Numerical methods for modeling heterogeneous materials 1

List of Figures2-2 Temperature distribution through the thickness of square FG plates jected to suddenly applied heat flux q = 106W/m2.. 332-2 Temperature distribution through the thickn

NUMERICAL METHODS FOR MODELING

HETEROGENEOUS MATERIALS

TRAN THI QUYNH NHU

A THESIS SUBMITTEDFOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009

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 tunity and the financial support to pursue the PhD study I am grateful to the staffs

oppor-of Dynamics Lab for their help during my time there

I would like to mention all of my Vietnamese friends in Singapore for our gettable friendship Hoang Quang Hung, Dau Van Huan, Le Ngoc Thuy, NguyenHoang Huy (the list would be very long) and the rest of Hoi Coc Oi I will alwaysremember the fun and the encouragement I have got from them

unfor-And finally, I owe my loving gratitude to my parents Tran Du Sinh and Ho ThiVan Nga, and my husband Huynh Dinh Bao Phuong, who have suffered very muchalong with me during my PhD study Without their love and support, I would not

be able to fulfil my dream To them I dedicate this thesis

In this thesis, the applications of the conventional finite element method (FEM) andits variations in modeling heterogeneous materials are presented At first the prelim-inary work on functionally graded materials (FGM) is presented in Chapter 2 andChapter 3 In these chapters, the FGM plates under thermal load are investigatedusing the conventional FEM with the aid of the FEM package ABAQUS The Voronoicell finite element method (VCFEM) is studied in Chapter 4 for analyzing the het-erogeneous materials In this chapter, various numerical examples from simple tocomplicated compositions of heterogeneous materials containing inclusions are stud-ied In some examples, the quadratic quadrilateral elements are introduced as the8-node elements for the VCFEM instead of the Voronoi cells Chapter 5 shows theapplication of the extended finite element method (XFEM) for heterogeneous mate-rials, including porous structures Instead of using the Heaviside enrichment functionfor 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 thesummary and the suggestions for future work

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

3 Transient thermal mechanical response of functionally graded thick

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

4.4.6 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

List of Figures

2-2 Temperature distribution through the thickness of square FG plates jected to suddenly applied heat flux q = 106W/m2 332-2 Temperature distribution through the thickness of square FG plates sub-jected to suddenly applied heat flux q = 106W/m2 342-3 Historical central displacement of simply supported FG square plates sub-jected to a suddenly applied heat flux q = 106W/m2 352-4 Historical central displacement of simply supported FG square plates sub-jected to a suddenly temperature rise 4Ttop = 200oK 362-5 Historical central displacement of simply supported FG square plates (n =2) under different temperature rises 372-6 Temperature distribution through the thickness of square FG plate: a com-parison between the dynamic solution and the quasi-static solution 383-1 Geometry of the plate 433-2 A 10 × 10 × 20 finite element mesh of the plate 443-3 Historical response of the FGM Al/SiC plate with the sinusoidal tempera-ture distribution over the top surface 473-4 Historical response of the FGM Al/SiC plate with the uniform temperaturedistribution over the top surface 49

sub-3-5 Historical response of an FGM Al/SiC plate with the sinusoidal heat fluxdistribution over the top surface 513-6 Historical response of an FGM Al/SiC plate with the uniform heat fluxdistribution over the top surface 523-7 Historical response of the FGM Al/SiC plate with the uniform heat fluxdistribution on partial area of the top surface 533-8 Historical response of the FGM Al/SiC plate with the uniform temperaturedistribution over the top surface 553-9 Through the thickness profile of a FGM Al/SiC plate with the uniformtemperature distribution over the top surface 564-1 A Voronoi cell finite element with an inclusion 604-2 Element sub-division and Gaussian integration points: (a) no additionalring (b) one additional ring with smaller Gaussian integration points ineach triangle Here (+) are Gauss points for domain integration; (◦) areGauss points for line integration 744-3 Meshes of the cantilever beam 784-4 Displacement of the cantilever beam: VCFEM (thick line) and FEM (thinline) Displacement solution is scaled by a factor of 1000 784-5 Von Mises stress distribution: VCFEM (top 794-6 Meshes of the unit cell containing an inclusion 804-7 Displacement of the unit cell containing an inclusion: VCFEM (thick line)and FEM (thin line) 804-8 Von Mises stress of the unit cell containing an inclusion: FEM (left) andVCFEM (right) 814-9 Meshes of a bi-unit cell containing 4 inclusions 824-10 Displacement solution: VCFEM (thick line) and FEM (thin line) 83

