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J. Mech. Phys. Solids. Vol. 45, No. 8, pp. 1281-1302, 19970 1997 Elsevier Science LtdPrinted in Great Britain. All rights reservedPI1 : SOO22-5096(97)00007-O 0022-5096/97 $17.00+0.00

MICROMECHANICAL MODELS FOR GRADEDCOMPOSITE MATERIALS?

THOMAS REITERS, GEORGE J. DVORAKCenter for Composite Materials and Structure, Rensselaer Polytechnic Institute, Troy,

NY 12180-3590, U.S.A.and

VIGGO TVERGAARDDepartment of Solid Mechanics, Technical University of Denmark, DK 2800 Lyngby, Denmark

(Received 9 May 1996; in revisedform It December 1996)ABSTRACT

Elastic response of selected plane-array models of graded composite microstructures is examined underboth uniform and linearly varying boundary tractions and displacements, by means of detailed finiteelement studies of large domains containing up to several thousand inclusions. Models consisting ofpiecewise homogeneous layers with equivalent elastic properties estimated by Mod-Tanaka and self-consistent methods are also analysed under similar boundary conditions. Comparisons of the overall andlocal fields predicted by the discrete and homogenized models are made using a C/Sic composite systemwith very different Youngs moduli of the phases, and relatively steep composition gradients.

The conclusions reached from these comparisons suggest that in those parts of the graded microstructurewhich have a well-defined continuous matrix and discontinuous second phase, the overall properties andlocal fields are predicted by Mori-Tanaka estimates. On the other hand, the response of graded materialswith a skeletal microstructure in a wide transition zone between clearly defined matrix phases is betterapproximated by the self-consistent estimates. Certain exceptions are noted for loading by overall transverseshear stress. The results suggest that the averaging methods originally developed for statistically homo-geneous aggregates may be selectively applied, with a reasonable degree of confidence, to aggregates withcomposition gradients, subjected to both uniform and nonuniform overall loads. 0 1997 Elsevier ScienceLtdKeywords: A. microstructure, A. voids and inclusions, B. layered material, B. particulate reinforcedmaterial.

1. INTRODUCTIONWe are concerned with graded composite materials, consisting of one or more dis-persed phases of spatially variable volume fractions embedded in a matrix of anotherphase, that are subdivided by internal percolation thresholds or wider transition zonesbetween the different matrix phases.

A detailed description of the geometry of actual graded composite microstructuresis usually not available, except perhaps for information on volume fraction dis-

t Dedicated to Professor Franz Ziegler on his 60th birthday.1 On leave from Institute of Lightweight Structures, Technical University of Vienna, Austria

1281

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1282 T. REITER et al .tribution and approximate shape of the dispersed phase or phases. Therefore, evalu-ation of thermomechanical response and local stresses in graded materials mustrely on analysis of micromechanical models with idealized geometries. While suchidealizations may have much in common with those that have been developed foranalysis of macroscopically homogeneous composites, there are significant differencesbetween the analytical models for the two classes of materials. It is well known thatthe response of macroscopically homogeneous systems can be described in terms ofcertain thermoelastic moduli that are evaluated for a selected representative volumeelement, subjected to uniform overall thermomechanical fields. However, such rep-resentative volumes are not easily defined for systems with variable phase volumefractions, subjected to nonuniform overall fields.

Regardless of such concerns, a variety of methods originally developed to describethe behavior of macroscopically homogeneous composites have been applied in ther-moelastic analyses of FGM components. At the most elementary level, rule-of-mixtureapproaches have been employed, for example, by Fukui et al. (1994), Lee and Erdogan(1994/1995), and Markworth and Saunders (1995) in elastic systems. Giannakopouloset al. (1995) and Finot and Suresh (1996) used this approach in elastic-plastic systems.Miller and Lannutti (1993) estimated elastic moduli and averages of the HashinShtrikman bounds for statistically homogeneous systems, while Hirano et al. (1990)introduced a fuzzy-set estimate based on the Mori-Tanaka (1973) method, with anassumed transition function to account for the effect of changes of the matrix andinclusion phases. The method was also used in investigations of thermoelastic behaviorof FGM structures (Tanaka et al., 1993a,b ; Hirano and Wakashima, 1995).Additional references appear in the review by Markworth et al. (1995) and inWilliamson et al. (1993).

The purpose of the present study is to test the proposition that for two-phasegraded materials with a single volume fraction gradient, the overall elastic responsecan be obtained from homogeneous layer models, with the layer moduli estimated interms of the local volume fractions, and matrix and inclusion moduli, by certainexisting micromechanical methods. Section 2 introduces the graded material modelsand describes the traction and mixed boundary conditions employed in the presentstudy. Both uniform and linearly varying overall stress and strain fields are considered.Section 3 explains the replacement of the discrete material models with piecewisehomogeneous layered materials with equivalent elastic constants. The Mori-Tanakaand self-consistent estimates of the effective overall moduli in two-phase particulateaggregates, and the related local field averages, are also summarized here. The resultsof the many comparative studies are illustrated by examples in Section 4. Finally,Section 5 provides a summary of conclusions that should be useful in applications ofthe approximate averaging methods to various graded material configurations.

