Micromechanics of composites/Average stress in a RVE

Average Stress in a RVE

edit

Let the average stress in the RVE be defined as

 

where   is the volume of  .

We would like to find out the relation between the average stress in a RVE and the applied tractions on the boundary of the RVE. To do that, recall the relation (see Appendix)

 

If we choose   such that  , we have

 

Therefore,

 

If we choose   to be the stress tensor  , and involve the symmetry of the stress tensor, we get

 

Now, the divergence of the stress is zero (from the conservation of linear momentum). Therefore,

 

Using the traction boundary condition, we have

 

Now   if  . Therefore, we have

 

Hence the average stress is given by

 

This implies that the average stress is completely determined by the applied tractions!

Symmetry of the average stress and the effect of rigid body translation

edit

Let us now assume that the applied tractions are self equilibrating. Then the resultant forces and moments due to the applied tractions vanish and we have

 

From the moment balance equation above we can show that (see Appendix)

 

Therefore the average stress tensor   is symmetric if the applied tractions are self equilibrated.

Now, if we translate the body by a constant amount   (rigid body translation), we get

 

or

 

Therefore, the average stress is not affected by a rigid body translation only if the applied tractions are self equilibrated.

We can conclude that the average stress   is an acceptable measure of stress in a RVE if the applied tractions are self equilibrated.