--FOURTH EDITION--

FRACTURE MECHANICS Fundamentals and Applications

T.L. Anderson

~ ~)1~r?Fr~~~~~p ~ Boca Raton London New York

CRC Press is an imprlnt of the Taylor & Francis Group, an lnforma business

Contents

~„.„ .•••..••......•.••••••...••........•.•..................................••...•..•. .......•.•...•.•••...........•...• ~·· · · · · · ····· · · ·XV

Section 1 Introduction

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 Sbips.„ .. „„ .. „„ .. „„ ... „ .. „ ... „ .. „„„„ .... „„„ .. „„„ .......... „„ ........ „ ...... 8 1.2.3 Postwar Fracture Mechanics Research .„„.„„„„„„„„„„„„„„„.„„„„„„„„„„ 9 1.2.4 Fracture Mechanics from 1960 through 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„.„ .. „ .. „.„„.„.„„.„.„„„„ .. „„„„.„„.„„„„„„„„„.„.„.„„„13 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 „.„„„.„„„„„„„„„„„„.„.„„„„„„„„.„.„„„ 17

1.5.1 The Buckingham TI-Theorem„.„„„.„„.„„.„„„„„„„„„„.„„.„ .. „„.„.„„„.„.„.18 1.5.2 Dimensional Analysis in Fracture Mechanics.„„„.„.„„„„„„„„„„„„„„„„19

References „„„„ ..... „.„.„ ...•... „ ..... „„„„„„„„ .. „.;:.„„„„„ ..... „.„ .....•. „ .........•........ „ .............. „ ... 21

Section II Fundamental Concepts

2. Linear Elastic Fracture Mechanics .„„„„„.„.„.„.„.„.„.„„„„„„„„„„„„„„„„„„„„„.„„„.„„ 25 2.1 2.2 2.3

2.4 2.5

2.6

An Atomic View of Fracture„„„„„.„.„„„„ .. „„.„.„„„„„„.„„„„„„„„„„„„„„„„„„„„. 25 Stress Concentration Effect of Flaws „„„„„„„.„.„„„„.„„„„.„„„„„„„„„„„„„„„.„.„ 27 The Griffith Energy Balance ....................................... „ ............ „ ..... „„„„„„„„„ ........ 30 2.3.1 Comparison with the Critical Stress Criterion „„„.„„.„„„„„„ .. „„.„„„„„. 32 2.3.2 Modified Griffith Equation .. „ .... „ .. „ ................. „ .... „„ .. „ ... „ ..... „„ .. „ ...... „ .. „ 33 Energy Release Rate.„ ... „ ........ „ .. „ ...... „ .. „ .. „ .... „„ .... „„ .. „„„ ... „ ... „ ................ „ .......... 35 Instability and the R Curve „„„.„„„„„„„„.„„„„„ .. „.„„„ .. „„„„„„„„ .. „„„„„„„.„„. „ 39 2.5.1 Reasons for the R Curve Shape„„ •.. „„„„„„.„„„„„.„.„„„„„„„„„„„.„„„.„.„40 2.5.2 Load Control versus Displacement Control....„„ .. „„ .. „.„„.„„ .. „„ .. „„„„„. 41 2.5.3 Structures with Finite Compliance ... „.„.„„.„ .. „„.„.„„„ .... „„„.„.„„„„„„.„42 Stress Analysis of Cracks „„„„„„ .. „„ .. „„„„.:„„„ .... „ ........ „„ ..• „.„ •.. „ .... „ .. „„ ...... „„„44 2.6.l The Stress Intensity Factot .. „ •.. „ .... „ ......... „„ .................. „ ............. „.„„„.„.„44 2.6.2 Relationship between K and Global Behavior „.„.„„„.„„„„„„.„.„.„„„„„. 47 2.6.3 Effect of Finite Size .„„„„„.„.„„.„„.„„ ... „.„„ .. „ .. „.„.„„ .. „.„„.„ .. „.„„ .. „ ... „.„ 51 2.6.4 Principle of Superposition „„„„.„„„.„„.„„ .. „„„„.„„„„ .. „„„„„„„„„„„„„„„ 55 2.6.5 Weight Functions ........................... „ ............ , .. „„ .. „ ...... „ ............................... 57

vii

viii

2.7 2.8

2.9

Contents

Relationship between K and q. ....................................... „ ...... „ .. „ •............... „ .... „„.„ 60 Crack Tip Plasticity ..................................................................................................... 62 2.8.1 The Irwin Approach ............................................................... „ ..................... 63 2.8.2 The Strip Yield Model ................................................................................... 66 2.8.3 Comparison of Plastic Zone Corrections .................................................... 68 2.8.4 Plastic Zone Shape ...................................................................................... „. 69 K-Controlled Fracture .„ .......................... „„ ........ „ ............. „ .... „ ................................. 71

