Hi All,
I think that the problem of R (Rotation Tensor) and C (Right Cauchy-Green Tensor) dependence is interesting enough to raise it once again. I have been reading the following paper
R.T.Shield The rotation associated with large strains. 1973
From this paper it can be concluded that if C is constant through the domain of the problem, R will be independent of C. Otherwise R and C vary through the domain with x (position of particle) and it is hard if not possible to compute the derivative of R with Respect to C. It is required to find an explicit relation connecting x to C (inverse of C = C(x)).
Mohsen
Dependence of R on C
Hi Mohsen,
It's me again :) This time I know the answer to your question! All you need is the so-called "nonlinear Cesaro formula". I give you some references where you can find it (may be under some other name):
W.Pietraszkiewicz, J.Badur, Finite rotations in the description of continuum deformation. 1983
A.Chiskis, A generalization of Cesaro's relation for plane finite deformations. 1995
If you don't have access to these papers, please tell me. I'll send them to you immediately on your request. Roughly speaking, if you want to compute x from C, you should learn first noncommutative calculus, particularly, the theory of path-ordered exponentials (sometimes also called matrizants or multiplicative integrals).
It's interesting that in Russia this topic appeared already in 1940! And it was Anatoly Lurie, who gave also very interesting geometrical interpretations in terms of the so-called "affine deformation tensors".
I will be very pleased to someone who could find English or may be German translations (if they ever exist) of the following papers, which I have fortunately in Russian:
A.I.Lurie, Determination of displacements via the given strain tensor (in Russian), PMM USSR, 1940.
V.A.Shamina, Determination of displacement vector from the components of the deformation tensor in nonlinear continuum mechanics (in Russian), MTT, 1974.
M.A.Zak, A generalization of Cesaro's formula (in Russian), MTT, 1976.
With best regards,
Svyatoslav
Cesaro Formula
Hi Svyatoslav,
Thanks for your helpful and interesting comments. I appreciate it so much. I had access to the first paper but I coudn't get the second one (A.Chiskis, A generalization of Cesaro's relation for plane finite deformations. 1995). I would be so grateful if you would sent it to me :) My email address is as follows:
jahanshahi [at] sharif.edu (jahanshahi[at]sharif[dot]edu)
With Kindest Regards
Mohsen