Numerical methods for modeling heterogeneous materials 1

NUMERICAL METHODS FOR MODELING
HETEROGENEOUS MATERIALS
TRAN THI QUYNH NHU
(B.Eng., HCMC University of Technology (2003))
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECH ANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2009
Acknowledgments
Foremost, I would like to express my deep and sincere gratitude to my supervisors,
Prof. Lee Heow Pueh and Prof. Lim Siak Piang, for their patient guidance, encour-
agement and advice during my time in NUS.
I would like to thank National University of Singapore for offering me the oppor-
tunity and the financial support to pursue the PhD study. I am grateful to the staffs
of Dynamics Lab for their help during my time there.
I would like to mention all of my Vietnamese friends in Singapore for our unfor-
gettable friendship. Hoang Quang Hung, Dau Van Huan, Le Ngoc Thuy, Nguyen
Hoang Huy (the list would be very long) and the rest of Hoi Coc Oi. I will always
remember the fun and the encouragement I have got from them.
And finally, I owe my loving gratitude to my parents Tran Du Sinh and Ho Thi
Van Nga, and my husband Huynh Dinh Bao Phuong, who have suffered very much
along with me during my PhD study. Without their love and support, I would not
be able to fulfil my dream. To them I dedic ate this thesis.
1
Summary
In this thesis, the applications of the conventional finite element method (FEM) and
its variations in modeling heterogeneous materials are presented. At first the prelim-
inary work on functionally graded materials (FGM) is presented in Chapter 2 and
Chapter 3. In these chapters, the FGM plates under thermal load are investigated
using the conventional FEM with the aid of the FEM package ABAQUS. The Voronoi

cell finite element method (VCFEM) is studied in Chapter 4 for analyzing the het-
erogeneous materials. In this chapter, various numerical examples from simple to
complicated compositions of heterogeneous materials containing inclusions are stud-
ied. In some examples, the quadratic quadrilateral elements are introduced as the
8-node elements for the VCFEM instead of the Voronoi cells. Chapter 5 shows the
application of the extended finite element method (XFEM) for heterogeneous mate-
rials, including porous structures. Ins tead of using the Heaviside enrichment function
for the strong discontinuity of the holes’ interfaces, the penalty method is introduced
to simulate the porous parts. Finally, the thesis is concluded in Chapter 6 with the
summary and the suggestions for future work.
2
Contents
1 Introduction 12
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Literature Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Review of Functionally graded materials . . . . . . . . . . . . . . . . 14
1.3.1 Review of micromechanical modeling for functionally graded
materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Review of the Voronoi Cell Finite Element Method . . . . . . 17
1.3.3 Review of the Extended Finite Element Method . . . . . . . . 20
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Thermal induced vibration of functionally graded thin plate 24
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 The finite element model . . . . . . . . . . . . . . . . . . . . . 27
2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Benchmarking of simulation results for vibration of FG plates 28
2.3.2 Thermal induced vibration of FG plates . . . . . . . . . . . . 31

2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3
3 Transient thermal mechanical response of functionally graded thick
plates 40
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 The finite element model . . . . . . . . . . . . . . . . . . . . . 43
3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Time-dependent prescribed temperature at the top surface . . 45
3.3.2 Time-dependent prescribed heat flux at the top surface . . . . 48
3.3.3 Time-dependent prescribe heat flux at partial top surface . . . 52
3.3.4 Comparison between continuous model and layered model . . 54
3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Voronoi Cell Finite Element Method 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Voronoi Cell Finite Element Method . . . . . . . . . . . . . . . . . . 62
4.3.1 Element formulation for homogeneous materials . . . . . . . . 62
4.3.2 Element formulation for heterogeneous materials . . . . . . . . 66
4.3.3 Interpolation stress function and the shape of heterogeneities . 69
4.3.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . 72
4.3.5 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.1 A Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 A unit cell containing a circular fiber . . . . . . . . . . . . . . 79
4.4.3 A unit cell containing elliptic inclusions . . . . . . . . . . . . . 81
4.4.4 A unit cell containing more circular inclusions . . . . . . . . . 82
4.4.5 A unit cell of a composite model with more elliptic inclusions 86
4

