effective thermal properties of viscoelastic composites having field-dependent constituent...

Acta Mech 209, 153–178 (2010)DOI 10.1007/s00707-009-0171-6

Kamran A. Khan · Anastasia H. Muliana

Effective thermal properties of viscoelastic compositeshaving field-dependent constituent properties

Received: 5 September 2008 / Revised: 18 February 2009 / Published online: 5 April 2009© Springer-Verlag 2009

Abstract This study introduces a micromechanical model for predicting effective thermal properties (linearcoefficient of thermal expansion and thermal conductivity) of viscoelastic composites having solid sphericalparticle reinforcements. A representative volume element (RVE) of the composites is modeled by a singleparticle embedded in the cubic matrix. Periodic boundary conditions are imposed to the RVE. The micro-mechanical model consists of four particle and matrix subcells. Micromechanical relations are formulated interms of incremental average field quantities, i.e., stress, strain, heat flux and temperature gradient, in the sub-cells. Perfect bonds are assumed along the subcell’s interfaces. Stress and temperature-dependent viscoelasticconstitutive models are used for the isotropic constituents in the micromechanical model. Thermal propertiesof the particle and matrix constituents are temperature dependent. The effective coefficient of thermal expan-sion is derived by satisfying displacement and traction continuity at the interfaces during thermo-viscoelasticdeformations. This formulation leads to an effective time–temperature–stress-dependent coefficient of thermalexpansion. The effective thermal conductivity is formulated by imposing heat flux and temperature continu-ity at the subcells’ interfaces. The effective thermal properties obtained from the micromechanical model arecompared with analytical solutions and experimental data available in the literature. Finally, parametric studiesare also performed to investigate the effects of nonlinear thermal and mechanical properties of each constituenton the overall thermal properties of the composite.

1 Introduction

Composite materials have been utilized in various heat transfer related applications ranging from low to ele-vated temperatures. Different composite performances can be achieved by varying thermal and mechanicalproperties of the constituents. However, when subjected to elevated temperatures, composites often experiencetime-dependent responses, and the material properties of the constituents at a specific location often vary withthe temperatures at that location. To gain fundamental understanding on the effects of constituents’ propertiesand microstructural geometries on the overall responses of composites, various micromechanical models havebeen formulated. The common concept of all micromechanical models is to provide equivalent homogeneousproperties of composites by first simplifying the heterogeneity condition and then averaging the properties ofthe constituents. It is also necessary to formulate micromechanical models that allow for continuous changesin the constituent properties due to external mechanical and thermal stimuli.

Micromechanical models have been developed to obtain elastic properties of composites, e.g., differentialmethod (DM), concentric cylinder assembly (CCA), Mori–Tanaka (MT), self-consistent method (SCM), andmethod of cells (MOC). Based on the uniform strain and stress approximations, Voigt [54] and Reuss [38]introduced simple micromechanical relations of the linear elastic material constants. Hill [20] has shown that

K. A. Khan · A. H. Muliana (B)Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USAE-mail: [email protected]

154 K. A. Khan, A. H. Muliana

the Voigt and Reuss models yield upper and lower bounds, respectively. Thus, effective responses derivedfrom micromechanical models should be within these bounds as they give a maximum error in predicting theoverall homogenized properties. A detailed discussion of various micromechanical models and bounds on theeffective mechanical properties can be found in Nemat-Nasser and Hori [36]. Several studies have been doneon formulating micromechanical models for predicting effective linear viscoelastic responses of composites,e.g., Christensen [12], Schapery [42], Brinson and co-authors [8,9,14], and Li and Weng [27]. Micromechan-ical models for obtaining effective viscoelastic behaviors with stress/strain dependent constituents have beenproposed, mainly for fiber reinforced plastic (FRP) composites by, e.g., Aboudi [2,3], Haj-Ali and Muliana[16,17], Muliana and Haj-Ali [31], Muliana and Sawant [35]. Recently, Muliana and Kim [34] formulated amicromechanical model for analyzing stress-dependent viscoelastic responses of solid spherical particle rein-forced polymers. Their micromechanical formulation is suitable for small deformation gradient problems. Thehomogenization schemes are formulated in terms of average stress–strain relations of the particle and matrixconstituents, which give approximated values of the effective properties.

Micromechanical models have also been formulated to obtain the effective coefficient of thermal expan-sion (CTE) and effective thermal conductivity (ETC). For an arbitrary phase geometry and composites withspherical inclusions, Turner [52], Kerner [22], Levin [25], Schapery [41], Fahmy and Ragai [13] and Rosenand Hashin [39] have derived equations for effective CTEs. The effective CTEs are functions of the CTE andelastic moduli of each constituent. Turner [52] considered one particle surrounded by the homogeneous matrixand satisfied the equilibrium of forces to derive the equation for effective CTE. Kerner’s [22] formulationutilized the assumption of a particle being suspended and bonded in the form of grains to a homogeneousmedium. Rosen and Hashin [39] used the thermo-elastic strain energy principle to evaluate bounds for theeffective CTE of composites having anisotropic phases. For the isotropic two phase composites, the calculatedCTE reduced to the bounds of Schapery [41], which was formulated using thermo-elasticity equations. Fahmyand Ragai [13] used linearized elasticity solutions to predict the effective CTE of composites. The compositesphere assemblage (CSA) model was used and appropriate boundary conditions were applied to derive theeffective CTE. Tseng [50] introduced a statistical micromechanics model to determine an effective CTE ofparticle reinforced composites. A volume averaging scheme was used to determine effective CTE. Stress andstrain concentration tensors were introduced to relate the local average fields to the global (effective) fields.Interaction effects among the particles were considered by introducing an interaction tensor. A closed formsolution of effective elastic moduli and effective CTE was derived in terms of the constituent’s propertiesand the interaction tensor. Sideridis et al. [43] developed an analytical model for the effective CTE for aparticle reinforced composite using a triphase model. The model consists of a composite particle embeddedin a homogeneous matrix. The composite particle is first obtained using a micromodel of matrix surroundedby a solid particle. Boundary value problems were solved with appropriate boundary conditions to derive theexplicit equation for effective CTE. Hsieh and Tuan [21] proposed a modified unit cell model composed ofa continuous matrix phase and an elongated particle. Upper and lower bounds for the effective CTE wereobtained by solving linear elasticity equations and applying appropriate boundary conditions.

Referring to the above discussion, the effective CTE of composites depends on the mechanical propertiesand CTEs of the constituents in the composites. In general, the mechanical and physical properties of materialschange with field variables such as stress, deformation, and temperature, resulting in nonlinear responses ofmaterials. This has been experimentally reported in several studies. Odegard and Kumosa [37], Rupnowskiet al. [40], and Marias and Villoutreix [28] have shown that the moduli of composites and their constituents varywith temperature. Cho et al. [11] studied the stress-dependent elastic modulus of particle composites, whileLai and Bakker [23,24] and Muliana et al. [32] are among researchers who experimentally showed the stress-dependent viscoelastic responses of polymers and composites. Available methods for predicting the effectiveCTE were formulated with constant constituent properties (independent of stress, strain, temperature etc.).Moreover, the effective CTE of composites having time-dependent constituent properties are not well under-stood. Although it is possible to extend available micromodels to incorporate nonconstant (nonlinear) con-stituent properties and form a set of equations to determine nonlinear effective CTE, it is not always easy toobtain the closed form solutions. To the best knowledge of the authors only Feltham et al. [15] have exper-imentally shown the variation of effective CTEs with temperature for a particle reinforced composite. Threecomposite systems having epoxy resin as matrix were considered. They employed the analytical models ofTurner [52], Kerner [22] and Fahmy and Ragai [13] to predict their experimental data. The temperature-dependent elastic moduli and CTE of particles and matrix were considered. Poisson’s ratios were assumed tobe independent of temperature. The comparisons showed that the CTE responses of Fahmy and Ragai [13]are in agreement with the experimental data. Fahmy and Ragai’s [13] model accounted for the interfacial

Effective thermal properties of viscoelastic composites 155

mismatch caused by the thermally induced dimensional changes, which was not considered thoroughly byothers.

