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...
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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 ixPreface to the Chinese Edition xiForeword to the Chinese Edition xvNomenclature xix1 Mathematical Preliminaries 11.1 Linear Space 11.2 Euclidean Space 61.3 Symplectic Space 91.4 Legengre's Transformation 261.5 The Hamiltonian Principle and the Hamiltonian Canonical Equations 281.6 The Reciprocal Theorems 301.6.1 The Reciprocal Theorem for Work 301.6.2 The Reciprocal Theorem for Displacement 321.6.3 The Reciprocal Theorem for Reaction 321.6.4 The Reciprocal Theorem for Displacement and Negative Reaction 33References 352 Fundamental Equations of Elasticity and Variational Principle 372.1 Stress Analysis 372.2 Strain Analysis 412.3 Stress-Strain Relations 442.4 The Fundamental Equations of Elasticity 482.5 The Principle of Virtual Work 512.6 The Principle of Minimum Total Potential Energy 522.7 The Principle of Minimum Total Complementary Energy 542.8 The Hellinger-Reissner Variational Principle with Two Kinds of Variables 552.9 The Hu-Washizu Variational Principle with Three Kinds of Variables 572.10 The Principle of Superposition and the Uniqueness Theorem 592.11 Saint-Venant Principle 60References 603 The Timoshenko Beam Theory and Its Extension 633.1 The Timoshenko Beam Theory 633.2 Derivation of Hamiltonian System 683.3 The Method of Separation of Variables 713.4 Reciprocal Theorem for Work and Adjoint Symplectic Orthogonality 743.5 Solution for Non-Homogeneous Equations 783.6 Two-Point Boundary Conditions 793.7 Static Analysis of Timoshenko Beam 843.8 Wave Propagation Analysis of Timoshenko Beam 873.9 Wave Induced Resonance 90References 944 Plane Elasticityin Rectangular Coordinates 974.1 The Fundamental Equations of Plane Elasticity 974.2 Hamiltonian System in Rectangular Domain 1014.3 Separation of Variables and Transverse Eigen-Problems 1064.4 Eigen-Solutions of Zero Eigenvalue 1094.5 Solutions of Saint-Venant Problems for Rectangular Beam 1174.6 Eigen-Solutions of Nonzero Eigenvalues 1234.6.1 Eigen-Solutions of Nonzero Eigenvalues of Symmetric Deformation 1254.6.2 Eigen-Solutions of Nonzero Eigenvalues of Antisymmetric Deformation 1284.7 Solutions of Generalized Plane Problems in Rectangular Domain 131References 1365 Plane Anisotropic Elasticity Problems 1395.1 The Fundamental Equations of Plane Anisotropic Elasticity Problems 1395.2 Symplectic Solution Methodology for Anisotropic Elasticity Problems 1415.3 Eigen-Solutions of Zero Eigenvalue 1455.4 Analytical Solutions of Saint-Venant Problems 1505.5 Eigen-Solutions of Nonzero Eigenvalues 1555.6 Introduction to Hamiltonian System for Generalized Plane Problems 158References 1626 Saint-Venant Problems for Laminated Composite Plates 1636.1 The Fundamental Equations 1636.2 Derivation of Hamiltonian System 1656.3 Eigen-Solutions of Zero Eigenvalue 1686.4 Analytical Solutions of Saint-Venant Problem 175References 1797 Solutions for Plane Elasticity in Polar Coordinates 1817.1 Plane Elasticity Equations in Polar Coordinates 1817.2 Variational Principle for a Circular Sector 1857.3 Hamiltonian System with Radial Coordinate Treated as "Time" 1877.4 Eigen-Solutions for Symmetric Deformation in Radial Hamiltonian System 1957.4.1 Eigen-Solutions of Zero Eigenvalue 1957.4.2 Eigen-Solutions of Nonzero Eigenvalues 1997.5 Eigen-Solutions for Anti-Symmetric Deformation in Radial Hamiltonian System 2027.5.1 Eigen-Solutions of Zero Eigenvalue 2027.5.2 Eigen-Solutions of μ = $$1 2057.5.3 Eigen-Solutions of General Nonzero Eigenvalues 2107.6 Hamiltonian System with Circumferential Coordinate Treated as "Time" 2137.6.1 Eigen-Solutions of Zero Eigenvalue 2167.6.2 Eigen-Solutions of μ = $$i 2197.6.3 Eigen-solutions of General Nonzero Eigenvalues 222References 2238 Hamiltonian System for Bending of Thin Plates 2258.1 Small Deflection Theory for Bending of Elastic Thin Plates 2258.2 Analogy between Plane Elasticity and Bending of Thin Plate 2328.3 Multi-Variable Variational Principles for Thin Plate Bending and Plane Elasticity 2398.3.1 Multi-Variable Variational Principles for Plate Bending 2408.3.2 Multi-Variable Variational Principle for Plane Elasticity 2488.4 Symplectic Solution for Rectangular Plates 2528.5 Plates with Two Opposite Sides Simply Supported 2578.6 Plates with Two Opposite Sides Free 2628.7 Plate with Two Opposite Sides Clamped 2698.8 Bending of Sectorial Plates 2748.8.1 Derivation of Hamiltonian System 2778.8.2 Sectorial Plate with Two Opposite Sides Free 280References 288About the Authors 291
