Prediction of crack growth based on a hierarchical diffusion model

Abstract

A general Bayesian approach for stochastic versions of deterministic growth models is presented to provide predictions for crack propagation in an early stage of the growth process. To improve the prediction, the information of other crack growth processes is used in a hierarchical (mixed-effects) model. Two stochastic versions of a deterministic growth model are considered. One is a nonlinear regression setup where the trajectory is assumed to be the solution of an ordinary differential equation with additive errors. The other is a diffusion model defined by a stochastic differential equation (SDE) where increments have additive errors. Six growth models in the two versions are compared with respect to their ability to predict the crack propagation in a large data example. Two of them are based on the classical Paris-Erdogan law for crack growth, and four are other widely used growth models. It turned out that the three-parameter Paris-Erdogan model and the Weibull model provide the best results followed by the logistic model. Suprisingly, the SDE approach has no advantage for the prediction compared with the nonlinear regression setup.