Establishment of bounds for the statistical moments of the crack size, for the collipriest model, using the fast crack bounds method

There are several mathematical models that describe the propagation of cracks. For many engineering applications, up to a certain point, it is not necessary to have great accuracy in predictions about the behavior of the evolution of a crack, but a reliable prediction, within certain limits, of such behavior. This work presents theoretical results consisting in obtaining lower and upper bounds that "envelop" the first and second order statistical moment estimators of the crack size function based on the fast crack bounds method. These bounds are polynomials defined in the variable "number of cycles" that consider the uncertainties of the parameters that describe the crack propagation models. The performance of the bounds for the statistical moments of the crack size is evaluated through the relative deviation between the bounds and the approximate numerical solutions of the initial value problems (IVP) that describe the crack evolution laws. For this work, the Collipriest model is used. The Monte Carlo simulation method is used to create samples of the selected parameters to obtain the crack size for both the bounds and the Runge-Kutta method.