Homogenization of Perforated Elastic Structures
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Homogenization of Perforated Elastic Structures Georges Griso1,2 · Larysa Khilkova2 · Julia Orlik2 · Olena Sivak2
Received: 6 September 2019 © The Author(s) 2020
Abstract The paper is dedicated to the asymptotic behavior of ε-periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as ε → 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 ε → 0 we use the periodic unfolding method. Keywords Homogenization · Periodic unfolding method · Dimension reduction · Linear elasticity · Perforated non-Lipschitz domains · Plate · Beam · Extension operators · Korn’s inequalities Mathematics Subject Classification (2010) 35B27 · 35J50 · 47H05 · 74B05 · 74K10 · 74K20
B G. Griso
[email protected] L. Khilkova [email protected] J. Orlik [email protected] O. Sivak [email protected]
1
Laboratoire Jacques-Louis Lions (LJLL), Sorbonne Université, CNRS, Université de Paris, 75005 Paris, France
2
Department SMS, Fraunhofer ITWM, 1 Fraunhofer Platz, 67663 Kaiserslautern, Germany
G. Griso et al.
1 Introduction This paper deals with the linearized elasticity problem posed in different periodic domains. These domains are obtained by reproducing a representative cell of size ε in such a way that one can get beam-like, plate-like or N -dimensional structures. It is assumed that a part of their exterior boundary denoted by Γε is fixed. The ε-cells are made of elastic materials. The reference cell is denoted by C (Fig. 1). We assume that C has a Lipschitz boundary and that the interior of the closure of the union of two contiguous cells is connected. Under these assumptions, the whole periodic structure might have a non-Lipschitz boundary. Throughout this article, the cell C is included in the unit parallelotope of RN (resp. R3 ), and one can replace this parallelotope by any bounded domain having the paving property with respect to a discrete group of rank N (resp. 3). Our aim is to investigate the asymptotic behavior of these elastic periodic structures as ε tends to 0. Since these structures might be non-Lipschitz, one of the main difficulties is to obtain a priori estimates. The classical extension approach (see [25])
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