2. THE TWO-PHASE GRADED MATERIAL MODEL2.1. Model geometry

The graded material models selected for the proposed comparative studies arebased on a planar hexagonal array of inclusions in continuous matrices. Figure 1

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Micromechanical models 1283

-bx2--_) _

+b

0.0 0.4 0.8Phase Volume Fraction

Fig. 1. A two-phase graded material Model 1 with a linear volume fraction gradient in the x,-direction anda distinct percolation threshold.

Fig. 2. Two examples of Models 2, the five graded material models used, with a linear volume fractiongradient in the x-,-direction and a wide transition zone.

shows a microstructure with a distinct threshold between the two matrix phases ; thismicrostructure will be referred to as Model 1. Figure 2 illustrates two of the fivemicrostructures of Model 2, with a wide transition zone of a skeletal microstructure,which were used in the comparisons. Figure 3 presents an arrangement incorporatingboth the skeletal transition zone (Model 3) and a threshold (Model 1.2) ; such a

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1284 T. REITER et al .

Fig. 3. Two-phase graded materials with a linear volume fraction gradient : Model 3 (left) with a transitionzone, and Model 1.2 (right) with a percolation threshold.

x3tI x2

(a) Double layer arrayc,=c,=os

(b) Single layer arrayFig. 4. (a) A double-layer array of hexagonal cells, and finite element discretization with 24 triangles percell ; (b) the two-phase model in a single layer array. The phase volume fractions are c, = 0.5, r = 1, 2, inboth arrays.

microstructure may result by mixing two phases of different grain size. Figure 4(a)shows in detail the two overlapping arrays used in generating the graded materialmodels, with a relative displacement in the x,-direction equal to one half width of onehexagonal cell. In contrast to the early (cr N 0.3) clustering observed in the single-layer array [Fig. 4(b)], the double-layer array preserves separation of the inclusionsto higher volume concentrations (cr N 0.6), while also offering an optional selectionof certain other inclusion shapes. Both arrangements provide for rows of hexagonalcells parallel to the x,-axis, and the number of inclusions in each row defines the rowvolume fraction. Volume fraction gradients in the x,-direction are simulated by

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Micromechanical models 1285changing the number of inclusions in subsequent rows ; many different gradientmagnitudes can be generated in this manner.The array of Fig. 1 has 20 hexagonal cells in each row, and 30 rows in the x3-direction. The first five top and bottom rows consist of pure phase 2 and phase 1,respectively. A constant volume fraction gradient was generated by adding a singleinclusion in each successive row. This provides for a rather steep gradient and thusfor a more stringent test of the model comparisons ; published micrographs of actualsystems suggest rates equal to l/5 to l/l0 of the present magnitude. The arrays ofFigs 2 and 3 have 50 rows and 40 hexagonal cells per row. Again, the five endrows are homogeneous, then one inclusion is added in each next row. The resultingcomposition gradients are thus half as steep as that of Fig. 1. Placement of inclusionsin the five Models 2 illustrated by Fig. 2 was made automatically, using a randomnumber generator. The Models 1, 1.2 and 3 of Figs 1 and 3 were generated manually.

As implied by the use of planar arrays, the typically particulate microstructure ofa graded material is replaced here by a graded fibrous system. The latter appears tobe more suitable for the intended comparisons, since realizations of fiber systems aremuch better understood than those of three-dimensional composites with randomlydispersed particles. The finite element mesh of Fig. 4(a) subdivides each fiber into 24three-noded triangular elements. The element properties are described in Section2.4. Convergence of the model with respect to the coarseness of the finite elementdiscretization was established by comparisons of selected results of the present modelwith a 96-element per fiber model, which showed no significant deviations in overallstiffness. Note in Fig. 4(a) the homogenized layer of elements aligned with the _Y~-direction ; such layers were used in finding volume averages of local fields computedin finite element analysis of the discrete two-phase graded material models ; they arealso employed in the homogenized models discussed in Section 3.Other meshing schemes may be implemented with automatic mesh generators.These may offer better approximations of the local fields, but should yield similarfield averages and overall response for a given phase density. The scheme selectedhere is more advantageous since it can also be used to generate certain gradedmicrostructures.2.2. Traction boundary conditions

Both traction and displacement, as well as mixed boundary conditions involvinguniform and linearly varying distributions were applied to the material models. Thesurface tractions can be converted into equivalent overall stresses which, in the caseof linear distributions, are subject to certain equilibrium conditions. These can beestablished by considering a volume V of an elastic homogeneous medium with anyphysically admissible material symmetry, such that with the possible exception of athin layer at the surface S, the stress field within V is a linear function of Cartesiancoordinates.

