Fracture Mechanics: Fundamentals and Applications Book

Part I

Introduction 1

Chapter 1

History and Overview 3

1.1 Why Structures Fail 3

1.2 Historical Perspective 6

1.2.1 Early Fracture Research 8

1.2.2 The Liberty Ships 9

1.2.3 Post-War Fracture Mechanics Research 10

1.2.4 Fracture Mechanics from 1960 to 1980 10

1.2.5 Fracture Mechanics from 1980 to the Present 12

1.3 The Fracture Mechanics Approach to Design 12

1.3.1 The Energy Criterion 12

1.3.2 The Stress-Intensity Approach 14

1.3.3 Time-Dependent Crack Growth and Damage Tolerance 15

1.4 Effect of Material Properties on Fracture 16

1.5 A Brief Review of Dimensional Analysis 18

1.5.1 The Buckingham P-Theorem 18

1.5.2 Dimensional Analysis in Fracture Mechanics 19

References 21

Part II

Fundamental Concepts 23

Chapter 2

Linear Elastic Fracture Mechanics 25

2.1 An Atomic View of Fracture 25

2.2 Stress Concentration Effect of Flaws 27

2.3 The Griffith Energy Balance 29

2.3.1 Comparison with the Critical Stress Criterion 31

2.3.2 Modified Griffith Equation 32

2.4 The Energy Release Rate 34

2.5 Instability and the R Curve 38

2.5.1 Reasons for the R Curve Shape 39

2.5.2 Load Control vs. Displacement Control 40

2.5.3 Structures with Finite Compliance 41

2.6 Stress Analysis of Cracks 42

2.6.1 The Stress Intensity Factor 43

2.6.2 Relationship between K and Global Behavior 45

2.6.3 Effect of Finite Size 48

2.6.4 Principle of Superposition 54

2.6.5 Weight Functions 56

2.7 Relationship between K and G 58

2.8 Crack-Tip Plasticity 61

2.8.1 The Irwin Approach 61

2.8.2 The Strip-Yield Model 64

2.8.3 Comparison of Plastic Zone Corrections 66

2.8.4 Plastic Zone Shape 66

2.9 K-Controlled Fracture 69

2.10 Plane Strain Fracture: Fact vs. Fiction 72

2.10.1 Crack-Tip Triaxiality 73

2.10.2 Effect of Thickness on Apparent Fracture Toughness 75

2.10.3 Plastic Zone Effects 78

2.10.4 Implications for Cracks in Structures 79

2.11 Mixed-Mode Fracture 80

2.11.1 Propagation of an Angled Crack 81

2.11.2 Equivalent Mode I Crack 83

2.11.3 Biaxial Loading 84

2.12 Interaction of Multiple Cracks 86

2.12.1 Coplanar Cracks 86

2.12.2 Parallel Cracks 86

Appendix 2: Mathematical Foundations of Linear Elastic

Fracture Mechanics 88

A2.1 Plane Elasticity 88

A2.1.1 Cartesian Coordinates 89

A2.1.2 Polar Coordinates 90

A2.2 Crack Growth Instability Analysis 91

A2.3 Crack-Tip Stress Analysis 92

A2.3.1 Generalized In-Plane Loading 92

A2.3.2 The Westergaard Stress Function 95

A2.4 Elliptical Integral of the Second Kind 100

References 101

Chapter 3

Elastic-Plastic Fracture Mechanics 103

3.1 Crack-Tip-Opening Displacement 103

3.2 The J Contour Integral 107

3.2.1 Nonlinear Energy Release Rate 108

3.2.2 J as a Path-Independent Line Integral 110

3.2.3 J as a Stress Intensity Parameter 111

3.2.4 The Large Strain Zone 113

3.2.5 Laboratory Measurement of J 114

3.3 Relationships Between J and CTOD 120

3.4 Crack-Growth Resistance Curves 123

3.4.1 Stable and Unstable Crack Growth 124

3.4.2 Computing J for a Growing Crack 126

3.5 J-Controlled Fracture 128

3.5.1 Stationary Cracks 128

3.5.2 J-Controlled Crack Growth 131

3.6 Crack-Tip Constraint Under Large-Scale Yielding 133

3.6.1 The Elastic T Stress 137

3.6.2 J-Q Theory 140

3.6.2.1 The J-Q Toughness Locus 142

3.6.2.2 Effect of Failure Mechanism

on the J-Q Locus 144

3.6.3 Scaling Model for Cleavage Fracture 145

3.6.3.1 Failure Criterion 145

3.6.3.2 Three-Dimensional Effects 147

3.6.3.3 Application of the Model 148

