ELASTICITY IN ENGINEERING MECHANICS SECOND EDITIONPDF电子书下载

CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS 1

Part Ⅰ Introduction 1

1-1 Trends and Scopes 1

1-2 Theory of Elasticity 4

1-3 Numerical Stress Analysis 4

1-4 General Solution of the Elasticity Problem 5

1-5 Experimental Stress Analysis 6

1-6 Boundary-Value Problems of Elasticity 6

Part Ⅱ Preliminary Concepts 8

1-7 Brief Summary of Vector Algebra 8

1-8 Scalar Point Functions 12

1-9 Vector Fields 14

1-10 Differentiation of Vectors 15

1-11 Differentiation of a Scalar Field 17

1-12 Differentiation of a Vector Field 18

1-13 Curl of a Vector Field 19

1-14 Eulerian Continuity Equation for Fluids 19

1-15 Divergence Theorem 22

1-16 Divergence Theorem in Two Dimensions 24

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

1-18 Stokes's Theorem 27

1-19 Exact Differential 27

1-20 Orthogonal Curvilinear Coordinates in Three-Dimensional Space 28

1-21 Expression for Differential Length in Orthogonal Curvilinear Coordinates 29

1-22 Gradient and Laplacian in Orthogonal Curvilinear Coordinates 30

Part Ⅲ Elements of Tensor Algebra 33

1-23 Index Notation: Summation Convention 33

1-24 Transformation of Tensors under Rotation of Rectangular Cartesian Coordinate System 37

1-25 Symmetric and Antisymmetric Parts of a Tensor 44

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

1-27 Homogeneous Quadratic Forms 47

1-28 Elementary Matrix Algebra 50

1-29 Some Topics in the Calculus of Variations 55

References 59

Bibliography 61

CHAPTER 2 THEORY OF DEFORMATION 62

2-1 Deformable, Continuous Media 62

2-2 Rigid-Body Displacements 63

2-3 Deformation of a Continuous Region.Material Variables. Spatial Variables 65

2-4 Restrictions on Continuous Deformation of a Deformable Medium 69

Problem Set 2-4 71

2-5 Gradient of the Displacement Vector. Tensor Quantity 72

2-6 Extension of an Infinitesimal Line Element 74

Problem Set 2-6 81

2-7 Physical Significance of ∈ii Strain Definitions 82

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

Problem Set 2-8 90

2-9 Tensor Character of ∈aβ. Strain Tensor 91

2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants 92

2-11 Determination of Principal Strains. Principal Axes 96

Problem Set 2-11 102

2-12 Determination of Strain Invariants. Volumetric Strain 104

2-13 Rotation of a Volume Element. Relation to Displacement Gradients 109

Problem Set 2-13 113

2-14 Homogeneous Deformation 115

2-15 Theory of Small Strains and Small Angles of Rotation 118

Problem Set 2-15 128

2-16 Compatibility Conditions of the Classical Theory of Small Displacements 130

Problem Set 2-16 135

2-17 Additional Conditions Imposed by Continuity 136

2-18 Kinematics of Deformable Media 138

Problem Set 2-18 144

Appendix 2A Strain--Displacement Relations in Orthogonal Curvilinear Coordinates 144

2A-1 Geometrical Preliminaries 144

2A-2 Strain-Displacement Relations 146

Appendix 2B Derivation of Strain--Displacement Relations for Special Coordinates by Cartesian Methods 150

Appendix 2C Strain-Displacement Relations in General Coordinates 153

2C-1 Euclidean Metric Tensor 153

2C-2 Strain Tensors 156

References 157

Bibliography 158

CHAPTER 3 THEORY OF STRESS 159

3-1 Definition of Stress 159

3-2 Stress Notation 162

3-3 Summation of Moments. Stress at a Point. Stress on an Oblique Plane 164

Problem Set 3-3 169

3-4 Tensor Character of Stress. Transformation of Stress Components under Rotation of Coordinate Axes 173