at~(x)= arj + VijkXk = Oji + Vjikxk, XE v. (1)To be admissible, this field must satisfy the equations of equilibrium in the absenceof body forces

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1286 T. REITER et al .&,(x)/dx, = 0 * qiij = 0, i, = 1,2,3. (2)

Inside a homogeneous medium, the strains caused by (1) are also linear functions ofcoordinates and, therefore, identically satisfy the equations of compatibility. Ofcourse, the linear strain distribution may not be preserved in graded and/or het-erogeneous systems, hence the stresses in V may not vary linearly.To find the overall stress field that supports (l), it is convenient to define the volumeV as a cuboid,

-a

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Micromechanical models 1287

Fig. 5. The overall stresses supporting a plane stress state involving the single applied shear stress gradientl&3 = $3.

-VII1 = YL2 $33, -11222 = r;,, +$33, 0--y/333 = yI3ll +rj;22. (12)Conservation of momentum and of the moment of momentum of the overall tractionson the surface S of V returns the expected symmetries aFli= sjitji, & = I$~, with?j& = 0.

As an example, we consider the overall linearly varying stress field involving asingle applied shear stress gradient q _13 = 7yiz3.According to (12) the normal stressgradient --z22 = 7$33 is induced by this application. The resulting equilibrium fieldof linearly varying tractions is applied as boundary conditions on a unit thicknesslayer with inhomogeneous properties in the x,x,-plane (Fig. 5). The model of Fig. 1is used in this illustration. In applying the in-plane stresses of Fig. 5 to the models ofa graded fibrous composite, the stress and moment resultants on the planes xl = const.were required to vanish, and the said planes remained plane during deformation.These boundary conditions are discussed in Section 2.4 below.

Note that the right-hand side of (12) includes only shear stress gradients. Therefore,applied normal stress gradients, such as y1i13or qi32, do not activate any additionalcomponents, but cause only simple fields CJ2 = GZ+rL3x3 or at3 = c33+~P32.yZ,respectively, continuous across planar boundaries; these can be easily added to thosein Fig. 5.

It is probably obvious that the overall stresses (4) and (7), and those on planesX, = CI, are in equilibrium and remain independent both of the properties of thematerial in V and of the size of V. Hence, overall stress fields defined by selected stressgradients that satisfy (12) may be applied to arbitrary volumes of both homogeneous

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1288 T. REITER et al.and graded solids. Of course, in the latter, the internal stresses may not be linearfunctions of coordinates except, for example, in layered materials under shear, asdiscussed below.

2.3. D i splacement and mi xed boundary condit i onsIn analogy to (1), consider a volume Vof a homogeneous solid with linearly varyingoverall and local strains,

s!O)x) = s!?) + K!!?)X11 II torn mv &jCX) = &j + KjmXmThe corresponding overall displacement field is (Zuiker, 1993)

(13)

up = &fXj+ (1/2)[@2 + KQ - r@]XjXk. (14)Taking the strain (13,) into the elastic constitutive relation with a constant stiffness,one finds the local stress,

oj(x) = LjklEkt(X),

which must satisfy the equilibrium equations(15)

acr,(x)/ax, +xi = 0, (16)so that the strain gradients (13,) are constrained by the Navier relations,

&J%/ + X = 0. (17)Consider again the cuboid volume (3) of a homogeneous solid subjected to the

displacements (14). The internal strains are linear functions of coordinates. Theinterior or in-plane components of the strain tensor in V must be continuous on S.The exterior or out-of-plane components may be discontinuous. Surface tractionsmust be continuous. For example, on any plane x, = c1one can verify that the interiorstrain gradient components are ICING,~~~~C*~~,CING,cjX2, nd rc333they are equal thereto their overall counterparts in (13,). The exterior components are IC~,~, ,,~, IC,~~,C,~~,K,~~,and IC,~~.Among the latter, the IC,~~ nd ICINGre the interior components onx2 = p, while the K,,* and x122are interior components on the plane x3 = y. On therespective planes, these components are again equal to those in (13) ; of the 18components, 15 are independent. The ~123, ~132, and ~231, do not appear as interiorcomponents on any plane. However, if the 15 interior strain gradients are prescribedon the surface planes, then these three internal gradients may be evaluated, if thesolution exists, from the three relations (17).

In conclusion, if the volume V that is subjected to (14) contains a homogeneous(or homogenized) elastic material with constant stiffness coefficients L,, there are15 constant internal strain gradients equal to the corresponding overall gradientsprescribed by (14). The remaining internal gradients ~123, IC,~~, ~~3, can be found from(17), if the solution exists. However, if the volume I/ subject to boundary conditionsthat agree with (14) contains a material of variable stiffness, the internal straingradients may no longer be constant, and since (17) by itself is not sufficient for theirevaluation within V, they cannot be found from (14) alone, except for the values ofthe continuous interior components on S.