3.6.4 Limitations of Two-Parameter Fracture Mechanics 149

Appendix 3: Mathematical Foundations

of Elastic-Plastic Fracture Mechanics 153

A3.1 Determining CTOD from the Strip-Yield Model 153

A3.2 The J Contour Integral 156

A3.3 J as a Nonlinear Elastic Energy Release Rate 158

A3.4 The HRR Singularity 159

A3.5 Analysis of Stable Crack Growth

in Small-Scale Yielding 162

A3.5.1 The Rice-Drugan-Sham Analysis 162

A3.5.2 Steady State Crack Growth 166

A3.6 Notes on the Applicability of Deformation Plasticity

to Crack Problems 168

References 171

Chapter 4

Dynamic and Time-Dependent Fracture 173

4.1 Dynamic Fracture and Crack Arrest 173

4.1.1 Rapid Loading of a Stationary Crack 174

4.1.2 Rapid Crack Propagation and Arrest 178

4.1.2.1 Crack Speed 180

4.1.2.2 Elastodynamic Crack-Tip Parameters 182

4.1.2.3 Dynamic Toughness 184

4.1.2.4 Crack Arrest 186

4.1.3 Dynamic Contour Integrals 188

4.2 Creep Crack Growth 189

4.2.1 The C * Integral 191

4.2.2 Short-Time vs. Long-Time Behavior 193

4.2.2.1 The C t Parameter 195

4.2.2.2 Primary Creep 196

4.3 Viscoelastic Fracture Mechanics 196

4.3.1 Linear Viscoelasticity 197

4.3.2 The Viscoelastic J Integral 200

4.3.2.1 Constitutive Equations 200

4.3.2.2 Correspondence Principle 200

4.3.2.3 Generalized J Integral 201

4.3.2.4 Crack Initiation and Growth 202

4.3.3 Transition from Linear to Nonlinear Behavior 204

Appendix 4: Dynamic Fracture Analysis 206

A4.1 Elastodynamic Crack Tip Fields 206

A4.2 Derivation of the Generalized Energy

Release Rate 209

References 213

Part III

Material Behavior 217

Chapter 5

Fracture Mechanisms in Metals 219

5.1 Ductile Fracture 219

5.1.1 Void Nucleation 219

5.1.2 Void Growth and Coalescence 222

5.1.3 Ductile Crack Growth 231

5.2 Cleavage 234

5.2.1 Fractography 234

5.2.2 Mechanisms of Cleavage Initiation 235

5.2.3 Mathematical Models of Cleavage Fracture

Toughness 238

5.3 The Ductile-Brittle Transition 247

5.4 Intergranular Fracture 249

Appendix 5: Statistical Modeling of Cleavage Fracture 250

A5.1 Weakest Link Fracture 250

A5.2 Incorporating a Conditional Probability

of Propagation 252

References 254

Chapter 6

Fracture Mechanisms in Nonmetals 257

6.1 Engineering Plastics 257

6.1.1 Structure and Properties of Polymers 258

6.1.1.1 Molecular Weight 258

6.1.1.2 Molecular Structure 259

6.1.1.3 Crystalline and Amorphous Polymers 259

6.1.1.4 Viscoelastic Behavior 260

6.1.1.5 Mechanical Analogs 263

6.1.2 Yielding and Fracture in Polymers 265

6.1.2.1 Chain Scission and Disentanglement 265

6.1.2.2 Shear Yielding and Crazing 265

6.1.2.3 Crack-Tip Behavior 267

6.1.2.4 Rubber Toughening 268

6.1.2.5 Fatigue 270

6.1.3 Fiber-Reinforced Plastics 270

6.1.3.1 Overview of Failure Mechanisms 271

6.1.3.2 Delamination 272

6.1.3.3 Compressive Failure 275

6.1.3.4 Notch Strength 278

6.1.3.5 Fatigue Damage 280

6.2 Ceramics and Ceramic Composites 282

6.2.1 Microcrack Toughening 285

6.2.2 Transformation Toughening 286

6.2.3 Ductile Phase Toughening 287

6.2.4 Fiber and Whisker Toughening 288

6.3 Concrete and Rock 291

References 293

Part IV

Applications 297

Chapter 7

Fracture Toughness Testing of Metals 299

7.1 General Considerations 299

7.1.1 Specimen Configurations 299

7.1.2 Specimen Orientation 301

7.1.3 Fatigue Precracking 303

7.1.4 Instrumentation 305

7.1.5 Side Grooving 307

7.2 K Ic Testing 308

7.2.1 ASTM E 399 309

7.2.2 Shortcomings of E 399 and Similar Standards 312

7.3 K-R Curve Testing 316