4.4.6 A composite plate containing a hole . . . . . . . . . . . . . . . 87
4.4.7 A FGM specimen . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4.8 A FG cantilever beam . . . . . . . . . . . . . . . . . . . . . . 93
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Extended Finite Element Method 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Level set method . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 General formulation . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.3 Choice of enriched nodes . . . . . . . . . . . . . . . . . . . . . 104
5.2.4 Enrichment function . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3.1 A unit cell containing a circular inclusion . . . . . . . . . . . . 106
5.3.2 A FGM specimen . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.3 A unit cell containing holes . . . . . . . . . . . . . . . . . . . 110
5.3.4 A specimen made of porous material . . . . . . . . . . . . . . 112
5.3.5 A unit cell of a bone model . . . . . . . . . . . . . . . . . . . 115
5.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Conclusions 122
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . 125
A List of Publications 127
B Material prope rties implementation into ABAQUS 128
C Interpolation Polynomial Matrices 130
5
List of Figures
2-1 FGM thin plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2-2 Temperature distribution through the thickness of square FG plates sub-
jected to suddenly applied heat flux q = 10

6
W/m
2
. . . . . . . . . . . . . 33
2-2 Temperature distribution through the thickness of square FG plates sub-
jected to suddenly applied heat flux q = 10
6
W/m
2
. . . . . . . . . . . . . 34
2-3 Historical central displacement of simply supported FG square plates sub-
jected to a suddenly applied heat flux q = 10
6
W/m
2
. . . . . . . . . . . . 35
2-4 Historical central displacement of simply supported FG square plates sub-
jected to a suddenly temperature rise T
top
= 200
o
K . . . . . . . . . . .
36
2-5 Historical central displacement of simply supported FG square plates (n =
2) under different temperature rises. . . . . . . . . . . . . . . . . . . . . 37
2-6 Temperature distribution through the thickness of square FG plate: a com-
parison between the dynamic solution and the quasi-static solution. . . . . 38
3-1 Geometry of the plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3-2 A 10 × 10 × 20 finite element mesh of the plate . . . . . . . . . . . . . . 44
3-3 Historical response of the FGM Al/SiC plate with the sinusoidal tempera-

ture distribution over the top surface . . . . . . . . . . . . . . . . . . . 47
3-4 Historical resp onse of the FGM Al/SiC plate with the uniform temperature
distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 49
6
3-5 Historical response of an FGM Al/SiC plate with the sinusoidal heat flux
distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 51
3-6 Historical response of an FGM Al/SiC plate with the uniform heat flux
distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 52
3-7 Historical response of the FGM Al/SiC plate with the uniform heat flux
distribution on partial area of the top surface . . . . . . . . . . . . . . . 53
3-8 Historical resp onse of the FGM Al/SiC plate with the uniform temperature
distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 55
3-9 Through the thickness profile of a FGM Al/SiC plate with the uniform
temperature distribution over the top surface . . . . . . . . . . . . . . . 56
4-1 A Voronoi cell finite element with an inclusion . . . . . . . . . . . . . . . 60
4-2 Element sub-division and Gaussian integration p oints: (a) no additional
ring (b) one additional ring with smaller Gaussian integration points in
each triangle. Here (+) are Gauss points for domain integration; (◦) are
Gauss points for line integration . . . . . . . . . . . . . . . . . . . . . . 74
4-3 Meshes of the cantilever b e am . . . . . . . . . . . . . . . . . . . . . . . 78
4-4 Displacement of the cantilever beam: VCFEM (thick line) and FEM (thin
line). Displacement solution is sc aled by a factor of 1000 . . . . . . . . . 78
4-5 Von Mises stress distribution: VCFEM (top . . . . . . . . . . . . . . . . 79
4-6 Meshes of the unit cell containing an inclusion . . . . . . . . . . . . . . . 80
4-7 Displacement of the unit cell containing an inclusion: VCFEM (thick line)
and FEM (thin line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4-8 Von Mises stress of the unit cell containing an inclusion: FEM (left) and
VCFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4-9 Meshes of a bi-unit cell containing 4 inclusions . . . . . . . . . . . . . . . 82
4-10 Displacement solution: VCFEM (thick line) and FEM (thin line) . . . . . 83

