Bayesian Modelling of Short Fatigue Crack Growth and Coalescence

Cathal Dominic Walsh

Thesis submitted for the degree of Doctor of Philosophy

Trinity College Dublin
Department of Statistics
October 1999

• Contents
• List of Figures
• List of Tables
• Introduction
  • Review of Current Models
  • Statistical Methodology
  • Growth Model
  • Coalescence
  • Joint Model
  • Overall Framework and Major Contributions
• Modelling Fatigue and Crack Propagation
  • Factors Important to Modelling Fatigue
    • Specimen Geometry
    • Material Properties
    • Nature of Loading
    • Statistics of Interest
    • Manufacturing Issues
  • Deterministic Models for Crack Growth
  • Wöhler or S-N Curve
  • Empirical Models
    • Paris-Erdogan Equation
    • Forman Equation
  • Theoretical Models
    • Cumulative Damage Theories
    • Elastic Fatigue Fracture
    • J Integral Methods
    • Empirical Stochastic Parameterisation
    • Stochastic Models
  • Fatigue Crack Growth Process
  • Current Research Questions
    • Growth Aspect
    • Coalescence Aspect
    • Other Considerations
• Statistical Methodology
  • Statistics
  • Bayesian Approach
    • Formal Bayesian Methodology
      • Prior Knowledge
      • Model or Likelihood
      • An Example
      • Posterior Distribution
    • Predictive Distribution
    • Kernel Density Estimation
    • A Simple Example -
    • Prior Elicitation and Non-informative Priors
  • Sampling from the Posterior Distribution
    • Stratified Sampling
    • Importance Sampling
    • Monte Carlo Method
    • Markov Chains
    • Markov Chain Monte Carlo
    • Metropolis Hastings
    • Gibbs Sampling
  • Issues of Convergence
  • Nature of Multidimensional Posterior
  • Graphical Representation of a Model
• Growth Model
  • Stages of Crack Growth
  • Characteristics of Microcracks
  • Experimental Details
  • Data
    • Treatment of Raw Data - Coalescence
  • Model
    • Hierarchical Population Model
    • Directed Graph
  • Analysis
    • Likelihood
    • Priors for the Parameters
      • Priors on , M and d.
    • Evaluation of the Posterior distribution
    • Evaluation of Other Quantities
  • Practicalities
    • Evaluation of
    • Sampling from Posterior
    • Algorithm
      • Preliminaries
      • Algorithm
      • Assessing Convergence
      • Transforming the Variables
    • Programming
  • Results
  • Conclusions
• Coalescence
  • Motivation
  • Fatigue in PMMA
  • Data
    • Preliminary Analysis
  • Model
  • Analysis
    • Likelihood
    • Priors
    • Evaluation of Posterior
  • Results
  • Conclusions
• Combining Coalescence and Growth
  • Background
  • Data
  • Model
  • Analysis
    • Likelihood
    • Priors
    • Evaluation of Posterior
  • Results
  • Conclusions
• Summary & Conclusions
  • Summary of Growth Model
  • Summary of Coalescence Model
  • Summary of Joint Model
  • Conclusions
  • Further Work
    • Spatial Aspect
    • Methodological Issues
    • Factors Influencing Coalescence
    • General Process of Failure
    • Alternative Growth Models
  • Closing Remarks
• Calculations
  • Bayes Theorem
  • Inverse Gamma
  • Constant Variance Growth Model
  • Multiplicative Variance Growth Model
  • Acceptance Probabilities
    • Growth Model for with Multiplicative Variance
    • Growth Model for Constant Variance
• References
• About this document ...