Topological elasticity of flexible structures

TL;DR

This paper introduces a continuum elasticity framework for flexible mechanical metamaterials, revealing new topological invariants that predict edge modes and are observable at macroscopic scales.

Contribution

It develops a micromorphic continuum model capturing nonuniform deformations and uncovers a novel topological invariant relating edge modes in such structures.

Findings

New topological invariant in continuum elasticity

Relation between microstructure and boundary modes

Potential for macroscopic observation of topological properties

Abstract

Flexible mechanical metamaterials possess repeating structural motifs that imbue them with novel, exciting properties including programmability, anomalous elastic moduli and nonlinear and robust response. We address such structures via micromorphic continuum elasticity, which allows highly nonuniform deformations (missed in conventional elasticity) within unit cells that nevertheless vary smoothly between cells. We show that the bulk microstructure gives rise to boundary elastic terms. Discrete lattice theories have shown that critically coordinated structures possess a topological invariant which determines the placement of low-energy modes on edges of such a system. We show that in continuum systems a new topological invariant emerges which relates the difference in the number of such modes between two opposing edges. Guided by the continuum limit of the lattice structures, we identify macroscopic experimental observables for these topological properties that may be observed independently on a new length scale above that of the microstructure.