Eduard Rohan, University of West Bohemia, Pilsen, Czech Republic Joint work with Vladimír Lukeš The two-scale homogenization is well suited for modeling periodic media described by linear PDEs. Nonlinear problems can also be treated, however, the separation of the local autonomous problems from the global ones relevant to the upscaled medium is not possible, in general. For problems related to deforming media, a linearization is always used to solve the problem using an incremental procedure, it is therefore natural to consider such an incremental problem for homogenization. We propose a homogenized model which is based upon the Eulerian rate formulation for large deforming fluid saturated porous medium. For problems characterized by moderate deformations, a weakly nonlinear homogenized model is proposed which involves linear expansions of the homogenized coefficients using their sensitivity w.r.t. macroscopic fields; the local problems and their sensitivities are solved for the initial configuration. In this case, computational costs are only slightly affected by the two-scale character of the problem, in contrast with solving a fully nonlinear problem requiring subsequent updates of local microstructures and, consequently, solving the local problems at almost any point of the macroscopic domain. Numerical illustrations are given.