Bounds for the dislocation densities and the stress intensity factors in elastic crack problems

Abstract

A method for obtaining bounds for the dislocations density along a crack and for the stress intensity factors at the tips of the crack inside an elastic medium under plane (or antiplane) conditions is proposed. This method is based on the reduction of such a crack problem to a Cauchy type singular integral equation and the transfer of well-known results for Fredholm integral equations of the second kind to the case of Cauchy type singular integral equations. The results of the paper are of particular interest in the problems where we wish to know if the values of the stress intensity factors at the tips of the crack exceed or not a critical value. Four applications to simple plane and antiplane elasticity problems are made for the illustration of the theoretical results.