7.3.1 Specimen Design 317

7.3.2 Experimental Measurement of K-R Curves 318

7.4 J Testing of Metals 320

7.4.1 The Basic Test Procedure and JIc Measurements 320

7.4.2 J-R Curve Testing 322

7.4.3 Critical J Values for Unstable Fracture 324

7.5 CTOD Testing 326

7.6 Dynamic and Crack-Arrest Toughness 329

7.6.1 Rapid Loading in Fracture Testing 329

7.6.2 KIa Measurements 330

7.7 Fracture Testing of Weldments 334

7.7.1 Specimen Design and Fabrication 334

7.7.2 Notch Location and Orientation 335

7.7.3 Fatigue Precracking 337

7.7.4 Posttest Analysis 337

7.8 Testing and Analysis of Steels in the Ductile-Brittle Transition Region 338

7.9 Qualitative Toughness Tests 340

7.9.1 Charpy and Izod Impact Test 341

7.9.2 Drop Weight Test 342

7.9.3 Drop Weight Tear and Dynamic Tear Tests 344

Appendix 7: Stress Intensity, Compliance, and Limit Load Solutions

for Laboratory Specimens 344

References 350

Chapter 8

Fracture Testing of Nonmetals 353

8.1 Fracture Toughness Measurements in Engineering Plastics 353

8.1.1 The Suitability of K and J for Polymers 353

8.1.1.1 K-Controlled Fracture 354

8.1.1.2 J-Controlled Fracture 357

8.1.2 Precracking and Other Practical Matters 360

8.1.3 Klc Testing 362

8.1.4 J Testing 365

8.1.5 Experimental Estimates of Time-Dependent Fracture Parameters 369

8.1.6 Qualitative Fracture Tests on Plastics 371

8.2 Interlaminar Toughness of Composites 373

8.3 Ceramics 378

8.3.1 Chevron-Notched Specimens 378

8.3.2 Bend Specimens Precracked by Bridge Indentation 380

References 382

Chapter 9

Application to Structures 385

9.1 Linear Elastic Fracture Mechanics 385

9.1.1 KI for Part-Through Cracks 387

9.1.2 Influence Coefficients for Polynomial Stress Distributions 388

9.1.3 Weight Functions for Arbitrary Loading 392

9.1.4 Primary, Secondary, and Residual Stresses 394

9.1.5 A Warning about LEFM 395

9.2 The CTOD Design Curve 395

9.3 Elastic-Plastic J-Integral Analysis 397

9.3.1 The EPRI J-Estimation Procedure 398

9.3.1.1 Theoretical Background 398

9.3.1.2 Estimation Equations 399

9.3.1.3 Comparison with Experimental J Estimates 401

9.3.2 The Reference Stress Approach 403

9.3.3 Ductile Instability Analysis 405

9.3.4 Some Practical Considerations 408

9.4 Failure Assessment Diagrams 410

9.4.1 Original Concept 410

9.4.2 J-Based FAD 412

9.4.3 Approximations of the FAD Curve 415

9.4.4 Estimating the Reference Stress 416

9.4.5 Application to Welded Structures 423

9.4.5.1 Incorporating Weld Residual Stresses 423

9.4.5.2 Weld Misalignment 426

9.4.5.3 Weld Strength Mismatch 427

9.4.6 Primary vs. Secondary Stresses in the FAD Method 428

9.4.7 Ductile-Tearing Analysis with the FAD 430

9.4.8 Standardized FAD-Based Procedures 430

9.5 Probabilistic Fracture Mechanics 432

Appendix 9: Stress Intensity and Fully Plastic J Solutions

for Selected Configurations 434

References 449

Chapter 10

Fatigue Crack Propagation 451

10.1 Similitude in Fatigue 451

10.2 Empirical Fatigue Crack Growth Equations 453

10.3 Crack Closure 457

10.3.1 A Closer Look at Crack-Wedging Mechanisms 460

10.3.2 Effects of Loading Variables on Closure 463

10.4 The Fatigue Threshold 464

10.4.1 The Closure Model for the Threshold 465

10.4.2 A Two-Criterion Model 466

10.4.3 Threshold Behavior in Inert Environments 470

10.5 Variable Amplitude Loading and Retardation 473

10.5.1 Linear Damage Model for Variable Amplitude Fatigue 474

10.5.2 Reverse Plasticity at the Crack Tip 475

10.5.3 The Effect of Overloads and Underloads 478

