4 /Si 3 N 4 oxidation Al 2 O 3 /SiC T<1500K

Fibers

20-30

CAS/SiC LAS/SiC Al2O3/SiC SiC/SiC processing SiC/C coating A12O3/A12O3 fibers

From these improvements a number of composite materials have recently found real applications, e.g.: -

SiC-glass ceramics: prostheses; ZrO 2 -Alumina: wear resistance; Carbon-Carbon: rockets, prostheses; SiC-SiC: special engine parts; SiC whisker-alumina: cutting tools.

5.5 References Akimune, Y, Katano, Y, Shichi, Y. (1988), Adv. Ceram. Mater. 3, 138-142. Anderson, C. A., Gupta, T. K. (1981), in: Advances in Ceramics, Vol. 3: Science and Technology of Zirconia. Heuer, A. H., Hobbs, L. W. (Eds.). Columbus, OH: American Ceramic Society, pp. 184-201. Anseau, M. R., Leblud, C , Cambier, F. (1983), J. Mater. Sci. Lett. 2, 366. Aveston, X, Cooper, G. A., Kelly, A. (1971), in: Proc. Properties Fiber Composites. London: Science and Technology Press, pp. 15-26. Baek, V. K., Kim, C. H. (1989), J. Mater. Sci. 24, 1589-1593. Baldoni, J. G., Buljan, S. T, Sarin, V. K. (1985), Inst. Phys. Conf. Series 75. Bristol, U.K.: Adam Hilger, pp. 427-438.

Baste, S., Gerard, A. (1987), C. R. Acad. Sci. (Paris) 305 (II). 1511-1516. Becher, P. F. (1981), /. Am. Ceram. Soc. 64, 37-39. Becher, P. F. (1986), Ada Metall. 34, 1185-1191. Becher, P. F. (1991), /. Am. Ceram. Soc. 74 (2), 255269. Becher, P. F , Tiegs, T. N. (1987), J. Am. Ceram. Soc. 70 (8), 651-654. Becher, P. R, Tiegs, T. N. (1988), Adv. Ceram. Mater. 3, 148-153. Becher, P. F, Wei, G. C. (1984), J. Am. Ceram. Soc. 12, C144-C145. Becher, P. F, Tiegs, T. N., Ogle, J. C. (1986), in: Fracture Mechanics of Ceramics, Vol. 7: Bradt, R. C. (Ed.). New York: Plenum Press, pp. 61-73. Becher, P. F , Hsueh, C. H., Angelini, P., Tiegs, T. N. (1988), J. Am. Ceram. Soc. 71 (12), 1050-1061. Becher, P. F , Fuller, E. R., Angelini, P. (1991), /. Am. Ceram. Soc. 74 (9), 2131-2135. Benoit, M., Guillard, M., Rouby, D., Fantozzi, G. (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, J. F, Massiah, A. (Eds.). Bordeaux: EACM, pp. 907-914. Berton, D., Bacos, M. P., Demange, D., Lahaye, J. (1990), in: Materiaux composites pour applications a hautes temperatures: Naslain, R., Lamalle, X, Zulian, J. L. (Eds.). Bordeaux: AMAC/CODEMAC, pp. 315-325. Bhatt, R. T. (1988), in: Whiskers- and Fiber-Toughened Ceramics: Bradley, R. A., Clark, D. E., Larsen, D. C , Stiegler, X O. (Eds.). Metalls Park, OH: ASM Int., pp. 199-207. Bleay, S., Scott, V. D. (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, X F , Massiah, A. (Eds.). Bordeaux: EACM, pp. 619-624. Boch, P., Chartier, T, Huttepain, M. (1986), J. Am. Ceram. Soc. 69 (8), C191-192. Bordia, R. K., Raj, R. (1986a), Mater. Sci. Research 20, 27-39. Bordia, R. K., Raj, R. (1986b), Adv. Ceram. Mater. 3, 122-126. Bordia, R. K., Raj, R. (1988), /. Am. Ceram. Soc. 71 (4), 302-310. Bouquet, M., Birbis, X M., Quenisset, X M. (1990), Compos. Sci. Technol. 37, 223-248. Bradley, S. A., Karasek, K. R., Martin, M. R., Yeh, H. C , Schienle, X L. (1989), J. Am. Ceram. Soc. 72 (4), 628-636. Braue, W., Holsher, A., Saruhan, B., Ziegler, G. (1989), Euroceramics, Vol.3: Engineering Ceramics: deWith, G., Terpstra, R. A., Metselaar, R. (Eds.). London: Elsevier Applied Science, pp. 3.263-3.267. Brennan, X, Prewo, K. (1982), /. Mater. Sci. 17, 2371-2383. Budiansky, B., Hutchinson, X W, Lambroupolos, X C. (1983), Int. J. Solids Struct. 19, 337-355.

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Llorca, X, Elices, M. (1990), Acta Metall. 38, 24852492. Lundberg, R., Pejryd, L. (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, X F., Massiah, A. (Eds.). Bordeaux: EACM, pp. 709-714. Lundberg, R., Kohlman, L., Pompe, R. (1987), Am. Ceram. Soc. Bull. 66, 330-333. Lundberg, R., Pompe, R., Carlsson, R., Goursat, P. (1990), Compos. Sci. Technol. 37, 165-176. Lutz, E. H., Claussen, N., Swain, M. V. (1991), J. Am. Ceram. Soc. 74, 11-18. Marshall, D. B., Cox, B. N. (1985), Acta Metall. 33, 2013-2021. Marshall, D. B., Evans, A. G. (1985), J. Am. Ceram. Soc. 68, 225-256. Marshall, D. B., Ratto, X X, Lange, F. F (1991), J.Am. Ceram. Soc. 74 (12), 2979-2987. Mathieu, P., Cales, B. (1987), in: Proc. 1st Int. Conf. Ceramic—Ceramic Composites, Mons, April 1987. Not published. McMeeking, R. M., Evans, A. G. (1982), /. Am. Ceram. Soc. 65, 242-246. McMurtry, C. H., Boecker, W. D. G. (1987), Am. Ceram. Soc. Bull. 66 (2), 325-329. Milewski, X V. (1986), Adv. Ceram. Mater. 1, 36-41. Miyoshi, T., Kodama, H., Sakamoto, H., Gotoh, A., Iijima, S. (1988), in: Whisker- and Fiber-Toughened Ceramics: Bradley, R. A., Clark, D. E., Larsen, D. C , Stiegler, X O. (Eds.). Metalls Park, OH: ASM Int., pp. 193-197. Monthioux, M., Cojean, D. (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, X F , Massiah, A. (Eds.). Bordeaux: EACM, pp. 729-734. Moya, J. S., Osendi, M. I. (1983), /. Mater. Sci. Lett. 2, 599. Mozdziierz, N., Backhaus-Ricoult, M. (1991), Rapport G. S. comportements thermomecaniques des composites ceramique-ceramique a fibres. Not published. Nakano, K., Kamiya, A. (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, X F , Massiah, A. (Eds.). Bordeaux: EACM, pp. 697-702. Naslain, R. (1986), J. Phys. Cl, 47, 703-715. Naslain, R., Hagenmuller, P., Christin, F., Heraud, L., Choury, X X (1980), in: ICCM-3, Adv. in Comp. Mater.: Bunsell, A. R. (Ed.). Paris: Pergamon Press, p. 2, pp. 1084-1097. Naslain, R., Rossignol, X Y, Quenisset, X M., Langlais, F. (1985), in: Introduction aux materiaux composites 2: Matrices metalliques et ceramiques: Naslain, R. (Ed.). Paris: Edition du CNRS, pp. 439-491. Navarre, G. (1990), Ph.D. thesis, INSA, Lyon, p. 212. Navarre, G., Rouby, D., Fantozzi, G. (1989), in: Euro-Ceramics, Vol. 3: Engineering Ceramics: de With, G., Terpstra, R. A., Metselaar, R. (Eds.).

5.5 References

London: Elsevier Applied Science, pp. 3.4483.452. Niihara, K. (1991), J. Ceram. Soc. Jpn. 99 (10), 974982. Niihara, K., Nakahira, A., Uchiyama, T., Hirai, T. (1986), Fracture Mechanics of Ceramics, Vol. 7: Bradt, R. C. New York: Plenum Press, p. 367. Nischik, C , Seibold, M. M., Travitzky, N. A., Claussen, N. (1991), /. Am. Ceram. Soc. 74, 24642468. Olagnon, C. (1990), Ph.D. thesis, INSA, Lyon, p. 148. Olagnon, C , Bullock, E., Fantozzi, G. (1991a), Ceram. Int. 17, 53-60. Olagnon, C , Bullock, E., Fantozzi, G. (1991b), /. Eur. Ceram. Soc. 7, 265-273. Orange, G. (1992), Thesis, INSA, Lyon. Orange, G., Fantozzi, G., Cambier, R, Leblud, C , Anseau, M., Leriche, A. (1985), J. Mater. Sci. 20, 2533-2540. Orange, G., Fantozzi, G., Homerin, P., Thevenot, K, Leriche, A., Cambier, F. (1988), Adv. Ceram. 24B, 1075-1082. Orange, G., Fantozzi, G., Homerin, P., Thevenot, R, Leriche, A., Cambier, F. (1992), / Eur. Ceram. Soc. 9, 177-185. Osmani, H., Rouby, D., Fantozzi, G., Lequertier, J. M. (1990), Compos. Sci. Technol. 37, 191-206. Outwater, J. D., Murphy, L. L. (1969), in: Proc. 24th Annu. Tech. Conf. on Composites. New York: The Society of Plastics Industry, Section l i e , pp. 1720. Peres, P. (1988), Ph.D. thesis, INSA, Lyon, p. 329. Petrovic, J. X, Milewski, J. V., Rohr, D. L., Gac, R D. (1985), /. Mater. Sci. 20, 1167-1177. Pezzotti, G., Tanaka, I., Okamoto, T, Koizumi, M., Miyamoto, Y. (1989), J. Am. Ceram. Soc. 72,1461 1464. Ponthieu, C. (1990), Ph.D. thesis, Universite Paris VI. Porter, J. R., Lange, F. R, Chokski, A. (1987), Am. Ceram. Soc. Bull. 66, 343-347. Pradel, T., Olagnon, C , Bullock, E. (1990), J. Mater. Sci. Lett. 9, 960-961. Prewo, K. M. (1987), J. Mater. Sci. 22, 2695-2701. Prewo, K. M., Brennan, J. I, Layden, G. K. (1986), Am. Ceram. Soc. Bull. 65, 305-313, 322. R'Mili, M., Rouby, D., Fantozzi, G. (1990), Compos. Sci. Technol. 37, 207-221. Raj, R., Bordia, R. K., Ada Metall. 32, 1003-1019. Requena, X, Moreno, R., Moya, J. X (1989), /. Am. Ceram. Soc. 72, 1511-1513. Reynaud, P. (1992), Ph.D. thesis, INSA, Lyon, p. 295. Reynaud, P., Rouby, D., Fantozzi, G. (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, J. R, Massiah, A. (Eds.). Bordeaux: EACM, pp. 597-602.

237

Rodel, X, Fuller, E. R., Lawn, B. R. (1991), J. Am. Ceram. Soc. 74 (12), 3154-3157. Rossignol, X Y, Quenisset, X M., Naslain, R. (1987), Composites 18 (2, April). Rouby, D., Navarre, G. (1990), in: Proc. 11th Riso Int. Symp. Metall. Mater. Sci.: Bentzen, X X, BildeSorensen, X B., Christiansen, N., Horsewell, A., Ralph, B, (Eds.). Ris0, Danmark: Riso National Laboratories, pp. 127-144. Rousseau, G., Saugnac, R, Boursereau, R (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5: Bunsell, A. H., Jamet, X F, Massiah, A. (Eds.). Bordeaux: EACM, pp. 511-516. Riihle, M., Heuer, A. H. (1984), in: Advances in Ceramics, Vol. 12: Science and Technology of Zirconiall: Claussen, N., Riihle, M., Heuer, A. H. (Eds.). Columbus, OH: American Ceramic Society, pp. 14-32. Saadaoui, M. (1991). Ph.D, thesis, INSA, Lyon, p. 170. Sakai, M., Yoshimura, X, Goto, Y, Inagaki, M. (1988), J. Am. Ceram. Soc. 71, 609-616. Sambell, R. A., Bowen, D. H., Phillips, D. C. (1972), /. Mater. Sci. 7, 663-675. Schioler, L. X, Stiglich, X X (1986), Am. Ceram. Soc. Bull. 65, 289-292. Scott, G. G. (1975), J. Mater. Sci. 10, 1527-1535. Seising, X (1961), /. Am. Ceram. Soc. 44, 419. Seyfert, D., Wiseman, G. H. (1984), /. Am. Ceram. Soc. 67, C-132-C-133. Seyfert, D., Wiseman, G. H., Prud'homme, C. (1983), J. Am. Ceram. Soc. 66, C-13-C-14. Shafry, N., Brandon, D. G., Terasaki, M. (1989), in: Euro-Ceramics, Vol. 3: Engineering Ceramics: de With, G., Terpstra, R. A., Metselaar, R. (Eds.). London: Elsevier Applied Science, pp. 3.4533.457. Shalek, P. D., Petrovic, X X, Hurley, G. R (1986), Am. Ceram. Soc. Bull 65, 1986, 351-356. Sheppard, L. M. (1992), Am. Ceram. Soc. Bull. 71, 617-631. Singh, X P., Goretta, K. C. Kupperman, K. S., Routbort, X L., Rhodes, X F. (1988), Adv. Ceram. Mater. 3, 357-360. Stevens, R. (1986), An Introduction to Zirconia. London: Magnesium Elektron Ltd., pp. 1-51. Stinton, D. P., Caputo, A. X, Lowden, R. A. (1986), Am. Ceram. Soc. Bull. 65, 347-350. Sutcu, M. (1989), Acta Metall. 37, 651-661. Swain, M. V. (1985), Acta Metall. 33, 2083-2091. Swain, M. V, Hannink, R. H. (1984), Adv. Ceram. 12, 225-239. Swain, M. V, Hannink, R. H., Garvie, R. C. (1983), in: Fracture Mechanics of Ceramics, Vol. 6: Bradt, R. C , Hasselman, D. P., Lange, R F. (Eds.). New York: Plenum Press, pp. 339-354. Swan, A. H., Brandt, X, O'Meara, C , Rundgren, K., Pompe, R. Lundberg, R., Pejryd, L. (1993), in:

238

5 Ceramic Matrix Composites

Proc. 2nd Eur. Ceram. Soc. Mtg., Augsburg, in press. Takemura, H., Miyamoto, Y., Koizumi, M. (1987), Proc. Int. Conf. Hot Isostatic Pressing, Lulea, Sweden. Talreja, R. (1990), in: Proc. 11th Riso Int. Symp. Metall. Mater. Sci.: Bentzen, I J., Bilde-Sorensen, J. B., Christiansen, N., Horsewell, A., Ralph, B. (Eds.). Ris0, Danmark: Ris0 National Laboratories, pp. 145-159. Thouless, M. D., Evans, A. G. (1988), Acta Metall. 36, 517-522. Thouless, M. D., Sbaizero, O., Sigl, L. S., Evans, A. G. (1989), J. Am. Ceram. Soc. 72, 525-532. Tiegs, T. N., Becher, P. F. (1987), Am. Ceram. Bull. 66 (2), 339. Torrecillas, R., Gros, H., Rahali, S., Fantozzi, R., Rodriguez, M. A., Moya, J. S., de Aza, S. (1990), in: Proc. 11th Riso Int. Symp. Metall. Mater. Sci.: Bentzen, J. J., Bilde-Sorensen, J. B., Christiansen, N., Horsewell, A., Ralph, B. (Eds.). Riso, Danmark: Riso National Laboratories, pp. 497-504. Tsukuma, K., Takahata, T., Shiomi, M. (1988), Adv. Ceram. 24B, 721-728. Tuffe, S. (1991), Ph.D. thesis, INSA, Lyon, p. 274. Wahi, R. P., Ilschner, B. (1980), J. Mater. Sci. 15, 875-885. Wallace, J. S., Petzow, G., Claussen, N. (1984), Adv. Ceram. 12, 436-442. Wang, I , Stevens, R. (1988), /. Mater. Sci. 23, 804808. Warner, D. A., Warner, K. A., Juul Jansen, D., Sorensen, O. T. (1992), Proc. Composites Advanced Ceramics. Meeting of the Am. Ceram. Soc, Coco Beach, FL: American Ceramic Society. Wei, G. C , Becher, P. F. (1984), /. Am. Ceram. Soc. 65, 257-267. Wei, G. C , Becher, P. F. (1985), Am. Ceram. Soc. Bull. 65, C144-145. Yonezawa, T., Saitoh, S., Minamizawa, M., Matsuda, T. (1989), in: Proc. 1st. Jpn. Int. SAMPE Symp., Nov. 28-Dec.L Zawada, L. P., Butkus, L. M., Hartman, G. A. (1991), J. Am. Ceram. Soc. 74, 2851-2858.

General Reading Bentzen, J. J., Bilde-Sorensen, J. B., Christiansen, N., Horsewell, A., Ralph, B. (Eds.) (1992), Proc. 11th Rise Int. Symp. Metall. Mater. Sci., Riso, Danmark: Ris0 National Laboratories. Bradley, R. A., Clark, D. E., Larsen, D. C , Stiegler, J. O. (Eds.) (1988), Whisker and Fiber-Toughened Ceramics. Materials Park, OH: ASM International. Bunsell, A. H., Jamet, J. F, Massiah, A. (Eds.) (1992), in: Developments in the Science and Technology of Composite Materials, ECCM5. Bordeaux: EACM. Chermant, J. L., Fantozzi, G. (1993), "Microstructure, comportements thermomecaniques et modelisation des composites ceramiques-ceramiques a fibres", in Revue des composites et des materiaux avances. Paris: Hermes. Claussen, N., Riihle, M., Heuer, A. H. (Eds.) (1984), Advances in Ceramis, Vol. 12: Science and Technology of Zirconia II. Columbus, OH: American Ceramic Society. Davidge, R. W. (Ed.) (1985), Survey of the Technological Requirements for High Temperature Materials R&D. Section 3: Ceramic Composites for High Temperature Engineering Applications, Luxembourg: Commission of the European Communities. Green, D. J., Hannink, R. H. X, Swain, M. V. (Eds.) (1989), Transformation Toughening of Ceramics, Boca Raton, FL: CRC Press. Harris, B. (Ed.) (1990), Ceramic Matrix Composites, Guest Ed.: Naslain, R.: Composites Science and Technology, Vol. 37, Essex: Elsevier Applied Science. Heuer, A. H., Hobbs, L. W. (Eds.) (1981), Advances in Ceramics, Vol. 3: Science and Technology of Zirconia. Columbus, OH: American Ceramic Society. Naslain, R., Lamalle, X, Zulian, X L. (Eds. (1990), Materiaux composites pour applications a hautes temperatures. Bordeaux: AMAC/CODEMAC.

6 Interfaces in Composites Jang-Kyo Kim and Yiu-Wing Mai Department of Mechanical Engineering, Centre for Advanced Materials Technology, University of Sydney, Sydney, NSW, Australia

List of 6.1 6.2 6.2.1 6.2.1.1 6.2.1.2 6.2.1.3 6.2.1.4 6.2.1.5 6.2.1.6 6.2.2 6.2.3 6.2.3.1 6.2.3.2 6.2.3.3 6.2.3.4 6.2.3.5 6.2.3.6 6.2.3.7 6.2.3.8 6.3 6.3.1 6.3.2 6.3.2.1 6.3.2.2 6.3.2.3 6.3.3 6.3.4 6.3.4.1 6.3.4.2 6.3.4.3 6.4 6.4.1 6.4.2 6.4.3

Symbols and Abbreviations Introduction Load Transfer at the Interface Theories of Adhesion Adsorption and Wetting Interdiffusion Electrostatic Attraction Chemical Bonding Reaction Bonding Mechanical Bonding Mechanics of Load Transfer at the Interface Characterization of Interface Properties Short-Beam Shear Test Iosipescu Shear Test Transverse Tensile Test Single-Fiber Compression Test Single-Fiber Fragmentation Test Fiber Pull-Out Test Fiber Push-Out (or Indentation) Test Slice Compression Test Surface Treatments of Fibers Glass Fibers (GFs) Carbon Fibers (CF's) Types of CF Surface Treatment Effects of Surface Treatment on Composite Properties Effects of Surface Treatment on Fiber Properties Aramid Fibers Other Fibers Alumina (A12O3) Fibers Boron (B/W) Fibers Silicon Carbide (SiC) Fibers Interface-Related Fracture Toughness Theories Fiber-Matrix Debonding Post-Debond Friction Stress Redistribution

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

241 244 244 245 245 245 246 246 246 248 248 249 250 250 251 251 252 254 256 257 257 258 261 261 263 263 265 266 267 267 268 269 270 271 271

240

6.4.4 6.4.5 6.4.6 6.5 6.5.1 6.5.1.1 6.5.1.2 6.5.2 6.5.3 6.6 6.6.1 6.6.2 6.7 6.8 6.9

6 Interfaces in Composites

Fiber Pull-Out Total Fracture Toughness Theory Fracture Toughness Maps Toughening Methods with Controlled Interfaces Fiber Coating and Intermittent Bonding Concept Intermittent Bonding Concept Enhanced Energy Absorption: Fiber Coating Technique Delamination Promoters Delamination Arresters Residual Stresses Origin of Residual Stresses Control of Residual Stresses Concluding Remarks Acknowledgements References

271 272 273 273 274 275 276 277 278 279 279 281 283 283 284

List of Symbols and Abbreviations

241

List of Symbols and Abbreviations a b b d £f,£m Glu Gic Gm K / /0 lc ld J po L m P Pmax q0 Q, Q' R Rd i?df Rf Rm Rpo Rv Rs Rt t Tg V{ Wa z

equivalent radius of the fiber equivalent radius of the matrix width of specimen in a shear test fiber diameter Young's moduli of fiber and matrix mode I and mode II interlaminar fracture toughness interfacial fracture toughness shear modulus of the matrix coefficient which depends on the variation of fiber fragment length and strength fiber length characteristic fiber length critical transfer length debond length fiber pull-out length embedded fiber length shape factor (Weibull modulus) applied load in a shear test applied maximum load (short-beam shear test) residual clamping stress non-dimensional coefficient in fiber pull-out and push-out fracture toughness interfacial debond toughness post-debonding friction toughness fracture toughness of fiber fracture toughness of matrix fiber pull-out toughness stress redistribution toughness specific surface energy total fracture toughness thickness of specimen in a shear test glass transition temperature volume fraction of a fiber work of adhesion fiber direction

af, am F ysv>yLv>7sL As £f 2m X

CTE of fiber and matrix gamma function surface energies of solid-vapor, liquid-vapor and solid-liquid interfaces differential strain of fiber and matrix fiber (fracture) strain matrix (fracture) strain reciprocal length giving effective frictional shear stress transfer

242

A \x vf, vm G0
6 Interfaces in Composites

parameter which represents the m e a n aspect ratio (l/l0) coefficient of friction Poisson ratio of fiber and matrix frictionless d e b o n d stress asymptotic debond stress applied compressive stress tensile b o n d stress composite strength (fiber) d e b o n d stress partial debond stress stress at the end of a fiber fiber axial stress fiber tensile strength at the m e a n fragment length initial frictional pull-out stress fiber tensile strength at the critical transfer length average fiber axial stress (tensile) strength of the matrix net-section compressive stress transverse strength of composites apparent interfacial shear b o n d stress (strength) interfacial shear b o n d stress (strength) interfacial friction shear stress (strength) interfacial shear stress maximum shear stress shear strength of the matrix Auger electron spectroscopy y-aminopropyltriethoxysilane boron fiber-epoxy matrix composites boron nitride carbon fiber carbon fiber-thermoset matrix composites carboxyl-terminated butadiene acrylonitrile coefficient of thermal expansion chemical vapor deposition fracture process zone Fourier transform infrared (spectroscopy) glass fiber glass fiber-thermoset matrix composites interlaminar shear strength interpenetrating network ion scattering spectroscopy Kevlar fiber-thermoset matrix composites JV-methylaminopropyltrimethoxysilane polyacrylonitrile

List of Symbols and Abbreviations

PBT PEEK PMMA PUV PVAL RoM SEM SIMS STM SVF TEM VS XPS

polybutyleneterephthalate polyetheretherketone polymethylmethacrylate polyurethane varnish polyvinyl alcohol rule of mixtures scanning electron microscopy scanning secondary ion mass spectroscopy scanning tunneling microscopy silicone vacuum fluid transmission electron microscopy vinyltriethoxysilane X-ray photoelectron spectroscopy

243

244

6 Interfaces in Composites

6.1 Introduction In fiber composites, both fiber and matrix retain their physical and chemical identities, yet they produce a combination of mechanical properties that cannot be achieved with either of the constituents acting alone, due to the presence of an interface between these two constituents. A classical definition of the interface in fiber composites is that "an interface is a surface formed by the common boundary of reinforcing fiber and matrix in contact which constitutes the bond in-between for transfer of loads. It has physical and mechanical properties which are unique and not either those of the fiber nor the matrix (Metcalfe, 1974)." An interface is usually considered to be zero thickness (or volume) and perfectly bonded. The concept of a two-dimensional interface has now been replaced by that of a three-dimensional region more properly termed an "interphase" (Drzal et al, 1983 a). This interphase includes the surface of the classical fiber-matrix contact as well as the region of finite thickness extending therefrom of which chemical, physical and morphological features are different from the bulk material. Imposed on this region are the processing conditions which allow chemical reactions, residual stresses and volumetric changes to take place. It is this latter definition of an interface that is used throughout this chapter. For analytical purposes the interface is still conveniently considered to be infinitely thin. To control the interface which can provide the composite with improved mechanical performance and structural integrity, it is essential to understand the mechanisms of adhesion (Sec. 6.2.1) which are specific to each fiber-matrix system and the mechanics of load transfer at the interface (Sec. 6.2.2). A number of theoretical and

experimental methods developed to assess the interface quality (in terms of bond strength in tension and shear) are summarized along with their applicability and limitations (Sec. 6.2.3). Particular emphasis is placed on the various techniques for fiber surface treatments on a range of technologically important fibers to improve bond strength as well as to enhance fiber-matrix compatibility and stability during processing or fabrication of the composites (Sec. 6.3). A review of the micro-failure mechanisms and their associated theories of fracture toughness of fiber composites (Sec. 6.4) shows that a high bond strength does not usually lead to a high fracture toughness. A compromise always has to be made in the bond strength to optimize the strength and toughness. Novel methods to improve the fracture toughness for both transverse and interlaminar fractures using controlled interfaces are described (Sec. 6.5). In particular, the effects of residual stresses are discussed (Sec. 6.6). Characterization of transverse and interlaminar fracture toughness in terms of the crack growth resistance curve (R-curve) as it may be affected by fibre-matrix and laminar interfaces is not discussed here. The interface/interphase behavior of semicrystalline thermoplastic composites is examined in Chap. 8 of this Volume.

6.2 Load Transfer at the Interface The structure and properties of the fibermatrix interface play a major role on the mechanical performance and structural integrity of composite materials. From the load-transfer mechanics viewpoint experimental and theoretical analyses dealing with the stress state at the interface are of special interest. In this section, the nature or origin of the bonding between fiber and

6.2 Load Transfer at the Interface

matrix is discussed in terms of the theories of adhesion. Examples of specific fiber-matrix systems are provided along with their corresponding mechanisms of adhesion. The micro-mechanics of load transfer based on a simple shear lag model is presented with relevant stress distribution solutions. Various experimental techniques which have been devised to assess the interfacial properties are also discussed with corresponding analytical models for evaluation of the experimental data. Advantages and limitations of each method are also presented. 6.2.1 Theories of Adhesion

Since the nature of the bonding is dependent on the atomic arrangement and chemical properties of the fiber as well as on the molecular conformation and chemical constitution of the matrix and diffusivity of elements in each constituent, it follows that the interface is specific to each fiber-matrix system. Adhesion can be attributed to the mechanisms described below which can occur at the interface region either in insolation, or most likely, in combination to produce the final bond. Reviews on these major mechanisms can be found in many references including Scolar (1974), Wake (1978), Hull (1981) and Metcalfe (1974), the last being particularly useful for metal matrix composites. 6.2.1.1 Adsorption and Wetting

Wetting of the fibers by the matrix during fabrication processes of composites is very important particularly for polymer matrix and metal matrix composites using molten metals. Bonding here involves short-range interactions of electrons on an atomic scale, which only develop when the atoms of the constituents approach within a few atom diameters of each other. Wet-

245

ting can be expressed in terms of the thermodynamic work of adhesion Wa which represents a physical bonding resulting from highly localized intermolecular dispersion forces between different phases: Wa = ySY + yLY — ySL

(6-1)

where y sv , yLV and ySh are the surface energies of the solid-vapor, liquid-vapor and solid-liquid interfaces respectively. The surface energy of the fiber (ysv) should be greater than that of the matrix (yLY) for proper wetting to occur. Thus, glass and carbon fibers (with surface energies 560 mJ/m 2 and 70 mJ/m2) can be readily wetted by thermoset resins like epoxy and polyester (with surface energies 43 mJ/m 2 and 35 mJ/m2) unless the viscosity of the resin is too high (Hull, 1981). However, it is difficult to wet polyethylene fibers (of surface energy 31 mJ/m2) with these resins unless the fibers are surface treated. For the same reason, carbon fibers are often coated with Ti-B (Amateau, 1976) using a chemical vapor deposition process to allow wetting by an aluminium (Al) matrix. 6.2.1.2 Interdiffusion

A bond between two surfaces may be formed by interdiffusion of atoms or molecules across the interface. The bond strength in polymer matrix composites will depend on the amount of molecular entanglement, the number of molecules involved and the strength of the bonding between the molecules. Interdiffusion may be promoted by the presence of solvents and the amount of diffusion will depend on the molecular conformation, constituents involved and the ease of molecular motion. For example, bonding between glass fibers and polymer resins through silane coupling agents by other than chemical bonding can be explained by inter-diffusion and

246

6 Interfaces in Composites

interpenetrating network (IPN) formation at the interface region (Plueddemann, 1974; Ishida and Koenig, 1978; Plueddemann and Stark, 1980). The interface region thus formed has a substantial thickness, and its chemical, physical and mechanical properties are different from those of either the bulk fiber and matrix (Garton and Daly, 1985; Thomason, 1990; Tsai etal., 1990; Williams et al., 1990). In metal matrix composites, interdiffusion is also necessary for proper reaction between elements of each constituent to take place. However, interdiffusion here may not always be beneficial because undesirable compounds are often formed, particularly when the oxide films present on the fibers are completely disrupted under extremely high temperature and pressure in a solid state process (Metcalfe, 1974; Whatley and Wawner, 1985). To prevent or reduce the interaction it is necessary to apply an effective diffusion barrier in the form of a coating on the fiber. The selection of an appropriate diffusion barrier relies on a detailed knowledge of the nature of the interaction which is specific to each system. 6.2.1.3 Electrostatic Attraction

A difference in electrostatic charge between constituents at the interface may contribute to the force of attraction bonding. The strength of the interface will depend on the charge density. Although this attraction is unlikely to make a major contribution to the final bond strength of the interface it could be important when the fiber surface is treated with some coupling agents. The bonding of this type will explain why silane finishes are especially effective for certain acidic or neutral reinforcements like glass, silica and alumina, but are less effective with alkaline surfaces like magnesium, asbestos and calcium carbonate (Plueddemann, 1974).

6.2.1.4 Chemical Bonding

The chemical bonding theory is the oldest and best known of all bonding theories. A chemical reaction at the interface is of particular interest for polymer matrix composites because it offers the major explanation for the use of coupling agents on glass fibers (GF's) and probably the surface oxidative treatments on carbon fibers (CF's) for application with most thermoset and amorphous thermoplastic matrices. A bond is formed between a chemical grouping on the fiber surface and a compatible chemical group in the matrix and the bond strength depends on the number and type of bonds, the formation of which are usually thermally activated chemical reactions. The chemical composition of the bulk fiber and of the surface for several widely used fiber systems are given in Table 6-1. It is interesting to note that except for GF, the chemical composition of the surface does not resemble that of the bulk fiber, and oxygen is common to all fiber surfaces. Further details on the types of surface treatments commonly applied to each fiber and their effects on the properties of the interface and composites are given in Sec. 6.3. 6.2.1.5 Reaction Bonding

In other than polymer matrix composites, chemical reaction between elements of constituents takes place in different ways. Reaction occurs forming new compound(s) at the interface region in metal matrix composites, particularly those manufactured in a molten metal infiltration process. Reaction involves transfer of atoms from one or both of the constituents to the reaction site near the interface and these transfer processes are diffusion controlled. Special cases of the reaction bond include the exchange reaction bond and

6.2 Load Transfer at the Interface

247

Table 6-1. Elemental composition of fibers. After Scola (1974). Fiber

E-glass

Bulk Analysis

Probable functional group

Si, O, Al, Ca, Mg, B, F, Fe, Na

Si, O, Al

-Si-OH, -Si-O-Si

W 2 B 5 , WB 4 (inner core), B (outer core)

B 2 O 3 as methyl borate

B-OH, B O B

Boron (B/W core) Silicon carbide (SiC/W core) Carbon

Surface

Si, W (inner core), C (outer core), O, N

Si, C

Si-O-Si, Si-OH

C, O, N, H, metal impurities

C, O, H

-COOH, C-OH, C = O

Table 6-2. Classification of fiber-metal matrix composite systems. After Metcalfe (1974). Class I

Class II

Class III

Tungsten-copper Alumina-copper Alumina-silver Boron-BN coating Boron-magnesium Boron-aluminum Stainless steel-aluminum Silicon carbide-aluminum

Tungsten-copper(chromium) Tungsten-columbium Carbon-nickel Tungsten-nickel

Tungsten-copper( titanium) Carbon-aluminum (> 700 °C) Alumina-titanium Boron-titanium Silicon carbide-titanium Silica-aluminum

the oxide bond. The exchange reaction bond occurs when a second element in the constituents begins to exchange lattice sites with the elements in the reaction product or compound, while the oxide bond occurs between oxide films present in the matching surfaces of fiber and matrix. The reaction bond makes a major contribution to the final bond strength of the interface for some metal matrix composites, depending on the fiber-matrix combination (which determines the diffusivity of elements from one constituent to the other) and the processing conditions (particularly temperature and exposure time). A general scheme for the classification of interfaces in metal matrix composites can be made based on the chemical reaction occurring between fiber and matrix according to Metcalfe (1974) and Table 6-2 gives exam-

ples of each type. In class I the fiber and matrix are mutually nonreactive and insoluble in each other; in class II the fiber and matrix are mutually nonreactive but soluble in each other; and in class III the fiber and matrix react to form compound(s) at the interface. Clear-cut definitions between the different classes are not always possible, but the grouping provides a systematic background to evaluate their characteristics. For the pseudo-class-I composites, which include B-Al, stainless steel-Al and SiC-Al, interaction hardly occurs in solid state diffusion bonding, but reaction does occur when Al matrix is melted for liquid infiltration. In general, in most ceramic matrix composites, chemical reaction hardly occurs between fiber (or whisker) and matrix. However, an extremely thin amorphous

248

6 Interfaces in Composites

film can be formed originating from the oxide present on the fiber surface (Sarin and Ruhle, 1987) or due to the limited fiber-matrix reaction, e.g., between alumina whisker and zirconia matrix (Becher and Tiegs, 1987). 6.2.1.6 Mechanical Bonding

Mechanical bonds involve solely mechanical interlocking at the surface. The strength of this interface is unlikely to be very high in transverse tension unless there are a large number of re-entrant angles on the fiber surface, but the strength in shear may be significant depending on the degree of roughness. In addition to the simple geometrical aspects of mechanical bonding, there are many internal or residual stresses in composite materials which develop during fabrication processes due to matrix shrinkage and differential thermal expansion between fiber and matrix. Among these stresses, the residual clamping stress on the fiber provides a major bonding at the interface of many ceramic matrix composites which plays a decisive role in controlling the fracture resistance of these materials. Further details of these residual stresses are discussed in Sec. 6.6. In addition to the major mechanisms described above, hydrogen bonding, van der Waals forces and other low energy forces may also be involved. 6.2.2 Mechanics of Load Transfer at the Interface

A phenomenon of great importance in composites is the transfer of load between fiber and matrix in the neighborhood of either a fiber break or a matrix crack. A number of models have been developed to predict the response of composite materials in terms of macromechanical, thermal and microfracture properties under vari-

ous loading and environmental conditions (Taya and Arsenault, 1989). The shear lag model (Cox, 1952) is most widely used to study the micromechanics of load transfer at the interface, particularly the stress distribution near the ends of a broken fiber. It is based on the assumptions that the fiber and matrix are elastic, the bond at the infinitely thin interface is perfect, and the fibers are arranged in a regular array. These assumptions are necessary to obtain simple approximate solutions. Consider a fiber of length / embedded in a matrix which is subjected to a strain 8m at its remote ends as shown in Fig. 6-1 (a). By assuming the differential displacement between fiber and matrix in the fiber direction is directly proportional to the shear stress at the interface, the fiber axial stress a{ and shear stress T{ at the interface are obtained along the fiber (z-) direction: [{(jJEfsJ-l]coshpz cosh OS//2) (6-2) [l-(a e /£ f 8 m )]sinh/?z cosh (j81/2) (6-3) where 2Gn 2 Efa ln{b/a)

1/2

(6-4)

ae is the fiber stress at its ends (z = ± 1/2% E{ and Em are Young's moduli offiberand matrix, Gm is shear modulus of the matrix, and a and b are equivalent radii of fiber and matrix respectively. The stress distributions of crf and Tj along the fiber length are illustrated in Fig. 6-1 (b). The tensile stress is a maximum in the center of the fiber while the shear stress is a maximum at the fiber ends and falls to almost zero in the center. However, cre may not be zero in the case of strongly bonded fiber ends (Nar-

6.2 Load Transfer at the Interface

(a)

249

of this type is not accurate nor completely adequate to predict the gross mechanical performance of composites for a given loading condition, it has provided a firm basis to help understand the micromechanics of load transfer at the interface of most fiber-matrix systems, and determine the interfacial properties in fiber fragmentation, fiber pull-out and fiber push-out (or indentation) tests.

(b)

/ c /2

6.2.3 Characterization of Interface Properties

0

Figure 6-1. (a) Schematic representation of deformation around a discontinuous fiber embedded in a matrix subjected to an axial loading, (b) Variation of fiber axial stress af and interfacial shear stress T{. After Hull (1981).

done and Prewo, 1986). The results shown in Fig. 6-1 (b) imply that there are regions at or near the fiber ends which do not carry the full load so that the average stress in the fiber of finite length / is always less than that for a continuous fiber subjected to the same external loading. The average axial fiber stress crf is given by: 1/2

(6-5)

J 0

iPl/2) The average fiber stress and thus the reinforcing efficiency decreases as the fiber length decreases because a greater proportion of the fiber length is not fully loaded. To achieve the maximum stress the fiber length should be longer than a critical value /c (Kelly and Tyson, 1965). The implication of the critical fiber length lc in evaluating interfacial properties is discussed in Sec. 6.2.3. Although the shear lag analysis

Many experimental techniques have been devised to assess the interface properties in fiber composites and some early works are included in reviews by Broutman (1969), Plueddemann (1974) and Hull (1981). Generally, there are two approaches depending on the nature of specimens employed and the scale of testing: one uses bulk materials in the form of laminates and the other involves testing in a microscopic scale on individual (sometimes multiple) fibers embedded in specially constructed matrix material of various shapes. The first approach uses invariably laminated composites (reinforced mainly with continuous fibers) measuring inplane or translaminar shear properties. While many different test geometries are available in the literature (e.g., off-axis tension, rail-shear, cross beam sandwich, thin-walled tube torsion, short beam shear, Iosipescu shear, etc., see Lee and Munro (1986) for details of some of these test methods), the results obtained from these tests are often inconsistent because the state of shear stress in the test sections of the specimens is neither pure nor uniform in most cases. Apart from the shear tests there is another simple test method to measure the bond quality of composites, namely the transverse tension test. The test methods in the second ap-

250

6 Interfaces in Composites

proach include the single fiber compression, fiber fragmentation, fiber pull-out, fiber push-out (or indentation) tests with a variety of specimen geometries and scales involved. The slice compression test is also finding increasing use for ceramic matrix composites. In the following the characteristic features of several popular test methods are discussed. 6.2.3.1 Short-Beam Shear Test

The short-beam shear test specified in ASTM D 2344 uses three-point bending where failure occurs in a shear mode parallel to the fibers in the midplane as shown in Fig. 6-2 (a). The maximum shear stress rmax (normally called the interlaminar shear strength, ILSS) is related to the applied maximum load Pmax and specimen dimensions, according to the simple elastic beam theory: rmax = 3 PmaJ4 b t, where b and t are width and thickness of specimen. It is noted that even in the absence of any bond strength at the fiber-matrix interface, ILSS of the composite still has a lower-bound value which is contributed solely by the shear strength of the matrix r m . By considering a brittle matrix with cylindrical pores (in place of fibers of volume fraction V{ in square array) ILSS is given approximately by Tm[l - (4 Vf/n)1/2]. Because of the simplicity of the test method and minimum complication in specimen preparation, it has become one of the most widely used methods to determine the quality of both polymer and metal matrices composites, and in particular to assess the effects of fiber finish and surface treatments, fibermatrix compatibility for the development of new fiber or matrix systems, etc. Unfortunately, this test has an inherent problem in that meaningful results cannot be obtained for very thin specimens. The high stress concentration and damage by crush-

(c)

j2/t P/2

P/2

(b)

Loading fixture

Specimen

Figure 6-2. Schematics of specimen geometry and loading configuration for (a) short-beam shear test, (b) Iosipescu shear test and (c) transverse tensile test.

ing in severe cases at the loading points with a very short span (normally five times the specimen thickness) may induce premature failure and complicates the interpretation of test results. To avoid this problem four-point bending of a long beam may be used. 6.2.3.2 Iosipescu Shear Test

The Iosipescu shear test originally developed for use with isotropic structural materials such as metals employs a doubleedge notched specimen which is subjected to two counteracting force couples (Iosipescu, 1967) (Fig. 6-2 (b)). A state of almost constant and pure shear can be achieved across the specimen section between the notches by selecting a proper combination of notch angle and notch depth (typically 90° and 22% of the full width (Adams,

6.2 Load Transfer at the Interface

251

1990)). The average shear stress in the notched section is simply given by the applied load P, divided by the cross-sectional area of the section: % = P/b t. However, many experimental and numerical studies have shown that there is a substantial stress concentration near the notch tip in orthotropic materials including most fiber composites, as opposed to isotropic materials where the stress concentration is almost negligible. The stress concentration is found to be a function of orthotropic ratio (i.e., Young's moduli ratio between two principle inplane directions which is governed by fiber orientation and volume fraction) and notch geometry, and can be reduced by incorporating a notch tip radius with a large notch angle depending on the type of composites (Adams and Walrath, 1987). The major advantage of this test is that there is a large region of uniform shear compared to the other shear tests including the short-beam shear test. It can measure both the maximum shear strengths and shear moduli in directions parallel and normal to the fiber with high accuracy and reproducibility.

brittle matrix with cylindrical pores, i.e., a ± = am [1 - (4 Vf/n)112]. For a good interface bond the transverse strength will increase accordingly due to the constraint of matrix material between reinforcing fibers. Since the transverse strength is governed by many factors including the properties of constituents, the interface bond strength, the presence and distribution of voids and foreign objects, etc., the test results may sometimes be misleading unless care is taken in interpretation. Details on the strength prediction and its relation with interface bond strength are given by Hull (1981). Both the ILSS and transverse tensile strength measured by the foregoing three test methods depend significantly on the fiber Vf. Therefore, they cannot be regarded as giving the true values of the bond strength. In particular, the mode of fracture in the transverse tension test may relate only indirectly to the interfacial bond. The significance of these test methods is that they provide some measure of the relative bond strength of different fibermatrix composite systems.

6.2.3.3 Transverse Tensile Test

6.2.3.4 Single-Fiber Compression Test

The transverse tensile test assesses the quality of composites by measuring the tensile strength of unidirectional fiber composites transverse to the fiber direction (Fig. 6-2 (c)). The most obvious feature of this test is that its strength is usually less than the strength of the matrix am so that in contrast to the effect of fibers on the transverse modulus (which is higher than that of unreinforced matrix) the fibers have a negative reinforcing effect. In the case of negligible interface bond, the lower-bound transverse strength o ±_ can also be estimated in the same way as for the interlaminar shear strength by considering a

Early work on direct measurements of interface bond strength for glass fiber composites employs specimens with a single fiber embedded completely in a transparent polymer resin matrix (Mooney and McGarry, 1959). The parallel sided specimen illustrated in Fig. 6-3 (a) represents a block of resin with uniform cross-section containing a fiber along the central axis. When the specimen is compressed in the longitudinal direction shear stresses are created at the fiber ends because of the difference in elastic properties between the fiber and matrix (see Eqs. (6-2) and (6-3) for tensile loading). The curved neck specimen

252

6 Interfaces in Composites

6.2.3.5 Single-Fiber Fragmentation Test

(a)

(b)

Figure 6-3. Single-fiber compressive tests with (a) parallel-sided and (b) curved-neck specimens.

shown in Fig. 6-3 (b) under compression produces tensile debonding rather than shear failure. A compressive longitudinal load on this specimen causes a radial expansion governed by the matrix Poisson ratio vm which is greater than that of the fiber vf. Therefore, the matrix expands transversely more than the fiber and an interfacial tensile stress is created. Solutions for the shear bond stress i b and tensile bond stress ah are (Broutman, 1969):

The single-fiber fragmentation test is based on the early work of Kelly and Tyson (1965) and was later given decisive impetus by Drzal and coworkers (Drzal etal., 1980; 1983a; 1983b). In this test, a dog-bone shaped specimen consisting of an isolated fiber embedded completely in a matrix is loaded in tension (or in compression (Boll et al., 1990)) (Fig. 6-4 (a)). As the applied load increases the embedded fiber breaks into increasingly smaller segments and load transfer occurs between the broken fibers and the matrix until the segments become too short to be broken. Fig. 6-4 (b) illustrates the schematic representation of fiber fragmentation under progressively increasing load and the corresponding fiber axial stress ai profile. The

(a)

(6-6)

Tb « 2.5 (7A

for the shear model and >-v f ) a(l-vf-2vf2)

(6-7)

for the tensile model, a = EJEf is Young's moduli ratio of matrix to fiber, <JA is applied compressive stress and aN is netsection compressive stress at the neck when debond occurs. The coefficient 2.5 in Eq. (6-6) arises from the empirically measured shear stress concentration factor. Both tests are inherently difficult and unreliable to measure accurate reproducible results because of the problems associated with specimen preparation which requires precise alignment of the fibers and visual detection of the onset of debonding.

(b)

4°' /I V

ii)

A

A

iii)

i

A

V

V

1

A/

V

h V

4 1

Figure 6-4. (a) Dog-bone shape fiber fragmentation test specimen; (b) fiber fragmentation under progressively increasing load from i) to iii) and the corresponding fiber axial stress o{ profile.

6.2 Load Transfer at the Interface

saturated fiber length corresponds to the critical transfer length lc which allows calculation of the apparent shear bond strength at the interface t a using the relationship (Kelly and Tyson, 1965):

daf

(6-8)

where of is fiber failure stress at the critical length and d is fiber diameter. The fragment length can be determined in transparent matrix composites with conventional optical, photoelastic (under polarized light) and acoustic emission techniques (Netravali et al., 1989 a). The photoelastic technique has an advantage over the optical method in that the spatial distribution of stresses can be examined in the matrix in the region around the fiber and near its broken ends. The acoustic emission technique can be easily adopted to non-transparent matrix materials including metal and ceramic matrix composites. For non-transparent polymer matrix composites, the matrix material can be dissolved chemically or burned off to obtain fragmented fibers. Alternatively, the specimen after loading can be polished to expose the fibers for metal matrix composites (Yang et al., 1991). A significant limitation of this test is that the matrix tensile strain must be much larger than that of the fiber. More importantly, the validity of i a determined based on the measurement of fragment length depends not only on r b but strongly on the properties of the constituents (such as matrix shear yield strength i m and the difference in Poisson ratios between fiber and matrix) which influence the failure mode (i.e., interfacial debonding versus fiber fragmentation without debonding). Bascom and Jensen (1986) argued that the shear stress transfer in this test is often limited by Tm of the matrix rather than xh of the interface. Fiber structural properties may also

253

cause complications in the interpretation of test results (e.g., splitting of Kevlar fibers into small fibrils within fiber surface (Kalantar and Drzal, 1990)). Therefore, the justification of using Eq. (6-8) to calculate i a from experimental lc values rather than calculating /c from the known t m as originally proposed is questionable since the equation requires a fully plastic matrix to be valid. Another important fact is that the actual fragment lengths and the fiber strength are not constants for a given composite system but are strongly dependent on the test gauge length. This requires Eq. (6-8) to be modified by:

*» = KYf

I6"9)

where
(6-11)

where the non-dimensional parameter A represents the mean aspect ratio (///0), IQ is the characteristic length varying between 1.34 (for m -• oo) and 1.96 (for m = 3 which is typically the smallest value in experiment). Therefore, K varies between 0.67 for very large m and 0.97 for m = 3.

254

6 Interfaces in Composites

6.2.3.6 Fiber Pull-Out Test

The fiber pull-out test uses a straight fiber(s) a portion of which is embedded in a matrix block of different geometries as shown in Fig. 6-5. This test measures the fiber force required to break the interfacial bond as well as to pull the fiber out against the frictional resistance after complete debonding as a function of embedded fiber length L (Fig. 6-5 (a)). Therefore, proper evaluation of all the relevant properties of the interface in a given system needs experimental results for a wide range of L. The permissible L is usually very short causing experimental difficulties and large data scatter especially for composite systems with high Tb and small fiber d. For example, for the carbon fiber-epoxy matrix system the maximum allowable L is less than 1.0 mm (Pitkethly and Doble, 1990; Marshall and Price, 1991; Kim et al., 1992 a). It is necessary to design a special jig to produce and hold the thin matrix block. Moreover, it is hard to avoid forming an elevated meniscus around the fiber which causes the evaluation of test results inaccurate. Therefore, to avoid this problem, a new approach has been devised by Miller etal. (1987) and Gaur and Miller (1989) where a small amount of liquid resin has been applied onto the fiber forming a concentrical microdroplet in the shape of an

V/////A V7///A

X7777\ V//A

(a)

(b)

(c)

Figure 6-5. Various experimental configurations of fiber pull-out tests: (a) disc-shaped, (b) micro-debond and (c) double pull-out with multiple embedded fibers.

ellipsoid after curing (Fig. 6-5 (b)). The microdebond technique has also been used for thermoplastic matrices (McAlea and Besio, 1988; Chuang and Chu, 1990). The fiber pull-out test has been extensively used not only for polymer matrix composites but also for some ceramic matrix (Goettler and Faber, 1988; Deshmukh and Coyle, 1988; Griffin et al., 1988; Buttler et al., 1990) and cement matrix composites (see Bartos (1981) for a useful review) as well as steel wire reinforced rubber matrix composites (Ellul and Emerson, 1988; Gent and Kaang, 1989). For cement matrix composites multiple fibers are preferred to the single fiber arrangement (Fig. 6-5 (c)) (Naaman and Shah, 1976; Mandel etal., 1987). Theoretical analysis of debonding and fiber pull-out processes were initially developed for ductile matrices (e.g., tungsten wire-copper matrix) assuming a uniform interface bond strength Tb or matrix shear strength Tm (Kelly and Tyson, 1965). However, for brittle matrix composites the distribution of interface shear stress x{ is not uniform and the functional dependence of the external stress for debonding on L is nonlinear. Theoretical relations for the debond and pull-out stresses and L have been obtained from two distinct approaches. One is based on a maximum shear stress criterion such that when T{ exceeds t b complete debonding occurs unstably; the other is based on the concept of fracture mechanics where a debonded zone is considered as an interfacial crack and its extension is dependent on a fracture energy criterion being satisfied. Both approaches use the shear lag model where a fiber is located in the center of a coaxial cylindrical shell of matrix, resembling the experimental specimen geometry. Since Gray's (1984) review on the shear strength approach to this problem there

6.2 Load Transfer at the Interface

have been significant recent advances. In particular, Hsueh (1988, 1990 a) modified the debond criterion and postulated a progressive stable debond including the effect of shear deformation in the matrix. Leung and Li (1990) studied a "two-way debonding" model suggesting that debond initiation may start from the embedded fiber remote end. This is different from the conventional analysis which assumes debond crack propagation only from the free fiber end. Other investigations include Banbaji (1988) who considered varying residual clamping stress along the fiber length and a thermomechanical analysis given by Sigl and Evans (1989). The fracture mechanics approach includes the work of Gurney and Hunt (1967) and Outwater and Murphy (1969) in which the rate of potential energy released from the fiber for complete debonding of embedded fiber length L is equated to the incremental interfacial fracture energy (which is the product of interface fracture toughness Gic and cylindrical debond area na2L). Therefore, a constant debond stress od is obtained in the fiber: Gd = 2[EfGJa]1/2. Later, Stang and Shah (1986) derived a debond criterion based on a compliance analysis, and Wells and Beaumont (1985 a) considered the effect of nonlinear friction in the debonded region and Poisson contraction of the fiber. Piggott (1987) equated the total elastic strain energy of fiber axial and matrix shear deformations in the bonded region directly to the interfacial fracture energy (which is not an appropriate fracture mechanics approach), proposing that the debond stress ad is approximately proportional to the embedded fiber length L or L1/2 for very short and long L, respectively. Penn and Lee (1989) cor-

255

rected Piggott's solution by including the energy stored in the debonded region and applying a proper rate of energy change with crack growth analysis. Recently, Gao et al. (1988) presented a new debond criterion, based on assumptions similar to those made previously (Wells and Beaumont, 1985 a). However, its significance is that once the relationship between the debond and post-debond frictional pullout stresses and L is established from experiments the model is able to determine the inherent interfacial properties including interfacial fracture toughness Gic, coefficient of friction \i and the residual clamping stress q0 at the debonded region of the interface (Kim et al., 1992 a). Other recent studies using fracture mechanics approach include those of Pally and Stevens (1989) and Hutchinson and Jensen (1990); the latter investigators also considered the effect of residual stresses in the fiber axial direction in a thermomechanical analysis. According to Gao et al. (1988) the partial debond stress ovA during stable debonding consists of two components: a frictionless debond stress component o0 and a friction stress component which is directly proportional to (a — o0) (Kim et al., 1991, 1992 a): <7pd = °o + (°-°o)Q

(6-12)

(7 is the asymptotic debond stress for long L which is related to q0 by:

where k = ( a Vf + y vj/{a(l - vf) + 1 + vm + 2y} and y = a2/(b2 — a2). The frictionless debond stress a0 is a function of Gic given by:

4 Ef Gic [(1 - 2 k vf) + (y/a) (1 - 2 k v j f l 1 / 2 a(l-2kv()

[ 4 Ef G{ a(l-2fcv f )

1/2

(6-14)

256

6 Interfaces in Composites

In a more exact fracture analysis taking into consideration the strain energy stored in the bonded region (Zhou et al., 1992), <J0 is shown to vary with debond length ld relative to total embedded fiber length L. The non-dimensional coefficient Q is: Q « 1 — exp (— X ld)

(6-15)

where X is the reciprocal length giving the effective frictional shear stress transfer and is related to the coefficient of friction fi by Therefore, depending on L for a given composite system, the maximum debond stress can be obtained at instability (Kim et al., 1991). The solution for the initial frictional pull-out stress oir after complete debonding can be determined by substituting <70 = 0 in Eq. (6-12) and the debond length by the total embedded length (i.e., ld = L) for the non-dimensional coefficient Qas: °fr = °Qld = L

(a)

(6-16)

6.2.3.7 Fiber Push-Out (or Indentation) Test

The identation technique (or push-out test as opposed to the pull-out test) utilizes a micro-hardness indenter of various tip shape and size to apply a force to push against the fiber end into the polished surface of a matrix block (Fig. 6-6 (a)). Initially, compression is applied to a thick specimen in a stepwise manner until debonding is first observed and the corresponding force and fiber displacement are measured. Although very difficult to prepare properly, a thin slice specimen of known embedded fiber length allows measurement of frictional properties after complete debonding. While this technique has been developed initially for polymer matrix composites (Mandell et al., 1980), its usefulness has been demonstrated largely for ceramic matrix composites where diffi-

(b)

i i i Top plate

Bottom plate

Figure 6-6. Schematics of (a) fiber push-out (or indentation) test and (b) slice compression test.

culties of specimen preparation and testing (e.g., fiber misalignment and breakage of high modulus fibers in grips) are often encountered in fiber pull-out tests. Another major advantage of this test over the single-fiber pull-out test is that it allows the effect of fiber interaction using specimens prepared from composites of high fiber Vf to be investigated. The fiber push-out problem can be analyzed exactly like fiber pull-out using a shear lag model but with some modifications. These include the change in sign of the interfacial shear stress and appropriately modified boundary conditions. Moreover, for fiber push-out the interfacial radial stress increases with fiber axial (compressive) stress due to Poisson expansion rather than fiber contraction as in fiber pull-out. Since in some composite systems this Poisson effect may dominate the whole debond and post-debond frictional

6.3 Surface Treatments of Fibers

push-out processes, meaningful calculations of the interface properties require a detailed stress analysis. There has been much current interest on the fiber push-out analysis (Shetty, 1988; Bright et al., 1989; Hsueh, 1990b; Lau and Mai, 1991 a; Weihs et al., 1991), as well as on the comparison of interface properties measured from both fiber pull-out and fiber push-out experiments (Bright etal., 1991; Hsueh, 1990b; Lau and Mai, 1991b). Indeed, solutions obtained for fiber pushout are quite similar to those for fiber pullout. Hence, the partial debond stress avd during progressive stable debonding and the frictional push-out stress
i

/ ~

i

\ r\i

a

d = ao + \a + Go) Si *fr = °Q'u = L

(c

\

H\

I""11) (6"18)

The non-dimensional coefficient Qf for fiber push-out is (Lau and Mai, 1991 b): -1

(6-19)

6.2.3.8 Slice Compression Test

The slice compression test, a modified version of fiber push-out test particularly for ceramic matrix composites, exploits the difference in elastic modulus between fiber and matrix. The test involves the compression of a polished slice of an aligned fiber composite between two plates (Fig. 6-6 (b)). When a critical load is applied, interfacial debonding occurs near the top surface of the specimen and the fibers protrude against the soft top plate (e.g., pure Al) of known work-hardening characteristics, while the hard bottom plate (e.g., Si3N4) ensures a constant strain in the specimen bottom. Upon removing the load, the fibers partially relax back into the matrix. Therefore, the interfacial properties can be estimated from the fiber protrusion under

257

a given load and the residual fiber protrusion after unloading. There are no comprehensive theoretical analyses available todate but Shafry etal. (1989) derived an approximate equation for the debond length ld and a constant interfacial friction stress Tf using the boundary conditions obtained for fiber pull-out (Gao et al., 1988) and the solutions for fiber stress relaxation after unloading (Marshall and Oliver, 1987).

6.3 Surface Treatments of Fibers There are several factors affecting the choice and application of surface treatments to different types of fibers. For example, sizes are usually applied as a very thin coating to glass and carbon fibers for polymer matrix composites. Depending on the intended application of the fibers the size serves several purposes: (a) to protect the fiber surface from damage; (b) to bind fibers together for ease of processing; (c) to lubricate the fibers so that they can withstand abrasive tension during subsequent processing operations; (d) to impart antielectrostatic properties; (e) to provide a chemical link between the fiber surface and the matrix and thus to improve the bonding at the interface. This section is concerned primarily with the surface treatment of high performance fibers, including glass, carbon (or graphite), aramid and some ceramic fibers. The methods of surface treatment and the exact mechanisms of improving the interfacial quality for a given fiber are different for different types of matrix material as thermal and chemical compatibilities are required. For polymer matrix composites, fiber surfaces are treated to enhance the interfacial bonding and preserve it in service, especially in the presence of moisture. For many metal and ceramic

258

6 Interfaces in Composites

matrix composite systems, chemical incompatibility is a severe problem due to inadequate or excessive reactivity at the interphase region. Therefore, fibers are usually treated with a diffusion barrier coating to protect them from damage by excessive reaction. Further, stability of the interface is an important requirement that is made critical by the high temperature service desired for these composites. Proper characterization of treated fibers and the modified interfaces and their evaluation are of prime importance in the design of advanced fiber composites. In addition to the mechanical testing methods discussed in Sec. 6.2.3, the use of conventional optical and (scanning and transmission) electron microscopy (SEM and TEM) as well as recently-developed scanning tunnelling microscopy (STM) and confocal scanning microscopy in composite fractography gives a valuable insight into the micro-mechanisms of interface related-failure to be discussed in Sec. 6.4. Moreover, most understanding of interfaces has been contributed by the use of sophisticated modern analytical techniques which allow researchers to probe and measure the chemistry and physics of interfaces. These include X-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES), ion scattering spectroscopy (ISS), scanning secondary ion mass spectroscopy (SIMS), Fourier transform infrared (FT-IR) and reflectance infrared spectroscopy, surfaceenhanced Raman spectroscopy, high-resolution solid-state nuclear magnetic resonance (NMR), etc. A list of additional techniques and their descriptions has been given by Adamson (1982) and Lee (1989). Discussions on surface treatments of fibers can also be found in Chap. 2 of this Volume.

6.3.1 Glass Fibers (GF's) The interfacial bonding between GF's and thermosetting resins has been greatly enhanced by the use of silane treatments. Since the early pioneering work of Plueddemann (1974), there is much information available on the silane coupling agents for GF's (Hull, 1981; Ishida, 1984; Cooke, 1987). A simple model for the function of silane coupling agents is schematically illustrated in Fig. 6-7, where the general chemical formula is shown as X 3 Si-R, multi-functional molecules which react at one end with the GF surface and the other end with the polymer phase. R is a group which can react with the resin and X is a group which can hydrolyze to form a silanol group in aqueous solution (Fig. 67 (a)) and thus to react with a hydroxy group of the glass surface. The R-group may be vinyl, y-aminopropyl, y-methacryloxypropyl, etc.; the X-group may be chloro, methoxy, ethoxy, etc. The trihydroxy silanols Si(OH)3 are able to compete with water at the glass surface by hydrogen bonding with the hydroxy groups at the surface (Fig. 6-7 (b)), where M may be Si, Fe, and/or Al (see Table 6-1).

(a) R-SiX 3 +H 2 O

HO-S'i-OH

Glass (b)

R-Si(0H) 3

R

+3HX

R

O-Si-O-Si-0 0 0

O-Si-O-Si-0 0 0 M M

(c)

(d)

Figure 6-7. Function of coupling agent: (a) hydrolysis of organo-silane to corresponding silanol; (b) hydrogen bonding between hydroxyl groups of silanol and glass surface; (c) polysiloxane bonded to glass surface; (d) organo-functional R-group reacted with polymer. After Hull (1981).

6.3 Surface Treatments of Fibers

composites made with silane treated fibers. However, a monolayer of silane usually does not yield an optimum mechanical strength and there must be other important mechanisms occurring near the interface. An established view is that bonding through silane by other than simple chemical reactivity are best explained by interdiffusion and interpenetrating network (IPN) formation in the interphase region (Plueddemann and Stark, 1980; Ishida and Koenig, 1980) (Fig. 6-8). In a study of yMPS with styrene matrix using FT-IR, Ishida and Koenig (1979) showed that the frequency of the carbonyl group of y-MPS shifted upon polymerization of the matrix. The frequency of the polymerized y-MPS was different from the homo-polymerized y-MPS without the matrix, indicating copolymerization. An indication of interpenetration was also observed on y-APS/ polyethylene (Sung et al., 1981). There are two kinds of intermixing which involve penetration of the resin into the chemisorbed silane layers and the migration of physisorbed silane molecules into the resin. The synergism of these two major bonding mechanisms with a silane coupling agent, namely the chemical reaction and

Chemically bonded interface

Diffused interface A

A

M-0-Si-OC Glass

When the treated GF's are dried, condensation occurs between the silanol and M - O H groups on the GF surface and between adjacent silanol molecules on the surface (Plueddemann, 1974) (Fig. 6-7 (c)). Therefore, the silane coated fiber presents a surface of R-groups to the uncured resins. During the curing process the R-groups react with the reactive functional groups present in the polymer resin (e.g., epoxy groups), forming a stable covalent bond with the polymer (Fig. 6-7 (d)). It is, therefore, essential that the R-group and the functional group be chosen so that they can react with each other under a given curing condition. Further, the X-groups must be chosen which hydrolyze to permit reactions between silane and the M - O H group on the GF surface. Several factors influence the structure of the coupling agent layers and subsequently the mechanical and physical properties of composites made therewith. These include the silane structure in the treating solution and its organo-functionality, the drying conditions, the topology of the fiber, and the chemical composition of the surface. Numerous experimental evidence exists to support the chemical reaction between GF and resin matrix in the presence of the silane coupling agent using various surface analysis techniques. For example, the chemical reaction between a highsurface-area silica and a vinyl functional silane using FT-IR spectroscopy (Ishida and Koenig, 1978), between amino-silanes (e.g., y-aminopropyltriethoxysilane (APS) and iV-methylaminopropyltrimethoxysilane (MPS)) on E-glass and an epoxy resin using FT-IR (Antoon and Koenig, 1981), and between vinyltriethoxysilane (VS) on E-glass and a polymethylmethacrylate (PMMA) resin (Koenig and Shih, 1971). The chemical bonding theory successfully explains many phenomena observed for

259

m m

°i

M-0-S,-| M-O-Si-gS

M-O-SiJ o

;

Coupling agent Polymer

Figure 6-8. A schematic model for interdiffusion and IPN in a silane treated glass fiber-polymer system. After Plueddemann (1988).

260

6 Interfaces in Composites

the IPN theories, seems to be particularly important in composites made with thermoset matrices. It is not known, however, the extent of chemical bonding in the thermoplastic matrices although there are appreciable improvements in flexural strength of composites containing silane treated fibers, particularly those that are compression molded (Table 6-3). Clearly, solubility or compatibility of the silane and the polymer matrix is more important in thermoplastic matrices, but chemical reaction can add additional strengths. The reactivity can be improved by tailoring the unreactive molecules in the thermoplastic so that it can contain special functional groups which can bond with the coupling agent. Another approach is to include chemicals in the size which induce local chain scission of the molecules near the fiber and allow chemical reaction to proceed so that coupling occurs directly with the molecules. In addition to the chemical reaction and the IPN discussed above, another important characteristics of silane treatment is Table 6-3. Improvement in flexural strength due to silane treatments in glass fiber-thermoplastic matrix composites. After Plueddemann (1988). Polymer-silane system

Percentage strength improvement Compression molded

Injection molded

Dry

Weta Dry

Nylon-aminosilane F Nylon-cationic silane H

55 85

40 40

36 45

PBT b-aminosilane F PBTb-cationic silane H

21 60

115 133 _ 47

23 28

24 11

Polypropylene-silane F Polypropylene-silane H

8 86

18 89

7 16

10 16

a

After 2 hours in boiling water; terephthalate.

b

Weta

polybutylene-

that it imparts to the GF a water resistant bond. The effect of water degradation on the untreated fiber-resin matrix interface is quite pronounced. Small molecules of water penetrate to the interface by diffusion and filtering through voids and cracks of the resin or by capillary migration along the fibers, and are absorbed by the glass. The water hydrolyzes the existing physical bonds at the fiber-matrix interface, destroying the adhesion, and ultimately resulting in mechanical failure of the system. With hot water, in addition to the above effects, resin swelling superseded by shrinkage due to leaching out of low molecular weight materials from the resin occurs. The silane film deposited on glass is subjected to extraction with organic solvents including water. There are three different structural regions in the interphase (Ishida and Koenig, 1980). The outermost layer, which is termed the physisorbed region consisting of weak oligomeric siloxanols, can be easily extracted with water, even at room temperature. The next layer, which can only be extracted by boiling water after prolonged immersion, is called the chemisorbed region. It consists mainly of higher oligomeric siloxanols which possess better resistance to hydrolysis than the lower form. The innermost region next to the glass surface is stable and resistant to extraction by hot water and may be regarded as the chemically reacted region where the interconnecting crosslinking is extensive and a three dimensional network of siloxane exists. This region is, most likely, responsible for the high resistance of the interfacial bond of silane treated composites to hydrothermal attack. This indicates that a thicker silane layer does not necessarily result in improvement in hydrothermal stability of the interfacial bond, but on the contrary, it may have an adverse effect on the strength of the interface. Koenig and

6.3 Surface Treatments of Fibers

Emadipour (1985) suggested that there is an optimum concentration of silane to impart the most favorable result on interfacial shear strength (e.g., 0.5% concentration for diaminosilane) and that partial removal of the thick layer prior to fabrication of the composite may enhance the mechanical performance (Fig. 6-9). The individual characteristics of each of these silane regions have a marked influence on the interfacial stability and mechanical properties of silane treated composites under both dry and hot/wet conditions.

261

The oxidation process can have a number of effects: increase the surface area of the fiber by roughening or pitting; remove a weak surface layer; and alter the chemical nature of the surface, thus improving wetting of the fiber by a resin, or increasing the possibility of hydrogen or covalent bonding. The mechanisms of bonding between CF's and polymer matrices are equally complex to that of GF's, and there are more complications with the CF surface which is highly active and readily absorbs gases. A range of active functional groups can be produced by the surface oxidative treatment. Reviews on important factors involved in CF surface treatment can be found in Scola (1974), Delmonte (1981), Riggs etal. (1982), Donnet and Bansal (1984), Ehrburger and Donnet (1985), Wright (1990) and Hughes (1991). 6.3.2.1 Types of CF Surface Treatment

2

U 6 8 10 Silane concentration in %

12

Figure 6-9. Effect of diamino-silane solution concentration and modification on interfacial bond strength i b of GFRP: o silane treated; • silane treated and partially removed. After Koenig and Emadipour (1986).

6.3.2 Carbon Fibers (CF's)

The poor shear strength of CF reinforced polymers is generally attributed to a lack of bonding at the fiber-matrix interface. Moreover, the ILSS decreases markedly when using fibers with increasing modulus, i.e., increasing crystalline ordering of the CF's. As a result, extensive work has been devoted to surface treatments of CF's by various oxidation processes to improve the fiber-matrix interface bonding.

Surface treatments of CF's can be classified into oxidative and non-oxidative treatments. Oxidative treatments can be further divided into dry oxidation in gases including plasma etching and wet oxidation carried out chemically or electrolytically. Non-oxidative treatments include deposition of more active forms of carbon, such as the highly effective whiskerization, plasma polymerization and polymer coatings on the CF surface. Dry oxidative treatments are carried out with air, oxygen or oxygen containing gases such as ozone and CO 2 at low or elevated temperatures. The treatment at a high temperature is very drastic and often causes excessive pitting of the CF surface which reduces the fiber strength (Novak, 1969). In effect, the surface layers burn away unevenly to create pits in lines which coalesce into channels giving rise to a high surface rugosity. These active sites could be

262

6 Interfaces in Composites

related to the edge plane of the CF surface and/or imperfections in the basal plane. Metallic impurities such as oxides of Cu, Pb, V and transition metals are known to enhance the degradation rate even at a low temperature (McKee, 1970). Plasma etching may involve submitting to a lowenergy microwave plasma in a reduced gas pressure. The atmosphere can be an inactive gas like argon or active gases like oxygen and nitrogen (Donnet et al., 1988; Yuan et al., 1991; Bascom and Chen, 1991). Several types of liquid-phase oxidizing agents, such as nitric acid, acidic potassium permanganate, acidic potassium dichromate, sodium hypochlorite, hydrogen peroxide and potassium persulphate, have been used with varying degree of success. These liquid-phase treatments are generally milder than the ones with a gaseousphase, and do not cause excessive pitting and hence degradation of the fiber strength. Several factors such as acid concentration, exposure time and temperature, and mode of treatment influence the effectiveness of these oxidative processes. Thus, depending on the type of CF, nitric acid treatment increases the surface area, surface functionality and surface oxide contents with increasing time and temperature of treatment and acid concentration (Herrick, 1968; Scola, 1974). This treatment generally causes an appreciable weight loss and smoothing of the fiber by removing surface irregularities (Donnet and Ehrburger, 1977). Electrolytic or anodic oxidation is fast, uniform and best suited to mass production processes and is probably the most widely used method for treatment of commercial CF's. Many electrolytes have been tried, including alkalis, nitric, sulphuric and phosphoric acids, dichromate permanganate and ammonium bicarbonate. The oxidation mechanism of most CF's is char-

acterized by a simultaneous formation of CO 2 and degradation products which are dissolved in the electrolyte (in alkaline solution) or adhering onto the CF surface (in nitric acid). Only minor changes in the surface topography and area of the fiber are obtained with a small weight loss (which is normally less than 2%). Several non-oxidative treatments of CF's have been developed, which include whiskerization, plasma polymerization and application of certain forms of organic and polymer coatings. Whiskerization involves a nucleation process and the growth of very thin and high strength single crystals of the chemical compounds, such as silicon carbide (SiC), titanium dioxide (TiO2) and silicon nitride (Si3N4), on the fiber surface perpendicular to the fiber axis (Goan and Prosen, 1969). The whiskers grow from individual fibers and their growth generally starts at the points of defects such as compositional heterogeneities, metallic inclusions or structural irregularities and imperfections. The technique of plasma polymerization utilizes polymerizable organic vapors like propylene (Jang et al., 1988), acrylonitrile and styrene monomers (Dagli and Sung, 1989). This treatment has shown to increase the polar component of surface free energy (Dagli and Sung, 1989; Donnet and Guilpain, 1991). Some organic polymer materials have also been used as a surface coating to improve the interfacial bonding, e.g., polyamide (Goan and Prosen, 1969), polyimide (Marks et al., 1975), organosilanes (Goan and Prosen, 1969) and other alternating and block polymers like styrene and maleic anhydride (Riess et al., 1974). The basic requirements for a polymer to be used as a coating are that it is capable of being fixed on the fiber surface by covalent or ionic bonds, and compatible with the resin matrix. If the polymer coating has functional groups capable of react-

6.3 Surface Treatments of Fibers

ing with the resin matrix, a better result can be obtained (Subramanian et al., 1978). 6.3.2.2 Effects of Surface Treatment on Composite Properties

The principal effects of CF surface treatment on composite properties are to increase the ILSS, flexural and tensile strengths. However, these are inevitably accompanied by a loss in the impact fracture toughness (of unidirectional laminates) or notched tensile strength (of multidirectional laminates) (Fig. 6-10) (Goan etal., 1973; Dauksys, 1973). Therefore, a careful balance is always necessary to ensure both adequate strength and toughness properties. The enhancement of the strength properties depends on the fiber elastic modulus (e.g., the greatest improvement in ILSS is obtained for high modulus fibers), the degree of surface treatment and in particular the resin matrix/curing agent used. Other composite properties have also been characterized: the compressive strength increases slightly (Norita et al., 1986) but the mode I interlaminar fracture toughness GIc for crack initiation is almost doubled (Ivens et al., 1991) with increasing

3.0 120*

°

100<

°

o - 2.0 .£

o o

ILSS in h>

a 8°

/

\

.

-1.0

*

20 0

i

0

r

i

i

i

i

.

1 2 3 Treatment time in min

Figure 6-10. Effect of CF surface treatment level on ILSS (o) and impact energy (•) for CFRP. After Goan et al. (1973).

263

degree of treatment. In general, an increase in the interfacial bond strength r b enhances the composite compressive strength by increasing the load necessary to cause transverse tensile failure of the interface due to the fiber Poisson effect. Also, delamination failure is reduced in favor of microbuckling of fibers (Drzal and Madhukar, 1991). The improvement in r b and Glc due to fiber surface treatments has been verified by direct measurements in fiber fragmentation tests (Drzal et al., 1983 a; 1983 b) and fiber pull-out tests (Baillie and Bader, 1991; 1992). 6.3.2.3 Effects of Surface Treatment on Fiber Properties

The increase in composite strength properties (e.g., ILSS, flexural, tensile and compressive strengths) due to various surface treatments can be related to the changes in surface area, rugosity and surface functional groups and the removal of weak outer layers. However, there is no simple dependence of the interfacial bond strength on these factors. Increase in the surface area by the formation of pits or enhancement of longitudinal striations promotes mechanical anchoring of the matrix and interpenetration between fiber and matrix. However, because the rugosity develops mainly in the fiber axial direction, the improvement of interfacial bonding is in the same direction as evidenced by the result that the adhesion is preferentially increased when a force is applied parallel to the fiber axis (Donnet et al., 1974). Yet, the rugosity is an ameliorating factor only when the fiber is perfectly wetted by the liquid resin. Therefore, unless the resin penetrates into the asperities of the fiber, polymerization of the resin is accompanied by the formation of interfacial cavities which causes premature failure of the inter-

264

6 Interfaces in Composites

facial bond. Another important characteristics of acid treated CF's is that an outer weak layer with various types of defects is removed by oxidation, resulting in a surface capable of supporting high shear loading (Drzal, 1983 a) and an incidental improvement of fiber strength with light anodic treatment (Bader et al., 1991). In contrast, plasma etching in nitrogen or oxygen causes excessive removal of the fiber surface layers, reducing the fiber diameter by up to 22% (Jang et al., 1988). Despite the considerable attention given to the evaluation of the nature of the chemical groups produced by surface treatments, there is still much controversy as to whether chemical bonding actually takes place at the interface. A schematic model for chemical reaction between oxidized CF's and an epoxy resin is illustrated in Fig. 6-11 according to Horie et al. (1977). The strong covalent bonds could be either ether (-COH) or ester (-COOH) bonds due to the reactions between the hydroxyl and carboxyl groups of CF and the epoxy group of the resin. Scola (1974) also argued that the large (about 100%) increase in ILSS is not mainly due to the increase in surface area but more likely is highly dependent on the surface reactivity of the treated fiber. Certainly, oxidative treatments increase the oxygen content (often more than double) and the nitrogen content (if nitric acid or ammonia is used as an oxidative medium) on the CF surface (Fig. 6-12), in the form of carboxyl, hydroxyl, carbonyl and phenolic groups (Scola, 1974; Hopfgarten, 1978). The postulation of chemical reaction has also been supported by the better retention of composite strength with oxidized fibers when subjected to hygrothermal ageing (Ko et al., 1982). In addition, the oxidized CF's may possess polar surface species (e.g., carboxylic group) which can provide good

C-O-CH 2 -CH-AA OH

CH-OA

i-CH 2 -CH-AA OH

Figure 6-11. A schematic model for chemical reaction between oxidized CF surface and epoxy matrix. After Horie et al. (1977).

wetting of the surface by the resin. In fact, a decrease in contact angle, an indication of better wetting, of different polymers has been reported upon fiber surface treatment (Yamamoto et al., 1971). Therefore, the influence of the surface groups has to be taken into account when the interface bonding is improved by the rugosity effect. Other effects of surface groups are to promote the bonding by establishing at the interface van der Waals forces, e.g., dispersion force, dipolar interaction and hydrogen bonds. From the available evidence, the following can be summarized of the effects of CF surface treatments: (a) there are substantial changes in fiber surface area with associ-

1

3

5 10 15 25 Treatment level

50 100 A00

Figure 6-12. Atomic percentage of oxygen (o) and nitrogen (•) contents for different levels of electrolytic oxidative surface treatment of XA CF's. Commercial treatment level regarded as 100. After Baillie et al. (1991).

6.3 Surface Treatments of Fibers

ated variations in rugosity depending on the oxidative treatment medium; (b) a weak surface layer may be removed, the removal being more serious in plasma etching; (c) there is an increase in the polar surface energy; (d) chemical modification takes place and carboxyl, hydroxyl and carbonyl groups are produced. Although the mechanisms of chemical bonding due to the functional groups have so far not fully been elucidated, they certainly change the energetics of CF surface. Better mechanical anchoring arising from the surface rugosity and the increased physical surface area as well as the beneficial effect of removing surface weak layers all contribute to an improved interfacial bond. The surface chemistry is obviously a complex process, particularly when CF's of different atomic structures and physical properties, and a large variety of surface treatment methods are involved. These factors have made it even more difficult to evaluate as to what extent each mechanism contributes to the total bond strength. Further, there must be a limit to the improvement in bonding by surface treatment and a decrease is produced by extended surface attack owing to the fiber damage. 6.3.3 Aramid Fibers

Significant improvement in interfacial bond strength with aramid fibers have not yet been developed as with coupling agent for GF's and oxidative treatments for CF's. Although aramid fibers could readily combine with most resins used in composite manufacture, the ILSS and transverse tensile strength of aramid fiber-epoxy matrix composites are only about half of the corresponding values for GF and CF counterparts. These properties are aggravated by the skin-to-core inhomogeneity and highly oriented microstructures of these fibers.

265

Many attempts have been made to enhance the interface bond with limited or little success. Various coupling agents or sizing have been tried: Takayanagi et al. (1982) modified Kevlar 29 by polymer reaction via the metalation reaction in dimethyl sulfoxide to provide the fiber surface with several functional groups such as n-octadecyl, carboxymethyl and acrylonitrile. They showed that the modified surface was much rougher than the original surface and improved mechanical properties of composites made with ionomer matrices. Similar roughening effects by bromine water attack on the composite ILSS have also been reported with a slight loss in fiber strength (Breznick et al., 1987). In a similar study, Vaughan (1978) applied several silane coupling agents developed principally for GF's to the Kevlar fibers achieving only a slight improvement in the composite strength compared to the unmodified fibers under both dry and water ageing conditions. Since Wake (1972) suggested that the aramid fiber surface may react with an isocyanate-linked polymer, Eagles et al. (1976) have studied the adhesion of Kevlar 49 fibers to thermoplastics using suitable agents and by immersing in boiling water. Andreopoulos (1989) treated Kevlar 49 fibers with reactive chemicals, such as acetic acid anhydride, methacryloyl chloride, sulfuric acid and acrylamide, to ensure grafting in the aramid chain while avoiding excessive loss in fiber weight. Tensile strengths of the fiber and the composite made therewith in a polyester matrix decreased as a result of fiber etching by most of the chemical agents used. The only exception is the methacryloyl chloride treated fibers which have an intermediate surface roughness and possibly chemical grafting considered to be favorable for interface bonding. Keller et al. (1981) also

266

6 Interfaces in Composites

treated Kevlar 49 fibers to produce surface chemical groups which might react with epoxy resins. They showed that acid (HC1, H 2 SO 4 ) or base (NaOH) hydrolysis of Kevlar fiber could yield reactive terminal amino groups, to which a small diepoxide molecule might be attached. If the process of hydrolysis was allowed too long, direct mechanical damage to the fiber would result and outweigh any advantage gained by the improved adhesion. A promising treatment that chemically modifies the aramid fiber surface is plasma treatment in vacuum, ammonia or argon. Plasma treatment in ammonia increases the amine concentration on the surface which is considered to be responsible for strong covalent bonding at the interface (Penn and Larson, 1979). This is accompanied by an increase in nitrogen content in the form of amine group but a decrease in oxygen content with increase in exposure time. The treatment increases the ILSS by 50% with associated dominant failure mechanisms of fiber splitting and matrix cracking in contrast to the apparent interface debonding in untreated counterparts (Fig. 6-13). Improvement in retainability of the composite flexural strength and ILSS

5 10 15 Treatment time in min

20

Figure 6-13. Surface amine concentration (o) of aramid fiber and ILSS (•) of epoxy matrix composites as a function of ammonia plasma treatment time. After Brown et al. (1991).

has also been shown under water ageing condition for plasma treated Twaron aramid fibers (Verpoest and Springer, 1988; Janssens et al., 1989). Quite clearly, modification of aramid fiber surface to produce chemical groups which can form covalent bonds with a resin matrix using chemical agents and plasma treatment warrants much further research. Though a significant improvement in interface bond is possible, the apparent failure strength at the interface region of the composite cannot be greatly improved unless the inherent microstructures of the fiber are modified to produce more homogeneous and isotropic material (Kalantar and Drzal, 1990). 6.3.4 Other Fibers

In the previous sections dealing with glass, carbon and aramid fibers the discussions are mainly concerned with polymeric matrices for which these fibers are commonly used. Here, surface treatments and coatings of fibers now popularly used for metal and ceramic matrices composites, particularly alumina (A12O3), boron (B/W) and silicon carbide (SiC) fibers are considered. Although high temperature graphite fibers have also been used to reinforce carbon (and magnesium (Mg)) they will not be discussed here because carbon/carbon composites technology is still very much in its development stage. Because of the accelerated reactivity of the surfaces of many fibers when in contact with metal and ceramic matrices at elevated temperatures, considerable care is needed to ensure fibermatrix compatibility and avoid fiber degradation. Furthermore, once fabricated, the service temperature must not exceed some maximum level, otherwise significant interdiffusion-controlled reaction occurs between the elements of fiber and matrix to

6.3 Surface Treatments of Fibers

form a compound layer of substantial thickness which is detrimental to the mechanical performance of the composites. In some cases, metal oxide fibers (e.g., A12O3) are not readily wetted by many metals unless they are very reactive (e.g., Zr). Consequently, a balance between the reaction which is necessary for efficient interfacial bonding and the fiber degradation is required. Several approaches have been used to achieve desired compatibility between fiber and matrix, which include fiber sizing and diffusion barrier coating, control of matrix composition and impurity contents and composite fabrication procedures. 6.3.4.1 Alumina (A1 2 O 3 ) Fibers

Alumina fibers are not readily wetted by most metals, due to their low surface energy, particularly if the fibers are in the form of short whiskers. The wettability of these fibers and whiskers can be increased by chemical vapor deposition (CVD) of a thin metallic coating such as nickel (Ni) (Sutton, 1966) or Ni alloys containing active metals like titanium (Ti) (Noone et al., 1969) for a molten silver matrix. A duplex Ti-Ni coating further promotes wetting and drastically improves the bonding as revealed by the improvement in the composite tensile strength. The fracture mode changes from interfacial failure to matrix shear failure with the coated fibers. The Ti-Ni coatings are also effective for other matrices like Al and Ni-Cr alloy (Nicholas, 1968). Alloying elements can have a significant effect on the reaction at the interface region. For example in an A12O3 fiber-Al matrix composite a small amount of Mg (normally less than 0.4 weight %) in the matrix is beneficial for metal infiltration and fracture resistance without causing a harmful reaction at the interface (Chapman etal., 1991). Increasing the Mg con-

267

tent, however, deteriorates the flexural strength due to a corresponding increase in the thickness of the reaction product MgAl 2 O 4 in the interface region (Johnston and Greenfield, 1991). 6.3.4.2 Boron (B/W) Fibers

Coatings on boron fibers have been studied to prevent the formation of compounds at the interface with common matrices such as Al and Ti and some ferrous metals including Fe, Co and Ni. The oxide films pre-existing on the fiber and the aluminum matrix can act to delay the reaction if the composite is made in the solid state. However, the oxides are useless in the presence of molten aluminum. Silicon carbide coatings on the boron fiber (so-called Borsic fiber) provide some protection so that direct contact with molten aluminum for a prolonged time does not degrade the fiber strength (Basche, 1969; Prewo and McCarthy, 1972). Boron nitride (BN) coatings of thickness 0.3 to 0.4 jim also appear to be very stable in the presence of molten aluminum at temperatures as high as 800 °C if the coating remains intact. This coating even improves slightly the average fiber strength (Vidoz etal., 1969; Ryder etal., 1970). Early work using Ti matrices showed limited success for diffusion barrier coatings on boron fiber reinforcements (Metcalfe, 1974). Coatings of boron nitride, boron carbide (B4C) and SiC on boron fibers result in a reaction rate similar to the uncoated fibers. The basic difference between these coatings is the reaction products formed: TiB2 was primarily formed in the case of boron-containing coatings whereas titanium silicides and carbides are formed with SiC coatings (Wawner, 1988). The B4C coating of thickness approximately 7 jam which is applied by CVD

268

6 Interfaces in Composites

from boron trichloride in methane gas improves substantially the tensile strength of the Ti matrix composite (Morin, 1976). 6.3.4.3 Silicon Carbide (SiC) Fibers

There are several forms of commercial SiC fibers depending on the types and thickness of coating applied onto the substrate. In the early development stage, SiC is deposited on a tungsten substrate or fiber producing SiC/W fibers. The surface of these fibers is bumpy and nodular, and sensitive to self-abrasion which may reduce the fiber tensile strength. More importantly, prolonged exposure to elevated temperature causes significant degradation of the fiber strength, presumably due to the deleterious tungsten core reaction. Later, the tungsten core is replaced with a carbon monofilament substrate producing SiC/C fibers. The carbon core is routinely coated with a thin (about 1 ^im) layer of pyrolytic graphite before deposition of SiC to minimize the reaction between the carbon core and SiC. To further minimize the extreme surface sensitivity of SiC/C fibers, a thin layer of carbon is deposited onto the surface. This surface carbon coating may also lower the stress concentrating ability of grain boundaries at the fiber surface, but the amorphous carbon-rich surface has a wetting problem with metal matrices (especially cast Al) and even epoxies causing poor adhesion and forming reaction products (e.g., aluminum carbide with Al matrix). To alleviate this detrimental situation, the fiber surface is modified by covering the carbon layer with additional SiC, designating the fiber SCS. Depending on the thickness of the final SiC coating designed for different metal matrices, three types of SCS, namely the SCS-2, SCS-6 and SCS-8, are available: SCS-2 and SCS-8 fibers are functional in Al matrices;

whereas SCS-6 fibers with a thicker final coating is utilized specifically for Ti matrices which require more protection from high temperature fabrication reactions (Strife and Prewo, 1982; Martineau et al., 1984). SCS-2 fibers have shown to retain their original strength for up to 1.5 h exposure in molten aluminum. But with continuing exposure the strength decreases systematically due to the fragmentation of the coating and formation of reaction products like aluminum carbides. SCS-6 fibers are normally coated with TiB2 which acts as a diffusion barrier with Ti alloys and Ti-Al intermetallic alloys (e.g., Ti3Al, TiAl and TiAl3) (Donnellan and Frazier, 1991). When subjected to tensile loading, the brittle TiB2 layer cracks easily promoting interfacial failure which enables the fiber strength to be preserved (Jeng et al., 1991). Additional surface modifications on vapor deposited SiC fibers, including WC, TaC, TiN, B4C, Al, Ni and Fe have been studied with varying degree of success (Wawner and Nutt, 1980; Debolt, 1982). Further details on SCS fibers can be found in a recent review by Wawner (1988). Different to the SiC fibers produced by the CVD technique onto tungsten or carbon substrates, production of continuous SiC fibers involves the transformation of an organic precursor (polycarbosilane) into a ceramic fiber by a route analogous to that taken in carbon fiber production from polyacrylonitrile (PAN). This technique was first developed by Yajima et al., (1978, 1979) and commercialized by Nippon Carbon Co., now called the Nicalon fibers. This fiber is also coated with pyrolytic graphite normally in argon to improve the strength and modulus of the composites based on ceramic matrices and to protect the fiber from chemical damage during composite processing (Hwang and Jang, 1991). There is, however, a weakened

6.4 Interface-Related Fracture Toughness Theories

interface bond due to the existence of this coating. Incorporation of oxygen into the SiC fiber by curing the precursor fiber in an oxidising atmosphere has been found to improve the fiber tensile strength. The beneficial effects of exposing the fibers to various types of radiation such as neutrons and electrons are also reported.

6.4 Interface-Related Fracture Toughness Theories The efficient use of advanced fiber composites in engineering applications requires a basic understanding of how the fracture process initiates and progresses to final failure. Of particular interest is the local response of the fiber-matrix interface within the composite during fracture. If the interface is to be effective in controlling damages in composites and enhancing the tolerance prior to failure, it is necessary to identify the basic failure mechanisms or origin of fracture toughness for a given fiber-matrix system and then to explore means for exploiting and/or controlling these mechanisms. There have been many theoretical and experimental studies of fracture in fiber composites and of mechanisms by which toughening is achieved. When a composite containing a defect or crack is loaded there is a highly strained region at the crack tip where failure mechanisms of various kinds take place before the crack grows. This localized zone is commonly called the fracture process zone (FPZ) or damage zone, the development of which gives rise to a crack growth resistance curve (or R-curve). Summaries of failure mechanisms in the FPZ of fiber reinforced polymer matrix composites can be found in many references including Marston et al. (1974), Atkins (1975) and Harris (1980), and these are reviewed re-

269

cently by Kim and Mai (1991a). Reviews on failure or toughening mechanisms are also available for metal matrix (Ochiai, 1989) and ceramic matrix composites (Davidge, 1989; Warren and Sarin, 1989). When a crack moves through a matrix containing fibers or whiskers, the following failure mechanisms may be expected to operate: matrix fracture, fiber-matrix interface debonding, post-debonding friction, fiber fracture, stress redistribution, fiber pull-out, etc. In conjunction with these mechanisms, fiber bridging, crack deflection and microcracking also take place depending on the strength of the constituents relative to that of the interface. All these microfailure mechanisms in principle apply to most composites reinforced with short and continuous fibers in polymer, ceramic, metal and cement matrices, but the extent to which and how they occur will be different for different fiber-matrix systems. It is not necessary for all these failure mechanisms to operate simultaneously for a given system, and in some cases one of these toughness contributions may dominate the total fracture toughness. There is sometimes disagreement about the relative contributions to the overall fracture toughness of any given composite. Indeed, no simple unified theory can be applied to predict the fracture toughness of all types of fiber composites. Many fracture toughness theories of fiber composites have to-date been concerned with unidirectional fibers (although they can be suitably modified to apply to short/randomly oriented fiber composites). Characterization of the various origins of fracture toughness in these composites may be obtained from a consideration of the sequence of microscopic fracture events that lead to the propagation of a macroscopic crack under monotonic increasing loads, Fig. 6-14 (Harris, 1980). In the following sections these vari-

270

(a)

/// 1

6 Interfaces in Composites

(b)

•I (c)

(d)

/ 7 (e)

Figure 6-14. Model of microscopic fracture events in a simple composite subjected to a monotonic tension. After Harris (1980).

ous toughening mechanisms, particularly the interface-related ones, and their associated equations are given. 6.4.1 Fiber-Matrix Debonding When the fiber fracture strain is greater than that of matrix (i.e., ef > em), a crack originating at a point of stress concentration in the matrix is either halted by the fiber if the local stress is not high enough or it may pass around the fiber without destroying the interfacial bond (Fig. 6-14 (b)). As the applied load is increased fiber and matrix attempt to deform differentially and a large shear stress is developed at the interface. When this shear stress exceeds the static interfacial shear strength interfacial debonding starting at the crack plane extends some distance along the fiber (Fig. 6-14(c)). The debond toughness Rd given by Outwater and Murphy (1969) may be obtained by dividing the total strain energy stored in the fiber over the debond length /d at its breaking stress of by the nominal cross-section area of the composite: 2£ f

matrix in shear, and Rd represented by Eq. (6-20) is a consequence of debonding rather than debonding itself as pointed out by Harris (1980). Following this latter argument then Rd should be rewritten with of replaced by the fiber debond stress od (which is a function of debond length ld as discussed in Sec. 6.2.3 and is normally smaller than of). The above discussion concerns fiber-matrix debonding in mode II shear. If the bond strength r b is sufficiently smaller than the matrix tensile strength
t

t

t J \

(6-20)

This debonding toughness is a major source of the total fracture toughness of glass fiber-thermoset matrix composites (GFRP). Implicit in the Outwater-Murphy analysis is the separation of fibers from

*1 •

Weak interface

t

f

Figure 6-15. Tensile debonding mechanisms at the weak interface. After Cook and Gordon (1964).

6.4 Interface-Related Fracture Toughness Theories

due to the tensile stress component under some favorable conditions, its contribution to the total fracture toughness is insignificant (Atkins, 1975). Consequently, the Cook-Gordon model is thought to apply essentially to laminate structures and its beneficial toughening mechanisms as a result of longitudinal splitting or delamination are crack tip blunting which reduces the stress concentration in the direction normal to the interface and crack arrest which increases further the amount of delamination. However, it is important to realize that the Cook-Gordon debonding mechanism also augments the fiber pullout length. 6.4.2 Post-Debond Friction The fiber matrix move relative to each other after interfacial debonding. This gives rise to post-debond frictional work which is equal to the frictional shear force times the differential displacement between fiber and matrix (Kelly, 1970). Approximating the displacement to be the product of the average ld and the differential strain Ae of fiber and matrix, the post-debonding friction toughness Rdf is: Rd{ = 2V{Tfl2dAs/d

(6-21)

where rf is the frictional shear strength at the interface. Since em can be neglected in brittle matrix composites Ae « ef (Harris, 1980). It is shown that Rdf contributes substantially to the total fracture toughness of GFRP and Kevlar fiber-thermoset matrix composites (KFRP) (Harris et al., 1975; Munro and Lai, 1988; Kim and Mai, 1991b). 6.4.3 Stress Redistribution When the continuous fiber is loaded to failure over the debonded length it may break at a weak point within this region

271

near the main fracture plane. Upon failure the fiber instantly relaxes back and regains its original diameter and then the ends are gripped by the matrix (Fig. 6-14(d)). This mechanism provides another source of toughness since after fiber fracture the strain energy stored in the fiber is redistributed to the matrix. Assuming that the stress builds up linearly from the broken end over a distance /c/2, the fracture toughness due to stress redistribution Rr has been derived by Piggott (1970) and FitzRandolph et al. (1972): r

3£ f

(6-22)

which is (2 Zc/3 Zd) times the Outwater-Murphy debond toughness in Eq. (6-20). It is shown that Rr makes a significant contribution to the total fracture toughness of boron fiber-epoxy matrix composites (BFRP) (Marston et al., 1974; Atkins, 1975). 6.4.4 Fiber Pull-Out When the crack propagates it pulls out the broken fiber from the matrix giving rise to a continuation of the post debonding frictional work (Fig. 6-14 (e)). This additional component of work of fracture is known as the Cottrell (1964)-Kelly (1970) pull-out work. Based on the work done by the frictional shear stress Tf which is assumed constant over a pull-out distance Zpo the fiber pull-out toughness Rpo is given by: Rpo = 2V(x(llJd

(6-23)

The magnitude of /po is difficult to determine with any accuracy because of the uneveness of the fracture surface, particularly when Vi is high. Therefore, it is desirable to express Rpo in terms of the inherent properties of composites. When the fiber length is less than the critical transfer length (i.e., / < Zc), all the fibers are pulled out. Assum-

272

6 Interfaces in Composites

ing Zpo varies between 0 and 1/2 with a mean value of 1/4 (Cooper and Kelly, 1969), Rpo becomes:

where Rs = Vt Rf + (1 - Vt) Rm + V{ (IJd) R, * *Vt(lJd-l)Rm

R Rpo

--LJ— for l
Rpo

=

Ylhk

for

/ = /c

(g.25)

If the composite containsfiberswith lengths greater than Zc, then the fraction of fibers which are pulled out is IJl (on the basis of normal probability) and Zpo ranges from 0 to IJ2. Hence R becomes: 6d

-M \l

for

1>L

(6-26)

For most continuous fiber composites with thermoset matrices, particularly those with CFs (CFRP), the fiber pull-out is a major source of fracture toughness (Munro and Lai, 1988; Kim and Mai, 1991 b). Even for composites with ductile metal matrices like carbon fiber-Al matrix systems (Taya and Daimaru, 1983; Kyono et al., 1986), # p o contribute significantly to the measured fracture toughness. 6.4.5 Total Fracture Toughness Theory Marston et al. (1974) and Atkins (1975) developed a total fracture toughness theory based on the coexistence of the three major sources of toughness, i.e., stress redistribution RT9 fiber pull-out Rpo and the creation of new surfaces Rs in fiber Rf, matrix Rm and at the interface R{. Thus, the total fracture toughness Rt is given by:

1/4 +

Ki-Ff)i?n

(6-28)

(6-24)

E{ (6-27)

It should be noted that the debond toughness Rd of Eq. (6-20) is included in Rs of Eq. (6-27) where Rf is neglected and R{ is approximated to Rm. Further, to a first approximation, the frictional shear strength rf is taken as the lower-bound estimate of the apparent bond strength in shear at the interface Ta for the critical transfer length lc(= daf/2ra) given in Eq. (6-8). In using Eq. (6-27) to predict Rt of a given system it is important that the failure mechanisms described above all do exist. If any one mechanism is absent the corresponding toughness term must not be included. In this way, Marston et al. (1974) showed that there is good agreement between the total fracture toughness theory and the experimental results for carbon and boron fiberepoxy matrix composites. Eq. (6-27) suggests that .Rt varies linearly with the reciprocal of the interfacial friction shear strength, i.e., l/r f , with a lower limit of (1 — Vf) Rm when rf -> oo. Figure 6-16 plots the variation of tensile strength of composites ac against Rt or l/t f according to Marston et al. (1974). For high xf (which is taken as the lower bound value of ra), ac is determined by the small Griffith cracks (i.e., ol oc i^t); whereas for low xf (or ra), ac decreases linearly with l/xf. They proposed, therefore, that both high composite strength (which requires a high bond strength T J and high fracture toughness (which needs a low bond strength ra) cannot be achieved simultaneously for a given brittle fiber-brittle matrix composite although these properties can be optimized as indicated by the point A in Fig. 6-16. Thus, it is important to realize that whilst many of the fiber surface treatment tech-

273

6.5 Toughening Methods with Controlled Interfaces

Rx or 1 / i f

Figure 6-16. Relationship between composite strength GC and total fracture toughness Rt or equivalently the inverse of frictional shear strength xf. After Marston etal. (1974).

niques described in Sec. 6.3 are useful in promoting the interface bond strength (and hence the composite strength) they are not desirable for high toughness. Controlled interfaces are required to produce high strength-high fracture toughness composites as discussed in Sec. 6.5.

toughnesses of thermoset matrix composites reinforced with unidirectional continuous glass, carbon and Kevlar fibers can be predicted based on the three principal failure mechanisms, i.e., interfacial debonding, stress redistribution and fiber pull-out. Fig. 6-17 is a toughness map of predicted fracture energy in kJ/m 2 for a GFRP as a function of fiber strength of and friction shear strength r f . The dashed line and arrows indicate respectively a change in dominant failure mechanisms from postdebond friction Rdf to interfacial debonding Rd and the effect of moisture on the changes of of and r f . The advantage of a reduction in rf is clearly indicated by an increase in the total fracture toughness in agreement with the theory of Marston et al. (1974). However, prolonged exposure to a humid environment reduces of leading to subsequent reductions in the total toughness of the composite (see Eq. (6-27)).

6.4.6 Fracture Toughness Maps

In analysing the energy absorption processes in unidirectional fiber composites, Wells and Beaumont (1982, 1985 b) have used "fracture toughness maps" to relate the composite toughness to the properties of the fiber, matrix and fiber-matrix interface. This approach is analogous to the well-known "material property charts" first proposed by Ashby (1989). These maps not only describe the effect of constituent material properties on fracture toughness, but they can also be used to characterize the effects of loading rate, fatigue, adverse environments, etc. on the practical performance of a given composite system. Having estimated /d, /po and other parameters such as fiber stresses during debonding and pull-out processes from the known properties of composite constituents (Beaumont and Anstice, 1980; Anstice and Beaumont, 1981), the total fracture

1.0 2

3 Tf In MPa

4

5

Figure 6-17. Toughness maps depicting contours of predicted fracture toughness (in kJ/m2) for GFRP as a function of fiber strength of and frictional shear strength t f . After Wells and Beaumont (1982).

6.5 Toughening Methods with Controlled Interfaces It is noted from the interface-related fracture toughness theories discussed in the

274

6 Interfaces in Composites

last section, the total composite toughness increases with increasing Vf and of but decreases with increasing r b (or T J because high Tb discourages interfacial debonding and fiber pull-out. A number of experimental techniques have been developed to improve the total fracture toughness of fiber composites without impairing other mechanical properties. There are two major approaches: one relies on the improvement of the intrinsic properties of composite constituents and the other depends on suitable fiber-matrix interface controls. The first approach includes the use of toughened matrices, fiber hybrids and large diameter fibers. Toughened matrices such as thermoplastics and rubber modified epoxies are particularly efficient for high toughness and damage tolerance of composites subjected to interlaminar fractures. However, they do not improve the transverse fracture toughness perpendicular to the fiber direction unless the fiber Vf is small and the matrix toughness is a major source of the total toughness of the composite (Kim et al., 1992 b). For a comprehensive review of the failure mechanisms of toughened epoxy resins see Garg and Mai (1988) and Low and Mai (1991). A hybrid fiber composite consists of two or more fiber types in the matrix. Common examples are glass/ carbon and Kevlar/carbon hybrid fiber composites to improve the impact fracture toughness while retaining high strength and modulus gained from carbon fibers. This toughening effect depends on the intimacy of fiber mixing and the sequence of ply lay-up. A review on this subject is given by Hancox (1981). The use of large diameter fibers can lead to much improved fracture toughness of brittle fiber-ductile metal matrix composites (Cooper and Kelly, 1967) (e.g., tungsten wire-copper matrix composites). This is because of the increased volume of ductile matrix material

involved in shear flow at the interface which gives rise to a fiber pull-out force proportional to the fiber diameter d. Fibers of large d are also beneficial for brittle fiber-brittle matrix composites (Piggott, 1970; Wells and Beaumont, 1985 b) because it increases ld and /po due to large lc (as envisaged from Eq. (6-8)) provided all other properties of fiber and interface remain the same. Unfortunately, the tensile strength and elastic modulus of many types of fibers show a systematic decrease with increasing d (Metcalfe and Schmitz, 1964; Kelly, 1970). This problem may be overcome by using bundle fibers impregnated with polymers prior to being incorporated in a resin matrix (Fila et al., 1972; Kim and Mai, 1991 c). The second approach includes fiber coatings with appropriate polymers, delamination arresters and promoters and reduction of shrinkage stresses in the matrix. These techniques are described in the following sections. A more detailed review on these thoughening methods is given by Kim and Mai (1991 a).

6.5.1 Fiber Coating and Intermittent Bonding Concept

One of the most effective methods in controlling the interface to enhance the fracture toughness of polymer matrix composites is the application of polymer coatings, either fully or intermittently along the fiber. The method is simple to use with many practical fiber composites and allows a direct evaluation of the fracture behaviors of composites with and without the coating. The principal effect of altering the interfacial properties by fiber coating is to modify the mode of failure and thus the potential energy absorption capacity, which in turn determines the fracture toughness of composites.

6.5 Toughening Methods with Controlled Interfaces

275

6.5.1.1 Intermittent Bonding Concept

The intermittent fiber bonding method follows the early work of Mullin et al. (1968) and Gatti etal. (1969) where coatings on boron fibers are found to be effective in isolating fiber fracture by causing debonding at the interface immediately adjacent to the failure site. The corn-cob shape surface of the boron fiber causes non-uniform application of coatings on the fiber with a resulting pattern of intermittently coated and uncoated regions. This condition provides bonding in the low spots, while high points are shielded from the matrix by the coating which inhibits a continuous and sudden crack growth at the interface. Marston et al. (1974) and Atkins (1975) further developed the intermittent bonding concept and argued that providing there were enough regions of high interfacial shear stress to ensure that the rule of mixtures (RoM) strength was picked up, the rest of the composite could have quite weak interfacial bonds which served to blunt the running crack by the Cook-Gordon debonding mechanism (1964). If a composite is laid up randomly with respect to weak and strong regions (Fig. 6-18) both high strength and high toughness should be simultaneously obtained. Atkins (1975) has achieved a remarkable 400% improvement in fracture toughness with less than 10% loss of tensile strength in the unidirectional BFRP using an 80% coating of polyurethane varnish (PUV), while the improvement with silicone vacuum fluid (SVF) coating is only 10 to 15% even though i b values of the fibers coated with the two coating materials were similar. Consequently, similar i b does not mean similar fracture toughness. The explanation lies with the Cook-Gordon debonding mechanism which takes place with PUV giving a

High strength region

Low strength region

l Figure 6-18. Crack propagates preferentially linking the coated fiber regions of low interfacial bond strength. After Atkins (1975).

longer fiber pull-out length, whereas this mechanism is absent with the SVF coating. Atkins and Mai (1976) confirmed that the intermittently bonded composites could be favorably used in adverse environmental conditions such as hygrothermal ageing. Later, Mai (1983, 1988) and Mai and Castino (1984, 1985) performed extensive studies with KFRP using coatings based on SVF and a blend of polyester-polyether resins (Estapol). They investigated the effects of hygrothermal ageing, percentage coating over a repeated fiber length, fatigue damage, strain rate and temperature on tensile strength, modulus, impact fracture toughness and pull-out toughness of the composite. They showed that the fracture toughness of composites with Estapol coated fibers could be increased by some 200 to 300%, particularly at high temperatures and low strain rates, without causing any significant reduction in strength properties.

