Mechanics of Materials Volume 43 issue 10 2011 [doi 10.1016_j.mechmat.2011.06.013] Kamran A. Khan; Romina Barello; Anastasia H. Muliana; Martin Lé -- Coupled heat conduction and thermal

8/10/2019 Mechanics of Materials Volume 43 issue 10 2011 [doi 10.1016_j.mechmat.2011.06.013] Kamran A. Khan; Romina

1/18

Coupled heat conduction and thermal stress analysesin particulate composites

Kamran A. Khan a, Romina Barello b, Anastasia H. Muliana a,, Martin Lvesque b

a Department of Mechanical Engineering, Texas A&M University, USAb CREPEC, Department of Mechanical Engineering, Ecole Polytechnique de Montreal, Canada

a r t i c l e i n f o

Article history:

Received 2 May 2010

Received in revised form 15 April 2011

Available online 13 July 2011

Keywords:

Heat conduction

Thermal stresses

Particulate composites

Finite element

Micromechanical model

a b s t r a c t

This study introduces two micromechanical modeling approaches to analyze spatial vari-

ations of temperatures, stresses and displacements in particulate composites during tran-

sient heat conduction. In the first approach, a simple micromechanical model based on a

first order homogenization scheme is adopted to obtain effective mechanical and thermal

properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic

constants, of a particulate composite. These effective properties are evaluated at each

material (integration) point in three dimensional (3D) finite element (FE) models that rep-

resent homogenized composite media. The second approach treats a heterogeneous com-

posite explicitly. Heterogeneous composites that consist of solid spherical particles

randomly distributed in homogeneous matrix are generated using 3D continuum elements

in an FE framework. For each volume fraction (VF) of particles, the FE models of heteroge-

neous composites with different particle sizes and arrangements are generated such that

these models represent realistic volume elements cut out from a particulate composite.

An extended definition of a RVE for heterogeneous composite is introduced, i.e., the num-

ber of heterogeneities in a fixed volume that yield the same expected effective response for

the quantity of interest when subjected to similar loading and boundary conditions. Ther-

mal and mechanical properties of both particle and matrix constituents are temperature

dependent. The effects of particle distributions and sizes on the variations of temperature,

stress and displacement fields are examined. The predictions of field variables from the

homogenized micromechanical model are compared with those of the heterogeneous

composites. Both displacement and temperature fields are found to be in good agreement.

The micromechanical model that provides homogenized responses gives average values of

the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the

particlematrix interface regions and local variations of the field variables within particle

and matrix regions.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

The existence of thermal stresses has always been a

subject of discussion when a body is subjected to coupled

heat conduction and mechanical loadings. In composites,

significant thermal stresses can arise due to the mismatch

in the coefficient of thermal expansions (CTEs) of the con-

stituents, affecting their overall performance. The use of

composite materials in structural components requires

understanding the variations in the field variables, such

as stress, strain, temperature and displacement, both at

micro- and macro scales. At the macro scale, composite

structures are often analyzed as homogeneous structures

through the use of effective properties, which allow per-

forming large-scale structural analyses. Composites are

heterogeneous materials that can exhibit large variations

and even discontinuities in the field variables. These

0167-6636/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmat.2011.06.013

Corresponding author.

E-mail address: [email protected](A.H. Muliana).

Mechanics of Materials 43 (2011) 608625

Contents lists available at ScienceDirect

Mechanics of Materials

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c at e / m e c h m a t
http://dx.doi.org/10.1016/j.mechmat.2011.06.013mailto:[email protected]://dx.doi.org/10.1016/j.mechmat.2011.06.013http://www.sciencedirect.com/science/journal/01676636http://www.elsevier.com/locate/mechmathttp://www.elsevier.com/locate/mechmathttp://www.sciencedirect.com/science/journal/01676636http://dx.doi.org/10.1016/j.mechmat.2011.06.013mailto:[email protected]://dx.doi.org/10.1016/j.mechmat.2011.06.013


2/18

variations and discontinuities at the micro-scale cannot be

captured if one treats composites as homogenous (homog-

enized) materials.

Different types of micromechanical models with simpli-

fied microstructures have been developed to obtain effec-

tive thermal and mechanical properties. The composite

spheres model (Hashin, 1962), the self consistent approach

(Budiansky, 1965 and Hill, 1965); the generalized self

consistent scheme (Christensen and Lo, 1979, 1986), the

Mori-Tanaka model (Mori and Tanaka, 1973), the probabi-

listic approach ofChen and Acrivos (1978), the differential

method (McLaughlin, 1977 and Norris, 1985) are some

examples. Detailed discussion of various micromechanical

models and bounds on the effective mechanical properties

can be found inAboudi (1991), Mura (1987),Nemat-Nas-

ser and Hori (1999). Khan and Muliana (2010) presented

Fig. 1. Schematic diagram of integration of micro-macro scale approach for particulate composites.

Fig. 2. Homogenization of the sub-regions of a composite component.

K.A. Khan et al. / Mechanics of Materials 43 (2011) 608625 609


3/18

a short review of the capabilities and shortcoming of few

analytical and numerical models for effective CTE and

effective thermal conductivity (ETC) of composites. Analyt-

ical expressions for ETC for two-phase composites made of

randomly distributed and dilute concentrations of spheres

in a homogeneous medium are given, for example, by Max-

well (1954), Verma et al. (1991) and Hasselman and John-

son (1987). Expressions for non-dilute concentrations of

spheres are provided by the models ofJeffrey (1973), Davis

(1986) and Sangani and Yao (1988). These models lead to

accurate ETC predictions, when compared to experimental

results, when the volume fraction (VF) is relatively small or

when the conductivities of the constituents are compara-

ble (Matt and Cruz, 2002). Bounds on the effective thermal

conductivity have been derived using variational principle

(Hashin and Shtrikman, 1962; Beran, 1965) and three point

probability function for the distribution of microstructures.

Several numerical modeling approaches have been

developed to estimate effective thermal and mechanical

behavior of a composite. To obtain effective heat conduc-

tion response and ETC of a composite the composites are

often described with random microstructures (Ostoja-Star-

zewski and Schulte, 1996; Nogales and Bohm, 2008) as

well as with periodic microstructures (Auriault, 1983; Ver-

ma et al., 1991; Jiang et al., 2002). Real composite micro-

structures are generally non periodic; however, the use

of periodic microstructures can give approximate effective

properties for a smaller computational cost when com-

pared to that of random microstructures. To obtain the

effective thermo-mechanical response, among the numer-

ical methods, the Finite Element (FE) technique has been

used for determining the micro- and macro-structural per-

formance of composites by meshing their detailed micro-

structures; see for example the works of Llorca and

Segurado (2002, 2003 and 2006), Lvesque et al. (2004,

2008) and Zohdi and Wriggers (2001). The composite

behavior was obtained from simulations of a 3D cubic Rep-

resentative Volume Element (RVE) containing several ran-

domly distributed non-overlapping identical spheres.

