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Math Help - Tensor partial differentiation question...

  1. #1
    Junior Member
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    Apr 2009
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    Tensor partial differentiation question...

    Hi I have the following tensor (the Cauchy-Green deformation tensor)

    C = \begin{bmatrix}<br />
\lambda_{1}^2 & 0 & 0 \\<br />
0 & \lambda_{2}^2 & 0 \\<br />
0 & 0 & \lambda_{3}^2 <br />
\end{bmatrix}<br />

    and I have the following energy function

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

    Where

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


    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

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

    D_{1} and D_{2} are constants

    (And 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 \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
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    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 \frac{\partial I_1}{\partial C}.
    So we know that this derivative is a 3 by 3 matrix \frac{\partial trC}{\partial C}=[\frac{\partial trC}{\partial C_{i,j}}].
    And you say you have the derivative of f(\lambda)
    So I think the rest should be doable?

    This may help also
    Matrix Calculus
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