4-11 Von Mises stress comparison: FEM (left) and VCFEM (right) . . . . . . . 83
7
4-12 Meshes of the composite unit cell containing 29 circular inclusions . . . . 84
4-13 Displacement of the composite unit cell containing 29 circular inclusions:
VCFEM (thickline) and FEM (thin line) . . . . . . . . . . . . . . . . . . 85
4-14 Von Mises stress distribution of the composite unit cell containing 29 circular
inclusions: FEM (left) and VCFEM (right) . . . . . . . . . . . . . . . . 85
4-15 Meshes of the composite unit cell containing 36 elliptic inclusions . . . . . 87
4-16 Displacement of the composite unit cell containing 36 elliptic inclusions:
VCFEM (thick line) and FEM (thin line) . . . . . . . . . . . . . . . . . 87
4-17 Von Mises stress of the composite unit cell containing 36 elliptic inclusions:
FEM (left) and VCFEM (right) . . . . . . . . . . . . . . . . . . . . . . 88
4-18 Meshes of a plate containing a hole . . . . . . . . . . . . . . . . . . . . 89
4-19 Displacement of a composite plate containing a hole: VCFEM (thick line)
and FEM (thin line). Displacement is scaled by a factor of 1/5 . . . . . . 89
4-20 Von Mises stress of a composite plate containing a hole: FEM (left) and
VCFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4-21 Meshes of the FGM specimen . . . . . . . . . . . . . . . . . . . . . . . 91
4-22 Displacement of the FGM specimen: VCFEM (thick line) and FEM (thin
line). Displacement is scaled by a factor of 1/5 . . . . . . . . . . . . . . 92
4-23 Von Mises stress of the FGM specimen: VCFEM (top . . . . . . . . . . . 92
4-24 Effect of the Young’s modulus on the maximum displacement of the FGM
specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4-25 Meshes of the FG cantilever beam . . . . . . . . . . . . . . . . . . . . . 95
4-26 Displacement of the FG cantilever beam: VCFEM (thick line) and FEM
(thin line). Dis placeme nt solution is scaled by a factor of 1000 . . . . . . 96
4-27 Von Mises stress distribution of the FG cantilever beam: VCFEM (top . . 96
4-28 Effect of the Young’s modulus on the tip displacement of the FG cantilever
beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8

5-1 Level set (a) zero level set (b) level set contour (c) level set function . . . 101
5-2 Level set (a) zero level set (b) level set contour (c) level set function . . . 102
5-3 Domain subdivision and integration points . . . . . . . . . . . . . . . . . 106
5-4 Meshes (a) FEM (b) XFEM: ◦ indicates enriched nodes , and enriched ele-
ments are those with thick line . . . . . . . . . . . . . . . . . . . . . . . 107
5-5 Total displacement of the unit cell containing one inclusion . . . . . . . . 108
5-6 Von-mises stress distribution of the unit cell containing one inclusion: FEM
(left) and XFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5-7 XFEM mesh of the FGM specimen: ◦ indicates enriched nodes, and enriched
elements are those with thick line . . . . . . . . . . . . . . . . . . . . . 109
5-8 Displacement of the FGM specim en: FEM solution(left) and XFEM solu-
tion(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5-9 Von-mises stress distribution of the FGM specimen: FEM (left) and XFEM
(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5-10 Meshes of the porous unit cell(a) FEM (b) VCFEM . . . . . . . . . . . . 111
5-11 XFEM mesh of the porous unit cell: ◦ indicates enriched nodes, and enriched
elements are those with thick line . . . . . . . . . . . . . . . . . . . . . 112
5-12 Displacement of the porous unit cell: VCFEM solution . . . . . . . . . . 113
5-13 Displacement of the porous unit cell: FEM solution(left) and XFEM solu-
tion(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5-14 Von-mises stress distribution of the porous unit cell: FEM (left) and XFEM
(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5-15 Meshes of the specimen made of porous material (a) FEM (b) XFEM: ◦
indicates enriched nodes, and enriched elements are those with thick line . 115
5-16 Displacement solution of the specimen made of porous material: FEM (left)
and XFEM (right). Displacement is scaled by a factor of 1/5 . . . . . . . 116
5-17 Von-mises stress comparison: FEM (left) and XFEM (right) . . . . . . . 116
9
5-18 Micrographs of some trab ecular bones (source: web.mit .edu) . . . . . . . 117
5-19 Level set conversion from a micrograph (a) original micrograph (b) de-noised