4-12 Meshes of the composite unit cell containing 29 circular inclusions 844-13 Displacement of the composite unit cell containing 29 circular inclusions:VCFEM (thickline) and FEM (thin line) 854-14 Von Mises stress distribution of the composite unit cell containing 29 circularinclusions: FEM (left) and VCFEM (right) 854-15 Meshes of the composite unit cell containing 36 elliptic inclusions 874-16 Displacement of the composite unit cell containing 36 elliptic inclusions:VCFEM (thick line) and FEM (thin line) 874-17 Von Mises stress of the composite unit cell containing 36 elliptic inclusions:FEM (left) and VCFEM (right) 884-18 Meshes of a plate containing a hole 894-19 Displacement of a composite plate containing a hole: VCFEM (thick line)and FEM (thin line) Displacement is scaled by a factor of 1/5 894-20 Von Mises stress of a composite plate containing a hole: FEM (left) andVCFEM (right) 904-21 Meshes of the FGM specimen 914-22 Displacement of the FGM specimen: VCFEM (thick line) and FEM (thinline) Displacement is scaled by a factor of 1/5 924-23 Von Mises stress of the FGM specimen: VCFEM (top 924-24 Effect of the Young’s modulus on the maximum displacement of the FGMspecimen 944-25 Meshes of the FG cantilever beam 954-26 Displacement of the FG cantilever beam: VCFEM (thick line) and FEM(thin line) Displacement solution is scaled by a factor of 1000 964-27 Von Mises stress distribution of the FG cantilever beam: VCFEM (top 964-28 Effect of the Young’s modulus on the tip displacement of the FG cantilever

5-1 Level set (a) zero level set (b) level set contour (c) level set function 1015-2 Level set (a) zero level set (b) level set contour (c) level set function 1025-3 Domain subdivision and integration points 1065-4 Meshes (a) FEM (b) XFEM: ◦ indicates enriched nodes, and enriched ele-ments are those with thick line 1075-5 Total displacement of the unit cell containing one inclusion 1085-6 Von-mises stress distribution of the unit cell containing one inclusion: FEM(left) and XFEM (right) 1085-7 XFEM mesh of the FGM specimen: ◦ indicates enriched nodes, and enrichedelements are those with thick line 1095-8 Displacement of the FGM specimen: FEM solution(left) and XFEM solu-tion(right) 1105-9 Von-mises stress distribution of the FGM specimen: FEM (left) and XFEM(right) 1115-10 Meshes of the porous unit cell(a) FEM (b) VCFEM 1115-11 XFEM mesh of the porous unit cell: ◦ indicates enriched nodes, and enrichedelements are those with thick line 1125-12 Displacement of the porous unit cell: VCFEM solution 1135-13 Displacement of the porous unit cell: FEM solution(left) and XFEM solu-tion(right) 1135-14 Von-mises stress distribution of the porous unit cell: FEM (left) and XFEM(right) 1145-15 Meshes of the specimen made of porous material (a) FEM (b) XFEM: ◦indicates enriched nodes, and enriched elements are those with thick line 1155-16 Displacement solution of the specimen made of porous material: FEM (left)and XFEM (right) Displacement is scaled by a factor of 1/5 116