The averaging procedures developed for the effective linear elastic material constants have been modifiedto homogenize nonmechanical (thermal, electrical, magnetic) properties of composites. Many analytical andnumerical models have been proposed to predict the ETC of the composites. Maxwell [29], Baschirow andSelenew [6] and Verma et al. [53] derived ETC for two phase composites. The particles were randomly dis-tributed in a homogeneous medium and there was no interaction among them. Benveniste [7] formulated ETCfor multiphase systems by determining the average flux in each constituent, and the homogenization was thenperformed using the Mori–Tanaka [30] and generalized SCM. The above models give good predictions of theoverall ETC when the volume fractions (Vf) are relatively small or when the conductivity of the particle iscomparable to the one of the matrix. Several empirical studies have also been conducted to investigate theeffect of shape and orientation of the dispersed particles on the overall ETC of the composites, e.g., Chengand Vachon [10] and Lewis and Nielsen [26].

It has been experimentally shown by Agri and Uno [4,5] and recently by Zhang et al. [57] that at highervolume fractions and high ratios of particle’s thermal conductivity to the one of the matrix, i.e., Vf > 20% andKp/Km > 100, there exists particle interaction in the form of a conductive chain mechanism. This mechanismaccelerates the heat conduction process, which was shown by an increase in the overall ETC. Agri and Uno[4] used an empirical approach to account for the chain conductive mechanisms. The volume fraction and thegeometry of the particle were responsible in forming the conductive chain mechanism. Zhou et al. [58] showedthat at higher particle concentration some particles flocculated to form conductive chains. They introduced theheat transfer passage (HTP) which takes the effect of local concentration fluctuation into account to evaluatethe ETC of the composites.

A previously developed micromechanical model of stress-dependent viscoelastic responses of solid spher-ical particle reinforced polymers [34] is modified to include the temperature-dependent material properties andto determine the effective CTE. Furthermore, a new micromechanical formulation is presented for the ETCof particle-reinforced composites. In this study, we derive a linearized micromechanical model for predictingeffective nonlinear thermal properties of viscoelastic composites reinforced with solid spherical particles. Thethermal and mechanical properties of the constituents are allowed to vary with temperature, stresses, and time.A unit-cell model with four particle and polymer subcells is generated. Simplified micromechanical relationsare formulated in terms of incremental average field quantities, i.e., stress, strain, heat flux and temperaturegradient, in the subcells of the micromodels. Perfect bonds are assumed along the subcell’s interfaces. Iterativeschemes are incorporated to solve the constitutive equations at the composite and constituents levels in orderto satisfy the micromechanical relations and the nonlinear constitutive equations. The effective CTE and ETCderived from the simplified micromodel can be implemented in FE framework and used for structural analyses.The paper is organized as follows. General thermo-viscoelastic constitutive models and heat flux equationsfor isotropic homogeneous material are presented in Sect. 2. Micromechanical formulations of the effectivethermal properties of the particle reinforced composites are described in Sect. 3. Verifications of the proposedmicromechanical models and comparisons with the available analytical models and experimental data areshown in Sect. 4. Parametric studies are also presented on understanding the effects of nonlinear thermal andmechanical properties of each constituent on the overall composite behavior.

2 Constitutive model for isotropic constituents

A thermo-viscoelastic constitutive model with stress and temperature dependent material properties is usedfor each constituent. The Schapery [42] nonlinear single integral equation is modified to include the time–temperature variations for non-aging materials and generalized for multi-axial stress–strain relations, which are:

εti j = et

i j + 13ε

tkkδi j + α

(T t − T0

)δi j ,

εM,ti j = et

i j + 13ε

tkkδi j , ε

T,ti j = α

(T t − T0

)δi j ,

(1)

eti j = 1

2 g0(σ̄t , T t )J0St

i j + 12 g1(σ̄

t , T t )∫ t

0 �J(ψ t − ψτ

) d[g2(σ̄

τ ,T τ )Sτi j

]

dτ dτ ,

εtkk = 1

3 g0(σ̄t , T t )B0σ

tkk + 1

3 g1(σ̄t , T t )

∫ t0 �B

(ψ t − ψτ

) d[g2(σ̄

τ ,T τ )σ τkk

]

dτ dτ ,


where εM,ti j and εT,t

i j are the total mechanical and thermal strains, respectively. The superscript ‘t’ indicatesa variable at time t . The parameters J0 and B0 are the instantaneous elastic shear and bulk compliances,respectively. The terms �J and �B are the time-dependent shear and bulk compliances, respectively. Thecorresponding linear elastic Poisson’s ratio, υ, is assumed to be time independent, which allows expressingthe shear and bulk compliances as:

J0 = 2(1 + υ)D0, B0 = 3(1 − 2υ)D0,

�Jψt = 2(1 + υ)�Dψ t

, �Bψt = 3(1 − 2υ)�Dψ t

. (2)

Here D0 and �D are the instantaneous elastic and transient compliances under uniaxial (extensional) creeploading and ψ is the reduced-time (effective time) given by:

ψ t ≡ ψ (t) =t∫

0

dξ

aσξ aT ξ, ψτ ≡ ψ (τ) =

τ∫

0

dξ

aσξ aT ξ. (3)

The uniaxial transient compliance, �D, is expressed in terms of Prony series as:

�Dψ t =N∑

n=1

Dn(1 − exp[−λnψ

t ]) . (4)

The parameter g0 is the nonlinear instantaneous elastic compliance and it measures the reduction or increase incompliance as a function of stress and temperature. The transient creep parameter g1 measures the nonlinearityeffect in the transient compliance. The parameter g2 accounts for the loading rate effect on the creep response.The parameters T and T0 are the current and reference temperatures, respectively. The linear CTE, α, alsovaries with temperature.

A recursive iterative method developed by Muliana and Khan [33] is used to solve the deviatoric andvolumetric components of the mechanical strains in Eq. (1). The formulation is derived with a constant incre-mental strain rate during each time increment, which is compatible with a displacement based FE analysis.The incremental form of the deviatoric and volumetric strains at a current time is expressed as:

deti j = et

i j − et−�ti j

= JtSt

i j − Jt−�t

St−�ti j − 1

2

N∑

n=1

Jn

(gt

1exp[−λn�ψ

t] − gt−�t1

)qt−�t

i j,n

−1

2gt−�t

2

N∑

n=1

Jn

[gt−�t

1

(1 − exp[−λn�ψ

t−�t ]λn�ψ t−�t

)− gt

1


t ]λn�ψ t

)]St−�t

i j , (5)

dεtkk = εt

kk − εt−�tkk

= Btσ t

kk − Bt−�t

σ t−�tkk − 1

3

N∑

n=1

Bn

(gt

1exp[−λn�ψ

t] − gt−�t1

)qt−�t

kk,n

−1

2gt−�t

2

N∑

n=1

Bn

[gt−�t

1


t−�t ]λn�ψ t−�t

)− gt

1


t ]λn�ψ t

)]σ t−�t

kk . (6)

The parameters J̄ t and B̄t are the effective shear and bulk compliances at the current time, respectively. Theparameters qt

i j,n and qtkk,n, n = 1, . . . , N are the hereditary integral for every term in the Prony series in the

form of deviatoric and volumetric strains, which are:

qti j,n = exp[−λn�ψ

t ]qt−�ti j,n +

(gt

2Sti j − gt−�t

2 St−�ti j

) 1 − exp[−λn�ψt ]

λn�ψ t,

qtkk,n = exp[−λn�ψ

t ]qt−�tkk,n +

(gt

2σtkk − gt−�t

2 σ t−�tkk

) 1 − exp[−λn�ψt ]

λn�ψ t.

