Asymptotic models for surface waves in coated elastic solids

Abstract

This thesis deals with surface wave propagation in elastic solids. It develops further the methodology of asymptotic hyperbolic-elliptic models for the surface elastic waves, aiming at two main areas, namely, accounting for the effects of a thin coating layer, as well as incorporating the influence of gravity.
The derived model for surface waves on a coated elastic half space subject to prescribed surface stresses reflects the physical nature of elastic surface waves, for which the decay over the interior is described by a "pseudo-static" elliptic equation, whereas wave propagation along the boundary is governed by a singularly perturbed hyperbolic equation, with the perturbation in the form of a pseudo-differential operator.
This perturbative term originates from the effect of a thin coating layer, which is modelled in terms of effective boundary conditions, derived within the long wave limit approximation of the corresponding problem in linear elasticity.
Various types of coatings are studied in this thesis, including anisotropic and vertically inhomogeneous thin layers. The analysis reveals a qualitatively similar hyperbolic equation, singularly perturbed by a pseudo differential operator, with the appropriate coeffcient incorporating the overall effect of the material properties of the coating and the substrate.
The established methodology is then illustrated for approximate treatment of several rather technical problems in elastodynamics, in particular, analysis of moving loads on a coated half-plane. The implementation of the hyperbolic-elliptic model allows a natural classification of scenarios and elegant approximations of the exact solution, with clear physical interpretation of the associated numerical illustrations of nearsurface dynamics for several types of vertical inhomogeneity.
Finally, the effect of gravity is embedded into the developed methodology of hyperbolicelliptic asymptotic models for surface waves. As a result, the wave equation on the surface is regularly perturbed by a pseudo differential operator, accounting for the effect of the gravitational field.