micrograph (c) black and white image (d) level set function . . . . . . . . 118
5-20 Level set conversion converted directly from the micrograph . . . . . . . . 118
5-21 XFEM meshes: ◦ indicates enriched nodes, and enriched elements are those
with thick line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5-22 Displacement and von-mises stress solutions for 128×128 XFEM mesh (left)
and 256 × 256 XFEM mesh (right). Displacement is scaled by a factor of
1/100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
10
List of Tables
2.1 Natural frequencies (Hz) of FG plates . . . . . . . . . . . . . . . . . . 29
2.2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Nondimentional frequencies of the FG plate in thermal environment . 31
3.1 Material properties of the constituents of the FGM . . . . . . . . . . . . 44
11
Chapter 1
Introduction
1.1 Motivation
Heterogeneous materials are very popular in nature. The constituents of heteroge-
neous materials can be two or more different material phases, compositions or states
in one length scale or multiple length scales . Common examples of heterogeneous
materials include bones, which are hierarchically heterogeneous, woods, composite
materials, polycrystalline materials, etc.
The effective properties of heterogeneous materials are usually not equal to the
averages of their constituents but are dependent on their microstructures of the ma-
terials such as the shape and size, the distribution, the interaction of the particles.
Therefore, a good understanding of the microstructure may provide a proper predict-
ing of the macroscopic behavior of the heterogeneous materials.
One of the recently developed heterogeneous materials is Functionally Graded
Material (FGM). FGM is a typical composite material in which the material properties
vary gradually in one or more directions in space. FGMs have been extensively used

because of their attractive properties, including a potential reduction of in-plane
and transverse through-the-thickness stress, an improved residual stress distribution,
12
enhanced thermal properties, etc. The spatial distribution of the FGMs can also be
designed to embed specific function to material properties for special purposes.
This study started with some preliminary research on the FGMs. The Finite
Element Method (FEM) was applied with the aid of the FEM package ABAQUS [1].
Although ABAQUS, together with its supporting user defined subroutines, was useful
in modeling the FGMs, there were certain limitations in new applications like FGMs.
Hence, there was a need to study the microstructure of the FGMs to better understand
the micromechanical behaviors of FGMs, which directly affects the macromechanical
properties of the materials.
In this study, the Voronoi Cell Finite Element Method (VCFEM) [2] was chosen
as the candidate for modeling the microstructure of the FGMs. This method is a vari-
ation of the conventional FEM in which a hybrid approximation of the displacement
field and the stress field is used to build up the formulation. The VCFEM would
be advantageous in modeling the particulate-matrix heterogeneous materials because
the nature of this method is that it does not require the conformal meshing at the
inclusion interfaces, which is a challenging for the conventional FEM. This property
of the VCFEM would be helpful in modeling the microstructure of FGMs where the
inclusions are distributed non-uniformly. However, the VCFEM would not predict
well the stress field, even though a correction of the shape effect of inclusions was
added into the stress interpolation function.
The Extended Finite Element Method (XFEM) [3], which is another variation of
the conventional FEM was also applied in this thesis to complement the disadvantages
of the VCFEM. The remarkable characteristic of the XFEM is that its mesh does not
need to conform with the discontinuities inside the material structures. Hence the
meshing difficulty could also be avoided like in the case of the VCFEM to model the
inclusions efficiently. Moreover, the XFEM could give a better stress resolution than
the VCFEM.