5-18 Micrographs of some trabecular bones (source: web.mit edu) 1175-19 Level set conversion from a micrograph (a) original micrograph (b) de-noisedmicrograph (c) black and white image (d) level set function 1185-20 Level set conversion converted directly from the micrograph 1185-21 XFEM meshes: ◦ indicates enriched nodes, and enriched elements are thosewith thick line 1195-22 Displacement and von-mises stress solutions for 128×128 XFEM mesh (left)and 256 × 256 XFEM mesh (right) Displacement is scaled by a factor of1/100 120

List of Tables

3.1 Material properties of the constituents of the FGM 44

The effective properties of heterogeneous materials are usually not equal to theaverages 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 GradedMaterial (FGM) FGM is a typical composite material in which the material propertiesvary gradually in one or more directions in space FGMs have been extensively usedbecause of their attractive properties, including a potential reduction of in-planeand transverse through-the-thickness stress, an improved residual stress distribution,

enhanced thermal properties, etc The spatial distribution of the FGMs can also bedesigned to embed specific function to material properties for special purposes.This study started with some preliminary research on the FGMs The FiniteElement 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 understandthe micromechanical behaviors of FGMs, which directly affects the macromechanicalproperties 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 ation of the conventional FEM in which a hybrid approximation of the displacementfield and the stress field is used to build up the formulation The VCFEM would

vari-be advantageous in modeling the particulate-matrix heterogeneous materials vari-becausethe nature of this method is that it does not require the conformal meshing at theinclusion interfaces, which is a challenging for the conventional FEM This property

of the VCFEM would be helpful in modeling the microstructure of FGMs where theinclusions are distributed non-uniformly However, the VCFEM would not predictwell the stress field, even though a correction of the shape effect of inclusions wasadded into the stress interpolation function

The Extended Finite Element Method (XFEM) [3], which is another variation ofthe 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 notneed to conform with the discontinuities inside the material structures Hence themeshing difficulty could also be avoided like in the case of the VCFEM to model theinclusions efficiently Moreover, the XFEM could give a better stress resolution than

In the next section, a brief literature review of FGMs and the micromechanicalmodeling of FGMs, the VCFEM and the XFEM will be presented.

The early research on Functionally Graded Materials (FGM) was on transient momechanics One of the first reported works was by Fuchiyama and Noda [4] Theydeveloped 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

ther-by a set of homogeneous layers Vel and Batra [5] also dealt with the transient thermalstresses in FG plates They developed an analytical solution for three-dimensionalplates using power law series method

Heat transfer was also considered widely Jin [6] derived a closed from asymptoticsolution to analyze the transient heat transfer in a FGM strip with the materialproperties varied through the thickness direction The surfaces of the FGM stripwere suddenly cooled to different temperatures Sladek et al [7] investigated a FGMstrip 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 subsequentlyderived the thermal induced stresses In another study, Chen et al [9] also used theGalerkin boundary element method for studying the transient heat transfer in a 3Dcube subjected to a prescribed heat flux regime The Galerkin boundary elementformulation was implemented to study the 3D heat transfer problems by Sutradhar

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 andforced vibration of three-dimensional FG plates, in which the exact solution for thenatural 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 theshape and the vibration control of FG plates with integrated piezoelectric sensorsand actuators The active control was processed via a constant velocity feedbackcontrol algorithm Yang and Shen [13] analyzed the dynamic response of initiallystressed functionally graded rectangular thin plates subjected to partially-distributedimpulsive lateral loads with or without an elastic foundation

