On a Mass Conservation Criterion in Micro-to-Macro Transitions

In order to characterize the macroscopic behavior of a microscopically heterogeneous material, a homogenization methodology is typically employed where a micromechanical sample is identified and tested. See Refs. 1,2,3 4 for details and extensive references. This micromechanical testing (micro-to-macro transition) procedure for solids at finite deformations is governed by certain criteria. The fundamental criterion to be satisfied is a micro-macro work balance in the incremental formthat will be referred to as the work criterion. Here, the macroscopic first Piola–Kirchhoff stress tensor $P¯$ and the macroscopic deformation gradient $F¯$ are the fundamental macroscopic kinetical and kinematical measures that are used to construct all remaining measures via the usual continuum mechanics relationships and they are defined in terms of the volume average of their microscopic counterparts...