13
In the next section, a brief literature review of FGMs and the micromechanical
modeling of FGMs, the VCFEM and the XFEM will be presented.
1.2 Literature Reviews
1.3 Review of Functionally graded materials
The early research on Functionally Graded Materials (FGM) was on transient ther-
momechanics. One of the first reported works was by Fuchiyama and Noda [4]. They
developed a finite element program to calculate the transient thermomechanical re-
sponse of functionally graded (FG) plates, in which the temperature dependent mate-
rial properties were accounted for. The variation of material properties was modeled
by a set of homogeneous layers. Vel and Batra [5] also dealt with the transient thermal
stresses in FG plates. They developed an analytical solution for three-dimensional
plates using power law series metho d.
Heat transfer was also considered widely. Jin [6] derived a c losed from asymptotic
solution to analyze the transient heat transfer in a FGM strip with the material
properties varied through the thickness direction. The surfaces of the FGM strip
were suddenly cooled to different temperatures. Sladek et al. [7] investigated a FGM
strip and an infinitely long FGM cylinder subjected to stationary thermal loads and
to thermal shock using a local boundary integral method. The transient heat transfer
in a thick FGM strip subjected to a nonuniform volumetric heat source was studied
by Ootao and Tanigawa [8] to get the temperature distribution and subsequently
derived the thermal induced stresses. In another study, Chen et al. [9] also used the
Galerkin boundary element method for studying the transient heat transfer in a 3D
cube subjected to a prescribed heat flux regime. The Galerkin boundary element
formulation was implemented to study the 3D heat transfer problems by Sutradhar
14
and Paulino [10]
Some other researchers studied the vibration characteristic of FG plates as well,
with or without the thermal effect. Vel and Batra [11] investigated the free and
forced vibration of three-dimensional FG plates, in which the exact solution for the

natural frequencies, the displacements and the stresses was reported. He et al. [12]
used a finite element formulation based on the classical plate theory to study the
shape and the vibration control of FG plates with integrated piezoelectric sensors
and actuators. The active control was processed via a constant velocity feedback
control algorithm. Yang and Shen [13] analyzed the dynamic response of initially
stressed functionally graded rectangular thin plates subjected to partially-distributed
impulsive lateral loads with or without an elastic foundation.
In the reported references, the thermal load was included beside the mechanical
load while investigating the vibration of FG plates [14–19]. Praveen and Reddy [14]
and Reddy [15] analyzed the response of thin FG plates under thermomechanical
loading. The plates were initially stressed by applying a temperature field. The static
and transient dynamic problems were presented and the material properties were
assumed to be temperature indep ende nt. Huang and Shen [16], Yang and Shen [17],
Shen [18] and Kim [19] presented a series of works that studied the vibration of
the FG plates in thermal environment. The thermal environment was imple mented
after a steady state heat transfer analysis. The problems were nonlinear due to the
thermal terms in the governing equations. The temperature dependent properties
were included in these reported studies.
1.3.1 Review of micromechanical modeling for functionally
graded materials
There have been several micromechanical models for estimating the material proper-
ties of composites in reported studies. The rule of mixture, the Mori-Tanaka model
15
and the self-consistence model were most widely used in modeling traditional com-
posite materials. Many authors have also applied those micromechanical models to
calculate the effective material properties of FGMs with the assumption that the ma-
terials remained homogeneous at the representative volume element (RVE) and the
size of the RVE was decided based on the gradient of gradation of the constituent
materials.
The authors who used the Mori-Tanaka as their homogenization technique for

FGMs included Tsukamoto [20] and Vel and Batra [5,11, 21]. Cho and Ha [22] used
the averagng techniques, i.e. the rule of mixture, the modified rule of mixture to
estimate the efficient material properties of FGMs. By comparison the standard mi-
cromechanical models to choose a suitable model for FGMs, Zuiker [23] investigated
the Mori-Tanaka, the self-consistent and the Tamura’s models, and a fuzzy logic tech-
nique. The self-consistent method was recommended for reliable first order estimates
over the entire range of volume fraction variations. In another comparison of Reuter
et. al. [24, 25], the Mori-Tanaka, the self consistent and the finite element simula-
tion were analyzed and the self-consistent model was recommended for the skeleton
microstructure while the Mori-Tanaka model was recommended for the particulate
microstructure with a “well-defined” matrix.
The micromechanical models mentioned above could be used conveniently. How-
ever, they could not describe fully the natural gradation of the FGMs and thus these
models could be applied where the material were relatively slow-changing functions
of spatial coordinates [26,27]. When the properties of the material vary rapidly with
coordinates, it is necessary to consider the heterogeneous nature at the RVE scale.
One of the earliest studies on the gradation of material distribution at grain size
level was reported by Dao et. al. [28] in which the residual stress of FGM at grain-
size level was studied. The domain was discretized into square cells, with each cell
representing one grain. Following this approach, the interaction between the adjacent
16
grains could be observed so that the residual stress was calculated more accurately.
However, the shape of the grains here was still assumed to be square while the real
grains normally have arbitrary shapes. Yin et al. [26] used the stress-strain analysis to
evaluate the effective properties along the gradation direction. The local interactive
between the particles was accounted for by using the integrated contributions between
each pair of particles to specify the averaged strains throughout the material. Another
study that accounted for the local material grading was by Aboudi et al. [27], in which
the constitutive modeling theory based on the higher-order generalized method of cells
was applied and subsequently extended to account for the incremental plasticity, the

