Transversely isotropic strain energy with physical invariants

In the past, different sets of invariants for strain energy function of transversely isotropic elastic solids were proposed for various reasons. In this paper, a strain energy function which depends on five independent variables that have immediate physical interpretation, is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The strain energy function, expressed in terms of these variables, has a symmetry almost similar to the symmetry of a strain energy function of an isotropic elastic solid written in terms of principal stretches. A mathematical simplicity is highlighted when the proposed strain energy function is applied to, for example, biaxial deformations. Since the proposed formulation is novel, the ground state and stress-strain relations are given. A class of strain energy functions is proposed which has an experimental advantage in the sense that pure shear tests can directly determine all the terms in the strain energy function. The stress response terms in the Cauchy stress are fully orthogonal when one on the principal directions of the right stretch tensor is the preferred direction and are mostly orthogonal in general. Finally, based on the class of energy functions given in this paper, we propose an anisotropic stress-softening Mullins model for a virgin material that is transversely isotropic.