The high order homogenization technique generates the so called infinite order homogenized equation. Its coefficients were widely discussed in composite mechanics literature because they are closely related to the so called high order strain gradients theories. However, it was not clear what is the correct mathematical setting for this equation and what are the asymptotically exact boundary conditions. In the present Note we give a variational formulation for the high order homogenized equation by the projection of the initial problem on the “ansatz subspace”. This formulation generates the appropriate boundary conditions for the high order homogenized equation. The error estimates for the solution of the original problem and the homogenized one are obtained.
La technique d'homogénéisation d'ordre élévé mène à l'équation homogénéisée d'ordre élévé. Ses coefficients ont été largement discutés dans la literature de la mécanique des composites parce qu'ils sont liés aux théories des gradients des déformations d'ordre élévé. Neanmoins, la nature mathématique de cette équation n'a pas été complètement clarifiée et les conditions aux limites asymptotiquement exactes n'ont pas été définies. Dans cette Note nous donnons la formulation variationnelle de l'équation homogénéisée d'ordre élévé. Cette formulation est dérivée par la projection du problème initial sur l'espace du dévéloppement asymptotique. Elle engendre les conditions aux limites appropriées pour l'équation homogénéisée d'ordre élévé. L'estimation de la difference entre la solution exacte et la solution approchée est obtenue.
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Mots-clés : Homogénéisation, Equation homogénéisée d'ordre élévé, Conditions aux limites d'ordre élévé, Théories des gradients des déformations
Grigory Panasenko 1
@article{CRMECA_2009__337_1_8_0,
author = {Grigory Panasenko},
title = {Boundary conditions for the high order homogenized equation: laminated rods, plates and composites},
journal = {Comptes Rendus. M\'ecanique},
pages = {8--14},
publisher = {Elsevier},
volume = {337},
number = {1},
year = {2009},
doi = {10.1016/j.crme.2008.10.008},
language = {en},
}
Grigory Panasenko. Boundary conditions for the high order homogenized equation: laminated rods, plates and composites. Comptes Rendus. Mécanique, Volume 337 (2009) no. 1, pp. 8-14. doi : 10.1016/j.crme.2008.10.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.10.008/
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