A Multi-scale Model for Coupled Heat Conduction and Deformations of Viscoelastic Functionally Graded...

7/23/2019 A Multi-scale Model for Coupled Heat Conduction and Deformations of Viscoelastic Functionally Graded Materials

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A multi-scale model for coupled heat conduction and deformations

of viscoelastic functionally graded materials

Kamran A. Khan, Anastasia H. Muliana *

Department of Mechanical Engineering, Texas A&M University College Station, TX 77843-3123, USA

a r t i c l e i n f o

Article history:

Received 4 December 2008Received in revised form 25 January 2009

Accepted 6 February 2009

Available online 7 May 2009

Keywords:

B. Creep

A. Particle reinforcement

C. Micro-mechanics

B. Thermomechanical Functionally graded

material (FGM)

a b s t r a c t

An integrated micromechanical-structural framework is presented to analyze coupled heat conduction

and deformations of functionally graded materials (FGM) having temperature and stress dependent vis-

coelastic constituents. A through-thickness continuous variation of the thermal and mechanical proper-

ties of the FGM is approximated as an assembly of homogeneous layers. Average thermo-mechanical

properties in each homogeneous medium are computed using a simplified micromechanical model for

particle reinforced composites. This micromechanical model consists of two isotropic constituents. The

mechanical properties of each constituent are timestresstemperature dependent. The thermal proper-

ties (coefficient of thermal expansion and thermal conductivity) of each constituent are allowed to vary

with temperature. Sequentially coupled heat transfer and displacement analyses are performed, which

allow analyzing stress/strain behaviors of FGM having time and temperature dependent material prop-

erties. The thermo-mechanical responses of the homogenized FGM obtained from micromechanical

model are compared with experimental data and the results obtained from finite element (FE) analysis

of FGMs having microstructural details. The present micromechanical-modeling approach is computa-

tionally efficient and shows good agreement with experiments in predicting time-dependent responses

of FGMs. Our analysis forecasts a better design for creep resistant materials using particulate FGM

composites.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are composite materials

in which the physical and mechanical properties of the materials

vary spatially along specific directions over the entire domain.

Changes in the composition of the constituents result in a nonuni-

form microstructure leading to gradual variations of the macro-

scopic material properties. Structures made of FGMs are often

subjected to high temperature gradient loadings. Under such con-

ditions, the properties of the constituents can vary significantly

with temperature accompanied by a non-negligible time-depen-

dent response. For example, FGMs composed of metal and ceramic

constituents tend to creep at high temperatures. In addition, non-

uniform temperature fields and mismatch in the properties of the

constituents in FGMs generate thermal/residual stresses that affect

overall performance of FGMs. Therefore, understanding nonlinear

thermo-viscoelastic behavior of FGMs plays a significant role in

evaluating the performance of structures made from such materi-

als. This study investigates the non-linear viscoelastic behavior of

FGMs during transient heat conduction process. The thermal and

mechanical properties of the constituents are allowed to change

with time, stress and temperature.

Extensive numerical and analytical models have been devel-

oped to study the macroscopic thermal, elastic and inelastic behav-

iors of FGMs. Noda [17] and Tanigawa [27] provided detailed

reviews on thermo-elastic and thermo-inelastic studies in FGMs

having temperature dependent/independent material properties.

Limited analytical models have been developed to study the linear

viscoelastic macroscopic behavior of FGMs, e.g., Yang[30], Paulino

and Jin[20] and Mukherjee and Paulino[13]. Finite element (FE)

formulations have also been developed to study the thermo-

mechanical behavior of cylinders and plates having graded mate-

rial properties. Examples are given in Reddy and Chin[22], Praveen

et al.[21], Reddy[23]and Shabana and Noda[26]. Few experimen-

tal studies have been performed to determine the variation of the

thermal as well as mechanical properties of FGMs and are mainly

limited to thermo-elastic behavior, e.g., Zhai et al.[32,33] and Para-

meswaran and Shukla[19].

The microstructural details of FGMs are complex and the distri-

bution of the inclusions in FGMs vary from one manufacturing pro-

cess to others. This drives the modelers to create idealized

inclusion distributions and particle geometries in order to analyze

the thermo-mechanical microscopic responses of FGMs. Simulat-

ing the microscopic responses of FGMs using detailed FE modeling

1359-8368/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.compositesb.2009.02.003

* Corresponding author. Tel.: +1 979 458 3579; fax: +1 979 845 3081.

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

Composites: Part B 40 (2009) 511521

Contents lists available at ScienceDirect

Composites: Part B

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 a t e / c o m p o s i t e s b
mailto:[email protected]://www.sciencedirect.com/science/journal/13598368http://www.elsevier.com/locate/compositesbhttp://www.elsevier.com/locate/compositesbhttp://www.sciencedirect.com/science/journal/13598368mailto:[email protected]


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of the graded microstructure with some idealized geometry is

computationally expensive and inadvisable for practical applica-

tions. Micromechanical models have an advantage over other mod-

eling techniques because they can give macroscopic properties of

non-homogeneous material while recognizing the microscopic

properties of each constituent.

Several micromechanical-modeling approaches have been used

to study the elastic behavior of composites including FGMs. De-tailed discussions of available micromechanical models to obtain

effective mechanical properties of the composites can be found

in Nemat-Nasser and Hori[18]. Micromechanical models like the

self-consistent method (SCM), Mori-Tanaka (MT), and Method of

Cells (MOC) have been used for evaluating the properties of FGMs.

In all these models, the macroscopic material point of a heteroge-

neous composite material is defined by a representative volume

element (RVE) consisting of the constituents of the composite,

which is a statistical representation of the microstructure in the

neighborhood of the RVE. The average properties of RVE represent

the overall properties of the homogenized composites. The FGM

was represented as a piece-wise layered material with uniform

effective properties. The homogeneous macroscopic properties in

each layer were evaluated using the micromechanical model. The

self-consistent scheme and Mori-Tanaka method have been widely

used by the authors to analyze the behavior of FGM, e.g., Zhai et al.

