A Multi-scale Model for Coupled Heat Conduction and Deformations of Viscoelastic Functionally Graded...
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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
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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.
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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
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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
Volume Fraction (VF)
0
1E-005
2E-005
3E-005
CoefficientofThermalExpansion
(K-1)
Experimental Data Geiger and Jackson (1989)
Micromechanical Model
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
Volume Fraction (VF)
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
Micromechanical 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.
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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
Micromechanical Model
Ti6Al-4v with Zr02inclusions
t =20s
t =50s
t =30s
t =5s
Steady State Time = 159 seconds
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)
Detail Microstructural Model
Micromechanical Model
Ti6Al-4v with Zr02inclusions
t =30s
t =50s
t =20s
Steady State Time = 159 seconds
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
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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)
Detail Microstructural Model
Micromechanical Model
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)
Detail Microstructural Model
Micromechanical Model
Al with SiC inclusions
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
Al with SiC inclusions
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.
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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.
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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.
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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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