(7)


The parameters qt−�ti j,n and qt−�t

kk,n are hereditary integral (history state variables) stored from the last converged

step at time (t −�t). The incremental reduced time is expressed as �ψ t ≡ ψ t − ψ t−�t . Equations (5), (6)define complete solutions for the current incremental strain tensors. The nonlinear parameters in Eqs. (5), (6)are expressed as functions of current temperature T t and effective stress σ t , which at the current time (t) arenot known. Linearized trial stress tensors are used as starting points for solving the stress tensor using Eqs. (5),(6). An iterative scheme is included in order to find the correct stress tensor for a given strain tensor. Theinitial approximation (trial) stress tensor is determined using the following approximation of the nonlinearparameters:

gt,trβ = gβ(σ̄

t−�t , T t−�t ), β = 0, 1, 2,(8)

at,tr = a(σ̄ t−�t , T t−�t ).

The trial current stress tensor is formed based on the given variables and history variables from the previousconverged step:

σt,tri j = σ t−�t

i j + dσ t,tri j ,

dσ t,tri j = d St,tr

i j + 1

3dσ t,tr

kk δi j ,(9)

where the trial incremental deviatoric and volumetric stresses are given by:

d St,tri j = 1

J̄ t,tr

[

deti j + 1

2gt,tr

1

N∑

n=1

Jn(exp[−λn�ψt ] − 1)qt−�t

i j,n

]

,

dσ t,trkk = 1

B̄t,tr

[

dεtkk + 1

3gt,tr

1

N∑

n=1

Bn(exp[−λn�ψt ] − 1)qt−�t

kk,n

]

.

(10)

The correct stress tensor at current time is solved by minimizing a residual tensor, which is defined in termsof incremental strains and expressed by:

Rti j = det

i j + 1

3dεt

kkδi j − dεM,ti j . (11)

Finally, the consistent tangent stiffness matrix is defined by taking the inverse of the partial derivative of theincremental strain with respect to the incremental stress at the end of the current time step. The consistenttangent stiffnesses, Ct

i jkl , at the converged state, are:

Cti jkl ≡ ∂dσ t

i j

∂dεM,tkl

=[∂Rt

i j

∂dσ tkl

]−1

;∥∥∥Rt

i j

∥∥∥ → 0. (12)

Equation (12) defines material properties at current time t for each subcell in the micromechanical model. Thecomponents of the consistent tangent stiffness tensor vary with time, temperature, and stress. Equation (1) canbe easily reduced to linear thermo-elastic constitutive models when the nonlinear and time-dependent effectsare negligible.

The total thermal strains are expressed by:

εT,ti j = ε

T,t−�ti j + dεT,t

i j , (13)

where for an isotropic material the incremental thermal strains are:

dεT,ti j = α(T t )dT tδi j , (14)

and dT t = T t − T t−�t . (15)

The Fourier heat flux equation having temperature-dependent thermal conductivity is used and expressed as:


qti = −Ki j

(T t)ϕt

j , where φtj = dT t

dx j, (16)

where qti and ϕt

j are the heat flux and temperature gradient, respectively. Ki j(T t

)is the temperature-dependent

thermal conductivity. The dependence of the thermal conductivity on the current temperature Ki j(T t

)will be

written as K ti j in the remainder of the manuscript. The solution for the heat flux is done incrementally and the

total heat flux at current time (t) is defined as:

qti = qt−�t

i + dqti , (17)

dqti = K t

i j dϕtj . (18)

In this case, K ti j is the consistent tangent thermal conductivity matrix, which varies with temperature at current

time t .

3 Formulations of the effective thermal properties

Previously developed micromechanical relations for the effective viscoelastic responses of a particle rein-forced composite [34] are modified to determine the effective CTE and ETC. Figure 1 illustrates the simplifiedmicromechanical model for the composite with randomly distributed solid spherical particles in a homoge-neous matrix. The gradient of the volume contents of the particles is assumed to be zero. The solid sphericalparticulate composites are idealized with uniformly distributed arrays of cubic particles. A representativevolume element (RVE) consists of a cubic particle embedded in the center of the matrix with cubic domain.

Fig. 1 Representative unit-cell model for the particulate reinforced polymers


A one-eight unit-cell consisting of four subcells is modeled due to symmetry. The first subcell is a particleconstituent, while subcells 2, 3, and 4 represent the matrix constituents. The micromechanical relations pro-vide equivalent homogeneous material responses from the heterogeneous microstructures and simultaneouslyrecognize nonlinear behaviors of the individual constituents from the overall composite responses. The mi-cromechanical formulations are designed to be compatible with general finite element structural analyses, i.e.,ABAQUS [1] in which the effective responses from the micromechanical relations are implemented at eachmaterial point (Gaussian integration point) within the finite elements.

Percolation theory is primarily related to random or disordered media. This notion is more important inthe modeling of the composite in which the dispersion of the inclusion is random in nature. In the proposedmicromechanical model, a statistically homogeneous distribution of inclusions is assumed. The micromechan-ical model is formulated by having all particles fully surrounded by the matrix, and an RVE with a singleinclusion embedded in a cubic matrix is selected. Periodic boundary conditions are imposed to the selectedRVE model. In reality, there often exists contact between particles. As the particle content in the compositesincreases contacts among particles become more pronounced and in some composites interpenetration betweenthe constituents (sometimes refer to as percolation) is also observed, making it difficult to distinguish the inclu-sion and matrix constituents. Interpenetration between constituents and percolation can significantly alter theeffective properties of composites [48]. Torquato [48] has also discussed that constituent/phase inversion incomposites is another crucial microstructural feature in estimating the overall performance of composites.Thus, the present micromechanical model, which requires defining inclusion and matrix constituents, is notsuitable to predicting overall composite responses with percolation effects.

3.1 Formulation of the effective coefficient of thermal expansion

The micromechanical relation for the effective CTE of the studied composite is derived in terms of the nonlineartime-dependent moduli and CTE of each constituent. Each unit-cell is divided into a number of subcells, andthe spatial variation of the displacement field in each subcell is assumed such that the stresses and deforma-tions are spatially uniform. The macroscopic effective properties of a heterogeneous medium are approximatedusing the volume average of the properties of the individual constituents.

For linear thermo-elastic problems, the effective stress and strain tensors of the composites are related bythe following constitutive equations:

σ i j = Ci jkl[εkl − αkl(T − T0)

]or (19)

ε̄i j = S̄i jkl σ̄kl + αi j (T − T0). (20)

Ci jkl and Si jkl are the components of the effective elastic stiffness and compliance tensors, respectively; andαi j are the components of the effective linear CTE tensor. The goal of micromechanical model formulation isto obtain Ci jkl or Si jkl based on the constitutive relations in each constituents and microstructural geometry,i.e., RVE. The constitutive relations for the linear thermo-elastic problem in each constituent (α) are expressedby:

σ(α)i j = C (α)

i jkl

[ε(α)kl − α

(α)kl (T

(α) − T0)]

or (21)

ε(α)i j = S(α)i jklσ

(α)kl + α

(α)i j (T

(α) − T0). (22)

The temperature T is measured at the boundary of the RVE. Due to the assumption that the RVE’s length scaleis much smaller than the macrostructural scale, this value is assumed uniform for the entire surfaces of oneRVE, while it can spatially vary within material points in the composite media.

