Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress

The stored energy functional of a homogeneous isotropic elastic body is inv ariant with respect to translation and rotation of a reference configuratio n. One can use Noether's Theorem to derive the conservation laws correspond ing to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibriu m of a homogeneous isotropic body. The resulting system is mathematically i dentical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the sys tem of equilibrium equations gives rise to another solution, which describe s the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugat e solutions for two models of elastic homogeneous isotropic bodies - the mo del of a simple material and the model of a material with couple stress (Co sserat continuum).