Periodic boundary conditions were imposed on the meshes

in order to obtain the effective properties. Linearly elastic,

elasticplastic and linearly (Lvesque and Barello, 2009)

and nonlinearly viscoelastic (Lvesque et al. 2004) re-

sponses were analyzed. Kwon and Kim (1998) developed

a 3D micromechanical model to compute thermal stresses

of particulate composites. Using a similar type of microme-

chanical model, Meijer et al. (2000) studied the influence of

inclusion geometry and thermal residual stresses- and

strains on the mechanical behavior of an aluminum alloy

reinforced with aluminum oxide particles. A unit-cell con-

sisting of eight subcells was considered. One cell repre-

sents the particle and the rest of the cells surrounding

the particle were considered as matrix. Both constituents

were assumed to be linearly elastic and the thermal prop-

erties were considered to be temperature independent. The

effects of the size of the subcells on the stresses and strains

Fig. 3. 3D FE models of (a) homogenized and (b) heterogeneous composites. (c) Schematic of thermal and mechanical loading directions and profiles along

which field variables are evaluated, i.e., AB, CD, EF and GH.

Table 1

Temperature dependent mechanical and physical properties of materials of Ti6Al4V and ZrO 2used in 3D FE analyses.

Property Ti6Al4V Zirconia (Zr02)

Young modulus (E) (Pa) 1.23 101156.46 106T 2.44 1011334.28 106T+ 295.24 103T2 89.79T3

Poisson ratio (t) 0.3 0.3Coefficient of thermal expansion (a)106,

1/K

7.58 106 + 4.93 109

T+ 2.39 1012 T21.28 105 19.07 109 T+ 1.28 1011

T2 8.67 1017 T3

Thermal conductivity, (k), W/m/K 1.21 + 0.0169T 1.7+ 2.17 104 T+ 1.13 105 T2

Specific heat (c), J/kg K 625.297 0.264T+ 4.49 104 T2 487.343 + 0.149T 2.94 105 T2

Density (q), kg/m3 4429 5700

T is temperature in Kelvin (K).

610 K.A. Khan et al./ Mechanics of Materials 43 (2011) 608625


4/18

were analyzed. It was found that the sharp corner and

edges of the cube resulted in localized stresses and strains.

Recently, Aboudi (2008)presented a micromodel for fully

coupled thermo-mechanical analysis for multiphase com-

posites where heat generation from the interconvertibility

of the mechanical and thermal energy was accounted for.

This study focuses on understanding the effects of con-

stituents properties and microstructural details on the

variation of stress, displacement and temperature in com-

posites. The results are used to justify the capability of

micromechanical models to analyze the overall response

of composites subjected to simultaneous mechanical and

thermal stimuli, within a certain degree of accuracy. A

simplified micromechanical model for predicting the effec-

tive thermo-mechanical behavior of particulate composites

subjected to both time and space varying temperature field

is introduced. The micromechanical model is called at each

integration point in a FE mesh to calculate thelocal effective

properties of the composite. The responses thus obtained

are then compared to FE simulations of composites using

the meshes ofBarello and Lvesque (2008). A sequentially

thermo-mechanical coupled problem, i.e., the temperature

field influences the deformation field, is considered. The

effects of particle volume contents and temperature depen-

dent constituent properties on the overall thermo-elastic

behavior of the composites are examined.

The paper is organized as follows. A brief outline of the

simplified micromechanicalmodel is presented in Section2.

The effective thermoelastic stress-strain relations, effective

heat flux equation and uncoupled energy equation for an

isotropic homogenized composite are also discussed.

Section 3 presents a micro-mechanical framework used

for computing the response of a real-size composite. FE

modeling of composites with microstructural details is

given in Section4. Coupled heat conduction and thermal

stress analysis of particulate composites are discussed in

Section5. A summary of the research findings is given in

Section6.

2. A simplified micromechanical model for particle

reinforced composites

Muliana and Kim (2007), Muliana (2008) and Khan and

Muliana (2010) developed a micromechanical model for

determining the effective viscoelastic responses and

300

400

500

600

700

Temperature(K)

Temperature(K)

a

t =12s

t=26s

t=2s

Steady State Time = 150 seconds

Detail Microstructural Model

Micromechanical Model(CPU Time = 277s)

Model-1 (CPU Time = 12785s)






(Volume fraction = 20%, 20 particles)

b

0 2 4 6 8 10

Distance (mm) Distance (mm)

300

400

500

600

700

300

400

500

600

700

Temperature(K)

Temperature(K)

300

400

500

600

700

t=12s

t =26s

t=2s


Micromechanical Model

Model-1

Model-2

Model-3

Model-4


t=12s

t=26s

t =2s

Detail Microstructural Model (Volume fraction = 20%, 20 particles)


c

0 2 4 6 8 10

0 2 4 6 8 10

Distance (mm) Distance (mm)0 2 4 6 8 10

t=12s

t=26s

t=2s




d

Fig. 4. Comparison of temperature profiles for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FEmodels with 3D microstructural detail (symbols) for volume fraction of 20%. (a) and (b) are actual values of temperature at top (corner) edge {( X1, 10, 10);

06X1 6 10}, (c) and (d), mean value of temperatures of different FE models measured at extreme top and bottom (corner) edges of the cubes along the

temperature gradient direction..

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


5/18

thermal properties (CTE, and thermal conductivity) of a par-

ticle reinforced polymer composite. The model is modified

for simulating sequentially coupled heat conduction and

deformation in particulate composites. The model idealizes

particles in the microstructure as cubes. The cubic particles

are arranged uniformly in a homogeneous matrix. The RVE

is defined as a single particle embedded in a cubic matrix.

Periodic boundary conditions are imposed to the RVE. Due

to the three-plane symmetry, a unit-cell model thatconsists

of four particle and matrix subcells is considered. Fig. 1

t =12s

t=26s

t =2s



Model-1

Model-2

Model-3

Model-4


t =12s

t=26s

t=2s



Model-1

Model-2

Model-3

Model-4


t =12s

t =26s

t =2s



Model-1

Model-2

Model-3

Model-4


t=12s

t =26s

t= 2s




t=12s

t=26s

t =2s




t=12s

t=26s

t=2s




300

400

500

600

700

Temperature(K)

a

300

400

500

600

700

Temperature(K)

d

0 2 4 6 8 10

Distance (mm)0 2 4 6 8 10

Distance (mm)

300

400

500

600

700

T

emperature(K)

b

300

400

500

600

700

T

emperature(K)

e

0 2 4 6 8 10


Distance (mm)

300

400

500

600

700

Temperature(

K)

c

300

400

500

600

700

Temperature(K)

f

0 2 4 6 8 10


Distance (mm)

Fig. 5. Comparison of temperature profiles for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE

models with 3D microstructural detail (symbols) for volume fraction of 30%. (a), (b) and (c) are actual values of temperature at top (corner) edge{(X1, 10,

10); 06X1 6 10}, (d), (e) and (f), mean value of temperatures of different FE models measured at extreme top and bottom(corner) edges of the cubes alongthe temperature gradient direction.



