Mechanics of Materials - New Mexico State Universitymae2.nmsu.edu/~igor/2012/TEC.pdfOn the thermal expansion of composite materials and cross-property connection between thermal expansion

Mechanics of Materials 45 (2011) 20–33

Contents lists available at SciVerse ScienceDirect

Mechanics of Materials

journal homepage: www.elsevier .com/locate /mechmat

On the thermal expansion of composite materials and cross-propertyconnection between thermal expansion and thermal conductivity

Igor SevostianovDepartment of Mechanical and Aerospace Engineering, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 April 2011Received in revised form 19 August 2011Available online 18 October 2011

Keywords:Composite materialThermal expansionCross-propertyMicrostructureThermal conductivity

0167-6636/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.mechmat.2011.10.001

E-mail address: [email protected]

The paper focuses on the quantitative characterization of heterogeneous microstructuresfrom the point of view of the material’s thermal expansion. First, we derive expressionfor the second rank thermal expansion contribution tensor of an inhomogeneity and spec-ify it for various inhomogeneity shapes. Case of a spheroidal inhomogeneity in an isotropicmaterial is discussed in detail. Thermal expansion contribution tensor is used as a basicbuilding block to calculate effective thermal expansion of a heterogeneous material andto derive explicit cross-property connection between thermal expansion and thermal resistivityof a composite. We compare our results with experimental data available in literature andwith other approaches.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction and overview of related results

The paper focuses on the quantitative modeling of het-erogeneous microstructures from the point of view of thematerial’s thermal expansion. The quantitative character-ization means identification of the microstructural parame-ters in whose terms this physical property is to beexpressed. The main requirement to the proper micro-structural parameters is that they should represent theindividual inhomogeneities in accordance with their contri-butions to the physical property considered – otherwise, theproperty cannot be uniquely expressed in their terms(Kachanov and Sevostianov, 2005). Thus, identification ofthe proper microstructural parameters requires knowledgeof an individual inhomogeneity contribution to the consid-ered property. Generally, such parameters are different fordifferent physical properties. If they are sufficiently similarfor certain pair of physical properties, this leads to explicitcross-property connections between the latter ones(Sevostianov and Kachanov, 2008). In the present paperwe illustrate this issue on example of cross-property con-

. All rights reserved.

nection between effective thermal expansion and thermalconductivity of a heterogeneous material.

Thermal expansion coefficient of a composite materialhas been discussed in literature from the middle of XX cen-tury. However, the issue of proper microstructural charac-terization of heterogeneous materials in the context ofthermal expansion has never been addressed to the bestof our knowledge. The earliest model with which theauthor is familiar was proposed by Turner (1946) who sug-gested heuristically the following relation for the effectivethermal expansion coefficient a of an isotropic two-phasecomposite of arbitrary microstructure:

a ¼ c1a1K1 þ c2a2K2

c1K1 þ c2K2; ð1:1Þ

where c1 and c2 are volume fractions of the two phases, K1

and K2 are bulk moduli, and a1 and a2 are thermal expan-sion coefficients of the phases. The basic assumptions ofTurner are that the shear deformation is negligible and that‘‘each component in the mixture is constrained to changedimension with temperature changes at the same rate asaggregate’’. The latter one means uniformity of strain anddue to that is equivalent to Voight bound in the mechanicsof composites (see Hill, 1963).

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 21

Kerner (1956) developed a so-called three-phase modelfor thermomechanical properties of composites. In thisone-particle approximation an average (spherical) inho-mogeneity surrounded by a shell of matrix material isembedded in the material with (unknown) effective prop-erties. The self-consistent field approach was used then tocalculate effective thermal expansion coefficient a (as wellas effective elastic properties):

a ¼ c1a1K1ð3K0 þ 4G0Þ þ c0a0K0ð3K1 þ 4G0Þc1K1ð3K0 þ 4G0Þ þ c0K0ð3K1 þ 4G0Þ

; ð1:2Þ

where subscripts ‘‘0’’ and ‘‘1’’ correspond to matrix andinhomogeneities respectively; and G is shear modulus.

Thomas (1960) suggested a phenomenological equation

a ¼ c0am0 þ c1am

1

� �1=m; ð1:3Þ

where exponent m varies from �1 to 1. At these values onecan get volume averages of either thermal expansion coef-ficients or their inverses. Holliday and Robinson (1973)stated that (1.3) at m = ± 1 represent bounds of Voigt andReuss. It is difficult to agree with this assessment, however,since Voigt and Reuss bounds are based on certain physicalassumptions (like Turner Eq. (1.1)) while (1.3) is a phe-nomenological equation with adjustable parameter. Forsmall m, (1.3) can be rewritten as

ln a ¼ c0 ln a0 þ c1 ln a1 ð1:4Þ

or

a ¼ ac00 ac1

1 : ð1:5Þ

Form (1.5) is known in literature as Thomas equation (see,for example, Feltham et al., 1982).

Arthur and Coulson (1964) discussing thermal expan-sion of uranium dioxide-stainless steel cermets used Black-burn equation that is given as private communicationwithout derivation. Holliday and Robinson (1973) men-tioned a misprint but miss-meshed the phases. After thecorrections, Blackburn equation has the following form:

a ¼ a0 � c1hða0 � a1Þ;

h ¼ 3E1ð1� m0Þ½ð1þ m0Þ þ 2c1ð1� 2m0Þ�E1 þ 2c0E0ð1� 2m0Þ

;ð1:6Þ

where E is Young modulus, m is Poisson’s ratio and sub-scripts ‘‘0’’ and ‘‘1’’ correspond to matrix and inhomogene-ities respectively. This equation was later derivedindependently by Wang and Kwei (1969) and Fahmi andRagai (1970).

Van Fo Fy (1965, 1971) obtained exact solutions forelastic properties and thermal expansion coefficients ofmaterials reinforced with doubly-periodic arrays of hollowand solid circular infinite fibers. However, since we arefocusing on three-dimensional problems, we do not usethis formulas in the present paper.

Break-through equation has been developed by Levin(1967) who established the connection between the effec-tive bulk modulus K and the effective thermal expansioncoefficient a of a two-phase (phases 0 and 1) isotropiccomposite with isotropic constituents:

a� hai ¼ ða0 � a1Þ1

K0� 1

K1

� ��1 1K� 1

K

� �� �; ð1:7Þ

where angle brackets denote volume averaging. It has beengeneralized to anisotropic two-phase composites by Rosenand Hashin (1970) as follows:

aij � haiji ¼ að0Þkl � að1Þkl

� Sð0Þklrs � Sð1Þklrs

� �1ðSrsij � hSrsijiÞ; ð1:8Þ

where Sijkl are elastic compliances and aij are thermalexpansion coefficients. Eq. (1.7) has been also re-derivedby several authors: Schapery (1968), Cribb (1968) andSteel (1968). Results of Levin (1967) and Rosen and Hashin(1970) were further extended to two phase electro-elasticcomposites by Dunn (1993). He related the effective ther-mal expansion and pyroelectric coefficients to the effectiveelastic, piezoelectric, and dielectric constants.

Schapery (1968) used extremum principles of thermo-elasticity to derive bounds on effective thermal expansioncoefficients of a composite in terms of exact solution foreffective bulk modulus of a composite.

a� 6 a 6 aþ; a� ¼ hai þhKai � hKihai

hKi

1KR� 1

Keff

� 1

KR� 1hKi

� � Da;

ð1:9Þ

where

Da¼1

Keff� 1hKi

� 1=21

KR� 1

Keff

� 1=2

1KR� 1hKi

� hKa2i�hKai2

hKi

!1

KR� 1hKi

� �"

� hKaihKi �hai

� �2#1=2

is non-negative. Note that, for a two-phase material Da = 0and (1.9) can be directly derived from Levin’s formula (1.7).If exact solution Keff is unknown, the Voigt–Reuss boundscan be obtained instead of (1.9):

hai 6 a 6 hai þ hKai � hKihaihKi : ð1:10Þ

Shapery illustrated his approach on examples of two- andthree-phase composites.

