Homogenization of an elasticity problem in domains with a net of slender bars near surface

Mariya Goncharenko; Leonid Pankratov

doi:10.1016/j.crme.2003.09.005

Homogenization of an elasticity problem in domains with a net of slender bars near surface

Presented by: Évariste Sanchez-Palencia

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

Comptes Rendus. Mécanique, Volume 331 (2003) no. 12, pp. 829-834

Abstract (Original language)
Résumé

We consider an elasticity problem in a domain $Ω^{(ϵ)} = Ω ⧹ F^{(ϵ)}$ , where $Ω$ is an open bounded domain in $R^{3}, F^{(ϵ)}$ 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 $Ω^{(ϵ)} = Ω ⧹ F^{(ϵ)}$ , où $Ω$ est un ouvert borné dans $R^{3}$ , 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).

Received: 2003-06-11
Accepted: 2003-09-23
Published online: 2003-11-07

DOI: 10.1016/j.crme.2003.09.005

Keywords: Computational solid mechanics, Elasticity problem
Mots-clés : Mécanique de solides numérique, Problème d'élasticité

Author's affiliations:

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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     title = {Homogenization of an elasticity problem in domains with a net of slender bars near surface},
     journal = {Comptes Rendus. M\'ecanique},
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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

References
Cited by

[1] N.S. Bakhvalov; G.P. Panasenko Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht–Boston–London, 1989

[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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