6/18

shows the RVE idealization, unit-cell and its integration

with the FE framework. The choice of the four-cell model

was primarily done to reduce computational costs; how-

ever, this turned out to cause extra effort in incorporating

material symmetry. It is noted that the chosen unit-cell

model, which is only one-eighth of the full model, was

due to the threeplanes of symmetrythat allows interchang-

ing the principal axes in formulating the stress, strain, tem-

perature gradient and heat flux quantities. As a result, the

unit-cell response is independent of the loading direction.

When both particle and matrix are isotropic, the outcome

of the homogenized micromechanical model does not

always fulfill the isotropic condition with regard to the

mechanical response. The micromechanical formulation

has been discussed in detail elsewhere (Muliana and Kim,

2007). Perfect bonds are assumed along the subcells inter-

faces. Thermo-elastic constitutive models with tempera-

ture dependent material parameters are used for both

isotropic constituents. Linearized micromechanical rela-

tions are formulated in terms of incremental average field

variables, i.e., stress, strain, heat flux and temperature

gradient, of the subcells. The effective CTE is derived by sat-

isfying total displacement compatibility and traction conti-

nuity at the interfaces during thermo-elastic deformations.

This formulation leads to an effective temperature-depen-

dent CTE. The effective thermal conductivity is formulated

by imposing heat flux and temperature continuities at the

subcells interfaces. This micromechanical model is

compatible with general displacement based FE software,

which can be used to perform thermo-mechanical analyses

of composites structures.

For linearly thermo-elastic problems, the effective

stress (rij) and strain (eij) are related through:

rij Cijklekl aklT T0 1

or

eij Sijklrkl aijT T0 2

whereCijkl and Sijkl are the components of the effective elas-

tic stiffness and compliance tensors, respectively. The aklare the components of the effective CTE tensor. The param-

eters TandT0 are the effective current and reference tem-

peratures, respectively.

The average heat flux equation for a homogeneous com-

posite medium is expressed by Fouriers law of heat con-

duction as:

qi Kij uj; where uj @T

@xj3

where qi and uj are the components of the average heatflux and temperature gradient vectors, respectively. In or-

der to obtain the temperature profiles during heat conduc-

tion in the composite, the energy equation needs to be

solved. For the thermo-elastic case, in the absence of inter-

nal heat generation and thermo-mechanical coupling ef-

fect, the energy equation can be written as:

qcxk_T qi;i i; k 1; 2; 3 4

whereq

cx

kis the effective heat capacity that depends on

the composition, density and specific heat of the two

constituents in the composite body. The effective heat

capacity is obtained using a volume average method.

The linearized thermo-elastic constitutive equations are

expressed in an incremental form. A macroscopic strain

and temperature gradient are known and used as input

variables to the micromechanical formulation. The current

microscopic strain and temperature gradient are expressed

as etij

etDtij

detij

anduti

utDti

duti

, respectively. For

simplicity, the superscript t indicating current time, will

be dropped. The macroscopic incremental strain (dekl) islinked to the average incremental strain of each sub-cell

(deaij ) using the strain concentration tensor (Baijkl), written

as

dea

ij Ba

ijkldekl 5

Similarly, the average temperature gradient in each

subcell duai ) is related through to overall temperaturegradient (d uj) by the concentration tensor (M

aij ), written

as

dua

i M

a

ij duj 6

where superscript (a) denotes the subcell number, i.e.,a = 1, 2, 3, 4. The strain concentration tensor (Baijkl) is

t=8s

t =26s

t=2s

Detail Microstructural Model (Volume fraction = 30%)

15 particles

30 particles

t=12st=18s

45 particles

t=8s

t=26s

t=2s


20 particles

40 particles

t =12s

t=18s

300

400

500

600

700

Tem

perature(K)

a

0 2 4 6 8 10

Distance (mm)

300

400

500

600

700

Temperature(K

)

b

0 2 4 6 8 10

Distance (mm)

Fig. 6. Mean temperature profiles for FE models with the unit cell

(micromechanicalmodel) at each integration point (solid line) and the FE

models with 3D microstructural detail (symbols) for volume fraction of(a) 20% and (b) 30%.



7/18

determined by imposing the constitutive relation of each

subcell and micromechanical relations such that the dis-

placement compatibility and traction continuity condi-

tions are satisfied. Two sets of equations are formed. The

first set of the equations are determined from the strain

compatibility equations which are given as:

fReg121

AM11224

e1

e2

e3

e4

8>>>>>:

9>>>=>>>;

241

DM1126

feg61

7

where {Re} is the strain residual vector. The second set of

the equations satisfies the traction continuity relations

within subcells:

fRrg121

AM21224

e1

e2

e3

e4

8>>>>>:

9>>>=

>>>;241

O126

feg61

8

where {Rr} is the stress residual vector. The matrixO is the

zero matrix and the components of matrix AM1 ;A

M2 andD

M1

can be found elsewhere (Muliana and Kim, 2007). For lin-

earized elasticity problems, the components of the residual

vectors are zero and thus the micromechanical relations

are exactly satisfied. When any of the constituents exhibit

nonlinear response, imposing linearized micromechanical

relations leads to non-zero residual vectors {Re} and {Rr}.

The Newton Raphson iterative method is used to minimize

these residual vectors. Upon minimizing the residual, the

strain concentration matrixB(a) is obtained from:

Ba;t246

A

M1

AM2

" #12424

DM1

O

" #246

9

To formulate the concentration tensor, Maij ;the micro-

mechanical relations and the constitutive equations for

heat flux are imposed. This requires forming twelve (12)

equations based on the temperature and heat flux continu-

ities at the interface of each subcell as:

-0.005

0

0.005

0.01

0.015

Di

splacement(mm)

t=26s

t=12s

t=2s



Model-1

Model-2

Model-3

Model-4

Model-5

Model-6

a

-0.005

0

0.005

0.01

0.015

Displacement(mm)

t=26s

t=12s

t=2s


Model-1

Model-2

Model-3

Model-4

b

-0.005

0

0.005

0.01

0.015

Displacement(mm)

t=12s

t=26s

t=2s




d

-0.005

0

0.005

0.01

0.015

Displacement(mm)

t =12s

t =26s

t=2s




cDetail Microstructural Model(Volume fraction = 20%, 20 particles)


0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

Fig. 7. Comparison of axial displacements for FE models with the unit cell (micromechanical model) at each integration point (solid line) andthe FE models

with 3D microstructural detail (symbols) for volume fraction of 20%.(a) and (b) are actual values of displacements at top (corner) edge {(X1, 10, 10);

06X1 6 10}, (c) and (d), mean value of displacements of different FE models measured at extreme top and bottom (corner) edges of the cubes along thetemperature gradient direction.