Tummala and Friedberg (1970) used approach analo-gous to first Eshelby problem (Eshelby, 1957) for sphericalinclusions and derived thermal expansion coefficient as

a ¼ a0 � c1hða0 � a1Þ;

h ¼ E1ð1þ m0Þð1þ m0ÞE1 þ E0ð1� 2m0Þ

:ð1:11Þ

No interaction between inhomogeneities has been consid-ered. Their approach was modified by Wakashima et al.(1974) – who accounted for the shapes of inhomogeneities(all the inhomogeneities are assumed to be of spheroidalshape of identical aspect ratio and strictly parallel) and ta-ken into account interaction between the inhomogeneitiesusing self-consistent method similar to one used by Kerner(1956). The resulting equation has the following form:

22 I. Sevostianov / Mechanics of Materials 45 (2011) 20–33

aij ¼ a0dij þ ða1 � a0Þfc1dij þ c1ð1� c1Þ½h1ðd1id1j

þ d2id2jÞ þ h3d3id3j�g: ð1:12Þ

Parameters h have been derived in explicit form for thecases of disks, infinite cylinders and spheres as follows:

Spherical inhomogeneities

h1 ¼ h2 ¼K1 � K0

ð1� cÞK0 þ cK1 þ ð3K0K1=4G0Þ:

Disc-shaped inhomogeneities

h1 ¼ð1� m0ÞE1 � ð1� m1ÞE0

ð1� cÞð1� m1ÞE0 þ cð1� m0ÞE1;

h3 ¼2ðm1E0 � m0E1Þ

ð1� cÞð1� m1ÞE0 þ cð1� m0ÞE1:

Fibers

h1 ¼A� B

AC � BDE1 � 1

�; h3 ¼

C � DAC � BD

E1 � 1 �

A ¼ ð1� cÞ2m0m1G1 þ 2ð1� 2m1ÞG0

ð1� m0Þþ 2cð1� m1ÞG1;

B ¼ ð1� cÞ m0G1 þ ð1� 2m1Þm0G0

ð1� m0Þþ 2cm1G1

C ¼ ð1� cÞG1 þ ð1� 2m1ÞG0

ð1� m0Þþ 2cG1;

D ¼ ð1� cÞ2m1G1 þ 2ð1� 2m1Þm0G0

ð1� m0Þþ 4cm1G1:

Note that for spherical inhomogeneities (1.12) coincideswith Kerner’s Eq. (1.2).

Predictions with different models have been comparedin reviews of Holliday and Robinson (1973) and Felthamet al. (1982). The main conclusion is that different modelswork well for certain specific composites. However, it isimpossible to estimate the accuracy of any model a’priori.Moreover, at low concentration of inhomogeneities, differ-ent models give different slopes of the curves. Probably it isthe main reason for appearance of new papers on this to-pic. Klemens (1986, 1988) used Eshelby’s results todevelop a model for thermal expansion of isotropic two-phase composites. His results, however can be easier de-rived from Levin’s formula (1.7). Eshelby’s solution wasalso used to develop Mori–Tanaka approximation for effec-tive thermal expansion coefficients in the papers of byTakahashi et al. (1980) (for particle reinforced composites)and Takao and Taya (1985) (for short fiber reinforced com-posites). Withers et al. (1989), in the part related to ther-mal expansion coefficients, repeated derivation ofTakahashi et al. (1980). Detailed analysis of the applicationof Mori–Tanaka scheme to calculation of the effective ther-mal expansion coefficients is given by Benveniste and Dvo-rak (1990). We also discuss it in Section 4.

Recently, Hsieh and Tuan (2006) suggested a model forquasi-periodic microstructure where a unit cell containsmultiple short fibers. Sevostianov (2007) used Levin’s for-mula to build Hashin–Shtrikman bounds for thermalexpansion and apply the latter to calculate thermal expan-sion coefficient of a composite with interphase layers be-tween the matrix and inhomogeneities (with applicationto nanocomposites).

Remark. Non-elastic effects in the context of thermalexpansion have been addressed in the papers of Wakashi-ma et al. (1974) (plasticity) and Berryman (2009) (visco-elastic composites). These effects, however, are beyond thescope of the present work, so we do not discuss them here.

Summing up, we can conclude that most of the existingmethods for calculating expansion coefficients of compos-ites reduce the problem to estimation of the effect ofinteraction between individual inhomogeneities based oncertain hypotheses accepted by authors. The problemabout contribution of individual inhomogeneities intothermal expansion has never been addressed in theliterature.

In the present paper we develop approach for properquantitative characterization of microstructure from thepoint of view of tensor of thermal expansion coefficients.For this aim microstructural parameters that describecontribution of an inhomogeneity into overall thermalexpansion are identified. Based on this parameter wedevelop a quantitative model for the tensor of effectivethermal expansion coefficients and verify it on two exam-ples (one of which involves material with negative thermalexpansion). Identification of the proper microstructuralparameter allows one to establish cross-property connec-tions between thermal expansion coefficient and othermaterial properties (in addition to Levin’s formula (1.7)).As example, we derive cross-property connection betweenthermal expansion and thermal conductivity of a hetero-geneous material and verify it by comparison with exper-imental data available in literature.

2. Thermal expansion contribution tensor

We consider a certain reference volume V of an infinitethree-dimensional medium with an inhomogeneity X ofvolume V⁄– a region possessing elastic and thermal proper-ties different from the ones of the surrounding material.The properties of the inclusion and of the matrix will bedenoted by superscripts ‘‘1’’ and ‘‘0’’, respectively. Thematerial is subjected to uniform temperature change Tand its boundary is traction free.

The thermal expansion contribution tensor HT of theinhomogeneity is defined by the following relation forthe overall strain per volume V:

eij ¼ a0ijT þ HT

ijT; ð2:1Þ

where the second term represents the strain change D eij

due to the presence of the inhomogeneity and dependson its shape and properties. To calculate Deij note thatthe average stress over the material is zero (since theboundary is traction free). Thus,

V�

VhrijiV� þ

V � V�

VhrijiV�V� ¼ 0; ð2:2Þ

where angle brackets mean average over the volume indi-cated in the subscript. Similar equation for strains is

eij¼V�

VS1

ijklhrkliV� þa1ijT

h iþV�V �

VS0

ijklhrkliV�V� þa0ijT

h i: ð2:3Þ

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 23

Combining (2.1), (2.2) and (2.3) yields

Deij ¼V�

VS1

ijkl � S0ijkl

� Akl þ a1

ij � a0ij

� h iT; ð2:4Þ

where second rank symmetric tensor Akl represents aver-age stresses in the inhomogeneity in response to the tem-perature change T:hrijiV� ¼ AijT: ð2:5ÞTensor Akl can be expressed in closed explicit form if theinhomogeneity has ellipsoidal shape. For this case, we uti-lize solution of Eshelby problem (Eshelby, 1957, 1961) inthe form suggested by Vakulenko (1968). The Hooke’slaw for the system of interest can be written as

rij ¼ Cijklðekl � aklTÞ

¼ C0ijkl þ dCijkl

� ekl � a0

kl þ dakl

� �T

� ; ð2:6Þ

where dCijkl ¼ C1ijkl � C0

ijkl

� vðXÞ; dakl ¼ a1

kl � a0kl

� �vðXÞ, and

v(X) is the characteristic function of domain X that isequal to unity in X and to zero outside X. After some alge-bra (2.6) is reduced to

rij ¼ C0ijkl ekl � a0

klT� �

þ C1ijkl � C0

ijkl

� ekl � C1

ijkla1kl � C0

ijkla0kl

� T

h ivðXÞ ð2:7Þ

and the equilibrium equations @rij/oxi = 0 take the form

C0ijkl@ekl

@xiþ Fj ¼ 0; ð2:8Þ

where

Fj ¼@

@xiC1

ijkl � C0ijkl

� ekl � C1

ijkla1kl � C0

ijkla0kl

� T

h ivðXÞ

n o:

