Chapter 1 Introduction to Piezoelectricity 1
1.1 Background 1
1.2 Linear theory of piezoelectricity 4
1.2.1 Basic equations in rectangular coordinate system 4
1.2.2 Boundary conditions 7
1.3 Functionally graded piezoelectric materials 8
1.3.1 Types of gradation 9
1.3.2 Basic equations for two-dimensional FGPMs 9
1.4 Fibrous piezoelectric composites 11
References 17
Chapter 2 Solution Methods 21
2.1 Potential function method 21
2.2 Solution with Lekhnitskii formalism 23
2.3 Techniques of Fourier transformation 28
2.4 Trefftz finite element method 31
2.4.1 Basic equations 31
2.4.2 Assumed fields 31
2.4.3 Element stiffness equation 33
2.5 Integral equations 34
2.5.1 Fredholm integral equations 34
2.5.2 Volterra integral equations 36
2.5.3 Abel's integral equation 37
2.6 Shear-lag model 39
2.7 Hamiltonian method and symplectic mechanics 41
2.8 State space formulation 47
References 51
Chapter 3 Fibrous Piezoelectric Composites 53
3.1 Introduction 53
3.2 Basic formulations for fiber push-out and pull-out tests 55
3.3 Piezoelectric fiber pull-out 59
3.3.1 Relationships between matrix stresses and interfacial shear stress 59
3.3.2 Solution for bonded region 61
3.3.3 Solution for debonded region 62
3.3.4 Numerical results 63
3.4 Piezoelectric fiber push-out 63
3.4.1 Stress transfer in the bonded region 64
3.4.2 Frictional sliding 66
3.4.3 PFC push-out driven by electrical and mechanical loading 69
3.4.4 Numerical assessment 70
3.5 Interfacial debonding criterion 76
3.6 Micromechanics of fibrous piezoelectric composites 81
3.6.1 Overall elastoelectric properties of FPCs 81
3.6.2 Extension to include magnetic and thermal effects 89
3.7 Solution of composite with elliptic fiber 94
3.7.1 Conformal mapping 94
3.7.2 Solutions for thermal loading applied outside an elliptic fiber 95
3.7.3 Solutions for holes and rigid fibers 104
References 105
Chapter 4 Trefftz Method for Piezoelectricity 109
4.1 Introduction 109
4.2 Trefftz FEM for generalized plane problems 109
4.2.1 Basic field equations and boundary conditions 109
4.2.2 Assumed fields 111
4.2.3 Modified variational principle 113
4.2.4 Generation of the element stiffness equation 115
4.2.5 Numerical results 117
4.3 Trefftz FEM for anti-plane problems 118
4.3.1 Basic equations for deriving Trefftz FEM 118
4.3.2 Trefftz functions 119
4.3.3 Assumed fields 119
4.3.4 Special element containing a singular corner 121
4.3.5 Generation of element matrix 123
4.3.6 Numerical examples 125
4.4 Trefftz boundary element method for anti-plane problems 127
4.4.1 Indirect formulation 127
4.4.2 The point-collocation formulations of Trefftz boundary element method 129
4.4.3 Direct formulation 129
4.4.4 Numefical examples 132
4.5 Trefftz boundary-collocation method for plane piezoelectricity 137
4.5.1 General Trefftz solution sets 137
4.5.2 Special Trefftz solution set for a problem with elliptic holes 138
4.5.3 Special Trefftz solution set for impermeable crack problems 140
4.5.4 Special Trefftz solution set for permeable crack problems 142
4.5.5 Boundary collocation formulation 144
References 145
Chapter 5 Symplectic Solutions for Piezoelectric Materials 149
5.1 Introduction 149
5.2 A symplectic solution for piezoelectric wedges 150
5.2.1 Hamiltonian system by differential equation approach 150
5.2.2 Hamiltonian system by variational principle approach 153
5.2.3 Basic eigenvalues and singularity of stress and electric fields 154
5.2.4 Piezoelectric bimaterial wedge 159
