Classifying topological floppy modes in the continuum

TL;DR

This paper develops a continuum framework for topological mechanics, enabling classification and prediction of edge states and Weyl modes in floppy mechanical systems without detailed lattice simulations.

Contribution

It introduces a continuum theory of topological mechanics with augmented elasticity, classifies topological states via invariants, and provides efficient tools for design and analysis.

Findings

Continuum invariants classify topological states.

Augmented elasticity models edge states and Weyl points.

Homogenization links microscopic lattices to continuum theories.

Abstract

In floppy mechanical lattices, robust edge states and bulk Weyl modes are manifestations of underlying topological invariants. To explore the universality of these phenomena independent of microscopic detail, we formulate topological mechanics in the continuum. By augmenting standard linear elasticity with additional fields of soft modes, we define a continuum version of Maxwell counting, which balances degrees of freedom and mechanical constraints. With one additional field, these augmented elasticity theories can break spatial inversion symmetry and harbor topological edge states. We also show that two additional fields are necessary to harbor Weyl points in two dimensions, and define continuum invariants to classify these states. In addition to constructing the general form of topological elasticity based on symmetries, we derive the coefficients based on the systematic homogenization of microscopic lattices. By solving the resulting partial differential equations, we efficiently predict coarse-grained deformations due to topological floppy modes without the need for a detailed lattice-based simulation. Our discovery formulates novel design principles and efficient computational tools for topological states of matter, and points to their experimental implementation in mechanical metamaterials.