In the present paper the homogenization problem of periodic composites is investigated, in the case of a Cosserat continuum at the macro-level and a Cauchy continuum at the micro-level. In the framework of a strain-driven approach, the two levels are linked by a kinematic map based on a third order polynomial expansion. The determination of the displacement perturbation fields in the Unit Cell (UC), arising when second or third order polynomial boundary conditions are imposed, is investigated. A new micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is proposed. The identification of the linear elastic 2D Cosserat constitutive parameters is performed, by using a Hill-Mandel-type macrohomogeneity condition. The influence of the selection of the UC is analyzed and some critical issues are outlined. Numerical examples referred to a specific composite with cubic symmetry are shown.
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