Problem Set 3-4 176

3-5 Principal Stresses. Stress Invariants. Extreme Values 177

Problem Set 3-5 181

3-6 Mean and Deviator Stress Tensors. Octahedral Stress 182

Problem Set 3-6 187

3-7 Approximations of Plane Stress. Mohr's Circles in Two and Three Dimensions 191

Problem Set 3-7 198

3-8 Differential Equations of Motion of a Deformable Body Relative to Spatial Coordinates 199

Problem Set 3-8 203

Appendix 3A Differential Equations of Equilibrium in Curvilinear Spatial Coordinates 204

3A-1 Differential Equations of Equilibrium in Orthogonal Curvilinear Spatial Coordinates 204

3A-2 Specialization of Equations of Equilibrium 206

3A-3 Differential Equations of Equilibrium in General Spatial Coordinates 208

Appendix 3B Equations of Equilibrium Including Couple Stress and Body Couple 210

Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory 212

3C-1 Material Derivative. Material Derivative of a Volume Integral 212

3C-2 Differential Equations of Equilibrium Relative to Material Coordinates 216

References 222

Bibliography 223

CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OF ELASTICITY 224

4-1 Elastic and Nonelastic Response of a Solid 224

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

4-3 Relation of Stress Components to Strain Energy Density Function 230

4-4 Generalized Hooke's Law 233

Problem Set 4-4 243

4-5 Isotropic Media. Homogeneous Media 243

4-6 Strain Energy Density for Elastic Isotropic Medium 244

Problem Set 4-6 250

4-7 Special States of Stress 254

Problem Set 4-7 257

4-8 Equations of Thermoelasticity 257

4-9 Differential Equation of Heat Conduction 259

4-10 Elementary Approach to Thermal-Stress Problem in One and Two Variables 261

4-11 Stress Strain-Temperature Relations 265

Problem Set 4-11 272

4-12 Thermoelastic Equations in Terms of Displacement 274

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

Problem Set 4-13 279

4-14 Thermoelastic Compatibility Equations in Terms of Components of Stress and Temperature. Beltrami-Michell Relations 279

Problem Set 4-14 284

4-15 Boundary Conditions 286

Problem Set 4-15 290

4-16 Uniqueness Theorem for Equilibrium Problem of Elasticity 290

4-17 Equations of Elasticity in Terms of Displacement Components 295

Problem Set 4-17 297

4-18 Elementary Three-Dimensional Problems of Elasticity. Semi-Inverse Method 298

Problem Set 4-18 304

4-19 Torsion of Shaft with Constant Circular Cross Section 308

Problem Set 4-19 312

4-20 Energy Principles in Elasticity 314

4-21 Principle of Virtual Work 315

Problem Set 4-21 320

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 Relationships 327

4B-1 Variable Stress-Strain Coefficients 328

4B-2 Higher-Order Relations 328

4B-3 Hypoelastic Formulations 328

4B-4 Summary 329

References 329

Bibliography 332

CHAPTER 5 PLANE THEORY OF ELASTICITY IN RECTANGULAR CARTESIAN COORDINATES 333

5-1 Plane Strain 334

Problem Set 5-1 338

5-2 Generalized Plane Stress 340

Problem Set 5-2 344

5-3 Compatability Equation in Terms of Stress Components 346

Problem Set 5-3 350

5-4 Airy Stress Function 351

Problem Set 5-4 361

5-5 Airy Stress Function in Terms of Harmonic Functions 368

5-6 Displacement Components for Plane Elasticity 369

Problem Set 5-6 373

5-7 Polynomial Solutions of TwoDimensional Problems in Rectangular Cartesian Coordinates 377

Problem Set 5-7 380

5-8 Plane Elasticity in Terms of Displacement Components 384

Problem Set 5-8 385

5-9 Plane Elasticity Relative to Oblique Coordinate Axes 386

Appendix 5A Plane Elasticity with Couple Stresses 390

5A-1 Introduction 390

5A-2 Equations of Equilibrium 391

5A-3 Deformation in Couple-Stress Theory 391

5A-4 Equations of Compatibility 393

5A-5 Stress Functions for Plane Problems with Couple Stresses 396

Appendix 5B Plane Theory of Elasticity in Terms of Complex Variables 398

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 Moment 403

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

5B-6 Plane Elasticity Boundary-Value Problems in Complex Form 408

5B-7 Note on Conformal Transformation 411

5B-8 Plane Elasticity Formulas in Terms of Curvilinear Coordinates 416

5B-9 Complex Variable Solution for Plane Region Bounded by Circle in the z Plane 419

