Hyperbolic-elliptic model for surface wave in a pre-stressed incompressible elastic half-space

Abstract

The paper aims at derivation of the asymptotic model for surface wave propagating in a pre-stressed incompressible elastic halfspace, subject to prescribed surface loading. The approach relies on the slow-time perturbation procedure, extending the previously known hyperbolic-elliptic formulations for surface waves in compressible linearly elastic solids. Within the derived model, the decay away from the surface is governed by a pseudo-static elliptic equation, whereas wave propagation is described by a hyperbolic equation on the surface. The effect of pre-stress, namely, the principal Cauchy stress σ 2 , is investigated. Finally, an illustrative example of the Lamb problem is considered, demonstrating the efficiency of the approach.

Explicit model for surface waves in a pre-stressed, compressible elastic half-space

“…The decay over the interior is governed by a scaled Laplace equation, whereas wave propagation is modelled by a hyperbolic equation. The results complement existing ones for isotropic media [15], as well as particular type of orthorhombic media [18] and pre-stressed incompressible media [19]. Scalar hyperbolic equations for surface displacements could potentially be beneficial for further development of methods of non-destructive evaluation.…”

“…In this paper, we extend the previous considerations to a pre-stressed compressible elastic half-space within the plane strain assumption, complementing the results in [19]. It is known that incorporation of compressibility may enrich the Int.…”

“…[17]. The approach has also been extended to a special case of anisotropy associated with attenuation without oscillations in [18], as well as pre-stressed incompressible half-space [19]. A parallel parabolic-elliptic formulation for dispersive bending edge Rayleigh-type waves has been presented in [20].…”

An asymptotic hyperbolic–elliptic model for flexural-seismic metasurfaces

“…Figure 3: The dispersion curves for surface waves on the half-space coated with simply supported beam like resonators, using physical parameters from Table 1. The solid blue lines show the dispersion curve of the full unimodal solution from (30) and the solid black lines the dispersion curve of the asymptotic solution from (33). The dashed green, orange and purple lines correspond to the longitudinal, shear and Rayleigh wave lines respectively.…”

Moving Load on an Elastic Half-Space Coated With a Thin Vertically Inhomogeneous Layer

“…The current work is based on the long-wave asymptotic formulation presented in [7], which has been applied to problems on a moving load for elastic half-space [17,10,8,26], see also [18] for a more systematic exposition of the methodology of hyperbolic-elliptic models for surface waves. Recent advances of the approach include in particular extensions incorporating the effects of anisotropy [13] and pre-stress [21], composite models for elastic layers [9], as well as development of seismic meta-surfaces [29] and the second-order refined model [30].…”