ð2:9Þ

This equation implies the following formulation of theproblem: homogeneous matrix with elastic constants C0

ijkl

contains fictitious body forces Fj distributed in domain X.Hence, the solution can be represented in terms of Green’sfunction for the matrix material:

eijðxÞ¼a0ijT� C1

mnkl�C0mnkl

� @

@xj

ZX

@Gin x�x0ð Þ@xm

ekl x0ð Þdx0

þ C1mnkla

1kl�C0

mnkla0kl

� T@

@xj

ZX

@Ginðx�x0Þ@xm

dx0: ð2:10Þ

Expression (2.10) is an integral equation for eij(x). How-ever, in the case of ellipsoidal shape of domain X, stressesand strains inside X are uniform, under the homogeneousboundary conditions (Eshelby, 1957, 1961):

einij ¼ a0

ijT � Pijmn C1mnkl � C0

mnkl

� ein

kl

þ Pijmn C1mnkla

1kl � C0

mnkla0kl

� T; ð2:11Þ

where

PmpijðxÞ �@

@xðp

ZX

@GmÞðjðx� x0Þ@x0iÞ

dx0 ð2:12Þ

is fourth-rank Hill’s tensor (Hill, 1963; Walpole, 1967).Parentheses in subscripts indicate symmetrization with re-spect to m M p and i M j. Hill’s tensor has the same symme-try as the tensor of elastic constants:

Pijkl ¼ Pjikl ¼ Pijlk ¼ Pklij: ð2:13Þ

Solving (2.11) for eij, we get the strains inside X as

einij ¼ Hijkl a0

kl þ Pklmn C1mnrqa

1rq � C0

mnrqa0rq

� h iT; ð2:14Þ

where

Hijkl ¼ Jijkl þ Pijrs C1rskl � C0

rskl

� h i�1ð2:15Þ

is the ‘‘strain concentration’’ tensor introduced by Wu(1966) for isothermal Eshelby problem and Jijkl ¼ðdikdlj þ dildkjÞ=2 is the fourth rank unit tensor; an inverseX�1

ijkl of fourth-rank tensor is defined by the relationX�1

ijmnXmnkl ¼ XijmnX�1mnkl ¼ Jijkl. Stresses inside the ellipsoidal

inhomogeneity can then be written as

rinij ¼ C1

ijkl einkl � a1

klT� �

¼ C1ijkl Hklmn Jmnpq � PmnrsC

0rspq

� a0

pq

h� Jklpq �HklmnPmnrsC

1rspq

� a1

pq

iT ¼ C1

ijklHklmn Jmnpq � PmnrsC0rspq

� a0

pq � a1pq

� T: ð2:16Þ

Thus tensor Akl entering (2.4) and (2.5) has the followingform:

Akl ¼ C1ijklHklmn Jmnpq � PmnrsC

0rspq

� a0

pq � a1pq

� : ð2:17Þ

So that Deij can be written, after some algebra as

Deij ¼ HTijT ¼ Hijkl S1

klmn � S0iklmn

� �1a1

mn � a0mn

� �T; ð2:18Þ

where Hijkl is compliance contribution tensor that describesextra strain per volume of the specimen due to the presenceof an inhomogeneity and is related to Hklmn by

Hijkl ¼ S1ijmn � S0

ijmn

� C1

mnqpHqprsS0rskl: ð2:19Þ

Remark 1. Compliance contribution tensor was first intro-duced in the form of tensor by Kachanov et al. (1994). Thebasic idea of such approach was, however formulated inclassical papers of Eshelby (1956, 1961).

Remark 2. It is seen from formula (2.18) that pores do notcontribute to the overall thermal expansion coefficient.

Remark 3. Levin’s formula (1.8) can be easily obtainedfrom (2.18) (or vice versa) if we represent effective proper-ties as aij ¼ a0

ij þ HTij and Srsij ¼ S0

rsij þ Hrsij.Similarly to compliance contribution tensor, HT

ij de-pends on shape and orientation of inhomogeneities, andis scaled as characteristic sizes of inhomogeneities cubed.This tensor identifies the general proper microstructuralparameter for effective thermal expansion coefficient of aheterogeneous material – namely, the sumX

p

HTðpÞ

ij : ð2:20Þ

This general parameter covers, in a unified way, mixturesof inhomogeneities of diverse shapes and orientations.Since tensors HT

ij represent individual inhomogeneities in

24 I. Sevostianov / Mechanics of Materials 45 (2011) 20–33

accordance with their actual contributions to the effectivethermal expansion coefficient, using parameter (2.20) in-sures that aij is a unique function of this parameter, at leastin the non-interaction approximation.

3. Particular case of isotropic phases

Substantial simplification of formula (2.18) is possiblein the case of isotropic phases and spheroidal shape ofthe inhomogeneity. In this section, we consider this caseto illustrate the shape dependence of the thermal expan-sion contribution tensor. For isotropic constituents

S1klmn � S0

iklmn

� �1a1

mn � a0mn

� �¼ 3K1K0ða1 � a0Þ

K0 � K1dij: ð3:1Þ

For spheroidal inhomogeneity (semiaxes are a1 = a2 and a3

and unit vector of the spheroid’s axis is m), we deriveexpressions for components HT

ij in terms of the aspect ratioc = a3/a1 and relevant material constants utilizing expres-sion for tensor Hijkl in the form given by Sevostianov andKachanov (2002):

H ¼ V�V½ðS� � S0Þ�1 þ Q ��1

; ð3:2Þ

where fourth rank tensors are represented in terms of acertain ‘‘standard’’ tensor basis T(1), . . . ,T(6):

H ¼ V�V

X6

k¼1

hkT ðkÞ; S ¼X6

k¼1

skT ðkÞ; Q ¼X6

k¼1

qkT ðkÞ; ð3:3Þ

so that finding these tensors reduces to calculation of fac-tors hk, sk, and qk. Tensors T(1), . . . ,T(6) are formed by com-binations of unit tensor dij and two orthogonal unitvectors (see Kunin, 1983 and Walpole, 1984):

Tð1Þijkl¼hijhkl; Tð2Þijkl¼ðhikhljþhilhkj�hijhklÞ=2; Tð3Þijkl¼hijnknl;

Tð4Þijkl¼ninjhkl Tð5Þijkl¼ðhiknlnjþhilnknjþhjknlniþhjlnkniÞ=4;

Tð6Þijkl¼ninjnknl; ð3:4Þ

where hij = dij � ninj and n = n1e1 + n 2e2 + n3 e3 is a unit vec-tor along the axis of transverse symmetry. Note that con-traction by two indices of any tensor X ¼

P6k¼1XkT ðkÞ with

second rank unit tensor yieldsXijkldkl ¼ ð2X1 þ X4Þhij þ ð2X3 þ X6Þninj: ð3:5Þ

We utilize representations for qi given by Sevostianov andKachanov (2002)

q1 ¼ G0½4j� 1� 2ð3j� 1Þf0 � 2jf1�;q2 ¼ 2G0½1� ð2� jÞf0 � jf1�;q3 ¼ q4 ¼ 2G0½ð2j� 1Þf0 þ 2jf1�;q5 ¼ 4G0½f0 þ 4jf1�; q6 ¼ 8G0j½f0 � f1�;

ð3:6Þ

where j = 1/[2(1 � m0)] and the geometrical factors g, f0,and f1 are functions of the aspect ratio c only:

f0 ¼c2ð1� gÞ2ðc2 � 1Þ f 1 ¼

c2

4ðc2 � 1Þ2½ð2c2 þ 1Þg � 3�;

g¼1

cffiffiffiffiffiffiffiffi1�c2p arctan

ffiffiffiffiffiffiffiffi1�c2p

c ; oblate shape ðc<1Þ

1cffiffiffiffiffiffiffiffic2�1p lnðcþ

ffiffiffiffiffiffiffiffiffiffiffiffiffic2�1

pÞ; prolate shape ðc>1Þ:

