Virtual configurations and constitutive equations for residually stressed bodies with material symmetry

The purpose of this paper is to construct the general forms for constitutive equations for residually stressed hyperelastic bodies that are composed of material with a specified symmetry. The effective mechanical properties of a material comprising a residually stressed body are dependent on both the intrinsic material properties of the underlying natural material and on the residual stress field itself. Thus, prediction of the mechanical behavior of a residually stressed body typically requires a constitutive model that explicitly includes the influence of the residual stress for deformations out of the residually stressed configuration. In this paper, constitutive equations are derived for the cases where the underlying natural material is transversely isotropic or orthotropic, two material symmetries of special interest in the description of biological tissues. In addition, it is established that the method used in the derivation can be applied to materials with any symmetry for which an integrity basis is known. The derivation requires that the constitutive equation for the underlying natural material be known and that the restriction of the response function to the set of positive definite symmetric tensors be locally invertible. The resulting constitutive equations are expressed as functions of the residual stress and the deformation gradient out of the residually stressed configuration. A major advantage of the method is that the only material parameters in the constitutive equation are those of the underlying stress free material.