Stress tensors associated with deformation tensors via duality

Abstract

The concept of dual variables, initially introduced by HAUPT and TSAKMAKIS [3], enables us to relate to each other strain and stress tensors, as well as associated rates, independently of particular material properties. Generally, it is different than the method of conjugate variables, as defined e.g. by MAcVEAN [2] or HILL [4-6]. The duality concept postulated by HAUPT and TSAKMAKIS [3] deals only with two classes of dual stress and strain tensors. The second Piola-Kirchhoff stress tensor and the Green strain tensor, as well as the negative convected stress tensor and the Piola strain tensor, are respectively the Lagrangean stress and strain tensors included in the two classes of dual stress and strain tensors. However, there are further (infinitely many) Lagrangean stress and strain tensors, which may be taken into consideration. The aim of the present paper is to develop further the concept of dual variables to take into account the whole set of Lagrangean stress and strain tensors. Doing this, we obtain a specific mathematical structure in the sets of all strain and stress tensors, which makes it possible to relate strain and stress tensors, as well as associated rates, independently of the particular material properties.