Micromechanics of composites/Average displacement in a RVE

Average Displacement in a RVE

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The average displacement in a RVE may be defined as

 

We would like to find the relation between the average displacement in a RVE and the applied displacements at the boundary of the RVE. To do that, recall the identity

 

where   and   are two vector fields.

If we choose   such that   in the above identity, then we can get an equation for  , i.e.,

 

Now,   if  . Therefore,

 

Using the above in the expression for the average displacement, we have

 

Applying the divergence theorem to the first term on the right, we get

 

There are two terms in the above expression: the first is a boundary term while the second requires information from the interior of the body. Hence, in general, the average displacement of a RVE cannot be determined solely on the basis of boundary displacements.

Incompressible materials

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In the material is incompressible, the balance of mass gives us

 

In that case,

 

It's only in this special case that the average displacement in the RVE can be expressed in terms of boundary displacements.