COMPOSITE MATERIAL MECHANICS: THERMOELASTIC MICROMECHANICS*

COMPOSITE MATERIAL MECHANICS: THERMOELASTIC MICROMECHANICS*

Charles W. Bert

School of Aerospace, hlechaniral and Niirlear Engineering University of Oklahoma Norman, Oklahoma 73069

Philip H. Francis?

Department of Mnterials Sciences Southwest Reseaich Institute San Antonio, Texas 78284

ABSTRACT

This paper is a survey of the principal analytical and experimental develop ments in tlie field of composite material mechanics. Emphasis is placed on filamentary-type composite materials. Topics covered are (1) elastic micromechanics, including a great variety of analytical approaches to the problem, and (2) thermal and otlier transport phenomena, including a number of useful analogies for such phenomena. The surrey concludes with a brief discussion of future trends in the field.

INTRODUCTION

The objective of this survey is to review briefly some of the principal analytical and experimental developments in the field of composite material mechanics. Here, a composite material is defined as a solid substance consisting of two or more distinct phases or constituent materials.

The reason for using any composite material is to combine the advantages of two or more constituent materials. Composite materials are not new, since they have existed in nature for many millennia. For example, wood has hard and soft regions oriented in certain ways; thus, it is a composite material with different properties in different directions. Other composite materials found in nature are the structure of teeth, bones, and blood vessels; bird feathers; leaves of plants. All of these are rather efficient structures for their intended uses.

Man-made composite materials are not really new either. For example, the Egyptians used laminated wood in approximately 1500 B.C., and in their exodus from Egypt, the Israelites used straw in the manufacture of bricks.$ Other early examples include Mongolian bows made of wood, animal tendons, and silk bonded together; medieval laminated armor; Japanese Samurai swords; and Damascan layered gun barrels. More recent examples in common use include ordinary con-

* Submitted May 7,1974; accepted June 17,1974. ?Work supported, in part, by U.S. Air Force Office of Scientific Research, Contract

$The chopped straw aided in drying the interior regions and in distributing the F44620-7 1 -C-0017.

drying-induced cracks more evenly.

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664

Crete (cement plu\ aggregate), \teel-reinforced concrete, plaster (which contains hairlike “si7ing”), met;illic alloys. paper laminates, plywood, carbon black-filled rubber, cord-reinforced ruhber, wire-wrapped high-pressure ront;iiners. arid fiberglass (glass-fil,er-reinforcetl plastic).

T h e two constituent materials of a two-phase composite m;iteri;il are usually known as the Feinforcernent, which is the stiffer material, and the other material, which is known as the rnntr ix . T h e geometric configurations of the reinforcement may take many forms, such as: (1) spheroitlal pellets, (2) platelets or flakes, (3) short staples or chopped fibers (oriented either randomly or unidirectionally). (4) woven fahric, (5) continuous unidirectional filaments. For nonstructural ap. plications all of these geometrical configurations are satisfactory: however, in structural applications, major interest is directed toward the last two configurations antl in compofites consisting of laminates (multiple layers bonded together) of these. Both these last two configurations consist of more or less continuous filaments or fibers.

By far the most common filament material in terms of current usage is glass, fol- lowed by carhon (or graphite), boron, and polymer fibers. Technical interest in metallic wire (steel or beryllium) is waning at present, but there is some interest in tungsten wire for elevated temperature applications. T h e use of fiber-reinforced polymers has expanded rapidly in the past decade. These materials have become popular for structural applications because they are much stronger and less brittle than their constituent materials in hulk form. Another advantage is that in fibrous form, the composite material can be rktignerl to have tlirectionally dependent m a terial properties. (Such a material is said to he nnislropic on a macroscopic basis, as opposed to most monolithic structural materials, which are essentially isotropic, i.e., their properties have no dependence on directional orientation.)

T h e function of the matrix material is to maintain the reinforcements in the desired spacing and directional orientation, provide a path of load transfer across noncontiguous filaments, and serve as a pressure sealant. Until recently, the most popular matrix materials have been nonmetallic, specially plastics (polymers). Polyesters are least expensive; epoxy resins are widely used for high-performance structures not required to operate much above room temperature. Considerable research attention is being devoted to the development of polymers less susceptible to high temperatures, such as the newly developed polyimide. For still higher temperatures, either ceramic or metallic matrix materials are required. I n the latter category, aluminum-matrix and titanium-matrix composites are currently undergoing intensive research,

T h e primary difference between structural design for filamentary composites iIIld for ordinary (monolithic, isotropic) metals is that the former are macroscopically anisotropic while the latter are essentially isotropic. Although this anisotropic behavior complicates structural analysis, i t can be used to great benefit in obtaining structures that are highly efficient on a strength-weight basis.