The average stresses and strains for linear thermo-elastic responses are defined by:

σ̄i j = 1

V

N∑

α=1

∫

V (α)

σ(α)i j (x

(α)k )dV (α) ≈ 1

V

N∑

α=1

V (α)σ(α)i j , (23)

ε̄i j = 1

V

N∑

α=1

∫

V (α)

ε(α)i j (x

(α)k )dV (α) ≈ 1

V

N∑

α=1

V (α)ε(α)i j . (24)


An overbar indicates effective material quantities. The superscript (α) denotes the subcell’s number and Nis the number of subcells. The stress σ (α)i j and strain ε(α)i j is the average stress and strain in each subcell,respectively. The unit-cell volume V is:

V =N∑

α=1

(α)V , N = 4. (25)

The volume of the unit-cell is taken as one. The volume of the subcell 1 is that of a cube of edge length b.Figure 1 represents the particle volume fraction of the composite systems. Thus, the magnitude of b is alwaysless than one. The volumes of the four subcells are then expressed as:

V (1) = b3, V (2) = b2(1 − b), V (3) = b(1 − b), V (4) = (1 − b). (26)

In this study, the elastic properties and CTE of the constituents are allowed to change with time, stress, andtemperature, which results in nonlinear stress–strain relations. To solve the nonlinear equations, linearizedpredictor and corrector schemes are developed and Eqs. (21)–(24) are then satisfied in incremental relations.Within the FE analyses, the total stress and strain at current time t are defined by:

σ ti j = σ t−�t

i j + dσ ti j ,

εi jt = εi j

t−�t + dεti j ,

σ(α),ti j = σ

(α),t−�ti j + dσ (α),ti j ,

ε(α),ti j = ε

(α),t−�ti j + dε(α),ti j .

(27)

The superscript, t − �t , indicates quantities at the previous converged time and the prefix d denotes incre-mental quantities at the current time increment. The linearized micromechanical relations in incremental formare first imposed in terms of the consistent tangent stiffness matrix at time t − �t (Eq. (12)). An iterationprocess is performed to minimize the residual strains from the linearization. The converged solution satisfiesboth the micromechanical relations and the nonlinear constitutive equations. This study also defines a straininteraction matrix (B(α),ti jkl ), which relates the incremental average strains in each subcell, dε(α),ti j , to the effectiveincremental strain, dεt

kl :

dε(α),ti j = B(α),ti jkl dεtkl . (28)

Using the strain in Eq. (28), the constitutive relation in the subcell (α) can be written as:

dσ (α),ti j = C (α),ti jkl B(α),tklrs dεt

rs . (29)

Substituting Eq. (29) into the incremental form of Eq. (23) gives the effective stresses:

dσ ti j = 1

V

N∑

α=1

V (α)C (α),ti jkl B(α),tklrs dεt

rs . (30)

The unit-cell effective tangent stiffness matrix C̄ ti jrs is then determined by:

C̄ ti jrs = 1

V

N∑

α=1

V (α)C (α),ti jkl B(α),tklrs . (31)

The B(α),ti jkl is a fourth order tensor, which can be obtained by satisfying the micromechanical relations and the

constitutive equations. The detailed formulation of B(α),ti jkl is described in Muliana and Kim [34].The micromechanical relations within the four subcells are derived by assuming perfect bond along the

interfaces of the subcells and imposing displacement compatibility and traction continuity at the subcells’ inter-face. In the case that both particle and matrix are isotropic, the outcome of the homogenized micromechanical


model is also isotropic. The homogenized incremental strain relations for the particle reinforced compositesare given as:

dεti j = 1

V (1) + V (2)

[V (1)dε(1),ti j + V (2)dε(2),ti j

]= dε(3),ti j = dε(4),ti j for i = j; i, j = 1, 2, 3, (32)

dγ ti j = V (1)dγ (1),ti j + V (2)dγ (2),ti j + V (3)dγ (3),ti j + V (4)dγ (4),ti j for i �= j. (33)

The homogenized stresses are written as:

dσ ti j = V (A)dσ (A),ti j + V (3)dσ (3),ti j + V (4)dσ (4),ti j for i = j,

dσ (A),ti j = dσ (1),ti j = dσ (2),ti j , (34)

dσ ti j = dσ (1),ti j = dσ (2),ti j = dσ (3),ti j = dσ (4),ti j for i �= j, (35)

where the total volume of subcells 1 and 2 in Eq. (34) is V (A) = V (1) + V (2).Using the micromechanical relations in Eqs. (32)–(35) and thermo-viscoelastic constitutive relations for

the particle and matrix subcells, the incremental form of the effective stress–strain relations for the particlereinforced composite are:

dσ ti j = 1

V

[V (A)C (A),t

i jkl + V(3)C (3),ti jkl + V(4)C (4),t

i jkl

]dεt

kl

−d�T

V

[V (A)C (A),t

i jkl α(A),tkl + V(3)C (3),t

i jkl α(3),tkl + V(4)C (4),t

i jkl α(4),tkl

]

for i = j and k = l; i, j, k, l = 1, 2, 3. (36)

Equation (36) can be rewritten as:

dσ ti j = C

ti jkldε

tkl − d�T

V

[V (A)C (A),t

i jkl α(A),tkl + V(3)C (3),t

i jkl α(3),tkl + V(4)C (4),t

i jkl α(4),tkl

]

= Cti jkl

[dεt

kl − d�Tαtkl

]. (37)

The effective consistent tangent CTE in Eq. (37), for the isotropic nonlinear responses, is then expressed as:

αti j = αtδi j = C

−1,ti jkl

V

[V (A)C (A),t

i jkl α(A),tkl + V (3)C (3),t

i jkl α(3),tkl + V (4)C (4),t

i jkl α(4),tkl

](38)

where α(A),tkl and C (A),ti jkl in Eq. (38) are given in the following equations:

α(A),ti j = α(A),tδi j = 1

V A

[V (1)α

(1),ti j + V(2)α(2),ti j

], (39)

C (A),ti jkl =

[1

V A

[V (1)C (1)−1,t

i jkl + V (2)C (2)−1,ti jkl

]]−1

. (40)

It is seen that the effective tangent CTE in Eq. (38) within the incremental time step depends on the time-dependent moduli and CTE of each constituent. Thus, for the stress and temperature dependent constituentmechanical and thermal properties, the effective CTE varies with stresses and temperatures. In the case thatviscoelastic constituents are considered, the effective CTE also changes with time.


3.2 Formulation of ETC

A volume averaging method based on the spatial variation of the temperature gradient in each subcell is adoptedto determine the ETC of the particle reinforced composites. The average heat flux and temperature gradientare:

qi = 1

V

N∑

α=1

∫

V (α)

q(α)i (x (α)k )dV (α) ≈ 1

V

N∑

α=1

V (α)q(α)i , (41)

ϕi = 1

V

N∑

α=1

∫

V (α)

ϕ(α)i (x (α)k )dV (α) ≈ 1

V

N∑

α=1

V (α)ϕ(α)i . (42)

The average heat flux equation for a homogeneous composite medium is expressed by the Fourier law of heatconduction as:

qti = −K

ti jϕ

tj where ϕt

j = dTt

dx j. (43)

It is noted that the components of the conductivity tensor, Kti j , vary with temperature as the thermal conduc-

tivity for each constituent is allowed to vary with temperature. The heat flux at the current time within an FEscheme is obtained numerically as:

qti = qt−�t

i + dqti . (44)

Due to the uncoupled thermo-mechanical problems in which the dissipation of energy is neglected, the tem-perature field can be found without the knowledge of the stress–strain fields. The incremental heat flux can bewritten as:

dqti = −K

ti j dϕ

tj . (45)

The micromechanical relations within the four subcells in Fig. 1 are derived by assuming perfect bond alongthe interfaces of the subcells. The homogenized temperature gradient and heat flux relations are summarizedas follows:

dϕti = 1

V (A)

