Derivative of Metric Tensor

Hi Mohsen,

I see you are a great fan of forth-order tensors :) The computation of the derivative of the square root is a very tough thing, isn't it?

The question is again of course too general. I suppose you have two distinct configurations of the same body and want to compute some "geometrical" stiffness tensor. Unfortunately, I don't see any possibility how to do it very shortly. Please, consult the following works:

J.E.Marsden, T.J.R.Hughes Mathematical foundations of elasticity. 1983

J.C.Simo, J.E.Marsden On the rotated stress tensor and the material version of the Doyle-Ericksen formula. 1984

M.Itskov The derivative with respect to a tensor: some theoretical aspects and applications. 2002

O.Kintzel Fourth-order tensors - tensor differentiation with application to continuum mechanics. Part II: Tensor analysis on manifolds. 2005

M.Itskov Tensor algebra and tensor analysis for engineers. 2009

Although I didn't find there exactly what you need, I think these works will be of some interest for you, especially those of Jerry Marsden and co-authors. In the works of Itskov you will find statements like "the derivative with respect to the metric tensor seems to be meaningless since in the Euclidean space g represents the identity tensor", but I wouldn't take them too seriously. Yes, it's true in each tangent space the metric tensor should act as the identity map, but a-priori it is not clear how the information about the global structure (the curvature of the manifold) may influence the derivative.

Svyatoslav

Dear Mohsen and Svyatoslav,

One situation in which a relationship like the R field being a 'function' of the C field arises is when, say in a simply-connected open set around  a point x of a reference, one considers a smooth deformation of the open set and the corresponding R and C fields of the resulting deformation gradient. In this case, the C *field* determines the R* field* up to a spatially uniform orthogonal tensor field. So, if one addtionally requires the value of the R field to be specified at a given point in the open set, then this situation results in the R field being uniquely determined by the C field (but only in this compatible case).

But even in this situation, the value of the constructed R field at a given point can only be considered as a function on the constrained space (perhaps an infinite dimensional manifold) of symmetric +ve definite tensor *fields*  that satisfy the finite-strain compatibility conditions on the open set. So, at best, one can think of doing a derivative of a function on an infinite dimensional space, which would  be a very complicated animal - the 'function' in this case is essentially a second order, nonlinear, differential operator.

I would be very surprised if someone could show me a function A that relates R(X) = A( C(X) ), where A is fixed over all X in some open ball and for all deformations of the open ball whose deformation gradient F pointwise yields the polar decomposition F(X) = R(X)U(X), where as usual U(X) is the tensor quare root of C(X).

May be this is of help in your discussion.

- Amit