In the reported references, the thermal load was included beside the mechanicalload while investigating the vibration of FG plates [14–19] Praveen and Reddy [14]and Reddy [15] analyzed the response of thin FG plates under thermomechanicalloading The plates were initially stressed by applying a temperature field The staticand transient dynamic problems were presented and the material properties wereassumed to be temperature independent Huang and Shen [16], Yang and Shen [17],Shen [18] and Kim [19] presented a series of works that studied the vibration ofthe FG plates in thermal environment The thermal environment was implementedafter a steady state heat transfer analysis The problems were nonlinear due to thethermal terms in the governing equations The temperature dependent propertieswere 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-and the self-consistence model were most widely used in modeling traditional posite materials Many authors have also applied those micromechanical models tocalculate the effective material properties of FGMs with the assumption that the ma-terials remained homogeneous at the representative volume element (RVE) and thesize of the RVE was decided based on the gradient of gradation of the constituentmaterials.

com-The authors who used the Mori-Tanaka as their homogenization technique forFGMs included Tsukamoto [20] and Vel and Batra [5, 11, 21] Cho and Ha [22] usedthe averagng techniques, i.e the rule of mixture, the modified rule of mixture toestimate the efficient material properties of FGMs By comparison the standard mi-cromechanical models to choose a suitable model for FGMs, Zuiker [23] investigatedthe 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 estimatesover 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 tion were analyzed and the self-consistent model was recommended for the skeletonmicrostructure while the Mori-Tanaka model was recommended for the particulatemicrostructure with a “well-defined” matrix

simula-The micromechanical models mentioned above could be used conveniently ever, they could not describe fully the natural gradation of the FGMs and thus thesemodels could be applied where the material were relatively slow-changing functions

How-of spatial coordinates [26, 27] When the properties How-of the material vary rapidly withcoordinates, 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 sizelevel 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 cellrepresenting one grain Following this approach, the interaction between the adjacent

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 realgrains normally have arbitrary shapes Yin et al [26] used the stress-strain analysis toevaluate the effective properties along the gradation direction The local interactivebetween the particles was accounted for by using the integrated contributions betweeneach pair of particles to specify the averaged strains throughout the material Anotherstudy that accounted for the local material grading was by Aboudi et al [27], in whichthe constitutive modeling theory based on the higher-order generalized method of cellswas applied and subsequently extended to account for the incremental plasticity, thecreep and the viscoplastic effects [29] The spatially varied thermal conductivity ofthe material at local level was investigated by Zhong and Pindera by reformulating

referred in a study by Yin et al [30], in which the effective thermal conductivity wasderived from the relationship between the gradient of temperature and the heat fluxdistribution

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 whichthe VCFEM was applied to simulate the microstructure of the FGMs Subsequently,Biner [32] and Vena [33] also used the VCFEM in modeling the microstructure ofFGMs The VCFEM was shown to be advantageous in micromechanical modeling ofFGM 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 ement formulation, in which the element is not restricted to 3 or 4-node elements

el-as with the conventional FEM but convex polygons with any number of edges, in

model The Voronoi cell finite element method was first introduced by Ghosh andMallett [2] based on an extension of the hybrid finite element method proposed byPian and his colleagues [34–36] Since then, it was quickly developed by Ghosh andhis colleague as an efficient method for 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 geneities (fiber, void, crack, etc.) with ease by including each heterogeneity insideeach “cell”, or “Voronoi element” [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 forregular heterogeneities from analytical micromechanics is also incorporated in to eachelement to enhance the convergence of the VCFEM Since the required mesh for theVCFEM no longer needs to conform the heterogeneities, the complexity or degree

hetero-of freedoms hetero-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 includethe works of Ghosh and Mukhopadhyay [43] and Ghosh and Moorthy [44] Voronoimesh generation, along with error estimation, still remain as some of the opened andinteresting topics in the development of VCFEM

