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Mat. Sb., 1999, Volume 190, Number 12, Pages 37–92 (Mi msb443)  

This article is cited in 24 scientific papers (total in 24 papers)

Homogenization of elasticity equations with contrasting coefficients

G. V. Sandrakov

M. V. Lomonosov Moscow State University

Abstract: Non-stationary problems of linearized elasticity theory in a periodic medium with pores filled with an easily deformable material are considered. The period of the medium is a small positive parameter. It is assumed that the density and the ratio of the minimum and the maximum values of the elasticity moduli of the material are also small positive parameters. Homogenized equations solutions of which approximate the solutions of the problems under consideration are derived. Estimates of the accuracy of this approximation as the parameters approach zero are proved. The form of the homogenized equations and the estimates of the accuracy depend strongly on the geometric properties of the pores and on the asymptotic behaviour of certain expressions containing these small parameters.


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English version:
Sbornik: Mathematics, 1999, 190:12, 1749–1806

Bibliographic databases:

UDC: 517.955.8
MSC: 73C99, 35B40, 35B27
Received: 24.03.1999

Citation: G. V. Sandrakov, “Homogenization of elasticity equations with contrasting coefficients”, Mat. Sb., 190:12 (1999), 37–92; Sb. Math., 190:12 (1999), 1749–1806

Citation in format AMSBIB
\by G.~V.~Sandrakov
\paper Homogenization of elasticity equations with contrasting coefficients
\jour Mat. Sb.
\yr 1999
\vol 190
\issue 12
\pages 37--92
\jour Sb. Math.
\yr 1999
\vol 190
\issue 12
\pages 1749--1806

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    This publication is cited in the following articles:
    1. Sandrakov G.V., “Homogenization of the filtration of a two-phase immiscible liquid flow”, Dokl. Math., 62:2 (2000), 186–189  mathnet  mathscinet  zmath  isi
    2. Sandrakov G.V., “Averaging of dynamic processes in composites periodically reinforced by fibers”, Dokl. Math., 61:3 (2000), 346–349  mathnet  mathscinet  zmath  isi
    3. Braides A., Piat V.Ch., Piatnitski A., “A variational approach to double-porosity problems”, Asymptot. Anal., 39:3-4 (2004), 281–308  mathscinet  zmath  isi  elib
    4. Sandrakov G.V., “Multiphase homogenized models for diffusion in highly nonhomogeneous media”, Dokl. Math., 70:1 (2004), 507–511  mathnet  mathscinet  zmath  isi  elib
    5. Pastukhova S.E., “Homogenization of elasticity problems on a periodic composite structure”, Dokl. Math., 69:2 (2004), 208–213  mathnet  mathscinet  zmath  isi  elib
    6. Pastukhova S.E., “Homogenization of elasticity problems on periodic rod frames of critical thickness”, Dokl. Math., 69:1 (2004), 20–25  mathnet  mathscinet  zmath  isi  elib
    7. Sandrakov G.V., “Multiphase diffusion models in homogenization”, Dokl. Math., 69:1 (2004), 50–53  mathnet  mathscinet  zmath  isi  elib
    8. G. V. Sandrakov, “Multiphase models of nonstationary diffusion in homogenization”, Comput. Math. Math. Phys., 44:10 (2004), 1741–1756  mathnet  mathscinet  zmath  elib
    9. S. E. Pastukhova, “Homogenization of elasticity problems on periodic composite structures”, Sb. Math., 196:7 (2005), 1033–1073  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. G. V. Sandrakov, “Multiphase homogenized diffusion models for problems with several parameters”, Izv. Math., 71:6 (2007), 1193–1252  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. Babych N.O., Kamotski I.V., Smyshlyaev V.P., “Homogenization of spectral problems in bounded domains with doubly high contrasts”, Netw. Heterog. Media, 3:3 (2008), 413–436  crossref  mathscinet  zmath  isi  elib
    12. Solci M., “Double-porosity homogenization for perimeter functionals”, Math. Methods Appl. Sci., 32:15 (2009), 1971–2002  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    13. Cherdantsev M., “Spectral convergence for high-contrast elliptic periodic problems with a defect via homogenization”, Mathematika, 55:1-2 (2009), 29–57  crossref  mathscinet  zmath  elib  scopus  scopus  scopus
    14. Smyshlyaev V.P., “Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization”, Mechanics of Materials, 41:4, The Special Issue in Honor of Graeme W. Milton (2009), 434–447  crossref  isi  elib  scopus  scopus  scopus
    15. Bellieud M., “Torsion Effects in Elastic Composites with High Contrast”, SIAM Journal on Mathematical Analysis, 41:6 (2010), 2514–2553  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Babych N.O., “The Bottom of the Spectrum in a Double-Contrast Periodic Model”, Integral Methods in Science and Engineering, 2010, 53–63  crossref  mathscinet  zmath  isi
    17. Shane Cooper, “Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions”, Applicable Analysis, 2013, 1  crossref  mathscinet  scopus  scopus  scopus
    18. Holovatyi Yu.D., Hut V.M., “Vibrating Systems with Rigid Light-Weight Inclusions: Asymptotics of the Spectrum and Eigenspaces”, Ukr. Math. J., 64:10 (2013), 1495–1513  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    19. V. M. Hut, “Asymptotic Expansions of Eigenvalues and Eigenfunctions of a Vibrating System With Stiff Light-Weight Inclusions”, J Math Sci, 198:1 (2014), 13  crossref  mathscinet  scopus  scopus  scopus
    20. Braides A. Piat V.Ch. Solci M., “Discrete Double-Porosity Models For Spin Systems”, Math. Mech. Complex Syst., 4:1 (2016), 79–102  crossref  mathscinet  zmath  isi  elib  scopus
    21. Bellieud M., “Homogenization of Stratified Elastic Composites With High Contrast”, SIAM J. Math. Anal., 49:4 (2017), 2615–2665  crossref  mathscinet  zmath  isi  scopus
    22. Bellieud M., Cooper Sh., “Asymptotic Analysis of Stratified Elastic Media in the Space of Functions With Bounded Deformation”, SIAM J. Math. Anal., 49:5 (2017), 4275–4317  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    23. Cooper Sh., “Quasi-Periodic Two-Scale Homogenisation and Effective Spatial Dispersion in High-Contrast Media”, Calc. Var. Partial Differ. Equ., 57:3 (2018), 76  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    24. Kamotski I.V. Smyshlyaev V.P., “Two-Scale Homogenization For a General Class of High Contrast Pde Systems With Periodic Coefficients”, Appl. Anal., 98:1-2, SI (2019), 64–90  crossref  mathscinet  isi  scopus
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