Micromechanics of composites/Average stress in a RVE with finite strain

Average stress in a RVE

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The average nominal (first Piola-Kirchhoff ) stress is defined as

 

Recall the relation (see Appendix)

 

In the above equation, let the volume integral be over   and let the surface integral be over  . Let the unit outward normal to   be  . Let the gradient and divergence operations be with respect to the reference configuration. Also, let   and let  . Then we have

 

If we assume that there are no inertial forces or body forces, then   (from the conservation of linear momentum), and we have

 

Let   be a self equilibrating traction that is applied to the RVE, i.e., it does not lead to any inertial forces. Then, Cauchy's law states that   on  . Hence we get

 

Given the above, the average Cauchy stress in the RVE is defined as

 

Note that, in general,  .

The Kirchhoff stress is defined as  . The average Kirchhoff stress in the RVE is defined as

 

In general,  .