Continuum Mechanics And Plasticity

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ISBN: 9781584883630 | 1584883634
Cover: Hardcover
Copyright: 12/20/2004

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Tremendous advances in computer technologies and methods have precipitated a great demand for refinements in the constitutive models of plasticity. Such refinements include the development of a model that would account for material anisotropy and produces results that compare well with experimental data. Key to developing such models-and to meeting many other challenges in the field- is a firm grasp of the principles of continuum mechanics and how they apply to the formulation of plasticity theory. Also critical is understanding the experimental aspects of plasticity and material anisotropy.Integrating the traditionally separate subjects of continuum mechanics and plasticity, this book builds understanding in all of those areas. Part I provides systematic, comprehensive coverage of continuum mechanics, from a review of Carteisian tensors to the relevant conservation laws and constitutive equation. Part II offers an exhaustive presentation of the continuum theory of plasticity. This includes a unique treatment of the experimental aspects of plasticity, covers anisotropic plasticity, and incorporates recent research results related to the endochronic theory of plasticity obtained by the author and his colleagues.By bringing all of these together in one book, Continuum Mechanics and Plasticity facilitates the learning of solid mechanics. Its readers will be well prepared for pursuing either research related to the mechanical behavior of engineering materials or developmental work in engineering analysis and design.

Preface

xiii

Author

xvii

Part I Fundamentals of Continuum Mechanics

1 Cartesian Tensors

(42)

1.1 Introduction

(2)

1.1.1 Notations

(1)

1.1.2 Cartesian Coordinate System

(1)

1.1.3 Special Tensors

(1)

1.2 Vectors

(3)

1.2.1 Base Vectors and Components

(1)

1.2.2 Vector Addition and Multiplication

(1)

1.2.3 The e-δ Identity

(2)

1.3 The Transformation of Axes

(4)

1.4 The Dyadic Product (The Tensor Product)

(1)

1.5 Cartesian Tensors

(7)

1.5.1 General Properties

(3)

1.5.2 Multiplication of Tensors

(2)

1.5.3 The Component Form and Matrices

(1)

1.5.4 Quotient Law

(1)

1.6 Rotation of a Tensor

(8)

1.6.1 Orthogonal Tensor

(2)

1.6.2 Component Form of Rotation of a Tensor

(1)

1.6.3 Some Remarks

(5)

1.7 The Isotropic Tensors

(6)

1.8 Vector and Tensor Calculus

(6)

1.8.1 Tensor Field

(1)

1.8.2 Gradient, Divergence, Curl

(3)

1.8.3 The Theorem of Gauss

(3)

References

(1)

Problems

(4)

2 Stress

(34)

2.1 Introduction

(1)

2.2 Forces

(1)

2.3 Stress Vector

(1)

2.4 The Stress Tensor

(3)

2.5 Equations of Equilibrium

(2)

2.6 Symmetry of the Stress Tensor

(1)

2.7 Principal Stresses

(2)

2.8 Properties of Eigenvalues and Eigenvectors

(4)

2.9 Normal and Shear Components

(15)

2.9.1 Directions Along which Normal Components of σij Are Maximized or Minimized

(1)

2.9.2 The Maximum Shear Stress

(4)

2.10 Mean and Deviatoric Stresses

(1)

2.11 Octahedral Shearing Stress

(1)

2.12 The Stress Invariants

(3)

2.13 Spectral Decomposition of a Symmetric Tensor of Rank Two

(2)

2.14 Powers of a Tensor

(1)

2.15 Cayley-Hamilton Theorem

(1)

References

(1)

Problems

(5)

3 Motion and Deformation

(62)

3.1 Introduction

(1)

3.2 Material and Spatial Descriptions

(3)

3.2.1 Material Description

(1)

3.2.2 Spatial Description

(2)

3.3 Description of Deformation

(1)

3.4 Deformation of a Neighborhood

(5)

3.4.1 Homogeneous Deformations

(1)

3.4.2 Nonhomogeneous Deformations

(2)

3.5 The Deformation Gradient

(10)

3.5.1 The Polar Decomposition Theorem

(2)

3.5.2 Polar Decompositions of the Deformation Gradient

(8)

3.6 The Right Cauchy-Green Deformation Tensor

(7)

3.6.1 The Physical Meaning

(3)

3.6.2 Transformation Properties of CRS

101

(2)

3.6.3 Eigenvalues and Eigenvectors of CRS

103

(1)

3.6.4 Principal Invariants of CRS

104

(1)

