Elasticity (eBook)

Front Cover 1
Elasticity: Theory, Applications, and Numerics 4
Copyright Page 5
Contents 6
Preface 10
About the Author 16
PART I: FOUNDATIONS AND ELEMENTARY APPLICATIONS 18
Chapter 1. Mathematical Preliminaries 20
1.1 Scalar, Vector, Matrix, and Tensor Definitions 20
1.2 Index Notation 21
1.3 Kronecker Delta and Alternating Symbol 24
1.4 Coordinate Transformations 25
1.5 Cartesian Tensors 27
1.6 Principal Values and Directions for Symmetric Second-Order Tensors 29
1.7 Vector, Matrix, and Tensor Algebra 33
1.8 Calculus of Cartesian Tensors 34
1.9 Orthogonal Curvilinear Coordinates 38
Chapter 2. Deformation: Displacements and Strains 48
2.1 General Deformations 48
2.2 Geometric Construction of Small Deformation Theory 51
2.3 Strain Transformation 55
2.4 Principal Strains 57
2.5 Spherical and Deviatoric Strains 58
2.6 Strain Compatibility 58
2.7 Curvilinear Cylindrical and Spherical Coordinates 63
Chapter 3. Stress and Equilibrium 72
3.1 Body and Surface Forces 72
3.2 Traction Vector and Stress Tensor 74
3.3 Stress Transformation 77
3.4 Principal Stresses 78
3.5 Spherical, Deviatoric, Octahedral, and von Mises Stresses 82
3.6 Equilibrium Equations 83
3.7 Relations in Curvilinear Cylindrical and Spherical Coordinates 85
Chapter 4. Material Behavior—Linear Elastic Solids 94
4.1 Material Characterization 94
4.2 Linear Elastic Materials—Hooke’s Law 96
4.3 Physical Meaning of Elastic Moduli 99
4.4 Thermoelastic Constitutive Relations 103
Chapter 5. Formulation and Solution Strategies 108
5.1 Review of Field Equations 108
5.2 Boundary Conditions and Fundamental Problem Classifications 109
5.3 Stress Formulation 114
5.4 Displacement Formulation 115
5.5 Principle of Superposition 117
5.6 Saint-Venant’s Principle 118
5.7 General Solution Strategies 119
Chapter 6. Strain Energy and Related Principles 130
6.1 Strain Energy 130
6.2 Uniqueness of the Elasticity Boundary-Value Problem 135
6.3 Bounds on the Elastic Constants 136
6.4 Related Integral Theorems 137
6.5 Principle of Virtual Work 139
6.6 Principles of Minimum Potential and Complementary Energy 141
6.7 Rayleigh-Ritz Method 145
Chapter 7. Two-Dimensional Formulation 152
7.1 Plane Strain 152
7.2 Plane Stress 155
7.3 Generalized Plane Stress 158
7.4 Antiplane Strain 160
7.5 Airy Stress Function 161
7.6 Polar Coordinate Formulation 162
Chapter 8. Two-Dimensional Problem Solution 168
8.1 Cartesian Coordinate Solutions Using Polynomials 168
8.2 Cartesian Coordinate Solutions Using Fourier Methods 178
8.3 General Solutions in Polar Coordinates 186
8.4 Example Polar Coordinate Solutions 189
Chapter 9. Extension, Torsion, and Flexure of Elastic Cylinders 232
9.1 General Formulation 232
9.2 Extension Formulation 233
9.3 Torsion Formulation 234
9.4 Torsion Solutions Derived from Boundary Equation 244
9.5 Torsion Solutions Using Fourier Methods 250
9.6 Torsion of Cylinders with Hollow Sections 254
9.7 Torsion of Circular Shafts of Variable Diameter 258
9.8 Flexure Formulation 260
9.9 Flexure Problems without Twist 264
PART II: ADVANCED APPLICATIONS 274
Chapter 10. Complex Variable Methods 276
10.1 Review of Complex Variable Theory 276
10.2 Complex Formulation of the Plane Elasticity Problem 283
10.3 Resultant Boundary Conditions 287
10.4 General Structure of the Complex Potentials 288
10.5 Circular Domain Examples 290
10.6 Plane and Half-Plane Problems 295
10.7 Applications Using the Method of Conformal Mapping 300
10.8 Applications to Fracture Mechanics 305
10.9 Westergaard Method for Crack Analysis 308
Chapter 11. Anisotropic Elasticity 314
11.1 Basic Concepts 314
11.2 Material Symmetry 316
11.3 Restrictions on Elastic Moduli 322
11.4 Torsion of a Solid Possessing a Plane of Material Symmetry 323
11.5 Plane Deformation Problems 329
11.6 Applications to Fracture Mechanics 342
11.7 Curvilinear Anisotropic Problems 345
Chapter 12. Thermoelasticity 354
12.1 Heat Conduction and the Energy Equation 354
12.2 General Uncoupled Formulation 356
12.3 Two-Dimensional Formulation 357
12.4 Displacement Potential Solution 360
12.5 Stress Function Formulation 361
12.6 Polar Coordinate Formulation 364
12.7 Radially Symmetric Problems 365
12.8 Complex Variable Methods for Plane Problems 369
Chapter 13. Displacement Potentials and Stress Functions 382
13.1 Helmholtz Displacement Vector Representation 382
13.2 Lamé’s Strain Potential 383
13.3 Galerkin Vector Representation 384
13.4 Papkovich-Neuber Representation 389
13.5 Spherical Coordinate Formulations 393
13.6 Stress Functions 398
Chapter 14. Nonhomogeneous Elasticity 408
14.1 Basic Concepts 409
14.2 Plane Problem of Hollow Cylindrical Domain under Uniform Pressure 413
14.3 Rotating Disk Problem 419
14.4 Point Force on the Free Surface of a Half-Space 424
14.5 Antiplane Strain Problems 432
14.6 Torsion Problem 435
Chapter 15. Micromechanics Applications 448
15.1 Dislocation Modeling 449
15.2 Singular Stress States 453
15.3 Elasticity Theory with Distributed Cracks 462
15.4 Micropolar/Couple-Stress Elasticity 465
15.5 Elasticity Theory with Voids 474
15.6 Doublet Mechanics 480
Chapter 16. Numerical Finite and Boundary Element Methods 490
16.1 Basics of the Finite Element Method 491
16.2 Approximating Functions for Two-Dimensional Linear Triangular Elements 493
16.3 Virtual Work Formulation for Plane Elasticity 495
16.4 FEM Problem Application 499
16.5 FEM Code Applications 501
16.6 Boundary Element Formulation 506
Appendix A Basic Field Equations in Cartesian, Cylindrical, and Spherical Coordinates 514
Appendix B Transformation of Field Variables Between Cartesian, Cylindrical, and Spherical Components 519
Appendix C MATLAB Primer 522
Appendix D Review of Mechanics of Materials 535
Index 550