8><>: ð3:7Þ

Now, accounting for (3.1) and (3.2), Eq. (2.18) can berewritten as

HTij

� �1¼ 1ða1 � a0Þ

dij þ Q ijkldklðK0 � K1Þ

3K1K0ða1 � a0Þ

��1

: ð3:8Þ

Since, according to (3.5), Qijkldkl = (2q1 + q4)hij + (2q3 + q6)ninj, substitution of (3.6) into (3.8) yields

HTij ¼ ða1 � a0Þ½M1hij þM2mimj�; ð3:9Þ

where

M1 ¼1

1þ sð1� f0Þ; M2 ¼

11þ 2sf0

;

and s ¼ ðK0 � K1ÞK1

1� 4m20

1� m20

; ð3:10Þ

Fig. 1a illustrates dependence of f0 on the aspect ratio ofspheroid. Fig. 1b and 1c show dependences of M1 and M2

for various values of s. It is seen that the dependence isweak for small values of dimensionless parameter s thatcorrespond to either (1) small elastic contrast (K1 � K0) or(2) m0 ? 0.5. In these cases the contribution of the inhomo-geneities into overall thermal expansion is shape indepen-dent and, therefore, no anisotropy can be produced by non-spherical inhomogeneities.

4. Effective thermal expansion of a heterogeneousmaterial

Thermal expansion contribution tensor derived in theprevious two sections can be used to calculate effectivethermal properties of heterogeneous materials (in particu-lar – multiphase composites) with various methods ofaveraging developed in the mechanics of composites. Inthe present section, we discuss non-interaction approxi-mation and effective field methods.

4.1. Non-interaction approximation

This approximation becomes rigorous at low concentra-tion of inhomogeneities (‘‘dilute limit’’) and produces rea-sonably accurate results for concentrations ofinhomogeneities up to 15% (see, for example, discussionof Sevostianov and Sabina, 2007). If interaction betweenthe inhomogeneities is neglected, contribution of the inho-mogeneity into the strain generated by temperaturechange T can be treated separately and the effective tensorof thermal expansion is

aij ¼ a0ij þ

Xi

HTij

!: ð4:1Þ

The summation over inhomogeneities may be changed bythe integration over orientations if convenient. In particu-lar, in the case of isotropic orientation distribution of sphe-roidal inhomogeneities of aspect ratio c (randomlyoriented spheroidal inhomogeneities of the same aspectratio)

eij ¼ a0 þ cða1 � a0Þ½2A1 þ A2�

3

�dijT: ð4:2Þ

0.01 0.1 1 10 1000.0

0.1

0.2

0.3

0.4

0.5f0

0.2=τ0.5=τ1.0=τ2.0=τ10.0=τ40.0=τ

γ

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0M1 M2

γ γ

(c)(b)

(a)

Fig. 1. (a) Dependence of the shape factor f0 on the aspect ratio of spheroid; (b), (c) dependences of M1 and M2 for various values of s. It is seen that thedependence is weak for small values of dimensionless parameter s that correspond to either (1) small ratio K1/K0 or (2) small values m0.

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 25

Using the methodology developed by Eroshkin and Tsuk-rov (2005) for effective elastic properties and Sevostianovet al. (2006) for effective elastic and conductive properties,one can introduce non-interaction thermal expansion con-tribution tensor HT

NI such that tensor of the effective ther-mal expansion coefficients aij is expressed in its terms as

aij ¼ a0dij þ HTNI

� ij: ð4:3Þ

For N sets of randomly distributed spheroidal inhomogene-ities of aspect ratios ci, i = 1,2, . . . ,N (N + 1 phase matrixcomposite)

HTNI

� kl¼

XN

i¼1

ciðai � a0Þ2MðiÞ

1 þMðiÞ2

h i3

24

35dklT

�XN

i¼1

ciðai � a0ÞgðiÞNIðc; sÞ" #

dklT; ð4:4Þ

where subscript i marks bulk modulus, thermal expansioncoefficient and concentration of ith phase. Fig. 2 illustratesdependence of parameter gNI on the aspect ratio c for var-ious values of s. This dependence is even weaker thanthose of M1 and M2 separately. In fact, for ðK1�K0Þ

K16

1�m20

1�4m20

ran-domly oriented spheroidal inhomogeneities can be treatedas spheres.

Note, that the non-interaction approximation, besidesbeing rigorous at small concentration of inhomogeneities,is of a fundamental importance since, as shown below, itserves as a basic build block for various commonly used

approximate schemes that place non-interacting inclu-sions into some sort of ‘‘effective environment’’ (effectivefield or effective matrix).

The easiest way to obtain formulas for effective thermalexpansion coefficients of a two-phase material accordingto various approximate schemes is to use formula (2.18).Then, if compliance contribution tensor HA

ijkl of the entireset of inhomogeneities according to certain scheme ‘‘A’’,is known, i.e.

eij ¼ S0ijkl þ HA

ijkl

� r1kl ð4:5Þ

the effective tensor of thermal expansion coefficients cal-culated according to the same scheme ‘‘A’’ is

aAij ¼ a0dij þ HA

ijkl S1klmn � S0

klmn

� �1a1

mn � a0mn

� �: ð4:6Þ

Note that some schemes like effective media of differentialallows explicit formulas for isotropic composites only, and,even in this case, require solution for a system of non-lin-ear algebraic equation (see, for example, review of Markov,2000). The only methods that allow one to derive analyti-cal expressions for general case are effective field schemes– either Mori–Tanaka scheme (MTS) or Kanaun–Levinscheme (KLS). Below, we discuss their application to thecalculation of the tensors of effective thermal expansioncoefficients.

In the effective field methods, each particle is treated asisolated one. All the inhomogeneities are considered asembedded into the matrix material and the effect of

0.01 0.1 1 10 100

0.02=τ0.2=τ0.5=τ2.0=τ5.0=τ20.0=τ

0.0

0.1

2.0

3.0

γ

η

Fig. 2. Dependence of parameter gNI entering (4.4) on the aspect ratio c for various values of s. This dependence is even weaker than those of M1 and M2

presented in Fig. 1.

26 I. Sevostianov / Mechanics of Materials 45 (2011) 20–33

interaction is accounted for through the assumption thateach particle lies within a certain effective field that differsfrom the applied macroscopic one. The basic idea of themethod has roots in works of Mossotti (see Feynmanet al., 1964, chapter 11). Two variants of the effective fieldmethod are used in the mechanics of composites.

4.2. Mori–Tanaka scheme

This approach (Mori and Tanaka, 1973) as interpretedby Benveniste (1986) is based on the assumption that theeffective field acting on each inhomogeneity is equal tothe average over the matrix. Then the macroscopic proper-ties may be calculated from the non-interaction approxi-mation with appropriate change of the remotely appliedfield. Contribution of the inhomogeneities into overallcompliance of the composite in the framework of MTS isdescribed as (Benveniste, 1986)

HMTijkl ¼ ð1� cÞ HNI

rskl

� �1þ S1

rskl � S0rskl

� �1 ��1

; ð4:7Þ

where HNIijmn is the fourth rank tensor describing contribu-

tion of the inhomogeneities into the overall compliancecalculated in the framework of non-interactionapproximation:

HNIijmn ¼

Xr

HðrÞijmn ð4:8Þ

and HðrÞijmn is given by (3.2). So that, according to (4.6), effec-tive thermal expansion coefficient is

aAij ¼a0dijþHNI

ijmn ð1�cÞ S1mnrs�S0

mnrs

� þHNI

mnrs

h i�1a1

rs�a0rs

� �:

ð4:9Þ

This expression coincide (with the accuracy of notations)with ones given in the papers of Takahashi et al. (1980)and Benveniste and Dvorak (1990). In the paper of Withers

et al. (1989) expressions for their tensors A and B are notspecified, so it is difficult to compare the results.

