Homogenization of Perforated Elastic Structures
Georges Griso, Larysa Khilkova, Julia Orlik, O. A. Sivak
Abstract
Abstract The paper is dedicated to the asymptotic behavior of $\varepsilon$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:math> . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> <mml:mi>D</mml:mi> </mml:math> to $2D$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> <mml:mi>D</mml:mi> </mml:math> or $1D$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mi>D</mml:mi> </mml:math> 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 $\varepsilon \to 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:math> we use the periodic unfolding method.