Fracture Mechanics 1 Analysis of Reliability and Quality Control

This first book of a 3-volume set on Fracture Mechanics is mainly centered on the vast range of the laws of statistical distributions encountered in various scientific and technical fields. These laws are indispensable in understanding the probability behavior of components and mechanical structures that are exploited in the other volumes of this series, which are dedicated to reliability and quality control.
The author presents not only the laws of distribution of various models but also the tests of adequacy suited to confirm or counter the hypothesis of the law in question, namely the Pearson (x2) test, the Kolmogorov-Smirnov (KS) test, along with many other relevant tests.
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.

Preface ix

Chapter 1. Elements of Analysis of Reliability and Quality Control  1

1.1. Introduction 1

1.1.1. The importance of true physical acceleration life models (accelerated tests = true acceleration or acceleration)  3

1.1.2. Expression for linear acceleration relationships 4

1.2. Fundamental expression of the calculation of reliability 5

1.3. Continuous uniform distribution 9

1.3.1. Distribution function of probabilities (density of probability)  10

1.3.2. Distribution function 10

1.4. Discrete uniform distribution (discrete U)  12

1.5. Triangular distribution 13

1.5.1. Discrete triangular distribution version  13

1.5.2. Continuous triangular law version  14

1.5.3. Links with uniform distribution  14

1.6. Beta distribution 15

1.6.1. Function of probability density  16

1.6.2. Distribution function of cumulative probability 18

1.6.3. Estimation of the parameters (p, q) of the beta distribution  19

1.6.4. Distribution associated with beta distribution 20

1.7. Normal distribution 20

1.7.1. Arithmetic mean 20

1.7.2. Reliability 22

1.7.3. Stabilization and normalization of variance error  23

1.8. Log-normal distribution (Galton)  28

1.9. The Gumbel distribution 28

1.9.1. Random variable according to the Gumbel distribution (CRV, E1 Maximum)  29

1.9.2. Random variable according to the Gumbel distribution (CRV E1 Minimum) 30

1.10. The Frechet distribution (E2 Max)  31

1.11. The Weibull distribution (with three parameters) 32

1.12. The Weibull distribution (with two parameters)  35

1.12.1. Description and common formulae for the Weibull distribution and its derivatives 37

1.12.2. Areas where the extreme value distribution model can be used 39

1.12.3. Risk model 40

1.12.4. Products of damage 41

1.13. The Birnbaum–Saunders distribution  42

1.13.1. Derivation and use of the Birnbaum–Saunders model 43

1.14. The Cauchy distribution 45

1.14.1. Probability density function  45

1.14.2. Risk function  48

1.14.3. Cumulative risk function  48

1.14.4. Survival function (reliability) 49

1.14.5. Inverse survival function  49

1.15. Rayleigh distribution 50

1.16. The Rice distribution (from the Rayleigh distribution) 52

1.17. The Tukey-lambda distribution 53

1.18. Student’s (t) distribution 55

1.18.1. t-Student’s inverse cumulative function law (T)  57

1.19. Chi-square distribution law (χ2)  57

1.19.1. Probability distribution function of chi-square law (χ2) 57

1.19.2. Probability distribution function of chi-square law (χ2) 58

1.20. Exponential distribution 59

1.20.1. Example of applying mechanics to component lifespan 63

1.21. Double exponential distribution (Laplace)  66

1.21.1. Estimation of the parameters  66

1.21.2. Probability density function  66

1.21.3. Cumulated distribution probability function 67

1.22. Bernoulli distribution 68

1.23. Binomial distribution 71

1.24. Polynomial distribution 75

1.25. Geometrical distribution 75

1.25.1. Hypergeometric distribution (the Pascal distribution) versus binomial distribution  76

1.26. Hypergeometric distribution (the Pascal distribution) 78

1.27. Poisson distribution 80

1.28. Gamma distribution 81

1.29. Inverse gamma distribution 85

1.30. Distribution function (inverse gamma distribution probability density) 85

1.31. Erlang distribution (characteristic of gamma distribution, Γ)  85

1.32. Logistic distribution 89

1.33. Log-logistic distribution 91

1.33.1. Mathematical–statistical characteristics of log-logistic distribution 91

1.33.2. Moment properties 92

1.34. Fisher distribution (F-distribution or Fisher–Snedecor)  92

1.35. Analysis of component lifespan (or survival) 95

1.36. Partial conclusion of Chapter 1 96

1.37. Bibliography 97

Chapter 2. Estimates, Testing Adjustments and Testing the Adequacy of Statistical Distributions 99