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Fig. 6. Overall stresses and displacements applied to cause linearly varying transverse normal strain

Fig. 7. Overall stresses and displacements applied with a transverse shear stress at xi = c

In what follows, we select both pure linear tractions boundary conditions andmixed conditions that simulate such deformation states as uniform normal strain,pure bending, and shear. These are displayed in Figs 5-7. Again, in each case, all

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1290 T. REI TER et al .stress resultants with components in the x,-direction are required to vanish, and theplanes x, = const. are required to remain planes during deformation (see Section2.4.).2.4. Generalized plane strain conditions

In actual solutions, the selected models of the graded fibrous material wereimplemented as two-dimensional finite element models of constant unit thickness inthe x,-direction, chosen as the diameter of a circle containing one hexagonal cellshown in Fig. 2, using the ABAQUS generalized plane strain elements. The underlyingtheory allows for a finite thickness solution domain, bounded in the thickness directionby two planes that are parallel before deformation. During deformation, the planescan move as rigid bodies relative to each other. However, it is assumed that thedeformation is uniform in the thickness direction, so that the relative motion of thetwo planes causes only normal strain in that direction. The relative motion of theplanes is completely described by the displacement Au, (xt, xy) of a selected point onone of the planes from its image on the other plane, and by the relative rotations A&A& about x2, x3, respectively, defined here as positive according to the right-handrule. In general, this adds at most three degrees of freedom to the system of equations ;implementation is accomplished with Lagrangian multipliers. The relative dis-placement of any point on one plane from its image on the other plane is then givenby

(18)In the context of the above strains and strain gradients, this implies that

&II z 0, &,* = &,3 = 0, I CI 12 z 0, I CI 13 z 0, Kl 2i = KI 3i = 0, (19)where i = 1, 2, 3. Similar relations hold for the overall components.

In the absence of constraints on the Au, displacement of the bounding planes,the boundary conditions (18) enforce a superposition of axial and pure bendingdeformations ; the resultants of the axial normal force and moments about the x2 andx3 axes being zero.

3. THE HOMOGENIZED GRADED MATERIAL MODEL3.1. The replacement scheme

Since detailed finite element analysis of a discrete two-phase or multi-phase gradedmaterial model may not be feasible in all applications, it is useful also to employ ahomogenized model of the graded material, consisting of parallel homogeneous layerswith certain effective elastic moduli. The layer properties are estimated with a suitableaveraging method, while the layer thicknesses and the phase volume fractions withinthe layer are selected to approximate the actual phase volume fraction variation inthe graded composite. The finite element mesh of Fig. 4(a) was retained in thehomogenized model and one homogeneous layer was located in each row of elements

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Micromechanical models 1291parallel to the x,-axis, even though such mesh refinement would not be needed inordinary applications. Note that three rows of elements separate the center lines ofthe hexagonal fibers, hence a total of 90 layers were used to represent the 30 rows ofhexagons in the x,-direction in Fig. 1, while 150 layers were needed for the arrays ofFigs 2 and 3. The volume fractions of the phases in each layer were determined interms of the number of fiber and matrix elements present. Also, both phase andoverall stress and strain field averages in each layer were found in terms of therespective element fields. Of course, the same boundary conditions were applied tothe models in each comparison.3.2. The Mori-Tanaka and self-consistent methods

Since these averaging methods have been extensively described in the literature, wepresent here only the results needed for their application to particulate composites.More extensive expositions of the Mori-Tanaka (1973) method can be found inBenveniste (1987) and Chen et al. (1992), and of the self-consistent method in Hill(1965) and Walpole (1969). Conditions limiting the use of these two methods in multi-phase systems have been identified by Benveniste et al. (1991), and their extensivesimilarities noted by Dvorak and Benveniste (1992).

Both methods provide estimates of the average local stress and strain fields in thephases of a composite material occupying a certain representative volume that issubjected to a uniform overall stress or strain. The local field averages are then usedto evaluate the overall elastic moduli of the composite aggregate. The results are first-order estimates, found in terms of phase elastic moduli and volume fractions, whileactual phase shapes are approximated by similar ellipsoidal shapes.

Although the size of the representative volume used by the two methods is notexactly specified, the expectation is that it should be much larger than the diameterof a typical inclusion. In the above replacement scheme, the methods are used toestimate the moduli of each of the 90 (or 150) layers representing the 30 (or 50) rowsof hexagonal cells. These are long layers of 20 (or 40) hexagonal cells, but theirthickness is equal to about l/3 of the inclusion spacing. However, recall that in thediscrete two-phase models, the composition gradient was generated by adding oneinclusion in each next row of hexagonal cells. The volume fraction thus remainsnominally constant in each three contiguous element layers, and then changes by 0.05(or 0.025) in the next three layers. Moreover, in the discrete two-phase models,the inclusions are added at distant locations, causing isolated local disturbances incomposition perceived by only a few adjacent inclusions. Therefore, with few excep-tions, the inclusions in the present microstructure (with a rather steep gradient) residein what appear to be sufficiently large volumes of constant composition.

