An approximate (Wiener-Hopf) kernel for dynamic crack problems in linear elasticity and viscoelasticity

Abstract

The analysis of dynamic crack problems often leads to results of such complexity that their physical significance cannot readily be apprehended; one must generally have recourse to ad hoc approximations, or to a numerical evaluation of the solutions, before such results can be interpreted. We present in this paper a more radical approximation in the form of an 'approximate kernel’ of the type first considered by Koiter in conjunction with the Wiener-Hopf technique. The approximate solutions to which this leads are compared with the exact solutions for a few basic problems concerning the elastic response to stationary, or moving, point forces applied to the faces of a semi-infinite crack. It is clear that all the essential features of the exact solutions are retained, and of course the approximate solutions are considerably simpler in form. The use of approximate kernels is not tied to the use of the Wiener-Hopf technique: they can also be used in con­junction with a method due to W. E. Williams, which is in fact much better suited to the analysis of crack problems. It is shown that the approximate kernel can also be used to simplify Kostrov’s solution of the arbitrarily moving crack problem and the analysis of crack problems in linear viscoelasticity.