Regression Analysis of Stochastic Fatigue Crack Growth Model in a Martingale Difference Framework

Abstract

In the present paper, we are mainly concerned with the degradation mechanism that arises in fatigue crack growth (FCG). The crack evolution mechanism is modeled by a first order stochastic differential system, composed by a deterministic FCG equation perturbed by a stochastic process. The main purpose is to investigate the estimators of the model parameters and establish some asymptotic properties by transforming the initial equation into a regression model. To this purpose, least squares estimation (LSE) is applied in the framework of the errors being martingale differences. The parametric conditional LSE are proved to be consistent. Our results are obtained in the general case and specified for the linear case with focus on a particular application model derived from the most widely used FCG law in fracture mechanics, i.e., the Paris model. We derive the asymptotic normality of the LSE for the nonlinear case with discussion for the linear case. Finally, we provide a numerical example to illustrate the performance of the proposed methodology, on a particular version of the stochastic model.