276

6 Interfaces in Composites

6.5.1.2 Enhanced Energy Absorption: Fiber Coating Technique

Novak (1969) and Bader et al. (1973), basing their work on surface treated CFRP, confirmed that a strong interface favored a brittle fracture mode with relatively low energy absorption, but a weak interface favored a multiple shear mode with high energy absorption. Similarly, Beaumont and Phillips (1972) found that carbon fibers coated with a silicone fluid resulted in the fibers being surrounded by an inert film which reduced i b and thus increased the toughness of CFRP. Hancox and Wells (1977) were also able to improve the toughness of CFRP by some 100% with a silicone rubber coating at the expense of up to 60% loss of flexural strength depending on the coating thickness (Fig. 6-19). Because fiber pull-out is a major source of fracture toughness for most continuous fiber-polymer matrix composites it is necessary to have a sufficiently high i f , in addition to a low r b , so that the work of frictional pull-out can be enhanced. The frictional pull-out work was investigated by Sung et al. (1977) where the reinforcing fibers were coated with a thin layer of a strain-rate sensitive viscous fluid, e.g., SVF.

0.1 0.2 Coating thickness in ^m

Figure 6-19. Fracture toughness (o) and flexural strength (•) of silicone rubber coated CFRP as a function of coating thickness. After Hancox and Wells (1977).

They proposed that at a given strain rate the viscous shear stress acting on the fibers during pull-out and hence the pull-out toughness could be maximized by selecting appropriate coatings of high fluid viscosity and small thickness. Other investigators successfully applied a range of other polymer coatings to many different types of composites: polysulfone, polybutadiene and silicone rubber on CFRP (Hancox and Wells, 1977; Williams and Kousiounelos, 1978); latex coatings (e.g., polybutyl acrylate, polyethyl acrylate, etc.) on GFRP (Pfeiffer and Nielson, 1979); poly vinyl alcohol (PVAL) on KFRP and CFRP (Kim and Mai, 1989 a, 1989 b, 1991b; Kim et al., 1989 a); anhydride copolymers (e.g., polybutadiene-co-maleic anhydride and polymethylvinylether-co-maleic anhydride (Crasto etal., 1988)) and acrylonitrile copolymers (e.g., acrylonitrile/methylacrylate and acrylonitrile/glycidylacrylate (Bell et al., 1987)) on CFRP. In particular, Pfeiffer and Nielsen (1979) achieved a significant 600% increase in impact toughness of GFRP with negligible strength reduction using colloidal latex particles which were attracted to glass fibers by electrostatic forces to form a rubbery acrylic polymer layer of uniform thickness. They proposed that the impact toughness is a function of both thickness and glass transition temperature (Tg) of the coating: the toughness was maximum when the coating had a low Tg and a thickness of about 0.2 |xm. Kim and Mai (1991 b) also reported that PVAL coating on fibers increases the impact toughness of KFRP and CFRP by more than 100%, particularly at sub-zero temperatures, without significant loss of flexural strength and interlaminar fracture toughness. The thermoplastic coatings have advantages over other coatings in that they would form a micro-ductile layer at the interface (Dauksys, 1973) which satisfacto-

6.5 Toughening Methods with Controlled Interfaces

rily function as a stress relief medium in reducing the residual stresses caused by matrix shrinkage occurring during curing (Arridge, 1975; Marom and Arridge, 1976); and as a crack inhibitor or arrester allowing large debonding and fiber pull-out to take place thus making substantial contributions to the total toughness of the composites. In summary, polyurethane coatings are effective in improving the fracture toughness of BFRP and KFRP. Other coatings, including silicone products on CFRP and GFRP; PVAL coatings on CFRP and KFRP, have also shown to be quite promising. However, the selection of an appropriate coating material for a given composite seems to be a black art. Though no rules are established, some points of generalization may be said about the conditions which determine a good coating for improving the total toughness of a composite with little loss of strength. These points are described below, (a) If the coating remains fluidic or rubbery at the interface after cure (e.g., SVF and Estapol) a high viscosity coating is preferred because it can increase the frictional shear work when the fibers are being pulled out (Sung et al., 1977). (b) If the coating becomes a rigid interlayer after cure it should be more ductile and compliant than the matrix (e.g., thermoplastic coatings in thermoset-based matrices) and should provide a weak bonding at the interface while retaining sufficiently high frictional bonding, (c) The thickness of the coating should be kept thin compared to the fiber diameter to eliminate any reductions of composite strength and modulus, particularly for coatings of which Tb are very low. Systematic reductions in flexural strength and ILSS with increasing coating thickness (e.g., silicone rubber coating (Hancox and Wells, 1977) and polyvinyl acetate (PVA) coating (Kim and

277

Mai, 1991b)) have been reported. Thick coatings also decrease the maximum V{ allowable for maximum composite strength without contact between neighboring fibers, (d) There are different views concerning the reactivity and miscibility of the coating with the resin matrix during curing. Sung etal. (1977) suggested that the coating should form and remain in a discrete layer at the interface without reaction with the composite constituents. However, a certain degree of chemical reaction between the coating and matrix may increase the frictional shear stress (Mai and Castino, 1984). Partial or complete mixing of the coating material with the matrix, e.g., carboxyl-terminated butadiene acrylonitrile (CTBN) rubber in epoxy (Gerard, 1988), produces composites with little modified interfaces and is undesirable. 6.5.2 Delamination Promoters The energy absorption of a laminate composite during fracture can be increased by promoting controlled delamination when the interlaminar bond strength is weakened. The effect of a weak interface is that, depending on the orientation of the interface relative to the running crack, the triaxial tension operative at a crack tip causes the crack to be arrested by delamination (Almond et al., 1969) thus reducing the stress concentration as envisaged from the Cook-Gordon (1964) mechanism. Following this idea Favre (1977) successfully increased the fracture toughness of CFRP to three times that of the laminates without delamination promoters by embedding nylon sheets between laminae; but this is at the expense of some 25% reduction in the ILSS (Fig. 6-20). Mai etal. (1982a) suggested that perforated films are more effective than full film due to the co-existence of weak and strong ILSS in the regions of film

278

6 Interfaces in Composites

200

100 80

H

I = 3 100

Is O Q.

60 § c

40 u

taken in applying it to composites for which delamination is a critical life-limiting failure mode. Excessive delamination growth may cause unacceptable reduction in the composite stiffness and impair the mechanical performance and structural integrity of the composite.

20 0 1

2 3 U 5 Number of nylon sheets

Figure 6-20. Impact fracture toughness (o) and ILSS (•) of CFRP with varying number of nylon sheets as delamination promoters. After Favre (1977).

and perforation respectively. In the weak regions delamination takes place, resulting in crack front bifurcation and sub-crack propagation along the laminar interface which promotes energy absorption (Jea and Felbeck, 1980). In contrast, delamination is discouraged and shear stress transfer is permitted between laminae in the strong regions which consequently enabled the original ILSS to be maintained. In a similar study using several other films as delamination promoters, Jang et al. (1987) concluded that the fracture mode, energy absorption mechanisms and hence the effectiveness of the toughening method depend largely on the loading directions relative to the laminar plane and other testing parameters like loading speed and span-todepth-ratio in bending. The intermittent interlaminar bonding concept has also been applied to cement mortars by inserting perforated papers as delamination promoters (Mai et al., 1982 b). The large improvement (200 to 800%) in fracture work is accompanied by a 30 to 50% reduction in flexural strength. Whilst this method is quite effective in improving the transverse fracture toughness of laminated composites, care must be

6.5.3 Delamination Arresters

The delamination resister or arrester technique uses a tough strip interleaved between plies to improve the interlaminar fracture toughness for the propagation of delamination cracks. Based on earlier crack arrest concepts using buffer strips for a through crack (Hess et al., 1977), Chan et al. (1986) have successfully applied adhesive layers having low modulus and high elongation at delamination-prone free edges (Fig. 6-21 (a)) to suppress delamination growth by reducing the transverse or through-thickness interlaminar stresses.

(a)

^ Adhesive \ / layers

(b)

Stitched fibers

Figure 6-21. Schematic configuration of laminate composites with (a) interleaved adhesive layers between plies and (b) stitched fibers in the throughthickness direction as delamination resisters. After Chan (1986) and Chung et al. (1986).

6.6 Residual Stresses

This effectively increases the interlaminar fracture toughness of CFRP by almost ten times that without delamination resisters depending on the thickness and types of interlayer used (Ishai et al., 1988; Sela et al., 1989) (Table 6-4). Delamination resisters made from ductile short fibers (e.g., Kevlar fiber mat) and an adhesive (Browning and Schwartz, 1986) require extra energy for delamination growth due to interfacial debonding and fiber pull-out which cannot occur in an interlayer made only with an adhesive. The effectiveness of the delamination arrester concept has been demonstrated quite convincingly under cyclic fatigue (Chan, 1986) and hygrothermal ageing conditions (Evans and Master, 1985; Ishai et al., 1988). Stitching fibers (Mignery et al., 1985; Chung et al., 1986) and addition of whiskers through the thickness of laminates (Fig. 6-21 (b)) is an alternative method to improve the ILSS and hence the delamination fracture resistance of fiber composites. In addition a high through-thickness strength can be produced by the third di-

Table 6-4. Mode I and mode II interlaminar fracture toughnesses (GY and Gn) of CFRP with interleaved adhesive layers as delamination arresters. After Sela et al. (1989). Types of adhesive layer

Adhesive thickness (mm)

GT (kJ/m2)

G n (kJ/m 2 )

0.527

Control

0

0.193

Tuff-ply Tuff-ply Tuff-ply

0.04 0.08 0.11

0.444 0.575 0.754

FM73

0.12

0.975

1.84

FM300 FM300 FM300 FM300 FM300

0.1 0.26 0.3 0.68 1.1

1.14 1.47 1.27 1.48 1.78

1.77 2.23 2.01 2.32 1.65

1.15 1.7 ' 2.61

279

rection fibers effectively giving a 3-D fiber composite. However, stitching may damage the inplane fibers and cause voids to be entrapped, particularly when carbon fiber prepregs are used (Chung etal., 1986). Both light and heavy stitchings are undesirable and an optimum amount of stitching is required to achieve good ILSS and high interlaminar fracture toughness. Recent developments of textile structural composite concept of totally integrated advanced fabric systems (e.g., weaving, knitting and braiding techniques) for primary structural applications have paved the way to achieve almost no delamination and perfect structural integrity in advanced fiber composites (Chou and Ko, 1989). The delamination resister concept is not identical to the delamination promoter approach with regard to both the toughening mechanisms and the primary direction of crack propagation relative to the laminar interfaces. Delamination resisters are intended to improve the interlaminar fracture toughness by suppressing delamination growth so that they should have high ductility and low modulus to help reduce the interlaminar stresses. However, delamination promoters are aimed at increasing the transverse fracture toughness through extra energy absorption required for the arrest and bifurcation of transverse cracks at the interface, and weak interlaminar bonding is needed to promote delamination. A common requirement for both methods to work is that the modifying layer must be thin enough as not to affect the inplane strength and stiffness.

6.6 Residual Stresses 6.6.1 Origin of Residual Stresses

Residual stresses are inherent in fiber composites but they are often ignored in

280

6 Interfaces in Composites

both analytical and experimental works on interfacial effects. This oversight may lead to incorrect interpretations of material properties and mechanical behavior. The primary origins of residual stresses in fiber composites are twofold: thermal and mechanical. The thermal origin is the most prevalent and arises from the different thermal expansion coefficient (CTE) of fiber and matrix. The differential shrinkage of the constituents upon cooling from processing temperature to ambient causes a high level of residual stresses to build up within the composite and this is especially so in those composites with ceramic and metal matrices. This source of residual stress also includes the shrinkage stress associated with the curing reaction of the polymer matrix and its crystallinity around the fiber. Sintering residual stresses also exist in ceramic matrix composites. The three-dimensional distribution of these stresses is very complicated and depends strongly on the fiber distribution and V{ and the elastic moduli of the constituents. Two-dimensional photoelastic study (Koufopoulos and Thecaris, 1969) shows that, for a square array of circular inclusions in a matrix, the residual stresses in the region between adjacent fibers are compressive while they are tensile in the resin pocket region surrounded by fibers. At very small fiber spacing (or high Vf) and small modulus ratio (Em/E{) the normally tensile stresses may become compressive, generating hydrostatic compression around the inclusion. A rough estimate of the compressive residual stress in the radial direction is obtained by calculating shrinkage fits for a single fiber embedded in a coaxial cylindrical matrix material (Harris, 1978): (am-af)AT£n af AT)(1 + a - vf)a

where am and af are the CTE of fiber and matrix, and AT is the temperature difference and is negative. In fact, the amount of residual stress is influenced not only by the cure temperature but by the whole cure cycle (Kim et al., 1989 b). The differential thermal contraction also results in the fibers being put into compression along the length which increases the tendency for fiber buckling and produces interfacial shear stresses leading to debonding (Rohwer and Jiu, 1986; Rodriguez, 1989). The compressive residual stresses in the fiber direction have been measured extensively in recent years for various combinations of fiber and polymer matrix, e.g., CF-polyetheretherketone (PEEK) matrix (Galiotis et al., 1988; Young et al., 1989) and Kevlar fiberepoxy matrix (Jahankhani and Galiotis, 1991) using a novel technique based on laser Raman spectroscopy. Since the peak positions of the Raman-active bands in the fiber are sensitive to the magnitude of the applied strain (Tuinstra and Koenig, 1970), the fiber residual strain can be estimated from the plots of fiber strain versus composite strain obtained on a model composite. Further extension of this technique allows the measurements of interfacial shear stress distributions along the fiber length at varying applied strain levels (Jahankhani and Galiotis, 1991). There is another type of thermal residual stress, namely the macro-residual stress or lamination residual stress (Chamis, 1969), in laminates where plies are oriented at different angles. They arise from the differences in the longitudinal and transverse thermal contraction within each individual ply: a tensile stress develops in the 90° plies with a corresponding compressive stress in

2y]

;(am-af)AT£n

(6-29)

6.6 Residual Stresses

the 0° plies. These lamination residual stresses promote the onset of transply cracking (Doner and Novak, 1969) and delamination (Jeronimidis and Parkyn, 1988), and causes serious warping in asymmetric laminates even in the absence of external loads (Kim and Hahn, 1989; Jun and Hong, 1990). The lamination residual stress can be reduced within a given temperature range by selecting the optimum stacking sequence and ply orientation (Ishikawa et al, 1989). The other major source of residual stress is mechanical in origin and is present mainly in ductile metal matrix composites due to the difference in flow stress between the components (Metcalfe, 1974). This is important when the composite is subjected to mechanical loading enough to plastically deform one or both components. Another possible source of residual stress in composites is due to phase transformation with accompanying volume changes. Since the transforming component is normally mechanically restrained, the surrounding material is cracked and/or locally strained by this volume change. A good example is the partially stabilized zirconia (ZrO2) containing small amounts of MgO, CaO or Y 2 O 3 which undergoes a 3.25% volume expansion during cooling below approximately 1000 °C due to transformation from the tetragonal phase to the monoclinic phase (Porter and Heuer, 1977). By careful control of particle size and processing a very high fracture toughness can be achieved.

281

model composite several times when compared to the control without coating. This is because the coating reduces the stress concentration around the inclusion which is a function of the shear modulus and Poisson ratio of the coating. They showed photoelastically that a plasticized epoxy resin and a silicone rubber coating (of thickness less than 10% of wire diameter) on the steel wire produce almost zero shrinkage stresses in the radial and circumferential directions, and a crack initiated within the soft coating is blunted and remain stable under the applied transverse loading. Similar effects of ductile interlayers on fiber composites have been reported (Kardos, 1981). Piggott (1981) has analyzed the fracture toughness R of composites due to pull-out of broken fibers at a constant frictional shear stress (if = \i qo% taking into account the residual clamping stress q0 in the radial direction caused by the shrinkage of matrix: R=

Vfdh2A2(-g0)

(6-30)

where A and h are parameters determined in the relationship between fiber strength <7* and length l\af = Al~h. From a plot of R versus q0 (Fig. 6-22), it is seen that the :600

-

200 -

6.6.2 Control of Residual Stresses In studying the stress distributions in an epoxy matrix surrounding a cylindrical inclusion, Marom and Arridge (1976) showed that a soft interlayer at the interface increases the transverse strength of a

\CFRP

10

\GFRP

20 30 -q0 in MPa

40

50

Figure 6-22. Predicted fracture toughness of CFRP and GFRP with varying matrix shrinkage stress q0. After Piggott (1981).

282

6 Interfaces in Composites

toughness can be enhanced by controlling q0 without deteriorating other mechanical properties. Indeed, in a series of experimental studies, Piggott and co-workers (Lim etal., 1984; Lam and Piggott, 1989, 1990) have successfully reduced shrinkage stresses by using special expanding monomers developed originally by Bailey etal. (1977). Epoxy resins, as matrices of CFRP pultrusions, were copolymerized with small amounts (5 to 10%) of dinorbornene (or tetramethyl) spiro ortho carbonates which could balance the shrinkage of matrix by expansion through ring opening reactions. The resulting composites are reported to have as little as one third of the usual residual stress and the Izod impact toughness of these composites are up to 50% higher than the equivalent composites without the spiro (Fig. 6-23) depending on the types of epoxy employed. Improved fatigue properties and better resistance to water absorption are also noted. Shrinkage stresses are also reduced using frozen aqueous solutions of soluble solids (e.g., sodium chloride glucose and

10 20 Spiro content in wt.%

20 30

Figure 6-23. Impact fracture toughness (o) and ILSS (•) of CFRP pultrusions as a function of expanding monomer (spiro) content. After Lim et al. (1984).

polyvinyl alcohol) and miscible liquids (e.g., glacial acetic acid and pure acetenyl acetone) as matrices for carbon fibers (Lim and Piggott, 1989). The maximum fracture toughness of these fiber composites is achieved when the interface is under slight compression. This observation is in agreement with the theory that the fracture toughness is maximum when the shrinkage stress is just equal to the Poisson contraction stress, i.e., the denominator of Eq. (6-30) is close to zero. In contrast to CFRP, the expanding monomer has little beneficial effect on GFRP and KFRP. It actually reduces the toughness by about 30% of GFRP (Lim et al, 1984). Whilst shrinkage stress control is useful to improve fracture toughness of fiber composites, many questions remain unanswered. Since the fracture toughness expression of Eq. (6-30) is based on the fiber pull-out mechanism, the effects of reduced shrinkage stress on other failure mechanisms such as interface debonding and post-debonding friction, have not been specifically addressed. Regarding the effect of fi on fracture toughness, there must be an optimum value which maximizes the pull-out toughness. If ji is too high fiber pull-out is inhibited, but if/x is too low only a small frictional energy is dissipated. The inefficiency of the expanding monomer in GFRP and KFRP needs much further work to explain the observed results. Further, the shrinkage stress is found to depend more on the state of matrix cure rather than on the matrix material used (Piggott etal, 1985). This means that a special expanding additive must be developed and its optimum amount determined for a given cure cycle if reduction in shrinkage stress is to be effective for a specific combination of matrix, hardener and reinforcing fiber. Apart from these effects, resins which cure with zero volume shrink-

6.8 Acknowledgements

age or expansion are highly desirable for various practical applications such as high performance adhesives, coatings, precision castings, dental fillings, binders for solid propellants, etc. Therefore, triggered by the early work of Bailey et al. (1977), shrinkage control in polymers especially epoxies (Shimbo et al, 1989; Ochi et al., 1989) and polyesters (Bucknall et al, 1991) has been a subject of much continuing research.

6.7 Concluding Remarks Much of the advancement in understanding and controlling interfaces has come from developments in modern analytical techniques applied to the fiber and composite fracture surfaces. In addition, many experimental techniques have been devised to assess the stress states and bond quality at the interface. Among these direct methods fiber fragmentation, fiber pull-out and fiber push-out tests are the most popular. However, correct interpretation of experimental results needs a proper analytical model of micromechanics developed for each test method. The failure of a composite involves fractures of the load-bearing fibers and the matrix as well as the complex combination of cracks propagated along the fiber-matrix and laminar interfaces. In a brittle fiberbrittle matrix composite a strong interfacial bond is necessary for efficient load transfer and high composite stiffness and strength. However, it leads to a more brittle and catastrophic failure where cracks propagate right through the fiber and matrix material with an energy dissipation being limited to the work of creating new fracture surfaces. In contrast, a weak bonding can enhance the fracture work absorbed through various failure mechanisms associated with interfacial debond-

283

ing. Therefore, depending on the end applications, a suitable compromise of interfacial bond (or composite) strength and composite toughness is always required in practical design of the interface to achieve the desired microfailure mechanisms. For example, surface treatments of GF's, CF's and aramid fibers, and reaction barrier coatings on ceramic fibers of A12O3, B/W and SiC, as discussed in Sec. 6.3, are designed primarily to enhance the interfacial bonding and ultimately to provide good composite strengths. In contrast, the experimental techniques including fiber coating and intermittent bonding with appropriate polymeric materials, delamination promoters, delamination resisters and shrinkage reduction in the matrix as presented in Sees. 6.5 and 6.6 are aimed at improving the composite fracture toughness without impairing strength properties whether in transverse or interlaminar fracture. However, the most important consideration in all forms of interface control is that the modified interface must be compatible with the composite fabrication processes and actual service environments. The study of interfaces in terms of the thermodynamic, physio-chemical and mechanical properties and their effects on the macro-mechanical performance and structural integrity of fiber composites, particularly those based on ceramic and metal matrices, will continue to be a major area of composites research towards new discovery and solving existing practical problems.

6.8 Acknowledgements We wish to thank the Australian Research Council (ARC) for continuing support of this work which is a part of the project "High Strength-High Fracture

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Toughness Fiber Composites with Controlled Interfaces." We are also greatly indebted to past and present colleagues: R. Asquith, A. G. Atkins, C. Baillie, F. Castino, B. Cotterell, D. K. Felbeck, R. M. L. Foote, Y. C. Gao, M. Hakeem, R. Lord, I. M. Low, and L. M. Zhou for their contributions to this chapter in the form of original work and/or stimulating discussions. J. K. K. was supported by an Australian Postgraduate Research Award, the Peter Nicol Russell Scholarship and a Junior Research Fellowship by the ARC when this work was done.

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Lim, J. T., Piggott, M. R. (1989), /. Mater. Sci. 24, 234-238. Lim, J. T., Piggott, M. R., Bailey, W. J. (1984), SAMPE Quarterly 15, 25-30. Low, I. M., Mai, Y. W. (1991), Handbook of Ceramics: Cheremisinoff, N. P. (ed.), Vol. 2. New York: Marcel Dekker; pp. 105-160. Mai, Y. W. (1983), /. Mater. Sci. Lett. 2, 723-725. Mai, Y. W. (1988), J. Mater. Sci. Lett. 7, 581-582. Mai, Y.W., Castino, F. (1984), /. Mater. Sci. 19, 1638-1655. Mai, Y. W, Castino, F. (1985), / Mater. Sci. Lett. 4, 505-508. Mai, Y.W., Cotterell, B., Lord, R. (1982a), Proc. Int. Conf. Composite Mater. IV, Progress in Science and Engineering of Composites: Hayashi, T., Kawata, K., Umekawa, S. (eds.). Amsterdam: North-Holland; pp. 271-277. Mai, Y. W., Hakeem, M., Cotterell, B. (1982 b), Cem. Concr. Res. 12, 661-663. Mandel, J. A., Wei, S., Said, S. (1987), ACI Mater. J. 84, 101-109. Mandell, J. R, Chen, J. H., McGarry, F. J. (1980), Int. J. Adhesion Adhes. 1, 40-44. Marks, B. S., Mauri, R. E., Bradshaw, W. G. (1975), 12th Bienn. Conf. on Carbon, Extended Abstracts. Pittsburgh, Penn.: American Carbon Committee; p. 337. Marom, G., Arridge, R. G. C. (1976), Mater. Sci. Eng. 23, 23-32. Marshall, D. B., Oliver, W. C. (1987), J. Am. Ceram. Soc. 70, 542-548. Marshall, P., Price, J. (1991), Composites 22, 53-57. Marston, T. U., Atkins, A. G., Felbeck, D. K. (1974), /. Mater. Sci. 9, 447-455. Martineau, P., Pailler, R., Lahaya, M., Naslain, R. (1984), J. Mater. Sci. 19, 2749-2770. McAlea, K. P., Besio, G.I (1988), Polym. Composites 9, 285-290. McKee, D. W. (1970), Carbon 8, 131-139. Metcalfe, A. G., Schmitz, G. K. (1964), ASTM Proc. 64, 1075. Mignery, L. A., Tan, T. M., Sun, C. T. (1985), Delamination and Debonding: Johnson, W. S. (ed.). ASTM STP876; 371-385. Miller, B., Muri, P., Rebenfeld, L. (1987), Composite Sci. Technol. 28, 17-23. Mooney, R. D., McGarry, F. J. (1959), 14th Ann. Tech. Conf, Reinf Plast./Comp. Washington, D.C.: Society of Plastic Industries; 12 E. Morin, D. (1976), J. Less-Common Met. 47, 207-214. Mullin, J. V., Berry, J. M., Gatti, A. (1968), J. Composite Mater. 2, 82-103. Munro, M., Lai, C. P. Z. (1988), /. Mater. Sci. 23, 3129-3136. Naaman, A. E., Shah, S. P. (1976), ASCE J. Struct. Div. 102, 1537-1548. Nardone, V. C , Prewo, K. M. (1986), Scripta Metall. 20,43-48.

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Netravali, A. N., Henstenburg, R. B., Phoenix, S. L., Schwartz, P. (1989 a), Polym. Composites 10, 226241. Netravali, A. N., Topoleski, L. T. T., Sachse, W. H., Phoenix, S. L. (1989 b), Composite Sci. Technol. 35, 13-29. Nicholas, M. (1968), J. Mater. Sci. 3, 571-576. Noone, M. X, Feingold, E., Sutton, W. H. (1969), Interfaces in Composites. ASTM STP452; 59-89. Norita, T, Matsui, X, Matsuda, H. S. (1986), Proc. Int. Conf. Composite Interfaces I, Composite Interfaces: Ishida, H., Koenig, X L. (eds.). New York: Elsevier Sci. Publ.; pp. 123-132. Novak, R. C. (1969), Composite Materials: Testing and Design. ASTM STP460; 540-549. Ochi, M., Yamazaki, K., Shimbo, M. (1989), /. Mater. Sci. 24, 3189-3195. Outwater, X D., Murphy, M. C. (1969), 24th Ann. Tech. Conf Reinf Plast./Comp. Washington, D.C.: Society of Plastic Industries; 11 D. Pally, I., Stevens, D. (1989), /. Adhesion Sci. Technol. 3, 141-153. Pan, T. Y, Robertson, R. E., Filisko, F. K. (1988), J. Mater. Sci. 23, 2553-2563. Peiffer, D. G., Nielsen, L. E. (1979), /. Appl. Polym. Sci. 23, 2253-2264. Penn, L. S., Larson, F. (1979), J. Appl. Polym. Sci. 23, 59-73. Penn, L. S., Lee, S. M. (1989), / Composite. Technol. Res. 11, 23-30. Piggott, M. R. (1970), J. Mater. Sci. 5, 669-675. Piggott, M. R. (1981), Proc. Int. Conf Fracture 5, Advances in Fracture Research: Francois, D. (ed.). New York: Butterworths; pp. 465-472. Piggott, M. R. (1987), Composite Sci. Technol. 30, 295-306. Piggott, M. R., Chua, P. S., Andison, D. (1985), Polym. Composites 6, 242-248. Pitkethly, M. X, Doble, X B. (1990), Composites 21, 389-395. Plueddemann, E. P. (1988), Proc. Int. Conf Composite Interfaces II, Interfaces in Polymer, Ceramic and Metal Matrix Composites: Ishida, H. (ed.). New York: Elsevier Sci. Publ.; pp. 17-33. Plueddemann, E. P., Stark, G. L. (1980), 35th Ann. Tech. Conf Reinf Plast./Comp. Washington, D.C.: Society of Plastic Industries; 20 B. Porter, D. L., Heuer, A. H. (1977), J. Am. Ceram. Soc. 60, 183-184. Prewo, K., McCarthy, G. (1972), /. Mater. Sci. 7, 919-928. Riess, G., Bourdeux, M., Brie, M., Jouquet, G. (1974), Carbon Fibres - Their Place in Modern Technology. London: Plast. Inst; p. 52. Rodriguez, E. L. (1989), /. Mater. Sci. Lett. 8, 116118. Rohwer, K., Jiu, X. M. (1986), Composites Sci. Technol. 25, 169-186. Ryder, C , Vidoz, A., Crossman, F , Camahort, X (1970), /. Composite Mater. 4, 264-272.

288

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Sarin, V. K., Ruhle, M. (1987), Composites 15, 16-24. Sela, N., Ishai, O., Banks-Sills, L. (1989), Composites 20, 257-264. Shafry, N., Brandon, D. G., Terasaki, M. (1989), in: Euro-Ceramics, Vol. 3 (Engineering Ceramics), de With, G., Terpstra, R. A., Metselaar, R. (eds.): London, Elsevier; pp. 453-457. Shetty, D. K. (1988), /. Am. Ceram. Soc. 71, C107C109. Shimbo, M., Ochi, M., Inamura, T., Inoue, T. (1985), /. Mater. Sci. 20, 2965-2972. Sigl, L. S., Evans, A. G. (1989), Mech. Mater. 8,1-12. Stang, H., Shah, S. P. (1986), /. Mater. Sci. 21, 953957. Strife, X, Prewo, K. (1982), J. Mater. Sci. 17, 65-72. Subramanian, R. V., Jakubowski, J. J. (1978), Polym. Eng. Sci. 18, 590-600. Sung, N. H., Jones, T. J., Suh, N. P. (1977), /. Mater. Sci. 12, 239-250. Sung, N. H., Kahl, A., Ni, S., Sung, C. S. P., Chin, I. J. (1981), 36th Ann. Techn. Conf. Reinf Plast.) Comp. Washington, D.C.: Society of Plastic Industries; 2 B. Sutton, W. H. (1966), 21st Ann. Tech. Conf. Reinf. Plast./Comp. Washington, D.C.: Society of Plastic Industries; 18 A. Takayanagi, M., Kajiyama, T., Katayose, T. (1982), /. Appl. Polym. Sci. 21, 3903-3917. Taya, M., Daimaru, A. (1983), J. Mater. Sci. 18, 3105-3116. Tetelman, A. S. (1969), Composite Materials: Testing and Design. ASTM STP 460; 473-502. Thomason, J. L. (1990), Polym. Composites ll,\05113. Tirosh, J. (1973), ASME Trans. E, J. Appl. Mech. 40, 785-790. Tsai, H. C , Arocho, A. M., Gause, L. W. (1990), Mater. Sci. Eng. A126, 295-304. Tuinstra, R, Koenig, J. L. (1970), J. Composite Mater. 4, 492-499. Vaughhan, D. J. (1978), Polym. Eng. Sci. 18, 167169. Verpoest, I., Springer, G. S. (1988a), /. Reinf Plast. Composites 7, 2-22. Vidoz, A., Camahort, X, Crossman, F. (1969), J. Composite Mater. 3, 254-261. Wake, W. C. (1978), Polym. 19, 291-308. Wawner, R, Nutt, S. (1980), Ceram. Eng. Sci. Proc. 1, 709-719. Weihs, T. P., Sbaizero, O., Luh, E. Y., Nix, W. D. (1991), J. Am. Ceram. Soc. 74, 535-540. Wells, J. K., Beaumont, P. W. R. (1982), Composite Materials: Testing and Design: Daniel, I. M. (ed.). ASTM STP 787; 147-162. Wells, J. K., Beaumont, P. W. R. (1985a), /. Mater. Sci. 20, 1275-1284. Wells, J. K., Beaumont, P. W. R. (1985 b), /. Mater. Sci. 20, 2735-2749. Whatley, W. X, Wawner, R E. (1985), J. Mater. Sci. Lett. 4, 173-175.

Williams, J. G., Donnellan, M. E., James, M. R., Morris, W. L. (1990), Mater. Sci. Eng. A126, 305312. Williams, X H., Kousiounelos, P. N. (1978), Fibre Sci. Technol. 11, 83-88. Wimolkiatisak, A. S., Bell, X P. (1989), Polym. Composites 10, 162-172. Yajima, S., Hasegawa, Y, Hayashi, X, Iimura, M. (1978), J. Mater. Sci. 13, 2569-2576. Yajima, S., Okamura, K., Matsuwa, X, Hasegawa, Y, Shishido, T. (1979), Nature 279, 706-707. Yamamoto, M., Yamada, S., Sakatani, Y, Taguchi, M., Yamaguchi, Y (1971), 1st Carbon Fibre Conf. London: Plast. Inst., p. 179. Yang, X M., Jeng, S. M., Yang, C. X (1991), Mater. Sci. Eng. A138, 155-167. Young, R. X, Day, R. X, Zakikhani, M., Robinson, I. M. (1989), Composites Sci. Technol. 34,243-258. Yuan, L. Y, Shyu, S. S., Lai, X Y (1991), J. Appl. Polym. Sci. 42, 2525-2534. Zhou, L. M., Kim, X K., Mai, Y W (1992), J. Mater. Sci. 27, 3155-3166.

General Reading Adamson, A. W. (1982), Physical Chemistry of Surfaces. New York: John Wiley. Chou, T. W, Ko, R K . (1989), Textile Structural Composites, Composite Materials Series Vol. 3. Amsterdam: Elsevier. Davidge, R. W (1989), in: Application of Fracture Mechanics to Composite Materials, Composite Materials Series Vol. 6: Friedrich, K. (ed.). Amsterdam: Elsevier; pp. 547-569. DeBolt, H. E. (1982), in: Handbook of Composites: Lubin, G. (ed.). New York: Van Nostrand Reinhold; pp. 171-195. Delmonte, X (1981), in: Technology of Carbon and Graphite Fiber Composites. New York: Van Nostrand Reinhold; pp. 171-197. Donnet, X B., Bansal, R. C. (1984), in: Carbon Fibers. New York: Marcel Dekker; pp. 109-161. Ehrburger, P., Donnet, X B. (1985), in: Strong Fibers, Handbook of Composites, Vol. 1: Watt, W, Perov, B. V. (eds.). Amsterdam: Elsevier Sci.; pp. 577603. Hull, D. (1981), An Introduction to Composite Materials. Cambridge: Cambridge University Press. Kelly, A. (1973), Strong Solids. Oxford: Clarendon Press. Lee, S. M. (1989), in: Reference Book for Composites Technology 2: Lee, S. M. (ed.). Lancaster, PA. USA: Technomic; pp. 81-103. Metcalfe, A. G. (1974), Interfaces in Metal Matrix Composites, Composite Materials Vol. 1. New York: Academic Press.

6.9 References

Ochiai, S. (1989), in: Application of Fracture Mechanics to Composite Materials, Composite Materials Series Vol. 6: Friedrich, K. (ed.). Amsterdam: Elsevier; pp. 491-545. Plueddemann, E. P. (1974), Interfaces in Polymer Matrix Composites, Composite Materials Vol. 6. New York: Academic Press. Riggs, D. M., Shuford, R. X, Lewis, R. W. (1982), Handbook of Composites, Lubin, G. (ed.). New York: Van Nostrand Reinhold; Chap. 11. Scolar, D. A. (1974), in: Interfaces in Polymer Matrix Composites, Composite Materials Vol. 6: Plueddemann, E. P. (ed.). New York: Academic Press; pp. 217-284.

289

Taya, M., Arsenault, R. J. (1989), Metal Matrix Composites

-

Thermomechanical

Behaviour.

Oxford:

Pergamon Press. Taylor, R. (1991), SAMPE J. 27, 6-8. Warren, R., Sarin, V. K. (1990), in: Application of Fracture Mechanics to Composite Materials, Composite Materials Series Vol. 6: Friedrich, K. (ed.). Amsterdam: Elsevier; pp. 571-614. Wawner, F. E. (1988), in: Fiber Reinforcements for Composite Materials, Composite Materials Series Vol. 2: Bunsell, A. R. (ed.). Amsterdam: Elsevier; pp. 371-425. Wright, W. W. (1990), in: Composite Polymer, Vol. 3: Dickin, P. (ed.). pp. 231-401.

7 Short Fiber Composites Michael G. Bader and Alistair R. Hill Composites Research Group, Department of Materials Science and Engineering, University of Surrey, Guildford, Surrey, U.K.

List of 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 7.2.7 7.2.8 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.3.7 7.4 7.4.1 7.4.1.1 7.4.1.2 7.4.1.3 7.4.2 7.4.2.1 7.4.2.2 7.4.2.3 7.4.2.4 7.4.2.5 7.5 7.5.1 7.5.2 7.5.3 7.5.4

Symbols and Abbreviations Introduction Short Fiber Reinforced Composites Classification Thermoplastic Molding Compounds Thermo-Formable Sheet Granular Thermosetting Compounds Thermosetting Sheet Molding Materials Thermosetting Bulk Molding Compounds Reinforced Reaction Injection Molding (RRIM) Summary The Principles of Reinforcement by Discontinuous Fibers Background Reinforcement by Short Fibers Stress Transfer Between Matrix and Fiber Fiber Length Distribution Fiber Orientation Distribution Models for Stiffness Models for Strength Control of Properties and Performance Constitutional Variables Choice of Matrix Reinforcement Other Additions Mechanical Properties Stiffness Ductility Strength Toughness Other Mechanical Properties Manufacturing Processes Introduction Rheological Considerations Thermal Considerations Injection Molded Thermoplastic Composites

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

293 295 296 296 296 296 297 298 298 298 299 299 299 299 301 307 308 310 312 313 313 314 314 316 317 318 320 320 321 323 324 324 325 327 329

292

7.5.4.1 7.5.4.2 7.5.4.3 7.5.5 7.5.6 7.5.7 7.5.8 7.6

7 Short Fiber Composites

Extrusion Compounding Injection Molding Process Control Injection Molding Thermosetting Composites Compression Molding Thermoforming Reaction Injection Molding (RIM) References

329 330 332 333 334 335 336 337

List of Symbols and Abbreviations

293

List of Symbols and Abbreviations af volume fraction of fibers oriented at angle 9 to direction of principal loading d fiber diameter E Young's modulus £ c , E{9 Em Young's modulus of composite, fiber, matrix Gc fracture energy Gm shear modulus of the matrix K constant Kc fracture toughness L, L c , L t fiber length, critical length, transfer length n exponent r, R radii of concentric cylinders Ra fiber aspect ratio Rac critical fiber aspect ratio Rc critical transfer aspect ratio Rt load transfer aspect ratio Tg glass transition temperature V volume fraction K> Vt> ym composite, fiber, matrix volume fraction xr distance from the fiber end j8 e ec, ef, em t], £ r]x r\0 9 a af aym <7(subcriticai) ^(supercritical) T imax

parameter strain strain in the composite, fiber, matrix factors efficiency factor factor based on the laminate transformation equations angle of the fibers to the principal stress direction applied stress tensile stress in the fiber stress in the composite at matrix yield average tensile stress in subcritical fibers average tensile stress in supercritical fibers interfacial shear stress maximum interfacial shear stress

BMC bulk molding compound DMC dough molding compound FLD fiber length distribution FOD fiber orientation distribution HDT heat distortion temperature LEFM linear elastic fracture mechanics MFD mold fill direction MW molecular weight

294

PEEK PP RIM RRIM SFC SMC SRIM

7 Short Fiber Composites

poly-ether-ether-ketone polypropylene reaction injection molding reinforced RIM short fiber composites sheet molding compound structural RIM

7.1 Introduction

7.1 Introduction Short fiber composites (SFC) comprise a wide and diverse range of materials. This discussion will be confined to systems with polymeric matrices and will include both thermoplastics and thermosets, but the main emphasis will be on short-fiber reinforced thermoplastics. Metal, inorganicglass and ceramic matrix materials are not addressed. However, many of the fundamental principles apply also to these inorganic matrix systems. The term "shortfiber" will be taken to be synonymous with "discontinuous-fiber". Plastics are commonly filled with fiber or particulate materials. In many cases the motivation is simply to extend a relatively expensive polymer with a cheap filler (e.g., chalk), but it is possible to exploit the filler to modify the properties of the polymer. When fibrous fillers are used it is also possible to enhance stiffness and strength and to modify other properties such as thermal expansion and toughness. This usage of a filler phase is termed "reinforcement" and such reinforced plastics are regarded as "composites". The most spectacular reinforcement may be achieved when the fibers are continuous, aligned and comprise more than 50% of the volume of the composite. These materials are, however, limited in their application to shapes which are formed by the "assembly" of sheetlike or rodlike feedstock. If the reinforcing fibers are discontinuous, the level of reinforcement must be reduced but many other processing routes become possible. A particular attraction of discontinuous-fiber composites is their ability to be molded into complex shapes using relatively cheap manufacturing routes. Discontinuous-fiber composites may be classified according to the relative length (more properly the aspect ratio - i.e.,

295

length/diameter) of the fibers in the final product. The length may vary from more than 100 mm to less than 0.1 mm in glass fiber containing systems, where the fibers are typically of 10-20 jim in diameter, that is aspect ratios of 5 to 10000. Within this range there is a very diverse range of materials and processes which form the subject matter of this chapter. Glass fiber is the dominant reinforcement. This is mainly for economic considerations, since glass is by far the cheapest of the presently available fibrous reinforcements and is produced in quantities which allow its use in commodity materials. Carbon fiber is used to a much more limited extent. It is in many respects more compatible with polymeric matrices than glass and much higher stiffness may be achieved but the increased cost permits it to be used only in the more specialized applications. By contrast the use of carbon fiber tends to dominate the high-performance end of the continuous-fiber composite market. Other fibers, such as the aramids, are used only to a very limited extent in short-fiber systems. In considering the manufacturing technology of SFC, it is well to note that the fibers are produced initially as continuous rovings and have to be cut, prior to, or are broken during the course of production of the SFC. The fiber format is very important in this respect. The weight (tex) of the roving and the finish or "size" applied to the fiber bundle may have a critical influence on both the economics of the manufacturing process and on the properties of the final product. The nature of the fiber finish is very important in the case of glass which, although produced in great quantities, is in many respects a rather delicate material.

296

7 Short Fiber Composites

7.2 Short Fiber Reinforced Composites 7.2.1 Classification

Short or discontinuous fibers are used to reinforce a wide range of polymeric materials with radically differing constitution, methods of manufacture, properties and applications. They may be divided into six main categories: (I) Thermoplastic molding compounds. (II) Thermoformable sheet materials. (III) Granular thermosetting molding compounds. (IV) Thermosetting sheet molding compounds. (V) Thermosetting bulk molding compounds. (VI) Reinforced reaction injection molding materials (RRIM). 7.2.2 Thermoplastic Molding Compounds

These materials are almost always processed by injection molding. An intermediate feedstock material is produced by melt compounding the base thermoplastic resin with fiber to produce a granular, or pelletized, product which consists of relatively short lengths of fiber dispersed in the thermoplastic matrix. This granular feedstock may then be fed into conventional injection molding machines. A wide range of base thermoplastics are utilized, notably the polyolefins, polyamides, thermoplastic polyesters and the more advanced engineering thermoplastics, typified by polyphenylene sulfide and the newer aromatic thermoplastics such as poly-ether-ether-ketone (PEEK) and related materials. They span a range from low-cost commodity materials to very expensive speciality polymers. The reinforcement is most commonly E-glass fiber,

which accounts for probably more than 90% of the market. Carbon fiber, usually the high strength version, is also used in premium variants, offering higher stiffness and strength. The injection molding process is capable of producing large numbers of intricate and dimensionally accurate moldings, typically in the weight range from a few grams up to about 10 kg. Within this group there are two sub-divisions based on the length of the fiber in the finished molding. Most molding compounds are manufactured by compounding either chopped glass strand or continuous rovings in single or twin-screw compounding extruders. During this process the fibers are broken up into very short lengths (discussed in more detail later), typically between 0.1 -1.0 mm. These lengths are well below the optimum for efficient reinforcement but the material offers a cost-effective advantage over the un-reinforced grades. More recently a number of processes for continuously infiltrating fiber roving have been introduced. These produce a thin rod of polymer reinforced with continuous filaments. This rod is then chopped into segments, typically 10 mm long, which are suitable feedstock for injection molding. With appropriate control of the injection molding operation, these grades allow the production of moldings containing much longer fibers, with significant improvements in stiffness, strength and toughness over the short-fiber grades. 7.2.3 Thermo-Formable Sheet

This is an alternative form of SFC using a thermoplastic matrix. It is manufactured using both discontinuous and continuous fiber reinforcement. Both may be processed by similar routes, the main difference being that the discontinuous fiber al-

7.2 Short Fiber Reinforced Composites

ternative (which is the only version appropriate to the present discussion) is capable of more extensive flow. A continuous sheet feedstock is manufactured either by extrusion of a short-fiber (glass)/polymer blend or, more usually, by corolling chopped strand glass mat between thin sheets of the thermoplastic. The sandwich of materials is passed between a series of heated rolls which force the polymer into the mat to produce a more-or-less homogeneous sheet of reinforced plastic. The sheet is typically about 5 mm thick, up to 2 m wide and contains up to about 30% by volume chopped glass strand. At this point it should be made clear that the reinforcement consists of bundles of glass fibers, typically 50 mm long and each containing several thousand individual filaments. The mechanics of reinforcement is based on the bundle rather than the fiber dimensions. The sheet feedstock may then be further processed by first heating to above the softening point * of the matrix in an oven and then pressing, or stamping, the softened sheet (or sheets) between cold metal dies. Considerable flow of the material may be induced in this operation to produce deep drawn products of considerable complexity. The process is rapid and is considered analogous to metal pressing operations. It was held to have great potential for production of parts for the automotive and allied industries but has, in fact, been taken up on a rather more modest scale than originally envisaged. Materials of this type are produced with both polypropylene and polyamide (nylon) based matrices reinforced with E-glass.

1 Semi-crystalline polymers must be heated above the crystalline melting point, whilst amorphous materials are heated until their viscosity is sufficiently low. The matrix needs to be virtually liquid.

297

7.2.4 Granular Thermosetting Compounds

These are analogous to the thermoplastic moulding compounds but utilizing thermosetting resin matrices. The materials are supplied in a coarse-granular or pelletized format for subsequent processing by compression or injection molding. Traditionally, thermosetting molding compounds have always utilized comparatively heavy loadings of fillers. The main purpose of these fillers has been to help control processing by reducing the proportion of reactive polymer precursors and thus reducing reaction exotherm and production of volatile condensates. They also markedly reduce the shrinkage on molding and contribute to dimensional stability. A negative effect is that they reduce the flow capability of the system. A delicate balance has to be struck between filler content and the degree of polymerization advancement, in order to combine good mechanical properties with adequate processability. Many fillers are also much cheaper than the polymer and hence are used to control cost. A number of the traditional fillers, notably mica and to a lesser extent talc, were also found to have a reinforcing effect. With the development of composite technology in other systems, the use of glass fibers as reinforcement was investigated and they were found to extend the performance range, especially at elevated temperatures and where dielectric properties were important. The production of thermosetting molding compounds incorporates mixing and grinding operations which allow reinforcement only by very short fibers. The most usual practice is to use milled glass where the fibers are all generally shorter than 0.1 mm. Fibers of such a short length have only very limited reinforcement potential. The main benefits are a reduction of mold-

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7 Short Fiber Composites

ing shrinkage and thermal expansion coefficient and small increases in strength and toughness, in materials which are inherently very brittle. 7.2.5 Thermosetting Sheet Molding Materials

These are the thermoset equivalent of the thermoformable sheet, but they must be finally compression molded in heated molds. The standard material, known as SMC (sheet molding compound), is manufactured as a wide sheet (e.g., 1 m x 5 to 10 mm thick) of an uncured unsaturated polyester resin filled with a random array of chopped glass strand, of 50 mm length. The resin also generally contains a mineral filler, usually calcium carbonate, and it is either chemically or physically thickened. This is a process which transforms the resin from a mobile liquid into a thick paste, without an increase in the degree of polymerization. The resultant sheet molding compound is easily handled and may be cut by shears or blanking press into shapes to be compression molded. A suitable preweighed and tailored pack of SMC is then placed between a pair of matched dies and hot pressed, when it first softens, then flows into the die cavity where it cures. The finished part is ultimately ejected, hot, from the die. The process cycle is typically between 2 and 20 minutes. As in the case of the thermoformable sheet material, SMC may also be fabricated using continuous fibers in the form of woven cloths or continuous random mat. Again it should be noted that the reinforcement is by bundles rather than individual filaments. SMC is very widely used in the automotive and consumer goods industries, the reinforcement is virtually always Eglass and the matrix mainly unsaturated polyester or vinylester resins.

7.2.6 Thermosetting Bulk Molding Compounds

These materials, variously known as bulk molding compounds (BMC) or dough molding compounds (DMC), are virtually identical in constitution to SMC, except that they are manufactured in a bulk form, usually as a flexible "rope" about 50 mm in diameter. The reinforcement is again chopped E-glass strand, but somewhat shorter (10-25 mm) than in the SMC. BMC is either compression or injection molded and it is the adaptation of the injection molding process to thermosets which accounts for the rapid growth in usage of this class of materials. 7.2.7 Reinforced Reaction Injection Molding (RRIM)

The RIM (reaction injection molding) process is based on the use of two highly reactive precursors which, when simultaneously injected into a die cavity, co-react to form a thermosetting polymer. RRIM is the version of the process which also exploits fiber reinforcement. The classic process is based on polyurethane systems, where a difunctional polyol is reacted with a di-isocyanate to form a polyurethane. In this case both components are mobile liquids and they may be formulated to obtain cure in 2 - 5 minutes from mixing. Pumping and metering units are arranged so that accurately metered quantities of the two components are simultaneously injected, at high velocity, through a turbulent mixing chamber, into the die cavity. After a 2 - 5 min dwell to allow for cure, the mold is opened and the part ejected. The polyurethane may be formulated to form soft elastomeric or hard glassy solids, or either rigid or flexible foams. It is, therefore, a very versatile process. In the RRIM version short glass fibers are suspended in

7.3 The Principles of Reinforcement by Discontinuous Fibers

the polyol component, so that a reinforced polymer is produced. In the urethane system it is not practicable to suspend the fibers in the isocyanate component due to the fact that premature chemical reactions are initiated, but in other systems the reinforcement may be added to both components. The need to maintain an easy flowing suspension limits both the proportion and the length of the glass fibers which may be added to the system, so that final fiber content rarely exceeds 10% by volume and fiber lengths are generally less than 0.1 mm. The degree of reinforcement is therefore very limited, but useful improvement in dimensional stability and performance under moderately elevated temperatures is achieved with polyurethanes. The RRIM process is now being adapted to other polymers; amongst those which are being developed are polyester and polyamide (nylon) based systems which have the possibility of providing a viable challenge to the other molding compounds. 7.2.8 Summary

Short fibrous reinforcement is utilized in a wide range of both thermoplastics and thermosetting polymers, an equally wide range of processes are used to compound and finally shape these materials. In the end the degree of reinforcement is a function of the fiber loading, the length and orientation distributions and the fibermatrix interface characteristics. There is also a fundamental difference in the expectations of reinforcement in different matrices, depending on whether they are fundamentally brittle, strong and ductile or elastomeric. This is discussed in detail in the section on the principles of reinforcement.

299

7.3 The Principles of Reinforcement by Discontinuous Fibers 7.3.1 Background

The mechanics of discontinuous, or short, fiber composites is considerably more complex than that of continuous aligned fiber systems. The complexity arises from the fact that if the fibers are discontinuous they are generally not of equal length nor are they all aligned in one direction, even within a small volume of the material. It is necessary, therefore, to consider the fiber length (or aspect ratio) distribution (FLD) and the fiber orientation distribution (FOD) when modeling the mechanical and physical properties. It should also be noted, at the outset, that the maximum achievable fiber volume fraction, Vf9 is lower in SFC than in continuous fiber systems so that the matrix and interface have relatively more influence on the properties of the composite. The fiber fraction is limited first by geometric constraints which impose a limit which decreases as the fiber orientation distribution becomes more random. In a fully random system the upper limit is about 0.3, whereas in a perfectly aligned system the theoretical upper limit is just over 0.9. However, practical considerations, especially the need for moldable materials to flow, set an upper limit of about 0.4 for most short-fiber systems. 7.3.2 Reinforcement by Short Fibers

In the consideration of the mechanisms of reinforcement of short fibers it is useful first to take another look at continuous fiber composites. The basic mechanics have been derived from the naive concepts of the celebrated "rules of mixtures equations". We consider the composite to con-

300

7 Short Fiber Composites

sist of an array of continuous, straight, aligned fibers embedded in a matrix in a perfect geometric array, usually hexagonal or square. The fibers are considered to be perfectly bonded to the matrix. It is then possible to derive the well known Voigt, or upper bound, equation for the stiffness of the composite in the direction of fiber alignment: E = E V -\- E V c

f

f

m

m

(7-1) V

to consider the failure strains of both fiber and matrix. The rule of mixtures approach to the prediction of properties in the direction transverse to the fibers is less well founded. The fibers and matrix are modeled as a series of lath-shaped lamellae (Fig. 7-1), the load being carried from one to the other through the notionally perfect interface. Not surprisingly the equation

/

or (7-la) where £ is a Young's modulus and V a volume fraction. The suffices f, m, and c refer to fiber, matrix, and composite, respectively. (In a two-component system This simple model predicts the stiffness, in the fiber direction, of a uniaxial composite remarkably well. This is because at high fiber contents, and when the fiber is much stiffer than the matrix, the stiffness is little influenced by the matrix or by the actual geometric arrangement of the fibers. Furthermore it is not necessary to consider the interface, since interfacial stresses exist only at the fiber ends. The equation is much less satisfactory, however, when applied to strength, because in this case interactions between fiber and matrix assume greater significance and it is also necessary

E

(7-2)

offers only a very crude approximation to experimentally observed stiffness and it is virtually useless for predicting strength. For a more comprehensive discussion see Kelly and Macmillan (1986). How does a short fiber reinforce a matrix? Clearly, if the fiber is completely surrounded by the matrix, stresses may only be introduced into the fiber from the matrix. This, in turn, is possible only if stresses can be transmitted across the interface. If the fiber is bonded to the matrix, then any displacement in the matrix must be transmitted to the fiber. As the fiber is stiffer than the matrix, it will tend to constrain it, thus, increasing its apparent stiffness. In real systems we cannot expect the bond between fiber and matrix to be "perfect", so there will be a limit to the constraining effect of the fiber which is determined by

Figure 7-1. A uniaxial laminate (A) is simplified to the form (B) in the simple rules-of-mixtures analysis. The response under longitudinal loading conforms closely to experiment and more sophisticated analyses. However under transverse loading the block model, which has no continuous matrix phase, is inadequate.

7.3 The Principles of Reinforcement by Discontinuous Fibers

V V

IQQOOOOl fi

the nature of the interface. The effectiveness of the reinforcement will also depend, as it does in continuous fiber systems, on the fiber content, Vf, on the fiber aspect ratio and on the fiber orientation. The interface has been identified as having a critical influence on the properties of SFC. Let us briefly consider some of the factors which determine the nature and properties of the interface. In a composite, depending on the local stress state, the interface may be ruptured by shear stresses parallel to the fibers or by tensile stresses normal to the fiber axes, as indicated in Fig. 7-2. It follows that both shear and transverse tensile strength of the interface will be significant factors. Bonding may be mechanical, physical and/or chemical in nature. Mechanical bonding arises from physical asperities on the fiber surface and from radial compressive stresses produced by shrinkage of the resin matrix, due to thermal contraction or volume changes during cure. These effects lead to frictional coupling between fiber and matrix which are effective under shear in the fiber direction. Purely frictional bonding may be vulnerable to drastic reduction if the temperature is raised, if the matrix absorbs water, or if viscoelastic relaxation can occur in the matrix. Physical bonding may also be developed if the resin matrix effectively wets the fiber and the frictional bond may be enhanced by dipole interactions. This may be effective in both shear and transverse

301

Figure 7-2. Under shear loading (A) the interface is subjected to shear stress in the direction parallel (and normal) to the fiber axis. Under transverse tension (B) a component of tensile stress normal to the fiber axis may lead to interface failure.

tensile directions. It is also probable that in wetting systems mechanical bonding will be more fully developed. The most effective bonding is considered to be chemical in nature with primary chemical bonds being developed between the fiber and matrix resin. Glass fibers do not naturally develop chemical bonds with polymers and chemical bonding is achieved only by the use of sophisticated coupling agents which form chemical "bridges" between fiber and matrix. These coupling agents are typically based on difunctional silanes. In glass fiber reinforced plastics systems the role of the coupling agent and the whole of the glass sizing technology is critical to the performance of the composite. Carbon fibers, on the other hand, are more compatible with polymer matrices and a simple oxidative surface treatment is all that is necessary to achieve an adequate interfacial bond. 7.3.3 Stress Transfer Between Matrix and Fiber

Consider a short stiff fiber completely embedded in a more compliant polymer matrix and perfectly bonded to it along its cylindrical surface. This may be modeled as a pair of concentric cylinders of radius r and R (Fig. 7-3), so that the inner cylinder represents the fiber and the outer the matrix. The ratio r/R may then be adjusted to give the desired fiber volume fraction, so that Vf = (r/R)2. If a tensile stress is applied

302

7 Short Fiber Composites

1 m

Figure 7-3. The composite is represented as a pair of concentric cylinders, perfectly bonded along the cylindrical interface. The central cylinder represents the fiber and the ratio r/R is adjusted to give the desired fiber volume fraction. The fiber aspect ratio is L/(2 r).

in the direction parallel to the fiber axis a strain will be induced in the matrix. Since the fiber is stiffer than the matrix, a shear stress will be developed along the cylindrical interface. This was treated by Cox (1952) using shear-lag analysis. This considers the differences in the strain displacements of matrix and fiber along the cylindrical interface. Perfect bonding is assumed and stress transfer from the fiber ends is disregarded. Both matrix and fiber are considered to be elastic. The tensile stress in the fiber was shown to build up from zero at the fiber ends to a value which approached E{ sc in the middle portion of the fiber. Equations (7-3) to (7-5) have been adapted from those published by Cox by using the parameters Ra (fiber aspect ratio) and V{ in place of length and r/R. The strain in the composite is denoted by ^c, the shear modulus of the matrix by Gm, and the interfacial shear stress by T. The distance from the fiber end is xr and is measured in units of the fiber diameter d. It is cosh[/?(R a -% r )] (7-3)

The shear stress along the interface is shown to be maximum at the fiber ends and to fall to virtually zero along the middle portion according to:

|_2£ f lnFf 1 / 2 J

cosh(jS-J?a) (7-4)

The parameter /? in Eqs. (7-3) and (7-4) is given by:

J

2

^

112

T

(7-5)

lEtr lnV{- ] This indicates that the stress distribution along the fiber is a function of the ratio of the shear modulus of the matrix and Young's modulus of the fiber (GJE{) and the fiber volume fraction (V{). In Figs. 7-4 a and 7-4 b curves for fiber tensile stress and interface shear stress are shown for two levels of applied strain for a fiber of aspect ratio 50. The fiber modulus £ f has been set at 100 GPa and the other parameters are typical of an epoxy resin. The plateau stress level Efe is not quite achieved in the middle of the fiber at the higher strain level. The predicted shear stress at the fiber end approaches 60 MPa at the 1 % strain level. This is higher than the shear strength of many polymers. The effect of fiber volume fraction is shown in Fig. 7-5 and indicates a lower fiber stress, and hence a lower degree of reinforcement at the lower level. In Fig. 7-6 the effect of reducing the shear modulus of the matrix Gm is shown. Again the effectiveness of stress transfer to the fiber is lower for the more compliant matrix. The important conclusions from the Cox analysis are that the stress in the fiber builds up from the ends towards the middle and that, in fibers of high aspect ratio, there is a uniformly stressed region in the middle. In a continuous fiber system the whole length of the fiber would be loaded to a similar extent to this middle portion. The inefficiency of the short fiber is, thus, reflected in the relative extent of the load transfer regions at the

7.3 The Principles of Reinforcement by Discontinuous Fibers

303

83 LJJ

tr

in 10

(a)

15 20 25 30 35 FIBER LENGTH / DIAMETER Strain 1 %

40

45

50

- Strain 0.5%

Figure 7-4. The tensile (a) and shear stress (b) distribution along a fiber of aspect ratio Ra = 50 according to the Cox model. The fiber modulus has been set at 100 GPa, Ff = 0.5, and the shear modulus of the matrix at 2 GPa. The two curves are for imposed strains of 0.5 and 1.0% respectively. At the higher strain the predicted shear stress at the interface exceeds the shear strength of the matrix.

-60

10 (b)

15 20 25 30 35 FIBER LENGTH / DIAMETER 1 % Strain

40

0.5% Strain

Figure 7-5. The predicted tensile stress distribution along a fiber loaded under similar conditions to that of Fig. 7-4. At the lower volume fraction the stress transfer is less effective according to this model: 10

15 20 25 30 35 FIBER LENGTH / DIAMETER Vf: 0.50

Vf: 0.25

40

45

50

304

7 Short Fiber Composites

1000

Figure 7-6. The predicted tensile stress distribution along the fiber is here shown for two different values of the ratio Gm/E{: stress transfer is more effective when the matrix is relatively stiffer. 10

15 20 25 30 35 FIBER LENGTH / DIAMETER

- Gm/Ef = 0.02

40

45

50

Gm/Ef = 0.002

fiber ends. This load transfer length, Rt (expressed in terms of the fiber diameter: i.e., Rt = Lt/d) may be defined by the point at which the stress in the fiber reaches 0.9 Ef sc (Fig. 7-7). If i?t is a significant proportion of R a , then the reinforcement efficiency of the system will be lower than for a continuous fiber system. This may be estimated from the ratio of the area under the curve of Fig. 7-7 and the rectangle Ef sc x L. It should be noted that Rt is independent of strain, but the maximum interfacial shear stress increases with increased strain. If the fiber aspect ratio is low (i.e., Ra<2Rt) then the stress in the middle zone of the fiber cannot

approach E(sc and reinforcement efficiency will be low. If, however, the fiber is very long (Ra 5> Rt) then the reinforcement efficiency will approach that of a continuous fiber. One obvious deficiency of the Cox analysis is that the maximum shear stress is predicted to occur at the fiber end. In fact it must fall to zero at the extremity of the fiber. Another problem is that the "strength" of the interface is not considered (the assumptions of perfect elasticity and interface bonding), a consequence of this is that interface shear stresses which exceed the yield strength of the matrix can be pre-

CO CO LLJ

Figure 7-7. The stress transfer length is defined as the length over which the stress increases to 0.9 x Ef sc (i.e., 90% of the maximum); it is shown here in terms of fiber diameter - i.e. Rt = LJ(2r).

az

5 -Rt-

10

15 20 25 30 35 FIBER LENGTH / DIAMETER

40

45 • R

t

-

50

7.3 The Principles of Reinforcement by Discontinuous Fibers

305

1000 900

800700600500400300200100

Figure 7-8. The tensile stress distribution for a fiber loaded under similar conditions to those of Fig. 7-4 according to the Kelly-Tyson model. The stress increases linearly from the fiber end to the maximum defined by E{sc. The transfer length increases proportionately to the applied strain. 5

10

15 20 25 30 35 FIBER LENGTH / DIAMETER 1 % Strain

40

50

0.5% Strain

dieted. In a real system the matrix, or interface (whichever were the weaker) would yield or debond and this would lead to a redistribution of the stresses. In spite of these limitations the Cox approach does allow a realistic, although not exact, appraisal of the principal variables of short fiber systems. An alternative treatment was proposed by Kelly and Tyson (1965) for a (metal) system in which the matrix yielded and stress transfer occured by friction at the interface. A simple force balance leads to an equation for the tensile stress in the fiber, which is applicable over the stress transfer region; i.e., from the end of the fiber to the point where the axial strains in the fiber and matrix become equal and there is no interfacial shear stress af = 2xRa

45

(7-6)

The resulting stress distribution is shown in Fig. 7-8. It will be noted that stress builds up from the fiber ends but in a linear manner so that the stress transfer length is precisely defined. The central plateau is where the strain in the fiber is equal to that of the matrix, so that there is no interfacial shear. The effects of applied strain and fiber

aspect ratio are similar to those predicted from the Cox model. In this model the interfacial shear stress, T, is constant over the stress transfer length and independent of the applied strain, hardly a realistic concept, but only a frictional coupling was assumed so that the shear modulus of the matrix and the strength of the bond do not feature in the model. The efficiency of reinforcement may be estimated using similar principles to those used in the Cox analysis and leads to a very similar relationship with fiber aspect ratio. This is shown in Figs. 7-9 and 7-10, where the effects of matrix shear modulus, in the Cox model, and interface shear stress, in the Kelly-Tyson model, are shown to have a similar general effect. It may be concluded that high levels of reinforcement efficiency may be obtained if the aspect ratio of the fiber is sufficiently high (> 50 typically) and there is an adequate interface bond. Both models thus give a reasonable qualitative understanding of reinforcement but neither address all the relevant parameters. Recent work by Galiotis et al. (1984, 1989) in which they have measured the actual tensile stress in a fiber embedded in a matrix has shown that the stress profile

306

7 Short Fiber Composites

Figure 7-9. The reinforcing efficiency as a function of fiber aspect ratio, calculated according to the Cox model for the two Gm/E{ ratios of Fig. 7-6. The efficiency increases with aspect ratio and for the stiffer matrix. 10

20

30

40 50 60 70 FIBER ASPECT RATIO

Gm/Ef = 0.02

80

90

100

Gm/Ef = 0.002

Figure 7-10. A similar pair of curves to Fig. 7-9, but using the Kelly-Tyson model at two values of interface shear stress. The reinforcing efficiency is higher at higher interface strength and at higher aspect ratio. 10

20

30

40 50 60 70 FIBER ASPECT RATIO 50 MPa

80

90

100

25MPa

is very similar to that predicted by Cox. However, when the applied strain is increased above some threshold value "failure" of the interface is observed and the stress profile changes dramatically. Failure is considered to be shear debonding at the fiber-matrix interface, or alternatively yielding of the matrix. It is initiated at the fiber end and propagates along the interface. There still appears to be some capacity for stress transfer, even in this debonded zone. Here the stress build up is linear (as would be predicted from the Kelly-Tyson analysis). There is then a transition to an

exponential profile consistent with the Cox approach. The general conclusion is that, in systems where a strong (chemical) bond is formed between fiber and matrix, Coxlike behavior will be observed under increasing applied strain until the shear stress at the fiber end exceeds the strength of the bond. Debonding will then occur, which dramatically decreases the shear which can be carried across the interface. Some capacity for stress transfer still remains but this is of a mechanical or frictional nature only. The overall effect is that a shallow stress gradient is established at

7.3 The Principles of Reinforcement by Discontinuous Fibers

the fiber end with a transition to a steeper region at the end of the debonded region. This is an important concept as it explains why the debond does not grow over the entire fiber length. It also indicates that the principles of both the Cox and KellyTyson analyses need to be combined for a more realistic understanding of short fiber systems. 7.3.4 Fiber Length Distribution The models discussed in the previous section have assumed that all the fibers in the system are aligned parallel to the direction of the applied stress and are of similar lengths. In a real system neither of these assumptions will be realistic. Fibers originate as tows or rovings of continuous filaments (typically 5000-20000 filaments of 5-15 jLim diameter in the case of glass or carbon), and during the course of processing these are broken up and dispersed, so that in the finished moulding the mean fiber length might be less than 1 mm. This fiber breakage is a random process so that fibers of a wide range of lengths will be found in the final material. In an injection moulded thermoplastic this range might span 10 jLim to 10 mm (three decades) for 10 jam diameter fibers. It is necessary, then, to consider the effect of this length distribution on the properties of the composite. It was shown in Sec. 7.3.3 that, if other factors remain unchanged, fibers of lower aspect ratio result in lower reinforcement efficiency. It is necessary, then, to consider the whole fiber aspect ratio distribution when estimating the degree of reinforcement. In practice it is usual to assume that the fiber diameter is constant and to simply measure the distribution of lengths. This may be done by releasing the fibers from a sample of the material, either by burningoff the matrix, or by use of a solvent or

307

chemical digestion. The released fibers are placed on a microscope slide and the lengths measured. This is most usually accomplished by use of an image analyzer, but even then it is a far from straightforward procedure and specialist software is required if accurate results are to be obtained (Lunt, 1980; Lung and Shortall, 1980). Once the fiber aspect ratio distribution is known it is then possible to compute an effective mean aspect ratio and an efficiency factor, Y\X , based on either the Cox or Kelly-Tyson model, which may then be applied to Eq. (7-1). The efficiency factor is simply the average stress in the short fiber divided by the stress in a continuous fiber subjected to a similar applied strain. It is f + Em Vm

(7-7)

where for the Cox model: tanh(j3fla/2) m

(7-8)

PRJ2

This approach is rather crude and a more effective model is based on summing the contributions to stiffness of all fiber aspect ratios in the distribution. This was done by Bader and Bowyer (1972) and Bowyer and Bader (1972 a, 1972 b) who also pointed out that the reinforcing efficiency of short fibers is a function of the applied strain. The principle of the Kelly-Tyson model may be extended so that a critical aspect ratio may be defined for any specified level of applied strain: «.c= —

(7-9)

and a fiber of any known Ra may then be designated "subcritical" or "supercritical". If .Ra JRac it is supercritical ^a (subcritical)

(supercritical)

(7-10)

308

7 Short Fiber Composites

The average tensile stress in sub- and supercritical fibers being given by: ^(subcritical) ~~

(7-11)

<Ti(supercritical)

(7-12)

At low strains Rac is small and it rises proportionately with strain. Thus, at very low strains even the shortest fibers are supercritical, but as the strain is increased an increasing proportion of the distribution will become subcritical and the reinforcement efficiency will fall. This means that the stress-strain curve will be nonlinear, even in the quasielastic region. This is certainly a common observation when tensile testing SFCs. The calculated reinforcement efficiencies for SFCs with fibers of different aspect ratios are shown in Figs. 7-9 and 7-10. If the aspect ratios are normalized with respect to Rac (Fig. 7-11) it is apparent that the ideal Ra should be between 5 and 10 times Rac, corresponding to a reinforcement efficiency of 90-95%. In typical glass fiber reinforced thermoplastics i?ac has been estimated at 20-50 (corresponding to fiber lengths of 200-500 jim), so that fibers 1-5 mm length would be desirable. In many commercial systems mean 2 fiber lengths are only 100-250 jum. In Fig. 7-12 the fiber length distributions determined from three different grades of a glass fiber reinforced polypropylene are shown. These were measured from burnt off samples using an image analyzing microscope with specially developed software (Hill, 1991). The mean fiber lengths are 730 jim for the 0.054 Ff sample and 600 and 540 jim respectively for the 0.108 and 0.144 V{ batches. It may be noted that the propor2

The mean length is usually taken as the number average of the fibers. This is identical to a volume average if all fibers are of equal diameter.

tion of longer fibers decreases as the fiber fraction is increased. This is due to the greater degree of fiber attrition when processing the more heavily filled material. The critical length (i.e., twice the transfer length) for this compound is of the order of 0.95 mm when a coupling agent is incorporated into the compound and 1.6 mm for uncoupled material. The mean fiber lengths are well below the optimum. This would be the case in most commercial systems. However, grades containing much longer fibers have recently become available (Gore etal, 1986). They offer enhanced stiffness, strength and toughness. 7.3.5 Fiber Orientation Distribution

In SFCs the fibers are seldom all oriented in a single direction. This is due to the flow processes used during fabrication. This is often extremely complex and gives rise to equally complex fiber (and molecular) orientation. Converging flow leads to a preferential orientation parallel to the flow direction, whereas in divergent flow the fibers orient normal to the flow. This behavior is further affected by interactions with the mold or die walls. The net result is that in a typical thin walled section of an injection molded component a layer structure is developed, with the fibers at the surface being aligned parallel to the principal flow direction and those in the center of the section lying predominantly normal to that direction. The elastic properties may be modeled in a similar way to those of a [0,90,0] cross-ply laminate, with appropriate factors for fiber aspect ratio (Halpin and Kardos, 1978). A more rigorous approach is to measure the fiber orientation distribution and then to apply a factor, based on the laminate matrix transformation equations (Tsai and Hahn, 1980), for the mean angle of the

7.3 The Principles of Reinforcement by Discontinuous Fibers

309

Figure 7-11. In this curve the fiber aspect ratio has been normalized with respect to the transfer length (Kelly-Tyson). This shows that if all fibers are of the critical transfer length (Rc) - the efficiency is 0.5. This rises to 0.9 when R = 5RC and 0.95 at 10 K . 2 3 4 5 6 7 8 RELATIVE FIBER ASPECT RATIO: R/Rc

8

1

-/' -X/ •' --

7-

FIBER 1VOLUME (

6-

...

\

543210J

. u. _

v

X--.. ^ ^

---// -----

'20

460

.... N _

^~ — . i i i i'M

900 1340 1780 FIBER LENGTH (MICROMETERS)

5.4% glass

— 10.8% glass

- COS4 —

i i'

2220

14.4% glass

fibers to the principal stress direction rj0 =

1 n

Figure 7-12. Fiber length distributions measured for three grades of glass fiber reinforced polypropylene injection moldings. The data have been reduced according to the method of Hill (1991) to correct for fibers mismeasured by the image analysis program.

(7-13)

where af is the fraction of fibers at each angle 9. Krenchel (1964) calculated values for the parameter rj0 for various fiber orientations; for the uniaxial case it is unity, for a random-planar distribution 0.375 and for three-dimensionally random distribution 0.2. This may then also be applied to Eq. (7-7) to give a relationship which cor-

rects for both fiber length and orientation distribution: E=nriEV-\-E

V

(7-14)

This principle was also used by Lees (1968). The orientation distribution may be measured by microradiography of thin sections (Darlington and McGinley, 1975; Bright and Darlington, 1981; Crowson et al., 1980) or by optical or SEM microscopy of polished sections, combined with manual (digitizer pad) or automatic computer im-

310

7 Short Fiber Composites

age analysis. To obtain the full distribution it is desirable to examine mutually perpendicular sections cut across the three principal axes of symmetry of the molding (Fig. 7-13). This is a straightforward but tedious procedure, as it is necessary to measure upwards of 1000 fibers on each of 2 or 3 sections to obtain an accurate distribution for a single sample (Hill, 1991). 7.3.6 Models for Stiffness The stiffness of a composite containing a distribution of short discontinuous fibers is controlled by a number of constitutional

parameters. These are the elastic properties of the fiber and matrix, the fiber fraction, the fiber aspect-ratio distribution, the fiber orientation distribution and the effectiveness of stress transfer across the fibermatrix interface. No single model has been proposed which addresses all of these parameters. The Cox model considers only ideal elastic behavior of fibers oriented in the principal stress direction. Interface strength is ignored and the boundary conditions are unrealistic. Nevertheless the proposed stress distribution in the fiber is close to that observed experimentally. The Kelly-Tyson model does not address elastic interactions or fiber fraction but, with a suitable choice of interface shear stress, also leads to quite realistic prediction of the stress distribution in the fiber and to reinforcement efficiency as a function of fiber length. The principal shortcoming is the assumption of an interface shear stress which does not vary with applied strain. (It should be noted that the model was originally proposed for a metal matrix systems with a yielding matrix where this objection is not valid.) Mittal and Gupta (1982) and Mittal et al. (1987) proposed a linear relationship between interface shear stress and the average applied stress on the composite: * = K
Figure 7-13. The conventions used by Hill (1991) for the determination of the fiber orientation distribution in injection molded sections. Fibers close to the mold walls tend to be aligned parallel to the mold fill direction (MFD), whilst those in the core lie normal to this direction. Sections taken perpendicular to the three major axes produce a good description of the overall fiber orientation distribution of the composite. However, the quantitative use of this data is highly complex.

(7-15)

and applied this to the Kelly-Tyson model and to the analysis proposed by Bowyer and Bader (1972 a). The constant K is determined experimentally from the slope of the stress-strain curve. This approach leads to a slightly different definition of critical length: L =

(7-16)

This can be seen to decrease proportionately to the slope of the stress-strain curve,

7.3 The Principles of Reinforcement by Discontinuous Fibers

rather than simply to applied strain as proposed by Bowyer and Bader. This is a realistic concept and can lead to quite accurate modeling of the stress-strain curve. Hill (1991) has carried out an extensive program of tensile tests on polypropylene/ glass fiber compounds, where both fiber length and orientation distributions have been experimentally determined, and interface strength has been estimated from single-fiber fragmentation tests (Drzal etal., 1980). Hill has assumed that the interface shear stress varies with applied stress in a nonlinear manner according to ^ W x W v ) " , where o is the applied stress, aym is the stress in the composite at matrix yield. It has been shown that n = 035 gives the best fit. Predicted stressstrain curves are compared with experimental curves in Fig. 7-14 and show good correlation. This is similar to the observations of Bowyer and Bader, although the fit is generally better, especially at higher strain. Deviation from the model is to be expected at high strain when the effects of progressive damage development, as the failure stress is approached, become dominant.

311

Halpin and Tsai proposed a modification of the "rules of mixtures" equations which included two empirical constants to account for fiber-matrix interactions (see Halpin and Kardos, 1976). (7-17)

F — F

where

^

to/EJ-l

(7-18)

The merit of this equation, as pointed out by Jones (1975), is that by suitable choice of the two constants it allows predictions anywhere between those of the Voigt, upper-bound and the Reuss, lower-bound rules of mixtures equations. By setting £ to infinity the equation becomes identical to the Voigt and when £ is zero to the Reuss equation. The constant rj takes the value of 1 for rigid inclusions, e.g., fibers, and 0 for voids. Wu and McCullough (1977) modeled the elastic properties of short fiber composites by adapting laminated plate theory with factors to allow for the fiber length and orientation distributions. Their approach

Figure 7-14. A comparison of predicted stress-strain curves (lines) (Hill, 1991) for rubber toughened, glass fiber filled polypropylene, with experimental data (points) at three glass contents and constant 30% rubber. The fiber fractions are by volume and correspond to 15, 30 and 40% by weight respectively. 0.5

1.5 2 STRAIN (%)

2.5

3.5

312

7 Short Fiber Composites

integrates the contributions of each fiber, or group of fibers, in the molding. The principles of the model are rigorous but it remains difficult to determine the necessary factors which describe interface strength, orientation and length, so that, ultimately, the model is still reliant on empirical parameters. See also Chap. 9. 7.3.7 Models for Strength

The models for the elastic behavior of SFCs, described above, provide a reasonable prediction of the shape of the stressstrain curve for practical systems, at least in the low-strain range (say, up to 1% strain), which is adequate for most design situations. Strength prediction is much more elusive. The reasons for this is inadequate understanding of the failure process. In continuous fiber systems, there is almost always a significant proportion of the fibers lying in the direction of the principal tensile stress. These fibers are dominant when strength is modeled and failure normally occurs when these fibers fail in tension. The failure strain of the composite is, thus, close to the failure strain of the fibers in the vast majority of composite systems. Significant deviations occur only when the fiber fraction is very low and when high local stress fields, e.g., around notches, lead to failure at lower strains. In SFCs it may be observed that, except when the fiber fraction is very low and the matrix very ductile or elastomeric, tensile failure occurs at strains much lower than the fiber failure strain (Bader et al., 1975; Bader and Curtis, 1977; Curtis et al., 1978). In general the higher the fiber fraction the lower the failure strain. The reasons for this are that the fibers constrain the processes which allow ductile flow in the matrix, act as internal stress raisers and they provide sites for the initiation of cracks.

The presence of a high proportion of fibers, which behave as rigid inclusions, in the matrix leads to a complex state of triaxial stresses in which shear flow processes in the matrix are inhibited (see also Sun, Chap. 10 of this Volume, for further discussion). This effectively raises the yield stress of the matrix but also embrittles it, so that failure occurs at a lower strain. It has also been observed, e.g., by Wetherhold (1980) that the actual fiber distribution is important and that greater strengthening is produced when there are many crossing fibers. In the models for stress transfer between fibers and matrix it is usually assumed that the fiber ends are not bonded to the matrix. This is not correct but in an aligned fiber under tensile loading the stresses at the fiber end would exceed the tensile strength of the interface or matrix (whichever were the higher) at very low applied strains. This will lead to the formation of a cylindrical void at the fiber end. As the strain is increased, the stresses around the fiber end will lead either to debonding along the fiber, if the interface strength is low, or to the formation of a disc shaped crack in the matrix, if the interface strength is high. A further possibility is the development of a plastically deformed, strain banded or crazed region in the matrix close to the fiber end. These effects are shown schematically in Fig. 7-15. In the more brittle matrix systems, matrix cracks develop from this damage zone at the fiber end, these progressively weaken the cross section leading to failure. These models of failure have been studied by a number of workers including Bader and Curtis (1977), Bader and Collins (1983), and Sato et al. (1982, 1984). It is noteworthy that even matrices which show great ductility in the unreinforced condition (e.g., polypropylene and nylon) are commonly observed to fail in an essentially brittle manner when reinforced

7.4 Control of Properties and Performance

Figure 7-15. Three failure processes may develop, under uniaxial tension, as a consequence of the stressconcentration in the matrix at a fiber end. Cavitation followed by debonding along the cylindrical interface (A), matrix cracking (B), and plastic flow in the matrix (C).

with short fibers and loaded at high strain rates. The effect of misaligned fibers on tensile strength can be even more dramatic. It has been shown that fibers aligned close to 90° to the principal tension direction will debond along their entire length. These comparatively large cracks then propagate through the matrix, again leading to a lowductility failure. Studies of fracture surfaces (Bader et al, 1975; Bailey and Bader, 1985), often show indications of extensive ductile flow in the matrix, but the strain to failure of the material is still observed to be very low (typically 1 -2%). It appears that ductile flow is induced in the matrix by stress concentrations around fibers but that it is confined to a very narrow macroscopic shear band across the critical section of the specimen, and that plastic flow is insignificant elsewhere. This localization of the plastic flow process means that macroscopic ductility is negligible and there is little energy contribution to the fracture process.

313

From the above we may conclude that the processes of failure in SFCs are complex. We may expect that they will be influenced by the fiber length and orientation distributions, by the strength of the fibermatrix interface and by the inherent ductility and notch sensitivity of the matrix. Failure occurs as a result of the accumulation of microscopic damage in the matrix (cracks, crazes, strain bands), which result in a modest reduction in stiffness and ultimately lead to failure across a critically weakened section. A model based on the concept that tensile failure occurs across a critically weakened section was proposed by Bader and Curtis (1977) and subsequently developed by Bader et al. (1978) and Fukuda and Chou (1982). Initially the load is carried by both fibers (some of which are aligned in the principal stress direction) and matrix. As the strain is increased damage is considered to originate at the ends of aligned fibers and along misaligned fibers. This damage accumulates as the strain is increased. The residual strength of the damaged composite is determined by fibers which bridge the damaged zones. The load is progressively transferred to these bridging fibers and failure occurs when the residual strength of the weakest section falls below the applied stress.

7.4 Control of Properties and Performance 7.4.1 Constitutional Variables

The mechanical and physical properties of short fiber composites are determined by the properties of the constituent matrix and reinforcement, the reinforcement geometry and the nature of the interface between matrix and reinforcement. In an

314

7 Short Fiber Composites

ideal system we could specify the requirements, but in real systems processing considerations usually limit what may be achieved and a compromise based on costeffectiveness has to be struck. Fiber length (aspect ratio) and volume fraction are limited according to the degree to which the compound is required to flow during molding. High fiber fraction and higher aspect ratio both increase the viscosity of the material, making it more difficult to process. The fiber length is often determined by type of equipment used, e.g., a screw pump, and by the tortuosity of the flow path during molding. Likewise the fiber orientation distribution will be determined by the flow pattern of the charge filling the mold and, whilst this can be influenced to some degree by the gateing arrangement and operating conditions, it is mainly controlled by the part geometry. A further factor is the degree of fiber dispersion: The fibers are initially bound together in the strand, but during the course of compounding and molding the bundle may be dispersed into individual fibers. Higher shear and more prolonged processing tend towards a greater degree of fiber dispersion but also result in greater fiber attrition, and hence shorter mean fiber length in the final product. The interface has a critical influence on the mechanical performance of SFCs. A higher interface strength ensures a more efficient utilization of the fiber reinforcement, it reduces the stress transfer length and it delays the onset of debonding and other damage which ultimately leads to tensile failure. The interface also contributes to the toughness and influences the resistance of the composite to environmental degradation. The performance of the material may thus be controlled by the choice of matrix and reinforcements, by control of the inter-

face and by the conditions specified for processing. 7.4.1.1 Choice of Matrix The choice of matrix polymer is intimately linked with the manufacturing process. The initial task will be to determine the most appropriate category of material and process route, the six categories set out in Sec. 7.2.1. The specific choice of polymer will be influenced by a number of further considerations: (I) Operating temperature range. (II) Mechanical property requirements. (III) Service environment. (IV) Component geometry and choice of fabrication process. (V) Production rate. (VI) Availability of feedstock and costs. The main influence of the choice of matrix on the mechanical performance of the composite will be on environmental factors, service temperature limitations and moisture and chemical resistance, and on the balance of stiffness, strength and toughness. More ductile matrices usually result in more damage resistant composites, but stiffness will be lower, especially when fiber content is low. 7.4.1.2 Reinforcement

The most common reinforcement is Eglass fiber. Its dominance is determined by its low cost and ease of handling. The stiffness of E-glass is, however, comparatively low (70 GPa) when compared with carbon (graphite) (230-350 GPa) or aramid fibers (100-200 GPa) but these are typically at least 5 times more expensive. Glass fiber is degraded by continuous exposure to moist environments, especially when under stress, and must be protected by a size, or finish. The size also serves to increase, the other-

7.4 Control of Properties and Performance

wise inadequate, adhesion of the matrix to the glass surface. The size must be applied as the fibers are drawn from the melt. It usually contains a protective coating, or film former, which coats the fibers and binds them together to form a strand. The fiber is thus presented to the processor as a more or less strongly bound bundle. Often several strands are combined to form a roving, which is a larger bundle. The degree of binding can have considerable influence on processing. It determines whether a chopped length of strand remains as a bundle or breaks up into individual filaments during compounding and molding operations. Strand integrity will be preserved if the melting point of the film former is higher than the process temperature and if it does not dissolve in the matrix polymer. It is also important if dry granules are to be mixed with chopped strand: A well bound strand will form a free-flowing mix whereas fibrillation will lead to clogging. The electrostatic characteristics of the finish are also influential in promoting uniform mixing. The chopped strand fragments should be attracted to the polymer granules rather than to each other. A coupling agent is also invariably incorporated in the size. This is usually a difunctional silane, one end of the molecule tends to bond to the glass surface whilst the other is designed to be compatible with the matrix. [See Plueddemann (1970,1982) for a full discussion.] The effect of the coupling agent (Ishida and Koenig, 1978) is to strongly increase the adhesion of the matrix to the glass. It also protects the interface from degradation when the composite is exposed to moisture, especially at higher temperatures. An effective coupling agent is essential in all short fiber systems and must be tailored for the individual matrices. Highly polar polymers, e.g., polyamides and epoxies, bond reasonably well

315

to glass, even without a coupling agent when the system is dry, but the bond is severely degraded under wet conditions. In less polar systems, such as the polyolefins, bonding is poor even when a coupling agent is used and further coupling must be arranged by use of reactive additives or, by the incorporation of polar comonomers into the matrix polymer. The exact formulation of sizing and coupling ingredients is seldom revealed by either the glass-fiber or polymer manufacturers. Carbon (graphite) fibers are also widely used in reinforced thermoplastics. Their much higher stiffness extends the range of mechanical properties attainable in a moldable product and very high stiffness and strength may be obtained in molded products. Carbon fiber is generally upwards of 5 times the cost of E-glass and at loadings of 20% by volume would increase the cost of a typical commodity thermoplastic by a factor of 2.5 or more. It is therefore necessary to assess the relative value of the enhanced properties when specifying carbon. Interface control is not as critical as in the case of glass. The fibers are surface treated by the manufacturers to promote adhesion but there appears to be no need for tailoring to individual polymer systems. The size (often an epoxy resin), if any, merely serves to maintain bundle integrity through the early stages of processing. Carbon fiber is also electrically conducting, so that loadings of as little as 10% will confer a measure of conductivity to the compound. Thermal conductivity is also high. Aramid fibers are not so widely used in moldable systems. This is partially due to the fact that fiber bundles are very difficult to cut and that the fiber is weak in compression. In comparison with glass and carbon the aramids have intermediate stiffness and they are nonconducting.

316

7 Short Fiber Composites

Particulate fillers, although beyond the scope of this discussion, are also widely utilized in both thermosetting and thermoplastic systems. In general they do not act as reinforcement in the sense that modulus and strength are significantly enhanced. They do, however, influence a number of other important properties, notably; to reduce the coefficient of thermal expansion, to reduce shrinkage during molding and to improve surface finish. Another important consideration is that when low aspect ratio particles are used, the degree of mechanical anisotropy, due to flow induced orientation, is much reduced. The practical effect is that thermally induced distortion of moldings is reduced. It is common practice in some systems to combine particulate fillers with fiber reinforcement in order to arrive at a costeffective compromise between stiffness and strength, on the one hand and, on the other, dimensional stability and surface finish. 7.4.1.3 Other Additions

The properties of the matrix may be further modified by the use of plasticizers, extenders or by the addition of an elastomeric phase. Plasticizers, which are compatible low molecular weight additives, act to increase the ductility of the polymer and to improve processability. They also usually reduce both stiffness and the glasstransition temperature, Tg and sometimes enhance toughness. In general their effect is opposite to that of fibrous reinforcement, so that their use is not often beneficial in reinforced systems. It is worth noting, however, that many polymers absorb moisture to a significant extent and that this acts as an effective plasticizer, reducing stiffness and Tg. This is especially the case in polyamides (nylons) and polycarbonates.

It is necessary to ensure that these polymers are dry when molded, to avoid problems due to the evolution of steam, which would lead to porosity and poor surface finish. The "as-molded" parts are often quite brittle, but over a period of time they absorb water from the atmosphere and become significantly tougher. In some cases the moldings are conditioned by immersion in hot water, or a steam treatment, immediately after molding to ensure adequate toughness. Rubber toughening is extensively utilized in unreinforced polymer systems. A fine disperion of elastomer particles is found to enhance toughness and ductility, again at the expense of stiffness and Tg, but a better balance is usually obtained than with simple plasticization. Rubber toughened systems are also more stable as plasticizers can migrate or be lost by volatilization over long periods of time. The rubber particles act by initiating crazes and/or shear bands in amorphous systems and by enhancing cold drawing in semicrystalline plastics. Elastomeric dispersions are most efficient (i.e., greater toughening for less reduction in stiffness and Tg) when the particles are small (~ 1 fim) and are chemically linked or grafted to the matrix molecules. This is typically achieved by selecting an elastomer which will dissolve in the molten resin or monomer (thermoplastic), or in the unreacted monomeric components of a thermosetting system. The elastomer also needs to have chemical groups which will react with and graft to the matrix molecules. During processing the elastomeric phase separates from the matrix to form a fine dispersion of elastomeric particles which are chemically linked to the matrix. The morphology is often very complex, with very fine scale precipitation of the matrix phase within the elastomer particles.

7.4 Control of Properties and Performance

In fiber reinforced systems the use of elastomeric dispersions (Hill, 1991; Bailey and Bader, 1985) has the opposite effect to the fibers: Stiffness and strength are reduced, whilst toughness and ductility are enhanced. It has, however, been demonstrated that more desirable combinations of stiffness, strength, toughness, ductility and environmental behavior may be achieved by a combination of elastomer and fiber than would be obtained by (a smaller proportion of) fiber alone. This is demonstrated in Fig. 7-16 in which the relative stiffness and Charpy impact performance for a range of reinforced polypropylene compounds is compared. The base polymer (PP), with 30 wt.% glass (PP + G30) and a similar grade using a chemical coupling system (PP + CG30) are compared with grades containing 15, 30 and 40%, by weight (5.4, 10.8 and 14.4% by volume) of glass-fiber and approximately 20 wt.% of a compatible rubber. The figure again emphasizes the beneficial effect of good interface bonding and also the higher toughness at the expense of some stiffness when the rubber is added. Increasing the glass from 30-40 wt.% and adding the rubber give almost the same

stiffness as the 30% grade without the rubber, but slightly better toughness. 7.4.2 Mechanical Properties

The mechanical properties of a shortfiber reinforced plastic will be determined by the interaction of a number of parameters. The principal influences are the stiffness, and to a lesser extent the strength, of the fiber, the fiber fraction and the fiber aspect ratio and orientation distribution. The properties of the matrix, as modified by any other additions (e.g., particulate fillers, elastomer dispersions, plasticizers) are also influential, especially when the fiber loading is low. The nature of the interface is critical in that it determines the effectiveness of the reinforcement and also influences resistance to environmental conditions, both elevated temperature and chemical hazards. In general the incorporation of short fibers results in a dramatic increase in stiffness, a significant increase in strength and decrease in ductility. The effect on toughness is more subtle and depends on the precise loading conditions and on the way in which it is measured. Short fiber systems

4.5

MODULUS

4-

CHARPY LLJ

3 2» !

KEY 1.5

PP: POLYPROPYLENE G : GLASS (Wt%) CG : COUPLED GLASS R .RUBBER

1 0.5 0

PP

PP+CG30 PP+R+CG30 PP+G30 PP + R+CG15 PP + R+CG40

MATERIAL

317

Figure 7-16. Young's modulus and notched Charpy energy are compared for six grades of polypropylene (PP), filled with 15, 30, and 40% by weight of glass fiber (e.g., G30), a coupling agent (e.g., CG30) and rubber additions. The coupling agent is very effective in improving both stiffness and toughness, whilst the rubber reduces the stiffness but improves the relative toughness. All data are normalized with respect to the property of the base polymer (from Hill, 1991).

318

7 Short Fiber Composites

have been demonstrated to have enhanced resistance to impact damage and to have higher measured fracture toughness. Elevated temperature strength and stiffness and general environmental resistance is usually enhanced by fiber reinforcement. The ultimate portfolio of properties in commercial reinforced systems is inevitably a compromise, in which stiffness and strength have been balanced against toughness, processability and other factors. This leads to an almost limitless range of possibilities based on the manipulation of the constituents. The most important properties of a selected group of reinforced thermoplastics are summarized in Table 7-1 and are further discussed in the following sections. 7.4.2.1 Stiffness The most direct effect of fiber reinforcement is enhancement of stiffness. An upper bound for stiffness may be readily calculated by assuming that the fibers are

aligned and that the interface bond is perfect. Application of the rules of mixtures equations will provide estimates for the stiffness in both the fiber and the transverse directions. A more realistic estimate may be obtained by consideration of the fiber length and orientation distributions, as outlined in Sec. 7.3. It is difficult to account quantitatively for the interface "strength" but an upper limit may be taken as the shear yield strength of the matrix. The Young's moduli of most unfilled plastics lie in the range 2.5-3.5 GPa so that the choice of matrix has comparatively little influence on the stiffness of the composite. The dominating factors are the stiffness of the fiber, its volume fraction, and length and orientation distribution. The variation of stiffness in a polyamide 6.6 thermoplastic reinforced with "short" (-250 jim) and "long" (-750 pm) discontinuous fibers is shown in Fig. 7-17 a. The rule-of-mixtures prediction for continuous glass fiber reinforcement is also shown. Even the "long" discontinuous fibers are

Table 7-1. Typical properties of selected short-fiber reinforced thermoplastics. Data has been abstracted from several sources and normalized to a standard fiber volume fraction of 0.20 (20%). These values should be taken as guidelines and not used for design purposes. Polymer

Polypropylene Polypropylene Polyamide 6.6 Polyamide 6.6 Polyamide 6.6 Polycarbonate Polyoxymethylene Polyphenylenesulfide Polyphenylenesulfide Polyetheretherketone a

Fiber

None Glass None Glass Carbon Glass Glass Glass Carbon Carbon

Relative density

0 0.20 0 0.20 0.20 0.20 0.20 0.20 0.20 0.20

E: Young's modulus; ' XT: tensile strength; thermal expansion.

0.91 1.14 1.14 1.46 1.28 1.45 1.58 1.65 1.45 1.45 c

Ea (GPa)

(MPa)

1.9 7.5 3.2 10 20 9.0 9.0 11 17 16

39 110 105 230 250 135 140 155 185 215

Charpy impact (kJ/m2) 2.7 8.0 >25 40 10 40 9.0 20 20 -

HDT: heat distortion temperature;

d

HDTC

CTE d

(°Q

(10- 5 /K)

60 155 100 250 255 160 165 263 263 310

15 2.7 8.5 2.5 1.9 2.3 3.5 2.7 1.1 -

CTE: linear coefficient of

7.4 Control of Properties and Performance

319

RoM aPaJ

50-


LONG 40

^ /

~

SHORT y/

8 » y*

o LLJ

yS

10n-

/

:

_

0.05

a)

:

:

"

0.2 0.1 0.15 FIBER VOLUME FRACTION

0.25

25 CARBON

20—

GLASS

/

Q

O

i

10~/

LLJ CO

^

0.05 b)

0.1 0.15 0.2 FIBER VOLUME FRACTION

less than half as effective as continuous aligned fibers. In Fig. 7-17 b the same "long" glass fiber material is compared with one reinforced with carbon fibers of similar aspect ratio. The stiffer carbon fibers result in much more effective reinforcement. As indicated in Sec. 7.3, the stress-strain curves of SFCs are markedly nonlinear, especially when the fibers are short. For this reason it is necessary to exercise caution when quoting values for stiffness, as the

Figure 7-17. The stiffness (0.1% secant modulus) enhancement of "long" and "short" carbon fibers in a polyamide 6.6 matrix is compared with the rule-of-mixtures prediction for continuous aligned fibers in Fig. 7-17 a. The longer fibers are seen to be much more effective. In Fig. 7-17 b the same "long" carbon fiber material is compared with one reinforced with glass fibers of similar aspect ratio. The stiffer carbon fibers result in much more effective reinforcement (data from Bowyer and Bader, 1972 a and b).

0.25

stiffness decreases with increasing strain. The rule-of-mixtures estimates predict the stiffness at zero strain, i.e., the tangent modulus at zero strain. A more practical value for design purposes is a secant modulus at either 0.5 or 1.0% strain (Fig. 7-18). The stiffness also decreases and the nonlinearity becomes more pronounced as the temperature is raised, and in addition in many systems viscoelastic effects become significant, so that a time dependent stiffness is a more appropriate parameter.

A

320

00 00

a: f— 00

0

/^^—

7 Short Fiber Composites

f

,

resistance, and also for tensile strength. These are discussed further in the following sections.

S 1

e,

7.4.2.3 Strength

e2

STRAIN

Figure 7-18. When the stress-strain curve (A) is nonlinear over its whole length, modulus may be defined by a secant modulus. This is depicted, in the figure, by the lines OSi and OS 2 , constructed at arbitrary strains st and e2 respectively.

7.4.2.2 Ductility In virtually all polymer systems the addition of rigid fillers of either particulate or fibrous form has the effect of greatly reducing the tensile ductility. The presence of a significant volume fraction of rigid filler inhibits the flow processes in the matrix associated with shear band formation and cold drawing. At the same time the particles act as internal stress concentrations, so that microcracks are more easily initiated. The inhibition of flow processes means that these cracks then tend to propagate in a more or less brittle manner. This effect tends to be more pronounced if the fiber fraction is higher and if stiffer fibers (e.g., carbon) are used. The fracture surfaces of fiber filled ductile matrices often show much evidence of ductile fracture but the measured tensile ductility (elongation) remains low. It appears that the ductile flow is confined to a narrow band either side of the fracture path so that the overall behavior is essentially brittle. This lack of ductility in filled systems has serious implications for toughness and

In a polymer reinforced with discontinuous fibers the load path must always pass through the matrix. The strength of such a system should therefore be considered in terms of a reinforced matrix rather than simply in terms of the fiber fraction which is the dominant consideration in continuous fiber systems. The strength of the matrix is enhanced by the presence of discontinuous fibers. They act to inhibit flow processes, and thus raise the yield stress of the matrix, and also act as failure barriers, forcing the failure path to follow a long, fiber avoiding, path across the section. The fibers also act as internal stress raisers and crack nuclei. There are, therefore, two competing effects: On the one hand the fibers inhibit flow and extend the failure path and, on the other, they embrittle the matrix and act as crack initiators. This is illustrated in Fig. 7-19 where the addition of glass fiber to a polyamide 6.6 thermoplastic results in progressive enhancement of stiffness and strength but a reduction of ductility to a very low level (~2%). In order for the maximum strength enhancement to be realized it is necessary that the fibers are well bonded to the matrix. Otherwise cracks are more easily initiated at the fiber interfaces and the embrittlement effect dominates. This was demonstrated by Bader and Collins (1983) who prepared well bonded and poorly bonded glass fiber/ polyamide-6 composites. The well bonded materials used a conventional E-glass fiber with a silane coupling agent (S), the other batch was identical except that the glass had not been treated with the silane (NS). The effect is dramatic, as shown in Fig.

7.4 Control of Properties and Performance

321

4

MODULUS

3.5 I'

STRENGTH

3

ELONGATION

g 2.5-1 ULJ

3

2

UJ

-j c

LU

1-

0.50

0.00

Figure 7-19. A comparison of the mechanical properties of a polyamide thermoplastic reinforced with increasing amounts of glass fiber. At 0.25 V{ the modulus is increased by a factor of almost 4, the strength by 3 but the elongation is reduced to less than a quarter of that of the unreinforced material (data from Bader and Bowyer, 1972; Curtis et al., 1978).

0.05 0.10 0.15 0.20 0.25 FIBER VOLUME FRACTION

DRY-S DRY-NS WET-S WET-NS

MODULUS

STRENGTH ELONGATION

Figure 7-20. This chart shows the effect of use of a silane coupling agent on the glass fiber in a reinforced polyamide. Stiffness, strength and impact energy are all reduced if the silane is not used. These properties are further reduced when the molding has been exposed to water, when the omission of the silane leads to a more severe degradation. Only the elongation, in the wet condition, is higher when the silane is not incorporated (data from Bader and Collins, 1983).

CHARPY

7-20, especially when the samples were conditioned by soaking in water prior to testing (WET). It was found that stiffness, strength, ductility and impact resistance were all lower when the interface bond was weaker. It is also observed that the strength is enhanced when the fibers are longer. The longer fibers result in less nonlinearity in the stress-strain curve so that the stiffness falls off less as the strain is increased. In these systems the tortuosity of the fracture

path is also greater and there are relatively fewer fiber ends to act as stress concentrators. All these effects combine to increase strength, and also toughness, although ductility remains low. 7.4.2.4 Toughness

Toughness in molded plastics is most often assessed by impact tests rather than by the more rigorous fracture mechanics procedures which have been adopted for most

322

7 Short Fiber Composites

engineering materials. The effects of fiber reinforcement in these systems are complex and often contradictory. Increases in one parameter, e.g., Charpy impact energy, may be observed, whilst another, e.g., fracture energy (Gc), may decrease when fibers are added to the matrix. Behavior at high strain rates may also differ dramatically from that at lower rates. Bailey and Bader (1985) observed that a rubber toughened, glass fiber reinforced polyamide appeared to be tougher at high strain rates than at low. This was attributed to a change in failure mode. At low strain rates the fibers pulled out "cleanly" from the matrix, whilst at the higher rates the pulled out fibers were sheathed with matrix. The additional energy was contributed by the adherent fibers tearing out of the matrix. Hill (1991) has observed similar effects in reinforced polypropylenes. Spohr et al. (1990) observed that the additional toughening effect of longer fibers was greater at high strain rates. It is necessary to distinguish the stress or energy threshold required to initiate fracture (i.e., below which no cracks are formed), the energy required to propagate fracture and the total energy of separation. The effect of fiber reinforcement is to increase both stiffness and tensile strength which leads to an increase in the parameter
inforcement is often observed to lead to a modest increase in the stress and energy thresholds to initiate failure but a large increase in the energy of propagation due to the energy dissipation processes associated with fiber pull-out. In more brittle matrix systems there is less improvement in the initiation thresholds but, again, significant improvements in propagation energy. The situation is further complicated by the fact that the introduction of fibers into the system also introduces additional modes for energy absorption during the fracture process. These include the extension of the fracture path necessitated by fiber avoidance, debonding at the fibermatrix interface as well as fiber pull-out. These processes will tend to increase the energy absorbed during the failure process but do not generally raise the stress or energy required to initiate failure. In well bonded systems it is usually observed that a low fiber fraction results in a reduction of impact energy and fracture toughness, due to the internal notching effect of the fiber, but higher fiber fractions enhance both these parameters due to their increased strength and stiffness and the additional energy dissipation modes which become available. This is shown in Fig. 7-21 in which the effects of "long" and "short" discontinuous fibers and glass "ballotini" (spheres with 30 jim diameter) on the notched Charpy impact energy of a polyamide 6.6 thermoplastic is compared. The ballotini (which had not been treated with a coupling agent) and the short glass fiber both result in a reduction of impact performance at low loadings. The impact behavior improves at higher fiber loadings but remains low at all loadings of the ballotini. The long fibers are particularly effective (see also Bailey et al., 1987). We may conclude that additional energy dissipation processes become available when the fibers

7.4 Control of Properties and Performance

323

LONG FIBER SHORT FIBER BALLOTINI

0.05 0.1 0.15 0.2 0.25 REINFORCEMENT VOLUME FRACTION

Figure 7-21. Charpy impact energy for a range of reinforced polyamides, containing ballotini (30 um diameter glass spheres), "short" glass fibers (< 250 um) and "long" glass fibers (> 500 um). The higher aspect ratio reinforcement is much more effective in enhancing the impact energy (unpublished data of Bader and Bowyer, 1972).

0.3

are used but that the dominant effect of the spheres is internal notching. Carling and Williams (1990) compared polyamides containing short and long fibers and found that there was little difference in fracture toughness when the materials were dry but the long-fiber grades were much better after exposure to a wet environment. This suggests that the effectiveness of the short fibers was reduced in the wet conditions, to the extent that the work of pull-out was greatly reduced. Toughness measured by established fracture mechanics tests (LEFM) which use low straining rates does not always correlate with the trends observed at higher strain rates (Charpy), as is indicated in Figs. 7-22 and 7-23, which also show how the overall balance of properties of a glass fiber/polypropylene system may be modified by rubber additions. 7.4.2.5 Other Mechanical Properties

Fiber reinforcement enhances both fatigue and creep performance of most thermoplastic systems. In the case of fatigue there are a number of interacting effects. On the one hand fatigue strength is increased by the increased stiffness and the

inhibition of microyielding processes, but this can also lead to high fatigue crack growth rates. The principal damage modes are cracking parallel to the fibers in the matrix or interface of regions where the fibers lie normal to the tension axis, and progressive debonding from the ends of fibers aligned in the loading direction. Both these phenomena lead to a progressive reduction in stiffness (Hitchen, 1991) and final failure will occur in a critically damaged section. At high fatigue strain amplitudes and at high loading frequency sufficient heat may be generated internally to cause local melting and thermal failure (Crawford and Benham, 1974). The overall performance is very sensitive to both fiber morphology and to the morphology of the polymer phase, especially in semicrystalline systems (see Karger-Kocsis and Friedrich, 1988; Karger-Kocsis, 1990). The heat distortion temperature (HDT) is significantly raised by the addition of fibers, especially when they are well bonded. This increases the maximum use temperature of the materials and also leads to improved creep performance. Dimensional stability is also improved notably by the reduction in the coefficient of thermal ex-

324

7 Short Fiber Composites

MODULUS CHARPY LEFM

KEY PP : POLYPROPYLENE G : GLASS (Wt%) CG : COUPLED GLASS R :RUBBER

PP

Figure 7-22. A comparison of stiffness and fracture behavior for the same range of glass fiber reinforced, rubber toughened polypropylenes shown in Fig. 7-16. Charpy energy and Gc measurements made on a compact tension test piece (LEFM) are compared. The relative toughness is improved by the combination of both glass fiber and the rubber (Hill, 1991).

PP + CG30 PP+R+CG30 PP + G30 PP + R+CG15 PP+R+CG40

MATERIAL

MODULUS STRENGTH ELONGATION

KEY PP : POLYPROPYLENE G : GLASS CG : COUPLED GLASS R :RUBBER

PP+CG PP+G

Figure 7-23. Stiffness,

strength and elongation compared for the same range of materials as shown in Figs. 7-16 and 7-22.

PP+R+CG PP+R

pansion, which also leads to much lower "in mold" shrinkage. This simplifies the manufacture of precisely dimensioned components. In many cases dimensional stability alone is a sufficient justification for selecting a fiber reinforced system. There is, however, a need for caution, since the fiber orientation distribution can also lead to considerable anisotropy and problems due to the development of differential shrinkage stresses.

