Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba–Jaumann objective derivative of the Cauchy stress $\sigma $ , i.e. \(\begin{aligned} \frac{\mathrm {D}^{\operatorname{ZJ}}}{ \mathrm {D}t}[\sigma ] = \frac{\mathrm {D}}{\mathrm {D}t}[\sigma ] - W \, \sigma + \sigma \, W, \qquad W = \mbox{skew}(\dot{F} \, F^{-1}) \end{aligned}\) and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F = \mathrm {D}\varphi $ , the left Cauchy-Green tensor $B = F \, F^{T}$ , and the strain-rate tensor $D = \operatorname{sym}(\dot{F} \, F^{-1})$ , we show that 1 \(\begin{aligned} & \forall \,D\in \operatorname{Sym}(3) \! \setminus \! \{0\}: ~ \left \langle \frac{\mathrm {D}^{\operatorname{ZJ}}}{ \mathrm {D}t}[\sigma ],D\right \rangle > 0 \\ & \quad \iff \quad \log B \longmapsto \widehat{\sigma}(\log B) \; \textrm{is strongly Hilbert-monotone} \\ &\quad \iff \quad \operatorname{sym} \mathrm {D}_{\log B} \widehat{\sigma}(\log B) \in \operatorname{Sym}^{++}_{4}(6) \quad \text{(TSTS-M$^{++}$)}, \end{aligned}\) where $\operatorname{Sym}^{++}_{4}(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality of (1) “corotational stability postulate” (CSP), a novel concept, which implies the True-Stress True-Strain strict Hilbert-Monotonicity (TSTS-M+) for $B \mapsto \sigma (B) = \widehat{\sigma}(\log B)$ , i.e. \( \left \langle \widehat{\sigma}(\log B_{1})-\widehat{\sigma}(\log B_{2}), \log B_{1}-\log B_{2}\right \rangle > 0 \qquad \forall \, B_{1}\neq B_{2} \in \operatorname{Sym}^{++}(3) \, . \) A similar result, but for the Kirchhoff stress $\tau = J \, \sigma $ has been shown by Hill as early as 1968. Leblond translated this idea to the Cauchy stress $\sigma $ but only for the hyperelastic case. In this paper we expand on the ideas of Hill and Leblond, extending Leblond calculus to the Cauchy elastic case.
