Homogenization of Elastic Media with Gaseous Inclusions

We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period $\varepsilon > 0$, and a perfect gas placed in each of these perforations, as $\varepsilon$ goes to zero. The model we use is linear, corresponding to deforma...

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Bibliographic Details
Published in:Multiscale modeling & simulation 2008-01, Vol.7 (1), p.432-465
Main Authors: Baffico, L., Grandmont, C., Maday, Y., Osses, A.
Format: Article
Language:English
Subjects:
Analysis of PDEs
Applied mathematics
Composite materials
Deformation
Geometry
Homogenization
Mathematics
Navier-Stokes equations
Respiratory system
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Summary:We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period $\varepsilon > 0$, and a perfect gas placed in each of these perforations, as $\varepsilon$ goes to zero. The model we use is linear, corresponding to deformations around a reference configuration. We apply both two-scale asymptotic expansion and two-scale convergence methods in order to identify the limit behaviors as $\varepsilon$ goes to 0. We state that in the limit, we get a two-scale linear elasticity-like boundary value problem. From this problem, we identify the corresponding homogenized and periodic cell equations which allow us to find the first corrector term. The analysis is performed both in the case of an incompressible and a compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we calculate numerically the homogenized coefficients in the incompressible case for different types of elastic materials.
ISSN:1540-3459
1540-3467
DOI:10.1137/070705714