Major symmetry of the induced tangent stiffness tensor for the Zaremba-Jaumann rate and Kirchhoff stress in hyperelasticity: two different approaches

We recall in this note that the induced tangent stiffness tensor ${\mathbb{H}}_{\tau }^{\text{ZJ}}(\tau )$ appearing in a hypoelastic formulation based on the Zaremba-Jaumann corotational derivative and the rate constitutive equation for the Kirchhoff stress tensor $\tau$ is minor and major symmetric if the Kirchhoff stress $\tau$ is derived from an elastic potential $\mathrm{W}(F)$. This result is vaguely known in the literature. Here, we expose two different notational approaches which highlight the full symmetry of the tangent stiffness tensor ${\mathbb{H}}_{\tau }^{\text{ZJ}}(\tau )$. The first approach is based on the direct use of the definition of each symmetry (minor and major), i.e., via contractions of the tensor with the deformation rate tensor $D$. The second approach aims at finding an absolute expression of the tensor ${\mathbb{H}}_{\tau }^{\text{ZJ}}(\tau )$, by means of special tensor products and their symmetrisations. In some past works, the major symmetry of ${\mathbb{H}}_{\tau }^{\text{ZJ}}(\tau )$ has been missed because not all necessary symmetrisations were applied. The approach is exemplified for the isotropic Hencky energy. Corresponding stability checks of software packages are shortly discussed.