Another concern pertains to the assumption of uniform overall fields applied tothe representative volume. As will be seen below, the stress and deformation fields inthe homogenized model are far from uniform. The role of field gradients in theresponse of heterogeneous solids was recently studied by Zuiker (1993) and Zuikerand Dvorak (1994a, b, c). The gradient effects were found to be important in fieldsof low average magnitude. However, when the field averages are high, their con-tribution to the energy of the inclusion far outweighs that of the gradients (Dvorakand Zuiker, 1995).

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1292 T. REITER et al.In conclusion, while the replacement scheme does not exactly comply with theusual assumptions of the homogenization methods, the departures do not appear to

seriously compromise its validity. This is borne out in the comparisons describedbelow.3.3. Est imat es of overal l moduli and local $el ds

A simple implementation of the methods in two-phase systems starts with evalu-ation of the overall moduli. In most functionally graded materials, the local effectivemoduli should be approximated by those of a matrix-based composite reinforced byspherical particles. Useful in applications are the following results for a randomdistribution of isotropic particles in an isotropic matrix. Let K,, G, denote the bulkand shear moduli, respectively, and c, the volume fraction of the matrix phase. TheK,, GZ, c2 denote the elastic constants and volume fraction of the particle phase ;c,+c, = 1.

The Mori-Tanaka estimate of the overall bulk and shear moduli K and G of sucha particle-reinforced composite was derived by Benveniste (1987) as(K-K,)l(K, -K,) = &I[(1 --y~)+wl,(G-G,)/(G -G,) = czB/[(l vz)+czPl~ (20)

where (Berryman, 1980) a = (K I +4G,/3)/(K2+4G,/3), b = (G, +F,)/(G2+F,), andF, = (G,/6)[(9K, +gG,)l(K, +2G) l .

The self-consistent estimate of the bulk and shear moduli of the above compositesystem was obtained by Hill (1965) ascr/K = c,/(K- Kz) +cJ(K- K ,),BlG = c,I(G-G,)+c,I (G-G,), (21)

where u = 3 -5p = K/(K+4G/3). Note that in contrast to (20), these are implicitexpressions for the unknown K and G, and that they are invariant with respect tophase exchange. It turns out that for K > 0, G > 0, j ? has the range 215 < /I < 315.After substituting for a, (21,) can be solved for K in terms of G :l/(K+4G/3) = c,/(K , +4G/-3)+c,/(K , +4G/3), (22)

while G can be obtained by solving (numerically) the quartic equation[c,K ,/(K, +~G /~)+~ ,K,/(K,+~G /~)]+~[C,G,/(G-G ,)+C,G,/(G-G ,)J+~ = 0.

(23)Analogous expressions for composites reinforced by aligned or randomly orientedneedle (= short-fiber) and disc-shaped (= platelets or flakes) isotropic inhomo-geneities were derived with the self-consistent method of Walpole (1969). The Mori-Tanaka estimates of the moduli for such systems are given by (20), with appropriatea and p taken from Berryman (1980), and for anisotropic reinforcement see Chenet al . (1992).The overall moduli of fiber composites are estimated in a similar manner. In thepresent work, the replacement of the two-phase model by the homogenized model

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Micromechanical models 1293utilized the Mori-Tanaka estimates by Chen et al. [ 1992, equations (8)-( 12)], and theself-consistent estimates of Hill [ 1965, equations (1.6))( 1.9)].

Evaluation of the local stress estimates in the phases can be easily accomplishedonce the overall moduli of a composite system are known (Hill, 1965). Suppose thata representative volume of the composite material is subjected to certain uniformoverall stress or strain fields, written as (6 x 1) vectors in the form

Go = {& rJ:* 0:x f& oY3 &)? E = {E:, &,& EiJ 2E;? 2Ey3 2Ey2sT, (24)and that the estimated phase (r = 1,2) volume averages of the local fields are

0, = {CJ,, CJ2 033 623 013 (T121h 8, = {GI ~22 ~33 2~23 2~13 24:. (25)The local averages and overall fields are connected as

co = C,d, +c,a2, &O= Cl&, +C2&2, (26)and the overall and local constitutive relations are then written in the form

6 = Leo, so = Ma, G, = L,E,, E, = Mrb,, (27)where the L, L,, and M = L-, M,. = L; are the (6 x 6) overall and local stiffnessand compliance matrices. The overall and average local fields can be related by (6 x 6)mechanical concentration factor matrices

E = A,&, tr, = B,aO, (28)where, for a two-phase system, the concentration factors are evaluated as

c,A, = (L, -LJ(L-L2), c,B, =(M, -M,))(M-Mz),c2A, = -(L, -L$(L-L,), c,B, = -(M, -M,))(M-M,), (29)

and, according to (26), must satisfy,c,A, +c,A, = I, c,B, +c,B, = I, (30)

where I is a (6 x 6) identity matrix. Note that the validity of (28) and (29) does notdepend on the method used to obtain the overall L or M ; several different approachesare available, including experimental measurements.