creep and the viscoplastic effects [29]. The spatially varied thermal conductivity of
the material at local level was investigated by Zhong and Pindera by reformulating
the higher-order theory for FGMs. The thermal conductivity of FGMs was also
referred in a study by Yin et al. [30], in which the effective thermal conductivity was
derived from the relationship between the gradient of temperature and the heat flux
distribution.
A finite element approach for the analysis of the FGMs without any assumption
of the gradation of the materials was suggested by Grujicic and Zhang [31] in which
the VCFEM was applied to simulate the microstructure of the FGMs. Subsequently,
Biner [32] and Vena [33] also used the VCFEM in modeling the microstructure of
FGMs. The VCFEM was shown to be advantageous in micromechanical modeling of
FGM. In the next section, a review of the VCFEM will be presented.
1.3.2 Review of the Voronoi Cell Finite Element Method
The Voronoi cell finite element method (VCFEM) is a special stress-based finite el-
ement formulation, in which the element is not restricted to 3 or 4-node elements
as with the conventional FEM but convex polygons with any number of edges, in
particular, Voronoi cells in a Voronoi diagram which conform the geometry of the
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model. The Voronoi cell finite element method was first introduced by Ghosh and
Mallett [2] based on an extension of the hybrid finite element method proposed by
Pian and his colleagues [34–36]. Since then, it was quickly developed by Ghosh and
his colleague as an efficient method f or micromechanical analysis of arbitrary hetero-
geneous microstructures [37] [38–42]. The most obvious advantage of the VCFEM
is that it can model complex microstructure models such as those contains hetero-
geneities (fiber, void, crack, etc.) with ease by including each heterogeneity inside
each “cell”, or “Voronoi eleme nt” [41]. Each Voronoi element with embedded het-
erogeneities represents the neighborhood or regions of influence for the heterogeneity.
That element is then treated as a single super–element. Known functional forms for
regular heterogeneities from analytical micromechanics is also incorporated in to each
element to enhance the convergence of the VCFEM. Since the required mesh for the

VCFEM no longer needs to conf orm the heterogeneities, the complexity or degree
of freedoms of VCFEM is significantly lower than the traditional FEM. As a result,
VCFEM offers great computational savings over conventional FEM. The VCFEM is,
thus, an attractive method to study composite/void structures.
Since the mesh in VCFEM is generated from Voronoi diagram which conforms
to the geometry, mesh generation in VCFEM is not a trivial problem. However,
very little studies were focused on this particular field. Most notable works include
the works of Ghosh and Mukhopadhyay [43] and Ghosh and Moorthy [44]. Voronoi
mesh generation, along with error estimation, still remain as some of the opened and
interesting topics in the development of VCFEM.
Special attentions were also paid by various VCFEM investigators to enhance
the convergence of the VCFEM by enhancing the approximation field inside each
heterogeneity embedded Voronoi cell. Li et al. [45] enhanced the stress approximation
in Voronoi cell containing cracks with branch functions and multi-resolution wavelet
functions in the vicinity of crack tips. Recently, Tiwary [46] developed a nonconformal
18
mapping and wavelet based functions to further enrich the stress approximation near
highly irregular heterogeneities such as crack and sharp inclusions. The new stress
correction accounted for the shape effect of the inclusions on the stress field so that
the inclusions could have different shapes and the microstructure of materials could
be modeled with more flexibility.
The VCFEM has enjoyed a great deal of applications over the last decade. Ghosh
and his colleagues applied this new method to various studies of composite materials
(Ghosh et. al. [37–39]) such as the elastic-plastic problem, the thermal mechanical
problem and the porous structure, etc. Their works showed the advantages of Voronoi
cell finite element method for analyzing comp osite material at micro level. Li et al. [45]
developed an Extended VCFEM to analyze cohesive crack propagation in brittle
materials. Hu et al. [47] developed a locally enhanced VCFEM to investigate crack
propagation in ductile microstructures. Li et al. [48] used the Extended VCFEM
to model interfacial debonding and matrix cracking in fiber reinforced composites.