10.5.4 Models for Retardation and Variable Amplitude Fatigue 484

10.6 Growth of Short Cracks 488

10.6.1 Microstructurally Short Cracks 491

10.6.2 Mechanically Short Cracks 491

10.7 Micromechanisms of Fatigue 491

10.7.1 Fatigue in Region II 491

10.7.2 Micromechanisms Near the Threshold 494

10.7.3 Fatigue at High DK Values 495

10.8 Fatigue Crack Growth Experiments 495

10.8.1 Crack Growth Rate and Threshold Measurement 496

10.8.2 Closure Measurements 498

10.8.3 A Proposed Experimental Definition of DKeff 500

10.9 Damage Tolerance Methodology 501

Appendix 10: Application of The J Contour Integral to Cyclic Loading 504

A10.1 Definition of D J 504

A10.2 Path Independence of D J 506

A10.3 Small-Scale Yielding Limit 507

References 507

Chapter 11

Environmentally Assisted Cracking in Metals 511

11.1 Corrosion Principles 511

11.1.1 Electrochemical Reactions 511

11.1.2 Corrosion Current and Polarization 514

11.1.3 Electrode Potential and Passivity 514

11.1.4 Cathodic Protection 515

11.1.5 Types of Corrosion 516

11.2 Environmental Cracking Overview 516

11.2.1 Terminology and Classification of Cracking Mechanisms 516

11.2.2 Occluded Chemistry of Cracks, Pits, and Crevices 517

11.2.3 Crack Growth Rate vs. Applied Stress Intensity 518

11.2.4 The Threshold for EAC 520

11.2.5 Small Crack Effects 521

11.2.6 Static, Cyclic, and Fluctuating Loads 523

11.2.7 Cracking Morphology 523

11.2.8 Life Prediction 523

11.3 Stress Corrosion Cracking 525

11.3.1 The Film Rupture Model 527

11.3.2 Crack Growth Rate in Stage II 528

11.3.3 Metallurgical Variables that Influence SCC 528

11.3.4 Corrosion Product Wedging 529

11.4 Hydrogen Embrittlement 529

11.4.1 Cracking Mechanisms 530

11.4.2 Variables that Affect Cracking Behavior 531

11.4.2.1 Loading Rate and Load History 531

11.4.2.2 Strength 533

11.4.2.3 Amount of Available Hydrogen 535

11.4.2.4 Temperature 535

11.5 Corrosion Fatigue 538

11.5.1 Time-Dependent and Cycle-Dependent Behavior 538

11.5.2 Typical Data 541

11.5.3 Mechanisms 543

11.5.3.1 Film Rupture Models 544

11.5.3.2 Hydrogen Environment Embrittlement 544

11.5.3.3 Surface Films 544

11.5.4 The Effect of Corrosion Product Wedging on Fatigue 544

11.6 Experimental Methods 545

11.6.1 Tests on Smooth Specimens 546

11.6.2 Fracture Mechanics Test Methods 547

References 552

Chapter 12

Computational Fracture Mechanics 553

12.1 Overview of Numerical Methods 553

12.1.1 The Finite Element Method 554

12.1.2 The Boundary Integral Equation Method 556

12.2 Traditional Methods in Computational Fracture Mechanics 558

12.2.1 Stress and Displacement Matching 558

12.2.2 Elemental Crack Advance 559

12.2.3 Contour Integration 560

12.2.4 Virtual Crack Extension: Stiffness Derivative Formulation 560

12.2.5 Virtual Crack Extension: Continuum Approach 561

12.3 The Energy Domain Integral 563

12.3.1 Theoretical Background 563

12.3.2 Generalization to Three Dimensions 566

12.3.3 Finite Element Implementation 568

12.4 Mesh Design 570

12.5 Linear Elastic Convergence Study 577

12.6 Analysis of Growing Cracks 585

Appendix 12: Properties of Singularity Elements 587

A12.1 Quadrilateral Element 587

A12.2 Triangular Element 589

References 590

Chapter 13

Practice Problems 593

13.1 Chapter 1 593

13.2 Chapter 2 593

13.3 Chapter 3 596

13.4 Chapter 4 598

13.5 Chapter 5 599

13.6 Chapter 6 600

13.7 Chapter 7 600

13.8 Chapter 8 603

13.9 Chapter 9 605

13.10 Chapter 10 607

13.11 Chapter 11 608

13.12 Chapter 12 609

Index 611