Probabilistic approach to growth and detection of a truncated distribution of initial crack lengths based on Paris' law

A method for incorporating probabilistic considerations into fatigue life prognosis based on experimental information for Paris' law is presented in this article. A truncated probability distribution of initial crack lengths is introduced to obviate a complication of the use of Paris' law. This formulation allows the calculation of several probabilities for various values of the truncation length, including the probability of the existence of a crack larger than a predetermined critical crack length and the probability of a crack in a prescribed domain of crack lengths, and the effect of an inspection on these probabilities. A probabilistic consideration for the fact that the Paris' law parameters are not constants but distributions is also addressed using a stochastic version of Paris' law and a Monte Carlo simulation. Finally, a novel Bayesian approach which highlights the effect of probability of detection, critical crack length, and applied stress versus the number of elapsed fatigue cycles is presented and ramifications explored. This article provides insight on how various factors affect diagnosis and prognosis in a probabilistic manner.