On the Stored Energy Functions of Hyperelastic Materials with Internal Constraints

Abstract

In this paper we continue our study [1] on elastic materials with internal constraints by formulating the stored energy functions as functions defined only on the constraint surfaces of the materials. Since the constraint surfaces generally have dimensions lower than that of the proper linear group, differentiations and integrations with respect to the deformation gradient are not applicable and must be replaced by suitable operators associated with the constraint surfaces. It turns out that many results for hyperelastic materials free of constraints can be carried over to similar results for hyperelastic materials with internal constraints. However, there are also some constraint-free results whose counterparts with constraints are more complex.