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

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. •Analytical results in nonlinear elasticity often require semi-invertibility of the Cauchy stress response.•The empirical inequalities ensure semi-invertibility but exclude many elastic models.•Weaker, mechanically motivated conditions sufficient for semi-invertibility are introduced.