Excess Floppy Modes and Multi-Branched Mechanisms in Metamaterials with Symmetries

TL;DR

This paper reveals that symmetric geometries in metamaterials can have unlimited excess floppy modes and multi-branched mechanisms, with a new counting algorithm to predict these features and enable design of complex folding behaviors.

Contribution

It uncovers the presence of unlimited excess floppy modes in symmetric geometries and introduces an accurate cluster counting algorithm for their prediction.

Findings

Excess floppy modes are extensive and peak at intermediate densities.

Symmetric geometries support multi-branched floppy modes.

A cluster counting algorithm accurately predicts floppy mode counts.

Abstract

Floppy modes --- deformations that cost zero energy --- are central to the mechanics of a wide class of systems. For disordered systems, such as random networks and particle packings, it is well-understood how the number of floppy modes is controlled by the topology of the connections. Here we uncover that symmetric geometries, present in e.g. mechanical metamaterials, can feature an unlimited number of excess floppy modes that are absent in generic geometries, and in addition can support floppy modes that are multi-branched. We study the number $\Delta$ of excess floppy modes by comparing generic and symmetric geometries with identical topologies, and show that $\Delta$ is extensive, peaks at intermediate connection densities, and exhibits mean field scaling. We then develop an approximate yet accurate cluster counting algorithm that captures these findings. Finally, we leverage our insights to design metamaterials with multiple folding mechanisms.