For the case of composites with nonmetallic fibers and matrix, there arise certain advmtages due to the nonmetallic nature of the entire material system, viz. low density, electromagnetic transparency, resistance to corrosion, and ease of m;inufiictiire and repair. However, these composites also have certain disadvan- tages, such as reduced resistance to creep (even at ordinary temperatures), low strength (especi;illy in shear antl bearing) a t elevated temperatures, susceptibility to weathering i I 1 I d rain erosion, and different properties in compression than in tension.

Numerous hooks devoted entirely to composite materials in general or composite-material mechanics have iippearetl since 1965. Those having a substantial

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Bert b Fiancia: Composite Mateiial Jlechanics 665

meclianic~ coiiteiit, rather than purely descriptive material, include REFERENCES

1-19.3 l’ievious w r t eys on micromechanics in particular include those of REFER- EXCES 20-22.

ELASTIC MICROMECHANICS

App1ic:itioris of micromecliaiiical elasticity approaches, in which detailed rein- forccnieii t-matrix interactions are accounted for, have been directed at two basic probki i i~. 0 1 1 the one liand, such approaches have been used to gain a thorougli knowledge of composite yield and failure behavior, where microstructural damage due to severe local stress concentrations controls the composite behavior. Our dis- cussioii of this problem area is deferred to a sequel to this paper, in which fracture plienoniena are discussed. T h e second basic concern of the elastic micromeclianical approach is that of predicting tlie effective elastic properties of the macroscopic continuum from some knowledge of tlie properties of its constituent materials. This iispect will he surveyed here. T h e emphasis on elastic behavior arises both from its importance in teclinological applications and because relatively little progress has been made in solving micromechanics problems under more general material behavior assumptions.

While various approaches have been used to predict tliermomeclianical properties. most emlmce a common set ol assirniptioils regarding constituent behavior. T h e ni;itroscopic properties are ;issumetl to be linearly elastic, homogeneous, and :inisotropic (u~uiilly orthotropic). ’llie constituent$ are corisidered homogeneous and elastic, 1,oidfree. and completely I)oiided. Initial stresses are not accounted for. Also, tlcpeiiding on the problem considered, some assumptions as to periodicity of coiistitirent geometry ;ire often invoked, e.g.. doubly periodic par;illel fibers.

Cli;iniis ;i i id Sendrckyj presented a critical review21 of the literature on elastic property determination (tlirough 1967). with reference to fiber reinforced composites.

l‘lic most elenieii tary method used to cnlculate effective elastic properties is /w//in,q u~rnlyi .>, i n which the films are assumed to provide all of the longitudinal stiffiicss and the matrix is assumed to provide tlie Poissoii effect and tlie transverse nornial i i i i d slieiir s.tiA’nesses.2:’ These assumptions have been found to be unreal- istic, and thus tlic I)retlictetl results can be Iiiglily inaccurate.

The rnechnnics-of-mntprinls nppronch was pioneered by Ekvall.*.* For the case 01 ihe longitudinal Young’s modulus (Ell) and major Poisson’s ratio (vI2), the resiilts iire:

(1) (2)

wlicre E,,, a n d f?,,,,, arc tlie longitutlinal Young’s moduli of the respective fiber and matrix materials. I’, ;ind V,,, are the fiber and matrix volume fractions, and vfLrv and V,,,,,T are the filler and matrix Poisson’s ratios associated with contraction in the triinsverse direction resulting from loading in the longitudinal direction. EQUATIOM 1 and 2 are often called Voigt estimates, in honor of the famous 19th- century crystallographic physicist. They are also examples of what is generally known ;IS the “rule of mixtures.” T h e longitudinal elastic modulus Ell of the

f i l l = Err, f’, + E e L v,, v12 = Vf1.T ‘ f + VmLT ‘,it

5 No survey or list of references can anywhere nearly cover such a wide field as composite material micromechanics. The REFERENCES cited give only a glimpse of the extensive literature in this field. The authors apologize for not citing a number of important con- tributions to the field.