4. COMPARISONS OF MODEL PREDICTIONS4.1. M ater i al system

The discrete two-phase models of a graded material shown in Figs 1-3, and thehomogenized model described in Section 3, were compared in their predictions oflocal stresses or strains and overall response, under the boundary conditions discussedin Sections 2.2-2.4. The material selected was the C/Sic system (Sasaki and Hirai,

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1294 T. REITER et al.Table 1. Phase propert i es of t he C/Si c system

Material E (GPa) V 5x (W/C)Phase 1 : carbon 28 0.3 9.3Phase 2 : silicon carbide 320 0.3 4.2

1991) with the elastic moduli E and v, and the coefficients of thermal expansion CI,given in Table 1. This system was chosen because of the relatively large difference inphase elastic moduli. Moreover, as discussed in Section 2.1, the composition gradientin the model material was made about five time steeper than that in the actual systemsdescribed in the literature. Both these features should elevate the heterogeneity of themodel material, thus making it more difficult to achieve satisfactory comparisons ofthe discrete and homogenized models, and reinforcing the validity of conclusionsbased on observed agreement between the model predictions. Also, due to the largephase moduli differences, the Mori-Tanaka estimates coincide with the Hashin-Shtrikman upper or lower bounds on elastic moduli, when the stiffer phase serves asa matrix or reinforcement, respectively.

The notation used in displaying the results employs the following shorthand :Phase 0 refers to the graded composite material itself. The stress or strain in thisphase is the volume average (26) in the layer of elements shown in Fig. 4(a).FEM indicates that the quantity in question was determined from local fields found

in element layers of the discrete two-phase model [Fig. 4(a)]. To minimize artificialoscillations, this is a three-point moving average of the values computed in the layers.

MTM I denotes results that were found by finite element analysis of the homo-genized model, with overall moduli and/or other properties estimated by the Mori-Tanaka method, such that the (carbon) phase 1 in Table 1 was regarded as thecontinuous matrix.MTM2 denotes results found as in MTMl, but with the (Sic) phase 2 serving asmatrix and phase 1 as the reinforcement.

SCS denotes results found by finite element analysis of the homogenized model,where the overall properties were estimated by the self-consistent method.4.2. Li near overal l shear st ress

The first comparative results were obtained for loading by the linearly changingoverall shear stress (Fig. 5). This is a relatively simple loading case, since the composite(phase 0) stress average is a linear function of coordinates in the model volume.Figure 8 shows the aFi(~~), s = 0, 1, 2, phase stress averages in the element layersobtained from the local fields computed with the finite element analysis of the discretemodel of Fig. 1. The averages are taken over the volumes of the respective phase 1and 2 elements residing in layers aligned with the x,-axis, as indicated in Fig. 4 ; theaverage shear stress in the composite itself (phase 0) is found from phase 1 and 2averages and volume fractions in (26). Note that, as expected, the average shear stressin the composite changes linearly in the x,-direction. This result is also recovered

