Odd Elasticity in Active Metamaterials

The theory of elasticity provides a foundation for describing the mechanics of deformations of continuous media. However, elastic theories of active matter must confront a fundamental challenge: the starting point of elasticity, the elastic energy, is not well defined due to microscopic activity. We introduce Odd Elasticity as a generalized theory of continuum mechanics that breaks key symmetries of the elastic (stiffness) tensor otherwise required by conservation of energy. We show that odd elasticity describes solids in which activity depends on the deformation of microscopic bonds. As a minimal model that produces odd elasticity upon coarse graining, we consider active metamaterials in which internal torques are actuated in response to compression or extension of the beams. Our odd-elastic theory, corroborated by simulations, sheds light on a rich phenomenology, including activity-induced auxetic behavior, active elastic waves and instabilities. Our work revisits the foundations of continuum mechanics and provides a blueprint for the design of active elastic engines, which utilize a strain cycle to convert microscopic activity into useful mechanical work.