Contents
1 Introduction 1
1.1 Functionally Graded Materials 1
1.2 Effective Material Properties of FGMs 2
1.2.1 The Ru1e of Mixtures 3
1.2.2 The Mori-Tanaka Scheme 4
1.2.3 The Self-Consistent Estimate 4
1.2.4 Mathematical Idealization of FGMs 5
1.3 A Review of Recent Research on FGM Structures 5
1.3.1 FGM Beams 5
1.3.2 Rectangu1ar FGM Plates 7
1.3.3 Circu1ar FGM Plates 10
1.3.4 FGM Cylmders 13
1.3.5 FGM Spheres 15
References 16
2 Fnndamentals for an Elastic FGM Body 29
2.1 Three-Dimensional Problems 29
2 1.1 Basic Assumptions 29
2 1.2 Equations in Cartesian Coordinates 30
2 1.3 Equations in Cylindrical Coordinates 36
2 1.4 Equations in Spherical Coordinates 37
2.2 Two-Dimensional Problems 40
2.2.1 Equations in Cartesian Coordinates 40
2.2.2 Equations in Polar Cosidinates 41
2.3 Boundary and Initial Conditions 42
2.3.1 Boundary Conditions 43
2.3.2 Initial Conditions 43
2.4 Mu1ti-field Coupling Media 44
2.5 Variational Principlsifsian FGM Body 45
2.5.1 Basic Equations 45
2.5.2 Principle of VJrtual Work 46
2.5.3 Unconventional Hamilton Variational Principles 47
2.5.4 Unconventional Hamilton Variational Principle in Phase Space 52
2.6 Uniqueness of Elasticity Solutions 53
2.7 Principle of Superposition 54
2.8 Principle of Saint-Venant 56
References 57
3 Governing Equations for Different Solution Schemes 59
3.1 Governing Equations in Terms of Displacements 59
3.1.1 Equations in Cartesian Coordinates 59
3.1.2 Equations in Cylindrical Coordinates 61
3.1.3 Equations in Spherical Coordinates 62
3.2 Governing Equations in Terms of Displacement Functions 63
3.2.1 Three-Dimensional Problem of Transversely Isotropic
3.2.2 Three-Dimensional Problem oflsotropic FGM Body 65
3.2.3 Plane Stress Problem of Orthotropic FGM Body 66
3.2.4 Axisymmetric Problem of Transversely Isotropic
3.2.5 Axisymmetric Problem oflsotropic FGM Body 68
3.3 Governing Equations in Terms of Stress Function 69
3.3.1 Two-Dimensional Problem in Cartesian Coordinates 69
3.3.2 Two-Dimensional Problem in Polar Coordinates 70
3.4 Governing Equations Expressed by State-Space Method 71
3.4.1 Equations in Terms of Mixed Variables in Cartesian Coordinates 71
3.4.2 Equations for Multifield Coupled Problem in Cartesian Coordinates 73
3.4.3 Equations in Terms of Displacements in Cylindrical Coordinates 74
3.4.4 Equations for Multifield Coupling Problem in Cylindrical Coordinates 76
References 78
4 Functionally Graded Beams 79
4.1 Analytical Solution of FGM Beams Using the Displacement Method 79
4.1.1 Formulation 79
4.1.2 Exact Solutions for FGM Beams 81
4.2 Analytical Solutions of FGM Beams Using the Displacement Function Method 83
4.2.1 Formulation 83
4.2.2 Solution 84
4.2.3 Boundary Conditions 87
4.2.4 Isotropic Cases 91
4.2.5 Numerical Examples 94
4.3 Analytical Solutions of FGM Beams Using the Stress Function Method 95
4.3.1 Formulation 95
4.3.2 Solution of Orthotropic FGM Beams 97
4.3.3 Examples of Orthotropic FGM Beams 102
4.4 Elasticity Solutions for FGM Beams Using the State Space Method 107
4.4.1 Formulation 107
4.4.2 Solution 108
4.5 Electroelastic Analysis of FGPM Beams Using the Stress Function Method 110
4.5.1 Formulation 110
4.5.2 Solution 111
4.5.3 Examples 117
References 121
5 Rectangular Functionally Graded Plates 123
5.1 Analytical Solutions oflsotropic Rectangular FGM Plates in Cylindrical Bending 123
5.1.1 Formulation 123
5.1.2 Solution Procedure 124
5.1.3 Numerical Examples and Discussion 129
5.2 Three-Dimensional Elastic Solution oflsotropic Rectangular FGM Plates 133
5.2.1 Formulation 133
5.2.2 Solution ， 134
5.2.3 Examples 140
5.3 Three-Dimensional Elastic Solution of Transversely Isotropic Rectangular FGM Plates 141
5.3.1 Formulation 141
5.3.2 Solution 144
5.3.3 Numerical Examples 150
5.4 Three-Dimensional Numerical Solutions of Orthotropic Rectangular FGM Plates Based on the Haar Wavelet Method 154
5.4.1 Formulation 154
5.4.2 Properties of the Haar Wavelet 155
5.4.3 Solution via the Haar Wavelet Method 157
5.4.4 Numerical Examples and Discussion 159
5.5 Dynamic Analysis of Rectangular FGM Plates Using Isoparametric Graded Elements and Time Subdomain Method 164
5.5.1 Unconventional Hamilton Variational Principle in Phase Space 164
5.5.2 Finite Elements in Symplectic Space 165
5.5.3 Numerical Results and Discussion 169
5.6 Exact Analysis of Simply Supported Rectangular FGPM Plates 172
5.6.1 Formulation 172
5.6.2 Solution 173
5.6.3 Examples 177
5.7 Vibration of Simply Supported Rectangular FGPM Plates 184
5.7.1 Formulation 184
5.7.2 Solution 186
5.7.3 Examples 192
Appendix: Transfer Matrix Calculation 197
References 200
6 Circular Functionally Graded Plates 201
6.1 Exact Solution for Axisymmetric Bending of Iso
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