[32,33], Reiter et al.[24], Tsukamoto[29]and Zhang et al. [34]. A

detailed review of the micromechanical-modeling approaches

used by the researchers to study the behavior of FGM can also be

found in Gasik[7]. Buryachenko and Rammerstorfer[3]developed

a micromechanical model based on the multiparticle effective field

method (MEFM), which constitutes the Green function and the the-

ory of function of random variables to evaluate the thermo-elastic

responses of FGM. Aboudi et al.[2]developed a generalized higher

order micromechanical theory using MOC. The coupling effects at

the micro and macro levels were considered to analyze the ther-

mo-mechanical behavior of the FGM graded in three directions.

Another approach to idealize the graded microstructure of FGM

is done by defining RVE in which functional spatial variations ofthe inclusions are assumed to analyze the thermo-mechanical

behavior of the FGM. Grujicic and Zhang[9] used the Voronoi cell

FE method (VCFEM), and Yin et al. [31]introduced pair wise parti-

cle interactions and Eshelbys equivalent inclusion solution to eval-

uate the effective thermo-elastic properties of graded RVE.

Recently, Fang and Hu [6] formulated the effective thermal con-

ductivity of functionally graded fibrous composite using the non-

Fourier heat conduction equation. Reiter et al.[24]and Dao et al.

[4] performed thermo-elastic FE analysis of FGM models. The

FGM consists of an idealized geometry of inclusions graded linearly

was considered. The results were also computed of FGM repre-

sented by a piece-wise layered model. They concluded that micro-

structural details should be considered to predict the residual

stresses at the interface of the inclusion and matrix. Muliana[14]

presented a micromechanical model for predicting effective ther-

mal properties and viscoelastic responses of FGM. A simplified

micromechanical model of particle reinforced composites is used

to obtain effective properties at each material point in the FGM.

Available micromechanical and FE based studies on FGM are lim-

ited to thermo-elastic behaviors. There is clearly a need to under-

stand the non-linear thermo-viscoelastic behavior of FGMs withnon-constant properties of the constituents.

The present study addresses coupled heat conduction and

deformation of viscoelastic FGM using a micromechanical-model-

ing approach. A through-thickness continuous variation of the

thermal and mechanical properties of the FGM is modeled as an

assembly of homogeneous layers. Average thermo-mechanical

properties in each homogeneous medium are defined using previ-

ously developed micromechanical models for particle reinforced

composites of Muliana and Kim[15], and Khan and Muliana [12].

This micromechanical model consists of two constituents, inclu-

sions and matrix. Both inclusions and matrix constituents are as-

sumed to have timestresstemperature dependent moduli. The

thermal properties (coefficient of thermal expansion and thermal

conductivity) of each constituent are allowed to vary with temper-

ature. Sequentially coupled heat transfer and displacement analy-

ses are performed, which allow analyzing stress/strain behaviors of

FGM having timetemperature dependent properties. Experimen-

tal data available in the literature are used to verify the model.

Numericalsimulations are also performed to analyze non-linear

thermo-viscoelastic responses of homogenized FGM using a micro-

mechanical model and comparisons are made with the results ob-

tained from FE analysis of two dimensional (2D) FGM models

having microstructural details (i.e., heterogeneous FGM). In the

2D FE model of heterogeneous FGM, the inclusions are idealized

as circles and their volume fraction vary from one end of geometric

model to the other. Comparisons of results show that the present

micromechanical model is capable of predicting the non-linear vis-

coelastic responses of FGMs.

2. Modeling of functionally graded material

In this study, a FGM consisting of two constituents whose

material properties change with time, stress, and temperature, is

considered. The FGM graded in one direction is idealized as a

piece-wise homogeneous medium whose macroscopic properties

are evaluated using a micromechanical model. The variation in

properties from a series of homogeneous layers along the graded

direction is shown in Fig. 1(a). The FGM is approximated as an

assembly of a fictitious layered medium to facilitate the process

of integration of micromechanical model with FE package, i.e.,

ABAQUS[1]. The variations in material properties with locations

a) Functionally graded

material idealizationb) Microstructure

detailsc) Microstructure

Idealizationd) RVE e) Unit Cell

Fig. 1. Illustration of modeling approach for FGM using a micromechanical model.

512 K.A. Khan, A.H. Muliana/ Composites: Part B 40 (2009) 511521


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are incorporated by introducing the multiple integration points

along the graded directions, i.e., thickness direction. Each integra-

tion point represents a material property of a discretized area.

Thus, the overall through-thickness FGM responses show zigzag

(discontinuous) variations. By increasing the number of integration

points along the graded directions, the discontinuities in the FGM

can be minimized. It is also possible to analytically solve the equa-

tions that govern the conduction of heat and deformation in theFGM body. The developed micromechanical model of particle rein-

forced composites can be used as material parameters in the gov-

erning equations.

Each layer of the FGM is composed of a homogeneous matrix

and spherical inclusion particles. The spherical particles are as-

sumed to be uniformly distributed in each macroscopic layer.

The particles in the microstructure are idealized as cubes and dis-

tributed uniformly in three dimensional periodic arrays. A repre-

sentative volume element (RVE) consisting of one particle

embedded in cubic matrix is considered. Due to the three plane

symmetry, one-eight unit-cell is assumed to consist of four sub

cells. The first subcell contains a particle constituent, while sub-

cells 2, 3, and 4 represent the matrix constituents, as shown in

Fig. 1(e). Perfect bonds are assumed at the subcells interfaces. Peri-

odic boundary conditions are imposed on the RVE. Micromechani-

cal relations are formulated in terms of incremental average field

quantities, i.e., stress, strain, heat flux and temperature gradient,

in the subcells. Stress, temperature and time dependent constitu-

tive models are used for the isotropic constituents. Temperature

dependent thermal properties are considered for particle and ma-

trix constituents. The effective properties of the unit-cell define the

macroscopic properties at a material point in the homogeneous

layer which in turn represents the effective response of each layer.

The present micromechanical model is compatible with general

displacement based FE software to perform the thermo-mechani-

cal analyses of FGM structures.