7.5 Manufacturing Processes 7.5.1 Introduction

It will be clear from Sec. 7.2 that a wide range of materials may be classified as short fiber reinforced plastics. The main distinctions lie in whether the matrix is a thermoset or a thermoplastic, whether the intermediate feedstock is granular or sheetlike and whether the reinforcement is indi-

7.5 Manufacturing Processes

vidual short (< 10 mm) filaments or longer bundles of chopped glass strand (1075 mm). It must be recognized that discontinuous fibers cannot reinforce as effectively as continuous fibers and, furthermore, the degree of reinforcement is limited by the fact that much lower fiber fractions must be employed in order that the flow characteristics of the material are adequate. It follows that SFCs will be considered for applications where ease of processing, and the associated lower costs, are more important than higher mechanical properties. Economic considerations also influence the choice of matrix and reinforcement. The short-fiber reinforced plastics market is dominated by cheap commodity plastics reinforced with E-glass fibers. One may argue that if it is necessary to have higher performance than can be obtained from these materials, it is more cost effective to go to continuous fibers, such as carbon or aramid, in a higher performance matrix. This is frequently the case, however, there are also many circumstances where the ability to shape intricate components by molding leads to the choice of engineering thermoplastics reinforced with high performance fibers (mainly carbon). These high performance composites command a significant cost premium over the commodity grades and are therefore used in much smaller (but increasing) quantities. 7.5.2 Rheological Considerations

The basis of all the processes to be discussed is that the materials are shaped by flow into molds of varying degrees of intricacy. The flow characteristics of the polymer and of suspensions of the reinforcement in the polymer are of the greatest significance. It is also necessary to consider whether the polymer is a thermoset or a

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thermoplastic and the condition, or format, of the material prior to the forming operations. Thermoplastics will always be solids at room temperature. They must be heated to above their melting or softening temperature so that they may be forced to flow into the mold. The pressure required will be a function of the viscosity of the compound and the intricacy of the mold, and to a lesser extent on the time available for the mold-filling process. Thermosets, on the other hand, may be solid, liquid or pasty at room temperature and prior to the final processing operation. When they are heated their viscosity is reduced, often to very low values, so that flow processes may operate. However, heating also initiates the curing reaction, which results in an increase in viscosity, leading eventually to gelation, when all flow processes cease. Heating to higher temperatures results in lower initial viscosity but much faster rates of cure. This may result in insufficient time for the charge to flow into the mold. With thermosets it is therefore necessary to establish a temperature and pressure cycle which allows an adequate processing window, as well as an economic production rate. In processing thermosets it is also necessary to consider the reaction exotherm, which can result in intense internal heating, and the disposal of any volatiles produced by the thermosetting reactions. The viscosity of molten polymers is influenced by their molecular weight (MW) and by interactions between molecules. In general thermoplastics are of much greater MW than the unreacted components of thermosets, so that molten thermoplastics have much higher viscosity. To some extent the viscosity may be controlled by the MW distribution. A relatively small proportion of lower MW polymer will

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7 Short Fiber Composites

reduce the overall viscosity by a significant amount. Plasticizers are also used to achieve similar effects. Viscosity may also be reduced by raising the temperature but there are practical limits imposed by the onset of thermal or oxidative degradation. The addition of solid fillers to the polymer leads to an increase in viscosity and this tends to be greater as the aspect ratio of the filler is increased. Fibers, thus, have the greatest effect and at volume loadings exceeding about 30% interactions between the suspended fibers cause an even greater increase. This usually imposes an upper limit to fiber content of about 40%, but particulate fillers can be tolerated to higher loadings. Polymer melts are pseudoplastic, i.e., their apparent viscosity drops as the shear rate is increased. This effect is attributed to the alignment of molecules under high shear rate conditions. The effect is large and polymer melts may be modeled as power law fluids with an exponent of about 0.5. This means that the viscosity drops by an order of magnitude for a two-order of magnitude increase in shear rate. Viscosities measured at low shear rates (e.g., melt flow index) can therefore be misleading. The addition of fibers, whilst tending to increase the viscosity at a given shear rate, leads to enhanced pseudoplasticity, so that the reinforced grades may flow almost as easily as the unfilled material at high shear rates. Such rates may apply in processes such as injection molding and reaction injection molding but not in compression molding and thermoforming. Another important consequence of flow is that the orientation of both the polymer molecules and the suspended fibers is controlled by the flow process. As a general rule convergent laminar flow, which occurs when the melt flows from a wide to a narrower section, results in the fibers becom-

ing oriented parallel to the flow direction. Under divergent flow the fibers become oriented normal to the direction of flow. Divergent flow is commonly encountered at the gate of a mold where the charge flows from the narrow sprue through the gate and into the wider mold cavity. At higher shear rates the degree of orientation is more pronounced. Filling a mold is, however, rather more complicated as the flow streams interact with the mold surfaces. Heat transfer (see below) considerations dictate that most practical moldings are based on thin platelike sections. Friction at the mold surfaces modifies the flow pattern so that in most cases a layered orientation distribution is developed. Close to the surface both fibers and the polymer molecules are oriented in the mold fill direction, whilst in the center the orientation tends to be normal to the mold fill. This is illustrated in Fig. 7-24. The overall effect is that the molding has some of the characteristics of a cross-plied laminate. The properties of a molding are strongly influenced by the fiber orientation distribution. Stiffness and strength are greater and coefficient of thermal expansion lower in the direction parallel to the fibers, and vice versa. Preferred fiber orientation leads to nonuniform shrinkage in the mold, which

MFD

Figure 7-24. Diagram illustrates the typical skin and core fiber orientation in an injection molding. The fibers near the mold wall tend to become aligned in the mold fill direction (MFD), whilst those in the core are normal to this direction. The ratio of core to skin thickness can be influenced by the injection speed and design of the mold.

7.5 Manufacturing Processes

may affect the final dimensions of the molding. Thermal stresses develop between the layers in a typical molding and, if there is any assymetry in the arrangement of surface and core layers, can lead to warping and distortion. It is therefore important to control the fiber orientation. This is achieved mainly by the design of the molding and especially by the positioning of the feed sprues. The charge temperature and the mold fill rate, which determine local shear rates are also important. The mold filling process may be modeled by finite element techniques and several commercial packages are available to mold designers. Another important aspect of the flow process is the formation of weld or knit surfaces in the molding (Cloud and McDowell, 1976). These may be formed wherever the melt stream has to split and then recombine. This will occur at any cored feature in the molding, e.g., a hole, and is illustrated in Fig. 7-25. The conditions during most molding operations are such that the melt flow is essentially laminar, so that when the streams recombine

(a)

327

downstream of the core there is no turbulence and the fibers tend to be aligned parallel to the flow direction. This results in a line of weakness. The effect is more severe if the core is situated towards the extremity of the molding. In practice some weld lines are inevitable in most moldings, their deleterious effects are minimized by ensuring that they lie in noncritical parts of the component. Weld lines can be virtually eliminated by use of the multiple live feed technique described by Allan and Bevis (1987, 1990). This is a modification of conventional injection molding, where, after the mold has been filled by a conventional injection process, a pair (or several) reciprocating pistons agitate the charge, breaking up weld lines and generally tending to promote a more random fiber distribution. Finally we must consider the effects of pressure. Most polymers melts are compressible to a significant extent, so that if very high pressures are used to fill a mold cavity the polymer is compressed and this offsets the shrinkage which occurs as the charge solidifies and cools in the mold. This effect may be exploited in the injection molding of thermoplastics to obtain moldings with lower apparent shrinkage from the nominal mold dimensions as shown schematically in Fig. 7-26. The degree of compressibility varies between polymers, in general the amorphous materials compress to a greater extent than the semicrystalline polymers. Optimum molding pressures are typically of the order 1-5 kbar. 7.5.3 Thermal Considerations

Figure 7-25. In a molding containing a core feature which splits the polymer flow a weld or knit line will form downstream of the obstruction. Fibers will not bridge this line and it is therefore a potential weakness. The effect tends to be more pronounced when the feature is remote from the gate [e.g., (b)].

Thermoplastics are processed by a meltfreeze cycle. They must first be plasticized by heating to temperatures above their melting point, then shaped and finally constrained in the desired shape whilst they are cooled to below their glass-transition

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7 Short Fiber Composites

INJECTION PRESSURE

= O

Figure 7-26. The mass of a molding is related to the injection pressure used. Higher pressure forces more material into the mold and compresses the liquid polymer. This tends to counteract the normal shrinkage associated with freezing and cooling down to ambient temperature. High pressure ensures that the molding conforms more closely to the cavity dimensions.

temperature 3, Tg. Heating and cooling are, thus, both vital aspects of processing. Polymers have poor thermal conductivity and their high melt viscosity means that convection can play no part in the heat transfer process. The melting or plasticizing stage of processing is therefore dominated by the need to get heat into the material. In compounding and injection molding operations it is almost always necessary to incorporate some mechanical mixing into this operation. The most efficient plasticization system is the screw pump, which forms an integral part of most extruders and injection molding machines. The granular charge is transported down the heated barrel, where it is heated by conduction and at the same time subjected to a vigorous mixing by the action of the screw. This constantly brings new material into contact with the hot barrel so that the whole charge is heated rapidly and 3 Semicrystalline polymers must be cooled to well below the crystalline melting point, whilst amorphous materials need to be cooled to below their Tg.

the temperature is equalized throughout the volume of the charge. The barrel may be maintained at a higher temperature than would be admissable in a static heat transfer situation where degradation of the polymer might occur. Very high shear rates can be developed in the screw pump and this provides an additional means of heating the polymer, simply through the mechanical work done during the mixing. In large compounding extruders most of the heat input can be provided by this means but it is seldom appropriate to use such high shear rates in fiber filled systems, owing to the amount of fiber attrition (and screw and die wear) which would occur. The output of a screw pump is a function of its size, barrel temperature, screw rotational speed, and the shear rate. The latter parameter being a function also of the screw design and the operating back pressure (Franzen et al., 1989; Bailey et al., 1987). The next stage of the process involves the injection of the charge into a cold mold, where the material must be allowed to cool until it is sufficiently dimensionally stable for the part to be removed from the tool. The heat extraction process must be entirely by conduction through the mold wall. This is an inherently slow process on account of the low thermal conductivity of the polymer and poor contact between the cooling, and shrinking, part and the mold surface. This is nearly always the rate controlling step of the injection molding process. Although the molds may be water cooled, it is often desirable to maintain them at moderately elevated temperatures in order to improve surface finish. The only practical means of control is to ensure that the component sections are kept thin. A similar strategy may be employed in other molding processes. It should be noted that thermoforming sheet materials

7.5 Manufacturing Processes

are made in thin sheet form so that they may be rapidly heated in radiation ovens. Rather different considerations apply when molding thermoset based materials. Here the polymer must first be plasticized by heating, but not be allowed to gel before all flow processes are complete, and then be cured in the hot mold. The part is ejected hot after the completion of the cure but attention must be paid to the exotherm heat produced by the reaction, which can result in overheating. This may restrict the maximum mold temperature and hence extend the processing time. 7.5.4 Injection Molded Thermoplastic Composites

Processing is in two distinct stages which will be considered separately. The first is a compounding operation in which the polymer and the fiber are combined and processed into granular feedstock for the final injection molding operation. 7.5.4.1 Extrusion Compounding

The polymer is plasticized in a single, or twin screw extruder and blended with the fiber. The compound is then extruded through a "spaghetti" die to form thin rods which are air or water cooled and then cut into pellets, typically 2 - 3 mm diameter and 3-6mm long. There are several variants to the basic process: (I) Polymer granules and chopped fiber strand are dry blended in the appropriate proportions and then charged into the feed hopper of the extruder. The blend is melted and mixed, and then possibly passed through a decompression zone where volatiles are removed by application of a vacuum. The plasticized charge is then recompressed and passed through the extruder die.

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(II) The polymer granules are charged separately into the feed hopper. They are plasticized and then the fiber is added at the decompression port, either as chopped strand or, more usually, as continuous roving, which is taken up by the screw and passed through a mixing zone, and possibly a further decompression zone, before passing through the die as before. In process (I) the fibers are originally chopped to lengths of between 5 and 10 mm before dry mixing with the polymer granules. The dry mixture is transported by the screw into the hot barrel section of the extruder where it starts to melt. Initially just a thin film of melt is formed adjacent to the extruder barrel and very high shear rates are developed in this zone. This results in a great deal of breakage of the fibers and few fibers of length greater than 0.5 mm survive this stage. Further fiber breakage occurs in the subsequent stages of the extruder and die so that the fibers in the final granules are generally of mean length of 0.2-0.4 mm. In process (II) the fibers are added to the already plasticized polymer. The extent of breakage is somewhat less severe but is still considerable, and even when continuous rovings are used the final fiber length is seldom above 0.8 mm. The degree of attrition can be reduced by operating the extruder under less severe conditions but this would seriously decrease output, reduce the degree of fiber dispersion and is not commercially acceptable. A number of alternative processes have been proposed, for example: Bader and Bowyer (1973) and U.K. patents of ICI Ltd. (1985 a, b), where a continuous fiber roving is impregnated with liquid polymer to form a thin rod filled with continuous aligned fibers. This may then be cut into pellets of 5-10 mm length but the fibers remain at this length (Gore et al., 1986).

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7 Short Fiber Composites

The essential differences in the two types of molding pellet are illustrated in Fig. 7-27. Various processes have used polymer melts, polymer solutions and cross-head dies to produce such a product. A current technique is based on a variation of the pultrusion process, where the continuous strand is passed through a fluidized bed of polymer powder and drawn through a tapered die, where the polymer melts and is forced to infiltrate the roving. These processes provide a "long fiber" feedstock for injection molding. Further attrition is bound to occur in the injection molding process, but this does not need to be operated at such high shear rates as the compounding extruder, so that this technique provides the possibility of retaining longer fibers in the final molded part (Bailey et al., 1987; Bailey and Kraft, 1987). The fiber strand, however, tends to be less well dispersed than in conventional systems, so that the improved mechanical properties must often be offset by poorer surface finish.

10mm

3mm

4mm •

Figure 7-27. Illustration of the "long" and "short" fiber molding pellets. The top sketch is of material manufactured by impregnating a continuous fiber roving and then cutting it into 10 mm pellets. The fibers run continuously through the entire 10 mm length. The lower sketch is of extrusion compounded pellets, in which the fibers have already been broken into submillimeter fragments.

7.5.4.2 Injection Molding

The basic principle of the injection molding process for thermoplastics is that a charge of the material is plasticized and then injected under high pressure into a cold mold. The charge cools in the mold and when sufficiently stable may be ejected from the mold, and the cycle repeated. Injection molding machines are highly sophisticated devices, designed for continuous automatic operation at high production rates. The machines and the tools require a high capital investment and their operation is relatively economic in the use of labor. The process is therefore most suited for long runs where many thousands of identical components are to be produced. The major attractions of the process are its high productivity and the fact that very complex shapes may be produced in a single automated operation. This often gives the further advantage of parts consolidation where a single injection molded component may replace a complex assembly of parts produced by competing processes, e.g., sheet metal pressings. The disadvantages of the process, as applied to short fiber composites, are that SFC's are intrinsically less stiff and strong than continuous fiber composites and that it is difficult to control the flow of material during molding to be able to exploit controlled fiber orientation. The molding of fiberreinforced thermoplastics is discussed by Folkes and Kells (1985). Most modern injection molding machines use a screw pump, similar to that on an extruder, to effect the plasticization. A typical machine layout is shown schematically in Fig. 7-28. Compounded granules are charged into the feed hopper from whence they enter the screw. The solid charge is moved forward by the rotation of the screw into the heated barrel, where the

7.5 Manufacturing Processes

CD

331

Q)

® Figure 7-28. A schematic diagram of an injection molding machine. The hydraulic ram (1) opens and closes the mold (2), which contains the mold cavity (3). The charge is fed from the hopper (6), and is plasticized as it is moved forward by rotation of the screw (5), where it is trapped by the nozzle valve (4), causing the screw to retract against a "back pressure" imposed by the ram (7). When sufficient charge has been plasticized, the carriage is moved forward by the ram (9) to locate the nozzle in the mold. The nozzle valve is opened and the ram (7) drives the stationary screw forward to inject the charge into the mold. After the charge has frozen in the gate the nozzle is disengaged from the mold, the hydraulic motor (8) turns the screw to plasticize the next charge. The mold is opened, the part ejected and the cycle is repeated.

charge is melted, by conduction from the hot barrel wall, and mixed by the action of the screw. The plasticized charge is transported forward through the compression and metering stages of the screw towards the nozzle which, at this stage, is closed by a valve. Here the charge sets up a back pressure which forces the screw to retract against a controlled hydraulic pressure in the main cylinder. This continues until a sufficient quantity of the charge has been built up in the nozzle region. At this stage the screw rotation is stopped. The carriage is moved forward to engage the nozzle in the mold, which is held closed by a hydraulic or mechanical locking device. The nozzle valve is opened and the charge forced into the mold cavity by the action of the main ram pushing the (nonrotating) screw forward. A considerable pressure is built up in the cavity (typically 1000-2000 bar, 100-200 MPa), and is maintained (dwell) until the charge has frozen in the

feed gate. This is the point at which the charge cannot flow out of the mold when the pressure is relaxed. At this time the carriage can be retracted and the next plasticization cycle initiated. Meanwhile the charge in the mold continues to cool. When it has cooled to well below Tg the mold is opened, the part ejected and the mold then closed again in readiness for the next injection cycle. This complex cycle must be precisely controlled if the output is to be consistent. In older machines this was accomplished by means of several temperature controllers on the barrel, nozzle, and mold, by timers to initiate the various actions and injection and back pressure settings were established by trial and error. Consistent operation was possible, once a steady state had been established, but problems can arise from minor variations in the feedstock and ambient conditions. Modern machines (see for example, Menges and

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7 Short Fiber Composites

Lutterbeck, 1979) are fitted with interactive controls: Sensors detect temperatures and pressures in vital parts of the system, monitor the component weight and adjust the operating parameters under the control of an expert system. 7.5.4.3 Process Control In view of the high capital cost of the equipment it is essential that productivity be maximized. The rate controlling process is most often the time required for the part to cool in the mold before it may be ejected. This may only be controlled, in any practical sense, by attention to component design. Ideally all sections should be thin and of uniform thickness. Sections of more than 6-10 mm thickness are generally to be avoided, although it is quite possible to handle thicker sections with the penalty of long cycle times. A further problem with thick sections is a tendancy for voiding to occur in the center of the section. This is due to shrinkage of the charge as it cools and solidifies. The problem tends to be more acute in fiber filled systems because, although the thermal shrinkage is lower than in unfilled materials, the shrinkage is nonuniform and greater in directions normal to the fiber orientation, i.e., the thickness direction. The fiber-matrix interface also provides a line of weakness which allows large voids to open up. In many cases, however, voiding confined to the core regions is not unduly detrimental and controlled core voiding is even promoted in the so-called structural foam materials. The fact that the cooling time is usually the critical parameter in determining the cycle time means that it is not necessary to maximize the plasticization capacity of the screw pump. Some fiber breakage will occur in the screw, but this may be minimized by use of moderate shear rates and low back pressure. Of course, this will not have

any useful effect if the fibers have already been broken up during the compounding operation but is vital when handling the molding materials containing longer fibers described in Sec. 7.4.4.1 above. Moderate injection rates also minimize fiber breakage as do larger gate dimensions. Unfortunately the operating conditions which limit fiber breakage all tend to increase cycle times and limit the degree to which the pseudoplastic nature of the charge can be exploited, by use of high shear rates. Undoubtably the most important aspect of injection molding of fiber reinforced materials is the control of fiber orientation distribution. The principles have been discussed in Sec. 7.3.5 above, but some further comment is relevant. In thin, platelike, sections a skin and core distribution is usual. Fibers near the surface are oriented in the mold fill direction and those in the core lie normal to this direction. In a complex part the material flow will be separated by cored out features and will recombine downstream, leading to the formation of weld or knit surfaces. At the best these result in differences in surface texture, at the worst they act as serious weaknesses. Fiber orientation distribution is controlled mainly by the design of the mold and its gating arrangements. This design is increasingly being based on computer simulation of the melt flow using sophisticated finite element based software. This is a very significant development since, previously, the only method of control was a trial and error process involving costly mold modification. It must be noted, however, that the mold flow pattern is also affected by the viscosity of the melt and the rate of injection, so that both mold design and machine operating parameters must be addressed during the design process. This is a strong argument for the subsequent use of machines with interactive control systems.

7.5 Manufacturing Processes

The phenomenon of jetting is of especial concern when molding fiber reinforced plastics. Jetting is the formation of a thin intense stream of the molten compound, usually at the gate. The stream will "squirt" right across the section, impinging on the far surface and then causing the section to fill in an uncontrolled manner. A "worm" of material folds resulting in a random array. This is highly undesirable. The problem is related to die-swell; when a stream of molten polymer emerges from a narrow die aperture, it swells due to elastic relaxation effects, so that the diameter of the stream is greater than that of the die. Ideally the gate diameter in an injection mold should be such that this expanded stream immediately makes contact with the mold walls (Fig. 7-29). If the die swell is insufficient a jet may form. Fiber filled polymers show much less die swell than unfilled materials and hence are more prone to jetting problems. The solution is to use gate locations which ensure that a jet will be immediately quenched by contact with the mold wall.

(a)

(b)

Figure 7-29. These figures illustrate the problem of jetting. The photograph (a) is of a deliberately "short" shot into an end-gated test piece mold. The charge has jetted right of the extremity of the cavity and has then started to fill in an irregular manner. The diagrams (b) and (c) illustrate alternative "side" and "tab" (or "submarine") gates. Both of these gates would obviate the jetting problem, as the jet would be quenched on the opposite face of the cavity. In this respect the tab gate is the most effective.

333

7.5.5 Injection Molding Thermosetting Composites

The application of the injection molding process to thermosetting polymers and composites is relatively recent. It had formerly been considered that these materials could only be molded by compression molding. The general principles of the process and the equipment are similar to those for handling thermoplastics but with some important modifications. Thermosets must be injected into hot molds where they cure. They may then be ejected whilst still at the cure temperature. This eliminates the rate controlling cooling phase from the process cycle but introduces the cure time as an alternative consideration. The plasticization process is rather more critical as the charge must be heated to reduce its viscosity to a level which will allow it to be injected effectively, but not sufficiently to initiate premature gelling-especially in the screw section of the machine! A further difference arises from the differences in the feedstock. Whilst fiber reinforced thermoplastics are invariably supplied as fully compounded solid granules, thermosets may also be in the form of pasty mixtures of resins and relatively long (25 mm) fiber bundles. The implications of these considerations on the molding operation will now be discussed. Some thermoset compounds, e.g., those based on phenolic resins, are supplied as solid free-flowing granules which may be conventionally charged into a screw plasticizer via a gravity feed hopper. However, more often the material is basically a dough or bulk molding compound (DMC, BMC) which consists of a pasty mixture of liquid resin, fiber, and filler. This must be fed to the screw by means of a stuffing box or a secondary screw designed to handle the compound.

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7 Short Fiber Composites

Once fed to the screw the compound is transported through a heated barrel section and injected in much the same way as in the case of thermoplastics. There are a number of differences. Firstly, the charge must be heated more gently, to avoid premature gelation whilst it is in the screw section. It is usual to use hot oil or steam jacketed barrels to avoid overheating. Secondly the plasticized material is usually much more viscous than the thermoplastics due to the temperature limitation and to high filler and fiber contents. This leads to the need for more robust machine design with more powerful drive motors. The charge is then injected into the mold which is hotter than the barrel, so that the charge must now be heated up to the cure (mold) temperature. With thick sections this may take a significant time. Finally the charge must reside in the mold at its cure temperature until the cure is complete. The heatup from barrel to cure temperature may be accelerated by use of fast injection speeds through restricted gates, which heats the charge adiabatically, at the expense of some fiber breakage. This can dramatically shorten the process cycle, especially when thick sections are formed. Two further considerations arise from the cure process itself. These are the effect of the cure exotherm, which might lead to uncontrolled overheating. This is not usually a serious matter if sections are not too thick and can be exploited to speed up the overall process. The other effect is that volatiles are often produced by thermosets as a consequence of condensation polymerization. If this occurs, provision must be made to vent the mold. It is sometimes even necessary to crack open the mold, for a second or so, before cure is complete to vent off excess volatiles. Overall, molding thermoset based composites requires a high level of control. The

rate controlling stage of the process tends to be the cure time of the resin and, since the part does not need to be cooled down before ejection, it is cost-effective to mold much thicker sections than is the case for thermoplastics. 7.5.6 Compression Molding

This is the most widely utilized process for molding thermoset based short fiber composites. The process is simple in principle: A charge of the compound is compressed in the cavity between a pair of heated matched dies. The pressure forces the charge to fill the cavity and the heat causes the thermoset to cross-link and cure. Production rate is controlled by the rate of heat transfer to the cold charge from the hot mold and by the time required for the resin to fully cure. As in other thermoset processing systems due regard must be paid to the cure exotherm and to the venting of any volatile condensation products. Three basic classes of composite are suitable for processing by compression molding. Granular fiber filled thermosets (e.g., phenolic resin based), bulk molding compounds (BMC) and sheet molding compounds (SMC), the latter may also be reinforced with continuous fibers. In the case of the granular thermosets the feedstock is usually a coarse dry powder formed by grinding a prepolymer which was melt compounded with the fiber (and other fillers). It is often convenient to precompress this material into larger pellets of up to 100 g or so. The charge is usually predried to remove any moisture, and then a weighed amount is placed in the hot open mold. The mold is then closed and pressure applied when the charge melts,flowsto fill the cavity and then cures. When cure is complete the mold is opened

7.5 Manufacturing Processes

and the part ejected. The cycle time is typically 5-10 min for parts of up to 200400 g weight. This process is generally used for relatively small parts (< 1 kg) and is suitable for relatively short runs (a few thousand). For longer runs injection molding would be preferred. Dimensional control is achieved by preweighing the charge and by the compaction pressure. The mold is designed so that the pressure is maintained throughout the curing cycle. (Alternative processes, in contrast, program the mold to close to fixed stops so that the excess charge may be bled away.) Controlled clearances are often designed into the mold to allow volatiles to escape. It is often useful to impose a dwell time between mold closure and application of the full pressure. This allows the charge to soften. The pressure will then force the charge into the extremities of the mold cavity. A material must be selected which has sufficient flow capacity for the intricacy of the molding to be produced. This is controlled by the degree of prepolymerization of the feedstock, its formulation and total filler content. More advanced polymers cure more rapidly and evolve less volatiles, but have less flow potential. It is desirable to use a system which cures as rapidly as possible but it is also necessary to ensure that the processing window is adequate (i.e., the time at low viscosity is sufficient to complete the flow processes before gelation intervenes). It is necessary, therefore, to establish an acceptable compromise. In general similar considerations apply when processing BMC and SMC materials. These are most commonly based on unsaturated polyester and vinylester resins which are liquids in the precured state. The bulk (or sheet) materials are preweighed and placed in the hot mold, as before, and then subjected to pressure so that the cavity is filled and the resin cures. It is not

335

always necessary to use as high a pressure as with the phenolic type material, so that larger components may be produced. Cure times of the order 5-20 min are common. In the case of SMC it is common practice to tailor the sheets of charge so that approximately 50% of the projected area of the mold is covered. This ensures the optimum degree of flow to fill the mold and produce an acceptable surface finish. SMC is widely used for the production of large panels for automotive applications where surface finish is one of the primary specification requirements. 7.5.7 Thermoforming

This process is applicable to thermoplastic based composites. The feedstock is usually in sheet form but variants using preplasticized "slugs" of compound are also used. The feedstock for the sheet process is manufactured by either extrusion of a short fiber filled thermoplastic compound, or by corolling or calendering a pack of alternate sheets of thin thermoplastic sheet (foil) and random fiber sheet. (Note that continuous mat is also processed by this route.) The final product is a sheet, perhaps 5 mm thick, consisting of thermoplastic resin with up to about 30% by volume of fiber. In the final shaping process suitable sized blanks are cut from the feedstock sheet and plasticized by passing through a radiant heated tunnel oven. The softened sheets are then placed between cold matched dies and pressed (slow) or stamped (fast). The soft charge flows under the pressure to fill the die cavity and is then rapidly chilled, when it may be removed from the tool. The critical aspect of the process is to ensure that the cavity is completely filled before the polymer is chilled to below its flow temperature. This is a

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7 Short Fiber Composites

function of the charge weight, the area of the die and the tool temperature. It may be necessary to keep the tool warm (just below Tg) in order to achieve adequate flow. This will extend the cycle time. Typical cycle time is of the order of 30 s and the process is therefore much faster than either the SMC/BMC or the injection molding processes. A considerable degree of draw may be achieved together with the ability to reproduce intricate detail, such as stiffening ribs. The main disadvantage of the process is that the maximum fiber content of the material is rather low and this limits the stiffness and strength of the product. In this respect thermoformed products tend to be inferior to their thermoset equivalents. In a further variant of this process the charge is produced by injection molding or extruding a blank, or roughly shaped part (a "slug"). This is then reheated and pressed between cold dies. In some cases the slug is extruded directly into the cold die, eliminating the reheating operation. The advantages claimed for this process is that tooling costs are much reduced in comparison with injection molding, but it is not able to produce components with an equivalent degree of intricacy. The process is not very widely used which is an indication of its limitations. 7.5.8 Reaction Injection Molding (RIM)

This may be considered as a special case of injection molding of thermosetting material. The basic process utilizes a fast curing thermoset system, in which two reactive components are mixed and then fully cured, usually without recourse to external heating, within a few minutes. The original formulations were based on polyurethanes, the two components being a difunctional polyol and a di-isocyanate. Various formu-

lations are available which cure to rigid or flexible solids, or even to rigid and flexible foams by utilizing a gaseous condensate as a blowing agent. Composite systems, known as reinforced reaction injection moldings (RRIM), are made by incorporating a suspension of short fibers in one, or both, of the prereacted components. Composites may also be formed by preplacing the reinforcement (short or continuous fiber preforms) in the mold and then injecting and infiltrating the reacting polymeric components. This is designated structural RIM (SRIM). The concept is now being developed for other types of thermosets. These include unsaturated polyester derivatives and polyamide systems based on the anionic polymerization of caprolactam (Nyrim). In the original polyurethane process the two components are low viscosity liquids. These are pumped from holding tanks to a pair of high pressure cylinders, from which precisely metered quantities of each component are injected at high velocity and high pressure into the mold cavity, through an intermediate turbulent mixing chamber, situated at the feed gate. This arrangement ensures that the liquids are thoroughly mixed as they enter the mold cavity, and not before! The curing reaction is immediately initiated and is typically complete within 30 s to 2 min. The reaction is exothermic so that the molds need to be cooled. In the RRIM version of the process, short glass fibers (usually very short milled fiber) are mixed into the polyol component. It is not practicable to put them in the isocyanate because moisture and other components in the glass finish (size) tend to initiate decomposition of the, highly reactive, isocyanate. The amount of fiber added to the polyol is limited by considerations of viscosity to about 20%, so that it is difficult

7.6 References

to get more than about 10% fiber in the finished product. This coupled with the fact that the fibers are very short places a severe limitation on the possible degree of reinforcement. The process is attractive on account of the fast cycle times and because tooling costs are significantly lower than for conventional injection molding. Polyurethane RRIM components have been successfully used for panels and moldings in automotive applications in situations where dimensional stability, rather than load bearing capability, is the principal requirement. In these circumstances the modest reinforcement confers enhanced stability at moderately elevated temperatures. This enables the moldings to be painted "on line" in conventional stoving ovens.

7.6 References Allan, P. S., Bevis, M. J. (1987), Plast. Rubb. Process. Appl. 7, 3. Allan, P. S., Bevis, M. J. (1990), Composites Manuf. 1, 79. Bader, M. G., Bowyer, W. H. (1972), /. Phys. D5, 2215. Bader, M. G., Bowyer, W. H. (1973), Composites 4, 150. Bader, M. G., Collins, J. F. (1983), Fibre Sci. and Tech. 18, 217-231. Bader, M. G., Curtis, P. T. (1977), in: Proc. Fall Meeting, Chicago, Failure Modes in Composites: Grossman, F. W. (Ed.). New York: TMS-AIME. Bader, M. G., Curtis, P. T., Chatwal, R. S. (1975), Proc. Int. Conf. Composite Materials, Vol. 2. New York: AIME, p. 191. Bader, M. G., Chou, T. W, Quigley, J. J. (1978), in: New Developments and Applications in Composites: Kuhlmann-Wilsdorf, D., Harrigan, W. C. (Eds.). Warrendale, PA: TMS-AIME, pp. 127-138. Bailey, R. S., Bader, M. G. (1985), in: Developments in the Science and Technology of Composite Materials: Bunsell, A., Lamicq, P., Massaih, A. (Eds.). Amsterdam: Elsevier, pp. 265-270. Bailey, R. S., Kraft, H. (1987), Int. Polymer Processing 2, 94-101. Bailey, R. S., Davies, M., Moore, D. R. (1987), Proc. Test. Eval. and Quality Control of Composites: Herriot, J. (Ed.). Sevenoaks, U.K.: Butterworths.

337

Bowyer, W. H., Bader, M. G. (1972 a), J. Mat. Sci. 7, 1315. Bowyer, W. H., Bader, M. G. (1972b), Chem. Soc. 2, 165. Bright, P. E, Darlington, M. W. (1981), Plast. Rubb. Process. Appl 1, 139. Carling, M. I, Williams, J. G. (1990), Poly. Composites 11, 6, 307. Cloud, P. J., McDowell, F. (1976), Plast. Tech., August. Cox, H. L. (1952), British J. Appl. Phys. 3, 72. Crawford, R. I, Benham, P. P. (1974), J. Mater. Sci. 9, 18-28. Crowson, R. J., Foulkes, M. I, Bright, P. F. (1980), Poly.Eng. Sci. 20(14), 925. Curtis, P. T., Bader, M. G., Bailey, J. E. (1978), /. Mater. Sci. 13, 317. Darlington, M. W, McGinley, P. L. (1975), / Mater. Sci. 10, 906. Drzal, L. T., Rich, M. X, Camping, J. D., Park, W. J. (1980), Proc. 35 th Tech. Conf. Reinforced Plastics I Composites Inst. New York: Soc. Plastics Ind., Paper 20-C. Folkes, M. I , Kells, D. (1985), Plast. Rubb. Process. 5, 125-130. Franzen, B., Klason, C , Kubat, X, Kitano, T. (1989), Composites 20, 65. Fukada, H., Chou, T. W. (1982), /. Mater. Sci. 17, 1003. Galiotis, C , Young, R. X, Yeung, P. H. X, Batchelor, D. N. (1984), /. Mater. Sci. 19, 3640. Galiotis, C , Jahankhan, H., Melanitis, I., Batchelder, D. (1989), Proc. 3rdEurop. Conf. Composite Materials, Bordeaux. Amsterdam: Elsevier. Gore, G. R., Cuff, G., Cianelli, D. A. (1986), Mat. Eng. 103.3, 47. Halpin, X C , Kardos, X L. (1976), Poly. Eng. Sci.

16(5), 344. Halpin, J. C , Kardos, X L. (1978), Poly. Eng. Sci. 18 (5), 496. Hill, A. R. (1991), Ph.D. Thesis, University of Surrey, U.K. Hitchen, S. (1991), Ph.D. Thesis, University of Surrey, U.K. ICI Ltd. (1985a), U.K. Patent GB 1334 702. ICI Ltd. (1985 b), U.K. Patent GB 1565 195. Ishida, H., Koenig, X L. (1978), Poly. Eng. Sci. 18 (2), 128. Jones, R. M. (1975), Mechanics of Composite Materials. New York: McGraw-Hill. Karger-Kocsis, X (1990), Composites 21, 3, 243. Karger-Kocsis, X, Friedrich, K. (1988), /. Thermoplast Compos. Mater. 1, \26. Kelly, A., Tyson, W. R. (1965), /. Mech. Phys. Solids

13(6), 329. Krenchel, H. (1964), Fibre Reinforcement. Copenhagen: Akademisk Forlag. Lees, X K. (1968), Poly. Eng. Sci. 8, 195. Lunt, X (1980), Ph.D. Thesis, Liverpool University, U.K.

338

7 Short Fiber Composites

Lunt, J., Shortall, 1 (1980), Plast. Rubb. Process. 5, No. 2, 37. Menges, G., Lutterbeck, J. (1979), Plast. Rubb. Int.

4(2), 59. Mittal, V. B., Gupta, R. K. (1982), /. Mater. Sci. 17, 3179. Mittal, V. B., Gupta, R. K., Sharma, P. K. (1987), /. Mater. Sci. 22, 1949. Plueddemann, E. P. (1970), 25 th Ann. Techn. Conf. Washington, DC: Soc. Plast. Ind. Plueddemann, E. P. (1982), Silane Coupling Agents. New York: Plenum Press. Sato, N., Sato, S., Kurauchi, T. (1982), Prog, in Science and Engineering of Composites, Proc. ICCMIV. Tokyo: Jpn. Soc. Comp. Mats., p. 1061. Sato, N., Kurauchi, T., Sato, S., Kamigaito, O. (1984), J. Mater. Sci. 19, 1145. Spohr, D. E., Friedrich, K., Shultz, J. M., Bailey, R. S. (1990), J. Mater. Sci. 25, 4427. Tsai, S. W, Hahn, H. T. (1980), Introduction to Composite Materials. Westport, CT: Technomic Publihing Co.

Wetherhold, R. (1987), /. Mater. Sci. 22, 663. Wu, C. D.,McCullough, R. L. (1977), in: Developments in Composite Materials: Holister, G. S. (Ed.). London: Applied Science Publishers.

General Reading Bader, M. G. (1983), Reinforced Thermoplastics, in: Handbook of Composites, Vol. IV, Fabrication of Composites: Kelly, A., Mileiko, S. T. (Eds.). Amsterdam: North-Holland. Folkes, M. J. (1982), Short Fibre Reinforced Thermoplastics. London: Research Studies Press. Kelly, A., Macmilan, N. H. (1986), Strong Solids, 3rd ed. Oxford: Clarendon Press. Chou, T. W. (1992), Micro structural Design of Fiber Composites. Cambridge: Cambridge University Press.

8 Morphology Control in Polymer Composites Hatsuo Ishida and Philippe Bussi Department of Macromolecular Science, Case Western Reserve University, Cleveland, OH, USA

List of Symbols and Abbreviations 8.1 Introduction 8.2 Classification of Substrate Activity 8.2.1 Empirical Approach 8.2.2 Theoretical Approach 8.3 Parameters Influencing the Appearance of a Transcrystalline Zone 8.3.1 Thermodynamic Parameters 8.3.2 Physico-Chemical Parameters 8.3.3 Presence of Stresses at the Substrate-Melt Interface 8.3.4 Molecular Weight 8.4 Experimental Techniques for Studying Transcrystallization 8.4.1 Optical Microscopy and Scanning Electron Microscopy 8.4.2 X-Ray Techniques 8.4.3 Differential Scanning Calorimetry 8.4.4 Dynamic Mechanical Spectroscopy 8.4.5 Infrared Spectroscopy 8.5 Influence of Transcrystallinity on the Mechanical Properties of Fiber-Reinforced Composites 8.5.1 Adhesion at the Polymer-Substrate Interface 8.5.2 Mechanical Properties of Transcrystals 8.6 The Induction-Time Approach 8.6.1 Theoretical Basis 8.6.2 Application of the Theory 8.7 Conclusions and Outlook 8.8 Acknowledgements 8.9 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

340 342 344 344 345 349 349 352 355 356 358 358 359 360 363 364 366 366 369 370 370 373 377 378 378

340

8 Morphology Control in Polymer Composites

List of Symbols and Abbreviations A At b0 E / f(g,t) g g* g0 G AG AG* AGC Agff A/zf / Io 7b Js k / Mn Mw n R t t{ T Tc T{ Tm T£ Tp AT

advantage for a polymer to crystallize at the substrate surface surface thickness of a layer dynamic Young's modulus correction factor for heat of fusion distribution function size of nucleus size of critical nucleus constant-growth rate shear modulus free enthalpy of crystallization critical free enthalpy bulk free enthalpy change bulk free energy of fusion per unit volume heat of fusion per unit volume rate of heterogeneous nucleation constant nucleation-rate bulk heterogeneous nucleation surface-nucleation rate Boltzmann constant unit thickness of a lamella number-average molecular weight weight-average molecular weight constant; Avrami exponent ratio of the intensities of crystalline bands time induction time absolute temperature temperature of crystallization initial melt temperature melting point equilibrium melting point characteristic crystallization temperature degree of supercooling

a' P y y ab y a , yb yc y cm ycs

supplemental transitions constant characterizing the regime of growth surface tension interfacial free energy dispersion components of surface tension of phases a, b dispersion component of crystal surface tension crystal-melt surface free energy crystal-substrate interfacial free-energy

List of Symbols and Abbreviations

ym y ms ys y^ 6 eY

341

Acp X (0

dispersion component of melt surface tension m e l t - s u b s t r a t e interfacial free energy dispersion component of substrate surface tension extrapolated infinite molecular weight surface tension degree of disregistry; loss factor contributions to surface energy other than those arising from dispersion or polar forces accommodation coefficient surface free energy fold surface free energy interfacial free energy difference function interfacial free energy difference function of a heterogeneities-crystallite system activation energy for a molecule to cross the phase boundary fraction of crystallized material at time t

ATR DMS DSC FTIR IR OM PE PEEK PEKK PP PPDT RAS SEM WLF ZBD

attenuated total reflection (spectroscopy) dynamic mechanical spectroscopy differential scanning calorimetry Fourier transform infrared (spectroscopy) infrared (spectroscopy) optical microscopy polyethylene poly(ether-ether-ketone) poly(ether-ketone-ketone) polypropylene poly(p-phenylene terephthalamide) reflection absorption spectroscopy scanning electron microscopy William-Landel-Ferry Zeldovich-Becker-Doring (theory of nucleation)

£ a ae ACT Aa'

342

8 Morphology Control in Polymer Composites

8.1 Introduction High-performance polymer-matrix composite materials are now past the stage of scientific curiosities that they were only a few decades ago. With the increase in availability of engineering data, polymer composites have gained wide acceptance as viable alternatives to metallic materials and have become relied on in critical applications such as those found in the aerospace industry. However, until now, these materials have been characterized mostly from a mechanical-property point of view and the molecular and chemical aspects have received little attention. Thus, to a great extent, the fundamental knowledge necessary to allow optimization of these systems is still missing. High-performance polymer-composite materials consist of two vastly different materials. The reinforcing fiber provides high specific modulus and specific strength to a relatively weaker matrix. The choice of the matrix is seldom based on its mechanical properties alone and is more often determined by its physico-chemical properties (glass transition temperature, potential bonding to the fiber surface, chemical resistance, etc.). The profound influence of the fiber-matrix interface and interphase on the mechanical and physical properties of such composites is now well recognized. An interface is a hypothetical plane of zero thickness which separates the reinforcement from the matrix. For a composite system where both components are immiscible and completely ignore each others' presence, a true interface might be observed. Short range chemical bonding between the fiber and the matrix is often considered as a practical example of an interface. However, most often the physical and chemical interactions between the fiber and the matrix lead to a gradient structure.

This interfacial region is known as the interphase. These two concepts are illustrated in Fig. 8-1. Three main types of interaction may be present in polymer-composite materials: physico-chemical interactions involving the surface energy of both components and the wetting behavior of the polymer matrix, chemical interactions or chemical bonding, and finally mechanical interactions or mechanical interlocking effects like those found in interpenetrating networks. Whereas chemical bonding affects mostly the interfacial adhesion, the presence of an interpenetrating network or the development of a particular morphology principally affects the properties in the interphase. Understanding the nature of the chemical and molecular phenomena involved is the first step of a systematic scientific approach. The knowledge gained enables one to control the interface-interphase structure and ultimately to tailor the composite properties for a particular application. In this respect, semicrystalline thermoplastic composites are of particular interest. Unlike thermoset composites, surface induced crystallization may result in a particular morphology at the interface, depending upon certain thermodynamic and

Figure 8-1. Schematic diagram showing concepts of interface and interphase in composite materials.

8.1 Introduction

physico-chemical conditions. Surface-induced crystallization, also referred to as transcrystallization, arises when the nucleation density at the fiber surface is so high that the lateral development of the spherulites is impeded and growth can only proceed perpendicular to the fiber direction. Thus, a columnar growth region is produced which extends to the bulk matrix where spherulitic morphology is present. Figure 8-2 shows a model for the formation of a transcrystalline zone, along with a typical example. Such a morphology is expected to have an influence on the adhesion at the interface, due to an increased nucleation density, and on the mechanical

(a) •y

Single nucleus

y

-

*,

Series of nuclei forcing perpendicular growth

Figure 8-2. (a) Model for the formation of a transcrystalline zone; because of the large density of surface nuclei, growth can only proceed perpendicular to the substrate surface (after Sharpies, 1966). (b) Typical example of transcrystallinity in an ultra-high modulus polyethylene fiber-reinforced polyethylene composite (T=126.5°C, after 15min) (Ishida and Bussi, 1991a; by permission).

343

properties of the interphase, due to a preferential orientation of the lamellae. Transcrystallization has been observed for a variety of substrates and crystallizing conditions; it has also been reported in cast metals (Barrett, 1952). With high fiber content or under special conditions (see Sec. 8.6.2), it is possible to develop an entirely transcrystalline sample. In this case, the notions of an "interphase" and a bulk matrix disappear and knowledge of the properties of the transcrystalline zone becomes essential. Although the first report on transcrystallization dates back to 1952 (Jenckel et al., 1952), it remains a poorly understood and highly controversial phenomenon. In particular, there is little or no mention of transcrystallization in well-recognized polymer morphology handbooks (Geil, 1963; Sharpies, 1966; Bassett, 1981) and, to the authors' knowledge, no other review article has been published on the subject. This lack of information is primarily due to the following situation. In most of the earlier reports, transcrystallization was not the main focus of the study and was reported as a side consequence of the substrate-polymer pair or the thermodynamic conditions chosen. However, with the development of thermoplastic composites, a better knowledge of the influence of the transcrystalline zone on the composite properties has become necessary. Several problems make this study difficult. First, transcrystallization is often observed as the result of the combined effect of several different parameters. It is therefore difficult to identify the most important contribution. Moreover, there is not always unanimity among authors concerning the influence of each parameter on transcrystallinity. All too often this is due to the fact that the investigator wants to reduce the appearance of transcrystalliza-

344

8 Morphology Control in Polymer Composites

tion to one essential parameter, whereas it is more likely that a combination of parameters is involved. This explains many of the seemingly contradictory reports found in the literature. However, most of the parameters influencing the appearance of a transcrystalline zone have been identified by now. Second, most of the published work has been qualitative in nature and little has been done in the direction of morphology control. Very few attempts have been made to quantitatively characterize the nucleating ability of a given substrate toward a particular polymer. As it will be shown later, this is mostly due to a practical limitation of the theory of heterogeneous nucleation. A quantitative approach is needed if one wants to compare different fiber polymer systems exhibiting transcrystallization, or if the extent of the transcrystalline zone is to be controlled. Also, it is very difficult to recognize and separate the contribution of transcrystallinity to the adhesion at the fiber-matrix interface and to the mechanical properties of the interphase region. This last point is of great importance because researchers have been arguing about the real role of transcrystallinity due to the difficulty of proving unambiguously any improvement in mechanical properties. This chapter reviews the parameters influencing the appearance of a transcrystalline zone and the experimental techniques which have been used in the study of transcrystallization. A significant example is given for each characterization technique. The influence of transcrystallinity on the mechanical properties of the composite is then examined. Last, a theoretical approach based on the theory of heterogeneous nucleation and on the induction time is presented, which allows control of the extent of the transcrystalline morphol-

ogy. Little has been published on the mechanical aspect of transcrystallinity, but this is an important factor because it will ultimately determine if it is desirable to induce transcrystallization in semicrystalline thermoplastic composites. Although the main application of transcrystallization seems to be in fiber-reinforced composites, results of studies with films as a substrate are directly applicable to fiberreinforced composites and have also been included in this chapter. In this chapter, the terms "surface-induced crystallization", "transcrystallization" and "epitaxial crystallization" have been used. Although the three terms refer to the same basic phenomenon, the terms do become increasingly restrictive. "Surface-induced crystallization" is the most general term whereas "transcrystallization" is more indicative of the visual phenomenon because growth proceeds transverse to the fiber direction. However "transcrystallization" only implies that the 6-axis of the polymer is perpendicular to the fiber. "Epitaxial crystallization" is the most restrictive because the orientation of the b- and oaxis are fixed. Whenever possible, the occurrence of melt epitaxy should be recognized because the degree of lamellar orientation at the surface is higher than for regular transcrystallization and may influence the mechanical properties of the interphase.

8.2 Classification of Substrate Activity 8.2.1 Empirical Approach

Figure 8-2 b is an example of a well-developed surface-induced morphology where large transcrystals (of the order of 20 |im) can be seen. However, depending on the

8.2 Classification of Substrate Activity

nature of the polymer-substrate pair and on the crystallization conditions, intermediate morphologies can be observed. In an effort to rationalize the study of surfaceinduced crystallization, Chatterjee and Price (1975 a) first introduced a classification of substrates with respect to their nucleating ability. Based on optical microscopy observations of over 43 polymer-substrate pairs, they distinguished three main types of substrates: • Type 1: Very active substrates: nucleation is favored at the substrate surface, not in the bulk polymer. Transcrystalline morphology is observed. • Type 2: Moderately active substrates: depending on the nucleation density, a mixed transcrystalline-spherulitic morphology is observed. As the nucleation density decreases, the surface morphology becomes mostly spherulitic. • Type 3: Inactive substrates: bulk nucleation is favored over surface-induced nucleation and little or no spherulites are seen at the interface. As a first step, this qualitative classification is very useful for comparing the activity of different substrates toward a particular polymer. Chatterjee and Price (1975 a) have also stressed that surface-induced crystallization is a competitive process. If the bulk nucleation density is very high, there is little chance to develop a large transcrystalline zone because the transcrystals will immediately impinge on bulk spherulites even if surface-nucleation density is high. This remark calls for two further observations. First, transcrystallization is a nucleation-controlled phenomenon. Therefore, it seems essential to study the nucleation step because once nucleation has occurred, growth in the matrix and on the fiber proceeds in exactly the same way. In particular, the extent of the

345

transcrystalline zone is not an indication of the nucleating ability of the fiber toward the melt. This is an important point because much of the literature has linked the substrate propensity to induce transcrystallization to the extent of the transcrystalline zone. A large transcrystalline zone can indicate a high surface activity or a weak bulk nucleation. Second, any quantitative study must take into account the bulk heterogeneous nucleation in the matrix, because this will determine the extent to which the transcrystalline growth front can proceed in the matrix. This qualitative approach, however, does not take into account the temperature dependency of these morphologies. It is well-known that nucleation density is highly dependent on the degree of supercooling. Because the surface nucleation density and the bulk nucleation density exhibit different temperature dependencies, their relative magnitudes and hence the resultant morphologies will vary with temperature. In the following section, a more quantitative approach is presented based on the value of the free-energy difference function as it appears in the theory of heterogeneous crystallization. 8.2.2 Theoretical Approach

The previous discussion has demonstrated the need for a more quantitative approach to surface-induced crystallization. A theoretical approach to transcrystallization is needed for several reasons. It is desirable to be able to compare the nucleating ability of different polymer-substrate pairs independently of temperature and on a more objective basis than the current evaluation from optical micrographs. One would also like to explain the observed or expected effects of such parameters as crystallization temperature,

346

8 Morphology Control in Polymer Composites

cooling rate, or surface energy. Last, one would like to be able to use the theory as a guideline to control the extent of the transcrystalline morphology. In a theoretical approach, it is desirable to determine the amount of energy necessary to create a stable nucleus at the fiber surface, and how this energy is affected by the various parameters mentioned above. In the classical nucleation-theory (Wunderlich, 1976), the overall free enthalpy of crystallization is written as: i

where AGC represents the bulk free enthalpy change (Gibbs free energy) involved in the melt-crystal phase transition and is related to the volume of growing crystal. AGC is negative and is the driving force in the crystallization process. The opposing force (positive contribution to the free enthalpy of crystallization) arises from the creation of new surfaces At having a specific surface free energy at. Initially, the driving force is not strong enough to overcome the amount of energy necessary to create new surfaces, and AG is positive. However, because the volume to surface area ratio increases as new crystallizable elements are added to the nucleus, AG exhibits a maximum (critical free enthalpy AG*) and eventually becomes negative. The nucleus then becomes stable (Fig. 8-3). When no impurities are present in the melt to initiate nucleation, the nucleation is termed homogeneous. However, in most cases, there are always some foreign surfaces provided by heterogeneities (catalyst residues, dust particles, etc.) or by remnant order in the melt (partially molten spherulites, local chain-alignment due to flow conditions, etc.). A foreign surface reduces the critical free enthalpy of formation of a nucleus, because the energy required to create an interface between the

foreign surface and the polymer crystal is often significantly less than that of the corresponding polymer crystal surface. In this case, the nucleation is termed heterogeneous. For a large substrate (film or fiber), a high nucleation density will result in a transcrystalline morphology. Compared to the dimensions of a heterogeneous nucleus (a few tens of nanometers) a fiber or a film can be treated as a flat surface. This model is very convenient, and it has been used in most of the published studies (Chatterjee and Price, 1975 c; Koutsky et al., 1967) dealing with heterogeneous nucleation. This enables comparison with the data found in the literature. Figure 8-4 presents the parameters defining the heterogeneous nucleus, a, b and c represent the dimensions of the nucleus, a is the side surface free energy and <7e is that for the fold surface (high-energy surface). A marked difference between heterogeneous and homogeneous nucleation is the introduction of ACT, the interfacial free energy difference-function. Aa is related to the creation of a new (ac) top surface. A better understanding of the nature of Aa is achieved when it is related to its three basic components (Cherry, 1981): 1

=

17

/cs

-4- V

'

/cm

— 17

/ms

(8-2)

AG

AG*. ,

Figure 8-3. Change in free enthalpy, AG, as a function of size, g (after Wunderlich, 1976).

8.2 Classification of Substrate Activity

Melt

347

Top Surface Aa Side Surface o Fold Surface c p Figure 8-4. Surface free energy parameters defining the heterogeneous nucleus (Ishida and Bussi, 1991 a; by permission).

where ycs is the crystal-substrate interfacial free energy, ycm is the crystal-melt surface free energy (ycm = a as defined earlier) and yms is the melt-substrate interfacial free energy. For hydrocarbons, where dispersion forces are predominant (no polar interactions), Fowkes (1964) further expressed the interfacial free-energy, yab, as: y

z^l/2

l/2\2

(8-3)

where ya and yh represent the dispersion components of the surface tension for phase a and b. In theory, Aa can be therefore expressed as a function of the surface tension properties of the substrate, polymer crystal and polymer melt. Because the surface (a c) is the only one to see the substrate, Aa is a convenient way to define the nucleating ability of a substrate toward a polymer melt. A lower value of Aa reduces the value of AG* and indicates a more favorable nucleation process. The limiting value of Aa, Aa = 0, is reached when secondary nucleation occurs, i.e. when there is no difference between the substrate and the polymer crystal. This is the basic process by which growth occurs in polymers. In this case, ycs = 0, ycm = yms and thus Aa = 0 [Eq. (8-2)]. However Aa should not be considered alone because, unless special precautions are taken, the transcrystallization process will have to compete with bulk heterogeneous nucleation in the matrix.

In the following section, Aa designates the interfacial free energy difference function in a substrate-crystallite system, whereas Aa' is that of a heterogeneities/ crystallite system in the melt. Clearly, the relative magnitude of Aa and Aa' influences the whole nucleation in the composite sample and its final morphology. In an effort to rationalize the classification of substrate activity first introduced by Chatterjee and Price (1975 a), Ishida and Bussi (1991a) defined the advantage, A, for a polymer to crystallize at the substrate surface rather than in the bulk as the ratio of Aa' and Aa: A = Aa'jAa

(8-4)

The following three situations are encountered (Table 8-1): Table 8-1. Parameter A is a measure of competition between surface induced nucleation and bulk nucleation. A« 0 Inactive substrate. The polymer melt is (ACT > ACT') unaffected by the presence of the fiber and there is no nucleation at the fiber/ film surface. 0
Moderately active substrate. Spherulitic surface morphology is observed. Transcrystallinity becomes more probable as A approaches 1.

A>1

Very active substrate. Nucleation is heavily favored at the fiber surface. Transcrystallinity is observed.

348

8 Morphology Control in Polymer Composites

In order to calculate A, one needs to estimate A a and ACT'. They may be practically obtained through a combined growth and nucleation rate experiment. According to the theory of polymer nucleation (Wunderlich, 1976), the rate of heterogeneous nucleation is given by: = /

oexp(-—)exp(- kT

(8-5)

where / 0 is a constant nucleation rate, Acp is the activation energy for a molecule to cross the phase boundary and AG* is the critical excess free energy due to the creation of a nucleus, k is Boltzmann's constant and T is the crystallization temperature. AG* can be further expressed as: 16 o
Once heterogeneous nucleation has occurred, growth will proceed by adsorption and crystallization of polymer molecules at the initial nucleus surface. This process is also referred to as secondary nucleation, as opposed to primary nucleation starting at the substrate surface. Intuitively, one can feel that Aa does not play any role in this case since no new (ac) surface is created. Two processes or regimes of growth can be distinguished (Hoffman et al., 1976; Fig. 8-5). In regime I, the completion of a new layer is rapid compared to the rate of nucleation, whereas in regime II, multiple nucleation occurs at the substrate surface before a layer is completed. These different processes are reflected in the expression of AG* and the net growth-rate is given by: A(?N

(8-6)

kT, where a, ae, and ACT have been defined previously. Ag{ is the bulk free energy of fusion per unit volume of crystal. In general, Agf can be expressed as (Hoffman et al., 1976): Ah{AT

•exp

-

kTATAhJ

(8-10)

(a)

(8-7)

1

m J

where A/if is the heat of fusion per unit volume of crystal at the equilibrium melting point T£ and AT is the degree of supercooling. Agf has to be corrected by a factor / since Ahf varies with the crystallization temperature: (b)

/ =- | ^ —

(8-8)

At low supercooling, / has little influence, but it must be taken into account as the crystallization temperature decreases (see for example Fig. 8-21). The heterogeneous nucleation rate can then be rewritten as: (8-9)

I = Io exp ( - ^ J exp ( -

kTAT2Ah}f2

Figure 8-5. Growth-front morphology for Regime I and Regime II growth, (a) Regime I: Single nucleus forms on surface, rapid completion of a new layer, (b) Regime II: New surface nuclei appear before a layer is completed (after Hoffmann et al., 1976).

8.3 Parameters Influencing the Appearance of a Transcrystalline Zone

where g0 is a constant growth-rate and b0 is the thickness of a new layer. b0 can be related to the Miller's indices of the polymer unit cell. /? is a constant characterizing the regime of growth: for regime I, p = 4 whereas for regime II, /? = 2. Since regime II requires a higher nucleation rate, it is observed at a higher degree of supercooling. It should be mentioned here that the activation energy, A
of lng + A(p/(kT)

versus

l/[T(ATf)]

should yield a straight line whose slope, K%, is proportional to a
349

tallization in the polymer melt, the rate of appearance of individual spherulites can be recorded without difficulty. This observation, coupled with growth rate measurements, enables us to determine ACT'. But in the study of surface-induced crystallization (determination of Aa), this approach is applicable only if the substrate shows a moderate activity. For a better nucleating substrate, the crowding of spherulites at the fiber surface makes any count nearly impossible. In the limiting case of transcrystallinity, the nucleation density cannot be measured. Consequently, the theory is restricted at this stage to a polymer-substrate pair with a certain range of ACT, that is, when A is much smaller than 1. In particular, this explains why, in their study of 43 polymer-substrate pairs, Chatterjee and Price (1975 c) had to select two adequate systems which showed only a partial transcrystalline character. Recently, however, Ishida and Bussi (1991 a, b) have proposed a different approach applicable to truly transcrystalline systems. This approach, based on induction time, is presented in Sec. 8.6.

8.3 Parameters Influencing the Appearance of a Transcrystalline Zone 8.3.1 Thermodynamic Parameters

In most of the early work on transcrystallization, the effect of various cooling rates was investigated, often with mixed results. Barriault and Gronholz (1955) found that for nylon 66 in contact with various media (air, oil, water, and silicone), the transcrystalline structure is not present only upon very rapid quenching. In the original report on transcrystallization, Jenckel et al. (1952) quenched

350

8 Morphology Control in Polymer Composites

polyamide in oil and water from about 20 °C above the melting point (i.e., around 240 °C) to various temperatures and showed that eventually, transcrystallization disappears when the quenching temperature is reduced to 40 °C (Fig. 8-6). A similar observation was made by Tung and Dynes (1987) for carbon fiber reinforced poly(ether-ether-ketone) composites quenched at various rates (from 1.5 °C/min to 2500°C/min). In contrast, Fitchmun and Newman (1970) found that the morphology was independent of the cooling rate and noted the importance of the presence of a temperature gradient. By rapidly cooling a polymer from a temperature above the melting point, a transient temperature gradient develops through the sample which favors a higher nucleation rate at the surface than in the bulk. For large gradients, the likelihood of a transcrystalline layer increases, independently of the nature of the substrate. The effect of a temperature gradient is also reflected in the size distribution of the bulk spherulites and in the extent of the transcrystalline zone, for example, a fast cooling rate produces a thinner transcrystalline layer. This

40

80

120

160

200

Temperature (°C)

Figure 8-6. Effect of the quenching temperature on the thickness of the transcrystalline zone (polyamide film quenched in water from 240 °C) (after Jenckel et al., 1952).

is often observed in compression-molded polymer films [polyethylene, polyamide, polypropylene with aluminum and Mylar: Fitchmun and Newman (1970); poly(vinylidene fluoride) with aluminum and Teflon: Weinhold et al. (1980)]. Recently, Hsiao and Chen (1990) were able to induce transcrystallization in various polymer-fiber pairs by simply immersing one end of the fiber in cool water. They pointed out that the ratio of thermal conductivity between carbon fibers and most polymeric matrices is about 1000. Thus, upon cooling, heat is dissipated more readily in the fiber than in the matrix and a surface gradient appears, which favors transcrystallization. It is important to note that this effect will be significant only if this transient temperature gradient is comparable in time to the induction period for massive nucleation to occur. Therefore, at high crystallization temperature, where the induction time is large, one should not expect such a short lived gradient to have any effect. The importance of this induction time is discussed in more detail in Sec. 8.6. A special case of transcrystalline morphology was recently reported by Ishida and Bussi (1991a), who reported a banded transcrystalline zone for high cooling rate in ultrahigh modulus polyethylene fiberreinforced polyethylene composites (Fig. 8-7). This morphology was detected through the presence of extinction lines in the transcrystalline zone parallel to the fiber direction. Banded structure results from cooperative twists of the lamellae induced by the presence of thermal stresses (Bassett, 1981). From the above discussion it is clear that one should distinguish the effect of a temperature gradient induced during severe quenching, which favors transcrystallinity, from the effect of the high cooling rate itself, which tends to decrease surface

351

8.3 Parameters Influencing the Appearance of a Transcrystalline Zone

125 °C, the morphology was spherulitic, whereas at 130°C a transcrystalline layer was observed. However, these two very different observations are not contradictory. Looking back at Eq. (8-9), one can see that the surface nucleation rate can be expressed as a function of Aa and AT: = Kexp

Figure 8-7. Banded transcrystalline zone observed as a result of rapid cooling of a polyethylenepolyethylene composite film in air. Notice the parallel extinction lines near the fiber surface (Ishida and Bussi, 1991a; by permission).

-

aAa TAT2

(8-11)

where a and K are two positive constants. Similarly, for bulk heterogeneous nucleation: (

^

)

(8-12)

where K' is a positive constant. Thus: nucleation. The effect of the cooling rate is better understood through a study of isothermal crystallization which can be found in the work of Chatterjee and Price (1975 a). The reported temperature ranges for which they observed transcrystallization in 43 polymer-substrate pairs. They found that the propensity to transcrystallize decreases with increasing crystallization temperature; at 130°C, isotactic polypropylene in contact with isotactic polystyrene exhibited transcrystallinity whereas the morphology was spherulitic at 135 °C. This effect was attributed to the reduced nucleation rate at smaller supercooling, as predicted by Eq. (8-9). A similar observation was made by Lopez and Wilkes (1989) who observed transcrystals at 200 °C but not at 240 °C in carbon fiber reinforced poly(/?-phenylene sulfide) composites. This was attributed to a steeper thermal gradient across the fiber-matrix interface as the supercooling increases. In contrast, Fitchmun and Newman (1970) observed the opposite effect for polypropylene in contact with aluminum: at

(8-13) where K" is a positive constant. It has been shown previously (see Table 8-1) that, when surface heterogeneous nucleation is favored, Ac is smaller than Aor. Therefore, one can rewrite Eq. (8-13) as: TAT:

(8-14)

where b is a positive constant. Since 1/[T(AT)2] increases with decreasing AT, it can be seen that IJIh increases with increasing crystallization temperature, that is, surface nucleation becomes more favorable at smaller supercooling. Therefore, even though both rates decrease with increasing temperature, their relative magnitude is such that transcrystallization is favored and will be observed provided that the absolute value of the surface nucleation rate is still high enough. One can postulate that, in the study of Chatterjee and Price (1975 a), / s was too small at higher temperature, whereas in the study of Fitchmun and Newman (1970), 7S was

352

8 Morphology Control in Polymer Composites

large enough even at higher temperatures so that the system could fully take advantage of a lower A
-,l/2\ /.,l/2

-,l/2\

/o j / ; \

For a given polymer, where yc (crystal) and ym (melt) are known, Eq. (8-16) predicts

that A
8.3 Parameters Influencing the Appearance of a Transcrystalline Zone

face tension of polyethylene single crystals (yc = 6 6 x l ( T 3 N / m ) and that of a completely amorphous surface layer (ym = 3 6 x l 0 ~ 3 N/m). This difference also indicates that the substrate should be equally well characterized if comparisons of nucleating ability are to be based on surface energy considerations. It is clear from the work of Schonhorn and Ryan (1966) that the degree of crystallinity of the substrate should be determined. It would also appear, due to the high value generally obtained for single crystals, that a high surface energy substrate is desirable. Indeed, it has been claimed in early investigation of transcrystallization that a high energy substrate is a necessary condition. Schonhorn (1966) noted that low-energy surfaces such as in polytetrafluoroethylene seem to be ineffective nucleating agents. However, transcrystallization has been observed by other workers for polymers in contact with low-energy surfaces (Chatterjee and Price, 1975 a; Fitchmun and Newman, 1970). Schonhorn (1968) also studied the connection between wettability and surface morphology and pointed out that in order for high energy surfaces to be fully effective, sufficient time must be allowed to achieve intimate contact with the surface. The importance of the true surface of contact leads to another characteristic of the substrate whose influence has been generally ignored - its roughness. On one hand, a rough surface might be more difficult to wet at the microscopic level, but on the other hand, it also provides such high energy features as dislocations which favors heterogeneous nucleation. Lopez and Wilkes (1989) found a correlation between the roughness of their carbon fibers (as determined by scanning electron microscopy) and their nucleating ability toward poly(/?-phenylene sulfide). Thus, determi-

353

nation of the substrate specific area by gasadsorption techniques seems advisable. Another problem with Eq. (8-16) is that it does not recognize the influence of lattice matching. The importance of lattice matching has been stressed very early by Turnbull and Vonnegut (1952) who predicted, based on theoretical considerations, that the nucleating efficiency should be inversely proportional to the degree of disregistry, <5, between the substrate and polymer unit cell. They further related the free energy of the phase transition to a function of 5. Binsbergen (1973) modified Eq. (8-16) to account for lattice matching and contributions other than those arising from dispersion or polar forces:

+ (l-0-sy

(8-17)

where £ is defined as an accommodation coefficient of value 0 < £ < 1 and sy refers to other contributions. For perfect latticematching, £ = 1. Equation (8-17) indicates that, in order to promote the transcrystallization of a given polymer, one can manipulate two variables: the surface energy of the substrate and the degree of lattice matching. In a study of carbon-fiber reinforced polypropylene, Hobbs (1971) observed a drastically different nucleating ability between fibers with large graphite planes that are highly oriented along the fiber axis (type I fiber) and fibers with small and disoriented graphite planes (type II fiber) (Fig. 8-8). Whereas, in general, type II fibers show no nucleation under the microscope, extensive transcrystalline growth was observed under the same thermodynamic conditions for type I fibers. Using a molecular model, Hobbs (1971) showed that the hydrogen atom of the helical polypropylene chain can be placed in con-

354

8 Morphology Control in Polymer Composites

(b) Figure 8-8. Influence of the surface morphology of carbon fiber on the crystallization of polypropylene, (a) Carbon fiber with large regular graphite planes (type I fiber), (b) Carbon fiber with small and disoriented graphite planes (type II fiber) [Hobbs (1971); reprinted by permission from Nature, Vol. 234, p. 12. Copyright © 1971 Macmillan Magazines Ltd.].

tact with the low-energy sites (C-C bonds) of the substrate. Such a process requires long range regularity of the adsorption plane, which is found only in type I fibers. Hartness (1984) arrived at a similar conclusion in a study of carbon fiberreinforced poly(ether-ether-ketone) composites. Theoretical calculations by Baukema and Hopfinger (1982) showed that the parallel orientation of polyethylene on the basal plane of graphite single crystals was

energetically favored. In a study of adsorption of ?7-alkanes onto graphitized carbon black and various ground graphites, Groszek (1970) found that the adsorption was confined entirely to the basal planes of graphite crystals. Lattice matching seems to be a predominant parameter as indicated by the recent work of Hsiao and Chen (1990). They first observed that poly(/?-phenylene terephtalamide) (PPDT) fibers could induce transcrystallization of poly(ether-etherketone), poly(/?-phenylene sulfide) and poly(/?-phenylene oxide), which was explained by the fact that these polymers and the fiber have very similar a and c unit cell dimensions. Moreover, they were able to induce transcrystallization of these polymers by coating previously inactive glass and polyacrylonitrile-based carbon fibers with PPDT In Sec. 8.5, it is shown, based on mechanical-properties considerations, that it might be desirable to induce transcrystallization. Coating of the reinforcement with a substance which accommodates the lattice of the crystallizing polymer might therefore be of great practical importance. Other researchers have suggested that transcrystallinity is affected by the substrate-surface chemistry and by the chemical similarity between the substrate and the crystallizing polymer (Fitchmun and Newman, 1970; Gray, 1974 a). The influence of particular chemical groups can be studied by two methods. First, one can relate the nucleating ability to a measure of the surface tension and look for a correlation between the value of the polar component of the surface tension, the presence of certain chemical groups, and the propensity to induce surface nucleation. Another approach is to study the effect of well-characterized heterogeneities on the degree of supercooling necessary to observe massive

8.3 Parameters Influencing the Appearance of a Transcrystalline Zone

nucleation. Extensive data have been compiled by Beck (1967) for seeded polypropylene melts for more than 150 nucleating agents. The chemical structure of the seeds was indeed found to strongly affect the crystallization temperature. The characteristic crystallization temperature, Tp, for sodium benzoate seeded polypropylene melt was 131 °C, whereas that for pure polypropylene was 106 °C. Beck (1967) proposed that a good nucleating agent for polypropylene should consist of two parts: an organic part which helps to solubilize the heterogeneity in the melt by reducing the interfacial surface free-energy and a polar group which ensures that the heterogeneity is still solid in the melt at all temperatures. Even though Beck (1967) could discern some trends (influence of ter t-butyl groups, of aromatic organic groups, of carboxylic acids, salts, etc.) it is still very difficult to tell exactly what makes a particular nucleating agent so effective based on its chemical structure alone. 8.3.3 Presence of Stresses at the Substrate-Melt Interface The relationship between stress and crystallization has been long recognized through the investigation of various phenomena: row nucleation in injectionmolded thermoplastics, shear-induced nucleation in dilute polymer solutions, and stress-induced crystallization in elastomers. Stresses present at the fiber/matrix interface can be mechanically or thermally induced, but they have the same effect the polymer molecules become oriented near the fiber surface. The entropy of the oriented melt is reduced, and there will be an additional decrease in the free energy. Thus, the melting temperature and the degree of supercooling are enhanced. Gray (1974 b) studied the effect of me-

355

chanical stresses on glass-fiber-reinforced polypropylene melts. When one of the two mutually perpendicular glass fibers was gently pulled, a transcrystalline region was observed under the microscope immediately along the length of the fiber, whereas nucleation did not occur on the undisturbed fiber. The appearance of this transcrystalline zone was similar to that observed in a polymer-substrate pair where transcrystallization occurs without shear. Gray (1974 b) also showed that the stress does not need to be external, but can be self-generated. For example, thermal contraction of air bubbles upon cooling of the melt produces stresses at the air-polymer interface and results in crystallization. In technological processing of fiber-reinforced thermoplastics, stress induced transcrystallization must be considered because flow will occur in the matrix prior to crystallization. Misra et al. (1983) found that the appearance of transcrystallization in injection-molded glass-fiber-reinforced polypropylene depends on a minimum fiber-weight fraction (i.e., the availability of internal stresses) of the order of 35%. They also noted that the extent of the transcrystalline zone was greater in the case of samples molded at lowest pressure, as low injection pressure leads to high internal stress. Kubat and Rigdahl (1975) have shown that internal stresses develop in the sample during injection molding. Addition of fibers will cause localized perturbations in the flow field. Mackley (1975) analyzed the flow past a fiber and showed that along the symmetry axis of the fiber, an extensional flow field can be generated with very high extension rate near the tip. Shearing flow also exists over the side surface of the fiber. This flow pattern suggests that transcrystallization should be seen not only at the fiber-matrix interface but also as a "tail" in the region where the fiber was

356

8 Morphology Control in Polymer Composites Transcrystallinity

Streamlines

FIBER

Extended Chain Backbone

pulled out of (Fig. 8-9). This phenomenon was observed experimentally by Burton et al. (1984) in injection molded bars of carbon-fiber-reinforced polypropylene and recently by Thomason and Van Rooyen (1990) in aramid-fiber-reinforced polypropylene. In this latter work, the aramid fiber was pulled at various speeds (5-5000 jim/min) through the melt. Transcrystallization was observed even at the lowest speed, and no lower boundary in pulling speed could be determined. However, they pointed out that stresses could have been built up during the cooling phase due to a mismatch in the thermal-expansion coefficient of the fiber and the melt. This large difference will tend to deform and align the polymer-melt molecules near the fiber interface. They noted a correlation between the axial-thermal-expansion coefficient of a given fiber and its ability to induce transcrystallization. They also offered an alternative explanation to that given in Sec. 8.3.1 for the observed temperature dependency of the transcrystalline morphology; at higher crystallization temperatures, these thermally induced stresses have sufficient time to relax, so that transcrystallinity is not observed. At

Figure 8-9. Influence of flow pattern on surface crystallization. Transcrystallization is not only seen at the fiber-matrix interface, but also as a "tail" in the region where extensional flow has occurred (after Burton et aL, 1984).

lower temperatures, the molecules are caught up by crystallization while still in an oriented state. 8.3.4 Molecular Weight

Although molecular weight is a very important parameter in polymer crystallization, little has been done to investigate its effect on transcrystallization. Moreover, there appears to be some controversy on its effect in the literature. Molecular weight is of particular interest because it influences the crystallization temperature (i.e., the nucleation rate) as well as other properties including wetting and viscosity. The wetting behavior influences the energetics at the interface, whereas the viscosity influences the processing conditions. Additionally, the polydispersity of the sample is also a factor because it leads to a distribution of crystallization temperatures. In this case, the higher molecular weight fraction may serve as a nucleating agent for the lower molecular weight material. Last, molecular weight also influences the final morphology of the sample, because it controls in part the size of the spherulites. Recently Ishida and Bussi

8.3 Parameters Influencing the Appearance of a Transcrystalline Zone

(1991a) studied the transcrystallization of a polyethylene fiber reinforced linear highdensity polyethylene and recorded the transcrystallization temperature for several molecular weight fractions under various cooling rates. As expected, the transcrystallization temperature decreases with molecular weight (5°C decrease from 31 300 to 3300 in molecular weight). For the larger chains, less configurational entropy is lost during the phase transition. This smaller loss corresponds to a lower value of the free energy for a critical nucleus (or to a smaller volume) and, therefore, to a higher crystallization temperature. For the lowest molecular weight fraction, the morphology in the matrix changes from mature spherulites to immature spherulites (sheaf-like spherulites). Namely, growth occurs along diverging radial fibrils before filling the space uniformly. In this case, the transcrystalline zone is expected to show a lower degree of order and to contain more defects. In particular, the transcrystalline growth front appears to be less regular in this case than for the unfractionated polymer. Similarly, Thomason and Van Rooyen (1990) studied the transcrystallization of two polypropylene samples with different melt indices and different molecular weights. Although both samples exhibited the same temperature dependency for the transcrystalline-growth rate, the maximum temperature at which transcrystallization was observed changed from 138 °C for the low-molecular weight fraction to 146 °C for the high-molecular weight fraction. They postulated that the relaxation time for the thermally induced stresses was larger in the case of the higher molecular weight material. Therefore, stress induced orientation of the polymer chains near the fiber surface can exist at higher temperature and give rise to transcrystallization. It appears that at a given

357

crystallization temperature there is a minimum molecular weight below which transcrystallization will not be observed. However, this view is in apparent contradiction to the results reported by Lopez and Wilkes (1989) who studied carbonfiber-reinforced poly(/?-phenylene sulfide) composites. They found that, under the same thermodynamic conditions, transcrystallization occurs in the lower molecular weight poly(p-phenylene sulfide) (Mw = 32 000) whereas no surface nucleation is observed for a higher molecular weight material (Mw = 63 000). Because an earlier study showed that the bulk nucleation density is greater for the higher molecular weight polymer, they speculated that the absence of transcrystallinity in their higher molecular weight sample is due to the increased competition from the matrix. This view is supported by the work of Folkes and Hardwick (1984) who also found an upper limit in molecular weight above which transcrystallization disappears in polyester fiber (Terylene) reinforced polypropylene composites. Assuming that nucleation is initiated at the attachment point of the polypropylene chain-ends to the fiber, they postulated that the observed effect is related to the decrease in chain-end concentration as molecular weight increases. Moreover, as higher molecular weight chains have a restricted mobility due to entanglements, they are also less available for rapid crystallization at the fiber surface. An interesting possibility was mentioned by Lovering (1970) who observed the transcrystallization of a trans-1,4-polyisoprene in contact with a piece of teflon or aluminum foil. He tentatively assigned the presence of a transcrystalline zone to segregated low molecular weight material. He also noted that fractions with a molecular weight in the 50 000-70 000 region showed

358

8 Morphology Control in Polymer Composites

the greatest propensity to transcrystallize. This concept of a region would reconciliate the partisans of an upper or lower limit. The idea of molecular weight segregation merits some more consideration and experimental work in light of the molecularweight dependency of the surface tension (Legrand and Gaines, 1969): (8-18) where y^ is the extrapolated infinite molecular weight surface tension, A: is a positive constant and Mn is the numberaverage molecular weight. As mentioned earlier, the surface free-energy of the substrate has been considered as an important parameter influencing the appearance of transcrystallinity. It was shown earlier that heterogeneous nucleation is favored when the free-energy difference function is minimized. This free-energy difference function depends on the surface free-energy of the different components in the system (substrate, melt, and polymer crystal). Therefore, there might be a range of molecular weights, which minimizes the value of this function and leads to selective adsorption of a particular fraction onto the substrate.

8.4 Experimental Techniques for Studying Transcrystallization 8.4.1 Optical Microscopy and Scanning Electron Microscopy Optical microscopy (OM) has been by far the most widely used experimental technique, because transcrystallization is easily discernible. The scale of observation is usually sufficient to study a typical transcrystalline zone whose thickness is of the order of a few tens of micrometers. The

requirements are quite simple; an optical microscope equipped with cross polarizers, since the transcrystalline zone is birefringent, and a good quality objective lens are needed. Since growth and nucleation are highly temperature dependent, a hot-stage with good temperature control is also desirable. Composite films are usually prepared in the following way: the polymer matrix is first cast from solution to produce a thin film with uniform thickness; the solvent is then evaporated and the fiber is deposited onto the molten film. Care must be taken to avoid orientation effects and stresses during the casting process, since these can affect crystallization. Similarly one should ensure thermal equilibrium and avoid the presence of thermal gradients at the film surface. Barriault and Gronholtz (1955) used polarized light microscopy to study the birefringence of crystalline and transcrystalline regions. Within experimental error they found a common value, of the order of 0.01. By comparing this value to the theoretical spherulite radial birefringence, they concluded that it was not likely that the polymer chains were oriented radially in the transcrystalline zone. On a more quantitative level, the transcrystallinegrowth rate has been monitored by OM to yield some of the interfacial energy parameters introduced in Sec. 8.2.2. However, the magnification and the resolution of a typical optical microscope are not high enough to recognize the individual nucleation sites at the fiber surface. In fact, in OM, one draws conclusions on the nucleation density and on the possible nature of the nuclei from the macroscopic effect that these microscopic nuclei have on the texture. Thus, recently, several studies have used scanning electron microscopy (SEM) as a characterization tool to gain further insight into the nucleation sites.

8.4 Experimental Techniques for Studying Transcrystallization

Besides enhanced resolution, SEM also enables observation of fracture surfaces following mechanical testing, in order to assess the quality of the adhesion at the fiber-matrix interface. Blundell et al. (1989) studied the etched cross section of carbonfiber-reinforced poly(ether-ether-ketone) composites by SEM. Because there is preferential etching of the amorphous region, crystalline features are highlighted. They found that nucleation predominantly occurs at the fiber surface, producing full fanlike half spherulites. Two types of nucleation sites were recognized; nucleation from discrete, well-separated points on the fiber surface and nucleation between very close fibers (i.e., where a thin polymer layer is sandwiched between two nearly touching fibers). It was postulated that nucleation is favored in these regions because thermally induced stresses are larger there. It can also be seen in Fig. 8-10 that, although fiber induced nucleation dominates the morphology, the nucleation density is not high enough to observe transcrystallinity. Recently, Guigon et al. (1989) studied the heterogeneous crystal-

359

lization of pultruded glass-fiber-reinforced polyamide rods, both parallel and perpendicular to the fiber direction. They found marked differences in morphology as a function of the observation distance to the center of the rod. Near the axis of the rod, spherulitic growth was observed initiating from glass fibers. The surface-nucleating density in this region was estimated to be in the range 0.05-0.20 nuclei/jim2. Near the surface of the rods where the stress level, the flow induced orientation, and the temperature gradient are expected to differ, a transcrystalline morphology was observed. After a flexural test, separation between the bulk matrix and the transcrystalline zone was observed, whereas adhesion was maintained at the fiber-transcrystalline zone interface. This test suggests an improvement in adhesion due to transcrystallinity. 8.4.2 X-Ray Techniques X-ray diffraction patterns provided, in a very early period of transcrystallization research, the first indications that the trans-

Figure 8-10. SEM reveals that crystallization predominantly occurs at the fiber surface in carbon fiber-PEEK composites. Also notice that numerous spherulites are initiated between very close fibers (A), where, presumably, thermally induced stresses are larger. (B) indicates nucleation from a fiber surface and (C) indicates nucleation from within the matrix (Blundell et al., 1989; by permission).

360

8 Morphology Control in Polymer Composites

crystalline layer was oriented. An isotropic polymer sample will scatter X-rays to give diffraction rings, whereas an oriented sample will give rise to arcs. Using X-ray microbeam analysis of cross sections of polyamide 66 transcrystalline films with the sample surface oriented parallel to the equator, Barriault and Gronholtz (1955) found that the patterns were always oriented with the reflection from the H-bonded sheet on the equator. They concluded that on the average the polymer chains were parallel to the surface and that the Hbonds were parallel to the normal surface. Eby (1964) also confirmed that the &-axis of the chains was mainly normal to the surface in transcrystalline polyethylene films. One of the advantages of the X-ray technique is its unique ability to distinguish simple transcrystallization from melt epitaxy. Indeed, in the interpretation of mechanical testing data, it is important to know whether the a- and oaxis are isotropically distributed at the substrate surface plane or have a preferred orientation. Moreover, it is also possible to evaluate the degree of crystallinity as well as the average crystallite size from an X-ray diffractogram. It is therefore possible, in principle, to estimate the thickness of the transcrystalline lamellae as a function of the crystallization conditions and to see if there is any difference to the lamellae thickness in the bulk matrix. The degree of crystallinity of the ordered transcrystalline structure could also be compared to that of the bulk matrix. Unfortunately, such a study has not been undertaken to the authors' knowledge. The X-ray diffractogram is also very sensitive to the presence of an oriented surface layer. Lovering (1970) showed that reflected X-rays are a convenient and rapid way to detect transcrystallinity. In the case

CO

>

20

10

30

20

10

(degrees) Figure 8-11. Transcrystallization strongly influences the relative intensity in a reflection X-ray diffractogram. (a) Transcrystalline /ra«j-l,4-polyisoprene fraction, (b) Spherulitic material (after Lovering, 1970).

of transmission, the transcrystalline layer has little or no effect because the information is blurred by the unoriented material. Transcrystallinity strongly affects the relative intensity of the diffraction peaks (Fig. 8-11) and Lovering (1970) suggested that it might be possible to estimate the amount of transcrystallinity by taking ratios from selected peaks. Using reflection X-ray, Clark and Starkweather (1962) were able to assign specific reflections to the oriented transcrystalline layer in a polytetrafluorethylene film by comparing the diffractogram of the initial film with that of a film where the surface layer had been removed. 8.4.3 Differential Scanning Calorimetry

Differential scanning calorimetry (DSC) is a useful tool because it enables isothermal crystallization study as well as kinetic

8.4 Experimental Techniques for Studying Transcrystallization

study. The extent of crystallization as a function of time can be recorded, and, using the Avrami analysis (reviewed by Schultz, 1974), information on the nucleation process can be obtained. It enables one to estimate the degree of crystallinity of the polymer sample, which can then be compared to the value obtained by X-ray analysis. The DSC thermogram is also very sensitive to the presence of heterogeneous nucleating agents. Kantz and Corneliussen (1973) found that in glass-fiber reinforced polypropylene, the crystallization exotherm is bimodal, thus indicating the occurrence of two types of nucleation events. The upper crystallization temperature (124°C) was assigned to fiber-induced crystallization, whereas the lower temperature exotherm (117°C) corresponds to the bulk matrix. It was also shown by Caramaro et al. (1988) and by Lee and Porter (1986) that surface induced crystallization is substantially enhanced when embryos in the bulk matrix have been eliminated by premelting at high temperature prior to crystallization. This observation is consistent with the OM observations of Fitchmun and Newman (1970). DSC thermograms of composites films exhibiting transcrystallization have been studied, but the results are sometimes conflicting. Recently, He and Porter (1988) have used DSC to study transcrystallization in ultra-high modulus polyethylene fiber-reinforced polyethylene composites. They observed a shoulder at 130°C in the main melting endotherm of the polyethylene matrix (132°C). This shoulder was attributed to the melting of the transcrystalline zone. A similar effect was reported by Matsuoka et al. (1968) for transcrystalline polyethylene films obtained by compression molding between copper sheets. They observed an even greater difference in melting point, of the

361

order of 5 °C, and found that the heat of fusion for the transcrystalline zone was about 20% lower than that for a regular spherulitic film. The lower heat of fusion suggests that the degree of order in the transcrystalline zone, or the degree of crystallinity, is lower than that of the more common bulk morphology. The lower melting point suggests that the crystallites are thinner in the transcrystalline zone. The melting point of a lamellae of unit thickness / depends on the fold surface free energy, the heat of fusion and the equilibrium melting temperature in the following manner: __ TO [ 4 _

Ahtl

(8-19)

One can also show that the initial lamellae thickness depends on the degree of supercooling, AT (Hoffman et al., 1976): - c2

(8-20)

where cx and c2 are approximately constant. He and Porter (1988) also showed that, in a constant cooling rate experiment, transcrystallization produced a shoulder at 124 °C, about 2 °C higher than the temperature at which the bulk matrix crystallizes. An isothermal-crystallization experiment further confirmed that transcrystallization occurs prior to bulk crystallization (Fig. 8-12). Since transcrystals are nucleated prior to bulk spherulites, that is, at lower supercooling, AT, Eq. (8-20) suggests that the transcrystalline-lamellae thickness should be greater than that of a bulk spherulite. Therefore, the transcrystalline zone should melt after the bulk matrix [see Eq. (8-19)] and not before, as it has been observed. Three explanations can be offered for this apparent discrepancy. First, the possibility must be considered that the true tempera-

362

8 Morphology Control in Polymer Composites

Time (min) Figure 8-12. DSC thermogram for the isothermal crystallization (124 °C) of (a) polyethylene-polyethylene composite (50wt.% fiber) and (b) pure polyethylene matrix (after He and Porter, 1988).

ture at the fiber surface is lower than that in the bulk matrix, because the heat is better dissipated by the fiber than by the matrix. Consequently, the true supercooling at the fiber surface is greater than that recorded and / is in fact lower than predicted. Thus, upon melting, the transcrystals will disappear before the bulk spherulites. The second possibility is to assume that the supercooling is uniform throughout the sample, but that the lower lamellae thickness arises from crowding of the nuclei at the fiber surface. It is indeed possible that, because of the high nucleation density at the fiber surface, the nuclei quickly impinge on each other and therefore cannot achieve their equilibrium dimensions as predicted by the theory. In the case of polyethylene-polyethylene composites, melt epitaxy is a distinct possibility. It would thus be relatively easy to nucleate at the fiber surface, and the transcrystal thickness could be lower than expected due to a crowding effect. Last, the lower melting point could be due to low molecular weight material which has been selectively

adsorbed for interfacial free-energy reasons. The degree of crystallinity of the transcrystalline zone is another point of interest. Indeed, it has been observed by He and Porter (1988) that the overall degree of crystallinity increases as the fiber fraction increases, that is, as the amount of transcrystals increases. This result suggests either that the presence of fiber allows more material to crystallize, or that the degree of crystallinity in the transcrystalline zone itself is higher. This latter possibility is in accordance with the simple idea that this zone possesses, a priori, a higher degree of orientation due to a likely epitaxial effect. However, this contradicts the earlier findings on the heat of fusion of Matsuoka et al. (1968). A possible explanation is that, upon nucleation and growth of the transcrystals, considerable uncrystallized material is trapped between transcrystals. Thus, the overall degree of crystallinity is lowered. The lower heat of fusion could also be due to the presence of low molecular weight material at the interface. The concentration of chain ends, which can be viewed as defects, is higher for low-molecular weight fractions, and therefore the heat of fusion is lowered. In any case, it remains that more experimental data are needed until one can provide an unambiguous answer to these two challenging questions. So far, DSC experiments have been analyzed in terms of lamellae thickness and heat of fusion. However DSC has also been used to study the development of crystallinity with time. This type of study is based on the Avrami equation: v f A — Pirn ( Jr tn\ A, \'-J — CAJJ \^ — AV I )

(R 91^ y(j~A±j

where x(0 is the fraction of crystallized material at time / and A: is a constant at a given temperature. The Avrami exponent n

8.4 Experimental Techniques for Studying Transcrystallization

can take values ranging from 0.5 to 4 depending on the growth geometry (rod, disk, or sphere), the nucleation mode (simultaneous or sporadic), and the ratecontrolling process (interface or diffusion control). This is an attractive approach in order to evaluate the effect of the introduction of fibers on the overall crystallization kinetic. The value of n for a composite sample is expected to differ from that for the pure matrix because of surface induced nucleation and restricted growth due to the presence of fibers. The Avrami equation is used for data acquired in isothermal experiments but it has been adapted by Ozawa (1971) to the case of non-isothermal study (at constant cooling rate). Recently, Echalier and Davis (1989) have studied glass-fiber-reinforced polypropylene composites exhibiting transcrystallization at various cooling rates (from 5°C/min to 40°C/min). They found a value of n = 2 (compared to n = 3 for pure polypropylene). However, they could not attribute this decrease solely to the change in growth geometry (from spherulite to disk). It should be noted that they did not mention the ratio of transcrystalline material to bulk spherulitic material. Changes due to growth geometry should be more noticeable for samples with very large transcrystals. The results must be interpreted very carefully, as deviations from the theory are observed when Ozawa's approach is applied to composites. Here, the crystallization kinetic is slowed down at a high degree of conversion and n increases continuously with temperature. As a consequence, the experimental data often tend to depart from the theoretical equation at high fractions of crystallized material. These problems arise because the Avrami equation was not developed for composites. Therefore, it does not take into account the geometrical constraints of such systems.

363

An interesting alternative approach has recently been taken by Billon et al. (1989) who used a numerical simulation to study the effect of restricted space on the value of the Avrami coefficient. Crystallization was simulated in a two dimensional film with and without surface nucleation. It was shown that the crystallization kinetic is slowed down when the available space decreases and that crystallization is substantially enhanced by surface nucleation. Both these phenomena contributed to changing the value of the Avrami exponent. 8.4.4 Dynamic Mechanical Spectroscopy Dynamic mechanical spectroscopy (DMS) is a very powerful and versatile tool because it allows not only a study of bulk properties such as the elastic or the loss modulus, but also observation of phenomena occurring at the molecular level, including molecular transitions and relaxations or chemical degradation. Commercial apparatus now enable one to investigate a wide range of frequencies, temperatures, and sample geometry and offer multiple test procedures (dynamic or steadystate tests, relaxation and creep measurements) for materials as different as polymer melts, rigid solids, composites systems, low viscosity liquids, or reactive systems. Matsuoka et al. (1968) studied the molecular relaxations of transcrystalline polyethylene films. They found that, at all temperatures, the transcrystalline material exhibits a higher loss factor, tan (5. Since the elastic part of the modulus also increases, this increase is mostly due to the much higher value of the loss modulus. They postulated that this change in tan<5 is either due to a greater number of non-crystalline regions within the transcrystalline

364

8 Morphology Control in Polymer Composites

structure or to peculiar but unresolved interactions between the transcrystalline lamellae. The transcrystalline material also exhibited a supplemental transition, a', found above 80 °C, which was absent in the spherulitic sample. They also noted marked differences in the low temperature region (-80°C to -160°C). A y-transition is observed at -140°C for the transcrystalline material, whereas a broader dispersion is seen at -125 °C for the bulk sample (Fig. 8-13). The interpretation of the data is complicated in this case by the fact that there is still controversy on the nature and the mechanisms associated with polyethylene transitions found in this region (Boyer, 1977). They concluded that the -140 °C peak is associated with defects confined within the crystals while the -110°C peak is a secondary transition of the amorphous glassy region. However, DMS results cannot be interpreted solely in terms of orientation. Indeed, the study of single crystal mats has revealed that the

o.i

0.01

0.001

-160

-120

-40

40

80

120

Temperature (°C) Figure 8-13. DMS spectrum of (a) transcrystalline poly(ethylene) film and (b) regular spherulitic film. Notice the higher loss factor, tan d, for the transcrystalline film (after Matsuoka et al., 1968).

crystal thickness strongly affects the magnitude of relaxation peaks. This particular study is beyond the scope of this chapter; however, an excellent review is available on the subject (Ferry, 1980). Finally, it should be emphasized that the DMS data available on transcrystallinity were obtained only for films. In the case of fiber-reinforced thermoplastic composites, one might expect additional effects on the dynamic mechanical spectrum due to restricted motion of the adsorbed chains at the interface. A priori, one would therefore expect the glass transition temperature to be shifted to higher temperature and the activation energy for this process to increase. However, until experimental data are available for composites, this must be considered just as a possibility. This lack of experimental data is not accidental and is rather revealing on the nature of the difficulties encountered in the characterization of transcrystallinity by DMS. Indeed, for DMS results to be meaningful, the behavior of a transcrystalline system must be compared to that of a system where there is no or little transcrystallinity. This assumes that the extent of the transcrystalline zone can be controlled in a systematic way. Studies by other techniques such as DSC would also be made easier if almost entirely transcrystalline composite samples could be produced without having to use a high fiber content. These kinds of experimental necessities are the reason for more quantitative studies aimed at controlling the morphology. 8.4.5 Infrared Spectroscopy

Infrared spectroscopy (IR) has long been recognized as a valuable analytical tool in the study of polymeric materials because of the wealth of informations provided on the conformation, orientation,

8.4 Experimental Techniques for Studying Transcrystallization

hydrogen bonding, molecular symmetry, and crystal forms in the sample. With the advent of Fourier transform infrared spectroscopy (FTIR), a considerable gain in sensitivity over regular dispersive spectroscopy has been achieved. This enhanced sensitivity enables one to detect surface regions with low signal intensity or even very thin layers of adsorbed material. FTIR is particularly useful for the study of surface layers, because with specific techniques, one can probe a predetermined thickness in the surface region, or obtain information on the specific orientation of an adsorbed species on a substrate. Moreover, these techniques are amenable to quantitative analysis. Luongo and Schonhorn (1968) have used attenuated total reflection (ATR) spectroscopy to characterize the surface region of polyethylene films nucleated on various substrates. By using ATR, they were able to selectively probe the surface region ( « 2 Jim). From the ratio, i?72o/73o> of the intensity of the crystalline bands at 720 cm" 1 and 730 cm" 1 , the degree of crystallinity in the surface region was measured. They found that the surface region of a polyethylene film nucleated on a high

1.3 a

1

365

surface-energy substrate (gold) was more crystalline than that of a film nucleated on a low surface energy substrate (polytetrafluoroethylene). By studying the dependency of ^720/730 o n the angle of incidence of the IR beam with the surface, they were also able to show that the polyethylene film nucleated against polytetrafluoroethylene has a higher degree of orientation (Fig. 8-14). Another technique, Reflection absorption spectroscopy (RAS), has been used by Nguyen and Ishida (1986) to study the degree of orientation at the interface of annealed poly(ether-etherketone) films and graphitic substrates. For randomly oriented films (quenched), no particular orientation was observed. However, it was shown, using parallel polarized light, that the carbonyl-stretching frequency and the aromatic C-H out-of-plane vibrations progressively disappear as surface orientation develops upon annealing. In this particular case, the crystalline polymer chains were found to be oriented normal to the substrate, which is rarely observed in surface induced crystallization. It was postulated that the intermolecular interactions are stronger than the interactions between the chains and the substrate. IR

1

1.1

0.9

-

0.8

^^~~ 5*-

A PTFE Film

O.t

A Gold Film

!

|

i

20

40

60

Angle of Incidence

80

Figure 8-14. ATR spectroscopy as a method of detection of surface orientation (polyethylene crystallized in contact with gold and polytetrafluoroethylene). The characteristic ratio, R12o/73o> depends on the angle of incidence for the oriented structure (after Luongo and Schonhorn, 1968).

366

8 Morphology Control in Polymer Composites

has also been used to characterize the changes in orientation for transcrystalline poly(vinylidene fluoride) films as well as uniaxially oriented and biaxially oriented films (Weinhold et al, 1984).

8,5 Influence of Transcrystallinity on the Mechanical Properties of Fiber-Reinforced Composites The presence of a transcrystalline zone is expected to improve both the adhesion at the fiber-matrix interface due to a higher nucleation density, and the mechanical properties of the interphase as a result of lamellae orientation. However, very few articles have been published, which address this issue. A reason for the scarcity of reports dealing with mechanical properties has been mentioned in Sec. 8.4.4. In order to provide an unambiguous answer, a sample with spherulitic-matrix morphology must be compared with an identical sample in which large transcrystals are present. In general, two such samples could be produced by varying the crystallization temperature and keeping all other parameters, such as the percentage of reinforcement, constant. However, this comparison cannot be valid because, while inducing transcrystallization by changing the crystallization temperature, the average size of the spherulites in the bulk matrix and the lamellae thickness are also modified. Because these two parameters are known to influence the mechanical properties, it is impossible to attribute any enhancement in properties solely to the presence of a transcrystalline zone. If such a test is to be successful, an entirely transcrystalline sample must be compared to a sample exhibiting no transcrystallinity. One possibility is to bring the fibers so close to each other

that only transcrystals can develop. However, a higher fiber content reduces proportionally the amount of transcrystalline material studied. The danger in this case is to test the effect of fiber volume rather than the effect of transcrystals. This discussion illustrates why morphology control is of central interest. The problem described affects more the study of the interphase than that of the interface because adhesion at the interface depends little on the extent of the transcrystalline zone. 8.5.1 Adhesion at the Polymer-Substrate Interface Much of the early work in this field was done by Schonhorn and co-workers (1964, 1968) who studied extensively the adhesion of polyethylene to various substrates. They found that the surface layer morphology strongly influences the adhesive joint strength. In a simple peeling test of bonded polyethylene-aluminum joints (Schonhorn, 1964), a relatively thick layer of polyethylene was found to adhere to the substrate surface. Optical microscopy revealed that this layer consisted of transcrystalline material. The thickness of the transcrystalline layer increased with the bond-preparation time at the selected temperature. Later, Schonhorn and Ryan (1968) studied the adhesion in epoxy adhesive-polyethylene film-epoxy adhesive pieces. It was shown that the presence of a large transcrystalline surface layer in the polyethylene film considerably enhances the adhesive-joint strength. From their work, it appears that the major interest of a transcrystalline layer is that it preserves the expected properties rather than brings any enhancement in itself. Indeed, when no transcrystalline zone is present, a weak boundary layer is formed, which decreases considerably the expected quality of adhe-

8.5 Influence of Transcrystallinity on the Mechanical Properties

sion. Transcrystallinity prevents such a detrimental effect by extending the bulk properties of the matrix to the surface region. If the transcrystalline zone is damaged, this protective effect is lost. Schonhorn and Ryan (1968) found that a similar result can be achieved by crosslinking the surface layer through activated species. They used an analogy between the behavior of a transcrystalline layer and that of a crosslinked layer to explain the increased cohesive strength. Because of extensive surface nucleation, there is a strong competition between the chains at the surface. Therefore, it can be speculated that considerable entanglements of the chains occur. These entanglements are very similar in effect to crosslinks. Other workers have indicated that transcrystallinity might contribute to enhanced adhesion. Hsiao and Chen (1990) have used a micro-debonding test on single filaments to measure the effect of transcrystallinity on the debonding force. They prepared single carbon-fiber-poly(etherketone-ketone) (PEKK) specimens with and without transcrystallinity. Transcrystallinity was induced by two methods: coating the fiber with a substance whose unit cell dimensions are similar to those of the PEKK matrix, and immersing the fiber end in a water bath so that a temperature gradient could develop at the fiber surface. The presence of a transcrystalline zone led to a 20% increase in debonding force in the first case and to a 30 % increase in the second case. One problem with this type of test, however, is that the uncertainty of the measurements is quite high (35%). Hsiao and Chen (1990) also found that the positive effect of transcrystallinity decreases as the fiber volume increases. As the fiber volume increases, the distance between fibers decreases sharply, and the transcrystalline zone becomes thinner. Consequently,

367

there is less and less difference between a regular sample and a transcrystalline sampie. Until now, the adhesion has been considered only in terms of what happens directly at the fiber surface. However, in light of some recent studies, the most important phenomenon might well occur away from the surface, at the bulk matrixtranscrystalline zone interface. Recently, Guigon et al. (1989) studied glass fibers polyamide composites by SEM and found that failure after flexural test occurs at the bulk matrix-transcrystalline zone interface rather than at the fiber-matrix interface (Fig. 8-15). The importance of this second interface has been generally ignored. A more complete picture of a fiber-reinforced composite system incorporating this interface is presented in Fig. 8-16. A similar observation was made earlier by Lee and Porter (1986) who studied the crystallization of PEEK in carbon fiber composites. They also measured the trans-

Figure 8-15. In polyamide-glass fiber composites, failure occurs at the transcrystalline zone-bulk matrix interface rather than at the fiber-transcrystalline zone interface. This suggests increased adhesion at the fiber surface [Guigon et al. (1989); reprinted by permission. Copyright Association pour les Materiaux Composites 1989].

368

8 Morphology Control in Polymer Composites

A: FIBER B: TRANSCRYSTALLINE ZONE C: INTERFACE TRANSCRYSTALLINE ZONE/BULK MATRIX D: BULK MATRIX

verse tensile strength of PEEK-carbonfiber composites. Samples exhibiting high surface-nucleation were found to have a much higher transverse tensile strength than the strength predicted by a model assuming no interfacial adhesion. The fracture toughness was also greatly increased. A very elegant explanation for these results was provided based on the theory of Keith and Padden (1963). Keith and Padden (1963) recognized that polymer melts contain numerous defects which can act as impurities: atactic species, branched or entangled molecules, and chain ends. They showed that during crystallization these impurities diffuse away from the crystal-growth front. Eventually, the concentration of such impurities become quite high in regions where the growing spherulites impinge on each other. Consequently, some uncrystallized material remains trapped between spherulites. This poorly crystallized zone acts as a weak point in the polymer sample. This

Figure 8-16. Schematic representation of a fiber reinforced semicrystalline thermoplastic system.

phenomenon is of great importance in fiber-reinforced composites. If matrix nucleation is dominant, the uncrystallizable impurities accumulate at the fiber-matrix interface. Consequently, a weak layer develops, and the adhesion is poor. However, if surface induced nucleation is favored, the transcrystalline-growth front can push these impurities away from the fiber surface. At the same time, spherulitic crystallization occurs in the matrix which also tends to reject these impurities. As a result, the concentration of uncrystallizable material will be much higher at the transcrystalline zone-bulk matrix interface than in the rest of the sample, and a weak zone will develop there. Thus, transcrystallinity enhances adhesion at the fiber-matrix interface by shifting the weak boundary layer toward the bulk matrix. Krautz (1971) has shown that the improvement in properties of composites is an order of magnitude greater for semicrystalline polymer matrices than for

8.5 Influence of Transcrystallinity on the Mechanical Properties

amorphous polymer matrices. The previous discussion has shown that transcrystallization might be responsible for much of this effect. Kardos (1973) has even proposed that preferential interfacial crystallization be preponderant in the case of difficult to crystallize engineering thermoplastics. In this later case, annealing might be of great interest to induce crystallization next to the reinforcement.

Kwei et al. (1967) studied the influence of an oriented transcrystalline zone on the real and imaginary part of the dynamic Young's modulus, E, in polyethylene and polypropylene films. By changing the thickness of the film, they were able to vary the ratio of transcrystalline material to regular spherulitic material. Eventually, the thickness of the film was decreased to the point where the film was entirely transcrystalline. The results were interpreted using a model in which the transcrystalline zone and the bulk region produces a parallel response to the applied stress. They found that the modulus reaches an asymptotic value as the thickness is decreased, corresponding to the value of the modulus for transcrystals (Fig. 8-17). This thickness effect is not an artefact, since a similar experiment for a noncrystalline epoxy system showed no change in the film properties with decreasing thickness. A two to three fold increase in the value of the elastic and loss moduli was noted for the transcrystalline polyethylene and polypropylene films. When the shear properties are tested, no major difference is observed between the level of the shear modulus, G, for the transcrystalline zone and that for regular

8.5.2 Mechanical Properties of Transcrystals

In polymer composites where transcrystallization takes place, the particular orientation of the lamellae is expected to influence the mechanical properties. Similarly, changes in the molecular relaxations (magnitude and temperature) are expected, due to the different mobility of the chains at the polymer-substrate interface. The transcrystalline structure is highly anisotropic, which means that the mechanical properties of the transcrystals will also be anisotropic. Therefore, in comparing results obtained by various investigators, special attention should be paid to the relative orientation of the transcrystalline lamellae with respect to the direction of testing. 2.2

•A

1.7 tr*

O

1.3

1

. . . . . . .| . —A—PP Between Cu —A—PE Between Cu

. . . . ...,

369

\

A^NJ

-

.

0.8

. . ....i

1

10 Thickness (|i,m)

.

.

Figure 8-17. Dependence of the real part of the Young's modulus, E, on the thickness of a polyolefin film. As the thickness decreases, the value of the modulus reaches a maximum, which corresponds to the value of the modulus for a purely transcrystalline film (after Kwei et al., 1967).