8/18

fRug91

AT1912

du1

du2

du3

du4

8>>>>>:

9>>>=>>>;

121

DT193

fd ug31

10 fRqg31

AT2312

du1

du2

du3

du4

8>>>>>:

9>>>=>>>;

121

O33

fdug31

11

-0.005

0

0.005

0.01

0.015

0.02

Displacement(mm)

t =26s

t =12s

t =2s



Model-1

Model-2

Model-3

Model-4

-0.005

0

0.005

0.01

0.015

0.02

Displacement(mm)

t =26s

t=12s

t=2s



Model-1

Model-2

Model-3

Model-4

-0.005

0

0.005

0.01

0.015

0.02

Displacement(mm)

t=26s

t =12s

t=2s



Model-1

Model-2

Model-3

Model-4

-0.005

0

0.005

0.01

0.015

0.02

Displacement(mm)

t=12s

t=26s

t=2s



(Volume fraction = 30%, 15 particles)(Volume fraction = 30%, 15 particles)

-0.005

0

0.005

0.01

0.015

0.02

Displacement(mm)

t=12s

t=26s

t=2s





-0.005

0

0.005

0.01

0.015

0.02

Displacement(

mm)

t=12s

t=26s

t=2s





0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

a d

b e

c f

Fig. 8. Comparison of axial displacements for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE models

with 3D microstructural detail (symbols) for volume fraction of 30%. (a), (b) and (c) are actual values of displacements at top (corner) edge {( X1, 10, 10);

06X1 6 10}, (d), (e) and (f), mean value of displacements of different FE models measured at extreme top andbottom(corner) edges of the cubes along thetemperature gradient direction.



9/18

where {R/} and {Rq} are the temperature gradient and heat

flux residual vectors, respectively. By substituting Eq. (6)

into Eqs.(10) and (11), theM(a) matrix can be determined

as:

Ma121

A

T1

AT2

" #11212

DT1

O

" #121

12

where AT1, A

T2 andD

T1 can be found elsewhere (Khan and

Muliana, 2009). The effective (homogenized) stress and

effective tangent stiffness matrix can be obtained from

the following equations:

rij1

V

X4a1

VaCa

ijklBa

klrsers a

a

kl DT Cijklekl aklDT 13

Cijkl1

V

X4a1

VaCa

ijklB

a

klrs 14

whereVandV(a) are the total volume of the unit-cell mod-

el and sub-cell volume, respectively. From Eq. (13), theeffective CTE, (aij) can be obtained and written as:

aijC1ijklV

X4a1

VaCa

klmnaamn 15

Using the heat conduction equation for each sub-cell

and the effective heat flux relation (Eq. (3)), the tangent

effective thermal conductivity matrix of the composite

can be expressed as:

Kik 1

V

X4a1

VaKa

ij Ma

jk 16

3. A multi-scale model for computing the response of a

homogenized composite

The micro-mechanical model of Section 2 has been inte-

grated into a multi-scale FE framework in order to com-

pute the field variables of a real-size composite.

Boundary conditions are imposed on the real-size compos-

ite model and initial values of the field variables are as-

sumed. Local effective properties (thermal, mechanical)

are computed at each integration point using the micro-

mechanical model. Computations of the effective proper-

ties are based on the assumption that the each integration

point is associated with a much smaller volume than that

of the whole composite. As a result, we assumed that each

volume associated with each integration point was at a

uniform temperature. Therefore, at the micro-scale, both

the matrix and reinforcement were assumed to be at the

same temperature. This allows determining temperature-

dependent thermal conductivities for the particle and ma-

trix constituents and calculating the effective thermal con-

ductivities of the composite at that instant of time in each

material point. Therefore, in this context, the periodic

boundary conditions are fully justified for obtaining the

effective properties. At macroscopic scale during the tran-

sient heat conduction, the temperature-dependent proper-

ties of the constituents lead to spatially dependent ETC.

To simulate the effective thermo-elastic response of the

particulate composite, the micromechanical model is inte-

grated with the ABAQUS/standard FE code. At each integra-

tion point in the FE mesh, the user subroutine UMATH is

first called to evaluate the effective thermal conductivity,

heat fluxes, and temperature gradient for solving the equa-

tion that governs the conduction of heat in the composite

body. The temperature distributions obtained from heat

transfer analyses at various instants of time in the compos-

ites are used as input transient thermal load to determine

the thermo-elastic deformation in the composite body.

UMAT and UEXPAN subroutines were used to evaluate

the effective mechanical response and CTE, respectively.

Fig. 1illustrates the whole multi-scale framework.

4. Three dimensional FE models of particulate

composites

4.1. General methodology for evaluating the performance of

the multi-scale framework

The multi-scale model of Section 3 can be considered as

an approximation to a complicated thermo-mechanical

-0.005

0

0.005

0.01

0.015

Displacement(m

m)

t =12s

t=26s

t=2s


15 particles

b

30 particles

45 particles

-0.005

0

0.005

0.01

0.015

Disp

lacement(mm)

t=12s

t=26s

t=2s


20 particles

a

40 particles

0 2 4 6 8 10

Distance (mm)

0 2 4 6 8 10

Distance (mm)

Fig. 9. Mean displacement profiles for FE models with the unit cell

(micromechanical model) at each integration point (solid line) and the FE

models with 3D microstructural detail (symbols) for volume fraction of(a) 20% and (b) 30%.

http://-/?-http://-/?-http://-/?-http://-/?-


10/18

problem: the problem of computing the temperature and

stresses inside a heterogeneous material subjected to both

thermal and mechanical loads. In order to evaluate the

reliability of the micro-mechanical model, it is therefore

required to compare its predictions against the numeri-

cally exact solution. The objective of this section is to

generate this so called numerically exact solution.

When dealing with effective properties, one of the key

issues is the definition of the RVE. A RVE can be seen as a

volume of material having the same behavior as any larger

volume of the same material. The size of the RVE is mea-

sured in terms of inhomogeneities it contains (e.g. the

number of particles meshed, like 5, 10, 15, etc., particles).

One of the techniques used for obtaining the RVE size is

to use numerical homogenization based on FE. The method

consists in generating many FE meshes of the composite

microstructure with a fixed number of reinforcements

(10, 15, 50, etc.). Since the particles are distributed accord-

ing to a statistical distribution, each mesh, or realization,

will be different. Therefore, each realization should lead

-400

-200

0

200

400

AxialStresses(

11

)MPa

t=2s



Model-1

Model-2

Model-3

Model-4

Model-5

Model-6


-400

-200

0

200

400

AxialStresses(

11

)MPa

t=12s



Model-1

Model-2

Model-3

Model-4

Model-5

Model-6


-400

-200

0

200

400

AxialStresses(

11

)MPa

t=26s



Model-1

Model-2

Model-3

Model-4

Model-5

Model-6


-400

-200

0

200

400

AxialStresses(

11

)MPa

t

=2s



Model-1

Model-2

Model-3

Model-4


-400

-200

0

200

400

AxialStresses(

11

)MPa

t=12s



Model-1

Model-2

Model-3

Model-4


-400

-200

0

200

400

AxialStresses(

11

)MP

a

t=26s



Model-1

Model-2

Model-3Model-4


0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10

Distance (mm)

0 2 4 6 8 10

Distance (mm)

0 2 4 6 8 10

Distance (mm)

0 2 4 6 8 10

Distance (mm)

a d

b e

c f

Fig. 10. Axial thermal stresses for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE models with 3Dmicrostructural detail (symbols) for volume fraction of 20% at different times.