After some algebra the latter equation can be rewrittenin terms of tensor HT

NI

� ij

introduced in (4.3):

aij ¼ a0dij þ ð1� cÞ HTNI

� �1

ijþ a1

ij � a0ij

� ��1

: ð4:10Þ

For isotropic mixture of inhomogeneities, expression forHT

NI

� ij

is simplified according to (4.4).

Kanaun–Levin scheme represents a more refined variantof the effective field method. This method is free from thedisadvantages of the Mori–Tanaka scheme discussed byQui and Weng (1990) and Ferrari (1991). This methodhas been proven on a variety of microstructures and, asshown by Markov (2001), its results coincide with the onesobtained later by Ponte-Castaneda and Willis (1995) usingvariational methods. The method was developed in worksof Levin (1975) and Kanaun (1977, 1982) (see book of Kan-aun and Levin, 2008) for elastic properties of composites.Note that predictions of Kanaun–Levin method for porousand microcracked materials coincide with ones obtainedby Kuster and Toksoz (1974) in application for porous rock.

Explicit expression of the effective elastic moduli in KLSinvolves HNI

ijmn as well as tensor Q that was introduced in(3.2). The latter tensor describes mutual orientation andmutual positions of the inhomogeneities. For example, inthe case of randomly oriented inhomogeneities we usetensor Q calculated for a sphere. In the case of strictly par-allel identical ellipsoids, we have to use tensor Q calculatedfor this ellipsoid. In more complex cases (like in the case ofinhomogeneities formed certain Brave lattice with givenorientation distribution) tensor Q has to be calculatednumerically. If there are two or more distinct families ofparallel ellipsoids then, instead of one tensor, several onesare used to describe mutual positions of the ellipsoids–four in the case of two families, nine for three families,etc. (Kanaun and Jeulin, 2001).

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 27

Contribution of the inhomogeneities into overall com-pliance of the composite in the framework of KLS is de-scribed by

HKLijkl ¼ HNI

ijkl

� �1� Qijkl

��1

: ð4:11Þ

So that the tensor of effective thermal expansion coeffi-cients can be written in terms of HT

NI

� ij

as follows:

aij ¼ a0dij þ HTNI

� �1

ij� Q T

ij

��1

; ð4:12Þ

where second-rank symmetric tensor Q Tij is

Q Tij ¼

1a1 � a0

Q ijkl S1klmm � S0

klmm

� : ð4:13Þ

For isotropic distribution of spheroidal inhomogeneities

Q Tij ¼

2a1 � a0

K0

K1 � K0

1� 2m0

1� m0: ð4:14Þ

Remark. In the case of spherical particles, as well as in thecase of aligned ellipsoids such that the overall patternrepeats the single inhomogeneity, Kanaun–Levin andMori–Tanaka scheme produce the same results (see dis-cussion in Chapter 7 of Kanaun and Levin (2008)).

Below we compare the predictions according to effec-tive field methods with the experimental data availablein literature and with calculations according to otherschemes discussed in Section 1.

For comparison, we use two composites – copper rein-forced with diamond particles and aluminum alloy rein-forced with eucryptic glass. Material constants for theconstituents are given in the Tables 1 and 2, respectively.

Experimental data for copper/diamond composite aretaken from the papers of Yoshida and Morigami (2004)and Sun and Inal (1996). In the latter one temperaturedependent properties are discussed, so we taken the datafor 40� C only to make them compatible with another set.The shape of the particles is reported to be almost sphericaland the thermal expansion coefficient is isotropic. Compar-

Table 1Material properties of copper and diamond (Sun and Inal, 1996).

a, �C K, GPa m

Copper 16.55 � 10�6 140 0.34Diamond 1.1 � 10�6 580 0.24

Table 2Material properties of aluminum alloy SAS-1 and eucryptite glass (Fridly-ander et al., 1974). Eucryptite glass has negative thermal expansioncoefficient.

a, �C K, GPa m

SAS � 1 16.0 � 10�6 104.17 0.34Eucryptic glass �9.0 � 10�6 68.2 0.28

ison of the calculations with experimental data is pre-sented in Fig. 3. It is seen that, for this composite, almostall the methods work reasonably well. NIA gives reason-ably accurate predictions for the range of the volume frac-tion of the filler up to 15%, as expected. Note, however, thatthe slopes of the theoretical curves built according toschemes discussed in Section 1 differ from that of NIAcurve at low concentration of inhomogeneities. Since NIAprovides a rigorous solution at low volume fraction of filler,it indicates certain qualitative inconsistency of the men-tioned approaches.

For the second material, we took a composite one con-stituent of which (eucryptic glass) shows negative thermalexpansion. The experimental data are reported by Fridlyan-der et al. (1974). The shape of the inhomogeneities wasapproximated by oblate spheroids of aspect ratio 1/3. Com-parison of analytical results with experimental data is pre-sented in Fig. 4. For this material, no one method except ofKLS provides reasonable predictions. NIA, again works wellin the expected interval of the filler volume fraction.

5. Cross-property connection between thermalexpansion and thermal resistivity

In the present section we derive the cross-property con-nection between effective thermal expansion and thermalresistivity. We utilize the methodology developed bySevostianov and Kachanov (2002, 2008) where cross-prop-erty connection is derived through elimination of propermicrostructural parameters if they are either the same orsimilar for the properties of interest. Both thermal expan-sion and thermal conductivity are described by second-rank tensors. Thermal resistivity contribution tensor hasthe form similar to (3.8) (Sevostianov and Kachanov,2002):

HR ¼ V�r0fA1I þ A2nng; ð5:1Þ

where factors A1and A2 are as follows

A1 ¼k0 � k1

k0 þ ðk1 � k0Þf0ðcÞ;

A2 ¼ðk0 � k1Þ2ð1� 3f 0ðcÞÞ

½k1 � 2ðk1 � k0Þf0ðcÞ�½k0 þ ðk1 � k0Þf0ðcÞ�

and f0(c) is given by (3.7). After some algebra (5.1) can berewritten as

ðHRÞij ¼V�Vðr1 � r0ÞfA1hij þ A2mimjg; ð5:2Þ

where

A1 ¼1

1þ kð1� f0Þ; A2 ¼

11þ 2kf0

;

and k ¼ r1 � r0

r0: ð5:3Þ

Comparison of (5.3) with (3.8) shows that dependences ofthe thermal resistivity and thermal expansion contributiontensors on inhomogeneity aspect ratios are the same. Theonly difference is in dimensionless material parameters sand k. Parameter s characterizes elastic contrast between

0.0 0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1.00αα

c

Kanaun-Levin(eq 4.12)

Non interaction(eq 4.4)

Mori-Tanaka(eq 4.10)

Tummala-Friedberg(eq 1.11)

Kerner(eq 1.2)

Blackburn(eq 1.6)

Turner(eq 1.1)

Fridlyander et al (1974)

Typical shape of eucryptic glass particles (Fridlyander et al., 1974).

Fig. 4. Effective coefficient of thermal expansion of SAS-1 aluminum alloy reinforced with eucryptic glassceramic particles (eucryptic glass has negativethermal expansion coefficient). Comparison of the calculations with experimental data of Fridlyander et al. (1974). No one method (except of effective field)provides reasonable predictions. Inset shows typical shape of the eucryptic glass inhomogeneities.

28 I. Sevostianov / Mechanics of Materials 45 (2011) 20–33

two phases of the composite, while k is the indicator of thecontrast in conductive properties.

