Random fatigue crack growth-a first passage problem

Abstract

The random residual in the logarithmic form of the Paris-Erdogan equation is for a single stress cycle modeled as a weighted average of a white noise material property process over the crack increment. This leads to a stochastic Paris-Erdogan equation that determines the crack increment implicitly in terms of a probability distribution of the smallest solution to the equation. This is a first passage problem in Brownian motion. For all weighting functions consistent with this model the solution of the first passage problem has the form as a randomized Paris-Erdogan equation simply with a multiplicative random variable on the right side of the equation. It is independently and identically distributed from stress cycle to stress cycle. For each of two specific weighting functions the probability distribution of this random factor is obtained. The simplest of the two is rather trivially leading to the lognormal distribution. The other and less trivial example leads to the inverse Gaussian distribution. This is interesting because it admits an exact description of a suitable transformation of the crack growth process as a function of the number of constant amplitude stress cycles in terms of an inverse Gaussian process with stationary and independent increments. Also the exact probability distribution of the number of stress cycles required to grow the crack a certain length can be calculated in this case. For any distribution of the factor the transformed crack growth process for constant amplitude loading will be asymptotically Gaussian with stationary and independent increments. © 1986.