Fracture Mechanics 2

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

• Preface xiGlossary xixChapter 1. Fracture Mechanisms by Fatigue  11.1. Introduction 11.2. Principal physical mechanisms of cracking by fatigue 21.2.1. Fracture mechanics 21.2.2. Criteria of fracture (plasticity) in mechanics 41.3. Modes of fracture 71.3.1. Directed works 111.4. Fatigue of metals: analytical expressions used in reliability 131.4.1. Wöhler’s law 141.4.2. Basquin’s law (1910) 151.4.3. Stromayer’s law (1914) 161.4.4. Palmgren’s law 161.4.5. Corson’s law (1949) 171.4.6. Bastenaire’s law 171.4.7. Weibull’s law 181.4.8. Henry’s law 181.4.9. Corten and Dolen’s law 191.4.10. Manson–Coffin’s law 201.5. Reliability models commonly used in fracture mechanics by fatigue 221.5.1. Coffin–Manson’s model for the analysis of crack propagation 241.5.2. Neuber’s relation (1958) 251.5.3. Arrhenius’ model 281.5.4. Miner’s law (1954) 291.6. Main common laws retained by fracture mechanics 311.6.1. Fost and Dugdale’s law 331.6.2. McEvily’s law (1979) 341.6.3. Paris’s law 351.6.4. G.R. Sih’s law 391.7. Stress intensity factors in fracture mechanics 401.7.1. Maddox’s model 401.7.2. Gross and Srawley’s model 411.7.3. Lawrence’s model 411.7.4. Martin and Bousseau’s model 421.7.5. Gurney’s model 431.7.6. Engesvik’s model 431.7.7. Yamada and Albrecht’s model 441.7.8. Tomkins and Scott’s model 451.7.9. Harrison’s model 461.8. Intrinsic parameters of the material (C and m) 461.9. Fracture mechanics elements used in reliability 481.10. Crack rate (life expectancy) and s.i.f. (Kσ) 511.10.1. Simplified version of Taylor’s law for machining 541.11. Elements of stress (S) and resistance theory (R) 551.11.1. Case study, part 2 – suspension bridge (Cirta) 551.11.2. Case study: failure surface of geotechnical materials 571.12. Conclusion 651.13. Bibliography 65Chapter 2. Analysis Elements for Determining the Probability of Rupture by Simple Bounds  692.1. Introduction 692.1.1. First-order bounds or simple bounds: systems in series 702.1.2. First-order bounds or simple bounds: systems in parallel 702.2. Second-order bounds or Ditlevsen’s bounds 702.2.1. Evaluating the probability of the intersection of two events 712.2.2. Estimating multinomial distribution–normal distribution 742.2.3. Binomial distribution 742.2.4. Approximation of ô2 (for m ≥ 3) 762.3. Hohenbichler’s method 782.4. Hypothesis test, through the example of a normal average with unknown variance 802.4.1. Development and calculations 822.5. Confidence interval for estimating a normal mean: unknown variance 842.6. Conclusion 852.7. Bibliography 85Chapter 3. Analysis of the Reliability of Materials and Structures by the Bayesian Approach 873.1. Introduction to the Bayesian method used to evaluate reliability 873.2. Posterior distribution and conjugate models 883.2.1. Independent events 913.2.2. Counting diagram 953.3. Conditional probability or Bayes’ law 993.4. Anterior and posterior distributions 1033.5. Reliability analysis by moments methods, FORM/SORM 1063.6. Control margins from the results of fracture mechanics 1073.7. Bayesian model by exponential gamma distribution 1103.8. Homogeneous Poisson process and rate of occurrence of failure 1123.9. Estimating the maximum likelihood 1133.9.1. Type I censored exponential model 1133.9.2. Estimating the MTBF (or rate of repair/rate of failure) 1133.9.3. MTBF and confidence interval 1143.10. Repair rate or ROCOF 1173.10.1. Power law: non-homogeneous Poisson process 1183.10.2. Distribution law – gamma (reminder) 1193.10.3. Bayesian model of a priori gamma distribution 1223.10.4. Distribution tests for exponential life (or HPP model) 1243.10.5. Bayesian procedure for the exponential system model 1263.11. Bayesian case study applied in fracture mechanics 1313.12. Conclusion 1373.13. Bibliography 138Chapter 4. Elements of Analysis for the Reliability of Components by Markov Chains  1414.1. Introduction 1414.2. Applying Markov chains to a fatigue model 1424.3. Case study with the help of Markov chains for a fatigue model 1454.3.1. Position of the problem 1464.3.2. Discussion 1494.3.3. Explanatory information 1494.3.4. Directed works 1544.3.5. Approach for solving the problem 1554.3.6. Which solution should we choose? 