A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain

Abstract

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 $\ddot{I}ƒ$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ] = \frac{\rm d}{{\rm d}{t}}[σ] - W \, σ+ σ\, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F=\mathrm{D}\ddot{I}†$, the left Cauchy-Green tensor $B=F{F}^T$, and the strain-rate tensor $D=\mathrm{sym}(\dot{F}{F}^{-1})$, we show that \begin{equation} \label{eqCPSdef} \begin{alignedat}{2} \forall \,D\in{\rm Sym}(3) \! \setminus \! \{0\}: ~ \langle{\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehatσ(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat σ(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M${}^{++}$)}, \end{alignedat} \tag{1} \end{equation} where ${\mathrm{Sym}}_4^{++}(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the \textbf{T}rue-\textbf{S}tress \textbf{T}rue-\textbf{S}train strict Hilbert-\textbf{M}onotonicity (TSTS-M${}^{+}$) for $B\mapsto \ddot{I}ƒ(B)=\hat{\ddot{I}}ƒ(\log B)$, i.e. \begin{equation} \langle \widehatσ(\log B_1)-\widehatσ(\log B_2),{\log B_1-\log B_2} \rangle> 0 \qquad \forall \, B_1\neq B_2\in{\rm Sym}^{++}(3) \, . \end{equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case.