Mechanical Vibration and Shock Analysis, Fatigue Damage

Christian Lalanne is a Consultant Engineer who previously worked as an expert at the French Atomic Energy Authority and who has specialized in the study of vibration and shock for more than 40 years. He has been associated with the new methods of drafting testing specifications and associated informatic tools.

Foreword to Series xiii

Introduction   xvii

List of Symbols   xix

Chapter 1. Concepts of Material Fatigue 1

1.1. Introduction 1

1.1.1. Reminders on the strength of materials 1

1.1.2. Fatigue  9

1.2. Types of dynamic loads (or stresses)  10

1.2.1. Cyclic stress 10

1.2.2. Alternating stress 12

1.2.3. Repeated stress 13

1.2.4. Combined steady and cyclic stress 13

1.2.5. Skewed alternating stress  14

1.2.6. Random and transitory stresses 14

1.3. Damage arising from fatigue   15

1.4. Characterization of endurance of materials 18

1.4.1. S-N curve 18

1.4.2. Influence of the average stress on the S-N curve 21

1.4.3. Statistical aspect 22

1.4.4. Distribution laws of endurance  23

1.4.5. Distribution laws of fatigue strength 26

1.4.6. Relation between fatigue limit and static properties of materials 28

1.4.7. Analytical representations of S-N curve    31

1.5. Factors of influence 41

1.5.1. General  41

1.5.2. Scale 42

1.5.3. Overloads 43

1.5.4. Frequency of stresses    44

1.5.5. Types of stresses 45

1.5.6. Non-zero mean stress   45

1.6. Other representations of S-N curves 48

1.6.1. Haigh diagram 48

1.6.2. Statistical representation of Haigh diagram   58

1.7. Prediction of fatigue life of complex structures 58

1.8. Fatigue in composite materials   59

Chapter 2. Accumulation of Fatigue Damage 61

2.1. Evolution of fatigue damage    61

2.2. Classification of various laws of accumulation 62

2.3. Miner’s method 63

2.3.1. Miner’s rule 63

2.3.2. Scatter of damage to failure as evaluated by Miner 67

2.3.3. Validity of Miner’s law of accumulation of damage in case of random stress 71

2.4. Modified Miner’s theory 73

2.4.1. Principle 73

2.4.2. Accumulation of damage using modified Miner’s rule    74

2.5. Henry’s method 77

2.6. Modified Henry’s method   79

2.7. Corten and Dolan’s method    79

2.8. Other theories 82

Chapter 3. Counting Methods for Analyzing Random Time History   85

3.1. General   85

3.2. Peak count method89

3.2.1. Presentation of method  89

3.2.2. Derived methods 92

3.2.3. Range-restricted peak count method 93

3.2.4. Level-restricted peak count method 93

3.3. Peak between mean-crossing count method 95

3.3.1. Presentation of method  95

3.3.2. Elimination of small variations  97

3.4. Range count method 98

3.4.1. Presentation of method  98

3.4.2. Elimination of small variations  100

3.5. Range-mean count method  101

3.5.1. Presentation of method  101

3.5.2. Elimination of small variations  104

3.6. Range-pair count method    106

3.7. Hayes’ counting method110

3.8. Ordered overall range counting method 112

3.9. Level-crossing count method  114

3.10. Peak valley peak counting method 118

3.11. Fatigue-meter counting method 123

3.12. Rainflow counting method    125

3.12.1. Principle of method 126

3.12.2. Subroutine for rainflow counting 131

3.13. NRL (National Luchtvaart Laboratorium) counting method    134

3.14. Evaluation of time spent at a given level 137

3.15. Influence of levels of load below fatigue limit on fatigue life  138

3.16. Test acceleration 138

3.17. Presentation of fatigue curves determined by random vibration tests 141

Chapter 4. Fatigue Damage by One-degree-of-freedom Mechanical System 143

4.1. Introduction 143

4.2. Calculation of fatigue damage due to signal versus time 144

4.3. Calculation of fatigue damage due to acceleration spectral density 146

4.3.1. General case 146

4.3.2. Particular case of a wideband response, e.g. at the limit r ?­ 0 151

4.3.3. Particular case of narrowband response 152

4.3.4. Rms response to narrowband noise G0 of width ?´f when G0 ?´ f ?­ constant 164

4.3.5. Steinberg approach 165

4.4. Equivalent narrowband noise  166

4.4.1. Use of relation established for narrowband response 167

4.4.2. Alternative: use of mean number of maxima per second  169

4.5. Calculation of damage from the modified Rice distribution of peaks 171

4.5.1. Approximation to real maxima distribution using a modified Rayleigh distribution 171

4.5.2. Wirsching and Light’s approach 175

4.5.3. Chaudhury and Dover’s approach 176

4.5.4. Approximate expression of the probability density of peaks   180

4.6. Other approaches 182

4.7. Calculation of fatigue damage from rainflow domains 185

4.7.1. Wirsching’s approach   185

4.7.2. Tunna’s approach 189

4.7.3. Ortiz-Chen’s method 191

4.7.4. Hancock’s approach 191

4.7.5. Abdo and Rackwitz’s approach 192

4.7.6. Kam and Dover’s approach   192

4.7.7. Larsen and Lutes (“single moment”) method 193

4.7.8. Jiao-Moan’s method 194

4.7.9. Dirlik’s probability density   195

4.7.10. Madsen’s approach 207

4.7.11. Zhao and Baker model    207

4.7.12. Tovo and Benasciutti method  208

4.8. Comparison of S-N curves established under sinusoidal and random loads  211

4.9. Comparison of theory and experiment 216

4.10. Influence of shape of power spectral density and value of irregularity factor 221

