Introduction

An important application of the reciprocity relation is its use to generate integral representations. With the aid of the basic singular elastodynamic solution for an unbounded solid, an integral representation can be derived that provides the displacement field at a point of observation in terms of the displacements and tractions on the boundary of a body. In the limit as the point of observation approaches the boundary, a boundary integral equation is obtained. This equation can be solved numerically for the unknown displacements or tractions. The calculated boundary values are subsequently substituted in the original integral representation to yield the desired field variables at an arbitrary point of observation.

The boundary element method is often used for the numerical solution of boundary integral equations. The advantage of the boundary element method for solving boundary integral equations is that the dimensionality of the problem is reduced by one. Rather than calculations in a two- or three-dimensional discretized space, we have calculations for discretized curves or surfaces. For detailed discussions of boundary element methods in elastodynamics we refer to the review papers by Beskos (1987) and Kobayashi (1987). These papers contain numerous additional references. We also mention the book edited by Banerjee and Kobayashi (1992), and a recent book by Bonnet (1995) that has several sections on dynamic problems.

We start this chapter with an exposition of the basic ideas for the simpler, two-dimensional, case of anti-plane strain in Section 11.2.