Generalized proper states for anisotropic elastic materials

Abstract

The main aim of this paper is to determine all the unit stresses ω (ω · ω = 1) for which the stored elastic energy Φ(ω) has the local extrema in some classes of stresses. Our consideration is restricted to two classes: Κ₁  – uniaxial tensions and then the directions for which the Young modulus assumes its extremal value are determined, and Κ₂ – pure shears in physical space. The problem is then reduced to the determination of the planes of minimal and maximal shear modulus. The idea of a generalized proper state for Hooke's tensor is introduced. It is shown that a mathematical treatment of the considered problem comes down to the problem of the generalized proper elastic states for the compliance tensor C. The problem has been effectively solved for cubic symmetry.