Symplectic Elasticity

by Weian Yao, Wanxie Zhong, Chee Wah Lim

Overview

Exact analytical solutions in some areas of solid mechanics, in particular problems in the theory of plates, have long been regarded as bottlenecks in the development of elasticity. In contrast to the traditional solution methodologies, such as Timoshenko's approach in the 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. Dissimilar to the conventional Euclidean space with one kind of variables, the symplectic space with dual variables thus provides a fundamental breakthrough. A unique feature of this symplectic approach is the classical bending problems in solid mechanics now become eigenvalue problems and the symplectic bending deflection solutions are constituted by expansion of eigenvectors. The classical solutions are subsets of the more general symplectic solutions.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 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.

Product Details

ISBN-13:	9789812778703
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/27/2009
Pages:	316
Product dimensions:	6.10(w) x 9.00(h) x 0.90(d)

Table of Contents

Preface ix

Preface to the Chinese Edition xi

Foreword to the Chinese Edition xv

Nomenclature xix

1 Mathematical Preliminaries 1

1.1 Linear Space 1

1.2 Euclidean Space 6

1.3 Symplectic Space 9

1.4 Legengre's Transformation 26

1.5 The Hamiltonian Principle and the Hamiltonian Canonical Equations 28

1.6 The Reciprocal Theorems 30

1.6.1 The Reciprocal Theorem for Work 30

1.6.2 The Reciprocal Theorem for Displacement 32

1.6.3 The Reciprocal Theorem for Reaction 32

1.6.4 The Reciprocal Theorem for Displacement and Negative Reaction 33

References 35

2 Fundamental Equations of Elasticity and Variational Principle 37

2.1 Stress Analysis 37

2.2 Strain Analysis 41

2.3 Stress-Strain Relations 44

2.4 The Fundamental Equations of Elasticity 48

2.5 The Principle of Virtual Work 51

2.6 The Principle of Minimum Total Potential Energy 52

2.7 The Principle of Minimum Total Complementary Energy 54

2.8 The Hellinger-Reissner Variational Principle with Two Kinds of Variables 55

2.9 The Hu-Washizu Variational Principle with Three Kinds of Variables 57

2.10 The Principle of Superposition and the Uniqueness Theorem 59

2.11 Saint-Venant Principle 60

References 60

3 The Timoshenko Beam Theory and Its Extension 63

3.1 The Timoshenko Beam Theory 63

3.2 Derivation of Hamiltonian System 68

3.3 The Method of Separation of Variables 71

3.4 Reciprocal Theorem for Work and Adjoint Symplectic Orthogonality 74

3.5 Solution for Non-Homogeneous Equations 78

3.6 Two-Point Boundary Conditions 79

3.7 Static Analysis of Timoshenko Beam 84

3.8 Wave Propagation Analysis of Timoshenko Beam 87

3.9 Wave Induced Resonance 90

References 94

4 Plane Elasticityin Rectangular Coordinates 97

4.1 The Fundamental Equations of Plane Elasticity 97

4.2 Hamiltonian System in Rectangular Domain 101

4.3 Separation of Variables and Transverse Eigen-Problems 106

4.4 Eigen-Solutions of Zero Eigenvalue 109

4.5 Solutions of Saint-Venant Problems for Rectangular Beam 117

4.6 Eigen-Solutions of Nonzero Eigenvalues 123

4.6.1 Eigen-Solutions of Nonzero Eigenvalues of Symmetric Deformation 125

4.6.2 Eigen-Solutions of Nonzero Eigenvalues of Antisymmetric Deformation 128

4.7 Solutions of Generalized Plane Problems in Rectangular Domain 131

References 136

5 Plane Anisotropic Elasticity Problems 139

5.1 The Fundamental Equations of Plane Anisotropic Elasticity Problems 139

5.2 Symplectic Solution Methodology for Anisotropic Elasticity Problems 141

5.3 Eigen-Solutions of Zero Eigenvalue 145

5.4 Analytical Solutions of Saint-Venant Problems 150

5.5 Eigen-Solutions of Nonzero Eigenvalues 155

5.6 Introduction to Hamiltonian System for Generalized Plane Problems 158

References 162

6 Saint-Venant Problems for Laminated Composite Plates 163

6.1 The Fundamental Equations 163

6.2 Derivation of Hamiltonian System 165

6.3 Eigen-Solutions of Zero Eigenvalue 168

6.4 Analytical Solutions of Saint-Venant Problem 175

References 179

7 Solutions for Plane Elasticity in Polar Coordinates 181

7.1 Plane Elasticity Equations in Polar Coordinates 181

7.2 Variational Principle for a Circular Sector 185

7.3 Hamiltonian System with Radial Coordinate Treated as "Time" 187

7.4 Eigen-Solutions for Symmetric Deformation in Radial Hamiltonian System 195

7.4.1 Eigen-Solutions of Zero Eigenvalue 195

7.4.2 Eigen-Solutions of Nonzero Eigenvalues 199

7.5 Eigen-Solutions for Anti-Symmetric Deformation in Radial Hamiltonian System 202

7.5.1 Eigen-Solutions of Zero Eigenvalue 202

7.5.2 Eigen-Solutions of μ = $$1 205

7.5.3 Eigen-Solutions of General Nonzero Eigenvalues 210

7.6 Hamiltonian System with Circumferential Coordinate Treated as "Time" 213

7.6.1 Eigen-Solutions of Zero Eigenvalue 216

7.6.2 Eigen-Solutions of μ = $$i 219

7.6.3 Eigen-solutions of General Nonzero Eigenvalues 222

References 223

8 Hamiltonian System for Bending of Thin Plates 225

8.1 Small Deflection Theory for Bending of Elastic Thin Plates 225

8.2 Analogy between Plane Elasticity and Bending of Thin Plate 232

8.3 Multi-Variable Variational Principles for Thin Plate Bending and Plane Elasticity 239

8.3.1 Multi-Variable Variational Principles for Plate Bending 240

8.3.2 Multi-Variable Variational Principle for Plane Elasticity 248

8.4 Symplectic Solution for Rectangular Plates 252

8.5 Plates with Two Opposite Sides Simply Supported 257

8.6 Plates with Two Opposite Sides Free 262

8.7 Plate with Two Opposite Sides Clamped 269

8.8 Bending of Sectorial Plates 274

8.8.1 Derivation of Hamiltonian System 277

8.8.2 Sectorial Plate with Two Opposite Sides Free 280

References 288

About the Authors 291