[v(1)dϕ(1),ti + v(2)dϕ(2),ti

]= dϕ(3),ti = dϕ(4),ti , (46)

dqti = 1

V

[v(A)dq(A),ti + v(3)dq(3),ti + v(4)dq(4),ti

], (47)

dq(A),ti = dq(1),ti = dq(2),ti , (48)

where the total volume of subcells 1 and 2 in Eqs. (46) and (47) is V (A) = V (1) + V (2).=

We introduce a concentration tensor that relates the average subcells temperature gradient to the overalltemperature gradient across the unit cell. Let M (α),t be the concentration tensor of the temperature gradient.The temperature gradient in each subcell is expressed by:

dϕ(α),ti = M (α),ti j dϕt

j . (49)

Using Eq. (49), the heat flux in each subcell is:

dq(α),ti = −K (α),ti j M (α),t

jk dϕtk . (50)

The average heat flux in the unit-cell model is approximated as:

dqti = 1

V

4∑

α=1

V (α)dq(α),ti . (51)


Substituting Eq. (50) into (51) gives:

dqti = − 1

V

4∑

α=1

V (α)K (α),ti j M (α),t

jk dϕtk . (52)

Comparing the above equation with Eq. (45) gives the tangent ETC matrix of the composite, which is:

Ktik = − 1

V

4∑

α=1

V (α)K (α),ti j M (α),t

jk . (53)

To formulate the M matrix that relates the average subcells temperature gradient to the overall temperaturegradient across the unit cell, the micromechanical relations and the constitutive equations are imposed. Thepresent micromodel consists of four (4) subcells with three (3) components of heat flux needed to be determinedfor every subcell. This requires forming twelve (12) equations. The first sets of equations are determined fromthe temperature continuity at the interface of each subcell in Eq. (46), which leads to the following equation:

[A1](9×12)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

dϕ(1),tidϕ(2),tidϕ(3),tidϕ(4),ti

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(12×1)

= [D1](9×3)

{dϕti }

(3×1). (54)

The second sets of equations are formed based on heat flux continuity relations, Eq. (48), which can beexpressed as:

[At

2

]

(1×12)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

dϕ(1),tidϕ(2),tidϕ(3),tidϕ(4),ti

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(12×1)

= [O](1×3)

{dϕti }

(3×1). (55)

By substituting Eq. (49) into Eqs. (54) and (55), the M matrix can be determined, which is:

[M(α),t

]

(12×1)

=[

A1At

2

]−1

(12×4)

[D1O

]

(4×1)

. (56)

The matrix O is the zero matrix and the components of the matrices A1,At2 and D1 are given as follows:

A1 =

⎡

⎢⎢⎢⎢⎣

V (1)

V (A) I(3×3)

V (2)

V (A) I(3×3)

0(3×3)

0(3×3)

0(3×3)

0(3×3)

I(3×3)

0(3×3)

0(3×3)

0(3×3)

0(3×3)

I(3×3)

⎤

⎥⎥⎥⎥⎦, (57)

At2 =

[K (1),t I(3×3)

−K (2),t I(3×3)

0(3×3)

0(3×3)

], (58)

D1 =

⎡

⎢⎢⎢⎢⎢⎢⎣

I(3×3)

I(3×3)

I(3×3)

I(3×3)

⎤

⎥⎥⎥⎥⎥⎥⎦

. (59)


4 Numerical simulation and verification of the micromechanical model

The effective CTE and ETC of the particle reinforced systems are verified using available analytical and exper-imental studies reported in the literature. Parametric studies on the effects of thermo-viscoelastic propertiesof the constituents and reinforcement compositions on the overall thermal properties of the composites arepresented.

4.1 Verification of effective CTE

The averaging procedure in the micromechanical model satisfies traction continuity and displacement com-patibility of the combined thermal and mechanical responses. Thus, the overall CTE of composites having aviscoelastic matrix may indeed exhibit time-dependent behavior. Furthermore, temperature changes in com-posites develop thermal stresses at each constituent due to different CTEs between the inclusion and the matrix.The existence of the thermal stresses can increase time-dependent deformations in the viscoelastic constitu-ents. Under a constant temperature change, the overall time-dependent deformation is attributed to the constantthermal stress, which is a creep problem. If the strain is fixed, the stress in the viscoelastic constituent canrelax with time and reduce the overall composite modulus. The viscoelastic characteristic may cause thermalstresses to relax, which may reduce the rate of creep deformations over a longer time period.

The effective CTE obtained from the presented micromodel is first compared with available analyticalmodels. Figure 2 shows effective CTE having linear thermo-elastic responses of the constituents. The studiedcomposite material consists of glass beads as inclusions and FM73 polymer as matrix. The mechanical andthermal properties of the constituents are given in Table 1. The proposed model is compared with analyticalmodels of Levin [25], Wakashima et al. [55], and Fahmy and Ragai [13], which are obtained from exact solu-tions. The present micromodel is a result of numerical approximation following a micromechanical averagingscheme. It is shown that the proposed model lies between the upper and lower bound which is acceptable asfar as the effective properties are concerned. The results of the proposed micro-models are comparable to theones reported by the Levin [25] and Fahmy and Ragai [13].

The effective CTEs of various composites are also compared with the experimental data available in lit-erature. Composite systems consisting of glass, zirconia and tin constituents are studied. The thermal andelastic properties of the glass, zirconia and tin constituents have been taken from Tummala and Friedberg [51],shown in Table 1. The material properties are assumed to be independent of temperature. Figure 3 illustratesthe effective CTE at various volume fractions of the inclusions for two composite systems. In both cases,the micromodel shows good prediction. The experimental data of Fahmy and Ragai [13] is also used forverification. Figure 4 shows the comparison of the effective CTE of composites having aluminum and siliconconstituents. The effective CTEs of the composite are plotted as functions of the silicon contents. The experi-ments were conducted for two temperature ranges, i.e., at low temperature (between −196 and 20◦C) and at

0 0.2 0.4 0.6 0.8 1

Volume Fraction (VF)

0

2E-005

4E-005

6E-005

8E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

Fahmy and Ragai (1970)Lower BoundWakashima (1974)

Present Micromodel

Upper BoundLevin (1967)

FM73 Polymer with Glass Bead

Fig. 2 Comparison of the effective CTE with analytical models


Table 1 Material properties used for CTE verification

Material Young’s modulus (E), MPa Poisson’s ratio (υ) Linear thermal expan-sion (α)10−6, 1/◦K

Glass bead 69,000 0.3 8.5FM73 2,700 0.35 75Zirconia 192,000 0.23 9.40Tin 42,000 0.36 23.5Glass 43,000 0.13 6.45Aluminum 69,000 0.33 24Silicon 107,000 0.17 3.2Ciba Geigy epoxy resin 3,380 0.39 81Silica flour 95,700 0.0775 10.5Solid glass 74,000 0.24 6.9Copper Powder 127,050 0.346 16.5

0 0.2 0.4 0.6 0.8 1


4E-006

8E-006

1.2E-005

1.6E-005

2E-005

2.4E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

Glass/ZrTin/Zr

Tin/Zr

Glass/Zr

Experimental Data Tummala and Friedberg (1970)

Proposed Micromechanical Model

Fig. 3 Comparison of the effective CTE with the experimental data of Tummala and Friedberg [51]

0 0.2 0.4 0.6 0.8 1


0

1E-005

2E-005

3E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

High TemperatureLow Temperature

High Temperature

Low Temperature

Experimental Data Fahmy and Ragai (1970)

Proposed Micromechanical Model

Aluminum matrix with Silicon inclusions

Fig. 4 Comparison of the effective CTE with the experimental data of Fahmy and Ragai [13]


0

5E-005

0.0001C

oeff

icie

nt o

f T

herm

al E

xpan

sion

(K

-1)

Ciba Geigy Epoxy Resin

α =7x10−10 T2 - 4x10−8 T + 3x10−5

100 ≤Τ ≤ 300

(a)

100 150 200 250 300100 150 200 250 300

100 150 200 250 300 100 150 200 250 300

Temperature (°K)Temperature (°K)

Temperature (°K) Temperature (°K)

0

1E-005

2E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

100 ≤Τ ≤ 300

Silica Flour

α =4x10−11 T2 - 1x10−10 T + 7x10−6

(b)

0

5E-006

1E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

Solid Glass

α = −6x10−11 T2 - 4x10−8 T + 3x10−7

(c)

100 ≤Τ ≤ 300

0

0.0002

0.0004

D(T

) ,

1/M

Pa

Temperature Dependent Compliancefor Ciba Geigy Epoxy Resin

(d)

Fig. 5 Nonlinear temperature-dependent constituents properties. a CTE for Ciba Geigy Epoxy Resin, b CTE for Silica Flour,c CTE for Solid Glass and d temperature-dependent compliance for Ciba Geigy Epoxy Resin

high temperature (between 20 and 400◦C). The temperature independent mechanical properties and thermalexpansions of silicon and aluminum are shown in Table 1. In both cases, the results of the proposed modelsare found in good agreement with the experimental data.