11/18

to different (within certain accuracy) effective properties.

For the same number of reinforcements and load history,

the effective responses are computed for each realization

and then averaged. Computing a confidence interval (for

example a two-tail 95% confidence interval) on this data

could give an estimation of the composites effective prop-

erties and its precision for a given number of reinforce-

ments. The relative precision (for example the Youngs

modulus is estimated to bexy%) can be adjusted by vary-

ing the number of realizations. If many realizations are

performed, the confidence interval can be adjusted to the

desired width. Kanit et al. (2003) mentioned that for

microstructures containing numerous reinforcements,

smaller numbers of realizations are required to estimate

0 2 4 6 8 10

Distance (mm)

-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=2s



-400

-300

-200

-100

0

100

200

300

400

Axial

Stresses(

11

)MPa

t =12s



-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=26s



0 2 4 6 8 10

Distance (mm)

-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=2s



-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=12s



-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=26s



0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

a d

b e

c f

Fig. 11. Axial thermal stresses for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE models with 3D

microstructural detail (symbols) for volume fraction of 20% at different times with C.I. of 95%. (a), (b) and (c) are mean value of stresses of different FE

models with 20 particles and (d), (e) and (f) with 40 particles, measured at extreme top and bottom (corner) edges of the cubes along the temperaturegradient direction.



12/18

the wanted overall property within desired precision. For

relatively small numbers of particles, the homogenized

properties vary statistically until a certain number of par-

ticles are meshed. The number of particles after which

the effective response does not change anymore is called

the representative volume element.

In this work, emphasis was put on field variables distri-

butions rather than on effective properties. For illustration

purposes, consider a macroscopic component made of a

nonlinear composite constituted of many reinforcements,

as shown in Fig. 2(a). Consider that the component is di-

vided into many sub-regions of length l(Fig. 2(b)). For each

(Volume fraction = 30%,, 15 particles)

-400

-200

0

200

400

AxialStresses(

11

)MPa

t=2s



Model-1

Model-2

Model-3

Model-4

-400

-200

0

200

400

AxialStresses(

11

)MPa

t=2s



Model-1

Model-2

Model-3

Model-4


-400

-200

0

200

400

AxialStresses(

11

)MPa

t=2s



Model-1

Model-2

Model-3Model-4


-400

-200

0

200

400

AxialStresses(

11

)MPa


t=12s



Model-1

Model-2

Model-3

Model-4

-400

-200

0

200

400

AxialStresses(

11

)MPa

t =12s



Model-1

Model-2

Model-3

Model-4


-400

-200

0

200

400

AxialStresses(

11

)MP

a

t=12s



Model-1

Model-2

Model-3Model-4


0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

a b

c d

e f

Fig. 12. Axial thermal stresses for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE models with 3Dmicrostructural detail (symbols) for volume fraction of 30% at different times.



13/18

sub-region, assume that the RVE size has been reached so

that effective properties could be calculated (Fig. 2(c)).

Therefore, the composite component can be divided into a

number of homogeneous sub-regions, as was done in Sec-

tion 3. If the component is subjected to an external loading

(thermal, mechanical, etc.), the effective properties of each

sub-region can be iteratively computed in order to estimate

the field variable distribution into the homogenized com-

ponent. It is recalled that each effective property is known

within a given accuracy and different realization of the

same composite component should lead to different field

variables distributions. As l decreases and the number of

sub-regions increases, the difference in the field variables

distributions between each realization should decrease un-

til an acceptable scatter is reached. Since each sub-region is

constituted of a finite number of reinforcements, there is a

finite number of reinforcements inside the composite com-

ponent for which the homogenized response from one real-

ization to the other stays within a given tolerance. In this

work, the meshed composites are considered as the macro-

scopic components mentioned above. The simulations

performed in Section5 aim at determining the number of

reinforcements required for obtaining field variables distri-

butions corresponding to the homogenized component, as

well as the distribution themselves.

4.2. Generation of the FE meshes used in this study

The FE meshes used in this study are those ofBarello

and Lvesque (2008). Their generation is recalled here.

The detailed composite consists of randomly distributed

identical spherical particles reinforced matrix. The micro-

structures were generated using the Random Sequential

Adsorption Algorithm (Segurado and LLorca, 2002). Thealgorithm consists in generating the position of a first

spherical particle center into a cubic volume using a uni-

form random number generator. Then, the center position

of a second sphere is generated. If the distance between the

centers of the first two particles and the distance from the

particle center from the cubes faces is smaller than a pre-

set value, then the second particle is rejected and a new

center position is generated until the minimum distance

criterion mentioned above is met. The other particles are

sequentially added, following the same process where

the distance criteria are checked with all the existing par-

ticles. The particles are added until the desired volume

fraction is reached.

The particles were allowed to cut the edges and the

faces of the cube. When this happened, the particles were

completed periodically on the corresponding faces and

edges. The realizations thus obtained were therefore peri-

odic and always had an integer number of complete

spheres. The minimum distance between two particles

centers was set to 2.07r, where ris the particle radius while

the minimum distance from a particle center to a cubes

face was set to 0.1r. These distance criteria were obtained

through trial and error with the meshing software until

elements of acceptable aspect ratios were obtained.

A Matlab program was used for generating the particle

centers. This program wrote an ANSYS command file for

generating the FE mesh of the microstructure. Finally, a

Matlab program was used for converting the ANSYS

model to ABAQUS. The mesh consisted of 10-noded

tetrahedra.

5. Simulation execution and results

5.1. Simulations performed in this study

For both the multi-scale framework and the detailed

models of Section 4, cubic models of dimensions10 10 10 mm were used.Fig. 3shows these models as

well as the axes used for defining the boundary conditions

below. The studied composite is a ZrO2 matrix reinforced by

randomly distributed Ti-6Al-4V spherical particles. The

heterogeneous composites directly incorporate nonlinear

thermo-elastic behaviors for the particle and matrix re-

gions. The thermal as well as the mechanical properties

used in the simulations can be found in Khan and Muliana

(2010)and are given inTable 1. Two volume fractions of

reinforcements were studied, namely 20% and 30%. For

the detailed FE meshes, cubes containing 15, 20, 30, 40

and 45 spheres were generated. The transient thermal anal-

ysis consisted in a problem where a composite was initiallyat 300 K, except for one face that was at 600 K. This tran-

sient heat transfer problem was solved until a steady state

was reached. A uniformstress of 10 MPa was applied on the

face that was at 600 K in order to simulate effective tran-

sient thermal stresses. The models were subjected to the

following mixed uniform boundary conditions:

