AVERAGING PROPERTIES FOR PERIODIC HOMOGENIZATION AND LARGE DEFORMATION

The main motivation of this paper consists of using the periodic homogenization theory to derive several relations between macroscopic Lagrangian (e.g., deformation gradient, Piola-Kirchhoff tensor) and Eulerian (e.g., velocity gradient, Cauchy stress) quantities. These relations demonstrate that these macroscopic quantities behave formally in the same way as their microscopic counterparts. We say therefore that these relations are stable with respect to the periodic homogenization. We also demonstrate the equivalence between the two forms of the macroscopic power density expressed in the Lagrangian and Eulerian formulations. Two simple examples illustrate these results, and indicate also that the Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor are not stable with respect to periodic homogenization.