  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.