4.11. Effects of peak truncation  221

4.12. Truncation of stress peaks    222

4.12.1. Particular case of a narrowband noise 223

4.12.2. Layout of the S-N curve for a truncated distribution 232

Chapter 5. Standard Deviation of Fatigue Damage 237

5.1. Calculation of standard deviation of damage: Bendat’s method  237

5.2. Calculation of standard deviation of damage: Mark’s method    242

5.3. Comparison of Mark and Bendat’s results    247

5.4. Standard deviation of the fatigue life 253

5.4.1. Narrowband vibration   253

5.4.2. Wideband vibration   256

5.5. Statistical S-N curves 257

5.5.1. Definition of statistical curves  257

5.5.2. Bendat’s formulation    258

5.5.3. Mark’s formulation. 261

Chapter 6. Fatigue Damage using Other Calculation Assumptions  267

6.1. S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit)   267

6.2. S-N curve defined by two segments of straight line on log-lin scales 270

6.3. Hypothesis of non-linear accumulation of damage 273

6.3.1. Corten-Dolan’s accumulation law 273

6.3.2. Morrow’s accumulation model  275

6.4. Random vibration with non-zero mean: use of modified Goodman diagram 277

6.5. Non-Gaussian distribution of instantaneous values of signal  280

6.5.1. Influence of distribution law of instantaneous values   280

6.5.2. Influence of peak distribution 281

6.5.3. Calculation of damage using Weibull distribution 281

6.5.4. Comparison of Rayleigh assumption/peak counting 284

6.6. Non-linear mechanical system 286

Chapter 7. Low-cycle Fatigue 289

7.1. Overview 289

7.2. Definitions  290

7.2.1. Baushinger effect 290

7.2.2. Cyclic strain hardening    291

7.2.3. Properties of cyclic stress–strain curves    291

7.2.4. Stress–strain curve 291

7.2.5. Hysteresis and fracture by fatigue 295

7.2.6. Significant factors influencing hysteresis and fracture by fatigue   295

7.2.7. Cyclic stress–strain curve (or cyclic consolidation curve)    296

7.3. Behavior of materials experiencing strains in the oligocyclic domain 297

7.3.1. Types of behaviors 297

7.3.2. Cyclic strain hardening    297

7.3.3. Cyclic strain softening   299

7.3.4. Cyclically stable metals    300

7.3.5. Mixed behavior 301

7.4. Influence of the level application sequence 301

7.5. Development of the cyclic stress–strain curve 303

7.6. Total strain  304

7.7. Fatigue strength curve 305

7.8. Relation between plastic strain and number of cycles to fracture   306

7.8.1. Orowan relation 306

7.8.2. Manson relation 307

7.8.3. Coffin relation 307

7.8.4. Shanley relation 317

7.8.5. Gerberich relation 318

7.8.6. Sachs, Gerberich, Weiss and Latorre relation 318

7.8.7. Martin relation 318

7.8.8. Tavernelli and Coffin relation  319

7.8.9. Manson relation 319

7.8.10. Ohji et al. relation 321

7.8.11. Bui-Quoc et al. relation  321

7.9. Influence of the frequency and temperature in the plastic field  321

7.9.1. Overview 321

7.9.2. Influence of frequency   322

7.9.3. Influence of temperature and frequency 322

7.9.4. Effect of frequency on plastic strain range 324

7.9.5. Equation of generalized fatigue 325

7.10. Laws of damage accumulation  326

7.10.1. Miner rule 326

7.10.2. Yao and Munse relation  327

7.10.3. Use of the Manson–Coffin relation 329

7.11. Influence of an average strain or stress 329

7.12. Low-cycle fatigue of composite material 332

Chapter 8. Fracture Mechanics    335

8.1. Overview 335

8.2. Fracture mechanism 338

8.2.1. Major phases 338

8.2.2. Initiation of cracks 339

8.2.3. Slow propagation of cracks   341

8.3. Critical size: strength to fracture 341

8.4. Modes of stress application    343

8.5. Stress intensity factor 344

8.5.1. Stress in crack root 344

8.5.2. Mode I  346

8.5.3. Mode II  349

8.5.4. Mode III 350

8.5.5. Field of equation use 350

8.5.6. Plastic zone 352

8.5.7. Other form of stress expressions 354

8.5.8. General form 356

8.5.9. Widening of crack opening   357

8.6. Fracture toughness: critical K value 358

8.7. Calculation of the stress intensity factor 362

8.8. Stress ratio  365

8.9. Expansion of cracks: Griffith criterion 367

8.10. Factors affecting the initiation of cracks 369

8.11. Factors affecting the propagation of cracks    369

8.11.1. Mechanical factors 370

8.11.2. Geometric factors 372

8.11.3. Metallurgical factors   373

8.11.4. Factors linked to the environment 373

8.12. Speed of propagation of cracks 374

8.13. Effect of a non-zero mean stress 379

8.14. Laws of crack propagation    379

8.14.1. Head law 380

8.14.2. Modified Head law 381

8.14.3. Frost and Dugsdale 381

8.14.4. McEvily and Illg 382

8.14.5. Paris and Erdogan 383

8.15. Stress intensity factor 396

8.16. Dispersion of results 397

8.17. Sample tests: extrapolation to a structure 398

8.18. Determination of the propagation threshold KS 398

8.19. Propagation of cracks in the domain of low-cycle fatigue    400

8.20. Integral J 401

8.21. Overload effect: fatigue crack retardation 403

8.22. Fatigue crack closure 405

8.23. Rules of similarity 407

8.24. Calculation of a useful lifetime 407

8.25. Propagation of cracks under random load 410

8.25.1. Rms approach 411

8.25.2. Narrowband random loads  416

8.25.3. Calculation from a load collective 422

Appendix    427

Bibliography  441

Index 487

Summary of Other Volumes in the Series 491