Elasticity: Tensor, Dyadic, and Engineering Approaches

PREFACE INTRODUCTION 1 ANALYSIS OF STRESS 1.1 Introduction 1.2 "Body Forces, Surface Forces, and Stresses" 1.3 Uniform State of Stress (Two-Dimensional) 1.4 Principal Stresses 1.5 Mohr's Circle of Stress 1.6 State of Stress at a Point 1.7 Differential Equations of Equilibrium 1.8 Three-Dimensional State of Stress at a Point 1.9 Summary Problems 2 STRAIN AND DISPLACEMENT 2.1 Introduction 2.2 Strain-Displacement Relations 2.3 Compatibility Equations 2.4 State of Strain at a Point 2.5 General Displacements 2.6 Principle of Superposition 2.7 Summary Problems 3 STRESS STRAIN RELATIONS 3.1 Introduction 3.2 Generalized Hooke's Law 3.3 Bulk Modulus of Elasticity 3.4 Summary Problems 4 FORMULATION OF PROBLEMS IN ELASTICITY 4.1 Introduction 4.2 Boundary Conditions 4.3 Governing Equations in Plane Strain Problems 4.4 Governing Equations in Three-Dimensional Problems 4.5 Principal of Superposition 4.6 Uniqueness of Elasticity Solutions 4.7 Saint-Venant's Principle 4.8 Summary Problems 5 TWO-DIMENSIONAL PROBLEMS 5.1 Introduction 5.2 Plane Stress Problems 5.3 Approximate Character of Plane Stress Equations 5.4 Polar Coordinates in Two-Dimensional Problems 5.5 Axisymmetric Plane Problems 5.6 The Semi-Inverse Method Problems 6 TORSION OF CYLINDRICAL BARS 6.1 General Solution of the Problem 6.2 Solutions Derived from Equations of Boundaries 6.3 Membrane (Soap Film) Analogy 6.4 Multiply Connected Cross Sections 6.5 Solution by Means of Separation of Variables Problems 7 ENERGY METHODS 7.1 Introduction 7.2 Strain Energy 7.3 Variable Stress Distribution and Body Forces 7.4 Principle of Virtual Work and the Theorem of Minimum Potential Energy 7.5 Illustrative Problems 7.6 Rayleigh-Ritz Method Problems 8 CARTESIAN TENSOR NOTATION 8.1 Introduction 8.2 Indicial Notation and Vector Transformations 8.3 Higher-Order Tensors 8.4 Gradient of a Vector 8.5 The Kronecker Delta 8.6 Tensor Contraction 8.7 The Alternating Tensor 8.8 The Theorem of Gauss Problems 9 THE STRESS TENSOR 9.1 State of Stress at a Point 9.2 Principal Axes of the Stress Tensor 9.3 Equations of Equilibrium 9.4 The Stress Ellipsoid 9.5 Body Moment and Couple Stress Problems 10 "STRAIN, DISPLACEMENT, AND THE GOVERNING EQUATIONS OF ELASTICITY" 10.1 Introduction 10.2 Displacement and Strain 10.3 Generalized Hooke's Law 10.4 Equations of Compatibility 10.5 Governing Equations in Terms of Displacement 10.6 Strain Energy 10.7 Governing Equations of Elasticity Problems 11 VECTOR AND DYADIC NOTATION IN ELASTICITY 11.1 Introduction 11.2 Review of Basic Notations and Relations in Vector Analysis 11.3 Dyadic Notation 11.4 Vector Representation of Stress on a Plane 11.5 Equations of Transformation of Stress 11.6 Equations of Equilibrium 11.7 Displacement and Strain 11.8 Generalized Hooke's Law and Navier's Equation 11.9 Equations of Compatibility 11.10 Strain Energy 11.12 Governing Equations of Elasticity Problems 12 ORTHOGONAL CURVILINEAR COORDINATES 12.1 Introduction 12.2 Scale Factors 12.3 Derivatives of the Unit Vectors 12.4 Vector Operators 12.5 Dyadic Notation and Dyadic Operators 12.6 Governing Equations of Elasticity in Dyadic Notation 12.7 Summary of Vector and Dyadic Operators in Cylindrical and Spherical Coordinates Problems 13 DISPLACEMENT FUNCTIONS AND STRESS FUNCTIONS 13.1 Introduction 13.2 Displacement Functions 13.3 The Galerkin Vector 13.4 The Solution of Papkovich-Neuber 13.5 Stress Functions Problems References INDEX