A natural requirement for objective corotational rates—on structure-preserving corotational rates

We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for

$$\begin{aligned} \frac{\textrm{D}^{\circ }}{\textrm{D}t}[B] = \mathbb {A}^{\circ }(B).D \end{aligned}$$

that the characteristic stiffness tensor \(\mathbb {A}^{\circ }(B)\) is positive-definite. Here, \(B = F \, F^T\) is the left Cauchy–Green tensor, \(\frac{\textrm{D}^{\circ }}{\textrm{D}t}\) is a specific objective corotational rate, \(D = {{\,\textrm{sym}\,}}\, D_\xi v\) is the Eulerian stretching and \(\mathbb {A}^{\circ }(B)\) is the corresponding induced characteristic fourth-order stiffness tensor. Well-known corotational rates like the Zaremba–Jaumann rate, the Green–Naghdi rate and the logarithmic rate belong to this family of “positive” corotational rates. For general objective corotational rates \(\frac{\textrm{D}^{\circ }}{\textrm{D}t}\), we determine several conditions characterizing positivity. Among them is an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers [84]. We also give a geometrical motivation for invertibility and positivity of \( \mathbb {A}^{\circ }(B)\) and highlight the structure-preserving properties of corotational rates that distinguish them from more general objective stress rates. Applications of this novel concept are indicated.