2.10 Plane Strain Fracture: Fact versus Fiction ................. „ .......................... „ ................ 75 2.10.1 Crack Tip Triax.iality .................................... „ .. „ ..... „ .......................... „„ ....... 76 2.10.2 Effect of Thickness on Apparent Fracture Toughness ....... „ .................... 78 2.10.3 Plastic Zone Effects„.„ ...... „ ............... „ ...... „„ ........ „ •.•....•..... „.„„ .. „ .. „„ ........ 81 2.10.4 Implications for Cracks in Structures ........... „ .......... „„ .. „.„ ... „ ............. „ ... 83

2.11 Mixed-Mode Fracture ................... „„ ...... „ .................... „ .... „„„ .... „ .•........ „ ....... „„„ ... 84 2.11.l Propagation of an Angled Crack .... „ ........................................................... 85 2.11.2 Equivalent Mode I Crack ....................... „ ..................................................... 87 2.11.3 Biaxial Loading ....................... „ ..................................... „ .............. „ .............. 88

2.12 Interaction of Multiple Cracks ........ „ .......... „ ......... „ ............ „ .................... „ ............ „ 90 2.12.1 Coplanar Cracks .............. „ ... „.„ .................... „„ ....•..................... „.„ .............. 90 2.12.2 Parallel Cracks .............................. „ ........................... „ .•. „ ... „„ ....................... 90

Appendix 2A: Mathematical Foundations of Linear Elastic Fracture Mechanics: Selected Results ..................................................... „ ............................... 92

References .............. „ ............................................ ................................................ „ ....... „„ .... 107

3. Elastic-Plastic Fracture Mechanics ................. „ ............................ „ .............. „ ................ 109 3.1 Crack Tip Opening Displacement ...... „„„„„ .......... „ ........ „ ... „ ............................... 109 3.2 The J Contour Integral.„„ .......................... „ .. „„„ .............. „ ........ „ .................. „ ....... 114

3.2.1 Nonlinear Energy Release Rate .„ ............. „ .......... „ ................. „ ................ 115 3.2.2 Jas a Path-Independent Line Integral... .................................. „„ ...... „„ .... 117 3.2.3 Jas a Stress Intensity Parameter ..... „ ..................... „ .................................. 118 3.2.4 The Large-Strain Zone ............ „ .................................................................. 119 3.2.5 Laboratory Measurement of f. .................................................................... 121

3.3 Relationships between J and CTOD ....... „ .............................................................. 127 3.4 Crack Growth Resistance Curves ........................................................................... 129

3.4.1 Stahle and Unstable Crack Growth ..... „ .... „ .................. „ .......... „ ....... „ ..... 131 3.4.2 Computing J for a Growing Crack ............................................................ 133

3.5 /-Controlled Fracture ...... „„ .......................................... „„ ........................................ 135 3.5.1 Stationary Cracks ......................................... „„ ............................................ 136 3.5.2 /-Controlled Crack Growth ...................................... „ .. „ ............... „ ... „ ...... 138

3.6 Crack Tip Constraint under Large-Scale Yielding ............................................... 141 3.6.1 The Elastic T Stress ............ „ ................ „ .............................. „ ........ „ ....... „ .... 145 3.6.2 J-Q Theory .................................................................................................... 147

3.6.2.1 The J-Q Toughness Locus .... „ .......... „ ........................... „„ ....... „ 149 3.6.2.2 Effect of Failure Mechanism on the J-Q Locus .............. „ ....... 150

3.6.3 Scaling Model for Cleavage Fracture ........................................................ 152 3.6.3.1 Failure Criterion ................................... „ ...................................... 152 3.6.3.2 The /0 Parameter ...................... „ ................................................... 153 3.6.3.3 Three-Dimensional Effects ..................... „ .................................. 154 3.6.3.4 Application of the Model ...................... „ .............. ...................... 155

3.6.4 Limitations of Two-Parameter Fracture Mechanics ............................... 157

Contents ix

Appendix 3A: Mathematical Foundations of Elastic-Plastic Fracture Mechanics: Selected Results .................................................................................... 160

References ............................................................................................................................. 178

4. Dynamic and Time-Dependent Fracture ....................................................................... 181 4.1 Dynamic Fracture and Crack Arrest. ..................................................................... 181