666 Transactions New York Academy of Sciences

a c t i i d composite i b never greater tlian the rule-of-mixtures estimate, whereas the actual value 01 v,2 may be oil either side of the estimate from E Q U A ~ I O S 2. EQUA- TIONS 1 and !? Iiitve I)een found to he generally i n good agreement with measured results, except wlieii consit1eral)le fiber mis;tlignnieiit is present, at which time ;I

correction suggestetl by .l’sai2.5 may be used. Finally. i t should hc mentioned tha t E y U A T l O N s I iind 2 are independent of the cross-sectional shape of the fibers and of the class of arrity ( i t . , rect;ingul:ir, hexagonal, and $0 on), and hrnre can not account for detailed micromechatiical behavior.

Assuming. i t rrct;~ngulai~ fi1)c .r ci,o\s-srction;il ~ I i a l x . Kkv:iltzl ;ip1)lird nietlt:inics- of-materials concepts to arrivr a t an expression for the transverse Young’s modulus EZ2. Making the following assitmptions rhat are typical of advanced filamentary

An expression such as EQUATION 4 is often referred to as a Keuss-type estimate, in honor of the early 20th-century mechanician. Except for the presence of the factor (1 - vznlLT), EQUATION 4 can be thought of as an “inverse rule of mixtures.”

In similar fashion Ekva1124 derived the following expression for the in-plane shear modulus GI2:

GI, = [(I’//G,) + (vm/Gm)l-l (5) where G, and Gr, are the fiber arid matrix shear moduli.

Unfortunately, EQUATIONS 4 and 5 do not give very good predictions of tlie properties of composites containing fibers of circular cross section. Thus, Ekvall assumed that EQUATIONS 4 and 5 hold for an elemental strip of fiber and matrix and then integrated over a typical repeating section of a composite containing a rectangular array of circular-cross-section fibers. This approach can be thought of as an improved mechanics-ol-maieria Is approach.

Another approach, which has a somewhat more rigorous appeal, is the so-called self-consistent model method. In this method a typical reinforcing element (inclu- sion or fiber) is regarded as imbedded in a second medium or a system of concen- tric media. Hill’Hv 27 considered the fiber to be imbedded in an infinite medium having the properties of the composite, and showed as a consequence that the effective composite moduli are related by simple expressions that are geometry- independent. Subsequently, Whitney and Riley28 modified Hill’s approach by treating the fiber as imbedded in an equivalent hollow cylinder of matrix material. KilChinskii29 and Hermanf” independently suggested a more refined model in which the fiber is imbedded in a hollow cylinder of matrix material, which in turn is surrounded by an infinite medium having the properties of the composite. A11 of these are variations of the basic theme of the self-consistent method, viz., to calculate the effective composite moduli from knowledge of tlie mechanical state, in both constituent phases, created by the external loading. Clearly, the self- consistent model method gives reasonable predictions for only low-to-moderate reinforcement volume fractions at which fiber-matrix interaction effects can be neglected.

The uariational approaches used in classic elasticity theory have been used to place upper and lower bounds on effective properties. The general approach is to

Bert k Francis: Composite Material Mechanics 667

define the effective moduli in terms of energy (the composite medium being macroscopically homogeneous), and then hound the energy for simple loading cases, thereby houiiding the efiective moduli. Hashin,*o. 31 a leading contributor to this school of thought, has succeeded in providing “best possible” upper and lower boii~ids on certain material systems. The minimum-complementary-energy theorem is used to establisli the lower bound, while tlie minimum-potential-energy theorem sets tlie upper bound. The bounds set, by these approaches converge at fiber volume fractions of 0 and 1 , but may be wide for moderate2to-high fractions of technological importance. Hasliin and Rosensl derived the effective elastic moduli for hexagonal fiber arrays of equal cross-section and for random arrays of fibers whose cross-sections may he unequal. The latter prob!em was treated by en- casing each fiber in the largest possible nonoverlapping equal circular cylinder and proceeding with the analysis of a circular fiber in a circular matrix. Use of this “composite cylinder assemhlage model” gave simple results for the moduli; in the case o f the random arriiy, the upper and lower hounds were equal for four of the five constitnts.

