Constitutive relations of linear elastic materials under various internal constraints

Abstract

The inverse representation of Hooke's law for linear elastic materials with internal constraints, such as incompressibility and inextensibility cannot be derived by inverting the elastic compliance tensor because of its singularities. The reverse form for incompresible isotropic materials has been well established, and that for incompressible anisotropic materials has been recently obtained under a certain restrictive assumption. In this paper, the explicit reverse representation of Hooke's law for materials, either isotropic or anisotropic, of general internal constraints, is derived with no restrictive assumptions. The results are valid for both Green elasticity and Cauchy elasticity. Extension to other physical linear constitutive laws with internal constraints is straight-forward.