Chapter 1 Mathematical Prerequisites 1.1 Index Notation 1.1.1 Range convention 1.1.2 Summation convention 1.1.3 The Kronecker delta 1.1.4 The permutation symbol 1.2 Vector Operations and Some Useful Integral Theorems 1.2.1 The scalar product of two vectors 1.2.2 The vector product of two vectors 1.2.3 The scalar triple product 1.2.4 The gradient of a scalar function 1.2.5 The divergence of a vector function 1.2.6 The curl of a vector function 1.2.7 Laplacian of a scalar function 1.2.8 Divergence theorem (Gauss's theorem) 1.2.9 Stokes' theorem 1.2.10 Green's theorem 1.3 Cartesian Tensors and Transformation Laws Problems 1 Chapter 2 Analysis of Stress 2.1 Continuum 2.2 Forces 2.3 Cauchy's Formula 2.4 Equations of Equilibrium 2.5 Stress as a Second-order Tensor 2.6 Principal Stresses 2.7 Maximum Shears 2.8 Yields Criteria Problems 2 Chapter 3 Analysis of Strain 3.1 Differential Element Considerations 3.2 Linear Deformation and Strain 3.3 Strain as a Second-order Tensor 3.4 Principal Strains and Strain Measurement 3.5 Compatibility Equations 3.6 Finite Deformation Problems 3 Chapter 4 Linear Elastic Materials, Framework of Problems of Elasticity 4.1 Introduction 4.2 Uniaxial Stress-Strain Relations of Linear Elastic Materials 4.3 Hooke's Law 4.3.1 Isotropic materials 4.3.2 Orthotropic materials 4.3.3 Transversely isotropic materials 4.4 Generalized Hooke's Law 4.5 Elastic Constants as Components of a Fourth-order Tensor 4.6 Elastic Symmetry 4.6.1 One plane of elastic symmetry (monoclinic material) 4.6.2 Two planes of elastic symmetry 4.6.3 Three planes of elastic symmetry (orthotropic material)
4.6.4 An axis of elastic symmetric (rotational symmetry) 4.6.5 Complete symmetry (spherical symmetry) 4.7 Elastic Moduli 4.7.1 Simple tension 4.7.2 Pure shear 4.7.3 Hydrostatics pressure 4.8 Formulation of Problems of Elasticity 4.9 Principle of Superposition 4.10 Uniqueness of Solution 4.11 Solution Approach Problems 4 Chapter 5 Some Elementary Problems 5.1 Extension of Prismatic Bars 5.2 A Column under Its Own Weight 5.3 Pure Bending of Beams 5.4 Torsion of a Shaft of Circular Cross Section Problems 5 Chapter 6 Two-dimensional Problems 6.1 Plane Strain 6.2 Plane Stress 6.3 Connection between Plane Strain and Plane Stress 6.4 Stress Function Formulation 6.5 Plane Problems in Cartesian Coordinates 6.5.1 Polynomial solutions 6.5.2 Product solutions 6.6 Plane Problems in Polar Coordinates 6.6.1 Basic equations in polar coordinates 6.6.2 Stress function in polar coordinates 6.6.3 Problems with axial symmetry 6.6.4 Problems without axial symmetry 6.7 Wedge Problems 6.7.1 A wedge subjected to a couple at the apex 6.7.2 A wedge subjected to concentrated loads at the apex 6.7.3 A wedge subjected to uniform edge loads 6.8 Half-plane Problems 6.9 Crack Problems Problems 6 Chapter 7 Torsion and Flexure of Prismatic Bars 7.1 Saint-Venant's Problem 7.2 Torsion of Prismatic Bars 7.2.1 Displacement formulation 7.2.2 Stress function formulation 7.2.3 Illustrative examples 7.3 Membrane Analogy 7.4 Torsion of Multiply Connected Bars 7.5 Torsion of Thin-walled Tubes 7.6 Flexure of Beams Subjected to Transverse End Loads 7.6.1 Formulation and solution 7.6.2 Illustrative examples Problems 7
Chapter 8 Complex Variable Methods 8.1 Summary of Theory of Complex Variables 8.1.1 Complex functions 8.1.2 Some results from theory of analytic functions 8.1.3 Conformal mapping 8.2 Plane Problems of Elasticity 8.2.1 Complex formulation of two-dimensional elasticity 8.2.2 Illustrative examples 8.2.3 Complex representation with conformal mapping 8.2.4 Illustrative examples 8.3 Problems of Saint-Venant's Torsion 8.3.1 Complex formulation with eonformal mapping 8.3.2 Illustrative examples Problems 8 Chapter 9 Three-dimensional Problems 9.1 Introduction 9.2 Displacement Potential Formulation 9.2.1 Galerkin vector 9.2.2 Papkovich-Neuber functions 9.2.3 Harmonic and biharmonic functions 9.3 Some Basic Three-dimensional Problems 9.3.1 Kelvin's problem 9.3.2 Boussinesq's problem 9.3.3 Cerruti's problem 9.3.4 Mindlin's problem 9.4 Problems in Spherical Coordinates 9.4.1 Hollow sphere under internal and external pressures 9.4.2 Spherical harmonics 9.4.3 Axisymmetric problems of hollow spheres 9.4.4 Extension of an infinite body with a spherical cavity Problems 9 Chapter 10 Variational Principles of Elasticity and Applications 10.1 Introduction 10.1.1 The shortest distance problem 10.1.2 The body of revolution problem 10.1.3 The hrachistochrone problem (the shortest time problem) 10.2 Variation Operation 10.3 Minimization of Variational Functionals 10.4 Illustrative Examples 10.5 Principle of Virtual Work 10.6 Principle of Minimum Potential Energy 10.7 Principle of Minimum Complementary Energy 10.8 Reciprocal Theorem 10.9 Hamilton's Principle of Elastodynamics 10.10 Vibration of Beams 10.11 Bending and Stretching of Thin Plates 10.12 Equivalent Variational Problems 10.12.1 Self-adjoint ordinary differential equations 10.12.2 Self-adjoint partial differential equations 10.13 Direct Methods of Solution
10.13.1 The Ritz method 10.13.2 The Galerkin method 10.14 Illustrative Examples 10.15 Closing Remarks Problems 10 Chapter 11 State Space Approach 11.1 Introduction 11.2 Solution of Systems of Linear Differential Equations 11.2.1 Solution of homogeneous system 11.2.2 Solution of nonhomogeneous system 11.3 State Space Formalism of Linear Elasticity 11.3.1 State variable representation of basic equations 11.3.2 Hamiltonian formulation 11.3.3 Explicit state equation and output equation 11.4 Analysis of Stress Decay in Laminates 11.5 Application to Two-dimensional Problems 11.5.1 Infinite-plane Green's function 11.5.2 Half-plane Green's functions 11.5.3 A half-plane under line load 11.5.4 Extension of infinite plate with an elliptical hole 11.6 Symplectic Characteristics of Hamiltonian System 11.6.1 Simpie and semisimple systems 11.6.2 Non-semisimple system 11.7 Application to Three-dimensional Elasticity Problems 11 References Appendix A Basic Equations in Cylindrical and Spherical Coordinates Appendix B Fourier Series Appendix C Product Solution of Biharmonic Equation in Cartesian Coordinates Appendix D Product Solution of Biharmonic Equation in Polar Coordinates Index
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