Effective boundary conditions for the scattering of two-dimensional SH waves from a curved thin elastic layer

Abstract

The problem of scattering of time-harmonic elastodynamic sh waves from a curved thin elastic layer is addressed in the present paper. In the literature there are a number of suggestions on how to model a thin elastic layer by means of approximate boundary conditions. The ones that seem to be most commonly used are the spring contact boundary conditions (BCS). These type of BCS are also used to model a number of other flaw types, including, e. g. a thin layer of viscous fluid or a distribution of microcracks over a surface. However, in some recent papers on the scattering from inclusions surrounded by thin elastic interphase layers, it has been shown that the spring contact BCS do not lead to correct results when used in the context of thin elastic layers. The reason for this is that the spring contact BCS do not incorporate certain effects, both physical and geometrical, which are of the same order of magnitude as the ones that are in fact taken into account by these BCS. In the present paper we analyse in detail what the appropriate BCS for two-dimensional (2D) SH wave scattering from a thin curved elastic layer are. We show that when all effects to lowest order in the layer thickness are included, we get BCS that have a number of nice features, including the one of implying that no scattering occurs when the layer has material parameters identical to the ones in the matrix medium. The jump in displacement is written in terms of the normal derivatives, and the jump in normal derivative is written in terms of displacements and their second-order tangential derivatives. The BCS are derived even for the case where the layer lies between two different media, and for the case of a planar layer with anisotropy. The energy conservation properties of the BCS are analysed. To test the approximate BCS we check them against the exact solution for the case of scattering from a layer of circular cross-section by a 2D point source outside the layer. We also compare the results against the commonly used spring contact model, as well as against an improved version of it.