Nonlinear isotropic odd elasticity

TL;DR

This paper develops a framework for large, nonlinear deformations in 2D isotropic odd elastic solids, revealing unique bifurcation behaviors and extending understanding of active, biological materials.

Contribution

It introduces a nonlinear elasticity framework for 2D isotropic odd elastic solids and analyzes bifurcations in the Rivlin problem, highlighting differences from passive elasticity.

Findings

Oddness suppresses bifurcations in 2D Rivlin squares.

Bifurcations in 3D Rivlin cubes persist despite oddness.

Large deformations exhibit unexpected nonlinear behaviors in active solids.

Abstract

The nonconservative elastic responses of active solids have driven a recent explosion of interest in two-dimensional "odd" elasticity: small, linear deformations of these Cauchy elastic solids enable new behaviour absent from classical, passive elasticity. Here, we establish the description of large, nonlinear deformations of isotropic two-dimensional Cauchy elastic solids. We apply our framework to the Rivlin problem, perhaps the simplest problem of elasticity lacking a linear analogue: a square deforms under dead load tractions. Surprisingly, we find that oddness suppresses the bifurcations of a passive Rivlin square. By contrast, we discover that the bifurcations of a three-dimensional Rivlin cube survive oddness even though there is no isotropic, odd linear elasticity in three dimensions. Our results thus form the basis for describing large deformations of active, biological solids while revealing their unexpected nonlinear behaviour that arises even in minimal problems.