The srtnifmpiriral inrthod wliicli has been most widely used to date was developed liy 7‘s;ii.sj who developed a linear interpolation between the lower and upper bounds obtained variationally. It is necessary to have test data available in order to determine the interpolation coefficient, which Tsai called the “contiguity factor.” Iater, Halpin iind T s a 9 developed a more refined nonlinear interpolation.

hIuch attention has heen given to approaches that may be considered exact within t h r f,nmercmrk of classic clnstirify theory. With reference to fiber-reinforced composites. an idealized mediuni is conceived of parallel, doubly periodic fibers bound to a matrix in a fixed array (usually rectangular or hexagonal), undergoing plane strain. The governing equations and boundary conditions for the fiber and matrix media are formulated. and the stress and displacement fields are calculated by any of a number of standard techniques. Haener and Aslibaugli33 used Papko- vitdi stress functions and an iterative solution technique. Sendeckyj3‘ outlined a complex-variable approach for calculating the effective shear modulus. Adams and Doneflz, :a developed a finite-difference solution in conjunction with a rectangular filament array to deduce tlie effective moduli. A number of investigators hare used boundary collocation (point-matching) methods, i n particular boundary- point least squares (overdetermined collocation) for problems of circular fiber geometry.RT-‘2 There are a variety of ways in which the technique can be applied. For example, Heaton.38 considering circular-cross-section fibers, satisfied the fiber- matrix interfacial conditions exactly and used point-matching only on the bound- aries. In contrast, Leissa and colleagues,41 treating fibers of circular, elliptic, and square cross-sections, point-matched at the fiber-matrix interface as well as the boundary. Hulbert and Rybicki42 used this method to show that tlie “free-edge’’ effect is confined essentially to the single row of fibers adjacent to the free edge, interior rows being unaffected. Advances in finite element techniques have been used to advantage in exploiting the classical elasticity approach for solving more difficult boundary value problems.43-as The approach of Adams and Tsai** is particularly interesting because it accounts for pseudo-random fiber arrays and concludes that randomness leads one to different moduli estimates from those derived from idealizations based on doubly periodic fiber arrays.

Recently. ittempts have been niiide to apply elasticity formulations to account for microstructure to the determination of stress and effective moduli.ao--‘R Micro- stmctur;il ett’ects, such as couple stresses, may hiwe some importance in mechanical

668 Transactions Xew York Academy of Sciences

response consitterations for problem5 of stiff, closely spaced fibers imbedtled in a low-modiilus niatrix. althoiiglt this Itas not Ixen firmly eatablishetl.

Another line of investigation has concerned the effects of anisotropic constituents (fiber or niatrix) on eflerti\,c moduli. ?'here l ix , been evidence that graphite fibers, in particular, are highly anistropic. specifically planar tr;insversely isotropic (PTI) sucli that the plane of isotropy is the cross-sectional plane. Recently, this has ;ictually beeii nic;ciirrd directly by Sniitli:'!' Transversely isotropic fibers have been considered by various investigators. '1. *7'1-R.j Herman+'* considered the case of PTI niatrix as well ;is I'TI tihers. Hiffle.5'; treated the case of PTI fibers and :I matrix that is cylintlriciilly traiisversely isotropic, i t . . one Iiaving the same properties in tlte ax id : i i t d circurnfcreitti;il directions but ;I different set of properties in the radial tlirectioit. ?-his is characteristic of the matrix of carbon-carbon composites, in wliicli tlie matrix is formed by chemic;il vapor deposition.

T h e problem of dynamic microstructural response has proved to be a very difficult one, to wlticli an increasing amount of research is (Ievoted.si Within the framework of classical elasticity theory, i t is known58 that tlie interfacial fiber/ matrix shear stress can be very high in tlie vicinity of ;I propagating wave front. especially for relatively low-fiber volume fractions. Also, geometric dispersion effects can be strong in liigli-fil,er-\~oliinie-fraction filu-our niaterids, which may be interpreted on the macroscopic scale as equivalent to high damping capacity.

