Conformal Elastodynamics in 2D Dilational Metamaterials

TL;DR

This paper investigates a 2D dilational metamaterial supporting conformal symmetries, revealing how conformal deformations influence wave responses and momentum conservation at various frequencies.

Contribution

It introduces a new framework linking conformal symmetry to wave dynamics in dilational metamaterials, with experimental insights into controlling nonlinear waves.

Findings

Low-frequency response dominated by boundary conformal deformations.

High-frequency conformal maps imply conserved complex momentum.

Experimental parameters can tune conformal momentum conservation.

Abstract

Flexible mechanical structures can undergo large deformations under small loads, enabling large, complex, and nonlinear wave responses under finite-frequency driving. Here, we study a dynamically driven canonical flexible mechanical metamaterial composed of rigid squares connected at their corners by flexible hinges. This metamaterial supports a uniform dilational mechanism and, in the limit of ideal joints, exhibits a Poisson ratio of -1. The presence of this dilational mode of deformation gives rise to a conformal symmetry, in which the dynamics are approximately invariant under a wide class of physical transformations -- conformal maps. We find that the low-frequency response of the system is dominated by conformal deformations consisting of spatially varying rotations and dilations concentrated at the boundary. Even at high frequencies, each conformal map implies a conserved spatially complex momentum. We explore how experimental parameters such as material stiffnesses and the geometry and number of unit cells allow experimental conformal momenta to approach this conservation, varying slowly compared to the non-conformal momenta of same order. These results constitute a new framework opening fundamental avenues for the study of conformal wave phenomena in dilational metamaterials as well as potential strategies for controlling nonlinear waves and vibrations.