Dear Colleagues:
I thought the following recent paper by Prof. Neff may be of interest to some of you.
http://arxiv.org/abs/1505.02203
This paper discusses the natural appearance of the Hencky-logarithmic strain tensor together with the Hencky strain energy, which can be motivated from some purely geometrical (kinematical) arguments based on the geodesic distance on the general linear group of all invertible tensors GL(n).
If you have any questions please directly email Prof. Neff: patrizio.neff [at] yahoo.de
Regards,
Arash