Micromechanics of composites/Proof 3

Surface-volume integral relation 1

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Let   be a body and let   be its surface. Let   be the normal to the surface. Let   be a vector field on   and let   be a second-order tensor field on  . Show that

 

Proof:

Recall the relation

 

Integrating over the volume, we have

 

Since   and   are constant, we have

 

From the divergence theorem,

 

we get

 

Using the relation

 

we get

 

Since   and   are constant, we have

 

Therefore,

 

Since   and   are arbitrary, we have