FRACTURE MECHANICS
1 1 INTRODUCTION 1 1 1 THE IMPORTANCE OF TRUE PHYSICAL ACCELERATION LIFE MODELS (ACCELERATED TESTS = TRUE ACCELERATION OR ACCELERATION) 1 1 2 EXPRESSION FOR LINEAR ACCELERATION RELATIONSHIPS 1 2 FUNDAMENTAL EXPRESSION OF THE CALCULATION OF RELIABILITY 1 3 CONTINUOUS UNIFORM DISTRIBUTION 1 3 1 DISTRIBUTION FUNCTION OF PROBABILITIES (DENSITY OF PROBABILITY) 1 3 2 DISTRIBUTION FUNCTION 1 4 DISCRETE UNIFORM DISTRIBUTION (DISCRETE U) 1 5 TRIANGULAR DISTRIBUTION 1 5 1 DISCRETE TRIANGULAR DISTRIBUTION VERSION 1 5 2 CONTINUOUS TRIANGULAR LAW VERSION 1 5 3 LINKS WITH UNIFORM DISTRIBUTION 1 6 BETA DISTRIBUTION 1 6 1 FUNCTION OF PROBABILITY DENSITY 1 6 2 DISTRIBUTION FUNCTION OF CUMULATIVE PROBABILITY 1 6 3 ESTIMATION OF THE PARAMETERS (P, Q) OF THE BETA DISTRIBUTION 1 6 4 DISTRIBUTION ASSOCIATED WITH BETA DISTRIBUTION 1 7 NORMAL DISTRIBUTION 1 7 1 ARITHMETIC MEAN 1 7 2 RELIABILITY 1 7 3 STABILIZATION AND NORMALIZATION OF VARIANCE ERROR 1 8 LOG-NORMAL DISTRIBUTION (GALTON) 1 9 THE GUMBEL DISTRIBUTION 1 9 1 RANDOM VARIABLE ACCORDING TO THE GUMBEL DISTRIBUTION (CRV, E1 MAXIMUM) 1 9 2 RANDOM VARIABLE ACCORDING TO THE GUMBEL DISTRIBUTION (CRV E1 MINIMUM) 1 10 THE FRECHET DISTRIBUTION (E2 MAX) 1 11 THE WEIBULL DISTRIBUTION (WITH THREE PARAMETERS) 1 12 THE WEIBULL DISTRIBUTION (WITH TWO PARAMETERS) 1 12 1 DESCRIPTION AND COMMON FORMULAE FOR THE WEIBULL DISTRIBUTION AND ITS DERIVATIVES 1 12 2 AREAS WHERE THE EXTREME VALUE DISTRIBUTION MODEL CAN BE USED 1 12 3 RISK MODEL 1 12 4 PRODUCTS OF DAMAGE 1 13 THE BIRNBAUM–SAUNDERS DISTRIBUTION 1 13 1 DERIVATION AND USE OF THE BIRNBAUM–SAUNDERS MODEL 1 14 THE CAUCHY DISTRIBUTION 1 14 1 PROBABILITY DENSITY FUNCTION 1 14 2 RISK FUNCTION 1 14 3 CUMULATIVE RISK FUNCTION 1 14 4 SURVIVAL FUNCTION (RELIABILITY) 1 14 5 INVERSE SURVIVAL FUNCTION 1 15 RAYLEIGH DISTRIBUTION 1 16 THE RICE DISTRIBUTION (FROM THE RAYLEIGH DISTRIBUTION) 1 17 THE TUKEY-LAMBDA DISTRIBUTION 1 18 STUDENT’S (T) DISTRIBUTION 1 18 1 T-STUDENT’S INVERSE CUMULATIVE FUNCTION LAW (T) 1 19 CHI-SQUARE DISTRIBUTION LAW (?2) 1 19 1 PROBABILITY DISTRIBUTION FUNCTION OF CHI-SQUARE LAW (?2) 1 19 2 PROBABILITY DISTRIBUTION FUNCTION OF CHI-SQUARE LAW (?2) 1 20 EXPONENTIAL