Elasticity - Theory, Applications, and Numerics

Front Cover

Elasticity: Theory, Applications, and Numerics

Copyright Page

Contents

Preface

About the Author

PART I: FOUNDATIONS AND ELEMENTARY APPLICATIONS

Chapter 1. Mathematical Preliminaries

1.1 Scalar, Vector, Matrix, and Tensor Definitions

1.2 Index Notation

1.3 Kronecker Delta and Alternating Symbol

1.4 Coordinate Transformations

1.5 Cartesian Tensors

1.6 Principal Values and Directions for Symmetric Second-Order Tensors

1.7 Vector, Matrix, and Tensor Algebra

1.8 Calculus of Cartesian Tensors

1.9 Orthogonal Curvilinear Coordinates

Chapter 2. Deformation: Displacements and Strains

2.1 General Deformations

2.2 Geometric Construction of Small Deformation Theory

2.3 Strain Transformation

2.4 Principal Strains

2.5 Spherical and Deviatoric Strains

2.6 Strain Compatibility

2.7 Curvilinear Cylindrical and Spherical Coordinates

Chapter 3. Stress and Equilibrium

3.1 Body and Surface Forces

3.2 Traction Vector and Stress Tensor

3.3 Stress Transformation

3.4 Principal Stresses

3.5 Spherical, Deviatoric, Octahedral, and von Mises Stresses

3.6 Equilibrium Equations

3.7 Relations in Curvilinear Cylindrical and Spherical Coordinates

Chapter 4. Material Behavior—Linear Elastic Solids

4.1 Material Characterization

4.2 Linear Elastic Materials—Hooke’s Law

4.3 Physical Meaning of Elastic Moduli

4.4 Thermoelastic Constitutive Relations

Chapter 5. Formulation and Solution Strategies

5.1 Review of Field Equations

5.2 Boundary Conditions and Fundamental Problem Classifications

5.3 Stress Formulation

5.4 Displacement Formulation

5.5 Principle of Superposition

5.6 Saint-Venant’s Principle

5.7 General Solution Strategies

Chapter 6. Strain Energy and Related Principles

6.1 Strain Energy

6.2 Uniqueness of the Elasticity Boundary-Value Problem

6.3 Bounds on the Elastic Constants

6.4 Related Integral Theorems

6.5 Principle of Virtual Work

6.6 Principles of Minimum Potential and Complementary Energy

6.7 Rayleigh-Ritz Method

Chapter 7. Two-Dimensional Formulation

7.1 Plane Strain

7.2 Plane Stress

7.3 Generalized Plane Stress

7.4 Antiplane Strain

7.5 Airy Stress Function

7.6 Polar Coordinate Formulation

Chapter 8. Two-Dimensional Problem Solution

8.1 Cartesian Coordinate Solutions Using Polynomials

8.2 Cartesian Coordinate Solutions Using Fourier Methods

8.3 General Solutions in Polar Coordinates

8.4 Example Polar Coordinate Solutions

Chapter 9. Extension, Torsion, and Flexure of Elastic Cylinders

9.1 General Formulation

9.2 Extension Formulation

9.3 Torsion Formulation

9.4 Torsion Solutions Derived from Boundary Equation

9.5 Torsion Solutions Using Fourier Methods

9.6 Torsion of Cylinders with Hollow Sections

9.7 Torsion of Circular Shafts of Variable Diameter

9.8 Flexure Formulation

9.9 Flexure Problems without Twist

PART II: ADVANCED APPLICATIONS

Chapter 10. Complex Variable Methods

10.1 Review of Complex Variable Theory

10.2 Complex Formulation of the Plane Elasticity Problem

10.3 Resultant Boundary Conditions

10.4 General Structure of the Complex Potentials

10.5 Circular Domain Examples

10.6 Plane and Half-Plane Problems

10.7 Applications Using the Method of Conformal Mapping

10.8 Applications to Fracture Mechanics

10.9 Westergaard Method for Crack Analysis

Chapter 11. Anisotropic Elasticity

11.1 Basic Concepts

11.2 Material Symmetry

11.3 Restrictions on Elastic Moduli

11.4 Torsion of a Solid Possessing a Plane of Material Symmetry

11.5 Plane Deformation Problems

11.6 Applications to Fracture Mechanics

11.7 Curvilinear Anisotropic Problems

Chapter 12. Thermoelasticity

12.1 Heat Conduction and the Energy Equation

12.2 General Uncoupled Formulation

12.3 Two-Dimensional Formulation

12.4 Displacement Potential Solution

12.5 Stress Function Formulation

12.6 Polar Coordinate Formulation

12.7 Radially Symmetric Problems

12.8 Complex Variable Methods for Plane Problems

Chapter 13. Displacement Potentials and Stress Functions

13.1 Helmholtz Displacement Vector Representation

13.2 Lamé’s Strain Potential

13.3 Galerkin Vector Representation

13.4 Papkovich-Neuber Representation

13.5 Spherical Coordinate Formulations

13.6 Stress Functions

Chapter 14. Nonhomogeneous Elasticity

14.1 Basic Concepts

14.2 Plane Problem of Hollow Cylindrical Domain under Uniform Pressure

14.3 Rotating Disk Problem

14.4 Point Force on the Free Surface of a Half-Space

14.5 Antiplane Strain Problems

14.6 Torsion Problem

Chapter 15. Micromechanics Applications

15.1 Dislocation Modeling

15.2 Singular Stress States

15.3 Elasticity Theory with Distributed Cracks

15.4 Micropolar/Couple-Stress Elasticity

15.5 Elasticity Theory with Voids

15.6 Doublet Mechanics

Chapter 16. Numerical Finite and Boundary Element Methods

16.1 Basics of the Finite Element Method

16.2 Approximating Functions for Two-Dimensional Linear Triangular Elements

16.3 Virtual Work Formulation for Plane Elasticity

16.4 FEM Problem Application

16.5 FEM Code Applications

16.6 Boundary Element Formulation

Appendix A Basic Field Equations in Cartesian, Cylindrical, and Spherical Coordinates

Appendix B Transformation of Field Variables Between Cartesian, Cylindrical, and Spherical Components

Appendix C MATLAB Primer

Appendix D Review of Mechanics of Materials

Index