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CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS1

Part Ⅰ Introduction1

1-1 Trends and Scopes1

1-2 Theory of Elasticity4

1-3 Numerical Stress Analysis4

1-4 General Solution of the Elasticity Problem5

1-5 Experimental Stress Analysis6

1-6 Boundary-Value Problems of Elasticity6

Part Ⅱ Preliminary Concepts8

1-7 Brief Summary of Vector Algebra8

1-8 Scalar Point Functions12

1-9 Vector Fields14

1-10 Differentiation of Vectors15

1-11 Differentiation of a Scalar Field17

1-12 Differentiation of a Vector Field18

1-13 Curl of a Vector Field19

1-14 Eulerian Continuity Equation for Fluids19

1-15 Divergence Theorem22

1-16 Divergence Theorem in Two Dimensions24

1-17 Line and Surface Integrals (Application of Scalar Product)25

1-18 Stokes's Theorem27

1-19 Exact Differential27

1-20 Orthogonal Curvilinear Coordinates in Three-Dimensional Space28

1-21 Expression for Differential Length in Orthogonal Curvilinear Coordinates29

1-22 Gradient and Laplacian in Orthogonal Curvilinear Coordinates30

Part Ⅲ Elements of Tensor Algebra33

1-23 Index Notation: Summation Convention33

1-24 Transformation of Tensors under Rotation of Rectangular Cartesian Coordinate System37

1-25 Symmetric and Antisymmetric Parts of a Tensor44

1-26 Symbols of δij and ∈ijk (the Kronecker Delta and the Alternating Tensor)45

1-27 Homogeneous Quadratic Forms47

1-28 Elementary Matrix Algebra50

1-29 Some Topics in the Calculus of Variations55

References59

Bibliography61

CHAPTER 2 THEORY OF DEFORMATION62

2-1 Deformable, Continuous Media62

2-2 Rigid-Body Displacements63

2-3 Deformation of a Continuous Region.Material Variables. Spatial Variables65

2-4 Restrictions on Continuous Deformation of a Deformable Medium69

Problem Set 2-471

2-5 Gradient of the Displacement Vector. Tensor Quantity72

2-6 Extension of an Infinitesimal Line Element74

Problem Set 2-681

2-7 Physical Significance of ∈ii Strain Definitions82

2-8 Final Direction of Line Element. Definition of Shearing Strain. Physical Significance of eij(i≠j)85

Problem Set 2-890

2-9 Tensor Character of ∈aβ. Strain Tensor91

2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants92

2-11 Determination of Principal Strains. Principal Axes96

Problem Set 2-11102

2-12 Determination of Strain Invariants. Volumetric Strain104

2-13 Rotation of a Volume Element. Relation to Displacement Gradients109

Problem Set 2-13113

2-14 Homogeneous Deformation115

2-15 Theory of Small Strains and Small Angles of Rotation118

Problem Set 2-15128

2-16 Compatibility Conditions of the Classical Theory of Small Displacements130

Problem Set 2-16135

2-17 Additional Conditions Imposed by Continuity136

2-18 Kinematics of Deformable Media138

Problem Set 2-18144

Appendix 2A Strain--Displacement Relations in Orthogonal Curvilinear Coordinates144

2A-1 Geometrical Preliminaries144

2A-2 Strain-Displacement Relations146

Appendix 2B Derivation of Strain--Displacement Relations for Special Coordinates by Cartesian Methods150

Appendix 2C Strain-Displacement Relations in General Coordinates153

2C-1 Euclidean Metric Tensor153

2C-2 Strain Tensors156

References157

Bibliography158

CHAPTER 3 THEORY OF STRESS159

3-1 Definition of Stress159

3-2 Stress Notation162

3-3 Summation of Moments. Stress at a Point. Stress on an Oblique Plane164

Problem Set 3-3169

3-4 Tensor Character of Stress. Transformation of Stress Components under Rotation of Coordinate Axes173

Problem Set 3-4176

3-5 Principal Stresses. Stress Invariants. Extreme Values177

Problem Set 3-5181

3-6 Mean and Deviator Stress Tensors. Octahedral Stress182

Problem Set 3-6187

3-7 Approximations of Plane Stress. Mohr's Circles in Two and Three Dimensions191