DISTRIBUTION 1 20 1 EXAMPLE OF APPLYING MECHANICS TO COMPONENT LIFESPAN 1 21 DOUBLE EXPONENTIAL DISTRIBUTION (LAPLACE) 1 21 1 ESTIMATION OF THE PARAMETERS 1 21 2 PROBABILITY DENSITY FUNCTION 1 21 3 CUMULATED DISTRIBUTION PROBABILITY FUNCTION 1 22 BERNOULLI DISTRIBUTION 1 23 BINOMIAL DISTRIBUTION 1 24 POLYNOMIAL DISTRIBUTION 1 25 GEOMETRICAL DISTRIBUTION 1 25 1 HYPERGEOMETRIC DISTRIBUTION (THE PASCAL DISTRIBUTION) VERSUS BINOMIAL DISTRIBUTION 1 26 HYPERGEOMETRIC DISTRIBUTION (THE PASCAL DISTRIBUTION) 1 27 POISSON DISTRIBUTION 1 28 GAMMA DISTRIBUTION 1 29 INVERSE GAMMA DISTRIBUTION 1 30 DISTRIBUTION FUNCTION (INVERSE GAMMA DISTRIBUTION PROBABILITY DENSITY) 1 31 ERLANG DISTRIBUTION (CHARACTERISTIC OF GAMMA DISTRIBUTION, ?) 1 32 LOGISTIC DISTRIBUTION 1 33 LOG-LOGISTIC DISTRIBUTION 1 33 1 MATHEMATICAL–STATISTICAL CHARACTERISTICS OF LOG-LOGISTIC DISTRIBUTION 1 33 2 MOMENT PROPERTIES 1 34 FISHER DISTRIBUTION (F-DISTRIBUTION OR FISHER–SNEDECOR) 1 35 ANALYSIS OF COMPONENT LIFESPAN (OR SURVIVAL) 1 36 PARTIAL CONCLUSION OF CHAPTER 1 1 37 BIBLIOGRAPHY CHAPTER 2 ESTIMATES, TESTING ADJUSTMENTS AND TESTING THE ADEQUACY OF STATISTICAL DISTRIBUTIONS 2 1 INTRODUCTION TO ASSESSMENT AND STATISTICAL TESTS 2 1 1 ESTIMATION OF PARAMETERS OF A DISTRIBUTION 2 1 2 ESTIMATION BY CONFIDENCE INTERVAL 2 1 3 PROPERTIES OF AN ESTIMATOR WITH AND WITHOUT BIAS 2 2 METHOD OF MOMENTS 2 3 METHOD OF MAXIMUM LIKELIHOOD 2 3 1 ESTIMATION OF MAXIMUM LIKELIHOOD 2 3 2 PROBABILITY EQUATION OF RELIABILITY-CENSORED DATA 2 3 3 PUNCTUAL ESTIMATION OF EXPONENTIAL LAW 2 3 4 ESTIMATION OF THE WEIBULL DISTRIBUTION 2 3 5 PUNCTUAL ESTIMATION OF NORMAL DISTRIBUTION 2 4 MOVING LEAST-SQUARES METHOD 2 4 1 GENERAL CRITERION: THE LSC 2 4 2 EXAMPLES OF NONLINEAR MODELS 2 4 3 EXAMPLE OF A MORE COMPLEX PROCESS 2 5 CONFORMITY TESTS: ADJUSTMENT AND ADEQUACY TESTS 2 5 1 MODEL OF THE HYPOTHESIS TEST FOR ADEQUACY AND ADJUSTMENT 2 5 2 KOLMOGOROV–SMIRNOV TEST (KS 1930 AND 1936) 2 5 3 SIMULATED TEST (1ST APPLICATION) 2 5 4 SIMULATED TEST (2ND APPLICATION) 2 5 5 EXAMPLE 1 2 5 6 EXAMPLE 2 (WEIBULL OR NOT?) 2 5 7 CRAMER–VON MISES (CVM) TEST 2 5 8 THE ANDERSON–DARLING TEST 2 5 9 SHAPIRO–WILK TEST OF NORMALITY 2 5 10 ADEQUACY TEST OF CHI-SQUARE (?2) 2 6 ACCELERATED TESTING METHOD 2 6 1 MULTI-CENSORED TESTS 2 6 2 EXAMPLE OF THE EXPONENTIAL MODEL 2 6 3 EXAMPLE OF THE WEIBULL MODEL 2 6 4 EXAMPLE FOR THE LOG–NORMAL MODEL 2 6 5 EXAMPLE OF THE EXTREME VALUE DISTRIBUTION MODEL (E-MIN) 2 6 6 EXAMPLE OF THE STUDY ON THE WEIBULL DISTRIBUTION 2 6 7 EXAMPLE OF THE BOX–COX MODEL 2 7 TREND TESTS 2 7 1 A UNILATERAL TEST 2 7 2 THE MILITARY HANDBOOK TEST (FROM THE US ARMY) 2 7 3 THE LAPLACE TEST 2 7 4 HOMOGENOUS POISSON PROCESS (HPP) 2 8 DUANE MODEL POWER LAW 2 9 CHI-SQUARE TEST FOR THE CORRELATION QUANTITY 2 9 1 ESTIMATIONS AND ?2 TEST TO DETERMINE THE CONFIDENCE INTERVAL 2 9 2 T_TEST OF NORMAL MEAN 2 9 3 STANDARD ERROR OF THE ESTIMATED DIFFERENCE, S 2 10 CHEBYSHEV’S INEQUALITY 2 11 ESTIMATION OF PARAMETERS 2 12 GAUSSIAN DISTRIBUTION: ESTIMATION AND CONFIDENCE INTERVAL 2 12 1 CONFIDENCE INTERVAL ESTIMATION FOR A GAUSS DISTRIBUTION 2 12 2 READING TO HELP THE STATISTICAL VALUES TABULATED 2 12 3 CALCULATIONS TO HELP THE STATISTICAL FORMULAE APPROPRIATE TO NORMAL DISTRIBUTION 2 12 4 ESTIMATION OF THE GAUSSIAN MEAN OF UNKNOWN VARIANCE 2 13 KAPLAN–MEIER ESTIMATOR 2 13 1 EMPIRICAL MODEL USING THE KAPLAN–MEIER APPROACH 2 13 2 GENERAL EXPRESSION OF THE KM ESTIMATOR 2 13 3 APPLICATION OF THE ORDINARY AND MODIFIED KAPLAN–MEIER ESTIMATOR 2 14 CASE STUDY OF AN INTERPOLATION USING THE BI-DIMENSIONAL SPLINE FUNCTION 2 15 CONCLUSION 2 16 BIBLIOGRAPHY CHAPTER 3 MODELING UNCERTAINTY 3 1 INTRODUCTION TO ERRORS AND UNCERTAINTY 3 2 DEFINITION OF UNCERTAINTIES AND ERRORS AS IN THE ISO NORM 3 3 DEFINITION OF ERRORS AND UNCERTAINTY IN METROLOGY 3 3 1 DIFFERENCE BETWEEN ERROR AND UNCERTAINTY 3 4 GLOBAL ERROR AND ITS UNCERTAINTY 3 5 DEFINITIONS OF SIMPLIFIED EQUATIONS OF MEASUREMENT UNCERTAINTY 3 5 1 EXPANSION FACTOR K AND RANGE OF RELATIVE UNCERTAINTY 3 5 2 DETERMINATION OF TYPE A AND B UNCERTAINTIES ACCORDING TO GUM 3 6 PRINCIPAL OF UNCERTAINTY CALCULATIONS OF TYPE A AND TYPE B 3 6 1 STANDARD AND EXPANDED UNCERTAINTIES 3 6 2 COMPONENTS OF TYPE A AND TYPE B UNCERTAINTIES 3 6 3 ERROR ON REPEATED MEASUREMENTS: COMPOSED UNCERTAINTY 3 7 STUDY OF THE BASICS WITH THE HELP OF THE GUMIC SOFTWARE PACKAGE: QUASI-LINEAR MODEL 3 8 CONCLUSION 3 9 BIBLIOGRAPHY