Problem Set 5B 423

References 424

Bibliography 425

CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES 427

6-1 Equilibrium Equations in Polar Coordinates 427

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

6-3 Strain Displacement Relations in Polar Coordinates 430

Problem Set 6-3 432

6-4 Stress-Strain-Temperature Relations 433

Problem Set 6-4 435

6-5 Compatibility Equation for Plane Elasticity in Terms of Polar Coordinates 435

Problem Set 6-5 436

6-6 Axially Symmetric Problems 438

Problem Set 6-6 449

6-7 Plane-Elasticity Equations in Terms of Displacement Components 451

6-8 Plane Theory of Thermoelasticity 455

Problem Set 6-8 458

6-9 Disk of Variable Thickness and Nonhomogeneous Anisotropic Material 460

Problem Set 6-9 465

6-10 Stress Concentration Problem of Circular Hole in Plate 465

Problem Set 6-10 472

6-11 Examples 473

Problem Set 6-11 478

Appendix 6A Stress-Couple Theory of Stress Concentration Resulting from Circular Hole in Plate 487

Appendix 6B Stress Distribution of a Diametrically Compressed Plane Disk 492

References 494

CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD 496

7-1 General Problem of Three-Dimensional Elastic Bars Subjected to Transverse End Loads 496

7-2 Torsion of Prismatic Bars. Saint-Venant's Solution. Warping Function 499

Problem Set 7-2 505

7-3 Prandtl Torsion Function 505

Problem Set 7-3 509

7-4 A Method of Solution of the Torsion Problem: Elliptic Cross Section 510

Problem Set 7-4 514

7-5 Remarks on Solutions of the Laplace Equation, ▽2F = 0 515

Problem Set 7-5 517

7-6 Torsion of Bars with Tubular Cavities 520

Problem Set 7-6 523

7-7 Transfer of Axis of Twist 523

7-8 Shearing-Stress Component in Any Direction 524

Problem Set 7-8 529

7-9 Solution of Torsion Problem by the Prandtl Membrane Analogy 529

Problem Set 7-9 538

7-10 Solution by Method of Series. Rectangular Section 538

Problem Set 7-10 543

7-11 Bending of a Bar Subjected to Transverse End Force 544

Problem Set 7-11 556

7-12 Displacement of a Cantilever Beam Subjected to Transverse End Force 556

Problem Set 7-12 559

7-13 Center of Shear 560

Problem Set 7-13 561

7-14 Bending of a Bar with Elliptic Cross Section 563

7-15 Bending of a Bar with Rectangular Cross Section 565

Problems Set 7-15 570

Review Problems 571

Appendix 7A Analysis of Tapered Beams 572

References 576

CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY 578

8-1 Introduction 578

Problem Set 8-1 579

8-2 Equilibrium Equations 579

Problem Set 8-2 581

8-3 The Helmholtz Transformation 581

Problem Set 8-3 583

8-4 The Galerkin (Papkovich) Vector 583

Problem Set 8-4 585

8-5 Stress in Terms of the Galerkin Vector F 585

Problem Set 8-5 586

8-6 The Galerkin Vector: A Solution of the Equilibrium Equations of Elasticity 586

Problem Set 8-6 588

8-7 The Galerkin Vector kZ and Love's Strain Function for Solids of Revolution 588

Problem Set 8-7 591

8-8 Kelvin's Problem: Single Force Applied in the Interior of an Infinitely Extended Solid 591

Problem Set 8-8 593

8-9 The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson's Ratio 593

8-10 Solutions of the Boussinesq and Cerruti Problems by the Twinned Gradient Method 597

Problem Set 8-10 600

8-11 Additional Remarks on ThreeDimensional Stress Functions 600

References 601

Bibliography 601

INDEX 603