1564.4. Conclusion 1574.5. Bibliography 157Chapter 5. Reliability Indices  1595.1. Introduction 1595.2. Design of material and structure reliability 1615.2.1. Reliability of materials and structures 1625.3. First-order reliability method 1655.4. Second-order reliability method 1655.5. Cornell’s reliability index 1665.6. Hasofer–Lind’s reliability index 1685.7. Reliability of material and structure components 1715.8. Reliability of systems in parallels and series 1725.8.1. Parallel system 1725.8.2. Parallel system (m/n) 1735.8.3. Serial assembly system 1735.9. Conclusion 1795.10. Bibliography 179Chapter 6. Fracture Criteria Reliability Methods through an Integral Damage Indicator 1816.1. Introduction 1816.2. Literature review of the integral damage indicator method 1856.2.1. Brief recap of the FORM/SORM method 1866.2.2. Recap of the Hasofer–Lind index method 1876.3. Literature review of the probabilistic approach of cracking law parameters in region II of the Paris law 1886.4. Crack spreading by a classical fatigue model 1906.5. Reliability calculations using the integral damage indicator method 1976.6. Conclusion 1996.7. Bibliography 201Chapter 7. Monte Carlo Simulation  2057.1. Introduction  2057.1.1. From the origin of the Monte Carlo method! 2057.1.2. The terminology 2067.2. Simulation of a singular variable of a Gaussian 2097.2.1. Simulation of non-Gaussian variable 2107.2.2. Simulation of correlated variables 2107.2.3. Simulation of correlated Gaussian variables  2107.2.4. Simulation of correlated non-Gaussian variables 2107.3. Determining safety indices using Monte Carlo simulation 2127.3.1. General tools and problem outline 2127.3.2. Presentation and discussion of our experimental results 2147.3.3. Use of the randomly selected numbers table 2157.4. Applied mathematical techniques to generate random numbers by MC simulation on four principle statistical laws 2207.4.1. Uniform law  2207.4.2. Laplace–Gauss (normal) law 2217.4.3. Exponential law 2227.4.4. Initial value control 2227.5. Conclusion 2317.6. Bibliography 232Chapter 8. Case Studies  2358.1. Introduction 2358.2. Reliability indicators (λ) and MTBF 2358.2.1. Model of parallel assembly 2358.2.2. Model of serial assembly 2368.3. Parallel or redundant model 2378.4. Reliability and structural redundancy: systems without distribution 2398.4.1. Serial model 2398.5. Rate of constant failure 2408.5.1. Reliability of systems without repairing: parallel model 2438.6. Reliability applications in cases of redundant systems 2488.6.1. Total active redundancy 2528.6.2. Partial active redundancy 2538.7. Reliability and availability of repairable systems 2588.8. Quality assurance in reliability 2648.8.1. Projected analysis of reliability 2648.9. Birnbaum–Saunders distribution in crack spreading 2688.9.1. Probability density and distribution function (Birnbaum–Saunders cumulative distribution through cracking) 2698.9.2. Graph plots for the four probability density functions and distribution functions 2708.10. Reliability calculation for ages (τ) in hours of service, Ri(τ) = ? 2708.11. Simulation methods in mechanical reliability of structures and materials: the Monte Carlo simulation method 2758.11.1. Weibull law 2778.11.2. Log-normal Law (of Galton) 2788.11.3. Exponential law  2788.11.4. Generation of random numbers 2798.12. Elements of safety via the couple: resistance and stress (R, S) 2848.13. Reliability trials 2868.13.1. Controlling risks and efficiency in mechanical reliability 2888.13.2. Truncated trials 2918.13.3. Censored trials 2928.13.4. Trial plan 2938.13.5. Coefficients for the trial’s acceptance plan 2968.13.6. Trial’s rejection plan (in the same conditions) 2978.13.7. Trial plan in reliability and K Pearson test χ2 2998.14. Reliability application on speed reducers (gears) 3008.14.1. Applied example on hydraulic motors 3038.15. Reliability case study in columns under stress of buckling 3058.15.1. RDM solution 3078.15.2. Problem outline and probabilistic solution (reliability and error) 3098.16. Adjustment of least squared for nonlinear functions 3118.16.1. Specific case study 1: a Weibull law with two parameters 3118.17. Conclusion 3148.18. Bibliography 314Appendix 317Index 333