4.1.1 Rapid Loading of a Stationary Crack. ....................................................... 182 4.1.2 Rapid Crack Propagation and Arrest ........................................................ 187

4.1.2.1 Crack Speed„ .................................................................................. 189 4.1.2.2 Elastodynamic Crack Tip Parameters ......... „ ............................ 190 4.1.2.3 Dynamic Toughness .................................................................... 193 4.1.2.4 Crack Arrest ................................................ .................................. 194

4.1.3 Dynamic Contour Integrals ....................................................................... 197 4.2 Creep Crack Growth.„ .............................................................................................. 198

4.2.1 The C* Integral ............................................................................................. 199 4.2.2 Short-Time versus Long-Time Behavior ................................................... 202

4.2.2.l The Ct Parameter .......................................................................... 203 4.2.2.2 Primary Creep .............................................................................. 205

4.3 Viscoelastic Fracture Mechanics ............................................................................. 206 4.3.1 Linear Viscoelasticity .................................................................................. 206 4.3.2 The Viscoelastic J Integral ................................ „„ ...................................... 209

4.3.2.1 Constitutive Equations ............................................... „„„ .. „ ....... 209 4.3.2.2 Correspondence Principle ....................... .................................... 210 4.3.2.3 Generalized J Integral... ............................................................... 210 4.3.2.4 Crack Initiation and Growth ...................................................... 212

4.3.3 Transition from Linear to Nonlinear Behavior ................................... „ .. 213 Appendix 4A: Dynamic Fracture Analysis: Selected Results ........................................ 216 References ...... .... ................................................................................................................... 223

Section III Material Behavior

5. Fracture Mechanisms in Metals ................................................................. „ .. „ .. „.„ ........ 229 5.1 Ductile Fracture.„ .. „.„„ ... „.„ ...... „ .......... „ ............................ „ ........ „ ........... „ ..... „ ..... 229

5.1.1 Void Nucleation ...... „ ............................... „ .. „ .. „ .. „ .................... „„„„„.„ .. „„ 231 5.1.2 Void Growth and Coalescence .„„„„„„„ ................... „ ............. „.„.„„„„„ .. 232 5.1.3 Ductile Crack Growth ...... „„.„„„„„„„ .... „„„ .. „„„„ ... „ ........ „ ................... „ 241

5.2 Cleavage .. „ ...... „„„„„„„„ .. „„„ ... „ .... „.„„„„„„ ........ „ ...... „ ...... „ .. „„„„ .. „ .............. „ .... 244 5.2.1 Fractography ................................................. „ .... „ .. „„ „ ....... .' .... „ .... „„ ...... „ .. 244 5.2.2 Mechanisms of Cleavage Initiation ............. „„ ................... „ .................. „. 244 5.2.3 Mathematical Models of Cleavage Fracture Toughness ........................ 249

5.3 The Ductile-Brittle Transition ............. „ .............. „ ...... „ .. „ .............. „ .. „ .... „ ............ 256 5.4 Intergranular Fracture.„ ................ „ .................. „.„ .... „ ........ „ .. „„„ .. „ ...... „ .. „„ .. „ .... 258 Appendix SA: Statistical Modeling of Cleavage Fracture .. „„„„„ .. „„.„ .... „„.„„„„ ... „ ... 259 References ............................................................................................................................. 264

6. Fracture Mechanisms in Nonmetals .... „ ................ „ ........... „ ................. ............... „ .. „ ..... 267 6.1 Engineering Plastics .......................... „„ ............ „ ........................ „ ........................... 267

X Contents

6.1.1 Structure and Properties of Polymers .. „ ...... „„ •.......... „ ........ „ •. „„ .. „ ..... „. 268 6.1.1.1 Molecular Weight .................. „ ......................... „ .......................... 268 6.1.1.2 Molecular Structure .... „ ............... „ ............ .......•.•.. ...•. „„.„ ...... „ .. 269 6.1.1.3 Crystalline and Amorphous Polymers ..................................... 269 6.1.1.4 Viscoelastic Behavior .. „ ............................................................... 271 6.1.1.5 Mechanical Analogs ....................•..... „ ........................................ 273

6.1.2 Yielding and Fracture in Polymers. .... „ ..•...............•. ~„ .. „ ...... „ .................. 274 6.1.2.l Chain Scission and Disentanglement ....................................... 275 6.1.2.2 .Shear Yielding and Crazing .......................• „ .......... „ ............•....• 276 6.1.2.3 Crack Tip Behavior ..•. „ ... „ ...........•..•.•..•.... „ ..•............................... 277 6.1.2.4 Rubber Toughening .. „ ..........•. „ .................... „ ............................. 279 6.1.2.5 Fatigue ..... „ .......... „ ............... „ ...... - ................................................ 279

