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Derivative of Logarithmic Strain
Some of you probably work on problems that involve moderately large strains. An useful strain measure for such problems in the logarithmic or Hencky strain. In particular, if you deal with the numerics of large strain simulations, you will often need to compute the material time derivatives of logarithmic strains.
In the mid 1980s, Anne Hoger , Donald Carlson , Morton Gurtin and their coworkers wrote a series of papers on issues involving such strain measures, how to take their derivatives, and how to solve linear equations containing tensor quantities. Most of this work originated at the Carnegie Mellon University in Pittsburgh, Pennsylvania.
This post is meant as a reminder that the formulae for the material derivatives of the logarithmic strain depend on the number of independent eigenvalues of the stretch tensors. It should also act as a quick reference for looking up some of those formulae.
Recall that the deformation gradient tensor, because it has a positive determinant, can be multiplicatively decomposed into a symmetric part and an orthogonal part, i.e.,
In terms of the stretches, the generalized n-th order material and spatial measures of strain are
The spectral decompositions of the stretches can be written as
Therefore we may also write the strain measures as
If you take the limit of these as n goes to zero, you will get the material and spatial logarithmic strains (prove this for fun)
Note that both the material and spatial strains have the same eigenvalues in this case.
Recall that the Cauchy stress and the rate of deformation tensor (better called the stretching tensor?) are power conjugate. So the question that arises is: what quantity is power conjugate to the material derivative of the Hencky strain, i.e., find the stress measures such that
It turns out that when the principal axes of strain remain fixed then
However, this is not true in general as the following formulae from  show. The detailed expressions are quite complicated and you can find them in Hoger's paper.
Please comment if you know of any further simplifications and developments in this regard in the last 20 years or if you need further clarification on any of the quantities used in these equations.