Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

A closed-form solution to propagation of guided waves in a layered half-space under a time-harmonic load: An application of elastodynamic reciprocity.

This article is concerned with the application of reciprocity in computing guided wave motions generated by a time-harmonic load in a layer of uniform thickness joined to a half-space. Explicit expressions for free Rayleigh waves and Love waves propagating in the layered half-space are introduced. Exact solutions of Rayleigh waves and Love waves are derived from reciprocity relations between an actual state - guided waves generated by a time-harmonic line load and a virtual state - an appropriately chosen free wave traveling in the structure. Scattered amplitudes of the wave motions are thus determined. The validation of the reciprocity approach is shown through the computation of the lowest Rayleigh wave mode in the layered half-space, which approaches the calculation of the Rayleigh surface wave in the half-space once the layer thickness approaches zero in the limit.

A theoretical approach to multiple scattering of surface waves by shallow cavities in a half-space.

The current theoretical work investigates multiple scattering of surface waves from shallow cavities at the surface of an elastic half-space. The multiple scattered field is shown to be equivalent to the field generated by the application of a distribution of tractions, obtained from the incident wave, on the surfaces of the cavities. Using a self-consistent method, an analytical approach is proposed to derive an explicit set of equations which approximate the multiple scattered field. Its far-field displacement amplitude is then calculated and verified by numerical calculation obtained by the boundary element method (BEM). The analytical and BEM results are graphically presented and show excellent agreement when the depths of cavities are small compared to the wavelength. The improvement of the proposed approach relative to a first order approximation is reported and its limitations are discussed based on the results of comparison.

Verification of surface wave solutions obtained by the reciprocity theorem.

Surface wave motions generated by a time-harmonic point load applied at the surface of an isotropic linearly elastic half-space are conventionally solved by the use of integral transform techniques. The inverse transforms, are often complicated and will not always yield closed-form solutions. In this paper expressions for the displacements for surface wave motions radiated from point-load excitation are determined in a simple manner by the use of the elastodynamic reciprocity theorem. It is shown that the radiated amplitudes of the surface displacements obtained by the reciprocity approach are identical to the corresponding results obtained by the use of Hankel transform and by Lamb in his classical paper.

Validity of the reciprocity approach for determination of surface wave motion.

Expressions for the displacements and the stresses for surface wave motion generated by a time-harmonic line load applied to the surface of an isotropic linearly elastic half-space are determined in a simple manner by the use of the reciprocity theorem. It is shown that their amplitudes show perfect agreement with the corresponding amplitudes obtained in the conventional manner by applying the Fourier transform technique. As an application of the reciprocity approach, the surface wave motion generated by uniform pressure over a cylindrical cavity located on the surface of a half-space has been determined. The analytical results have been verified by comparison with boundary element method (BEM) results. For a prescribed frequency and depth of the cavity, the analytical and BEM results are graphically displayed versus the surface length of the cavity, and show excellent agreement.