u10;x2;x3; t 0:0; 0 6x2 6 10; 0 6x3 6 10; tP 0

u2x1;0;x3; t 0:0; 0 6x1 6 10; 0 6x3 6 10; tP 0

u3x1;x2;0; t 0:0; 0 6x1 6 10; 0 6x2 6 10; tP 0

p110;x2;x3; t 10:0 MPa; 0 6x2 6 10; 0 6x3 6 10; tP 0

p2x1;10;x3; t 0:0 MPa; 0 6x1 6 10; 0 6x3 6 10; tP 0

p3x1;x2;10; t 0:0 MPa; 0 6x1 6 10; 0 6x2 6 10; tP 0

18

Tx1;x2;x3; 0 300 K; 0 6x1 6 10; 0 6 x2 6 10; 0 6 x3 6 10

T10;x2;x3; t 600 K; 0 6x2 6 10; 0 6 x3 6 10; tP 0

@Tx1; 0;x3; t

@x2

@Tx1; 10;x3; t

@x2 0:0; 0 6x1 6 10; 0 6 x3 6 10; tP 0

@Tx1;x2; 0; t

@x3

@Tx1;x2; 10; t

@x30:0; 0 6x1 6 10; 0 6 x2 6 10; tP 0

17

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


14/18

whereui and pi (i= 1, 2, 3) are the components of the dis-

placements and the surface tractions, respectively. It is re-

called that the set of boundary conditions affects the size of

the representative volume element, and hence, that of the

representative component. It is therefore expected that the

same component subjected to different boundary condi-

tions requires a different number of heterogeneities in or-

der to lead to converged field variables distributions.

-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=2s

Detail Microstructural Model(Volume fraction = 30%, 45 particles)


-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=12s



-400

-300

-200

-100

0

100

200

300

AxialStresses(

11

)MPa

t=2s



-400

-300

-200

-100

0

100

200

300

400

400

AxialStresses(

11

)MPa

t=2s



-400

-300

-200

-100

0

100

200

300

AxialStresses(

11

)MPa

t =12s



-400

-300

-200

-100

0

100

200

300

400

400

AxialStresses(

11

)MPa

t=12s



0 2 4 6 8 10Distance (mm) 0 2 4 6 8 10Distance (mm)

0 2 4 6 8 10


Distance (mm)

0 2 4 6 8 10


Distance (mm)

a b

c d

e f

Fig. 13. Axial thermal stresses for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE models with 3D

microstructural detail (symbols) for volume fraction of 30% at different times with C.I. of 95%. (a), (b) and (c) are mean value of stresses of different FE

models with 15 particles and (d), (e) and (f) with 30 particles, (g), (h) and (i) with 45 particles, measured at extreme top and bottom (corner) edges of thecubes along the temperature gradient direction.



15/18

Due to the boundary conditions, the field variables

distribution on the four cube segments oriented along x1(see Fig. 3) should be identical for a large number of

spheres. For the detailed models, the field variables were

extracted at identical x1 coordinates and then averaged.

At each x1 coordinate, 95% confidence intervals on the

mean value were computed. Finally, the averaged distribu-

tions of the detailed models were compared to the

distributions of the multi-scale model.

In the following subsections, distributions of the field

variables predicted from the multi-scale framework are

compared to those of the detailed FE meshes of Section4.

5.2. Temperature distribution

Fig. 4(a) and (b) show the temperature distributions ob-

tained from the homogenized model as well as from the

heterogeneous composite reinforced by 20% of Ti6Al4V

particles for model sizes of 20 and 40 particles, respec-

tively, for different times.Fig. 4(c) and (d) show the mean

responses of the various realizations, along with 95% con-

fidence intervals for models of 20 and 40 particles, respec-

tively. For the 20 particle model, the largest width of the

confidence interval is 1.74% of the mean value while it is

of 3.42% for the 40 particles model. The width of the confi-

dence interval decreases as the time increases. Fig. 5(a)(f)

show temperature profiles from similar type of analyses

but for a composite reinforced by 30% of Ti6Al4V parti-

cles and for models containing 15, 30 and 45 particles.

The largest widths of the confidence intervals are of

3.25%, 1.88% and 3.7% for the 15, 30 and 45 particle models,

respectively. It is interesting to note that the confidence

interval widths did not decrease as the number of particles

increased, as would have been expected. The reasons for

this phenomenon are unclear at this time. Conducting

more realizations and/or simulating larger number of par-

ticles might lead to the expected tendency and this phe-

nomenon might be of statistical nature.

Fig. 6(a) shows the mean temperature curves for a com-

posite with 20% of Ti6Al4V particle volume content and

for models containing 20 and 40 particles. Considering

their relatively narrow confidence intervals, it can be seen

that the RVE has been reached for these microstructures

since both the 20 and 40 particle models lead to very sim-

ilar results. Fig. 6(b) shows the mean temperature curves

for the 15, 30 and 45 particle models for composites with

30% particle volume content. It can also be concluded that

the RVE size has been reached and overcome. Figs. 4 and 5

show that the micromechanical model predicts fairly well

the temperature profiles for the range of material proper-

ties simulated.

5.3. Displacement distribution

Fig. 7(a) and (b) show the displacement distributions

obtained from the homogeneous and heterogeneous mod-

els for a composite containing 20% of Ti6Al4V particles

for models having 20 and 40 particles, respectively.

Fig. 7(c) and (d) show the average response of the various

realizations, along with 95% confidence intervals on the

mean value for models containing 20 and 40 particles,

respectively. For the 20 particles model, the largest width

of the confidence interval is 96% of the mean value while

it is of 52% for the 40 particles model.

Fig. 8(a)(f) show displacements from similar analyses

butfor a composite reinforced by 30% of Ti6Al4V particles

for models containing 15, 30 and 45 particles. The largest

-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t =2s


40 particles

20 particles

-400

-300

-200

-100

0

100

200

300

400

Axial

Stresses(

11

)MPa

t =12s


40 particles

20 particles

-400

-300

-200

-100

0

100

200

300

400

AxialStresses(

11

)MPa

t=26s


40 particles

20 particles

0 2 4 6 8 10

Distance (mm)

0 2 4 6 8 10

Distance (mm)

0 2 4 6 8 10

Distance (mm)

a

b

c

Fig. 14. Mean stress profiles for FE models with the unit cell (microme-

chanical model) at each integration point (solid line) and the FE models

with 3D microstructural detail (symbols) for volume fraction of (a) 20% atdifferent times.

http://-/?-http://-/?-


16/18

widths of the confidence intervals were of 30%, 65% and

143% for the 15, 30 and 45 particle models, respectively.

