An employment of the catastrophe theory in fracture mechanics as applied to brittle strength criteria

Abstract

I n the present paper, catastrophe theory is applied to the analysis of strength criteria delivered by the mechanics of brittle fracture. Introduction of a potential function of a specific kind enables one to examine brittle strength criteria in a broad class of multiparametric problems. Two example problems of fracture mechanics with two independent loading parameters are considered in detail. The first one concerns the development of two symmetric cracks issuing from a circular hole under the effect of biaxial compression-extension at infinity. The second one is that of an isolated rectilinear crack stretched by two concentrated forces and tension at infinity. The first one provides a node-type catastrophe, and the second a fold-type catastrophe. For any problems of fracture mechanics with two loading parameters, these are the only modes allowed by catastrophe theory. It is shown by these examples that the set of states of unlimited unstable crack growth (i.e. failure) in the two-dimensional space of loading parameters forms a certain area called herein the fracture domain, so that depending on the loading path, failure can either occur or not occur at the given point of a fracture domain. It is deduced that in the case of two or more loading parameters the brittle strength criterion depends on the loading process prehistory. The classical approach to strength criteria is confirmed by fracture mechanics only if there are no catastrophes at all (i.e. in extreme cases). In conclusion, two brittle strength criteria are suggested for use, one of which characterizes the possibility of failure and the other guarantees it. In the intermediate domain between these criteria, failure can either occur, or not, depending on the loading path.