370

8 Morphology Control in Polymer Composites

spherulites (Eby and Colson, 1966; Crissman and Passaglia, 1966). This observation can be rationalized as follows. What is tested in a shear experiment is the response of the weakest component, i.e. the bulk matrix; the transcrystalline zone and the bulk matrix are parallel with respect to strain. In a tensile test, the response of the strongest component is tested; the transcrystalline zone and the bulk matrix are in series with respect to strain. Moreover, even though transcrystalline lamellae are preferentially oriented perpendicular to the fiber direction, there is a randomization effect because they are also uniformly distributed around the fiber. Thus, the anisotropy is reduced. In shear testing where the strain is perpendicular to the fiber direction, this effect tends to minimize the differences between an oriented and an unoriented structure. The magnitude of the mechanical relaxation is affected by the presence of oriented material. Both Eby and Colson (1966) and Crissman and Passaglia (1966) found that, in oriented polyethylene films, the oc-relaxation (relaxation found above room temperature) decreases whereas the |3- and y-processes are essentially not affected. However, it is difficult to attribute this effect solely to orientation, as the lamellae thickness of their samples was affected by the crystallization and annealing temperature chosen. As mentioned earlier (Sec. 8.4.4), lamellae thickness is known to influence the magnitude of the relaxation peak. Another property of transcrystals which might be of practical interest is the mode of plastic deformation. More specifically, the elongation at break for the transcrystalline zone can be expected to differ from that for the bulk spherulites when the strain is applied along the fiber direction. In particular, Campbell and Qayyum (1977) have

proposed that transcrystallization leads to increased elongation at break by restraining necking of the fiber. Indeed, necking would require lateral extension of the lamellae along the a-axis, which is very unlikely. Rather, they postulated that the strain at the interface is accommodated by separation of the transcrystalline lamellae along the c-axis (which is along the fiber axis for transcrystals). This way the incidence of fracture is postponed.

8.6 The Induction-Time Approach From Sec. 8.4 on experimental techniques and Sec. 8.5 on mechanical properties, it is clear that the study of transcrystallization would be made easier and more rigorous if some method to control the extent of the transcrystalline morphology were available. Furthermore, such a method should have some theoretical basis. However, the current theory of heterogeneous nucleation has a very serious practical limitation when applied to a purely transcrystalline system; because of the large number of spherulites nucleating at the interface, the nucleation rate cannot be measured and therefore Aa cannot be obtained (Sec. 8.2.2). This difficulty explains why the characterization of the energetics of the transcrystallization process using the classical theory has been attempted in only one major study (Chatterjee and Price, 1975 c). In this section, an alternative approach is presented, based on the induction time. Because this approach might help to characterize more quantitatively the transcrystallization process, it is presented in a more detailed fashion. 8.6.1 Theoretical Basis In Sec. 8.2.2, the temperature dependency of the rate of heterogeneous nucleation,

8.6 The Induction-Time Approach

/, has been described by Eq. (8-5). /represents a number of nuclei per unit time and volume. In an isothermal experiment, a constant value is predicted. Thus, the number of nuclei observed under the microscope in a given volume should be linear with time and nucleation should start at £ = 0. However, this effect is not what is typically observed. Generally an S-shaped curve is obtained with a certain delay before the onset of massive nucleation occurs (Fig. 8-18). This delay is often called induction time, ti9 and is defined as the intercept of the time axis of the nucleation curve. Just as /varies with temperature, so does t{. However, these variations are opposite in direction. As the supercooling is decreased, the size of the critical nucleus becomes larger and the time required to create a stable nucleus increases. At the same time, the nucleation rate decreases. Thus, intuitively, one can assume that the induction time and the nucleation rate might be related and carry the same information on the energetics of the system.

Number of nuclei

time Figure 8-18. Definition of the nucleation rate / and the induction time t{. (a) Rapid nucleation, I-> co and ^-+0. (b) Slow nucleation, /-•() and t^co (Ishida and Bussi, 1991 a; by permission).

371

This relationship is of particular interest in the case of surface-induced crystallization because the time at which massive nucleation occurs can still be recorded even though individual spherulites cannot be distinguished at the fiber surface. Because the absolute number of nuclei is very important, the observed transition will be sharp. Thus, if one can find a way to relate / and t{, Aa can be obtained, whereas / cannot be measured. Because / varies inversely with t{, Ishida and Bussi (1991b) have proposed that the product of these two quantities is a constant. Mathematically:

I(T)t{(T) = K= constant

(8-22)

This relationship holds for the boundary conditions. For infinitely long nucleation (very low supercooling), the induction time goes to infinity as the nucleation rate is almost zero, but the product of these two can still be a finite constant. Inversely, for almost immediate nucleation (very high supercooling), the nucleation rate goes to infinity as the induction time goes to zero (Fig. 8-18). By using Eqs. (8-22) and (8-9), one can see that a plot of \nI+Acp/(kT) versus l/[71[Ajy)2] or aplot of 111(1/0 +A

372

8 Morphology Control in Polymer Composites

(ZBD) theory of nucleation (Zeldovich, 1942; Becker and Doring, 1935). Why is an induction time observed? The nucleation rate derived earlier is a steadystate nucleation rate. It is assumed that, at the crystallization temperature considered, there is an equilibrium size distribution of subcritical nuclei (embryos). Following the notations used by Frisch (1957) f (g,t) represents the distribution function of an embryo of size g at time t. As suggested by Wunderlich (1976), this distribution can be represented by a decreasing exponential (Fig. 8-19). It is much more probable to find numerous small embryos in the melt than large embryos, because the free enthalpy necessary for a large embryo to survive is greater. In an isothermal-nucleation experiment, generally the sample is cooled down quickly from T{ (initial melt temperature, far above the equilibrium melting point) to Tc (temperature of crystallization). The induction time, t{, is the time necessary to reach the steady state distribution at TG9 / s , T c (0), starting from an initial steady-state distribution at T{, / s T .(g,0). Using the ZBD theory of nucle-

ation, Frisch (1957) showed that:

(8-23)

i CD =

where G is the size of the embryo for which the probability of decomposition is essentially zero (G is slightly larger than #*, the size of the critical nucleus, as defined in Sec. 8.2.2). The ZBD theory was initially derived for what is termed self-nucleation, that is, nucleation which arises in the melt from its own, previously grown, crystals. Partially molten, high molecular weight crystals, can serve as seeds for self-nucleation. However, this theory can be applied to heterogeneous nucleation if the following remarks are taken into account: In the case of heterogenous nucleation, it is assumed that the subcritical nucleus initiates at the surface of the substrate, or at the heterogeneities present in the melt. This assumption is justified because it is easier for an embryo to survive on a foreign surface than in its own melt. In the case of transcrystallization, the assumption of subcritical nuclei preferentially growing at the fiber surface, rather than in the melt, is even more justified because of the particular affinity between the matrix and the fiber. Therefore, if the concentration of heterogeneities is assumed to be constant for the temperature range [7\, T2] investigated, one can write: = constant,

Tc e [7\, T2] (8-24)

Sizeg Figure 8-19. Steady-state distribution of subcritical nuclei as a function of size at a given temperature (after Wuncterlich, 1976).

Equation (8-24) basically expresses that the number of embryos of size g can change from one temperature to the other, but the total number of stable nuclei which are generated is constant because there is only a fixed number of heterogeneities in the

8.6 The Induction-Time Approach

melt. This is equivalent to a normalization condition for a probability. The second assumption made is that at the initial temperature T{\ /s,ri(flf,0) = 0,

gs[O9G]

(8-25)

When T{ is far above T£ and the melt is kept at this temperature for a sufficient period of time, this assumption is reasonable. This means that at temperature Ti9 no embryo can survive at the melt-heterogeneities interface. With these two assumptions, Eq. (8-26) can be derived as follows:

G

J/..T.

constant

Hence,

was assigned to k and n, and no correlation to the surface free energy parameters was attempted. It should be noted that there is little reference in the literature to the induction time as a way to gain information on the energetics of the nucleation process because, generally, nucleation rates can be measured. However, another particular variable, the half-time of crystallization has been used in the past to study homogeneous nucleation in the droplet experiment (Gornick et al., 1967). It is interesting to note that, in this case, the equation used to correlate the nucleation rate and the halftime of crystallization is derived from the fact that there is only a fixed number of crystallizable droplets. It is an assumption similar to the one made in Eq. (8-22). 8.6.2 Application of the Theory

1 ( 1 ) t i ( l ) = K.

(o-zoj

The ZBD theory has been originally applied to the study of phase transitions in gases. In a study of the condensation of droplets upon very rapid expansion of a gas, Kantrowitz (1951) predicted that the induction time should vary inversely with the fourth power of the degree of supersaturation, which is the equivalent of the degree of supercooling in a polymer nucleation experiment. However, this relationship was derived for very small induction times, of the order of 0.1 |is. Nonetheless, Magill (1962) used a generalized expression of the same type to study the heterogeneous crystallization of Nylon 6: ti = k(AT)~n

373

k,n = constants

(8-27)

He was able to fit his data with n — 3 and n = 7, depending on the initial melt temperature and on the final crystallization temperature. However, no physical meaning

The goal of a theoretical approach is to obtain the value of the free-energy difference function, ACT, for the heterogeneous nucleation of transcrystals at the fiber surface. Ac can then be compared to ACT', the value of the free-energy difference function for the bulk matrix/heterogeneities in the melt system. This way, one can judge which type of nucleation is favored, independently from the temperature. Furthermore, the nucleating ability of a given fiber toward different polymers can be compared quantitatively. The first step in the determination of ACT involves a classical growth-rate study. This step does not present any difficulties even though it is performed on the transcrystalline zone. Once nucleation at the fiber surface has occurred, growth in the matrix or in the transcrystalline zone should proceed in exactly the same way. Therefore, one can apply Eq. (8-10) to the transcrystalline-growth rate data and the calculated

374

8 Morphology Control in Polymer Composites

value of a o-e should not depend on the type of crystals studied. Nonetheless, it was not until recently that extensive transcrystalline-growth rate data were compiled by Thomason and Van Rooyen (1990) and Ishida and Bussi (1991a, b) [Chatterjee and Price (1975 b) first observed that the two growth rates were identical, but for a single temperature measurement]. The thickness of the transcrystalline-growth front generally increases linearly with time. Measurements on the transcrystalline zone are made more easily and more precisely, since a uniform growth front is observed. This advantage is especially valuable when the growth rate is very important (high supercooling), that is, when quick focus on the growth front must be achieved. Ishida and Bussi (1991a) selected polyethylene as a matrix because extensive data on its spherulitic growth rate were available for comparison. Moreover, polyethylene is well known for exhibiting a transition in growth regime at roughly 127 °C (Hoffman et al., 1976; see Sec. 8.2.2 and Fig. 8-5). Also, the ability of the transcrystalline-growth rate to recognize this transition can be tested. For bulk spherulites, the occurrence of a certain regime can be deduced from the observed morphology. At higher temperatures, axialites are observed in the polymer melt but disappear at lower temperatures. Axialites, which appear as immature spherulites with thicker and more diffuse branches, are characteristic of a regime I behavior. However, this change in morphology is not seen in the transcrystalline zone. Once the transcrystalline-growth rate has been recorded as a function of temperature, the analysis of the data is performed as outlined in Sec. 8.2.2. The plot of In g versus i/(TAT) for polyethylene-polyethylene composites is shown in Fig. 8-20 and exhibits a regime II behavior for the first 10 temperatures.

From the slope of the curve, o ae was calculated to be 1.38 x 10" 3 J 2 /m 4 , which is in good agreement with other independent estimates for bulk spherulites, where an average value of 1.31 x 10 ~ 3 J 2 /m 4 has been reported (Hoffman et al., 1976). The change in growth regime is also noticeable. The values of the growth rate at higher temperatures are believed to follow a regime I behavior where 127.3 °C is the breaking point. Indeed, the calculated slope using these points is roughly twice the value for regime II as expected from the theory. In a further study of poly(gcaprolactone)-polyethylene-fiber composites, Ishida and Bussi (1991 b) found that the transport term becomes predominant as the crystallization temperature decreases, and that the transcrystalline-growth rate eventually reaches a maximum (Fig. 8-21). At higher supercooling, the melt viscosity is higher, and it becomes more difficult for the polymer chains to diffuse toward the transcrystalline-growth front. Such an effect had not been revealed by a regular growth-rate study on bulk spherulites (Phillips et al., 1989). At high temperatures, the term due to the free energy rather than the transport term controls the growth process. During a growth-rate experiment, the induction time for transcrystallization to appear can also be recorded. As shown in the previous section, this time carries the same information on the energetics of the system as the nucleation rate. A plot of ln(l/*i) + A(p/(/cr) versus l/[T(ATf)2] should yield a linear curve whose slope is proportional to <7(7eAcr, and from which ACT can be deduced. Such a curve is shown in Fig. 8-22. However, one would like to test the validity of Eq. (8-22) on a system where both the induction time and the nucleation rate can be measured at the same time. In their work, Ishida and Bussi

8.6 The Induction-Time Approach

-2

i

•

i

375

i

,,= 12.2xlO 4

XH-

- qx

4 h

"X- \ L

6 h

-

A

4

(Kg,= 26.1xlO K

-

\

|A

j

i

i

1

1.4

1.5

1.6

1.7

-10 1.3

>

2

1.8

Figure 8-20. Transcrystalline growth rate analysis for polyethylene-polyethylene composite. Plot of ln# versus 1/(TAT). The change in growth regime is indicated by the arrow (Ishida and Bussi, 1991 a; by permission).

104/TAT

1 -2 k a

-1 \

T* = 297 K =0.65 um/s

g &

max

r

'

-4

-6 h

50

1

1

i

i

40

30

20

10

0

Figure 8-21. Transcrystalline growth rate analysis for poly(fi-caprolactone) - polyethylene composite. As the supercooling increases, the transport term becomes predominant and the growth rate reaches a maximum value gmax at the temperature T* (Ishida and Bussi, 1991b; by permission).

Supercooling

8 H H

1 0

+

-1 -2 -3 10

12

10 6 /T(ATf) 2

14

16

Figure 8-22. Determination of the value of the free-energy difference function, ACT, from the temperature dependency of the induction time (Ishida and Bussi, 1991 b; by permission).

376

8 Morphology Control in Polymer Composites

(1991b) selected the bulk poly(Vcaprolactone) matrix as a test system, because it also provides an estimate of Aa' (Fig. 8-23). They noticed that the value of the induction time was much less sensitive to the melt history than the value of the nucleation rate was. Their data shows that the plot of ln(l/fj) + A

Chatterjee and Price (1975 c) also collected Aa and Aa' data using the value of the surface nucleation rate. For their study they had to work with a partially transcrystalline system in order to distinguish the individual spherulites at the substrate surface. Their results along with those of Ishida and Bussi (1991a, b) are presented in Table 8-2, using the characteristic parameter A introduced in Sec. 8.2.2. It can be seen that advantage A is larger than 1 for truly transcrystalline systems. The difference in nucleating ability between heterogeneous crystallization in the melt and at the fiber surface is more strik-

*4—(K. = 0.94xl06 K3)

6

fK.=0.91xl0 K

Figure 8-23. Comparison of the induction time and the nucleation rate approach for the poly(e-caprolactone) melt. The similar slope of these two curves suggests that Eq. (8-22) is valid for the polymer considered (Ishida and Bussi, 1991 b; by permission).

:

106/T(ATf)2 2000

1500
1000

-

500

-

| O

44

48

52

Temperature (°C)

56

Figure 8-24. Temperature dependency of the induction time for (a) massive bulk nucleation and (b) transcrystallization [poly(e-caprolactone)-polyethylene composite]. Notice the large difference for the temperature associated with each process, at a given induction time (Ishida and Bussi, 1991b; by permission).

8.7 Conclusions and Outlook

377

ticular temperature, there is no competition from the matrix and transcrystals can grow unrestricted. Thus, it is possible to produce an entirely transcrystalline sample without having to use a high percentage of reinforcement (Fig. 8-25). The interest of such samples has been stressed earlier.

8*7 Conclusions and Outlook

Figure 8-25. Growth of a highly transcrystalline film with low percentage fiber [poly(e-caprolactone)polyethylene composite at 52 °C, after 20 h].

Table 8-2. Correlation between the observed morphology and the value of the parameter A defined by Eq. (8-4). Polymer/Substrate

A = Acr'/Acr

Morphology

poly(butene-l) polystyrene (film)

0.5

spherulitic

poly(butene-l) polypropylene (film)

0.6

spherulitic

polyethylene polyethylene (fiber)

1.3

transcrystalline

poly(fi-caprolactone) polyethylene (fiber)

2.0

transcrystalline

ing when the induction time for each process is reported as a function of the crystallization temperature (Fig. 8-24). Such a diagram enables one to predict and control the morphology when it is coupled with transcrystalline-growth rate data. If a temperature such as 52 °C in Fig. 8-24 is selected, transcrystallization will appear after roughly 500 s. But for the same temperature, the induction time for massive nucleation to occur in the bulk matrix approaches infinity, which indicates that the nucleation rate becomes negligible. For this par-

Although transcrystallization has been observed for a long time in polymer films, it is a relatively new subject in the field of fiber-reinforced semicrystalline thermoplastics composites. A look at the references in Sec. 8.9 confirms that most of the work which has been done in this area is less than five years old. There is substantial evidence that transcrystallization improves the adhesion at the fiber-matrix interface and the mechanical properties of the interphase. Transcrystallization enhances interfacial adhesion by preventing the formation of a weak layer at the fiber surface. This weak layer is thought to consist of various impurities, which are unable to crystallize. The transcrystalline growth front pushes these impurities away from the fiber surface, into the bulk matrix. The improvement in adhesion might also be an intrinsic property. It has been postulated that increased competition at the fiber surface during the nucleation step forces entanglements of the chains, which can act as crosslinks with the fiber. The transcrystals have also been found to possess a higher Young's modulus. However, this type of improvement in mechanical properties depends on the orientation of the transcrystals with respect to the strain direction. These positive effects suggest that it might be advisable to induce transcrystallization in practical applications. Several

378

8 Morphology Control in Polymer Composites

methods can be suggested: coating of the fiber with a substance whose unit cell parameters match those of the polymer matrix, chemical treatment of the fiber surface to create favorable surface energy conditions, selection of a proper (i.e., higher) crystallization temperature, mechanical treatment of the fiber to increase roughness and thus to provide more potential nucleation sites, etc. A number of questions merit further study. Among them are: the degree of crystallinity of the transcrystalline zone compared to that of the spherulitic bulk matrix, the thickness of the transcrystalline lamellae (and effect of the crystallization temperature on this thickness) compared to that of the bulk spherulites. More experimental data are also needed to evaluate the effect of the extent of the transcrystalline zone on the mechanical properties (i.e., is there an optimum thickness?). Recent theoretical developments suggest that it might be possible to compare the nucleating ability of various polymer-substrate pairs on a quantitative basis. Last, the knowledge of the temperature dependency of the transcrystalline-growth rate and the induction time enables one to control the extent of the transcrystalline zone to yield samples with known morphology.

8.8 Acknowledgements This work was in part supported by the Office of Naval Research.

8.9 References Barrett, C. S. (1952), Structure of Metals. New York: McGraw-Hill, p. 510. Barriault, R. I, Gronholz, L. F. (1955), /. Polym. Sci. 18, 393.

Baukema, P. R., Hopfinger, A. J. (1982), J. Polym. Sci., Polym. Phys. Ed. 20, 399. Beck, H. N. (1967), /. Appl. Polym. Sci. 11, 673. Becker, R., Doring, W. (1935), Ann. Physik 24, 1935. Bessell, T., Shortall, J. B. (1975), J. Mater. Sci. 10, 2035. Billon, N., Monasse, B., Haudin, J. M. (1989), in: Annales des Composites: Association pour les Materiaux Composites (Ed.). Paris: Edition Pluralis, p. 117. Binsbergen, F. L. (1973), J. Polym. Sci., Polym. Phys. 11, 117. Blundell, D. I, Crick, R. A., Fife, B., Peacock, I , Keller, A., Waddon, A. (1989), /. Mater. Sci. 24, 2057. Boyer, R. F. (1977), in: Encyclopedia of Polymer Science and Technology, Suppl. Vol. II. New York: Wiley, p. 745. Burton, R. H., Day, T. M., Folkes, M. J. (1984), Polym. Commun. 25, 361. Caramaro, L., Chabert, B., Chauchard, J. (1988), C. R. Acad. Sci. Paris 306, Serie II, 887. Campbell, D., Qayyum, M. M. (1977), /. Mater. Sci. 12, 2427. Chatterjee, A. M., Price, F. P. (1975 a), /. Polym. Sci., Polym. Phys. Ed. 13, 2369. Chatterjee, A. M., Price, F. P. (1975b), J. Polym. ScL, Polym. Phys. Ed. 13, 2383. Chatterjee, A. M., Price, F. P. (1975c), /. Polym. Sci., Polym. Phys. Ed. 13, 2391. Clark, E. S., Starkweather, H. W. (1962), /. Appl. Polym. Sci. 6, 41. Crissman, J. M., Passaglia, E. (1966), J. Res. NBS 70A (3), 225. Eby, R. K. (1964), /. Appl. Phys. 35, 2720. Eby, R. K., Colson, J. P. (1966), J. Acoust. Soc. Am. 39, 506. Echalier, B., Davis, P. (1989), in: Annales des Composites: Association pour les Materiaux Composites (Ed.). Paris: Edition Pluralis, p. 75. Fitchmun, D. R., Newman, S. (1970), J. Polym. Sci. A-2 8, 1545. Folkes, M. I, Hardwick, S. T. (1984), /. Mater. Sci. Lett. 3, 1071. Fowkes, F. W. (1964), Ind. Eng. Chem. 56, 40. Frisch, H. L. (1957), J. Chem. Phys. 27, 90. Goldfarb, L. (1980), Makromol. Chem. 181, 1757. Gornick, F., Ross, G. S., Frolen, L. J. (1967), J. Polym. Sci. C 18, 79. Gray, D. G. (1974a), J. Polym. Sci., Polym. Lett. 12, 509. Gray, D. G. (1974b), J. Polym. Sci., Polym. Lett. 12, 645. Groszek, A. J. (1970), Proc. Roy. Soc. Lond. A. 314, 473. Guigon, M., Laporte P, Echalier, B. (1989), in: Annales des Composites: Association pour les Materiaux Composites (Ed.). Paris: Edition Pluralis, p. 91.

8.9 References

Hartness, I T. (1984), SAMPE J. 20, 26. He, T, Porter, R. S. (1988), /. Appl. Polym. Sci. 35, 1945. Hobbs, S. Y. (1971), Nature Physical Sci. 234, 12. Hsiao, B. S., Chen, E. J. (1990), in: Controlled Interphases in Composite Materials: Ishida, H. (Ed.). New York: Elsevier Science, p. 613. Ishida, H., Bussi, P. (1991a), Macromolecules 24, 3569. Ishida, H., Bussi, P. (1991 b), J. Mater. Sci., in press. Jenckel, E., Teege, E., Hinrichs, W. (1952), Kolloid-Z. 129, 19. Kantrowitz, A. (1951), J. Chem. Phys. 19, 1097. Kantz, M. R., Corneliussen, R. D. (1973), / Polym. Sci., Polym. Lett. 11, 279. Kardos, J. L. (1973), /. Adhesion 5, 119. Keith, H. D., Padden, F. J. (1963), J. Appl. Phys. 34, 2409. Koutsky, J. A., Walton, A. G., Baer, E. (1967), /. Polym. Sci., Polym. Lett. 5, 185. Kubat, X, Rigdahl, M. (1975), Polymer 16, 925. Krautz, F. G. (1971), SPE J. 27, 74. Kwei, T. K., Schonhorn, H., Frisch, H. L. (1967), /. Appl. Phys. 38, 2512. Lee, Y, Porter, R. S. (1986), Polym. Eng. Sci. 26, 633. Legrand, D. G., Gaines, G. L. (1969), J. Colloid Interface Sci. 31, 162. Lopez, L. C , Wilkes, G. L. (1989), Polym. Preprints 30, 207. Lovering, E. G. (1970), J. Polym. Sci. A-2 8, 1697. Luongo, J. P., Schonhorn, H. (1968), J. Polym. Sci. A-2 6, 1649. Mackley, M. R. (1975), Colloid Polym. Sci. 253, 373. Magill, J. H. (1962), Polymer 3, 43. Matsuoka, S., Daane, J. H., Bair, H. E., Kwei, T. K. (1968), / Polym. Sci., Polym. Lett. 6, 87. Misra, A., Deopura, B. L., Xavier, S. F., Hartley, F. D., Peters, R. H. (1983), Angew. Makromol. Chem. 113, 113. Nguyen, H. X. (1986), Ph. D. Thesis, Case Western Reserve University. Ozawa, T. (1971), Polymer 12, 150. Phillips, P. I , Rensch, G. J., Taylor, K. D. (1989), J. Polym. Sci., Polym. Phys. 25, 1725.

379

Schonhorn, H. (1964), J. Polym. Sci., Polym. Lett. 2, 465. Schonhorn, H. (1968), Macromolecules 1, 145. Schonhorn, H., Ryan, F. W. (1966), J. Phys. Chem. 70, 3811. Schonhorn, H., Ryan, F. W (1968), J. Polym. Sci. A-2 (5,231. Thomason, J. L., Van Rooyen, A. A. (1990), in: Controlled Interphases in Composite Materials'. Ishida, H. (Ed.). New York: Elsevier Science, p. 423. Tung, C. M., Dynes, P. J. (1987), J. Appl. Polym. Sci. 33, 505. Turnbull, D., Vonnegut, B. (1952), Ind. Eng. Chem. 44, 1292. Weinhold, S., Litt, M. H., Lando, J. B. (1980), /. Appl. Phys. 51, 5145. Weinhold, S., Litt, M. H., Lando, J. B. (1984), Ferroelectries 57, 277. Zeldovich, J. (1942), /. Exptl. Theoret. Phys. 12, 525.

General Reading Bassett, D. C. (1981), Principles of Polymer Morphology. Cambridge, U.K.: Cambridge University Press, Chap. 6. Cherry, B. W. (1981), Polymer Surfaces. Cambridge, U.K.: Cambridge University Press. Geil, P. H. (1963), Polymer Single Crystals. New York, London: Interscience Ferry, J. D. (1980), Viscoelastic Properties of Polymers. New York: Wiley, Chap. 16. Hoffman, J. D., Davis, G. T, Lauritzen, J. I. (1976), in: Treatise on Solid State Chemistry, Vol. 3: Hannay, N. B. (Ed.). New York: Plenum Press, Chap. 7. Schultz, J. M. (1974), Polymer Materials Science. Englewood Cliffs, NJ: Prentice Hall, pp. 380-390. Sharpies, A. (1966), Introduction to Polymer Crystallization. New York: St. Martin's Press, Chap. 1. Wunderlich, B. (1976), Macromolecular Physics, Vol. 2. New York: Academic Press, Chap. 5.

9 Elastic Properties of Composites Rushad E Eduljee and Roy L. McCullough Center for Composite Materials, University of Delaware, Newark, DE, U.S.A.

List of 9.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.4.5 9.5 9.5.1 9.5.1.1 9.5.1.2 9.5.2 9.5.3 9.5.3.1 9.5.3.2 9.5.4 9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.7 9.8

Symbols and Abbreviations Introduction Elements of Anisotropic Elasticity Generalized Hooke's Law Orientation Transformations Micromechanical Models for Continuous Fiber Composites Mechanics of Materials Models Self-Consistent Field or Embedding Models Bounding Methods Semi-Empirical Models: The Halpin-Tsai Relationship Summary Micromechanical Models for Particulate Reinforced Composites Mechanics of Materials Bounding Approach Embedding Models Semi-Empirical Models Summary Micromechanical Models for Discontinuous Fiber Composites Bounds on the Effective Elastic Properties of Discontinuous Fiber Composites Dispersed Ellipsoidal System Aggregated System Influence of Inclusion Geometry Configurational Averaging Orientation Averaging Aspect Ratio Averaging Summary Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems Aligned Inclusions Planar Orientation Three-Dimensional Orientations Summary Summary References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

382 385 393 397 403 406 409 412 413 421 423 424 425 426 429 431 433 434 435 435 439 441 445 446 449 452 452 453 458 462 468 469 472

382

9 Elastic Properties of Composites

List of Symbols and Abbreviations a a al ac a* ae aef aeq A Ao c C C* Cpq Cijkl d d d dt dtj D(a) e i 9 e'{ e, e'i £, £ L , ET Efjkl f f / F F / a , Fa fd ft /•, Ft fij / p , Fp g G, Gtj Gfjkl G| T hk

aspect ratio (subscript) aggregated texture (subscript) aligned inclusion effective aspect ratio of cylindrical inclusions critical aspect ratio (cylindrical fiber) effective aspect ratio of ellipsoidal inclusions effective aspect ratio (general case) equivalent ellipsoidal aspect ratio area initial area stress tensor (alternative contraction rules) stiffness tensor, elastic constant tensor effective elastic constant of composite components of contracted elastic tensor elements of elastic constant tensor diameter (subscript) dispersed alternative strain vector f = 1-6, components of contracted strain tensor shear strain in terms of extension of the diagonal i, j face aspect ratio distribution unit vectors along ith direction matrix of unit vectors et Young's modulus; £ of a laminate longitudinal and transverse, respectively, to reinforcing fibers (i.e., perpendicular to laminate) component of directional average of Gfjkl orientation parameter (subscript) fiber phase, reinforcing phase alternate stress vector force matrix of forces Ft axial orientation parameters "diagonal" stress i = 1,..., 6, components of contracted stress tensor traction acting at faces perpendicular to i axis projection of stress along the direction of the diagonal planar orientation parameters (subscript) grain shear modulus correlation weighting function transverse shear modulus shape functions (i = 1,..., 6)

List of Symbols and Abbreviations

Hijkl fc* K kT KE L L Lo Lc lb m n

383

inverse of the deviatoric elasticity tensor Rijkl effective plane strain bulk modulus bulk modulus plane strain bulk modulus fiber efficiency factor (subscript) longitudinal direction length initial length ineffective (or critical) length (subscript) lower bound (subscript) matrix phase 6 x 6 matrix that relates the elements of the traditional contracted stress or strain vector to the elements of the alternative stress or strain vector N((j)9 6, \jj) orientation distribution P, Pt any mechanical property (of the zth component in a composite) P* effective property of a composite ptj(r) component of stress polarization tensor r Pkm( ')> Pif(r') deviation of polarization tensor (/cth cell of matrix or /th fiber) from average polarization field R rotational transform for stress and strain using alternate contraction rules Rf orthogonal vectors, axis vectors ARd diagonal strain R(j components of axis vectors R'd°, R'd (initial) vector of diagonal length Rijki deviatoric elasticity tensor s alternate strain tensor S compliance tensor S* effective compliance constant of composite Si(r — X) characteristic structure function Sij components of contracted compliance tensor t (superscript) trial T (subscript) transversal direction T rotational transform for stress and strain in contracted rotation Tg glass-transition temperature ut displacement function «(7» strain energy density U strain energy U° equilibrium strain energy of reference system Ub equilibrium strain energy u'{ fluctuations in the displacement field u'ij partial derivative of u\ with respect to coordinate xj9 du'JdXj ub (subscript) upper bound v volume fraction fit parameter [Eq. (9-117)] V volume vt volume fraction of the ith component in a composite

384

V{ v™ax

9 Elastic Properties of Composites

overall volume of the ith component in a composite maximum packing fraction

a p A/? y A A{ s e e° e®, 4 et 8tj s[j v, v(j £ G cr a* <xal(oo) <jj cr£, axk Gtj crzz T 4>c X \j/(vf)

contracted stress or strain vector stress or strain vector contracted using alternate rules angle of shear deformation shear strain, "engineering" strain, tangential strain matrix of generalized deformation vectors general deformation tensile strain strain tensor far field strain uniform strain field, trial strain field, respectively components of contracted strain tensor components of strain tensor fluctuations in the strain field Poisson's ratio reinforcing factor tensile stress stress tensor average stress of composite average stress of composite (infinitely long fibers) components of contracted stress tensor uniform stress field trial stress field, respectively components of stress tensor axial stress shear stress critical volume fraction "reduced" property proportionality factor for the McGee-McCullough model for particulate reinforced composites Q (T, , \j/) rotational transform for second rank stress and strain tensors cotj components of Q Msi SMC

million pounds per square inch sheet molding compound

9.1 Introduction

9.1 Introduction The opportunity to tailor material characteristics to match performance requirements is a key feature of composite materials. In continuous fiber systems, mechanical behavior is tailored by matching the orientation of stiff and strong fibers to anticipated load paths. The matrix component serves as a binder to promote the integrity of the fiber assembly. The fibers dominate the elastic behavior while the matrix component establishes the environmental characteristics and, in particular, the temperature range in which composite structures can operate. Ceramics and metals are used as the matrix for high temperature applications. The lower cost and more easily formed polymeric matrices are used for operating temperatures below about 300 °C. The focus of this treatment will be on polymeric based composites; however, the models presented to describe elastic behavior are applicable to ceramic and metallic matrix composites. The reinforcing fibers used in conjunction with polymeric matrices are derived from the "glass", "metallic", "carbon" and "organic" families. Within each family a range of properties can be generated through variations in compositions and the processing conditions used during fiber formation. The elastic properties of frequently used fibers are summarized in Table 9-1 a. The glass and metallic fibers are isotropic so that only two of the elastic descriptions are independent. However, as a consequence of molecular alignments, the carbon and organic fibers are transversely isotropic and require five independent descriptions for the specification of elastic behavior. Reinforcing fibers and whiskers are treated in detail in Chapter 2 of this volume.

385

Polymeric matrix materials may be distinguished as thermosetting and thermoplastic. Thermoplastic polymers are similar to metals in the sense that they can be "melted" to form structures and solidified by cooling. In contrast, thermosetting polymers are solidified by irreversible chemical curing reactions that lock the molecular structures into three-dimensional networks. These materials cannot be melted and reformed. The upper range of temperature for which polymeric materials can be used is determined by the "glasstransition" temperature, Tg. At temperatures above Tg, the rigidity of the polymers is decreased by several orders of magnitude. The elastic properties of some common thermoplastic and thermosetting polymers are given in Table 9-1 b. Both thermosetting and thermoplastic polymeric matrices are isotropic so that only two of the elastic descriptors are independent. Since the range of properties of the polymeric matrix is not as great as the range of fibers, it will be convenient to refer to a "typical" polymer with the elastic properties listed in Table 9-1 b. In addition to the selection of reinforcing fibers and matrix material, tailoring is accomplished through the control of reinforcing geometry (or microstructure). The microstructure may be deliberately constructed by laminations of collimated fibers or the selection of various woven or braided architectures. However, for discontinuous ("short") fibers, the microstructure is strongly dependent upon processing conditions and processing history. Variations in fiber orientation and fiber lengths induced during the fabrication of molded parts can cause significant variations in the overall mechanical behavior. The richness of possible reinforcing geometries through these various arrangements can give rise to a wide range of properties for composite

386

9 Elastic Properties of Composites

Table 9-1. Elastic properties of selected components of composite materials. a. Reinforcing

materials Poisson's ratio

Moduli in G P a (Msi)

Material

Plane strain

Shear modulus

Young's modulus

k in GPa (Msi) a 111

IV

ET

EL Inorganic Carbon (AS-4) Copper Glass (E) Steel Quartz Organic Aramid(PRO-29) Aramid(PRO-49) Nylon 66 Poly(ethylene terephthalate) Polyvinyl alcohol

GLT

? 241 (35) 21 (3) 105 (15.2) 105 (15.2) 40.4 71.2(10.3) 71.2(10.3) 29.2 190 (27.5) 190 (27.5) 73.1 69.0(10.0) 69.0(10.0) 29.5

(5.84) (4.08) (10.6) (4.27)

vTT

GTT

8.3 40.4 29.2 73.1 29.5

(1.20) (5.84) (4.08) (10.6) (4.27)

72.0(10.4) 124 (18) 6.6(0.95) 13.8(2.0)

5.5(0.80) 6.9(1.0) 6.6(0.95) 13.8(2.0)

? ?

2.5 (0.36) 5.18(0.75)

2.0 (0.29) 2.6 (0.38) 2.5 (0.36) 5.18(0.75)

21 (3.0)

21 (3.0)

7.9 (1.14)

7.9 (1.14)

Source

\J

A

t*,

0.25

0.27 a

0.3

0.3

0.22 0.30 0.17

0.22 0.30 0.17

14.6(2.11) 101 (14.6) 52.1 (7.53) 183 (26.5) 44.7(6.47)

0.36 0.33 0.33 b 0.33 b

0.38 a 0.33 a 0.33 b 0.33 b

4.6(0.66) 5.2(0.75) 7.3(1.05) 15.3(2.21)

f

0.33 b

0.33 b

22.9(3.32)

g

d

d f

8 8

b. Polymeric matrix tnaterials Material

Moduli Young's in G P a (Msi) EL = ET = E

Shear in G P a (Msi)

GTT = G

Thermoset Epoxy (Epon 828) Epoxy Epoxy (Epon 828/z) Polyester

3.8 2.5 3.25 3.25

Thermoplastic Nylon 12 Polycarbonate Poly(etheretherketone) Poly(methyl methacrylate)

1.24(0.18) 2.38 (0.34) 3.7 (0.50) 2.6 (0.38)

0.466 (0.07) 0.895 (0.128)

Typical

3

(0.55) (0.36) (0.47) (0.51)

(0.4)

1.4

(0.20)

Poisson's

Plane

ratio

strain

vLT = vTT = V

h in G P a (Msi) a

0.35 0.35 0.35

0.3

4.7 3.1 4.0 3.1

(0.68) (0.45) (0.58) (0.45)

0.98 (0.14)

0.33 b 0.33 b 0.36 0.33 b

1.38 2.63 4.9 2.9

(0.20) (0.37) (0.71) (0.42)

1

0.33

3.3 (0.48)

1.28 1.38

(0.185) (0.20)

(0.16)

Source

d h

* 8 j k g

a Computed; b assumed; c Whitney and Drzal (1978); d Berthelot et al. (1978); e Carman and Reifsnider (1992); f Camacho et al. (1990); 8 Blumentritt et al. (1974); h Kacir et al. (1978 a); J McCullough et al. (1983); j Eduljee (1991); k Bozarth et al. (1987).

9.1 Introduction

materials of equivalent composition. Short fiber composites are also discussed in Chapter 7 of this volume. The wide range of possible combinations of materials, concentration and microstructures available to composite materials precludes a total reliance on direct experimental characterization to obtain essential design data. This situation provides a strong practical motivation for the development of relationships which can be used to predict and design materials performance in terms of (1) the properties of the components, (2) the concentration (e.g., volume fraction) of components and (3) the internal microstructure. Indeed, such relationships are emerging as an essential component of the technology of composite materials. A brief discussion on micromechanics can also be found in Chapter 3 of this volume. The central problem in the development of such micromechanical models is the specification of an internal stress (or strain) field consistent with the external field imposed on the macroscopic body. The load deformation field acting on a homogenous macroscopic body can be obtained from the specification of the load-deformation pattern imposed on the boundaries of the body. The resulting internal surface tractions and deformations acting on the internal boundaries of the microscopic phase regions are not so readily obtained. The internal stress-strain fields of a heterogeneous system are locally influenced by (i) the relative magnitudes of the properties of the components, (ii) the size, shape and relative orientation of the phase regions, and (iii) the packing geometry of the phase regions. The general strategy for dealing with such complex features of heterogeneous materials is based on the notion of "effective" or "equivalent" homogeneity. This

387

approach presumes that a length scale exists over which the variation in properties of the fiber and matrix can be appropriately averaged. This length scale must be large enough to capture the important structural fluctuations yet smaller than the dimensions of the macroscopic body. Under this approach, the heterogeneous material is replaced by an effective homogeneous material that exhibits the appropriately averaged properties of the heterogeneous system. This continuum hypothesis provides the basis for the utilization of the powerful tools of continuum elasticity theory. The selection of the appropriate scale for the application of the continuum hypothesis is important in composite materials since a rich structural hierarchy is revealed at different levels of magnification. At the low levels of magnification associated with lamination theory, a macroscopic body is subdivided into apparent layers with each layer treated as a homogeneous (but anisotropic) material The internal structure of the individual layers is ignored; the averaged properties of an individual layer (or laminae) are frequently determined by direct experimental characterizations. At a higher level of magnification, the heterogeneous nature of the individual layer is revealed. The reinforcing fibers are usually provided in the form of tows (or bundles) or individual filaments; the remnants of this original structure are frequently manifest as fluctuations in the packing geometry and packing density of the filaments. Layered structures are also encountered in injection molded shortfiber composites. In this case the non-uniform flow fields within the mold cavity establish varying fiber orientations that give rise to "skin-core" layers. At this scale, micromechanical models which take

388

9 Elastic Properties of Composites

into account the properties and concentrations of components and the microstructure provide the basis for predicting the "homogeneous equivalent" properties to serve as input to the lamination models. At yet a higher level of magnification, the heterogeneous nature of individual fibers and matrix can be revealed. For example, carbon fibers are heterogeneous materials comprised of rigid crystalline graphitic inclusions embedded in an amorphous carbon matrix; organic aramid fibers are similarly comprised of rigid crystalline and compliant amorphous components. Semi-crystalline thermoplastic polymer matrices are comprised of various "spherulitic", "lamellar", and "fibrillar" crystalline morphologies embedded in an amorphous component. A focus on this reduced scale takes into account inhomogeneities in the internal structure of the reinforcing agents and matrix, as well as the possibility of a perturbed interphase region near the surface of the reinforcing agent. The nature of the interphase region can be dependent upon the chemical composition of the reinforcing agent and matrix, as well as coupling agents and additives. The behavior of the interphase region is expected to be sensitive to concentration and thermal gradients encountered during processing. The nature, extent, and influence of the interphase region is not well understood. Both the reinforcing components and the matrix phase may be broadly distinguished as either "glassy" or crystalline materials. For example, glass filaments, thermosetting resins and ceramics may be viewed as glassy materials; carbon fibers, aramid fibers, engineering structural thermoplastic, and metals may be classified as crystalline materials. On this reduced scale, crystalline materials may be viewed as a "microcomposites"

comprised of homogeneous reinforcing agents associated with ordered "crystalline aggregates" embedded in a homogeneous matrix and assigned properties which reflect the behavior of the less ordered noncrystalline material (see Chapter 7, Volume 12). Under this abstraction, the macrocomposite models for the short-fibers, particles, and platelets may be applied with the shape and aspect ratio parameters selected as appropriate for particular observed morphologies (e.g., fibrillar, spherulitic, or lamellar; McCullough, 1977). This analogy includes the class of "self-reinforcing" or "molecular" composites in which highly extended fibrillar morphologies assume the role of a reinforcing agent embedded in a chemically identical, but physically distinct, matrix. The next lower level of organization is envisioned as a network system. The nodal points of the network are associated with chemical crosslinks, physical entanglements, and/or regions of localized order stabilized by intermolecular interactions. The links between nodal points are associated with the flexible character of weakly interacting disordered structures. The mechanical analog is a three-dimensional collection of springs subject to varying constraints at the nodal points. The influence of the constraint is dependent upon the nature of the nodal pinpoints; the behavior of the intervening links reflects the flexibility and mobility of intervening molecular segments. This analogy provides a representation of the internal structure of "glassy" materials, as well as the internal structure of the morphological and non-crystalline components of crystallitic materials. Network models have a long and rich history in polymer physics. Indeed, network models provide the foundations for the theory of rubber-like elasticity, one of the major accomplishments of polymer phys-

9.1 Introduction

ics. The application of network theory to ideal rubbers and swollen systems has been quite successful. Unfortunately, the theory in its current form contains deficiencies which make it inapplicable to glassy and noncrystalline materials. As a consequence, the elastic behavior of "glassy" materials is usually determined by direct experimental characterization of the bulk material. Further magnification reveals the unit cell structure of the crystalline morphologies and the molecular structure of the amorphous components. At this structural level the continuum mechanics approach employed at the preceding levels must be replaced by "lattice dynamics" and "molecular mechanics". These atomistic based approaches provided the link between mechanical behavior and molecular structure. It is well accepted that the physical properties of polymeric materials have their origin in the interactions within and between molecules. Continuum properties such as stiffness, strength, damping, etc., are gross descriptions of response characteristics which are the consequence of the reaction of molecules to external actions such as an external force, a change in temperature, etc. These reactions are manifest as changes in the structure and packing of the molecules to relieve or accommodate the imposed actions. In principle, a knowledge of the energy associated with a manifold of structures and structural conversions should suffice for a description and/or prediction of these fundamental molecular response mechanisms. In practice, the rigorous assessment of the energetics of polymeric solids can be untenable. The structural variations and the number of important intra- and intermolecular interactions can be so extensive that even with the use of high-speed computers an exhaustive con-

389

sideration of the manifold structures and structural conversions is impractical. However, continued advances in numerical techniques of multivariable optimization offer the potential of a significant reduction in computational effort. At the present time, the elastic behavior is determined by direct experimental characterization of the bulk material. The various elements of the structural hierarchy are summarized in Fig. 9-1. Structures at the supramolecular microsystem level (and below, bottom half of Fig. 9-1) may have an important influence on ultimate properties such as strength, fracture toughness, fatigue life, etc. Properties belonging to this important group are sensitive to the "weak links" generated by small scale variations in structures and properties. These local variations tend to be submerged by the averaging processes associated with the description of the effective elastic response. As a consequence, structural features in the micron to centimeter range (top half of Fig. 9-1) are sufficient to account for the elastic response characteristics of composite materials. At this scale, the microstructure is dominated by the geometrical features of the reinforcing agents. The periodic structure associated with woven and braided architectures permits a formulation based on precisely specified geometries. The elastic behavior of these systems has been treated by Chou and Ko (1989) and will not be considered here. Aboudi (1983) has treated continuous and discontinuous composites as triply periodic arrays of rectangular fibers. The treatment presented here will focus on systems in which the microstructure must be characterized in terms of distributions rather than precise geometries. An important problem arises in specifying the appropriate features of the distribu-

390

MACRO-COMPOSITES REINFORCING AGENT

J-"V1

FIBERS

FILLER

• Cont. Bundles • Textile Weaves • Short Fiber Aggr.

• Particulate • Platelet • Irregular

MATRIX

X

FILAMENT • Packing •Twisting

JL

UNIT GEOMETRY • Aspect Ratio • Orientation

• Shape • Packing

UNIT STRUCTURE

MATRIX STRUCTURE

INTERPHASE • Crystalline • Glassy • Other

• Crystalline • Glassy

• Crystalline • Glassy

MICROSYSTEMS

GLASSY SYSTEMS

CRYSTALLINE SYSTEMS

f

MICRO-COMPOSITE

J.

REINFORCING MORPHOLOGY

AHF

• Fibrillar • Spherulitic •Lamellar

I

<+***•++++Mr

TRANS. CRYST. MATERIALS CRYSTALLINE MATERIALS

**+++++**+J

1 NETWORK COMPOSITE

W»s G • Entanglements •Crosslinks •Local Order

1

I

Figure 9-1. Various elements of the structural hierarchy in composite materials.

9.1 Introduction

tion of microstructures to serve as the dominant structural descriptors. A focus on the average response characteristics of a heterogeneous material will submerge localized variations in the microstructure and associated internal fields so that any distribution of these quantities is replaced by statistically equivalent averages and/or higher moments of the structural distributions. Although of lesser importance for continuous fiber-reinforced composites, the identification of proper statistical descriptors, which are accessible to experimental evaluation, becomes a critical issue for short-fiber reinforced composites. The emphasis of this chapter is on the identification of the appropriate statistical measures of structural distributions. It can be anticipated that certain moments of the aspect ratio and orientation distributions of the reinforcing short fibers will emerge as parameters in micromechanical models which relate the composition and processing dependent microstructural features to the elastic properties. It will be shown in subsequent sections that the dominant geometrical features of the microstructure may be quantitatively described by (i) orientation parameters, / , which characterize the average state of orientation of the reinforcing agent and (ii) an

TYPE

SYMMETRY

391

effective aspect ratio, aef, which reflects the role of the average size and shape of the reinforcing agent. The specific roles that these structural parameters play in micromechanical models will be presented in a subsequent section. For the present purpose, it is sufficient to note that the orientation parameter, / , assumes a value of "0" for three-dimensionally random orientations and " 1 " for perfect collimation of the fibers; states of partial orientation are indicated by intermediate values of / . The effective aspect ratio can range from a value approaching zero for plate-like inclusions, 1 for spherical inclusions, to a value approaching infinity for continuous fibers. Discontinuous fibers fall in the range from one to infinity. As illustrated in Fig. 9-2, various types of composite materials may be distinguished as special cases of these structural features. This comparison emphasizes the simplicity of the micro-structure of continuous fiber composites. For this special material system, the continuity (aef = oo) of the perfectly collimated fibers ( / = 1) assures the simplifying condition that the strain field parallel to the aligned fibers is essentially uniform. As a consequence of the special condition, the longitudinal properties follow the simple rule of mixtures. The

MICROSTRUCTURE

Orientation: Aligned (f=1) Aspect Ratio: a ef = 1 Orientation: Independent Othotropic, 1 < Aspect Ratio < <*> (aef) Transversely Planar Random Orientation Isotropic (f=0), Aligned (f=1) Injection/ Transfer Molded Materials

Othotropic, Transversely Isotropic, Isotropic

I < Aspect Ratio < <*> (aef) Random Orientation (f=0), Aligned (f=1)

Figure 9-2. Schematic definition of the structural features that distinguish various classes of composite materials.

392

9 Elastic Properties of Composites

variation of the stress (and strain) fields transverse to the fiber direction present a major difficulty in the development of micro-models for the prediction of transverse properties and shear moduli. Particulate filled systems represent the next level of complexity beyond continuous fibers. In these systems, the reinforcing particles are spherical (or near spherical) so that the effective aspect ratio is unity. The marked discontinuity of particulate filled systems introduces significant fluctuations in the internal stress-strain field which complicate the analysis of mechanical behavior. On the other hand, the simple spherical reinforcing geometry precludes a dependence on orientation so that consideration of this structural feature is not required in the analysis of these systems. Discontinuous fiber reinforced materials represent yet a higher level of complexity since variable aspect ratios, shape and fiber orientation must be taken into consideration in the development of micro-models. The processing conditions used to fabricate short-fiber composites can cause fiber fragmentation resulting in a distribution of fiber lengths and a variety of end shapes which influence the capacity of the fiber to accept load transfer from the matrix. Varying flow fields carry the fibers into complicated states of orientation. The discontinuous fiber systems can be further distinguished according to the orientations induced by fabrication. Sheet molding compound (SMC) is produced by dropping chopped glass rovings (of approximately 1 to 2 inches [2.5-5 cm] in length) onto a moving belt where the rovings are impregnated with the uncured resin. The material is usually used in compression molding to produce a composite structure. Under these conditions, the fibers are essentially confined to lie within a plane but may take on various orienta-

tions within the plane. In this case a twodimensional "planar" orientation distribution can be employed to describe the state of orientation. Three-dimensional states of orientation can be generated in the core of injection molded or transfer molded materials if the mold cavity is significantly larger than the fiber length. The flow fields within the mold cavity carry the shortfibers into a three-dimensional orientation with "out-of-plane" tilting of thefiberspermitted. In addition, two distinct textures have been observed for discontinuous fiber systems: dispersed and aggregated. These textures are illustrated schematically in Fig. 9-3. The dispersed texture is associated with injection molded components in which the milling action of the injection process fully disperses the individual filaments from the bundles within the rovings or tows (usually ~10 4 filaments). These processing methods require fibers of substantially shorter lengths (1/8 to 1/4 inch

Dispersed texture (e.g. injection molded composites)

Aggregate texture exhibiting a domain structure (e.g. sheet molding compounds) Figure 9-3. The two distinct textures of discontinuous fiber systems.

9.2 Elements of Anisotropic Elasticity

[0.3-0.6 cm]) which may become fragmented during processing to produce a broad distribution of aspect ratios. The domain structure of the aggregate texture is observed in compression molded sheet molding compounds. In this case the integrity of the initial bundle of filaments tends to be preserved so that aggregated bundles of filaments can be readily distinguished. Such aggregates of close packed filaments tend to behave as a coherent unit. The variety of structures encountered in discontinuous fiber systems present a major challenge to the development of micromechanical models. It is clear that such models should reduce to established micromechanical models for the special case of fully aligned ( / = 1) continuous fibers (
393

to distinguish dispersed and aggregate textures are given. The subsection on configurational averaging provides the key relationships that link elastic behavior to the appropriate statistical descriptors of the orientation and aspect ratio distribution. The relationships developed in Sec. 9.5 are used in Sec. 9.6 to illustrate the role that the microstructure plays in determining elastic behavior. Approximations are introduced to give relatively simple relationships for the prediction of elastic behavior. Comparisons between predicted and measured elastic moduli emphasize the importance of the orientation distribution, the aspect ratio distribution and the dispersed or aggregate texture of discontinuous fiber systems. The final section provides a summary of the key microstructural features that are required to estimate the elastic constants of composite materials.

9.2 Elements of Anisotropic Elasticity Attention will be directed in this section to the review of key concepts and associated relationships to (i) specify the minimum number and kinds of material descriptors required to specify the initial load-deformation characteristics of various classes of anisotropic materials and (ii) describe the load-deformation response characteristics along an arbitrary direction in terms of these unique material descriptors. The relationships developed in this section serve as the basis of the analysis of elastic behavior developed in subsequent sections. It is common practice to distinguish between two types of deformations: elongation (or contraction) and shear. The elongation of a body involves a change in dimensions without a change in shape; shear

394

9 Elastic Properties of Composites

deformation is associated with a change in shape without a change in volume. Thus, as illustrated in Fig. 9-4, a rectangular body of original dimensions L o and Ao will undergo an elongation to a length L (and area A) when subjected to a tensile loading of force F. In order to emphasize the intrinsic properties of the material, the absolute dimensions of the specimen are removed by considering the force per unit area (the stress) and the relative deformation (the strain). Accordingly, the load is described by a tensile stress and the deformation by a tensile strain: Tensile stress: a = F/A Tensile strain: s = (L — Lo)/Lo

(9-1 a) (9-1 b)

For the case of a shear deformation, a force applied to parallel faces of a sample results in the deformation of a rectangle into a parallelogram. The change in angle (A/?) associated with a shear deformation is illustrated in Fig. 9-5.

Again, the intrinsic properties of the material may be emphasized by considering the force acting over a unit area of the face and the relative deformation: Shear stress: T = F/A Shear strain: y = sin(A^) « Ap

(9-2 a) (9-2 b)

A geometrical interpretation of strain can be obtained through the consideration of vector relationships which describe the state of deformation. These relationships are illustrated for a two-dimensional case. The original shape of a face of the body can be specified by two orthogonal vectors: (9-3)

R°22e2

where the unit vector e1 and e2 specify a cartesian reference frame. The deformed body is characterized by R'2 =

AR21e1+(R°22

For a unit cube such that R®l=Ri: then ARX = R\ - R<1 = AR1X e1+AR12

e2

AR2 = R'2 -R°2 = AR21 el + AR22 e2

(9-5)

For a pure tensile deformation with no change in the shape of the body (A/3 = 0), ARl2 = AR21 = 0, so that the tensile strains are given by Figure 9-4. Elongation under tensile load.

= AR11 = AR2 2

(9-6)

For a pure shear deformation AR±1 = AR22 = 0, so that A/J « sin AjS - COS(TI/2 - Aj8) =

(9-7 a) Thus for y = A/?, the "engineering" shear strain is given by Figure 9-5. Deformation under shear load.

= AP = AR 12

AR 21

(9-7 b)

9.2 Elements of Anisotropic Elasticity

395

a) shear deformation characterized by change in face angle

Figure 9-6. Alternative description of deformation resulting from shear loads.

b) shear deformation characterized by change in length of the face diagonal

An alternate geometrical description of shear strain may be developed by directing attention to the change in length of the diagonal element as illustrated in Fig. 9-6. The vector along the diagonal is given by Rd = R\

(9-8)

R2

so that (9-9 a) and |AR d \ 2 = {ARX +AR2)• (AR, +AR2) (9-9b) For small deformations such that (ARij)2 = 09 the pure shear condition (ARli = AR22 = 0) is described by the change in length of the diagonal by AR

(9-10)

but R^ = Hence,

for a face of a unit cube. )

(9-11 a)

Thus the "diagonal strain" is related to the "engineering" strain by ARd =

1

(9-1 l b )

The corresponding projection of the force F causing the change in the shape of the body provides a relationship between the "diagonal stress", / d , and the shear stress, T; viz.

/ d = V2z

(9-12)

These geometrical interpretations of strain evolved as the historical basis for describing strain prior to the development of tensor analysis. A more general basis for describing non-uniform infinitesimal strain has been developed through the consideration of displacement functions Ui(xux2,x3\ with i = 1, 2, 3. In this notation, the subscript index refers to one of the three directions in a Cartesian coordinate system. In terms of these displacement functions, the components of a strain tensor are defined as 1 /dut

(9-13)

Comparison of these definitions of strain with the geometrical definitions shows that the tensile strains are mutually consistent. However, the shear strains differ by a constant factor. The engineering or tangential components of shear strain are designated by the symbol y and are related to the ten-

396

9 Elastic Properties of Composites

sorial shear strains by a factor of 2; viz., (9-14) If the shear strain described in terms of the extension of the diagonal in the i,j face is designated dij9 then (9-15) The observation given by Eq. (9-15) will become useful in describing transformation characteristics of the strain tensor. In order to generalize the loading of a material, it is convenient to consider a unit cube of a material so that an applied force may be treated as a stress (i.e., the force per unit area). Directional effects are treated by constructing a Cartesian coordinate system with axes parallel to the edges of the cube, as shown in Fig. 9-7. It will be useful to label these axes as " 1 " , "2" and "3" rather than the traditional x, y and z axes. Consider an arbitrary traction F1 acting against the faces of the material cube which are perpendicular to the 1 axis. The vector Fx could be inclined at some angle with respect to the face of the cube of material. The force can be resolved into components along the 1,2 and 3 axes so that the loading condition could be described F1 = a11e1 + (T12e2 + (T13e3, where the e^ are the unit vectors along the axes 0' = 1, 2, 3) and the Gtj are the resolved components of F1 along the axes. Arbitrary forces acting against the faces perpendicular to the 2 axis and/or the

Figure 9-7. Definition of stresses and resultant forces.

3 axis could be similarly resolved. Indeed, any arbitrary combination of the forces could be specified by the following collection of equations: F2 = G21e1 + G22 e2 + G23 e3

(9-16 a)

This collection of three equations can be compactly represented by the relationship Ft=

(9-16 b)

Alternately, the collection of three equations can be conveniently represented in a matrix format as

(9-16 c)

or symbolically as F = ae

(9-16 d)

The symbol F represents the column of F/s9 and the symbol a represents the twodimensional table of the Gtj. The symbol e represents the column of e/s. Juxtaposition of symbols such as ae implies that the elements of crand e are to be multiplied (in the matrix sense) so as to generate the three equations. The two-dimensional array, a; is called the stress tensor; the nine entries in the table, aij9 are called the components of the stress tensor. The first subscript of the stress component, aij9 indicates the direction normal to the area on which the stress component acts, and the second subscript indicates the direction of the resolved component of stress. Thus, a normal stress is indicated by repeated subscripts (e.g., GX x, °"229 033X a n d a shear stress is indicated by mixed subscripts (e.g., a129 a 13 , a23). It should be noted that if the diagonal exten-

397

9.2 Elements of Anisotropic Elasticity

sion, dtj, is used to describe strain, the associated projection of stress, fij9 along the direction of the diagonal is fa = J2a{j

(9-17)

The general deformations of a cube of material (Al9 A2, and A3) can be described in a similar fashion in matrix format; viz., A = ee

(9-18)

The quantity € is called the "strain tensor". The nine components of the strain tensor are the quantities s-7- given in Eq. (9-13). The subscripts have an analogous meaning to those of the components of the stress tensor; consequently, components with repeated indices represent elongational deformations (e.g., s11? s 22 , 833), whereas components with mixed indices represent shear deformations (e.g., e12, £13, £23)Both the stress and strain tensors are symmetric with atj =
"strain" should be proportional. Hooke's law requires that the body recover completely from a deformation. If the force is removed (cr = O), the body returns to its original dimensions, e = 0, so that AL = 0. Solid bodies which follow Hooke's law for all loads and deformations are called perfectly elastic solids. Real solids tend to obey Hooke's law over only a limited range of deformations. The significance of this observation can be readily appreciated by considering a Maclaurin's series expansion of the energy density (i.e., the energy per unit volume) of the system in terms of the strains U = Uo

B

12

e12

B

13

e13

12 + 2

(9-19a) In the compact summation format, the series expansion takes on the form (9-19b) 3

9.2.1 Generalized Hooke's Law The intrinsic load-deformation characteristics of a material are established by specifying a relationship between the load (the stress) and the relative deformation (the strain). Traditionally, this relationship has been expressed as Hooke's law. Hooke's law asserts that the force required to achieve a deformation is directly proportional to the extent of the deformation. The mechanical analog frequently used to illustrate this principle is a simple spring. If a load F is attached to a spring, the spring elongates by an amount Ax. The force, F, and elongation, Ax, are related by a proportionality constant, k. The proportionality constant k is a parameter characteristic of the spring. By analogy to Hooke's law for a simple spring, the "stress" and

3

3

3

I I E E Ci i = l

j = l

k = l

1 = 1

with Bij = dU/deij

(ate o . = 0)

(9-19 c) (9-19 d)

For small deformations, terms involving e3 and higher powers may be neglected. At equilibrium, U = Umin and £,7 = 0 (the energy is at minimum). Since stress = force/unit area, cri;- = dU/ dEtj. Hence, 3

S t = i

3

X Cijkiski

(9-20)

i = i

Thus a four-index quantity (i.e., a fourth order tensor) is required to describe the general load-deformation characteristics of

398

9 Elastic Properties of Composites

an elastic material. The Cijkl represent the elements of the elastic constant tensor and are related in a fundamental way to the energy of the material. The energy content is in turn closely associated with the internal microstructure of the material. This observation provides a valuable tool for the prediction of values for the various Cijkl in terms of the composition of the material and the microstructure. The fundamental relationship shown in Eq. (9-20) is an extension of Hooke's law to general materials; the approximation that stress is proportional to strain(s) is maintained. The indices i, h K and / can each take on values of 1, 2, and 3. This implies that 3 4 = 81 parameters (e.g., C l l l l 9 C 1 1 1 2 , C 1 1 2 i, etc.) could be required to describe the general load-deformation response characteristics of a material. However, both the components of the stress and strain tensors are symmetric; viz. aij = aji and 8kl = slk. Thus it follows that Cijkl = Cm = Cjiki = Cjuk s o ^ a t a maximum of only thirty-six of the possible eighty-one parameters could be independent. The symmetry in the stress and strain tensors can be further exploited to simplify the generalized expression for Hooke's law. Due to the symmetry that atj = ajt, only six independent components can appear in the stress tensor. These six independent components of stress and strain can be "contracted" to a single index notation by the following standard convention: O' = l> 2, 3) T

2 3 = <*23 =

723 = 2^23 =

stant tensor may be contracted to a twoindex notation by the application of the following standard conventions: C(ij){kl) -+CipHq)

where (ii) -> (i) for i = 1, 2,

or 3; [(23),(32)H(4); [(31),(13)]->(5); and [(12),(21)]->(6). For example, C^^-^C^, C n 2 2 - > C 1 2 , and C 1 1 1 2 ->C 1 6 . Following this convention, the generalized Hooke's law relationship between the elements of the stress and strain tensor (represented as six element column vectors) can be compactly written in matrix notation as a = CG

(9-22 a)

where C is a six-by-six matrix array of the contracted elements, Cpq, called the stiffness coefficients. An expanded form of the matrix notation is

("A

'12 ^2

'-'21 '-'22 C 23 C 2 4


C31

C

32 C 3 3 C 3 4

c25 c 2 6 c 35 c 36

e2 £3

c 41 c 4 2 C 4 3 C 44 C45 C46 c c C 5 3 C 5 4 c 55 C 5 6 5 1 '-'52 k / '^•61 ^ 6 2 c63 c 65 c / <74

U

e4 £

5

(9-22 b)

This matrix relationship represents six equations (/ = 1, 2,..., 6) of the form 3

The number of constants required to describe the behavior of the material may be further reduced by considering the energy density function expressed in the contracted form 1

(9-23 a) (9-21) This contraction of the stress and strain tensors to vectors (with six components) means that the fourth order elastic con-

or alternatively (9-23 b)

9.2 Elements of Anisotropic Elasticity

maintain a specific state of deformation

so that (i = l, 2 , . . . , 6)

then the stiffness matrix C of must be symmetric: Ctj = Cjt. be illustrated by differentiating and substituting the results into which yields + CjJsj j=1

(i = l,2

(9-24)

Eq. (9-22) This can Eq. (9-23) Eq. (9-24),

6) (9-25)

Equations (9-25) and (9-22 c) can be made compatible only if the elastic constant matrix is symmetric, i.e., Ctj = C^Uj = 1, 2,..., 6). Thus, the number of independent elastic constants is reduced from thirty-six to twenty-one. As indicated previously, it is frequently useful to express the strain in terms of the stress: £ = S(T

(9-26)

where S is the compliance tensor comprised of the 36 elements S^(i = 1, 2,..., 6; j = 1, 2,..., 6). The quantity S is the inverse of C in the matrix sense S = C'1 so that CS = 1

399

(9-27)

For this relationship, the unit matrix, 1, is a six-by-six matrix with 1 on the diagonals and zeros elsewhere. Standard techniques are available for executing such matrix inversions. It should be evident that if the elastic constant tensor C is symmetric, then its inverse S should also be symmetric (i.e., Stj = Sji). Since the elastic constants and compliance constants are related, either collection can serve as generalized material descriptors. The compliance constants are used to describe the ability of a material to transform loads into deformations (g = S
These results imply that, in general, twenty-one separate experimental measurements would be required to characterize the load-deformation response characteristics of a material. Fortunately, very few materials of practical interest require twenty-one unique material descriptors to specify the load-deformation response characteristics of the material. Materials symmetry induced by the nature of the microstructure can dramatically reduce the number of independent parameters required to describe the behavior of materials. An inspection of common forms of microstructures will show that if the material undergoes certain specific rotations, the rotated material is indistinguishable from the original one. Under these circumstances, the material descriptions should be identical to the original set. In order for the rotated and unrotated sets of descriptors to be equivalent for these special orientations, certain of the Ctfs must vanish or become related to combinations of the other Ct/s. Orthotropic symmetry is usually the lowest symmetry exhibited by practical composite materials. The structure of the elastic constant array associated with orthotropic symmetry is summarized in Fig. 9-8; nine independent material descriptors are required to completely specify the mechanical response characteristic of this symmetry class. Transversely isotropic symmetry is the next higher symmetry class. The general structure of the orthotropic array is preserved for this symmetry class; however, the special conditions listed in the figure reduce the number of independent descriptors to five. Special relationships are given for two choices of unique axis: the 1 and 3 axes. For this material class, the properties in a plane perpendicular to the unique axis are independent of

400

9 Elastic Properties of Composites Orthotropic Materials

Special Conditions

Transverse Isotropy in 1-2 Plane

Transverse Isotropy in 2-3 Plane

C11 = C 2 2 2 3 = C13

^22 = £33 C 1 2 = C13

£44 = ^55

C44=(1/2)(C22.C21)

\ ' ^/\

^55=^66

C

66

11" 22 /

o

Isotropic Materials £11= c22=

cZ3

c12= c 1 3 = c23 C 4 4 = C 5 5 = C 6 6 = (1/2)(C

r

C12)

Figure 9-8. Materials symmetry.

the direction within the plane. In order to emphasize the fact that only five independent parameters are required, certain "convenience moduli" are introduced.

^44

pliance constant array can be simply obtained by replacing the C{j with Stj. However, as a consequence of the difference in the standard contraction rules, the special conditions on compliance constants for isotropic and transversely isotropic materials must be generated from the conditions given in Fig. 9-8 by the substitution Q, -> - ^ -

=

(9-28) And, finally, isotropic symmetry is exhibited by many important materials. Again, the structure of the elastic constant array is preserved, but the special conditions listed in Fig. 9-8 reduce the number of independent material descriptors to two. No unique material axis exists for isotropic materials, i.e., properties are independent of direction. Equivalent reductions can be obtained for the compliance constant array. Indeed, the structure of the corresponding com-

(9-29)

with mp = l forp = 1, 2, 3 and mp = 2 for p = 4, 5, 6. Traditionally, material response has been characterized by "engineering constants" rather than the elements of the elastic constant or compliance constant array. According to custom, Young's modulus, E, is used to describe the ability of a material to transfer a pure extensional strain into a pure extensional stress; Poisson's ratio, v, is used to indicate the extent to which the lateral dimensions of a body decrease (or increase) in response to a pure extensional

9.2 Elements of Anisotropic Elasticity

(or compressional) strain; and the shear modulus, G, is used to describe the ability of a material to transfer pure shear strain into a pure shear stress. In addition, the bulk modulus, X, is used to describe the deformation of a body subject to a uniform hydrostatic pressure. These engineering constants are the conventional quantities reported as a result of characterization experiments. Specific engineering constants are required to characterize the behavior associated with each unique direction of an isotropic material. For example, a continuous fiber, unidirectional laminate would be expected to exhibit a longitudinal Young's modulus (in a direction parallel to the reinforcing fibers), EL(=E11) that is different from the Young's modulus transverse to the fibers, ET(=E22). Furthermore, the Young's modulus perpendicular to the laminate, Ep (= £ 33 ), could differ from both EL and £ T . Similarly, three distinct shear moduli could be expected. These quantities are schematically defined in Fig. 9-9. The engineering constants £ L , ET, £ p , ^125 G13, G23 are readily interpreted in terms of response characteristics of a material; the theoretical descriptors (Cfj- or Stj)

401

are more amenable to analysis. Consequently, it is necessary to transfer back and forth among these various sets of descriptors of material behavior. The purpose of the following is to relate these alternate descriptions of material behavior. The relationships will be developed for a material with orthotropic symmetry. The relationships for materials with higher symmetry (transverse isotropy and complete isotropy) can be obtained as special cases which result from the application of special symmetry conditions. The usual expression of Hooke's law in terms of the elastic constants, a = C s, suggests that the strain s is the specified quantity so that the stress G is the dependent variable. In most practical situations, the converse is true: stress is specified and strain is to be determined. Consequently, the compliance form of Hooke's law reflects the experimental situation normally encountered; viz., s = S
Figure 9-9. Definition of moduli for orthotropic materials.

402

9 Elastic Properties of Composites

with

1

"f3-

£1 ^12 —

£2

£, — V

£3

1

c

^22

-

£22

c

C ^23 —

^32

_

V

32

£3

1 ^33 c

^33

1

O44

1

c

1

(9-30)

^66

with vtj = v^EJEj) with ij = 1, 2, 3. The elastic constants, Cij9 can be obtained through the matrix relationships: C = S~1. The results are summarized below.

1 x

=G

v 23 + £ 2 v12 v13)D

23

(9-31)

23

— V 12

As was the case for the Ctj and 5^, materials symmetry reduces the number of independent engineering constants required to describe the load-deformation response characteristics of a material. It should be noted that under the current formulation, the values that would be obtained for the Stj (or the Ctj) are dependent upon the assignment of the 1, 2, and 3 axes along the edges of an arbitrary cube of material. Thus, if a particular cube section had been selected from a unidirectional ply of continuous fiber composite such that the fibers were parallel to the 1 axis, the S1X compliance constant would represent the compliance of the material in the fiber direction. Alternately, if the cube had been selected so that the fibers were parallel to the 2 direction, the quantity Sx x would represent the compliance of the material in a direction perpendicular to the fibers. The values for S1X obtained under these two arbitrary assignments of the axes could differ by an order of magnitude. Indeed, the cube could have been selected so that the fibers were inclined at some arbitrary angle to the axes. In this case yet another value would be obtained for the S±1 term (as well as all other Stj). These considerations point out that the specific values obtained for the Stj (or Ctj) are dependent upon the particular association of the el9 e2, e3 body axes of the cube with unique directions of the material (e.g., the fiber direction). This conclusion could give the erroneous impression that characterization experiments must be conducted for all possible orientations of the material. Clearly, the fact that

9.2 Elements of Anisotropic Elasticity

the unique material directions have been rotated with respect to the body axes cannot alter the intrinsic behavior of the material. Consequently, the values for the material descriptors developed from material sampled at different orientations with respect to the body axes must be related to certain intrinsic materials descriptors. These relationships are developed in the next subsection. 9.2.2 Orientation Transformations

The approach to developing orientation relationships can be simply summarized. Two reference cubes of an equivalent material will be considered. In one cube unique material directions will be assigned to lie along the body axes. The body axes will in turn be used to describe a general collection of force vectors which specify the applied loads. In the second cube, the unique material directions will be allowed to assume some arbitrary orientation with respect to the forces applied which are equivalent to those acting on the first cube. The elastic constants for the two situations will be determined and compared. Since the same material is used in both situations, the comparison will yield the required relationships. A situation in which the body axes and material directions are coincident can be taken to correspond to the results developed in the preceding subsection. Hence, the stress tensor and strain tensor associated with the loading field, F, and the deformations, A can be stated as F = ae

(9-32)

and A = ee

(9-33)

where e is the common coordinate (e1? £ 2 ,£ 3 ).

403

Consider an alternate situation in which the material cube has some arbitrary orientation with respect to the axes set, e, so that vectors e'l9 e'2, and e3 (associated with certain material directions) make some angles with the axes e±, e2, e3. The vector forces, / i , F2 and F3 are maintained in the same directions with respect to el9 e2, e3, as in the first situation, so that the loading is equivalent for the two situations. Let the equivalent force field described in the e' coordinates be represented by F[, F2\ and F3. Consequently, the set F/, F2, and F3 will give a new stress tensor, a': F' = &er

(9-34)

Similarly, the decomposition of the deformations (expressed in the e' set) will give a new strain tensor, €'\ Af = ere'

(9-35)

In order to compare these situations, the quantities developed with reference to e' must be transformed to equivalent quantities in e. These relationships can be developed from the rotational transformation that takes e' to e, 0(0,0,^), so that e' = O(6,(f),\l/)e

(9-36)

The orientation of the axes and the elements of O(9,(f),il/) are given in Fig. 9-10. Different forms for the matrix could be obtained by using alternate sequences of rotations. Indeed, there are six ways to represent such general states of orientation and consequently six ways of defining orientational angles. Care must be exercised so that the set of angles used to describe orientation is specified and the corresponding form of O specified. The form given in Fig. 9-10 is the form customarily used in the field of physics (Goldstein, 1957). Since O1 = Ol so that OlO = 1, the elements of O must satisfy the following

404

9 Elastic Properties of Composites

Following similar procedures, the strain tensors can be related through an equivalent transformation

31

€=QX€ Q

(9-40 a)

which represents nine simultaneous equations of the form a>tq e'st s=i

cou = cos^ cos0-

^

{ p 9 q = 1 , 2 ,3)

(9-40b)

*=

The generalized Hooke's law relating a to e may be written in expanded form as

CL>12 = cosip s'mcj) + cosO cos0 sin ij)

- cos0 sin0 cos

ou= £ £ C H M e M (fc, / = 1, 2, 3) (9-41)

= cos^sinfl = sin0sin0

P

= i « = i

Substitution of this expression for crin Eq. (9-39 b) yields

Figure 9-10. Definition of Eulerian angles for orientation transformations.

E Z

E Z

p=l

(9-42)

q=l

Subsequent substitution for e in terms of e' yields relationships:


(9-37) k=l

'I

From Eq. (9-36), F = &Qe. But the load field F' is related to the original load field F by the same orientation transformation: F = QF. Consequently, QF = & Qe so that F = Q1 & Qe or since Q~x = Q\

(9-43)

l

3

3

3

Z

Z

Z

\*=1 *=1

3

l =ll pl= l

Z <*>ikjlspU>tklpq qC l

However, the generalized Hooke's law relating & to e! is of the form 3

3

ij = Z

Z

G

(9-44)

s = l r =

F = Qta/Qe

(9-38)

However, in the e frame F=ae. Comparison of these results (expressed in the common e axes system) shows that the stress tensors are related by a "similarity" (or "congruent") transform: & = Q<jQi

(9-39 a)

This matrix expression represents nine simultaneous equations of the form (U = U , 3 ) (9-39 b)

Comparison of these two equivalent expressions for & in terms of e' shows that 3

3

3

C'ijst= Z Z

3

Z

Z

k = l 1 = 1 p = l q = l

^ik^jiCOspCOtqCklpq (9-45)

This relationship provides a means of relating the elastic constants determined at a given state of orientation of the material to the elastic constants that would describe the load deformation behavior of the material at any other state of orientation of the material in the field.

405

9.2 Elements of Anisotropic Elasticity

Analogous expressions can be developed for the compliance constants: 3

S!jst=

Z

3

23_

2

3

Z Z Z

(9-46)

A more compact form may be obtained for the various orientation relationships by the use of the contracted notation. In the contracted notation, the stress transformation takes on the form o=To

(9-49) and *„ = ./;

(9-47) °23 =

where the er's are the one-by-six column matrices of the independent components of <j, and T is a six-by-six matrix. The elements of 7 can be obtained by comparing the expanded forms of the stress tensor, o; and the contracted stress "vector" a. The elements of T are given in Table 9-2. A similar expression is obtained for the strain vector, viz., sf = Ts

(9-48)

The six-by-six matrix 7 exhibits an inconvenient feature in that the transpose is not equal to the inverse. This situation may be remedied by introducing an alternative set of contraction rules for the components of the stress and strain tensor.

(i = l , 2 , 3 )

h

A G12

=

A

(9-50)

The quantities dtj and fijii^j) are associated with the description of strain and the associated stress projection in terms of the extension of the diagonal of the ij face. Under this set of contraction rules, the independent components of the strain tensor may be represented by a 1 x 6 vector, d, with components dt\ the stress is represented by a 1 x 6 vector,/, with components ft so that (9-51)

d' = Rd Table 9-2. Elements of the rotation matrix T.

co\2

^22

co12col3

co22ca23

^32

2co 2 2 cy 3 2

Ico,

2

2coi

2co12co22

co32co33 co31co12

»11%

406

9 Elastic Properties of Composites

The incorporation of the ^/2 factor into the transformation 7 yields a new 6 x 6 matrix, ft, that exhibits the useful property that ft"1 = ftl. The statement for Hooke's law for the new contracted stress and strain tensors becomes f =cd

trix with n11 = n22 = ^33 = 1; n44 = n55 = = n66 = 1/^/2; and all other components equal 0. The rotational transformation for the standard contracted vector is a' = 7 a; thus, n /T = 7nj?, so that (9-55) However, /T = R ft so that

or equivalently d = sf

T=nRnx (9-52)

The components of c and s are related to the fourth rank tensors through the alternative contraction rules

(9-56) 1

1

1

The quantity 7" = nR~ n~ Rl. From Eq. (9-56),

with R1 =

A T 1 = Rl = n 1 P i n ' 1 ) 1 = n T n

1

(9-57)

Substitution yields ijkl

?

. rijJ

9 53

( ~ ) Tl

with mfc = l for /c = l,2,3 and mk = s/l for k = 4, 5, 6. The rotational transformations of the Elastic Constant Matrix and the Compliance Constant Matrix is given by ' = RARl

(9-54)

with A = c or s. Although the mathematical simplicity obtained from the use of the alternative contraction rules has merit, the introduction of new definitions for the components of the contracted elastic constant and compliance constant arrays could lead to confusion. Consequently, we will retain the traditional definitions obtained from the standard contraction rules. However, the results obtained from the alternative contraction rules can be fruitfully utilized to obtain a simplification for the evaluation of T1. In order to obtain this simplification, we observe that the elements of the traditional contracted stress or strain vector (a = s or a) are related to the elements of the alternative stress or strain vector (ft = d or / ) by a = n ft where n is a diagonal 6 x 6 ma-

=N T x N

1

(9-58)

where N = nn with N11=N22 = N33 = 1; N44r = N55=N66 = l/2; and all other components equal 0. The relationship given in Eq. (9-58) yields a straightforward device for obtaining the elements of 7" 1 . In summary, orientation transformations are given by the following matrix relationships: a' = Ta e' = Ts S' = TST

1

(9-59)

9.3 Micromechanical Models for Continuous Fiber Composites The central problem of a micromechanical analysis of heterogeneous materials is the specification of an internal stress (or strain) field consistent with the external field imposed on the macroscopic body. These internal fields are locally influenced by (i) the relative magnitudes of the properties of the components; (ii) the size, shape, and relative orientation of the phase re-

9.3 Micromechanical Models for Continuous Fiber Composites

gions; and (iii) the packing geometry of the phase regions. A second and more subtle problem arises in specifying the appropriate features of the microstructure to serve as the dominant structural descriptors. Although of lesser importance for continuous fiber reinforced composites, the selection of proper statistical descriptors, which are accessible to experimental evaluation, becomes a major problem for short-fiber and particulate reinforced composites. As a consequence of the second problem, it will be useful to direct initial considerations to the simpler case of continuous fiber reinforced composites. The background developed in this initial treatment can be extended to develop general relationships which contain a wide variety of heterogeneous materials as special cases, viz., continuous fiber reinforced composites, short-fiber reinforced composites, particulate reinforced composites, platelet reinforced composites, as well as "grainy" metals and semicrystalline polymers. Attention can be further restricted to a unidirectional ply of continuous fiber reinforced materials. This unit serves as the basic building block of composites materials from the point of view of both manufacture and analysis. The laminates encountered in most practical applications of composite materials are fabricated by laying up several continuous fiber plies with the fiber directions of each ply at prescribed angles. In analysis, as in fabrication, the unidirectional ply is recognized as the basic structural element of a composite structure. Well-established methods of "laminate analysis" have been developed for predicting the behavior of complex composite structures in terms of the properties of the individual plies. The starting point for a laminate analysis is a set of properties for the unidirectional ply. Con-

407

sequently, the micromechanical models reviewed in this section provide the connecting link between the performance of a composite structure and the composition of the composite material. The unidirectional ply will exhibit different properties along and perpendicular to the fiber direction: properties along the fiber direction will more nearly reflect fiber properties, whereas properties perpendicular to the fiber direction will tend toward the properties of the matrix phase. Consequently, it is necessary to distinguish the various directions of the ply in terms of a set of coordinate axes. The 1 axis will be taken parallel to the fibers; the 2 axis will be taken within the plane of the ply and perpendicular to the fiber direction; and the 3 axis is taken perpendicular to the plane of the ply. The " 1 " index (used to denote the direction along the 1 axis) is frequently replaced by the symbol "L" to indicate the longitudinal direction (i.e., fiber direction) of the ply; the symbol "T" is used to denote the transverse direction within the plane of the ply (i.e., the "2" direction). A subscript "f" will be used to denote a property of the fiber (Pf = £ f , G f ,...), and the subscript "m" will be used to denote properties associated with the matrix phase (Pm = £ m , G m ,...). The matrix phase is usually isotropic so that only two independent parameters (e.g., £ m , Gm) are required to describe its mechanical behavior; however, the presence of filler may induce anisotropy in the matrix phase. As a further complication, the elastic properties of resins determined from castings and films may not always be the same as the resin properties in the composite system due to residual stress and other processing factors that can influence their in situ properties. Glass fibers are usually isotropic and hence require only two independent material descriptors (e.g.,

408

9 Elastic Properties of Composites

£ f and Gf). However, other fibers such as carbon fibers and organic fibers can exhibit transverse isotropy so that the properties perpendicular to the fiber axis differ from the properties along the fiber axis. In this casefiveindependent materials parameters would be required to describe the behavior of the fiber components. Unfortunately, transverse properties of fibers are difficult to measure so that transverse fiber properties often are not well characterized. In some instances the fiber cross sections may not be circular. However, if the ply has been manufactured under conditions such that the fibers have been randomly rotated about the fiber axis, the average effective cross section of the fiber can be taken as circular without introducing significant error in the prediction of properties. Even in the case of isotropic fiber and resin components, the number of independent parameters required to describe the behavior of the ply is dictated by the packing geometry of the fibers. Thus, if fibers are packed in a rectangular pattern, unidirectional composites will exhibit orthotropic materials symmetry and hence require nine independent parameters to describe the mechanical behavior of the ply. Consequently, nine relationships are required to predict the ply properties in terms of the properties and concentration of the components. In the event of the higher symmetry square packing, the number of independent relationships required to predict the behavior reduces to six. Both hexagonal (close) packing and random packing give a transversely isotropic material for which five independent parameters are required to describe the behavior. Hexagonal or random packing is the common geometry encountered in practical composites. In principle, the average elastic properties of a ply can be obtained by specifying

the details of the geometry (fiber shape, packing geometry, and spacing); the distribution of loads on the surfaces of the ply; and the connectivity between the fiber and matrix phases. The average bulk response can be determined by taking appropriate volume averages, thereby relating the volume fraction of the components and their respective properties to the average properties of the ply. In practice, either simplifying assumptions must be introduced to make an analysis tractable, or detailed numerical analyses must be performed for special cases. A wide range of relationships has evolved as the consequence of applying a variety of simplifying assumptions. The majority of the approaches share the following common assumptions: (i) The phase surfaces are assumed to be in direct contact and bonded (either chemically or physically) so that slip does not occur at a phase interface, and (ii) attention is directed toward the overall average response of the material to loads (or deformations). These assumptions are appropriate for material properties associated with small deformation behavior and hence do not seriously limit the application of the relationships to the prediction of elastic properties. Mathematical models for mechanical properties have been developed by employing assumptions concerning the nature of the internal stress-strain fields and/or the internal geometry. Assumptions concerning the explicit nature of the phase geometry are frequently justified on the grounds that detailed experimental data for size, shape, orientation, and spacing distributions are rarely available. The various approaches to predicting properties are distinguished by the nature of the assumptions introduced to obtain tractable solutions. The methods can be broadly grouped into the following categories: (1) numerical techniques, (2) mechanics of materials,

409

9.3 Micromechanical Models for Continuous Fiber Composites

(3) self-consistent field (or embedding), (4) bounding, and (5) semi-empirical. The numerical techniques rely heavily on series expansions and/or numerical analysis. A regular packing of the reinforcing phase is required so that considerations can be focused on a typical repeating unit of the system. The analysis of Adams and Tsai (1969) is illustrative of the basic features of this approach. The results from such numerical analyses are specific to the materials and geometries covered by the analysis. Such specialized studies yield limited insights toward quantitative generalizations and consequently will not be further explored. Nonetheless such work serves as a reference check for the simplifying assumptions used to develop "closed-form" algebraic relations. The mechanics of materials approach is the simplest and, consequently, the most commonly encountered method. The rule of mixtures relationships result from the mechanics of materials approach. In contrast to the mechanics of materials approach, the self-consistent field method introduces simplifications in the microstructure of the composite in order to deal in a more realistic fashion with the internal distribution of stress and strain. The simplification introduced in the description of the microstructure is of lesser significance for continuous fiber reinforced composites; however, it becomes particularly important for short-fiber and particulate reinforced composites. The bounding approach avoids the problems of specifying the explicit nature of the microstructure and internal stress-strain fields by employing variational principles to establish upper and lower bounds on the properties. The results from the application of a bounding approach provide practical guides to material behavior only if the upper and lower bounds are reasonably

close together so as to bracket the properties to within experimental error. The upper and lower bounds on longitudinal properties are very close and, indeed, converge to the simple results of the longitudinal rule of mixtures. Unfortunately, the upper and lower bounds on transverse properties and shear moduli are too far apart to provide useful brackets. These difficulties with the theoretical approaches, as well as the computational complexities introduced by some of the models, have prompted the development of semi-empirical models. Of the various semi-empirical models, the mixing rules (proposed by Halpin and Tsai, 1969) offer the greatest potential for providing useful predictive relationships subject to rather wide generalizations. In addition, the Halpin-Tsai relationship provides a convenient format for comparing the results from various theoretical models. 9.3.1 Mechanics of Materials Models

The simplest models for predicting elastic properties may be developed by treating response to mechanical loads by analogy to springs. The longitudinal response characteristics are treated as if the components of the ply react to the loads by parallel response. For mechanical loads, the central feature of this analogy is the assumption that each component is subject to the same strain. This analogy yields the following simple linear relationship: (9-60) i= 1

where P{ is any one of the pertinent mechanical properties of the ith component. The concentration terms, vt, are the volume fractions of the ith component given by vt = VJV

(9-61)

410

9 Elastic Properties of Composites

where Vt is the overall volume occupied by the ith component and V is the total volume of the ply. The volume fractions must satisfy the relationship

J>, = 1

(9-62)

For a two-component (n = 2) fiber/resin composite the relationship is of the form p * = vfP{ + vm Pm

(9-63)

with vf + vm = 1 so that vm = 1 — v{. If the property of the fiber, P f , is considerably greater than the corresponding property of the matrix, Pm, then the contribution of the matrix to the composite property, P*, is negligible so that the longitudinal properties of the composite are dominated by the fiber contributions, The transverse response of the ply to mechanical loads is modeled by considering that the components react to these loads as if they were connected in series. The central feature of this analogy is the assumption that each component is subject to the same stress. This analogy yields the simple reciprocal relationship (9-64)

P*

where, again, P{, is any one of the pertinent properties of the ith component and vt is the volume fraction of the ith component. For a two-component (n = 2) fiber/resin composite, the relation reduces to the form P*

Pf

Pm

(9-65)

In this case, if the fiber properties, P f , are considerably greater than the matrix properties, Pm, the term (vf/Pf) is negligible as compared to (vJPJ; hence, (1/P) « ( v J P J or P « (Pm/vm) so that the transverse prop-

erties are dominated by the contributions of the matrix phase. These simple relationships have become popularly known as the "Rule of Mixtures". It has been demonstrated that the rule of mixtures relationship for longitudinal properties (Eq. (9-63)) gives excellent agreement between the predicted and observed properties of continuous fiber composites. This result is not surprising in light of the internal structure of a continuous fiber reinforced ply: the continuity of the fiber assures that the strain field in a direction parallel to the fibers is essentially uniform, i.e., both the fiber and resin components are strained by the same amount. In contrast, the rule of mixtures relationship for transverse properties (and shear moduli) expressed in Eq. (9-64), gives a serious underestimate of the performance of composite materials - the transverse properties (and shear moduli) are always greater than those predicted by the transverse rule of mixtures relationship. This discrepancy is a direct consequence of the assumption of uniform stress. It is evident that the stress field transverse to the fiber direction must fluctuate between the fiber phase and the matrix phase. Consequently, the major difficulty in predicting the behavior of continuous fiber reinforced composites is associated with predicting the transverse properties and shear moduli. It can be anticipated that the situation for short-fiber and particulate reinforced composites will tend toward that encountered in dealing with the transverse properties; even if all the short fibers were perfectly aligned, the discontinuities of the fiber ends would give rise to fluctuations in the internal stress and strain fields parallel to the (aligned) fiber axis. Accordingly, improved models are required to obtain realistic predictions for the transverse properties of continuous fiber composites, as well as

9.3 Micromechanical Models for Continuous Fiber Composites

for the properties of short-fiber and particulate reinforced systems. Evidently, the transverse properties are sensitive to structural features ignored by the simple mechanics of materials approach. Several models have been proposed to compensate for the loss of structural features. In these models certain prescribed coupling configurations of the components are assumed to be subject to a uniform stress, while other configurations are assumed to be subject to uniform strains. Specification of particular configurations is rationalized in terms of the packing geometry. The abstraction employed by Shaffer (1964) to construct such combination models is typical of this approach. Under this model, the matrix phase is recognized to be intermittently distributed between the fiber cross sections contained within a column of material. Consequently, this portion of the matrix phase is assumed to be coupled in series with the fiber phase. The remaining portion of the matrix phase which separates the columns is assumed to react in parallel with the material of the column. The volume fractions of components in these series and coupled regions can be specified in terms of spacing parameters of the fiber packing and the fiber diameter. Consequently, this approach tends to distinguish among various packing geometries. Unfortunately, these decorations give only slightly improved results from the simple transverse rules of mixtures. In summary, the central feature of the mechanics of materials approach is the simplifying assumption of uniform stress or uniform strain. This simplification is retained in the elaborations of the "combination models". As pointed out by Hill (1964), none of the models based on uniform stress or uniform strain can be strictly correct: the interfacial forces are not in equilibrium for constant strain, while inter-

411

facial discontinuities must exist under the assumption of constant stress. Notwithstanding these limitations, these models provide useful physical insights and give reasonable guidelines for materials design with a minimum of computational effort. Furthermore, these models can be easily extended to multiphase materials. In spite of the lack of rigor, the simple rule of mixtures relationships adequately predict longitudinal properties of continuous fiber composites. Since these relationships do not explicitly contain terms associated with fiber shape, packing geometry, and fiber spacing, it is reasonable to conclude that these factors do not exert an appreciable influence on the longitudinal elastic properties of continuous fiber composites. According to these relationships, the influence of the fiber and matrix phases is directly proportional to their volume fraction concentrations and the appropriate properties of the components. If the properties of the fiber and matrix phases are appreciably different, then the component whose property is the largest dominates the longitudinal properties of the composite. In particular, the contribution of the modulus of the matrix phase to the modulus of high-performance composites is usually negligible so that the longitudinal modulus is independent of the choice of resin systems. In contrast, the shear moduli and transverse properties appear to be quite sensitive to the geometry of the fibers, fiber packing, and fiber spacings, as well as to the models and assumptions involved in deriving relationships between the composite properties and the properties of the components. The shear modulus and transverse properties of composites are strongly dependent upon the behavior of the matrix phase; hence, these properties

412

9 Elastic Properties of Composites

must be taken into consideration when selecting resin systems. In lieu of other relationships, the simple transverse rule of mixtures can be used to obtain conservative estimates. As will be shown, the effective properties of unidirectional composites will always be greater than the value predicted by this simple relationship. 9.3.2 Self-Consistent Field or Embedding Models

In order to overcome the approximations introduced by the assertion of uniform stress or uniform strain fields in the mechanics of materials models, several workers have developed methods to obtain improved descriptions of the internal stress-strain field by introducing gross simplifications in the phase geometry. The work of Hill (1965 a), Whitney and Riley (1966), Whitney (1967), and Hermans (1967) is particularly directed to continuous fiber reinforced composites. The principal abstractions used to simplify phase geometry are illustrated in Fig. 9-11. The "embedded" model illustrated in Fig. 9-11 a is the basis for the analysis reported by Hill (1965 a). Hill represented the phase geometry as a typical (average) fiber embedded in a medium whose properties were taken to be equivalent to the overall average (or "effective") properties of the

(a)

Singly embedded

(b)

Doubly embedded

Figure 9-11. Simplifications of the geometrical model for the self-consistent field approach.

composite material. An elasticity problem can be formulated for this simple geometry such that a "self-consistent" stress field can be identified and the effective properties of the medium determined. The results of Hill's analysis are not convenient from a computational point of view (e.g., the effective shear moduli must be extracted as roots of polynomials). Furthermore, Hill's results have been shown to be limited in application to systems of low concentration of fibers (Christensen, 1979). The "doubly embedded" model (Hermans, 1967; Whitney, 1967) illustrated in Fig. 9-11 b has proven to be more realistic and mathematically tractable. The doubly embedded model consists of a typical (average) fiber embedded in a microregion of the continuous matrix phase. This composite element is, in turn, considered to be embedded in a homogeneous medium. The homogeneous outer material is taken to be a transversely isotropic material whose properties are identical to the effective properties of the composite material. The volume fraction of the fiber in the concentric cylinder of matrix material is taken to be the same as the unidirectional composite. Specification of the volume fraction of fiber establishes the relative size of the matrix cylinder to the fiber cylinder. Upon assertion of a uniform strain field only in the longitudinal (fiber) direction, a self-consistent stress field may be obtained and the required relationships obtained for the effective composite properties. The central feature of these embedding models is the simplifying reduction of the microstructure to the system of embedded cylinders illustrated in Fig. 9-11 b. Under this simplification, the internal distributions of stress and strain can be dealt with in a more rigorous fashion than that used for the mechanics of materials approach. The basic formulation of the elasticity

9.3 Micromechanical Models for Continuous Fiber Composites

problem in terms of cylindrical orthotropy limits attention to composite materials which are transversely isotropic. Under this restriction, properties perpendicular to the plane of a ply are assumed to be the same as the transverse properties within the plane of the ply. Since the behavior of transversely isotropic materials can be described by any collection of five independent material descriptors, only five relationships are required to predict the behavior of the ply in terms of the corresponding properties of the components. It is mathematically convenient to use the plane-strain bulk modulus, feT, as one of the independent descriptors. The value for E ? 9 can be obtained from E22=-

1 • 4/cT

+

1 1 4G 23

(9-66) "12

Eli

The predictions, based on this doubly embedded model, for the plane strain bulk modulus, /cT, the longitudinal Young's modulus, £ 11? the Poisson's ratio, v 12 , and the shear modulus, G 12 , coincide with the improved lower bounds for arbitrary transverse phase geometry reported by Hashin (1965) and Hill (1964). The predictions for these four elastic constants also coincide with the results obtained from the composites cylinder assemblage model (Hashin and Rosen, 1964). In this geometrical model, infinitely long cylindrical fibers of radius a are embedded in a cylinder of continuous matrix phase with radius b. The ratio a/b is required to be constant for each cylinder; however, the absolute values of a and b are allowed to vary down in infinitesimal dimensions so that a volume filling assemblage is obtained. An exact solution for G23 from the composites cylinder assemblage model does not exist; however, upper and lower

413

bounds can be obtained for this particular geometry (Hashin and Rosen, 1964). The transverse shear modulus, G23, derived from Hermans' treatment of the doubly embedded model also coincides with the improved lower bound; however, Christensen (1979; Christensen and Lo, 1979) points out that an error is involved in this derivation. The correct form for G23 is given by Christensen (1979). The resulting expression takes on an involved analytical form that yields results that approach the improved lower bound results for both low and high concentration of fibers and marginally higher values of G23 for intermediate concentrations of fibers. Christensen recommends the use of the lower bound expression as a reasonable approximation for G23. It should be noted that terms related to variations in the cross-sectional shape of the fibers and specific fiber packing geometries (e.g., rectangular, square, hexagonal) do not appear in the relationships. This result is a direct consequence of the loss of microstructural details resulting from the geometrical simplifications introduced in Fig. 9-lib. Furthermore, the simplifications introduced by the embedded cylinder model tend to mask the extent to which neighboring fibers mutually influence their response characteristics (e.g., correlation effects). 9.3.3 Bounding Methods

The problems associated with specifying the detailed features of the microstructure as well as the explicit natures of the internal stress-strain fields are avoided in this approach by directing attention to upper and lower bounds on the response characteristics of composite materials. The results from the bounding approach can serve as guides to material behavior, provided the

414

9 Elastic Properties of Composites

that the actual energy content of a heterogeneous material can never exceed a value that is computed by the Us form; the theorem of minimum potential energy implies that the actual energy can never exceed a value that may be computed by the Uc form. The actual energy is taken as the energy stored by a homogeneous material whose elastic constants, C*, and compliance constants, S*, are the effective elastic constants and compliance constants of the composite material. Accordingly, the two energy theorems provide upper limits for the elastic constants, C*, as well as the compliance constants, S*; however, due to the reciprocal nature of the compliance constants, an upper limit on S* is equivalent to a lower bound on the engineering constants. Thus, the two energy theorems provide bounds on the engineering constants from above (through C*) and from below (through S*). (9-67) Explicit relationships for the bounds are obtained by proposing a statically equivaand lent trial stress field (compatible with the (9-68) surface tractions) for use in the Us form. Similarly, equilibrium displacement fields where the term mk = 1 (for k = 1, 2, 3) and (compatible with the displacements on the mk = 2 (for k = 4, 5, 6) has been introduced surface of the body) are proposed as trial to compensate for the contraction of the forms for the strain fields for use in comfourth order compliance tensor (Sijkl ->Spq). puting Uc. Selection of the appropriate trial The Uc form emphasizes the role of the stress and trial strain fields is the central strain field, whereas the Us form emphaissue of the bounding approach. The simsizes the role of the stress field. These alterplest form of compatible fields corresponds nate expressions are equivalent for homoto uniform stress or uniform strain. These geneous materials; however, the two forms trial fields give rise to primitive bounding can be significantly different for heterogerelationships that are too far apart to be neous materials, when simplifying assumpuseful. Variational methods have been detions are made concerning the nature of veloped which provide a more realistic the strain and stress fields. choice for the compatible trial fields and Bounding methods exploit this differhence yield tighter bounds. The improved ence in the Uc and Us forms through appli- bounding methods use the techniques of cation of the theorem of "least work" and variational calculus to assure that the trial the theorem of "minimum potential enerfields not only are compatible with the surgy". The theorem of "least work" implies face tractions and displacements but also

upper and lower bounds are sufficiently close together (or "tight") so as to bracket the true behavior to within the experimental error inherent in the various characterization methods. Consequently, considerable effort has been made to develop improved or "tight" bounds. The starting point for the bounding approach is a consideration of the energy content of a material undergoing deformation. The effective elastic constants, C*, and compliance constants, S*, of a heterogeneous system are defined as the properties of a homogeneous material which has an equivalent free energy content when both systems are subject to identical surface displacements and tractions. For linearly elastic materials, the stored elastic energy can be expressed in alternative forms:

415

9.3 Micromechanical Models for Continuous Fiber Composites

yield the smallest possible values for the energy computed by the Uc and Us forms. The variational procedures used for continuous fiber composites are somewhat specialized. More general methods provide the basis for developing relationships for a wide class of heterogeneous materials which contain continuous fiber composites as a special case. Specification of improved trial fields to tighten the bounds requires progressively greater knowledge concerning the detailed features of the microstructure. Since detailed knowledge of the microstructure may not be available, the improved bounding techniques for unidirectional continuous fiber composites have focused on establishing the "best" possible bounds based on information limited to the volume fraction concentration and properties of the components. As a consequence, the improved bounds on some of the properties still remain too far apart to serve as brackets. Fortunately, the improved bounds on the longitudinal properties are extraordinarily tight and tend to converge for all practical purposes to the rule of mixtures. Consequently, the bounding approach provides rigorous theoretical justification for this relationship without resorting to the assumption of specific models. For the application of bounding methods, it is more useful to direct attention to the average energy content per unit volume than to the total energy of the deformed body. Thus, the "energy density" of the material is given as

«l/» = y JJJ U(x,y,z)dx dy dz

point in the material. This normalized form of the energy removes the dimensions of the specimen from consideration and therefore directs attention to the intrinsic ability of the material to store elastic energy. Volume averaged forms such as Eq. (9-69) are frequently written in a more compact notation: (9-70) where r = r1e1 + r2e2 + r3e3 *s a vector that locates position with respect to an arbitrary origin; the rt are components of the vector along the et axis. The integration symbol means that the integration is to be conducted over the entire volume of the body with a volume element of dV. The simplest trial field that can be proposed for the internal strain, ek(r), such that it is compatible with surface deformations characterized by £°, is a uniform strain: (9-71) Under this assertion, (9-72) The elastic constants are uniform within each component; thus, for r falling within a fiber, Cfj(r) = Cij{, and for r falling within the matrix phase, Ctj(r) = Cijm. Under these circumstances the integral is given by

V JCy

(9-69) so that

where x, y, z are Cartesian coordinates which specify the position within the material with respect to some point of origin. The notation U(x,y,z) is used to indicate that the energy may vary from point to

y

( 9 _ 73a )

^ | Ci](r)dV = t>f + » m v (9-73 b) where v{ and vm are the respective volume fractions of the fiber and matrix.

416

9 Elastic Properties of Composites

This relationship is the classical Voigt model (1910) > + t>m (9-74a) or in matrix notation « C V » = vf + vm

(9-74 b)

The terms (x = f, m) represent the average of the elastic constants. If the aligned fibers and matrix are transversely isotropic or isotropic, then (CX} = CX; (x = f, m). Otherwise, techniques of orientation averaging must be used to obtain these quantities. The double brackets « » signify averaging over location and microstructure; see Eq. (9-128). The Voigt model is a straightforward extension of the simple mechanics of materials model. Focusing on the elastic constants rather than on engineering constants compensates for the differential Poisson's contraction ignored in the mechanics of materials approach. Under the simple trial field of Eq. (9-71), the upper bound on the effective elastic constants becomes C*<«Cv»(=i;fCf

(9-75)

The fact that a uniform strain field would result in an upper bound can be appreciated by the following considerations. Since the elastic field C^iy) of a heterogeneous material is not uniform, a constant strain field will result in a nonuniform stress field. If we envisage a heterogeneous system under constant strain, the correspondent nonequilibrium stress field would tend to displace the components of the system; therefore, the situation of constant strain corresponds to an energy state higher than the situation for static equilibrium. Accordingly, the energy computed under the Voigt model would be greater than the energy content of a real material. Conse-

quently, the Voigt model serves as an upper limit on the elastic constants. The lower bounds can be obtained by asserting a simple uniform stress field as the trial field: atr(r) = a°;

(k = 1, 2 , . . . , 6)

(9-76)

Under this assertion, (9-77) Again 1

(9-78)

where the <Sijx>; (x = f, m) represent the average compliance constants. This expression is the classical Reuss model (1929) (9-79 a) or in matrix notation (9-79 b) As before for transversely isotropic or isotropic aligned fibers, the averaged compliance constants are the same as the compliance constants. Under the simple trial field of Eq. (9-76), the bounds on the effective compliance constants become

S$ < « S g » (=vfSij{ + vmSijm)

(9-80)

for isotropic or transversely isotropic aligned fibers. Since S1X = 1/£L, S22 = 1/ET, etc., the Reuss model is equivalent to the simple transverse mechanics of materials models. Consequently, this model provides an absolute lower bound on the properties of the composite. This result is not surprising since both treatments are based on the assumption of a uniform stress field. Any result obtained by assuming a constant stress field certainly satisfies force equilibrium conditions. However, because of the

9.3 Micromechanical Models for Continuous Fiber Composites

heterogeneous compliance field, S^ir), the associated strain field is no longer compatible in the sense that the displacement field is not continuous. If we envisage a multiphase system under a constant stress field, there could be material overlaps and voids in the vicinities of the phase boundaries. This situation is not permitted for perfectly connected phase regions. In order to adjust the displacement field so as to eliminate overlaps and voids, additional energy would be required to bend or distort the material elements to the suitable positions. Both the Reuss model and the mechanics of materials model ignore this effect. Consequently, energies computed by these models will be lower than the real situation so that these models provide lower bounds. These primitive bounds are rigorous and assure that the effective properties of a composite material will never exceed the upper bound nor fall below the lower bound. Unfortunately, these limits of behavior are too far apart to provide useful brackets on the response characteristics of composite materials. The generality of the bounding approach provides considerable theoretical appeal, which has motivated continuing efforts to develop improved bounding methods to tighten the brackets on the response characteristics of multiphase materials. The basis for these improved bounds is the recognition that an additional criterion can be placed on the selection of the trial fields: Compatible trial fields should be selected which give the smallest possible value for the energy. A significant reduction in the energy will provide tighter bounds. This optimization of the trial fields can be achieved through variational methods. A further refinement in methodology is accomplished by considering deviations of the real material away from some homogeneous reference system.

417

The introduction of an initially unspecified reference system provides considerable flexibility in the application of the results. The reference system is taken to be a homogeneous material with unspecified elastic constants C° which are uniform throughout the reference material. If the homogeneous reference material is subjected to the same load-deformation patterns as the composite specimen, it will experience uniform stress and strain fields. The analysis focuses attention on the fluctuations of the true stress and strain fields,
M|(r) = ut(r) - uf

£ C% ew(r)

(9-81 a)

1=1

(9-81 b) (9-81 c) (9-81 d) (9-81 e)

The quantities p o (r) are the components of the stress polarization tensor that was introduced by Eshelby (1957, 1961) and subsequently treated by Hashin and Shtrikman (1962 a, b, 1963) as the major variable of a variational treatment. The term Rijkl gives the fluctuations in the elastic constants with respect to the reference material; &r gives the fluctuations in the strain field; u't gives the fluctuations in the displacement field. This improved version of the bounding method focuses on specification of a trial stress polarization tensor p\j that simultaneously satisfies all continuity and equilibrium conditions, as well as provides the smallest possible value, for the energy density of the system. The variational problem formulated in terms of these deviatoric

418

9 Elastic Properties of Composites

quantitatives yields the following results (Hashin and Shtrikman, 1962 a, b, 1963): Objective Function: (9-82)

Uh=

where U° is the equilibrium strain energy of the reference system. Subsidiary Conditions: (9-83 a) u},,)

(9-83 b) (9-83 c)

where S symbolically denotes the surface of the body. Stationary Condition: (9-84) The objective potential of Eq. (9-82), subject to the subsidiary conditions, is stationary when Eq. (9-84) is satisfied. The equilibrium strain energy of Eq. (9-82), Uh9 satisfies the following inequalities: Uh > Uh when Rijkl is positive definite (9-85 a) Uh < Uh when /? ijkl is negative definite (9-85 b) This result has a highly desirable feature: Both the upper and lower bounds can be obtained by adjusting the arbitrary reference system. Unfortunately, all the trial approximation fields for ptj and &ftj have to be made admissible to the subsidiary conditions before they can be substituted into the objective potential for the estimation of the bounds. This difficulty can be eliminated by constructing a constraint free variational problem (Wu and McCullough, 1977). The basis for this refinement is the observation that if ptj is treated as an independent variable tensor function, the 8ftj

field can be completely determined by the subsidiary conditions of Eqs. (9-83). These three equations can be regarded as a static elastic problem for a system of homogeneous elasticity, Cfjkl9 subject to homogeneous boundary conditions and body force fiels, piJ9 j. The resulting strain field s[j can be expressed in terms of ptj j through an integral operator with the Green's tensor function as a weighting factor. The &ftj so determined for a given ptj field statisfies the subsidiary conditions automatically. Substitution of this relationship into the objective function, Eq. (9-82), to eliminate the presence of 8'tj yields a constraint free objective with only one variable, viz. pijm The results obtained by employing this approach state that for a system under load such that the average strain is given by s?j, the equilibrium strain energy per unit volume is given by the stationary value of the objective potential

«t/R» = \ 4 cfjkl 4 - \ « Pu Hij (9-86) where the stationary value, denoted by « C / R » , corresponds to a maximum when Rijki i s positive definite, and to a minimum when Rijkl is negative definite. The integral term in Eq. (9-86) involves the two-point correlation function iip'ij{r') V'ki(r'— r ) » - ^ has been shown (Wu and McCullough, 1977) that if the two-point correlation function exhibits ellipsoidal or cylindrical symmetry, the last integral of Eq. (9-86) is given by

where E° is a constant tensor. The stationary value of the energy is given by » « o (9-88)

419

9.3 Micromechanical Models for Continuous Fiber Composites

and the optimal trial field is obtained from the relationship

Table 9-3. Components of the correlation tensor, E°, for a composite of aligned continuous fibers with random transverse packing.

