Table Of ContentFatigue Damage,
Crack Growth and
Life Prediction
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Fatigue Damage,
Crack Growth and
Life Prediction
Fernand Ellyin
Professor of Mechanical Engineering
NOVA CORP and NSERC Senior Industrial Research Chair
University of Alberta
Edmonton, Alberta, Canada
I~nl
CHAPMAN & HALL
London· Weinheim . New York· Tokyo· Melbourne· Madras
Published by Chapman & Hall, 2-6 Boundary Row, London SE18HN, UK
Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK
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First edition 1997
© 1997 Chapman & Hall
Softcover reprint of the hardcover 1s t edition 1997
Typeset in 10/12 Palatino by Thomson Press (India) Ltd, New Delhi
Edmunds, Suffolk
ISBN-13:97S-94-010-7175-S e-ISBN-13:97S-94-009-1509-1
DOl: 10 .1 007/97S-94-009-1509-1
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I§
Printed on permanent acid-free text paper, manufactured in accordance
with ANSI/NISO Z39.48-1992 and ANSI/NISO Z39.48-1984 (Permanence
of Paper).
To my wife, Suzanne, and our
children, Lise and Christopher
I dedicate this book
Contents
Preface xiii
1 Some general concepts concerning fatigue 1
1.1 Introduction 1
1.2 Types ofloading 3
1.2.1 Approximate methods of fatigue load description 5
1.2.2 The rainflow counting method 8
1.3 Fatigue failure mechanisms 8
1.3.1 Microstructural changes during deformation processes 10
1.3.2 Initiation of fatigue cracks 12
1.3.3 Crack propagation 15
1.3.4 Effect oftemperature 20
1.3.5 Effect of oxidation 21
1.4 Factors affecting fatigue life 22
1.4.1 Effect of microstructure 23
1.4.2 Effect of processing techniques 23
1.4.3 Environmental factors 24
1.4.4 Effect of load spectrum 24
1.4.5 Effect of geometry 25
1.5 Fatigue design methodology 26
1.5.1 Safe-life design method 26
1.5.2 Fail-safe design method 28
1.5.3 Damage tolerance design method 28
1.6 Probabilistic approach 29
References 30
2 Cyclic stress-strain response 33
2.1 Introduction 33
2.2 Monotonic behaviour under tension or compression 33
2.3 Material response to cyclic deformation or
loading - transient behaviour 38
2.4 Stable cyclic response 43
2.4.1 Microstructural changes during cyclic loading 45
2.4.2 Determination of the cyclic curve 48
2.4.3 Mathematical description of the stress-strain relationship 53
2.5 Analysis of hysteresis loops 55
2.6 Description of the master curve 57
2.7 Slope of the stress-strain curve during load reversal 62
2.8 Effect of temperature on the cyclic stress-strain relationship 64
viii Contents
2.9 Effect of environment on the stable cyclic stress-strain relationship 67
2.10 Effect of rate of loading on the stable cyclic response 69
2.11 Cyclic stress-strain relationship for multiaxial
stress states - proportional loading paths 69
References 73
3 Phenomenological approach to fatigue life prediction under
uniaxial loading 77
3.1 Introduction 77
3.1.1 General approach 78
3.2 Stress-based approach 80
3.3 Strain-based approach 82
3.4 Energy-based approach 85
3.4.1 Hysteresis energy approach 87
3.4.2 Plastic strain energy approach 91
3.4.3 Total strain energy approach 93
3.4.4 Note on the mean-stress effect 96
3.5 Cumulative damage 97
3.5.1 Description of the concept 97
3.5.2 Multi-level cyclic loading 100
3.5.3 Specification of the damage function with reference
to other approaches 103
3.5.4 Determination of the critical damage curve 108
3.5.5 Note on the damage controlling variable, t/J 113
3.6 Time-dependent fatigue 115
3.6.1 Effect of the wave form 116
3.6.2 Life prediction methods 118
3.7 A mechanism-based damage function for time-dependent fatigue 119
3.7.1 Correlation with experimental data 126
3.8 Effect of environment on crack initiation and fatigue life 129
3.8.1 Correlation with plastic strain energy 132
3.9 Effect of mean stress and ratcheting strain on fatigue life 135
3.9.1 A fatigue criterion with mean-stress and
ratcheting strain effects 136
References 139
4 Fatigue failure under multiaxial states of stress 145
4.1 Introduction 145
4.2 Previous investigations 148
4.2.1 Stress-based criteria 149
4.2.2 Strain-based criteria 150
4.2.3 Energy-based criteria 154
4.3 A general approach to multiaxial fatigue 155
4.3.1 Elastic strain energy 157
4.3.2 Cyclic plastic strain energy 161
4.3.3 Cyclic total strain energy 164
4.4 The multiaxial fatigue failure criterion 165
4.5 Multiaxial fatigue life prediction 168
Contents ix
4.6 Effect of mean-stress - proportional loading 170
4.7 Non-proportional cyclic loading 172
4.8 Effects of mean stress and ratcheting deformation 173
References 176
5 Multiaxial experimental facilities 179
5.1 Introduction 179
5.2 Specimen geometry 179
5.3 Analysis of thin-walled cylindrical specimens 181
5.3.1 Computed stresses and strains in the gauge length 182
5.4 The test system 187
5.5 Measuring devices 189
5.6 Test procedure 192
5.7 Typical multiaxial test results 194
5.7.1 In-phase loading conditions (¢ = 0) 194
5.7.2 Out-of-phase loading conditions (¢ oF 0) 196
