The paper is dedicated to the asymptotic behavior of $\varepsilon$ ε -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ ε → 0 . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ 3 D to $2D$ 2 D or $1D$ 1 D 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 $\varepsilon \to 0$ ε → 0 we use the periodic unfolding method.

In this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

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In our previous papers (Griso et al. in J. Elast. 141:181–225, 2020; J. Elast., 2021, 10.1007/s10659-021-09816-w), we considered thick periodic structures (first paper) and thin stable periodic structures (second paper) made of small cylinders (length of order $\varepsilon $ ε and cross-sections of radius $r$ r ). In the first paper $r=\kappa \varepsilon $ r = κ ε with $\kappa $ κ a fixed constant, $\varepsilon \to 0$ ε → 0 , while in the second $\varepsilon \to 0$ ε → 0 and ${r/ \varepsilon }\to 0$ r / ε → 0 . In this paper, our aim is to give the asymptotic behavior of thin periodic unstable structures, when $\varepsilon \to 0$ ε → 0 , ${r/ \varepsilon }\to 0$ r / ε → 0 and $\varepsilon ^{2}/ r\to 0$ ε 2 / r → 0 .Our analysis is again based on decompositions of displacements. As for stable periodic structures, Korn type inequalities are proved. Several classes of unstable and auxetic structures are introduced. The unfolding and limit homogenized problems are really different of those obtained for the thin stable periodic structures. The limit homogenized operators are anisotropic, the spaces containing the macroscopic limit displacements depend on the periodicity cells. It was not the case in the two previous studies. Some examples are given.

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