Introduction .................................................... 1
1 Fundamentals and Basic Relations ............................. 9
1.1 Energy Release and Energy Criterion ..................... 9
1.1.1 What is a Crack? ................................. 9
1.1.2 How Do Cracks Arise? ............................ 10
1.1.3 Energy Release .................................. 12
1.1.4 Energy Criterion ................................ 14
1.1.5 Surface Energy and the Failure Energy of a
Sample .......................................... 15
1.1.6 Strength and Weaknesses of the Energy
Criterion ....................................... 16
1.1.7 Surface Tension and Surface Energy .............. 19
1.1.8 Nucleation of a Crack and Strength of
a Material ...................................... 20
1.2 Some Methods for Determination of Energy Release ....... 23
1.2.1 Variational Approach ............................ 23
1.2.2 Convolution Formula ............................. 25
1.2.3 J-integral ...................................... 28
1.2.4 J-integral for Steady-state Motion .............. 31
1.3 Other Examples of the Energy Release Phenomenon ........ 33
1.3.1 Shock Wave ...................................... 34
1.3.2 Moving Ship ..................................... 35
1.3.3 Vehicle Moving Along a Beam on an Elastic
Foundation ...................................... 35
1.4 Stress Intensity Criterion ............................ 39
1.4.1 Fracture Process Zone ........................... 39
1.4.2 Irwin Fracture Criterion ........................ 40
1.5 Some Fracture-Associated Phenomena ..................... 43
1.5.1 Size Effect ..................................... 43
1.5.2 Difference Between Crack Initiation and
Propagation Criteria ............................ 44
1.5.3 Instabilities in Crack Propagation .............. 44
2 Fourier Transform and Related Topics ........................ 47
2.1 Continuous Fourier Transform ........................... 47
2.1.1 Definitions ..................................... 47
2.1.2 The Inverse Fourier Transform ................... 49
2.1.3 Cauchy-Type Integral. Continuous Case ........... 52
2.1.4 Laplace Transform ............................... 53
2.1.5 Fourier Transform of a Convolution .............. 53
2.1.6 Some Asymptotes ................................. 54
2.2 Wiener-Hopf Technique .................................. 56
2.2.1 The Equation .................................... 56
2.2.2 Factorization ................................... 56
2.2.3 Solution in Terms of the Transform .............. 58
2.2.4 Delta-function as a Generalized Limit ........... 58
2.2.5 Solution ........................................ 59
2.2.6 Considerations in Terms of Original Functions ... 60
2.3 Laplace and Fourier Transform .......................... 60
2.3.1 Straightforward Inversion Formula ............... 61
2.3.2 Double Fourier Transform and Hankel Transform ... 63
2.4 Discrete Fourier Transform ............................. 64
2.4.1 Definition ...................................... 64
2.4.2 Inverse Transform ............................... 65
2.4.3 Cauchy-Type Integral. Discrete Case ............. 67
2.4.4 Convolution ..................................... 67
2.4.5 Some Asymptotes ................................. 68
2.4.6 Power Asymptotes and the Related Continuum ...... 69
2.4.7 Wiener-Hopf Technique for the Discrete
Transform ....................................... 69
3 Waves ....................................................... 71
3.1 Waves of Sinusoidal and Exponential Types .............. 71
3.1.1 Equations ....................................... 71
3.1.2 Complex Wave and Dispersion Relations ........... 72
3.1.3 What is a Uniform Waveguide? .................... 76
3.1.4 Phase and Group Velocities ...................... 77
3.1.5 Energy Flux in a Wave ........................... 78
3.1.6 Additivity of Energy Fluxes in Different
Waves ........................................... 81
3.2 Waves in Periodic Structures ........................... 84
3.2.1 Discrete Chain .................................. 84
3.2.2 General System of Periodic Structure ............ 88
3.3 Forced Waves ........................................... 90
3.3.1 Complex Wave and Fourier Transform .............. 90
3.3.2 Causality Principle for Steady-state
Solutions ....................................... 91
3.3.3 Pre-Limiting Location of a Zero Point and the
Group Velocity .................................. 94
