功能梯度材料结构的解析和半解析解（英文版）

Chapter 1 Introduction
　　This chapter introduces the concept of functionally graded material (FGM) and presents a brief review of recent developments in modeling and analysis of FGM structures.
　　1.1 Functionally Graded Materials
　　Functionally graded material (FGM) is a novel concept in material design and manu- facture in which the volume fractions of material constituents vary continuously with spatial positions. The concept of FGM was first proposed by Japanese scientists in 1984 for preparing thermal bamer coatings [1， 2]. Initially， FGMs were made from a mixture of ceramics and metal， or a combination of different metals. The brittle ceramic constituent of the mixture， with low thermal conductivity， provides high-temperature resistance， while the ductile metal constituent prevents the onset of cracking at stress concentration sites caused by rapid development of a high temperature gradient. The advantage of FGM is that no distinct internal boundaries exist， and failures due to interfacial stress concentrations developed in conventional components can be avoided. Featuring gradual transitions in their microstructure and composition.FGMs are designed to meet functional requirements that vary with the position of a structure component， and to optimize the overall performance of the component. As a new concept in material design， FGMs have found immediate or potential applications in the aerospace， electronics， chemical， optics， and biomedical industries [3， 4].
　　Although the concept of FGM is new in the industrial field， these materials exist extensively in nature. Some examples of natural FGMs are shown in Fig. 1.1. Bones， skin， and the bamboo tree have graded microstructures and display obvious graded material properties. Generally， artificially produced FGMs are composed

《功能梯度材料结构的解析和半解析解（英文版）》主要阐述了功能梯度材料与结构的基本理论及分析方法，包括：功能梯度材料的概念及其相关力学问题的研究进展；功能梯度材料弹性体的基本方程、各种不同的求解技术以及功能梯度材料梁、板、壳结构静、动力问题的解析解或半解析解等。《功能梯度材料结构的解析和半解析解（英文版）》可供致力于非均匀材料结构研究的学者和高等院校研究生参考。

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