On the relationship between the logarithmic strain rate and the stretching tensor

In this paper we investigate the relationship between the stretching tensor 2 and the logarithmic (Hencky) strain Any, with V the left stretch tensor. We establish the simple formula (AnV)- sym (j3l), which holds for arbitrary three-dimensional motions. Here F is the deformation gradient, (1n))0 is the time derivative of Any measured in a coordinate system which rotates with the left principal strain axes, and Nr is the spin of the right principal strain axes. we use this formula to show that 2- (tnV), (or, equivalently, p- (Int) * , the Jaumann derivative of ln,V) , if and only if the characteristic spaces of the right stretch tensor are constant on any time interval in which the number of distinct principal stretches is constant. Finally, we show that the asymptotic approximation j-(.) * + O(e3 holds whenever the displacement gradient! satisfies ow.- 0(1)