Feltham et al. [15] have experimentally studied the effects of temperature on the effective CTE of theparticle reinforced epoxy composites. The capability of the presented micromodel to predict the temperature-dependent effective CTE is evaluated using Feltham’s et al. [15] experimental data. Effective CTEs of threecomposite systems, i.e., silica flour/epoxy, glass/epoxy and copper/epoxy, were measured for a temperaturerange of 100–300◦K and with particle volume fractions of 10, 20, 30 and 40%. The elastic modulus of theepoxy resin at room temperature is taken from Feltham et al. [15], while the temperature-dependent elasticmodulus of the epoxy resin is obtained from Hartwig and Wuchner [18]. Temperature-dependent CTEs of theepoxy resin and solid glass are taken from Feltham et al. [15] while the one for the silica flour is given inTouloukian et al [49]. The elastic moduli of the particles are assumed to be independent of temperature and aretaken from Simmons and Wang [44]. The temperature-dependent properties of the constituents are shown inFig. 5, while the properties at reference temperature (300◦K) are given in Table 1. Figure 6 shows predictionsof the effective CTE of silica flour/epoxy composites. Analytical models of Fahmy and Ragai [13] and Turner[52] are also reported. It is observed that the present micromodel predicts the experimental data very well andis also comparable to the Fahmy and Ragai [13] model. Figure 7a shows CTE of a composite system consistingof Ciba-Geigy epoxy resin with solid glass microspheres, while Fig. 7b illustrates CTE for copper powderparticles dispersed in epoxy resin. In both composites, responses obtained using the micromechanical modelsare in good agreement with the experimental data.

The present micromodel is derived for predicting the effective CTE of composites having nonlinear thermo-viscoelastic responses. Several numerical simulations have been performed to analyze the time-dependenteffective CTEs of the composite. The time-dependent behavior of the effective CTE is found to be dependent


(b)

(d)

0

2E-005

4E-005

6E-005

8E-005

Experimental data Feltham and Martin (1982)

Present Micromodel

Turner (1946)

Fahmy and Ragai (1970)

(a)

100 150 200 250 300

Temperature (°K)

100 150 200 250 300

Temperature (°K)

100 150 200 250 300

Temperature (°K)

100 150 200 250 300

Temperature (°K)

0

2E-005

4E-005

6E-005

8E-005


Present Micromodel

Turner (1946)


(c)

0

2E-005

4E-005

6E-005

8E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)


Present Micromodel

Turner (1946)


0

2E-005

4E-005

6E-005

8E-005


Present Micromodel

Turner (1946)


Fig. 6 Temperature-dependent effective CTE of Ciba-Geigy epoxy resin containing silica flour at different volume fraction,a 10, b 20, c 30 and d 40%

on the elastic moduli and the CTE of the constituents and the roles of the constituents as inclusion and matrix,i.e., stiffer inclusions dispersed in softer matrix or softer inclusion in stiffer matrix. Four parametric studieshave been done to determine the effects of the particle and matrix thermal expansion and instantaneous elasticmoduli on the overall time-dependent CTE of the particulate composite. In all studies, the matrix modulusfollows a nonlinear viscoelastic model, while the particle has linear elastic behavior. Nonlinear and time-dependent properties of the epoxy matrix can be found in Muliana and Khan [33]. Temperature loading witha constant rate is applied. In the first two studies, both constituents have thermal properties independent oftemperature, but due to the temperature-dependent viscoelastic behavior of the matrix the overall CTE showstime-dependent behavior. In the third and fourth studies, temperature dependent CTE of particle and/or matrixis considered and their effects on the overall CTE of the composites are examined.

Let Ep be the elastic modulus of the particle, Em be the elastic modulus of the matrix and αm and αp bethe linear thermal expansion coefficients of the matrix and particle, respectively. In the first study, the ratio ofαp/αm = 0.5 is considered and the time-dependent effective CTE are determined for composites having 10,25 and 50% volume fractions, as shown in Fig. 8. It is shown that for Ep < Em (softer inclusions dispersed ina stiffer matrix) the effective CTE decreases tremendously with time as compared to the case when Ep > Em(stiffer inclusions dispersed in a softer matrix). In the second study, whenαp/αm = 5, the time-dependent effec-tive CTE increases prominently for the case when Ep < Em as illustrated in Fig. 9. The percentage changesin the value of the effective CTE after 1,800 s are shown in Tables 2 and 3. In both cases, for Ep < Em,a large amount of changes in the effective CTE is observed with the increase of the volume fraction of theparticle. It is noted that in all cases the particle responses are assumed linear elastic, while the matrix exhibitstemperature-dependent viscoelastic behavior. Softer particles result in less resistance to the microstructuralchanges, which make matrix and the micro-constituents easier to deform, while stiffer inclusions provide betterresistance to deformations. Furthermore, adding softer inclusions increases the ductility of the composite.


100 150 200 250 300

Temperature (°K)

100 150 200 250 300

Temperature (°K)

0

2E-005

4E-005

6E-005

8E-005

Coe

ffic

ient

of

The

rmal

Exp

ansi

on (

K-1

)C

oeff

icie

nt o

f T

herm

al E

xpan

sion

(K

-1)

Experimental data Feltham and Martin (1982)Present Micromodel

Turner (1946)Fahmy and Ragai (1970)

(a)

0

2E-005

4E-005

6E-005

8E-005


Present Micromodel

Turner (1946)


(b)

Fig. 7 Temperature-dependent effective CTE of Ciba-Geigy epoxy resin containing a solid glass microspheres and b copperpowder at a volume fraction of 20%

The last two studies deal with the cases when the CTEs of the matrix and particle are temperature depen-dent. The following cases are considered for each of the parametric studies: (a) αp is a linear function oftemperature, αp(T ), and αm is constant; (b) αm is a linear function of temperature, αm(T ), and αp is con-stant; and (c) both αp and αm are linear functions of temperature. Figure 10a–d shows the variation oftime-dependent CTEs of the composite under a linear temperature loading from reference temperature(300◦K) to 600◦K at different volume fractions with the ratio of αp/αm = 0.5. Composites having Ep > Emand Ep < Em are studied. It is found that with the increase of volume fractions the change in effective CTEafter 1,800 s increases for composites having temperature dependent CTE of the particle. For composites withαm(T ) and both αm(T ) and αp(T ) the percent change in effective CTE decreases after 1,800 s with increasingvolume contents of the inclusion. Figure 11a–d shows the time-dependent CTEs of the composite subject toa linear temperature loading from reference temperature (300◦K) to 600◦K at different volume fractions withthe ratio of αp/αm = 5. The results follow a similar trend as observed in the previous study. Tables 4 and 5show the percent changes in the effective CTE for all the cases described earlier.

