Chiral nonreciprocal elasticity and mechanical activity

There has been significant recent interest in creating, modeling, and exploiting the novel functionality afforded by odd elastic solids, which are a specific class of active matter whose behavior cannot be described by a free energy function. As a result, the mechanical behavior of such solids can be described by a non-symmetric elasticity tensor which means they can be mechanically active, and thus do work on their surroundings through quasistatic deformation cycles without energetic gain or loss terms explicitly appearing in the solid’s equation of state. However, previous incarnations of such solids have required the usage of active elements coupled with robotic machinery powered by independent external energy sources to operate. As such, it is unclear whether the non-symmetric elasticity of these solids can be developed using only passive elements that do not require the usage of energy sources, and furthermore how nonreciprocity in elastic media enables non-symmetric elasticity in elastic solids or mechanical activity. In this work, we propose the notion of chiral, nonreciprocal elasticity, which represents a generic route to enabling 2D, isotropic elastic solids exhibiting non-symmetric elasticity. Chiral, nonreciprocal elasticity describes elastic behaviors that result from coupling chirality with nonreciprocity—specifically, (1) the modulation of the elastic properties depending on the mode and direction of deformation and (2) the nonreciprocal coupling of different deformation fields, both of which enable the solid to exhibit a non-symmetric elasticity tensor. To motivate this, we introduce an isotropic 2D chiral metamaterial made of passive chiral elements that, by exploiting local geometric asymmetry, behaves in a chiral, nonreciprocal elastic fashion. We derive, based on the mechanics of a discrete model of this chiral element, the resulting continuum field equations and constitutive relationships that capture the chiral, nonreciprocal elastic behavior. Then, we establish a thermodynamic framework of energy balance and conservation of chiral, nonreciprocal elastic solids, based on which we demonstrate the ability of the proposed chiral metamaterial to act as a source of mechanical work when used in specific quasistatic deformation cycles, though no energy is dissipated by its passive elements. Finally, we demonstrate through numerical finite element simulations the practical implementation of the deformation cycles, while elucidating the specific conditions needed for the chiral metamaterial to exhibit linear, chiral nonreciprocal elastic behavior throughout the deformation cycle, and thus reveal mechanical activity.