Do we need Truesdell's empirical inequalities? On the coaxiality of stress and stretch

TL;DR

This paper questions the necessity of Truesdell's empirical inequalities in nonlinear elasticity, exploring weaker conditions and their implications for stress and stretch relations, and clarifies common misconceptions about matrix eigenvector properties.

Contribution

It analyzes the relationships between empirical inequalities, weaker constitutive conditions, and bi-coaxiality, offering insights into their roles in stress-stretch relations in elasticity.

Findings

Weaker constitutive conditions can replace empirical inequalities in certain contexts.

The phrase 'same eigenvectors' often leads to misconceptions about tensor commutativity.

Connections between bi-coaxiality and matrix properties are clarified.

Abstract

Truesdell's empirical inequalities are considered essential in various fields of nonlinear elasticity. However, they are often used merely as a sufficient criterion for semi-invertibility of the isotropic stress strain-relation, even though weaker and much less restricting constitutive requirements like the strict Baker-Ericksen inequalities are available for this purpose. We elaborate the relations between such constitutive conditions, including a weakened version of the empirical inequalities, and their connection to bi-coaxiality and related matrix properties. In particular, we discuss a number of issues arising from the seemingly ubiquitous use of the phrase "$X,Y$ have the same eigenvectors" when referring to commuting symmetric tensors $X,Y$.