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1295I I I

.__ .---

E

~~~~~~~~~~~.......__________\ \ .= I ...(II- 1 --._c AI_ --. -radient Shear-Stress Loading-r I I IO .4 .8x3 lcFig 8. Average stresses ~~~ n the phases of the two-phase model of Fig. 1, under the linear overall shearstress state of Fig. 5. The averages were obtained from the computed fields in element layers illustrated inFig. 4. The applied stress uy?(x,) = 1(x,/c) MPa.I I I I I

/Sic-FGM-System

-. :.+\... ...

..__

\

.._. .._.

Gradient Shear-Stress Loading7 I I I I-.a -.4 .O .4 .a

x3 lcFig. 9. Comparison of average stresses oz3 computed by several different methods in phase 2 (Sic) of thegraded composite material model of Fig. 1. MTMl is applicable at x2 < 0, and MTM2 at xj > 0. Theaverages were obtained from the computed fields in element layers illustrated in Fig. 4. The applied stress& (xl) = 1 q/c) MPa.when the phase volume average stresses are estimated with the Mori-Tanaka method,with either phase serving as matrix in -c d x3 d c, and also when the phase stressaverages are estimated with the self-consistent scheme. Figure 9 provides comparisons

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1296 T. REITER et al.a I I/Sic-FGM-Systemr

Gradient Shear-Stress LoadingI I I I I-.8 -.4 .O .4 3

Fig. 10. Average strains 2~~~ computed in the phases of the two-phase model of Fig. 1, under the linearoverall shear stress state of Fig. 5. The averages were obtained from the computed fields in element layersillustrated in Fig. 4. The applied stress u!j3 x3) = 1 x3/c) MPa.

of several estimates of the &)(x3) stress average in phase 2 (Sic) with the FEM resultfound from the finite element analysis of the discrete graded material model. Both theself-consistent (SCS) estimate, and the MTM 1 and MTM2 estimates in the respectiveparts of the model volume, are in good agreement with the FEM result. Note thatMTMl is applicable at xj < 0, and MTM2 at xj > 0.Figure 10 presents the distribution of the element layer-averaged 2&)(.x3) straincomponent in the composite (phase 0) and phases 1 and 2, where the underlyingstrain field was obtained by finite element analysis of the discrete model of Fig. 1. Asexpected, the O-phase average converges to that in the respective matrix phase andthe two curves coincide in the homogeneous end layers. A comparison of the layer-averaged 2&(x3) strains in phase 1 was also made with the different estimates ; intheir respective regions, the MTMl and MTM2 results followed the FEM resultsrather closely, somewhat better than those of the self-consistent scheme.4.3. Overall transverse strain

This loading configuration, shown schematically in Fig. 6, was first applied as auniform overall strain, to the two-phase models of Figs l-3. Five models of the typeshown in Fig. 2 were analysed ; the results presented below are averages of the stressescomputed in the five models. The purpose here was to study the effect of graded phasedistributions on the overall and local stresses. Of course, the estimates depend onlyon the phase volume fractions in each element layer, which are determined by thecomposition gradient. Therefore these estimates are not sensitive to the actual dis-

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Micromechanical models 1297I I I I

Uniform transverse overall strain

---. 522 - Ph: 0, FEM_hQAV- S22. Ph: 0, FEM_M1_2~~- s22-Ph:O.tm?JP I--- !322_pp--- -. sz2-Pt

-.8 -.4 :o .4 .8x3 lc

Fig. 11. Comparison of the normal stress averages u#(x,) in the composite (phase 0) obtained underuniform transverse overall strain E$J x,) = 1.0, for the Models 2, I .2 and 3 of Figs 2 and 3, with several

different methods.

tribution. However, as the following comparisons show, the predictions of the discretemodels with different microstructural arrangements are best approximated by differ-ent estimates.In Fig. 11, we compare the composite-averaged stresses r&)(x3) evaluated by thefinite element method in the five different two-phase models of the type shown in Fig.2 (Models 2), and in the model of Fig. 3 (Model 3), with the Mori-Tanaka and self-consistent estimates. In Fig. 12, a similar comparison is presented of the #(xj)stresses found in Phase 2 (Sic). The results suggest that if the particle distributionprovides for a rather wide skeletal zone between the respective matrix phases, thenthe self-consistent estimate of the stress distribution approximates rather well theresults obtained by averaging the transverse composite stresses computed in thediscrete two-phase models. This is illustrated in Fig. 12, both for the models of Fig,2 and also in the transition from phase 1 toward the threshold at x3/c N 0.2 in themodels of Fig. 3.Additional comparisons of the discrete and homogenized models were made underthe linearly varying transverse overall strains of Fig. 6, applied to the model of Fig.1. The applied displacement boundary conditions produced two distinct overall strainstates, one with the distribution &(x~) = 0.5(1 +x,/c), and another with &(x3)= 0.5x,/c, where -c Q x3 d c. For these overall strains, Figs 13 and 14 show theresulting distributions of the average stress #(x3) in the (Sic) phase 2. In both cases,the FEM local stress average follows closely the MTMl and MTM2 estimates in theregions where they are valid. Of course, the same is true for the composite (phase 0)and phase 1 (C) stress averages.

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1298 T. REITER et al.

Uniform transverse overall

YSiC-FGM

Fig. 12. Comparison of the normal stress averages &22)(x3) in phase 2 (Sic) obtained under uniformtransverse overall strain #(x3) = 1 O, for the Models, 2, 1.2 nd 3 of Figs 2 and 3, with several different

I I I I I