2.1. Constitutive model for isotropic constituents

At the constituent level, the modified viscoelastic constitutive

model of Schapery [25]is used for each subcell. The generalized

three dimensional constitutive equations with stress and tempera-

ture dependent behavior for non-aging materials can be written as:

etij eijt g0rt; TtSijkl0r

tklg1r

t; Tt

Z t0

DSijklwt ws

d g2r

s; Tsrskl

dsds

Z t0

aijTt

dDTs

ds ds: 1

The superscript t indicates a variable at time t.Sijkl0 are the com-

ponents of the instantaneous elastic compliance, DSijklwt ws are

the components of the transient compliances, and aij are the com-ponents of coefficient of thermal expansion (CTE) tensor. The

parameters Tt and T0 are the current and reference temperatures.

The linear coefficient of thermal expansion, a, also varies with tem-peratures.w is the reduced-time (effective time) given by:

wt wt

Z t0

dn

arnaTn ws ws

Z s0

dn

arnaTn; 2

whereg0; g1; g2in Eq.(1)and a in Eq.(2)are nonlinear parameters

and defined as functions of current temperature Tt and effective

stress rt. For isotropic materials, the total strain can be separatedinto deviatoric and volumetric parts. A recursiveiterative method

developed by Muliana and Khan[16]is used to solve the deviatoric

and volumetric components of the mechanical strains. The formula-

tion is derived with a constant incremental strain rate during each

time increment, which is compatible with a displacement based FEanalysis. Linearized trial stress tensors are used as starting points

for solving the stress tensor using the incremental strains. An iter-

ative scheme is included in order to minimize residual from the lin-

earization. A detailed discussion about the non-linear parameters

and recursive algorithm formulation of Eq. (1) can be found in

Haj-Ali and Muliana [10] and Muliana and Khan [16]. An outline

of the recursiveiterative algorithm for the nonlinear isotropic vis-

coelastic material is given AppendixA.

The conduction mode of heat transfer is considered. The Fourier

law of heat conduction with temperature dependent thermal con-

ductivity is used and can be expressed as:

qti Ktiju

tj ; whereu

tj

@Tt

@xj; 3

whereq ti and ut

j are the heat flux and temperature gradient. Ktij is

the temperature dependent thermal conductivity. Ktij is also called

the consistent tangent thermal conductivity, which varies with

temperature at current time t.

2.2. Effective thermo-viscoelastic properties

By satisfying the displacement and traction continuities at the

interfaces during thermo-viscoelastic deformations, the expression

for the effective time dependent stiffness matrix and coefficient of

thermal expansion are formulated. This formulation leads to an

effective timetemperaturestress-dependent coefficient of ther-

mal expansion. The effective thermal conductivity is formulated

by imposing heat flux and temperature continuities at the subcell

interfaces.

The method of volume averaging is used to evaluate the effec-

tive response of a unit-cell (micromechanical model). The average

stresses and strains are defined by:

rij 1

V

XNa1

ZVa

raij xak dV

a 1

V

XNa1

Varaij and 4

eij 1

V XN

a1 ZVa eaij x

ak dV

a 1

VXN

a1

Vaeaij ; 5

where an over bar indicates average material quantities. The super-

script a denotes the subcells number andNis the number of sub-cells. Stressraij and straine

aij are the average stress and strain in

each subcell. The unit-cell volume Vis:

VXNa1

Va; N 4: 6

For non-linear stressstrain constitutive relations, the total average

of the stresses and strains at current time t are solved incremen-

tally, which are:

rtij rtDtij Dr

tij 7

etij etDtij De

tij 8

The superscript t Dt represents the converged field quantities

from the previous steps, which are stored as history variables. Detijand Drtij are the incremental average strains and stresses of theunit-cell at current time. The unit-cell length scale is assumed much

smaller compared to the macroscopic geometry of FGM, so that the

total temperature in each unit-cell of a material point is defined by:

Tt;a TtDt;a DTt;a and DTt;a DTt 9

The micromechanical model is designed to be compatible with dis-

placement based FE software, in which the incremental strains are

chosen as independent variable. We introduce a strain interaction

matrix Bt;a, which relates the subcell average strains, Det;aij , tothe unit-cell average strain, Detij , and it is written as:

Det;aij Ba;tijkl Detkl 10

K.A. Khan, A.H. Muliana / Composites: Part B 40 (2009) 511521 513
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Using the incremental strains in Eq.(10),constitutive equation for

each subcell, and volume average of the incremental stress, the sca-

lar components of the incremental effective stress are:

Drtij 1

V

XNa1

VaCa;tijkl B

a;tklrs

Detrs aa;tkl

DTth i

Ctijkl Detkl a

tklDT

t

11

From the above equation, the effective tangent stiffness matrix, Ctijrs,

is:

Ctijrs1

V

XNa1

VaCa;tijkl B

a;tklrs ; 12

where Ba;t

klrs is a scalar component of the fourth order interaction

tensor, which can be obtained by satisfying the micromechanical

relations and the constitutive equations. The micromechanical rela-

tions within the four subcells are derived by assuming perfect

bonding at the interfaces of the subcells and imposing displacement

compatibility and traction continuity at the subcell interfaces. De-

tailed formulations of Ba;tklrs

and Ctijrs are described in Muliana and

Kim[15]. From Eq.(11), the effective coefficient of linear thermal

expansion is given as:

atij C1;tijkl

V

XNa1

VaCa;tklmna

a;tmn ; 13

The previously developed micromechanical relations for the effec-

tive viscoelastic response of particle reinforced composite (Muliana

and Kim[15]) is modified to determine the effective CTE. Using the

micromechanical relations and thermo-viscoelastic constitutive

relations for the particle and matrix subcells, the effective CTE for

the isotropic nonlinear responses can be expressed as:

atij atdij

C1;tijklV

VACA;tklmna

A;tmn V

3C3;tklmna

3;tmn V

4C4;tklmna

4;tmn

h i;

14

where the total volume of subcells 1 and 2 in Eq. (14) isVA V1 V2, and whereaA;tij andC

A;t

ijkl in Eq.(14)are the effec-

tive thermal expansion and stiffness expressions for subcells 1 and

2. The scalar components ofaA;tij and CA;t

ijkl can be expressed in the

following equations:

aA;tij aA;tdij

1

VA V1a1;tij V

2a2;tij

h i; and 15

CA;t

ijkl X1;tijkl 16

where Xtijkl 1

VA V

1C11;t

ijkl V2C

21;t

ijkl

h i: 17

The effective CTE in Eq.(14)depends on the moduli and CTE of each

constituent. Thus, for the stress, temperature and time dependent

constituent mechanical and thermal properties, the effective CTE

also varies with stress, temperature and time. A detailed formula-tion for the effective coefficient of thermal expansion can be found

in AppendixB.

