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Mat. Zametki, 2004, Volume 75, Issue 6, Pages 818–833 (Mi mz73)  

This article is cited in 4 scientific papers (total in 4 papers)

Homogenization of Elasticity Problems with Boundary Conditions of Signorini type

G. A. Iosif'yan

A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: In a perforated domain $\Omega^\varepsilon =\Omega\cap\varepsilon \omega$ formed of a fixed domain $\Omega$ and an $\varepsilon$-compression of a 1-periodic domain $omega$, we consider problems of elasticity for variational inequalities with boundary conditions of Signorini type on a part of the surface $S^\varepsilon _0$ of perforation. We study the asymptotic behavior of solutions as $\varepsilon\to0$ depending on the structure of the set $S^\varepsilon _0$. In the general case, the limit (homogenized) problem has the two distinguishing properties: (i) the limit set of admissible displacements is determined by nonlinear restrictions almost everywhere in the domain $\Omega$, i.e., in the limit, the Signorini conditions on the surface $S^\varepsilon _0$ can turn into conditions posed at interior points of $\Omega$ (ii) the limit problem is stated for an homogenized Lagrangian which need not coincide with the quadratic form usually determining the homogenized elasticity tensor. Theorems concerning the homogenization of such problems were obtained by the two-scale convergence method. We describe how the limit set of admissible displacements and the homogenized Lagrangian depend on the geometry of the set $S^\varepsilon _0$ on which the Signorini conditions are posed.

DOI: https://doi.org/10.4213/mzm73

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English version:
Mathematical Notes, 2004, 75:6, 765–779

Bibliographic databases:

UDC: 517.958
Received: 14.01.2002

Citation: G. A. Iosif'yan, “Homogenization of Elasticity Problems with Boundary Conditions of Signorini type”, Mat. Zametki, 75:6 (2004), 818–833; Math. Notes, 75:6 (2004), 765–779

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Stelzig Ph.E., “Homogenization of Many-Body Structures Subject to Large Deformations”, ESAIM-Control OPtim. Calc. Var., 18:1 (2012), 91–123  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Jaeger W., Neuss-Radu M., Shaposhnikova T.A., “Homogenization of a Variational Inequality for the Laplace Operator with Nonlinear Restriction for the Flux on the Interior Boundary of a Perforated Domain”, Nonlinear Anal.-Real World Appl., 15 (2014), 367–380  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Capatina A., Timofte C., “Homogenization results for micro-contact elasticity problems”, J. Math. Anal. Appl., 441:1 (2016), 462–474  crossref  mathscinet  zmath  isi  elib  scopus
    4. Ptashnyk M., “Homogenization of Some Degenerate Pseudoparabolic Variational Inequalities”, J. Math. Anal. Appl., 469:1 (2019), 44–75  crossref  mathscinet  zmath  isi  scopus
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