Macrohomogeneity condition in dynamics of micropolar media

We determine the macrohomogeneity (Hill-Mandel type) condition in the dynamic response of inhomogeneous micropolar (Cosserat) materials. The setting calls for small deformation gradients and curvatures, but without restrictions on the constitutive behavior and without any requirements of spatial periodicity. The condition gives admissible boundary loadings, along with extra terms representing kinetic energy contributions of both classical type and micropolar type. The said loadings involve various combinations of average stresses and strains, along with couple-stresses and curvature-torsion tensors. If applied to a specific microstructure in a computational mechanics approach, these boundary loadings will allow one to determine scale-dependent homogenization toward a representative volume element (RVE) of an equivalent homogeneous micropolar medium in either elastic or inelastic settings. By restricting the continuum model to an inhomogeneous Cauchy continuum and/or a quasi-static setting, the macrohomogeneity condition simplifies to conventional versions.