It is therefore concluded that for temperature independent thermal and mechanical properties the effectiveCTE is strongly dependent on the elastic moduli of the constituents. With any ratio of the CTE of the constit-uents, the softer inclusion allows the composite to deform more easily under temperature changes as compareto the ones with stiffer inclusion. Changes in the effective CTE with time increase with the increase of volumecontents. When temperature-dependent properties are considered, with the increase in volume fractions ofthe softer inclusion the effective CTE is strongly dependent on αp(T ). This is valid only for the case when


Fig. 8 Time-dependent effective CTE at different volume fraction with thermal expansion ratio less than one (αp < αm)

αp > αm. For all other cases, the effective CTE of the composite is strongly dependent on the function of thetemperature-dependent CTE of the matrix, i.e., αm(T ).

4.2 Verification of ETC

The ETC obtained from the micromechanical formulation is verified using analytical models and experimentaldata available in the literature. Figure 12 shows ETC obtained from the present micromodel at various volumefractions. High density polyethylene based composite systems with tin particles as inclusion are used. Thethermal conductivities for all constituents are constant and given in Table 6. It is shown that the ETC obtainedfrom the present micromodel is comparable to the results obtained by Maxwell [29], Hashin and Shtrikman[19] and Benveniste [7] and is closer to the lower bound. The experimental data of Sugawara and Yoshizawa isalso considered to validate the ETC. Figure 13 shows the ETC of air saturated sandstone with porosity. Ther-mal conductivities of the studied materials are given in Table 6. The porous sandstone is assumed containingmacroscopically series of consolidated spheres of pores, and the air in the pores is considered as the inclusion.In this case, the conductivity of the inclusion is less than that of the matrix with a ratio of Kp/Km equal to1:80. The ETC determined from the present micromodel agrees well with the experimental data.

Highly conductive materials have been added as fillers into the polymeric based matrix composites toincrease the overall thermal conductivity of the composites. In most cases, fillers have less than 15% volumefractions. It has been observed that with the increase of the inclusion volume fractions and the ratios of thethermal conductivity of inclusions to the ones of the matrix the ETC of the composites increased tremendously.Tavman [47] experimentally investigated the ETC of tin powder filled high density polyethylene composites.


Fig. 9 Time-dependent effective CTE at different volume fraction with thermal expansion ratio larger than one (αp > αm)

Table 2 Changes in effective CTE (%) after 1,800 s with αp/αm = 0.5

Vf (%) Ep < Em Ep > Em

10 4.0 0.325 10.4 0.950 19.7 1.4

Table 3 Changes in effective CTE (%) after 1,800 s with αp/αm = 5.0


10 28 1.325 61 2.350 83 1.9

Figure 14 shows the comparison of the present micromodel and the Maxwell model [29] with the experimentaldata. Thermal properties of tin and polyethylene are given in Table 2. Good predictions are shown for com-posites with volume content less than 10%. The results of the proposed model deviate from the experimentaldata as the volume fraction of the tin particles increases as the present micromodel does not account for theconductive chain mechanism. Next, the experimental data of Zhang et al. [57] on composites with variousratios of Kp/Km is also used in this study, as illustrated in Fig. 15. It is seen that the micromodel predictionsare in good agreement with experimental data for composites with low values of Kp/Km. As Kp/Km increases,the micromodel shows good predictions only for lower volume contents (less than 15%). To capture better


(b)

(c) (d)

6E-005

8E-005

0.0001

0.00012

0.00014

0.00016

EpEm= 72.0GPa

2.7GPa

VF=10%

VF=50% VF=50%

VF=10%

αm(T) and αp(T)

αm(T)

αp(T)

(a)

4E-005

6E-005

8E-005

0.0001

EpEm= 72.0GPa

2.7GPa

αm(T) and αp(T)

αm(T)

αp(T)

4E-005

6E-005

8E-005

0.0001600500400300

Temperature (°K)

600500400300

Temperature (°K)

600500400300

Temperature (°K)

600500400300

Temperature (°K)

EpEm= 0.72GPa

2.7GPa

αm(T) and αp(T)

αm(T)

αp(T)

0 300 600 900 1200 1500 1800

Time (sec)

0 300 600 900 1200 1500 1800

Time (sec)

0 300 600 900 1200 1500 1800

Time (sec)0 300 600 900 1200 1500 1800

Time (sec)

6E-005

8E-005

0.0001

0.00012

0.00014

0.00016

(1/°

K)

αm(T) and αp(T)

EpEm= 0.72GPa

2.7GPaαm(T)

αp(T)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

Fig. 10 Time-dependent effective CTE at different volume fraction for CTE ratios less than one, i.e., αp < αm, with and/orwithout linear temperature-dependent CTE of each constituent

predictions at higher volume fractions, there is a need to develop a micromodel that incorporates the chainconductive mechanism of the particles. Nevertheless, the present micromodel is suitable for predicting ETCfor filler composite or for composite with low Kp/Km (thermal conductivity of the particle is comparable tothe one of the matrix).

As mentioned earlier, the proposed model is developed for the class of particulate composite material con-sisting of equi-sized spheres of one constituent arranged in a simple cubic array throughout a continuous secondconstituent. Therefore the effect of percolation (the formation of thermal conductive strings of the particles)cannot be considered in the present study. It is shown by Zhang et al. [57] that the larger Kp/Km gives the largerETC/Km for a certain volume fraction (Vf ), the increase of ETC is not significant when Vf < 0.3. Accordingto the percolation theory given in Stauffer [45], the exact value of the threshold for cubic percolation, Vfc, is0.3117–0.3333. When Vf < Vfc, the conductive particles are mainly dispersed, so the effect of the particles’conductivity on ETC is small. When Vf goes up to Vfc, the connections of the particles increase exponentiallyand the formations of the conductive chains dominate the change of ETC. Yin et al. [56] mentioned that thethreshold limit of percolation can go as high as 78%. It is observed that for the present model the thresholdlimit is found to be near 80%, as shown in Fig. 16.

The parametric studies are done to determine the variations in the ETC of the composite when the particleand/or matrix conductivities are temperature dependent. The thermal conductivities of the particle and matrixare assumed to vary linearly with temperature. Consider the thermal conductivities of the particle and matrix asKp and Km, respectively. Three cases have been considered: (a) Kp is a linear function of temperature and Kmis constant; (b) Kp is constant and Km is a linear function of temperature; and (c) both Kp and Km are linear


0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

EpEm= 72.0GPa

2.7GPa

VF=10%

VF=50%

VF=50%

VF=10%

αm(T) and αp(T)αm(T)

αp(T)

8E-005

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

EpEm= 0.72GPa

2.7GPa

αm(T) and αp(T)

αm(T)

αp(T)

0.00026

0.00028

0.0003

0.00032

EpEm= 72.0GPa

2.7GPa

αm(T) and αp(T)

αm(T)

αp(T)

8E-005

0.00012

0.00016

0.0002

0.00024

0.00028

0.00032

EpEm= 0.72GPa

2.7GPa

αm(T) and αp(T)

αm(T)

αp(T)

(b)(a)

(c) (d)

5.0αm

αp

5.0αm

αp

5.0αm

αp 5.0αm

αp

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

αp = 8.5x10−6 + 0.01x10−6 (T-To)αm = 75x10−6 + 0.3x10−6 (T-To)

0 300 600 900 1200 1500 1800

Time (sec)0 300 600 900 1200 1500 1800

Time (sec)

0 300 600 900 1200 1500 1800

Time (sec)0 300 600 900 1200 1500 1800

Time (sec)

600500400300

Temperature (°K)

600500400300

Temperature (°K)

600500400300

Temperature (°K)

600500400300

Temperature (°K)

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

(1/°

K)

