  Odd elasticity describes the unusual elastic response of solids whose
stress-strain relationship is not compatible with an elastic potential. Here,
we present a study of odd elasticity in a driven granular matter system
composed of grains with ratchet-like interparticle friction and activated by
oscillatory shear. We find that the system permits a time-averaged elasticity
theory featuring nonzero odd elastic coefficients. These coefficients are
explicitly measured using molecular dynamics simulations and can be predicted
and tuned from microscopics. In the presence of disorder, our driven granular
material displays distinctive properties ranging from self-healing grain
boundaries in polycrystalline systems to chiral plastic vortices and force
chain deflection in amorphous packings. Beyond granular matter, our work
motivates the search for microscopic transduction mechanisms that convert
periodic nonuniform drive into uniform elastic properties of active solids.
