  Capturing and predicting the effective mechanical properties of highly porous
cellular media still represents a significant challenge for the research
community, due to their complex structural interdependencies and known size
effects. Micromorphic theories are often applied in this context to model the
inelastic deformation behavior of foam-like structures, in particular to
incorporate such size effect into the investigation of structure-property
correlations. This raises the problems of formulating appropriate constitutive
relations for the numerous non-classical stress measures and determining the
corresponding material parameters, which are usually difficult to assess
experimentally. The present contribution therefore alternatively employs a
concurrent micromorphic multi-scale approach within the Direct FE$^2$ framework
to simulate the complex irreversible behavior of foam-like porous solids. The
predictions of Cosserat (micropolar) and a fully-micromorphic theory are
compared with conventional FE$^2$ results and direct numerical simulations
(DNS) for complex loading scenarios with elastic, elastic-plastic, and creep
deformations.