'I'IIF.RAIAI. AND 01 HER TKA;VSPOK.I~ PHENOMENA

I t is generally recognized that, in a strict sense, the problems of heat conduction and elasticity are coupled together. In fact, in the original development of thermoelasticity in tlte 1 MOs, D u h a m e P derived i i coupled energy expression. However, tliermomechanical coupling effects are generally small aiitl thus are neglected in most cases, When this is done, the heat conduction problem can be solved separately to obtain the temperature distribution through a body. Then, the thermal stress probIem can be solved separately also, by using the temperature distribution as input information.!/

Just as in the case of elasticity. in heat conduction and thermal expansion of composite materials the actual nonhomogeneous material can be considered to be macroscopically homogeneous but anisotropic. To provide an analytical basis for determining the necessary macroscopic anisotropic properties, various micro- analyses of heat conduction and thermal expansion in filamentary composites have been developed.

It has been known for a long time that the two-dimensional Laplace equation, which governs steady-state heat conduction in two-dimensional isotropic media, also governs a variety of other seemingly different physical phenomena in two dimensions. These include diffusion, electrostatics, electrical conduction, fluid flow, gravitational field, longitudinal shear, magnetostatics, membrane deflection, neutron diffusion, and torsion warping.

Although governed by the same partial. differential equation, some of the phenomena listed require different boundary conditions on the primary dependent variable; thus, they are not true analogues of heat conduction. However, many are

T[ Biota1 presented a variational method for solving thermal stress problems directly without first determining the temperature distribution. However, since it is often desir- able to know the temperature distribution itself in conncction with material-property degradation, Biot's method is not presented here.

Bert % Frailcis: Composite h1;iterial h1eclr;inics 669

true ; I I I ; I ~ ( I ~ I I ~ S , ;is listed in T A H I X I , i i n d permit application of viirious tecliniqiies clevelop(xl in other fields to be applied to the present prol,lem.

It should be mentioned that tlierm;il conductivity, ;IS well as all of the other pl iysid toethcierits listed in tlie last column of TABLE 1, are second-rank tensors. Tliii\. t l i ey have known tr;insformation properties under rotations, i.e., by the ~ I o l i r ' s circle for planar rotations. Generally. i t is assumed tha t tlie fibers are ori- eiitetl either in a square array or in a statistical c1istril)iition. so that the properties i11 the filler cross-sectional plane (tlie plane perpendicular to the fiber axis) are rantloni. Then. tlie macroscopic I)ehavior is transversely isotropic, with the fiber cross-scctional plane as the plane of isotropy. In other words. i t is assumed that the conducti\.ity (denoted by the symbol k,) in all directions i n this plane is the same. I[ the cotidiicti\.ity tensor is rei;ited to tlie orthogonal axes formed by the inter- sections o f the planes of material symmetry, then he material is orthotropic at most, i.c.., all of the off-tliagoniil terms in the conductivity tensor vanish. Further- 1:iorc'. i l l \ iew of tlie transversely isotropic assumption, only two conductivities arc Iieccwiry: and fil, the conductiiity in tlie direction of the fibers.

TABLE 1 ANALOGIES AMONG PHYSICAL FIELDS*

Field T - V T f k - Diffusion con cent rat ion concentration co n cent ra t ion diffusion

electrical dielectric Electrostatic?. electrical electrical potential FI induction constant

gradient flux coefficient

Electrical electrical electrical current conduction potential FI density

Heat temperature thermal heat conduct ion gradient flux

Longitudinal shear dis- shear shear placement strain

shear stress

electrical conductivity

thermal conductivity

shear modulus

Magnetostati cs magnetic magnetic magnetic permeability

"FI denotes field intensity; bold face type denotes a vector quantity; Vdenotes the grad-

potential FI induction

ient; f = kVT.

'I'lic self-consistent model method has been applied to analysis of electrical conductivity of composite mediii.~~~--"-l These works were concerned with media containing splieroitl;il inclusions. rather than filaments. T h e thermal model approach was ;ippliecl to fil;iment;rry composites by Thornlxirgli and Pears.G.5 Assuming that the fibers and matrix are connected in parallel, the major thermal conductivity k, is given by the following rule-of-mixtures expression:

(6) f : , = / i f J'J + fi,,, 17,,1

where k,. k,,, are the filler and matrix conductivities. Similarly, as in the determination of elastic coiist;ints for miiltiphiise materials. this expression may he considered as ;in tipper bound for Ii.

Similarly, assuming t h t the fibers and matrix are ;irr;inged in series, the following iuverse-rille-of-mixtures expression for the minor thermal conductivity k, is obtainetl:


Again, as in the case of elastic constants, EQUATION 7 may be considered a lower bound for k . Expressions equivalent to EQUATIONS 6 and 7 were derived by Tsaos6 in the context ol upper and lower bounds for conductivity of particulate composites, and by Hashin and Shtrikmansi for magnetic permeability.

