Fracture Mechanics 2 Applied Reliability

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2012-12-26
  • Publisher: Wiley-ISTE

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Supplemental Materials

What is included with this book?


The science of engineering is soiled by uncertainties. The experimental data cost. The design stumbles at random. The objective of the design is to maximize the chances of success of a dimensioning. The objective of this work is to allow the users to understand the main methods. This volume is centered on a vast range of statistical distributions met in reliability. The aim is to run statistical measures, to present a report of enhanced measures in mechanical reliability and to evaluate the reliability of the repairable or not repairable systems. To reach these purposes, we present an approach theory /practice based on these themes: Criteria of failures; Bayesian Applied Probability; Chains of Markov; Simulation of Monte Carlo and finally many solved cases of studies.

Author Biography

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

Table of Contents



Chapter  1 Fracture Mechanisms by Fatigue


Principal physical mechanisms of cracking by fatigue

Fracture mechanics

Criteria of fracture (plasticity) in mechanics

Modes of fracture

Directed works

Fatigue of metals: analytical expressions used in reliability

Wöhler’s law

Basquin’s law

Stromayer’s law

Palmgren’s law

Corson’s law

Bastenaire’s law

Weibull’s law

Henry’s law                   

Corten and Dolen’s law

Manson–Coffin’s law

Reliability models commonly used in fracture mechanics by fatigue
Coffin–Manson’s model for the analysis of crack propagation

Neuber’s relation

Arrhenius’ model

Miner’s law

Main common laws retained by fracture mechanics

Fost and Dugdale’s law

McEvily’s law

Paris’s law       

GR Sih’s law

Stress intensity factors in fracture mechanics  

Maddox’s model

Gross and Srawley’s model

Lawrence’s model 

Martin and Bousseau’s model   

Gurney’s model  

Engesvik’s model

Yamada and Albrecht’s model

Tomkins and Scott’s model

Harrison’s model

Intrinsic parameters of the material (C and m)

Fracture mechanics elements used in reliability

Crack rate (life expectancy) and sif (Kσ)

Simplified version of Taylor’s law for machining

Elements of stress (S) and resistance theory (R)

Case study, part  – suspension bridge (Cirta)

Case study: failure surface of geotechnical materials



Chapter 2 Analysis Elements for Determining the Probability of Rupture by Simple Bounds


First-order bounds or simple bounds: systems in series 

First-order bounds or simple bounds: systems in parallel

Second-order bounds or Ditlevsen’s bounds

Evaluating the probability of the intersection of two events
Estimating multinomial distribution–normal distribution

Binomial distribution

Hohenbichler’s method

Hypothesis test, through the example of a normal average with unknown variance      

Development and calculations                      

Confidence interval for estimating a normal mean: unknown variance                              


Chapter 3 Analysis of the Reliability of Materials and Structures by the Bayesian Approach

Introduction to the Bayesian method used to evaluate reliability

Posterior distribution and conjugate models

Independent events

Counting diagram

Conditional probability or Bayes’ law

Anterior and posterior distributions

Reliability analysis by moments methods, FORM/SORM

Control margins from the results of fracture mechanics

Bayesian model by exponential gamma distribution

Homogeneous Poisson process and rate of occurrence of failure

Estimating the maximum likelihood                     

Type I censored exponential model                   

Estimating the MTBF (or rate of repair/rate of failure)

MTBF and confidence interval

Repair rate or ROCOF  

Power law: non-homogeneous Poisson process

Distribution law – gamma (reminder)

Bayesian model of a priori gamma distribution

Distribution tests for exponential life (or HPP model)

Bayesian procedure for the exponential system model

Bayesian case study applied in fracture mechanics



Chapter 4 Elements of Analysis for the Reliability of Components by Markov Chains


Applying Markov chains to a fatigue model

Case study with the help of Markov chains for a fatigue model

Position of the problem


Explanatory information

Directed works

Approach for solving the problem

Which solution should we choose?



Chapter 5 Reliability Indices 


Design of material and structure reliability

Reliability of materials and structures

First-order reliability method

Second-order reliability method

Cornell’s reliability index

Hasofer–Lind’s reliability index

Reliability of material and structure components   

Reliability of systems in parallels and series

Parallel system

Parallel system (m/n)                           

Serial assembly system



Chapter 6  Fracture Criteria Reliability Methods through an Integral Damage Indicator


Literature review of the integral damage indicator method

Brief recap of the FORM/SORM method

Recap of the Hasofer–Lind index method

Literature review of the probabilistic approach of cracking law

parameters in region II of the Paris law

Crack spreading by a classical fatigue model

Reliability calculations using the integral damage indicator method                               


Chapter 7 Monte Carlo Simulation


From the origin of the Monte Carlo method!

The terminology                              

Simulation of a singular variable of a Gaussian

Simulation of non-Gaussian variable                  

Simulation of correlated variables                    

Simulation of correlated Gaussian variables

Simulation of correlated non-Gaussian variables           

Determining safety indices using Monte Carlo simulation        

General tools and problem outline                    

Presentation and discussion of our experimental results       

Use of the randomly selected numbers table   

Applied mathematical techniques to generate random numbers

by MC simulation on four principle statistical laws   

Uniform law             

Laplace–Gauss (normal) law

Exponential law

Initial value control



Chapter 8  Case Studies


Reliability indicators (λ) and MTBF 

Model of parallel assembly

Model of serial assembly        

Parallel or redundant model

Reliability and structural redundancy: systems without distribution                
Serial model

Rate of constant failure    

Reliability of systems without repairing: parallel model

Reliability applications in cases of redundant systems

Total active redundancy      

Partial active redundancy 

Reliability and availability of repairable systems

Quality assurance in reliability

Projected analysis of reliability

Birnbaum–Saunders distribution in crack spreading

Probability density and distribution function

(Birnbaum–Saunders cumulative distribution through cracking)      

Graph plots for the four probability density functions and

distribution functions                    

Reliability calculation for ages (τ) in hours of service, Ri(τ) = ?    

Simulation methods in mechanical reliability of structures and materials: the Monte Carlo simulation method 

Weibull law                

Log-normal Law (of Galton)

Exponential law                              

Generation of random numbers                     

Elements of safety via the couple: resistance and stress (R, S)     

Reliability trials                                

x Fracture Mechanics

Controlling risks and efficiency in mechanical reliability

Truncated trials

Censored trials    

Trial plan

Coefficients for the trial’s acceptance plan       

Trial’s rejection plan (in the same conditions)

Trial plan in reliability and K Pearson test χ

Reliability application on speed reducers (gears)

Applied example on hydraulic motors

Reliability case study in columns under stress of buckling

RDM solution                               

Problem outline and probabilistic solution

(reliability and error)

Adjustment of least squared for nonlinear functions

Specific case study : a Weibull law with two parameters




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