The paper is dedicated to the asymptotic behavior of e-periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as eRT0. In case of plate-like or beam-like structures the asymptotic reduction of dimension from 3D to 2D or 1D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary ""flattened"" ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn's inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn's inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as eRT0 we use the periodic unfolding method.