Recently, Ghosh et al. extended the VCFEM to three dimensional problems to model
microstructures with ellipsoidal heterogeneities [44].
Some researchers applied this method to study the microstructure of FGMs. One
of the first studies was the work of G rujicic and Zhang [31]. They used the Voronoi cell
finite element method to investigate the micromechanical response of FGM and then
determine the effective material properties. Their work considered the elastic response
of FGM only. The thermal mechanical problem was later studied by Biner [32] and
Vena [33]. In these studies, the shape effect of the inclusions was not mentioned.
Thus far, it is obvious that there have been just a few studies on the application
of Voronoi cell finite element method for FGMs in spite of its many advantages. Since
each element can contain an inclusion, the material distribution can be modeled easily
by arranging the inclusions in space.
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1.3.3 Review of the Extended Finite Element Method
The Extended Finite Element Method (XFEM) is a well known method that is use-
ful in modeling structures with discontinuities and/or singularities such as cracks,
dislocations, and phase boundaries. Perhaps the most important advantage of this
method is that the finite element mesh can be completely independent of the geome-
try of the discontinuities, thus the mesh needs not conform with the entities as in the
conventional FEM [3, 49]. The XFEM improves the versatility of the conventional
FEM by introducing a local enrichment of the approximating space [50, 51].
The early research of XFEM was on fracture mechanics. The concept of the lo-
cal partition of unity (PUM) was first introduced by Babuska [52]. The XFEM was
first introduced based on the PUM concept to study the stress intensity factor of
various cracks settings and crack propagation (without using level set) by Black and
Belytschko [3]. The method was further refined in Dolbow et al. [53] and became
a standard methodology of investigating crack models without explicitly model the
cracks themselves. The Heaviside enrichment in conjunction with a near-tip enrich-
ment was applied in the research of Daux et al. [49] and Dolbow [53]. Arbitrary
curved cracks in higher-order elements was modeled by Heaviside by Stazi et al.

in [54]. Daux et al. [55] extended the methodology to study the branched cracks by
superposed the Heaviside function at the junctions of the cracks. Solution near the
crack tip was also corrected using crack tip enrichment, which was investigated by
a great deals of publications from a lot researchers [53, 56–60]. The XFEM was also
used to model discontinuities of materials, such as holes and inclusions by Sukumar
et al. [61]. Recently, Huynh and Belytschko [62] applied the XFEM to investigate the
fracture in 3D composite materials.
Discontinuities in the XFEM were modeled using the level sets method, which
now becomes a key ingredient in the XFEM. The level s ets method was advantageous
in describing the discontinuities implicitly instead of explicitly mathematical curves
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or surfaces [63,64]. Complicated discontinuity shapes or geometries such as cracks or
curved interface s were described by simple discrete representation [65]. Calculation
of enrichment functions also benefited from level sets representations, which could
be done in a straightforward manner with the use of finite element shape functions.
Furthermore, the level sets method was shown to be useful in tracking the evolution
of holes and inclusions, which proved to be very useful in application such as phase
change simulations or crack propagations in both two dimensions and three dimen-
sions [59, 60, 66, 67]. Applications of the XFEM and level sets method to problems
with complex geometries such as multiple intersecting and branching cracks can be
found in Budyn et al. [68], Zi et al. [69] and Loehnert et al. [70].
Investigation on accuracy and convergence of the XFEM also received a lot of
attentions recently. Sukumar et al. [61] and Chessa et al. [71] showed that the XFEM
convergence rate was degraded by the approximation in the so-called “blending ele-
ments” in the transition of the enriched elements and normal finite elements. Much
attentions were paid to improve the convergence rate by correcting the approxima-
tion in the blending areas, such as the work of Gracie et al. [72] by the discontinuous
Galerkin and Fries [73] by smoothing the enrichment in the blending elements by a
weight function.
Another important issue in the implementation of the XFEM was the compu-

