# Tensor partial differentiation question...

• Jul 10th 2010, 12:15 PM
Bwts
Tensor partial differentiation question...
Hi I have the following tensor (the Cauchy-Green deformation tensor)

$\displaystyle C = \begin{bmatrix} \lambda_{1}^2 & 0 & 0 \\ 0 & \lambda_{2}^2 & 0 \\ 0 & 0 & \lambda_{3}^2 \end{bmatrix}$

and I have the following energy function

$\displaystyle W = D_{1}(I_{1}-3) + D_{2}(I_{2}-3) + f(\lambda)$

Where

$\displaystyle I_{1} = tr C = \lambda_{1}^2+\lambda_{2}^2+\lambda_{3}^2$

$\displaystyle I_{2} = {(tr C)^2 - tr C^2 = \lambda_{1}^2\lambda_{2}^2+\lambda_{2}^2\lambda_{3 }^2+\lambda_{3}^2\lambda_{1}^2$

$\displaystyle \lambda_{3} = \frac{1}{\lambda_{1}\lambda_{2}}$

$\displaystyle D_{1}$ and $\displaystyle D_{2}$ are constants

(And $\displaystyle f(\lambda)$ is a piecwise function who's derivitives are already known but I can supply this information if it is needed.)

How do I calculate $\displaystyle \frac{\partial W}{\partial C}$?

Some version of the chain rule?

Thanks for any help in this matter, Nic
• Jul 12th 2010, 09:22 PM
MattMan
Since no one else has posted anything, I figure I'll give it a try.
Note: I'm not the strongest at matrix calculus so take this with a grain of salt.

You're taking the derivative of a scalar function with respect to a matrix (3 x 3). So our result is going to be a (3 x 3) matrix as well.
It looks like the difficulty here is finding $\displaystyle \frac{\partial I_1}{\partial C}$.
So we know that this derivative is a 3 by 3 matrix $\displaystyle \frac{\partial trC}{\partial C}=[\frac{\partial trC}{\partial C_{i,j}}]$.
And you say you have the derivative of $\displaystyle f(\lambda)$
So I think the rest should be doable?

This may help also
Matrix Calculus