Symplectic elasticity

Solid mechanics problems have long been regarded as bottlenecks in the development of elasticity. In contrast to traditional solution methodologies, such as Timoshenko's theory of elasticity for which the main technique is the semi-inverse method, this book presents a new approach based on the Hamiltonian principle and the symplectic duality system where solutions are derived in a rational manner in the symplectic space. Departing from the conventional Euclidean space with one kind of variable, the symplectic space with dual variables thus provides a fundamental breakthrough. This book explains the new solution methodology by discussing plane isotropic elasticity, multiple layered plate, anisotropic elasticity, sectorial plate and thin plate bending problems in some detail. A number of existing problems without analytical solutions within the framework of classical approaches are solved analytically using this symplectic approach. Symplectic methodologies can be applied not only to problems in elasticity, but also to other solid mechanics problems. In addition, it can also be extended to various engineering mechanics and mathematical physics fields, such as vibration, wave propagation, control theory, electromagnetism and quantum mechanics.

Preface	p. ix
Preface to the Chinese Edition	p. xi
Foreword to the Chinese Edition	p. xv
Nomenclature	p. xix
1 Mathematical Preliminaries	p. 1
1.1 Linear Space	p. 1
1.2 Euclidean Space	p. 6
1.3 Symplectic Space	p. 9
1.4 Legengre's Transformation	p. 26
1.5 The Hamiltonian Principle and the Hamiltonian Canonical Equations	p. 28
1.6 The Reciprocal Theorems	p. 30
1.6.1 The Reciprocal Theorem for Work	p. 30
1.6.2 The Reciprocal Theorem for Displacement	p. 32
1.6.3 The Reciprocal Theorem for Reaction	p. 32
1.6.4 The Reciprocal Theorem for Displacement and Negative Reaction	p. 33
References	p. 35
2 Fundamental Equations of Elasticity and Variational Principle	p. 37
2.1 Stress Analysis	p. 37
2.2 Strain Analysis	p. 41
2.3 Stress-Strain Relations	p. 44
2.4 The Fundamental Equations of Elasticity	p. 48
2.5 The Principle of Virtual Work	p. 51
2.6 The Principle of Minimum Total Potential Energy	p. 52
2.7 The Principle of Minimum Total Complementary Energy	p. 54
2.8 The Hellinger-Reissner Variational Principle with Two Kinds of Variables	p. 55
2.9 The Hu-Washizu Variational Principle with Three Kinds of Variables	p. 57
2.10 The Principle of Superposition and the Uniqueness Theorem	p. 59
2.11 Saint-Venant Principle	p. 60
References	p. 60
3 The Timoshenko Beam Theory and Its Extension	p. 63
3.1 The Timoshenko Beam Theory	p. 63
3.2 Derivation of Hamiltonian System	p. 68
3.3 The Method of Separation of Variables	p. 71
3.4 Reciprocal Theorem for Work and Adjoint Symplectic Orthogonality	p. 74
3.5 Solution for Non-Homogeneous Equations	p. 78
3.6 Two-Point Boundary Conditions	p. 79
3.7 Static Analysis of Timoshenko Beam	p. 84
3.8 Wave Propagation Analysis of Timoshenko Beam	p. 87
3.9 Wave Induced Resonance	p. 90
References	p. 94
4 Plane Elasticity in Rectangular Coordinates	p. 97
4.1 The Fundamental Equations of Plane Elasticity	p. 97
4.2 Hamiltonian System in Rectangular Domain	p. 101
4.3 Separation of Variables and Transverse Eigen-Problems	p. 106
4.4 Eigen-Solutions of Zero Eigenvalue	p. 109
4.5 Solutions of Saint-Venant Problems for Rectangular Beam	p. 117
4.6 Eigen-Solutions of Nonzero Eigenvalues	p. 123
4.6.1 Eigen-Solutions of Nonzero Eigenvalues of Symmetric Deformation	p. 125
4.6.2 Eigen-Solutions of Nonzero Eigenvalues of Antisymmetric Deformation	p. 128
4.7 Solutions of Generalized Plane Problems in Rectangular Domain	p. 131
References	p. 136
5 Plane Anisotropic Elasticity Problems	p. 139
5.1 The Fundamental Equations of Plane Anisotropic Elasticity Problems	p. 139
5.2 Symplectic Solution Methodology for Anisotropic Elasticity Problems	p. 141
5.3 Eigen-Solutions of Zero Eigenvalue	p. 145
5.4 Analytical Solutions of Saint-Venant Problems	p. 150
5.5 Eigen-Solutions of Nonzero Eigenvalues	p. 155
5.6 Introduction to Hamiltonian System for Generalized Plane Problems	p. 158
References	p. 162
6 Saint-Venant Problems for Laminated Composite Plates	p. 163
6.1 The Fundamental Equations	p. 163
6.2 Derivation of Hamiltonian System	p. 165
6.3 Eigen-Solutions of Zero Eigenvalue	p. 168
6.4 Analytical Solutions of Saint-Venant Problem	p. 175
References	p. 179
7 Solutions for Plane Elasticity in Polar Coordinates	p. 181
7.1 Plane Elasticity Equations in Polar Coordinates	p. 181
7.2 Variational Principle for a Circular Sector	p. 185
7.3 Hamiltonian System with Radial Coordinate Treated as "Time"	p. 187
7.4 Eigen-Solutions for Symmetric Deformation in Radial Hamiltonian System	p. 195
7.4.1 Eigen-Solutions of Zero Eigenvalue	p. 195
7.4.2 Eigen-Solutions of Nonzero Eigenvalues	p. 199
7.5 Eigen-Solutions for Anti-Symmetric Deformation in Radial Hamiltonian System	p. 202
7.5.1 Eigen-Solutions of Zero Eigenvalue	p. 202
7.5.2 Eigen-Solutions of ¿ = $$1	p. 205
7.5.3 Eigen-Solutions of General Nonzero Eigenvalues	p. 210
7.6 Hamiltonian System with Circumferential Coordinate Treated as "Time"	p. 213
7.6.1 Eigen-Solutions of Zero Eigenvalue	p. 216
7.6.2 Eigen-Solutions of ¿ = $$i	p. 219
7.6.3 Eigen-solutions of General Nonzero Eigenvalues	p. 222
References	p. 223
8 Hamiltonian System for Bending of Thin Plates	p. 225
8.1 Small Deflection Theory for Bending of Elastic Thin Plates	p. 225
8.2 Analogy between Plane Elasticity and Bending of Thin Plate	p. 232
8.3 Multi-Variable Variational Principles for Thin Plate Bending and Plane Elasticity	p. 239
8.3.1 Multi-Variable Variational Principles for Plate Bending	p. 240
8.3.2 Multi-Variable Variational Principle for Plane Elasticity	p. 248
8.4 Symplectic Solution for Rectangular Plates	p. 252
8.5 Plates with Two Opposite Sides Simply Supported	p. 257
8.6 Plates with Two Opposite Sides Free	p. 262
8.7 Plate with Two Opposite Sides Clamped	p. 269
8.8 Bending of Sectorial Plates	p. 274
8.8.1 Derivation of Hamiltonian System	p. 277
8.8.2 Sectorial Plate with Two Opposite Sides Free	p. 280
References	p. 288
About the Authors	p. 291

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