Elastic and Inelastic Stress Analysis

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ISBN: 9781560326861 | 1560326867
Cover: Hardcover
Copyright: 2/1/1997

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Presents clearly certain key aspects of inelastic solid mechanics centered around viscoelasticity, creep, viscoplasticity and plasticity. Covers fundamentals of elasticity, useful constitutive laws and applications to simple structural members. DLC: Elasticity.

Preface

xiii

Part I FUNDAMENTALS

(98)

Chapter 1 Introduction to Cartesian Tensors

(38)

1.1 Introduction

(1)

1.2 The Free Index

(1)

1.3 The Dummy Index

(2)

1.4 Vectors and Tensors

(5)

1.5 The Transformation Matrix; Orthogonal Matrices

(4)

1.6 Simple Operations on Tensors

(3)

1.7 Symmetry and Skew Symmetry

(1)

1.8 Vector Algebra in Index Notation

(3)

1.9 Differential Operators for Cartesian Tensors in Index Notation

(2)

1.10 Additional Properties of Second-Order Tensors

(3)

1.11 Integral Theorems

(3)

1.12 Stokes' Theorem

(4)

1.13 Closure

(7)

Chapter 2 Stress

(31)

2.1 Force Distributions

(1)

2.2 Stress

(3)

2.3 Transformation Equations for Stress

(3)

2.4 Cauchy's Formula

(1)

2.5 Equations of Motion

(4)

2.6 Principal Stresses

(5)

2.7 The Stress Quadric Surfaces

(3)

2.8 Three-Dimensional Mohr's Circles

(6)

2.9 Closure

(6)

Chapter 3 Strain

(29)

3.1 Introduction

(1)

3.2 The Displacement Gradient Matrix

(4)

3.3 Superposition; Small Deformation

(2)

3.4 Strain and Rotation Tensors for Small Deformation

(10)

3.5 Cubical Dilatation and Distortion

(2)

3.6 Strain as a Second-Order Tensor

(1)

3.7 Equations of Compatibility

(3)

3.8 Closure

(6)

Part II USEFUL CONSTITUTIVE LAWS

(298)

Chapter 4 Behavior of Engineering Materials

(32)

4.1 Introduction

(1)

PART A One-Dimensional Macroscopic Behavior

100

(13)

4.2 The Tensile Test

100

(6)

4.3 Strain Hardening

106

(2)

4.4 Other Properties Pertaining to the Tensile Test

108

(2)

4.5 Creep

110

(3)

PART B One-Dimensional Idealized Material Behavior

113

(18)

4.6 Idealized One-Dimensional Stress-Strain Diagrams

113

(2)

4.7 Phenomenological Model Classification of Materials

115

(8)

4.8 Closure

123

(8)

Chapter 5 Linear Elastic Behavior

131

(30)

5.1 Introduction

131

(1)

5.2 Isotropic Hooke's Law via Superposition

131

(5)

5.3 Interdependence of Engineering Elastic Constants for Isotropic Materials

136

(3)

5.4 Generalized Hooke's Law; Anisotropic Behavior

139

(7)

5.5 Alternative Forms of Isotropic Hooke's Law; Relations between Elastic Constants

146

(4)

5.6 Introduction to Thermoelasticity

150

(4)

5.7 Closure

154

(7)

Chapter 6 Linear Viscoelastic Behavior

161

(41)

6.1 Introduction

161

(1)

PART A One-Dimensional Models

162

(16)

6.2 Simple Models and the Standard Test Procedures

162

(5)

6.3 More Complex One-Dimensional Models

167

(5)

6.4 The Creep Compliance and Relaxation Modulus Functions

172

(3)

6.5 Useful Results from Differential and Integral Forms of Viscoelastic Constitutive Laws

175

(3)

PART B Multidimensional Linear, Isotropic Viscoelastic Laws

178

(24)

6.6 Introduction

178

(1)

6.7 Postulates from Simple Tests

179

(3)