6.1.3 Fiber-Reinforced Plastics ....•............. „ ....................................... „„ ............. 280 6.1.3.1 An Overview of the Failure Mechanisms ................................ 281 6.1.3.2 Delamination ............................................ „ ..... „ ........................... 282 6.1.3.3 Compressive Failure .................................................................... 286 6.1.3.4 Notch Strength ................................................... - ......................... 288 6.1.3.5 Fatigue Damage .............................................. „ ............................ 291

6.2 Ceramics and Ceramic Composites .............................. „ ..... ~ ............... „ ................ 291 6.2.1 Microcrack Toughening ................... „ ........ „ ................ „ .•........•.................. 295 6.2.2 Transformation Toughening .............. „ ..... „ ............................................... 297 6.2.3 Ductile Phase Toughening „ .............. „.„ ................................ „ .................. 298 6.2.4 Fiber and Whisker Toughening ........ „ ...... „ ............. ... „ ............... „.„ ......... 299

6.3 Concrete and Rock .„ ............ „„„ .. „ ................ „„„ ........ „ ........ „„ ........ „ ................ „ ... 301 References ............ „ ...... „ ................... „ .............. i· ·· · · ······ · · ····~·· ········„·-··· · ·· · ... ···········„ ........ „ .. 304

Section IV Applications

7. Fracture Toughness Testing of Metals ............................. „.„.„ .....................•.. „ ....... - •.. 309 7.1 General Considerations ............ „ •. „ ...... „ .. „ ........... „ .. „ ...... „ ...... „ ..... „ ...................... 309

7.1.1 Specimen Configurations .... „ ....... „ .. „.„ ................... „ ........ „ ............... „ ... „. 310 7.1.2 Specimen Orientation .......................... „ ............................... „ ......•........ „ .... 310 7.1.3 Fatigue Precracking ........ „ ........................................•.............. „ .................. 314 7.1.4 Instrumentation „ ............................... „ ......... „ ........ „ ................. „„ ...... „ ...... 315 7.1.5 Side Grooving.„ .................................... „„ .... „ ............... „ .. „ ........ „ .. „ .... ~ ...... 316

7.2 K1c Testing .„ ......... „ ...............................................................................................•..... 317 7.2.1 ASTM E399 .......... „ ...... „ .... „ .... „ .............. „ .................. „ ......•...... , .. „ .• „ .. .... .... 318 7.2.2 Limitations of E399 and Similar Standards ................ „ ..... „ .................... 322

7.3 K-R Curve Testing ....... „ .... „ ....... „ ................ „.„ .•. „ ..................... ~.„ ..... ,.„ .. „ ........... 326 7.3.1 Specimen I>esign .............. „ ........ „ .. „ .................... „ ...... „, .......... „ .............. „. 327 7.3.2 Experimental Measurement of K-R Curves ....................................... , .... 328

7.4 J Testing of Metals .....................................................•. , ................. „ ........................ „ 330 7.4.1 The Basic Test Procedure and J1c Measurements ..................................... 330 7.4.2 J-R Curve Testing ............ „ ......... „ ............................................................... 333 7.4.3 Critical J Values for Unstable Fracture ............................... „ ..................... 335

7.5 CTOD Testing ..... „ ................. „ ............................ „ •.......• „„.„ .•••• „ •..• „ .••........•.........•. 336 7.6 Dynamic and Crack Arrest Toughness .............. „ .. „ .. „ .. „.„ .. „ . ••.......... ••................ 338

Contents xi

7.6.1 Rapid Loading in Fracture Testing ............................................................ 339 7.6.2 K1• Measurements ........................................................................................ 340

7.7 Fracture Testing of Weldments ...... „ ..•.•...............................................••.••.............. 344 7.7.1 Specimen Design and Fabrication .................. „ ..........•..............•............... 344 7.7.2 Notch Location and Orientation ................................... „ ........................... 345 7.7.3 Fatigue Precracking ..................................................................................... 347 7.7.4 Post-Test Analysis .............. „ ........................................ „ .............................. 347

7.8 Testing and Analysis of Steels in the Ductile-Brittle Transition Region .. „ ...... 348 7.9 Component Fracture Tests „ ................................................................................. „ .. 350

