Anti-plane surface waves in media with surface structure: Discrete vs. continuum model

Abstract

We present a comparison of the dispersion relations derived for anti-plane surface waves using the two distinct approaches of the surface elasticity vis-a-vis the lattice dynamics. We consider an elastic half-space with surface stresses described within the Gurtin-Murdoch model, and present a formulation of its discrete counterpart that is a square lattice half-plane with surface row of particles having mass and elastic bonds different from the ones in the bulk. As both models possess anti-plane surface waves we discuss similarities between the continuum and discrete viewpoint. In particular, in the context of the behaviour of phase velocity, we discuss the possible characterization of the surface shear modulus through the parameters involved in lattice formulation.

Wave transmission across surface interfaces in lattice structures

“…Mishuris et al (2007Mishuris et al ( , 2009 ;Slepyan (2002); Sharma (2015aSharma ( , 2017c; Porubov and Andrianov (2013); Porubov et al (2018); Gorbushin et al (2018), where the efficiency of discrete lattice models is demonstrated. As an example, we recently provided the comparison of the surface wave propagation within the Gurtin-Murdoch model and the lattice dynamics which shows similarities in dispersion relations for both models after some re-scaling, see Eremeyev and Sharma (2019). Thus, the lattice dynamics can serve as a counterpart of continuum surface elasticity models.…”

“…Following Eremeyev and Sharma (2019) here we discuss the anti-plane surface waves propagation in an elastic half-space with a piece-wise inhomogeneity in surface properties. We consider a square lattice which consists of material particles of mass M and stiffness of bonds K, where the surface consists of material particles of two kind which both are different from the ones in the bulk.…”

“…Figure 1: Geometry of the three dimensional lattice half space is shown schematically by placing a cutting section along x-y plane. On the right side (by zooming into the central portion), an atomic/particle based discrete model of an elastic half space with surface interface Eremeyev and Sharma (2019) is shown. The kinematics assumed is that if anti-plane motion, i.e.…”

“…The question is answered in the present paper and an exact expression is provided. Moreover, an additional question becomes an interesting follow up of this paper concerning the continuum limit: to elastic half space following the work of Eremeyev and Sharma (2019); Sharma (2015cSharma ( , 2017a as well as the limit to an elastic strip; this question has been deferred to another paper elsewhere while the solution of the continuum problem itself is under investigation presently.…”

Transmission of waves across atomic step discontinuities in discrete nanoribbon structures

“…Within this context of surface induced phenomena, the present paper extends the applications of recent framework developed for discrete scattering problems [33,34,43]. In contrast to the recent investigation of the wave transmission of surface localized anti-plane shear wave [42] in lattice structures, motivated by certain discrete counterpart [8] of Gurtin-Murdoch model [11,12,45,46], the present paper applies similar methodology to reveal the transmission characteristic associated with a close-up of discrete form of surface roughness.…”