Keele Research Repository

The thesis is concerned with asymptotic analysis of static as well as dynamic problems for a solid coated by a thin isotropic elastic layer with a high contrast in their material parameters. First, the static anti-plane shear deformation problem is considered with the layer being relatively soft or stiff in order to investigate the behaviour of the coating. The two-parametric asymptotic procedure is introduced motivated by the scaling obtained from the exact solution of a model problem. As a result, Winkler-type behaviour appears for a relatively soft coating, whereas for a relatively stiff one, the equations of plate shear are valid. Further, the formulation is extended to a 3D case with vertical force applied at the surface aiming at asymptotic investigation of the area of validity of the Winkler-Fuss hypothesis and the Kirchhoff plate theory. It is established that the aforementioned theories are valid only at a rather high contrast in stiffness of the layer and the half space. However, a uniformly valid formula is deduced in case of a layer being soft, and for low contrast with the coating being stiff, several approximate formulations are suggested based on the reduced problems for the half space. Then, the problem is considered in dynamic formulation yielding the higher order effective boundary conditions modelling the presence of the coating layer. The obtained results indicate again the inconsistency of the conditions earlier presented in [22]. The validity of the asymptotic results is demonstrated by comparison with the long-wave expansion of the exact solutions of plane and anti-plane time-harmonic problems for the coating. Finally, the dynamic problem for a coated half-space with clamped surface is considered. The exact solutions are obtained in an anti-plane and plane strain formulations resulting in the range of material parameters for which the sought for localised wave exists. The effective boundary conditions are obtained and the model for Rayleigh wave field is applied for the layer being thin and soft, leading to an explicit correction to the classical Rayleigh wave speed.