2.3. Effective thermal conductivity

A volume averaging method based on spatial variation of the

temperature gradient in each subcell is adopted to determine the

effective thermal conductivity of particle reinforced composites.

The average heat flux and temperature gradient are:

qi1

V

XNa1

ZVa

qai x

ak

dVa

1

V

XNa1

Vaqai ; and 18

ui

1

VXN

a1 ZVa uai xak dVa 1VXN

a1 V

a

u

a

i : 19

The average heat flux equation for a homogeneous composite med-

ium is expressed by the Fourier law of heat conduction as:

qti Ktijutj where u

tj

dTt

dx j: 20

It is noted that the components of the conductivity tensor,Ktij,

vary with temperature as the thermal conductivity for each con-

stituent is allowed to vary with temperature. The micromechanicalrelations within the four subcells inFig. 1(e) are derived by assum-

ing perfect bonding at the interfaces of the subcells. Using the heat

conduction equation for each subcell and the effective heat flux

relation, the tangent effective thermal conductivity matrix of the

composite can be expressed as:

Ktik 1

V

X4a1

VaKa;tij M

a;tjk

; 21

where, the Ma;t matrix is the concentration tensor that relates the

average subcell temperature gradient with the overall temperature

gradient across the unit-cell. A detailed formulation ofMa;t tensor

and the effective thermal conductivity can be found in AppendixB.

In the following sections, Eqs. (12), (14) and (21) are used to

determine the variations of effective time dependent stiffness ma-trix, coefficient of linear thermal expansion, and thermal conduc-

tivity along the graded direction of FGMs.

3. Numerical simulations and discussion

A micromechanical model of particulate composite having time,

temperature and stress dependent material properties at the con-

stituent level is utilized for predicting the thermo-mechanical

behavior of FGM. To demonstrate the capability of the proposed

micromechanical model, the thermo-mechanical responses of

FGM from the micromechanical model are compared with ones

from existing experimental data, and FE model of FGMs having

microstructural details.

3.1. Effective thermo-mechanical properties

The experimental data of Zhai et al.[32,33]is used to compare

the variations of elastic modulus in the FGMs. TiC=Ni3 Al FGMs

were prepared with Ni3Al particles dispersed in the continuous

TiC matrix. The elastic properties of the constituents are shown

inTable 1. Comparisons of the predicted elastic modulus distribu-

tion along the gradation direction with the experimental data are

shown in Fig. 2. The proposed model provides relatively good

agreement with the experimental data. The responses are charac-

terized at fixed temperatures.

The FGMs consisting of metal-matrix systems are widely used

in high temperature applications. The metallic components having

a high coefficient of thermal expansion (CTE) are generally dopedwith constituents having a low CTE to tailor the overall CTE. Thus,

the FGM can be used in applications requiring low CTE and high

thermal conductivity. The CTE is one of the main features of FGM

needed to be analyzed when designing FGM for high temperature

applications. Geiger and Jackson[8]measured the CTE distribution

in Al/Si FGM with a volume fraction of Si particles that varies from

0% to 40%. The material properties of Al and Si are given in Table 1.

Comparisons of the effective CTE obtained from the micromechan-

ical model with the ones obtained from the experimental data are

shown inFig. 3. The results are found to be in good agreement with

the experimental data. Geiger and Jackson[8] also measured the

thermal conductivity of the Al-6061(T6)/SiC FGM with a SiC parti-

cle volume fraction varies from 0% to 60%. The material properties

of Al-6061(T6) and SiC are given inTable 1.Fig. 4shows the com-parison of experimental data and the results obtained from the

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micromechanical model for the thermal conductivity variations

along the graded direction. Comparing the experimental data with

the results obtained from the micromechanical model, as shown in

Figs. 24, it is suggested that the proposed micromechanical model

is capable of predicting the thermo-mechanical behaviors of FGM

along the graded direction.

3.2. Sequentially coupled heat conduction and deformations of FGM

Experimental data of FGMs having timetemperature depen-

dent behavior is currently not available. FE analysis is performed

to determine the effects of timetemperature dependent constitu-

ent properties on the thermo-viscoelastic behaviors of FGMs. All

simulations are performed using the ABAQUS FE software. The re-

sults obtained from the micromechanical model are compared

with the ones from FE analysis of the FGM model having micro-

structural details. The FGM panel of 16 mm length 10 mmheight 1 mm depth is studied. The volume fraction of inclusions

varies along the length direction. A 2D FE model of FGM having a

gradation of the particle in one direction is shown in Fig. 5(a)

and (b). The particles in the form of circles are dispersed randomly

with a gradient of volume fractions of particle from 0% to 40%. In

Fig. 5(a), large diameter particles are distributed while Fig. 5(b)

contains small size particles. Small size particles show more uni-

form distribution as compare to the ones with large size particle.

Fig. 5(c) illustrates the simplified piece-wise homogenized model

with sixteen (16) layers. Each layer represents the macroscopic

material point with homogeneous properties varying with the gra-

dient of volume fraction of the particles. The heat transfer thermal

analysis is first performed to obtain the temperature distribution

along the graded direction. Using the temperature distribution,

the stress analyses are carried out to determine the timetemper-

ature dependent deformations of FGM along the graded directions.

In order to obtain the temperature profiles, the equation governing

the heat conduction in an FGM body needs to be solved. This equa-

tion is written as:

qcxk T

qi;i i; k 1; 2; 3 22

where qcxk is the effective heat capacity that depends on the com-position, density, and specific heat of the two constituents in the

FGM body. The effective heat capacity is obtained using a volume

average method.

The FGM consisting of Ti6Al4V and ZrO2 is first considered.

The temperature dependent mechanical and physical properties

of these materials are given in Table 2. The properties are takenfrom Praveen et al. [21]. First, a transient heat transfer analysis is

Table 1

Mechanical and physical properties of materials used in FGM.