7.9.l Surface Crack Plate Specimens ........................... „ ................... „ ................ 351 7.9.2 SENT Specimens ............................ „ ............................................................ 353

7.10 Qualitative Toughness Tests ................................................................ „ .................. 353 7.10.1 Charpy and Izod Impact Test.. .................... „ ................... „„„ .................... 355 7.10.2 Drop Weight Test ............................ „ ............... „ .................... „ .................... 356 7.10.3 Drop Weight Tear and Dynamic Tear Tests.„ .............. „„ ................. „ ..... 358

Appendix 7: Stress Intensity, Compliance, and Limit Load Solutions for Laboratory Specirnens ................. „ ..................................................................... 358

References ............................................................................................................................. 364

8. Fracture Testing of Nonmetals ..... „ .. „ .... „ .... „ ......... „ ... „ ................................................. 369 8.1 Fracture Toughness Measurements in Engineering Plastics .. „ ............... „ ......... 369

8.1.1 The Suitability of K and J for Polymers ................................ „ .................. 369 8.1.1.1 K-Controlled Fracture .......... „ ........................................... „„„ ..... 370 8.1.1.2 /-Controlled Fracture ................................................................... 373

8.1.2 Precracking and Other Practical Matters ................................................. 376 8.1.3 K1c Testing ....... „ ...•....... „ .. „„ .. „ ...... „„„ ..• ; .•.•..•......•. „ .....................•.••............ 378 8.1.4 J Testing ......................................................................................................... 382 8.1.5 Experimental Estimates of Time-Dependent Fracture Parameters ...... 384 8.1.6 Qualitative Fracture Tests on Plastics ....................................................... 387

8.2 Interlaminar Toughness of Composites ........................................ „ .. „ .......... „ ....... 389 8.3 Ceramics ..................................................................................................................... 393

8.3.1 Chevron-Notched Specimens .................................................................... 394 8.3.2 Bend Specimens Precracked by Bridge Indentation ............................... 396

References .................. „ ...•.................................... „ ...................................... ..... .................... 398

9. Application to Structures .................................................................................................. 401 9.1 Linear Elastic Fracture Mechanics ............................ „ •••• „ ..................... „„ ........... .401

9.1.1 K1 for Part-Through Cracks ............ ; .. „ ............ „ ......•.......•..............•............ 403 9.1.2 Influence Coefficients for Polynomial Stress Distributions .. „ .•..•......... .404 9.1.3 Weight Functions for Arbitrary Loading .. „ ...•........•.• „ ..•....... „ .... „„ ....... .408 9.1.4 Primary, Secondary, and Residual Stresses ................. „ .......................... 410 9.1.5 A Warning about LEFM ....... „ ...................... „ ............................................. 411

9.2 The CTOD Design Curve ....................................................... „„„ ............................ 412 9.3 Elastic-Plastic /-Integral Analysis ....................................................... „ .••.........•••.. 414

9.3.1 The EPRI /-Estimation Procedure ............. „.„ ............................................ 414 9.3.1.1 Theoretical Background .............................................................. 415 9.3.1.2 Estimation Equations .................. „„ .....................•.....•..........•...... 416 9.3.1.3 Comparison with Experimental J Estimates ............................ 418

9 .3.2 The Reference Stress Approach ........... „ ....................................... „ ........... 420

xii Contents

9.3.3 Ductile Instability Analysis ..................... „ .•.................. „ ........................... 422 9.3.4 Some Practical Considerations ...... „.„ .......... „ .. „ .•.. „ ...... „ ••.•.... „ ........ „„ .... 425

9 .4 Failure Assessment Diagrams ..•... „ ...... „ ....... „„.„ ..•.... „.„ .•........ „„„ ...... „„ ... „ ........ 427 9.4.1 Original Concept„„ ..•... „ ...• „ ....... „.„ ... „ .. „ .•... „„.„ ...... ~ .• „m„„ ........ „ .. „.„ .. „427 9.4.2 J-Based FAD .. „„.„ ...... „ ... „ .. „.„ .. „ .. „„.„.„ .. „ .... „ .•............. „ ... „ ... „.„.„„ ........ 430 9.4.3 Approximations of the FAD Curve ... „„„.„.„.„ ...... „ .... „ .. „.„ ... „.„„„„„„ . .433 9.4.4 Fitting Elastic-Plastic Finite Element Results to a FAD Equation „„ .. „434 9.4.5 Application to Welded Structures „ .. „„„ .. „ .. „.„.„„„„ ... „.„.„„„ .. „.„„.„.„.441

