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 Mat. Sb., 2005, Volume 196, Number 7, Pages 101–142 (Mi msb1402)

Homogenization of elasticity problems on periodic composite structures

S. E. Pastukhova

Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: Elasticity problems on a plane plate reinforced with a thin periodic network or in a 3-dimensional body reinforced with a thin periodic box skeleton are considered. The composite medium depends on two parameters approaching zero and responsible for the periodicity cell and the thickness of the reinforcing structure. The parameters can be dependent or independent.
For these problems Zhikov's method of ‘two-scale convergence with variable measure’ is used to derive the homogenization principle: the solution of the original problem reduces in a certain sense to the solution of the homogenized (or limiting) problem. The latter has a classical form. From the operator form of the homogenization principle, on the basis of the compactness principle in the $L^2$-space, which is also established, one obtains for the composite structure the Hausdorff convergence of the spectrum of the original problem to the spectrum of the limiting problem.

DOI: https://doi.org/10.4213/sm1402

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English version:
Sbornik: Mathematics, 2005, 196:7, 1033–1073

Bibliographic databases:

UDC: 517.9
MSC: 35B27, 74Kxx, 74Q05

Citation: S. E. Pastukhova, “Homogenization of elasticity problems on periodic composite structures”, Mat. Sb., 196:7 (2005), 101–142; Sb. Math., 196:7 (2005), 1033–1073

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1402
• https://doi.org/10.4213/sm1402
• http://mi.mathnet.ru/eng/msb/v196/i7/p101

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This publication is cited in the following articles:
1. V. V. Zhikov, S. E. Pastukhova, “On the Trotter–Kato Theorem in a Variable Space”, Funct. Anal. Appl., 41:4 (2007), 264–270
2. S. E. Pastukhova, “Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems”, Sb. Math., 198:10 (2007), 1465–1494
3. Braides A., Chiadò Piat V., “Non convex homogenization problems for singular structures”, Netw. Heterog. Media, 3:3 (2008), 489–508
4. Cancedda A., “Spectral Homogenization For a Robin-Neumann Problem”, Boll. Unione Mat. Ital., 10:2 (2017), 199–222
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