/o The optimal energy is given by E°

\°

Comparison yields the following expression for the trial elastic constants: E

(9-91 a)

with

0 kE-mE kE + mE 0 0 0

0 0 0 mE 0

fiF

0

0

0 0 0 0

0 \ 0 0 0 0 HE 1

1 (4X° + m°) -(kl

or, in compact matrix notation, (9-91 b)

0 kE + mE kE-mE 0 0 0

0 0 0 0

D

+ m°) c° + m°)

-1

Transversely isotropic reference:

= m,f The various C's are the appropriate elastic constant arrays. The quantities M°, H°, and R° are constructed from the C arrays as indicated above. The term E° is similar to the "Eshelby" tensor; the elements of E° are dependent upon the shape of the phase regions, as well as the particular choice of the reference material represented by C°. The elements of E° for the special case of aligned continuous fibers packed in a random transverse fashion are given in Table 9-3 (Wu and McCullough, 1977). As before, the averaged properties for transversely isotropic systems of aligned fibers are identical to the properties so that « > = MX°. The relationship given in Eq. (9-91) provides multiple service through the flexibility introduced by the C° terms associated with the unassigned reference material. The relationship can be used for bounding or modeling approaches, depending upon assertions made for the C° term (Korringa, 1973).

2[(1 -v°T)E°-\ -2£?(v»T)2]

(v? T = v°2:?; LT = m° = — a0 = J V

V

= v? 3 )

?2

.Go

)

3

Isotropic reference:

E° 2(1 + v°) ( 1 - 2 V°) m0--= G° = G°

Bounds are provided by the following general conditions: Lower Bounds: If C - C° > 0 (positive definite) C*>C°

1

+ E0) ' 1

(9-92 a)

Upper Bounds: If C - C° < 0 (negative definite) C* < C° 4- (((M 0 ))" 1 + E 0 )" 1

(9-92b)

where C* is the effective elastic constant array for the composite material. The positive definite condition can be simply assured by taking C° as the null matrix, 0. For this choice of a reference

420

9 Elastic Properties of Composites

system, Eq. (9-92) reduces to the primitive Reuss bound

°=

(9-93)

Similarly, the negative definite condition can be assured by taking C° infinitely large. For this choice, Eq. (9-92 b) reduces to the primitive Voigt bound c

* < °°c —

f

f

(9-94)

c m

m

As indicated previously, these bounds are too far apart to be useful. The positive definite condition can be alternately assured by taking the reference material as the pure matrix phase. Clearly, the composite material will have properties which are greater than the matrix so that C — Cm > 0. Under this selection of a reference material, an improved lower bound is obtained: Improved lower bound: C° = Cm C*>Cm + (({M0}}-1

+ E 0 )" 1

(9-95)

E° is determined with the matrix as the reference phase. Similarly, the negative definite condition can be assured by taking the reference material as the fiber phase. In this case the diluted composite would not be expected to exhibit properties greater than a material composed of the pure fiber component so that C — Cf < 0. This selection for the reference material gives an improved upper bound: Improved upper bound: C° = Cf C*
(9-96)

E° is determined with the fiber as the reference phase. These results are consistent with the particular bounds reported by Hill (1964), Hashin and Shtrikman (1962 a, b, 1963), and Walpole (1966 a, b, 1969).

The longitudinal Young's modulus and the major Poisson's ratio are tightly bounded by the improved relationships. Indeed for the case of nearly equal Poisson's ratios (v m ^v f ) the bounds converge to the simple rule of mixtures. Thus, for all practical purposes, the simple rule of mixtures adequately predicts the values for Ell and v12 for continuous fiber composites. Unfortunately, the remaining properties are not so tightly bounded. Whereas the bounds on the remaining properties are significantly closer than the primitive bounds (predicted by the rule of mixtures), the range is too broad to serve as a useful bracket on these properties. The upper-lower bound bracket can be further decreased by the introduction of additional information concerning the microstructure. This information may be developed in terms of three-point and higher correlation functions. A recent review of the current status of this approach is given by Torquato (1991). The nature of the threepoint correlation function has been studied through computer simulation of a variety of microstructures: impenetrable infinitely long cylinders of uniform and distributed radii, penetrable infinitely long cylinders of uniform and distributed radii, and "coated" cylinders. In general, the introduction of additional structural information via threepoint and four-point correlations gives a significant tightening of the upper-lower bound bracket. This tightening of bounds is achieved primarily through the significant reduction of the upper bound representations. The improvements in the lower bound representations yield results marginally larger than the Hashin-Shtrikman lower bound. Torquato (1991) presents strong arguments to support the contention that these improved lower bound results provided useful estimates of effective elastic constants and effective conductivities.

9.3 Micromechanical Models for Continuous Fiber Composites

9.3.4 Semi-Empirical Models: The Halpin-Tsai Relationship The difficulties associated with specifying details of the microstructure and the associated features of the internal distribution of stress (and/or strain) prompted the development of a variety of semi-empirical relationships to connect the effective properties of composite materials to the properties of the components. Of the various semi-empirical forms, the following equation proposed by Halpin and Tsai (1969; Ashton et al., 1969) provides a simple relationship which is widely applicable: (9-97) with

As before, P* is an appropriate property of the composite, with Pm and Pf representing the associated property of the matrix and fiber, respectively; v{ is the volume fraction of the fiber. The dimensionless quantity x may be viewed as the "reduced" properties of the components. The central feature of the Halpin-Tsai relationship is the recognition that the

421

properties of a composite material must lie somewhere between the extremes of the primitive Voigt upper bound and the Reuss lower bound. The arbitrary scaling parameter, £, serves to adjust the effective properties, P*, between these two limits. As illustrated in Fig. 9-12, various assignments for £ give values of P* intermediate between the Voigt upper bound (corresponding to the mechanical analog of "springs in parallel") and the Reuss lower bound (associated with a "springs in series" analogy). In Fig. 9-12, component A is taken to be the compliant phase (e.g., matrix phase) with property PA; component B is taken as the rigid phase with property PB such that PB> PA. The £ parameter serves to indicate the extent to which the applied load is transmitted to the reinforcing phase. Accordingly, the £ parameter may be interpreted as a measure of the "reinforcing efficiency" of the composite material system. The reinforcing factor, £, may be treated as an adjustable parameter whose value can be obtained from an experimental determination of the property of a composite, P*x, at a specific volume fraction of fiber. For this application, Eq. (9-97) can be solved for the unknown parameter, £, in terms of a known property, P*, and the specified volume fraction, v{, and proper-

1.0

Figure 9-12. Illustration of the role of the reinforcing factor, £, of the Halpin-Tsai equation.

422

9 Elastic Properties of Composites

ties of the component to give

*

PAiPf-p^-a-^ip^pj]

(9 98)

'

verse Young's modulus wasfitby £ values given by £ = 2 (a/b). These results prompted the generalizations for predicting values of £ in terms of fiber geometry. Continuous fiber composites can be treated as if the longitudinal aspect ratio (e.g., L/d) is infinite so that the Halpin-Tsai relationship reduces to the simple rule of mixtures for longitudinal properties. The use of the transverse values for £ presented in the preceding paragraph predict two more of the five required properties for transversely isotropic materials. A remaining property could be obtained by assuming that the £ = 1, to give the value associated with G23. This value is obtained from the Hermans' doubly embedded model (Hermans, 1967). However, Kardos (1973) recommends that a remaining property, v 23 , be computed by an equation proposed by Foye (1972):

Once a value for £, has been established for one set of experimental values (P£,v{), the property P* could be predicted at any other volume fraction fiber - provided that £ does not change with volume fraction. When the Halpin-Tsai relationship was developed, reproducible data for continuous fiber composites was limited to longitudinal properties for which the rule of mixtures (corresponding to the trivial case of £ -»oo) was known to adequately correlate composite properties to the composition and concentration of the components. Hence, it was difficult to test the validity of the relationship for transverse properties. In lieu of experimental data, theoretical results from detailed numerical analyses (9-99) v23 = were used as values for P*x in Eq. (9-98) in order to obtain estimates for £. These comwith parisons revealed that £ was dependent , = l + vm-v12(£m/£1) upon the cross-sectional shape of the fibers 1 -vi + vmv<7(Em/E*) and the packing geometry. Specifically, the numerical results obtained by Adams and where v12 and E1 are values computed Doner (1967 a, b; Adams et al, 1967) for from Eq. (9-97) with £=oo. The shear continuous circular filaments packed in modulus, G 23 , can be obtained by the relaa square array could be fit with a value tionship G23 = E2/(2 + 2 v23). of £ = 1 for the effective shear modulus Some uncertainty remains concerning (P* = G12) and £ = 2 for the effective trans- the proper choice for a value of £ associverse Young's modulus (P* = £T). The inated with E2. Slightly improved fits for cirfluence of the cross-sectional shape of the cular fibers can be obtained by assigning fiber was examined through the numerical a value of £ = 1 if fibers are packed in a results obtained by Foye (1966 a, b) for hexagonal (or random) pattern. A value of continuous ribbon-like fibers of width a £ = 2 provides a better fit for circular fibers and thickness b packed in a "diamond" packed in a square array. (i.e., hexagonal) array. The cross-sectional The Halpin-Tsai relationship can be dimensions of the fiber were varied and fruitfully applied to provide a common bavalues of £ fit to the transverse aspect sis for comparing the closed form expresratio, a/b. In this case, the effective shear sions obtained from the mechanics of mamodulus (P* = G_12) wasfitby £ values of terials models, the doubly embedded modthe form £ = ^ 3 In (a/b), while the transels, and the improved bounds. In this ap-

423

9.3 Micromechanical Models for Continuous Fiber Composites

plication, the theoretical results are used for P*x in Eq. (9-98) so that an expression for £ is obtained in terms of the properties of the components. The expressions thus obtained for £ provide a convenient basis for comparing and interpreting the results obtained from the various approaches. The convenient descriptors for these comparisons are the plane-strain bulk modulus, fcT, and the shear moduli, G12 = G13 —\x and G23 = m- The relationships obtained for £ through these comparisons are summarized in Table 9-4. These comparisons again emphasize that the doubly embedded self-consistent field model gives results which are comparable to the improved lower bound results. The value of £ associated with G23 corresponds to the value obtained from the fit to numerical analyses for circular fibers. With this exception, these comparisons suggest that the value of £ for transverse properties is influenced by the properties of the components as well as the cross-sectional shape and packing geometry of the fibers. In particular, the values of £ for the lower bounds (and doubly-embedded model) results are determined by the prop-

erties of the matrix component. In most cases of practical interest, the matrix phase is isotropic. Under these circumstances, the £ parameters for the lower bound and the doubly embedded model reduce to a simple dependence on the Poisson's ratio of the matrix phase: for kT

1 3-4vm

(9-100)

for G,

<* = 1

for G12

9.3.5 Summary

In light of experimental errors involved in characterizing composite materials and scatter in properties, it is prudent to base design parameters for composites on reasonably conservative estimates of the properties. The Reuss model (£ = 0) of Eq. (9-65) and Eq. (9-79) provides an ultraconservative estimate that would lead to inefficient use of expensive materials. The improved lower bound offers a more sensible conservative estimate that incorporates the assurance that the observed properties of the material will be as good as or slightly

Table 9-4. Comparisons of the various models for the elastic properties, P*, or the "reduced" properties, %, of two-component transversely isotropic continuous fiber composites. The body of the table gives the equivalent reinforcing factor, fp. m: matrix, f: fiber; P* = Pm(l + £ px v f )/(l - xvf); x-ftPJUPt + ZO Treatment

p*.

Variational: improved upper bound

co

improved lower bound

oo

Self-consistent field (doubly embedded) Mechanics of materials

Computed from other properties.

00

mfk{ mm{kf + 2m f ) km

00

424

9 Elastic Properties of Composites

Table 9-5. Summary of the improved bounds on the engineering constants for composites of aligned continuous fibers with random transverse packing. Upper bound: P* (+) = P* (a = f, p = m) Lower bound: P* (-) = P* (a = m, p = f) m: matrix property; f: fiber property Property, P*

Bounding function, P (a,

[*.*«

- mg)] 2mamp + m,) + 2m m v (m m j (k a a p p p a + 2mJ]

GTT(=m)

+ ^ J + ^^ ^q (kp ~ K)

Auxiliary: ET(±) =

4/c(±)G TT (±)£ L (±) EL(±) [k(±) + G Tr (±)] + 4/c(±) GTT(±) v^T(+)

vTT(+) = - i

[(fc(±) - GTr{±)] - 4 vgT(+) GTT(±) fc(±)

JEL(±) [k(±) + G TT (±)]

/c(±)

better than the predicted values. Hence, it is recommended that the improved lower bound relationships (or the equivalent doubly embedded relationships, or the composite cylindrical assemblage relationships) be used to predict transverse properties and shear moduli of continuous fiber composites. These results are summarized in Table 9-5. Alternatively, these quantities may be computed by the Halpin-Tsai equation summarized in Table 9-6 with the appropriate values of £p computed from the properties of the matrix as indicated by the relationships summarized in Table 9-6 b. Values for ET and vTT must be computed from the auxiliary relationships given in these tables. As shown in Eq. (9-100), the relationships for the various £p

reduce to simple functions of the Poisson's ratio for an isotropic matrix material. Since the Poisson's ratio for most polymeric matrices is in the vicinity of 1/3, these relationships can be further simplified to the values in parentheses in Table 9-6 b.

9.4 Micromechanical Models for Particulate Reinforced Composites Particulate filled systems represent the next level of complexity beyond continuous fiber composites. In these systems, the reinforcing particles are spherical (or near spherical). The marked discontinuity of particulate filled systems introduces significant fluctuations in the internal stress-

9.4 Micromechanical Models for Particulate Reinforced Composites

425

Table 9-6. Mixing rules for conservative estimates of properties of continuous fiber composites

Systems reinforced with spherical inclusions exhibit material isotropy with

P* = composite property, P{ = fiber property, Pm = matrix property, v{ = volume fraction fiber.

c

—c

—c

C12 = C13 = C23 C44 = C55 = C66 = 2 55

£p = reinforcing factor a. Longitudinal properties. P * = vtPf + (1 -vt)Pm

Property, P*

£p

Young's modulus, EL Major Poisson's ratio, vLT Transport properties, q - Thermal conductivity - Electrical conductivity - Diffusion coefficients

00

oo 00

so that only two material constants are needed to characterize elastic behavior. The two independent elastic constants are taken as the bulk, K, and shear, G, modulus with K = ±(C11 + C12) G = -(C

—C

(9-101) )=C

=C

9KG 3K + G 3K-2G v = 2(3K + G)

E=

Property, P* In-plane shear modulus, GLT

=C

(9-102)

(9-103) (9-104)

As was the case for continuous fiber systems a variety of approaches has been proposed to develop relationships to predict the Young's modulus, shear modulus, Poisson's ratio and bulk modulus of particulate filled materials in terms of the properties and concentration of the polymeric and filler components. These approaches range from empirical curve fitting techniques to sophisticated analytical treatments. This review will focus on the predictive models.

Plane strain bulk modulus, kT Transverse shear modulus, GTT Transport properties, q

Transverse Poisson's ratio, vTT

—C12)

The Young's modulus, E, and Poisson's ratio, v, may be obtained from K, G, viz.

b. Transverse and shear properties

Transverse Young's modulus, ET

66

4fc T G T T £ L

ET

strain fields which complicate the analysis of mechanical behavior. However, the simple spherical reinforcing geometry precludes a dependence on orientation so that consideration of this structural feature is not required in the analysis of these systems.

9.4.1 Mechanics of Materials

The mechanics of materials approximation is the simplest and, consequently, the most commonly encountered approach. This approach is based on the assumption that each phase component (or prescribed combination of phase components) is subject to either the same stress or the same strain. These gross simplifications in the internal distributions of stress (or strain)

426

9 Elastic Properties of Composites

mitigate the influence of the shape, size, and packing of the phase components so that the only descriptors of the phase regions that are retained in this approach are (1) the elastic properties of the components and (2) the volume fraction of the components. The classical results of Voigt (1910) and Reuss (1929) belong to this category. Voigt assumed that each component was subject to the same strain. By analogy to parallelconnected springs, this assumption leads to the following combining rule: Voigt (constant strain): (9-105) pv =

Vmpm

+

V{p{

(with PW = K or G)

Alternatively, Reuss assumed that each phase component was subject to the same stress. For this situation, an analogy to series-connected springs leads to the following combining rule on the compliance constants: Reuss (constant stress): j_

=

^

v±_ (p =

£ G

x )

(9-106)

The estimates of the effective composites moduli determined from the Voigt and Reuss models represent extremes in behavior. The Voigt model attributes more significance to the "stiffer" phase, whereas the Reuss model emphasizes the compliant phase. Thus, if the properties of the components differ by an order of magnitude, the results obtained from these models will differ by an order of magnitude. Several models have been proposed to give results intermediate between the extremes of the Voigt and Reuss models. In these models certain prescribed combinations of phase components are assumed to be subject to constant stress while other combinations are subject to constant strain. Such specifications introduce additional parameters which may be adjusted to fit observed data.

As pointed out by Hill (1964), none of the models based on uniform stress or uniform strain fields can be strictly correct: the interfacial forces are not in equilibrium for constant strain; interfacial discontinuities must exist for constant stress. In contrast to the mechanics of materials approach, the embedding (or "self-consistent field") methods introduce gross simplifications in the microstructure in order to deal in a more realistic manner with the internal distribution of stress and strain. For example, Hill (1965 b) represents the phase geometry of a particulate filled system as a single (typical) spherical particle embedded in a medium whose properties are taken to be equivalent to the overall average properties of the composite material. An elasticity problem can be formulated for this simple geometry such that a "self-consistent" stress field can be identified and the effective properties of the medium can be determined. The difficulties associated with specifying the explicit nature of the internal stress-strain field, can be avoided by directing attention to the upper and lower bounds on the response characteristics of heterogeneous materials. 9.4.2 Bounding Approach

All of the approaches may be reconciled through the general bounding methods described in Sec. 9.3.3. In this case, the system exhibits a spherical symmetry so that the tensor E° is of the form F° -- E

T7° — ^ 3 3 ~~

30(1 - v o ) G o

F° -- F° — F° — 12" ~ ^ 3 1 "-

1 — E° — E° 30(1 —vo)Go ^21 — ^ 1 3 —

— Z?0

1 5 ( 1 - vo)Go with

all other £° == 0.

9.4 Micromechanical Models for Particulate Reinforced Composites

The quantities Go, v0 are the shear modulus and Poisson's ratio of the reference phase. This treatment leads to constitutive relationships of the form 0

E ))"

1

(9-108)

with

The term C* is the array of predicted elastic constants for the composite material. The term C° represents the elastic constant array for an arbitrary reference material and is the origin of the versatility of the treatment. The relationship expressed in Eq. (9-108) provides dual service as bounding relationships or a model relationship. The choice of these applications depends upon assertions concerning the arbitrary reference material, C°. In the application as a bounding relationship, the values assigned to C° are constrained by the following conditions:

can never exceed the upper bound (Pv) nor fall below the lower bound (PR): PK
P v with P = G or K

(9-109 a)

Lower bound: C - C° > 0 (positive definite)

(9-109 b)

The simplest choice for C° that assures the conditions of Eq. (9-109 a) is C° = oo. Under this selection, Eq. (9-108) reduces to the classical Voigt result of Eq. (9-105). Alternately, the simplest choice for C° that guarantees the conditions of Eq. (9-109 b) for the lower bound is C° = 0. For this choice, Eq. (9-108) reduces to the classical Reuss model of Eq. (9-106). These simple results correspond to the bounds determined by Paul (1960) and represent the extremes in behavior for the material in the sense that the properties. P*

(9-110)

Hashin and Shtrikman (1963), as well as Hill (1963, 1964), noted that more realistic choices for C°, which assure the conditions of Eq. (9-109), may be made by assigning to C° the properties of the average (isotropic) particle and isotropic matrix, respectively. These improved choices for C° reflect the fact that the fluctuations in the internal elastic field, C, can never exceed the values of the most rigid component nor fall below the values of the most compliant component. Consequently, the following improved bounds are obtained: Cub = Cf + ^ M V

1

+ E 0 )" 1 1

0

Clb = Cm + ( ( ( M V + E )-

1

(9-111 a) (9-111 b)

E° is computed with the matrix and the fiber as the reference phase for the lower and upper bound, respectively. The bounds in the bulk and shear modulus are obtained as (9-112a)

vmAK

Upper bound: C - C° < 0 (negative definite)

427

vfAK

vmAG 1

-

(9-112 b)

(9-113 a)

77 K (9-113 b) Vf AG

15/

with AP =

Pf-Pm.

m

\

1-vm

/ \ ^m

428

9 Elastic Properties of Composites

These relationships suggest that the lower bound corresponds to particles embedded in a continuous matrix phase, while the upper bound relationship may be viewed as representing a situation in which the matrix material is embedded in a continuous phase that possesses the properties of the filler. It is reasonable to expect that the behavior of particulate filled polymers will tend toward the lower bound representation. These relationships can be cast into the compact form of the Halpin-Tsai Equation (Ashton et al., 1969): (9-114)

p* = with P —P

for P = K or G. The Young's modulus, £, and Poisson's ratio, v, can be computed from the auxiliary relationships given in Eqs. (9-103) and (9-104). The factor fp for the Reuss lower bound, the improved lower bound, the improved upper bound, and the Voigt upper bound is summarized in Table 9-7. Although in a different form,

Table 9-7. Values for the reinforcing factor,
K

Bounds

Reuss bound

Lower bound

Upper bound

Voigt bound

0

2(1 -2v m ) (1+vJ

2(l-2v f )K f (l+v f )X m

oo

7-5v m 8-10v m

(7-5v f )G f (8-10vf)Gm

these results correspond to the improved bounds reported by Hill (1963, 1964), Hashin and Shtrikman (1963), and Walpole (1966 a, b). These bounds are the most restrictive that can be made without information (or assertions) concerning the distributions of size, shape, and packing of the inclusions. Whereas these improved bounds offer a significant improvement over the primitive Voigt and Reuss bounds, they remain too far apart to bracket the properties to within the margins of experimental error. This separation in the bounds implies that the properties of filled systems are sensitive to detailed features of the microstructure that are not included in the information used to develop the bounds. Thus further tightening of the bounds would be expected upon the introduction of additional information concerning the microstructure. Additional information concerning the microstructure can be introduced through the use of "n-point" correlation functions. Nomura (1979) has shown that the incorporation of three-point correlations yields some further tightening of the bounds. Unfortunately, these further improvements do not tighten the bounds sufficiently to bracket the behavior to within experimental error. Incorporation of yet higher order correlations leads to tighter bounds. A review of this approach is given by Torquato (1991). In general, the introduction of additional structural information via threepoint and four-point correlations gives a significant tightening of the upper-lower bound bracket. As was the case for continuous fiber composites, the tightening of bounds is achieved primarily through the reduction of the upper bound representation. The improvement in the lower bound yields results marginally larger than the improved lower bound of Eq. (9-111 b).

9.4 Micromechanical Models for Particulate Reinforced Composites

Development of the information necessary for the determination of n-point correlations will require structural data (or structural models) of progressively greater detail. In order to obtain such information, detailed microstructural characterizations will be required to identify and evaluate the appropriate structural parameters that can be used to predict thermomechanical behavior. In effect, this approach trades direct mechanical characterization of a composite material for more complicated structural characterization experiments. Consequently, a theory based on the use of higher order correlation functions will have limited value in practical applications. These considerations suggest that the search for further improvements in the bounds should be abandoned in favor of a modeling approach that yields relationships based on limited and readily accessible structural data. 9.4.3 Embedding Models

In the application of Eq. (9-108) as a model relationship, the reference material parameters C° may be viewed as parameters that incorporate the influence of the various features of the microstructure. Indeed, the use of C° in the bounding approach may be interpreted from this point of view. Assigning to C° the values of 0 and oo may be viewed as a total lack of information concerning the nature of the composite specimen. With no information available, preliminary predictions must be limited to the statement that the properties are bounded from below by 0 and from above by oo. The use of these preliminary predictions as values for C° gives the Voigt and Reuss bounds. Improved bounds are obtained by introducing information concerning the properties of the components. With a knowledge of the properties of

429

the components, the preliminary prediction can be made that the composite specimen will be at least bounded from below by the properties of the compliant component and bounded from above by the rigid component. The subsequent use of these preliminary predictions for C° leads to the improved bounds of Eqs. (9-111). This interpretation of the role of C° suggests that a further tightening of the brackets on behavior could be obtained by assigning the value of Cub and Clb as preliminary estimates for C° from which tighter upper and lower brackets could be obtained. Such trials could be proposed as an iterative scheme where the C° of the (i + l)th iteration is assigned the value of the C* computed from the zth iteration. Continued iterations could force a convergence of the bounds so that the final assignment for C° becomes the unknown effective elastic constant array C*. Such an assignment corresponds to the "self-consistent field" model originally proposed by Hershey (1954) and Kroner (1958). The assertion that C* = C° leads to the following relationships:

r-

(9-115)

where the various elements of C° appear in both M° and E°. This relationship represents a series of coupled equations for the elements of C°. Solution of these relationships gives the elements of C° and hence the elements of C*. These solutions for the special case of an isotropic material correspond to the self-consistent field result for spherical inclusions reported by Hill (1965 b) and Budiansky (1965). Smith (1976) has compared the self-consistent field results for spherical inclusions to experimental values and has shown that the self-consistent field approach can seriously overestimate the effective properties in the

430

9 Elastic Properties of Composites

practical ranges of volume fraction concentrations. Christensen (1979) argues that the selfconsistent field approach should not be viewed as a geometric model but rather as an iterative scheme, as described in the previous paragraphs. The iterative replacement of C° by successive approximations will lead to unique relationships by artificially requiring coalescence of the bounds. There is no a priori reason to believe that the bounds should converge for information limited to the properties and concentration of the components. The self-consistent field scheme remains popular in spite of these deficiencies. Further progress in developing relationships to predict the properties of particulate filled materials can be achieved by introducing specific geometrical models to take into account the influence of the microstructure. The composite sphere model proposed by Hashin (1962) and the threephase (or doubly embedded) model introduced by Kerner (1956) and van der Poel (1958) provide the basis for this approach. The basic element of the composite sphere model is a spherical inclusion of radius a embedded in a sphere of the matrix material of radius b. The ratio a3/b3 is taken to be independent of the absolute size of the element and equal to the volume fraction of particles. Hashin (1962) further proposed that a particulate filled system could be modeled as a collection of such elements with a particular gradient in the size of the spheres so that the collection of spherical elements could be space filling. Consequently, this model presumes a broad distribution of particle size with a range extending to infinitesimal particles. Bounds on the effective moduli for the composites spheres model were obtained by variational techniques (Hashin, 1962). The upper and lower bounds on the bulk

modulus, K, coincide and are given by the relations obtained from the improved lower bound, viz. Eq. (9-112). Aboudi (1983) modeled particulate composites as triply periodic array of cubes. His results for the bulk modulus, K*, are essentially equivalent to the results obtained from the composite spheres model and hence the improved lower bound given by Eq. (9-112). This correspondence suggests that the bulk modulus is insensitive to the shape of the inclusion. The bounds on the shear modulus - and hence on the Young's modulus and Poisson's ratio - remain distinct. The form obtained for the upper and lower bounds on the shear modulus is somewhat involved with the lower bound of the composite sphere model falling slightly below the improved lower bound of Eq. (9-113). The three-phase (or doubly embedded) model provides the basis for an extension of the self-consistent field treatment. In this model, the phase geometry is represented by a single (typical) spherical inclusion of radius a embedded in a sphere of the matrix material with a radius b. As was the case with the composite sphere model, the ratio a 3 /b 3 can be taken to be equal to the volume fraction of filler. The composite sphere is in turn considered to be embedded in a medium that is assigned the unknown effective properties of the filled material. An elasticity problem can be formulated for this simplified geometry such that a self-consistent stress field can be identified and the effective properties of the medium determined. The results reported by Kerner (1956) for this model correspond to the relationships given for the improved lower bounds, Eq. (9-1 lib). Unfortunately, the treatments by Kerner (1956) and van der Poel (1958) are in error. Smith (1974 a, b) and Christensen and Lo

9.4 Micromechanical Models for Particulate Reinforced Composites

(1979) have pointed out the specific error van der Poel made in his work. The explicit error in Kerner's work cannot be pinpointed because of the lack of details in the derivation (Christensen, 1979). The correct treatment of the three-phase model, as reported by Smith (1974 a, b) and Christensen and Lo (1979), results in involved expressions which yield values for the shear modulus that are slightly higher than those obtained from the lower bound. These analyses show that the assumption of a specific geometrical model enters into the prediction of the shear modulus of composite materials. Evidently, the error made by Kerner gave results which fortuitously coincided with the lower bound results. A comparison of these various models to observed elastic behavior is given by McGee and McCullough (1981). These comparisons show the deficiencies of the models at high (v{ > 0.40) volume fractions of particles. All the models tend to converge to the same predictions for low (vf < 0.40) volume fractions. 9.4.4 Semi-Empirical Models

The complexities and inadequacies of the theoretical models have prompted recourse to semi-empirical relationships. An evaluation of the host of semi-empirical relationships that rely upon the determination of adjustable parameters by curve fitting techniques is beyond the intent of this work. Rather, attention will be directed to those forms for which the parameters can be fixed by other considerations so that the resulting relationships can serve in a predictive capacity. The Halpin-Tsai form (Ashton et al, 1969), Eq. (9-114), has emerged as the most popular semi-empirical model. In this application of Eq. (9-114), the symbol P de-

431

notes an appropriate property (P = K, G); vf is the volume fraction of the rigid component (i.e., Pf > Pm), and £ can be treated as an adjustable parameter. The central feature of the Halpin-Tsai relationship is the recognition that the properties of a composite material must lie between the extremes of the primitive Voigt upper bound and the Reuss lower bound. As pointed out in Table 9-7, the parameter £ serves to adjust the effective properties between the limits of the Reuss (£ = 0) and Voigt (£ = oo) bounds. The extension of the Halpin-Tsai equations to particulate filled systems (with / = d) corresponds to £E = 2 and £G = 1. The Halpin-Tsai relationship (with £E = 2) tends to underestimate the Young's modulus for volume fractions of filler in excess of 0.4. In an analogy to the Mooney equation for the viscosity of suspensions, Lewis and Nielsen (1970) proposed a modification of the Kerner equation (1956) to account for the limits imposed by the maximum packing that can be achieved for uniform spherical particles. Since the Kerner equation corresponds to the improved lower bounds, the Lewis-Nielsen relationship can be expressed as a modification of Eq. (9-114): P = K o r G

with

with the factor £P1 given in Table 9-7 under the improved lower bound column. The term v is introduced to compensate for the fact that close-packed spheres cannot exceed a certain maximum packing fraction, vmax p o r e x a m pie 5 hexagonal and facecentered close packing are both limited to a maximum packing fraction of 0.74; body-

432

9 Elastic Properties of Composites

centered packing is limited to 0.60, while simple cubic packing is limited to rjnax = 0.52. In a real particulate system, it may be assumed that these various packing forms occur at random and with equal probability so that the maximum volume fraction for such "random" close packing is approximated by the average of these four forms, viz., « § . To specify v, Lewis and Nielsen (1970) required that (1) i7i;f = 0 at vf = 0 d{vv{) (2) ^ =

at v{ = 0 and

(3) vvf = .

at vf =

They proposed the following form (among others) that would satisfy these conditions:

A classic approach to developing constitutive relationships for composite materials is based on series expansions in concentrations (Christensen, 1979). The application of such asymptotic relationships is usually restricted to the extremes of concentration (e.g., vf« 0). However, as noted by Christensen (1979), the range of applicability can be extended by a judicious choice for the basis of the expansion. McGee and McCullough (1981) proposed that the effective properties may be modeled as a series expansion in terms of the difference between the upper and lower bounds: — O j — if/ \^Oj — o

u

j

T

•••

[y-i

lo)

where S is any one of the diagonal elements of the compliance matrix (S = Sii9 j = 1, 2,..., 6). For convenience, only the linear term was retained. The proportionality factor if/ was approximated by a truncated series expansion in v{.

il/(v{) = a + Pvf + yvf + ...

(9-119)

Since the behavior of the actual material must fall between the extremes of the improved upper and lower bounds, the term \l/(v{) must satisfy the condition 0 < il/(v{) < 1. Alternatively, as v{ approaches 1 so that the filler assumes the hypothetical role of the continuous phase, the behavior should tend to the upper bound. This condition can be assured by requiring that ^(0) = 0 and ^(1) = 1. These conditions on \p leave the term y of Eq. (9-119) as the only undetermined parameter, and Eq. (9-118) takes on the form of the S-mixing rule (or S-combining rule) proposed by McGee and McCullough (1981); viz. S* = vmSl + v(Su + yvfvm(Sl-Su)

(9-120)

The parameter y may be fixed by considering the limiting aspects of phase continuity. As previously noted, the maximum volume fraction that can be achieved for equivalent close-packed spheres corresponds to hexagonal (or face centered) close packing with v™ax = 0.74. Any value for v{ > 0.74 would correspond to a continuous filler phase containing polymeric inclusions. Alternatively, for a hypothetical system of equivalent spherical particles of the matrix material embedded in a continuous "filler" phase, the maximum packing fraction of matrix in excess of 0.74 would correspond to a system in which the polymeric phase can be treated unequivocally as the continuous phase. In terms of the notion of "contiguity," as described by Hashin (1962), the situation for vm>v™x corresponds to "zero" inclusion contiguity in the sense that the average contact area of neighboring inclusions is zero; the situation for vf > v™ax corresponds to "zero" matrix contiguity. Intermediate states of contiguity are obtained between these limits of volume fraction, viz., 1 — v™ax
9.4 Micromechanical Models for Particulate Reinforced Composites

These considerations imply that at some critical volume fraction, c, a transition in phase contiguity must occur so that in the neighborhood of vf = <j)c9 the behavior tends toward the limits of the upper bound, whereas in the vicinity of vm = 4>c, the behavior will tend toward the limits of the lower bound. It is reasonable to require, for volume fractions of matrix material ^m = 0c ( o r vf = 1 — 0c X that S* be within s of Sx. Similarly, for vf = 4>c, S* will be within an equivalent range of Su. For PfpPm, these considerations lead to the following approximate specification of 7: (9-121)

0c

The results of Eq. (9-120) can be alternatively arranged into a form similar to the Lewis-Nielsen modification of the HalpinTsai relationship:

(9-122)

with ZPI = -

For the case of interest with Pf J> Pm and £pu > 1, the correction factor ij/ reduces to

(9-123a) If the critical volume fraction, 0C, is taken as the maximum packing fraction for "random" close packing (0C « § ) then tfr«l + | ( l - i ; f 2 )

(v{>0)

(9-123 b)

as compared to the Lewis-Nielsen correction * = l + (!)t>f (9-124) Although the two correction factors give essentially equivalent values for the com-

433

pensated volume fractions (v v{ or \\i v{) for 0 < Vf < | , the Lewis-Nielsen factor causes vvf to increase more rapidly than ij/v{ in the region v{ > \. This behavior is a consequence of the rapid adjustment of v to satisfy the requirement that v vf = 1 at v{ = cj)c. This requirement was imposed to normalize the volume fraction of filler with respect to the maximum packing fraction. It is not evident that such a simple scaling reflects the role of the microstructure; indeed, it appears to overemphasize the contribution of the filler at higher volume fractions. McGee and McCullough (1981) show that the S-mixing rule as reflected in Eq. (9-120) yields predictions in good agreement with the available data. 9.4.5 Summary

The comparison given by McGee and McCullough (1981) shows that the S-combining rule provides predictions in agreement with experimental data over a wide range of concentrations and for a variety of materials. This combining rule was developed by making use of the symmetry of the improved bounding expressions, with respect to the interchange of the role of the components for the continuous phase: the lower bound expression. The relationships developed from this approach are summarized in Table 9-8. The notion of a critical packing concentration, cf)c, is used to designate transitions in phase continuity (or "contiguity"). When the volume fraction of either component is greater than this critical packing concentration, it is possible to unambiguously identify this phase as the continuous phase. Under these conditions of "zero" contiguity for the second component, the material system tends toward the behavior described by the bounding expressions in which the first (major) component is as-

434

9 Elastic Properties of Composites

Table 9-8. Approximate relationships for estimating the thermoelastic properties of participate reinforced systems. P*: composite property; Pf: filler property; Pm: matrix property; v{: volume fraction filler. Conditions: Pf > Pm; 0 < v{ < 1/2

1-^fXpi

1 2

2 f

p* Primary elastic properties: Bulk modulus, K Shear modulus, G

2 ( 1 - -2vm) 1 H 7 - 5vm

8 - 10v m

Indeed, most of the models discussed will adequately predict the behavior for particulate filled systems in the concentration range, 0 < v{ < | ; however, only the S-combining rule, Eq. (9-120), and the Lewis-Nielsen Equation, Eq. (9-116), are in reasonable agreement with the data at higher volume fraction concentrations of filler. A clear distinction between the milder dependence on filler concentration predicted by the S-mixing rule and the substantial dependence predicted by the LewisNielsen equation could be made with data for particulate filled systems at high volume fractions of fillers (vf ^0.5-0.6) for which the properties of the components differ by a wide margin (e.g., diamond particles in an elastomeric matrix).

Auxiliary elastic properties: Young's modulus, E Poisson's ratio, v

9KG 3K + G 3K - 2 G 2(3X4-G)

signed the role of the continuum. The behavior of the material at intermediate concentrations is assumed to follow a smooth transition between the limiting behavior when either phase is identified as the continuum. An assignment of 0C = § (corresponding to "random" close packing) appears to be an appropriate choice for the critical packing fraction. Since the contiguity of the particulate inclusions is "zero" for vf < 1 — (/>c, this choice suggests that the polymeric component can be assigned unequivocally the role of the continuous phase for vf < | . It is interesting to note that all models developed under the primary assumption that the compliant phase is the continuous phase give essentially equivalent predictions for volume fractions of filler in the range 0 < v{ < f.

9.5 Micromechanical Models for Discontinuous Fiber Composites The special microstructure of a lamina of aligned continuous fibers introduced certain simplifying conditions that allowed reasonable estimates to be obtained for the elastic behavior. The situation for discontinuous fibers is much more complex. This complexity is reflected through variable states of fiber orientation and distributions of fiber lengths coupled with dispersed or aggregate textures. These complex and variable features of the microstructure point out the futility of attempting detailed numerical analyses. Similarly, a rigorous treatment of the classical boundary value problems of elasticity theory is hopeless. Bounding methods based on variational treatments have emerged as a fruitful approach to understanding the behavior of heterogeneous material systems (Milton and Kohn, 1988). However, for large differences in the components of the heterogeneous system, the bounds are too far apart

9.5 Micromechanical Models for Discontinuous Fiber Composites

to serve as useful brackets on the elastic properties. Consequently, considerable effort has been devoted to obtaining improved or "tighter" bounds. The central issue in obtaining tighter bounds is the systematic introduction of additional information concerning the microstructure. This information is usually developed in terms of two-point, three-point and higher order correlation functions. A review of the current status of this approach is given by Torquato (1991). This approach yields valuable insights into the role that microstructure plays in influencing a wide range of properties; viz., elastic behavior, dielectric behavior, electrical conductivity and thermal conductivity. However, the assessment of the resulting models requires a large number of structural descriptors in addition to the information available from the direct examination of micrographs; scattering experiments are frequently required to characterize the correlation functions. From a practical point of view, if the number of structural characterizations required to evaluate microstructural descriptors exceeds the number of independent effective properties, then the direct characterization of the mechanical behavior of the heterogeneous system could just as well be used to describe the system. As will be subsequently shown, the number of structural descriptors associated with orientation and aspect ratio distributions approaches the number of independent elastic properties. Consequently, the introduction of additional parameters associated with correlation functions requires the replacement of relatively straightforward mechanical characterization techniques with rather complicated structural characterizations. The introduction of additional information via higher order correlation functions should give a significant tightening of the upper-lower bound

435

bracket. As was the case for continuous fiber systems, the tightening of bounds is achieved primarily through a significant reduction of the upper bound representation while the improvement in the lower bound representation is marginally affected (Chou et al., 1980). These observations suggest that improved lower bounds of the Hashin-Shtrikman type may provide useful estimates of the effective elastic constants for discontinuous fiber systems. Accordingly, we will focus our attention on a bounding approach and introduce simplifying assumptions in order to avoid the introduction of higher order correlation functions. A key issue is the identification of the appropriate statistical descriptors of the discontinuous microstructure that can serve as parameters in combining rules for estimating elastic properties. Consequently, attention will be focused on developing improved bounds for discontinuous fiber systems that are subject to distributions of fiber orientation and fiber lengths. A new formulation (Eduljee et al., 1992 b) for treating dispersed systems will be presented first. This treatment introduces the role of configurational averaging which is subsequently applied in the treatment of aggregated systems.

9.5.1 Bounds on the Effective Elastic Properties of Discontinuous Fiber Composites 9.5.1.1 Dispersed Ellipsoidal System

The starting point for this analysis is the refined variational treatment (Wu and McCullough, 1977) described in Sec. 9.3.3. Using matrix notation, the average strain energy per unit volume for a heterogeneous system, as given by Eq. (9-86), can be

436

9 Elastic Properties of Composites

written as (9-125)

and is stationary at

H(r)p(r) = s° +

\G0{r-r')p'{r')dV

It is necessary to solve the stationary condition, Eq. (9-126), for the average value, «/?», of the stress polarization vector, p. In order to do this, it is first necessary to make certain statements about the structure of the material, so that the integral in Eq. (9-126) can be evaluated. A characteristic structure function St(r — Xt) is defined such that

(9-126)

$(»•-*;)= (9-129) 1 if r is within the /th inclusion 0 if r lies outside the /th inclusion

Substitution of the stationary condition, Eq. (9-126), into Eq. (9-125) gives the stationary value of the strain energy density as

If the microstructure is independent of position, then

This definition is illustrated in Fig. 9-13. For example, if the inclusions are approximated as ellipsoids, then the function St will be the equation of an ellipsoid centered at Xt. Similarly, a structure function Sk(r — Xk) can be defined for the matrix phase. However, the appropriate partitioning of the continuous matrix phase into cells of shape Sk is not obvious. Eduljee etal. (1992 b) have shown that the exact nature of the matrix partitioning is unimportant. For any space vector r,

« f » = i;f<0f> + vm(qm}

ZSk(r-Xk) + ZSl(r-Xl) = \

with

= (C(r)-C°°r1

= \ *~° C° s° + \ S°

(9-127)

The double brackets, «#», signify averaging over location and microstructure of the quantity q: « « » = lim v -»oo y

jqdV

(9-128 a)

y

(9-128 b)

where vf and vm are the volume fractions of the reinforcement phase and the matrix respectively, and <#> is the configurational average of q (a second rank tensor, or a vector). Configurational averaging includes both orientation averaging as well as averaging over the fiber aspect ratio distributions. Consequently, all structural descriptors are embedded in the quantity <#>. The key features of configurational averaging will be treated in a subsequent section. It is evident that the condition (specified through Eq. (9-128)) that the microstructure be independent of position is a key restriction. Relaxation of this condition would require specific details of the coupling of microstructure to position.

(9-130)

1 if r lies within the nth ellipsoid 0 if r lies outside the nth ellipsoid Figure 9-13. Schematic definition of the structure function for the fiber.

9.5 Micromechanical Models for Discontinuous Fiber Composites

Eqs. (9-129) and (9-130) are now applied to the stationary condition, Eq. (9-126). The left hand side of Eq. (9-126) can be written as H(r)p(r) = S T k Hmp'kJr) Sk(r-Xk) + k

+ ZTlHtp'lt(r)Sl(r-Xl)

437

matrix cell is piece-wise constant with a unique value for each ellipsoid. In addition, it has been shown (Wu and McCullough, 1977) that for a function/(r) having ellipsoidal symmetry, the integrals behave as point functions at the origin, viz.

(9-131) (9-133)

where Tk and Tt are the transformation matrices for transforming from the local coordinate system of the fcth matrix cell or the /th fiber respectively, into the global coordinate system. With the help of Eqs. (9-129) and (9-130), the integral on the right hand side of Eq. (9-126) can be written as J G°{r-r')p'{r')&Vf

=

where E° is a modified Eshelby tensor, and is a function of the fiber aspect ratio and the reference phase. Based on Eq. (9-133), the integrals in Eq. (9-132) can be evaluated for ellipsoidal shapes as

(9-132)

r'

= 1 ZTk G0{r-r')p'km{r')Sk(r'-Xk)dV'

+

r' k

+ J ETi G°(y-r')p'l{(r')Sy

E°km Tk

-Xt)dV

r' I

1

«/>»Sk(r-Xk)

Z.TlEftPl[Sl(r-Xl)-

where p'knSf) is the deviation of the polarization tensor in the fcth cell of the matrix from the average polarization field

(9-134) The quantities E?f and E£m are defined by

and p'l{ (r') is the deviation of the polarization tensor in /th fiber from the average polarization field

At this point it will be useful to restrict the treatment to ellipsoidal inclusions. The distinctions between ellipsoidal inclusions and right cylinders will be discussed in a subsequent section. The ellipsoidal geometry offers the advantage of approximating a variety of shapes: for example, continuous fibers (aspect ratio oo), and disc (aspect ratio 0). For an isolated ellipsoidal inclusion, the stress polarization is constant within the inclusion (Eshelby, 1957). Hence it is reasonable to propose, as a trial field, that the stress polarization in each fiber and each

J G°(r-r') Stf -Xt)dVf

= E?f St(r - * i ) (9-135 a)

and $

k

(9-135 b) Eff can be evaluated if the shape of the fiber is approximated as an ellipsoid of known aspect ratio. The components of the resulting transversely isotropic E° are given in Table 9-9. The tensor Ekm cannot be evaluated since the cell distribution of the matrix phase is unknown. It will be shown that for the lower bound on elastic properties (when the matrix properties are assigned to the reference phase), the evaluation of Ekm is not required.

438

9 Elastic Properties of Composites

Substituting Eqs. (9-136) into Eq. (9-137) and simplifying yields

Table 9-9. Components of the tensor £°. = (3oe/i3-4jS

«#»» = 1 + of <Mf £?> + (9-138 a) + iv<M m E° >"1(ff <Mf> + vm<Mm))s°

i/2

E°32 = 4(och

or

2fih2)

(9-138 b) C

22

~

where 1 is the identity matrix. The strain energy density of a heterogeneous material can be expressed in terms of its effective properties as

^ ~" 4C° C° h5 = 1 For aspect ratio greater than one (a > 1):

Y-1

X 2 = (« 2 ~ i)/«2

\n(l

+ X

l-x

)

«{yR*» = f£°C*£ 0

Inserting Eq. (9-138) into Eq. (9-127), and comparing with Eq. (9-139) gives the effective elastic properties of the composite as

For aspect ratio equal to one (a = 1): hi =2/3 h3 = S/15

M

1 + M E° = M H

Therefore, from Eqs. (9-131) and (9-134), for the vector r in the fcth element of the matrix Tk l «/>» (9-136 a)

where Mfcm = (Hm - £ t ° J " '• Similarly, for the vector r in the /th fiber, plt = Ml( Tf1 s° - Ml( E,°f Tf1 « p » (9-136 b) where Mn = {H( — Eft)~1. The average value, «/*», of the stress polarization tensor,/?, is defined by

«/»» = lim i J ( 2 T, /»If S,(r «-oo

V

r'\l

XTkPkmSk(r-Xk))dV

(9-140)

From the definition of M = (H - Eo)~\ it follows that

For aspect ratio less than one (0 < a < 1): Z 2 = a2/(I - a2) ^ =Z[(l+Z2)tan-1(l/Z)-Z] fc3 = Z[Z(1 + Z2) + (1 - 5Z 3 + 2Z 4 ) tan

Pkm = Mkm Tk~ls°-MkmE°km

(9-139)

(9-137)

(9-141)

In order to obtain the lower bound, it is useful to first rewrite Eq. (9-140) with the help of Eq. (9-141) as C* = C° + (vf <Mf #*f> + vm <Mm> H J " 1 • •(t;f<Mf> + ^ < M m » (9-142) The improved Hashin-Shtrikman type lower bound can be obtained by selecting the reference phase to be the matrix, i.e., C° = Cm. Under these conditions, <Mm> = 0, <Mm> Hm = 1, and Eq. (9-142) reduces to: Cfb,d = Cm + (v{ <Mf ##f> + vj-1^

<M f » (9-143)

The subscript "lb" denotes the lower bound, while "d" denotes a dispersed system in the sense that the individual inclusions are considered. Note that the evaluation of Eq. (9-143) requires the configurational averaging of the matrix product

9.5 Micromechanical Models for Discontinuous Fiber Composites

Mf H{. Now, Hf = (Cf - C J " 1 for the lower bound, and behaves like a compliance tensor; Mf = (Hf — Ef y1, and transforms as a stiffness tensor. The quantity, Hf, does not involve the aspect ratio. Further, if the fiber is taken to be isotropic, then the deviatoric quantity Hf can be excluded from the configurational averaging. The effective elastic properties are then given by (t>f <M f >f*f

^ <M f » (9-144)

In all further discussion presented herein, the fiber and the matrix are taken to be isotropic. The quantity <Mf> must be obtained by appropriate averaging over the orientation and aspect ratio distributions. The upper bound for dispersed systems can be obtained by setting the reference phase to be the fiber, i.e., C° = Cf. Under these conditions, <Mf> = 0, <Mf>Hf = l, and Eq. (9-142) reduces to (9-145) However, <Mm> = <(Hm - E J " 1 ) cannot be evaluated directly since E^ is not known. However, by expressing Eq. (9-126) in terms of strain and using the condition that the averaged strain is given by s° = v{ <«f > + vm <8m >, it can be shown (Eduljee et al., 1992 b) that: \-i

(9-146)

Thus, the quantity <Mm> can be evaluated indirectly in terms of <E°>. The evaluation of the improved bounds for dispersed systems requires the determination of configurational averages of <M> and <E°>. The evaluation of these configurational averages requires a knowledge of the fiber orientation distribution and the fiber aspect ratio distribution. These quantities are discussed in a subsequent section.

439

9.5,1.2 Aggregated System

It was pointed out in the introduction that a domain structure is observed in sheet molding materials. This texture arises from portions of the chopped rovings persisting as bundles of filaments in the final material. These agglomerates of filaments tend to behave as locally correlated units so that it is reasonable to expect that this texture could exhibit properties that differ from those of fully dispersed filaments. Micrographs of these materials are reminiscent of "grainy" metals in which the "grains" assume a variety of orientations with respect to the external axes. Consequently, it is useful to partition the domains for this texture into such "grains" with each grain treated as an apparent homogeneous (but anisotropic) material which may be described by an array of elastic constants Cg or compliance constants S g . This abstraction reduces the analysis to that of a one component grainy metal in which the grains assume a variety of orientations with respect to the external axes. The application of this model requires (i) the specification of the properties of a representative grain and (ii) the specification of the appropriate procedures for averaging over the various orientations of the grain. The majority of reported "short-fiber" models fall into the category of aggregate models in the sense that the properties for a collection of aligned inclusions are estimated and then subjected to orientational averaging. The various models differ in the relationships used to specify the properties, Cg, of the aligned system. Specification of the appropriate averaging of the properties of the characteristic grain is dependent upon the distribution of stress and strain throughout the aggregate. If the stress is uniform, the Reuss average

440

9 Elastic Properties of Composites

for a one-component system is applicable, so that the effective properties of the aggregate are given as S* = <S>, where S is the compliance array for the characteristic grain; the angle brackets denote orientation averaging. Alternatively, if the strain is uniform, the Voigt average for a one-component system is applicable; C* = , where C is the elastic constant array for the typical grain. The condition of uniform strain could be envisioned by a parallel connection of the grains. Under this portrayal, the grains assume the role of a microlaminate so that the composite material could be viewed as an aggregate of microlaminates stacked one upon the other. This parallel construction constitutes the laminate analogy of Halpin and Pagano (1969), which corresponds to a condition of uniform strain and thereby implies a prejudice toward the Voigt average. If the typical grain is anisotropic, the Voigt average and the Reuss average can differ by an order of magnitude. As before, these two averages represent extremes in behavior. If the stress and strain fields fluctuate throughout the aggregate, the behavior of the composite material will lie somewhere between the extremes: <S>~ 1 . Christensen (1979) resolves the specification of the orientation averaging procedure by restricting attention to the special case of infinitely long fibers. He envisions a representative volume element as a section of material containing sections of the fibers in a random array. Since the fibers are infinitely long, they are continuous through the surface of the volume element and connect all partitioned regions. This situation gives a condition of uniform strain so that the Voigt average over C is applicable. Christensen's model, although developed from a different point of view, is equivalent to the laminate analogy proposed by

Halpin and Pagano (1969). The differences in the results obtained from these models are due to the choice of relationships used to predict the properties, C, of the aligned continuous fiber laminate. In both cases, the results are specific to the special case of continuous fibers in a random or planar random state of orientation. Halpin and Kardos (1976) attempt to take into account the effects of finite fiber length through an extension of the laminate analogy. In this treatment, the characteristic structural unit is taken as a microlaminate comprised of a collection of parallel short fibers with a finite aspect ratio, a, for all fibers. The properties of the microlaminate are predicted by the Halpin-Tsai relationship with £ = 2 a for the longitudinal modulus EL. These workers proposed that the effective properties could be obtained from an orientation average conducted on the elastic constant array, C, of the aligned short fibers of the microlaminate. It is not apparent in this case that an averaging of the elastic constant array is generally warranted. The discontinuities due to the finite length of the short fibers cause fluctuation in the stress and strain fields, so that the effective properties of the short-fiber composite could be intermediate between the averages of the compliance constants and the averages of the elastic constants. For sheet molding materials with a high concentration of fibers, a domain structure is exhibited that is much smaller in length than the initial length of the chopped roving. At high concentrations the increased contact and compaction of the rovings produce a marked "swirling". As a consequence, the dimensions of the domains over which a local collection of parallel filaments can be identified are substantially reduced with domain aspect ratios approaching 1-2 (McCullough etal., 1983).

9.5 Micromechanical Models for Discontinuous Fiber Composites

It must be emphasized that the apparent discontinuities of the observed domains are artifacts resulting from polishing the surface of the specimen in preparation for microscopy. In the actual material, the domains are connected by the filaments of the initial roving that weave through the surface of the domains. As a consequence of this connectivity of the domains, the strain field may tend toward a uniform condition so that the Voigt averaging of the elastic constants, C, could provide a reasonable approximation. Under this approximation the effective elastic constants of discontinuous fiber systems with aggregated texture, C*, is given by C* = , where the <> symbol represents configurational averaging over the orientation distribution and the aspect ratio distribution associated with the domains. The elastic constants, Cg, may be estimated by constructing a microlaminate consisting of aligned inclusions with the principal axis of the microlaminate taken to be coincident with the filament direction of the domain. The improved bounding models for the special case of aligned ellipsoidal inclusions can be employed to obtain bounds on the effective elastic constants for the domain or "grain", C*j. The subscript "al" is used to denote the special case of aligned inclusions. As with the continuous fiber composites and dispersed short-fiber composites, the lower bound result will be taken as a reasonable conservative estimate for the elastic constants for the domain. Accordingly, Cfb,a = and: CjU = Cm + (vf Mf Hf + vj'1

441

Comparison of Eqs. (9-144) and (9-147 b) show the distinct differences between the aggregate and dispersed models. However, aggregate averaging, and averaging for dispersed systems are equivalent under two conditions: - Spherical reinforcement - the properties are not directionally dependent, and therefore the predictions are independent of averaging method. - Fully aligned fibers - the microstructural geometry for the aggregated system is identical to the dispersed system, and therefore both averaging formulations yield identical results. The determination of the average of C?b,ai over a distribution of aspect ratios may be somewhat obscure. Under this model for aggregated textures (Eq. (9147 b)), the filaments within the grain are approximated by ellipsoidal inclusions with the minor axis taken as the radius of the filament and the major axis taken as one-half the length of the filament within the inclusion. McCullough et al. (1983) present arguments for taking the effective aspect ratio as the aspect ratio of the domain rather than the individual filaments within the domain. In any case, subjective judgments are required to generate an aspect ratio distribution from measurements taken from micrographs. Results presented in a subsequent section will highlight the differences in the predicted properties of aggregated systems and dispersed systems.

(9-147 a)

(vfMf) (9-147 b) For an infinite aspect ratio, Eqs. (9-147) are equivalent to the Christensen model (1979).

9.5.2 Influence of Inclusion Geometry

The treatment of Sec. 9.5.1.1 was restricted to inclusions with ellipsoidal shapes. This focus was motivated by mathematical tractability and the opportunity to deal with a wide range of inclusion types

442

9 Elastic Properties of Composites

(ranging from continuous and discontinuous ellipsoidal inclusions, spherical inclusions and plate-like inclusions) in terms of a single geometric parameter, the aspect ratio. The ellipsoidal shape assumption has been used in the self-consistent field approaches of Wu (1966), Korringa (1973) and Chou et al. (1980), as well as the bounding approaches of Wu and McCullough (1977) and Nomura and Chou (1984). Although the ellipsoidal shape approximation is appealing, it is unlikely that precise ellipsoidal shapes will be encountered in short-fiber composites. The milling actions encountered in injection molding are expected to generate a variety of fiber lengths and end geometries - ranging over various end bevel angles, cone-like ends, parabolic ends, and right circular cylinders. The influence of end-geometry has been investigated by several authors. In particular, Russel and Acrivos (1972) and Russel (1973) used a slender body approximation to examine various geometries for dilute rigid inclusions. Their analysis predicts that blunt geometries, such as cylinders, are less efficient as reinforcing agents than ellipsoidal shapes. Aboudi (1983) has treated the case for aligned rectangular shapes in a triply periodic array. Recently, Carman and Reifsnider (1992) reported a refined treatment for right circular cylinders at finite concentrations. Comparison of their results for aligned inclusions of steel fibers (with an aspect ratio of 50) in an epoxy matrix shows that the predictions of the longitudinal Young's modulus for the ellipsoidal shape are considerably larger than the Young's modulus of the cylinder and that cylindrical shapes give a higher longitudinal Young's modulus than the rectangular shape of the Aboudi model; for all inclusion geometries the transverse Young's modulus and shear modulus were

comparable. In all the shapes considered, the aspect ratio emerges as the dominant geometric parameter governing the elastic behavior of the discontinuous composite. The dependence of the elastic behavior on inclusion shape and end geometry introduces additional complications to the development of models to predict elastic behavior. It can be anticipated that at sufficiently high aspect ratios, the distinctions among the various shapes and end geometries will be suppressed. The following discussion will focus on the treatment of blunt end-geometry of right circular cylinders with the anticipation that the influence of cone-like and parabolic geometries will fall between the ellipsoidal and cylindrical geometries. Stress distributions around and within discontinuous fibers represented by right circular cylinders have been estimated by several workers (Cox, 1952; Rosen, 1964; Kelly and Tyson, 1965) through a "shear lag" analysis; a discussion of these treatments is given by Piggott (1980) and Chou (1992). In these treatments load is transferred from the matrix to the fiber by shear stresses acting over the fiber-matrix interface; an axial distance equivalent to several fiber diameters from the end is required for the build up of a significant load on the fiber. Whitney and Drzal (1987) treated the case for an isolated cylindrical inclusion. This analysis was conducted to provide a basis for interpreting single-fiber fragmentation results. Consistent with this purpose, these workers imposed a free end boundary condition that decreases the reinforcing efficiency of the inclusion and thus provides a lower limit to the cylinder efficiency. They treated the case for transversely isotropic fibers and expansional strains; the results specialized to isotropic fibers for isothermal conditions are summarized be-

443

9.5 Micromechanical Models for Discontinuous Fiber Composites

low for the axial {ozz\ radial (arr), and shear (orz) stress distribution within a cylindrical inclusion of radius R and length L: GZZ{Z) = Ax 8° [1 - (1 + Xz) exp(-

Az)]

(0 < z < L/2) arr(r,z) = <7Z2(r,z) = (9-148) = 80^2-^li82(2-(r/R)2)(l-^z)•exp(-Az)] arz(r,z) = -A± s°Xpx(r/R) exp(- /lz) where s° is the far-field strain and X = The material constants are given as

• — vn

Ef

2U 2R

V cv

These relationships describe the variation of the stress distribution within the inclusion. In particular, the axial stress is zero at the fiber ends and increases to an asymptotic value azz(co) = A1£°, at the midsection, as the length of the fiber increases. This behavior reflects the transfer of load to the fiber by matrix shear stresses acting over the surface of the fiber. The section of the fibers near the ends is referred to as the "ineffective" or "critical" length, L c , and is arbitrarily defined as the length required for the stress to achieve 95% of the asymptotic value. This specification relates the critical length to the material parameter /?; viz., 1.-1375

|

(9-149)

Thus a critical aspect ratio for a cylindrical fiber can be defined as

(9-150)

Critical aspect ratios for selected fibers in a polymer matrix are given in Table 9-10. In order for the fiber to provide significant reinforcement, the fiber aspect ratio must be greater than the critical aspect ratio. Carman and Reifsnider (1992) extended the treatment of Whitney and Drzal to finite concentrations of fibers subject to general unsymmetrical loading cases and included the possibility of load transfer at the ends of finite-length fibers. These workers propose that the local stress in the matrix near the fiber ends, although important for the treatment of failure stress, has negligible influence on the average state of stress of the system. Under this assertion, the average state of stress of the composite is given by

l

l-4vf

2.375

°°) + * m M + <xe (r)]dVr dVc (9-151)

where the quantity J[*]dFc denotes configc

urational averaging over the orientation and aspect ratio distributions, and j[#]dFr V

denotes volume averaging over the posiTable 9-10. Material and aspect ratio parameters for various reinforcing materials. Gm = 0.924 GPa (0.134 Msi), vm = 0.35. Reinforcing material

Carbon (AS-4) Steel Aramid (Kevlar49) Copper Glass

Material parameters

Critical

f

P

aspect ratio <

241 (35.0)

0.25

0.0620

38

190 (27.5) 124 (18.0)

0.30 0.33

0.0699 0.0868

34 27

105 (15.2) 71.6(10.4)

0.3 0.22

0.0944 0.115

25 20

E{ in GPa (Msi)

v

444

9 Elastic Properties of Composites

tion vector r. The quantity <xx(oo) represents the far-field stress for the fiber (x = f) and matrix (x = m) associated with an infinitely long fiber. The term <xe(r) denotes the transient stresses within the fiber associated with the end-geometry. The averaging scheme is of the form of an aggregate average: (9-152) with ff*l=^al(°°)-

The term (ol\(a)y is the average stress induced by the end effects for an individual fiber of volume V{ and aspect ratio a = L/(2R). Carman and Reifsnider based their treatment on concentric cylinder elements in the fashion of the "concentric cylinder assemblage" model discussed in Sec. 9.3. Accordingly, the terms <xal(oo) represent the average stress state for an assemblage of infinitely long fibers, as predicted by the concentric cylinder assemblage model (Hashin and Rosen, 1964). These workers provide sets of expressions containing material parameters and certain coefficients that are determined from boundary conditions and continuity relationships. Upon specification of these coefficients, the desired elastic constants are obtained by the systematic imposition of unit strains. They conducted parametric studies on aligned cylinders which show that the transverse moduli are essentially independent of aspect ratio; a strong dependence on the aspect ratio was obtained for the axial Young's modulus. Based on these observations, Eduljee et al. (1992 a) introduced reasonable approximations to obtain simplified expressions.

Attention will be directed to aligned fibers packed in a random fashion around the alignment direction so that the assemblage is transversely isotropic. The subscript "al" will be dropped with the understanding that the resultant expressions will be subjected to configurational averaging over the orientation and aspect ratio distributions. Consistent with the observations of Carman and Reifsnider, it is reasonable to propose that for aspect ratios greater than the critical aspect ratio, the transverse elastic constants may be approximated by the far-field values associated with an infinite aspect ratio; i.e., for a^> a*, Cfx (a) = Cf^oo) and Cf2(a) = Cf2(oo). The aspect ratio dependence of the remaining elastic constants is obtained by considering the stresses ozz = 0-33, arr9 and arz. For a transversely isotropic array of randomly packed aligned cylinders, the transient stresses developed by Carman and Reifsnider reduce to the parent forms of Eqs. (9-148) proposed by Whitney and Drzal. In order to take into account finite volume fractions, Carman and Reifsnider evaluated the coefficient X (in Eqs. (9-148)) by techniques proposed by Rosen (1964) to obtain R

(9-153)

with 2(1B2 ^ — Averaging the end-stresses over the volume of the cylinder of length L and radius R yields < = -A33-l—-

(9-154)

9.5 Micromechanical Models for Discontinuous Fiber Composites

= -(f)An

e x p ( - 2(pa)

The quantities A are dependent upon the material properties and can be evaluated in terms of the strain by applying boundary conditions and traction continuity at the fiber-matrix interface. For a > a * , the terms involving exp(— 2 fa (la) can be treated as negligible. The quantities,
the models presented, configurational averages are required for the specific quantities <M>, , and <E°> which will be represented by the general symbol <4>. The configurational average can be obtained from a knowledge of the orientation distribution function of the Eulerian angles and an aspect ratio distribution function. Presumably, these distribution functions can be obtained through microscopic examination of sections of the composite material. From such observations, an orientation distribution N (0, 6, \jj) dVQ can be constructed that gives the fraction of fibers at an orientation specified by the angles 0, 0, x// within a volume element of the orientation space, dVQ, as defined in Fig. 9-14. Similarly, an aspect ratio distribution, D (a) da, can be constructed that gives the fraction of fibers with aspect ratio "a" that falls within a±da. Both distributions are normalized in the sense =1

AC 33 C*5(a ) = CUa ) = C*4(oo) - t>f

with AP = Pf - Pm. The quantities C*(oo) correspond to the elastic constants for the improved lower bound for aligned continuous fiber systems. Relationships for computing these terms are given in Sec. 9.3. The simple expressions of Eqs. (9-155) give results within approximately 2% of the values reported by Carman and Reifsnider (1992).

(9-156) (9-157)

(9-155) A Q 4

445

The configurational average is of the form Qa

(9-158) At high volume fractions of inclusions with large aspect ratios, the fiber orientation may be dependent upon the aspect

9.5.3 Configurational Averaging

The role of the microstructure of discontinuous fiber composites is embedded in the specification of the configurational averages of certain tensor quantities. For

Figure 9-14. Definition of axes for fiber orientation.

446

9 Elastic Properties of Composites

ratio; certain orientations may be excluded in order to accommodate fiber packing. In this event no further progress can be made in the reduction of Eq. (9-158) unless a specific relationship coupling the orientation with aspect ratio can be developed. Such relationships are not currently available. At low volume fraction concentrations of fibers with small aspect ratios, the fiber orientation can become independent of the aspect ratio. In order to make further progress, we will assume that the orientation and aspect ratio distributions are independent. Under this condition

with {A > a = | A (, 9,
(9-162 b) (9-162 c)

N((f), 6) = Np(
(9-163)

0

•sin0d(^d0diA

(9-160 b)

These conditions permit orientation averaging and aspect ratio averaging to be treated separately. 9.5.3.1 Orientation Averaging

For simplicity in notation the subscript Q will be omitted for this section with the understanding that (A} refers to orientation averaging. If the tensor A is transversely isotropic, then A is independent of \//. Hence,

(A}= f . ° with

»)

and under the symmetry conditions satisfies the normalization conditions

7f7 0

(9-162 a)

These symmetries impose the condition that the "up" and "down" as well as the "left" and "right" properties of the bulk (9-159) material are on the average equivalent. A tractable model is obtained by further assuming that the orientation distribution (9-160 a) is separable, viz., or

0

axis of the element, then transverse isotropy is induced in the sense that certain elements of A appear in specific combinations. The fiber orientation distribution may be subjected to the following symmetry conditions:

sin0d<£d0 (9-161)

o o

Alternatively, if a uniform (random) distribution in \j/ is imposed about the unique

(9-164a) o 71/2

(9-164b) The quantities A (0, 90)=T(90)AT-1(9e)

(9-165)

Expansions of these matrix relationships yield terms containing sinm 6 cos" 9 • sin0d0 and sinm(/> cosncf) d, >, , and . Munson-McGee and McCullough (1992) introduced four scaled orientation parameters to characterize the state of orientation.

9.5 Micromechanical Models for Discontinuous Fiber Composites

(9-166a) F p = 8 } - 8
(9-166 b)

2-1

fa = 12( 3
1

_ l) = 1 ( 3
(9-167 b) \ '

J

= -(15
<(cos m ) = j ATp (0) c o s m (/> d(p o 7t/2

= j JVa(0)sin0cosM0d0 0

The values of / p range from zero to one. Values of / p = Fp = 0 indicate a totally random distribution in the 1-2 plane. Values of fp = Fp = 1 indicate a totally aligned col-

447

lection of fibers along the 1 axis. The quantity fa corresponds to the Hermans' orientation function (Hermans et al., 1946) that has been widely used to characterize crystalline orientation in semicrystalline polymers. The values of fa range from —1/2 to 1. The case for fa = —1/2 and Fa = l corresponds to all the fibers perpendicular to the 3 axis and contained within the 1-2 plane; values of fa = Fa = 0 correspond to a uniform distribution in the angle 9; fa = Fa = ^ corresponds to all the fibers aligned along the 3 axis. Under the restriction specified previously, the components of the orientation averaged tensor (A} are given in Table 9-11 (Munson-McGee and McCullough, 1992). Two important special cases can be generated from the results summarized in Table 9-11. Planar orientations associated with compression molded sheet molding

Table 9-11. Orientation averages for second rank orthotropic tensors3.

=(k

- / „ ) ( „ -k-n)-

AGO[(1 - / p ) ( l -F.) + |(1 -Q(\

- Fp) - i(l - F.) (1 - Fp)]

= / - i ( l - /„) (1 - /„) (/ - k + ft) + AGO(1 + /„) (1 - Fa)

(A2i) = X - 1(1 + /„) (1 - / . ) ( / - k + n) + AGO(1 -fp) (1 - Fa) - M) + AG O (I - /„) (i - F.)

/a=i[3

a

Replacement of Atj by CV} gives the elastic constant array. Replacement of Atj by i = l,2,3 and mi = 2 for i = 4, 5, 6, gives the compliance constant array.

mj), where mi = 1 for

448

9 Elastic Properties of Composites

materials correspond to fibers confined to lie in a plane. This situation corresponds to a specified value of 9 = n/2 so that fa=—l/2 and Fa = 1. Substitution of these values yields expressions for orientation averaging subject to a planar orientation distribution summarized in Table 9-12. Axial orientations may be associated with rod-like extrusion processes. In this case, the orientation may be random about the extrusion direction. Setting fp = Fp = 0 generates this special axial orientation distribution summarized in Table 9-13. These results show that four independent statistical quantities / p , Fp9 fa, Fa are required to characterize the general state of orientation for a transversely isotropic body (or an orthotropic body subject to

Table 9-13. Axial orientation averages for second rank orthotropic tensorsa.

- f fa


= (A55}a = G0 + | / a [y - A - AG0 Fa

Table 9-12. Planar orientation averages for second rank orthotropic tensors3. a

- y)

y

=

kp+np-±fp[n-k-ri-

Replacement of Atj by Ctj gives the elastic constant array. Replacement of Atj by Sy^mj), where m- = 1 for / = 1,2, 3 and m,. = 2 for i = 4, 5, 6, gives the compliance constant array.

, ^/Cp-^-Fp^p-y)

^C = T [

= 2-1 l = i\

fc« = 4

a uniform distribution about the unique axis). However, the number of independent orientation descriptors can be reduced for certain special distributions. A distribution describing a uniform outof-plane variation away from the condition of a planar distribution (9 = n/2) is of particular interest. For this situation the distribution Na(9) takes on the form Na(9) = JV°; Na(9) = 0;

a

Replacement of Atj by Ctj gives the elastic constant array. Replacement of Atj by Sij^nij), where mi = 1 for i = 1, 2, 3 and mt = 2 for i ~ 4, 5, 6, gives the compliance constant array.

0<9
For this distribution, Fa is related to fa: (9-169 a) <5)

(9-169 b)

9.5 Micromechanical Models for Discontinuous Fiber Composites

In the event that the out-of-plane variations are small such that cos 25 ^(1^25 2 % then, (1-/J=i(3-<S2) 2

(1-F f l ) = §<5

(9-170 a) (9-170 b)

The uniform variation away from the condition of alignment with the 1 axis ((/> = 0) can be described by a special form

449

9.5.3.2 Aspect Ratio Averaging

Under the assumption that the orientation and aspect ratio distributions are independent, the aspect ratio average can be conducted on any state of orientation. It is convenient to use the aligned state as the basis for aspect ratio averaging; viz. (A}a = f Aal D{a) da

(9-176)

Again for simplicity in notation, the sub£<(/)< n

A/p(0 = O;

(9-171)

Thus Fp = / p cos2e p

(9-172 a)

sin2fi 2e

(9-172 b)

In the event that the variations are small such that sin2£ = 2s — (|)fi3, then (l_/p) = | £ 2

(9-173 a)

( 1 - F p ) = §£2

(9-173 b)

A general relationship between the / ' s and F s can be obtained for a unimodal distribution of the form (9-174 a) k

Na(9) =Aacos «9

(9-174b)

A k value of zero corresponds to a uniform (or random) distribution; as k approaches infinity, the distribution becomes a delta function centered at zero. As a consequence of these forms, the F's can be obtained in terms of the / ' s ; viz.,

2-/p

(9-175a) (9-175 b)

scripts "a" and "al" will be dropped for this section with the understanding that <4> refers to the aspect ratio average of aligned inclusions. Aspect ratio distribution data will normally be in the form of a histogram which gives the number of fibers, nk, with aspect ratios in the range ak — Aak/2 to ak + Aak/2 as a function of ak. For this form of data, the aspect ratio average of Atj is given by (9-177 a)

If the intervals are taken as equal, Aak = Aa, then = Z^yfak)/*

(9-177 b)

k

where fk = nk/Y,nk is the fraction of fibers with aspect ratios ak in the interval ak — Aa/2 to ak + Aa/2.

For quantities A^a) that exhibit a complicated dependence on a, accurate aspect ratio averaging will require techniques of numerical integration specified by Eq. (9-176). However, some insights may be gained by introducing simplifying assumptions. For aligned cylindrical inclusions with a > a* the aspect ratio average of elastic constants based on the approximate rela-

450

9 Elastic Properties of Composites

tionships of Eqs. (9-155) take on the simple form

on a simpler function of the variable; viz.

(9-178)

with

<3c

a

(9-179) where g(a) is a simpler function such as a polynomial in the variable; the quantities F , F", ... are the derivatives of F with respect to g(a) evaluated at <#>. With this approximation, the average of F becomes

ak

and Ac4 4 = V55 = v f

This simple result shows that rather than is the proper statistical descriptor of the aspect ratio distribution of cylinders. This result gives heavier weighting to the lower tails of the aspect ratio distribution so that > l/ or ( l / ^ ) " 1 < . As a consequence, the lower tails of an aspect ratio distribution have more influence on the elastic behavior than the higher tails of the distribution. The form of Eq. (9-178) is particularly convenient since the aspect ratio descriptor can be obtained directly from the aspect ratio distribution and the effective aspect ratio taken as (X/a^'1 for direct substitution for the variable ac in Eqs. (9-155). The situation for the quantities required for the aggregate average, the dispersed average and the upper bound average (i.e., C*, M and E°) of ellipsoidal inclusions is not as straightforward. The dependence on aspect ratio of these quantities through the shape functions, hk, yields complicated forms. Nonetheless, the identification of the proper statistical descriptor of an ellipsoidal aspect ratio distribution can be obtained by a replacement approximation described below. A useful approach to obtaining averages of complicated functions involves approximating the function as a Taylor series based

= F«<7» + \F»(g(a) - + ... (9-180) since <{g(a)- <#» = 0. If the function, g (a), is selected so that the terms involving the derivatives are relatively small, then = F « # » so that the average of the function may be estimated by replacing the variable a in the function, F, with its value obtained from <#>. The aspect ratio dependence of C* and M is contained in E° and specified in terms of the shape functions ht. For an isotropic reference phase the components of the transversely isotropic E° given in Table 9-9 can be expressed in terms of two of the shape functions, h2 and /i 4 , as:

• [(1 - x0) - (1 - xo)h2

+ xoh4]

• [(2 - x 0 ) - (2 - x 0 ) h2 + x0 nE =

= - ( 7 ^ J[(l ~ *o) hi + *o' = -(~)x

1/ 1

h

[1+/Z2-4X0/J4]

(9-181)

9.5 Micromechanical Models for Discontinuous Fiber Composites

with x 0 = 1/[2(1 — v0)] and where Go and v0 are the shear modulus and Poisson's ratio for the reference phase. The shape functions h2 and h4 contain a common logarithmic term so that they are related by l/2a2 1 h2 =0 2 2(a2-l) 2Va - 1 (9-182 a) For a i 1, h4 can be expressed in terms of h2 as

451

The deviation of the true average from the replacement estimate is given by

a»] «a 3 > - 3) (9-185)

If 5 is small (in the range of 0.01 to 0.02), the replacement F « a » becomes a good estimate for . The value of S is depen(9-182 b) dent upon the breadth and skewness of the distribution as well as the strength of the and dependence of F on a as reflected by F" (9-182 c) and F'". Eduljee and McCullough (1992) have shown that for a typical glass fiber/ polymer system with a "worst-case" uniform distribution of aspect ratios, 5 is less than 1 % in the range 20 < a < 200. Thus, A series expansion of the logarithmic for systems of interest, the replacement apterm in Eq. (9-182 c) yields a polynomial of proximation should give a reasonably acthe form curate portrayal of the influence of aspect fc2(a) = a £ (2n + 3)"1 (1 - a)" (9-183) ratio distributions. n=O The replacement approximation prowith a = I/a 2 . vides a convenient mechanism for evaluatFor a = 1, h2 = | ; as a->oo, Jz2->0. This ing the role of the distribution of aspect expansion suggests the use of the variable ratios. The effective average aspect ratio a (=l/a2) as the appropriate descriptor ae = < a >- 1 / 2 = - 1 / 2 can be directly of the aspect ratio distribution. Indeed, obtained from histograms of aspect ratio regression analyses show that for a > 10, distributions and used to obtain estimates all of the components of E°, M and C* of <E°>, <M>, and through the recan be accurately represented by a third placement approximation: <E°> = E°«a»; order polynomial in a. Thus for the choice <M> = M«a»; and = C*«a». As g(a) = ot = I/a 2 , the averages of Efj9 Mtj and with the case for cylindrical inclusions, the Cfj = F are of the form emergence of as the appropriate statistical descriptor of the aspect ratio dis = F « a » + f (F" - F") • (9-184) 2 2 3 3 • «a > - ) + | F " K a > - ) + ... tribution of ellipsoidal inclusions emphasizes the importance of the "lower tails" of where F" and F'" are the second and third the aspect ratio distribution in establishing derivatives of F with respect to a evaluatthe elastic properties of discontinuous sysed at . The quantity «a 2 >- 2 ) is a tem; viz. measure of the spread of the distribution; the term «a 3 > —3) is a measure of the skewness.

452

9 Elastic Properties of Composites

9.5.4 Summary

The relationships for the elastic constants for discontinuous fiber composites presented in this section offer the advantage of identifying appropriate descriptors of the complex microstructure of these materials. The analyses of the configuration averages show that the important statistical descriptors of the microstructure are obtained from appropriate moments of the orientation distribution and the aspect ratio distribution. The scaled descriptors of the state of orientation (/ a , F a , / p , Fp) involve the second and fourth moments of the cosines of the angles 9 and 0, or the first and second moments of 2 6 and 2 0. The descriptors of the aspect ratio distribution involve and higher order moments of the variable a = I/a2. To a reasonable approximation the aspect ratio averages of {A = M, E°, C*x) can be obtained by replacing the variable a in the components of E°, given in Table 9-9, by the quantity ~ 1/2 . A formulation capable of distinguishing between dispersed and aggregated microstructures in discontinuous fiber composites was presented. The dispersed microstructure was defined through the use of a structure function. It was shown that dispersed averaging and aggregate averaging were equivalent for the cases of spherical inclusions and for perfectly aligned inclusions. The results predict that dispersed microstructures possess higher elastic properties than equivalent aggregated microstructures due to the more efficient reinforcement of a dispersed system. Most workers focus on the aggregate texture through the orientation averaging procedures employed to account for orientation effects. Aggregate averaging may be applicable to sheet-molding compounds which exhibit a domain microstructure, while the

dispersed averaging technique is more suited to extruded, transfer molded and injection molded composites. It would be useful to compare the model predictions with experimental observations of the elastic properties of discontinuous fiber composites. Unfortunately, data regarding fiber orientation distributions and aspect ratio distributions are rarely reported concurrently with mechanical characterizations. In the following section, the predictions of the various models are contrasted and comparisons made with limited microstructural and mechanical property data.

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems The aligned continuous fiber systems, treated in Sec. 9.3, represent the simplest microstructure with the orientation state specified and the aspect ratio taken as infinite. Particulate systems comprised of spherical inclusions (treated in Sec. 9.4), although subject to more complex internal stress-strain fields, possess a relatively simple microstructure described by a unit aspect ratio which yields an orientation independent isotropic system. The discontinuous fiber systems, treated in the preceding section, possess more complex microstructures that introduce additional parameters into the micromechanical models. The evaluation of the configuration averages, under the restrictions given in the preceding section, introduces four orientation parameters (/ a , / p , Fa, Fp) and at least one "effective" aspect ratio parameter (ae for ellipsoidal inclusions, ac for cylindrical inclusions). With the incorporation of these structural parameters, the models account

453

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

for various states of fiber orientation and a distribution of aspect ratios. Further, the models provide a distinction between dispersed and aggregated textures. The arguments presented in previous sections suggest that the lower and bound expressions can serve as a reasonable (conservative) estimate of the elastic behavior. In this section, certain assumptions will be introduced to simplify the equations relating elastic properties to the composition, concentration and microstructure of discontinuous fiber systems. These simplifications provide a more direct means of assessing the influence of the various structural parameters on elastic behavior and serve as a convenient basis for comparing predicted elastic behavior with experimentally determined elastic constants and structural parameters. An important feature of these models is the reduction to specific models for special values of the structural parameters. For example, the specification of fa = Fa = \, / p = Fp = 0, a = cc generates the relationships for aligned continuous fibers; the specification of a = 1 removes the dependency on the orientation parameters and generates the relationships for spherical inclusions. As described in Sec. 9.5, selected specifications of the orientation parameters generate special states of orientation. It will be useful to consider further reductions in terms of three special states of orientation: aligned, planar and three-dimensional. These reductions are developed below. 9.6.1 Aligned Inclusions

An aligned system consists of perfectly collimated inclusions with the long axis of the inclusions all parallel to the 3 direction. There is no distinction between the dispersed and aggregated models for this situ-

ation; however, the elastic constants developed for this case serve as the basis for aggregate averaging. Following the analyses of Eduljee et al. (1992 a) the elastic constants for aligned ellipsoidal inclusions will be treated first and then compared to the elastic constants for aligned cylindrical inclusions. The lower bound relationship for ellipsoidal inclusions with uniform or effective aspect ratio ae can be written as -0\-l

(9-186)

For simplicity in notation, the subscript "lb" will be dropped in the subsequent analysis. As a consequence of the assumed random packing parallel to the 3 axis, the elastic constant array is transversely isotropic with 22?

23 ~~ ^ 1 3 ?

44

55

C* \ — f^

(C*

Q

~~^2

where L and T stand for longitudinal and transverse, respectively. For isotropic inclusions in an isotropic medium the matrix relationships reduce to five equations of the form [vf AP + vf vm Afc p Hp]/Dp (9-187)

with

5

GLT, C 1 3 , C 3 3 , AP = P{ - Pm

and AE

A/cA£ The quantities Hp are given in terms of the shape functions h (defined in Table 9-8). TT

TT

_i_ i | 2 / ' J

Y

\

He 13 — ^0^4 H

Gur =

H

GTT = 0

(9-188)

454

9 Elastic Properties of Composites

G*T (ae > 20) = GT*T (oo) = Gm +

with

(9-190 a)

»fAGGm(l-vJ The quantities D are given by £GTT=

(9-190 b) 2t;fGmAG

[2 Gm + vJ2 - xo)AG] (9-189a)

vmAG(2-x0) m + vm(2-x0)AG

fe*(«.>20)

(9-190c)

=.

with l-2vn (9-189 b)

vmAG <2LT =

Ak (l+vmpHD)

+ vmQD

-xo)h2

+

(9-189 c)

2Av = AG l-2v The axial elastic constants, C*3 and CJ3, exhibit a dependence on the aspect ratio. The familiar engineering constant £* a n d v^T can be obtained from these quantities through the relationship A/c(l-2v f ) =

v*T =

C*13

3xoh4]-pHD

The term QD can contribute at most 1 % to D and therefore may be neglected. The transverse elastic constants, k* and GfT, are weakly dependent upon aspect ratio. Indeed for aspect ratios greater than ae > 25[(1 - 2vJ/(3 - 4VJ] 1 ' 2 , k* and G& are within 1 % of their asymptotic values at ae->oo. For a typical polymer matrix this corresponds to aspect ratios larger than 10. For aspect ratios ae > 22 [(Gf — Gm)/ (Gf + GJ] 1 / 2 , GfT is well within 1% of the asymptotic value for a continuous fiber system (ae->oo). Thus for ac > 20, the quantities fc*, G*T and G^T can be closely approximated by the expression for continuous fiber systems.

(9-191)

2/c*

= Cf3 -

(9-192)

For aspect ratios greater than 20, the shape functions can be approximated as (9-193 a) = -T = ch2

(9-193 b)

(3 = 2.9)

2g

= 0.83^ (9-193 c)

Using these approximations for the shape function and the conditions E(>Em, Gf>Gm yields the following relationship for v£T and E£: E*{ae>20) = £*(oo) -

(9-194)

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

(9-195) V Vm

< lkc-vmAk(l-Xo)}-

T (AE/GJfff (AE/GJH? 1 2 la e+vm(AE/G (AE/Gm)H°D] v*T(oo) - v{ vf + u m v m + uf r m • AkAv(lkc-vmAk(l

*o)

1

-*o)J

- Hf, — xnc \q =

1 [2vf(l-2vm)-c(l-2vf)]0 2(1-vj The quantities El and v£T are within 1 % of the values for a continuous fiber system for aspect ratios ae > 9[(£f - £ m )/GJ 1 / 2 . For a typical glass polymer system, the limit for continuous fiber behavior is ae > 80. Thus for aspect ratios in this range, an aligned system of ellipsoids can be treated as a continuous fiber composite. Similar simplifications can be developed for cylindrical inclusions of effective aspect ratio aG. For cylindrical aspect ratios ac > (3/2)a*, the quantities fc*(ac) and G%T(ac) are independent of aspect ratio and equal tofc*(oo)and G*T(oo). For aspect ratios ac > (3/2)a*, the engineering constants £J, v£T, and G^T can be closely approximated by neglecting terms involving exp(- 2/?0f aG) [in Eqs. (9-154)]. The elastic descriptions for aligned cylindrical inclusions reduce to the following simple forms E*(ac)

=EZ(co)-v{-j^-

) =k*(oo) G*T(ac)=G*T(oo)

455

(9-196)

These simplified relationships give exceptionally good agreement for values of E*(ac) and GlT(ac) reported by Carman and Reifsnider (1992) for an epoxy/steel composite with aspect ratios of 50 and 100. It is evident from these relationships that the appropriate descriptor for a distribution of cylinder aspect ratios is (\/d) as compared to for ellipsoidal aspect ratios. Both the ellipsoid and cylinder give essentially equivalent values for the in-plane descriptors fc* and GfT. As would be expected, the major differences due to shape are reflected in the axial descriptors ££, G£T, and v£T. The in-plane engineering constants £f and vfT can be obtained from standard auxiliary relationships 1

1 4GTT

1 4/c

v?T £L

£ L (/c-G T T )-4/cG T T v 2 T vTT-

(9497) (9-198)

The transverse Young's modulus, £ T , is dominated by the terms involving GTT and k. Accordingly, ET should be essentially equivalent for aligned cylinders and aligned ellipsoids. In both cases, ET is essentially independent of aspect ratio. The transverse Poisson's ratio, v| T , exhibits a dependence on shape and aspect ratio through the dependency on Ef. The reproducibility of experimental measurements of the elastic descriptions of discontinuous fiber systems is in the range of 5 to 10%. Consequently, it is difficult to distinguish between the shape models when the predictions differ by less than

456

9 Elastic Properties of Composites

10%. The lower limits of aspect ratios for which the predictions of the cylindrical and ellipsoidal shape models differ by greater than 10% are summarized in Table 9-14. For a typical glass/polymer, shape effects are indistinguishable (to within 10%) for aspect ratios greater than ~ 150. For larger differences in E{ and Gm (such as steel/polymer) the limiting aspect ratio increases to -400. The relative reinforcing efficiency of ellipsoidal and cylindrical inclusions can be obtained by comparing the expression for El. In order for the inclusions to reinforce to the same extent such that El(ac) = ££(a e ), then

The quantity acq is the equivalent ellipsoidal aspect ratio that provides longitudinal reinforcement equivalent to a cylinder with aspect ratio ac. Clearly, aeq < ac so that the ellipsoidal shape is a more efficient reinforcing geometry. Experimental verification of the model relationships for aligned discontinuous fiber composites is difficult since perfectly aligned fibers are never achieved under usual fabrication conditions. The following studies illustrate typical behavior for "nearly" aligned systems. Berthelot etal. (1978) and Berthelot (1982) report the elastic properties of steel and copper fibers in an epoxy based polymer matrix. Starting with isolated wires of diameter 0.1 and 0.2 mm, they cut short cylindrical fibers ranging in length from 5 to 20 mm to give cylindrical aspect ratios in the range 50 to 200. They constructed "hand aligned" composite plaques with volume fractions of fibers in the range 0.05 to 0.4. From optical microscopy, these workers concluded that the fibers were aligned to within ±10°.

Table 9-14. Estimates of limiting aspect ratios for which the predictions of the ellipsoidal and cylindrical shape models differ by less than 10%. Elastic property

Limiting aspect ratio

k*

3.5j(Es-EJ/Gm 3.5 J(E,-EJ/Gm

G*T V£T

5V(£ f -£ m )/G m 2(E,-Em)/Gm

E*

lSj{E,-EJ/Ga

Carman and Reifsnider (1992) show that the predictions of the cylindrical model for perfectly aligned (0 = 0) and fibers all misaligned by (j) = 10° closely bracket the experimental values reported by Berthelot et al. (1978) for steel fibers. Application of the simplified relationships of Eqs. (9-196) to (9-198) yields comparable results for the steel and copper systems. Carman and Reifsnider (1992) compare the predictions of the verified cylindrical model to the predictions of E L based on Russel's (1973) ellipsoidal model for infinitely dilute systems, the Halpin-Tsai relationship and Aboudi's (1983) periodic rectangular fiber model. The ellipsoidal model and the Halpin-Tsai relationship give predictions larger than the cylindrical model; the periodic rectangular fiber model yields lower values for EL. Carman and Reifsnider conclude that the cylindrical model is the proper representation for discontinuous fiber systems. Kacir et al. (1978 a, b) examined the elastic behavior of short-glass fibers in an epoxy based polymeric matrix. Composites were fabricated from 3.1 mm (0.125 inches) and 6.2 mm (0.25 inches) chopped tows containing 400 filaments with filament diameters of 10 microns. The volume fractions of fiber were maintained at v{ = 0.5.

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

These workers measured the fiber orientation in compression molded plaques by counting tracer fibers. No degradation of the lengths of the fibers was observed. The highest level of fiber alignment achieved in this study corresponds to a fiber orientation distribution in which 90% of the fibers were within +15° of the alignment axis. Cumulative fiber orientations were reported for 5° intervals. From these data, the highest level of orientation corresponds to an orientation factor of fp = 0.88. The isolated filament aspect ratios correspond to 310 and 620. Aspect ratios in this range imply that the longitudinal Young's modulus is essentially equivalent to that of a continuous fiber system and is given by the simple rule of mixtures. However, the rule of mixtures prediction gives results significantly larger than the observed El. Kacir et al. (1978a) argue that the fibers "... react to the load as bundles and not as individual fibers; therefore it is the bundle thickness and not fiber diameter that should be considered ...". Examination of photomicrographs indicated populations of close-packed bundles with a range of diameters distributed among some isolated filaments. Evidently, some close-packed regions of the starting fiber bundles persisted through the processing steps employed to disperse the bundles. From these observations, Kacir et al. (1978 a) proposed that effective bundle aspect ratios of 53 and 136 be used to replace the filament aspect ratios of 310 and 620. Using these effective aspect ratios and the reported orientation distribution data, Carman and Reifsnider (1992) show that the predictions of the cylindrical model are in excellent agreement with experimental values of El reported by Kacir et al. (1978 b). These effective aspect ratios are somewhat arbitrary and should be viewed as

457

adjustable parameters which account for coherency affects not included in the formulation of the micromechanical models. Fibers aggregated into closely packed bundles would be expected to behave more coherently than fibers separated by larger matrix regions. However, there is no reason to require that the coherent regions assume cylindrical shapes. Indeed, shearing actions introduced during processing to disperse the fiber tow could tend to produce remnants of the bundle structure in a variety of shapes of which ellipsoidal shapes would tend to be stable. The experimental values of El reported by Kacir et al. (1978 b) can be just as well reconciled by assuming effective ellipsoidal aspect ratios of 30 and 45. A wide range of short-fibers in a variety of thermoplastic polymers was studied by Blumentritt et al. (1974,1975). The Young's modulus of the fibers ranged from 6.4 GPa (nylon 6-6) to 230 GPa (carbon fiber) with filament aspect ratios ranging from 200 to 1250. The Young's modulus of the thermoplastic polymers ranged from 0.16 GPa to 2.66 GPa. Using a dry blending process, these workers mixed powdered thermoplastic polymers and chopped bundles of fibers. Specimens with "nearly aligned" fibers were prepared by hand aligning the fibers. The specimens were compression molded in a heated press to "melt" the thermoplastic polymer and consolidate the fiber/polymer mixture. The resultant plaques were 1 mm (0.04 inches) thick; the fiber volume fraction ranged from 0.1 to 0.5. Microscopic examination of the plaques indicated a high degree of fiber alignment and minimal breakage. The workers indicate that the filaments appear to be dispersed; however variable fiber-tofiber spacings were observed. Blumentritt et al. (1974, 1975) reported results for El in terms of a fiber efficiency

458

9 Elastic Properties of Composites

factor, KE9 based on the rule of mixtures; viz. E* = v E K + v E

(9-200)

Comparison of Eq. (9-196) with Eq. (9-200) gives K

-1

(9-201)

In this study values of /? ranges from 0.0077 to 0.42. Thus for these discontinuous fibers with filament aspect ratios in the range 200 to 1250, the ideal values for KE should be greater than 0.90. Blumentritt et al. reported values of KE in the range 0.17 to 0.84. These workers attribute this discrepancy to "imperfections". They concluded that none of the existing models could provide accurate estimates for the observed E£. In addition they conclude that: "A correlation between the fiber efficiency and the fiber or matrix was not found ...". If the quantity a in Eq. (9-201) is treated as an adjustable effective aspect ratio, aef, then a strong linearly increasing relationship is obtained between the aef computed from the reported KEs and the quantity v / ( £ f - £ m ) / G m . In addition, the computed ae's show a systematic decrease with increasing volume fraction. Values for the computed aef's range from 4 to 250. The "bundle effect" (Kacir et al., 1978 a) interpretation suggests that fairly large regions of close-packed filaments persisted in the composite. The strong correlations indicate that the thickness of these bundles tends to increase with volume fraction and decrease as the ratio of (E{ — Em)/ Gm increases. Both of these trends are reasonable. At higher fiber concentrations, the break-up of the starting bundles is more difficult to achieve. Low values for Gm in the solid state are indicative of a lower viscosity of the polymer in the melt. Lower viscosities would facilitate flows to break-

up of bundles during the compaction of the material in the heated press. In summary, the cylindrical model appears to be an appropriate model for dispersed filaments of known cylindrical shapes. However, based on filament aspect ratios, both the cylindrical model and the ellipsoidal model yield predictions of EL which are considerably higher than those experimentally observed by several workers. Both the cylindrical and ellipsoidal models are based on lower bound formulations, and therefore these predictions should fall below the observed values. The incorporation of three-point and higher correlations will cause the predictions to give yet higher values and hence their omission from the cylindrical and ellipsoidal models cannot be the source of discrepancy. Evidently, additional structural features, not considered in the formulation of the micromechanical models, must play a significant role in establishing elastic behavior. The notion of an effective aspect ratio based on clusters of filaments responding as a coherent reinforcing unit appears to provide a means for reconciling a wide range of experimental observations. Unfortunately, the use of effective aspect ratios introduces an adjustable parameter into the model relationships and reduces them to semiempirical status. Hopefully future considerations will provide means for predicting those conditions which promote the formation of coherent clusters which reduce reinforcing efficiency. 9.6.2 Planar Orientation Planar orientations are associated with sheet molding materials and thin compression molded plaques in which the fibers are confined to lie in plane with minimal outof-plane tilting. Under these restrictions

459

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

the planar orientation distribution is a special case of a three-dimensional distribution in which all elements are required to be contained in 1-2 planes such that the angle 6 with respect to the 3 axis is n/2. The 1 axis is taken as the in-plane alignment axis for the specification of the planar orientation variable 0. For symmetric distributions [n (<£) = n(—(j))], the planar orientation averages are obtained by setting the axial orientation parameters of Table 9-11 at / a = —1/2 and Fa = 1. The two surviving parameters, / p and F p , are sufficient to characterize the effect of planar orientation distribution. Systems described by symmetric planar orientation distributions are generally orthotropic so that nine independent material descriptors are required to specify elastic behavior. The special case of random inplane orientation distribution (/ p = Fp = 0), i.e., "quasi-isotropic" systems, is transversely isotropic with the 3 (or throughthe-thickness) axis. In this case, only five independent material descriptors are required to describe elastic behavior. Various planar averaging schemes may be generated from the planar averaging relationships given in Table 9-12. Dispersed averaging is obtained by replacing all A^s in Table 9-12 by M y ; compliance averaging is obtained by replacing Atj (i = 1, 2, 3; j = 1, 2, 3) with Stj and Akk (k = 4, 5, 6) with (l/4)Sfcfe. Most workers treat planar systems through an aggregate averaging of the elastic constant array derived from the consideration of aligned inclusions. This averaging scheme is generated by replacing the (A{jy of Table 9-12 with and specifying the right hand side terms with C* determined from the relationships for the aligned ellipsoidal or cylindrical models given in the previous section. These replacements are summarized in Table 9-15. The five quantities

0 correspond to the

Table 9-15. Aggregate average of elastic constants for a planar distribution. / p = 2 >-l Fp = 2 < c o s 2 2 0 > - 1

=

-L[Et-l

=

G*

= '

= 0 -F p [ 0 -G* T ]

13 > 0

= [/c*(l+v* T )-G* T ]

transversely isotropic (quasi-isotropic) case for a random in-plane distribution. The term is associated with the throughthe-thickness modulus and is independent of the state of in-plane orientation; this quantity is also independent of aspect ratios for ellipsoidal inclusions with ae > 10 and cylindrical inclusions with a c >(3/2)aj. The cylindrical inclusion model predicts that the elastic constants and are independent of aspect ratio for ac > (3/2)a*; the ellipsoidal inclusion model exhibits a weak dependence on aspect ratio through the quantity v£T. Alternatively, the ellipsoidal inclusion model predicts that the shear moduli and are independent of aspect ratio for ae > 20 while the cylindrical inclusion model predicts aspect ratio dependence through the quantity G*T. Both shape models predict a strong dependence on aspect ratio for ( C n ) , and as reflected

460

9 Elastic Properties of Composites

through E£. As was the case for aligned inclusions, the ellipsoid and cylindrical model predictions differ by less than 10% for aspect ratios greater than the limiting aspect ratio. Comparison of model predictions with experimental results is hampered by the lack of reports which include simultaneous measurements of elastic properties and the state of fiber orientation. Based on qualitative inspection, many workers assume a random planar distribution. This approach can be misleading since / p values ~0.3 can give the qualitative appearance of a random distribution (McGee and McCullough, 1981). Even those workers that report quantitative measurements of the state of orientation tend to limit the evaluation of elastic behavior to the Young's modulus. Camacho et al. (1990) examined discontinuous quartz and Kevlar 29 fibers embedded in a dicyclopentadiene Novalac dicyanate thermosetting resin. Composites were prepared by a process similar to paper processing to give mats of well-dispersed filaments. The mats and polymer were consolidated and cured in a compression mold to produce plaques approximately 3.2 mm (0.125 inches) thick. The individual filaments were 6.4 mm (quartz) and 12.7 mm (Kevlar 29) in length with diameters of 9.0 and 12.0 jam respectively. Fiber volume fractions were in the range 0.06 to 0.20. These workers measured the orientation distribution of the isolated filament in the plaques by a procedure described by Konicek (1987). These measurements showed that the filaments were uniformly distributed within the plane (/ p = 0) with a slight out-of-plane tilt of 5-10°. Camacho et al. (1990) used the Halpin-Tsai relationships to obtain the quantities P* of Table 9-15. Upon accounting for void fractions in the range of 7-15%, these workers re-

ported good agreement between predicted and measured elastic moduli in the sense that the differences generally fell within the experimental error of 10 to 20%. The dispersed filament aspect ratios of 710 (quartz) and 1,060 (Kevlar 29) place the systems within the "continuous fiber" range, with E£ = vfEf-\- vmEm. Accordingly, both the cylindrical model and ellipsoidal model are consistent with aspect ratios in this range. Kacir et al. (1978 b) provide a rare study in which measurements of nonrandom fiber orientations are concurrently reported with mechanical property evaluations. These workers used a special converging flow technique to obtain various orientations of strands of E-glass chopped into 3.2 mm (0.125 inches) and 6.35 mm (0.25 inches) length. Each strand contained 400 filaments of 10 jim diameter. Fiber orientation was determined by counting colored tracer fibers and counting X-ray photographs of nylon coated steel tracer fibers. The orientation was determined to be essentially planar with minimal out-of-plane tilting. Orientation distribution data was reported as the cumulative percent of aligned fibers in increments of 5°. These data can be used to generate the quantities > and > required for the determination of / p and Fp. These values for selected systems which cover the range of orientation states are given in Table 9-16. The quantity o corresponds to values obtained via Eq. (9-175 a) for an orientation distribution of the form n((f)) oc cosk(j). The good correspondence between the observed > and >0 suggests that only one independent orientation parameter is required; viz., / p . The quantity Fp° can be obtained as ro

2-/ p

461

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

Table 9-16. Comparison of effective aspect ratios for cylindrical and ellipsoidal regions in various states of planar orientation. Orientation parameters

Aspect ratios

Young's modulus, (E± !> in GPa

Effective

a

4

>



o

/P

Filament

Cylinder

Ellipsoid

Cylinder a

Ellipsoid

Measured b

0.55 0.71 0.90 0.94

0.42 0.60 0.86 0.90

0.42 0.58 0.82 0.89

0.10 0.42 0.80 0.88

635 317 635 317

136 53 136 53

45 30 45 30

21.0 24.6 32.5 30.7

21.0 23.3 30.5 30.6

20.6 23.6 32.5 30.6

Carman and Reifsnider (1992);

b

Kacir et al. (1978 b), as reported by Carman and Reifsnider (1992).

As discussed in the previous section, Kacir et al. (1978 a) proposed that an effective aspect ratio be used rather than the filament aspect ratio to account for populations of close-packed bundles that would tend to behave as coherent reinforcing agents. Using the effective aspect ratios, Carman and Reifsnider (1992) obtained excellent agreement between predicted and measured values for <En>, based on cylindrical inclusion. Again, there is no compelling reason to require that the coherent region retain a cylindrical shape. The data of Kacir et al. (1978 b) can be reconciled in terms of effective ellipsoidal aspect ratios. The results obtained for effective cylindrical aspect ratios and effective ellipsoidal aspect ratios are compared in Table 9-16. Sheet molding compounds are a common form of discontinuous fiber composites exhibiting planar orientation. Sheet molding compounds are made by dropping 25 to 50 mm (1 to 2 inch) bundles (or "roving") containing about 400 glass filament onto a moving belt and impregnating the mat with an unsaturated polyester resin. Composites are fabricated by heating the material in a compression mold to cure the thermosetting resin. Although the sheet molding process is intended to generate nearly random in-plane orientation,

the belt movement and flow within the mold cavity can induce some orientations. Microscopic examination of low fiber content SMC-25 material (weight fraction = 0.25, i;f = 0.2) shows that the bundle structure is preserved. However, the bundle aspect ratio is approximately 200. As a consequence, the material can be approximated by a continuous fiber system. Using an average over elastic constants, McCullough et al. (1983) show that an orientation factor of / p ~ 0.2 and ae -> oo bring predicted values into good correspondence with measured values of {EX1} and (E22y. For the higher fiber content material SMC-65 (weight fraction = 0.65; vf = 0.5), the dense packing and overlapping of the rovings produce a grainlike structure of locally parallel filaments. Within a given plane these grainlike domains have small length-to-width ratios as compared to the original tow. These domains are connected by common filaments which are much longer than the length of the domain. McCullough etal. (1983) show that the experimental and predicted values of <£n> and (E2iy could be reconciled with / p = 0.3 and ae w 2. These workers propose that a value ae« 2 is consistent with the observed in-plane domains of denselypacked parallel filaments.

