# Controlling the Deformation of Metamaterials: Corner Modes via Topology

## TL;DR

This paper introduces a new class of topological mechanical metamaterials that utilize corner modes for controlling deformation, achieved through a topological degree map, enabling targeted boundary modes in rigid structures.

## Contribution

The work demonstrates how rigid periodic structures can host topologically protected corner modes using a generalized topological degree, expanding the design principles of metamaterials beyond Maxwell lattices.

## Key findings

- Bulk structure generates a topological deformation mode localized at a corner.
- Experimental prototype confirms the existence of corner modes.
- Single mode selection enables variable stiffness and mechanical amplification.

## Abstract

Topological metamaterials have invaded the mechanical world, demonstrating acoustic cloaking and waveguiding at finite frequencies and variable, tunable elastic response at zero frequency. Zero frequency topological states have previously relied on the Maxwell condition, namely that the system has equal numbers of degrees of freedom and constraints. Here, we show that otherwise rigid periodic mechanical structures are described by a map with a nontrivial topological degree (a generalization of the winding number introduced by Kane and Lubensky) that creates, directs and protects modes on their boundaries. We introduce a model system consisting of rigid quadrilaterals connected via free hinges at their corners in a checkerboard pattern. This bulk structure generates a topological linear deformation mode exponentially localized in one corner, as investigated numerically and via experimental prototype. Unlike the Maxwell lattices, these structures select a single desired mode, which controls variable stiffness and mechanical amplification that can be incorporated into devices at any scale.