Two-scale modelling of micromorphic continuaA numerical homogenization scheme

According to their peculiar mechanical properties, the description of cellular materials is of high interest. Modelling aspects to be considered are, e.g. pronounced size depending boundary layer effects as well as a deformation-driven evolution of anisotropy or porosity. In the present contribution, we pay special attention to the description of size-dependent microtopological effects on the one hand. On the other hand, we focus on the relevance of extended continuum theories describing the local deformation state of microstructured materials. We, therefore, introduce a homogenization scheme for two-scale problems replacing a heterogeneous Cauchy continuum on the microscale by a homogeneous effective micromorphic continuum on the macroscale. The transitions between both scales are obtained by appropriate projection and homogenization rules which have to be derived, on the one hand, by kinematic assumptions, i.e. the minimization of the macroscopic displacement field, and, on the other hand, by energetic considerations, i.e. the evaluation of an extended Hill–Mandel condition.