Simple views on symmetries and dualities in the theory of elasticity

TL;DR

This paper explores how microscopic and non-spatial symmetries, including dualities, impose constraints on the elastic properties of crystalline solids and metamaterials, reducing independent elastic moduli.

Contribution

It introduces a general framework for understanding how non-spatial symmetries and dualities constrain elastic tensors in mechanical structures.

Findings

Non-spatial symmetries reduce the number of elastic moduli.

Duality transformations impose additional constraints, especially at self-dual points.

Constraints persist away from self-dual points, similar to critical phenomena.

Abstract

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden non-spatial symmetries can occur microscopically in special classes of mechanical structures. Examples of such non-spatial symmetries occur in families of mechanical metamaterials where a duality transformation relates pairs of different configurations. We show on general grounds how the existence of non-spatial symmetries further constrains the elastic tensor, reducing the number of independent moduli. In systems exhibiting a duality transformation, the resulting constraints on the number of moduli are particularly stringent at the self-dual point but persist even away from it, in a way reminiscent of critical phenomena.