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Micromechanics of composites/Proof 14

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Question

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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

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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,