3.7 Deformation of Volume and Area of a Material Element

105

(3)

3.8 The Left Cauchy-Green Deformation Tensor

108

(1)

3.9 The Lagrangian and Eulerian Strain Tensors

108

(26)

3.9.1 Definitions

108

(4)

3.9.2 Geometric Interpretation of the Strain Components

112

(3)

3.9.3 The Volumetric Strain

115

(3)

3.10 Other Strain Measures

118

(1)

3.11 Material Rate of Change

119

(3)

3.11.1 Material Description of the Material Derivative

119

(1)

3.11.2 Spatial Description of the Material Derivative

120

(2)

3.12 Dual Vectors and Dual Tensors

122

(2)

3.13 Velocity of a Particle Relative to a Neighboring Particle

124

(1)

3.14 Physical Significance of the Rate of Deformation Tensor

125

(3)

3.15 Physical Significance of the Spin Tensor

128

(1)

3.16 Expressions for D and W in Terms of F

129

(2)

3.17 Material Derivative of Strain Measures

131

(1)

3.18 Material Derivative of Area and Volume Elements

132

(1)

References

133

(1)

Problems

134

(7)

4 Conservation Laws and Constitutive Equation

141

(64)

4.1 Introduction

141

(1)

4.2 Bulk Material Rate of Change

142

(3)

4.3 Conservation Laws

145

(5)

4.3.1 The Conservation of Mass

145

(1)

4.3.2 The Conservation of Momentum

146

(2)

4.3.3 The Conservation of Energy

148

(2)

4.4 The Constitutive Laws in the Material Description

150

(14)

4.4.1 The Conservation of Mass

150

(1)

4.4.2 The Conservation of Momentum

151

(12)

4.4.3 The Conservation of Energy

163

(1)

4.5 Objective Tensors

164

(2)

4.6 Property of Deformation and Motion Tensors Under Reference Frame Transformation

166

(3)

4.7 Objective Rates

169

(5)

4.7.1 Some Objective Rates

169

(3)

4.7.2 Physical Meaning of the Jaumann Stress Rate

172

(2)

4.8 Finite Elasticity

174

(19)

4.8.1 The Cauchy Elasticity

175

(2)

4.8.2 Hyperelasticity

177

(4)

4.8.3 Isotropic Hyperelastic Materials

181

(4)

4.8.4 Applications of Isotropic Hyperelasticity

185

(8)

4.9 Infinitesimal Theory of Elasticity

193

(386)

4.9.1 Constitutive Equation

193

(2)

4.9.2 Homogeneous Deformations

195

(2)

4.9.3 Boundary-Value Problems

197

(1)

4.10 Hypoelasticity

197

(3)

References

200

(1)

Problems

200

(5)

Part II Continuum Theory of Plasticity

5 Fundamentals of Continuum Plasticity

205

(60)

5.1 Introduction

205

(1)

5.2 Some Basic Mechanical Tests

205

(26)

5.2.1 The Uniaxial Tension Test

205

(11)

5.2.2 The Uniaxial Compression Test

216

5.2.3 The Torsion Test

215

(10)

5.2.4 Strain Rate, Temperature, and Creep

225

(6)

5.3 Modeling the Stress-Strain Curve

231

(3)

5.4 The Effects of Hydrostatic Pressure

234

(3)

5.5 Torsion Test in the Large Strain Range

237

(23)

5.5.1 Introduction

237

(4)

5.5.2 Experimental Program and Procedures

241

(5)

5.5.3 Experimental Results and Discussions

246

(10)

5.5.4 Determination of Shear Stress-Strain Curve

256

(4)

References

260

(2)

Problems

262

(3)

6 The Flow Theory of Plasticity

265

(57)

6.1 Introduction

265

(1)

6.2 The Concept of Yield Criterion

265

(9)

6.2.1 Mathematical Expressions of Yield Surface

265

(6)

6.2.2 Geometrical Representation of Yield Surface in the Principal Stress Space

271

(3)

6.3 The Flow Rule

274

(2)

6.4 The Elastic-Perfectly Plastic Material

276

(10)

6.5 Strain-Hardening

286

(20)

6.5.1 Drucker's Postulate

287

(3)

6.5.2 The Isotropic-Hardening Rule

290

(6)

6.5.3 The Kinematic-Hardening Rule

296

(5)

6.5.4 General Form of Subsequent Yield Function and Its Flow Rule

301

(5)

6.6 The Return-Mapping Algorithm

306

(2)

6.7 Combined Axial-Torsion of Strain-Hardening Materials

308

(6)