Material Young modulus (E), GPa Poisson ratiot Linear thermal expansiona 106 ; 1=K Thermal conductivity (K), W/m/K.

Ni3Al 199 0.295 11.90

TiC 460 0.19 7.20

Al-6061 (T6) 70.3 0.34 23.40 173

SiC 400 0.20 3.4 120

Si 112.4 0.42 3.0 100

Al 72 0.33 23.6 234

0 0.2 0.4 0.6 0.8 1

Volume Fraction (VF)

100

200

300

400

500

Effe

ctiveYoungModulus(GPa)

Experimental Data Zhai et. al. (1993)

TiC filled with Ni3Al Particles

Micromechanical Model

Fig. 2. Comparison of Youngs modulus for FGM consisting of TiC and Ni 3Al.

0 0.1 0.2 0.3 0.4 0.5


0

1E-005

2E-005

3E-005

CoefficientofThermalExpansion

(K-1)

Experimental Data Geiger and Jackson (1989)


Aluminum with Silicon inclusions

Fig. 3. Comparison of the coefficient of thermal expansion for FGM consisting of Alwith Si inclusions.

0 0.1 0.2 0.3 0.4 0.5 0.6


100

120

140

160

180

200

EffectiveThermalConductivity(W/m.K

)

Experimental Data Geiger and Jackson (1989)

Kp/Km = 120:173Aluminum Filled with Silicon Carbide Particles

Present Model

Kp = Thermal Conductivity of SiC

Km = Thermal Conductivity of Al

Fig. 4. Comparison of the thermal conductivity of FGM consisting of Al and SiC

particles.



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performed by applying the uniform temperature of 1000 K at one

end. The entire FGM is initially at constant temperature of 300 K.

After 159 s, the temperature distributions reach to steady state

condition. Temperature profile is plotted along the graded direc-

tion. The results obtained from our micromechanical formulation

are compared with the ones obtained from FE model having coarse

and fine microstructural details. Comparison of the results inFig. 6

shows that profiles obtained from fine microstructural details and

micromechanical model are in good agreement, while results ob-

tained from coarse microstructural details show some deviations.

The deviation is mainly due to uneven and sparse distribution of

inclusions. In all further analyses, FE model having fine microstruc-

tural details will be considered. The computational (CPU) time ta-

ken to analyze the FE model of FGM having fine microstructural

details is 118 s which is about 13 times higher than the one taken

by the analysis using our micromechanical formulation.

Next, the stress analyses are performed based on the tempera-

ture distribution obtained from the heat conduction analyses. A

uniaxial stress of 10 MPa is applied along the graded direction.

The temperature distribution obtained from transient heat transfer

analysis is considered as a field dependent temperature loading.

The elastic properties of the constituents change with temperature

as shown inTable 2.Fig. 7shows the profiles of the displacement

fields. The results obtained from FE analysis of FGM models having

fine microstructural details and the ones using a micromechanicalmodel are in good agreement. In the above analyses, the effect of

temperature on the deformation is incorporated through the tem-

perature dependent elastic properties while the effect of thermal

expansion is neglected. However, with such a high temperature

changes, the effects of free thermal expansion of each constituents

can be very significant. Mismatches in the thermal expansion coef-

ficient of the constituents can generate thermal stresses. Fig. 8

shows the results of variations of displacement along the graded

direction with temperature dependent thermal expansion and

elastic properties. The FE model with microstructural details incor-

porates thermal stresses due to mismatches in the thermal expan-

sion coefficients. At the beginning of the heat transfer analysis,

there is a high rate of change of the temperature gradient which

causes generation of high thermal stresses at the interfaces ofthe constituents. These stress fields in turn affect the displacement

fields of the entire FGM. The thermal stress effect is currently not

being included in the present micromechanical model, which is

shown by the deviation in the two responses inFig. 8. As time pro-

gresses, the temperature gradient decreases which reduces ther-

mal stresses and for a zero temperature gradient both results

agree quite well.

Next, coupled heat conduction and deformation analysis is per-

formed to investigate the effect of viscoelastic constituents on the

overall thermo-mechanical responses of FGMs. The time-depen-

dent behavior of metal-matrix composites is of importance at high

temperatures. The creep behavior of Al with silicon carbide inclu-

sions is thus numerically studied as an example. The temperature

dependent elastic modulus of aluminum is taken from Kaufman

[11]. The properties of SiC are taken from Geiger and Jackson[8].

The temperature dependent mechanical and physical properties

of Al and SiC are given in Table 3. The present model requires

the creep parameters which can be obtained from a series of exper-

imental data performed at constant stress and different tempera-

tures. Because of the unavailability of such data, creep propertiesof aluminum at 573 K and 28.5 MPa are taken from the experimen-

tal work of Tjong and Ma[28]. The time dependent and non-linear

temperature dependent parameters of Al are given in Table 4.

(a) (b) (c)

Fig. 5. Illustration of the geometry of the finite element models for a volume fraction that varies from 0% to 40%. (a) Coarse and (b) fine microstructural details; (c) piece-wise

homogeneous macroscopic layers.

Table 2

Temperature dependent mechanical and physical properties of materials of Ti6Al4V and ZrO 2.

Property Ti6Al4V ZirconiaZrO2

Young modulus (E), Pa 1:23 1011 56:457 106T 2:44 1011 334:28 106T 295:24 103 T2 89:79T3

Poisson ratiot 0.3 0.3Coefficient of thermal expansiona 106; 1=K 7:58 106 4:927 109T 2:388 1012T2 1:28 105 19:07 109T 1:28 1011T2 8:67 1017T3

Thermal conductivity (K), W/m/K 1.2095 + 0.01686T 1:7 2:17 104T 1:13 105 T2

Specific heat (C), J/kg K 625:2969 0:264T 4:49 104T2 487:3427 0:149T 2:94 105T2

Densityq kg=m3 4429 5700

Tis temperature in K.

