Diffraction of elastic waves by a penny-shaped crack: analytical and numerical results

Abstract

The diffraction of time-harmonic stress waves by a penny-shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials. Martin ( Proc. R. Soc. Lond . A 378, 263 (1981)) has proved that the corresponding linear boundary-value problem has precisely one solution, and that this solution can be constructed by solving a two-dimensional Fredholm integral equation of the second kind. However, this integral equation has a complicated matrix kernel and the components of its vector solution are coupled. The main purpose of the present paper is to show how Martin’s integral equation can be explicitly solved in terms of a sequence of functions, each of which satisfies a very simple scalar integral equation of the second kind; this simplification may be made for any incident wave. For an incident plane wave, further simplifications are possible. We show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence, namely grazing incidence. The resulting computational procedure is especially attractive if only the stress-intensity factors or the far-held displacements are required. Finally, we present some numerical results for the scattering of a P-wave at normal incidence and an SV-wave at oblique incidence, and compare these with those of other authors.