9.4.5.1 Incorporating Weld Residual Stresses ... „ ... „ .... „ .. „ ... „ ... „ .. „.„„442 9.4.5.2 Weld Misalignment and Other Secondary Stresses .... „ .... „ ... 445 9.4.5.3 Weld Strength Mismatch „„.„ ... „ .... „ ...•..... „ .................. „„ ....... „446

9.4.6 Primary versus Secondary Stresses in the FAD Method„„ ... „ .. „ .... „ .... 447 9.4.7 Ductile Tearing Analysis with the FAD .. „„ ..........• „„ ........... „ ..... „„ .. „ .... 449 9.4.8 Standardized FAD-Based Procedures .„.„ ... „ .. „ ... „ .....• „.„ .......... „.„ ... „ .. .450

9.5 Probabilistic Fracture Mechanics .... „.„ ...... „.„ ..... „ .. „ ....•.... „ ........ „„„ ...... „„ ..... „ .. 451 Appendix 9: Stress Intensity and Fully Plastic J Solutions for Selected

Configurations „ ........ - ....................... „ ..................... „ .... „ .............. „ ........................ 4.53 References ..... „ ..•..........•.... „„„„„„ .... „ .. „ ..........................•...•.............. „ .....••............... „ ..... „ 469

10. Fatigue Crack Propagation-.„ ............. „ . .••••.•••••.. „ ....... „ .; •• „ .............• ~•··············· · ··· ···· · ··· 471 10.1 Similitude in Fatigue •. „ ...... „ •.. „ ....... „ ........•........................... „ ..... „ .... „ ...... „ ........... 471 · 10.2 Empirical Fatigue Crack Growth Equations .. „„ ........ „ ............... „ ......... „.„.„ ... „ ... 473 10.3 Life Prediction „ ........ „.„ .•... „ •..• „ .•......... „„ ...................... „ ........ .....••......................•.. 476 10.4 Crack Closure ............. „„ ........... „„ .. „ ......................................................... „ ............. 478

10.4.1 A Closer Look at Crack Wedging Mechanisms ........................•...... „ .. „.483 10.4.2 Effects of Loading Variables on Closure ..... „ .. „ .... „ .. „ ...•. „ ..... „ .•............. 484

10.5 The Fatigue Threshold ·-························································„····························· 487 10.5.1 The Closure Model for the Threshold.„ ............ „ ...................................... 488 10.5.2 A Two-Criterion Model.„ ..•.......... „ ...... „ ... „ .......... „„ ..•. „„ .......... „ .............. 490

10.6 · Variable-Amplitude Loading and Retardation ...• „ .................. „ .... „ ... „ .•.......... „„ 493 10.6.1 Linear Damage Model for Variable-Amplitude Fatigue ........................ 493 10.6.2 Cycle Counting and Histogram Construction „„ ...... „ .... „ .........• „„„ ...... 497 10.6.3 Reverse Plasticity at the Crack Tip ........... „ ............................. „.„ .......... „. 501 10.6.4 The Effect of Overloads and Underloads „ ....... „„ ........ „ ....... „.„ ... „ ......... 505 10.6.5 Modeling Retardation and Variable-Amplitude Fatigue .. „„ ................. 510

10.7 Growth of Short Cracks„ .. „ ..•...• „ ...... „ ...... „„.-.... „ ........ „ .••• „„ ............. : . .i„ •• „„.„ .. „ 512 10.7.1 Microstructurally Short Cracks ......................... „ .......... „„ ....... „ ............... 514 10.7.2 Mechanically Short Cracks ......................... „ ... „.„ .....•... „ .•....... „.„ ..... „ ...... 515

10.8 Micromechanisms of Fatigue ...................................... „ .....•..• „ ............. „ ................ 516 10.8.1 Fatigue in Region II •...... „ .. „ ........................... „„„.„ ...... „ ...... „„ ... „„.„ .... „„ 517 10.8.2 Micromechanisms near the Threshold„„.„ .... „.„ .... „ .•...... „ ..................... 518 10.8.3 Fatigue at High tJ< Values .•....... „ .. „„ ..... „.„„ ... „.„ ..... „„ ... „.„ ............ „ ...... 520

10.9 Fatigue Crack Growth Experiments.„ .....•.......•. „ ...... „ ............ „„ .................... „„ ... „ 521 10.9.1 Crack Growth Rate and Threshold Measurement .. „„.„„ .. „„.„ .... „ ..... „.521 10.9.2 Closure Measurements .„ ... „„ .. „ .• „.„ •......................... „ .„ .............•.............. 523 10.9.3 A Proposed Experimental Definition of ~ ... „ ........ „.„ ..................... „.525