For a solid with many inhomogeneities, utilizingrelations (3.8) and (5.3), we obtain the following effectiveproperties in the framework of non-interaction approxi-mation:

0αα

0.0

0.25

0.50

0.75

1.0

0.0 0.2 0.4 0.6 0

Yoshida and Mirugami (2004)

40-60 µm 20-30 µm

Sun and Inal (19

Fig. 3. Effective coefficient of thermal expansion of diamond-copper matrix comInal (1996) and Yoshida and Morigami (2004). For this composite, almost all tscheme) and (4.12) (Kanaun–Levin’s scheme) coincide in this case. Note, howediscussed in Section 1 differ from that of NIA curve at low concentration of inh

aij ¼a0dijþ1Vða1�a0Þ dij

Xk

ðV �M1ÞðkÞ þX

k

½V � ðM2�M1Þninj�ðkÞ( )

;

ð5:4Þ

rij ¼ r0dijþ1Vðr1� r0Þ dij

Xk

ðV �A1ÞðkÞ þX

k

½V � ðA2�A1Þninj�ðkÞ( )

;

ð5:5Þ

c

.8 1.0

96)

Kanaun-Levinand Mori-Tanaka(eq 4.10 and 4.12)

Non interaction(eq 4.4)

Tummala-Friedberg(eq 1.11)

Kerner(eq 1.2)

Blackburn(eq 1.6)

Turner(eq 1.1)

posite. Comparison of the calculations with experimental data of Sun andhe methods work reasonably well. Calculation by (4.10) (Mori–Tanaka’sver, that the slopes of the theoretical curves built according to schemesomogeneities.

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 29

where coefficients Mi and Ai are given by (3.8) and (5.3),respectively. The general structure of these formulas (aswell as the logic of the derivation to follow) is discussedby Sevostianov and Kachanov (2002) and Kachanov et al.(2001).

These formulae apply to an arbitrary mixture of inhomo-geneities of diverse aspect ratios and orientations and con-tain factors Ai, and Mi that depend on the shapes andmaterial properties of the inhomogeneities. If inhomogene-ities’ aspect ratios are not correlated with either orienta-tions of the inhomogeneities or their volumes (note thatvolumes and orientations may be correlated), coefficientsAi, and Mi can be replaced by their averages and taken outof the summation signs (if all the inhomogeneities havethe same orientation n, this requirement reduces to thecondition that the distributions over shapes and over vol-umes of the inclusions are uncorrelated). This importantcase appears to be relevant for realistic microstructures.

Then tensors aij and rij are expressed in terms of thesame second rank symmetric tensor

x ¼ 1V

Xi

ðV�nnÞðkÞ ð5:6Þ

(its trace trx ¼ ð1=VÞP

V� is the volume fraction of inclu-sions c) as follows:

aij ¼ a0dij þ ða1 � a0Þðm1cdij þ ðm2 �m1ÞÞxij; ð5:7Þrij ¼ r0dij þ ðr1 � r0Þða1cdij þ ða2 � a1ÞÞxij: ð5:8Þ

Coefficients mi and ai – average shape factors for the ther-mal expansion and thermal conductivity problems, respec-tively – are averages (over all the inhomogeneities) ofcoefficients Mi and Ai:

mi ¼Z 1

0MiðcÞf ðcÞdc ai ¼

Z 1

0AiðcÞf ðcÞdc; ð5:6Þ

where f(c) is the shape distribution density.To derive cross-property connection between thermal

resistivity and thermal expansion, it is convenient to repre-sent tensors aij and rij as the sums of their volumetric anddeviatoric parts. Then

aij ¼13

trðaijÞdij þ a0ij ¼ a0 þða1 � a0Þ

3ð2m1 þm2Þc

�dij

þ ða1 � a0Þðm2 �m1Þx0ij; ð5:7Þ

rij ¼13

trðrijÞdij þ r0ij ¼ r0 þðr1 � r0Þ

3ð2a1 þ a2Þc

�dij

þ ðr1 � r0Þða2 � a1Þx0ij; ð5:8Þ

where prime indicates the deviatoric part and tr( ) stays forthe trace of a tensor. Eliminating now c and x0ij we get ex-plicit cross-property connection between thermal resistiv-ity and thermal expansion coefficient:

rij � r0dij ¼ðr1 � r0Þða1 � a0Þ

� 2a1 þ a2

2m1 þm2

13

trðaijÞ � a0

� �dij þ

a2 � a1

m2 �m1a0ij

�: ð5:9Þ

This connection is exact in the framework of non-interac-tion approximation. Generally, coefficients (2a1 + a2)/(2m1 + m2) and (a2 � a1)/(m2 �m1) are shape dependent.

This shape sensitivity depends on the ratio betweendimensional material parameters s ¼ ðK0�K1Þ

K1

1�4m20

1�m20

andk ¼ r1�r0

r0. Obviously, when these parameters are equal,

(5.9) is shape independent. However, in most cases theseparameters are significantly different. This situation is typ-ical, for example for polymer matrix composites. In thiscase shape dependence may be very strong. Fig. 5 illus-trates this dependence for k = �0.9 and k = �0.5 and vari-ous positive values of s. Note that this strong shapedependence is in contrast with one observed for elastic-ity-conductivity cross-property connections, where shapedependence is moderate. It is associated with negative signof k. For positive values of this parameter the shape depen-dence is much weaker (see Fig. 6).

We emphasize that the derived cross-property connec-tion cover all inhomogeneity shapes (including mixtures ofdiverse shapes) and orientation distributions of inhomoge-neities in a unified way. It contains elastic constants, ther-mal expansion coefficients and thermal resistivities of theconstituents and average shapes of the inhomogeneitiesand does not contain any adjustable parameters.

Cross-property connection (5.9) expresses the changes inthe thermal resistivity tensor due to inhomogeneities as alinear function of the changes in the tensor of thermal expan-sion coefficients. The utility of this connection can be viewedas follows. If the tensor of effective thermal expansion coef-ficients aij is known, then the only additional informationneeded to find the thermal conductivities is approximateestimate of the average inclusion shapes – combinations offactors mi and ai – and not the orientation distribution.

If the composite material is isotropic (inhomogeneitiesare either spherical or randomly oriented), tensor xij is iso-tropic (xij = cdij) and the cross-property connection con-tains only one shape factor

r � r0

r0¼ ðr1 � r0Þ

r0

ða� a0Þða1 � a0Þ

2a1 þ a2

2m1 þm2

�¼ W

a� a0ð Þa0

;

ð5:10Þ

where W ¼ ðr1�r0Þr0

a0ða1�a0Þ

2a1þa22m1þm2

h i.

Remark. The cross-property connection based on similar-ity between microstructural parameters was first proposedby Bristow, 1960 for a material containing randomlyoriented microcracks. Chen (1993) expressed thermoelas-tic and conductive properties of a composite reinforcedwith spherical particles in terms of the same microstruc-tural parameters (formula (47) in the mentioned paper),but he did not eliminate it to connect different physicalproperties. The detailed review of various approaches tocross-property connections is given in Chapter 2 of thereview of Sevostianov and Kachanov (2008).

We now verify this cross-property connection on theexperimental data of Wong and Bollampally (1999) forthree composites with epoxy matrix reinforced with silica,alumina, or SCAN particles. Material properties of theconstituents are given in Table 3. Shapes of the inhomo-geneities were estimated from the microphotographsprovided in the paper according to methodology discussed

3

11

2

2

0.01 0.1 1 10 10010

100

1000

1

1 2

2

3

3

3

0.01 0.1 1 10 100

10

100

1

1

2

3

Fig. 5. Dependence of shape factors 2a1þa22m1þm2

and a2�a1m2�m1

on the aspect ratio of identical spheroidal inhomogeneities for negative values of parameter k. Notevery strong shape dependence of a2�a1

m2�m1for 0.1 6 c 6 10 in the case of k = �0.9.