It was noted t h a t EQUATION 7 is independent of both fiber cross-sectional shape and geometrical arrangement. Springer and TsaifiA proposed a thermal model that accounts, in an approximate manner, for both of these factors. They presented closed form expressions for k , for square filaments in a square array and for cylin- drical filaments in a square array. I t was further pointed out, by analogy with composite stiff’ness results, that a hexagonal packing array yields lower values for composite thermal conductivity than does a square array for the same filament volume content. .4dditional attention to the conductivity prohlem was given by Jackson and C:oriell,Rg who presented general integrals for electrical conduction which are analogous to those of Springer and Tsai.

Springer and TsaiRR also used the analogy with longitudinal shear loading to apply the numerical results obtained by Adams and DonerqE to prediction of k,. For both circular and square filaments in a square array, their numerical results were slightly higher than predicted by the thermal model for circular filaments.

More recently, Asliton and colleague+* applied the Halpin-Tsai interpolation to transport properties. The predicted values agreed quite well with experimental data for graphite fabric/plastic. Behrensio applied Born’s “method of long waves” to computation of thermal conductivity in composites. Recently, Ben-AmoP presented a direct interactive solution and also suggested a weighted geometric for- mula but gave no evaluation of its success. Rybicki and Hopper72 have presented a refined finite-element for the analysis of transient heat conduction in composite media.

It has long been known that problems involving thermal stresses, i.e., stresses induced by virtue of temperature changes, can be treated hy adding an additional thermal-expansion term to the generalized Hooke’s law:

{er} = [S4J {a,} + { e i } ; i , j = 1,. . . , 6 (8) Here e, is the thermal strain, or may. in general, be some other kind of environ- mentally induced strain, such as shrinkage due to curing or phase transformations, and swelling caused by ahsorbed liquids (moisture), gases, or fission products (Wigner effect); et, a, strain and stress components; Sij E compliance matrix.

(9) where T is the temperature change measured from the strain-free temperature. In the case of absorption swelling, the local concentration of the absorbed medium plays the same role as T. In cases in which the depends upon T, experimentally determined average values are generally used. I t is noted that the thermal strains, e0 appearing in EQUATIONS 8 or 9 need not satisfy the strain compatibility conditions. Thus, the temperature distribution induced nearly instantaneously within a body by a laser, for example, may he discontinuous. Since thermal expansion is also a second-rank tensor, as in the case of heat conduction, it is generally assumed that only three macroscopic expansions are required, at most, to describe the thermal-expansion behavior of the continuum.

In principle. each of the analysis methods described under elastic micromechanics (see the preceding section) can be used. However, in their extensive survey paper covering elastic and thermal-expansion properties, Chamis and Sendeckyj21

It is generally assumed that:

ei = aiT; i, = 1,. . . , 6

Bert PC Francis: Composite Material Mechanics 67 1

mentioned only a few analyses of tliermal-expansion properties of composites. More recent Kussion work has since been reviewed by Sendeckyj.73

The inecliaiiics-of-materials approach has been applied to thermal expansion by C.reszcruk.7-4 Kil6hinskii'g used ;I version of the self-consistent model method, while the variational approach was used i n REFERENCES 75-78. Levin's paper75 is of particular iniportaiice, for i t provided a niethod for expressing tlie effective thermal expamioil coefficienta 0 1 the composite in terms of the elastic moduli of the phase constituents. ' l l ie general results were given in terms of upper and lower bounds, which, i i i certain restricted cases, coincide. So-called exact classic solutions have been obtained by Herrni;iiin aiitl Pister,'o taking full advantage of geometric symnierry, by V;ui Fo FyH('. 81 tising the Kolosov-Xlusklielislivili complex-variable method, and by Atlains antl ;issociates.R3 using a finite-different solution.

For engineering development work, the following equations, due to Schapery'a and verified experimentally by Halpin and Pagan0.a' are most frequently used. For iliermal expansion parallel to tlie fibers:

For expansion perpendicular to the fibers:

where c, q T , e,,, (Y,,,T, and v, and v,,, are the fiber and matrix Poisson's ratios. These results were based upon tliermoelastic energy princip!es together with a method for minimizing the difference between upper and lower bounds. EQUA- TION 10 is exact when tlie Poisson's ratio of tlie matrix is equal to that of the fiber. EQUATION 11 is valid for any fiher geometry antl distrihution, provided the composite is statistically homogeneous and transversely isotropic.