10.10 Damage Tolerance Methodology .......... „„ •.•..• „ .. „ ........ „ ........................................ 527 Appendix lOA: Application of the J Contour Integral to Cyclic Loading ..... „„ ........ „ 529 References .......................... „ ... _ ........ „ .......•...............•. „„ ........ „ ............. „ ............................ 534

Contents xiii

11. Environmentally Assisted Cracking in Metals .......... „.„„ .... „ ..................................... 537 11.1 Corrosion Principles ......................... .. ...................................................................... 537

11.1.1 Electrochemical Reactions ........................................................... „.„„ .... „ .. 537 11.1.2 Corrosion Current and Polarization ..... „„ .. „ ... „.„ .................... „„ .... „„.„ .. 540 11.1.3 Electrode Potential and Passivity „ ......................................................... „.541 11.1.4 Cathodic Protection ..................................................................................... 541 11.1.5 Types of Corrosion„ ...... „ ....................................................... „ .......... „ ........ 542

11.2 Environmental Cracking Overview „ ..................................................................... 542 11.2.1 Terminology and Classification of Cracking Mechanisms .............. „ .... 543 11.2.2 Occluded Chemistry of Cracks, Pits, and Crevices .... „ ...... „.„ ... „.„ ........ 544 11.2.3 Crack Growth Rate versus Applied Stress Inten~ity ..................... „ ... „„544 11.2.4 The Threshold for EAC .„„„„„ .................... ... .... .. .. ............. ... ..................... 546 11.2.5 Small Crack Effects ........................................................................... „ .. „ ..... 547 11.2.6 Static, Cyclic, and Fluctuating Loads ........ „ ................................. „.„ .... „ .. 549 11.2.7 Cracking Morphology ................................................................................. 549 11.2.8 Life Prediction ........................................................................................... „.550

11.3 Stress Corrosion Cracking ......................................................................... „ ...... „ .... 551 11.3.1 The Film Rupture Model ....................... „ ................................. „ ................ 553 11.3.2 Crack Growth Rate in Stage II„„„„„„„ .... „.„„ ... „.„ ... „ ....... „ ...... „ ............. 554 11.3.3 Metallurgical Variables That Influence SCC. ........................................... 554 11.3.4 Corrosion Product Wedging .... „ .. „„ ......... „ .. „ .................. „ .. .. „ ................. 555

11.4 Hydrogen Embrittleinent ........................... „ ........ „ .. „ ...... „ ............................ „ ... „ ... 556 11.4.1 Cracking Mechanisms ........................................... „ ................................... 556 11.4.2 Variables That Affect Cracking Behavior „„„„ .... „ .. „ ......... „ ....... „ .... „ ..... 557

11.4.2.1 Loading Rate and Load History .......... „ .. „„ ............................... 557 11.4.2.2 Strength ..................... „ .................................................................. 560 11.4.2.3 Amount of Available Hydrogen ...... „ ...... „ ................................. 561 11.4.2.4 Temperature .................................................................................. 561

11.5 Corrosion Fatigue ............ „ ... „„ ... „ ................. „ ........... „ ... „.„ .......... „ .. „ ....... „.„ ..... „.564 11.5.l Time-Dependent and Cycle-Dependent Behavior .................... „„„„ .. „„564 11.5.2 Typical Data .. „„ ........................ „ ....... „„„„„ .... „ ............ „„ .. „ ........ „ ....... „ .... 566 11.5.3 Mechanisms.„ .................................................... „ ......................... „ .. „„ .. „„„ 569

11.5.3.1 Film Rupture Models ....... „ ........... „ ... „„ ..... „ ................... „ .. „ .. „„ 569 11.5.3.2 Hydrogen Environment Embrittlement...„ .. „ ...... „ ................. „569 11.5.3.3 Surface Films ..... , ................... „ ...... „ ........... „ ........................... „„ .. 570

11.5.4 The Effect of Corrosion Product Wedging on Fatigue ....... „ ........ „„.„„. 570 11.6 Experimental Methods .. „ .... „ ..... „ ... „ .............. „„ ........ „„ ................ „ .. „„ .... „ .. „„ .. „. 571

11.6.1 Tests on Smooth Specimens ......... „„.„ .. „ ................. „ ....... .. „ ...... „„ „ .. „ „ „. 571 11.6.2 Fracture Mechanics Test Methods ..... „„ .. „ ... „ ....... „ .. „.„.„ ........ „ .... „ .... „„ 573

