Comptes Rendus
Homogenization of an elasticity problem in domains with a net of slender bars near surface
Comptes Rendus. Mécanique, Volume 331 (2003) no. 12, pp. 829-834.

We consider an elasticity problem in a domain Ω(ϵ)=ΩF(ϵ), where Ω is an open bounded domain in R3,F(ϵ) is a connected nonperiodic set in Ω like a net of slender bars, and ε is a parameter characterizing the microstructure of the domain. We consider the case of a surface distribution of the set F(ε), i.e., for sufficiently small ε, the set F(ε) is concentrated in arbitrary small neighbourhood of a surface Γ. Under a hypothesis on the asymptotic behaviour of the energy functional, we obtain the macroscopic (homogenized) model.

On étudie un problème d'élasticité dans un domaine Ω(ϵ)=ΩF(ϵ), où Ω est un ouvert borné dans R3, F(ε) est un ensemble non-périodique connexe dans Ω de type un réseau de barres fines et ε est un paramétre qui caractérise la microstructure du domaine. On considère le cas d'une distribution surfacique de l'ensemble F(ε), c'est-á-dire, lorsque ε→0, cet ensemble se concentre dans un voisinage d'une surface Γ aussi petit que l'on veut. Sous une hypothèse sur le comportement asymptotique de la fonctionelle d'énergie on obtient le modèle macroscopique (homogénéisé).

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Accepted:
Published online:
DOI: 10.1016/j.crme.2003.09.005
Keywords: Computational solid mechanics, Elasticity problem
Mot clés : Mécanique de solides numérique, Problème d'élasticité

Mariya Goncharenko 1, 2; Leonid Pankratov 1, 2

1 Laboratoire Jacques-Louis Lions de l'Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France
2 Département de mathématiques, B.Verkin institut des basses températures (FTINT), 47, av. Lénine, 61103 Kharkov, Ukraine
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Mariya Goncharenko; Leonid Pankratov. Homogenization of an elasticity problem in domains with a net of slender bars near surface. Comptes Rendus. Mécanique, Volume 331 (2003) no. 12, pp. 829-834. doi : 10.1016/j.crme.2003.09.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2003.09.005/

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[2] D. Cioranescu; J. Saint Jean Paulin Homogenization of Reticulated Structures, Applied Mathematical Sciences, 136, Springer-Verlag, New York–Berlin–Heidelberg, 1999

[3] E. Khruslov; V. Marchenko Boundary Value Problems in Domains with Fine-Grained Boundaries, Naukova Dumka, 1974 (in Russian)

[4] É. Sanchez–Palencia Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys., 127, Springer-Verlag, New York, 1980

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