3.3.4 Contributions of Singular Points ................ 97
3.3.5 Resonant Waves .................................. 99
3.4 Waves in Homogeneous Space and Half-Space ............. 101
3.4.1 Linear Elastic Isotropic Space ................. 101
3.4.2 Longitudinal and Shear Waves ................... 103
3.4.3 Rayleigh Wave .................................. 104
3.5 Nonlinear Waves in a String ........................... 106
3.5.1 The Wavefront Conditions ....................... 106
3.5.2 Two-Step-Wave Configuration .................... 109
3.5.3 Some Asymptotic Results ........................ 110
4 One-dimensional Models ..................................... 111
4.1 String Model .......................................... Ill
4.1.1 String Attached to a Rigid Foundation .......... 111
4.1.2 Cohesive Zone Model for a String ............... 113
4.1.3 String on a Linear Elastic Foundation .......... 115
4.1.4 Nonlinear Post-peak Softening Cohesive
Forces ......................................... 115
4.1.5 Discrete Bonds ................................. 116
4.1.6 Soundless Crack ................................ 117
4.1.7 Nonlinear String Model ......................... 120
4.1.8 Nonuniform Crack Propagation ................... 121
4.1.9 Dynamic Fracture Under a Fracture Criterion .... 126
4.1.10 Tearing of a String from a Solid Under an
Impact ......................................... 127
4.1.11 Nonlinear Dynamic Problem ...................... 129
4.2 Bending Beam Model .................................... 132
4.2.1 Splitting of a Beam in Half .................... 132
4.2.2 Size Effect .................................... 134
4.2.3 Steady-State Dynamic Problem ................... 135
4.2.4 Thread—Beam Problem ............................ 137
4.2.5 Wave Resistance in Crack Propagation ........... 139
5 Static Cracks in a Linearly Elastic Body ................... 143
5.1 Field Representations ................................. 143
5.2 Kolosov-Muskhelishvili Representation ................. 145
5.2.1 Opening Mode ................................... 145
5.2.2 Shear Mode ..................................... 146
5.2.3 Anti-plane Mode ................................ 146
5.2.4 Boundary Conditions, Harmonic Function and
Integral Equation .............................. 147
5.3 Papkovich Representation .............................. 148
5.3.1 Opening Mode ................................... 149
5.3.2 Shear Mode ..................................... 149
5.3.3 Opening Mode in Cylindrical Coordinates ........ 150
5.3.4 Shear Mode in Cylindrical Coordinates .......... 151
5.3.5 Axially Symmetric Case ......................... 152
5.4 Crack in an Unbounded Plane ........................... 154
5.4.1 Finite Plane Crack ............................. 154
5.4.2 Nonlinear Condition for Mode I ................. 158
5.5 Asymptotes ............................................ 160
5.5.1 Stress Intensity Factors ....................... 160
5.5.2 Crack Tip Singularity .......................... 160
5.5.3 Stresses in Polar Coordinates .................. 161
5.5.4 Stress Intensity Factors and Energy Release ... 164
5.6 Homogeneous Solutions ................................. 165
5.6.1 Homogeneous Solution as a Limit ................ 165
5.6.2 Other Homogeneous Solutions .................... 166
5.7 Integral Equations for a General Crack System ......... 167
5.7.1 Field Induced by a Dislocation ................. 167
5.7.2 Superposition .................................. 170
5.8 Crack Interaction ..................................... 172
5.8.1 Collinear Crack Array. General Distribution .... 172
5.8.2 Periodic Collinear Crack Array ................. 173
5.8.3 Parallel Cracks ................................ 176
5.8.4 Collinear Cracks Do Not Like to Meet Each
Other .......................................... 179
5.9 Energy Release Under Crack Kink ....................... 181
5.10 Cohesive Zone Model ................................... 184
5.10.1 Formulation and Solution ....................... 185
5.10.2 Energy Release. Large and Small Cracks ......... 187
5.11 Penny-Shaped Crack .................................... 189
5.11.1 Crack Under Normal Traction .................... 189
5.11.2 Axially Symmetric Problems ..................... 194
5.11.3 Harmonic Green's Function ...................... 197
5.12 Betti's Theorem and the Weight Functions .............. 197
5.12.1 Betti's Reciprocity Theorem .................... 197