FUTURE TREKDS

In sucli a fast-moving field as composite material micromechanics, it is very difficult to predict significant future trends with much confidence. However, the authors helieve tliat considerahly more mechanics research is needed in certain problem areas and tliiis venture a guess tliat future developments may be along the following lines:

1. hlicromedianics studies wiil increasingly be concerned with the effects of various types of flaws and thus will tie in very closely with fracture on one hand and nondestructive inspection on the other.

2. Stochastic aspects will be included much more comprehensively in future analyses to permit a sound statistical basis for structural reliability prediction and control.

3. More attention will be paid to predicting the strength of composites reinforced with short fibers arranged in either planar random or three-dimensional random arrays. Such composites hold great promise for low-cost, high-production products made in matched-die molds, for example.

4. More attention will be directed to the mechanical behavior of composites at large strains. In tlie case of a composite containing a highly elastic elastomer, these strains could be within the elastic range.

These, as well as other possible lines of research, will continue to develop as the technology of composite materials comes closer to fruition. At the present time,


materials and processing costs are tlic major obstacle to expaiisiw K a n d D ac- tivity in composites. As these costs continue to decline, however, it may he expected that utilization of advanced composites i n aerospace, marine, nuclear, and other industries will increase, which will br ing abou~ ii colicomitant development of the mechanical foundations.

1. 2. 3.

4.

5.

6.

7.

8.

9.

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11.

12.

13.

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ing, Mass. SCHWARTZ, R. T. & H. S. SCHWARTZ Eds. 1968. Fundamental Aspects of Fiber Rein-

forced Plastic Composites. Wiley-Interscience. New York, N.Y. TSAI, S. W., J. C. HALPIN & N. J. PAGANO. Eds. 1968. Composite Materials Workshop.

Technomic Publishing Co., Inc. Stamford, Conn. ASHTON, J. E., J. C. HALPIN & P. H. PETIT. 1969. Primer on Composite Materials:

Analysis. Technomic Puhl. Co., Inc. Stamford, Conn. CALCOTE, L. R. 1969. T h e Analysis of Laminated Composite Structures. Van Nostrand

Reinhold Co. New York, N.Y. COMFQSITE MATERIALS: TESTING AND DESIGN. 1969. ASTM STP 460, American Society

for Testing and Materials. Philadelphia, Pa. DIETZ, A. G. H., Ed. 1969. Composite Engineering Laminates. MIT Press, Cambridge,

Mass. See especially Chap. 1, Mechanics of Composite Structures, by Y. Stavsky & N. J. Hoff.

LUBIN, G., Ed. 1969. Handbook of Fiberglass and Advanced Plastics Composites. Van Nostrand Reinhold. New York, N.Y.

WENDT, F. W., H. LIEBOWITZ & N. PERRONE, Eds. 1967. Mechanics of Composite Ma- terials. Proc. 5th Symposium on Naval Structural Mechanics. Philadelphia, Pa., May 8-10. Pergamon Press. Oxford, England, 1970.

FRANTSEVICH, I. N. & D. M. KARPINOS, Eds. 1970. Kompozitsionnye Materialy Volok- nistogo Stroeniya. Kiev. Engl. translation: Fibrous Composites. NASA TT F-675, 1972.

14. COMP&ITE MATERIALS: TESTING AND DESIGN. 1971. Second Conference, ASTM STP 497, American Society for Testing and Materials. Philadelphia, Pa.

15. MCCULLOUGH, R. L. 1971. Concepts of Fiber-Resin Composites. Marcel Dekker. New York, N.Y.

16. NOTON, B., Ed. 1973. Composhe Materials in Engineering Design. American Society for Metals. Metals Park, Ohio.

17. SENDECKYJ, G. P., Ed. Micromechanics. Vol. 2 of Treatise on Composite Materials. L. J. Broutman & R. H. Krock, Eds. Academic Press. New York, N.Y.

18. CHAMIS, C. C., Ed. Structural Design, Parts I & 11, Volumes 7 & 8 of Treatise on Com- posite Materials. L. J. Broutman & R. H. Krock, Eds. Academic Press. New York, N.Y.

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