On the treatment of non-solvable implicit constitutive relations in solid mechanics

Abstract

We report results concerning the treatment of elastic bodies described by implicit constitutive relations which are not solvable, in the sense of expressing the stresses as functions of the strains or vice versa. Motivated by the theory of generalized hyperelastic materials, which applies also to polyatomic crystals, the field equations to be solved for a body described by a non-solvable implicit constitutive relation in the above sense are lied down. In addition to the momentum equation, auxiliary variables come into play. These variables are accompanied by an equation governing them. We specialize to the case of an isotropic constitutive relation and give some conditions in order for a broad subclass of an isotropic body to be a generalized material. Then, we lay down the field equations for such a body.