Abstract
Elastodynamics of a half-space coated by a thin soft layer with a clamped upper face is considered. The focus is on the analysis of localized waves that do not exist on a clamped homogeneous half-space. Non-traditional effective boundary conditions along the substrate surface incorporating the effect of the coating are derived using a long-wave high-frequency procedure. The derived conditions are implemented within the framework of the earlier developed specialized formulation for surface waves, resulting in a perturbation of the shortened equation of surface motion in the form of an integral or pseudo-differential operator. Non-uniform asymptotic formula for the speeds of the sought for Rayleigh-type waves, failing near zero frequency and the thickness resonances of a layer with both clamped faces, follow from the aforementioned perturbed equation. Asymptotic results are compared with the numerical solutions of the full dispersion relation for a clamped coated half-space. A similarity with Love-type waves proves to be useful for interpreting numerical data. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.
Highlights
It is well known that the Rayleigh wave on a homogeneous elastic half-space [1] exists only for a traction-free surface
In contrast to traditional derivations usually dealing with low-frequency vibrations of a coating with a traction-free upper face (e.g. [3,4,5,6]), we develop an asymptotic procedure covering the highfrequency vibrations, similar to what has been done for a thin interfacial layer in [7], and earlier for thin-walled structures in [8,9,10], see [11,12,13,14,15]
Our goal is to demonstrate that the presence of inhomogeneity in the form of a coating supports a family of solutions, corresponding to Rayleigh-type surface waves, as follows from the mathematical analysis in [2]
Summary
It is well known that the Rayleigh wave on a homogeneous elastic half-space [1] exists only for a traction-free surface. The main focus is on a physical insight into the peculiarities of the observed localized dynamic phenomena using an asymptotic approach oriented to a soft coating with a stiffness much lower than that of a substrate In this case, we should expect that the associated counterparts of the Rayleigh wave could be treated as its perturbations. The asymptotic formulae for the corrections to the Rayleigh wave speed due to the effect of a soft coating readily follow from the pseudo-differential equation along the substrate surface They obviously fail near zero frequency, where the localized wave is not yet generated, and in the vicinities of the thickness resonances in [8], as may be observed from numerical comparison with the exact dispersion relation, which is studied in great detail making use of the analogy with better understood Love-type waves
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