30 I. Sevostianov / Mechanics of Materials 45 (2011) 20–33

by Prokopiev and Sevostianov (2007) and Sevostianov et al.(2006). We first calculated cross-property coefficient W aswell as parameters s and k for the materials we consider.These coefficients are given in Table 4 for illustration togive the reader an idea about their values. We emphasizethat these parameters are not adjustable ones, but arecombinations of standard thermoelastic constants of theconstituents and shapes of the inhomogeneities. Fig. 7illustrates workability of the derived cross-property con-nections: from the data for the variation of thermalexpansion coefficient with the volume concentration offiller we calculated thermal resistivity according to (5.10)and compared the results with the reported data. It is seenthat the agreement is quite good for all three materialsconsidered. In particular it means that interaction betweeninhomogeneities affects both thermal conductivity andthermal expansion in the same way, since the cross-property (5.10) has been derived in the framework of non-interaction approximation.

6. Conclusions

The paper addresses thermal properties of compositeswith microstructures that comprise a mixture of inhomo-

geneities of diverse shapes and orientations. The existingapproaches to this problem are mostly focused on thedescription of interaction between the inhomogeneities.We emphasize that the problem of calculation of the effec-tive thermal expansion is closely related to the one ofquantitative characterization of composite microstructure– identification of microstructural parameters, in whoseterms the tensor of thermal expansion coefficients is tobe expressed. These proper microstructural parametersshould represent the individual inhomogeneities in accor-dance with their contributions to the thermal expansioncoefficients. The advantages of such approach include(Kachanov and Sevostianov, 2005)

� Quantitative characterization of microstructures involv-ing mixtures of diverse inhomogeneities.� Explicit cross-property connections can be established

between two physical properties, if the proper parame-ters for these properties are sufficiently similar.

We explicitly derived this parameter, expressed effec-tive thermal expansion of a composite in its terms andcompared the results with experimental data available inliterature. Using this parameter, we also derived cross-property connection that interrelates, in the closed form,

Table 4Dimensionless material parameters entering cross-property connections.

Silica/Epoxy SCAN/Epoxy Alumina/Epoxy

s ¼ ðK0�K1ÞK1

1�4m20

1�m20

� 0.356 �1.161 �1.163

k ¼ r1�r0r0

�0.87 �0.999 �0.995

W 0.96 1.62 1.64

1=λ

10=λ

1

2

3

0.01 0.1 1 10 100

0.01 0.1 1 10 100

10

100

1

0.1

10

1

0.1

12

3

1

2 3

1

2

3

Fig. 6. Dependence of shape factors 2a1þa22m1þm2

and a2�a1m2�m1

on the aspect ratio of identical spheroidal inhomogeneities for positive values of parameter k. Noteshape insensitivity for prolate shapes of inhomogeneities.

Table 3Thermoelastic properties of materials used for verification of cross-property connections.

Material a, ppm/�C K, GPa m k, W/mK

Epoxy 88 3.75 0.41 0.195Silica 0.5 39 0.19 1.5SCAN 4.4 220 0.25 220Alumina 6.6 247 0.24 36

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 31

the effective thermal expansion coefficients and the ther-mal conductivities of anisotropic composites. This cross-property connection may be valuable for applications sincethermal expansion coefficients are easier to measure thananisotropic thermal conductivities. Comparison withexperimental data for composites with extremely contrastconstituents (negative k) shows that the connections arequite accurate even in this case. The agreement withexperimental data is in the limits of experimental error.

Our results are given in closed form that explicitly re-flects shapes of inhomogeneities. They are derived in thenon-interaction approximation. However, they can bereformulated, in a straightforward way, in the frameworkof the commonly used approximate schemes (self-consis-tent, differential, Mori–Tanaka’s) that place non-interact-ing inhomogeneities into some sort of ‘‘effectiveenvironment’’ (see Sevostianov et al., 2006 for cross-prop-erty between elastic and conductive properties). The de-rived connections contain factors that depend on theaverage shapes of inhomogeneities. Their presence reflectsthe fact that inhomogeneities generally affect the thermalexpansion and the thermal conductivity differently; other-wise, the cross-property connections would be indepen-dent of microstructures. The shape dependence becomesespecially important if parameter k, characterizing contrastin conductivities of two phases becomes close to its limit-ing value-1. However, even in this limiting case, the infor-mation on the microstructure that is reflected in the shapefactors is much less detailed than the one required for a di-rect expression of the effective properties in terms of themicrostructure (for example, knowledge of the orientationdistribution of inhomogeneities is not needed).

Acknowledgement

The research is sponsored by NASA (Grant No.GR0002488). The author is grateful to the anonymous

0.0 0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1.0

0.0

c

Silica particles

SCAN particles

Alumina particles

Fig. 7. Verification of cross-property connection (5.10) for epoxy matrix composites reinforced with three types of ceramic particles. Experimental data aretaken from Wong and Bollampally (1999).

32 I. Sevostianov / Mechanics of Materials 45 (2011) 20–33

reviewer for pointing at papers of Benveniste and Dvorak(1990) and Kuster and Toksoz (1974).

References

Arthur, G., Coulson, J.A., 1964. Physical properties of uranium dioxide-stainless steel cermets. Journal of Nuclear Materials 13, 242–253.

Benveniste, Y., 1986. On the Mori–Tanaka method for cracked solids.Mech. Res. Comm. 13 (4), 193–201.

Benveniste, Y., Dvorak, G.J., 1990. On a correspondence betweenmechanical and thermal effects in two-phase composites. In: Weng,G.J., Taya, M., Abe, H. (Eds.), Micromechanics and InhomogeneitiesThe Toshio Mura 65th Anniversary Volume. Springer.

Berryman, J.G., 2009. Frequency dependent thermal expansion in binaryviscoelastic composites. Mech. Materials 41, 463–480.

Chen, T., 1993. Thermoelastic properties and conductivity of compositesreinforced by spherically anisotropic particles. Mech. Materials 14,257–268.

Cribb, J.L., 1968. Shrinkage and thermal expansion of a 2 phase material.Nature 220, 576–577.

Dunn, M.L., 1993. Exact relations between the thermoelectroelasticmoduli of heterogeneous materials. Proc. Roy. Soc. L. A441, 549–557.

Eroshkin, O., Tsukrov, I., 2005. On micromechanical modeling ofparticulate composites with inclusions of various shapes. Int. J.Solids Struct. 42, 409–427.

Eshelby, J.D., 1956. The continuum theory of lattice defects. Solid StatePhys. 3, 79–144.

Eshelby, J.D., 1957. The determination of the elastic field on an ellipsoidalinclusion and related problems. Proc. Roy. Soc. L. A 241, 376–392.

Eshelby, J.D., 1961. Elastic inclusions and inhomogeneities. In: Sneddon,I.N., Hill, R. (Eds.), Progress in Solid Mechanics, vol. 2. Norht-Holland,Amsterdam, pp. 89–140.

Fahmi, A.A., Ragai, A.N., 1970. Thermal expansion behavior of two-phasesolids. J. Appl. Phys. 41, 5108–5111.

Feltham, S.J., Yates, B., Martin, R.J., 1982. The thermal expansion ofparticulate-reinforced composites. J. Materials Sci. 17, 2309–2323.

Ferrari, M., 1991. Asymmetry and high concentration limit of the Mori–Tanaka effective medium theory. Mech. Mater. 11, 251–256.

Feynman, R.P., Leighton, R.B., Sands, M., 1964. The Feynman Lectures onPhysics, vol. 2. Addison-Wesley, Reading, MA.

Fridlyander, I.N., Klyagina, N.S., Tykachinskii, I.D., Dain, E.P., Gordeeva,G.D., 1974. Composite materials with a low coefficient of thermalexpansion. Metal Sci. Heat Treat. 16, 495–497.

Hill, R., 1963. Elastic properties of reinforced solids: some theoreticalprinciples. J. Mech. Phys. Solids 11, 357–372.

