The influence of the invariant I8I8 on the stress–deformation and ellipticity characteristics of doubly fiber-reinforced non-linearly elastic solids

In this paper we analyze some aspects of the mechanics of anisotropic non-linearly elastic solids for which the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement. Particular attention is focused on the role of the invariant I8I8, which couples the fiber kinematics in the constitutive law, and results are illustrated for a model strain–energy function that includes this invariant. The distinctive effects of this invariant on the stress–strain behaviour of the material and on the failure of ellipticity are discussed separately for pure homogeneous deformation and for a simple shear deformation. For pure homogeneous deformation it is shown that ellipticity fails with an associated weak surface of discontinuity that bisects the fiber directions. Moreover, ellipticity fails close to the reference configuration in either tension or compression. For simple shear it is found that loss of ellipticity occurs both for positive and negative shearing, and the orientation of the weak surface relative to the fiber directions is characterized in each case. Again ellipticity fails close to the reference configuration, and inclusion of the invariant I8I8 in the constitutive law therefore has a pronounced destabilizing influence.