Micromechanics of composites/Proof 13

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