Bayesian prediction of crack growth based on a hierarchical diffusion model

In many research areas of engineering it is of particular interest to understand the
process of material fatigue and to predict the failure time. Experiments are often very
expensive because they take a long time and the constructions are costly. To extract
as much information as possible from the available experiments and to predict fatigue,
statistical models are a valuable tool.
A general Bayesian approach for stochastic versions of deterministic growth models
is presented to provide prediction 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 compared. 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 the
increments have additive errors. While Bayesian prediction is known for the hierarchical
model based on nonlinear regression, we propose a new Bayesian prediction method for
the hierarchical diffusion model.
The presented methods are applied on the degradation data set of Virkler et al.
(1979) where sixty-eight replicate constant amplitude tests in aluminum alloy were
carried out to investigate the fatigue crack propagation.