6.8 Extrapolation to Three Dimensions; Isotropy

182

(4)

6.9 Bulk and Distortional Behaviors

186

(3)

6.10 The Viscoelastic Poisson Effect

189

(4)

6.11 Nonisothermal Linear Viscoelastic Behavior

193

(3)

6.12 Closure

196

(6)

Chapter 7 Introduction to Nonlinear Viscoelastic Behavior: Creep

202

(46)

7.1 Introduction

202

(1)

PART A Nonlinear Viscoelastic Behavior in One Dimension

203

(26)

7.2 Creep Laws for Constant Stress; General Aspects

203

(2)

7.3 Steady Creep; Constant Stress

205

(4)

7.4 Transient and Total Creep; Constant Stress

209

(3)

7.5 Nonlinear Viscoelastic Laws for Variable Stress; Strain-Hardening Hypothesis

212

(9)

7.6 Time-Hardening Hypothesis

221

(3)

7.7 Comparison of Hypotheses Using a Step-Stress Test

224

(5)

PART B Nonlinear Viscoelastic Behavior in Three Dimensions

229

(19)

7.8 A Comment

229

(1)

7.9 Introductory Considerations

229

(2)

7.10 Powers of Tensors

231

(2)

7.11 Three-Dimensional, Nonlinear Viscoelastic Law for Constant Stress

233

(3)

7.12 Neglecting the Effect of the Third Invariant

236

(1)

7.13 Experimental Input from Tensile Test; Some Three-Dimensional Constant Stress Laws

237

(2)

7.14 Three-Dimensional, Nonlinear, Viscoelastic Laws for Variable Stress

239

(3)

7.15 Closure

242

(6)

Chapter 8 Plasticity

248

(83)

8.1 Introduction

248

(1)

PART A One-Dimensional Plasticity and Viscoplasticity

249

(13)

8.2 Rate-Independent Plasticity Models

249

(4)

8.3 A Rate-Sensitive Viscoplastic Mechanical Model

253

(4)

8.4 The Case of Stress History Varying Linearly with Time

257

(3)

8.5 Some Generalizations for Rate-Sensitive Viscoplastic Materials

260

(2)

PART B Three-Dimensional Nonhardening Plasticity Theory

262

(36)

8.6 General Yield Criteria in Three Dimensions

262

(2)

8.7 The General Yield Surface

264

(4)

8.8 Tresca and Mises Yield Criteria for Ductile Materials

268

(4)

8.9 Maximum Distortional Strain-Energy and Octahedral Stress Theories

272

(2)

8.10 Yield Surfaces for the Tresca and Mises Criteria

274

(10)

8.11 Drucker's Postulate; Normality Principle and Convexity

284

(6)

8.12 Prandtl-Reuss Equations; Flow Laws

290

(8)

PART C Three-Dimensional Plasticity with Strain and Strain-Rate Hardening

298

(33)

8.13 Preliminary Remarks

298

(1)

8.14 Isotropic Strain Hardening; Universal Plastic Stress-Strain Curve

299

(3)

8.15 Flow Law for Isotropic Hardening; Case of Power-Law Hardening

302

(5)

8.16 Additional Hardening Hypotheses and Flow Laws

307

(6)

8.17 Three-Dimensional Viscoplasticity

313

(6)

8.18 Closure

319

(12)

Chapter 9 Boundary Value Problems

331

(66)

9.1 Introduction

331

(1)

PART A Summary of Governing Field Equations

332

(9)

9.2 Basic Laws and Geometric Conditions

332

(3)

9.3 Constitutive Laws

335

(6)

PART B Key Boundary Value Problems

341

(13)

9.4 Boundary Value Problem Formulations for Linear Elasticity

341

(5)

9.5 Correspondence Principle of Linear Viscoelasticity

346

(5)

9.6 Nonlinear Viscous Flow; Analogies with Nonlinear Elasticity and Plasticity

351

(3)

PART C Uniqueness Theorems

354

(5)