2.1. Introduction to assessment and statistical tests 99

2.1.1. Estimation of parameters of a distribution 100

2.1.2. Estimation by confidence interval 102

2.1.3. Properties of an estimator with and without bias  103

2.2. Method of moments 106

2.3. Method of maximum likelihood 106

2.3.1. Estimation of maximum likelihood  107

2.3.2. Probability equation of reliability-censored data 108

2.3.3. Punctual estimation of exponential law  109

2.3.4. Estimation of the Weibull distribution  110

2.3.5. Punctual estimation of normal distribution 111

2.4. Moving least-squares method  113

2.4.1. General criterion: the LSC  114

2.4.2. Examples of nonlinear models 118

2.4.3. Example of a more complex process  122

2.5. Conformity tests: adjustment and adequacy tests  123

2.5.1. Model of the hypothesis test for adequacy and adjustment  125

2.5.2. Kolmogorov–Smirnov Test (KS 1930 and 1936)  126

2.5.3. Simulated test (1st application)  131

2.5.4. Simulated test (2nd application)  131

2.5.5. Example 1 132

2.5.6. Example 2 (Weibull or not?)  135

2.5.7. Cramer–Von Mises (CVM) test  139

2.5.8. The Anderson–Darling test 140

2.5.9. Shapiro–Wilk test of normality  145

2.5.10. Adequacy test of chi-square (χ2) 145

2.6. Accelerated testing method 151

2.6.1. Multi-censored tests 152

2.6.2. Example of the exponential model  152

2.6.3. Example of the Weibull model  152

2.6.4. Example for the log–normal model  153

2.6.5. Example of the extreme value distribution model (E-MIN) 153

2.6.6. Example of the study on the Weibull distribution  154

2.6.7. Example of the BOX–COX model  156

2.7. Trend tests 157

2.7.1. A unilateral test 158

2.7.2. The military handbook test (from the US Army)  160

2.7.3. The Laplace test  160

2.7.4. Homogenous Poisson Process (HPP)  160

2.8. Duane model power law 164

2.9. Chi-Square test for the correlation quantity  166

2.9.1. Estimations and χ2 test to determine the confidence interval 167

2.9.2. t_test of normal mean  170

2.9.3. Standard error of the estimated difference, s  171

2.10. Chebyshev’s inequality 171

2.11. Estimation of parameters 173

2.12. Gaussian distribution: estimation and confidence interval 174

2.12.1. Confidence interval estimation for a Gauss distribution 175

2.12.2. Reading to help the statistical values tabulated 175

2.12.3. Calculations to help the statistical formulae appropriate to normal distribution 175

2.12.4. Estimation of the Gaussian mean of unknown variance 175

2.13. Kaplan–Meier estimator 178

2.13.1. Empirical model using the Kaplan–Meier approach 179

2.13.2. General expression of the KM estimator 180

2.13.3. Application of the ordinary and modified Kaplan–Meier estimator 181

2.14. Case study of an interpolation using the bi-dimensional spline function 181

2.15. Conclusion 183

2.16. Bibliography 184

Chapter 3. Modeling Uncertainty 187

3.1. Introduction to errors and uncertainty 187

3.2. Definition of uncertainties and errors as in the ISO norm 189

3.3. Definition of errors and uncertainty in metrology 191

3.3.1. Difference between error and uncertainty 192

3.4. Global error and its uncertainty 202

3.5. Definitions of simplified equations of measurement uncertainty 204

3.5.1. Expansion factor k and range of relative uncertainty 206

3.5.2. Determination of type A and B uncertainties according to GUM 208

3.6. Principal of uncertainty calculations of type A and type B 229

3.6.1. Standard and expanded uncertainties  231

3.6.2. Components of type A and type B uncertainties 232

3.6.3. Error on repeated measurements: composed uncertainty 232

3.7. Study of the basics with the help of the GUMic software package: quasi-linear model 239

3.8. Conclusion 245

3.9. Bibliography 245

Glossary  249

Index 257

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