On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues

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The paper discusses relations related to the dual variables of the Cauchy stress tensor and the logarithmic strain tensor. We give a new proof that the logarithmic strain is dual to the Cauchy stress tensor in the isotropic hyperelastic case. Corresponding relations are given in the case of multiplicative finite strain elasto-viscoplasticity for the material Eshelby-like stress tensor and the rotated material stress tensor as well. The general case of anisotropic response is discussed. The dual variable of the logarithmic strain tensor is derived and some new expressions are given. Specifically it is proven that the skew-symmetric part of the Eshelby tensor is determined by the commutator of the logarithmic strain tensor and its conjugate variable. The question of the existence of dual variables for the Cauchy stress tensor and its material counterparts is rigorously answered, where it is shown that, in the general case, a dual variable does not exist.