Thermodynamic basis for odd matter

Continuum-type constitutive relations of odd matter need to be formulated according to the second law of thermodynamics. Based on the primitive thermodynamics of Edelen (1973-1974), a procedure admitting most general relations, is outlined for heat-conducting fluids and solids. For viscous responses of odd matter, the theory accounts for the irreversible and non-dissipative forces, besides the hyperdissipative ones. For quasi-static responses, the theory grasps the elastic and non-conservative forces besides those derivable from the free energy, this being the realm of Cauchy elasticity beyond hyperelasticity. The obtained constitutive relations satisfy the principle of material frame-indifference and the angular momentum conservation. Two applications of odd fluid mechanics in laminar flow show the velocity fields to be unchanged, with the stress fields altered by the odd viscosity. In Cosserat-type odd matter, primitive thermodynamics also accounts for curvature-torsion and couple-stress besides, respectively, deformation and force-stress tensors. In both, classical and micropolar cases, the theory grasps all possible couplings between the thermodynamic velocity and force vectors, along with a full range of anisotropies of hyperdissipative and hyperelastic responses (both linear and nonlinear).