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

Abstract

We recall in this note that the induced tangent stiffness tensor $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ appearing in a hypoelastic formulation based on the Zaremba-Jaumann corotational derivative and the rate constitutive equation for the Kirchhoff stress tensor τ is minor and major symmetric if the Kirchhoff stress τ is derived from an elastic potential $W\left(F\right)$ . 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 $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ . 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 $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ , by means of special tensor products and their symmetrisations. In some past works, the major symmetry of $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ has been missed because not all necessary symmetrisations were applied. The analogous tangent stiffness tensor $H^{\mathrm{ZJ}}\left(\sigma \right)$ , relating the Cauchy stress tensor σ to the Zaremba-Jaumann corotational derivative is also obtained, with both methods used for $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ . The approach is exemplified for the isotropic Hencky energy. Corresponding stability checks of software packages are shortly discussed.