10 - Forced motion of an elastic layer

Summary

Introduction

As discussed in Chapter 9, the modes of wave propagation in an elastic layer are well known from Lamb's (1917) classical work. The Rayleigh–Lamb frequency equations, as well as the corresponding equations for horizontally polarized wave modes, have been analyzed in considerable detail; see Achenbach (1973) and Mindlin (1960). It appears, however, that a simple direct way of expressing wave fields due to the time-harmonic loading of a layer in terms of mode expansions, and a suitable method to obtain the coefficients in the expansions by reciprocity considerations, has so far not been recognized. Of course, wave modes have entered the solutions to problems of the forced wave motion of an elastic layer, at least in the case of surface forces applied normally to the faces of the layer, but via the more cumbersome method of integral transform techniques and the subsequent evaluation of Fourier integrals by contour integration and residue calculus. For examples, we refer to the work of Lyon (1955) for the plane-strain case, and that of Vasudevan and Mal (1985) for axial symmetry.

In this chapter the displacements excited by a time-harmonic point load of arbitrary direction, either applied internally or to one of the surfaces of the layer, are obtained directly as summations over symmetric and/or antisymmetric modes of wave propagation along the layer. This is possible by virtue of an application of the reciprocity relation between time-harmonic elastodynamic states.