Fracture Mechanics 2: Applied Reliability

This second book of a 3-volume set on Fracture Mechanics completes the first volume through the analysis of adjustment tests suited to correctly validating the justified use of the laws conforming to the behavior of the materials and structures under study. This volume focuses on the vast range of statistical distributions encountered in reliability. Its aim is to run statistical measurements, to present a report on enhanced measures in mechanical reliability and to evaluate the reliability of repairable or unrepairable systems. To achieve this, the author presents a theoretical and practice-based approach on the following themes: criteria of failures; Bayesian applied probability; Markov chains; Monte Carlo simulation as well as many other solved case studies. This book distinguishes itself from other works in the field through its originality in presenting an educational approach which aims at helping practitioners both in academia and industry. It is intended for technicians, engineers, designers, students, and teachers working in the fields of engineering and vocational education. The main objective of the author is to provide an assessment of indicators of quality and reliability to aid in decision-making. To this end, an intuitive and practical approach, based on mathematical rigor, is recommended.

Table of Contents:
Preface xi Glossary xix Chapter 1. Fracture Mechanisms by Fatigue  1 1.1. Introduction 1 1.2. Principal physical mechanisms of cracking by fatigue 2 1.2.1. Fracture mechanics 2 1.2.2. Criteria of fracture (plasticity) in mechanics 4 1.3. Modes of fracture 7 1.3.1. Directed works 11 1.4. Fatigue of metals: analytical expressions used in reliability 13 1.4.1. Wöhler’s law 14 1.4.2. Basquin’s law (1910) 15 1.4.3. Stromayer’s law (1914) 16 1.4.4. Palmgren’s law 16 1.4.5. Corson’s law (1949) 17 1.4.6. Bastenaire’s law 17 1.4.7. Weibull’s law 18 1.4.8. Henry’s law 18 1.4.9. Corten and Dolen’s law 19 1.4.10. Manson–Coffin’s law 20 1.5. Reliability models commonly used in fracture mechanics by fatigue 22 1.5.1. Coffin–Manson’s model for the analysis of crack propagation 24 1.5.2. Neuber’s relation (1958) 25 1.5.3. Arrhenius’ model 28 1.5.4. Miner’s law (1954) 29 1.6. Main common laws retained by fracture mechanics 31 1.6.1. Fost and Dugdale’s law 33 1.6.2. McEvily’s law (1979) 34 1.6.3. Paris’s law 35 1.6.4. G.R. Sih’s law 39 1.7. Stress intensity factors in fracture mechanics 40 1.7.1. Maddox’s model 40 1.7.2. Gross and Srawley’s model 41 1.7.3. Lawrence’s model 41 1.7.4. Martin and Bousseau’s model 42 1.7.5. Gurney’s model 43 1.7.6. Engesvik’s model 43 1.7.7. Yamada and Albrecht’s model 44 1.7.8. Tomkins and Scott’s model 45 1.7.9. Harrison’s model 46 1.8. Intrinsic parameters of the material (C and m) 46 1.9. Fracture mechanics elements used in reliability 48 1.10. Crack rate (life expectancy) and s.i.f. (Kσ) 51 1.10.1. Simplified version of Taylor’s law for machining 54 1.11. Elements of stress (S) and resistance theory (R) 55 1.11.1. Case study, part 2 – suspension bridge (Cirta) 55 1.11.2. Case study: failure surface of geotechnical materials 57 1.12. Conclusion 65 1.13. Bibliography 65 Chapter 2. Analysis Elements for Determining the Probability of Rupture by Simple Bounds  69 2.1. Introduction 69 2.1.1. First-order bounds or simple bounds: systems in series 70 2.1.2. First-order bounds or simple bounds: systems in parallel 70 2.2. Second-order bounds or Ditlevsen’s bounds 70 2.2.1. Evaluating the probability of the intersection of two events 71 2.2.2. Estimating multinomial distribution–normal distribution 74 2.2.3. Binomial distribution 74 2.2.4. Approximation of ô2 (for m ≥ 3) 76 2.3. Hohenbichler’s method 78 2.4. Hypothesis test, through the example of a normal average with unknown variance 80 2.4.1. Development and calculations 82 2.5. Confidence interval for estimating a normal mean: unknown variance 84 2.6. Conclusion 85 2.7. Bibliography 85 Chapter 3. Analysis of the Reliability of Materials and Structures by the Bayesian Approach 87 3.1. Introduction to the Bayesian method used to evaluate reliability 87 3.2. Posterior distribution and conjugate models 88 3.2.1. Independent events 91 3.2.2. Counting diagram 95 3.3. Conditional probability or Bayes’ law 99 3.4. Anterior and posterior distributions 103 3.5. Reliability analysis by moments methods, FORM/SORM 106 3.6. Control margins from the results of fracture mechanics 107 3.7. Bayesian model by exponential gamma distribution 110 3.8. Homogeneous Poisson process and rate of occurrence of failure 112 3.9. Estimating the maximum likelihood 113 3.9.1. Type I censored exponential model 113 3.9.2. Estimating the MTBF (or rate of repair/rate of failure) 113 3.9.3. MTBF and confidence interval 114 3.10. Repair rate or ROCOF 117 3.10.1. Power law: non-homogeneous Poisson process 118 3.10.2. Distribution law – gamma (reminder) 119 3.10.3. Bayesian model of a priori gamma distribution 122 3.10.4. Distribution tests for exponential life (or HPP model) 124 3.10.5. Bayesian procedure for the exponential system model 126 3.11. Bayesian case study applied in fracture mechanics 131 3.12. Conclusion 137 3.13. Bibliography 138 Chapter 4. Elements of Analysis for the Reliability of Components by Markov Chains  141 4.1. Introduction 141 4.2. Applying Markov chains to a fatigue model 142 4.3. Case study with the help of Markov chains for a fatigue model 145 4.3.1. Position of the problem 146 4.3.2. Discussion 149 4.3.3. Explanatory information 149 4.3.4. Directed works 154 4.3.5. Approach for solving the problem 155 4.3.6. Which solution should we choose? 