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Thread: Tensor partial differentiation question...

  1. #1
    Junior Member
    Joined
    Apr 2009
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    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
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  2. #2
    Junior Member
    Joined
    May 2010
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    Texas
    Posts
    48
    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
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