Slepyan L.I. Models and phenomena in fracture mechanics (Berlin; Heidelbrg; New York, 2002). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаSlepyan L.I. Models and phenomena in fracture mechanics. - Berlin; Heidelbrg; New York: Springer-Verlag, 2002. - xvii, 576 p.: ill. - (Foundations of engineering mechanics). - Ref.: p.559-579. - Ind.: p.573-576. - ISBN 978-3-540-43767-3
 

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Оглавление / Contents
 
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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