An asymptotic hyperbolic–elliptic model for flexural-seismic metasurfaces

Abstract

We consider a periodic array of resonators, formed from Euler-Bernoulli beams, attached to the surface of an elastic half-space. Earlier studies of such systems have concentrated on compressional resonators. In this paper we consider the effect of the flexural motion of the resonators, adapting a recently established asymptotic methodology that leads to an explicit scalar hyperbolic equation governing the propagation of Rayleigh-like waves. Compared with classical approaches, the asymptotic model yields a significantly simpler dispersion relation, with closed form solutions, shown to be accurate for surface wave-speeds close to that of the Rayleigh wave. Special attention is devoted to the effect of various junction conditions joining the beams to the elastic half-space which arise from considering flexural motion and are not present for the case of purely compressional resonators. Such effects are shown to provide significant and interesting features and, in particular, the choice of junction conditions dramatically changes the distribution and sizes of stop bands. Given that flexural vibrations in thin beams are excited more readily than compressional modes and the ability to model elastic surface waves using the scalar wave equation (i.e. waves on a membrane), the paper provides new pathways toward novel experimental set-ups for elastic metasurfaces. arXiv:1904.07690v2 [physics.class-ph]

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].…”

Reduced model for the surface dynamics of a generally anisotropic elastic half-space

“…Although the reduced model is derived with a surface wave in mind, our last two examples concerning surface resonators and coated half-spaces serve to demonstrate that the reduced model may also apply to many other situations where surface wave type behaviour is involved. In particular, the recently considered problem for an isotropic half-space subject to an array of Euler-Bernoulli beams attached to the surface (Wootton et al , 2019) can be readily extended to the generally anisotropic case. Also the developed model may be implemented as a short-wave limiting behaviour in the composite hyperbolic equations for bending and extension of anisotropic strips, in a similar manner to what has been done earlier within the isotropic context (Erbas et al , 2018(Erbas et al , , 2019.…”

Near-Resonant Regimes of a Moving Load on a Pre-Stressed Incompressible Elastic Half-Space

“…where 1 is given by (38), and 2 = 1 −2 + 2 −2 + 3 −2 , see also a recent contribution by (Zhou et al, 2018).…”

“…The advantage of this approach is related to the representation of the surface wave field in terms of a single harmonic function, providing reduction of the vector problem of elastodynamics to a scalar formulation. Recent developments in this area include the incorporation of effects of anisotropy (Fu et al, 2020), a refined second-order model (Wootton et al, 2020), explicit formulations for seismic meta-surfaces in the form of an array of resonators attached to the surface (Ege et al, 2018;Wootton et al, 2019) and formulations for surface wave on a coated halfspace with non-classical boundary conditions (Kaplunov et al, 2019).…”