6.8 Flow Theory in the Strain Space

314

(2)

6.9 Remarks

316

(1)

References

317

(1)

Problems

318

(4)

7 Advances in Plasticity

322

(77)

7.1 Introduction

322

(2)

7.2 Experimenal Determination of Yield Surfaces

324

(12)

7.2.1 Factors Affecting the Determination of Yield Surface

322

(6)

7.2.2 A Summary of Experiments Related to the Determination of Yield Surfaces

328

(4)

7.2.3 Yield Surface Versus Loading Surface

332

(3)

7.2.4 Yield Surface at Elevated Temperature

335

(1)

7.3 The Direction of the Plastic Strain Increment

336

(4)

7.4 Multisurface Models of Flow Plasticity

340

(11)

7.4.1 The Mroz Kinematic-Hardening Model

340

(4)

7.4.2 The Two-Surface Model of Dafalias and Popov

344

(7)

7.5 The Plastic Strain Trajectory Approach

351

(6)

7.5.1 The Theory of Ilyushin

351

(5)

7.5.2 The Endochronic Theory of Plasticity

356

(1)

7.6 Finite Plastic Deformation

357

(36)

7.6.1 The Stress and Strain Measures

358

(1)

7.6.2 The Decomposition of Strain and Strain Rate

358

(6)

7.6.3 The Objective Rates

364

(4)

7.6.4 A Theory of Finite Elastic-Plastic Deformation

368

(6)

7.6.5 A Study of Simple Shear Using Rigid-Plastic Equations with Linear Kinematic Hardening

374

(10)

7.6.6 The Yield Criterion for Finite Plasticity

384

(9)

References

393

(4)

Problems

397

(2)

8 Internal Variable Theory of Thermo-Mechanical Behaviors and Endochronic Theory of Plasticity

399

(56)

8.1 Introduction

399

(1)

8.2 Concepts and Terminologies of Thermodynamics

399

(4)

8.2.1 The First Law of Thermodynamics

399

(1)

8.2.2 State Variables, State Functions, and the Second Law of Thermodynamics

400

(3)

8.3 Thermodynamics of Internal State Variables

403

(7)

8.3.1 Irreversible Systems

403

(2)

8.3.2 The Clausius-Duhem Inequality

405

(1)

8.3.3 The Helmholtz Formulation of Thermo-Mechanical Behavior

406

(2)

8.3.4 The Gibbs Formulation of Thermo-Mechanical Behavior

408

(2)

8.4 The Endochronic Theory of Plasticity

410

(40)

8.4.1 The Concepts of the Endochronic Theory

410

(2)

8.4.2 The Simple Endochronic Theory of Plasticity

412

(9)

8.4.3 The Improved Endochronic Theory of Plasticity

421

(4)

8.4.4 Derivation of the Flow Theory of Plasticity from Endochronic Theory

425

(2)

8.4.5 Applications of the Endochronic Theory to Metals

427

(14)

8.4.6 The Endochronic Theory of Viscoplasticity

441

(9)

References

450

(2)

Problems

452

(3)

9 Topics in Endochronic Plasticity

455

(88)

9.1 Introduction

455

(1)

9.2 An Endochronic Theory of Anisotropic Plasticity

455

(13)

9.2.1 An Endochronic Theory Accounting for Deformation Induced Anisotropy

455

(6)

9.2.2 An Endochronic Theory for Anisotropic Sheet Metals

461

(7)

9.3 Endochronic Plasticity in the Finite Strain Range

468

(19)

9.3.1 Corotational Integrals

469

(5)

9.3.2 Endochronic Equations for Finite Plastic Deformation

474

(2)

9.3.3 Application to a Rigid-Plastic Thin-Walled Tube Under Torsion

476

(11)

9.4 An Endochronic Theory for Porous and Granular Materials

487

(19)

9.4.1 The Endochronic Equations

490

(10)

9.4.2 Application to Concrete

500

(2)

9.4.3 Application to Sand

502

(1)

9.4.4 Application to Porous Aluminum

503

(3)

9.5 An Endochronic Formulation of a Plastically Deformed Damaged Continuum

506

(31)

9.5.1 Introduction

506

(1)

9.5.2 The Anisotropic Damage Tensor

507

(5)

9.5.3 Gross Stress, Net Stress, and Effective Stress

512

(4)

9.5.4 An Internal State Variables Theory

516

(5)

9.5.5 Plasticity and Damage

521

(2)

9.5.6 The Constitutive Equations and Constraints

523

(3)