Fig. 9(a) shows the superimposed curves for 20 and 40

particles model for a sphere volume fraction of 20%. Consid-

ering thewidth of theconfidence intervals ofFig. 7, itcan be

seen that the average responses are reasonably close and

hence that the RVE has been reached. These observations

allow to conclude that the micro-mechanical model pre-

dicts relatively well the macroscopic response of the

composite for this specific microstructure. Fig. 9(b) shows

the superimposed curves for 15, 30 and 45 particles model

for a sphere volume fraction of 30%. For times t= 12 s and

t= 26 s (seconds), the huge widths of the confidence inter-

vals (see Fig. 8(d)(f)) do not allow to conclude whether the

size of the RVE has been reached or not within a reasonable

precision and hence render these RVE analyses meaning-

less. However, fort= 2 s, the confidence intervals are rela-

tively narrow and it is possible to conclude fromFig. 9(b)

that for this time, the RVE size has been reached. For

t= 2 s, it seems that the micro-mechanical model predicts

relatively well the homogenized displacement distribution,

although it is less accurate than the microstructure having

20% of reinforcements. Moreover, it seems that performing

simulations with more than 45 reinforcements might lead

to narrower confidence intervals for a better assessment

of the RVE size. Finally, it can be observed that the micro-

mechanical model predicts with more accuracy the tem-

perature distribution than the displacement field, for the

cases studied here.

5.4. Thermal stresses distributions

The contrast in the CTEs values of the constituents and

high temperature gradient are the main cause for the gen-

eration of high thermal stresses.Figs. 1013show the var-

iation of thermal stresses for spheres volume fractions of

20% and 30% at different times for the homogenized and

heterogeneous composites, respectively. For all the figures,

except for t= 2 s over a certain distance, the width of the

confidence intervals cannot be used to determine if the

RVE size has been reached with a high degree of confi-

dence. For t= 2 s, it seems that the micro-mechanical

TOP ELEMENTS

BOTTOM ELEMENTS

-400

-200

0

200

400

AxialStresses(

11

)MPa

t =26s

a

TOP ELEMENTS

BOTTOM ELEMENTS

-400

-200

0

200

400

AxialStresses(

11

)MPa

t=26s


Model-1

b


Model-2

0 2 4 6 8 10


Distance (mm)

Fig. 15. Axial thermal stresses for FE models with 3D microstructural detail for volume fraction of 20% att= 26 s. (a) and (b) are actual values of stresses at

top (corner) edge {(X1, 10, 10); 06X1 6 10} for model-2 and model-1, respectively.



17/18

model can predict reasonably well the thermal stresses

distribution. However, for all the other cases, the results

suggest that the micromechanical model is not capable of

capturing the thermal stresses with good accuracy. To cor-

roborate the above-mentioned hypothesis, the mean val-

ues of axial thermal stress at different times are shown

in Fig. 14(a)(c) for the microstructures having 20% of rein-

forcements. More realizations, and possibly with models

having more reinforcements, are required for confirming

this hypothesis with more confidence.

The localized stresses are found in some models which

are generally due to the specific micro-geometrical fea-

tures and the high fluctuation about the mean stress pro-

file is due to the presence or absence of the particle

along the profile where the stresses are computed. These

high compressive stresses are found in those matrix ele-

ments which surround the particle region. In this study

the thermal expansion of the particle is higher than the

surrounding matrix at all temperatures. Therefore, during

transient heat conduction the free expansion of the particle

is constrained by the surrounding matrix elements. The

larger CTE mismatch of particle/matrix elements results

in such high values of compressive stresses in the neigh-

boring elements of particle.

For example, consider model-2 shown in Fig. 15(a) for

which the high compressive stresses are found in the ma-

trix region that restraints the free expansion of two parti-

cle regions. Similar behavior is found for the elements

neighboring the particle region approximately at 2.5 mm

and 8.3 mm, respectively. For the same temperature differ-

ence the particle expands more than the matrix but the

surrounding restraints provided by the matrix elements

are the main cause for the generation of such high values

of compressive stresses. The same description is applicable

to other models where such micro-geometrical features

are found; for example, see belowFig. 15(b) of model-1.

5.5. Effective displacement

The effective displacement, (d1), is defined asd1 e11 L,wheree11 is the volume average of the strains in x1 direc-tion and L is the length of the cube. For both the multi-scale

and the detailed models, (d1) was computed at the face of

loading (BDHF inFig. 3(c)) for composites having a sphere

volume fraction of 20% and 30%, respectively. The

d1 as afunction of time is plotted inFig. 16(a) and (b). The mean

values of effective displacements (along with 95%

confidence intervals) for heterogeneous composite models

0

0.005

0.01

0.015

0.02

Displacements(mm

)

Displacements(mm)



Model-1

Model-2

Model-3

Model-4

0

0.005

0.01

0.015

0.02




Model-1

Model-2

Model-3

Model-4

Model-5

Model-6

300K

600KCenter line

0

0.005

0.01

0.015

0.02

Displacements(mm)



0 5 10 15 20 25 30

Time (seconds)

0 5 10 15 20 25 30

Time (seconds)

0 5 10 15 20 25 30

Time (seconds)

0 5 10 15 20 25 30

Time (seconds)

0

0.005

0.01

0.015

0.02

Displa

cements(mm)



a c

b d

Fig. 16. Effective Axial displacements for FE models with the unit cell (micromechanical model) at each integration point (solid line) and the FE models

with 3D microstructural detail (symbols) for volume fraction of (a) 20% and (b) 30%. Mean values of effective displacements for (c) 20% and (d) 30% with C.Iof 95%.



18/18

having 20% and 30% reinforcement particles are shown in

Fig. 16(c) and (d). Agreement of these results corroborate

that the present micromechanical formulation is suitable

for the prediction of effective responses of composites

through the incorporation of a nonlinear thermo-elastic

constitutive material model.

6. Summary

The transient responses of the homogenized and heter-

ogeneous composites due to coupled heat conduction and

mechanical loading have been studied. For the tempera-

ture response, the RVE size was reached for both models

having 20% and 30% reinforcements. It was found that

the temperature distribution is relatively well predicted

with the multi-scale model. The width of the confidence

intervals for the displacements were larger than those for

the temperature but allowed nevertheless to conclude that

the multi-scale framework can also predict with a reason-

able accuracy the displacement field inside the composite.

The RVE size was not reached for the thermal stresses andit is not possible to conclude that the multi-scale frame-

work is suitable for representing accurately these stresses.

Larger RVEs or many more simulations for the same RVE

sizes would be required in order to narrow the confidence

intervals. However, the mean results obtained are encour-

aging and running more simulations might reveal that the

multi-scale framework is also suitable for evaluating the

thermal stresses. Finally, the multi-scale model reasonably

predicts the effective displacement. Therefore, the main

contribution of this work was the development and the

partial validation of a multi-scale framework that allows

predicting the field variables of a temperature dependent

thermo-mechanical problem.

Acknowledgements

This research is sponsored by the Air Force Office of Sci-

entific Research (AFOSR) under Grant No. FA 9550-10-1-

0002. We also thank the Texas A&M Supercomputing Facil-

ity (http://sc.tamu.edu/) for providing computing resources

useful in conducting the research reported in this paper.

