Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis

Smart structures that contain embedded piezoelectric patches are loaded by both mechanical and electrical fields. Traditional plate and shell theories were developed to analyze structures subject to mechanical loads. However, these often fail when tasked with the evaluation of both electrical and mechanical fields and loads. In recent years more advanced models have been developed that overcome these limitations. Plates and Shells for Smart Structures offers a complete guide and reference to smart structures under both mechanical and electrical loads, starting with the basic principles and working right up to the most advanced models. It provides an overview of classical plate and shell theories for piezoelectric elasticity and demonstrates their limitations in static and dynamic analysis with a number of example problems. This book also provides both analytical and finite element solutions, thus enabling the reader to compare strong and weak solutions to the problems. Key features: compares a large variety of classical and modern approaches to plates and shells, such as Kirchhoff-Love , Reissner-Mindlin assumptions   and higher order, layer-wise and mixed theories introduces theories able to consider electromechanical couplings as well as those that provide appropriate interface continuity conditions for both electrical and mechanical variables considers both static and dynamic analysis accompanied by a companion website hosting dedicated software MUL2 that is used to obtain the numerical solutions in the book, allowing the reader to reproduce the examples given as well as solve problems of their own The models currently used have a wide range of applications in civil, automotive, marine and aerospace engineering. Researchers of smart structures, and structural analysts in industry, will find all they need to know in this concise reference. Graduate and postgraduate students of mechanical, civil and aerospace engineering can also use this book in their studies. www.mul2.com

Table of Contents:
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, , σn) 24 2.5.2 RMVT(u, , Dn) 26 2.5.3 RMVT(u, , σn, 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, , σn) for the electromechanical plate case 122 6.5 RMVT(u, , Dn) for the electromechanical plate case 130 6.6 RMVT(u, , σn, 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, σn) 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, , σn) for the electromechanical shell case 175 7.5 RMVT(u, , Dn) for the electromechanical shell case 181 7.6 RMVT(u, , σn, 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, σn) 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,, σn) for the electromechanical plate case 231 8.5 RMVT(u,,Dn) for the electromechanical plate case 238 8.6 RMVT(u,, σn,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