9.5.7 A Brief Summary of Wu and Nanakorn's Endochronic CDM

526

(4)

9.5.8 Application

530

(5)

9.5.9 Concluding Remarks

535

(2)

References

537

(4)

Problems

541

(2)

10 Anisotropic Plasticity for Sheet Metals

543

(1)

10.1 Introduction

543

(2)

10.2 Standard Tests for Sheet Metal

545

(1)

10.2.1 The Uniaxial Tension Test

545

(1)

10.2.2 Equibiaxial Tension Test

545

(1)

10.2.3 Hydraulic Bulge Test

545

(1)

10.2.4 Through-Thickness Compression Test

545

(1)

10.2.5 Plane-Strain Compression Test

546

(1)

10.2.6 Simple Shear Test

546

(1)

10.3 Experimental Yield Surface for Sheet Metal

546

(2)

10.4 Hill's Anisotropic Theory of Plasticity

548

(7)

10.4.1 The Quadratic Yield Criterion

548

(2)

10.4.2 The Flow Rule and the R-Ratio

550

(2)

10.4.3 The Equivalent Stress and Equivalent Strain

552

(1)

10.4.4 The Anomalous Behavior

553

(2)

10.5 Nonquadratic Yield Functions

555

(3)

10.6 Anisotropic Plasticity Using Combined Isotropic-Kinematic Hardening

558

(17)

10.6.1 Introduction

558

(2)

10.6.2 The Anisotropic Theory Using Combined Isotropic-Kinematic Hardening

560

(6)

10.6.3 Results and Discussion

566

(6)

10.6.4 Summary and Conclusion

572

(3)

References

575

(2)

Problems

577

(2)

11 Description of Anisotropic Material Behavior Using Curvilinear Coordinates

579

(30)

11.1 Convected Coordinate System and Convected Material Element

579

(1)

11.2 Curvilinear Coordinates and Base Vectors

580

(4)

11.3 Tensors and Special Tensors

584

(6)

11.4 Multiplication of Vectors

590

(1)

11.5 Physical Components of a Vector

591

(1)

11.6 Differentiation of a Tensor with Respect to the Space Coordinates

592

(7)

11.6.1 Derivative of a Scalar

593

(1)

11.6.2 Derivatives of a Vector

593

(1)

11.6.3 Derivatives of a Tensor

594

(5)

11.7 Strain Tensor

599

(4)

11.8 Strain-Displacement Relations

603

(3)

11.9 Stress Vector and Stress Tensor

606

(3)

11.10 Physical Components of the Stress Tensor

609

(1)

11.11 Other Stress Tensors and the Cartesian Stress Components

610

(2)

11.12 Stress Rate and Strain Rate

612

(5)

11.13 Further Discussion of Stress Rate

617

(2)

11.14 A Theory of Plasticity for Anisotropic Metals

619

(11)

11.14.1 The Yield Function

621

(7)

11.14.2 The Flow Rule

628

(1)

11.14.3 The Strain Hardening

628

(1)

11.14.4 Elastic Constitutive Equations

629

(1)

References

630

(1)

Problems

631

(2)

12 Combined Axial-Torsion of Thin-Walled Tubes

633

(35)

12.1 Introduction

633

(1)

12.2 Convected Coordinates in the Combined Axial-Torsion of a Thin-Walled Tube

634

(3)

12.3 The Yield Function

637

(5)

12.3.1 The Mises Yield Criterion

637

(1)

12.3.2 A Yield Criterion Proposed by Wu

638

(4)

12.4 Flow Rule and Strain Hardening

642

(4)

12.5 Elastic Constitutive Equations

646

(1)

12.6 Algorithm for Computation

647

(2)

12.7 Nonlinear Kinematic Hardening

649

(1)

12.8 Description of Yield Surface with Various Preloading Paths

649

(16)

12.8.1 Path (1) - Axial Tension

652

(4)

12.8.2 Path (2) - Torsion

656

(4)

12.8.3 Path (3) - Proportional Loading

660

(1)

12.8.4 Tor-Ten Path (4)

660

(2)

12.8.5 Tor-Ten Path (5)

662

(2)

12.8.6 Tor-Ten Path with Constant Shear Strain

664

(1)

12.9 A Stress Path of Tension-Unloading Followed by Torsion

665

(3)

12.10 Summary and Discussion

668

(1)

References

669

(1)

Problems

670

(1)

Answers and Hints to Selected Problems

671

(6)

Author Index

677

(4)

Subject Index

681

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Continuum Mechanics And Plasticity

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