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Micromechanics of composites/Proof 13

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Question

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Let be the Cauchy stress and let be the velocity gradient in a body with boundary . Let be the normal to the boundary. Let be the volume of the body. If the skew-symmetric part of the velocity gradient is zero, i.e., , or if the stress field is self equilibrated, i.e., , show that

Proof

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Taking the trace of each term in the identity

the difference between the average stress power and the product of the average stress and the average velocity gradient can be written as (using either the symmetry of the stress or of the velocity gradient)

Recall that

Also, from the divergence theorem

Therefore,

Since and are independent of , we can take these inside the integrals to get

Using the identity

we get

Also, using the identity

we get

Since , we have (we could alternatively use the symmetry of to arrive at the following equation). Hence,

Plugging these back into the original equation, we have

Hence