9.7 Uniqueness Theorem for Linear Elasticity

354

(3)

9.8 Comments on Uniqueness for Inelastic Materials

357

(1)

9.9 Saint-Venant's Principle

358

(1)

PART D Closed-Form Solutions to Problems for Different Constitutive Laws

359

(38)

9.10 Linear Elastic Prismatic Bar Hanging by Its Own Weight

359

(5)

9.11 Linear Elastic Beam in Pure Bending

364

(6)

9.12 Application of the Linear Viscoelastic Correspondence Principle

371

(7)

9.13 Nonlinear Viscous Behavior; Examination of Superposition

378

(3)

9.14 Elastic, Perfectly Plastic Deformation of a Thick-Walled Sphere

381

(6)

9.15 Closure

387

(10)

Part III APPLICATIONS TO SIMPLE STRUCTURAL MEMBERS

397

(307)

Chapter 10 Flexure of Beams

397

(4)

10.1 Introduction

397

(4)

PART A Linear Elastic Beams

401

(9)

10.2 Euler-Bernoulli Theory of Bending

401

(1)

10.3 Deflection Computations Using Discontinuty Functions

402

(8)

PART B Linear Viscoelastic Beams

410

(11)

10.4 Basic Considerations; Stresses

410

(3)

10.5 Deflections in Linear Viscoelastic Beams; Basic Equations

413

(1)

10.6 Case of Suddenly Applied Constant External and Support Forces

414

(3)

10.7 Statically Indeterminate Beams with Deformable Supports

417

(4)

PART C Nonlinear Viscous Beams

421

(13)

10.8 Stress and Strain Considerations

421

(6)

10.9 Deflection of Nonlinear Viscous Beams

427

(3)

10.10 Extension to Case of Inhomogeneous Viscosity

430

(4)

PART D Plastic Behavior of Beams

434

(26)

10.11 The Linear Elastic, Perfectly Plastic Case with Pure Bending

434

(3)

10.12 Limit Analysis of Engineering Structures

437

(4)

10.13 Limit Analysis of Indeterminate Beams

441

(3)

10.14 General Elastoplastic Behavior of Beams

444

(3)

10.15 Closure

447

(13)

Chapter 11 Torsion of Shafts

460

(63)

11.1 Introduction

460

(1)

PART A Circular Shafts

460

(24)

11.2 General Considerations

460

(2)

11.3 Linear Elastic Circular Shafts

462

(2)

11.4 Consideration of Exactness of Elastic Solution

464

(4)

11.5 Linear Viscoelastic Circular Shafts

468

(5)

11.6 Nonlinear Viscous Circular Shafts; Simple Torsion Test Results

473

(2)

11.7 Useful Relations for Nonlinear Viscous Circular Shafts

475

(3)

11.8 Linear Elastic, Perfectly Plastic Torsion

478

(3)

11.9 Residual Stress and Strain

481

(3)

PART B Noncircular Shafts

484

(39)

11.10 Saint-Venant Theory of Torsion; Linear Viscoelastic and Linear Elastic Shafts

484

(7)

11.11 Alternative Formulation in Terms of Conjugate Warping Function

491

(2)

11.12 Alternative Formulation in Terms of Prandtl Stress Function

493

(6)

11.13 The Elastic Rectangular Shaft

499

(6)

11.14 Nonlinear Viscous Noncircular Shafts

505

(4)

11.15 Closure

509

(14)

Chapter 12 Plane Strain

523

(50)

12.1 Introduction

523

(2)

PART A Elastic Plane Strain

525

(20)

12.2 Basic Equations

525

(4)

12.3 Airy Stress Function

529

(4)

12.4 Elastic Plane Strain in Polar Coordinates

533

(2)

12.5 Problems with Axial Symmetry

535

(4)

12.6 Stress Field of an Edge Dislocation in an Elastic Medium

539

(6)

PART B Linear Viscoelastic Plane Strain

545

(9)

12.7 Governing Equations

545

(3)

