A probabilistic criterion for evaluating the goodness of fatigue crack growth models

Abstract

The fatigue crack growth (FCG) phenomenon is usually modelled by means of a differential equation relating the crack growth rate, da dN , to the instantaneous crack size, a( N). This equation usually contains two or more parameters, which are used to “fit” the model to some experimental result given as pairs ( a, N). As a rule, the fitting procedure is such that the differences between the solution of the differential equation and experimental points are minimized according to, let's say, the least square criterion. This procedure implies the interpretation of the differential equation as an equation for the mean growth curve, which leads to difficulties when considering the random fluctuations observed in the growth rate. In this paper, the meaning of the crack growth equation is reconsidered and the interpretation as a local relation adopted. Coherently, model parameters may no longer be fitted to experimental curves as a whole: they must be locally evaluated along the crack path. A statistical analysis of these local values can be used to investigate whether or not a differential equation is an appropriate model for the FCG phenomenon. In particular, the homogeneous random field (HRF) criterion is shown to be very useful in the search for improved FCG equations. The improvements that can be attained with this criterion are demonstrated with an application to the Bogdanoff-Kozin(BK) model, where the FCG process is represented as an equivalent Markov chain. Finally, conclusions about the correlation length of an inferred random field are drawn. Considering that the magnitude of this correlation length is of utmost importance when the Markovian approximation is adopted, it is demonstrated that the FCG process can be regarded as Markovian only if the crack tip is observed at relatively large time intervals.