# Odd elasticity

## TL;DR

This paper introduces odd elasticity, a generalization of linear elasticity for active media with non-conservative interactions, revealing new moduli and phenomena like activity-induced auxetic behavior and self-sustained wave propagation.

## Contribution

It extends classical elasticity to include active, non-reciprocal interactions, uncovering additional elastic moduli and novel mechanical behaviors in active materials.

## Key findings

- Discovery of two additional elastic moduli in active media.
- Identification of activity-induced auxetic behavior.
- Demonstration of wave propagation powered by active elastic cycles.

## Abstract

Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. In this Letter, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity. This generalized framework, which we dub odd elasticity, reveals that two additional moduli can exist in a two-dimensional isotropic solid with active bonds. Such an odd-elastic solid can be regarded as a distributed engine: work is locally extracted, or injected, during quasi-static cycles of deformation. Using continuum equations, coarse-grained microscopic models, and numerical simulations, we uncover phenomena ranging from activity-induced auxetic behavior to wave propagation powered by self-sustained active elastic cycles. Besides providing insights beyond existing hydrodynamic theories of active solids, odd elasticity suggests design principles for emergent autonomous materials.