Special attentions were also paid by various VCFEM investigators to enhancethe convergence of the VCFEM by enhancing the approximation field inside eachheterogeneity embedded Voronoi cell Li et al [45] enhanced the stress approximation

in Voronoi cell containing cracks with branch functions and multi-resolution waveletfunctions in the vicinity of crack tips Recently, Tiwary [46] developed a nonconformal

mapping and wavelet based functions to further enrich the stress approximation nearhighly irregular heterogeneities such as crack and sharp inclusions The new stresscorrection accounted for the shape effect of the inclusions on the stress field so thatthe 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 Ghoshand 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 mechanicalproblem and the porous structure, etc Their works showed the advantages of Voronoicell finite element method for analyzing composite material at micro level Li et al [45]developed an Extended VCFEM to analyze cohesive crack propagation in brittlematerials Hu et al [47] developed a locally enhanced VCFEM to investigate crackpropagation 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 modelmicrostructures with ellipsoidal heterogeneities [44]

Some researchers applied this method to study the microstructure of FGMs One

of the first studies was the work of Grujicic and Zhang [31] They used the Voronoi cellfinite element method to investigate the micromechanical response of FGM and thendetermine the effective material properties Their work considered the elastic response

of FGM only The thermal mechanical problem was later studied by Biner [32] andVena [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 Sinceeach element can contain an inclusion, the material distribution can be modeled easily

by arranging the inclusions in space

1.3.3 Review of the Extended Finite Element Method

The Extended Finite Element Method (XFEM) is a well known method that is ful in modeling structures with discontinuities and/or singularities such as cracks,dislocations, and phase boundaries Perhaps the most important advantage of thismethod 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 theconventional FEM [3, 49] The XFEM improves the versatility of the conventionalFEM by introducing a local enrichment of the approximating space [50, 51]

use-The early research of XFEM was on fracture mechanics use-The concept of the cal partition of unity (PUM) was first introduced by Babuska [52] The XFEM wasfirst introduced based on the PUM concept to study the stress intensity factor ofvarious cracks settings and crack propagation (without using level set) by Black andBelytschko [3] The method was further refined in Dolbow et al [53] and became

lo-a stlo-andlo-ard methodology of investiglo-ating crlo-ack models without explicitly model thecracks 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] Arbitrarycurved 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 bysuperposed the Heaviside function at the junctions of the cracks Solution near thecrack 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 alsoused 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 thefracture in 3D composite materials

Discontinuities in the XFEM were modeled using the level sets method, whichnow becomes a key ingredient in the XFEM The level sets method was advantageous

in describing the discontinuities implicitly instead of explicitly mathematical curves

or surfaces [63, 64] Complicated discontinuity shapes or geometries such as cracks orcurved interfaces 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 phasechange 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 problemswith complex geometries such as multiple intersecting and branching cracks can befound 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 ofattentions recently Sukumar et al [61] and Chessa et al [71] showed that the XFEMconvergence rate was degraded by the approximation in the so-called “blending ele-ments” in the transition of the enriched elements and normal finite elements Muchattentions 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 discontinuousGalerkin and Fries [73] by smoothing the enrichment in the blending elements by aweight function

Another important issue in the implementation of the XFEM was the tation of the quadrature in the weak form, such as the stiffness matrix of enrichedelements For enriched elements, since the approximation space is enriched and theapproximation functions are no longer polynomials, special attentions must be paid

compu-to calculate the quadrature accurately Notable works on this subject included theuse of quadrature on subdivided elements [3, 49], the polar mapping of the integrandfunctions and domains by Laborde et al [74] and Bechet et al [75], and the fastmethod recently proposed by Ventura et al [76]

cie et al studied dislocations in references [77–79] The dislocations in the XFEMwere treated as discontinuities which were very similarly to crack models The methodallowed treatments for complex micromechanics models such as dislocations in car-bon nanotubes and thin films recently [77–81].The phase boundaries were studied byChessa and Belytschko [82–84].

