Geodesic
Homogenization of nonlinear problems in the mechanics of composites
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2012), pp. 58-62 
S. V. Sheshenin; M. I. Savenkova. Homogenization of nonlinear problems in the mechanics of composites. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2012), pp. 58-62. http://geodesic.mathdoc.fr/item/VMUMM_2012_5_a12/
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2012_5_a12/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider the method of homogenization that helps to solve the nonlinear equilibrium problems for layered plates and FGM plates. The homogenization method uses the superposition idea to combine the solution on the global domain with the fluctuations obtained as the solutions to local problems on the representative volume element domain. Superposition is not correct in the case of any nonlinearity. Nevertheless, it is quite possible to use the linearization of a nonlinear problem by differentiating the variational equation with respect to time or with respect to a loading parameter and by considering the constitutive equations formulated in terms of strain and stress rates. These are equations of hypo-elastic materials. As an example, a symmetric laminated plate subjected to bending is studied when a load uniformly depends on time.

[1] Pobedrya B.E., Mekhanika kompozitsionnykh materialov, Izd-vo MGU, M., 1984

[2] Bakhvalov N.S., Panasenko G.P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR

[3] Hui-Shen Shen., Functionally graded materials: nonlinear analysis of plates and shells, CRC Press, Boca Raton, 2009

[4] Sheshenin S.V., Fu M., Ivleva E.A., “Ob osrednenii periodicheskikh v plane plastin”, Tr. Mezhdunar. konf. “Teoriya i praktika rascheta zdanii, sooruzhenii i elementov konstruktsii. Analiticheskie i chislennye metody”, Izd-vo MGSU, M., 2008, 148–158

[5] Muravleva L.V., Sheshenin S.V., “Effektivnye svoistva zhelezobetonnykh plit pri uprugoplasticheskikh deformatsiyakh”, Vestn. Mosk. un-ta. Matem. Mekhan., 2004, no. 3, 62–65

[6] Muravleva L.V., “Effektivnye svoistva ortotropnykh kompozitov pri uprugoplasticheskikh deformatsiyakh”, Uprugost i neuprugost, Mat-ly Mezhdunar. nauch. simp. po problemam mekhaniki deformiruemykh tel, posvyaschennogo 95-letiyu so dnya rozhdeniya A. A. Ilyushina, URSS, M., 2006, 371–378

[7] Ilyushin A.A., Plastichnost, v. I, OGIZ, M., 1948 | MR

[8] Sheshenin S.V., “Primenenie metoda osredneniya k plastinam, periodicheskim v plane”, Vestn. Mosk. un-ta. Matem. Mekhan., 2006, no. 1, 47–51 | MR