462

9 Elastic Properties of Composites

In summary, experimental evidence suggests that orientation averaging of the elastic constant array C captures the essential features of planar orientation distribution for systems with large filament aspect ratios. Comparison of orientation parameters / p and Fp with measured orientation distributions imply that Fp is dependent on / p so that only one independent orientation parameter is required to describe planar orientation. Comparison for other orientation distributions are needed to substantiate this simplification. With the exception of systems with isolated filaments, predictions based on the filament aspect ratio yield values for <£1X> which are larger than observed. Since lower bound results are used as the basis for both the ellipsoidal and cylindrical inclusion models, the predicted < £ n ) should be smaller than the observed values. As was the case for aligned systems, the notion of an effective aspect ratio can be employed to reconcile model predictions with experimental observation. The smaller value of the effective aspect ratio is rationalized in terms of close-packed bundles of filaments that appear to respond to loads and deformation as coherent units. At this time no model is available to predict effective aspect ratios. Accordingly, resort has been made to using the effective aspect ratio as an adjustable structural parameter. As shown in the previous section, the notion of the effective aspect ratio serves as a basis for establishing data reduction schemes that appear to provide systematic correlations. 9.6.3 Three-Dimensional Orientations

Three-dimensional states of orientation are associated with section thickness much larger than the lengths of the fibers. In this case, dimensional constraints no longer

confine the fibers to a plane. Composite structures of this type are usually produced by injection or transfer molding processes in which mixtures of polymer and chopped fibers are forced through gates and runners into a mold cavity in the shape of the desired part. The varying flow fields within the cavity carry the fibers into complex states of three-dimensional orientation. Converging flows through constricted regions tend to enhance fiber alignment; diverging flows from constricted regions tend to promote random orientations. For complex mold geometries, the state of orientation can vary with location. Even simple rectangular bars can exhibit spatially dependent orientations. The fibers near the surface of the cavity tend to align in the flow direction; fibers near the midsection approach random orientations. The thickness of the "skin" of nearly aligned fibers is dependent on flow-rates and the temperature of the mold surface. The development of processing models to predict the state of local fiber orientation induced in mold cavities with complex geometry remains a major challenge to the advancement of the technology of short-fiber composites. The milling actions encountered while forcing the polymer/fiber mixture into the mold cavity tend to produce dispersed filaments. Consequently, the aggregate averaging scheme may not be applicable to these materials. Injection and transfer molding processes require fibers of substantially smaller lengths and lower volume fractions of fibers than those used in SMC compounds. In addition, the mechanisms used to force the material into the mold cavity cause extensive fiber fragmentation resulting in a wide range of aspect ratios and end-shape geometries. These complex and varying structural features present a considerable challenge to both the development of micromechani-

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

cal models and the experimental characterization of the materials. Even in the ideal situation of spatially independent states of orientation, four orientation parameters are required to characterize three-dimensional states of orientation (fa9 Fa9 / p , Fp). As a consequence of fiber fragmentation, at least two aspect ratio parameters may be needed to describe the mean and spread of the aspect ratio distribution. These structural characterizations are difficult to conduct and as a consequence insufficient experimental data are available to test the validity of micromechanical models for discontinuous fibers in three-dimensional states of orientation. Semiempirical correlations with four to six adjustable parameters are expected to be successful but will give limited insights into the systematic development of optimum processes. Alternately, such semiempirical correlations may be the only means of relating processing conditons to the mechanical performance of short-fiber composites. In this application, the micromechanical model provides a rational basis for identifying the key structural parameters and the proper form for the correlations. In the remainder of this section, simplifying conditions will be introduced to facilitate the identification of significant trends and provide a framework for data reductions to establish process-property correlations. Attention will be directed to representative volume elements which are large enough to sample the local homogeneous elastic characteristics of the material but small enough to avoid significant spatial variations in the state of orientation. This approach has been employed by Bozarth et al. (1987) to model the skin-core behavior of injection molded bars. Fiber orientation distributions were measured for sections of the skin and the core. These structural parameters were used to predict

463

the elastic properties of the skin and core layers. These predicted properties were used as input to a traditional laminate analysis to predict the over-all behavior of the bar. Good correspondence was obtained between the observed and predicted longitudinal Young's modulus as well as the longitudinal and transverse coefficients of thermal expansion. Under the restriction of spatially independent fiber orientation, the general three-dimensional state of orientation within the representative volume element can be characterized by the relationships summarized in Table 9-12. Various averaging schemes may be generated from these relationships. The dispersed averaging scheme is obtained by replacing all Atj by Mij9 compliance averaging is obtained by replacing Atj (i = 1, 2, 3; j = 1, 2, 3), Su and Akk (k = 4, 5, 6) with (1/4) Skk; the Stj on the right-hand side are obtained from the predictions of a selected micromechanical model (e.g., ellipsoidal or cylindrical). Aggregate averaging is obtained by replacing (Atjy by and specifying the righthand side terms with Ctj determined for a selected micromechanical model for aligned inclusions. As will be shown, the various averaging schemes yield quite different results. In general, four orientation parameters (/«, Fa> fP, FP) are required to specify the state of orientation for symmetric distributions. For the current purposes, it will be useful to assume that the fiber orientations can be represented by functions of the form cosk* x (x = 9, (j)) so that only two independent parameters are required (/fl, / p ). The quantities Fp and Fa are related to fp and fa by the relationships given in Sec. 9.5.3.1. Axial orientations are a special case of three-dimensional orientations obtained by setting / p = Fp = 0 so that the distribution is random about the 3 axis. Such dis-

464

9 Elastic Properties of Composites

tribution could be encountered in rod-like extrusion processes. The reductions of the general Atj to the axial form are given in Table 9-17. This special distribution is transversely isotropic with respect to the 3 axis so that five independent elastic constants are required to describe the elastic behavior. For fa = l9 the relationships revert to the aligned case treated in Sec. 9.6.1. The quantities Ko and Go correspond to the bulk modulus and shear modulus of an isotropic material generated by a random three-dimensional distribution (i.e., fa = f^ = Fa = Fp = 0). Since an isotropic system can be described by two independent elastic constants, the Young's modulus and Poisson ratios for an isotropic three-dimensional random distribution are given by 1

Vn =

1 1 3Xn 9K« 3 Ko — 2 Go

(9-202 a)

(9-202 b)

The elastic behavior of random three-dimensional orientations is strongly dependent upon aspect ratio as reflected by the dependence of Ko, Go (or £ 0 , v0) on ££. This dependence of Eo and Go on aspect ratio is illustrated for a typical glass/polymer system in Figs. 9-14 and 9-15. For these examples, the aspect ratio dependence is based on an ellipsoidal model. The ellipsoidal aspect ratio dependence depicted in Figs. 9-15 and 9-16 can be converted to cylindrical aspect ratios by treating Eq. (9-199) as a scaling relationship relating ellipsoidal and cylindrical aspect ratios. These plots show that the change in moduli with respect to change in aspect ratio is largest in the region 1 < a < 10 and decreases for a > 10 to become vanishingly small for a > 100. In all cases, for a > 100 the moduli are essentially equivalent to the corresponding moduli for continuous fiber systems. The behavior of biased three-dimensional distributions for which fa ^ 0, / p ^ 0

Table 9-17. Aggregate average of elastic constants for an axial distribution. /. = j[3-l] Fa = i[15-7] o_/J10/o-71 5-2fa = = K0+ | G 0 - f faKk}0 = Ko + | G o + §/„ [Et -k*{\-

+ 0 - k* - G| T ]

f AG 0 Fa

4v*T2) - GTT] + 2 AG 0 Fa

= Ko - f Go - f/J 0 - 0 -fe*+ G*T] + | A G 0 J = = Ko - f Go - i / . [ k * ( l - 2v*T) - G M - AG 0 1 = = = G o + i / . [ G & - Gft.] - AG0 Fo = = Go - f = i [ >0 - GfJ + {

15G o =

6(Gf

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems 80 a=1 O=10

60 Ui

T3 O 2

0=100

J

40

20

>i

0.2

1 0.4 0.6 0.8 Volume fraction. Vi Figure 9-15. Elastic modulus of a polycarbonate/ glass composite with a three-dimensional random fiber orientation distribution as a function of fiber volume fraction and aspect ratio.

35 3 30

i\

CD

«/> 2 0

].

/

i

h «

0.2

0.4

0.6

0.8

bution (n((j)) = A cosk<}) (j)\ this condition

occurs at /p* = 2 - ^ 3 = 0.27. The quantity decreases from the maximum at (/ a = - 1 / 2 , / p = 0.27) to the G*T for the aligned case (/fl = 1, / p = 0). The term <^455>, associated with , has a maximum value for / p = 1 subject to

I 5 0

dominates the behavior, <£*!> uniformly decreases with increasing fa and decreasing / p . The minimum value of <£*!> occurs at (/ p = 0, fa = 1) corresponding to E% for the aligned case. The maximum value for (^A17y occurs at (/ p = 0, fa = —1/2). This corresponds to the value of <£*2> f° r a random planar distribution; <£*2> decreases as / p and fa increase to the minimum at fa = i corresponding to Ef for an aligned system. The shear terms attain maximum values at an intermediate state of orientation. The quantity (A66), associated with , has a maximum value at fa = —1/2 subject to dfp° = 0. For the assumed distrifP=fl

o=5 o=10 o=100

T25

465

1

= 0. For the assumed coske(9)

Volume fraction, Vf

Figure 9-16. Shear modulus of a polycarbonate/glass composite with a three-dimensional random fiber orientation distribution as a function of fiber volume fraction and aspect ratio.

requires quantitative values of fa and / p . However, general trends can be inferred from the relationships given in Table 9-11. The term <^ 33 > in Table 9-11 is independent of / p , but strongly dependent upon fa. Since the Young's modulus, <£33>, is dominated by this term, variations in fiber orientation in the 1-2 plane exert minimal influence on <£ 33 >. The quantity < i n ) of Table 9-10 takes on a maximum value at fa = —1/2 and / p = l, corresponding to the case for aligned inclusions, £J\ Since this term

form of the orientation distribution this occurs at f* = 1/2(5 - 3 y/2) = 0.38; G13 decreases from G13 (fa = 0.38; / p = 1) to the value for the planar random distribution (fa = - I/ 2 , /p = 0). The shear term {A44}, associated with , has a maximum value at fp = 0 subIn this case 3AGn the location of the maximum is dependent on the combination of materials. For a typical glass/polymer system, this maximum will occur in the range / / = 0.4-0.6. The shear modulus G23 decreases from a maximum value at G12 (fa = f*9 fp = 0) to the value for the planar aligned case (/ = —1/2,

ject to

a= f

466

9 Elastic Properties of Composites

The off-diagonal elements , <^413>, are summarized in Table 9-18. These results suggest that optimum states of orientation could be designed to generate a well balanced collection of elastic constants in the sense that some sacrifice in the Young's modulus could lead to enhanced performance in shear. Currently, a technology is not available to manipulate fiber orientations into prescribed states of orientation. The rationalizations used to justify the aggregate averaging scheme may not apply to highly dispersed systems. In this situation, the dispersed averaging scheme may provide a more accurate prediction of elastic behavior. For the lower bound model the dispersed averaged elastic constants are given by

Table 9-18. Location of maxima and minima in three-dimensional state of orientation. Maxima

General Aggregate tensor model

(9-205)

= {Ct-Cm)

Applying the approximations used to simplify the expression for C* yields the following expressions for the components of the transversely isotropic tensor M:

km(oo) =

Ak

(l-x o )(A/c/GJ

/.

/P

-1/2 -1/2 1

1 0 0

1 1 -1/2

0 0 1



(A23>



-1/2 0.38 0.4-6.6

0.27 1 0

1 -1/2 -1/2

0 0 1

O*55> <^66>



0.4-0.6 0.38 -1/2

0 1 0.27

-1/2 -1/2 1

1 0 0

M 1 3 (a) = M 1 3 (oo)-

1 -

(AE/GJ(2v(HD-x0h4) t[l + (AE/Ga)HD] AC

M 13 (oo) =

(9-206 b)

13

(AE/GJHD l+(AE/GJHD/ (9-206 c)

(AE/Gm)(l-x0)Ak

M33(co) =

(l-xo)(Afc/GJ

M 6 6 (a) = - [Mt x (a) - Mt2 (a)] = M 6 6 (oo) 1 +

M 66 (OD)

2 Gm - M 6 6 (oo) [(2 - x 0 ) h2 - x0

M66(oo) =

2G m AG

1+

(AE/GJ(l-xo)h22 1+(AE/GJHD

/P



(9-203)

(9-204)

fa <^n>

C,*b,d = Cm + vt(v,Hf + »m<Mf > " 1 ) " 1

where the quantities <M f > y are obtained by replacing all Atj in Table 9-11, with the My. The Mi} are generated from the shape tensor E^ and elastic constants of the components.

(9-206 d)

(2-x 0 )AG M44(oo)

2Gm-M44(oo)(4x0h-h2) 2GmAG M 44 (oo) = (9-206 e) with AP = P f - P m , xo =

(9-206 a)

Minima

HD = (1 —xo)h1h2

1

2(1-vj

+ xoh4..

, and

467

9.6 Relationship of Microstructure to Elastic Behavior of Discontinuous Fiber Systems

Comparisons between the Young's moduli and shear moduli obtained under the dispersed and aggregate averaging schemes for axial orientations (/ p = 0) are given in Figs. 9-17 and 9-18. These comparisons are based on a glass/polycarbonate system with 0.4 volume fraction glass fiber and an ellipsoidal aspect ratio of 100. An ellipsoidal aspect ratio of 100 is equivalent to a continuous fiber and was selected to emphasize the difference between the two averaging schemes. The differences will decrease with decreasing aspect ratio and converge to equivalent predictions for a = \. The comparisons displayed in Figs. 917 a and 9-17 b show that the dispersed av-

eraging scheme yields consistently higher values for the Young's moduli. The largest difference in the axial Young's modulus, E 3 , occurs for a random planar orientation (fa = - 1/2, / p = 0). This difference decreases with increasing axial alignment and vanishes for an aligned system (fa = 1, / p = 0). The dispersed and aggregate averages of the in-plane Young's modulus ET = E2 are equivalent for both the planar random state (fa = -1/2, fp = 0) and the aligned state (/fl = l, / p = 0). However, at intermediate states of axial orientation, the dispersed average moduli are significantly larger than the aggregate averaged moduli. The Young's modulus for dispersed averaging for a random three dimensional state

35 30

Dispersed averaging Aggregate averaging

(a)

Dispersed averaging Aggregate averaging

(a)

\

to 3 i -0.5

0.5

-0.5

0.5

18 — CO

16

Dispersed averaging Aggregate averaging

(b) :

O

uf

12

LJlUS,

14

10 —

-0.5

0.5

Figure 9-17. (a) Axial modulus (E3) and (b) in-plane modulus (El=E2) predictions for aggregated and dispersed polycarbonate/glass systems; / p = 0.0, v{ = 0.4, aspect ratio = 100.

Dispersed averaging Aggregate averaging

1.5 -0.5

0.5

1

Figure 9-18. Shear modulus predictions, (a) G12 and (b) G13 = G23, for aggregated and dispersed polycarbonate/glass systems; fp = 0.0, vf = 0.4, aspect ratio = 100.

468

9 Elastic Properties of Composites

of orientation (/R = / p = 0) is about 25% larger than the aggregate averaged Young's modulus. The comparison of the dispersed averaging scheme and the aggregate averaging scheme for the shear moduli is shown in Fig. 9-18. Similar to the in-plane Young's modulus, the largest differences in the transverse modulus, GTT, occur for a random planar orientation (fa = —1/2, fp = 0), decreases with increasing axial orientation and vanish for an aligned system (/a = l, /„ = 0). The dispersed and aggregate averaged axial shear moduli, GLT, are equivalent for both the planar random state (fa = -1/2, / p = 0) and the axially aligned state (fa = 1, / p = 0). As expected from the previous discussion, the largest differences occur at the maximum at fa « 0.4 with the dispersed averaging result approximately 50% larger than the aggregate averaging result. These comparisons show that dispersed textures provide significantly enhanced reinforcing efficiencies over aggregated textures. Indeed, if the effective aspect ratio for clusters is used as a basis for predicting the aggregate elastic properties and the filament aspect ratio for dispersed elastic properties, the difference between the dispersed and aggregate textures is further increased.

9.6.4 Summary

Simplifications were introduced to provide a direct means of assessing the influence of various structural features on the elastic behavior of discontinuous fiber systems. The structural features treated were the state of orientation as specified by scaled orientation parameters (fa, Fa, / p , Fp), aspect ratios, and shape effects (associated with ellipsoidal and cylindrical inclu-

sions), as well as dispersed and aggregated textures. Comparison of the predictions from the dispersed averaging scheme and from the aggregate averaging scheme show that the dispersed textures provide significantly enhanced reinforcing efficiencies over aggregated textures. If the effective aspect ratio for clusters is used as a basis for predicting the aggregate elastic properties, and the filament aspect ratio for dispersed elastic properties, the difference between the dispersed and aggregated textures is further increased. The cylindrical shape of Carman and Reifsnider (1992) appears to be an appropriate model for the dispersed filaments of known cylindrical shapes. However, based on filament aspect ratios, both the cylindrical model and the ellipsoidal model yield predictions of EL which are considerably higher than those experimentally observed by several workers. Both the cylindrical and ellipsoidal models are based on lower bound formulations, and therefore these predictions should fall below the observed values. The incorporation of three-point and higher correlations will cause the predictions to give yet higher values, and hence their omission from the cylindrical and ellipsoidal models cannot be the source of discrepancy. Evidently, additional structural features, not considered in the formulation of the micromechanical models, must play a significant role in establishing elastic behavior. The notion of an effective aspect ratio based on clusters of filaments responding as a coherent reinforcing unit appears to provide a means for reconciling a wide range of experimental observations. At this time no model is available to predict effective aspect ratios. Accordingly, resort has been made to using the effective aspect ratio as an adjustable parameter which can

9.7 Summary

serve as a basis for establishing data reduction schemes that appear to provide systematic correlations. Experimental evidence suggests that orientation averaging of the elastic constant array C captures the essential features of planar orientation distribution for systems with large filament aspect ratios. Comparison of orientation parameters, / p and Fp, with measured orientation distributions imply that Fp is dependent of / p so that only one independent orientation parameter is required to describe planar orientation. Comparisons for other orientation distributions are needed to substantiate this simplification.

9.7 Summary The prediction of effective elastic moduli is a central problem underlying the development of quantitative structure-property relationships for a wide range of heterogeneous materials - e.g., composites, polycrystalline aggregates, and partially crystalline polymers. Indeed, as illustrated in Fig. 9-1, all of these material systems can be encountered at various levels in the structural hierarchy of composite materials. In this treatment, attention has been directed to the highest microstructural level consisting of reinforcing inclusions in a matrix phase. A rigorous prediction of effective elastic moduli requires a complete description of the microstructure. Rigorous treatment can be developed for woven or braided fiber architectures that can be competely described by periodic functions. However, most heterogeneous systems do not possess a precise deterministic microstructure. The stochastic nature of the microstructure of many important heterogeneous sys-

469

tems requires a statistical description of the microstructure in terms of distribution of shape and orientations. As illustrated in Fig. 9-2, the dominant geometrical features of the microstructure may be quantitatively described by orientation parameters which characterize the average state of orientation of the reinforcing agents and an effective aspect ratio which reflects the role of the average shape of the reinforcing agent. The effective aspect ratio can range from a value approaching zero for plate-like inclusions, to 1 for spherical inclusions, to a value approaching infinity for continuous fibers. Discontinuous fibers fall in a range between one and infinity. An essential step toward an understanding of the behavior of heterogeneous materials is the identification of the appropriate fundamental descriptions of the system which are susceptible to experimental characterization. It is apparent that the following information is essential input to any model to describe the behavior of composite materials: - the elastic properties of each of the components - the volume fraction concentration of each component In order to distinguish among the behaviors of widely different classes of composite materials (e.g., continuous fiber composites, particulate reinforced composites, and short-fiber composites), the following additional information is required: - a measure of the load transfer efficiency associated with various reinforcing geometries (e.g., the effective aspect ratio, - a measure of the state of fiber orientation (e.g., the orientation parameter, / ) In view of the complex and variable features of the internal microstructure, a rig-

470

9 Elastic Properties of Composites

orous treatment was abandoned in favor of the development of bounding models which attempt to capture the dominant features of the reinforcing geometry and the state of orientation. The bounding approach avoids the problems of specifying the explicit nature of the microstructure and the internal stress-strain field by employing variational principles to establish upper and lower bounds on the properties. The results from these bounding methods can provide practical guides to material behavior only if the upper and lower bounds are reasonably close together so as to bracket the properties to within experimental error. The Hashin-Shtrikman upper and lower bounds on longitudinal properties of continuous fiber composites are very close and converge to the simple results of the rule of mixtures. Unfortunately, the upper and lower bounds on transverse properties, shear moduli, and longitudinal properties of aligned discontinuous fiber composites are too far apart to be of much practical value. In light of experimental errors involved in characterizing composite materials and scatter in properties, it is prudent to base design parameters for composites on reasonably conservative estimates of the properties. The traditional Reuss model of Eq. (9-65) provides an ultraconservative estimate that would lead to inefficient use of expensive materials. The HashinShtrikman improved lower bound offers a more sensible yet adequate conservative estimate that incorporates the assurance that the observed properties of the material will be as good as or slightly better than the predicted values. Hence, it is recommended that the improved lower bound relationships (or the equivalent doubly embedded relationships or the composites cylindrical assemblage relationships) be used to predict transverse properties and shear mod-

uli of continuous fiber composites. These results are summarized in Table 9-5. Alternatively, these quantities may be computed by the Halpin-Tsai equation summarized in Table 9-6 a with the appropriate values of £p computed from the properties of the matrix as indicated by the relationships summarized in Table 9-6 b. Values for ET and vTT must be computed from the auxiliary relationships given in these tables. Particulate filled systems represent a higher level of complexity beyond continuous fiber composites. The discontinuities of particulate filled systems introduce significant fluctuations in the internal stressstrain fields which complicate the analysis. However, the ample spherical reinforcing geometry precludes a dependence on orientation so that consideration of this structural feature is not required for the analysis of these systems. Since these systems are isotropic, only two independent material constants (e.g., the bulk modulus K and the shear modulus G) are required to characterize elastic behavior. The improved Hashin-Shtrikman upper and lower bounds for K and G remain too far apart to bracket the behavior for particulate systems. However, the HashinShtrikman lower bound and the concentric spheres model adequately predict elastic behavior in the concentration range 0 < v{ < l / 3 and yield conservative estimates for vt > 1/3. The semi-empirical S-combining rule (McGee and McCullough, 1981) provides predictions in agreement with experimental data over a wide range of concentrations and for a variety of materials. This combining rule was developed by making use of the symmetry of the improved bounding expressions, with respect to the interchange of the role of the components for the continuous phase - the lower bound expression. The relationships devel-

9.7 Summary

oped from this approach are summarized in Table 9-8. The notion of a critical packing concentration, 0C, is used to designate transitions in phase continuity (or "contiguity"). When the volume fraction of either component is greater than this critical packing concentration, it is possible to unambiguously identify this phase as the continuous phase. Under these conditions of "zero" contiguity for the second component, the material system tends toward the behavior described by the bounding expressions in which the first (major) component is assigned the role of the continuum. The behavior of the material at intermediate concentrations is assumed to follow a smooth transition between the limiting behavior when either phase is identified as the continuum. An assignment of c = 2/3 (corresponding to "random" close packing) appears to be an appropriate choice for the critical packing fraction. Discontinuous fiber reinforced materials represent a higher level of complexity than continuous fiber arid particulate systems. The processing conditions used to fabricate discontinuous fiber composites can cause fiber fragmentation resulting in a distribution of fiber lengths and a variety of end shapes which influence the capacity of the fiber to accept load transfer from the matrix. Varying flow fields carry the fibers into complicated states of orientation. In addition, two distinct textures have been observed for discontinuous fiber systems: dispersed and aggregated. A formulation capable of distinguishing between dispersed and aggregated microstructures in discontinuous fiber composites was presented. The dispersed microstructure was defined through the use of a structure function. It was shown that dispersed averaging and aggregate averaging were equivalent for the cases of spherical

471

inclusions and for perfectly aligned inclusions. The results predict that dispersed microstructures possess higher elastic properties than equivalent aggregated microstructures due to the more efficient reinforcement of a dispersed system. Most workers focus on the aggregate texture through the orientation averaging procedures employed to account for orientation effects. Aggregate averaging may be applicable to sheet-molding compounds which exhibit a domain microstructure, while the dispersed averaging technique is more suited to extruded, transfer molded and injection molded composites. A key issue in the analysis of discontinuous fiber composites is the identification of appropriate statistical parameters, representative of the microstructural features of the composite. The analysis reveals that the important statistical descriptors of the microstructure are obtained from appropriate moments of the fiber orientation distribution and the aspect ratio distribution. The scaled descriptors of the state of orientation (fa9 Ffl, / p , Fp) involve the second and fourth moments of the cosines of the angles 6 and 4>, or the first and second moments of 26 and 20. The descriptors of the aspect ratio distribution involve for ellipsoids and for cylinders. To a reasonable approximation the aspect ratio averages can be obtained by replacing the variable a by the quantity ~ 1/2 or (I/a)"1. The elastic behavior of discontinuous fiber composites is further influenced by the inclusion shape. Ellipsoidal inclusions yield higher estimates for the elastic properties than cylindrical inclusions of the same aspect ratio. Therefore, ellipsoidal inclusions provide more efficient reinforcement than cylindrical inclusions. The distinction between inclusion shapes and

472

9 Elastic Properties of Composites

geometries is suppressed with increasing inclusion aspect ratios. The cylindrical shape model of Carman and Reifsnider (1992) appears to be an appropriate model for dispersed filaments of known cylindrical shape. However, based on filament aspect ratios, both the cylindrical shape model and the ellipsoidal shape model yield predictions for longitudinal Young's modulus which are considerably larger than those experimentally observed by several workers. Both the cylindrical and ellipsoidal models are based on lower bound formulations, and therefore these predictions should fall below the observed values. Evidently, additional structural features, not treated in the formulation of these models, must play a significant role in establishing elastic behavior. The notion of an effective aspect ratio based on clusters of filaments responding as a coherent reinforcing unit appears to provide a means for reconciling a wide range of experimental observations. At this time no model is available to predict the effective aspect ratios for clusters. Finally, the general model for the effective elastic properties of discontinuous fiber composites can be reduced to the special cases of aligned fiber orientation, planar fiber orientation and three-dimensional orientation by appropriate assignments of the fiber orientation and fiber aspect ratio descriptors. Simplifications are introduced to provide a direct means of assessing the influence of various structural features on the elastic behavior of discontinuous fiber systems. The resulting models can be extended to predict the elastic behavior of partially crystalline fibers or matrices by assigning various crystalline morphologies the role of reinforcing inclusions in a matrix of "amorphous" material.

9.8 References Aboudi, J. (1983), Int. J. Solids Struct. 19, 693. Adams, D. K, Doner, D. R. (1967a), J. Compos. Mater. 1, 4. Adams, D. R, Doner, D. R. (1967b), J. Compos. Mater. 1, 52. Adams, D. R, Tsai, S. W. (1969), J. Compos. Mater. 3,6%. Adams, D. R, Doner, D. R., Thomas, R. L. (May 1967), Mechanical Behavior of Fiber-Reinforced Composte Materials, Air Force Technical Report AFML-TR-67-96. Ashton, J. W, Halpin, J. C , Petit, P. H. (1969), Primer on Composite Materials: Analysis. Lancaster, PA: Technomic. Berthelot, J. M. (1982), Fibre Sci. Technol. 17, 25. Berthelot, I M., Cupice, A., Maufras, J. M. (1978), Fibre Sci. Technol. 11, 367. Blumentritt, B. R, Vu, B. T., Cooper, S. L. (1974), Polym. Eng. Sci. 14, 633. Blumentritt, B. R, Vu, B. X, Cooper, S. L. (1975), Polym. Eng. Sci. 15, 482. Bozarth, J. M., Gillespie, J. W, Jr., McCullough, R. L. (1987), Polym. Compos. 8, 74. Budiansky, B. J. (1965), /. Mech. Phys. Solids 13, 223. Camacho, C. W, Tucker III, C. L., Yalvac, S., McGee, R. L. (1990), Polym. Compos. 11, 229. Carman, G. P., Reifsnider, K. L. (1992), Compos. Sci. Technol. 43, 137. Chou, T-W. (1992), in: Microstructural Design of Fiber Composites: Cahn, R. W., Davis, E. A., Ward, T. M. (Eds.). Cambridge: Cambridge University Press. Chou, T-W, Ko, R K. (1989), Textile Structural Composites, Composite Materials Series, Vol. 3. Amsterdam: Elsevier. Chou, T-W, Nomura, S., Taya, M. (1980), /. Compos. Mater. 14, 178. Christensen, R. M. (1979), Mechanics of Composite Materials: New York: Wiley. Christensen, R. M., Lo, K. H. (1979), J. Mech. Phys. Solids 27, 315. Cox, H. L. (1952), Br. J. Appl. Phys. 3, 72. Eduljee, R. R (1991), Ph.D. Thesis, University of Delaware, Newark, Delaware. Eduljee, R. R, McCullough, R. L. (1992), in: Computer Aided Design in Composite Materials Technology HI, Proc. 3rd Int. Conf. on Computer Aided Design in Composite Material Technology (CADCOMP 92). University of Delaware, Newark, Delaware, pp. 571-579. Eduljee, R. R, McCullough, R. L., Gillespie, J. W, Jr. (1992a), Polym. Eng. Sci., in press. Eduljee, R. R, McCullough, R. L., Gillespie, J. W, Jr. (1992b), Compos. Sci. Technol., in press. Eshelby, J. D. (1957), Proc. R. Soc. London A 241, 376.

9.8 References

Eshelby, J. D. (1961), in: Progress in Soil Mechanics, Vol. 2: Sheddon, N., Hill, R. (Eds.). Amsterdam: North-Holland, Chap. III. Foye, R. L. (1966 a), 10th National Symp. and Exhibit., San Diego, SAMPE 10, G-31. Foye, R. L. (1966 b), Quarterly Progress Reports Nos. 1, 2, AFML Cont. No. AF33/615-5150. Foye, R. L. (1972), /. Compos. Mater. 6, 193. Goldstein, H. (1957), Classical Mechanics. Reading, MA: Addison-Wesley. Halpin, J. C , Kardos, J. L. (1976), Polym. Eng. Sci. 16, 344. Halpin, J. C , Pagano, N. J. (1969), J. Compos. Mater. 3, 720. Halpin, J. C , Tsai, S. W. (June 1969), Air Force Technical Report AFML-TR-67-423. Hashin, Z. (1962), J. Appl Mech. 29, 143. Hashin, Z. (1965), 1 Mech. Phys. Solids 13, 119. Hashin, Z. (1983), /. Appl. Mech. 50, 481. Hashin, Z., Rosen, B. W. (1964), J. Appl. Mech., Trans. ASME 31, 233. Hashin, Z., Shtrikman, S. (1962a), J. Mech. Phys. Solids 10, 335. Hashin, Z., Shtrikman, S. (1962b), J. Mech. Phys. Solids 10, 343. Hashin, Z., Shtrikman, S. (1963), /. Mech. Phys. Solids 11, 127. Hermans, J. J. (1967), Proc. K. Ned. Akad. Wet. B 70(1), 1. Hermans, J. I, Hermans, P. H., Vermaas, D., Weidinger, A. (1946), Reel. Trav. Chim. Pays-Bas 65, 427. Hershey, A. V. (1954), /. Appl. Mech. 21, 236. Hill, R. J. (1963), /. Mech. Phys. Solids 11, 357. Hill, R. I (1964), /. Mech. Phys. Solids 12, 199. Hill, R. J. (1965 a), J. Mech. Phys. Solids 13, 189. Hill, R. J. (1965b), /. Mech. Phys. Solids 13, 213. Kacir, L., Ishai, O., Narkis, M. (1978 a), Polym. Eng. Sci. 18, 45. Kacir, L., Narkis, M., Ishai, O. (1978 b), Polym. Eng. Sci. 18, 234. Kardos, J. L. (1973), CRC Crit. Rev. Solid State Sci. 3, 419. Kelly, A., Tyson, W. (1965), /. Mech. Phys. Solids 13, 329. Kerner, E. H. (1956), Proc. Phys. Soc. London 69 B, 808. Konicek, T. S. (1987), MS Thesis, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL. Korringa, J. (1973), /. Math. Phys. 14, 509. Kroner, E. (1958), Z. Phys. 151, 104. Lewis, T. B., Nielsen, J. (1970), J. Appl. Polym. Sci. 14, 1449. McCullough, R. L. (1977), in: Treatise on Materials Science and Technology, Vol. 10, Part B, Properties of Solid Polymeric Materials. New York: Academic.

473

McCullough, R. L., Jarzebski, G. J., McGee, S. H. (1983), in: The Role of Polymer Matrix in the Processing and Structural Properties of Composite Materials: Seferis, J. C , Nicolais, L. (Eds.). New York: Plenum, p. 261. McGee, S. H., McCullough, R. L. (1981), Polym. Compos. 2, 149. Milton, G. W, Kohn, R. V. (1988), /. Mech. Phys. Solids 36, 597. Munson-McGee, S. H., McCullough, R. L. (1992), Polym. Eng. Sci., in press. Nomura, S., Chou, T-W. (1984), J. Appl. Mech. 51, 540. Paul, B. (1960), Trans. Metall. Soc. AIME 218, 36. Piggot, M. R. (1980), Load Bearing Fibre Composite. New York: Pergamon. Reuss, A. Z. (1929), Angew. Math. Mech. 9, 49. Rosen, B. W. (1964), Fiber Composite Materials. Papers presented at a Seminar of the American Society for Metals. Russel, W. B. (1973), /. Appl. Math. Phys. 24, 581. Russel, W. B., Acrivos, A. (1972), J. Appl. Math. Phys. 23, 434. Shaffer, B. W. (1964), AIAA J. 2, 348. Smith, J. C. (1974a), /. Res. Natl. Bur. Stand. 78A, 335. Smith, J. C. (1974b), / Res. Natl. Bur. Stand. 79A, 419. Smith, J. C. (1976), J. Res. Natl. Bur. Stand. 80 A, 45. Torquato, S. (1991), Appl. Mech. Rev. 44, 37. van der Poel, C. (1958), Rheol. Ada 1, 198. Voigt, W. (1910), Lehrbuch der Kristallphysik. Leipzig: Teubner. Walpole, L. J. (1966 a), J. Mech. Phys. Solids 14, 151. Walpole, L. I (1966 b), J. Mech. Phys. Solids 14, 289. Walpole, L. J. (1969), J. Mech. Phys. Solids 17, 235. Whitney, J. M. (1967), J. Compos. Mater. 1, 88. Whitney, J. M., Drzal, L. T. (1987), in: Toughened Composites, ASTM STP 937: Johnston, N. J. (Ed.). Philadelphia, PA: ASTM, p. 179. Whitney, J. M., Riley, M. B. (1966), AIAA J. 4, 1537. Wu, T. T. (1966), Int. J. Solids Struct. 2, 1. Wu, C. D., McCullough, R. L. (1977), in: Developments in Composite Materials: Holister, G. S. (Ed.). London: Applied Science.

General Reading Broutman, L. J. (Ed.) (1974), Fracture and Fatigue. New York: Academic Press. Chamis, C. C. (Ed.) (1974), Structural Design and Analysis, Parts I and II. New York: Academic Press. Chou, T-W, McCullough, R. L., Pipes, R. B. (1986), Scientific American 255(4), 192.

474

9 Elastic Properties of Composites

Hull,D. (1981), An Introduction to Composite Materials: Cahn, R. W, Thompson, M. W, Ward, I. M. (Eds.). Cambridge: Cambridge University Press. Jones, R. M. (1975), Mechanics of Composite Materials. Washington, DC: Scripta Book Co. Kelly, A., Macmillan, N. H. (1986), Strong Solids: 3rd ed. Oxford: Clarendon Press.

Lubin, G. (Ed.) (1982), Handbook of Composites. New York: Van Nostrand Reinhold. McCullough, R. L. (1971), Concepts of Fiber-Reinforced Composites. New York: Marcel Dekker, Inc. Tsai, S. W, Hahn, H. T. (1980), Introduction to Composite Materials. Westport, Connecticut: Technomic.

10 Inelastic Properties of Composites C. T. Sun School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, U.S.A.

List of Symbols and Abbreviations 10.1 Introduction 10.2 A 3-D Plasticity Model 10.3 Off-Axis Tension Test 10.4 One-Parameter Plasticity Model 10.5 AS4/PEEK Thermoplastic Composite 10.6 Metal-Matrix Composites 10.6.1 Boron/Aluminum 10.6.2 Titanium Matrix Composite 10.7 Overstress Viscoplasticity Model 10.8 Symmetric Laminates 10.9 Conclusion 10.10 Acknowledgements 10.11 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

476 478 479 481 482 482 485 485 486 488 490 492 492 492

476

10 Inelastic Properties of Composites

List of Symbols and Abbreviations atj a66 A d 44 Ex f(<Jij) F G 12 h H Hp k K m n {dN} [Qe] [<2ep] [Qep] Setj dWp Wp

coefficients that describe the amount of anisotropy parameter (function of temperature and strain rate) coefficient (function of temperature and strain rate) numerical value apparent elastic modulus of the off-axis yield function yield surface in-plane shear modulus laminate thickness effective overstress plastic modulus state variable viscoplastic parameter viscoplastic parameter coefficient (function of temperature and strain rate) incremental in-plane resultant force reduced elastic stiffness matrix elastic-plastic stiffness matrix transformed matrix of [Qep] elastic compliances increment of plastic work per unit volume rate of plastic work

y 12 sx dsfj dsp y de^ dsij deij de p ep etj ex 8P sex sp efj efj {de} {dep} {de0} {s1}

engineering shear strain elastic strain in loading direction plastic part of the incremental strain plastic strain increment in x, y-direction incremental strain in loading direction incremental strain elastic part of the incremental strain effective plastic strain increment total effective plastic strain increment total strain rate total axial strain rate axial plastic strain rate axial elastic strain rate effective plastic strain rate elastic part of the total strain rate plastic part of the total strain rate increment elastic strain (in matrix notation) incremental plastic strain (in matrix notation) incremental mid-plane strain vector initial strain

List of Symbols and Abbreviations

477

9 angle between x-axis (uniaxial loading direction) and fiber direction jx^-axis dA proportionality factor k proportionality factor v£ plastic Poisson ratio a effective stress ax applied uniaxial stress atj stresses that refer to the principal material direction {da} increment elastic stress (in matrix notation) {a1} initial stress

polyetheretherketone (PEEK) matrix reinforced by AS4 carbon fibers glass transition temperature titanium matrix reinforced by SCS 6 ceramic fibers

478

10 Inelastic Properties of Composites

10,1 Introduction Fiber composites exhibit physically nonlinear behavior. This nonlinearity is especially pronounced in metal-matrix and thermoplastic composites. A number of models have been used to describe the nonlinear stress-strain relationship. Hahn and Tsai (1973) employed a complementary elastic energy density function that contained a biquadratic term for in-plane shear stress. As a result, the strain-stress relation is linear in simple longitudinal and transverse extensions, and the shear strain is a third-order polynomial function in the shear stress, which is unsuitable for describing the nonlinear behavior exhibited by many composites. Assuming that fibers are linearly elastic, Sun et al. (1974) modeled the composite by employing nonlinear matrix layers alternating with effective linearly elastic fibrous layers. This approach is limited by the fact that the nonlinearity was obtained by including third order terms in strains in the stress-strain relations. The physical nonlinearity in fiber composites can also be considered a plastic behavior. Dvorak and Bahei-El-Din (1979, 1982) used a microchemical model consisting of elastic filaments of vanishing diameters and an elastic-plastic matrix. An elastic-plastic continuum model was developed for which the constitutive relations depend on the constituent properties, their volume fractions, and mutual constraints between the phases associated with the geometry of the microstructure. A few authors (Chamis and Sullivan, 1973; Adams, 1970; Foye, 1973; Lin et al., 1972) used finite element method to analyze a representative volume of a unidirectional composite to understand the elastic-plastic behavior of the composite.

Inherent in these micromechanical models is the need to know the constituent material properties, fiber array, and fiber matrix interfacial condition. As noted in Bahei-El-Din and Dvorak (1980), the insitu matrix property can be quite different from that of the bulk matrix material. Consequently, adjustment of the matrix property in the micromechanical model is often necessary to yield good results. Another difficulty encountered in formulating micromechanical models lies in the 3-D elastic constants of the fiber, which are extremely difficult to measure because of their small size. Recently, Johnson et al. (1988) indicated that the fiber-matrix interfacial bond in some metal-matrix composites is very weak as opposed to the perfect bond assumed in developing these micromechanical models. To overcome the aforementioned uncertainties present in the micromechanical level of the composite, an alternative approach is to regard the composite as a homogeneous elastic-plastic continuum and measure its macromechanical properties directly. This approach enables one to employ well documented classical plasticity theories and many existing finite element codes for structural analysis. For unidirectional composites, orthotropy exists in their plastic behavior under monotonic loading. Griffin et al. (1981) proposed a three-dimensional flow theory based on Hill's (1948) orthotropic yield criterion to describe the nonlinear behavior of a boron/epoxy composite. The present author and co-workers have recently employed a more general plastic potential function, which is a quadratic form of the stress components, to characterize the orthotropic plasticity in a boron-aluminum metal-matrix composite (Kenaga et al., 1987; Rizzi et al, 1987; Sun and J. K. Chen, 1987; Sun and J. L. Chen, 1989). A simpli-

10.2 A 3-D Plasticity Model

fied version of the plastic potential function with a single parameter proposed by Sun and Chen (1989) was successfully employed by Sun and Yoon (1989) to characterize the elastic-plastic behavior of AS4/ PEEK thermoplastic composite at various temperatures. This plasticity model is simple in form and can easily be obtained experimentally using off-axis specimens. Because of its simplicity, this testing procedure can easily be performed in elevated temperature environments. Many polymer matrix composites exhibit plastic behavior when they are loaded in the matrix dominant direction. A few models were proposed to describe the viscoplastic behavior of orthotropic materials. Most of them were modified from the classical viscoplasticity models of metallic materials, such as the overstress model (Malvern, 1951; Eisenberg and Yen, 1981) and Bodner and Partom's model (1975). The overstress concept was proposed by Malvern (1951) to explain the high strain rate effect on wave propagation in metallic bars. This overstress approach was discussed by Cristescu and Suliciu (1982) in several alternative forms of overstress function. Subsequently, Eisenberg and Yen (1981) incorporated an anisotropic hardening law in the overstress theory to describe the movement of the yield surface. Following a different approach, Bodner and Partorn (1975) developed a set of constitutive equations for elastic-viscoplastic behavior of isotropic materials which require neither a yield criterion nor the distinction between loading and unloading. The hypothesis of their model was partially motivated from dislocation dynamics which indicates that the dislocation velocity is a function of stress. To describe the viscoplastic effect in fiber reinforced composites, Gates and Sun (1989) and Yoon and Sun (1991) proposed

479

an elastic-viscoplastic constitutive model using a one-parameter plastic potential function in conjunction with the overstress concept. They verified the validity of the model using AS4/PEEK thermoplastic composite. Krempl and Hong (1989) introduced a simple lamination theory using an orthotropic viscoplastic theory based on overstress to describe the in-plane stressstrain relationship for metal matrix composites. In their model, at least 17 constants and two functions need to be determined experimentally. Additional viscoplasticity models include one by Ha and Springer (1989) and one by Robinson et al. (1987). Ha and Springer (1989) developed a rate-dependent model using viscoelastic and viscoplastic constitutive equations for each lamina. In their model, twenty-three material constants are needed for the viscoelastic part, and nine constants are needed for the viscoplastic part of the constitutive equation. The viscoplasticity model proposed by Robinson et al. (1987) involves the use of a potential function, flow law and evolutionary law. Four constants and one function need to be determined experimentally. The model has not been verified by experiments.

10.2 A 3-D Plasticity Model A yield criterion is assumed for the general 3-D fiber composite as

2a
480

10 Inelastic Properties of Composites

tions. The coefficients atj describe the amount of anisotropy in the inital plasticity. This expression for /(^y) satisfies the orthotropy condition. The values of atj can be determined from the experimental data. This criterion reduces to von Mises yield criterion for isotropic solids when the atj have the values

The effective plastic strain increment de p can be defined such that dWp = (Jrdsf-= ads9

(10-7)

Substitution of Eqs. (10-5) and (10-6) into (10-7) yields (10-8) and

= <*22 = %3 = 2/3

(10-2)

The Hill-type yield criterion for orthotropic materials (1948) is a special case of Eq. (10-1) when the function /(o^y) is assumed to be independent of the hydrostatic stress. Hill's yield function is obtained if = fl33 - « 1 1 - 0 22 2023=011 -022-033

(10-3)

2\da

Since the incremental strain de,-y is assumed to be small, we are able to decompose it linearly into the elastic part defy and the plastic part defy as

Consider a state of plane stress parallel to the x1-x2 plane. The plastic potential function reduces to = a11o1

= -^-dX

ij = 11,22,12

(10-4)

0(7:ij

where the superscript p denotes plasticity, and dX is a proportionality factor. In general, the yield function given by Eq. (10-1) implies neither the incompressibility of plastic strains nor the assumption that hydrostatic stresses result in no plastic deformation. The increment of plastic work per unit volume is given by dWp = Gijdefj = 2/d/l

(10-5)

Let the effective stress c be defined as (10-6)

(10-10)

e;y = defy + defy

2a 13 = 022-011 -033

By using the associated flow rule, the yield function is taken as the plastic potential function from which the incremental plastic strains can be derived as

(10-9)

) \ 5

<j +a22<j 22

22

(10-11) Without loss of generality, we further set 022 = 1-

The effective plastic strain increment can be derived using Eqs. (10-5)-(10-7) and the inversion of Eq. (10-3). It is explicitly written as

- 2a ±1

12

2a

(10-12) 66

From Eqs. (10-4) and (10-11), the plastic strain increments are obtained as (10-13) 12

1 0

o-

0 <0-22 2a 6 6 _ 1 ^12

481

10.3 Off-Axis Tension Test

10,3 Off-Axis Tension Test

Comparison of Eq. (10-19) with Eq. (10-8) leads to

The complete orthotropic plastic flow rule is defined if the parameters all9 a12, a66 and dX are determined. To determine dA, the effective stress-effective plastic strain relation must be established. This can be accomplished from the results of tension tests on off-axis specimens. Let the x-axis be the uniaxial loading direction which makes an angle 9 with the fiber direction xx-axis. The stresses referring to the material principal axes (xx and x2) are related to the applied uniaxial stress
dsp = dspx/h{9)

a 22 = sin2 9 ax

a = h{9)ax

(10-15)

where h(9) = , / f [alt cos4 9 + sin4 9+ 2

(10-16)

+ 2(a 12 + a 66 )sin 0cos 6>] 1/2

2

Using the coordinate transformation for strains, the plastic strain increment in the loading direction is given by 2 p dep = cos2 ede?, l i + sin 9ds ?? -

(10-17)

Substituting Eq. (10-13) into Eq. (10-17) and then using the relations (10-14), we obtain del = \h (9) ax dX

8p = spx/h(9)

(10-21)

The desired relation between a and i? can now be obtained from the experimentally obtained relation between ax and 8P. From Eqs. (10-15) and (10-21), it follows that ^L = h2(9)-^ Ui>

(10-22)

Ubx

(10-14)

Substitution of Eqs. (10-13) into Eqs. (10-10) and (10-5) yields

2

For monotonic loading, the above equation is integrable. Thus,

and

all = cos2 9 ax

— sin9cos9dyp12

(10-20)

(10-18)

In view of Eq. (10-15), we thus obtain (10-19)

Since the a-sp relation should be unique in monotonic loading for the given material, the parameters all9 a12 and a66 must be chosen so that the resulting a-sp relation is independent of 9. The above procedure can only determine the sum a12 + a 66 , as is evident from Eq. (10-16). To separate the two parameters, the plastic Poisson's ratio must also be used. Using the coordinate transformation law on strain components and Eqs. (10-13) and (10-14), we obtain the plastic strain increments in the x-direction and the y-direction as deP = [axl cos4 9 + sin 4 9 + (10-24) + 2(a12 +

a66)sin29cos29]dXax

and dep = [(1 + a±! - 2 a 66 ) sin2 9 cos2 9 + + a12 (sin4 9 + cos 4 9)] dX ax

(10-25)

respectively. The plastic Poisson's ratio is defined as

(1 +alt — 2 a 66 )sin 2 9 cos2 9 + a 12 (sin 4 9 + cos4 9) cos4 9 + sin 4 0 + 2(a 12 + a 66 )sin 2 9

(10-26)

482

10 Inelastic Properties of Composites

For 9 = 0°, 90°, and 45°, the plastic Poisson's ratios reduce to a12

-2(a12-a66) In the off-axis tension test, the transverse strain is measured, from which the transverse plastic strain e£ can be obtained, and, thus, the plastic Poisson's ratio. The additional data allow us to separate a12 and

10.4 One-Parameter Plasticity Model Numerous experimental data show that fiber composites behave linearly when loaded in the fiber direction. Thus, it is reasonable to assume that de?1=0

(10-28)

which leads to the condition fli

9 =

= [f (a\2 + 2a66a212)]1>2dX

respectively, and h(9) becomes (10-27)

# 1 1 =

and

(10-29)

'

As a result, we obtain a one-parameter plastic potential 2 / = a! 2 + 2fl66<7

2

(10-30)

2

From this plastic potential, the plastic strain increments are derived: (10-31) where y12 is an engineering shear strain. The corresponding effective stress and effective plastic strain increments reduce to 1/2

(10-32)

(10-33)

,.Q ^A\

h (9) = [f (sin4 9 + 2 a66 sin2 9 cos2 9)]1/2 This simplification results in a single parameter a66 to be determined. The procedure for determining the one-parameter plasticity model becomes very simple. In theory, only two off-axis specimens with different 0's are required to determine a66. Note that for 9 = 90°, h(9) = ^ 3 / 2 , and thus, the stress-strain relation for the 90°specimen is not affected by the choice of a66 and can be conveniently used to construct the master effective stress-effective plastic strain curve for the composite. Moreover, there is no need to measure the plastic Poisson's ratio to aid in determining the parameters as in the 3-parameter model. The plastic Poisson's ratio corresponding to the one-parameter plastic potential is given by (2a 6 6 -l)cos 2 6> sin2 9 + 2 a6fi cos2 9

(10-35)

As 9 -* 0°, the plastic Poisson's ratio approaches v0o = 1 - J -

(10-36)

-*66

10.5 AS4/PEEK Thermoplastic Composite The one-parameter plasticity model was employed by Sun and Yoon (1989) to characterize the plastic behavior of the AS4/ PEEK (APC-2) thermoplastic composite. Specimens with 9 = 0°, 15°, 30°, 45°, and 90° were tested. The stress-strain curves obtained at 24 °C (75 °F), 66 °C (150°F),

483

10.5 AS4/PEEK Thermoplastic Composite

121 °C (250°F)9 and 177 °C (350 °F) are presented in Figs. 10-1-10-4, respectively. Experimental results indicate that for the thermoplastic fiber composite there is no well defined yield point. In view of this, the master effective stress-effective plastic strain curve can be fitted as a power law, = A (a)n (
300 200 ' 30°

\

100

^ k

•

0

0.5 1.0 1.5 Strain in %

0° ,300

2.0

2.5

3.0

^z

"200

/

30°

!

M00 —•

-1.0 -0.5

0

2.0

-0.5 1.0 1.5 Strain in %

2.5

3.0

Figure 10-3. Stress-strain curves for AS4/PEEK offaxis specimens at 250 °F (121 °C). ( : Strain gage debonded but strength obtained.)

——Long.

0.5 1.0 1.5 Strain in %

2.0

/

0

,300

Moo

ID"

15°

\

n -1.0 -0.5

•

-Long.

Trans.-

^ 100 ^.30°

/

15°

45°

Figure 10-1. Stress-strain curves for AS4/PEEK offaxis specimens at 75 °F (24 °C). ( : Strain gage debonded but strength obtained.)

400

^200 .,15° (/)

:2oo

^90°

n -1.0 -0.5

O 300 Q_

0°

30°

/

45°^\ 1

/0°

400

1

0°

0°l

Trans. ^—

—— Long.

Trans.—

—^Long.

(10-37)

The parameter a 66 , coefficients A and n may be functions of temperature, strain rate and other environmental factors. The effective stress-effective plastic strain relations of all off-axis specimens at 24 °C (75 °F), 66 °C (150°F), 121 °C (250 °F), and

400

Trans.——

400

2.5

3.0

Figure 10-2. Stress-strain curves for AS4/PEEK offaxis specimens at 150°F (66 °C). ( : Strain gage debonded but strength obtained.)

—

^

^

15°.....

^

30°

\

0 45°, -1.0 -0.5

0.5 1.0 1.5 Strain in %

2.0

2.5

3.0

Figure 10-4. Stress-strain curves for AS4/PEEK offaxis specimens at 350 °F (177 °C). ( : Strain gage debonded but strength obtained.)

177 °C (350 °F) are shown in Fig. 10-5. It is seen that the effective stress-effective plastic strain data for all off-axis angles more or less collapse into one curve with a66 = 1.5 for all temperatures. This indicates that the orthotropy of plasticity in AS4/PEEK can be regarded as independent of temperature. The coefficients A and n can be determined from fitting the data into the power law of Eq. (10-37). The results, presented in Table 10-1, indicate that n = 7.0 is adequate for temperatures up to 121 °C (250 °F). It is

10 Inelastic Properties of Composites

75°F (24°C) 150°F (66°C) ._ 250°F |121°C) . 350°F (177°C)

Figure 10-5. Effective stress and effective plastic strain curves for AS4/PEEK offaxis specimens.

2.5

1.0 1.5 2.0 Effective plastic strain in %

3.0

Table 10-1. Coefficients A and n for AS4/PEEK. ^ ^ \ ^ ^ Temperature Coefficient

24 °C (75 °F)

66 °C (150°F)

121 °C (250°F)

177 °C (350 °F)

7.0 5.14xlO~ 18 -39.8

7.0 2.18 xlO~ 1 7 -38.4

7.0 1.70xl0~ 16 -36.3

2.9 7.06 xlO" 8 -13.5

^ ^ ^ ^

n A In (A)

also noted that coefficient A is related to temperature as

i

(0.036 T - 40.7 (Tin°C)

(1 38)

t

°

for temperatures below the glass transition point. At 177 °C (350 °F), which is above the T.G. (glass transition temperature), n = 2.9, and the value of A drops significantly. These data are presented in Fig. 10-6. The total incremental strain in the loading direction can be decomposed into the elastic part d&ex and the plastic part dex as dsx = del + dsx

In Eq. (10-40), sex is the elastic strain in the loading direction defined as

(10-39)

where Ex is the apparent elastic modulus of the off-axis specimen which can be obtained from the transform equation 1 _ 1

4

12

+ — sin 4 0 E~,

(10-42)

In the monotonic loading case Eq. (10-39) can be integrated to yield

Using Eqs. (10-15), (10-21) and (10-37), sx can be expressed as

(10-40)

(10-43)

X

X

I

485

10.6 Metal-Matrix Composites

Therefore, the stress-strain curve in monotonic loading can be predicted using the following equation.

1.0 Predicted Experimental • 75°F

o

'pO.8 u)

§0.6

sx = £ + eg = £ + [h (9)f ^x

+1

A < (10-44)

en in

^x

Figure 10-7 shows the total stress-strain curves for the off-axis specimens. Agreement between the experimental data and predictions according to the model is excellent. The value of a66 can be checked by the comparison of the plastic Poisson's ratios obtained from theory and experiment. Figure 10-8 shows the predicted plastic Pois-

° 02 Q_

0

15.0

30.0 45.0 60.0 8 in degrees

75.0

90.0

Figure 10-8. Measured and predicted plastic Poisson's ratios for AS4/PEEK.

son's ratio (with a66 = 1.5) and the experimental data up to 121 °C (250 °F). The present plasticity model seems to yield reasonably accurate plastic Poisson's ratios.

10.6 Metal-Matrix Composites

50

100

150

200

250

300

350

400

Temperature in °F

Figure 10-6. Logarithm of the coefficient A vs. temperature for AS4/PEEK composite.

Metal-matrix composites display pronounced stress-strain curves under off-axis loading. The one-parameter plasticity model was shown to be very successful in describing nonlinear behavior for boron/ aluminum by Sun and Chen (1989), and for SCS6/Ti-6-4 titanium matrix composite by Sun etal. (1990). It is important to note that in boron/aluminum the fiber/matrix interface bond is very good and the observed nonlinear behavior is originated from the matrix yielding. However, for the titanium matrix composite, the apparent plasticity is mostly due to fiber/matrix separation. 10.6.1 Boron/Aluminum

• 1.0

-.Measured Data Predicted Curve

1.5 2.0 Strain in %

2.5

3.0

Figure 10-7. Measured and predicted stress-strain curves for AS4/PEEK off-axis specimens at 75 °F (24 °C).

For a boron/aluminum composite, Sun and Chen (1989) found that a66 = 2.0 was able to bring all the off-axis plastic stressstrain curves together in the effective stress-effective plastic strain plot, see Fig. 10-9. This master curve was fitted into

486

10 Inelastic Properties of Composites

the power law given by Eq. (10-37) with the result: A=

J0.41 x 10 - 1 5 10.3 x l O " 1 0

a in MPa a in kpsi (10-45)

This master curve is also shown in Fig. 10-9. Using a66 = 2.0, the plastic Poisson's ratio vs. off-axis angle 6 is shown in Fig. 10-10. It is evident that experimental data fall on the predicted curve nicely except for the 10° off-axis case for which plasticity is not pronounced.

1

o

2 3 4 5 6 Effective plastic strain in %

7

Figure 10-9. Effective stress-effective plastic strain relation for boron/aluminum composite. Solid line is predicted with a66 = 2.0.

10.6.2 Titanium Matrix Composite Sun et al. (1990) investigated the nonlinear stress-strain behavior of a titanium matrix composite (SCS6/Ti-6-4 with 40% fiber volume) using off-axis specimens. The value of a66 was found to be 0.85. The master curve is shown in Fig. 10-11. This master curve can be fitted into the power law with

0

15

28

A=

0.134 xlO~ , a in MPa 0.325 x 10" 20 , a in kpsi

30 45 60 Fiber angle in degrees

75

90

Figure 10-10. Comparison of measured and predicted plastic Poisson's ratios for boron/aluminum.

(10-46) Although the off-axis plastic stress-strain curves do not collapse exactly into a single curve, the recovered total off-axis stressstrain curves match with the experimental curves very well, as shown in Fig. 10-12. Johnson et al. (1988) and Sun et al. (1990) found that in titanium matrix composites fiber could separate from the matrix under loading. Comparing the off-axis stressstrain curves for the SCS6/Ti-6-4 composite with those of the titanium matrix as shown in Fig. 10-13, it is evident that the apparent yielding is not caused by plasticity in the matrix. Sun etal. (1990) employed a micromechanical model to analyze the interfacial

100 • 90 *60 • 45 *30 0 15

0.6 0.5 0.4

50

r

0.3 0.2 0.1 n

0

0.1 0.2 0.3 0.4 Effective plastic strain in %

0.5

Figure 10-11. Effective stress-effective plastic strain relation for SCS 6/Ti-6-4 composite.

487

10.6 Metal-Matrix Composites

stresses. A fiber/matrix interfacial failure criterion was proposed for this composite as {oi2)2 + 2d66{GA12)2

= a2

(10-47)

where ai7 and oA7 are the interfacial stresses, and (10-48) as = 106.9 MPa (15.5 kpsi)

0.2

0.6

0.3 0.4 Strain in %

Figure 10-12. Comparison of the predicted and experimental off-axis stress-strain curves for SCS 6/Ti-6-4 composite.

/

°

Fiber- - /

/

/

15°

The values of d66 and 20°, fiber/matrix debonding would take place before the matrix yields. For small off-axis angles, no fiber/matrix debonding would occur before total failure of the specimen.

- • — Fiber-Matrix Separation

• 1.0

/ /

- * — Matrix Yielding

1.5

100 Matrix /

50

/

/

S 1.0

30° y

0

0.5

0

0.1

0.2 0.3 (U Strain in %

0.5

Figure 10-13. Comparison of Ti-6-4 matrix, SCS 6 fiber, and the composite off-axis stress-strain curves.

u

0

30 60 Fiber angle in degrees

90

Figure 10-14. Off-axis stresses corresponding to the onsets of fiber/matrix separation and matrix yielding in SCS 6/Ti-6-4 composite.

488

10 Inelastic Properties of Composites

10.7 Overstress Viscoplasticity Model The following development of an overstress viscoplasticity model follows that presented by Sun and Yoon (1991). Within the scope of small strain assumptions, the total strain rate is composed of an elastic part and a plastic part as

assuming that the effective stress-effective plastic strain relation is strain rate dependent. The effective plastic strain rate £p is assumed as a function of effective overstress if, i.e., (10-56) in which (10-57)

(10-49)

pP

The elastic components e^- are related to the stress rate as e^ = S^(JlV

(10-50)

where S£ are the elastic compliances. The plastic strain rate is derived from the associated flow rule as

where cr* is the quasistatic effective stress corresponding to vanishingly small strain rates. The overstress function

\(H/K)

1 o

if

<7>
if
(10-58)

where K and m are the viscoplasticity parameters. Consequently,

_ _„

(10-51) where k is a proportionality factor. The rate of plastic work can be expressed as

(10-59) if

<7<<J*

For off-axis loading in the x-direction, the total strain rate can be expressed as

The effective plastic strain rate ep is defined such that

sx = el +

Wp = alp

where the elastic and the plastic strain rates are given by

(10-53)

From the definition of a and Eqs. (10-51) to (10-53), we have (10-54) With Eqs. (10-52) and (10-53) we obtain l =

3

-j

(10-55)

The one-parameter plasticity model is combined with the overstress concept by

°x

fiP

(10-60)

(10-61)

and s =

(10-62)

respectively. In a monotonic loading with a constant strain rate, the effective stress is always higher than the quasistatic effective stress for a given strain. From Eqs. (10-59) through (10-62), the total axial strain rate

10.7 Overstress Viscoplasticity Model

can be expressed as (10-63)

If we let 1/m

(10-64) then the total strain rate is given by £x = — + P(
(10-65)

X

By solving the above nonlinear differential equation using a numerical program, the total stress-strain curves for different strain rates can be predicted. The above overstress-based viscoplasticity model was used by Yoon and Sun (1991) to describe the viscoplastic behavior of AS4/PEEK thermoplastic composite. From the stress-strain curves of off-axis specimens tested at a very slow strain rate of about 10" 6 s per second and a fast strain rate of about 10 ~ 3 e per second, the effective overstress-effective plastic strain rate relation can be established. The total

489

stress-strain curves of 15°, 30° and 45° offaxis coupon specimens for low and high strain rates at room temperature are shown in Fig. 10-15. The stress-strain relations obtained from the slow strain rate test were taken as quasistatic stress-strain relations. From the quasistatic curve and higher strain rate curve, the effective overstress and the effective plastic strain rate relationship for each off-axis angle are calculated and are shown in Fig. 10-16. Even though there is some scatter, the curves of all cases considered collapsed into one master curve which appears to be linear on the logarithmic plot. This means that the viscoplastic behavior of the composite can be characterized by the constant viscoplasticity parameters K and m. The parameters K and m for 24 °C (75 °F) were obtained by linear curve fitting on the log-log scale, and the resulting values are shown in Fig. 10-16. Thus, the overstress function H is given by Eq. (10-58) with m -0.37 K =196

(
MPa)

•.Experimental : Curve fitted with A and n 1.0 x10" 3 /sec. 1.0 *1(T 6 /sec.

15°

U*1Cr 3 /sec. l 3 0 o 1.0x1(T6/sec. / 0.7 x10"Vsec. 1.0*i0- 6 /sec.

1.2

1.6 2.0 Strain in %

7.U

2.8

Figure 10-15. Stress-strain curves for AS4/PEEK offaxis specimens for low and high strain rates at 75 °F (24 °C). 3.2

490

10 Inelastic Properties of Composites

£1000.0 2 ^ 100.0 in in O)

V>

10.8 Symmetric Laminates

•15° •45°

Elastic-plastic deformation in composite laminates can be formulated using classical laminated plate theory. Each lamina is assumed to be orthotropic elastic-plastic. In matrix notations, Eq. (10-10) can be written in the form

10.0

CD

Z 1.0

m 3 0.37

/C=196.A o f i f i = 1.5

o Qi

S

0.1 1 1010"' 10" 10" Effective plastic strain rate §[H) in s"1

{de} = {dse} + {de»}

Figure 10-16. Effective overstress vs. effective plastic strain rate at 75 °F (24 °C).

It is interesting to note that the value of a66 is the same as in the plasticity model. By solving the nonlinear differential equation (10-65) with the obtained viscoplasticity parameters, the strain rate-dependent stress-strain curve can be predicted. Figure 10-17 shows the total stress-strain curves of different strain rates, experimental and predicted, for off-axis specimens at 24 °C (75 °F). We can see good agreement between experimental data and predicted curves.

(10-66) T

where {ds} = {da 11 ,d£ 22 ,d'y 12 } . The incremental elastic stress-strain relations are given by (10_6?) where [Qe] is the reduced elastic stiffness matrix (Jones, 1975). To obtain the incremental stress-strain relations, we start from the yield surface defined as

= 2f(aij)-k(s») =

(10-68)

The consistency condition which imposes the restriction on the increment between d<Tij and dfc at work-hardening can be expressed as

dF(aipk) = 0

(10-69)

500 •

A : Experimental : Predicted with m and K

400 1.0*i0" 3 /sec. l 1 5 o 1.Q*10"6/sec. /

:AA

0.5

1.0

1.5 Strain in %

1.0*10-6/sec. 0.7 *10-V see 1.0 * 1 0"6/sec.

2.0

2.5

Figure 10-17. Measured and predicted (overstress model) stress-strain curves of AS4/PEEK off-axis specimens for different strain rates at 75 °F (24 °C). 3.0

10.8 Symmetric Laminates

Considering that the current stress components satisfy the yield criterion at the plastic stage, and substituting yield function F into Eq. (10-69), we obtain

491

where the elastic-plastic stiffness matrix is defined as [Qep] = [Qe] ~

do-70)

(10-77)

[Ge]

Since k = 2 / = 2
= lJL=

Hpa

(10-71)

where H p is the plastic modulus, i.e.,

Substitution of Eq. (10-67) into Eq. (10-70) yields 8/


2

Following the formulation of classical laminated plate theory, the incremental stress-strain relations given by Eq. (10-76) can be used to derive the relations between the plate incremental resultant forces and the mid-plane displacements. By using the proper coordinate transformation, this formulation can be achieved with an arbitrary coordinate system x-y. For a symmetric laminate, we have (10-78)

(10-73) .dAL

Using Eq. (10-8), we obtain from Eq. (10-73) the following. 074)

in which {dN} is the incremental in-plane resultant force vector, {de0} is the incremental mid-plane strain vector, and fc/2

8/ V

8/

Therefore, the incremental plastic strain can be expressed as {d e p } = dX

8/

(10-75)

[A]=

f [<2 ep ]dz

(10-79)

-fc/2

where h is the laminate thickness, and [Qcp] is the transformed matrix of [Qep]. The total stresses and strains can be obtained by accumulating stress and strain increments as (10-80)

8/ Y

(10-81)

8/

9^

P

8/

^V9W V f f i l ^

From Eqs. (10-67) and (10-75), the relations between stress increment and strain increment can be expressed as {da} = [Q*"] {de}

(10-76)

where {CJ1} and {e1}, respectively, are initial stresses and strains such as thermal residual stresses and strains. When residual stresses are present, it is necessary to include them in the plastic analysis. Using the above procedure, Sun and Yoon (1990) investigated the elastic-plastic

492

10 Inelastic Properties of Composites

behavior of an AS4/PEEK [± 45] 3s laminate at room temperature. Because of its high processing temperature (380 °C), the laminate contained significant thermal residual stresses at room temperature. The residual stresses in the laminas were obtained by measuring the warpage in a [04/904] unsymmetric laminate from which the effective stress-free temperature was estimated to be 252 °C. The thermal residual stresses were calculated using the elastic lamination theory. The result was ai11 = - 5 8 . 3 M P a
MPa

(10-82)

Figure 10-18 presents the comparison of the measured and predicted strains in the [±45] 3s AS4/PEEK laminate. The solid line is the prediction that includes thermal residual stresses, and the dotted line is the prediction that omits thermal residual stresses. It is evident that thermal residual stresses cannot be neglected in the plastic analysis of laminates.

10.9 Conclusion Thermoplastic and metal matrix composites display significant nonlinear inelastic behaviors. These orthotropic nonlinear stress-strain relations can be modeled very well by an orthotropic plasticity model which contains a single orthotropy parameter. The simplification of the model is achieved by assuming that plasticity in the longitudinal (fiber) direction is negligible. The one-parameter plasticity model and the overstress viscoplasticity model were verified experimentally to be suitable for the description of plastic and viscoplastic behaviors in the AS4/PEEK thermoplastic composite and boron/aluminum metal matrix composite. For AS4/PEEK composite, the orthotropy parameter a66 and the power index n can be regarded as constants for temperatures below the glass transition temperature. For some metal matrix composites with a weak fiber/matrix interface, the apparent yielding could be caused by debonding between the fiber and the surrounding matrix.

10.10 Acknowledgements

300 Experimental data • Predicted with residual stresses • Predicted without residual stresses

250

This work was supported by NASA Langley Research Center under Grant No. NAG-1-825 and by the National Science Foundation under Grant CDR 8803017 to the Engineering Research Center for Intelligent Manufacturing Systems, Purdue University, West Lafayette, Indiana.

10.11 References 0

0.4

0.8 1.2 1.6 Longitudinal Strain in %

2.0

Figure 10-18. Uniaxial stress-strain curves for AS4/ PEEK [±45] 3s laminate at room temperature.

Adams, D. F. (1970), /. Comp. Mats. 4, 310-328. Bahei-El-Din, Y. A., Dvorak, G.I (1980), /. Appl. Mech. 47, 827-832. Bodner, S. R., Partom, Y. (1975), J. Appl. Mech. 42, 385-389.

10.11 References

Chamis, C. C , Sullivan, T. L. (1973), Theoretical and Experimental Investigation of the Nonlinear Behavior of Boron Aluminum Composites. NASA-Lewis Research Center, Cleveland, OH: NASA TM X-68-205. Cristescu, N., Suliciu, I. (1982), Viscoplasticity. The Hague, Boston, London: Martinus Nijhoff Publishers. Dvorak, G. J., Bahei-El-Din, Y. A. (1979), /. Mech. Physics Sols. 27, 51-72. Dvorak, G. J., Bahei-El-Din, Y. A. (1982), /. Appl. Mech. 49, 327-335. Eisenberg, M. A., Yen, C. F. (1981), J. Appl. Mech. 48, 276-284. Foye, R. L. (1973), /. Comp. Mats. 7, 178-193. Gates, T. S., Sun, C. T. (1989), in: Proc. 30th AIAA Struct., Struct. Dynamics and Mats. Conf, Mobile, AL, pp. 845-851. Griffin, O. H., Kamat, M. P., Herakovich, C. T. (1981), /. Comp. Mats. 5, 543-560. Ha, S. K., Springer, G. S. (1989), /. Comp. Mats. 23, 1130-1158. Hahn, H. T., Tsai, S. W. (1973), J. Comp. Mats. 7, 102-118. Hill, R. (1948), Proc. the Royal Society, Math, and Phys. Sci. 193, No. 1033, 281-297. Johnson, W. S., Lubowinski, S. X, Highsmith, A. L., Brewer, W. D., Hoogstraten, C. A. (1988), Mechanical Characterization of SCS6/Ti-15-3 Metal Matrix Composites at Room Temperature. NASALangley Research Center, Hampton, VA: NASP Technical Memorandum 1014. Jones, R. M. (1975), Mechanics of Composite Materials. New York: McGraw-Hill Company. Kenaga, D., Doyle, J. F., Sun, C. T. (1987), /. Comp. Mats. 21, 516-531. Krempl, E., Hong, B. Z. (1989), Comp. Sci. & Tech. 35, 53-74. Lin, T. H., Salinas, D., Ito, Y M. (1972), /. Appl. Mech. 39, 321-326. Malvern, L. E. (1951), /. Appl. Mech. 18, 203-208. Rizzi, S. A., Leewood, A. R., Doyle, X F , Sun, C. T. (1987), /. Comp. Mats. 21, 734-749.

493

Robinson, D. N., Duffy, S. F , Ellis, X R. (1987), A Viscoplastic Constitutive Theory for Metal Matrix Composites at High Temperature, in: Thermal Stress, Material Deformation and Thermo-Mechanical Fatigue, Vol. 23: Sehitoglu, H., Zamrik, S. Y. (Eds.). New York: ASME, PVP. Sun, C. T., Chen, X K. (1987), J. Comp. Mats. 21, 969-985. Sun, C. T., Chen, X L. (1989), /. Comp. Mats. 23, 1009-1020. Sun, C.T., Yoon, K. X (1989), in: Proc. 7th Int. Conf. on Comp. Mats. (ICCM VII), Vol. 2, November 22-24. Guangzhou, China, pp. 185-191. Sun, C. T., Yoon, K. X (1990), in: Proc. 5th Japan-US Conf. on Comp. Mats., June 24-27. Tama City, Tokyo, Japan. Sun, C. T, Feng, W. H., Koh, S. L. (1974), Int. J. Eng. Sci. 12, 919-935. Sun, C.T., Chen, XL., Sha, G. T., Koop, W E . (1990), J. Comp. Mats. 24, 1029-1059. Yoon, K.X, Sun, C. T. (1991), /. Comp. Mats. 25, 1277-1296.

General Reading Christensen, R. M. (1979), Mechanics of Composite Materials. New York: Wiley. Dvorak, G. X (Ed.) (1991), Inelastic Deformation of Composite Materials. New York: Springer. Hill, R. (1950), The Mathematical Theory of Plasticity. London: Oxford University Press. Johnson, W, Mellor, P. M. (1962), Plasticity for Mechanical Engineers. Princeton, NJ: Van Nostrand. Malvern, L. E. (1969), Introduction to the Mechanics of a Continuous Medium. Englewood Cliffs, NJ: Prentice-Hall, Chap. 6. Whitney, X M., Daniel, I. M., Pipes, B. R. (1982), Experimental Mechanics of Fiber Reinforced Composite Materials. Brookfield Center, CT: The Society for Experimental Stress Analysis.

11 Strength of Fiber Composites Ran Y. Kim Research Institute, University of Dayton, Dayton, OH, U.S.A.

List of Symbols and Abbreviations 11.1 Introduction 11.2 Strength Analysis of Unidirectional Laminates 11.2.1 Longitudinal Tensile Strength 11.2.2 Longitudinal Compressive Strength 11.2.3 Transverse Tensile Strength 11.2.4 Shear Strength 11.3 Strength Analysis of Multidirectional Laminates ,11.3.1 Failure Modes 11.3.2 Failure Theories 11.3.2.1 Maximum Stress Theory 11.3.2.2 Maximum Strain Theory 11.3.2.3 Maximum Distortional Energy Theory 11.3.2.4 Quadratic Theory 11.3.3 First Ply Failure 11.3.4 Ultimate Failure 11.3.5 Free Edge Effect 11.3.6 Notched Strength 11.4 Test Methods 11.4.1 Constituent Materials 11.4.1.1 Fibers 11.4.1.2 Matrix 11.4.1.3 Interfacial Bond or Fiber-Matrix Adhesion 11.4.2 Laminates 11.4.2.1 Tension 11.4.2.2 Compression 11.4.2.3 Shear Tests 11.4.3 Statistical Treatment of Data 11.5 Conclusions 11.6 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

496 498 498 498 501 502 503 505 505 506 506 507 507 507 509 510 512 514 516 516 516 519 519 520 520 522 524 528 530 531

496

11 Strength of Fiber Composites

List of Symbols and Abbreviations A cross-sectional area of a fiber Aij9 Qij reduced laminate and ply stiffnesses a0 distance ahead of a hole b beam width b, W specimen width c half crack length of a center crack d, t specimen thickness df fiber diameter E Young's modulus Ef m, £ fy , my fiber, matrix Young's modulus, in longitudinal and transverse directions Es shear modulus ej9 e] laminate mid-plane total strains, ply thermal strains measured from the stressfree state FhFtj strength parameters (failure theory) Fx (x) cumulative distribution function Fx* dimensionless parameter /0 amplitude of fiber deflection G shear modulus G m , Gx matrix, composite shear modulus h laminate or beam thickness / sectional modulus of beam / y , J* percentage point for confidence interval K stress concentration factor Kms K in shear Kmy matrix stress concentration factor KT finite plate stress concentration factor Kj infinite plate stress concentration factor KT/Kj finite width correction factor L gage length L half wavelength of fiber buckling L(a, P) likelihood function /c critical length Nt,N? applied laminate stress resultants, due to thermal strains p, P applied load at failure PT probability Critical critical buckling load R radius of a hole in an infinite orthotropic plate S, S' positive a n d negative shear strength in longitudinal direction Sj interface or interlaminar shear strength ^n matrix shear strength Us testing system displacement Ut total displacement Vfm fiber, matrix volume fraction

List of Symbols and Abbreviations

w

X,X' X

Y,Y' z Z

r 3 3 £

c,f,m

width Weibull distribution random variable tensile and compressive strength in longitudinal direction static strength (random variable) fiber tensile strength interface strength matrix tensile strength tensile and compressive strength in transverse direction thickness coordinate ultimate strength in z-direction shape, scale parameter estimate of a, fi values of the confidence interval for a, /? gamma function ineffective length deflection at the mid-point of the specimen composite, fiber, matrix strain axial, transverse strain strain at failure strain at failure in longitudinal and transverse directions stress partitioning parameter ply angle mean value Poisson's ratio = R/(R + a0) = c/(c + a0)

far field stress, along the x-axis stress components in the material axis fiber, matrix strength applied stress shear stress <jf m in transverse direction variance ply residual stresses notched strength of the infinite width laminate average fiber strength average interfacial shear strength fraction of undisturbed stress value FPF IITRI LEFM PEEK SACMA

first ply failure Illinois Institute of Technology Research Institute linear elastic fracture mechanics poly(ether ether ketone) Suppliers of Advanced Composites and Materials Association

497

498

11 Strength of Fiber Composites

11.1 Introduction In this chapter, representative theories for predicting the strength of unidirectional laminae and multidirectional laminates and test methods for determining ultimate strength of composites are discussed. The discussion deals with continuous fiber reinforced resin matrix composites. Unidirectional strengths are discussed on the basis of micromechanics theories for predicting longitudinal, transverse, and shear strengths including the effect of fiber-matrix interface on composite strength. Representative failure theories for laminates are discussed for predicting the first-ply failure as well as the ultimate failure in conjunction with the classical laminated plate theory and compared with experimental results for a variety of material systems and laminates. The effects of free-edge stresses and stress concentration around a notch are also discussed. In the last section, representative test methods for determining strength of constituents, unidirectional and multidirectional laminates are described, including statistical treatment of test data for the evaluation of design allowables and reliability analysis.

11.2 Strength Analysis of Unidirectional Laminates A unidirectional laminate consists of continuous fibers embedded in a resin matrix and is a basic building block of structural laminates. The fibers are bonded to the matrix at the interface. The fibers are very stiff and strong compared with the matrix. Thus, a unidirectional composite displays a large degree of anisotropy in its properties. The unidirectional composite is very strong in the fiber direction but is

weak in the direction perpendicular to the fibers. It is also important to note that longitudinal tensile response is mainly governed by the fiber properties, whereas the transverse and shear responses are strongly dependent upon the matrix and the interfacial bond. Prediction of unidirectional composites on the basis of constituent properties has not been as successful as prediction of their stiffness. Consequently, a number of micromechanics models for prediction of unidirectional strengths are available in the literature. An important role of the micromechanics analysis is to establish how these macroscopic properties can be controlled by the geometric and material properties of the constituents. In this section, longitudinal tensile and compressive strengths, transverse tensile and compressive strengths, and shear strength are discussed in continuous fiber reinforced resin matrix composites following a brief description of failure modes. 11.2.1 Longitudinal Tensile Strength

Fiber break and interface debonding are two major failure modes in a unidirectional composite in a longitudinal tensile loading state, and differ widely, depending on the properties of the constituent materials and fiber defects. In most advanced composites, such as boron and graphite fiber with polymer matrix, the strain to failure is greater in the matrix than in the fibers. Therefore, fibers which contain flaws or defects can break before the interface or matrix fails. Representative fracture surfaces of the unidirectional composites loaded under tension along the fiber direction are illustrated in Fig. 11-1 (Chamis, 1974). The crack created by a fiber break grows along a path varying with matrix, fiber-matrix interface, or adjacent fibers as the load increases. If bonding is strong, the

11.2 Strength Analysis of Unidirectional Laminates

Axial crack or ||/interf acial failure

Filament pullout

.Fiber break

499

where subscripts c, f, and m refer to composites, fiber and matrix, respectively. The use of force equilibrium along the fiber direction results in the following expression for composite longitudinal tensile strength, X (11-2)

Figure 11-1. Longitudinal tensile failure modes, (a) Brittle, (b) brittle with fiber pullout, (c) irregular (Chamis, 1967).

when the fiber controls the composite strength, and (11-3)

crack grows into the matrix resulting in a fairly smooth surface across the section. With a weak bond, the crack is more likely to lead to interfacial debonding and extensive fiber pullout. These failure mechanisms occur under static as well as fatigue loading. A variety of analytical models in the literature are available for evaluating the longitudinal tensile strength. We will present two of the most popular approaches, the rule-of-mixture model and the statistical model. The rule-of-mixture model is simple and reasonably accurate for estimation of composite strengths from the constituent properties. Furthermore for resin matrix composites reinforced with stiff fibers such as carbon and boron fibers, fiber strength and its volume fraction are only required for calculation of composite strength. This is very convenient for material development, quality control, material screening and selection, etc. This model also assumes that the constituents (fiber and matrix) carry loads proportional to their stiffness as schematically shown in Fig. 11-2; the composite fails when the average fiber strength is reached, and the ply is in a state of constant strain, i.e., 8 = 8

f

=

8n

(11-1)

when the matrix controls the composite strength. The symbols V and E are volume fraction and Young's modulus, respectively. Equations (11-2) and (11-3) are known as the rule of mixture equations. Although these equations are crude approximations to the longitudinal tensile strength of a unidirectional laminate, these equations display the significant contribution of the fiber strength to the ply strength. Tensile strengths of all resin matrix composites reinforced with carbon, boron and glass fibers are controlled by fibers, and

Fiber

Matrix

Strain

Figure 11-2. Typical stress-strain relations of fiber, matrix and composite. The composite failure strain is close to the fiber failure strain. The matrix is nonlinear above the fiber failure strain (Tsai and Hahn, 1980).

500

11 Strength of Fiber Composites

thus are calculated by Eq. (11-2). For the cases where EJEf <£ 1 Eq. (11-2) becomes X = Vt*f

(11-4)

Fiber strength can be determined by either the fiber bundle strength or the single filament strength. It is noted that the fiber bundle strength is generally smaller than that of the single filament strength. Experiments show that most composite materials exhibit large scatter in strengths. The degree of scatter may depend mainly on flaws or cracks in fibers which were formed in the course of fiber manufacturing as well as in composite processing. Since the size and distribution of the above defects are generally statistical in nature, many investigators adopted a statistical approach that can describe the prediction of the longitudinal strength. Rosen (1964) treats the composite failure based on the statistical distribution of flaws or imperfections in the fibers. The fibers containing critical flaws will fail first at the stress level below the average fiber strength. The accumulation of such fiber failures causes composite failure when the remaining unbroken fibers are unable to withstand the applied load. In this treatment of the tensile failure, Rosen introduced the ineffective fiber length concept. The ineffective length is defined as the portion of a broken fiber which is ineffective as reinforcement at the stress below the average stress of unbroken fibers. Outside this ineffective length, the broken fiber is still stressed equally with adjacent fibers due to the load transfer in the matrix. The most probable failure strength X for the composite was calculated by using Eq. (11-5) derived by Rosen (1964) X= V{(oLp5e)~1/p

(11-5)

where V{ is the fiber volume fraction, a and B are the statistical constants and are de-

termined by experiment for the different lengths of fibers, 3 is the ineffective length, and e is the base of natural logarithms. The statistical constants a and /? can be determined from test data on average strength as a function of fiber length. Assuming fiber strength is characterized by the Weibull distribution function, the average fiber strength is given by (11-6) where L is the gage length of the specimen and F is the gamma function. For a straight line log-log plot of <7f versus L, /? is estimated by log (L 2 /L t )

(11-7)

log((T fl /(J f2 )

Subscripts 1 and 2 are two stress levels, and a can be calculated from Eq. (11-6) after estimating /?. The ineffective length 3 is determined by a one-dimensional shear-lag theory. Rosen derived the following equation for the ineffective length, 3

V<1/2Gm cosh - 1

J

2(1-.

(11-8)

where df is fiber diameter, Vf fiber volume fraction, Ef fiber modulus, Gm matrix shear modulus, and cp the fraction of the undisturbed stress value below which the fiber is considered to be ineffective. The determination of q> value can be found in the reference: Rosen (1964). These ineffective lengths are relatively small. The predicted composite failure stress is plotted in Fig. 11-3 for the range of ineffective lengths of 1 to 100 fiber diameters. Experimental investigation shows that the typical failure of the high fiber content

11.2 Strength Analysis of Unidirectional Laminates 10 000

5 000

1000 10 Ineffective Length Ratio, 6/df

100

Figure 11-3. Statistical model of glass fiber-plastic composite failure stress (Rosen, 1964).

composite closely resembles the suggested failure model, characterized by random fracture of the fibers prior to ultimate failure of the composite. This model predicts considerably higher strength than the experimental data obtained from E-glass/ epoxy (Chamis, 1967). 11.2.2 Longitudinal Compressive Strength

In longitudinal compression, unlike tension, the matrix plays a significant role in compressive strength. The matrix provides lateral support for the fibers to carry compressive load without buckling. Without such matrix support fibers cannot resist compressive load. Fiber microbuckling and shear failure are considered to be two major failure mechanisms in longitudinal compression. There is also constituent debonding followed by fiber microbuckling, and ply separation or vertical splitting by transverse tension. Fiber microbuckling with an elastic matrix usually occurs in composites with low fiber volume content. In most composites in which the fiber volume fraction is greater than 0.5, fiber microbuckling, matrix yielding and constituent debonding occur.

501

Experimental investigation indicates that in most cases the microbuckling failure occurs at a considerably lower stress than the anticipated compressive strength. The compressive strength with shear failure mode approaches that for longitudinal tension in graphite, boron and s-glass fiber reinforced composites. Rosen (1964) proposed an analytical model for compressive strength of fibrous composites based on the microbuckling mechanism. Two possible failure modes considered in this analysis are the extension mode and shear mode. In the extension mode the fibers may buckle in opposite directions (symmetric) in adjacent fibers, and the major deformation of the matrix is an extension in the direction perpendicular to the fibers. In the shear mode the fibers buckle in phase with one another, so that the matrix deformation between fibers is due primarily to a shear. The condition for instability is given by equating the strain energy change to the work done by the external loads during buckling. The results for the compressive strength, X\ for the extension mode are given by X' = 2 F f {[F f £ m £ f /3(l - V,)]}1'2

(11-9)

The result for the shear mode is given by ' = Gm(l-7f)

(11-10)

As shown in Eqs. (11-9) and (11-10), the matrix shear modulus is an important factor in compressive strength. Hahn also derived a similar expression that the upper bound of compressive strength becomes equal to the effective shear modulus of the composite (Hahn and Williams, 1986). He notes that a correction should be made for the nonlinearity in shear of a unidirectional composite in order to improve prediction. The compressive failure of the composite is caused by a local shear failure,

502

11 Strength of Fiber Composites

cussed earlier, internal defects, such as voids, can also have an important effect on transverse strength property. In transverse tension or compression the load sharing by the matrix is of the same order of magnitude as that of the fiber. Since the matrix is much weaker than the fiber, the matrix will fail first and lead to the failure of the composite. The composite stress is related to the average stress in the constituents, and the transverse strength is given by

and the resulting compressive strength is X' =

(11-11)

1+(TI/0/L)/(S/GX)

where L is half wavelength of fiber buckling, / 0 the amplitude of fiber deflection, S the shear strength of the composite, and Gx the shear modulus of the composite. The compressive strength also can be calculated by the rule-of-mixture method as shown in tension in a previous section. Although the rule-of-mixture equation is a rough approximation for the longitudinal compressive strength, some experimental data indicate that the equation might predict reasonable results.

Y=Vfafy+Vm
(11-12)

where subscript "y" denotes the transverse direction. The transverse matrix strength and transverse fiber strength are related to the stress partitioning parameter, rjy as follows (Tsai and Hahn, 1980)

11.2.3 Transverse Tensile Strength

Depending upon the matrix properties and interface bond, the unidirectional composites would fail in the following modes under transverse tensile load as illustrated in Fig. 11-4: (1) matrix tensile failure, and (2) mixture of matrix tensile failure and fiber-matrix debonding. For the case where bond strength and fiber transverse strength are high, the failure surface tends to run primarily through the matrix (Fig. 11-4 a). When the fiber-matrix bond strength is low, debonding occurs and the fracture surface includes many debonded resin surfaces (Fig. 11-4 b). When fiber transverse strength is low, and matrix and bond strength are high, transverse fiber failure may occur (Fig. 11-4 c). As dis-

J

fy

(11-13)

where 0 < rjy < 1. By substituting Eq. (11-13) into (11-12), the average transverse stress of the composite can be expressed as follows y = [l + Vf(l/rjy- l)]ermy

(11-14)

The stress amy in the matrix is not uniformly distributed, but its maximum value, (o-my)max, occurs at the fiber-matrix interface. Therefore when the (crmy)max reaches the matrix tensile strength, X m , at the interface, the failure occurs; that is, ((7my)max ~

(11-15)

I—^-e~^—i Matrix failure

Bond failure

Fiber failure

(a)

(b)

(c)

Figure 11-4. Typical transverse tensile failure modes, (a) Matrix failure, (b) bond failure, (c) fiber failure.

11.2 Strength Analysis of Unidirectional Laminates

It is noted that if the interface strength X{ is smaller than Xm, X{ should be used in place of Xm. Introducing a stress concentration factor Kmy defined by Kmy

=

((Jm

y)max

( 1 1 1 6 )

Tsai (1968) and Adams et al. (1967) used finite difference to solve the plane elasticity problem for obtaining the stress concentration factor. The results of stress concentration from the analysis are shown in Fig. 11-5 (Adams et al., 1967). This is because the fibers and matrix have different transverse stiffness properties. The results indicate that the transverse stress in the material is significantly higher than the applied transverse stress. The magnitude of the stress concentration factor increases as fiber volume increases. The results also show that the stress concentration factor increases as the ratio Ef/Emy increases. This occurs when fiber volume fraction Vl increases as shown in Fig. 11-5. Experimental results indicated that only qualitative agreement exists between experiment and theory. The major reasons for this discrepancy are as follows. The distribution of

o o 3.4 LL

c 3.0 - circular filaments o in a square array "o 2.6 S 2.2 o o 1.8

503

fiber in a real composite is greatly different from that of a regular square array. Because of the progressive nature of composite failure, local failure at some region of the specimen prior to the final failure will redistribute the stress, and the elastic analysis is no longer valid. Nevertheless, analysis of this type provides a valuable insight into factors affecting composite strength. By combining Eqs. (11-14), (11-15) and (11-16), we obtain the transverse tensile strength of the composite, Y K my

(11-17)

An exact determination of rjy is difficult mainly because the behavior of the matrix is nonlinear near failure. If the matrix is linear elastic up to failure, then the factor [1 + Vf(l/rjy - i)]/Kmy is known to be less than unity and decreases with increasing fiber volume fraction. Therefore, the transverse tensile strength will always be less than the matrix tensile strength, and the difference increases with V{. On the other hand, if Kmy becomes close to unity, Y can be greater than the matrix tensile strength. Figure 11-6 shows the experimental evidence for the observation just described (Tsai and Hahn, 1980). The fracture surface in transverse tension is normal to the loading as shown in Fig. 11-4. However, in transverse compression, it is approximately 45° to the loading. The transverse compressive strength in graphite/epoxy composites is four to six times as high as the transverse tensile strength for advanced composites.

S 1.4 in 1.0 1

2 4 6 810 20 4060100 200 400 1000 Constituent Stiffness Ratio, Ef/Em

Figure 11-5. Matrix transverse stress concentration factor for various composite systems (Adams et al., 1967).

11.2.4 Shear Strength

A composite could fail in the following modes under in-plane shear stress: (a) matrix shear failure, (b) constituent debond-

504

11 Strength of Fiber Composites

pears to be the interfacial bond strength. In other words, the interlaminar strength can be increased only by improving interfacial bond strength. Voids can also have a considerable effect on the shear strength as illustrated in Fig. 11-9 (Brelant and Petker, 1970). The voids cause severe internal stress concentration in the material.

Figure 11-6. The composite-to-matrix strength ratio in transverse tension decreases as the matrix strength increases. Note that the composite strength can be higher than the matrix strength. The data are for E-glass and S-glass/epoxy composites (Tsai and Hahn, 1980).

ing, and (c) matrix shear failure and constituent debonding. The mechanisms of shear failure are similar to those of transverse tension failure. The shear strength S can be studied using an equation similar to Eq. (11-14)

- circular filaments in a square array

2

4 6 810 20 4060100 200 400 1000 Shear Modulus Ratio, Gx/Gm

Figure 11-7. Stress concentration factor subjected to in-plane shear loading (Adams et al., 1967).

(11-18) IDU "

D Q_

B

a

I

C

£ 100-

B / B

cn c (U

B jL

(/)

J"


g 50-

0 0

B

B

B^/H

B/

Resin

where Sm and Kms are the matrix shear strength and the matrix stress concentration factor in shear, respectively. Analytical results (Adams et al., 1967) on the stress concentration factor shown in Fig. 11-7 are similar to the transverse case discussed in the previous section. Again, if the interface shear strength Sj is smaller than Sm, Sj should be used in place of Sm. The significance of resin strength on interlaminar shear strength is illustrated in Fig. 11-8 (Brelant and Petker, 1970). Interlaminar strength increases as resin tensile strength until about 80 MPa and levels off afterward. The limitation factor for this ap-

B

20 40 60 Interlaminar Shear Strength in MPa

80

Figure 11-8. Interlaminar shear strength versus resin tensile strength (Brelant and Petker, 1970).

11.3 Strength Analysis of Multidirectional Laminates

505

tion. The analytical results are compared with experimental results obtained from a number of laminates. The effect of delamination and notch on the ultimate strength of laminates is given. 11.3.1 Failure Modes

5

10 Void Content in %

Figure 11-9. Effect of void on interlaminar shear strength (Brelant and Petker, 1970).

11.3 Strength Analysis of Multidirectional Laminates Most composite laminates for structural application consist of multidirectional layers to meet the structural integrity requirement. In-plane strength analysis of a multidirectional laminate is of great importance in design of the laminates as well as development of material systems. In this section we first describe briefly the damage modes, in terms of transverse crack and delamination for selected laminates, and present some of the failure theories currently employed in composite strength analysis. The prediction of the first-ply failure and the ultimate tensile strength of multidirectional laminates is described as an engineering approxima-

Fiber reinforced polymer matrix composites such as graphite/epoxy, boron/epoxy, and glass/epoxy exhibit complex damage mechanisms under static and fatigue loading because of anisotropic characteristics in their strength and stiffness. Instead of a predominant single crack, often observed in most isotropic brittle materials, extensive damage throughout the specimen usually accompanies static and fatigue failure in composites. Basic failure mechanisms in a multidirectional laminate are matrix cracking, delamination, fiber breakage, and interfacial debonding. Any combination of these causes damage, which may result in reduction of strength and stiffness. Damage varies widely, depending on material properties, lamination (including stacking sequence), type of loading, etc. Unidirectional laminates with the fiber along the loading direction do not show any matrix cracking in the polymer matrix composite, since the strain to failure for the matrix is much larger than that of the fiber (Fig. 11-2). However matrix cracking occurs in multidirectional laminates prior to ultimate failure. To illustrate matrix cracking, consider the matrix of a [0/90/±45] s graphite/epoxy laminate subjected to uniaxial tension. Successive transverse cracks in the respective off-axis layer are expected as the applied load to the laminate increases. The first cracking occurs in the 90° layers; with greater load, the number of cracks increases and is initially confined to the 90° layers. As the load increases, new

506

11 Strength of Fiber Composites

cracks occur in the adjacent 45° layers, appearing at the tip of the 90° cracks and extending to the interface of the + 45°/-45° layers. Subsequently, the number of cracks increases with the load level until final laminate failure (Kim, 1980). However, some laminates reach a crack density limit, after which no new cracks occur before final failure, despite additional loading (Reifsnider et al., 1979). The crack-density limit for a given layer varies with its thickness and constraints but appears to be independent of laminate type (Wang et al., 1980; Garret and Bailey, 1977; Flaggs and Kural, 1982; Reifsnider et al., 1979). The 0° layers in a multidirectional laminate are also susceptible to cracking along the fiber. In composites, free-edge delamination is caused by interlaminar stresses localized around the free edge under in-plane loading (Pagano, 1978). Among the interlaminar stresses, tensile normal stress is a significant factor in most delaminations (Pagano, 1978; Kim and Soni, 1984). The laminate stacking sequence determines whether interlaminar normal stress produces tension or compression at the free edges. For example, a quasi-isotropic laminate with two stacking sequences, [0/90/±45]s and [0/±45/90] s , is subjected to uniaxial tensile loading. The latter stacking sequence produces tensile normal stress along the free edges of a coupon, whereas the former produces compression. Consequently, the [0/±45/90] s laminate shows extensive delamination under tension before final failure but shows no delamination under compression. In addition to interlaminar tensile stress, other mechanisms such as transverse cracking and interlaminar shear stresses appear to be significant in the onset and growth of delamination under fatigue loading (in some cases).

11.3.2 Failure Theories

There exist a number of failure theories available in composite strength analysis, although it is important to have a unique failure theory that can be predict the strength of laminated composites accurately. We will discuss failure theories that are widely used in composite strength prediction. 11.3.2.1 Maximum Stress Theory

Maximum stress theory assumes that failure of the structural element subjected to a combined state of stress occurs when loading has reached such a value that one of the principal stresses becomes equal to the uniaxial strength in tension or compression. If one of the following three principal stresses reaches the uniaxial strength, then the failure occurs: <jx = X , if (j x > 0 or

ay = X

=X'

if (Jx < 0

if 0 or \ay\ = Y

f

if
where X9X' = tensile and compressive strength in longitudinal direction, X Y' = tensile and compressive strength in transverse direction, and S, S" = positive and negative shear strength in longitudinal direction. From orthotropic symmetry condition, shear strength is independent of sign. There are five independent failure modes as shown in Eq. (11-19). Since failure processes in composite laminates are highly interacting among the stress components and more complicated than the simple mode, the agreement between test result and theory is poor for certain combinations of stresses. Figure 11-10 represents the maximum stress theory for two dimensional conditions expressed by Eq. (11-19).

11.3 Strength Analysis of Multidirectional Laminates

507

parameters. The relations between the strength parameters, F , . . . , AT, and actual strength values, X, Y and S, can be determined simply by applying individual uniaxial loadings, i.e.:

X'

Y'

Figure 11-10. Maximum stress criteria for an orthotropic ply in stress space.

G + H = 1/X\

F + H = 1/72,

F + G = 1/Z2,

2iV = l/S 2 ,

where Z is the ultimate strength in the z-direction. Consequently,

11.3.2.2 Maximum Strain Theory

1

This is very similar to the maximum stress theory. This theory assumes that failure of the structural element subjected to a combined state of stresses occurs when the maximum value of the principal strains equals the value of the strain at failure (e*) in uniaxial tension or compression. If one of the following conditions is met, then the failure occurs:

A

1 1

1

2F = 2 +

T

1 Z^

1

1

(11-22)

Y*~Y2

Assuming that Y = Z and plane stress conditions (cz = erxz =
X' or Y

X2 Y'

X2

xy

~

(11-23)

11.3.2.3 Maximum Distortional Energy Theory

Equation (11-23) is known as the Tsai-Hill failure criterion. A comparison with experimental results (and with the two previous failure criteria) is shown in Fig. 11-11 (Tsai, 1968). Clearly the Tsai-Hill failure criterion seems to agree very well with experimental results obtained for glass/epoxy.

Hill (1950) has formulated the von Mises criterion for orthotropic materials as

11.3.2.4 Quadratic Theory

(11-20)

S

F (
This failure criterion proposed by Tsai and Wu (1971), has been published in numerous reports and journal articles. The mathematical formulation of this model was originally presented by Tsai (1968). Details on experimental procedures, optimal experimental measurements, and mathematical formulations are given by Tsai and Wu (1971) and Wu (1972) and it is

508

11 Strength of Fiber Composites

To ensure that the failure surface will intercept each stress axis, the magnitude of the Fxy term is constrained by the following inequality F

F

—F

> i

(11-27)

In view of the difficulty in experimental determination of Fxy, Tsai introduced a dimensionless parameter (11-28)

30 45 60 Fiber Orientation, d

Figure 11-11. Comparison of maximum stress, maximum strain and Tsai-Hill failure criteria (Tsai, 1968).

recommended to refer to these in order to become more familiar with this model. Tsai-Wu (1971) failure theory for anisotropic materials in stress space is given by FtjWj + Fta^l

(11-24)

where Ftj and Ft are strength parameters associated with the failure theory, and o{ and a • are the stress components in the material axis. When the problem is limited to orthotropic materials under the plane stress state, Eq. (11-24) assumes the following form

Tsai assigned — 1/2 to the value of Fx* based on a generalization of the von Mises yield criterion for isotropic materials. Details can be found in: Tsai and Hahn (1980). The quadratic failure criterion takes into account the interaction among the stress components and offers a great convenience in operation. This failure criterion is widely accepted and is used in the composite community. Figure 11-12 shows the failure envelope constructed by the quadratic failure criterion in stress space. Strength tests on off-axis and angle-ply laminates offer useful information for the evaluation of the failure theory and failure mechanisms. Figures 11-13 and 11-14 show comparison of Tsai-Wu quadratic failure theory with experimental results obtained from the off-axis laminates and angle-ply laminates, respectively (Kim, 1981 a).

- 2 + 2Fxy<7x<7y + Fss<7s2 + L P ff + F a = 1 (11-25) + F,

Fxxa2x+FT

The strength parameters are determined from material strength and are given by 1 XX

yy

xx" I YY"

J

'

x

y

1 ss

ss"

J s

1

I

X

x'

1

l

y

r

l

l

s

(11-26) Figure 11-12. Failure envelope constructed by the quadratic failure criterion.

509

11.3 Strength Analysis of Multidirectional Laminates

Gr/Ep As/3501

1600

1 600

Gr/Ep As/3501 Angle ply

Off-axis

F12*=-0.5

F 12 =-0.5 1200

1200

800

800

400

400

.£

i

i

15 30 9 in Degree

15 30 6 in Degree o -400

-400

-800

-800

-1200

-1200

-1600L

-1600 L

45

60

75

90

Figure 11-13. Tensile and compressive strength of offaxis laminates as a function of fiber orientation.

Figure 11-14. Tensile and compressive strength of angle-ply laminates as a function of ply angles.

11.3.3 First Ply Failure

chanical stresses and curing residual stresses. The analytical model proposed by Pagano and Hahn (1976) can be used to calculate the residual stresses resulting from cure. The basic assumption of this model is that the material behavior is elastic. Thus the total ply stresses, af, are related to the applied laminate stress resultants, Ni9 and ply residual stresses, of, and given by Ot-QtjAjSNt + of (11-29)

Because of the inherent inhomogeneity, the ultimate failure of multidirectional composite laminates is invariably preceded by the failure of weaker plies. The first ply failure (FPF) is defined as the first occurrence of a crack within a laminate subjected to a loading. The applied stress at the moment of the FPF depends on the type of laminate, constituent ply properties, constraint by the adjacent plies, and residual stress present within the laminate. The first ply failure can be predicted by applying a failure theory in conjunction with calculation of stresses in the ply of interest. The stresses consist of the me-

where Qtj and Atj are reduced ply stiffnesses and laminate stiffnesses, respectively. Note that the superscript — 1 denotes matrix inversion.

510

11 Strength of Fiber Composites

The ply residual stresses, of, within a laminate can be calculated by (11-30)

of = QtJ(eJ - e]) where

ment for first ply failure. Wet specimens contain approximately 1.4% of moisture by weight. 11.3.4 Ultimate Failure

e°j = mid-plane total strains measured from the stress-free state, e] = ply thermal strains measured from the stress-free state. The laminate thermal strains e? then follow from the classical laminated plate theory as e°i=A^N?

(11-31)

N? are the equivalent stress resultants due to the thermal strains, i.e., -h/2

Que]dz

(11-32)

where h is the laminate thickness and z the thickness coordinate. Upon substitution of Eq. (11-31) into Eq. (11-32), the residual stresses of in each ply are obtained as follows of = QijA^1

JVfeT - Qtje]

(11-33)

Equation (11-33) gives the residual stress components in the ply of interest. By applying an appropriate failure theory with the total stresses calculated by Eq. (11-29), we can calculate the first ply failure stress for a laminate. Table 11-1 shows comparison of theory with experiTable 11-1. First ply-failure stresses in MPa (Kim

and Hahn, 1979). Dry

Laminate

Theory Experiment [0/903]s [±45/90 2 ] s [0/± 45/90],

119 79 153

159 73 174

Wet Theory Experiment 139 84 197

168 69 205

When the applied load exceeds the FPF load, the laminate may or may not carry the additional load. Most laminates, especially those containing the 0° ply under tension, have a capability to carry the load beyond the FPF and reach ultimate failure. The ultimate failure or last-ply failure will be examined in this section for applied tensile loading. Most laminates reveal a progressive type of failure as described in Sec. 11.3.1 when subjected to an applied stress. In view of this progressive failure in multidirectional laminates, a number of approaches have been reported to develop a reliable methodology to predict the in-plane tensile strength. These can be divided into two major approaches that are with and without considering the damage occurred prior to the last failure. The approaches with consideration of damage can be divided into again two groups, depending upon complete elimination of the damaged ply or employing damage functions in calculation of strength. One of the main obstacles in accepting the damage function approach in strength prediction is complexity in the calculation process. Therefore, success of this approach appears to remain in question until one has a proper understanding of the damage process and its influence on the strength change. In this section we will discuss the simple approach without considering the damage within the laminate as examples. Figures 11-15 and 11-16 show analytical prediction of the failure stress of the respective layers for the laminate family of [0 2 /± 0]s, [0/90/± 6]s and [0/+ 9/90]s as a function of

11.3 Strength Analysis of Multidirectional Laminates

T300/5208 Gr/Ep [O 2 /+0] s

uoo 1200

a

1000

Q_

•£ 800 en c

600 400 200 0

0

15

30 45 60 8 in Degree

75

90

Figure 11-15. Comparison of calculated in-plane tensile strength with experimental. The solid line and dotted line indicate the predicted strengths of 0° and 90° plies, respectively.

1000

T300/5208 Gr/Ep o[O/±0/9O]s MO/90/10],

800 600

200

0

15

30 45 60 6 in Degree

75

90

Figure 11-16. Comparison of calculated in-plane tensile strength with experiment. Predicted 90°, off-axis and 0° plies are shown for two stacking sequences.

angle 9 (Kim, 1981 b). The dotted line and solid line indicate the predicted strength for the first ply failure and the last ply failure, respectively. Various symbols in Figs. 11-15 and 11-16 are experimental results which represent the mean of 10 to 11 specimens for a given laminate. In the [02/ ± 0]s laminate the experimental results compare fairly well with the analytical results of the ultimate failure (0° layer failure

511

in this case). Analysis gives a slightly conservative prediction in all cases except for 0 = 10° and 15°. Analytical prediction shows almost identical strength for the 0° and off-axis layers up to approximately 30°, and thereafter the difference increases as angle 9 increases as shown in Fig. 11-15. This implies that the laminates for 9 < 30° will fail without revealing progressive failure; that is, no off-axis ply failure will precede the final failure. On the other hand, off-axis ply failure will be anticipated before final failure when 9 is greater than 30°. The theory also appears to underestimate the off-axis ply (± 9) failure for the family of the [0 2 /± 0]g laminate for 9 > 30°. In spite of the numerous cracks extending to the entire off-axis layers in the [O 2 /±0] s and [0 2 /± 0]s laminates, the experimentally-determined strengths are greater than those of prediction. This implies that the transverse cracks in the off-axis ply which is greater than 60° do not influence the tensile strength of this laminate group. Experimentally-determined strength is significantly different between two families of the [0/90/ ±9]s and [0/+ 0/9O]s laminates for 9 < 45° in spite of identical prediction as shown in Fig. 11-16. For those laminates which did not show any delamination until final failure, the experimental results compared fairly well with the analytical results of the 0° layer. Experimentally-determined strength for the laminates experienced in delamination is markedly smaller than the predicted strength. Quadratic failure theory in conjunction with laminated plate theory has been applied to predict the in-plane tensile strength of multidirectional composite laminates. In analytical prediction, the ultimate failure of a laminate is assumed to occur upon the failure of the strongest ply in the laminate by ignoring the contribution of interlaminar stresses and transverse cracks preced-

512

11 Strength of Fiber Composites

ing final failure. The experimental results obtained from various types of laminates of T300/5208 graphite/epoxy were found to be in good agreement with prediction with consistent exceptions. The exceptions are that all the laminates show delamination at the free edges, which substantially reduces the load-carrying capability. The experimental result indicates that the effect of transverse cracking on the strength appears to be insignificant for those laminates considered in this study. This approach has the advantage of simplicity and quickness in approximation of the in-plane tensile strength of multidirectional laminates. 11.3.5 Free Edge Effect Delamination has long been recognized as one of the important failure modes exhibited in composite laminates. The growth of delamination under load may result in a reduction of strength and stiffness of a laminate. Such a detrimental effect of delamination on laminates has been demonstrated in: O'Brian (1981), Kim (1983).