1 ILinear transverse overall strain

---- 522 - Ph: 2, MTMP...... S22 - Ph: 2, MTM1-.-. S22 - Ph: 2. SCS- S22 - Ph: 2, FEM

1

-.a -4 I.O .4 .ax3 ic

Fig. 13. Comparison of the normal stress averages u$(xg) in phase 2 (Sic) obtained under linear transverseoverall strain &$02)(x3) 0.5(1 +x,/c), for the model of Fig. 1, with several different methods.

4.4. Uniform transverse shear stressThe boundary conditions that create this loading state are shown in Fig. 6. The

graded layer is loaded by a uniform surface shear stress, while being supported on a

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pi2 -Linear transverse overall strain

Micromechanical modelsI I I 1 ,f

/

---- SZ-Ph:2,MTM2..... S22 - Ph: 2, MTMl__-- _Hs -.-. 92 - Ph: 2, SCS__-- - S22 - Ph: 2, FEM

I I I I-.8 -.4 .o .4 .ax3 lc

1299

Fig. 14. Comparison of the normal stress averages &2(x,) in phase 2 (SIC) obtained under linear transverseoverall strain E$$)(x~) = 1 x,/c), for the model of Fig. 1, with several different methods.

Uniform overall shear stress, I

.-

---- Sk?2.-FlY1.FEM_M3-~~ SZI-pII:I.Mlw--- ~-R:t.MTMl- s23-m1,scs

I I I I-.8 7.4 .O .4 .0x3 fc

Fig. 15. Comparison of the shear stress averages &2)(x1) in phase I (C) obtained under the constanttransverse shear stre ss a ,? = I .O MPa (Fig. 7). using several models and approximate methods.

rigid substrate. The average stress in the composite (phase 0) is uniform and equal tothe applied stress in --c < x3 d c. The two-phase model of Fig. 1 was used here,together with its homogenized counterpart. Figure 15 provides a comparison between

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T. REITER et al.I I I I I I 1

/Sic-FGM-System

Uniform overall shear stress

1 I I I I-.8 -.4 .O .4 .8x3 lc

Fig. 16. Comparison of the shear strain averages 2s$y(x3) in the composite (phase 0) obtained under theconstant transverse shear stress #32= 1 O MPa (Fig. 7), using several different models and methods.

the phase 1 (C) stresses 0$:)(x3) found from the different models. Figure 16 shows thecomposite (phase 0) strains 24)(x3). As before, the self-consistent method approxi-mates quite closely the computed average response of the five Model 2 microstructures.The discrete Models 1.2 and 3 also appear to be well represented by this method inthe part of the microstructure with the carbon matrix. However, both models predicthigher stresses in the SIC phase and higher stiffness than the self-consistent andMTM2 estimates in the other part of the microstructure with the Sic matrix.

5. CLOSUREThe elastic response of several plane-array models of graded composite micro-structures, under different traction and mixed boundary conditions, has been exam-ined by detailed finite element studies, and the results compared with those provided

by equivalent piecewise homogeneous models. In the case of the one-directionalgradient of the phase volume fractions used here, the actual microstructure wasreplaced by a material consisting of thin layers, and the effective elastic constants ofthe layers were estimated by the Mori-Tanaka and self-consistent methods, in termsof phase volume fractions and approximated ellipsoidal inclusion shape. To make theresults of the comparisons more convincing, a C/Sic composite system with thephases assumed to be isotropic was chosen because of the large difference in theYoungs moduli of the phases. Also, relatively steep composition gradients wereselected. The expectation is that the conclusions derived will remain valid in systemswith less dissimilar phase moduli, and both smaller and moderately variable gradients.The overall applied loads (Figs 5-7) involved linearly varying as well as constant

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Micromechanical models 1301shear stresses, and uniform and linearly changing overall strains, transverse to thedirection of the composition gradient.

The conclusions reached from these studies indicate that in those parts of thegraded microstructure that have a well-defined continuous matrix and discontinuousreinforcement, the overall properties and local fields are closely predicted by Mori-Tanaka estimates. For example, in the model of Fig. 2 with a definite thresholdseparating the respective matrix phases, the two Mori-Tanaka estimates gave a veryclose approximation of the response of the discrete two-phase model under identicalboundary conditions. On the other hand, the response of the model graded materialswith a skeletal microstructure that does not have a well-defined matrix, was betterapproximated by the self-consistent estimates. For example, both the overall and localfield averages evaluated by finite element analysis of the discrete two-phase Models 2and 3 of Figs 2 and 3, showed distinct transitions toward the respective estimates, inresponse to the changing configurations of the graded microstructure.Although limited to the particular cases studied, the results suggest that the mic-romechanical methods originally developed for statistically homogeneous aggregatescan be applied with a reasonable degree of confidence to composites with relativelysteep composition gradients and very dissimilar phase moduli. However, certainexceptions should be noted, as observed under transverse shear loading in Figs 1.5and 16.

ACKNOWLEDGEMENTSThe work of TR was supported by a grant from the Max Kade Foundation, and that of

GJD by the DARPAjONR University Research Initiative project on the mechanism-baseddesign of composite structures at Rensselaer. This effort was initiated under partial fundingfrom the Direktor Ib Henriksens Fulbright Grant to GJD at the Department of Solid Mech-anics, Technical University of Denmark.

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