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

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 $X,Y$.

References in corpus (9)

• Methodical Fitting for Mathematical Models of Rubber-like Materials
• Geometry of logarithmic strain measures in solid mechanics
• Simple shear is not so simple
• Shear, pure and simple
• The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers
• Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations
• Injectivity of the Cauchy-stress tensor along rank-one connected lines under strict rank-one convexity condition
• On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity
• Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations

Cited by in corpus (4)

• Shear, pure and simple
• Stretch formulations and the Poynting effect in nonlinear elasticity
• Modelling of initially stressed solids: structure of the energy density in the incompressible limit
• A commented translation of Hans Richter's early work "The isotropic law of elasticity"