Analysis has shown the existence of interlaminar stresses along the free-edge region of composite laminates in the presence of an in-plane uniaxial loading (Pipes and Pagano, 1970; Pagano and Rybicki, 1974; Pagano and Soni, 1983). Figure 11-17 illustrates the state of stress at the free edge region of a laminate under uniaxial loading. Of the interlaminar stresses, tensile normal stress plays a significant role in initiation and propagation of delamination. The nature (tensile or compressive) and determination of the interlaminar normal stress has received extensive analysis. The strength reduction due to delamination varies, depending upon the delaminated area and type of loading for a given laminate. Table 11-2 shows the tensile strength of a quasi-isotropic laminate with four different stacking sequences. The first three stacking sequences underwent considerable delamination before ultimate failure, but the fourth stacking sequence, [0/90/± 45]s, failed without exhibiting delamination. The ultimate strength of the undelaminated [0/90/±45] s is greater than the strength of all three that had delami-

Free Edge Interface

Figure 11-17. Models for studying free-edge effects laminate geometry.

513

11.3 Strength Analysis of Multidirectional Laminates

Stacking sequence

Unreinforced

Reinforceda

Delamina- Ultimate tion thresh- strength old level (MPa) (MPa)

Ultimate strength

[0/±45/90]s 360 [±45/0/90]8 270 [+45/90/0], 370 [0/90/+45]s No delamination

446 431 498 573

(MPa) 577 572 576 585

No delamination occurred in reinforced specimens.

1 000 900

Prediction: Experiment; unreinforced A(O/9O/i0]s O[O/i0/9O]s reinforced #(O/±0/9O]s

\ O

800 o Q_ IE

700

JZ

\ \ A

_

C

6ue

nated. These specimens were also tested with a reinforced free edge to prevent delamination. None of the reinforced-specimen free edges exhibited delamination. The strength of the first three laminates with reinforcement was substantially increased and was practically equal to the strength of the [0/90/±45] s laminate. In other words, all the reinforced laminates show practically the same strength. Figure 11-18 shows the effect of delamination on the in-plane tensile strength of the [O/±0/9O]s and [0/90/±0] 8 laminates as a function of the angle 6 (Kim, 1981b). The solid line indicates the analytical prediction under the assumption that ultimate failure of the laminate occurs upon the strongest ply failure for a given externally applied stress resultant, ignoring the damages incurred in the course of loading. To construct the solid line, the in-plane stress components in each constituent ply for a given laminate were calculated using laminated plate theory. The quadratic failure criterion is then applied to the respective ply. In Fig. 11-18 the triangles and circles are the observed strengths for the [0/90/ ±0] s and [0/ + 6/90]s laminates, respectively. For 9 less than 45°, the [O/±0/9O]s laminates

revealed substantial delamination and failed at a much lower stress than the [0/90/ ±0] 8 laminates. However, with the free-edge reinforcement (solid circle in Fig. 11-18), the strengths increased substantially and were almost the same as those of the [0/90/±0] s laminate. It is also noted that for those laminates which did not delaminate until final failure, the experimenal results agree fairly well with the prediction. The strength reduction appears to be more pronounced in those laminates that do not contain 0° plies. Figure 11-19 shows the predicted strain response of the boron/epoxy [ ± 30]s angle-ply laminate along with the experimentally observed response (Pipes et al., 1973). The prediction of the strain response agrees quite well with the observed response, but the observed strength was approximately onehalf of the predicted strength. This discrepancy between prediction and observation of the strength is mainly attributed to the presence of a high interlaminar shear

Str

Table 11-2. Tensile strength of quasi-isotropic laminate with four different stacking sequences.

O

600

-

500

-

#

\ \ \

1 0 \a

400

a

^

^

O

300 1

15

1

1

30 45 60 6 in degrees

1

I

75

90

Figure 11-18. Comparison of prediction and experiment for ultimate failure for quasi-isotropic laminates of T300 graphite/5208 epoxy laminates.

514

11 Strength of Fiber Composites

NARMCO 5505 B0R0N-EP0XY 130° COUPON Predicted Experimental

Figure 11-19. Comparison of prediction and experiment for strength and strain for the [±30] s laminate of boron/epoxy (Pipes etal., 1973).

0.0077 I i

0.002

0.004

0.006 0.008 0.010 c x , Strain

0.012

0.0U

stress at the +30/ —30 interface. A similar strength reduction was found for the graphite/epoxy [±30/90] s laminate (Kim, 1986). The effect of delamination can be much more severe under compressive loading than under tensile loading. Buckling is the major failure mode of a delaminated laminate under applied compression. When a laminated composite with delamination is subjected to compressive loads, the delaminated region buckles, as illustrated in Fig. 11-20 (Whitcomb, 1984). This buckling causes a high interlaminar stress concentration at the delamination front (crack tip); and under increased load, the buckled area increases to a critical size which can lead to a loss of global stability or total collapse of the plate. This occurs usually at load levels far below the undamaged compressive strength of the material or stability of the plate. There are some

Figure 11-20. Illustration of through width buckling of a composite plate (Whitcomb, 1984).

0.016

analytical models available to predict the buckling growth and the effect of various parameters on the delamination growth (Chai, 1981; Whitcomb, 1984; Donaldson, 1987). The models are based on the fracture criterion or on the strain energy release rate. However, because of the complexity of the problem, due to factors such as geometric shape of the delaminated region, thickness of the buckled laminate, fracture criterion, the highly nonlinear character of deformation, type of laminate (including stacking sequence), one-dimensional or two-dimensional growth, multiple delamination, stiffness and toughness of the laminate, etc., the models cannot be considered to be validated at this time. 11.3.6 Notched Strength

The presence of a discontinuity or notch such as a hole or a crack in a laminated structure introduces stress concentration around the notch. This stress concentration can be high, which results in initial localized failure. Problems of stress concentration in an anisotropic plate are much more complicated than in an isotropic plate. We consider an anisotropic homogeneous plate of any shape which is weakened by a circular hole and deformed by

11.3 Strength Analysis of Multidirectional Laminates

Figure 11-21. Plate containing a circular hole subjected to an axial tension.

515

size effect in anisotropic materials. The problem of the notch size effect has been studied by many investigators. Two approaches used are concepts of linear elastic fracture mechanics (LEFM) (Waddaoups etal., 1971; Cruse, 1973) and stress criterion (Whitney and Nuismer, 1974). In advanced composites, the model developed by Whitney and Nuismer is widely used for predicting the strength of laminated composites containing a circular hole or straight crack using stress criteria rather than LEFM. Consider a hole of radius R in an infinite orthotropic plate, as shown in Fig. 11-21. If a uniform stress a is applied parallel to the x-axis at infinity, then the stress, o"x, along the y-axis can be approximated by (Patterson, 1974)

f - ,,,-34, forces acting on the edges as shown in Fig. 11-21. When the opening is small in comparison with the plate size, then the problem becomes simple. We can assume the plate as being infinite and will disregard the effect of the finite width. Unfortu-

? =1 +

where Kj is the orthotropic stress concentration factor for an infinite width plate as determined from the following relationship ^ 2 2 ~~

L

n

nately, in most cases, the plate width is finite and finite width correction is required. In anisotropic materials the problems for finite width have not been fully studied, and many investigators have been using the correction for the isotropic case, which may result in some errors. Unlike isotropic material, experiment shows that the fracture strength of notched anisotropic plates is dependent upon the notch sizes, even though the stress concentration at the edge of the notch is independent of the notch size. This is called notch

2A 66

(11-35)

where A(j are the in-plane laminate stiffness as determined from laminated plate theory. Subscripts 1 and 2 denote the directions parallel and perpendicular to the applied stress at infinity, respectively. The average stress criterion introduced by Whithney and Nuismer (1974) has been widely accepted and used for calculation of the notched strength of composite laminates. Figure 11-22 illustrates this failure theory. The average stress criterion assumes failure to occur when the average value of <JX over some distance, a 0 , ahead of

516

11 Strength of Fiber Composites

and circles are experimental data. In data reduction the notch strength for an infinite plate is obtained from the experimental notched strength aN using the relationship

d

Figure 11-22. Stress distribution around a circular hole.

where KT/Kj is finite width correction factor. Approximate expressions for these factors can be found in refs. (Patterson, 1974; Paris and Sih, 1965) and are given by = T/

the hole first reaches the unnotched tensile strength of the laminate; that is, when \

-

(11-39)

T

2 + (l-2R/w)3 3(l-2i?/w)

for circular hole and KT/K? = v/(w/7ic)tan(7ic/w)

R+a0

J ox(y,0)dy = a0

(11-36)

Using this criterion with Eq. (11-34) results in the notched to unnotched strength ratio

(11-37) where a0) and O"N is the notched strength of the infinite width laminate. For the case of center crack with crack length 2 c, the strength ratio is given by (Whitney and Nuismer, 1974)

(11-40)

(11-41)

for center crack. Equations (11-37) and (11-38) predict the proper trend of observed strength behavior, that is, decreasing strength of a laminate with increasing hole size. It is clear that the utility of the model would be greatly increased if the characteristic distance a0 is shown to remain constant for all laminates of a particular materials system. Otherwise, the utility of the model decreases considerably. Tan and Kim (1991) observed that in general the characteristic distance remains constant unless the predominant failure mode changes from fiber dominant mode to matrix dominant mode and vice versa.

(11-38)

11.4 Test Methods £2 = c/(c + a0) Figures 11-23 and 11-24 show the results of Eqs. (11-37) and (11-38) compared with experimental results obtained from a quasiisotropic laminate of graphite/epoxy (Whitney and Nuismer, 1974). The solid line represents prediction for a0 = 3.81 mm

11.4.1 Constituent Materials 11.4.1.1 Fibers

Test methods for determination of tensile strength and tensile modulus of reinforcing fibers will be described. Accurate measurement of tensile properties of rein-

11.4 Test Methods

517

(0/±45/90) 2 S T300/5208

Figure 11-23. Comparison of failure theory and experimental results for circular holes in [0/±45/90]2s laminate (Whitney and Nuismer, 1974). 5.0 7.5 Hole Radius R in mm

10.0

12.5

1.0 (0iA5/90]2S T300/5208 0.8 -

0.6 0.4 --

0.2 -

\

-c=±=

o0 = 3.8 mm

J

U-2C-*J

1° 2.5

" """"—•

-

+~ y 1

1

1

5.0 7.5 10.0 Crack Half Length C in mm

forcing fibers is useful from the standpoint of product development, quality control, materials selection, and design and stress analysis of composite materials. Test methods commonly applied to measure tensile properties of fibers are single filament test and resin impregnated yarn or strand test. The details of these test methods are given in ASTM Standards D 3379-75 for single filament test and D 4018-81 for yarn test. Sometimes fiber strength is calculated from the value of unidirectional composite laminates using Eq. (11-4) in Sec. 11.2.1.

1 12.5

Figure 11-24. Comparison of failure theory and experimental results for center cracks in [0/±45/90]2s laminate (Whitney and Nuismer, 1974).

The single filament test method has been widely used to determine tensile strength, modulus, and strain to failure on a variety of fibers. The single fiber drawn from yarn or tow is placed on a paper mounting tab as shown in Fig. 11-25 (ASTM, 1987). The fiber must be handled with special care to avoid any visible damage. Both ends of the gage length are bonded to the paper mound using an adhesive (sometimes wax). The specimen gage is recommended to be greater than 2000 times the fiber diameter and 25 mm long in most cases. After grip-

518

11 Strength of Fiber Composites This section burned or cut away after gripping in test machine

Cement or wax

50 Test specimen .Grip area

A

,40-

/ E

Width

/

.£ 30-

/

5: 5

Grip area

20-

Figure 11-25. Tab showing typical specimen mounting method (ASTM, 1987).

a. t

10y

ping the specimen in the testing machine, the middle section of the paper mounting tab is cut away. The tensile strength and tensile modulus are calculated from the following equations:

y

A

/

/

*1

2.5 )U /P System Cc mpliance S 0

1 10

20 30 Gage Length in mm

50

Figure 11-26. System compliance calibration method (ASTM, 1987).

Tensile strength .

(11-42)

Tensile modulus Ef = LIA (Ut/P - UJP)...

(11-43)

where P = applied load at failure, A = cross-sectional area of fiber L = gage length, Ut = total displacement, Us = testing system displacement. The cross-sectional area can be determined by a variety of methods such as laser diffraction, microscopy, etc. The testing system compliance UJP must be determined experimentally by using several different gage lengths as shown in Fig. 11-26. Instead of the specimen displacement, the specimen strain can be directly measured with considerable accuracy using a laser extensometer. The fiber strength determined by this method varies widely dependent upon the gage length as illustrated in Fig. 11-27 (Kim, 1986). Impregnated yarn test method is widely used by many fiber producers. The test

1 10 Gage Length in mm

100

Figure 11-27. Tensile strength versus gage length for AS4 single fibers. Solid line represents linear fit of data.

specimen is prepared by impregnating the sample yarn or tow with suitable binder materials. The purpose of the impregnating resin is to provide yarn, roving or tow, when cured, with sufficient mechanical strength to produce a rigid test specimen capable of sustaining uniform loading of the individual filaments in the specimen. If the impregnating resin-to-fiber ratio is ex-

11. 4 Test Methods

W = G = L = R =

O

Specimen Dimensions in mm

CD O O O O

cessively high or low, erratic test results will be obtained. Tension test specimens shall consist of whole-number multiples of the yarn, tow, etc. The total length of the specimen is usually 330 mm. The specimen is usually tested by either a special castresin end tab with gage length of 152 mm or no end tabs. Figure 11-28 (ASTM, 1987) illustrates the test specimen with end tabs. The gage length of the specimen is 250 mm. The specimen strain is usually measured by an extensometer. Sometimes the crosshead displacement is used for calculation of elastic modulus. In this case the modulus and strength are calculated by Eqs. (11-42) and (11-43), respectively. The cross-sectional area (A) in this case is given by dividing weight per unit length of dry yarn or tow (g/m) by density of the dry yarn or tow (g/m3).

519

Figure 11-29. Tensile test specimen of neat resin.

strain can be obtained using a strain gage mounted perpendicular to the loading axis. Special care for strain measurement for plastics and composites is given in Sec. 11.4.2.1.

-330 mm-152 mm-

11.4.1.3 Interfacial Bond or Fiber-Matrix Adhesion

16 mm

t

1 End tab

Figure 11-28. Tensile yarn test specimen with castresin tabs (ASTM, 1987).

11.41.2 Matrix The tensile strength, modulus, and failure strain can be determined by using flat panel or die cast specimens as shown in Fig. 11-29 (ASTM, 1987). The details of the test method are given in ASTM D 638M-84 for a thick panel (1.0-10.0 mm) and D 63872 for a thin panel (< 1.0 mm). Because of the high degree of strain rate dependency on the loading rate, test results must be carefully interpreted for application. The strain can be measured using either a strain gage or an extensometer. If Poisson's ratio is desirable, the transverse

Several test methods are available for measuring the strength of bond at the interface. The most common test in measuring interfacial bond strength is the fiber pullout test. This fiber pullout method is extensively used for glass fibers but found to be very difficult to use for graphite fibers because of their mainly brittle nature (they usually break before pullout). In order to overcome this difficulty, Drzal et al. (1980) developed an indirect test method using a single fiber imbedded in a resin dogboneshaped specimen as illustrated in Fig. 11-30. A tensile force is applied to the specimen and is then transmitted to the fiber by shear at the fiber-matrix interface. As shown in Fig. 11-31 a the fiber axial stress rises from the ends of the fiber until the fiber fracture stress oi is reached. At that point the fiber will fracture at some point

520

11 Strength of Fiber Composites -A

11.4.2 Laminates 11.4.2.1 Tension

L Section A-A

Figure 11-30. Schematic diagram of single fiber interfacial shear strength specimen (Drzal et al, 1980).

where the fiber stress is a maximum depending on fiber defect and probability. Continued application of stress to the specimen will result in repetition of fracture until all remaining fiber lengths are equal to or less than the critical length (Fig. 11-31 c). This lc or critical length can then be measured and an average interfacial shear strength (T) calculated according to the relationship (11-44) The specimen is examined under a polarizing microscope to measure the critical length. The details of the method can be found in ref. (Drzal et al., 1980). More extensive discussions of the interfacial strength behavior can be found in Chap. 6 of this Volume. !a)

The uniaxial tension test is the most fundamental method for the determination of data such as material specification, screening, research and development, and design of structural components. For unidirectional and woven composites, we can measure £ x , Ey, vxy, X and Y. For multidirectional symmetric laminates, we can measure El9 v 12 , and tensile strength of laminates. Straight-sided, constant cross section specimens with tabs bonded to the ends are widely accepted. Figure 11-32 shows the specimen geometry and dimensions recommended by ASTM D 3039-76. End tabs that are made of fabric or cross-ply E-glass/epoxy and aluminum plates are widely used. The 90° specimen is often tested without end tabs. The length of the tab is determined by the adhesive shear strength and tensile strength of the composite. It should range between 25 and 38 mm. Tab thickness varies with specimen thickness and can range between 1.5 and 2.5 mm. Any high elongation adhesive that will meet the environmental conditions is recommended. The bond surface must be prepared by sanding and cleaning with an appropriate solvent such as acetone. A diamond impregnated saw with water cooling is recommended for cutting the specimen. This should prevent rough or un-

<J1 • -

\. (b)

a2

\

Ic)

a3

^-

a3>a2>

Figure 11-31. Fiber fracture and stress distribution in an interfacial shear strength specimen, (a) At low stress, (b) after fiber break at defects, and (c) after critical length is reached (Drzal et al., 1980).

11.4 Test Methods

T b

" 5-10 deg. taper Figure 11-32. Tensile coupon.

even cut edges and surface scratches. Specimen edges should be parallel to within 0.125 mm. The following is the general test procedure. Width and thickness are measured at several places, and the minimum values are used to calculate the cross-sectional area. Apply load to the specimen through a set of wedge section grips in order to provide sufficient lateral pressure to prevent slippage. A serrated grip surface is desirable (always keep the serrated surface clean). Alignment of the specimen can be checked by strain gages mounted as shown in Fig. 11-33. A constant strain rate of 16.7 10" 6 to 33.7- K T S " " 1 is recommended. A constant crosshead speed or loading rate corresponding to the strain rate is also acceptable. Strain measurement during test can be done using either an extensometer with a gage length of 12.5 or 25 mm or electrical resistance strain gages having 350 or 120 ohm with a gage length in the range of 3.3 to 6.5 mm. It is suggested that manufacturer's recommendations be followed for strain gage mounting technique and selection of adhesive. How-

ever, precaution must be taken for probable stiffening of the test specimen by the strain gage and adhesive. This often occurs in certain composites such as the low-density carbon-phenolic composites. Misalignment of the strain gage axis to the specimen axis is a more serious problem with anisotropic material than the conventional isotropic material. Low voltage for the strain gage circuit is desirable to minimize heat generation. One volt is recommended. Flexural Test This is a quality control and material specifications test. It is not a data generation test. This test is used to determine the outer fiber tensile strength and Young's modulus of homogeneous composite and polymeric materials. For flexural testing of multidirectional composite laminates, the strength and stiffness interpretations are not as simple because of the complex state of stress in the specimen. The most common three-point and four-point flexural tests are illustrated in Figs. 11-34 and 11-35, respectively. The three-point test is

•L/2

p/2

p/2

Figure 11-34. Three-point flexural test. p/2

p/2

-L/2—H 1 1

1

2i

i

Figure 11-33. Specimen alignment, gage 3 is on the back. | [ e 3 - ( e x + e2)/2]/e11 < 0.05 and \(e1-s2)/e11 < 0.05.

521

p/2

p/2

Figure 11-35. Four-point flexural test.

522

11 Strength of Fiber Composites

principally designed for materials that break at relatively small deflection. The strength and modulus for this test are given by X =

3PL 2bh2

(11-45)

E =

PL3 4bh38

(11-46)

where 3 is deflection at the mid-point of the specimen. The four-point test is primarily designed for materials that undergo large deflection. The strength and modulus for this test are given by X = E =

3PL 4bh2 11PL 3

64bh35

(11-47) (11-48)

In flexure test the specimen should be deflected until failure occurs. Deflection should not exceed 10 percent of the span length. Correction factors for large deflection will be needed. A cylindrical surface is used for loading and supporting noses. The diameter of the nose is usually greater than 6.4 mm to minimize local indentation and stress concentration. In many cases of advanced composite testing, the indentation due to the nose cannot be completely eliminated. The span-to-depth ratio (L/h) depends on the ratio of the tensile to interlaminar shear strengths and a guideline for suitable L/h ratio will be given in Sec. 11.4.2.6. Recommended ratios are 16, 23, 40 and 60 for composite laminates. When a specimen is tested at low span-todepth ratio, shear deflection influences the apparent modulus. The 60:1 ratio is recommended for modulus determination.

11.4.2.2 Compression

Compressive elastic constants and strengths are determined by this test. It is well known that it is exceedingly difficult to determine accurate, reliable, and reproducible compressive strengths for unidirectional composites. Because of the high anisotropy of these materials, compressive strength is sensitive to the test method and procedure employed. A slightly eccentric load will cause premature buckling rather than the intrinsic compressive failure. A number of test fixtures have been developed to measure true compressive strength by various investigators. As a result, a number of techniques have evolved and are described in the literature. The method of load application, either direct end loading or shear loading such as ASTM D3410, significantly influences the compressive strength and the failure mode. Direct end loading causes end crushing and vertical splitting in orthotropic materials. This vertical splitting occurs at a stress considerably lower than the compressive strength of the materials system. To avoid these undesirable features, shear loading has been commonly employed and offers the most attractive and feasible approach to determine composite compression strength. The most widely-used test method in the composite is ASTM D 3410, which uses either a Celanese fixture or an IITRI test fixture. These fixtures are designed for a specimen gage length less than 12.6 mm and are primarily used for the 0°, 90°, and cross-ply laminates. Figure 11-36 illustrates the Celanese fixture and specimen. Recommended specimen thicknesses for various materials are: Boron Graphite Glass

1.5-2.0 mm 1.5-3.0 mm 3.2-4.0 mm

11.4 Test Methods

523

Fixture

Dimensions in mm: a = 3.99 b = 6.35 c = 57.15 d = 12.7 e = 3.18 f = 63.5

TEdge front

Euler's column buckling formula can be used for the specimen thickness calculation. For pinned-end column with length L critical

= n2EI/L2

(11-49)

For a fixed-end column the buckling load is four times that for a pinned-end. The test fixture is placed between two platens, and the compressive load is applied. One of the platens normally has a spherical seat to minimize the eccentricity in the load introduction. The short gage length of this test has several undesirable features. Probable end effects for measuring elastic constants and strengths will be encountered. This test method cannot be applied to laminates with off-axis angles and cannot be used for tension-compression fatigue-testing. The failure normally occurs at the gripping area rather than the gage section. These specimens (Celanese and IITRI) frequently fail prematurely, due to an instability arising from misalignment of the specimen with respect to the loading axis. To overcome this instability, SACMA (Sup-

Figure 11-36. IITRI compression test fixture with specimen.

pliers of Advanced Composites and Materials Association) advocates a modified ASTM D 695 method where the specimen has a gage length of 5 mm. End effects are of significant concern with such a short gage length, and furthermore the instability still remains a problem. In addition to these drawbacks, a majority of the specimens tested by these techniques usually fail within the gripped section rather than the gage section. For the above reasons, the ultimate failure stress determined in these tests may not represent the true compressive strength of the material system. Recently, a promising test method has been developed using a mini-sandwich specimen (Crasto and Kim, 1990). A sandwich was constructed in which the conventional honeycomb core was replaced with a core of neat resin, similar to the matrix of the composite system to be tested. The specimen geometry and other dimensions (except for thickness) are in accordance with ASTM D3410. Specimens are tested in the IITRI compression test fixture. The compressive stress in the

524

11 Strength of Fiber Composites

composite skins is calculated by applying the rule of mixtures and given by Xf = — (
(11-50)

where V is volume fraction, o stress, e strain, and E elastic modulus. The subscripts "a", "1", and "2" represent the entire specimen, skin (composite), and core, respectively. The results of sandwich specimens and ASTMD3410 (all-composite) are compared in Table 11-3. The compressive strengths obtained from the sandwich specimen are significantly greater than those determined from the ASTM specimen and comparable to the tensile strengths. The most significant result of this test method is that these mini-sandwich specimens display no evidence of premature failure caused by instability of the test specimen, and specimen failure is confined to the gage section in most cases. The details of this method can be found in ref. (Crasto and Kim, 1990).

Table 11-3. Compressive strength of sandwich and all-composite specimens. Material system S-glass/1034 AS-4/3501-6 AS-4/PEEK

Fiber volume

Minisandwich (MPa)

All composite (MPa)

60 62 57

2284 2020 1573

1400 1280 1100

where p is applied load at failure, b specimen width, d specimen thickness, as shear stress, and ax and e2 axial and transverse strains, respectively. The classical laminated theory indicates that a combined state of stress rather than a pure shear state of stress exists in the specimen. The results of this specimen compared well with the results of the cylindrical torsion specimen that has the pure shear. This method has the advantage of utilizing the relatively inexpensive straight-sided tensile coupon and conventional test method without any special fixture. A nonlinear stress-strain relation usually occurs in shear. Interpretation of shear strength requires care and judgement.

11.4.2.3 Shear Tests

There are a number of test methods available for the in-plane shear modulus, in-plane shear strength, and interlaminar shear strength. [±45] s Coupon This is a simple test that follows the same procedure as the tensile test. Figure 11-37 shows the specimen and the locations of the two strain gages. From the measured longitudinal and transverse strain data, we can deduce the shear strength (S) and shear modulus (£s), as follows: S

=p/2bd

(11-51) (11-52)

Rail Shear

A flat rectangular plate is tested in a two-rail or three-rail fixture. The latter is shown in Fig. 11-38. An analysis of the load and strain gages data is as follows:

P 2A

(11-53)

= 2 8'45

(11-54) (11-55)

Strain gages 1 and 2 Figure 11-37. [±45] 2s specimen for in-plane shear test.

11.4 Test Methods

525

Center rail slides through guide

Dimensions in mm: a = 203.20

—

Bolts

b = 152.40 c = 1 84.15 d=

31.75

e =

12.70

f =

44.45

g =

31.75

/ =

19.05

} =

9.53 dia

Figure 11-38. Three-rail shear test fixture.

where A = b h, b is the length, and h the thickness. Special recommendation on the rail shear: drill oversized holes to prevent stress risers when bolts are tightened. Abrasive paper or cloth should be adhered to the rails for improved gripping. Recommendation on the specimen: holes should be drilled and reamed. Use carbide tipped drill bit. Holes should be larger than the bolts. Apply torque of 7 to 70 Nm to each bolt. Tighten bolts evenly. A fixed pattern of tightening should follow finger tightening. The final level should be reached in two or three stages. On testing; unless otherwise specified, use a loading rate of 1 to 1.5 mm/min. Apply preload and release to align the heads and rails. Alignment can be improved by using a spherical seat between the load head and the center rail. For unidirectional laminates, the fiber orientation can be parallel or perpendicular to the longitudinal axis of the rails. The perpendicular orientation produces higher shear

strength than the parallel orientation. Failure in the parallel orientation often starts in the corners of the rail caused by local stress concentration. Iosipescu Test

The specimen is loaded to produce a shear stress acting between the notch ends with zero bending moment and zero normal stress. The shear test is achieved by application of two counteracting moments produced by two force couples as illustrated in Fig. 11-39 (Adams et al., 1990). Under such a loading configuration, a state of constant shear stress is induced through the middle section of the test specimen and given by P Wt

(11-56)

where t is specimen thickness. The shear loading is achieved by a shear fixture

526

11 Strength of Fiber Composites

to three decimal figures:

P

(7^ = 0.970(7! (7yy = 0.030 *! (7^ = 0.171(7!

(11-58)

For the shear failure, the following conditions must be satisfied if the combined stress interaction is not taken into account: Figure 11-39. Schematic of the loading fixture for the Iosipescu shear test.

a

S X (11-59)

which prevents the rotation of the ends of the specimen while undergoing shear loading. A number of different specimen designs and test fixtures have been used in the Iosipescu test. 10° Off-Axis Test The 10° off-axis unidirectional tension coupon has been used for the in-plane shear characterization of continuous fiber reinforced composite materials. This test method was proposed by Chamis and Sinclair (1976), and later adopted as an ASTM standard test method for in-plane shear. The state of the stress in the off-axis tension coupon is given by the following transformation equations which are derived from force equilibrium considerations: axx = GX cos2 9 ayy = ax sin 2 9 avv = a, sin 2 9

(11-57)

al denotes applied stress, axx,
yy

These inequalities are true in most continuous fiber reinforced composite materials. Furthermore, in most advanced composite materials, the computed longitudinal and transverse stresses are approximately 60 percent and 40 percent of their respective failure stresses when the computed shear stress reaches its shear strength. Since the contribution of longitudinal stress and transverse stress to the shear failure is so small, the use of the 10° off-axis specimen is justifiable. The effect of the combined stress interaction on the shear failure appears to be negligible. Short Beam Shear This test is for estimating the interlaminar shear strength only. The test is shown in Fig. 11-34 or 11-35 with an appropriate span-to-depth ratio to produce interlaminar failure instead of tensile failure. The analysis for this test for the interlaminar shear strength Sj is:

= 3P/4bh

(11-60)

where P is applied load, and b and h are beam width and thickness, respectively. This test is not suitable for design data generation because failure often occurs at a location other than the expected neutral

527

11.4 Test Methods

plane. It is therefore important to know the actual location of the fracture and the failure mode. The span-to-depth ratio should be selected so as to induce interlaminar shear fracture mode. The ratio between the maximum outer fiber stress and the interlaminar stress as a function of the span-todepth ratio is shown in Fig. 11-40. The selection of span-to-depth ratio above the straight line for a given strength ratio would lead to shear failure. For example, for the strength ratio 10, the span-to-depth ratio for interlaminar shear failure should be smaller than 5 and 10 for three-point bend and four-point bend, respectively.

30 3pt/ .20 4-pt^ 10

0

5

10 L/h

15

20

Figure 11-40. Strength ratio vs. span-to-depth ratio.

Double Notch Shear A double notch specimen for interlaminar testing is shown in Fig. 11-41. The two notches are cut to half thickness in each face and are spaced a distance L apart along the length of the specimen. Under either applied tension or compression, shear stress is distributed along the central plane section between the two notches and given by 1

(11-61)

WL

where W is specimen width. The specimen geometry must be chosen which guarantees that only tangential stresses will act and that failure due to the interlaminar shear will take place. It is well recognized that there exists a considerable stress concentration around the region of the notch tip. The variation of shear stress, as, between the notches is shown in Fig. 11-42. The shear stress ratio crs/Sl varies with the value of 9 as given by (Markham and Dawson, 1975) '2

(11-62)

Figure 11-41. Schematic of the double notch specimen for interlaminar shear test, t is the specimen thickness.

where t is thickness, G shear modulus, and E Young's modulus. The stress concentration factor decreases with 9 value and reaches to roughly unity at 9 = 0.5. In addition, the bending moment (M = P t/2) produces normal stress at the notch tip which in turn significantly influences the measured value of shear strength. This normal stress is tension for applied tensile load, while it is compression for compressive load. The specimen may fail due to the tensile normal stress rather than the interlaminar shear when tested under tension. On the other hand, the specimen may fail due to the shear stress when tested under compression. This is the main explanation for the experimental observation that the failure load in compression is almost two times greater than in tension. Whitney (1989) performed a stress analysis on the double notch specimen and suggested that

11 Strength of Fiber Composites

Figure 11-42. Variation of interlaminar shear stress along the center plane between notches.

a relatively small value of L/t is desirable to assure that the interlaminar stresses are distributed over majority of the specimen, rather than being isolated to a region near the notch tip. For a large value of L/t, a pure tension field results in a major portion of the specimen.

Xx, X2, • • •, Xn is a random sample of size n (data) from the Weibull distribution, W(a,j3), the likelihood function, L(a,/J), is given by: L{pL9P)= ft/(*,-,a, jB)

(11-65)

11.4.3 Statistical Treatment of Data For static strength testing of composite materials, the two-parameter Weibull distribution is used widely for failure estimation. A random variable, X (static strength), has a Weibull distribution with shape parameter a and scale parameter /? if the cumulative distribution function, Fx (x\ is:

x>0

(11-63)

or its density function, fx (x), is given by

The maximum likelihood estimations of a and ft maximize L (a, /?) and therefore it is necessary to solve the equations: 8 — L(ocJ) = O (11-66) 6a and 8 — L(ot,P) = O

(11-67)

That is, a and /? are solutions of the maximum likelihood equations:

£ xflnx,.

£ lnx,i= 1

(11-64) A maximum likelihood estimation method is applied for the parameter estimation. If

= I"1 v *T

=o

(11-68)

(11-69)

529

11.4 Test Methods

The symbol " A " denotes the estimate of the parameters. The Weibull parameters a and P are related with mean, \i, and variance,
^.^{r(. + ?)-r»(.+I)}(«-7i, An efficient iterative technique for obtaining a, the solution off (A) = 0 [Eq. (11-68)], is the Newton-Raphson method, in which the (j + l)th successive approximation, &j+1 to &j, is given by:

f'(&j)

(11-72)

Design Allowables A statistical procedure will be described for obtaining material allowables for use in evaluating composite material capabilities in structural design. The allowable is a value determined from a specified probability of survival with a 95% confidence in the assertion. A-basis (-allowables) and B-basis (-allowables) are defined by 99% and 90% probabilities of survival. The confidence intervals for a and /? can be computed from Tables 11-4 and 11-5 given by Thoman et al. (1969), that is, =7

(11-73)

for a and

Table 11-4. Percentage points Iy so that PT{d/a
5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 40 50

for /?. In the Tables, n and y are sample size and confidence level. Using the values of the confidence interval, aL for a in Eq. (11-73) and fih for /? in

0.95

0.98

2.277 2.030 1.861 1.747 1.665 1.602 1.513 1.452 1.406 1.356 1.343 1.320 1.301 1.284 1.269 1.257 1.211 1.182

2.779 2.436 2.183 2.015 1.896 1.807 1.682 1.597 1.535 1.487 1.449 1.418 1.392 1.370 1.351 1.334 1.273 1.235

3.518 3.067 2.640 2.377 2.199 2.070 1.894 1.777 1.693 1.630 1.579 1.538 1.504 1.475 1.450 1.429 1.351 1.301

Table 11-5. Percentage points I* so that

\ n 5.. 6 7. 8 Q

10 12 14 16 18 70

(11-74)

0.90

99

?4 96 78

30 40 50

y

0.90

0.95

0.98

1.107 0.939 0.829 0.751 0.691 0 644 0.572 0.520 0.480 0.447 0.421 0.398 0.379 0.362 0.347 0.334 0.285 0.253

1.582 1.291 1.120 1.003 0.917 0.851 0.752 0.681 0.627 0.584 0.549 0.519 0.494 0.472 0.453 0.435 0.371 0.328

\ .. 0.772 0 666 0 598 0 547 0 507 0 475 .. 0.425 .. 0.389 .. 0.360 0.338 0.318 . .0.302 . 0.288 ,.. 0.276 ... 0.265 ... 0.256 ... 0.222 ... 0.195

530

11 Strength of Fiber Composites

Eq. (11-74), A-basis and B-basis can be determined from the following equations ln

is = ^L [in ( ^ ] 1 M L

B-basis

(11-75) (11-76)

Example The tensile strength data for the [0/ ± 45/ 90]s laminate of graphite/epoxy in MPa are 497, 499, 528, 532, 533, 547, 555, 562, 565, 566, 567, 569, 575, 578, 586, 591, 592, 596, 597, 598, 607, 610, 613, 618, 622, 630, 632, 637, and 644. The following estimates are obtained using the Newton-Raphson approximation: dc= 18.01 £=598 To find the 95% confidence intervals for a and /?, let a 0 95 = 1.342 (n = 29) and let I* 95 = 0.34 (n = 29) from Tables 11-4 and 11-5. Therefore, 95% confidence intervals for a and /?, are a

18.01 = 13.42 1.342 598 = 587 exp (0.34/18.01)

With these values A-basis and B-basis can be calculated from Eqs. (11-75) and (11-76) and are 417 MPa and 496 MPa, respectively.

11.5 Conclusions The prediction of the static strength of unidirectional composite laminates from the properties of their constituents is not reliable in many cases because there are many unknown variables which influence the strength. These variables range from

constituent properties and inherent material flaws to fabrication processes. The formulas derived from micromechanics, however, provide valuable insight in identifying important variables, and their influence on the strength, and give direction to material improvement. There exists a serious need for more analytical and experimental work to improve the accuracy of the predictions and in the characterization of the constituents and fiber-matrix interface properties. The prediction of strength for multidirectional laminates after first ply failure is complicated by a multitude of independent and interacting damage mechanisms of fiber breaks, matrix cracks, delamination and crack propagation. These complicated failure mechanisms with successive ply failures make it difficult to calculate the stress redistribution in the laminate at the moment of ultimate failure. Consequently, the stresses calculated on the basis of the intact laminate significantly differ, in general, from the actual state of stresses at failure, and this results in an unreliable prediction of strength in many cases. More experimental data must be generated to fully verify the existing failure criteria under a wide range of combined loading. Thus, strength prediction may be regarded as a useful tool for material characterization and design, which is a relatively simple method of estimating the load carrying-capacity of a structural component. An effort must be made to develop a methodology to quantify the degree of degradation caused by progressive failure in a multidirectional composite laminate. Finally, we need more systematic research to develop methodology that can integrate micromechanics and macromechanics and predict the load-carrying capacity of a structure from the properties of the constituents.

11.6 References

11.6 References Adams, D. F. (1990), Polymer Composites 11, 287. Adams, D. R, Doner, D. R., Thomas, R. L. (1967), AFML-TR-67-96: Wright-Patterson Air Force Base (Ohio). ASTM (1987), ASTM Standards and Literature References for Composite Materials. Philadelphia (PA): ASTM. Brelant, S., Petker, I. (1970), Mechanics of Composite Materials: Wendt, F. W, Liebowitz, H., Perrone, N. (Eds.). Oxford: Pergamon Press. Chai, H., Babcock, C. D., Knauss, W. G. (1981), Int. J. Solids and Struct. 17, 1069-1083. Chamis, C. C. (1967), DMSMD Rep. No. 9. Cleveland, Ohio: Case Western Reserve University. Chamis, C. C. (1974), Composite Materials Vol. 6: Plueddemann, E. P. (Ed.). New York: Academic Press, pp. 31-74. Chamis, C. C , Sinclair, J. H. (1976), NASA TN D8215. Houston (TX): NASA. Crasto, A., Kim, R. Y. (1990), International SAMPE Technical Conference 22, 264-277. Cruse, T. A. (1973), J. Compos. Mater. 7, 218. Donaldson, S. L. (1987), Compos. Sci. and Technol. 28, 34. Drzal, L. T, Rich, M. J., Camping, J. D., Park, W. J. (1980), 35 th Annual Technical Conference: Society of Plastic Industries: p. 20-c. Flaggs, D. L., Kural, M. H. (1982), J. Compos. Mater. 16, 103-116. Garrett, K. W, Bailey, J. E. (1977), J. Mater. Sci. 12, 157. Hahn, H. T, Williams, J. G. (1986), Composite Materials: Testing and Design, ASTM STP 893, 115 — 139. Hill, R. (1950), The Mathematical Theory of Plasticity. Oxford: Oxford University Press. Kim, R. Y. (1980), Third International Conference on Composite Materials: Paris (France): p. 1015. Kim, R. Y. (1981a), ASTM STP 734, 91-108. Kim, R. Y (1981b), Technical Report UDR-TR-81-84. Dayton (Ohio): University of Dayton Research Institute. Kim, R. Y. (1983), 28th National SAMPE Symposium and Exhibition: Anheim (CA): pp. 200-209. Kim, R. Y. (1986), Unpublished data. Kim,R. Y, Hahn, H. T. (1979), /. Compos. Mater. 13, 3 Kim, R. Y, Soni, S. R. (1984), /. Compos. Mater. 18, 70. Markham, M. R, Dawson, D. (1975), Composite, July, 173. O'Brien, T. K. (1981), NASA Technical Memorandum 81940. Houston (TX): NASA. Pagano, N. J. (1978), Int. J. Solids and Struct. 14, 385. Pagano, N. J., Hahn, H. T (1976), ASTM STP 617, 317-329.

531

Pagano, N. J., Rybicki, E. F. (1974), J. Compos. Mater. 8, 214. Pagano, N. J., Soni, S. R. (1983), Int. J. Solids and Struct. 19(3), 207. Paris, P. C , Sih, G. C. (1965), ASTM STP 381, 84113. Parvizi, A., Bailey, J. E. (1978), J. Mater. Sci. 13, 2131. Patterson, R. E. (1974), Stress Concentration Factors. New York: Wiley, pp. 110-111. Pipes, R. B., Pagano, N. J. (1970), J. Compos. Mater. 4, 538. Pipes, R. B., Kaminski, B. E., Pagano, N. J. (1973), ASTM STP 521, 218. Reifsnider, K. L., Henneke, E. G., Stinchcomb, W. W. (1979), AFML-TR-76-81 Part IV: Wright-Patterson Air Force Base (Ohio). Rosen, B. W. (1964), J. Am. Inst. Aero. Astron. 2, 198. Tan, S. C , Kim, R. Y. (1991), ASTM STP 1120, 414-427. Thoman, D. R., Bain, L. J., Antle, C. E. (1969), Technometrics II, 445-450. Tsai, S. W. (1968), Fundamental Aspects of Fiber Reinforced Plastic Composites: Schwartz, R. T, Schwartz, H. S. (Eds.). New York: Wiley Interscience, pp. 3-11. Tsai, S. W, Hahn, H. T. (1980), Introduction to Composite Materials. Lancaster (PA): Technomic, Chap. 7. Tsai, S. W, Wu, E. M. (1971), /. Compos. Mater. 5, 58-80. Waddoups, M. E., Eisenmann, J. R., Kaminski, B. E. (1971), J. Compos. Mater. 5, 446. Wang, A. S. D., Crossman, F. W (1980), / Compos. Mater. Suppl, 71. Whitcomb, J. D. (1984), NASA Technical Memorandum 86301. Houston (TX): NASA. Whitney, J. M. (1989), 4th Technical Conference, American Society for Composites. Lancaster (PA): Technomic. Whitney, J. M., Nuismer, R. J. (1974), /. Compos. Mater. 8, 254. Wu, E. M. (1972), J. Compos. Mater. 5, All.

General Reading Ashbee, K. H. G. (1989), Fundamental Principles of Fiber Reinforced Composites. Lancaster (PA): Technomic. ASM International (1987), Engineered Materials Handbook, Vol. 1, Composites. Metal Park (OH): ASM International. Broutman, L. J. (Ed.) (1974), Composite Materials, Vol. 5, Fracture and Fatigue. New York (NY): Academic Press. Datoo, M. H. (1991), Mechanics of Fiberous Composites. London: Elsevier Applied Science.

532

11 Strength of Fiber Composites

Hahn, H. T, Zweben, C. (1989), Mechanical Behavior and Properties of Composite Materials in Delaware Composite Design Encyclopedia, Vol. 1. Lancaster (PA): Technomic. Jones, R. M. (1975), Mechanics of Composite Materials. Washington (DC): Scripta Book. Tsai, S. W. (1988), Composite Design, 4th Ed. Dayton (OH): Think Composites.

Vinson, I R., Sierakowskie, R. L. (1987), The Behavior of Structures Composed of Composite Materials. Dordrecht: Kluwer. Whitney, I M., Daniel, I. M., Pipes, R. B. (1982), Experimental Mechanics of Fiber Reinforced Composites. Brookfield Center (CT): Society of Experimental Mechanics.

12 Fracture of Fiber Composites Leif A. Carlsson Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, U.S.A.

List of Symbols and Abbreviations 12.1 Introduction 12.2 Basic Fracture Mechanics Concepts 12.2.1 Stress-Intensity Factors for Isotropic Solids 12.2.2 Stress-Intensity Factors for Anisotropic Solids 12.2.3 Strain-Energy Release Rate 12.2.4 Mixed-Mode Fracture Mechanics 12.2.5 Cracks at Bimaterial Interfaces 12.2.6 Cracks in Bonded Structures 12.3 Fracture Specimens 12.3.1 Double Cantilever Beam Specimen 12.3.2 End-Notched Flexure Specimen 12.3.3 Mixed-Mode Fracture Specimens 12.3.4 Test Specimens for Interleaved Composites 12.4 Fracture Data 12.4.1 Mode I Fracture Data 12.4.2 Mode II Fracture Data 12.4.3 Mixed-Mode Fracture Data 12.4.4 Mode I and Mode II Fracture Data for Interleaved Composites 12.5 Acknowledgements 12.6 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

534 536 537 538 539 544 548 556 559 561 561 562 564 566 572 573 575 576 578 580 580

534

12 Fracture of Fiber Composites

List of Symbols and Abbreviations a semi-crack length A crack area 5a crack extension c distance between point of load application and midspan C compliance Cij stiffness elements Cf modified shear-compliance E Young's modulus; axial modulus of beam E1,E2 Young's modulus in fiber direction, in transverse direction F c ,F d normal forces to close crack FH,FV horizontal and vertical crack-closure forces FT crack-closure force applied in the z-direction G strain-energy release rate G shear modulus Gc work required to create a unit crack area Gc fracture toughness Gio GIIC fracture toughnesses for mode I, II loadings GIC (init.) initiation fracture toughness GIC (prop.) steady-state fracture toughness GI? G n , G in strain-energy release rate components for mode I, II, and III loadings G*, G^ asymptotic energy-release rates G12 shear modulus in 1-2 plane h thickness H potential energy; experimentally determined parameter hl9h2 distances from top (1) and bottom (2) surfaces / moment of inertia of a cross-section k shear correction-factor Kl9KU9Km stress intensity factors for mode I, II, and III loadings K±,K2 bimaterial interfacial stress-intensity factors ^ic> ^IIC stress intensity factors L half span length M moment Ml9 M 2 , M 3 , Ml9 Mu moments (local quantities) M* moment P applied load Pc critical load Pt upward force Pu P2, P3 axial loads Pl9 Pu P components for mode I, II loadings r radial distance from the crack-tip Tc, Td tangential forces to close crack u x-axis displacement uc,ud tangential nodal displacements

List of Symbols and Abbreviations

elastic-strain energy stored in a body y-axis displacement normal nodal displacements specimen width work supplied by movement of external forces material-dependent factor complex stress function

u v w W Y Z a yxy

V

angle shear strain displacement at the critical load size of a crack-tip element oscillation index strains in x-, y-, z-directions angle Poisson's ratio thickness ratio stress stress along x, y, and z-axes yield strength of matrix far-field tensile stress shear stresses far-field shear stress angle Airy stress-function; Westergaad function; angle; mode mixity angle; dimensionless function Laplacian operator

ASTM CLS DCB ELS ENF GMPS IDCB IENF ksi LMPS LVDT MMB NDE PEEK PTFE TPI TSI

American Society for Testing and Materials cracked-lap shear double cantilever beam end-load split end-notch flexure global mode partitioning scheme interleaved DCB specimen interleaved ENF specimen 1000 lb/sq. inch local mode partitioning scheme linear variable differential transformer mixed-mode bending nondestructive evaluation poly(ether-ether-ketone) poly(tetrafluoroethylene) thermoplastic interleaf thermoset interleaf

<5C A 8 X'

V'

Z

9 V

G

Gx, Gy,

Gz

G

y

Xxr

TXZ,

CO

Xy z

535

536

12 Fracture of Fiber Composites

12.1 Introduction With the development of improved manufacturing methods, composite materials have become the materials of choice whenever high stiffness and strength at minimum weight are desired. Consequently, many advanced structural applications such as airplane wings, fuselages, and aerospace structures now involve composites. Although advanced nondestructive evaluation (NDE) techniques applicable for detecting flaws and defects in composite structures have been developed (ASTM STP696, 1979; Pipes and Blake, 1990), there is still a possibility that material deficiencies in the form of preexisting flaws may pass inspection. Another threat of great concern for composite structures is service-induced damage, where low velocity, hard-object impact damage is most important (ASTM STP936, 1987; Masters, 1985; Guynn and O'Brien, 1985; Bostaph and Elber, 1982). This type of damage may be caused by dropped tools on an airplane structure or runway rocks and debris hitting the airframe during take-off or landing. There are many possible types of damage in composites as discussed by Beaumont and Schultz (1990 a, b). They list fiber fracture and debonding, fiber pullout, matrix microcracking and matrix cracking parallel to the fibers as failure processes in composites. On a macroscopic level, damage can be grouped into fiber breakages, matrix cracks parallel to the fibers, and "delamination" or "interlaminar cracks", which represent cracks between the plies in a composite laminate (Wilkins, 1983). The delamination mode of failure has received considerable attention in the literature, (e.g., ASTM STP 876, 1985; Bascom et al., 1980; O'Brien, 1982; Russell and Street,

1982; Wang, 1983; Wilkins etal., 1982), because delaminations may significantly reduce the compressive load-bearing capability of a composite structure and may grow during cyclic loading (Martin and Murri, 1990; Mohlin et al., 1985; O'Brien, 1982; Trethewey, 1988; Wilkins, 1982). Fracture mechanics was introduced after World War II to analyze fracture, which occasionally occurred at low stresses, in high strength steel structures (Broek, 1982). Fracture mechanics is a science developed to analyze the tendency for a preexisting crack in a structure to grow as a result of applied external loads. Due to the preexistence of a crack, the strength of the structure may be decreased and, depending on the crack size, may fall below the designed limit load. Under these conditions, high loads may cause crack propagation and structural collapse. The assessment of defect criticality on structural performance is one of the applications of fracture mechanics. Another common situation where fracture mechanics, in combination with NDE techniques, has been found extremely useful is in the prediction of the maximum crack size that can be allowed in a structure. Furthermore, fracture mechanics may be employed to formulate laws on crack extension with an elapsed number of load cycles (fatigue). This information enables structural inspection schedules to be defined for monitoring the actual growth of the crack. The field of fracture mechanics traditionally involves applied mechanics and materials science. Applied mechanics relates external loads applied to a flawed, or cracked, structural component to crack-tip stress fields and elastic deformations of the material in the vicinity of the crack tip. By contrast, materials science views fracture mechanics as a means of characterizing the fracture resistance of a material on a mi-

12.2 Basic Fracture Mechanics Concepts

crosopic scale. In this manner, material parameters of importance to crack resistance can be isolated and materials with improved fracture toughness can be devised. Both aspects of fracture mechanics have provided improved analytical capabilities and damage tolerant structural materials. Most work on fracture mechanics is related to isotropic materials, such as metals or polymers. Several textbooks (e.g., Broek, 1982; Ewalds and Wanhill, 1983; Hellan, 1984; Kinloch and Young, 1983; Knott, 1973) dealing with fracture in isotropic materials have been published. Fracture mechanics of anisotropic materials has been treated to a much lesser extent, although basic solutions to plane-crack problems are available (Sih et al., 1965; Sih and Liebowitz, 1968). It is the objective of this chapter to review some important developments in linear elastic fracture mechanics of composite materials. Because of the linear elastic and self-similar crack growth restrictions, the analysis and experiments presented are limited to crack propagation parallel to the fibers, such as matrix cracking and delamination. Crack propagation perpendicular to the fibers is not considered. Both applied mechanics and materials science aspects of fracture will be emphasized. The chapter is organized in three

537

major sections. In the first section, basic fracture mechanics concepts are reviewed. The second section deals with the analysis of some contemporary fracture test specimens employed for composite materials. The final section considers experimental fracture toughness data for composite materials and also discusses methods to improve the fracture resistance of composites. Only high-performance continuous fiber composites are considered in this review, because these materials are employed in highly stressed structures. Fracture of short fiber composites is considered in, for example, Mandell et al. (1981) and Friedrich (1989).

12.2 Basic Fracture Mechanics Concepts In this section we will introduce the concepts of crack-tip stress intensity factors and the elastic-strain energy release rate. Relations between these fracture parameters will be discussed for the three basic modes of crack surface displacement illustrated in Fig. 12-1. A common coordinate system has the x-axis normal to the crack contour, the y-axis perpendicular to the crack plane, and the z-axis directed along the edge of the crack, see Fig. 12-1. Stress

Figure 12-1. Modes of crack loading and definition of stress components: (a) Mode I (opening), (b) mode II (forward shear), (c) mode III (tearing).

538

12 Fracture of Fiber Composites

components defined according to this doordinate system are also illustrated in Fig. 12-1. Subsequent to this section, mixed-mode fracture will be discussed, that is, situations where combinations of the basic modes illustrated in Fig. 12-1 prevail. Following this, cracks between dissimilar elastic solids and cracks in bonded structures will be discussed. 12.2.1 Stress-Intensity Factors for Isotropic Solids

As an introduction, the stress field for a mode I loaded crack in an isotropic solid will be discussed. Consider a cracked body in Fig. 12-2 subject to mode I loading. For plane problems, the equilibrium equations in the absence of body forces are 9TV

9x

+

dy

(12-1 a)

•=0

(12-1 b)

dx

With u and v denoting x- and y-axis displacements, the expressions for the strains are

_dv "y~d~y xy

(12-2b)

_ du dy

dv dx

(12-2c)

The equilibrium equations are satisfied if x= | | a

>

=

(12-4a)

8V

(12-4 b) (12-4c)

8x6);

where i// = ifr (x, y) is the Airy stress function (Timoshenko and Goodier, 1970). Combination of Eqs. (12-2), (12-3), and (12-4) yields, after differentiation twice,

8>

8V

9x4+ or

8x28

8V 8/

4

V iA = 0

l/£ = [ -v/E 0

- v/E \/E

(12-6)

where "Re" and "Im" denote the real and imaginary parts of a complex quantity, respectively, and Z and Z are defined by

0

where E and G are Young's modulus and the shear modulus, respectively, and v is Poisson's ratio; G = £/[2(l + v)].

(12-5)

Equation (12-5) is the biharmonic equation of elasticity theory. As will be shown later, a similar equation may be derived for plane problems in anisotropic solids. As shown by Timoshenko and Goodier (1970) many plane problems in linear elasticity can be solved by finding a stress function i// that satisfies Eq. (12-5). The stress components obtained from Eqs. (12-4) must satisfy the particular boundary conditions of the problem. For the crack problem under consideration, Fig. 12-2, a complex stress function Z (z), where z — y + i y and i = -v/— 1, has been shown to provide a solution (Broek, 1982). The Westergaard (Ewalds and Wanhill, 1989) function is \j/ = Re Z + y Im Z

The constitutive relation is

'

az dZ

(12-7 a) (12-7 b)

12.2 Basic Fracture Mechanics Concepts

539

t t t t t t t t t

I I I I I I I I I It can be shown that Eq. (12-6) satisfies the biharmonic Eq. (12-5). The stress components obtained from Eqs. (12-4) are -ylmZ' ay = ReZ + ylmZ'

(12-8 a) (12-8 b)

^ v =

(12-8 c)

- y

where Z' = dZ/dz. For the cracked plate shown in Fig. 122, the stress function is (Broek, 1982)

f(z) Z(z) =

(12-9)

where / (z) is a real function that attains a constant value at the origin. In the limit (12-10)

Figure 12-2. Mode I crack under equal biaxial stress.

For plane stress oz = 0, and for plane strain oz = v (ax +ffy).KY is called the "stress intensity factor" for mode I loading. As r -• 0 the stresses become unbounded. KY is thus a measure of the stress singularity at the crack tip. For tension a at infinity, K{ is proportional to a. From dimensional arguments, KY must also be proportional to the square root of crack length. Most commonly KY is given by (12-12) Equations for the displacements can also be determined, see Broek (1982). Similar procedures can be used to achieve stress solutions for mode II and mode III crack problems (Broek, 1982; Ewalds and Wanhill, 1989).

/ 2 7TZ

In polar coordinates, z = re 10 , the asymptotic crack-tip stresses can be calculated from Eqs. (12-8) and (12-10) . e . 30

1 — sin - sin — 3 2 (12-11 a) KY

9 . 09 .. 3 0 cos -I 1 + sin - sin 2 2 (12-11 b)

. 9 9 36 K ^ ^ sin-cos-cos — 2 n r i l l

(12-11 c)

12.2.2 Stress-Intensity Factors for Anisotropic Solids The complex stress potential theory thoroughly described by Lekhnitskii (1981) is used to find stress solutions (Sih et al., 1965; Sih and Liebowitz, 1968) for cracked anisotropic solids. The off-axis constitutive relation for an orthotropic solid under plane stress is (12-13)

540

12 Fracture of Fiber Composites

where the overbars denote off-axis properties. For orthotropic materials with the principal coordinate directions in the x ysystem (on-axis), the shear coupling terms S16 = S26 = 0, and

where the elements btj are

s -J- s -

0

^11 — r- '

b22

r-

"21

^11 ^33 ~~ ^13

(12-14)

(12-19b)

bl2 = -

S33

'12

where E1 and E2 are the principal Young's moduli in the fiber (1) and transverse (2) directions, respectively, G12 is the shear modulus in the 1 —2 plane, and v12 and v 21 are the major and minor Poisson's ratios. Relations between on-axis and off-axis properties are provided by Jones (1975), Lekhnitskii (1981), and Tsai and Hahn (1980). Plane Strain For thick sections (large dimension along the z-axis), plane strain conditions are generally assumed

(12-19a)

S33

°12 —

s -J- s - J -

b26

c _ e »°33 °13 (

(12-19d) (12-19e) °33

S 3 3 -i

(12-19f)

For the case where the xy-system coincides with the principal material axes (onaxis)

(12-20a)

(12-15)

= lyz = 7x2 =

(12-20b)

For this situation, a normal stress oz will develop in the thickness direction:

The constants btj are given by

^=_(S^ax

b.j^Sij-^j^1

+ S23Gy + S36xxy)

(12_16)

33

b6e = S66

where expressions for S 13 , S 23 , and S36 are provided by Lekhnitskii (1981). For the on-axis situation — v3 1

— V13

(12-19c)

(ij = l,2) (i=j = 6)

(12-21 a) (12-21 b)

As recognized by Sih et al. (1965) and Sih and Liebowitz (1968), Eqs. (12-13) and (12-18) are of the same form, and the solution of a plane-stress problem can thus be transformed to the corresponding plane strain problem by replacing Stj with btj.

(12-17) Stress Analysis for Plane Problems The off-axis constitutive relation for plane strain is:

With the stress function \\i (x, y) defined in exactly the same manner as for isotropic materials, Eqs. (12-4), and using the consti-

12.2 Basic Fracture Mechanics Concepts

tutive relation, Eq. (12-13), the governing differential equation for two-dimensional anisotropic materials is obtained as S ^2

8x4

b± 4= b2. The stress function thus be expressed as

541 may

(12-26)

' 8x3 8y

where zx = x + fi1 y, z2 = x + jn2 y9 and \//1 '8x 2 8y 2

-

(12-22)

'8x8y 3

It can be easily verified that this equation reduces to the biharmonic equation, V4i/f = 0, for isotropic materials. The solutions of Eq. (12-22) are most conveniently expressed in terms of the complex variable (Lekhnitskii, 1981): (12-23) z = x jay where ji = a + i b, where a and b are real numbers and i = ^/ — 1. ja is obtained from the characteristic equation =0

(12-24)

This can be readily verified by recognizing that I,IV

and \j/2 are analytical functions of z± and z 2 , respectively. Because the solutions appear in terms of conjugate pairs, it is possible to express the stress function as (A(x,};) = 2Re[iA1(z1) + ^ 2 (z 2 )]

(12-27)

For isotropic materials both roots ji1 and \i2 are equal to i (Williams, 1989) and the stress function becomes (12-28)

iA (x, y) = 2 Re [z

It is common to introduce two new functions (j> and F A,l,

A,I,

dZl'

dz,

(12-29)

Combination of Eqs. (12-4), (12-28), and (12-29) yields ux =

(12-30 a)

a =

(12-30b)

(12-25 a)

IV

(12-25 b)

and so on. Substitution of these results into Eq. (12-22) yields Eq. (12-24). It has been shown (Lekhnitskii, 1981) that the four roots to Eq. (12-24) are either complex or purely imaginary and occur as two conjugate pairs:

(12-30c) Integration of the strain-displacement relations [Eqs. (12-2)] yields the displacement components u and v: (12-31 a)

]

(12-31 b)

where

+

where overbars here denote the complex conjugate (e.g., ju3 = a± —ibj and al9 a2, bl9 b2 are positive real numbers and

}

ri2

_

(12-32)

542

12 Fracture of Fiber Composites

is satisfied provided that Thus, the plane problem of anisotropic elasticity is reduced to identifying the two 82w 62w functions <j>{z^) and F(z2) that satisfy the boundary conditions specified. Longitudinal Shear Longitudinal shear of cylindrical bodies possessing at least one plane of symmetry normal to the generators, which are parallel to the z-axis, see Fig. 12-3, has been treated by Sih et al. (1965) and Sih and Liebowitz (1968). This mode of loading corresponds to the action of loads directed along the generators of the cylinder and has been denoted "antiplane" loading (Sih et al, 1965; Sih and Liebowitz, 1968). The elastic displacements, strains and shear stresses in the cylinder are given by

(12-37) 82w

^8/

=0

The solution of Eq. (12-37) is again conveniently expressed in terms of the complex variable z = x + \i y. \i is obtained from the characteristic equation C

ii2 A-1C

u A- C

—0

M?-^

Because w must be real, it follows that the two roots \x 5 and ji6 are (12-39) where

(12-40)

~

45

->2

•-7^^

•^45

-^44

U = V

= o,

: = 7Xy

8X = £ T

yz

=

w -= w(x,y) = 0

C44y yz + Q 5 yxz

(12-33) (12-34 a) (12-34 b)

where the C{j are the stiffness elements (Lekhnitskii, 1981) and the shear strains are 9w

9w

8?

(12-36)

dy

w(x, y) = U3 (x + M5 y) + U3 (x + fi5 y) = = 2Re[t73(z3)]

(12-41)

For convenience, a new function W(x,y) is introduced:

(12-35)

[cf. Eq. (12-33)]. The equilibrium condition dx

The displacement w can be expressed in an arbitrary function of the complex variable z 3 , that is,

W'(z3) = i

- C

Generators

?xz

=

\

-2Re[pi6W(z 2lm[W(z3)]

Ic c - r2

/^44L55

Figure 12-3. Longitudinal shear of a cylindrical body.

5

^

(12-42)

With this definition, the stresses and displacement become 2Re[Pr (z3)]

ay

2

3)]

(12-43 a) (12-43 b) (12-43 c)

^45

In contrast to the plane problem previously discussed, longitudinal shear requires only one function, W(z3), that satisfies the boundary conditions prescribed.

543

12.2 Basic Fracture Mechanics Concepts

Solutions for Crack Problems

II —

For a central crack in an infinite sheet as shown in Fig. 12-2, but with the origin of the x j/-system placed at the center of the crack, any point is defined as = rsin6

(12-44)

As shown previously, the stresses and the displacements are given by three functions:
KTTI =

T

yz

na

(12-46b)

na

(12-46c)

Based on the above stress functions, Sih etal. (Sih etal., 1965; Sih and Liebowitz, 1968) derived expressions for the stresses corresponding to mode I, mode II, and mode III loadings of single crack geometries. For mode I loading the near tip stresses are ax=

(12-47 a)

sin 6) F'(z2) =

O(r1/2) "(cos 9 + fi2sm9)

(12-47 b)

(12-45 b) Hi V^/cos 9 + ji2 sin 9

0{r112) ' (cos 9 -\- ji6 sin 9)

Hi

(12-45c)

The constants X{ are dependent on the stress intensity factors Kl9 Ku, Km (Sih etal, 1965; Sih and Liebowitz, 1968), which for single crack geometries loaded with the remote stresses of, %™y, and T*Z, are (12-46 a)

Figure 12-4. Anisotropic plate with crack at an angle cc to the loading direction.

(12-47c) '2nr

\_Hi~Hi "•

^/cos 9 + fi2 sin 0/ Expressions for stresses and displacements for this and the other modes of crack loading are provided by Sih et al. (1965) and Sih and Liebowitz (1968). Note that the authors defined stress intensity factors without including n inside the square roots in Eqs. (12-46). Here n is included in the definition of the stress intensity factors in order to retain the conventions of isotropic fracture. Inspection of the near crack-tip stress field shows that, similar to isotropic crack problems, the stress singularity is of the order of r~1/2. As an illustration, consider the anisotropic plate in Fig. 12-4 loaded by a remote

544

12 Fracture of Fiber Composites

tensile stress of magnitude p. The crack front may be considered as subjected to a far-field tensile stress of magnitude aO0=psin2a

(12-48)

and a far-field shear stress of magnitude Too

=p sin a cos a

(12-49)

The stress intensity factors KY and Kn become KY =/7sin 2 a v / 7ra

(12-50 a)

Ku = p sin a cos a ^ / n a

(12-50b)

where a is the semi-crack length. Notice that the stress intensity factors are the same for plane stress and plane strain. Wu (1967 and 1974) was probably the first to experimentally study fracture of anisotropic materials. Balsa wood and glass/ polyester plates with cracks aligned with the principal material directions (along the fiber directions) were fractured under combined tensile and shear loading. Figure 12-5 shows critical stress for onset of crack propagation plotted versus critical crack length (crack length at the onset of fast crack growth) for a glass/polyester composite. The linearity of the data with a slope of approximately —1/2 is experimen-

4.0

0.2

tal support that the critical stress intensity factors defined according to the concepts above [Eqs. (12-46)] are indeed constants.

0.3 0.4 0.5 0.6 0.8 1.0 1.5 Critical Half Crack Length, a c (in.)

Figure 12-5. Critical stress versus critical crack length at onset of crack propagation in glass/polyester composite (Wu, 1974).

12.2.3 Strain-Energy Release Rate In the application of fracture mechanics to anisotropic materials it has become customary to consider the rate of input of work (or strain-energy release rate) into the fracture process. The strain-energy release rate, G, is a quantity based on energy considerations. It is mathematically well defined and may be physically measurable in experiments. The energy approach, which stems from the original Griffith treatment (Griffith, 1921), is based on a thermodynamic criterion for fracture. One considers the mechanical work and strain energy of the system available for crack growth on one hand, and the energy consumed to extend an existing crack on the other hand. In brittle systems, Griffith found that the dominant source of energy consumption during crack propagation was the generation of additional crack surface energy. In composites, numerous energy absorbing micromechanisms contribute to the macroscopic fracture work: (i) formation of the fracture surfaces of the main crack, (ii) plastic deformation of the matrix in the cracktip region, (iii) microcracking and secondary cracks, (iv) crack bridging by fibers resulting in peeling and fracture. The strain energy release rate, G, is conveniently defined in terms of a potential energy, H, which quantifies the energy available for the formation of new crack surfaces: H= W- U

(12-51)

where W is the work supplied by the movement of the external forces, and 17 is the elastic-strain energy stored in the body.

12.2 Basic Fracture Mechanics Concepts

545

Many fracture specimens employed for experimental fracture characterization are of constant width, w. The amount of new crack surface is 5^4 = w 8a, where 5a is the crack extension that is measured in fracture testing. The energy release rate becomes w 9a Figure 12-6. Cracked plate of thickness w, with load P and displacement u.

The energy release rate becomes

where dA is the increase in crack area. If Gc is the work required to create a unit crack area, a criterion for crack growth may be stated as 5H > Gc 5 A

(12-53 a)

or equivalently, G>GC

(12-53 b)

Critical conditions occur when the net energy supplied just balances the energy required, that is, bH = Gc5A

(12-54)

The equilibrium becomes unstable when the net energy supplied exceeds the work of fracture (crack growth resistance)

8H>Gn8A

(12-55)

Unstable crack growth is thus expected to occur when 8G

As discussed elsewhere (Broek, 1982; Carlsson and Pipes, 1987), it is possible to determine G from the compliance of a cracked body. The compliance, C, Fig. 12-6, is defined as C=^

(12-52)

(12-56)

if Gc is independent of crack extension.

(12-57)

(12-58)

where P is the applied load and u is the displacement of the point of load application, see Fig. 12-6. The energy release rate becomes (12-59) G= — — 2w da Equation (12-59) is very convenient for analytical and experimental fracture studies of composites because it relies on the global specimen compliance. The alternative stress intensity approach previously discussed requires quite elaborate stress analysis. For many fracture specimens the compliance C can be derived from strength of materials principles. Alternatively, C can be measured experimentally at various crack lengths and the C versus a relation can be differentiated numerically to obtain the energy release rate G from Eq. (12-59). This is essentially the basis for experimental compliance calibration methods. Fracture Mode Separation As for the stress intensity factors Kl9 Klu and Km, it is possible to separate G into

546

12 Fracture of Fiber Composites

K2

three components: G = GY + G n + Gm

Theoretically, the mode separation is based on Irwin's contention that if the crack extends by a small amount, Aa, the energy absorbed in the process is equal to the work required to close the crack to its original length (Irwin, 1958). For the three basic modes shown in Fig. 12-1, this method (12-61 a) = lim —— Aaf oyx (Aa — r, 0) t; (r, n) dr i, Aa^o2Aa 1

G n = lim —-— | txy(Aa

(12-61 b) — r,O)u(r,n)dr

Aa -> 0 2. Afl o !

Aa

(12"61C)

G i n = lim — - J T yz (Aa-r,0)w(r,7i)dr where r is the radial distance from the crack tip, cy,xxy, and iyz are normal and shear stresses ahead of the crack tip, and v, w, and w are the relative opening and sliding displacements between points on the crack faces behind the crack tip. As pointed out by Sih et al. (1965) and Sih and Liebowitz (1968), calculation of the components Gl5 Gn, and Gm of the energy release rate, according to Eqs. (12-61) is somewhat academic because cracks in anisotropic materials do not generally extend in a planar fashion. If the crack plane, however, coincides with the direction of minimum crack resistance, the direction of crack propagation will be collinear with the original crack, and the computed values of G,, G n , Gm are physically meaningful. Substitution of the crack-tip stresses and displacements into Eqs. (12-61) yields relationships between energy release rate components and stress intensity factors (Sih et al., 1965; Sih and Liebowitz, 1968): ~2'

(12-62b)

(12-60)

(12-62 a)

KUI Im [C 45 + JX6 C 44 ]

(12-62c)

Equations (12-62) were derived under the assumption that each fracture mode can be analyzed separately, that is, in the absence of the other two modes (Sih et al., 1965; Sih and Liebowitz, 1968). If more than one mode is present at the crack tip, cross product terms will appear in the expressions for the components of the energy release rate. For orthotropic materials with the crack plane coinciding with one of the planes of material symmetry, the three fracture modes are independent and Eqs. (12-62) become 1/2

G, = Kt

'11 °22

(12-63 a)

1/2

11/2

H/2

(12-63 b) G

m =

r

W2

(12-63c)

where C 4 4 and C55 are shear stiffnesses defined by = Z

CAr4ryy

(12-64 a) (12-64b)

Equations (12-63 a) and (12-63 b) for Gx and Gn are defined for plane stress. For plane strain, substitute btj given by Eqs. (12-21) into Eqs. (12-63). It may be readily verified that Eq. (12-63 a) applied to a mode I crack in an

547

12.2 Basic Fracture Mechanics Concepts

Table 12-1. Material properties of graphite-PEEK and glass-epoxy composites (Salpekar et al., 1988).

isotropic materials yields G,= G,=

E (l-v2)K,2

(plane stress)

(12-65 a)

(plane strain) (12-65 b)

where E is Young's modulus and v is Poisson's ratio. Similar results are obtained for mode II and mode III. These are the classical relations between stress intensities and energy release rates for isotropic materials (Broek, 1982). To illustrate the difference between plane stress and plane strain in an orthotropic material, Eq. (12-63 b) is applied to a highly anisotropic graphite-PEEK composite and a moderately anisotropic glass-epoxy composite under pure mode II loading (Salpekar, 1988). Material properties for the two composites are provided in Table 12-1. In the calculations of the plane stress and plane strain constants it is assumed that: v12 = v 23 = v 13 , G12 = G 13 , and E2 = E3. The material has the fiber direction aligned with the x-axis as shown in Fig. 12-1. The ratio between the energy release rates, Gn (plane stress)-G n (plane strain), is 1.016 for graphite-PEEK and 1.023 for glass-epoxy. This example shows that plane stress and plane strain yield similar results, and that the difference between plane stress and plane strain diminishes with increasing anisotropy (EJE2) ratio.

(a)

(b)

Material AS4-PEEK S2-SP 250-epoxy

Et

E3

G 13

(GPa)

(GPa)

(GPa)

146 43.5

10.3 17.2

4.62 4.14

0.37 0.25

Finite-Element Crack-Closure Method The finite-element method is widely used to analyze fracture mechanics problems. A commonly employed technique to separate the components Gl9 Gn, and Gm of the energy release rate has been proposed by Rybicki and Kanninen (1977). This technique is called "the crack-closure method" and constitutes a numerical solution of Eqs. (12-61). Figure 12-7 illustrates the crack-closure method applied to a finite-element mesh of a mixed mode I and mode II problem. A certain crack extension Aa may be introduced in the finite-element mesh by releasing duplicate nodes at the crack tip. The resulting deformations are the relative opening and sliding crack-tip deformations. By applying nodal forces to the released nodes in two orthogonal directions, it is possible to close the crack tip to its original state. The products of crack-tip nodal displacements and forces enable the work of fracture to be evaluated numeri-

(c)

Figure 12-7. The crack-closure technique in finite elements. FH and F v are horizontal and vertical forces applied to close the crack, (a) Original configuration, (b) configuration after release of crack-tip nodes, (c) the released nodes are brought back to their initial position by application of nodal forces.

548

12 Fracture of Fiber Composites

cally. For the plane situation illustrated in Fig. 12-7, GY and Gn become =

FvAv 2wAa 2 wAa

(12-66a) (12-66 b)

where FH and Fy are the horizontal and vertical crack-closure forces and Aw and At; are the horizontal and vertical increments of displacements required to bring the released nodes to their original positions. Notice that the far-field load applied is constant in the steps illustrated in Figs. 12-7 a through c. For mode III, Gm is similarly obtained as FTAw 2wAa

(12-67)

where JFT is the magnitude of the crackclosure forces applied in the z-direction (Fig. 12-1), and Aw is the z-directional incremental displacement required to close the crack. Implicit in the derivation of Eqs. (12-66) and (12-67) is the requirement that the elements enclosing the crack-tip are identical in size. Consequently, Aa is equal to the distance between adjacent nodes. This constraint can be relaxed, but Eqs. (12-66) and (12-67) must be modified to evaluate the crack closure forces accurately, see Rybicki and Kanninen (1977) for details. Further details on the finite-element crack-closure technique as applied to anisotropic fracture problems can be found in Wang and Crossman (1980) and Raju (1987).

It should be observed that the released nodes are assumed to displace in opposite directions with the same magnitude (antisymmetry), see Fig. 12-8. This is the situation encountered in symmetric crack geometries and specimens. For asymmetric crack geometries or cracks between dissimilar materials, the displacements of the released nodes may no longer be of equal magnitude. 12.2.4 Mixed-Mode Fracture Mechanics Cracks in isotropic materials tend to grow under mode I conditions (Sih and Liebowitz, 1968). Due to the presence of weak planes between the layers of a composite laminate, however, delaminations often grow under the influence of a mixed mode stress field. In a mixed mode, both normal and shear stresses act across the interface ahead of the crack tip, and both opening and sliding displacements occur on the crack faces behind the crack tip. In two-dimensional crack problems involving mixed mode I and mode II loading, fracture toughness characterization must account for the relative amounts of mode I and mode II loadings. Most analysis of mixed-mode crack problems employ finite elements to compute the energy-release rate components based on nodal forces and displacements [e.g., Eqs. (12-66) and (12-67)]. In order to obtain accurate results with finite-element solutions to crack problems, highly refined meshes around the crack tip are required. For each new crack geometry or change in material

Figure 12-8. Displacements behind the crack tip in symmetric specimens; vl = v2 and w1 = u 2 .

12.2 Basic Fracture Mechanics Concepts

549

M1

Figure 12-9. Beam-type fracture specimen subjected to bending moments.

M o = M1 + M2

properties, a new finite-element solution is necessary. To circumvent this inconvenience, analytical schemes for fracturemode separation in beamlike specimens have been recently proposed by Williams (1988 and 1989), Suo and Hutchinson (1990) and Suo (1990 b).

Global-Mode Partitioning Method Williams (1988 and 1989) proposed a scheme for determining G and the components Gj and Gn for fracture specimens without the need of a detailed analysis of the crack-tip stress and displacment fields. This analysis is termed the "global mode partitioning scheme" (GMPS). A beamlike specimen of total thickness 2 h with a crack located a distance h1 from the top surface and a distance h2 from the bottom surface is considered (see Fig. 12-9). Initially, it is assumed that only bending moments are acting on the cracked beam, see Fig. 12-9. The definition of the energy

release rate, Eq. (12-57) is _ 1 w \ da

(12-68)

da /

To calculate the work performed by the applied moments, consider the cross-sections AB and CD at crack length a and a + 5a, respectively, in Fig. 12-10. The moments will perform work when rotating the cross-sections. If the crack tip is originally at 0 on AB and extends to 0' on CD, the changes in slopes in the upper and lower beams are, according to Fig. 12-10, (upper beam)

da

d0 o da

da

8a (lower beam)

Consequently,

d0 da l

\da

da da

5a

(12-69)

Crack

Figure 12-10. Rotations of cross-sections before and after crack growth.

550

12 Fracture of Fiber Composites

From classical theory of beam bending (Gere and Timoshenko, 1984), the magnitude of the change in slope (beam curvature) is given by d(f) _M

(12-70)

dx " " £ /

where E is the axial modulus of the beam and / is the moment of inertia of the crosssection. For the beam shown in Figs. 12-9 and 12-10 M1 + M 2 da

~d~a~ d^2 da

w{2hf 12

(12-71 a)

wh\ 12

(12-71 b)

wh\

Mi

=

(12-71 c)

Ex 12

Introducing a dimensionless parameter 3

Z = h1/{2h) and I = wh /12,

M2

1

M2f

(12-72)

The strain energy per unit length stored in a beam is given by Gere and Timoshenko (1984) as dU dx

Mz :IEI

(12-73)

For the beam under consideration, the rate of strain energy change is (i? 74 a ) dU da or

M\ :IE1I1

dU da

1

leE^

J

M\ ' 2E1I2

YM\

- (M, + M 2 ) 2

(12-75)

This equation allows for the computation of G based only on local (crack tip) quantities (M\ and M2) and eliminates the need to compute the global specimen compliance. Axial loading of the beam is also considered by Williams (1988 and 1989). If, instead of the moments M x and M 2 , axial loads Px and P2 act on the neutral plane of each beam in Figs. 12-9 and 12-10, a similar analysis as above yields G=

4w2E1h

the rate of

work change is dW "da"

Combination of Eqs. (12-68), (12-72), and (12-74) yields

(M1 + •M2f 2 £ Jo

Ml

-a3

- ( M , 4 •M,)21

(12-74 b)

Notice that the load Po = Px + P2 is applied from the at a distance h0 = hx + h2 bottom surface in order to maintain moment balance. In structural applications of composite materials, cracks will most likely propagate under mixed-mode stress fields. The most commonly studied fracture mode in composite materials is delamination, or interlaminar fracture, where crack propagation occurs in the resin-rich region between the plies in a composite laminate (Wilkins et al., 1982). Studies of delamination propagation under various modes have shown that the critical values of G are significantly different for the various fracture modes, (e.g., Hashemi, 1990; Johnson and Mangalgiri, 1987; Jordan and Bradley, 1987; Russell and Street, 1985). This is illustrated by the data in Table 12-2, which lists critical values of G determined in pure mode I, mode II, and mixed-mode I/II delamination tests. It is, therefore, necessary to sep-

551

12.2 Basic Fracture Mechanics Concepts

Table 12-2. Critical values of the strain-energy release rate in kJ/m 2 for toughened graphite-epoxy composites (Jordan and Bradley, 1987). Mode Mode I 20% mode II 43% mode II Mode II

AS4-3502

T6T145-F155

T6T145-HX205

T6T145-F185

0.189 0.264

0.520 0.525 0.548 1.27

0.455 0.796 0.789 1.05

2.21

0.570

arate (partition) the total G into its components, Gl9 Gn, and Gm. For the plane situation considered Gm = 0

(12-77 a)

Mn =

M2 + M1 1 +a

(12-77 b)

When both modes are present, a suitable mixed mode fracture criterion has to be established by testing. This will be discussed further in Sec. 12.4.3. Williams (1988 and 1989) states the following condition for pure mode II loading:

(12-80b)

Substitution of these expressions in Eq. (12-75) gives

m

Gu

2.44

(1 + a)

Mf

a—

_aMj)2

1

(12-81 a)

r(i 3 MS

c

16w£lJ

r -M 2 ) 2

#2

da

da

that is, equal curvatures of the beams. This condition will be fulfilled if a moment Mn is applied to the upper beam and a moment oc Mu on the lower beam so that

EJ2

(12-81 b)

(pure mode II)

or a =

Lzi

Axial loads, according to Williams (1988 and 1989), correspond to pure shear (mode II). A similar analysis as above applied to the axially loaded situation yields Pi

(12-78)

Pure mode I, according to Williams (1988 and 1989), corresponds to opposite moments of the same magnitude, that is, — M, on the upper beam, and MY on the lower beam, see Fig. 12-11. Superposition of the two cases in Fig. 12-11 gives

=

Pi +

(12-82 a)

Pu

P2 = {~r®Pl + Pll

(12-82b)

n

aM T

±J (a) Mode II

(12-79 a) M2 = a M n H

(12-79 b)

or

(b) Mode I

M7-

(12-80 a)

Figure 12-11. Moments corresponding to (a) pure mode II, and (b) pure mode I.

552

12 Fracture of Fiber Composites

For the ENF test, Fig. 12-12b,

and substitution into Eq. (12-76) yields G, = 0

(12-83 a)

G,,=

(12-83 b)

(12-86 a) £ = 0.5, a = 1.0

(12-86b)

and from Eqs. (12-81) Examples

G, = 0

The global mode partitioning scheme will be applied on some commonly employed fracture specimens, namely the double cantilever beam (DCB), the endnotch flexure (ENF) and the end-loaded split (ELS) specimens illustrated in Fig. 12-12. These specimens were introduced to characterize delamination fracture by Bascom et al. (1980) and Wilkins et al. (1982) (DCB specimen), Russell and Street (1982) (ENF specimen), and Hashemi et al. (1990) (ELS specimen). For the DCB test shown in Fig. 12-12 a t= -M2=

-Pa

(12-84a)

(12-87 a) 2

G,,=

3P a

2

(12-87 b)

which agrees with results derived from classical beam theory (Russell and Street, 1982). As pointed out by Williams (1989), the GMPS yields Gl = 0 even if the crack is located asymmetrically (h1 # h2). For the ELS test, shown in Fig. 12-12 c, M± =

(12-88 a)

-Pa

(12-88 b)

M2 =

Substitution of Eqs. (12-88) into Eq. (12-75) yields

(12-84b)

= 0.5, a = 1.0

Substitution into Eqs. (12-81) yields GT =

P2a2 wEI

(12-85 a) (12-85 b)

Equations (12-81) yield G, =

Notice that this analysis yields Gu = 0 even if the crack is located asymmetrically

P/2

ENF (Mode II) (a)

i

h h

•pt DCB (Mode I) (b)

(Pa a)2

(12-90 a) (12-90b)

P/2 P

1

\ h2

ELS (Mixed Mode) (c)

Figure 12-12. Fracture specimens for composite materials, (a) ENF specimen (mode II), (b) DCB specimen (mode I), (c) ELS specimen (mixed mode).

12.2 Basic Fracture Mechanics Concepts

553

Local Mode Partitioning Method

1.0

Figure 12-13. Mode mix (Gj/G) versus thickness ratio [h1/(2h)] for ELS specimen.

Consequently, both fracture modes are acting on the crack front. The fraction of the opening mode is 1

(12-91)

This equation shows that the fraction of GY is only dependent on the thickness ratio ?; = hj(2h). From an experimental point of view this is an advantage, because the ratio between the two modes does not vary with crack extension. The ratio Gx/G is plotted versus £ in Fig. 12-13. It is observed that the ELS test is pure mode I for near top surface cracks, and pure mode II as the crack plane approaches the bottom surface.

Alternative analyses of stress intensity factors and strain energy release rates for mixed mode I and mode II problems have been presented by Suo and Hutchinson (1990) and Suo (1990 b). Suo and Hutchinson (1990) deal with isotropic interfacial fracture, while Suo (1990 b) deals with cracked orthotropic materials, pertinent for this chapter. The superposition scheme shown in Fig. 12-14 is employed to construct the solution for an edge dislocation (introduced to represent the crack tip singularity) embedded in an infinite strip of orthotropic material. A solution without singularity in the strip, obtained by the Airy stress function i//{x,y), see Sec. 12.2.2 and Suo (1990), is superposed onto the solution for a strip containing an edge dislocation to account for tractions on the strip boundaries. The solution employs the complex potential method of Lekhnitskii (1981), see also Sec. 12.2.2, and Fourier integral equations. The analysis provides analytic expressions for the stress intensity factors with only one parameter undetermined. This parameter is obtained from numerical solutions to integral equations, and the results are tabulated by Suo (1990 b). The results are, despite the involved mathematical procedure, relatively

-2-*p3 M

Figure 12-14. Superposition scheme for reduction of variables.

554

12 Fracture of Fiber Composites

simple to apply, as will be shown by analysis of some fracture specimens. This analysis is termed the "local mode partitioning scheme" (LMPS). Analysis For a crack in an orthotropic body, where the crack plane is one of the principal planes of symmetry and the crack front is in a principal direction, the crack tip stresses are given asymptotically by (12-92 a) (12-92b) [cf. Eq. (12-47 b)]. The energy-release rate obtained from the crack-tip stresses and the displacements behind the crack tip [see Eqs. (12-61)] is

Kf

(12-93)

where b1± is given by Eq. (12-19 a), and n and X are material parameters given by

same for the beams in Fig. 12-14 a and 12-14 c. Static equilibrium of forces and moments yields M* = M -\-P(H +•h)/2 P = Pi- -CtP3- -C2M

(12-95 a) (12-95 b)

M = Mt -C3M

(12-95c)

where

22

Q)/2

(12-94b)

with '66 Q =

(12-94c)

These results are consistent with those derived by Sih et al. (1965), except that the stress intensities are defined including y/n, as in Eqs. (12-46). With consideration of the problem shown in Fig. 12-14, it is recognized that the beam shown in Fig. 12-14 b contains no crack. Consequently, the stress intensity factors and energy release rates must be the

6<

1

r C3 = Cl,

n=-

(12-96)

The solution is now focused on the beam shown in Fig. 12-14c. For this beam the energy release rate is

where (12-98 a)

A=

(12-98 b)

12

(12-94 a) y

3

Al

Hi+n)

(12-98 c)

where |y| < n/2 and h < H {t] < 1). By solving the two expressions for G, Eqs. (12-93) and (12-97), Suo (1990) found COS CO +

(12-99 a)

(12-99 b)

555

12.2 Basic Fracture Mechanics Concepts

where the dimensionless function co depends only on rj and Q, and not on A, that is, co = (D(Y],Q). The magnitude of co is restricted by: 0 < CD < n/2 to recover KY > 0 and Ku > 0 for the case P > 0 and M = 0. The function co {r\, Q) can be determined exactly for a symmetric DCB fracture specimen, Fig. 12-12 a with h = H, 17 = 1, by considering a special loading: P = 0 (P1 = p3 = M 3 = 0), and M = 1 (Suo, 1990 b). For this case Eqs. (12-98) give A = 1/14,

1 = 1/24,

Table 12-3. Values of a) (rj, Q) indegrees (Suo, 1990 b). \

-0.5

Q

1

2

3

4

52.1 50.9 49.1

52.2 51.1 49.1

52.2 51.1 49.1

52.3 51.7 49.1

\

51.0 51.7 50.4 50.4 49.1 49.1

0 0.5 1

Substitution of Eq. (12-101) into (12-102) yields M2

G,=

sin y = V 3/7

0

Eq.

(12-103)

with / = 1/24, and M = Pa/w9 GY becomes Because Kn = 0 for this specimen, Eq. (12-99 b) gives cos(co 4- y) = 0 with the solution co = 90° - sin" x y/3/7 « 49.1°. Consequently, co(l,g) = 49.1°, independent of g. Numerical results for the general case co(rj9g) are presented in Table 12-3. If the (weak) dependency of co{rj,g) on g is neglected (Suo and Hutchinson, 1990; Suo, 1990 b), co is approximated by co « 52.1 — 3f|

(in degrees)

(12-100)

G, =

12 P2 a2

(12-104)

h3w2E1

which is the classical beam theory result for the DCB specimen [cf. Eq. (12-85 a)]. For the ENF specimen (Fig. 12-12 b) Pi =P2 = Pi = 0

(12-105 a)

M j = — M 2 = P a/(4 w)

(12-105 b) (12-105 c)

Eqs. (12-95) yield

P = -C2M3/h= M =M1-C3M3

Examples The DCB, ENF, and ELS specimens shown in Fig. 12-12 are considered. For the DCB specimen, Fig. 12-12a, co(l9g) = 49.1°, from above. The angle co + y required for Eq. (12-99 a) is co + y = 49.1° + sin" x ^ 3 / 7 = 90° Consequently,

= M1-2C3M1 C2 = 3/4,

The strain energy release rate Gx = G is (for plane stress) given by Eq. (12-93)

C 3 = 1/8

(12-106 c)

Consequently, P = - 3MJ(2h), M = 3 Mi/4. Substitution into Eqs. (12-99) with
Ka =

- 4.2422 A

-

1/8

M1

(12-107 b)

The strain energy release rate Gu is (plane stress)

nK\ (12 102)

(12-106a) (12-106b)

K, = 0 (12-101)

&

-2C2MJh =

9 Mf

9P2a2 16WE!?!3

(12-108)

556

12 Fracture of Fiber Composites

which is the classical beam theory result, Eq. (12-87 b). The mixed mode ELS specimen, Fig. 12-12 c, may be analyzed in a similar manner as the DCB and ENF specimens. This has been performed by Hashemi et al. (1990), who also compared the GMPS with the LMPS. The results are shown in Table 12-4. Inspection of the results in Table 12-4 shows that the global and local mode partitioning schemes predict widely different results, except when the crack is in the center of the beam thickness. For example, for bending of a thin arm, hj(2h) -> 0, the LMPS predicts about 38% mode II while the GMPS predicts 100% mode I behavior for this configuration. The discrepancy between these methods is currently the subject of investigation by Williams (1990) and Hutchinson (1990). The global mode partitioning scheme appears to produce more consistent data from an experimental point of view (Hashemi, 1990). On the other hand, Hutchinson (1990) has provided examples on crack geometries where a beam analysis would not be adequate. It appears beyond doubt that the approach by Suo and Hutchinson (Suo, 1990 b; Suo and Hutchinson, 1990) is correct within the limits of linear elasticity. 12.2.5 Cracks at Bimaterial Interfaces

Figure 12-15 shows a bimaterial plate with a central crack subjected to uniform tension. Initially both materials are as-

t t t t t Material 1

Material 2

I I I I I Figure 12-15. Bimaterial plate with an interfacial crack.

sumed to be isotropic. Because the materials are joined at the interface (y = 0) shear stresses xxy develop along the bondline to maintain continuity in displacements along the interface, and this creates a local mixed mix at the crack tip. Analyses of the stress state near the crack tip (Williams, 1959; England, 1965; Erdogan, 1965; Erdogan and Gupta, 1971) show that the stresses share the inverse square-root singularity of homogeneous fracture [Eqs. (12-11) and (12-47)]. In addition, the stresses exhibit an oscillatory behavior as the crack tip is approached. The stresses can be expressed in a compact complex form as (Hutchinson, 1989):

oy + i r ^

+

l52)^

(12-109)

where K1 and K2 are the bimaterial interfacial stress intensity factors. Because rie = cos (s In r) + i sin (e In r), the stresses <jy and T will change sign infinitely often as

Table 12-4. GJG ratios for ELS specimens of various thickness ratios [hj(2h)] computed by global (Williams, 1988 and 1989) and local (Suo, 1990 b) mode partitioning schemes (Hashemi et al., 1990).

hj(2h)

0

0.2

0.4

0.5

0.6

0.8

1.0

GJG (global) GJG (local)

1.00 0.623

0.992 0.611

0.824 0.588

0.571 0.571

0.292 0.549

0.031 0.481

0 0.377

12.2 Basic Fracture Mechanics Concepts

r -• 0. Consequently, s is called the oscillation index (Hutchinson, 1989) and is given by (12-110) where /? is (12-111) where subscripts 1 and 2 refer to materials 1 and 2 in Fig. 12-15, x = (3- v)/(l + v) for plane stress and 3-4 v for plane strain (v is Poisson's ratio), and F = GJG29 where G is the shear modulus (Suo and Hutchinson, 1990; Dundurs, 1969). For plane strain the relative crack face displacements 8{ = u{(r, 6 = n)-u{(r,0 (u1 = u and u2 = v)

= -n)

are given by Hutchinson (1989): \-v\

Sy-idx =

l-v\

+ 2 i e) cosh (7i e) (12-112)

For plane stress, replace the factors (1 - vf)/E{ in Eq. (12-112) with l/£ s . Consequently, the displacements also show oscillatory behavior as the crack tip is approached. In fact, Eq. (12-112) indicates crack face interpenetration behind the crack tip, which is physically inadmissible. As pointed out by several investigators, (England, 1965; Erdogan, 1965; Erdogan and Gupta, 1971; Hutchinson, 1989; Rice, 1988; Williams, 1959), the contact region is generally exceedingly small compared to the plastic zone or fracture-process zone, and may therefore be disregarded. K1 and K 2 , however, cannot be directly linked to the remote normal and shear stresses and

557

the dimensional form of a complex quantity (K1, K2) will cause the relative proportion of Kx and K2 to change when the units of stress and crack length are changed (Hutchinson, 1989; Rice, 1988). In order to circumvent these complicating features of interfacial fracture mechanics, Hutchinson (1989) and Suo and Hutchinson (1989) suggest that a rational approach is to take /? = 0 = e in the analysis and application of interfacial fracture mechanics. With e = 0, ri£ = l in Eqs. (12-109) and (12-112), the conventional meanings of K± and K2 will be retained. An interfacial crack between dissimilar anisotropic materials has been pursued by many authors (Ma and Luo; Raju et al., 1988; Sun and Manoharon, 1987; Suo, 1990; Ting, 1986 and 1990; Wang, 1983; Willis, 1971). These studies provide useful information on the structure of the near-tip fields, analogies with the isotropic-bimaterial interface problem, the oscillation index, stress intensity factors and displacement jumps across the crack faces. As mentioned in Sec. 12.2.3, it has become customary to approach damage tolerance of composite structures by computing the energy release rate, G, and subsequently separate it into its components. The most commonly applied approach utilizes finiteelement modeling of the cracked geometry and fracture-mode separation based on the crack-closure method discussed in Sec. 12.2.3. As will be discussed in this section, however, the analysis of cracks in composite structures is greatly complicated by the bimaterial nature of many crack geometries. A delamination, for example, is likely to exist between plies of different fiber orientation angles, which disrupts the symmetry in material properties across the crack plane and represents a "bimaterial" crack problem, even if the principal material properties of the plies are identical.

558

12 Fracture of Fiber Composites

Sun and Manoharan (1987) were perhaps the first to illustrate the complications that may arise in the strain-energy release rate analysis of cracks between orthotropic materials of different ply orientation. For two orthotropic plates joined together (Fig. 12-15), Sun and Manoharan (1987) calculated the energy release rate components GY and Gu for remote tensile and shear loadings: (Aa^-[2e CI.r(l+4e-2 t. F lim — ± 47\L2 cosh \^7^ (71 e) Aa -> 0 \ a Aa -> 0 \ a

Aa

C 11 = — \ —

-i2e

— -F! lim a 4 |_2cosh(7i£) " Aa -> 0 - is lim — Aa-

where i = 1,2 for tensile and shear loading, respectively. The constants Cl9 C2, Fl9 F2 are functions of the elastic constants of the materials and the magnitude of the applied loads, e is the oscillatory index, which is a function of the orthotropic elastic constants only. As the crack-closure distance (Aa) approaches zero, the exponential functions have no definite limits. This means that in the finite-element analysis, the computed values of Gj and Gn will depend on crack-tip element size and will not converge with decreasing element size. If the components Gx and Gn are added the result is 4 cosh (n e)

(12-114)

The total energy release rate will thus not oscillate as the crack closure distance approaches zero. To further investigate the influence of the oscillating singularity on the components of the energy-release rate, Sun and Manoharan (1987) performed

finite element studies of a bimaterial plate using different crack increments Aa. The results for tensile and shear loading are presented in Fig. 12-16. The results show that the total energy release rate remains unchanged as the size of the crack-tip elements decreases, which is indicative of a convergent numerical solution. The components Gr and Gn, however, do not converge with decreased crack increment. At larger crack increments (or crack-tip element sizes) constant values of Gj and Gn are approached. The nonconvergent behavior is apparently due to the oscillatory stress singularity. Raju et al. (1988) studied convergence of finite element solutions for free-edge delamination problems where the delamination crack is located between different ply angles (-35°/90°). They found that the total energy release rate converged to a constant value with decreasing size of the crack-tip elements. However, the individual components Gj and Gn did not con-

o.oo

0.06

Figure 12-16. Strain-energy release rate (arbitrary units) for bimaterial plate with interfacial crack under remote tension (top), and remote shear (bottom). Symbols •, •, and A represent G, GI? and Gn respectively (computed by finite element analysis). Solid line represents closed-form analytical solution (Sun and Manoharan, 1987).

12.2 Basic Fracture Mechanics Concepts

verge with decreasing element size. Raju et al. (1988) similarly showed that the oscillatory component of the singularity is the cause of the nonconvergent behavior of Gx and Gu. To further study the accuracy of the results they incorporated a thin (0.1 mm) layer of resin between the plies and located the crack centrally within this layer. Because the crack exists in a homogeneous (and isotropic) material there is no oscillatory component of the singularity, that is, the classical singularity of the type r -i/2 j s r e c o v e r e c j Strain energy release rates for the "bare interface" crack and the crack within the resin layer were computed by finite-element analysis (Raju et al., 1988) and are shown over a wide range of cracktip element sizes, A, normalized with the composite ply thickness, /z, in Fig. 12-17. Comparison of the results shows that the "bare interface" models converge to the resin layer case with increased crack-tip element size. Raju et al. (1988) recommend a minimum size of the crack-tip elements in the range h/4 to h/2. Based on the insight provided by the works discussed above, however, the partitioning of the energy release rate into tensile and shear components is associated with fundamental difficulties for bimaterial crack problems, and a stress intensity approach seems better justified on physical grounds.

559

12.2.6 Cracks in Bonded Structures

A recently introduced concept to improve the delamination resistance of laminated composites is to bond "interleave", thin layers of ductile polymer films to the plies in the laminate (Evans and Masters, 1987; Masters, 1987 a and b). As a consequence of the improved delamination toughness of the material, the impact strength has drastically improved (Evans and Masters, 1987; Masters, 1987 a and b). The reason for the improved delamination toughness appears to be that the ductile resin film allows for more expansion of the crack-tip yield zone between the plies (Aksoy and Carlsson, 1991). In composites without interleaves, the rigid elastic layers provide severe constraints to the expansion of the plastic and process zones around the crack tip (Bradley and Cohen, 1985; Friedrich et al., 1989; Hunston, 1984). The presence of an interlayer, however, alters the local stress field, and may complicate the data reduction procedure for the fracture toughness (Aksoy and Carlsson, 1992; Sela et al, 1989; Suo and Hutchinson, 1989). There are two main reasons for the alteration of the stress field. First, the crack commonly propagates at the film-composite interface, see Fig. 1218, which renders the situation of a bimate-

0.06

0.05

Results of the resin layer model "bare" interface model / Total

QU-—o-o

0.04

0.03

0.02

0.01

Mode III

0.00

log (A/h)

Figure 12-17. Strain-energy release rates for interfacial crack and for a crack within a thin resin layer between the plies in a composite laminate as a function of crack-tip element size (Raju et al., 1988). e0: applied axial strain, t: total laminate thickness, and E1: major Young's modulus of a ply.

560

12 Fracture of Fiber Composites

where p = V(l - a)/(l - ,

(12-117)

in which a = •

Figure 12-18. Interface crack in bonded structure.

rial fracture problem with an oscillatingstress singularity as discussed in Sec. 12.2.5. Second, the asymmetry of the geometry of interleaved specimens may lead to a local mixed-mode crack loading situation, even if the far-field loading is symmetric. The local mixed-mode phenomenon has been termed "phase-shifting" of the stress field by Suo and Hutchinson (1989). The local stress field for test specimens with isotropic constituents has been investigated (Suo and Hutchinson, 1989). The authors considered a semi-infinite crack problem and derived asymptotic (layer thickness -> 0) expressions for the near-tip stresses and displacement fields based on an integral equation formulation for an edge dislocation located at the interface. The far field loading is expressed by the stress intensity factors K{ and Kn that would be present in a homogeneous specimen (without an interlayer) K

= /Cj + 1 Ku

(12-118)

P is given by Eq. (12-111), and r = GJG2 where G is the shear modulus, x is defined in connection with Eq. (12-111). The function co (a, jS) is obtained from a numerical solution of the integral equations for the interface crack problem and is tabulated by Suo and Hutchinson (1989). The interfacial stress intensity is thus scaled by a factor, /?, and phase shifted by an angle, co, with respect to the far field stress intensities, see Fig. 12-19 (notice that the factor tj~iB has magnitude of unity). For material combinations where ft vanishes (and thus s = 0), the stress intensities K1 and K2 can be interpreted in terms of the crack opening and forward shearing modes of crack propagation, and Eq. (12-116) becomes ,.*

K1 + \K2 =

iKu)eic

The phase shift o>, see Fig. 12-19, means that, for example, a mode I specimen with an interlayer will not remain mode I at the crack tip. As shown by Suo and Hutchinson (1989), however, the amount of shift will not be large.

(lz-lljj

For the interface crack shown in Fig. 1218, the interfacial stress intensities are denoted by Kx and K2. The stresses oy and Txy are given by Eq. (12-109). By energy and dimensional arguments, Suo and Hutchinson (1989) derived a relationship between the far field and interfacial stress intensities (12-116)

Figure 12-19. Relation between far field and interfacial stress intensities.

12.3 Fracture Specimens

It is expected that analysis of crack propagation in interleaved composite specimens with anisotropic or orthotropic constituents would also be complicated by the issues discussed above. To date, no analytical results have been presented for thin bondlines between anisotropic or orthotropic materials, but numerical studies provide some insight, see Sec. 12.3.4.

12.3 Fracture Specimens In this section various contemporary fracture specimens employed for composite materials will be described. Most of the attention in the evaluation of damage tolerance of advanced laminated composite structures has been devoted to delamination failure, because in these structures there are no reinforcing fibers holding the plies together. In particular, hard object impact loading has been a much studied loading case for quality assurance of composite structures; this loading case typically results in delamination (ASTM STP 936, 1987; Bostaph and Elber, 1982; Dost et al, 1991; Guynn and O'Brien, 1985; Masters, 1985, 1987a and 1987b; Wilkins, 1983). The description is limited to plane fracture specimens (mode I, mode II, and mixed mode I and II). A discussion of mode III specimens for composite materials is found in Gillespie Jr. and Carlsson (1990). Mode I, mode II, and mixed mode composite fracture specimens are also discussed by Carlsson and Pipes (1987), Friedrich (1989), Gillespie Jr. and Carlsson (1990), and Wilson and Carlsson (1991). These references provide information on specimen design, testing and data reduction methodology. The description presented herein will therefore be relatively brief.

561

Because the strain energy release rate, G, is experimentally accessible for many of the beam-like fracture specimens employed for fracture characterization of composite materials, most fracture studies focus on this parameter. The presentation in this section essentially follows this convention. 12.3.1 Double Cantilever Beam Specimen

The double cantilever beam (DCB) specimen was discussed in Sec. 12.2.4. The geometry and loading of the DCB specimen is shown schematically in Fig. 12-12. A more comprehensive illustration is provided in Fig. 12-20. Commonly a 24 ply unidirectional composite is split in the center, end tabs are attached, and the specimen is loaded in a testing machine with a sensitive load cell. The precrack is achieved by placing a thin inert film, typically 1 5 75 |im in thickness at the mid-plane of the laminate prior to processing. The choice of material and thickness for the insert film has been the subject of an ASTM committee on interlaminar fracture testing (ASTM D30.06) and several publications, for example, Martin (1988). Procedures for load tab attachment and specimen preconditioning are discussed by Carlsson and Pipes (1987) and Gillespie Jr. and Carlsson (1990). The crack may be grown continuously, as suggested in the preliminary ASTM procedure (Proposed Standard Test

Figure 12-20. DCB test geometry.

562

12 Fracture of Fiber Composites

Method, ASTM, 1990), while the load-displacement response is recorded and crack length is monitored by inspection of the specimen edge. Alternatively, the crack may be grown incrementally while monitoring several loading-unloading increments (Wilkins et al., 1982). This procedure has the advantage over continuous loading of providing opportunity for more precise crack-length measurements (since the cross-head is stopped). Furthermore, information on crack initiation mechanisms is obtained for the entire load history, see also Ozdil and Carlsson (1992 a).

expressions for the energy-release rate of the DCB specimen discussed previously in Sec. 12.2.3 [Eqs. (12-85 a) and (12-104)] are strictly valid only for infinitely long crack lengths. End effects tend to decay at a slow rate in orthotropic specimens (Horgan, 1982; Carlsson et al., 1986 c) and long precrack lengths may be necessary to achieve satisfactory agreement with the infinite beam solutions. Finite-element calculations presented by Suo et al. (1991) and reviewed by Hutchinson and Suo (1991) yield the following calibration for finite geometry DCB specimens,

DCB Data Reduction

+

ASTM (Proposed Standard Test Method, ASTM, 1990) recommends adoption of the experimental compliance calibration method, based on the approach originally suggested by Berry (1963). In this approach the beam compliance, C, is expressed as

where G* is the asymptotic energy release rate for DCB specimens given by Eq. (12-104), 7 is a material-dependent factor, Y(Q) = 0.667 + 0.149 ( e - 1 ) -0.013(e-l)2

C=—

(12-120)

where a is the crack length, and the exponent n and factor H are experimentally determined parameters. Substitution of Eq. (12-120) in Eq. (12-59) yields the critical value of the strain energy release rate associated with a given crack length, nPG(a)6e(a) 2wa

(12-121)

in which Pc and Sc are the critical load and the displacement at the critical load, and w is the specimen width, see Fig. 12-20. The empirical basis for the Berry method is necessitated by the fact that the loaded beam ends are elastically built into the remainder of the specimen and that interlaminar shear deformations may affect the results for the finite geometry specimens typically employed in actual testing. The analytical

(12-122)

(12-123)

and Q and X are defined in Eqs. (12-94b and c). For plane stress k=^

(12-124a)

2G 12

(12-124b)

For DCB specimens with A1/4 a/h > 1 and 0 < Q < 5, Eq. (12-123) is accurate within 1% (SuoetaL, 1991). 12.3.2 End-Notched Flexure Specimen

Several mode II fracture specimens have been suggested for interlaminar fracture testing of composites, see Gillespie Jr. and Carlsson (1990). The most widely adopted mode II fracture specimen, however, is the so-callled end-notched flexure (ENF) specimen illustrated in Fig. 12-12. The ENF ge-

12.3 Fracture Specimens

ometry was originally proposed for mode II fracture studies of wood by Barrett and Foshi (1977), and was adopted for delamination studies of composites by Russell and Street (1982). The ENF specimen employs a 24 ply unidirectional composite with a starter crack that is achieved by placing a thin inert film, 7 to 25 jim in thickness at the center of the layup. Because the ENF specimen is inherently unstable (Carlsson et al., 1986 a), precracking becomes an important issue. Initially, the proposed ASTM standard (Test Procedure for the ENF Specimen, ASTM D 30.02, 1987) suggested mode I precracking, but more recent experimental evidence and physical arguments (O'Brien et al., 1987) favor mode II precracks or very thin films with no precracking over the tensile precrack. The issue is still under investigation (Minutes of the ASTM D 30.06 subcommittee). The fracture test uses a conventional 3point flexure test fixture with a total span, 2 L, of 50.8 mm between the outer specimen support pins, see Fig. 12-12. The crack length, a, is 25.4 mm (a/L = 0.5) so that the crack tip is positioned midway between the outer support pin and midspan-load point. The beam deflection, 5, should be monitored at the midspan (beneath the load point) by an extensometer or a LVDT (linear variable differential transformer) or from the cross-head displacement if corrections for machine compliance are made. Further experimental details are provided in the preliminary ASTM test procedure (Test Procedure for the ENF Specimen, ASTM D 30.02, 1987) and by Carlsson and Pipes (1987) and Gillespie Jr. and Carlsson (1990). After the crack propagates and is arrested, the fracture test is completed. The compliance and critical load are employed in the data reduction for the fracture toughness and should be recorded.