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 ofthe inclusions would be added to the stress approximation to increase the solutionaccuracy 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 Inthis study, the quadrilateral meshes would be investigated as an alternative for theVoronoi diagram

The XFEM thereafter would be applied in examples similar to the VCFEM forfacilitating the comparative studies Moreover, the porous structure would also bestudied 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 bonewould be studied to show the advantages of the XFEM In this study, the level setmethod would be applied so that the XFEM model could be built directly from themicroimage of the bone

1.5 Thesis Outline

In this thesis, the applications of the conventional FEM and its variations in modelingheterogeneous materials 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 thermalload 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 heterogeneousmaterials In this chapter, various numerical examples from simple to complicatedcompositions of heterogeneous materials containing inclusions are studied In someexamples, the quadratic quadrilateral elements are introduced as the 8-node elementsfor the VCFEM instead of the Voronoi cells Chapter 5 shows the application of theXFEM for heterogeneous materials, including porous structures Instead of using theHeaviside enrichment function for the strong discontinuity of the holes’ interfaces,the penalty method is introduced to simulate the porous parts Finally, the thesis isconcluded in Chapter 6 with the summary and the suggestions for future work

Chapter 2

Thermal induced vibration of

functionally graded thin plate

The transient thermomechanics of functionally graded materials (FGM) was widelyinvestigated One of the first reported works was by Fuchiyama and Noda [4], in whichthe temperature dependent material properties were accounted Vel and Batra [5] alsodealt with the transient thermal stresses in FG plates They developed an analyticalsolution for three-dimensional plates using power law series method

The vibration characteristic was studied by some other authors [11, 12] Yangand Shen [13] analyzed the dynamic response of initially stressed functionally gradedrectangular thin plates subjected to partially-distributed impulsive lateral loads with

or without an elastic foundation In the reported references [14–19], the authorsincluded the thermal load beside the mechanical load while investigating the vibration

of FG plates Praveen and Reddy [14] and Reddy [15] analyzed the response ofthin 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 independent.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 Thethermal environment was implemented after a steady state heat transfer analysis.The problems were nonlinear due to the thermal terms in the governing equations.However, none of the above reports studied the thermal induced vibration of

FG plates FGMs are frequently applied in aircraft structures that usually work insevere temperature environments and the heat sources change quite often from time

to time A significant variation of temperature in a solid may cause deformation due

to thermal expansion or contraction Depending on the constraint of the solid, itcan be bended, elongated or subjected to thermal stresses If the temperature variesrapidly, vibration may also occur, which may affect the dynamics and the stability ofthe structures Tauchert [85], Chang et al [86], Chen and Lee [87], Tran et al [88]presented their studies on thermally induced vibration of thin composite plates

In this chapter, the study of thermally induced vibration of a functionally graded(FG) thin plate is presented The plate was subjected to an external thermal load,which was either a heat flux or a temperature rise at the top surface The finiteelement package ABAQUS/Standard was used for the simulation

2.2.1 Material properties

At macro-scale, the effective moduli of a composite material are normally evaluatedbased on the volume fraction of the constituents FGM is a special composite thusthe schemes that are applied to conventional composites can be applied to FGM aswell The rule of mixture was adopted here for estimating the effective moduli of

where P stand for the material moduli such as Young’s modulus, Poisson’s ratio, massdensity, etc The subscripts T and B refer to the top and the bottom surfaces of theplates V stands for the volume fraction of the ceramic constituent of the FGM Asstated earlier, the volume fraction of the constituent of an FGM can vary smoothly as

a function of position through certain directions For the study of thermally inducedvibration of rectangular thin plates here, this variation was assumed through thethickness of the plate and the distribution of the constituents was assumed to follow

a simple power law as:

2h

(2.2)

In Equation (2.2), V is the volume fraction of the ceramic, h is the total thickness

of the plate, z is the coordinate along the thickness of the plate and the power n

is the parameter that decides the material profile of that FG plate Following thispower law, the plate was ceramic rich at the top surface and metal rich at the bottomsurface

