Fracture Mechanics 1

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

• Preface ixChapter 1. Elements of Analysis of Reliability and Quality Control  11.1. Introduction 11.1.1. The importance of true physical acceleration life models (accelerated tests = true acceleration or acceleration)  31.1.2. Expression for linear acceleration relationships 41.2. Fundamental expression of the calculation of reliability 51.3. Continuous uniform distribution 91.3.1. Distribution function of probabilities (density of probability)  101.3.2. Distribution function 101.4. Discrete uniform distribution (discrete U)  121.5. Triangular distribution 131.5.1. Discrete triangular distribution version  131.5.2. Continuous triangular law version  141.5.3. Links with uniform distribution  141.6. Beta distribution 151.6.1. Function of probability density  161.6.2. Distribution function of cumulative probability 181.6.3. Estimation of the parameters (p, q) of the beta distribution  191.6.4. Distribution associated with beta distribution 201.7. Normal distribution 201.7.1. Arithmetic mean 201.7.2. Reliability 221.7.3. Stabilization and normalization of variance error  231.8. Log-normal distribution (Galton)  281.9. The Gumbel distribution 281.9.1. Random variable according to the Gumbel distribution (CRV, E1 Maximum)  291.9.2. Random variable according to the Gumbel distribution (CRV E1 Minimum) 301.10. The Frechet distribution (E2 Max)  311.11. The Weibull distribution (with three parameters) 321.12. The Weibull distribution (with two parameters)  351.12.1. Description and common formulae for the Weibull distribution and its derivatives 371.12.2. Areas where the extreme value distribution model can be used 391.12.3. Risk model 401.12.4. Products of damage 411.13. The Birnbaum–Saunders distribution  421.13.1. Derivation and use of the Birnbaum–Saunders model 431.14. The Cauchy distribution 451.14.1. Probability density function  451.14.2. Risk function  481.14.3. Cumulative risk function  481.14.4. Survival function (reliability) 491.14.5. Inverse survival function  491.15. Rayleigh distribution 501.16. The Rice distribution (from the Rayleigh distribution) 521.17. The Tukey-lambda distribution 531.18. Student’s (t) distribution 551.18.1. t-Student’s inverse cumulative function law (T)  571.19. Chi-square distribution law (χ2)  571.19.1. Probability distribution function of chi-square law (χ2) 571.19.2. Probability distribution function of chi-square law (χ2) 581.20. Exponential distribution 591.20.1. Example of applying mechanics to component lifespan 631.21. Double exponential distribution (Laplace)  661.21.1. Estimation of the parameters  661.21.2. Probability density function  661.21.3. Cumulated distribution probability function 671.22. Bernoulli distribution 681.23. Binomial distribution 711.24. Polynomial distribution 751.25. Geometrical distribution 751.25.1. Hypergeometric distribution (the Pascal distribution) versus binomial distribution  761.26. Hypergeometric distribution (the Pascal distribution) 781.27. Poisson distribution 801.28. Gamma distribution 811.29. Inverse gamma distribution 851.30. Distribution function (inverse gamma distribution probability density) 851.31. Erlang distribution (characteristic of gamma distribution, Γ)  851.32. Logistic distribution 891.33. Log-logistic distribution 911.33.1. Mathematical–statistical characteristics of log-logistic distribution 911.33.2. Moment properties 921.34. Fisher distribution (F-distribution or Fisher–Snedecor)  921.35. Analysis of component lifespan (or survival) 951.36. Partial conclusion of Chapter 1 961.37. Bibliography 97Chapter 2. Estimates, Testing Adjustments and Testing the Adequacy of Statistical Distributions 992.1. Introduction to assessment and statistical tests 992.1.1. Estimation of parameters of a distribution 1002.1.2. Estimation by confidence interval 1022.1.3. Properties of an estimator with and without bias  1032.2. Method of moments 1062.3. Method of maximum likelihood 1062.3.1. Estimation of maximum likelihood  1072.3.2. Probability equation of reliability-censored data 1082.3.3. Punctual estimation of exponential law  1092.3.4. Estimation of the Weibull distribution  1102.3.5. Punctual estimation of normal distribution 1112.4. Moving least-squares method  1132.4.1. General criterion: the LSC  1142.4.2. Examples of nonlinear models 1182.4.3. Example of a more complex process  1222.5. Conformity tests: adjustment and adequacy tests  1232.5.1. Model of the hypothesis test for adequacy and adjustment  1252.5.2. Kolmogorov–Smirnov Test (KS 1930 and 1936)  1262.5.3. Simulated test (1st application)  1312.5.4. Simulated test (2nd application)  1312.5.5. Example 1 1322.5.6. Example 2 (Weibull or not?)  1352.5.7. Cramer–Von Mises (CVM) test  1392.5.8. The Anderson–Darling test 1402.5.9. Shapiro–Wilk test of normality  1452.5.10. Adequacy test of chi-square (χ2) 1452.6. Accelerated testing method 1512.6.1. Multi-censored tests 1522.6.2. Example of the exponential model  1522.6.3. Example of the Weibull model  1522.6.4. Example for the log–normal model  1532.6.5. Example of the extreme value distribution model (E-MIN) 1532.6.6. Example of the study on the Weibull distribution  1542.6.7. Example of the BOX–COX model  1562.7. Trend tests 1572.7.1. A unilateral test 1582.7.2. The military handbook test (from the US Army)  1602.7.3. The Laplace test  1602.7.4. Homogenous Poisson Process (HPP)  1602.8. Duane model power law 1642.9. Chi-Square test for the correlation quantity  1662.9.1. Estimations and χ2 test to determine the confidence interval 1672.9.2. t_test of normal mean  1702.9.3. Standard error of the estimated difference, s  1712.10. Chebyshev’s inequality 1712.11. Estimation of parameters 1732.12. Gaussian distribution: estimation and confidence interval 1742.12.1. Confidence interval estimation for a Gauss distribution 1752.12.2. Reading to help the statistical values tabulated 1752.12.3. Calculations to help the statistical formulae appropriate to normal distribution 1752.12.4. Estimation of the Gaussian mean of unknown variance 1752.13. Kaplan–Meier estimator 1782.13.1. Empirical model using the Kaplan–Meier approach 1792.13.2. General expression of the KM estimator 1802.13.3. Application of the ordinary and modified Kaplan–Meier estimator 1812.14. Case study of an interpolation using the bi-dimensional spline function 1812.15. Conclusion 1832.16. Bibliography 184Chapter 3. Modeling Uncertainty 1873.1. Introduction to errors and uncertainty 1873.2. Definition of uncertainties and errors as in the ISO norm 1893.3. Definition of errors and uncertainty in metrology 1913.3.1. Difference between error and uncertainty 1923.4. Global error and its uncertainty 2023.5. Definitions of simplified equations of measurement uncertainty 2043.5.1. Expansion factor k and range of relative uncertainty 2063.5.2. Determination of type A and B uncertainties according to GUM 2083.6. Principal of uncertainty calculations of type A and type B 2293.6.1. Standard and expanded uncertainties  2313.6.2. Components of type A and type B uncertainties 2323.6.3. Error on repeated measurements: composed uncertainty 2323.7. Study of the basics with the help of the GUMic software package: quasi-linear model 2393.8. Conclusion 2453.9. Bibliography 245Glossary  249Index 257