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

TL;DR

This paper establishes a new constitutive condition for isotropic nonlinear elasticity involving the logarithmic strain, linking the corotational derivative of the Cauchy stress to Hilbert-monotonicity properties of the stress-strain relation.

Contribution

It introduces the corotational stability postulate (CSP) and connects it to strict Hilbert-monotonicity of the stress-strain relation, extending previous work to the Cauchy elastic case.

Findings

Defines the CSP as a novel stability criterion.

Shows equivalence between CSP and TSTS-M++ monotonicity.

Extends Leblond's calculus to Cauchy elasticity.

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 $\sigma$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[\sigma] = \frac{{\rm d}}{{\rm d}{t}}[\sigma] - W \, \sigma + \sigma \, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} \varphi$, the left Cauchy-Green tensor $B = F \, F^T$, and the strain-rate tensor $D = {\rm sym}(\dot F \, F^{-1})$, 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}[\sigma]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehat\sigma(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat \sigma(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M$^{++}$)}, \end{alignedat} \tag{1} \end{equation} where ${\rm Sym}^{++}_4(6)$ 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 $B \mapsto \sigma(B) = \widehat \sigma(\log B)$, i.e. \begin{equation} \langle \widehat\sigma(\log B_1)-\widehat\sigma(\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.