Eff

ectiv

e C

oeff

icie

nt o

f L

inea

r T

herm

al E

xpan

sion

Fig. 11 Time-dependent effective CTE at different volume fraction for CTE ratios larger than one, i.e., αp > αm, with and/orwithout linear temperature-dependent CTE of each constituent

Table 4 Changes in effective CTE (%) with αp/αm = 0.5


αp(T ) αm(T ) αp(T ) and αm(T ) αp(T ) αm(T ) αp(T ) and αm(T )

10 2.66 51.64 51.74 1.52 113.71 114.2950 16.35 31.20 32.40 5.22 59.36 63.55

functions of temperature. Since the present micromodel does not incorporate the chain conductive mechanism,the ETC is analyzed for a low ratio of thermal conductivities, i.e., Kp/Km = 7.7 : 1. Figure 17 shows thevariation of ETC with temperature for composites with 5, 15 and 50% volume fractions. For low volumecontents (<15%), with only temperature dependent Kp(T ), the ETC insignificantly varies with temperature.For composites with 50% volume contents, the ETC with the temperature-dependent Kp(T ) shows signifi-cant variations with temperature, although the changes are less than the ones with the temperature-dependentKm(T ). It is noted that the rate of change of the thermal conductivity with temperature for the particle ishigher than the one of the matrix. It is observed that the ETC is strongly dependent on the rate of change inthe thermal conductivity of the matrix with temperature, i.e., with only temperature-dependent Kp(T ), lessvariations of ETC at all volume contents are shown, while higher variations of the ETC with temperature are


Table 5 Changes in effective CTE (%) with αp/αm = 5.0


αp(T ) αm(T ) αp(T ) and αm(T ) αp(T ) αm(T ) αp(T ) and αm(T )

10 18.18 52.38 52.46 3.90 60.02 60.3650 36.36 43.51 43.83 0.55 8.53 9.25

0 0.2 0.4 0.6 0.8 1


0

20

40

60

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

Upper BoundLower BoundGeometric MeanHashin & Shtrikman (Upper Bound)Benveniste (Moritanaka Method)Hashin & Shtrikman (Lower Bound)Maxwell ModelProposed Model

High Density Polyethylene filled with Tin Particles Kp/Km = 120:1

Fig. 12 Comparison of the ETC with different analytical models

Table 6 Material properties used for ETC verification

Material Thermal conductivity (K), W/m/◦K

Air 0.02Sandstone 1.6Firebrick 1.2High density polyethylene 0.532Polyethylene 0.29Polystyrene 0.14Polyvinyl chloride 0.17Polyamide 0.19Tin 64Graphite 209

shown for temperature-dependent Km(T ). These parametric studies indicate that ETC responses depend onthe microstructural constituents (inclusion and matrix) and also on the properties of the constituents. To betterunderstand the effects of temperature-dependent thermal conductivities and their variations on ETC with theincrease of the volume fraction, it is required to analyze the ETC of the composites with the micromodel thatincorporates the chain conductive mechanism.

5 Conclusions

The effective thermal properties of particulate composites have been studied. The micromechanical modelpreviously developed by Muliana and Kim [34] for stress-dependent viscoelastic responses has been modi-fied to derive the effective CTE and ETC. A stress and temperature-dependent viscoelastic model is used for


0 0.2 0.4 0.6 0.8 1


0

0.4

0.8

1.2

1.6

2

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

Proposed model

Experimental Data Sugawara and Yoshizawa (1962)

Kp/Km = 1:80Saturated stone with porosity

Fig. 13 Comparison of the ETC with the experimental data of Sugawara and Yoshizawa [46]

0 0.05 0.1 0.15 0.2


0.4

0.6

0.8

1

1.2

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

Maxwell Model (1954)

Experimental Data Tavman (1998)

Kp/Km = 120:1

High Density Polyethylene filled with Tin Particles

Present Model

Fig. 14 Comparison of the ETC with the experimental data of Tavman [47]

the isotropic constituents in the micromechanical model. Thermal properties of the particle and matrix con-stituents are temperature dependent. Perfect bonds are assumed along the subcell’s interfaces. The effectiveviscoelastic responses and linear thermal expansion coefficient are derived by satisfying displacement andtraction continuity at the subcells’ interfaces. This formulation leads to a time–stress–temperature-dependentthermal expansion coefficient of the composites. The ETC is formulated by imposing heat flux and temperaturecontinuity at the subcells’ interfaces. The effective thermal properties are comparable with analytical solutionsand available experimental data in the literature. Parametric studies are also performed to investigate the effectof nonlinear thermal and mechanical properties of the constituents on the effective CTE of the composites.For temperature independent CTE of each constituent, the effective CTE is strongly dependent on the elasticmoduli of the constituents. With temperature dependent properties and increase in volume fractions of thesofter inclusion, the effective CTE is strongly dependent on αp(T ) for the case when αp > αm. Otherwise,the effective CTE strongly depends on the temperature-dependent function of αm. Micromodel predictions forthe ETC are also in good agreement with the available experimental data for composites with low values of


0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

Volume Fraction (VF) Volume Fraction (VF)

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5


0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5


0.1

0.2

0.3

0.4

0.5

0.6

Proposed model

Experimental Data

Silica filled with Epoxy, Kp/Km = 7.7:1

0.1

0.2

0.3

0.4

0.5

0.6

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

0

0.4

0.8

1.2

1.6

0.1

0.2

0.3

0.4

0.5

0.6

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

0

0.4

0.8

1.2

1.6

2

0

0.4

0.8

1.2

1.6

2

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

.K)

(a)

Proposed model

Experimental Data

CaO Filled with Polystyrene, Kp/Km = 97:1

Proposed model

Experimental Data

Alumina Filled with Epoxy, Kp/Km = 185:1

Proposed model

Experimental Data

MgO Filled with Polystyrene, Kp/Km = 354:1

Proposed model

Experimental Data

Scan filled with Epoxy, Kp/Km = 1128:1

Proposed model

Experimental Data

Graphite Filled with Polyvinyl Chloride, Kp/Km = 1240:1

(b)

(c) (d)

(e) (f)

Fig. 15 Comparison of the ETC with the experimental data of Zhang et al. [57]


0 0.2 0.4 0.6 0.8 1


0.1

1

10

100

1000

10000

ET

C /

Km

104

102

103

10

Kp:Km

Fig. 16 Numerical results of ETC/Km for different volume fraction and Kp/Km

Kp = 1.5 - 0.002 (T-To)Km = 0.195 - 0.0005 (T-To)

Kp = 1.5 - 0.002 (T-To)Km = 0.195 - 0.0005 (T-To)

Kp = 1.5 - 0.002 (T-To)Km = 0.195 - 0.0005 (T-To)

0.08

0.12

0.16

0.2

0.24

Km(T)

Km(T) and Kp(T)

Kp(T)

500400300

Temperature (°K)500400300

Temperature (°K)

500400300

Temperature (°K)

0.12

0.16

0.2

0.24

0.28

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

/°K

)E

ffec

tive

The

rmal

Con

duct

ivity

(W

/m/°

K)

Eff

ectiv

e T

herm

al C

ondu

ctiv

ity (

W/m

/°K

) VF=15%VF=5%

VF=50%

Km(T)

Km(T) and Kp(T)

Kp(T)

0.25

0.3

0.35

0.4

0.45

0.5

Km(T)

Km(T) and Kp(T)

Kp(T)

(a) (b)

(c)

Fig. 17 ETC with and/or without linear temperature variation of the thermal conductivities of each constituent for a ratio ofKp/Km = 7.7 :1


Kp/Km. As Kp/Km increases, the micromodel shows good predictions only for lower volume contents (lessthan 15%). Better results at higher volume fractions are expected with the incorporation of the chain conductivemechanism in the micromodel. Nevertheless, the present micromodel is quite suitable for predicting ETC forfillers composite.

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