5.8 Other test facilities 201
References 201
6 Constitutive laws for transient and stable behaviour of inelastic solids 205
6.1 Introduction 205
6.2 Requirements for a constitutive model 206
6.2.1 Initial yield surface 206
6.2.2 Hardening rule - subsequent yield surfaces 208
6.2.3 Flow rule 212
6.3 Experimental definition of yield point and yield loci 213
6.4 Experimental observations 216
6.4.1 Subsequent yield loci 217
6.4.2 Hardening modulus curve 219
6.4.3 Evolution of yield and memory surfaces 221
6.5 A constitutive model for transient non-proportional plasticity -
rate-independent behaviour 223
6.5.1 Some experimental observations regarding transient
hardening 223
6.5.2 Description of the constitutive model 224
6.6 Correlation with some experimental results 233
6.6.1 Effect of strain range 234
6.6.2 Effect of strain history 235
6.6.3 Effect of non-proportional strain path 238
6.6.4 Effect of out-of-phase loading 240
6.6.5 Ratcheting under cyclic loading with mean stress 241
6.7 Extension to rate-dependent behaviour 245
6.7.1 Background 246
6.7.2 Formulation of a rate-dependent elastic-plastic
constitutive model 247
6.8 Correlation with some rate-dependent experimental observations 252
6.8.1 Stress-strain response at load or deformation controlled
modes 252
x Contents
6.8.2 Response to repeated strain-rate changes 253
6.8.3 Effect of strain-rate history 253
6.8.4 Stress-strain response during step-up creep tests 256
6.8.5 Strain-controlled cyclic tests with hold-time
(step-up relaxation tests) 257
6.8.6 Transient hardening behaviour for
alternating axial-torsional cycling 257
6.8.7 Material response due to change in strain trajectory and
strain-rate 259
6.9 A constitutive model for creep deformation including
prior plastic strain effects 260
6.9.1 Description of the creep model 261
6.9.2 Biaxial stress condition 269
6.9.3 Correlation with experimental results 269
6.10 Concluding remarks 272
References 273
7 Fatigue crack growth 278
7.1 Introduction 278
7.2 Linear elastic fracture mechanics 279
7.2.1 Description of stress, strain and deformation in cracked
bodies 279
7.2.2 Energy release rate 284
7.3 Nonlinear fracture mechanics 287
7.3.1 Yielding on discrete surfaces-plane stress condition 287
7.3.2 Crack fields for plastically deformed solids - HRR singular-
ity fields 290
7.3.3 Relationship between the J-integral and energy release rate 292
7.4 The concept of small-scale yielding 294
7.4.1 Elastic-plastic solutions in small-scale yielding 294
7.4.2 Plastic zone size 298
7.5 Initiation of crack growth 300
7.5.1 J-dominance region 302
7.6 Mechanics of fatigue crack growth 303
7.6.1 Elastic-plastic response to cyclic loading 308
7.6.2 Fatigue crack propagation models 311
7.7 A low-cycle fatigue-based crack propagation model 313
7.7.1 Particular cases 317
7.7.2 Comparison with experimental data 319
7.7.3 Process zone size 320
7.7.4 Load ratio effect 323
7.7.5 Analysis of the stress ratio in the crack tip area 327
7.7.6 Comparison with experimental data, R of 0 330
7.8 The crack closure phenomenon 332
7.8.1 Effect of variable amplitude loading 339
7.9 Crack closure models 346
7.9.1 Budiansky and Hutchinson model 347
7.9.2 Other models based on yielding on a discrete surface 349
Contents xi
7.9.3 Finite element studies of crack closure 351
7.10 Time-dependent crack growth - temperature effects 355
7.10.1 Time-dependent stationary crack tip fields 356
7.10.2 Creep crack growth fields 359
7.10.3 Correlation with experimental data 360
7.10.4 Combined cycle and time-dependent crack growth 361
7.11 Time-dependent crack growth - environmental effects 363
7.11.1 Proposed mechanisms 364
7.11.2 Crack growth models 366
References 370
8 Fatigue of notched members 381
8.1 Introduction 381
8.2 Notch analysis 382
8.2.1 Elastic analysis of an elliptical hole in a plate 383
8.2.2 Stress distribution around slender notches 387
8.2.3 Nonlinear analysis 388
8.2.4 Approximate methods 390
8.2.5 A general approach 391
8.2.6 The finite element method 394
8.3 Life to crack initiation 397
8.3.1 Fatigue notch factor approach 397
8.3.2 Local strain approach 398
8.3.3 Energy approach 399
8.4 Growth of cracks initiated from notches 400
8.4.1 Crack growth rate 402
8.4.2 Fatigue limit stress range in notches with short cracks 404
8.5 Initiation and growth of cracks from notches subject to
far-field cyclic compressive load 408
References 412
9 Growth and behaviour of small cracks 415
9.1 Introduction 415
9.2 Small crack regimes 417
9.3 Mechanisms of small crack growth 420
9.4 Experimental data on small crack behaviour 423
9.4.1 Initiation of small cracks 423
9.4.2 Growth of corner cracks 424
9.4.3 Crack closure development 427
9.5 Models describing small crack behaviour 429
9.5.1 Slip band models 430
9.5.2 Surface strain redistribution model 432
9.5.3 Surface layer yield stress redistribution model 433
9.5.4 Effective stress intensity factor range models 437
References 438
10 Probabilistic fatigue crack growth 442
10.1 Introduction 442