12.8 Alternative Form for the Constitutive Law; Application to Thick-Walled Viscoelastic Cylinders

548

(6)

PART C Nonlinear Viscous and Perfectly Plastic Plane Strain

554

(19)

12.9 Formulation of Governing Equations

554

(2)

12.10 Formulations for the Case of Axial Symmetry

556

(2)

12.11 Solutions to Nonlinear Viscous Axially Symmetric Governing Equations

558

(4)

12.12 Closure

562

(11)

Chapter 13 Plane Stress

573

(68)

13.1 Introduction

573

(2)

PART A Elastic General Plane Stress

575

(11)

13.2 Basic Equations

575

(3)

13.3 Airy Stress Function for General Plane Stress

578

(5)

13.4 Plane-Stress Problems; Case of Symmetric Loading

583

(3)

PART B Elastic Two-Dimensional Plane Stress

586

(21)

13.5 Two-Dimensional Plane Stress Boundary Value Problem

586

(5)

13.6 Stretched Plate with a Small Hole

591

(7)

13.7 Pure Bending of a Curved Beam

598

(3)

13.8 Axially Symmetric Problems with Body Forces

601

(6)

PART C Inelastic Two-Dimensional Plane Stress

607

(34)

13.9 Linear Viscoelastic Plane Stress

607

(4)

13.10 Vsicoelastic Solution via the Correspondence Principle

611

(4)

13.11 Constitutive Laws for Perfectly Plastic Plane Stress

615

(6)

13.12 Plastic Behavior of the Rotating Disk

621

(5)

13.13 Nonlinear Viscous Plane Stress

626

(2)

13.14 Closure

628

(13)

APPENDIXES

641

(63)

Appendix I Sufficiency Conditions for Compatibility

641

(6)

I.1 Simply Connected Domains

641

(2)

I.2 Multiply Connected Domains

643

(4)

Appendix II Discontinuity Functions

647

(10)

II.1 Introduction

647

(1)

II.2 The Unit Step Function and the Ramp Function

647

(1)

II.3 The Dirac Delta Function

648

(3)

II.4 The Doublet Function

651

(2)

II.5 Vehicle Functions for Obtaining Delta and Doublet Functions

653

(3)

II.6 Closure

656

(1)

Appendix III The Laplace Transform

657

(8)

III.1 Introduction

657

(1)

III.2 Basic Considerations

657

(1)

III.3 The Laplace Transform of Discontinuity Functions

658

(2)

III.4 Transform Pairs

660

(1)

III.5 Transforms of Integrals

661

(1)

III.6 Limit Theorems

662

(2)

III.7 Closure

664

(1)

Appendix IV Note on the Flexure Formula for Beams

665

(4)

Appendix V Traction Conditions in Terms of for Plane Strain and Two-Dimensional Plane Stress

669

(5)

Appendix VI Cesaro Integrals in Terms of for Plane Strain and Two-Dimensional Plane Stress

674

(4)

Appendix VII Microscopic Considerations of Properties of Solids

678

(1)

VII.1 Introduction

678

(1)

PART A Atomic and Molecular Structure

678

(9)

VII.2 Introduction to Interatomic Energies and Forces

678

(3)

VII.3 Lattice Geometry

681

(1)

VII.4 Crystalline Structure of Engineering Materials

682

(4)

VII.5 Polycrystalline Structure of Engineering Materials

686

(1)

PART A Mechanisms of Mechanical Deformation

687

(17)

VII.6 Nature of Elastic Deformation in Crystals

687

(2)

VII.7 Introduction to Plastic Deformation in Crystals

689

(3)

VII.8 Dislocations in Crystals

692

(2)

VII.9 Stress Field of Dislocations

694

(3)

VII.10 Dislocations and Plastic Deformation

697

(2)

VII.11 Creep in Polycrystalline Materials

699

(2)

VII.12 Deformation of Amorphous Polymers

701

(2)

VII.13 Closure

703

(1)

Index

704

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Elastic and Inelastic Stress Analysis

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