tation of the quadrature in the weak form, such as the stiffness matrix of enriched
elements. For enriched elements, since the approximation space is enriched and the
approximation functions are no longer polynomials, special attentions must be paid
to calculate the quadrature accurately. Notable works on this subject included the
use of quadrature on subdivided elements [3,49], the polar mapping of the integrand
functions and domains by Laborde et al. [74] and Bechet et al. [75], and the fast
method recently proposed by Ventura et al. [76].
The XFEM was also applied to study a vast class of discontinuities problem. Gra-
21
cie et al. studied dislocations in references [77–79]. The dislocations in the XFEM
were treated as discontinuities which were very similarly to crack models. The method
allowed treatments for complex micromechanics models such as dislocations in car-
bon nanotubes and thin films recently [77–81].The phase boundaries were studied by
Chessa and Belytschko [82–84].
1.4 Objectives
The aim of the study reported in this thesis was to apply the VCFEM and the XFEM
to study the microstructure of particulate type heterogeneous materials.
The VCFEM would be applied in various examples of heterogeneous materials,
including two examples of FGMs. The stress correction for the effect of the shape of
the inclusions would be added to the stress approximation to increase the solution
accuracy and thus the inclusions could be circular or elliptical. One of the difficulties
of the VCFEM is to generate a Voronoi diagram that is applicable as a mesh. In
this study, the quadrilateral meshes would be investigated as an alternative for the
Voronoi diagram.
The XFEM thereafter would be applied in examples similar to the VCFEM for
facilitating the comparative studies. Moreover, the porous structure would also be
studied using the XFEM. A penalty method would be utilized so that strong discon-
tinuities could be treated as weak discontinuities. An example of the trabecular bone
would be studied to show the advantages of the XFEM. In this study, the level set
method would be applied so that the XFEM model could be built directly from the

microimage of the bone.
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1.5 Thesis Outline
In this thesis, the applications of the conventional FEM and its variations in modeling
heterogeneous mate rials are presented. At first the preliminary work on FGMs is pre-
sented in Chapter 2 and Chapter 3. In these chapters, the FGM plates under thermal
load are investigated using the conventional FEM with the aid of the FEM pack-
age ABAQUS. The VCFEM is studied in Chapter 4 for analyzing the heterogeneous
materials. In this chapter, various numerical examples from simple to complicated
compositions of heterogeneous materials containing inclusions are studied. In some
examples, the quadratic quadrilateral elements are introduced as the 8-node elements
for the VCFEM instead of the Voronoi cells. Chapter 5 shows the application of the
XFEM for heterogeneous materials, including porous structures. Instead of using the
Heaviside enrichment function for the strong discontinuity of the holes’ interfaces,
the penalty method is introduced to simulate the porous parts. Finally, the thesis is
concluded in Chapter 6 with the summary and the suggestions for future work.
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Chapter 2
Thermal induced vibration of
functionally graded thin plate
2.1 Introduction
The transient thermomechanics of functionally graded materials (FGM) was widely
investigated. One of the first reported works was by Fuchiyama and Noda [4], in which
the temperature dependent material properties were accounted. Vel and Batra [5] also
dealt with the transient thermal stresses in FG plates. They developed an analytical
solution for three-dimensional plates using power law series method.
The vibration characteristic was studied by some other authors [11, 12]. Yang
and Shen [13] analyzed the dynamic response of initially stressed functionally graded
rectangular thin plates subjected to partially-distributed impulsive lateral loads with
or without an elastic foundation. In the reported references [14–19], the authors

included the thermal load beside the mechanical load while investigating the vibration
of FG plates. Praveen and Reddy [14] and Reddy [15] analyzed the response of
thin FG plates under thermomechanical loading. The plates were initially stressed
by applying a temperature field. The static and transient dynamic problems were
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