Problem Set 3-7198

3-8 Differential Equations of Motion of a Deformable Body Relative to Spatial Coordinates199

Problem Set 3-8203

Appendix 3A Differential Equations of Equilibrium in Curvilinear Spatial Coordinates204

3A-1 Differential Equations of Equilibrium in Orthogonal Curvilinear Spatial Coordinates204

3A-2 Specialization of Equations of Equilibrium206

3A-3 Differential Equations of Equilibrium in General Spatial Coordinates208

Appendix 3B Equations of Equilibrium Including Couple Stress and Body Couple210

Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory212

3C-1 Material Derivative. Material Derivative of a Volume Integral212

3C-2 Differential Equations of Equilibrium Relative to Material Coordinates216

References222

Bibliography223

CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OF ELASTICITY224

4-1 Elastic and Nonelastic Response of a Solid224

4-2 Intrinsic Energy Density Function (Adiabatic Process)228

4-3 Relation of Stress Components to Strain Energy Density Function230

4-4 Generalized Hooke's Law233

Problem Set 4-4243

4-5 Isotropic Media. Homogeneous Media243

4-6 Strain Energy Density for Elastic Isotropic Medium244

Problem Set 4-6250

4-7 Special States of Stress254

Problem Set 4-7257

4-8 Equations of Thermoelasticity257

4-9 Differential Equation of Heat Conduction259

4-10 Elementary Approach to Thermal-Stress Problem in One and Two Variables261

4-11 Stress Strain-Temperature Relations265

Problem Set 4-11272

4-12 Thermoelastic Equations in Terms of Displacement274

4-13 Spherically Symmetrical Stress Distribution (The Sphere)277

Problem Set 4-13279

4-14 Thermoelastic Compatibility Equations in Terms of Components of Stress and Temperature. Beltrami-Michell Relations279

Problem Set 4-14284

4-15 Boundary Conditions286

Problem Set 4-15290

4-16 Uniqueness Theorem for Equilibrium Problem of Elasticity290

4-17 Equations of Elasticity in Terms of Displacement Components295

Problem Set 4-17297

4-18 Elementary Three-Dimensional Problems of Elasticity. Semi-Inverse Method298

Problem Set 4-18304

4-19 Torsion of Shaft with Constant Circular Cross Section308

Problem Set 4-19312

4-20 Energy Principles in Elasticity314

4-21 Principle of Virtual Work315

Problem Set 4-21320

4-22 Principle of Virtual Stress (Castigliano's Theorem)321

4-23 Mixed Virtual Stress-Virtual Strain Principles (Reissner's Theorem)323

Appendix 4A Application of the Principle of Virtual Work to a Deformable Medium (Navier-Stokes Equations)324

Appendix 4B Nonlinear Constitutive Relationships327

4B-1 Variable Stress-Strain Coefficients328

4B-2 Higher-Order Relations328

4B-3 Hypoelastic Formulations328

4B-4 Summary329

References329

Bibliography332

CHAPTER 5 PLANE THEORY OF ELASTICITY IN RECTANGULAR CARTESIAN COORDINATES333

5-1 Plane Strain334

Problem Set 5-1338

5-2 Generalized Plane Stress340

Problem Set 5-2344

5-3 Compatability Equation in Terms of Stress Components346

Problem Set 5-3350

5-4 Airy Stress Function351

Problem Set 5-4361

5-5 Airy Stress Function in Terms of Harmonic Functions368

5-6 Displacement Components for Plane Elasticity369

Problem Set 5-6373

5-7 Polynomial Solutions of TwoDimensional Problems in Rectangular Cartesian Coordinates377

Problem Set 5-7380

5-8 Plane Elasticity in Terms of Displacement Components384

Problem Set 5-8385

5-9 Plane Elasticity Relative to Oblique Coordinate Axes386

Appendix 5A Plane Elasticity with Couple Stresses390

5A-1 Introduction390

5A-2 Equations of Equilibrium391

5A-3 Deformation in Couple-Stress Theory391

5A-4 Equations of Compatibility393

5A-5 Stress Functions for Plane Problems with Couple Stresses396

Appendix 5B Plane Theory of Elasticity in Terms of Complex Variables398

5B-1 Airy Stress Function in Terms of Analytic Functionsψ(z) and χ(z)398

5B-2 Displacement Components in Terms of Analytic Functions, ψ(z) and χ(z)399

5B-3 Stress Components in Terms of ψ(z) and χ(z)400

5B-4 Expressions for Resultant Force and Resultant Moment403

5B-5 Mathematical Form of Functions ψ(z)and χ(z)404

5B-6 Plane Elasticity Boundary-Value Problems in Complex Form408

5B-7 Note on Conformal Transformation411

5B-8 Plane Elasticity Formulas in Terms of Curvilinear Coordinates416

5B-9 Complex Variable Solution for Plane Region Bounded by Circle in the z Plane419

