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 \(\sigma \), 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

$$\begin{aligned} & \forall \,D\in \operatorname{Sym}(3) \! \setminus \! \{0\}: ~ \left \langle \frac{\mathrm {D}^{\operatorname{ZJ}}}{ \mathrm {D}t}[\sigma ],D\right \rangle > 0 \\ & \quad \iff \quad \log B \longmapsto \widehat{\sigma}(\log B) \; \textrm{is strongly Hilbert-monotone} \\ &\quad \iff \quad \operatorname{sym} \mathrm {D}_{\log B} \widehat{\sigma}(\log B) \in \operatorname{Sym}^{++}_{4}(6) \quad \text{(TSTS-M$^{++}$)}, \end{aligned}$$

(1)

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.