The paper is dedicated to the asymptotic behavior of $\varepsilon$ -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ . 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 $\varepsilon \to 0$ we use the periodic unfolding method.

中文翻译：

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穿孔弹性结构的均质化

该论文致力于 $\varepsilon$ - 周期性穿孔弹性（3 维，板状或梁状）结构的渐近行为，如 $\varepsilon \to 0$ 。在板状或梁状结构的情况下，尺寸从 $3D$ 逐渐减少到 $2D$ 或 $1D$ 分别发生。所考虑的结构的一个例子可以通过周期性重复基本的“扁平”球或圆柱体来获得，用于板状或梁状结构，使得两个相邻球/圆柱体之间的接触表面具有非零措施。由于结构占据的域可能具有非 Lipschitz 边界，因此不能使用基于扩展的经典均质化方法。因此，为了获得用于推导先验估计的 Korn 不等式，我们使用基于插值的方法。在板状和梁状结构的情况下，Korn 不等式的证明分别基于板或梁的位移分解。为了将极限作为 $\varepsilon \to 0$ 传递到极限，我们使用周期性展开方法。