5.12.2 Weight Function Method ......................... 199
6 Nonlinear Elastic Body ..................................... 205
6.1 Some Data from Nonlinear Elasticity ................... 205
6.1.1 Geometrical Relations .......................... 205
6.1.2 Physical Relations ............................. 207
6.2 Lagrangian and Eulerian Interpretation of Linear
Elasticity ............................................ 213
6.2.1 Boundary Conditions ............................ 213
6.2.2 Lagrangian Interpretation ...................... 214
6.2.3 Eulerian Interpretation ........................ 216
6.3 Strains in the Neighborhood of a Singular Point ....... 221
6.3.1 Lagrange Variables ............................. 221
6.3.2 Euler Variables ................................ 223
6.3.3 Logarithmic Singularity is the Lower Bound ..... 225
6.4 Exact Relationships for the Energy Release and Some
Consequences .......................................... 226
6.4.1 J-integral ..................................... 226
6.4.2 Crack Opening and Stresses on the Crack Line ... 227
7 Viscoelastic Fracture ...................................... 229
7.1 Some Data from Viscoelasticity ........................ 229
7.1.1 General Formulations ........................... 229
7.1.2 Standard Viscoelastic Material ................. 231
7.1.3 Stability and Passivity ........................ 233
7.1.4 Correspondence Principle ....................... 234
7.1.5 Static Problems. Time-dependent Boundary
Regions ........................................ 235
7.2 Stationary Crack and Collinear Crack System ........... 237
7.3 Growing Crack ......................................... 238
7.3.1 Steady-state Formulation ....................... 238
7.3.2 Energy Release and Crack Growth Paradox ........ 240
7.4 Cohesive Zone for Viscoelastic Material ............... 241
7.4.1 Elastic Cohesive Zone .......................... 241
7.4.2 Viscoelastic Cohesive Zone ..................... 244
7.4.3 Global-to-Local Energy Release Ratio ........... 245
8 Elastic-Plastic Fracture ................................... 249
8.1 Elastic-Plastic Fields ................................ 250
8.1.1 Some Basic Relations ........................... 250
8.1.2 Stress Fields .................................. 252
8.1.3 Continuity Conditions .......................... 256
8.1.4 Strain Fields .................................. 258
8.1.5 Moving Strain Fields ........................... 259
8.1.6 Unloading Domain ............................... 262
8.2 Fixed Cracks .......................................... 267
8.2.1 Proportional Loading ........................... 267
8.2.2 Mode III Crack ................................. 268
8.2.3 Crack Under Plane Strain ....................... 271
8.2.4 Barenblatt-Dugdale Model for Plane Stress
Crack .......................................... 273
8.3 Growing Cracks ........................................ 275
8.3.1 Mode III Growing Crack ......................... 276
8.3.2 Mode I Growing Crack ........................... 278
8.3.3 Mode II Growing Crack .......................... 283
8.3.4 A Note on the Logarithmic Singularity .......... 284
8.3.5 Modified Barenblatt-Dugdale Model for Crack
Under Cyclic Loading ........................... 285
8.4 Elastic-Plastic Dynamic Fracture ...................... 289
8.4.1 Mode III Crack Propagation ..................... 290
8.4.2 Mode I Crack Propagation ....................... 293
9 Dynamic Fracture in a Homogeneous Elastic Medium ........... 297
9.1 Some Basic Relations .................................. 297
9.1.1 Mode III and Hydrodynamic Analogue ............. 297
9.1.2 Mode III Fundamental Solution .................. 298
9.1.3 Plane Problem Fundamental Solutions ............ 299
9.2 Crack Tip Asymptotes and the Energy Release ........... 301
9.2.1 Validity of the Steady-State Formulation ....... 301
9.2.2 Subsonic Crack ................................. 302
9.2.3 Intersonic Crack ............................... 305
9.3 Factorization of the Fundamental Solutions ............ 307
9.3.1 Singular Points, Convolutions and Supports ..... 308
9.3.2 Factorization for Transient Problems ........... 309
9.3.3 Factorization for Uniform Crack Propagation .... 314
9.4 Uniform Crack Propagation ............................. 316
9.4.1 Steady-State and Static Solutions .............. 316
9.4.2 Transient Problem with a Constant Crack
Speed .......................................... 318
9.5 Nonuniform Crack Speed Problem ........................ 319