This position dependent property was implemented into ABAQUS/Standard via afield variable The value of this field variable was determined at each integration pointand equaled to the third coordinate of that point Finally, the material properties weredefined as field dependent By this way, the material properties would be represented

as a function of position along the thickness of the plate

Moreover, the temperature dependent properties of materials was accounted here

as well There is an option in ABAQUS/Standard, which allows us to define thematerial properties dependent on the temperature field According to Touloukian [89],

2.2.2 The finite element model

The considered plate was a square plate with side a and thickness h The platewas simply supported at four edges and subjected to either a thermal load or amechanical load or both The thin plate was modeled in ABAQUS/Standard by thinshell elements

For an uncoupled thermomechanical problem, the simulation process was dividedinto two stages The first one is the heat transfer analysis It was a steady state or

a transient step depending on the specific problem The thermal load was applied atthis step to get the temperature field over the plate The element type DS8, which

is a quadratic 8-node heat transfer shell element was used The temperature field

of the plate at each different time instance would then be implemented to the nextstep, which was a mechanical analysis to calculate the historical displacements Theelement type S8R, which is a quadratic 8-node shell element, was used in this secondstep The S8R element accounts for the shear effect of the thin plate

2.3 Results and discussion

2.3.1 Benchmarking of simulation results for vibration of FG

plates

Before analysing the thermally induced vibration of the thin plate, some exampleswere conducted first and compared with published works to assure that the proposedprocess was applicable to FG plates Three benchmarks were investigated One

of them studied the vibration characteristic of a simply supported FG thin plateand compared with the work reported by He et al [12] Another benchmark alsostudied the vibration characteristic of a simply supported FG thin plate but in thermalenvironment This benchmark was compared with the report by Yang and Shen [17].The final benchmark analysed the dynamic response of an FG plate under mechanicalload and in thermal environment It was compared with the report by Praveen andReddy [14]

Example 1 The vibration characteristic of a thin square FG plate of side a =0.4m, thickness h = 0.005m was studied The plate was made of two constituent

thickness of the plate followed Equation (2.2), but V was the volume fraction of themetal instead of the ceramic The material properties of the constituents as provided

A 20 × 20 mesh was used for this example A linear perturbation analysis step wasused to extract the natural frequencies of the plate The first ten natural frequencies(Hz) were considered The results for various volume fraction index n are presented

in Table 2.1 It shows that the results from this study is comparable to the indicatedreference The differences are less than 10%

Table 2.1: Natural frequencies (Hz) of FG plates

Table 2.2: Material properties

Example 2 This example also studied the vibration characteristic of an FG thin

the ceramic and stainless steel SUS304 as the metal In this example the plate wasceramic-rich at the top surface and metal-rich at the bottom surface The volumefraction power index in Equation 2.2 was chosen to be 2 The material properties

in Table 2.2 The mass density and the thermal conductivity were assumed to be

The plate had the side a = 0.2m and the thickness h = 0.02m The plate wasclamped at all edges and the whole plate was subjected to a uniform temperature rise

A 20 × 20 mesh and a linear perturbation step were also used in this example.Here the natural frequencies of the plate under different thermal conditions werenormalized to compare with the reference [17] as the following:

steel SUS304 plate of the same dimension with the current investigated plate at stress

Table 2.3: Nondimentional frequencies of the FG plate in thermal environment

2.3.2 Thermal induced vibration of FG plates

The examples presented in section 2.3.1 show that the proposed method was cable for analyzing the vibration of FG plates in thermal environment Thus it wasalso applicable for studying the thermally induced vibration of FG plates as well, inwhich the plates were excited due to an external heat source only The heat sourcecould be a suddenly applied surface heat flux or a sudden temperature rise

appli-In the following studies of thermal induced vibration of FG plates, the plate hadthe side a = 0.2m and the thickness h = 0.001m (Figure 2-1) The constituent

Figure 2-1: FGM thin platevolume fraction of the ceramic varied as per Equation 2.2 The effect of temperaturefield on the material properties was accounted and follow Equation 2.3 The coefficient

