Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

Abstract

In this paper, general relations between two different stress tensors T f and T g, respectively conjugate to strain measure tensors f( U) and g( U) are found. The strain class f( U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T (0) which is the stress conjugate to the logarithmic strain tensor ln U.