References ......................................................................................... „ ...... „ .......................... 578

12. Computational Fracture Mechanics .... „ .... „ „ „ .... „ .............. „„ ...................... „ ................ 581 12.1 An Overview of Numerical Methods „„„ .. „„ .. „ .. „„ ..... „ ..... „.„„ ....................... „ .. 581

12.1.1 The Finite Element Method ............ „ .. „„„.„„ ............................. „„ ............ 582 12.1.2 The Boundary Integral Equation Method.„„„„„ ................... „ ...... „ ........ 584

12.2 Traditional Methods in Computational Fracture Mechanics ............. „.„ ........... 586 12.2.1 Stress and Displacement Matching .......... „„„„„„„.„„„ .. „.„. „ ......... „„.„ .. 587 12.2.2 Elemental Crack Advance„„ .... „ ... „ .. „ ....... „ .. „.„ .... „ .... „ .... „„ .. „ „„„ ...... „ .. 588 12.2.3 Contour Integration ................................. „ ............................ „ .. „ ................ 588

xiv Contents

12.2.4 Virtual Crack Extension: Stiffness Derivative Formulation .................. 589 12.2.5 Virtual Crack Extension: Continuum Approach ............... „ ............. „ ..... 590

12.3 The Energy Domain Integral. ...... „ .... „ .•.... „.„ •.•... „„ ..... „ •. _ ........•.......•................... 592 12.3.1 Theoretical Background .. „ .................................... „ ...........•... „ ................ „. 592 12.3.2 Generalization to Three Dimensions„„„.„„ ........... „.„ ... „ ... „ ............... „„ 595 12.3.3 Finite Element Implementation ................................... ,.„ .......................... 597

12.4 Mesh Design ......•............ „„ .•.....•................ „ ... „ ..•.. - ... „ ...... „.„_.,„ .•.. „ .. „ .. „.„ .......... 599 12.5 Linear Elastic Convergence Study .•................................. „ ....... „ .... „ .................... „ 606 12.6 Analysis of Growing Cracks •... „ ••.•...... „„.„ ............•..•. - ......• „ ......• „ ....... „ .• - ........... 614 Appendix 12: Properties of Singularity Elements .... „ ....... „ .............. „„ .. „.„ ..•................ 618 References ... „ .. „ .. „.„ .. „ ... „ ... „ ... „„ .. „ .•. „.„ ... „ .. „ ...• „ .............................. _ ... „ ...... „ .. „„ .......... 622

13. Practice Problems.„ ... „ .... „ ..... „ .... „ ......... „ ... „ .... „ ......•... „„ .• „„_ ... „.„_.„„ ......... „ •............. 625 13.1 Chapter 1 ........................... „„.„ ... „ ................ „„ .•.. „ ....... „ ............ - •..........•....•........... 625 13.2 Chapter 2 ..........................•........ „ ..•......... „ ..................................•.•............... „ ......... „ 626 13.3 Chapter 3 ... „ .... „ .. „ ... „ ... „ .... „ .... „ .. „„„„ ••..... „ ..... „ ... „ ... „„.„„.„ •.. „„ ......... „ ........ „„„. 629 13.4 Chapter 4 „„.„ ..... „ ..............................................• „ .............•.. „ ................. „„ ........ _„ .. 631 13.5 Chapter 5 ................................................ „ .... „ •••..••• „ ...•... „ ... _ •.. „„„.„ ... „.„ ................ 632 13.6 Chapter 6 ............................................. „.„.„ .......•.. „ ....•.•..... „„„„ .. „.„ ............•........... 633 13.7 Chapter 7 .......... „ ...................... -„.„.„ ...................•...............•.•........•................... „ .... 634 13.8 Chapter 8 ..........................................................................•......... „ ......... „ ................... 637 13.9 Chapter 9 ...... „ ................................... „ ...... „ ..........•.......• „ ....... „„ ..•.... „ .•. „ .... „ ........... 639 13.10 Chapter 10 ....................................................•... „ ............................ „.„ ...... „ ... „ ........... 640 13.11 Chapter 11 ............................... „ .......................... „ •.. „ ................ „.-.......................... 642 13.12 Chapter 12 ......................•.... „ ••.••.••••. „ .... „ ....... „ ......... „ ... „ .......... „ ............................. 643

lndex .......................................... „ •••••.•••••••••••••••.••••.•••••.•••••••••••••••••••••••••••••• „ ...... , •........................ 647