Zero-frequency and extremely slow elastic edge waves in mechanical granular graphene

Abstract

We report here the theoretical description and analysis of edge elastic waves in a semi-infinite mechanical granular graphene structure. The granular graphene, composed of spherical beads arranged in a single-layer honeycomb structure, is studied for two types of edge configurations: zigzag and armchair. Due to the existence of the rotational degrees of freedom of the grains, rotation-associated shear, bending and torsional couplings between the neighbor beads are activated. The dispersion curves of the edge waves are theoretically derived and numerically analyzed for various configurations of bead couplings, as well as the existence of edge states when the torsional or/and bending rigidities are weak/vanishing. Quasi-flat edge mode dispersion curves with near zero frequency are observed for both the zigzag and armchair edges. These quasi-flat dispersion curves, supporting the propagation of waves with extremely slow group velocity, tend to be perfect zero-frequency modes for zero torsional rigidity or vanish for zero bending rigidity, indicating that weak bending and torsional interbead interactions are critical in the transformation of zero-frequency modes into extremely slow propagating modes. These results on edge waves in mechanical granular graphene structures with rotational degrees of freedom are the necessary preliminary step for the design of granular meta-graphenes with artificial symmetry breaking for inducing topologically protected unidirectional edge states.