Holliday, L., Robinson, J., 1973. Review: the thermal expansion ofcomposites based on polymers. J. Materials Sci. 8, 301–311.

Hsieh, C.-L., Tuan, W.-H., 2006. Elastic and thermal expansion behavior oftwo-phase composites. Materials Sci. Eng. A-425, 349–360.

Kachanov, M., Sevostianov, I., Shafiro, B., 2001. Explicit cross-propertycorrelations for porous materials with anisotropic microstructures. J.Mech. Phys. Solids 49, 1–25.

Kachanov, M., Tsukrov, I., Shafiro, B., 1994. Effective moduli of solids withcavities of various shapes. Appl. Mech. Rev. 47 (1), S151–S174.

Kachanov, M., Sevostianov, I., 2005. On quantitative characterization ofmicrostructures and effective properties. Int. J. Solids Struct. 42, 309–336.

Kanaun, S.K., 1977. Self-consistent field approximation for an elasticcomposite medium. J. Appl. Mech. Tech. Phys. 18, 274–282.

Kanaun, S.K., 1982. Effective field method in the linear problems of staticsof composite media. Appl. Math. Mech. 46, 655–665.

Kanaun, S.K., Jeulin, D., 2001. Elastic properties of hybrid composites bythe effective field approach. J. Mech. Phys. Solids 49, 2339–2367.

Kanaun, S.K., Levin, V.M., 2008. Self-consistent methods for composites.Static Problems, vol. 1. Springer.

Kerner, E.H., 1956. The elastic and thermoelastic properties of compositemedia. Proc. Phys. Soc. (B), 808–813.

Klemens, P.G., 1986. Thermal expansion of composites. Int. J.Thermophys. 7, 197–206.

Klemens, P.G., 1988. Theory of thermal expansion of composites. Int. J.Thermophys. 9, 171–177.

Kunin, I.A., 1983. Elastic Media with Microstructure. Springer Verlag,Berlin.

Kuster, G.T., Toksoz, M.N., 1974. Velocity and attenuation of seismic-waves in 2-phase media. 1. Theoretical formulations. Geophys. 39,587–606.

Levin, V.M., 1975. Determination of effective elastic-moduli of composite-materials. Sov. Phys. Dokl. 220, 1042–1045.

Levin, V.M., 1967. On the coefficients of thermal expansion ofheterogeneous material. Mech. Solids 2, 58–61, English transl. ofIzvestia AN SSSR, Mekhanika Tverdogo Tela.

Markov, K.Z., 2000. Elementary micromechanics of heterogeneous media.In: Markov, K.Z., Preziozi, L. (Eds.), Heterogeneous Media:Micromechanics Modeling Methods and Simulations. Birkhauser,Boston, pp. 1–162.

Markov, K.Z., 2001. Justifcation of an effective field method in elasto-statics of heterogeneous solids. J. Mech. Phys. Solids 49, 2621–2634.

Mori, T., Tanaka, K., 1973. Average stress in matrix and averageelastic energy of materials with misfitting inclusions. Acta Met. 21,571–574.

Ponte-Castaneda, P., Willis, J.R., 1995. The effect of spatial distribution onthe effective behavior of composite materials and cracked media. J.Mech. Phys. Solids 43 (12), 1919–1951.

Prokopiev, O., Sevostianov, I., 2007. Modeling of porous rock: digitizationand finite elements versus approximate schemes accounting for poreshapes. Int. J. Fract. 143, 369–375.

I. Sevostianov / Mechanics of Materials 45 (2011) 20–33 33

Qui, Y., Weng, G.J., 1990. On the application of Mori–Tanaka’s theoryinvolving transversely isotropic spheroidal inclusions. Int. J. Eng. Sci.28, 1121–1137.

Rosen, B.W., Hashin, Z., 1970. Effective thermal expansion coefficients andspecific heats of composite materials. Int. J. Eng. Sci. 8, 157–173.

Schapery, R.A., 1968. Thermal expansion coefficients of compositematerials based on energy principles. J. Compos. Materials 2, 380–404.

Sevostianov, I., 2007. Dependence of the effective thermal pressurecoefficient of a particulate composite on particles size. Int. J. Fract.145, 333–340.

Sevostianov, I., Kachanov, M., 2002. Explicit cross-property correlationsfor anisotropic two-phase composite materials. J. Mech. Phys. Solids50, 253–282.

Sevostianov, I., Kachanov, M., 2008. Connections between elastic andconductive properties of heterogeneous materials. In: van derGiessen, E., Aref, H. (Eds.), Advances in Applied Mechanics, vol. 42.Academic Press, pp. 69–252.

Sevostianov, I., Sabina, F., 2007. Cross-property connections for fiberreinforced piezoelectric materials. Int. J. Eng. Sci. 45, 719–735.

Sevostianov, I., Kovácik, J., Simancík, F., 2006. Elastic and electricproperties of closed-cell aluminum foams. Cross-propertyconnection Mater. Sci. Eng. A-420, 87–99.

Steel, T.R., 1968. Determination of the constitutive coefficients for amixture of two solids. Int. J. Solids Struct. 4, 1149–1160.

Sun, Q., Inal, O.T., 1996. Fabrication and characterization of diamond/copper composites for thermal management substrate applications.Materials Sci. Eng. B 41, 261–266.

Takahashi, K., Harakawa, K., Sakai, T., 1980. Analysis of the thermalexpansion coefficients of particle filled polymers. J. Compos. Materials14, 144–159.

Takao, Y., Taya, M., 1985. Thermal expansion coefficients and thermalstresses in an aligned short-fiber composite with application to ashort carbon fiber/aluminum. J. Appl. Mech. 52, 806–810.

Thomas, J.P., 1960. Effect of inorganic fillers on coefficient of thermalexpansion of Polymeric Materials. Gen. Dyn./Fort Worth, Texas, AD,287–826.

Tummala, R.R., Friedberg, A.L., 1970. Thermal expansion of compositematerials. J. Appl. Phys. 41, 5104–5107.

Turner, P.S., 1946. Thermal expansion stresses in reinforced plastics. J.Res. (Nat. Bureau Stand.) 37, 239–250.

Vakulenko, A.A. 1968. Private communication.Van Fo Fy, G.A., 1965. Elastic constants and thermal expansion of certain

bodies with inhomogeneous regular structure. Dokl. Akad. Nauk SSSR166, 817–820 (in Russian).

Van Fo Fy, G.A., 1971. Theory of Reinforced Materials. Naukova Dumka,Kiev.

Wakashima, K., Otsuka, M., Umekawa, S., 1974. Thermal expansions ofheterogeneous solids containing aligned ellipsoidal inclusions. J.Compos. Mater. 8, 391–404.

Walpole, L.J., 1967. On overall elastic moduli of composite materials. J.Mech. Phys. Solids 17, 235–251.

Walpole, L.J., 1984. Fourth-rank tensors of the thirty-two crystal classes:multiplication tables. Proc. Roy. Soc. L. A 391, 149–179.

Wang, T.T., Kwei, T.K., 1969. Effect of induced thermal stresses oncoefficients of thermal expansion and densities of filled polymers. J.Poly. Sci. (A-2) 7, 889–896.

Withers, P.J., Stobbs, W.M., Pedersen, O.B., 1989. The application of theEshelby method of internal stress determination to short fibre metalmatrix composites. Acta Metallurgica 37, 3061–3084.

Wong, C.P., Bollampally, R.S., 1999. Thermal conductivity, elasticmodulus, and coefficient of thermal expansion of polymercomposites filled with ceramic particles for electronic packaging. J.Appl. Poly. Sci. 74, 3396–3403.

Wu, T.T., 1966. The effect of inclusion shape on the elastic moduli of atwo-phase material. Int. J. Solids Struct. 2, 1–8.

Yoshida, K., Morigami, H., 2004. Thermal properties of diamond/coppercomposite material. Microelectro. Reliab. 44, 303–308.