The FG plate subjected to a suddenly applied heat flux was studied first The fluxwas uniformly distributed over the top surface of the plate and had the magnitude of

and the plate was assumed to be stress free at this temperature

Figure 2-2 presents the temperature profile through the thickness of the FG platewith various volume fractions and at different times It shows that the temperatureprofile varied with the material profiles The temperature increased with the decrease

of n or the increase of the proportion of the ceramic This is because the thermalconductivity of metal is higher than that of ceramic, thus the heat flux was conductedfaster from the top to the bottom of the plate in the metal rich environment than inthe ceramic environment Hence the volume fraction of the constituent materials can

be designed to achieve the desired temperature profile through the thickness of the

FG plates

The transient central displacement of the FG plate under this thermal condition ispresented in Figure 2-3 The volume fraction of the constituent materials was changedand the results were put together in Figure 2-3 It shows that the vibration amplitude

n=1 n=0.5 n=0.2

Temperature

n=1 n=0.5 n=0.2

(b) t = 0.006s

n=1 n=0.5 n=0.2

(c) t = 0.01s

to suddenly applied heat flux q = 106W/m2

of the metal plate is the highest and the ceramic plate is the lowest Although thetemperature decreased, the deflection still increased with the increase of the volumefraction of the metal It is because the coefficient of thermal expansion of metal ishigher than that of ceramic Also from Figure 2-3, while the amplitude increased, thevibration frequency decreased with the increase of the metal proportion The reason

is that the density of metal is higher than that of ceramic Therefore, the materialproportion in an FG plate can be designed to control the vibration under an externalheat flux

The vibration of the FG plate due to a sudden increase of the temperature fieldwas also examined The temperature on the top surface of the plate was suddenly

dis-placement of the plate is presented in Figure 2-4 Similar to the case of heat flux

0 0.002 0.004 0.006 0.008 0.01

−2 0 2 4 6 8 10 12

Time (s)

Ceramic n=0.2 n=0.5 n=1 n=2 Metal

to a suddenly applied heat flux q = 106W/m2

distribution, the metal plate also had the highest amplitude of vibration and the ramic had the lowest The frequency of the plates also decreased with the increase ofthe volume index n or the increase of the metal proportion

ce-A study on temperature variations showed that the temperature field affected thevibration amplitude as well as the vibration frequency of the FG plate Figure 2-5shows a comparison of the central displacement of the plate under different temper-

dynamic solution and the quasi-static solution were compared to see the effect ofthe inertia on the movement of the FG plate under thermal load, which is shown

in Figure 2-6 The quasi-static solution also captured the transient response of theplate subjected to external thermal load but it neglected the inertia effect Figure 2-6shows that there is a difference between the dynamic solution and the quasi-staticsolution Hence it is necessary to include the inertia force in analyzing the thermallyinduced vibration of an FG plate as this difference would be significant when the

0 1 2 3 4 5

−1 0 1 2 3 4

Time (s)

Ceramic n=0.2 n=0.5 n=1 n=2 Metal

to a suddenly temperature rise 4Ttop = 200oK

0 1 2 3 4 5

−1 0 1 2 3 4

under different temperature rises

0 0.002 0.004 0.006 0.008 0.01

−2 0 2 4 6 8 10

12x 10

Time (s)

n=2 n=0.2

(a) suddenly applied heat flux

x 10−3

−0.5 0 0.5 1 1.5 2 2.5 3

(b) suddenly applied temperature rise

compar-ison between the dynamic solution and the quasi-static solution

The finite element package ABAQUS has been shown to be a useful tool to

gradual change of the material properties through the thickness of the plate via thefield dependent variable at the integration points However, for thin plate elementsthe maximum number of integration points through the thickness of the plate is 21.Hence, for the plate with a complicated function of material distribution that needsmore than 21 interpolation points, ABAQUS cannot work Then it is necessary touse other methods in this case