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

TL;DR

This paper studies objective corotational rates in continuum mechanics, identifying conditions for positivity and structure preservation, and highlights their significance in modeling material behavior.

Contribution

It introduces conditions for positive definite corotational rates, including a geometric perspective, and distinguishes structure-preserving rates from general objective stress rates.

Findings

Well-known corotational rates are part of the positive family.

Explicit conditions for positivity of corotational rates are provided.

Structure-preserving properties of these rates are emphasized.

Abstract

We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for \begin{equation*} \frac{{\rm D}^{\circ}}{{\rm D} t}[B] = \mathbb{A}^{\circ}(B).D \end{equation*} that $\mathbb{A}^{\circ}(B)$ is positive definite. Here, $B = F \, F^T$ is the left Cauchy-Green tensor, $\frac{{\rm D}^{\circ}}{{\rm D}t}$ is a specific objective corotational rate, $D = {\rm sym} \, {\rm D} v$ is the Eulerian stretching and $\mathbb{A}^{\circ}(B)$ is the corresponding induced fourth order tangent 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 $\frac{{\rm D}^{\circ}}{{\rm D} t}$ we determine several conditions characterizing positivity. Among them an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers (2004). We also give a geometrical motivation for invertibility and positivity 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.