References

Aboudi, J., 1991. Mechanics of Composite Materials. A Unified

Micromechanical Approach. Elsevier, New York.

Aboudi, J., 2008. Thermomechanically coupled micromechanical analysisof multiphase composites. J. Eng. Math. 61 (24), 111132.

Auriault, J.L., 1983. Effective macroscopic description for heat conduction

in periodic composites. Int. J. Heat Mass Transfer 26, 861869.

Barello, R.B., Lvesque, M., 2008. Comparison between relaxation spectra

obtained from homogenization model and FE simulation. Int. J. Solids

Struct. 45, 850867.

Beran, M., 1965. Nuovo Cimento 38, 771.

Budiansky, B., 1965. On the elastic moduli of some heterogeneous

materials. J. Mech. Phys. Solids 13, 223227.

Chen, H.S., Acrivos, A., 1978. The effective elastic moduli of composite

materials containing spherical inclusions at non-dilute

concentrations. Int. J. Solids Struct. 14 (5), 349364.

Christensen, R.M., Lo, K.H., 1979. Solutions for effective properties of

composite materials. J. Mech. Phys. Solids 27, 315.

Christensen, R.M., Lo, K.H., 1986. J. Mech. Phys. Solids 34, 639.

Davis, R.H., 1986. The effective thermal conductivity of a composite

material with spherical inclusions. Int. J. Thermophys. 7 (3), 609620.

Hashin, Z., 1962. The elastic moduli of heterogeneous materials. In: Trans.

ASME 84 (E) Ser. J. Appl. Mech., vol. 29. pp. 143150.

Hashin, Z., Shtrikman, S., 1962. J. Mech. Phys. Solids 10, 343352.

Hasselman, D.P.H., Johnson, L.F., 1987. Effective thermal conductivity of

composites with interfacial thermal barrier resistance. J. Compos.

Mater. 21, 508515.

Hill, R., 1965. A self consistent mechanics of composite materials. J. Mech.

Phys. Solids 13, 213222.

Jeffrey, D.J., 1973. Conduction through a random suspension of spheres.

Proceedings of the Royal Society of London Series A. Mathematical

and Physical Sciences 335 (1602), 355367.Jiang, M., Jasiuk, I., Ostoja-Starzewski, M., 2002. Apparent thermal

conductivity of periodic two-dimensional composites. Comput.

Mater. Sci. 25, 329338.

Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D., 2003.

Determination of the size of the representative volume element for

random composites: statistical and numerical approach. Int. J. Solids

Struct. 40, 36473679.

Khan, K.A., Muliana, A.H., 2010. Effective thermal properties of

viscoelastic composites having field-dependent constituent

properties. Acta Mech. 209 (12), 153178.

Khan, K.A., Muliana, A.H., 2009. A multi-scale model for coupled heat

conduction and deformations of viscoelastic functionally graded

materials. Compos Part B: Eng 40 (6), 511521.

Kwon, Y.W., Kim, C., 1998. Micromechanical model for thermal analysis of

particulate and fibrous composites. J. Therm. Stress. 21, 2139.

Lvesque, M., Barello, R.B., 2009. On the homogenization of nonlinearly

viscoelastic composite materials. In: ASME International MechanicalEngineering Congress and Exposition, Proceedings, vol. 12. pp. 303304.

Lvesque, M., Derrien, K., Mishnaevski, L., Baptiste, D., Gilchrist, M.D.,

2004. A micromechanical model for nonlinear viscoelastic particle

reinforced polymeric composite materials: undamaged state.

Composites A 35, 905913.

Lvesque, M., Derrien, K., Baptiste, D., Gilchrist, M.D., 2008. On the

development and parameter identification of Schapery-type

constitutive theories. Mech. Time-Depend. Mat. 12 (2), 95127.

Maxwell, J.C., 1954. A Treatise on Elec. and Magnetism, third ed. Dover,

New York.

Matt, C.F., Cruz, M.A.E., 2002. Application of a multiscale finite-element

approach to calculate the effective thermal conductivity of particulate

media. Comput. Appl. Math. 21, 429460.

McLaughlin, R., 1977. A study of the differential scheme for composite

materials. Int. J. Eng. Sci. 15, 237.

Meijer, G., Ellyin, F., Xia, Z., 2000. Aspects of residual thermal stress/strain in

particlereinforcedmetalmatrixcomposites. Composites,Part B 31,2937.Mori, T., Tanaka, K., 1973. Average stress in matrixand average elasticenergy

of materials with misfitting inclusions. Acta Metall. 231, 571574.

Muliana, A.H., Kim, J.S., 2007. A concurrent micromechanical model for

predicting nonlinear viscoelastic responses of composites reinforced

with solid spherical particles. Int. J. Solids Struct. 44 (21), 68916913.

Muliana, A.H., 2008. Multi-scale framework for the thermo-viscoelastic

analyses of polymer composites. Mech. Res. Commun. 35 (12), 8995.

Mura, T., 1987. Micromechanics of Defects in Solids. Martinus Nijhoff, The

Hague.

Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall Properties of

Heterogeneous Materials, second ed. Elsevier, Amsterdam.

Nogales, S., Bohm, H.J., 2008. Modeling of the thermal conductivity and

thermomechanical behavior of diamond reinforced composites. Int. J.

Eng. Sci. 46 (6), 606619.

Norris, A.N., 1985. A differential scheme for the effective moduli of

composites. Mech. Mater. 4, 116.

OstojaStarzewski, M., Schulte, J., 1996. Bounding of effective thermalconductivities of multiscale materials by essential and natural

boundary conditions. Phys. Rev. B 54 (1), 278285.

Sangani, A.S., Yao, C., 1988. Bulk thermal conductivity of composites with

spherical inclusions. J. Appl. Phys. 63, 1334.

Segurado, J., Gonzalez, C., Llorca, J., 2003. A numerical investigation of the

effect of particle clustering on the mechanical properties of

composites. Acta Mater. 51, 23552369.

Segurado, J., Llorca, 2002. A numericalapproximationto the elastic properties

of sphere-reinforced composites. J. Mech. Phys. Solids 50, 21072121.

Segurado, J., Llorca, 2006. Computational micromechanics of composites:

The effect of particle spatial distribution. Mech. Mater. 38, 873883.

Verma, L.S., Shrotriya, A.K., Singh, R., Chaudhary, D.R., 1991. Thermal

conduction in two phase materials with spherical and non spherical

inclusions. J. Phys. D 24, 17291737.

Zohdi, T.I., Wriggers, P., 2001. Aspects of the computational testing of the

mechanical properties of microheterogeneous material samples. Int. J.

Numer. Methods Eng. 50, 25732599.

http://sc.tamu.edu/http://sc.tamu.edu/

Documents

Mechanics of Materials Volume 43 issue 10 2011 [doi 10.1016_j.mechmat.2011.06.013] Kamran A. Khan; Romina Barello; Anastasia H. Muliana; Martin Lé -- Coupled heat conduction and thermal