On the use of universal relations in modeling nonlinear electro-elastic materials

In this article we employ the nonlinear constitutive framework of isotropic electro-elasticity to derive universal relations. These are connections between the components of the total stress, the electric field and the deformation and, for a class of materials, are independent of the specific free energy function. Universal relations are derived by investigating the coaxiality of the total stress tensor and the corresponding deformation, but only the universal manifold method gives the general set of universal relations for a given material class. Universal relations must hold independently of the constitutive law for a given family of materials and can be used by the experimentalist to determine if a particular material should be included in such a family, i.e. the universal relations must be satisfied by the experimental data. To inform the experimentalist, we illustrate the universal relations for the full constitutive relation and show the consequences if the number of constitutive functions is reduced. In particular, we consider the homogeneous deformation known as simple shear and a non-homogeneous deformation of a cylindrical solid with circular cross-sectional area. The latter is one of the controllable states proposed by Singh and Pipkin