Micromechanics of composites/Proof 14

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,