Zero-frequency corner modes in mechanical graphene

TL;DR

This paper designs a mechanical graphene lattice with an elastic foundation to exhibit zero-frequency corner modes, demonstrating their origin, conditions for emergence, and robustness against distant defects.

Contribution

It introduces a novel elastic foundation modulation in mechanical graphene to realize zero-frequency corner modes, linking them to diatomic chain dynamics and boundary conditions.

Findings

Zero-frequency corner modes are enabled by elastic foundation stiffness modulation.

Corner modes depend on the shape and corner angles of the lattice.

Robustness of corner modes against distant structural defects is demonstrated.

Abstract

In an unconstrained elastic body, emergence of zero natural frequencies is an expectable outcome on account of the body's ability to purely translate or rotate with no structural deformation. Recent advances in literature have pushed such conventional definition and demonstrated properties transcending typical zero-frequency modes, such as localization of deformation at a structural edge or corner. In this paper, a spring-mass honeycomb lattice with an elastic foundation, referred to here as mechanical graphene, is designed to exhibit zero-frequency corner modes. A central element in the proposed design is the elastic foundation, and the zero-frequency corner modes are enabled by intricate modulation of the elastic-foundation's stiffness. These modes are proven to have their origins from the dynamics of a diatomic chain, made from a single strip of the mechanical graphene with free boundaries. Different shapes of finite mechanical graphene with free boundaries are considered and conditions leading to the manifestation of corner modes are correlated with the angle of corners and stiffness of springs supporting them. Finally, the effect of defects on zero-frequency corner modes is briefly discussed, demonstrating robustness against structural defects that are distant from corners.