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