0 4 8 12 16

Distance (mm)

200

400

600

800

1000

1200

T

emperature(K)

Fine Microstructural Model


Ti6Al-4v with Zr02inclusions

Volume fraction varies from 0 to 40%with maximum at distance 16 mm.

t =5s

t =10s

t =15s

t =3s

Coarse Microstructural Model

t =1s

Steady State Time = 159 seconds

Fig. 6. Temperature profiles at different times along the graded direction of FGM

having constituents with temperature dependent thermal conductivities, for Ti

6Al4V/ZrO2.



7/11

Moreover, DiCarlo and Yun[5]reported that the SiC does not show

any creep up to 1073 K. This temperature is far above the temper-

atures considered in this study. Therefore, SiC is assumed to be-

have linearly elastic. The axial creep data of pure aluminum is

shown inFig. 9. The axial creep of composites and FGMs consisting

of aluminum and SiC is computed using the micromechanical mod-

el. A constant stress of 28.5 MPa is applied at one end followed byholding for 12,000 s at a constant temperature of 573 K. The

boundary conditions of the specimens are shown inFig. 9. Compos-

ites having a uniform distribution and linear gradation of SiC along

the graded direction are considered. The results of axial creep

deformations (measured at point B) inFig. 9show that, the graded

material shows more creep resistance than the composites having

uniform distribution of the SiC particles.

A sequentially coupled analysis of Al/SiC FGM is then performed

to analyze its timetemperature dependent behavior. The Al is as-

sumed to have timetemperature dependent properties while SiC

behaves linearly elastic. Initially the entire FGM is assumed to be

at 300 K. Transient heat transfer analysis is performed by applying

a constant temperature of 573 K at one end. The analysis reached

steady state heat transfer conditions after 14.8 s. Fig. 10 showsthe temperature profile at different times along the graded direc-

tion. The results obtained from heat transfer analyses of FGM mod-

els having fine microstructural details and the ones using

micromechanical model are in good agreement. Next, the stress

analyses are performed based on the temperature distribution ob-

tained from the thermal analyses. A constant stress of 28.5 MPa is

applied in the graded direction. The stress is held constant for up to

2000 s. Creep deformations at different times are plotted along the

graded direction. Though, steady state time is reached after 14.8 s,

because of the presence of viscoelastic Al, the deformation contin-

ues to grow under a constant stress of 28.5 MPa at a temperature of

573 K.Fig. 11shows that the results obtained from FE analysis of

FGM models having fine microstructural details and the ones using

a micromechanical model are in good agreement. The comparisonsof these results are strong evidence that the present micromechan-

ical model is capable of predicting non-linear viscoelastic behavior

of FGM with a reasonable level of uncertainty 10%.

4. Conclusion

The coupled thermo-viscoelastic analysis of FGM is performed

using a micromechanical-modeling approach. The proposed model

has a capability to analyze the heat conduction and thermo-visco-

elastic deformations of FGMs having timetemperature and stress

dependent field dependent properties. When the gradients of tem-

perature in the FGM are substantially large, the effect of thermal

stresses due to the mismatch in the coefficient of thermal expan-

sions of the constituents on the overall mechanical responses ofFGM is significant. The thermal stresses are localized at the inter-

face between inclusion and matrix, which can potentially cause

debonding. In the present micromechanical model, the effect of

0 4 8 12 16

Distance (mm)

0

0.0002

0.0004

0.0006

0.0008

Displacem

ent(mm)

Detail Microstructural Model



t =20s

t =50s

t =30s

t =5s


t =40s

Fig. 7. Variations of displacement field at different times along the graded direction

of FGM having constituents with temperature dependent elastic properties, for Ti

6Al4V/ZrO2.

0 4 8 12 16

Distance (mm)

0

0.05

0.1

0.15

0.2

Displacement(mm)




t =30s

t =50s

t =20s


t =159s

t =40s

Fig. 8. Variations of displacement field at different times along the graded direction

of FGM having constituents with temperature dependent elastic and thermal

properties, for Ti6Al4V/ZrO2.

Table 3

Temperature dependent mechanical and physical properties of materials of Al and SiC.

Property Aluminum (Al) Silicon carbide (SiC)

Young modulus (E), MPa 65144 73:432T 0:1618T2 406,783 22.61T

Poisson ratiot 0.33 0.2Thermal conductivity (K), W/m/K 235 0:0305T 0:0003T2 6E 07T3 3E 10T4 183.78 0.1569T

Specific heat (C), J/kg K 900 750

Densityq; kg=m3 2700 3210Coefficient of thermal expansiona106; 1=K 2 105 6 109T 3 1012T2 1 1014T3 3 106 3 109T 6 1013T2

Tis temperature in K.

Table 4

Prony series coefficients and non-linear temperature dependent parameters for Al.

n kn s1 Dn 10

6 MPa1

1 1 0.1

2 101 0.15

3 102 20

4 103 30

5 104 160

6 105 1100

gT0 exp 0:36 TT0

T0

2 ; gT1 g

T2 aT 1



8/11

stress concentration is not incorporated, which is shown to be a

limitation of the current model. Based on creep analysis of Al/SiC,it is concluded that better creep resistant material for high temper-

ature applications can be obtained by proper distribution of the

particles along the graded direction of the composites. The present

micromechanical-modeling approach is computationally efficient

and quite accurate in predicting time-dependent responses of

FGMs.

Acknowledgements

This research is sponsored by the National Science Foundation

(NSF) under Grant No. 0546528 and the Air Force Office of Scien-

tific Research (AFOSR) under Grant number FA9550-09-1-0145.