9.5.1 Solution for a Free Sector ..................... 319
9.5.2 Mode III Explicit Solution ..................... 322
9.5.3 Crack Tip Asymptotes for Plane Problem ......... 325
9.5.4 Energy Release Versus Current Crack Speed ...... 330
9.5.5 Crack Speed Crosses the Critical Speed ......... 332
9.6 Self-Similar Dynamic Problems ......................... 337
9.6.1 Formulation and Basic Relations ................ 337
9.6.2 Homogeneous Solutions .......................... 339
9.6.3 Solution to the Problem ........................ 340
9.6.4 Stress Intensity Factors for Symmetric Case .... 342
9.7 Dynamic Crack in a Plate Under Bending ................ 343
9.7.1 Formulation .................................... 344
9.7.2 Dynamic Fracture Equations ..................... 346
9.7.3 Bending Waves Under Plate-Fluid Interaction .... 348
9.7.4 Edge Bending Waves ............................. 349
9.7.5 Crack Tip Asymptotes and the Local Energy
Release ........................................ 351
9.8 Principle of Maximum Energy Dissipation Rate .......... 353
9.8.1 Introductory Remarks ........................... 353
9.8.2 The Dynamic Fracture Criterion ................. 355
10 Cracks in a Bending Plate .................................. 359
10.1 Asymptotic Solution for a Single Crack ................ 359
10.1.1 Crack Closure Phenomenon ....................... 359
10.1.2 Plane-Bending Problem .......................... 361
10.1.3 Contact Problem ................................ 362
10.1.4 Energy Release ................................. 363
10.1.5 Limiting Cases and Asymptotes .................. 365
10.1.6 Closure Force and Moment ....................... 365
10.1.7 Contact Stress Distribution .................... 367
10.1.8 Asymptotic Closure Width ....................... 367
10.2 Radial Cracking with Closure .......................... 370
10.2.1 Few Versus Many Cracks ......................... 371
10.2.2 Formulation of the Coupled Problem ............. 372
10.2.3 Crack Closure, Open Crack and Intact Regions ... 375
10.2.4 Solutions ...................................... 377
10.2.5 Energy Release ................................. 379
10.3 Self-Similar Dynamic Problem .......................... 382
10.3.1 Formulation .................................... 382
10.3.2 General Solution ............................... 383
10.3.3 Energy Criterion and the Number of Cracks ...... 387
10.3.4 Concluding Remarks ............................. 388
11 The Square-Cell Lattice .................................... 389
11.1 Preliminaries ......................................... 389
11.2 Some Introductory Remarks ............................. 390
11.3 Elastic Lattice: Formulation and the Governing
Equation .............................................. 392
11.3.1 Formulation ................................... 392
11.3.2 Derivation of the Governing Equation .......... 393
11.3.3 Zero Points of the Functions h(k) and r(k) .... 395
11.4 Factorization ......................................... 398
11.4.1 Direct Factorization .......................... 398
11.4.2 Other Type of Factorization ................... 399
11.5 Solutions ............................................. 400
11.5.1 General Homogeneous Solution ................... 400
11.5.2 Macrolevel-Associated Solution ................. 403
11.5.3 Layered and Homogeneous Material ............... 406
11.5.4 Microlevel Solutions ........................... 409
11.5.5 Structure of Waves in the x, y-Plane ........... 414
11.5.6 Wave Amplitude in the x, y-Plane ............... 416
11.5.7 Existence of Real Solutions .................... 420
11.6 Viscoelastic Lattice .................................. 422
11.6.1 Introductory Remarks ........................... 422
11.6.2 Formulation and Basic Relations ................ 424
11.6.3 Stress-Strain Relation in Terms of Fourier
Transform ...................................... 425
11.6.4 Local Energy Release ........................... 427
11.6.5 Unbounded Lattice .............................. 428
11.6.6 Lattice Strip .................................. 436
11.6.7 Quasi-static Crack Growth ...................... 439
12 Triangular-Cell Elastic Lattice ............................ 445
12.1 Introductory Remarks .................................. 445
12.2 General Properties of Fundamental Solutions ........... 447
12.2.1 Lattice and Coordinates ........................ 447
12.2.2 Plan of the Solution ........................... 447
