Closed-form solutions for wave propagation in hexagonal diatomic non-local lattices

Abstract

Periodic mass–spring lattices are commonly used to investigate the propagation of waves in elastic systems, including wave localisation and topological protection in phononic crystals and metamaterials. Recent studies have shown that introducing non-neighbouring (i.e., beyond nearest neighbour) connections in these chains leads to multiple topologically localised modes, while generating roton-like dispersion relations. This paper focuses on the theoretical analysis of elastic wave propagation in hexagonal diatom mass–spring systems in which both neighbouring and non-neighbouring interactions occur through linear elastic springs. Closed-form expression for the dispersion equations are derived, up to an arbitrary order of beyond-the-nearest connections for both in-plane and out-of-plane mass displacements. This allows to explicitly determine the influence of the order of non-neighbouring interactions on the band gaps, the local minima and the slope inversions in the first Brillouin zone for the considered unit cell. All analytical solutions are numerically verified. Finally, examples are provided on how non-neighbouring connections can be exploited to enhance the localisation of topologically-protected edge modes in waveguides constructed using mirror symmetric diatomic lattices constituted by two regions with different unit cell orientations. The study provides further insight on how to design phononic crystals generating roton-like behaviour and to exploit them for topologically protected waveguiding.