Anti-plane waves in an elastic thin strip with surface energy

Abstract

We consider anti-plane motions of an elastic plate taking into account surface energy within the linear Gurtin–Murdoch surface elasticity. Two boundary-value problems are considered that describe complete shear dynamics of a plate with free faces or with free and clamped faces, respectively. These problems correspond to anti-plane dynamics of an elastic film perfectly or non-perfectly attached to a rigid substrate. Detailed analysis of dispersion relations is provided. In particular, the influence of the ratio of a plate thickness to characteristic length on the dispersion curves is analysed. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space

“…For example, it may describe a thin film with modified surface properties attached to a substrate. The considered layered medium generalizes recent results by Mikhasev et al (2022) towards more realistic behaviour of substrate, which is now deformable and can transmit waves. The paper is organized as follows.…”

“…In the recently published contribution by Mikhasev et al (2022), an analysis of the wave equation for a thick plate with at least one free surface revealed the existence of two different regimes of anti-plane shear waves: (a) the TE regime for which waves decay exponentially from the upper and lower surfaces of the plate and (b) the TH regime with the harmonic behaviour of waves in the transverse direction. A similar analysis of Eqs.…”

“…They are similar to Love waves in classic elastodynamics (Achenbach, 1973). The provided analysis of these waves for a half-space and for a layer of finite thickness given by Zhu et al (2019); Eremeyev (2020); Mikhasev et al (2021Mikhasev et al ( , 2022 demonstrated an essential role of surface energy on the wave propagation. In particular new regimes of surface waves in an elastic layer were discussed by Mikhasev et al (2022).…”

“…The provided analysis of these waves for a half-space and for a layer of finite thickness given by Zhu et al (2019); Eremeyev (2020); Mikhasev et al (2021Mikhasev et al ( , 2022 demonstrated an essential role of surface energy on the wave propagation. In particular new regimes of surface waves in an elastic layer were discussed by Mikhasev et al (2022). Similar to the case of plane geometry, surface elasticity introduces new phenomena in waves in cylinders, see Chen et al (2014); Xu & Fan (2016); Huang (2018); Eremeyev et al (2020).…”

“…Following Eremeyev et al (2016); Mikhasev et al (2022), in this paper we discuss anti-plane surface waves in a multilayered medium which consists of thin coating modelled within the Gurtin-Murdoch surface elasticity and an elastic layer of finite thickness perfectly attached to an elastic half-space (Fig. 1).…”