Appendix A

This appendix outline the basic concepts of solving the Eq.( 1)

using the recursiveiterative algorithm. For isotropic materials,

the total strain in Eq.(1) can be written as:

etij etij

1

3etkkdij aT

t T0dij

eM;tij

etij1

3etkkdij; e

T;tij

aTt T0dij

etij1

2g0r

t; TtJ0Stij

1

2g1r

t; TtZ t

0

DJwt wsd g2rs; TsSsijh i

ds ds

etkk1

3g0r

t; TtB0rtkk1

3g1r

t; Tt

Z t0

DBwt wsd g2r

s; Tsrskk

ds ds

A:1

where eM;tij and eT;tij are the total mechanical and thermal strains,

respectively. The superscript t indicates a variable at time t. The

parameters J0 and B0 are the instantaneous elastic shear and bulk

compliances, respectively. The terms DJand DBare the time-depen-

dent shear and bulk compliances, respectively. The corresponding

linear elastic Poissons ratio,t, is assumed to be time independent,which allows expressing the shear and bulk compliances as:

J0

21 tD0 B0 31 2tD0

DJwt

21 tDDwt

DBwt

31 2tDDwt A:2

HereD0 and DDare the instantaneous elastic and transient compli-

ances under uniaxial (extensional) creep loading. The uniaxial tran-

sient compliance, DD, is expressed in terms of Prony series as:

DDwt

XNn1

Dn 1 expknwt

A:3

A recursiveiterative method developed by Muliana and Khan[16]

is used to solve the deviatoric and volumetric components of the

mechanical strains in Eq. (A.1). Outline of the recursiveiterative

algorithm for the nonlinear isotropic viscoelastic material is given

inFig. A1.

The parameters qtD

tij;n and qtD

tkk;n are hereditary integral (historystate variables) stored from the last converged step at time

t Dt. The parametersqtij;nand qtkk;n; n 1 . . . Nare the hereditary

integrals at current time for every term in the Prony series in the

form of deviatoric and volumetric strains. The superscript tr

means trial value of that variable. The gb b 0; 1; 2 represent

the nonlinear parameters g0; g1 and g2, given in Eq. (1). The

parametersJt andBt are the effective shear and bulk compliances

at the current time, respectively. The initial approximation (trial)

incremental stress tensor is determined using the trial nonlinear

parameters, incremental deviatoric stress tensor, DSt;trij , and volu-

metric stress tensor, Drt;trkk

. The trial current stress tensor is formed

based on the given variables and history variables from the previ-

ous converged step. An iterative scheme is then employed to find

the correct stress tensor for a given strain tensor. The correct stresstensor at current time is solved by minimizing a residual tensor,

0 4 8 12 16

Distance (mm)

300

400

500

600

Temperature(K)



Al with SiC inclusions

Volume fraction variesfrom 0 to 40% withmaximum at distance16 mm.

t =1s

t =2s

t =3s

t =0.5s

t =0.1s

Steady State Time = 14.8 seconds

Fig. 10. Temperature profiles at different times along the graded direction of FGM

having constituents with temperature dependent thermal conductivities, for Al/SiC.

0 4 8 12 16

Distance (mm)

0

0.01

0.02

0.03

0.04

AxialCreepDeformation(mm)




t =14.8s

t =1000s

t =2000s

t =350s

t =0.5s

Steady State Time = 14.8 seconds

=28.5

(MPa)

t2000

A B

x

Fig. 11. Variations of axial creep deformations of FGM having constituents with

temperature dependent elastic and thermal properties, for Al/SiC.

0 4000 8000 12000

Time (seconds)

0

0.002

0.004

0.006

0.008

0.01

AxialCreepDeformation(mm)

Pure Aluminum Experimental Creep Data ( Tjong and Ma,1999)

25% inclusions by volume distributed uniformly


25% inclusions by volume graded linearly with a maximum of 40%.

=28.5

(MPa)

t12000

T(K)

t12000

T=573

A B

x

Fig. 9. Comparison of axial creep deformations for Al, Al/SiC composite and FGM.



9/11

given in 3.3 ofFig. A1. Finally, the consistent tangent stiffness ma-

trix is defined by taking the inverse of the partial derivative of the

incremental strain with respect to the incremental stress at the end

of the current time step. The consistent tangent stiffness,Ctijkl, at

the converged state, are:

Ctijkl

@drtij@deM;t

kl

@Rtij

@drtkl

" #1; Rtij

! 0 A:5Eq. (A.5) defines material properties at current time tforeach subcell

in the micromechanical model. The components of the consistenttangent stiffness tensor vary with time, temperature, and stress.

Appendix B

This appendix outlines the basic concepts require in formulat-

ing the effective thermal properties of the homogenized composite

medium having spherical particle inclusions. The detailed micro-

mechanical formulations are presented in the manuscript Khan

and Muliana[12]which is currently under review.

B.1. Periodic boundary conditions

To reduce complexity in the micromechanical formulation, a

composite having a periodic microstructure is assumed. A repre-sentative volume element (RVE) is then defined by a single particle

embedded in a cubic matrix (Fig. B1). Periodic boundary conditions

(BCs) are imposed to the RVE. After solving boundary value prob-

lems (BVPs) for the RVE with the prescribed periodic BCs, the effec-

tive field quantities and material properties are obtained using the

volume averaging relations. Detailed description about the peri-

odic structure can be found in Nemat-Nasser and Hori[18].

Letx i be the local coordinate system of the RVE. Let ai denote

the dimensions of the cube in local coordinate system, i.e.,

xi i 1; 2; 3 that can be reduced to ai a. In this study, we as-

sumed that the component of displacementuxi at any pointxkinside

the RVE can be written as follows:

uxi xk ux;

i uXi;i xkx

k

~uxi xk B:1

The superscriptx is for local field quantities while overbar and Xis

for global field quantities. Where ux;0i is the component of displace-

ment at a reference point in the ith direction, which can be picked

arbitrarily, e.g., point (*); ~uxi xk is the component of the displace-

ment inside the RVE due to the prescribed macroscopic boundary

condition. In this study, due to the nature of the prescribed bound-

ary condition, i.e., prescribed displacement on the boundary, the

volume integral of~uixk vanishes. Thus, the periodic boundary con-

ditions are given by:

uxi x0k u

xi x

aik u

Xi;i x

0k x

aik B:2txi x0k txi xaik txi rxijnj B:3

Fig. A1. Recursiveiterative algorithm for the nonlinear isotropic viscoelastic material.



10/11

wheretxi represents the traction on the surfaces of the RVE, with njbeing the corresponding unit normal vectors.xokand x

aik

are the coor-

dinates of arbitrary points on faces x i 0 andx i ai.

For heat conduction equation, the similar procedure is adopted

and the total temperature fieldTx at any pointxkinside RVE can be

written as follows:

Txxk Tx; Ti;i xkxk eTxk B:4This form of expansion of the temperature field gives a similar set of

periodic boundary conditions for temperature and heat flux, as pre-

viously obtained for displacement field and traction.

