Symplectic Elasticity

Contents......Page 6
Preface......Page 10
Preface to the Chinese Edition......Page 12
Foreword to the Chinese Edition......Page 16
Nomenclature......Page 20
1.1. Linear Space......Page 24
1.2. Euclidean Space......Page 29
1.3. Symplectic Space......Page 32
1.4. Legengre’s Transformation......Page 49
1.5. The Hamiltonian Principle and the Hamiltonian Canonical Equations......Page 51
1.6.1. The Reciprocal Theorem for Work......Page 53
1.6.3. The Reciprocal Theorem for Reaction......Page 55
1.6.4. The Reciprocal Theorem for Displacement and Negative Reaction......Page 56
References......Page 58
2.1. Stress Analysis......Page 60
2.2. Strain Analysis......Page 64
2.3. Stress-Strain Relations......Page 67
2.4. The Fundamental Equations of Elasticity......Page 71
2.5. The Principle of Virtual Work......Page 74
2.6. The Principle of Minimum Total Potential Energy......Page 75
2.7. The Principle of Minimum Total Complementary Energy......Page 77
2.8. The Hellinger–Reissner Variational Principle with Two Kinds of Variables......Page 78
2.9. The Hu–Washizu Variational Principle with Three Kinds of Variables......Page 80
2.10. The Principle of Superposition and the Uniqueness Theorem......Page 82
References......Page 83
3.1. The Timoshenko Beam Theory......Page 86
3.2. Derivation of Hamiltonian System......Page 91
3.3. The Method of Separation of Variables......Page 94
3.4. Reciprocal Theorem for Work and Adjoint Symplectic Orthogonality......Page 97
3.5. Solution for Non-Homogeneous Equations......Page 101
3.6. Two-Point Boundary Conditions......Page 102
3.7. Static Analysis of Timoshenko Beam......Page 107
3.8. Wave Propagation Analysis of Timoshenko Beam......Page 110
3.9. Wave Induced Resonance......Page 113
References......Page 117
4.1. The Fundamental Equations of Plane Elasticity......Page 120
4.2. Hamiltonian System in Rectangular Domain......Page 124
4.3. Separation of Variables and Transverse Eigen-Problems......Page 129
4.4. Eigen-Solutions of Zero Eigenvalue......Page 132
4.5. Solutions of Saint–Venant Problems for Rectangular Beam......Page 140
4.6. Eigen-Solutions of Nonzero Eigenvalues......Page 146
4.6.1. Eigen-Solutions of Nonzero Eigenvalues of Symmetric Deformation......Page 148
4.6.2. Eigen-Solutions of Nonzero Eigenvalues of Antisymmetric Deformation......Page 151
4.7. Solutions of Generalized Plane Problems in Rectangular Domain......Page 154
References......Page 159
5.1. The Fundamental Equations of Plane Anisotropic Elasticity Problems......Page 162
5.2. Symplectic Solution Methodology for Anisotropic Elasticity Problems......Page 164
5.3. Eigen-Solutions of Zero Eigenvalue......Page 168
5.4. Analytical Solutions of Saint–Venant Problems......Page 173
5.5. Eigen-Solutions of Nonzero Eigenvalues......Page 178
5.6. Introduction to Hamiltonian System for Generalized Plane Problems......Page 181
References......Page 185
6.1. The Fundamental Equations......Page 186
6.2. Derivation of Hamiltonian System......Page 188
6.3. Eigen-Solutions of Zero Eigenvalue......Page 191
6.4. Analytical Solutions of Saint–Venant Problem......Page 198
References......Page 202
7.1. Plane Elasticity Equations in Polar Coordinates......Page 204
7.2. Variational Principle for a Circular Sector......Page 208
7.3. Hamiltonian System with Radial Coordinate Treated as “Time”......Page 210
7.4.1. Eigen-Solutions of Zero Eigenvalue......Page 218
7.4.2. Eigen-Solutions of Nonzero Eigenvalues......Page 222
7.5.1. Eigen-Solutions of Zero Eigenvalue......Page 225
7.5.2. Eigen-Solutions of µ = ±1......Page 228
7.5.3. Eigen-Solutions of General Nonzero Eigenvalues......Page 233
7.6. Hamiltonian System with Circumferential Coordinate Treated as “Time”......Page 236
7.6.1. Eigen-Solutions of Zero Eigenvalue......Page 239
7.6.2. Eigen-Solutions of µ = ±i......Page 242
7.6.3. Eigen-solutions of General Nonzero Eigenvalues......Page 245
References......Page 246
8.1. Small Deflection Theory for Bending of Elastic Thin Plates......Page 248
8.2. Analogy between Plane Elasticity and Bending of Thin Plate......Page 255
8.3. Multi-Variable Variational Principles for Thin Plate Bending and Plane Elasticity......Page 262
8.3.1. Multi-Variable Variational Principles for Plate Bending......Page 263
8.3.2. Multi-Variable Variational Principle for Plane Elasticity.......Page 271
8.4. Symplectic Solution for Rectangular Plates......Page 275
8.5. Plates with Two Opposite Sides Simply Supported......Page 280
8.6. Plates with Two Opposite Sides Free......Page 285
8.7. Plate with Two Opposite Sides Clamped......Page 292
8.8. Bending of Sectorial Plates......Page 297
8.8.1. Derivation of Hamiltonian System......Page 300
8.8.2. Sectorial Plate with Two Opposite Sides Free......Page 303
References......Page 311
About the Authors......Page 314