12.2.3 Some Properties of the Fundamental Solutions ... 449
12.3 Equations and General Solutions ....................... 451
12.3.1 Dynamic Equations .............................. 451
12.3.2 General Solution for the Intact Lattice ........ 453
12.3.3 Symmetry and the Modes ......................... 455
12.3.4 Dynamic Equation for a Particle with n = 0 ..... 456
12.3.5 Green's Function L(k) and Dispersion
Relations ...................................... 457
12.3.6 General Solution ............................... 461
12.4 Macrolevel-Associated Solution ........................ 462
12.4.1 Various Asymptotes ............................. 462
12.4.2 Asymptotes for L(k) ............................ 463
12.4.3 Asymptotes for L±(k) ........................... 466
12.4.4 Energy Release ................................. 468
12.4.5 Mode II Intersonic Crack Speed. Inhomogeneous
Problem ........................................ 471
12.4.6 Dissipative Waves .............................. 473
12.5 Microlevel Solutions .................................. 474
12.5.1 General Characterization ....................... 474
12.5.2 Sub-Rayleigh Crack Speed ....................... 475
12.5.3 Super-Rayleigh Crack Speed ..................... 476
12.5.4 Intersonic Crack Speed ......................... 477
12.5.5 Supersonic Crack Speed ......................... 478
12.6 Concluding Remarks .................................... 478
13 Phase Transition Waves ..................................... 481
13.1 Introductory Remarks .................................. 481
13.2 Macrolevel Solution ................................... 483
13.3 Discrete Chain ........................................ 485
13.3.1 Formulation .................................... 485
13.3.2 Derivation of the Governing Equation ........... 486
13.3.3 Factorization .................................. 488
13.3.4 General Homogeneous Solution ................... 490
13.3.5 Macrolevel-Associated Solution ................. 492
13.3.6 Chain-Based Macrolevel Solution ................ 494
13.3.7 Dissipative Waves .............................. 499
13.3.8 Microlevel Solutions ........................... 499
13.4 Higher-Order-Derivative Model ......................... 503
13.4.1 Some General Considerations .................... 503
13.4.2 Theorem on the Highest Modulus ................. 505
13.4.3 Equation of the Fourth Order ................... 507
13.4.4 Subsonic Speed ................................. 509
13.4.5 Intersonic Speed ............................... 511
13.4.6 Supersonic Speed ............................... 513
13.5 Concluding Remarks .................................... 514
14 Dynamic Amplification Factor in Fracture and Phase
Transition ................................................. 517
14.1 Introductory Remarks .................................. 517
14.2 Line of Viscoelastic Oscillators ...................... 519
14.3 DOR and SAR Domains for Viscoelastic Oscillator ....... 521
14.4 Viscoelastic Square-Cell Lattice ...................... 523
14.4.1 Superposition .................................. 523
14.4.2 Derivation of a Governing Equation ............. 525
14.4.3 Factorization .................................. 527
14.4.4 Division of the Right-Hand Side ................ 527
14.4.5 Solution ....................................... 528
14.5 Slow Phase Transition Wave in a Chain ................. 532
14.5.1 Formulation .................................... 532
14.5.2 Superposition .................................. 532
14.5.3 Solution ....................................... 533
14.5.4 Some Remarks ................................... 535
14.6 Triangular-Cell Lattice. Irregularities in Fracture ... 537
14.6.1 Introductory Remarks ........................... 537
14.6.2 Superposition .................................. 538
14.6.3 Superposition Paradox .......................... 539
14.6.4 Transient Problem for an Intact Viscoelastic
Lattice ........................................ 540
14.6.5 Lattice with a Crack ........................... 543
14.6.6 Solution of the Auxiliary Problem .............. 545
14.6.7 Solutions for Statics .......................... 546
14.6.8 Some Results of Numerical Simulations .......... 549
14.6.9 Concluding Remarks ............................. 556
References ............................................ 559
Index ......................................................... 573
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