Txx0k Txx

aik T

Xi;i x

0k x

aik B:5

qxi x0k ni q

xi x

aikni B:6

B.2. Formulation of effective coefficient of thermal expansion

The micromechanical relations within the four subcells in

Fig. 1(e) are derived by assuming perfect bond along the interfaces

of the subcells and imposing displacement compatibility and trac-

tion continuity at the subcells interface. The homogenized incre-

mental strain relations for the particle reinforced composites are

given as:

detij 1

V1 V2 V

1de1;tij V2de2;tij

h i de3;tij de

4;t

ij

for i j; i;j 1; 2; 3 B:7

dctij V1dc1;tij V

2dc2;tij V3dc3;tij V

4dc4;tij forij

B:8

The homogenized stresses are written as:

drtij V1 V2drA;tij V

3dr3;tij V4dr4;tij for i j

drA;tij dr

1;tij dr

2;tij

B:9

drtij dr1;tij dr

2;tij dr

3;tij dr

4;tij for ij B:10

Using the thermo-viscoelastic constitutive relations for the particle

and matrix subcells, volume averaging schemes for the incremental

stress and strain, and micromechanical relations in Eqs. (B.7)

(B.10), the effective CTE is obtained. The effective CTE in Eq.(14)re-

quires defining the effective tangent stiffness matrix. Formulations

of the effective tangent stiffness matrix, which also require formu-

lating the strain interaction matrix Ba;t, are given in Muliana and

Kim[15].

In order to formulate the strain interaction matrix Ba;t, intro-

duced in Eq.(10),the micromechanical relations and the constitu-

tive equations are imposed. The micromechanical models consistsof four subcells with six components of strains need to be deter-

mined for every subcell. This requires forming 24 equations. The

first sets of equations are determined from the strain compatibility

equations which are given as:

f Re121

g AM1

h i1224

e1

e2

e3

e4

8>>>>>:

9>>>=>>>;241

DM1

h i126

feg61

B:11

where Re is the residual vector arising from imposing strain com-

patibility relations. In the case of linear elastic responses are exhib-

ited for all subcells, the vector Re is zero. The second sets of

equations are formed based on traction continuity relations. The

equations based on the traction continuity relations within

subcells:

Rrf g

121

AM;t2

h i1224

e1

e2

e3

e4

8>>>>>:9>>>=>>>;

241

O 126

feg61

B:12

The residual vector Rr results from satisfying traction continuityrelations. For linear elastic constituents, the components ofRr are

zero. The matrixO is the zero matrix and the components of matrix

AM1;AM;t2 andD

M1 are given as follows:

AM1

V1

VAI

33

033

V2

VAI

33

033

033

033

033

033

033

033

033

033

I33

033

033

033

033

033

033

033

033

033

I33

033

033

V1I33

033

V2I33

033

V3I33

033

V4I33

26666666664

37777777775B:13

AM;t2

C1ax33

033 C2ax

33033 033 033 033 033

033

C1

sh33

033

C1

sh33

033

033

033

033

033

C1

sh33

033

033

033

C3

sh33

033

033

033

C1

sh33

033

033

033

033

033

C4

sh33

2666666666664

3777777777775B:14

where:

Cax

C1111 C1122 C1133

C2211 C2222 C2233

C3311 C3322 C3333

2

64

3

75 Csh

C1212 0 0

0 C1212 0

0 0 C1212

2

64

3

75B:15

DM1

I33

033

I33

033

I33

033

033

I33

266666664

377777775 B:16

TheBa;t matrices in Eq.(10)are then formed using Eqs.(B.11) and

(B.12), which in linearized relations are.

Ba;th i24x6 A

M1

A

M;t

2" #24x241 D

M1

O" #24x6 B:17

x3

x1

x2

(*)

a1

a3

a2

Fig. B1. Representative volume element of the periodic microstructure.



11/11

Once theBa;t matrices are determined, the effective homogenized

stresses and stiffness matrix can be solved using Eqs.(11) and (12),

respectively.

B.3. Formulation of effective thermal conductivity

The homogenized temperature gradient and heat flux relations

for unit-cell are summarized as follows:

duti 1

VA v

1du1;ti v2du2;ti

h i du3;ti du

4;t

i B:18

dqti 1

V v

AdqA;t

i v3dq

3;t

i v4dq

4;t

i

h i B:19

dqA;t

i dq1;t

i dq2;t

i B:20

where the total volume of subcells 1 and 2 in Eqs.(B.18) and (B.19)

isVA V1 V2

: .

We introduce a concentration tensor that relates the average

subcells temperature gradient with the overall temperature gradi-

ent across the unit-cell. LetMa;t be the concentration tensor of the

temperature gradient. The temperature gradient in each subcell is

expressed by:

dua;ti Ma;tij d u

tj B:21

To formulate the Ma;t matrix, the micromechanical relations and

the constitutive equations are imposed. The present micromodel

consists of four (4) subcells with three (3) components of heat flux

need to be determined for every subcell. This requires forming

twelve (12) equations based on the temperature and heat flux con-

tinuities at the interface of each subcell which are written as:

A1 912

du1;tidu2;tidu3;ti

du4;ti

8>>>>>>>:

9>>>>=>>>>;

121

D193

fd uti31

g B:22

At2

112

du1;tidu2;tidu3;tidu4;ti

8>>>>>>>:

9>>>>=>>>>;121

O13

fd uti31

g B:23

By substituting Eq.(B.21)to Eqs.(B.22) and (B.23), theMa;t matrix

can be determined, which is:

Ma;t

h i121

A1

At2

124

1 D1

O

41

B:24

The matrix O is the zero matrix and the components of matrixA1;At

2andD1 are given as follows:

A1

V1

VAI

33

V2

VAI

33

033

033

033

033

I33

033

033

033

033

I33

2666664

3777775 B:25A

t2 K

1;tI

33K2;tI

330

330

33

h i B:26

D1

I33

I33

I33

I33

266666664

377777775

B:27

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A Multi-scale Model for Coupled Heat Conduction and Deformations of Viscoelastic Functionally Graded...

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