Problem Set 5B423

References424

Bibliography425

CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES427

6-1 Equilibrium Equations in Polar Coordinates427

6-2 Stress Components in Terms of Airy Stress Function F = F(γ, θ)428

6-3 Strain Displacement Relations in Polar Coordinates430

Problem Set 6-3432

6-4 Stress-Strain-Temperature Relations433

Problem Set 6-4435

6-5 Compatibility Equation for Plane Elasticity in Terms of Polar Coordinates435

Problem Set 6-5436

6-6 Axially Symmetric Problems438

Problem Set 6-6449

6-7 Plane-Elasticity Equations in Terms of Displacement Components451

6-8 Plane Theory of Thermoelasticity455

Problem Set 6-8458

6-9 Disk of Variable Thickness and Nonhomogeneous Anisotropic Material460

Problem Set 6-9465

6-10 Stress Concentration Problem of Circular Hole in Plate465

Problem Set 6-10472

6-11 Examples473

Problem Set 6-11478

Appendix 6A Stress-Couple Theory of Stress Concentration Resulting from Circular Hole in Plate487

Appendix 6B Stress Distribution of a Diametrically Compressed Plane Disk492

References494

CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD496

7-1 General Problem of Three-Dimensional Elastic Bars Subjected to Transverse End Loads496

7-2 Torsion of Prismatic Bars. Saint-Venant's Solution. Warping Function499

Problem Set 7-2505

7-3 Prandtl Torsion Function505

Problem Set 7-3509

7-4 A Method of Solution of the Torsion Problem: Elliptic Cross Section510

Problem Set 7-4514

7-5 Remarks on Solutions of the Laplace Equation, ▽2F = 0515

Problem Set 7-5517

7-6 Torsion of Bars with Tubular Cavities520

Problem Set 7-6523

7-7 Transfer of Axis of Twist523

7-8 Shearing-Stress Component in Any Direction524

Problem Set 7-8529

7-9 Solution of Torsion Problem by the Prandtl Membrane Analogy529

Problem Set 7-9538

7-10 Solution by Method of Series. Rectangular Section538

Problem Set 7-10543

7-11 Bending of a Bar Subjected to Transverse End Force544

Problem Set 7-11556

7-12 Displacement of a Cantilever Beam Subjected to Transverse End Force556

Problem Set 7-12559

7-13 Center of Shear560

Problem Set 7-13561

7-14 Bending of a Bar with Elliptic Cross Section563

7-15 Bending of a Bar with Rectangular Cross Section565

Problems Set 7-15570

Review Problems571

Appendix 7A Analysis of Tapered Beams572

References576

CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY578

8-1 Introduction578

Problem Set 8-1579

8-2 Equilibrium Equations579

Problem Set 8-2581

8-3 The Helmholtz Transformation581

Problem Set 8-3583

8-4 The Galerkin (Papkovich) Vector583

Problem Set 8-4585

8-5 Stress in Terms of the Galerkin Vector F585

Problem Set 8-5586

8-6 The Galerkin Vector: A Solution of the Equilibrium Equations of Elasticity586

Problem Set 8-6588

8-7 The Galerkin Vector kZ and Love's Strain Function for Solids of Revolution588

Problem Set 8-7591

8-8 Kelvin's Problem: Single Force Applied in the Interior of an Infinitely Extended Solid591

Problem Set 8-8593

8-9 The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson's Ratio593

8-10 Solutions of the Boussinesq and Cerruti Problems by the Twinned Gradient Method597

Problem Set 8-10600

8-11 Additional Remarks on ThreeDimensional Stress Functions600

References601

Bibliography601

INDEX603