Micromechanics of composites/Proof 14
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Question
[edit | edit source]Let be the first Piola-Kirchhoff stress and let be the time rate of the deformation gradient in a body whose reference configuration is with boundary . Let be the normal to the boundary. Let be the volume of the body. Let represent the position of points in the reference configuration. Let be the material time derivative of . Let represent the unweighted volume average of a quantity . Show that
Proof
[edit | edit source]Recall the identity
Therefore,
We want express the volume integrals above in terms of surface integrals. To do that, recall that
Therefore,
Collecting the terms, we have
Therefore,
From the above, clearly
Therefore,
Thus we can alternatively write the expression for the difference as
or,
Hence,