156 4.4. Conclusion 157 4.5. Bibliography 157 Chapter 5. Reliability Indices  159 5.1. Introduction 159 5.2. Design of material and structure reliability 161 5.2.1. Reliability of materials and structures 162 5.3. First-order reliability method 165 5.4. Second-order reliability method 165 5.5. Cornell’s reliability index 166 5.6. Hasofer–Lind’s reliability index 168 5.7. Reliability of material and structure components 171 5.8. Reliability of systems in parallels and series 172 5.8.1. Parallel system 172 5.8.2. Parallel system (m/n) 173 5.8.3. Serial assembly system 173 5.9. Conclusion 179 5.10. Bibliography 179 Chapter 6. Fracture Criteria Reliability Methods through an Integral Damage Indicator 181 6.1. Introduction 181 6.2. Literature review of the integral damage indicator method 185 6.2.1. Brief recap of the FORM/SORM method 186 6.2.2. Recap of the Hasofer–Lind index method 187 6.3. Literature review of the probabilistic approach of cracking law parameters in region II of the Paris law 188 6.4. Crack spreading by a classical fatigue model 190 6.5. Reliability calculations using the integral damage indicator method 197 6.6. Conclusion 199 6.7. Bibliography 201 Chapter 7. Monte Carlo Simulation  205 7.1. Introduction  205 7.1.1. From the origin of the Monte Carlo method! 205 7.1.2. The terminology 206 7.2. Simulation of a singular variable of a Gaussian 209 7.2.1. Simulation of non-Gaussian variable 210 7.2.2. Simulation of correlated variables 210 7.2.3. Simulation of correlated Gaussian variables  210 7.2.4. Simulation of correlated non-Gaussian variables 210 7.3. Determining safety indices using Monte Carlo simulation 212 7.3.1. General tools and problem outline 212 7.3.2. Presentation and discussion of our experimental results 214 7.3.3. Use of the randomly selected numbers table 215 7.4. Applied mathematical techniques to generate random numbers by MC simulation on four principle statistical laws 220 7.4.1. Uniform law  220 7.4.2. Laplace–Gauss (normal) law 221 7.4.3. Exponential law 222 7.4.4. Initial value control 222 7.5. Conclusion 231 7.6. Bibliography 232 Chapter 8. Case Studies  235 8.1. Introduction 235 8.2. Reliability indicators (λ) and MTBF 235 8.2.1. Model of parallel assembly 235 8.2.2. Model of serial assembly 236 8.3. Parallel or redundant model 237 8.4. Reliability and structural redundancy: systems without distribution 239 8.4.1. Serial model 239 8.5. Rate of constant failure 240 8.5.1. Reliability of systems without repairing: parallel model 243 8.6. Reliability applications in cases of redundant systems 248 8.6.1. Total active redundancy 252 8.6.2. Partial active redundancy 253 8.7. Reliability and availability of repairable systems 258 8.8. Quality assurance in reliability 264 8.8.1. Projected analysis of reliability 264 8.9. Birnbaum–Saunders distribution in crack spreading 268 8.9.1. Probability density and distribution function (Birnbaum–Saunders cumulative distribution through cracking) 269 8.9.2. Graph plots for the four probability density functions and distribution functions 270 8.10. Reliability calculation for ages (τ) in hours of service, Ri(τ) = ? 270 8.11. Simulation methods in mechanical reliability of structures and materials: the Monte Carlo simulation method 275 8.11.1. Weibull law 277 8.11.2. Log-normal Law (of Galton) 278 8.11.3. Exponential law  278 8.11.4. Generation of random numbers 279 8.12. Elements of safety via the couple: resistance and stress (R, S) 284 8.13. Reliability trials 286 8.13.1. Controlling risks and efficiency in mechanical reliability 288 8.13.2. Truncated trials 291 8.13.3. Censored trials 292 8.13.4. Trial plan 293 8.13.5. Coefficients for the trial’s acceptance plan 296 8.13.6. Trial’s rejection plan (in the same conditions) 297 8.13.7. Trial plan in reliability and K Pearson test χ2 299 8.14. Reliability application on speed reducers (gears) 300 8.14.1. Applied example on hydraulic motors 303 8.15. Reliability case study in columns under stress of buckling 305 8.15.1. RDM solution 307 8.15.2. Problem outline and probabilistic solution (reliability and error) 309 8.16. Adjustment of least squared for nonlinear functions 311 8.16.1. Specific case study 1: a Weibull law with two parameters 311 8.17. Conclusion 314 8.18. Bibliography 314 Appendix 317 Index 333

About the Author :
Ammar Grous is Teacher of Mechanical Engineering at CéGEP de l'Outaouais (Academic College), Gatineau, Quebec, Canada.