About the Authors ix
Preface xi
1 Introduction 1
1.1 Direct and inverse piezoelectric effects 2
1.2 Some known applications of smart structures 3
References 6
2 Basics of piezoelectricity and related principles 9
2.1 Piezoelectric materials 9
2.2 Constitutive equations for piezoelectric problems 14
2.3 Geometrical relations for piezoelectric problems 18
2.4 Principle of virtual displacements 20
2.4.1 PVD for the pure mechanical case 23
2.5 Reissner mixed variational theorem 23
2.5.1 RMVT(u, , ?n) 24
2.5.2 RMVT(u, , Dn) 26
2.5.3 RMVT(u, , ?n, Dn) 28
References 30
3 Classical plate/shell theories 33
3.1 Plate/shell theories 33
3.1.1 Three-dimensional problems 34
3.1.2 Two-dimensional approaches 34
3.2 Complicating effects of layered structures 37
3.2.1 In-plane anisotropy 38
3.2.2 Transverse anisotropy, zigzag effects, and interlaminar
continuity 38
3.3 Classical theories 41
3.3.1 Classical lamination theory 41
3.3.2 First-order shear deformation theory 42
3.3.3 Vlasov?Reddy theory 45
3.4 Classical plate theories extended to smart structures 45
3.4.1 CLT plate theory extended to smart structures 45
3.4.2 FSDT plate theory extended to smart structures 56
3.5 Classical shell theories extended to smart structures 58
3.5.1 CLT and FSDT shell theories extended to smart structures
59
References 60
4 Finite element applications 63
4.1 Preliminaries 63
4.2 Finite element discretization 64
4.3 FSDT finite element plate theory extended to smart
structures 68
References 87
5 Numerical evaluation of classical theories and their
limitations 89
5.1 Static analysis of piezoelectric plates 90
5.2 Static analysis of piezoelectric shells 92
5.3 Vibration analysis of piezoelectric plates 98
5.4 Vibration analysis of piezoelectric shells 101
References 104
6 Refined and advanced theories for plates 105
6.1 Unified formulation: refined models 105
6.1.1 ESL theories 106
6.1.2 Murakami zigzag function 108
6.1.3 LW theories 110
6.1.4 Refined models for the electromechanical case 113
6.2 Unified formulation: advanced mixed models 113
6.2.1 Transverse shear/normal stress modeling 113
6.2.2 Advanced mixed models for the electromechanical case
115
6.3 PVD(u, ) for the electromechanical plate case 117
6.4 RMVT(u, , ?n) for the electromechanical
plate case 122
6.5 RMVT(u, , Dn) for the electromechanical plate
case 130
6.6 RMVT(u, , ?n, Dn) for the
electromechanical plate case 137
6.7 Assembly procedure for fundamental nuclei 148
6.8 Acronyms for refined and advanced models 150
6.9 Pure mechanical problems as particular cases, PVD(u)
and RMVT(u, ?n) 151
6.10 Classical plate theories as particular cases of unified
formulation 153
References 154
7 Refined and advanced theories for shells 157
7.1 Unified formulation: refined models 157
7.1.1 ESL theories 158
7.1.2 Murakami zigzag function 160
7.1.3 LW theories 162
7.1.4 Refined models for the electromechanical case 165
7.2 Unified formulation: advanced mixed models 165
7.2.1 Transverse shear/normal stress modeling 166
7.2.2 Advanced mixed models for the electromechanical case
168
7.3 PVD(u, ) for the electromechanical shell case 169
7.4 RMVT(u, , ?n) for the electromechanical
shell case 175
7.5 RMVT(u, , Dn) for the electromechanical shell
case 181
7.6 RMVT(u, , ?n, Dn) for the
electromechanical shell case 188
7.7 Assembly procedure for fundamental nuclei 197
7.8 Acronyms for refined and advanced models 200
7.9 Pure mechanical problems as particular cases, PVD(u)
and RMVT(u, ?n) 200
7.10 Classical shell theories as particular cases of unified
formulation 202
7.11 Geometry of shells 202
7.11.1 First quadratic form 204
7.11.2 Second quadratic form 204
7.11.3 Strain?displacement equations 205
7.12 Plate models as particular cases of shell models 208
References 210
8 Refined and advanced finite elements for plates 213
8.1 Unified formulation: refined models 213
8.1.1 ESL theories 215
8.1.2 Murakami zigzag function 217
8.1.3 LW theories 219
8.1.4 Refined models for the electromechanical case 222
8.2 Unified formulation: advanced mixed models 222
8.2.1 Transverse shear/normal stress modeling 223
8.2.2 Advanced mixed models for the electromechanical case
225
8.3 PVD(u,) for the electromechanical plate case 226
8.4 RMVT(u,, ?n) for the electromechanical plate
case 231
8.5 RMVT(u,,Dn) for the electromechanical plate case
238
8.6 RMVT(u,, ?n,Dn) for the electromechanical plate
case 244
8.7 FE assembly procedure and concluding remarks 252
References 252
9 Numerical evaluation and assessment of classical and advanced
theories using MUL2 software 255
9.1 The MUL2 software for plates and shells: analytical
closed-form solutions 256
9.1.1 Classical plate/shell theories as particular cases in the
MUL2 software 264
9.2 The MUL2 software for plates: FE solutions 269
9.3 Analytical closed-form solution for the electromechanical
analysis of plates 276
9.4 Analytical closed-form solution for the electromechanical
analysis of shells 283
9.5 FE solution for electromechanical analysis of beams 290
9.6 FE solution for electromechanical analysis of plates 296
References 302
Index 303
