On scattering of waves on square lattice half-plane with mixed boundary condition

Discrete scattering by a pair of parallel defects

“…Furthermore, within in the geometrically reduced diffraction problem on the lattice half-plane, a limiting case coincides with that studied recently [47]. When N = 1 but β = 0, γ = 0, the problem reduces to that for a variant of mixed boundary condition at y = 1.…”

“…The above expression results after simplification of −e −iκ y − c B e iκ y + β(e iκ y + c B e −iκ y ) − γ(1 + c B ) = 0. The general case of the scattering problem on a lattice half-plane S H with the above boundary condition (3.4) involving β and γ can be solved using the complex analysis as developed in [32,47]. Details, using similar notation, are provided below while also following the technique introduced in […”

Wave transmission across surface interfaces in lattice structures

“…In the same time discrete models can describe in all details the material behaviour in the vicinity of crack tips, changes in geometry and other geometrical singularities, see, e.g. 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).…”

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

“…An alternative approach uses the numerical technique of molecular dynamics simulations, see, e.g., Miller and Shenoy (2000); Shenoy (2005), where the surface elastic moduli are determined from direct atomistic simulations. Let us note that the lattice dynamics as described by Brillouin (1946); 20 Born and Huang (1985) provides the possibility to solve various dynamical problems involving waves with attention focussed on the influence of microstructure in as much detail as possible, including even the surface microstructure, see Slepyan (2002); Mishuris et al (2007Mishuris et al ( , 2009 ;Porubov and Andrianov (2013); Sharma (2017b); Porubov et al (2018) and the references therein.…”