Continuum mechanics/Relations between surface and volume integrals

Surface-volume integral relation 1Edit

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

 


Surface-volume integral relation 2Edit

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

 

Proof:

Recall that

 

where   is any second-order tensor field on  . Let us assume that  . Then we have

 

Now,

 

where   is any second-order tensor. Therefore,

 

Rearranging,