Diffraction of elastic waves by a penny-shaped crack

Consider an infinite elastic solid containing a penny-shaped crack. We suppose that time-harmonic elastic waves are incident on the crack and are required to determine the scattered displacement field ui. In this paper, we describe a new method for solving the corresponding linear boundary-value problem for ui, which we denote by S. We begin by defining an ‘elastic double layer’; we prove that any solution of S can be represented by an elastic double layer whose ‘density’ satisfies certain conditions. We then introduce various Green functions and define a new crack Green function, Gij, that is discontinuous across the crack. Next, we use Gij to derive a Fredholm integral equation of the second kind for the discontinuity in ui across the crack. We prove that this equation always has a unique solution. Hence, we are able to prove that the original boundary-value problem S always possesses a unique solution, and that this solution has an integral representation as an elastic double layer whose density solves an integral equation of the second kind.