5.2.5 Multi-piezoelectric material wedge 162
5.3 Extension to include magnetic effect 166
5.3.1 Basic equations and their Hamiltonian system 166
5.3.2 Eigenvalues and eigenfunctions 167
5.3.3 Particular solutions 170
5.4 Symplectic solution for a magnetoelectroelastic strip 171
5.4.1 Basic equations 171
5.4.2 Hamiltonian principle 172
5.4.3 The zero-eigenvalue solutions 175
5.4.4 Nonzero-eigenvalue solutions 179
5.5 Three-dimensional symplectic formulation for piezoelectricity 182
5.5.1 Basic formulations 182
5.5.2 Hamiltonian dual equations 183
5.5.3 The zero-eigenvalue solutions 184
5.5.4 Sub-symplectic system 187
5.5.5 Nonzero-eigenvalue solutions 190
5.6 Symplectic solution for FGPMs 192
5.6.1 Basic formulations 192
5.6.2 Eigenvalue properties of the Hamiltonian matrix H 194
5.6.3 Eigensolutions corresponding to μ＝0 and-α 194
5.6.4 Extension to the case of magnetoelectroelastic materials 197
References 201
Chapter 6 Saint-Venant Decay Problems in Piezoelectricity 205
6.1 Introduction 205
6.2 Saint-Venant end effects of piezoelectric strips 206
6.2.1 Hamiltonian system for a piezoelectric strip 206
6.2.2 Decay rate analysis 211
6.2.3 Numerical illustration 216
6.3 Saint-Venant decay in anti-plane dissimilar laminates 218
6.3.1 Basic equations for anti-plane piezoelectric problem 218
6.3.2 Mixed-variable state space formulation 219
6.3.3 Decay rate of FGPM strip 220
6.3.4 Two-layered FGPM laminates and dissimilar piezoelectric laminates 226
6.4 Saint-Venant decay in multilayered piezoelectric laminates 231
6.4.1 State space formulation 231
6.4.2 Eigensolution and decay rate equation 234
6.5 Decay rate of piezoelectric-piezomagnetic sandwich structures 237
6.5.1 Basic equations and notations in multilayered structures 238
6.5.2 Space state differential equations for analyzing decay rate 239
6.5.3 Solutions to the space state differential equations 241
References 246
Chapter 7 Penny-Shaped Cracks 249
7.1 Introduction 249
7.2 An infinite piezoelectric material with a penny-shaped crack 250
7.3 A penny-shaped crack in a piezoelectric strip 255
7.4 A fiber with a penny-shaped crack embedded in a matrix 258
7.5 Fundamental solution for penny-shaped crack problem 263
7.5.1 Potential approach 263
7.5.2 Solution for crack problem 265
7.5.3 Fundamental solution for penny-shaped crack problem 266
7.6 A penny-shaped crack in a piezoelectric cylinder 268
7.6.1 Problem statement and basic equation 269
7.6.2 Derivation of integral equations and their solution 272
7.6.3 Numerical results and discussion 277
7.7 A fiber with a penny-shaped crack and an elastic coating 279
7.7.1 Formulation of the problem 279
7.7.2 Fredholm integral equation of the problem 285
7.7.3 Numerical results and discussion 286
References 287
Chapter 8 Solution Methods for Functionally Graded Piezoelectric Materials 291
8.1 Introduction 291
8.2 Singularity analysis of angularly graded piezoelectric wedge 292
8.2.1 Basic formulations and the state space equation 292
8.2.2 Two AGPM wedges 298
8.2.3 AGPM-EM-AGPM wedge system 301
8.2.4 Numerical results and discussion 303
8.3 Solution to FGPM beams 308
8.3.1 Basic formulation 308
8.3.2 Solution procedure 308
8.4 Parallel cracks in an FGPM strip 312
8.4.1 Basic formulation 313
8.4.2 Singular integral equations and field intensity factors 315
8.5 Mode Ⅲ cracks in two bonded FGPMs 318
8.5.1 Basic formulation of the problem 318
8.5.2 Impermeable crack problem 320
8.5.3 Permeable crack problem 324
References 325
Index 329