Homogenization of an elastic media with gaseous inclusions

Léonardo Baffico (1) , Céline Grandmont (2) , Yvon Maday (3) , Axel Osses (4)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 REO - Numerical simulation of biological flows
3 LJLL - Laboratoire Jacques-Louis Lions
4 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]

We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period ε > 0, and a perfect gas placed in each of these perforations, as ε 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 ε 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 allows us to find the first corrector term. The analysis is performed both in the case of an incompressible and compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we numerically calculate the homogenized coefficients in the incompressible case, for different types of elastic materials.

Mots clés: Mathematical modeling; Periodic homogenization; Asymptotic analysis; Fluidstructure interaction.; Fluidstructure interaction

Domaines: Equations aux dérivées partielles [math.AP]

Léonardo Baffico, Céline Grandmont, Yvon Maday, Axel Osses. Homogenization of an elastic media with gaseous inclusions. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2008, 7 (1), pp.33. ⟨inria-00180307⟩