A branching model for crack propagation

Abstract

A branching model for crack propagation is proposed, a ‘branch’ corresponding to an existing microfissure or flaw in the material, and the propagation of the crack to the coalescence of such branches. Increase in external stress increases the probability that a given branch will link into more than a specified number of further branches. Such increases can continue until a critical state is reached when the mean number of branches linking into a given branch is equal to unity; beyond this point, the system becomes unstable, and any slight movement is likely to lead to catastrophic rupture. The distribution of the sums of the lengths of the branches linked together in a cracking episode is investigated, and shown to lead, in the critical case, to a Gutenberg-Richter type relation with parameterb=0.75. Departures from this value are attributed to the influence of the distribution of the lengths of preexisting fissures, this distribution varying with the strength of the material and its stress history. Some difficulties with the theoretical model of Scholz are raised, and it is suggested that a more complete analysis of Scholz's model should lead to results qualitatively similar to those obtained for the branching model.