563

ENF Data Reduction The data reduction scheme presented here is essentially the one recommended by the preliminary ASTM standard (Test Procedure for the ENF Specimen, ASTM D 30.02,1987). The ENF beam compliance is given by the following expression (Carlsson et al., 1986 a; Carlsson and Pipes, 1987; Gillespie Jr. and Carlsson, 1990), 1.2 L +0.9 a 4whG 13

(12-125)

where L is the half span length, a is the crack length, E1 is the axial Young's modulus of the beam, w is the beam width, h is the semi-thickness of the beam (Fig. 12-12) and G13 is the interlaminar shear modulus. Substitution of Eq. (12-125) in the expression for G, Eq. (12-59), yields after some algebraic manipulations, the mode II fracture toughness, 9 a2 Pc2 (C - Cs*) 4 w L 3 [ l + 1.5 (a/L) 3 ]

(12-126)

in which Pc is the critical load for propagating the crack and Cf is a modified shear compliance given by C.* = -

0.6a-0.2L 3 /a 2

20whG 13

(12-127)

For experimental fracture toughness determination, C in Eq. (12-126) represents the experimentally determined ENF compliance. Equation (12-126) represents an expression valid for the finite width geometries employed in fracture testing (Salpekar, 1988). The expressions (12-87b) and (12-108) are infinite beam solutions strictly valid only for long crack lengths and beams with high shear modulus G 13 . An alternative expression for G that applies to finite geometry orthotropic ENF

564

12 Fracture of Fiber Composites

specimens (Hutchinson and Suo, 1991) is (12-128) where G^ is the asymptotic energy release rate for the ENF specimen, Eq. (12-108), and Y(Q) = 0206 + 0.078 (Q - 1) -0.008(£-l)2

shearing displacements of the crack flanks. As argued by Hutchinson and Suo (1991), systems with /? + 0 may be analyzed by taking /J = 0, because p is generally small (<1) and has negligible influence on fracture parameters. A convenient measure of the mode mixity at the crack tip is given by the angle ij/ (Hutchinson and Suo, 1991):

(12-129)

where X and Q are defined in Eqs. (12-94 a and c) and are given for plane stress in terms of engineering constant in Eqs. (12-124). Alternative data reduction schemes based on compliance calibration for the fracture toughness evaluation using the ENF specimen (Test Procedure for the ENF Specimen, ASTM STP 936,1987, Carlsson etal., 1986 b; Gillespie Jr. and Carlsson, 1990; O'Brien, 1986) have been examined, but they demand more work from the experimenter than the procedure outlined herein, and appear to produce more scatter in the data (minutes of the ASTM, D 30.02.02 Task Group). The ASTM task group does not recommend compliance calibration. 12.3.3 Mixed-Mode Fracture Specimens As mentioned in Sec. 12.2.4, cracks in orthotropic fiber-reinforced materials are often trapped in mixed-mode-stress fields. Furthermore, the toughness for a delamination or an interface depends on the mode of loading. For situations when the delamination is located between similar isotropic or orthotropic materials, or for bimaterial interfacial cracks with the oscillation index s = 0 (and thus /? = 0), see Sec. 12.2.5, the interface stress intensity factors Kx and K2 measure the amplitudes of the tensile and shear stress singularities respectively, and the relative opening and

(12-130) where the interfacial stress intensity factors Kx and K2 are replaced by the conventional Kx and Ku. Thus, for pure mode I problems \jj = 0, while pure mode II is represented by = ± 90°, with the sign given by the direction of shear. Many studies, for example, Hashemi etal. (1990), Johnson and Mangalgiri (1987), Jordan and Bradley (1986), Jurf and Pipes (1982), Leichti and Chai (1992), and Wang and Suo (1990) have shown that the fracture toughness, Gc, depends on the mode of loading: Gc = Gc(iA)

(12-131)

These studies show that both the delamination and interfacial toughness generally increase with increasing amount of mode II loading, that is, with increased magnitude of \j/. This topic will be discussed further in Sec. 12.4.3. Consequently, delamination and debonding fracture characterizations require mixed-mode fracture testing. Several fracture specimens have been designed for the study of mixed-mode delamination fracture of composites. Figure 12-21 shows some of the more popular mixedmode fracture specimens. The earliest mixed mode composite specimen is the cracked-lap shear (CLS) specimen introduced by Wilkins et al. (1982). The CLS specimen (Fig. 12-21 a) was a candidate for a standard ASTM mixed-mode composite

12.3 Fracture Specimens

(a)

(c)

(d)

fun mi i

(e)

(f)

Figure 12-21. Mixed-mode fracture specimens for composite materials: (a) cracked-lap shear (CLS), (b) modified Arcan, (c) asymmetric DCB, (d) mixed-mode flexure, (e) end-loaded split (ELS), and (f) mixed mode bending (MMB).

test specimen, but it suffers from testing difficulties, geometric nonlinearities, and a narrow range of mode mixities, \j/9 and is, therefore, not a popular specimen anymore. Jurf and Pipes (1982) modified the Arcan specimen (Arcan et al., 1978) to produce mixed-mode fracture data by using a single-edge notch specimen in place of the original V-notched specimen, see Fig. 12-21 b. The asymmetric DCB specimen shown in Fig. 12-21 c was introduced by Bradley and Cohen (1985). This specimen, however, demands a complex loading arrangement. The mixed-mode flexure specimen, Fig. 12-21 d, devised by Russell and Street (1985) is easier to test than the previously mentioned specimens but does not allow much variation in the mode mixity. The

565

end-loaded split (ELS) specimen devised by Hashemi et al. (1990), Fig. 12-21e, was discussed at some length in Sec. 12.2.4. It was demonstrated, based on global-mode partitioning, that the mode I fraction, GYIG, could be varied between zero and unity (Fig. 12-13) by varying the ratio of the thickness of the loaded beam to the total beam thickness. Notice, however, that the local-mode partitioning predicts a ratio GJG that varies only between 0.38 and 0.62 for the full range of thickness ratios, see Table 12-4. Besides the limited range of mode mixities realized for most specimens discussed above, some of the specimens suffer also from having a varying mode mixity as the crack grows. A specimen that does not possess these limitations is the recent mixedmode bending (MMB) test suggested by Crews and Reeder (1988), see Fig. 12-21 f. The MMB test is currently a candidate for becoming an ASTM standard because of simplicity of testing and the wide range of mode mixities. Figure 12-22 depicts the geometry parameters of the specimen. The MMB test is a superposition of the DCB and ENF tests discussed previously. As shown in Fig. 12-22, the loading lever will add an opening load to the mid-span loaded ENF specimen. The distance, c, between the point of load application and the

, Loading lever

Figure 12-22. Geometry and loading of the MMB test.

566

12 Fracture of Fiber Composites

mid-span, see Fig. 12-22, determines the ratio of the downward force, Pc, to upward force, Pt, and hence the mode mixity. Pure mode II corresponds to c = 0 with the ratio GJGn increasing with increased distance c. Crews and Reeder (1988) derived the following asymptotic beam theory expressions for Gl5 Gn, and G 3a2P2 GT = 4w . 2,h3, , L„2 £i „ (3c-L)2

(12-132a)

9 a2 P2 Gu = 16w2h3L2E1

(12-132b)

G =

3 a2 P2

16w2h3L2E1 + 3(c + L)2]

L)2 [4(3c-L)2

(12-132c)

where the geometry symbols are defined in Fig. 12-22, and Ex is the longitudinal beam modulus. The mode mixity is Gj _ 4 / 3 c -

(12-133)

Thus, Gi/Gn is a function only of load position, c, and half-span length, L. For c < L/3, Eq. (12-133) is invalid because crack-face contact may occur that is not accounted for in the beam analysis. The asymptotic beam expressions above are strictly valid for long crack lengths and beams with high shear modulus G 13 . Crews and Reeder (1988) provide modified expressions for GY and Gn that are quite accurate for the finite geometries employed in actual fracture testing

3P 2 (3c-L) 2 4w2h3L2E1

2a

10G 13 16 w2

1

(12-134a)

, h2E, (12-134b) 5G 13,

where (12-135) Inspection of Eqs. (12-134) reveals that the ratio Gl/Gu is slightly dependent on crack length. Detailed analysis of a typical MMB test presented by Crews and Reeder (1988), however, shows that within the practical range of crack lengths (a = 25 to 45 mm), the Gl/Gll ratio displays only 5% variation. The design of the MMB apparatus is straightforward for stiff beams with low toughness values. Ductile thermoplastic composites, however, tend to fracture at relatively large displacements, and rotation of the lever arm of the original test apparatus introduced a geometrically nonlinear axial load in the beam that reduced the strain-energy release rate of the specimen (Reeder and Crews, 1991). The beam analysis presented above is valid only for small displacements and significant errors may appear if fracture toughness data recorded in the nonlinear range are reduced based on the above linear small displacement analysis. Reeder and Crews (1991) recently presented a redesigned MMB apparatus that possesses negligible geometrical nonlinearities even for extremely tough composites. It is thus recommended that mixed-mode fracture testing be performed using the redesigned MMB apparatus, see Reeder and Crews (1991). 12.3.4 Test Specimens for Interleaved Composites

As discussed in Sec. 12.2.6, interleaved composites (Fig. 12-18) have aroused considerable interest because of their improved delamination resistance. Delamination test specimens for interleaved composites are similar to the DCB and ENF specimens employed for mode I and

12.3 Fracture Specimens

mode II fracture testing of homogeneous composites, see Sees. 12.3.1 and 12.3.2. Figure 12-23 shows the geometry of the interleaved DCB and ENF specimens, here denoted by "IDCB" and "IENF", respectively. An asymptotic solution for bonded fracture specimens with isotropic constituents by Suo and Hutchinson (1989) was discussed in Sec. 12.2.6. For interleaved specimens with moderately thick interlayers or orthotropic constituents, no closed-form solution for the crack-tip fields exists, but it is reasonable to expect that the presence of an interlayer in a composite laminate will influence crack-tip field parameters in a manner similar to that in bonded specimens with isotropic constituents. The influence of a compliant interlayer on the compliance and energy-release rate of composite fracture specimens has been investigated by Sela et al. (1989), Ozdil and Carlsson (1992 b) and Carlsson and Aksoy

567

(1991). In the investigations of the IDCB and IENF specimens presented by Ozdil and Carlsson (1992 b) and Carlsson and Aksoy (1991), particular emphasis was placed on the detection of any mixed-mode effects. In the next section we will review recent finite element and beam theory analyses of these specimens. IDCB Specimen Ozdil and Carlsson (1992 b) present finite element analysis of the IDCB specimen illustrated in Fig. 12-23 a. Material properties of the composite are representative for graphite-epoxy composites: E = 149 GPa

E2 = E3 = 9.4 GPa

v 2 , = 0.54 v12 = v 13 = G12 = G 13 = 7.5 GPa G 23 = 3.1 GPa Material properties of the interlayers listed in Table 12-5 are typical for two commer-

Piono hinge Insert film (Starter crack) Precrack y—Crack extension Interleaf film

>—Composite yf

(a)

B

L

C

IP/2

\

Figure 12-23. Interleaved fracture specimens, (a) IDCB specimen, (b) IENF

(b)

specimen.

568

12 Fracture of Fiber Composites

Table 12-5. Mechanical properties of thermoplastic and thermoset interleaves. Property

Units

Thermoplastic

Thermoset

E v

GPa

4.57 0.40

2.45 0.38

dally available interleaves made from thermoplastic and thermoset polymers. These materials are denoted TPI and TSI, respectively. The IDCB specimen was modeled in plane strain using four-node isoparametric elements. Two configurations of the crack plane were considered, one (symmetric) where the crack is located centrally within the interleaf, and the other (asymmetric) where an interfacial crack is located between the interleaf and the composite. As will be discussed in Sec. 12.4.4, fracture of interleaved composites commonly occurs between the interleaf and composite. Therefore, the analysis of the asymmetric specimen may be more pertinent for the actual fracture process. In the finite-element analyses, a homogeneous DCB specimen (no interleaf), and IDCB specimens with 16, 40, 80, and 128 (im thick thermoplastic and thermoset interleaves were considered. The total specimen length and thickness were kept constant at 120 and 3.4 mm irrespective of the interlayer thickness. The crack length was 20 mm for all cases and the specimen was loaded by 1 N (1 mm wide specimen). Although the analysis is limited to specific loading and specimen dimensions, the results should be of some general character. Changes in the magnitude of load and specimen dimensions will only alter the magnitude of the stresses and displacements, but not the character of the distributions.

Compliance Compliance, C, of the IDCB specimen was calculated from the vertical deflection, <5, of the points of load application, divided by the applied load, P. Table 12-6 lists IDCB compliance as a function of interleaf thickness for the symmetric and asymmetric crack configurations, Ozdil and Carlsson (1992 b). For interleaf thicknesses up to about 16 (im, the presence of an interlayer does not significantly affect the IDCB compliance. For thicker interleaves, however, the increased compliance is due to the decreased thickness of the composite plies (total beam thickness is constant). The asymmetry has a negligible influence on the IDCB compliance. Slightly larger compliance values were obtained for the more compliant TSI. Strain-Energy Release Rate Components of the strain-energy-release rate, GY and Gn, were computed for the IDCB specimen using the finite element crack closure method outlined in Sec.

Table 12-6.

Compliance C of IDCB specimen. a = 20 mm, w = \ mm. Interleaf thickness (urn)

Crack configuration a

C (mm/N) TPI

TSI O.128Ob

0

S

16 40 80 128

S S S S

0.1294 0.1320 0.1362 0.1416

0.1297 0.1325 0.1373 0.1431

16 40 80 128

A A A A

0.1294 0.1319 0.1361 0.1414

0.1297 0.1325 0.1372 0.1429

a

A: asymmetric, S: symmetric; b without interleaf; TPI: thermoplastic interleaf, TSI: thermoset interleaf.

12.3 Fracture Specimens

12.2.3. Equations (12-66) were modified to incorporate the asymmetric crack configuration according to

2Aa

(Fcvc-Fdvd)

(12-136a)

(Tcuc-Tdud)

(12-136b)

where Fc and Fd, and Tc and Td are the normal and tangential forces, respectively, required to close the crack tip (bring nodes c and d together). Similarly, the quantities vc, vd and uc ud are the normal and tangential nodal displacements (crack closure distances). For the symmetric crack configuration: Fc = Fd, vc = — vd and Tc = Td = 0. Substitution in Eq. (12-136b) yields Gn - 0, which shows that the symmetric crack configuration is pure mode I, and G = Gx. As

569

discussed previously, the asymmetric configuration is inherently mixed mode and the decomposition of G into tensile and shear components, G, and Gn, may suffer from convergence problems due to the bimaterial oscillatory singularity. Therefore, the results for GY and Gn presented herein should be viewed as qualitative. The total energy-release rate, however, is a well-defined physical quantity. Table 12-7 presents components of the energy-release rate for symmetric and asymmetric IDCB specimens. Inspection of the results in Table 12-7 reveals that the energy-release rate is only slightly affected by the presence of thin interlayers. However, for thicker interleaves, the energy-release rate increases significantly due to the addition of a compliant interlayer. The asymmetric configuration provides somewhat larger energy-release rate than the

Table 12-7. Strain-energy release rate components for symmetric (S) and asymmetric (A) IDCB specimens. a = 20 mm, P = 1 N/mm. Interleaf thickness (^im)

Interleaf material a

Crack configuration

Gl (J/m2)

Gu (J/m2)

G (J/m2)

None

S

7.72

0

7.72

16 40 80 128

TPI TPI TPI TPI

S S S S

8.16 8.29 8.49 8.82

0 0 0 0

8.16 8.29 8.49 8.82

16 40 80 128

TSI TSI TSI TSI

S S S S

8.27 8.36 8.56 8.90

0 0 0 0

8.27 8.36 8.56 8.90

16 40 80 128

TPI TPI TPI TPI

A A A A

8.49 8.67 8.97 9.30

0.09 0.13 0.17 0.33

8.58 8.80 9.14 9.63

16 40 80 128

TSI TSI TSI TSI

A A A A

8.45 8.65 8.99 9.35

0.06 0.12 0.20 035

8.51 8.77 9.19 9.70

0

TPI: thermoplastic interleaf, TSI: thermoset interleaf.

570

12 Fracture of Fiber Composites

symmetric one. Furthermore, the asymmetric geometry is mixed mode, although the magnitude of Gn is very small for thin interlayers. Stress-Intensity Factor The mode I stress-intensity factor Kl9 was determined for the symmetric IDCB geometry. Kx was calculated from the global energy-release rate and relations between KY and GI? provided in Sec. 12.2, see Ozdil and Carlsson (1992 b). Table 12-8 lists calculated Kx values for the various symmetric IDCB specimens considered. It is observed that the intensity of stress is much reduced by the presence of a compliant layer, and that KY is almost independent of layer thickness. The TSI reduced KY more than the TPI because of its lower Young's modulus, see Table 12-5. Similar features have been observed in adhesively bonded specimens (Crews Jr. et al., 1988; Sela et al., 1989; Wang, 1978). The reduction of KY due to the compliant bondline material has been termed "elastic shielding" by Hutchinson and Suo (1991). The apparent fracture toughness of bonded specimens, Klc, may thus increase as a result of the shielding effect.

Table 12-8. Stress intensity factor for homogeneous and interleaved symmetric DCB specimens, a = 20 mm, P = \ N/mm. Interleaf thickness

Xj(kNm" 3/2 )

TPI 368.1l

0 16 40 80 128 a

TSI

210.6 212.5 215.0 219.1

154.0 154.6 156.5 159.7

Without interlayer; TPI: thermoplastic interleaf, TSI: thermoset interleaf.

The mixed mode stress intensity factors, Kx and K 2 , for the asymmetric IDCB configuration have not been determined. In principle, K1 and K2 could be determined from the stress field ahead of the crack, but this procedure requires an extremely refined finite-element mesh. Based on the analysis of bonded fracture specimens with isotropic constituents by Suo and Hutchinson (1989) it is reasonable to expect that K1 and K2 would diminish in the IDCB specimens with orthotropic adherends as a result of the compliant interlayer. Data Reduction for Fracture Toughness of IDCB Specimen The results presented for the IDCB specimen show that this specimen essentially remains pure mode I in the presence of a thin compliant interlayer. Furthermore, as shown in the previous section, a thin interlayer was found not to change the compliance and energy release rate very much, as compared to the homogeneous specimen. Compliance and energy-release rate are global quantities that are dominated by the stiffness of the composite material in the cantilever beams. Consequently, compliance calibration procedures for determination of GIC from experimental fracture data, derived for homogeneous and symmetric DCB specimens, are adequate also for the IDCB specimen with thin interleaves. At larger interleaf thicknesses, however, mixed mode effects may influence the results. IENF Specimen The IENF specimen, Fig. 12-23 b, has been investigated with both shear deformation beam and finite-elements theory by Carlsson and Aksoy (1991). Below, we will review these results.

571

12.3 Fracture Specimens

Beam Theory Analysis With the geometry symbols in Fig. 12-23b, the IENF compliance is Q

L (^ll)BC , M ^ 5 5 ) B C

2w/c

6w

12w

(12-137)

where L, w, and k are the half span length, beam width, and shear correction factor (k = 5/6), respectively. D'11 and A%5 are elements of the inverted bending and shear stiffness matrices, defined by Carlsson and Aksoy (1991). The subscripts AB and BC denote delaminated and intact beam sections, see Fig. 12-23 b. The energy release rate is given by G =

8w 2

(12-138)

Notice that, due to the approximations involved, the beam analysis makes no distinction between symmetric and asymmetric crack configuration. Finite-Element Analysis A two-dimensional plane strain finite element analysis of the IENF geometry was performed by Carlsson and Aksoy (1991) in order to numerically calculate compliance and energy-release rate components. Stress-intensity factors were not considered by the authors. As for the IDCB specimen, both symmetric and asymmetric (bimaterial) crack configurations were investigated. The material properties of the composite and thermoset interleaf are identical to those provided in Table 12-5 for the IDCB specimen discussed in the previous section. Homogeneous and interleaved specimens with 12.7, 25.4, 76.2, and 127 \xm thick interleaves were considered. For all cases,

the crack length (a) was 25.4 mm and the semispan length (L) was 50.8 mm, (a/L = 0.5). The total beam thickness was kept constant at 3.43 mm for all specimens, and the beam width (w) was 25.4 mm. Compliance of the IENF specimen was determined from the vertical beam deflection, (5, at the beam center, divided by the applied load, P. Strain energy release rate components GY and Gu were calculated using the finite element crack closure method, Eqs. (12-136). In all symmetric and asymmetric configurations considered, no crack opening was found which shows that the IENF geometry is pure mode II. Table 12-9 provides compliance and energy release rate for symmetric and asymmetric IENF specimens with 12.7 and 25.4 jim thick interlayers. The asymmetry of the crack has negligible influence on compliance and energy release rate for this range of interleaf thicknesses. Table 12-10 provides a comparison between finite-element (symmetric crack location) and beam theory results, Eqs. (12-137) and (12-138), for the IENF compliance and energy release rate. Finite element analysis gives slightly larger compliance values than the beam theory. Also the strain-energy release rates predicted by the two approaches are in close agreement. Consequently, the more convenient beam theory approach could be employed for ex-

Table 12-9. Compliance and energy-release rate for IENF specimens with symmetric (S) and asymmetric (A) (interfacial) crack location; a = 25A mm, L = 50.8 mm, w = 25.4 mm, P = 4.44 N. Interleaf thickness (Mm) 12.7 25.4

C[(m/N)x 10"5] S 4.190 4.222

A 4.189 4.220

Gu[(J/m 2 ) x l 0 " 2 ] S

A

1.320 1.330

1.330 1.360

572

12 Fracture of Fiber Composites

Table 12-10. Compliance and energy-release rate for IENF specimen (thermoset interleaf, symmetric crack location) calculated from beam theory (BT) and finite elements (FE); a = 25 A mm, L = 50.8 mm, w = 25.4 mm, p = 4.44N. Interleaf

cFE

thickness (urn)

[(m/N)x 10~5]

0 25.4 76.2 127

4.157 4.222 4.330 4.431

CBT

4.137 4.153 4.193 4.239

GFE

GBT

[(J/m 2 ) x l 0 " 2 ] 1.260 1.330 1.530 1.730

1.230 1.340 1.620 1.720

perimental evaluation of mode II fracture toughness. Data Reduction for Fracture Toughness of IENF Specimen Inspection of Table 12-10 reveals that the compliance and energy release rate of the interleaved ENF specimen may be significantly larger than for the homogeneous specimen. Crack growth is inherently unstable for the ENF and IENF specimens (Carlsson and Pipes, 1987; Aksoy and Carlsson, 1992) and fracture toughness evaluation is based on the compliance and critical load at one crack length, as opposed to the compliance calibration method recommended for the IDCB specimen. Aksoy and Carlsson (1992) outline the recommended data reduction methodology for the IENF specimen. The appropriate composite modulus E1 is obtained by iterating Eq. (12-137) until the calculated compliance agrees with the experimentally measured. The modulus E± so obtained is then substituted into Eq. (12-138) together with the critical load for crack propagation, Pc, to obtain the fracture toughness. For further details, see Aksoy and Carlsson (1992).

12.4 Fracture Data The experimental fracture characterization of composites will be discussed in this section. The description is limited to crack propagation parallel to the fibers in homogeneous orthotropic beam specimens and fracture of interleaved composites. With the above restrictions, the experimental determination of fracture toughness appears straightforward, but experimenters have found that the interlaminar fracture toughness is extremely sensitive to numerous experimental parameters. Figure 12-24 illustrates micromechanisms that may influence the resistance of a growing crack. The fracture energy measured in a test could be related to the following mechanisms: formation of the fracture surface of the main crack, plastic dissipation due to plastic-zone formation in the matrix around the crack tip and translation of the plastic zone, crack bridging by intact fibers left behind the crack front, and formation of microcracks and secondary cracks that may parallel the main crack. Several of these mechanisms develop with crack extension. For example, fiber briding tends to build up and increase with crack propagation and a resistance curve (K-curve) is necessary to describe the fracture process. Crack tip plasticity may also increase with crack extension. FihprQ Mbers \

Fiber breakage

^ Main crack

Side

1

••>

>r^2

Matrix

Bridging fibers

A

s

Plastic zone

Figure 12-24. Illustration of energy absorbing mechanisms in interlaminar fracture (Friedrich et al., 1989).

12.4 Fracture Data

The size of the crack-tip plastic zone is sensitive to the volume fraction of matrix material because the fibers generally are linear elastic and constrain the development of crack-tip plastic zones. Significant increases of the plastic-zone size and fracture toughness are achieved by interleafing, as will be discussed later in this section. In addition, the matrix may display viscoelastic and viscoplastic effects, in which case the fracture resistance is deformationrate and temperature dependent. Other complications that may influence the i^-curve for composites are the methods of forming a precrack during processing of the composite and the subsequent extension of the precrack prior to the actual fracture testing (precracking). Figure 12-25 illustrates the situation at the crack tip after processing of the composite. The starter film is placed between the plies at the desired crack plane during layup of the composite. Typically, the insert is a 7-25 jim thick PTFE or polyimide film coated with a release agent. As illustrated in Fig. 12-25, a wedge-shaped resin zone is formed between the plies ahead of the insert film, and fracture testing where the crack is propagated directly from the tip of the starter film may lead to artifically large toughness values because of the locally high resin content (Martin, 1988; Murri and O'Brien, 1985). In order to minimize the influence of the insert film, its thickness should be kept at a minimum. ASTM recommends a maximum film thickness of 13 pm for the DCB test (minutes of the ASTM Task Group). On the other hand, fiber bridging, Fig. 12-26, may develop during mode I loading already at short crack lengths and, therefore, the precracking distance should be kept at a minimum. In fact, ASTM recommends no precracking for the DCB test (minutes of the ASTM Task Group). Fiber bridging does not oc-

Starter film 10-50 [im thick

573

Resin-rich region 200-500 \im in length

Figure 12-25. Starter film and resin-rich region in composite laminate.

cur in multidirectional composite laminates and fracture toughness values incorporating contributions from fiber bridging should thus not be used in an assessment of the damage tolerance of composite structures. This discussion brings up yet another issue, namely the appropriate mode of loading during precracking. Initially, mode I precracking was suggested for most fracture tests, but the extensive fiber bridging that occurs in some system has shifted the attention to shear-driven precracks or thin films without precracks (Meeting between the ASTM Task Group D 30.02.02, European Group on Fracture, and Japan Industrial Standards in 1989; minutes of the ASTM Task Group). The issue is under investigation by standard issuing organizations.

Bridged fibers

Figure 12-26. Schematic of fiber bridging in mode I loading.

12.4.1 Mode I Fracture Data A comprehensive review of mode I interlaminar fracture testing and data has been presented by Davies and Benzeggagh

574

12 Fracture of Fiber Composites

(1989). In this section only some illustrative experimental fracture data generated in mode I testing of polymer matrix composites will be presented. R-Curve Effect As discussed previously in this section, pull-out and fracture of bridged fibers behind the crack front (Figs. 12-24 and 12-26) is a common occurrence in mode I fracture testing. At the first loading increment (Aa = 0), when the delamination grows from the tip of the thin film insert starter crack, Fig. 12-25, there is no fiber bridging and the crack flank displacements behind the crack tip are at a minimum. As the crack grows, the crack surfaces become more and more separated and bridged fibers may fracture or become pulled out from the matrix, which causes the apparent fracture toughness to increase. With further crack extension, a steady-state toughness is usually reached corresponding to an equilibrium number of bridged fibers per unit crack area, see Fig. 12-27. The #-curve illustrated in Fig. 12-27 may be characterized by an initiation value, GIC (init.) corresponding to crack extension after precracking from the film starter, and a steady-state propagation

Graphite/ rubber toughened epoxy

60

70

80

90

Crack length (mm)

Figure 12-27. incurve describing mode I fracture resistance of a graphite-rubber toughened epoxy composite.

value, GIC(prop.). As demonstrated by Hashemi et al. (1990), however, GIC(init.) is not a unique value for a composite system, because GIC(init.) depends on the amount of precracking beyond the tip of the insert film. To obtain the full #-curve, Hashemi et al. (1990) showed that the extent of precracking (crack growth beyond the tip of the insert film) should be kept at a minimum. To propagate the delamination directly from the starter crack may lead to an erroneous overestimate of the initiation toughness, GIC (init.), due to a resin-rich region at the tip of the film insert (see Fig. 12-25). The increase in fracture toughness with crack extension illustrated in Fig. 1227, may (besides fiber bridging) also be due to an increase in the degree of plastic deformation and microcracking around the crack tip with crack extension. Temperature and Rate Effects Because polymer matrices generally are viscoelastic or viscoplastic, changes in temperature and strain rate will influence the fracture toughness. Shivakumar and Crews (1987) proposed that the contribution of the plastic energy dissipation to the fracture toughness, GIC, is directly proportional to the height of the crack-tip plastic zone. Precracking leads to the formation of a plastic zone while fracture testing on precracked specimens leads to translation of the plastic zone. The volume of the newly formed plastically deformed material should therefore be proportional to the dimension of the plastic zone perpendicular to the crack plane ("plastic-zone height"). To examine rate and temperature effects on fracture toughness, we will thus consider the factors contributing to the plastic-zone height. The height of the mode I crack-tip plastic zone in the absence of fiber bridging is for plane strain (Ozdil and

575

12.4 Fracture Data 10'

Carlsson, 1992 c) (12-139) in which K1 is the mode I stress intensity factor and oy is the yield strength of the matrix. It should be pointed out that Eq. (12-139) is valid only for the situation when the plastic zone is fully contained in the resinrich layer between plies in a composite. As this layer is thin, 8-32 jim in thickness (Bradley and Cohen, 1985; Friedrich et al, 1989; Hunston, 1984), severe constraints may be imposed on the plastic zone development by the rigid fiber-reinforced layers, which will limit plastic zone height, and thus the fracture toughness (Bradley and Cohen, 1985; Friedrich et al., 1989; Hunston, 1984). The plastic zone model should thus be used with caution, but is useful for the interpretation of rate and temperature effects on fracture toughness. For a viscoelastic matrix, the fracture toughness Klc generally decreases strongly with strain rate while the yield strength increases, see, for example, Friedrich et al. (1991). A higher strain rate is thus expected to lead to a smaller plastic zone height in the composite and hence diminished fracture toughness, GIC. Increased temperature will soften the polymer and reduce the yield strength. The influence of temperature on Klc is usually less pronounced and Eq. (12-139) would predict an increased plastic zone height and fracture toughness GIC with temperature. Figures 12-28 and 12-29, obtained from Friedrich et al. (1989), Smiley and Pipes (1987 a), and Hashemi et al. (1990) provide examples on rate and temperature dependence of GIC for graphite fiber composites with thermoset and thermoplastic matrices. The trends of the data are consistent with the plastic zone arguments above.

/Graphite/PEEK

i

^, •i-i-

o,0-i

Graphite/Epoxy

HO

10

ICf

IO"V

IO'° 10"' yCT (m/sec)

Figure 12-28. Mode I fracture toughness, GIC, versus crack tip opening velocity for graphite-epoxy and graphite-PEEK composites (Friedrich et al., 1989; Smiley and Pipes, 1987).

80

100

120

140

Temperature, C

Figure 12-29. Mode I fracture toughness, GIC (prop.) versus temperature for a graphite-PEEK composite (Hashemi et al., 1990).

12.4.2 Mode II Fracture Data Mode II interlaminar fracture has been the subject of many investigations (Carlsson etal., 1986 b; Hashemi et al., 1990; Martin and Murri, 1990; Murri and O'Brien, 1985; O'Brien et al., 1987; Russell and Street, 1982 and 1985; Trethewey et al., 1988). It has generally been observed that the mode II fracture toughness, GIIC, is larger than GIC for composites with brittle resins, while increased matrix ductility makes GIC approach GIIC. The micromechanisms responsible for the fracture tough-

576

12 Fracture of Fiber Composites

G

..C<

__o_^, o trcP—

3

E

3

V 50mm

•

o"

2 / i 1r j L_I 1

0

O a p = 63mm

o V 68mm

GIIC (init.)

A ap

=

82mm

I

I

i

i

I

10

20

30

40

50

60

Increment of crack propagation, Aa (mm) Figure 12-30. R-curve describing mode II fracture resistance of a graphite-PEEK composite (Hashemi etal., 1990).

10.00

/Graphite/PEEK

E 100 pox Graphite/Epoxy0.10

0.01

ioy

I07

icr

u CT (m/sec)

Figure 12-31. Mode II fracture toughness, GIIC (init.), versus crack tip sliding velocity for graphite-epoxy and graphite-PEEK composites (Friedrich et al., 1989; Smiley and Pipes, 1987).

20

40

60

80 100 120 140

Temperature, T (°C) Figure 12-32. Mode II interlaminar fracture toughness, GIIC (prop.), versus temperature for a graphitePEEK composite (Hashemi et al., 1990).

ness were discussed early in Sec. 12.4 and are illustrated in Fig. 12-24. Plastic deformation around the crack tip appears to be the major energy-absorbing mechanism because fiber bridging is much less pronounced in shear (mode II) than in tension (mode I) (Hashemi etal., 1990). Consequently, the plastic zone argument presented above for mode I fracture testing should be appropriate for explaining the mode II fracture toughness. Figure 12-30 shows a mode II R-curve for a graphite-PEEK composite (Hashemi et al., 1990). The various symbols represent various precracking lengths and the unique R-curve observed indicates that fiber bridging is not a major factor in mode II fracture of this composite. The primary mechanism explaining the rising portion of the R-curve is crack-tip plasticity that develops in the intense crack tip shear field and evolves with crack growth. It should be pointed out that PEEK is an engineering thermoplastic with extraordinary toughness. For a more brittle matrix such a pronounced incurve is not expected. Rate effects on mode II fracture were examined by Friedrich et al. (1989), Carlsson etal. (1986b), and Smiley and Pipes (1987b). Figure 12-31 displays GIIC(init.) versus crack tip sliding velocity. Similarly to the mode I fracture toughness, Fig. 12-28, GIIC drops substantially with rate at high crack tip sliding velocities. The effect of temperature on GIIC of a graphite-PEEK composite is illustrated in Fig. 12-32. Similar to GIC, Fig. 12-29, GIIC increases with increasing temperature. 12.4.3 Mixed-Mode Fracture Data This section will address data compiled for fracture due to combinations of tensile (mode I) and shear (mode II) stresses ahead of the crack tip. The micromechanisms dis-

577

12.4 Fracture Data

cussed early in Sec. 12.4 and the plastic zone argument detailed in Sec. 12.4.1 should be applicable also for mixed mode fracture. As discussed in Sec. 12.3.3, a complete fracture characterization of composites requires mixed mode testing. This was first recognized by Wu (1967), who studied intralaminar fracture of balsa wood and a glass-polyester composite, and formulated a "best fit"-type fracture criterion in terms of stress intensity factors: K ic

(12-140)

K IIC

More customary in interlaminar fracture characterization of composites is to use the components of the strain-energy release rate and to employ the following fracture criterion (Russell and Street, 1985)

ic,

G

=1

(12-141)

IIC,

where the exponents m and n are determined by curve-fitting of experimental data. Johnson and Mangalgiri (1987) collected data on mixed-mode interlaminar fracture toughness for composites with a wide variety of resin toughnesses. As shown in Fig. 12-33, a straight line, corresponding to m = n = 1 in Eq. (12-141) provides a good fit to the experimental data, that is, -^- + - ^ = 1

2.0

0.4

0.8

1.2

G,|,kJ/m

1.6

2.0

2

Figure 12-33. Mixed mode interlaminar fracture toughness data for various graphite fiber composites. Only the matrix material is indicated in the figure (Johnson and Mangalgiri, 1987).

with caution because GIC is rarely equal to GIIC. More recent mixed-mode fracture studies of graphite-PEEK composites have been presented by Reeder and Crews (1991) and Hashemi et al. (1990). Figure 12-34 shows delamination toughnesses of a graphite-PEEK composite determined from the MMB test. The results in Fig. 12-34 support the validity of the linear mixed-mode criterion, Eq. (12-142). The characterization of mixed-mode fracture of unidirectional composites is obscured by the i^-curve effects, see Fig. 12-35. The difference between GIC(init.) and GIC(prop.) in Fig. 12-35 can be attributed

(12-142)

It was also found that the ratio Glc/Gnc increases from about 0.1 to about 1.0 as the resin ductility increases. For ductile resins with G I C « GIIC, the total strain-energy release rate controls the propagation of a crack under pure or mixed-mode conditions. Generally, however, the total-fracture-energy approximation should be used

G, (kJ/m2)

Figure 12-34. Mixed mode interlaminar fracture toughness of a graphite-PEEK composite (Reeder and Crews, 1991).

12 Fracture of Fiber Composites

578

AS4/PEEK

Propagation

ness in a similar manner as the mode I and mode II fracture toughnesses discussed previously. Except at room temperature (20 °C), the data in Fig. 12-36 are reasonably approximated by the linear mixedmode criterion, Eq. (12-142). 12.4.4 Mode I and Mode II Fracture Data for Interleaved Composites

I

2 G,,,(kJ/m2)

Figure 12-35. Mixed mode interlaminar fracture toughness data for a graphite-PEEK composite (Hashemi et al., 1990).

to various degrees of crack-tip plasticity and fiber bridging, see the earlier discussion in this section. The data on mixedmode fracture toughness for initiation toughness presented in Fig. 12-35 are somewhat larger than those presented for the graphite-PEEK system in Figs. 12-33 and 12-34, especially at high Gl/Gll ratios. This may be due to differences in materials processing and precracking methods. Figure 12-36 shows that temperature influences the mixed-mode fracture tough-

-

AS4/PEEK TCC) • 130 D 60 O 20

\v

2 3

I 9

0

1

2

3

4

5

G,, (prop.), (kJ/m*) Figure 12-36. Mixed mode interlaminar fracture toughness data for a graphite-PEEK composite at various temperatures (Hashemi et al., 1990).

Fracture toughness characterization of interleaved composites has been reported in several studies (Aksoy and Carlsson, 1991 and 1992; Evans and Masters, 1987; Masters, 1987 a and b; Ozdil and Carlsson, 1992 a; Sela et al., 1989). All these investigations report tremendous elevations of the mode I and mode II fracture toughnesses, GIC and GIIC. In this section we will briefly review the experimental results of Aksoy and Carlsson (1990) and Ozdil and Carlsson (1992 a). The mode I and mode II fracture investigations (Aksoy and Carlsson, 1992; Ozdil and Carlsson, 1992 a) employed a unidirectional graphite-epoxy composite with a large range of thermoplastic (TPI) and thermoset (TSI) interleaf thicknesses. The mode I (IDCB) and mode II (IENF) fracture specimens, Fig. 12-23, used in the experimental fracture studies, were of similar geometry and material properties as those analyzed in Sec. 12.3.4. The mode I fracture investigation covered interleaf thicknesses between 16 and 129 jim for TPI and 59 and 214 jam for TSI. The corresponding TPI and TSI thickness ranges for the mode II fracture investigation (Aksoy and Carlsson, 1992) are 12.7 to 127 and 43 to 429 fim. It should be pointed out that several of the interleaved fracture specimens suffered from poor adhesion, which preempted the potentially large fracture toughness. In this section, we will limit the presentation to specimens with adequate

12.4 Fracture Data

579

5.0

2.0r

4.0 -

Thermoplastic interleaf (Film E)

3.0

20

40 Interleaf

60

80

I00

I20

I40

thickness,/i.m

Figure 12-37. Mode I fracture toughness versus interleaf thickness for a graphite-epoxy composite interleaved with thermoplastic (TPI) and thermoset (TSI) interleaves (Ozdil and Carlsson, 1992 a).

adhesion. For a more complete discussion, see references (Aksoy and Carlsson, 1992; Ozdil and Carlsson, 1992 a). Data reduction for mode I fracture toughness evaluation followed the empirical compliance calibration procedure outlined for DCB specimens by ASTM (proposed standard test method), while mode II fracture toughness evaluation was based on beam theory, see Sec. 12.3.4 and Aksoy and Carlsson (1992). Figure 12-37 shows GIC plotted versus interleaf thickness. The TPI enhanced the fracture toughness about five-fold over the baseline value at a thickness of only 16 jam. Thicker interleaves did not provide any further elevation of GIC. The 59 \xm TSI provided about 30% elevation of GIC compared to the baseline. Thicker TSI were investigated, but the specimens failed at low loads due to adhesion problems. It should be noted that the thermoset (FM 300) interleaved composites investigated by Sela et al. (1989) displayed large enhancements of GIC, which indicates that thermoset interleaves also have toughening potential. Thick thermoset interleaves may, however, be required. The thinnest

0.00

0.05 0.10 Interleaf thickness, mm

0.15

4.0

E

U)

in (D

C JZ

Thermoset interleaf (FM300I)

b) 0 0.0

0.1 0.2 0.3 0.4 Interleaf thickness, mm

0.5

Figure 12-38. Mode II fracture toughness versus interleaf thickness for an interleaved graphite-epoxy composite (Aksoy and Carlsson, 1992). (a) TPI, (b) TSI.

FM 300 interlayer studied by Sela et al. (1989) was 100 pm. Microscopic investigation of the fracture process (Ozdil and Carlsson, 1992 a) indicated that the large toughness of the IDCB specimens is associated with a significant degree of plastic deformation of the interleaf. This observation supports the plastic

580

12 Fracture of Fiber Composites

energy dissipation arguments for the toughness discussed in Sec. 12.4. Detailed estimates of the plastic zone height in interleaved DCB specimens presented by Ozdil and Carlsson (1992 c) support the experimentally observed trends in Fig. 12-37. Figure 12-38 shows GIIC plotted versus interleaf thickness. Notice the small interleaf thickness range and the rapid development of GIIC for TPI, Fig. 12-38 a. The extremely large GIIC value at 127 jum film thickness is largely attributed to fracture of fibers bridging the crack surfaces. The development of GIIC versus film thickness for the TSI is less rapid than that for the TPI, see Fig. 12-38 b, although large GIIC values can be achieved at large TSI thicknesses. Inspection of the fracture process under optical and scanning electron microscopes (Aksoy and Carlsson, 1992) revealed plastic deformation and limited microcrakking in the TPI and significant microcrakking in the TSI. Detailed estimates of the plastic zone height in interleaved ENF specimens presented by Aksoy and Carlsson (1991) are in agreement with the toughness versus interleaf thickness trends shown in Fig. 12-38.

12.5 Acknowledgements This work was supported in part by the National Science Foundation (Grant MSM-88-22229), and by the Department of Mechanical Engineering, Florida Atlantic University (FAU). Part of the review builds on my previous cooperation with graduate students of FAU and Dr. John W. Gillespie Jr. and Dr. R. Byron Pipes of the University of Delaware, as well as several graduate students I was fortunate enough to work with during may stay at the University of Delaware. In addition, I would like to acknowledge cooperation with Prof.

Klaus Friedrich and Dr. Joseph KargerKocsis, both of the University of Kaiserslautern. More recently, I have had much fruitful communication with Prof. John W. Hutchinson of Harvard University. This chapter would not have been possible without the assistance from Rosemarie Chiucchi, Susan Cury, and Evelyn Hoffman, who typed and retyped many iterations of the manuscript. The graphics work by Mark Deshon of the University of Delaware and Shawn Pennell and Sherri Vonhartman of FAU is gratefully acknowledged.

12.6 References Aksoy, A., Carlsson, L. A. (1991), Eng. Fracture Mech. 39, 525. Aksoy, A., Carlsson, L. A. (1992), Compos. Sci. Tech. 43, 55. Arcan, M., Hashin, S., Voloshin, A. (1978), Exp. Mech. 18, 141. ASTM STP696 (1979). Philadelphia, PA: Am. Soc. Test. Mat. ASTM STP876 (1985). Philadelphia, PA: Am. Soc. Test. Mat. ASTM STP936 (1987). Philadelphia, PA: Am. Soc. Test. Mat. Barrett, J. D. Foshi, R. O. (1977), Eng. Fracture Mech. 9, 371. Bascom, W. D., Bitner, J. L., Mouton, R. J., Siebert, A. R. (1980), Composites 11, 9. Beaumont, P. W. R., Schultz, J. M. (1990a), in: Delaware Composites Design Encyclopedia, Vol. 4: (Eds.). Westport, CT: Technomic, p. 21. Beaumont, P. W. R., Schultz, J. M. (1990b), in: Delaware Composites design Encyclopedia, Vol. 4: (Eds.). Westport, CT: Technomic, p. 39. Berry, J. P. (1963), J. Appl. Phys. 34, 62. Bostaph, G. M., Elber, W. (1982), paper presented at: AS ME Winter Conference. Bradley, W. L., Cohen, R. N. (1985), in: Delamination and Debonding of Materials: ASTM STP876. Philadelphia, PA: Am. Soc. Test. Mat., p. 389. Broek, D. (1982), Elementary Engineering Fracture Mechanics. Dordrecht: Martinus Nijhoff. Carlsson, L. A., Aksoy, A. (1991), Int. J. Fracture 52, 67. Carlsson, L. A., Pipes, R. B. (1987), Experimental Characterization of Advanced Composite Materials. Englewood Cliffs, NJ: Prentice-Hall. Carlsson, L. A., Gillespie, Jr., J. W, Pipes, R. B. (1986 a), J. Compos. Mater. 20, 594.

12.6 References

Carlsson, L. A., Gillespie, Jr., J. W, Trethewey, B. R. (1986 b), /. Reinf. Plast. Compos. 5, 170. Carlsson, L. A., Sindelar, P., Nilsson, S. (1986c), Compos. Sci. Tech. 26, 307. Crews, Jr., J. H., Reeder, J. R. (1988), NASA TM 100662. Crews, Jr., J. H., Shivakumar, K. N., Raju, I. S. (1988), in: Adhesively Bonded Joints: Testing, Analysis and Design: ASTM STP981. Philadelphia, PA: Am. Soc. Test. Mat., p. 119. Davies, P., Benzeggagh, M. L. (1989), in: Application of Fracture Mechanics to Composite Materials. Friedrich, K. (Ed.). Amsterdam, New York: Elsevier, p. 81. Dost, E. E, Ilcewicz, L. B., Avery, W. B., Coxon, B. R. (1991), Composite Materials: Fatigue and Fracture, 3rd Vol. ASTM STP 1110. Philadelphia, PA: Am. Soc. Test. Mat., p. 476. Dundurs, J. (1969), /. Appl. Mech. 36, 650. England, A. H. (1965), /. Appl. Mech. 32, 400. Erdogan, F. (1965), J. Appl. Mech. 32, 403. Erdogan, F , Gupta, G. D. (1971), Int. J. Solids Struct. 7, 1089. Evans, R. E., Masters, J. E. (1987), in: Toughened Composites: ASTM STP 937. Philadelphia, PA: Am. Soc. Test. Mat. Ewalds, H. L., Wanhill, R. J. H. (1989), Fracture Mechanics. London: Edward Arnold. Friedrich, K. (Ed.) (1989), Application of Fracture Mechanics to Composite Materials. Amsterdam, New York: Elsevier. Friedrich, K., Carlsson, L. A., Smiley, A. J., Walter, R., Gillespie, Jr., J. W. (1989), J. Mater. Sci. 24, 3387. Friedrich, K., Carlsson, L. A., Gillespie, Jr., J. W, Karger-Kocsis, J. (1991), in: Thermoplastic Composite Materials: Carlsson, L. A. (Ed.). Amsterdam, New York: Elsevier, p. 233. Gere, J. M., Timoshenko, S. P. (1984), Mechanics of Materials, 2nd ed. PWS Engineering. Gillespie, Jr., J. W, Carlsson, L. A. (1990), in: Delaware Composites Design Encyclopedia, Vol. 6. Westport, CT: Technomic, p. 111. Griffith, A. A. (1921), Phil Trans. Roy. Soc. of London A 221, 163. Guynn, E. G., O'Brien, T. L. (1985), in: Proceedings AIAA/ASME/ASCE/AHS 26th Structures, Structural Dynamics and Materials Conference (AIAA85-0646), Orlando, FL. Hashemi, S., Kinloch, A. I , Williams, J. G. (1990), J. Compos. Mater. 24, 918. Hellan, K. (1984), Introduction to Fracture Mechanics. New York: McGraw-Hill. Horgan, C. O. (1982), J. Compos. Mater. 16, 411. Hunston, D. L. (1984), Compos. Tech. Rev. 4, 176. Hutchinson, J. W (1989), Report MECH-139, Division of Applied Sciences, Harvard University, MA. Hutchinson, J. W. (1990), personal communication. Hutchinson, J. W, Suo, Z. (1991), Adv. Appl Mech. 28.

581

Irwin, G. R. (1958), in: Handbuch der Physik, Vol. 6: Flugge, (Ed.). Berlin, Heidelberg: Springer, p. 551. Jones, R. M. (1975), Mechanics of Composite Materials. New York: McGraw-Hill. Johnson, W. S., Mangalgiri, P. D. (1987), in: Toughered Composites. ASTM STP 875. Philadelphia, PA: Am. Soc. Test. Mat., p. 295. Jordan, W. M., Bradley, W. L. (1987), in: Toughened Composites. ASTM STP 937. Philadelphia, PA: Am. Soc. Test. Mat., p. 95. Jurf, R. A., Pipes, R. B. (1982), J. Compos. Mater. 16, 386. Kinloch, A. I , Young, R. J. (1983), Fracture Behavior of Polymers. Amsterdam, New York: Elsevier. Knott, J. F. (1973), Fundamentals of Fracture Mechanics. London, Washington: Butterworths. Leichti, K. M., Chai, Y.-S. (1992), /. Appl. Mech. 59, 295. Lekhnitskii, S. G. (1981), Theory of Elasticity for an Anisotropic Body. Moscow: MIR Publishers. Ma, C.-C, Luo, J.-X, Theoretical Analysis of the Interfacial Crack for General Anisotropic Material, Part I: Theory, to be published. Mandell, J. F., Darwish, A. Y, McGarry, F J. (1981), Test Methods and Design Allowables for Fibrous Composites. ASTM STP 734. Philadelphia, PA: Am. Soc. Test. Mat., p. 73. Martin, R. H. (1988), in: Proc. Am. Soc. Composites: Westport, CT: Technomic, p. 688. Martin, R. H., Murri, G. B. (1990), in: Composite Materials: Testing and Design. ASTM STP 1059. Philadelphia, PA: Am. Soc. Test. Mat. Masters, J. E. (1985), Presented at: ASTM Symposium on Fractography of Modern Engineering Materials. Nashville, TN. Masters, J. E. (1987 a), Final Report for Period Sept. 1984-June 1987, AFWAL/MLBC. Wright Patterson Air Force Base, Ohio. Masters, J. E. (1987b), Key Eng. Mater. 37, 317. Mohlin, T, Blom, A. F , Carlsson, L. A., Gustavsson, A. I. (1985), in: Delamination and Debonding of Materials. ASTM STP 876. Philadelphia, PA: Am. Soc. Test. Mat., p. 168. Murri, G. B., O'Brien, T. K. (1985), in: AIAA-85-0647, Proceedings of the 26th AIAA/ASME/ASCE/ASH Conference on Structures, Structural Dynamics and Materials, Orlando, FL. O'Brien, T. K. (1982), in: Damage in Composite Materials. ASTM STP 775. Philadelphia, PA: Am. Soc. Test. Mat., p. 140. O'Brien, X K., Murri, G. B., Salpekar, S. A. (1987), NASA TM 89157. Ozdil, F , Carlsson, L. A. (1992 a), J Compos. Mater. 26, 432. Ozdil, F , Carlsson, L. A. (1992b), Eng. Fracture Mech. 41, 475. Ozdil, F , Carlsson, L. A. (1992c), Eng. Fracture Mech. 41, 645.

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12 Fracture of Fiber Composites

Pipes, R. B., Blake, Jr., R. A. (1990), in: Delaware Composites Design Encyclopedia, Vol. 6. Westerville, CT: Technomic, p. 55. Proposed Standard Test Method for Mode I Interlaminar Fracture Toughness of Continuous Fiber Reinforced Composite Materials, ASTM (1990). Philadelphia, PA: Am. Soc. Test. Mat. Raju, I. S. (1987), Eng. Fracture Mech. 28, 251. Raju, I. S., Crews, Jr., J. H., Aminpour, M. A. (1988), Eng. Fracture Mech. 30, 383. Reeder, J. R., Crews, Jr., J. H. (1991), in: Proc. ICCM-8: Springer, G. S. (Ed.). Honolulu, HI. Rice, J. R. (1988), /. Appl. Mech. 55, 98. Russell, A. J., Street, K. N. (1982), in: Progress in Science and Engineering of Composites: Hayashi, T., Kawata, K., Umekawa, S. (Eds.). Tokyo: ICCM-IV, p. 279. Russell, A. X, Street, K. N. (1985), in: Delamination and Debonding of Materials. ASTM STP876. Philadelphia, PA: Am. Soc. Test. Mat., p. 349. Rybicki, E. R, Kanninen, M. F. (1977), Eng. Fracture Mech. 9, 931. Salpekar, S. A., Raju, I. S., O'Brien, T. K. (1988), /. Compos. Tech. Res. 10, 133. Sela, N., Banks-Sills, L., Ishai, O. (1989), Eng. Fracture Mech. 32, 533. Sela, N., Ishai, O., Banks-Sills, L. (1989), Composites 20, 257. Sih, G. C , Liebowitz, H. (1968), in: Mathematical Theories of Brittle Fracture, Fracture, Vol. 2: Liebowitz, H. (Ed.). London: Academic Press. Sih, G. C , Paris, P. C , Irwin, G. R. (1965), Int. J. Fracture Mech. 1, 189. Shivakumar, K. N., Crews, Jr., J. H. (1987), Eng. Fracture Mech. 28, 319. Smiley, A. I, Pipes, R. B. (1987 a), J. Compos. Mater. 21, 670. Smiley, A. X, Pipes, R. B. (1987b), Compos. Set Tech. 29, 1. Sun, C. T, Manoharan, M. G. (1987), in: Am. Soc. for Composites. Technomic, p. 49. Suo, Z. (1990a), Proc. R. Soc. London A 427, 331. Suo, Z. (1990b), /. Appl. Mech. 57, 627. Suo, Z., Hutchinson, J. W. (1989), Mater. Sci. and Eng. A 107, 135. Suo, Z., Hutchinson, J. W. (1990), Int. J. Fracture 43, 1. Suo, Z., Bao, G., Fan, R, Wang, T. C. (1991), Int. J. Solids Struct. 28, 235. Test Procedure for the End Notched Flexure (ENF) Specimen (1987), ASTM D 30.02 Round Robin. Philadelphia, PA: Am. Soc. Test. Mat. Timoshenko, S. P., Goodier, J. N. (1970), Theory of Elasticity, 3rd ed. New York: McGraw-Hill. Ting, T. C. T. (1986), Int. J. Solids Struct. 22, 965. Ting, T. C. T. (1980), / Mech. Phys. Solids 38, 505.

Trethewey, B. R., Gillespie, Jr., J. W., Carlsson, L. A. (1988), /. Compos. Mater. 22, 459. Tsai, S. W, Hahn, H. T. (1980), Introduction to Composite Materials. Westerville, CT: Technomic. Wang, S. S. (1983), /. Compos. Mater. 17, 210. Wang, S. S., Choi, I. (1983), NASA CR 172269 and 172270. Wang, A. S. D., Crossman, F. W (1980), /. Compos. Mater. Suppl. 14, 71. Wang, S. S., Mandell, J. R, McGarry, R J. (1987), Int. J. Fracture 14, 39. Wang, J. S., Suo, Z. (1990), Acta Met. 38, 1279. Wilkins, D. J. (1983), in: Proc. AGARD meeting on Characterization, Analysis and Significance of Defects in Composite Materials, Proceeding No. 355, London. Wilkins, D. X, Eisenmann, X R., Camin, R. A., Margolis, W S., Benson, R. A. (1982), in: Damage in Composite Materials. ASTM STP 775. Philadelphia, PA: Am. Soc. Test. Mat., p. 168. Williams, M. L. (1959), Bull. Seismol. Soc. America 49, 199. Williams, J. G. (1988), Int. J. Fracture 36, 101. Williams, X G. (1989), in: Application of Fracture Mechanics to Composite Materials: Friedrich, K. (Ed.). Amsterdam, New York: Elsevier, p. 3. Williams, X G. (1990), personal communication. Willis, X R. (1971), J. Mech. Phys. Solids 19, 353. Wilson, D., Carlsson, L. A. (1991), in: Physical Methods of Chemistry: Rossiter, B. W (Eds.). New York: Wiley, p. 139. Wu, E. M. (1967), J. Appl. Mech. 34, 967. Wu, E. M. (1974), in: Strength and Fracture of Composite Materials, Fatigue and Fracture, Vol. 5: Broutman, L. X (Ed.). London: Academic Press.

General Reading Application of Fracture Mechanics to Composite Materials (1989): Friedrich, K. H. (Ed.). Amsterdam: Elsevier. Delamination and Debonding of Materials: ASTM STP 876 (1985). Philadelphia, PA: Am. Soc. Test. Mat. Ewalds, H. L., Wanhill, R. H. X (1989), Fracture Mechanics. London: Edward Arnold. Hutchinson, X W, Suo, Z. (1991), Mixed Mode Cracking in Layered Materials, in: Adv. Appl. Mech. 28. Lekhnitskii, S. G. (1981), Theory of Elasticity for an Anisotropic Body. Moscow: MIR Publishers. Mathematical Theories of Brittle Fracture, in: Fracture, Vol. 2 (1968): Liebowitz, H. (Ed.). New York: Academic Press.

13 Fatigue of Fiber Composites Ramesh Talreja School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, U.S.A.

List of Symbols and Abbreviations 13.1 Introduction 13.2 Short Fiber Composites 13.3 Long and Continuous Fiber Composites 13.3.1 Unidirectional Composites 13.3.1.1 Loading Parallel to Fibers 13.3.1.2 Loading Inclined to Fibers 13.3.2 Laminates 13.3.2.1 Angle Ply Laminates 13.3.2.2 Cross Ply Laminates 13.3.2.3 General Laminates 13.3.3 Woven Fabric Composites 13.4 Conclusion 13.5 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

584 585 585 592 592 593 597 599 599 601 603 605 605 606

584

13 Fatigue of Fiber Composites

List of Symbols and Abbreviations a, c A, m E fc (c) Fc (c) ge (9) Gd {9) K Kc Xj Kth / N JVf V{

crack length constants Young's modulus density function for crack length cumulative distribution function for crack length density function for crack orientation cumulative distribution function for crack orientation stress intensity factor stress intensity factor, critical value stress intensity factor, mode I stress intensity factor, threshold value length of sampling line number of cycles number of cycles to failure volume fraction of fibers

8C gm ed b £f L e max 9 a amax (T U T S

composite failure strain fatigue limit of matrix fatigue limit, fiber debonding fatigue limit m a x i m u m strain crack orientation stress maximum stress ultimate tensile stress

CDS CSM SMC UTS

characteristic damage state chopped strand mats sheet molding compounds ultimate tensile strength

585

13.2 Short Fiber Composites

13.1 Introduction Fatigue of single-phase materials, e.g. metals and polymers, is a classical field. For fiber composites, i.e. composites in which fibers play a significant role in determining the properties, the process of fatigue is far from fully understood. Indeed, this field may be considered in its infancy since the basic concepts concerning how to characterize and describe the fatigue process are still being developed. It is often the case with fatigue work that test data are presented without satisfactory interpretations and at best some remarks are presented concerning fatigue life of a material being better or worse than another. Since for fiber composites numerous possibilities exist for combining the constituents, the need for rational interpretations of the basic fatigue characteristics is critical. This exposition will not attempt to conduct an exhaustive survey of fatigue of fiber composites. Rather, selected results will be presented and discussed to bring out the essential characteristics of the fatigue process in this class of materials. Since the fiber geometry has a major influence on the process the treatment will be divided into sections pertaining to short fiber composites, long and continuous fiber composites, laminates and woven fabric composites. Only polymeric matrices will be treated. The reader is referred to a number of reviews for supplementary information (Boiler, 1969; Owen, 1974; Hahn 1979; Stinchcomb and Reifsnider, 1979; Harris, 1986; Konur and Matthews, 1989).

polyester was reported by Owen and associates in the period 1967-79 which has been summarized in Owen (1982). The material consisted of 50 mm glass multifilament strands randomly arranged in a plane and surrounded by a polyester resin. Flat plates of the composite were made with wet lay-up technique. Fiber strands could be observed by optical microscopy with transmitted light in thin plates. Combining this with observations of polished edges of specimens the mechanisms of debonding and resin cracking were detected. Debonding of strands from matrix was the first mechanism observed in static as well as fatigue loading. The first group of strands to debond were those with their fiber axes at approximately normal to the applied load direction. With increased load level or number of cycles of a maximum load debonding of strands deviating from the normal to the load axis occurred. The debond cracks extended into resin rich areas causing resin cracking which grew with further load cycling. Figure 13-1 140 120 100 CO Q_

Separation

"o. 60

I 40

A substantial amount of work on fatigue of chopped strand mat reinforced

Debonding

20

0.1

13.2 Short Fiber Composites

Cracking v ^

1

10

102

103 104 Cycles

105

106

Figure 13-1. Stress amplitude versus cycles for polyester resin reinforced with chopped strand mats of glass indicating the onset of debonding and cracking under cyclic loading with zero mean stress (from Owen etal. 1969).

586

13 Fatigue of Fiber Composites

shows when debonding, cracking and separation of specimen occurred at various stress amplitudes in a cyclic loading with zero mean stress (Owen et al., 1969). It is noted that the stress threshold below which no debonding occurs may be usable as a conservative estimate of the fatigue limit. It was thought that improvement in the polyester resin ductility might delay the onset and development of fatigue damage in the reinforced resin. A flexibilizing agent was added to the resin to increase its strain to failure. The static strain to failure of the composite showed, however, only slight improvement, see Fig. 13-2 (Owen and Rose, 1970). The elastic modulus of the resin decreased with increasing flexibilizer content causing the composite modulus also to decrease. However, it turned out that the strain for onset of debonding remained unaffected at about 0.3% under static loading irrespective of the flexibilizer 100r

#10 CD

CO

C.S.M. and fabric laminates

0

20

40

Flexible resin addition (wt %)

Figure 13-2. Strain to failure of polyester resin with polypropylene maleate adipate flexibilizer and of its composites with chopped strand mats (CSM) and with fabric (from Owen and Rose, 1970).

content. More importantly, the stress-cycles plots for onset of debonding, e.g. that in Fig. 13-1, for all composites tested, with varying contents of flexibilizer in resin, superimposed on a single curve (or band) when strain was plotted instead of stress. Also, the threshold strain for debonding in fatigue, referred to as fatigue limit for transverse fiber debonding, came out to be a fairly constant value of 0.12%. As far as resin cracking is concerned, Owen and Rose (1970) found that no significant delay in its occurrence was observed with the addition of flexibilizer in resin. For high contents of the flexibilizer, however, it was found that cracks appeared at the ends of strands that were aligned with the loading axis slightly before initiation of transverse fiber debonding. An investigation of fatigue damage in sheet molding compounds (SMC) was reported by Wang et al. (1986) (see also Wang et al., 1987). The material investigated is denoted SMC-R50 and consists of chopped glass fibers of length 25.4 mm, calcium carbonate filler and polyester resin. The fibers are randomly oriented in the plane of the sheet and constitute 50 per cent by weight of the composite. The main difference from the material of the work described above is that the fibers are about half as short here and are more evenly spread rather than being clumped together as thick strands. The observations of damage were made by taking replicas of the specimen surface periodically during the cyclic tensile loading. The replicas were coated by sputtering and examined in a scanning electron microscope. Observations showed that in a resin-rich area microcracks formed nearly normal to the loading direction. In a fiber-dominant region with fibers aligned with the loading direction, cracks appeared between fibers,

587

13.2 Short Fiber Composites

also normal to the loading direction and were limited in length by the interfiber spacing. Debonding of fiber strands roughly normal to the loading axis was also seen. Typical micrographs showing the cracks are reproduced in Fig. 13-3. All crack types were grouped together for the purpose of statistical descriptions. This was achieved by a crack sampling procedure where cracks intersecting a straight line parallel to the loading axis were counted. The crack size and orientation were represented by the parameters c and 6, see Fig. 13-4. From the sampling statistics, distributions of the two parameters were derived assuming Weibull-type functions. The cumulative distribution function for crack length, denoted FC(Q, and its density function/c (C) are shown in Fig. 13-5 for a fatigue loading with the maximum stress amax at 60% of the ultimate tensile stress aUTS. For the same loading the cumulative distribution and density functions for crack orientation, Ge (6) and ge (9), respectively, are shown in Fig. 13-6. The main observations of interest concerning the microcrack geometry are that most cracks remain less than 1 mm in length during fatigue and that their orientations are mainly within 30° of the normal to the loading direction. The multitude of microcracks within the volume of a SMC degrade the elastic constants. Figure 13-7 shows the longitudinal Young's modulus versus the number of cycles applied. The predicted curve in the figure is calculated by a so-called self-consistent estimation scheme using the measured crack statistics (Wang et al., 1986). The preferential cracking resulting from the applied uniaxial loading leads to anisotropy in the initially isotropic SMC material. This necessitates calculation of more elastic constants than those needed to characterize the virgin state of the material.

(a)

(b)

Figure 13-3. Microcracking in a short fiber reinforced SMC-R50 composite, (a) Cracks in a resinrich region and (b) cracks in a fiber-dominant region with fibers oriented at an angle to the loading direction (from Wang et al. 1987).

ACT

ACT

Figure 13-4. (a) Specimen of a short fiber composite with a sampling line of length / parallel to the axial direction, (b) A crack intersecting the sampling line is described by length 2c and angle 6 (from Wang et al. 1987).

588

13 Fatigue of Fiber Composites

40

Fitted Exptl curves data

35

<W=0-6o" UTS

30 _

D

---

O A

25 -

o

Cycles

J.Q — Q — ^—D3-D

20 15

o

5000 Cycles 3000 1000 100

— —

10

C5)

O-C0D"O<3> O ' O O - - u u -

10 5

5000 1000 100 10

-o

.

w

I

l

1 2 Crack length (mm)

Crack length (mm)

Figure 13-5. The cumulative distribution function FG(c) and the density function fc(c) of crack length at various cycles with maximum stress
by weight of glass or graphite fibers. The matrix materials were engineering thermoplastics, including semicrystalline Nylon 66, amorphous polycarbonate, polysulfone, amorphous polyamideimide and semicrystalline polyphenylene sulfide. The fibers were reported to seldom exceed 1 mm in length and have typical length of 0.2 mm, giving a typical length to diameter aspect ratio of 20.

Studies related to fatigue of injection molded reinforced thermoplastics have been reported by Mandell et al. (1982) and (1985). These authors have studied growth of a precut crack under cyclic tensile loading. The plane of the precut crack was normal to the dominant fiber direction which was also the direction of loading. The materials tested were commercial injection molding compounds containing 30 to 40% 40 35

Fitted curves

I 30

Exptl data n 5000 Cycles o 1000

ou a

max=0-6auTs

40 ^ ^ ^

•g 2 5

E

CO

_^^K

"S 30 o

""* # '*s fc#

CO &

^15

Z

20

•••

— 5 0 0 0 Cycles 3000 1000

100 ^^w

10

">-v^

10 a> 10

5 TT/4

Crack orientation (rad)

TT/2

TT/4

Crack orientation (rad)

Figure 13-6. The cumulative distribution function G9 (6) and the density function ge (9) of crack orientation at various cycles with maximum stress on . of 60% of the ultimate tensile stress al]TS for a short fiber SMC-R50 composite (from Wang et al., 1987).

13.2 Short Fiber Composites

Predicted

growth with local failure zones ahead of the crack. The band of width (or height) h about the crack plane is suggested by the authors to be a characteristic of the material microstructure and independent of the loading mode. The authors further suggest that only processes within this band govern crack growth (i.e. a plastic zone-like concept). The crack growth rate is taken to be given by the Paris law:

—

Exptl data D O 102 103 104 Number of cycles

105

Figure 13-7. Change in the longitudinal Young's modulus of a short fiber SMC-R50 composite under fatigue at the maximum stress of 60% of the ultimate tensile stress (from Wang et al., 1987).

Optical and scanning electron microscopy were done to study the mode of crack growth. This mode was described as a fiber avoidance mode. Although differences were found from material to material, the dominant feature of the crack growth process was described as a main crack growing in a zigzag pattern avoiding cutting through fibers and instead finding local paths of lower resistance along the fibers and between the fibers. Ahead of the main crack tip local failure zones were found. These zones contained isolated crazes or isolated shear yielding. It was thought that these zones coalesce and merge into the main crack. The schematic in Fig. 13-8 depicts the fiber avoidance mode of crack Isolated yield zones or craze / cracks

___-Ar_

Fibers

589

Main crack tip zone

Figure 13-8. Schematic of crack zone in fiber avoidance mode of crack growth (from Mandell et al., 1985).

(13-1) where a is the crack length, N is the number of cycles, K is the stress intensity factor and A and m are constants. For a zigzagging crack the measurement of crack length is not simple. The zigzag pattern observed on the specimen surface does not necessarily hold through the specimen thickness. The authors give no details concerning this and it therefore appears that they use the projected length on the extended initial crack plane measured on the specimen surface. In any case, their crack growth data are reproduced in Figs. 13-9 and 13-10. It is noted that the crack growth exponent increases from the value of 4 for unreinforced matrix to 7 (for nylon 6,6) or 8 (for polysulfone). Assuming the Paris law to hold until the critical crack length to failure, the fatigue life can be obtained by integrating Eq. (13-1). The slope of the <7max — N{ line on a log-log scale would then be the inverse of the exponent in the Paris law. Figure 13-11 shows the Gmax — Nf data for unreinforced and reinforced polysulfone with the corresponding slopes indicated. The deviation at high stresses is attributed to general yielding of the materials. From the work of Mandell et al. (1982, 1985) summarized above it may be concluded that for sufficiently small fibers a

590

13 Fatigue of Fiber Composites

Kc (MNrrr3/2) -1 p Material a Unreinforced 4.1 A 4 0 % Glass 9.3 -2 O40% Carbon 10.9 o -3

E -4

-5

-6

-7

o

? -0.6

-0.4

0 0.2 0.4 0.6 Log (maximum Klt MNnr3/2)

-0.2

crack emanating in a resin-rich area may grow under cyclic loading, avoiding fibers in its path by zigzagging around them, until a critical size is attained beyond which further growth is unstable and failure reKc (MNnr 3 / 2 )

Material

D Unreinforced A 4 0 % Glass o 4 0 % Carbon

K th (MNrrr 3/2 )

1.37 2.35 2.70

5.82 11.7 17.3*

/

A

Some non colinear crack growth

A/

7

• a

E

CD •D

s

AA /

-6

m =7

o

-7

? , -0.2

0

0.2 0.4 0.6 0.8 Log (maximum Kx, MNm372)

1.0

Figure 13-10. Fatigue crack growth data for unreinforced and injection molded fiber reinforced nylon 6,6 (from Mandell et al., 1985).

1,2

0.8

1.0

Figure 13-9. Fatigue crack growth data for unreinforced and injection molded fiber reinforced polysulfone (from Mandell et al., 1985).

suits. The question remains as to what role the fibers play other than being in the way of a growing matrix crack and thereby slowing down its rate of growth. Based on some fracture toughness considerations Mandell et al. (1985) suggest that the ultimate tensile strength (UTS) of a composite, determined by the fiber strength, geometry and volume fraction, determine also the fatigue strength. The schematic of relationship between fatigue crack growth and fatigue life, suggested by Mandell et al. (1985), is reproduced in Fig. 13-12. The difficulties related to characterization of fatigue crack growth data in composites are illustrated by a systemic investigation on composites with spherical fillers reported by Gadkaree and Salee (1983). They studied fatigue crack growth in cantilever beam specimens of a thermoplastic bisphenol A-terephalate/isophthalate copolyester filled with a fly-ash filler with mean diameter of 20 jim. Standard fracture mechanics techniques of compliance calibration of crack length were used. Tests were carried out with composites

591

13.2 Short Fiber Composites 2.5

£ 2.0^

W

40 % Carbon 40 % Glass 20 % Glass

OO,

Yielding Fracture

1.5 Figure 13-11. Gm&x-Nf data

1.0

for unnotched polysulfone in unreinforced and various injected molded fiber reinforced conditions (from Mandell et al., 1985).

3 4 5 Log (Cycles to fail, Nf)

containing 10, 20, 30 and 40% by weight of filler. The crack growth data for unfilled resin are shown in Fig. 13-13 and the Paris law constants derived from best-fit are also indicated. Note that the exponent value is

fairly close to the value of 4 generally reported for polymers. The data for composite with 10% by weight of filler are shown in Fig. 13-14. The exponent here takes the value 8.78, which is not far from the values of 7 and 8 found by Mandell etal. (1985) for their injection molded composites. The data for composites with

Glass Graphite i

•a

m= 8

3

-2.60

^~

r

Corr.coeff .=• 0.9698 JO

=A(AK) f t

— -2.80 - d / V

y

o Log /Cmax

-3.00 - -3.20

o^o 7

3

^

7°

-3.40 -

/1

-3.60 -

LogA/f

Figure 13-12. Schematic of relationship between fatigue crack growth andtfmax—N{behavior of injection molded fiber reinforced composites (from Mandell etal., 1985).

-3.80 2.30

5.57108 • 10' 14 3.97990

/

i

2.50 LogAK

2.70

Figure 13-13. Fatigue crack growth data for unfilled resin (from Gadkaree and Salee, 1983).

592

13 Fatigue of Fiber Composites 4 2 - = 2.42795x10" 2 4 (A/<) 8 ' 7 7 9 4 5 dA/

-3.60

Corr. coeff.= 0.976

-3.50

u

o/ / /o

oo

B " 4 - 10

a) -3.70

3 -4.60 -3.90 -5.10 / -5.60 2.0

2.1

2.2 LogAK

-4.10 2.20

2.3

Figure 13-14. Fatigue crack growth data for resin with 10% by weight filler (from Gadkaree and Salee, 1983).

20 and 30% by weight of filler are shown next in Figs. 13-15 and 13-16. The data here refuse to be characterized by the Paris law. Finally a comparison of the data for unfilled resin and the composite with 40% by weight of filler are shown in Fig. 13-17. Once again, the composite data show lower growth rates but cannot be described by the Paris law. Apparently, there are certain mean interfiber or interparticle spacings beyond which crack growth becomes erratic and cannot be described by the stress intensity parameter.

2.22

2.24 LogAK

2.25

2.28

Figure 13-15. Fatigue crack growth data for resin with 20% by weight filler (from Gadkaree and Salee, 1983). -3.40 r" o

o

-3.80 £ -4.20 3 -4.60

_ \

°.oo

•v °\ .'/

-5.00 I-

-5.40 2.20

2.25

2.30 LogAK

i

I

2.35

2.40

Figure 13-16. Fatigue crack growth data for resin with 30% by weight filler (from Gadkaree and Salee, 1983). -2.60

13.3 Long and Continuous Fiber Composites

o .o>°

-3.10

cf°

Unfilled

B -3.60 -o o ^oo o

13.3.1 Unidirectional Composites In this section composites with aligned fibers only will be considered. Laminates with plies of aligned fibers and woven fabric composites will be treated in the subsequent sections. Mechanisms of fatigue damage in aligned fiber composites depend significantly on the mode of loading, i.e. whether

O)

o

-3 -4.10

/ Filled

-4.60

-5.10

2.38

i

2.48

2.58

2.68

Log A K

Figure 13-17. Comparison of fatigue crack growth data for unfilled resin and for resin with 40% by weight filler (from Gadkaree and Salee, 1983).

13.3 Long and Continuous Fiber Composites

it is tensile, compressive, shear or combinations of these. The mechanisms are usually first studied in individual loading modes and effects on these due to presence of the other loading modes are then investigated. Due to difficulties involved in fatigue testing, most studies reported have been for uniaxial tensile loading. The testing techniques for other loading modes are still being developed. The mechanisms of damage will be discussed separately for tensile loading parallel to and inclined to the fiber axis. 13.3.1.1 Loading Parallel to Fibers

The mechanisms of damage may be separated as those governed by fibers and those determined by the matrix properties. Since fibers are usually much stronger than matrix fiber damage occurs at high load levels, restricted mainly within the scatterband of the composite failure load. At first application of the maximum load, fibers fail at weak points which are dispersed throughout the volume of composite. An individual fiber failure leads also to failure of the fiber/matrix interface. This interface is usually strong in polymeric matrix composites and the resulting debond zone is therefore small. The cavity-like zone created by a fiber failure and the associated debond raises the local stress on the neighboring fibers, one or more of which will also fail if the local stress exceeds the strength. In a perfectly brittle composite the local stress redistributions caused by fiber and interfacial failures at the first load maximum in a cyclic load would be achieved instantaneously and would remain unchanged at subsequent load applications, assuming no frictional energy dissipation to take place. In principle, therefore, no fatigue failure is possible in perfectly brittle composites with no friction

593

mechanism. However, neat polymeric resins are ductile and although their ductility is subdued by stiff fibers under loading parallel to fibers, some local flow around broken fibers would occur. This flow would allow changes in the local stress state with repeated load application. Thus, failure of more fibers in the vicinity of fibers broken in the first load application would occur. For relatively brittle thermosetting resins, e.g. polyester and epoxy, the very limited local flow combined with the statistical nature of fiber failure suggests that the fiber failure process may, for all practical purposes, be assumed non-progressive. In other words, a given site of broken fiber would not nucleate more broken fibers at a well-defined rate (per cycles of load). Instead, failure would occur at random from a broken fiber site which happens to find sufficient weak fibers in its neighborhood to form a core of fiber failures large enough to grow unstably at the applied maximum load (see Fig. 13-18). Let us now consider fatigue damage at cyclic loads with a maximum load at sufficiently low value to cause essentially no failure of fibers. The fibers would now cyclically deform and the matrix would be cycled between the strain limits allowed by the fibers. This would result in a straincontrolled fatigue of the matrix. The polymeric matrix would initiate cracks if the strain range is sufficient for the purpose. If cracks initiate, they would grow until the crack fronts encounter fibers (see Fig. 1319). The strain in the composite at the maximum cyclic load below which the matrix cracks remain arrested by fibers would then be the fatigue limit of the composite. If the maximum cyclic load subjects the matrix to higher strain than the fatigue limit, the matrix cracks will either fail fibers at their fronts or bypass fibers by

594

13 Fatigue of Fiber Composites N =1

N =Ni

N =Nf

Jxrndii Figure 13-18. A schematic describing the non-progressive fiber failure in fatigue of unidirectional composites under loading parallel to fibers.

debonding the interfaces, in which case, fibers could fail by pullout from the matrix. A matrix crack is shown schematically during its growth process in Fig. 13-20. The final failure (separation) is expected to result from unstable growth of a matrix crack. The mechanisms of fatigue damage described above are based on logical deductions rather than direct observations. Techniques of direct observations, e.g. microscopy, are less easily performed on polymeric matrix composites than on metals and the observations are often intractable due to the complexity of the operative mechanisms. To facilitate interpretation of the fatigue behavior, therefore, a

conceptual framework in the form of a fatigue life diagram has been proposed (Talreja, 1981). The diagram for tension-tension loading of unidirectional fiber composites is shown in Fig. 13-21. The axes of the diagram are strain and logarithm of the load cycles to failure. Although fatigue testing is done under controlled load, the variable on the vertical axis of the diagram is the maximum strain attained in the first load cycle. The significance of the maximum strain is that this quantity represents the state of damage reached in the first load cycle and it is reasonable to expect that any progression of damage in subsequent load cycles will be determined by this state of damage. Furthermore, the two

o O ( o o o c G O o c# ( fO (w? o o o \o c o c I/O o o §1 D O o

i ki

W°

U

Matrix cracks

Figure 13-19. Matrix cracking in unidirectional composites arrested by fibers such that the crack planes are confined to resin-rich areas.

13.3 Long and Continuous Fiber Composites

I

1 1

11i 1 I n

;•

Ii

Pi

i

595

\ Crack front (A/ = A / f )

Crack front (N =A/ 1 )

P

Fiber - bridged matrix crack

extreme states in fatigue, i.e. the static failure and the fatigue limit, are given generically in terms of strain. The static failure occurs at the strain to failure of fibers irrespective of the fiber volume fraction and the fatigue limit is governed by the matrix which is undergoing strain-controlled fatigue, as noted above. The scatter-band located horizontally about the composite failure strain represents the assumed non-progressive fiber failure mechanism likely to occur in composites with thermosetting resins, as discussed above. The sloping scatter-band stands for the progressive cracking of matrix and the associated fiber/matrix interfacial failure. The role of fibers in this mechanism is to suppress the rate of matrix crack growth by bridging the crack planes. The location of the scatter-band, i.e., where it deviates from the fiber failure scatter band, is governed by the effectiveness of the fiber bridging mechanism. The lower bound to the fatigue life diagram is the fatigue limit, which, as discussed above, is a matrix property affected by the geometric configuration in which the fibers are distributed in the cross-sectional plane. Experimental data (for a glass-epoxy composite) seem to indicate that the fatigue limit of composite is practically the same as that of the neat resin (Dharan, 1975). In any case, the fatigue

Figure 13-20. Matrix cracking in unidirectional composites aided by fiber/matrix debonding, fiber pullout and fiber failure.

limit of the neat resin would be the lowest limit and therefore it may be useful to operate with this quantity for design purposes. In order to appreciate the usefulness of the fatigue life diagram consider two unidirectional composites with the same matrix but different fibers. Let the fibers be of relatively low stiffness in one case and of relatively high stiffness in the other. Figure 13-22 illustrates schematically stress-strain curves for the two cases. The composite failure strain, denoted by ec and given by fiber failure strain, is different in the two cases while the fatigue limit, denoted by Fibre breakage, interfacial debonding

Matrix cracking, interfacial shear failure

Y///////////////////////A V

log/V

Figure 13-21. Fatigue life diagram of unidirectional composites in tension-tension fatigue under loading parallel to fibers (from Talreja 1981).

596

13 Fatigue of Fiber Composites

Fibre

I

(a) Figure 13-22. Stress-strain characteristics of unidirectional composites: (a) low stiffness fibers, (b) high stiffness fibers.

8m, would remain the same in the two cases if it is assumed to be equal to the neat resin fatigue limit. The two strain values, am and ec, are further apart in the first case than in the second. Indeed, for fibers of very high stiffness the composite failure strain may even be less than the fatigue limit strain. To illustrate the significance of fiber stiffness (and the resulting failure strain) on fatigue behavior, data for a glass-epoxy composite are shown in Fig. 13-23. Note first that fatigue lives for composites of three different volume fractions fall within the same scatter-band, while three different and separate scatter-bands result when the usual stress-life plots are made (Dharan, 1975). The fatigue limit of neat resin was found by Dharan (1975) to be at 0.6%

Figure 13-23. Fatigue life diagram of glass-epoxy under loading parallel to fibers; data from Dharan (1975).

strain, which is the value set in the fatigue life diagram. Remarkably different behavior is displayed by a graphite-epoxy composite whose strain to failure is only about 0.5%. Assuming that the fatigue limit of epoxy is at least 0.6% strain, the fatigue life diagram takes the shape shown in Fig. 13-24. The sloping scatter-band of progressive matrix and interfacial damage disappears, as the fatigue limit of the resin becomes unreachable. Thus, a logical interpretation of the fatigue damage tolerance of ultra-high modulus graphite fiber composites emerges. Figure 13-25 shows the fatigue life diagram of a high modulus graphite-epoxy composite. The horizontal scatter-band (limited by 5% and 95% failure probability levels) is unusually wide here, presumably due to poor quality of fibers and/ or composite, and the fatigue limit is once again set at 0.6% for epoxy. The apparent fatigue damage tolerance may now be interpreted differently from the behavior displayed by data in Fig. 13-24. Data for a lower modulus graphite/epoxy composite are shown plotted in the fatigue life diagram, Fig. 13-26. Note here that the sloping scatter-band is further out to the right than that for glass-epoxy composites, Fig. 13-23. This is due to more effective fiber bridging action by graphite fibers than is possible by the less stiff glass fibers.

13.3 Long and Continuous Fiber Composites

597

13.3.1.2 Loading Inclined to Fibers

0.006

•->

;

T

•

>v k • •

•

0.004

0.002 0

4

8

log N

Figure 13-24. Fatigue life diagram of an ultra-high modulus graphite-epoxy under loading parallel to fibers; data from Sturgeon (1973).

0.010

0.006

y/////////////////////////

0.002 0

4 log N

Figure 13-25. Fatigue life diagram of high modulus graphite-epoxy under loading parallel to fibers; data from Awerbuch and Hahn (1977).

When loading is not parallel tofibersthe fiber/matrix interface is subjected to normal and shear stresses. The interfacial region (interphase) initiates cracks under these stresses which then grow on repeated load application aided by irreversible dissipative processes at crack fronts. The mixed-mode crack growth in interfacial region becomes the primary progressive fatigue damage mechanism. The role of fibers under loading inclined to fibers is only significant for small inclination angles (probably less than 5°). As the inclination angle increases the normal stress in fibers falls rapidly. At the same time, normal and shear stresses in the interfacial region increase. The openingmode contribution to the mixed-mode crack growth also increases with increasing inclination angle. Since the crack growth resistance is expectedly less for opening-mode growth, the threshold strain for crack growth will decrease with increasing inclination angle. Consequently, the fatigue limit will decrease as the inclination angle increases. The above-mentioned considerations are reflected in the schematic fatigue life diagram, Fig. 13-27. The fatigue life dia-

Figure 13-26. Fatigue life diagram of a lower modulus graphite-epoxy under loading parallel to fibers; data from Curtis (1987).

598

13 Fatigue of Fiber Composites

_I_I_I_I_I_I_I

(0°< 6 <90°)

Mixed-mode matrix and interfacial damage

Figure 13-27. Fatigue life diagram for unidirectional composites under loading inclined to fibers. Dottedline diagram is for loading parallel to fibers.

Transverse fibre debonding

gram for inclination angle 9 = 0°, i.e. for loading parallel to fibers, is shown as a reference by dashed lines. For an inclination angle of more than a few degrees, where fiber failure ceases to be a governing mechanism, the sloping scatter-band starts at the static failure strain and ends at the fatigue limit. The horizontal scatter-band of fiber failure is lost and, consequently, the mechanism for delaying the progressive damage is also not effective. In the limiting condition of inclination angle, 0 = 90°, the opening-mode crack growth, described as transverse fiber debonding, is the predominant mechanism of damage. Test data for a glass-epoxy unidirectional composite subjected to tension-tension fatigue at various inclination angles have been reported by Hashin and Rotem (1973). Their data, reported in terms of the maximum stress, have been replotted with maximum first-cycle strain in Fig. 13-28. The lines drawn through the data-points have been started at the static failure strain. The fatigue limit of epoxy, reported by Dharan (1975), is shown as a reference fatigue limit for 0 = 0°. The fatigue limit for transverse fiber debonding was reported by Owen and Rose (1970) for various

composites of reinforced polyester to be at 0.12% strain. The data for glass-epoxy shown in Fig. 13-28 suggest that this fatigue limit is approximately at the same value. In Fig. 13-29 the strains at fatigue limits for various inclination angles are plotted. The strain values are taken from the fa^/deg

0.008

°5

> = 5°)

+

•

0.006

10 30 60

/ / / / /

/ / /v / / / / / X

= 10°) 0.004

v O

cc(0 = 30°) 0.002 ec(d = 60°)

•^—A—_

/ / / / / / / / /, 1

0

1

2

1

1

4 log N

6

Figure 13-28. Fatigue life diagram of a unidirectional glass-epoxy under loading inclined to fibers; data from Hashin and Rotem (1973).

13.3 Long and Continuous Fiber Composites

0.006

0.004

0.002 -

30

60

90

0/deg

Figure 13-29. Variation of the fatigue limit ef#1# with the inclination angle, 6, based on data of Hashin and Rotem (1973).

599

In order to achieve rational and efficient design procedures for laminates under fatigue the first step is to generate basic understanding of the damage development process. In the following this aspect will be reviewed and the role of damage mechanisms will be interpreted with the aid of fatigue life diagrams. For the sake of clarity and systematic description, laminates of angle ply and cross ply types will be considered before discussing more general laminates. 13.3.2.1 Angle Ply Laminates

tigue life diagrams, Fig. 13-28. A smooth curve drawn through the points is made to pass through the reference values at the two extremes, i.e. 6 = 0° and 0 = 90°, at 0.6% and 0.12%, respectively. 13.3.2 Laminates A primary motivation for constructing laminates is to achieve desired combination and distribution of properties. As far as the elastic response is concerned, wellestablished plate theories exist for making assessment of the stiffness properties. A number of strength criteria have been proposed and, although a single criterion has not been found to be universally applicable, sufficient basis exists for evaluation, at least a preliminary one, for selection of laminate configurations. For long-term performance of laminates, in particular under cyclic loads, the situation is far from satisfactory. A selection of laminate configuration on the basis of fatigue performance cannot yet be made. The design approach, instead, has been to select laminate configurations on the basis of stiffness and strength considerations and then to assure that these properties do not degrade unacceptably under fatigue.

This class of laminates is widely used in filament wound structures. It is also suited for systematic studies of the so-called constraint effect in laminates. For illustration of this effect, consider the fatigue damage of unidirectional composites under loading inclined to fibers. As discussed above, the predominant damage mechanism here is the mixed-mode cracking in the fiber/ matrix interfacial region. Indeed, failure will occur when a single crack grown from a preexisting flaw reaches a critical size, at which stage further growth becomes unstable, see Fig. 13-30 a. Now consider a unidirectional ply placed in an angle ply laminate subjected to loading along the symmetry axis, see Fig. 13-30b. Under the effect of the stresses in the plane of a ply mixed-mode cracking will also occur here. However, growth of a crack here will be under constraint of the adjacent ply (or plies) and failure of the laminate will not depend on the length of the crack since the constraining ply (or plies) will locally (i.e. at crack fronts) carry the load shed by the cracking ply. If fibers in a constraining ply sustain the additional load, which acts over a distance from the crack planes, called the shear-lag distance, more cracks can form further away. This multiple

600

13 Fatigue of Fiber Composites

(a)

- - Initial flaw Fatigue crack

(b)

- Matrix crack Delamination

Figure 13-30. Comparison of fatigue damage mechanisms in unidirectional and angle ply laminates, (a) Cracking from initial flaw in a unidirectional ply under loading inclined to fibers, (b) Multiple matrix cracking in one ply of an angle ply laminate under the constraint of an adjacent ply of opposite orientation. The intraply cracks cause delaminations along the crack fronts.

cracking process has been treated by Aveston et al. (1971) and since then by many others. In angle ply laminates loaded along a symmetry axis the multiple cracking process takes place in plies of both orientations since the plies of one orientation apply constraint to the plies of the other ori-

entation and simultaneously crack under the constraint of plies of that orientation. Under cyclic tensile loads of a constant amplitude the multiple cracking process in angle ply laminate progresses causing increase of the crack number density. If no other damage mechanism is initiated the multiple intraply cracking process will stop when saturation spacings between cracks in plies of both orientations have been reached. However, the intraply crack fronts can locally debond the ply interfaces if the debond energy is exceeded by the local strain energy release rate. The delaminations thus caused (see Fig. 13-3Ob) can grow under cyclic loads. The stress states in plies in the delamination regions will then change and may cause fiber failures and thereby failure of laminate. The delaminations can also merge together by growth and cause laminate failure by separating plies from each other sufficiently such that linkage of intralaminar cracks throughout the thickness is achieved. The process of damage in angle ply laminates has not been quantitatively analyzed for predicting fatigue life. It can, however, be appreciated, from description of the damage process given above, that the constraint effect (determined by ply thickness, orientation and stiffness) governs initiation, progression and eventual saturation of intraply cracking. The associated local delamination process is also affected by the ply constraint since the surface displacements of the intraply cracks are subject to this constraint. For given ply material and stacking sequence the constraint effect can be made to vary by varying the off-axis angle of angle ply laminates. Fatigue life data for angle ply laminates of glass-epoxy with the off-axis angle (denoted 6 in Fig. 13-30) of various values between 30° and 60° were reported by Rotem and Hashin (1976). The fatigue limits tak-

13.3 Long and Continuous Fiber Composites

en from their data have been plotted as strains (by dividing their stress by their reported elastic moduli) and are shown in Fig. 13-31. Also shown in the figure are fatigue limit data, connected by dotted line, for unidirectional composites of same material under loading inclined to fibers. Comparison of the two sets of data gives a good illustration of the constraint effect discussed above. A significant improvement in the fatigue performance due to the constraint effect (or simply the fiber architecture) is thus displayed. (The nature of variation of the fatigue limit of angle ply laminates of glass-epoxy, shown in Fig. 13-31, has been found to be slightly different by a recent study, yet to be published. The improvement over unidirectional composites remains of about the same magnitude.)

0.006 -

0.004

0.002

30

60 0/deg

Figure 13-31. Variation of the fatigue limit s{l of an angle ply laminate of glass-epoxy. Fatigue limits are taken from the data reported by Rotem and Hashin (1976). The dotted line reproduces fatigue limit variation for unidirectional composites, Fig. 13-28.

13.3.2.2 Cross Ply Laminates This class of laminates, although of less practical application than the angle ply laminates, has been the subject of extensive analytical, numerical and experimental studies. The reason is that the multiple transverse cracking in these laminates is

601

constrained by longitudinal plies that remain uncracked in most cases on loading in longitudinal direction, while in angle ply laminates both sets of off-axis plies crack simultaneously. Thus, the multiple transverse cracking can be studied under the constant constraint of the longitudinal plies. The constraint can, however, only be varied by varying the ratio of thickness of the two sets of plies, if the same ply material is used throughout the thickness. The studies related to the cracking behavior of cross ply laminates have been concerned with three aspects: the initiation of cracking, also called the first ply failure, the development, i.e. multiplication, of cracking, and the changes (reduction) of stiffness properties due to cracking. A recent stateof-the-art review of this subject is given in Han and Hahn (1989). Most studies reviewed there have treated monotonic tensile stressing of cross ply laminates along longitudinal plies. A thorough study of tensile fatigue of cross ply laminates of graphite-epoxy has been reported by Jamison et al. (1984). The evolution of fatigue damage was described by separating it in three stages, the reflection of which was shown by the modulus reduction curve, Fig. 13-32, where the three stages are marked as I, II and III. Stage I consists of transverse crack formation which progresses with the cycles applied until a fairly constant spacing between cracks is attained. Figure 13-33 shows the transverse crack density with cycles and the associated reduction in the longitudinal Young's modulus of the laminate. Damage in Stage II is exemplified by an X-ray radiograph shown in Fig. 13-34. The horizontal lines are the transverse cracks in 90° plies and the vertical lines represent longitudinal cracks between fibers in 0° plies. These cracks initiate in Stage I but are fewer in number and short-

lus

602

0.99

o 0.97

13 Fatigue of Fiber Composites

M.

c 0.95 o 0.93 ized

§

0.91

P3

o

0.89 0.87

5

10

15 20 25 30 35 40 45 Thousands of cycles

Figure 13-32. Development of damage in (0, 902)s laminate of graphite-epoxy reflected as Stages I, II and III in the modulus reduction curve (from Jamison et al., 1984).

1.00 r Normalized secant modulus

Transverse crack density

0

10

cut at various places along a delamination it was found that a delamination was formed by diversion of a longitudinal crack into the interlaminar plane. Stage III of damage was found to be dominated by growth and coalescence of the delaminations and fiber failures in the delaminated regions. The delamination growth was primarily in the longitudinal direction. The coalescence of such delaminations formed strips of width equal to the spacing between adjacent longitudinal cracks. Failure of such strips by fiber breakage in them induced the laminate failure. It may be argued that if the maximum strain attained in the first application of a tensile load on a given cross ply laminate does not exceed the threshold strain for multiple transverse cracking, subsequent application of the same load will not cause fatigue failure (in, say, 106 cycles). Thus, the strain to onset of multiple cracking

-

20 30 40 Thousands of cycles

f:

Figure 13-33. Development of transverse crack density and modulus under fatigue of (0, 902)s graphiteepoxy laminate (from Jamison et al., 1984).

er in length. The initiation of these cracks is attributed to the tensile stress normal to fibers resulting from the difference of the Poisson's ratio in the two ply directions. Subsequent to the appearance of the transverse and longitudinal cracks delaminations of roughly elliptical shape were found located around the crossover points of the two sets of cracks. These delaminations appear as shadow-like regions in the X-ray radiographs, e.g. Fig. 13-34. By scanning electron microscopy of sections

Figure 13-34. X-ray radiograph showing transverse cracks (horizontal lines), longitudinal cracks (vertical lines) and local delaminations (shadow-like zones) in (0, 902)s graphite-epoxy laminate under fatigue, Stage II (from Jamison et ah, 1984).

13.3 Long and Continuous Fiber Composites

may be taken as a good estimate of the fatigue limit. The static (or one cycle) failure limit is given by the fiber failure strain, which is also the composite failure strain. A fatigue life diagram of a cross ply laminate will thus have these two strains (i.e. strain to multiple transverse cracking and strain to fiber failure) as the two limits. Figure 13-35 shows an example of the fatigue life behavior of a cross ply laminate. The data taken from Grimes (1977) are interpreted on the basis of the fatigue

Debonding in 90° plies.delamination

603

life diagram concept. The fatigue limit is placed at the strain of 0.43% which was the strain at which the plot of longitudinal stress versus transverse strain showed a proportionality limit indicating onset of (multiple) transverse cracking. A scatter band is placed horizontally about the static failure strain to indicate presence of the nonprogressive fiber failure mechanism discussed above in connection with tensile fatigue of unidirectional composites under loading parallel to fibers. In a cross ply laminate this mechanism will also exist due to presence of the 0° plies stressed parallel to fibers. 13.3.2.3 General Laminates

0.008

0.004

0

Figure 13-35. Fatigue life diagram for a cross ply laminate of graphite-epoxy; data from Grimes (1977). emc, the strain to multiple cracking, is the fatigue limit.

I - matrix cracking

Numerous laminate configurations can be obtained by combining the unidirectional, angle ply and cross ply arrangements. A number of these have been studied by various workers. A summary of an extensive range of investigations has been reported by Reifsnider et al. (1983). Figure 13-36 shows the schematic description of

Controlling damage modes during fatigue life 3 - delamination

2 - crack coupling - interfaciat debonding Percent of life

5 - fracture

4 - fiber breaking

100

Figure 13-36. Development of damage in composite laminates (from Reifsnider etal., 1983).

604

13 Fatigue of Fiber Composites

the controlling damage mechanisms suggested by them for a wide class of laminates. The initial stage of damage development consists of multiple matrix cracking along fibers in plies inclined to the applied load direction. The nature of this cracking process is as that described above for angle ply laminates and cross ply laminates. The saturation of the intraply cracking process in individual plies has been found to lead to a cracking state which is a generic property of the given laminate and is independent of the loading amplitude or in general the path of loading. This state has been called the characteristic damage state (CDS) and signifies termination of the initial stage of damage development. Further damage on continued cycling occurs by local debonding of interply bonds at the fronts of the intraply cracks. This causes delamination which grows in the interlaminar planes leading to coalescence of adjacent delamination zones and stress enhancement in the separated plies. This enhances fiber breakage and induces instability of the damage developments leading to final failure. The complexity of the damage development process has not yet allowed a quanti1.8

[

Graphite / epoxy

T 800 / 5245

1.5

1.2

•

•o c

D Unidirectional 0.9

tative description of the rates of the process. A pragmatic approach at present is therefore to search for trends in the fatigue lives induced by changes of ply orientations from a basic configuration. As an example of this consider three laminate configurations: a undirectional, (0, ±45) s and (0, ±45, 90)s all of the same graphiteepoxy. Fatigue life data for these laminates obtained under tension-tension loading have been kindly given to the author by Dr. P. T. Curtis of Royal Aerospace Establishment, U.K. In Fig. 13-37 the data have been plotted together on axes of the maximum strain in first cycle and logarithm of the number of cycles to failure. The fatigue life diagram has been drawn in light of the data for the unidirectional composite. The usual characteristic features of the diagram discussed above are present. It is of interest to see that fatigue lives for (0, ±45) s and (0, ±45, 90)s laminates fall within the lower part of the sloping scatterband of the unidirectional composite. This suggests that at relatively low strains the progressive damage mechanism of matrix cracking and the associated interfacial failure in unidirectional composites dominate and grovern the fatigue life. At higher strains the deviation in fatigue lives of the two laminates from the sloping scatterband of the unidirectional composite may now be interpreted as due to the delamination growth in the laminates causing enhanced stresses in the 0° plies and the consequent lower fatigue lives.

o (0,±45)s • (0, ±45,90 )s

0.6 log N

Figure 13-37. Fatigue data (P.T. Curtis, Royal Aerospace Establishment, U.K.) of three graphite-epoxy composites. The drawn fatigue life diagram is for unidirectional composite only.

605

13.4 Conclusion

13.3.3 Woven Fabric Composites

Woven fabric composites are a useful class of composites in particular for structures with large thickness. Little work on fatigue behavior of these composites has, however, been reported. A systematic study has been conducted by Schulte et al. (1987), which will be summarized below. Schulte et al. (1987) tested 8-harness satin weave of graphite fibers impregnated by an epoxy resin. Laminates of this material were made by stacking the prepregs and curing them. For comparison continuous fiber cross ply laminates of the same material were made. Development of damage in the laminates under tension-tension fatigue was studied by conducting X-ray radiography intermittently during fatigue and by post-failure scanning electron microscopy of the interlaminar regions. The development of damage in the initial stage showed basically the same features as have been observed in cross ply laminates. Cracks were found to initiate in the warp (transverse to loading) direction along the fibers. On continued loading longitudinal cracks appeared between fibers in the fill direction in the undulation regions where the fill fibers cross over the warp fibers. The crack densities increased with load cycling and a uniformly distributed pattern of orthogonal cracks appeared to develop. On further cycling delaminations were observed confined primarily to the undulation regions. Thus a uniform distribution of delamination areas in the interlaminar planes developed. Toward the end of fatigue life fiber bundles were found to fail in the undulation regions. A progressive weakening of the laminate strength then led to final failure. Figure 13-38 shows the Young's modulus change E/Eo with fatigue life for the woven fabric composite. Data for the

o Continuous fibre

1.0

N f = 799700 Cyc UJ

Matrix cracking

'O-O-O

I

• 8 - harness satin fabric \

0.8

N f = 763475 Cyc Delamination at interlaced regions

55 0.7

20

40 60 Fatigue life, %

80

100

Figure 13-38. Change in the longitudinal Young's modulus of a woven fabric composite and a cross ply laminate of graphite-epoxy under fatigue; data from Schulte etal. (1987).

straight fiber cross ply laminate are also plotted in the figure for comparison. The differences in the two systems appear beyond about 50% of fatigue life suggesting that the delaminations at the undulating regions in the woven fabric laminate are responsible for the extra reduction of stiffness. The fatigue limit of the woven fabric laminate was found at 0.62% strain as against the value of 0.85% for the straight fiber cross ply laminate. Although the reasons for this difference have not been explored, it is likely that the cause of this may lie in the reduced constraint to transverse cracking provided by the curved fibers in the woven fabric composite.

13.4 Conclusion This chapter has been a selected exposition of the fatigue behavior of fiber reinforced composites. Of major focus have been the mechanisms of damage - multitudes of cracks of various size, shape and orientation - operating in composites with different fiber reinforcement configura-

606

13 Fatigue of Fiber Composites

tions. For short fiber composites matrix crack growth characteristics dependent on the fiber size have been described. The role of fiber/matrix debonding has also been discussed. For long and continuous fiber composites the fiber orientation with respect to the loading direction has been seen as the major factor governing initiation and growth of cracking. The role of fibers has been described as the primary loadbearing constituent and as a constraining element in growth of matrix cracks. The method of plotting fatigue life diagrams as a means of interpreting the governing mechanisms of fatigue failure has been discussed. Various examples have been given that elucidate the use of these diagrams. A particular detail of practical use is the fatigue limit whose variation with fiber orientation and configuration has been described. A large emphasis has recently been placed on toughened polymers, e.g. modified epoxies and some thermoplastics. Fatigue behavior of composites of these polymers has not been treated here. Many interesting features have already emerged and more are being reported as work on these composites continues.

13.5 References Aveston, I , Cooper, G. A., Kelly, A. (1971), in: Conference on the Properties of Fibre Composites, National Physical Laboratory. Guildford, Surrey: IPC Science and Technology Press, pp. 15-26. Boiler, K.H. (1969), in: Composite Materials: Testing and Design, ASTM STP 460, Philadelphia: American Society for Testing and Materials, pp. 217235. Dharan, C.K.H. (1975), in: Fatigue of Composite Materials, ASTM STP 569, Philadelphia: American Society for Testing and Materials, pp. 171188. Gadkaree, K. P., Salee, G. (1983), Polymer Composite 4, 19.

Grimes, G. C. (1977), in: Composite Materials: Testing and Design, ASTM STP 617. Philadelphia: American Society for Testing and Materials, pp. 106-119. Hahn, H. T. (1979), in: Composite Materials: Testing and Design, ASTM STP 674: Tsai, S.W. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 383-417. Han, Y.-M., Hahn, H.T. (1989), Ply Cracking in Composite Laminates, CMTC-8937. University Park, Pennsylvania: Composite Manufacturing Technology Center. Harris, B. (1986), Engineering Composite Materials. London: Institute of Metals. Hashin, Z., Rotem, A. (1973), J. Comp. Mats. 7, 448. Jamison, R. D., Schulte, K., Reifsnider, K. L., Stinchcomb, W.W. (1984), in: Effects of Defects in Composite Materials, ASTM STP 836. Philadelphia: American Society for Testing and Materials, pp. 21-55. Konur, O., Matthews, EL. (1989), Composites 20, 317. Mandell, I E , Huang, D.D., McGarry, F.J. (1982), in: Short Fiber Reinforced Composite Materials, ASTM STP 772: Sanders, B.A. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 3-32. Mandell, IF., McGarry, E l , Li, C.G. (1985), in: High Modulus Fiber Composites in Ground Transportation and High Volume Application, ASTM STP 873: Wilson, D. W. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 36-50. Owen, M.I (1974), in: Composite Materials, Volume 5, Fatigue and Fracture: Broutman, L.I (Ed.). New York: Academic Press, pp. 341-369. Owen, M.I (1982), in: Short Fiber Reinforced Composite Materials, ASTM STP 772: Sanders, B.A. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 64-84. Owen, M.I, Rose, R.G. (1970), Modem Plastics 47, 130. Owen, M.J., Smith, T.R., Dukes, R. (1969), Plastics and Polymers 37, 227. Reifsnider, K. L., Henneke, E. G., Stinchcomb, W. W., Duke, I C . (1983), in: Mechanics of Composite Materials, Recent Advances: Hashin, Z., Herakovich, C.T. (Eds.). New York: Pergamon, pp. 399-420. Rotem, A., Hashin, Z. (1976), AIAA J. 14, 868. Schulte, K., Reese, E., Chou, T.-W. (1987), in: Sixth International Conference on Composite Materials, Second European Conference on Composite Materials, Volume 4: Matthews, F.L., Buskell, N.C.R., Hodgkinson, IM., Morton, I (Eds.). London: Elsevier, pp. 89-99. Stinchcomb, W.W, Reifsnider, K.L. (1979), in: Fatigue Mechanisms: ASTM STP 675: Fong, IT. (Ed.). Philadelphia: American Society for Testing and Materials, pp. 762-787. Talreja, R. (1981), Proc. R. Soc. Lond. A378, 461.

13.5 References

Wang, S.S., Chim, E.S.-ML, Suemasu, H. (1986), J. Appl. Mech. 53, 339. Wang, S.S., Suemasu, H., Chim, E.S.M. (1987), /. Comp. Mats. 21, 1084.

General Reading Fong, J. T. (Ed.) (1979), Fatigue Mechanisms, ASTM STP 675. Philadelphia: American Society for Testing and Materials.

607

Hahn, H. T. (Ed.) (1986), Composite Materials: Fatigue and Fracture, ASTM STP 907. Philadelphia: American Society for Testing and Materials. Lagace, P. A. (Ed.) (1989), Composite Materials: Fatigue and Fracture, 2nd Conf, ASTM STP 1012. Philadelphia: American Society for Testing and Materials. Lauraitis, K. N. (Ed.) (1981), Fatigue of Fibrous Composite Materials, ASTM STP 723. Philadelphia: American Society for Testing and Materials. Reifsnider, K. L. (Ed.) (1991), Fatigue of Composite Materials, in: Composite Materials Series, Vol. 4: Pipes, R. B. (Ed.). Amsterdam: Elsevier. Talreja, R. (1987), Fatigue of Composite Materials. Lancaster, PA: Technomic Publishing Co.