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Mat. Sb., 2011, Volume 202, Number 8, Pages 41–80 (Mi msb6358)  

This article is cited in 10 scientific papers (total in 10 papers)

Homogenization of a thin plate reinforced with periodic families of rigid rods

S. A. Nazarova, G. H. Sweersbc, A. S. Slutskijad

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences
b Delft University of Technology
c Mathematical Institute, University of Cologne
d St. Petersburg State University of Service and Economics

Abstract: The asymptotics of the solution to the elastic bending problem for a thin plate reinforced with several periodic families of closely spaced but disjoint rods are constructed and justified, the result of homogenization being substantially different from the case when the rods are welded together into a single periodic mesh. The material in the rods is assumed to be appreciably more rigid than that in the plate. An averaged fourth-order differential operator is obtained from summing the nonelliptic operators generated by each of the families of the rods. This operator is shown to be elliptic if and only if the rods from at least two families are nonparallel. As a simplified example, the paper examines a similar stationary heat conduction problem.
Bibliography: 24 titles.

Keywords: thin plate, homogenization, asymptotics, composite material.

DOI: https://doi.org/10.4213/sm6358

Full text: PDF file (1134 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2011, 202:8, 1127–1168

Bibliographic databases:

Document Type: Article
UDC: 517.956.8+517.958:539.3(5)
MSC: Primary 74K20, 35B27; Secondary 74H10, 74B05, 35Q74
Received: 01.05.2008 and 21.04.2011

Citation: S. A. Nazarov, G. H. Sweers, A. S. Slutskij, “Homogenization of a thin plate reinforced with periodic families of rigid rods”, Mat. Sb., 202:8 (2011), 41–80; Sb. Math., 202:8 (2011), 1127–1168

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Nazarov S.A., Slutskij A.S., Sweers G.H., “Korn inequalities for a reinforced plate”, J. Elasticity, 106:1 (2012), 43–69  crossref  mathscinet  zmath  isi  elib  scopus
    2. V. A. Kozlov, S. A. Nazarov, “Asymptotic models of the blood flow in arterias and veins”, J. Math. Sci. (N. Y.), 194:1 (2013), 44–57  mathnet  crossref  mathscinet
    3. V. A. Kozlov, S. A. Nazarov, “A simple one-dimensional model of a false aneurysm in the femoral artery”, J. Math. Sci. (N. Y.), 214:3 (2016), 287–301  mathnet  crossref  mathscinet
    4. V. A. Kozlov, S. A. Nazarov, “One-dimensional model of viscoelastic blood flow through a thin elastic vessel”, Journal of Mathematical Sciences, 207:2 (2015), 249–269  crossref  mathscinet  zmath  scopus
    5. V. A. Kozlov, S. A. Nazarov, “Transmission conditions in a one-dimensional model of bifurcating blood vessel with an elastic wall”, J. Math. Sci. (N. Y.), 224:1 (2017), 94–118  mathnet  crossref  mathscinet
    6. F. Berntsson, M. Karlsson, V. Kozlov, S. A. Nazarov, “A one-dimensional model of viscous blood flow in an elastic vessel”, Appl. Math. Comput., 274 (2016), 125–132  crossref  mathscinet  isi  elib  scopus
    7. Berchio E., Buoso D., Gazzola F., Zucco D., “A Minimaxmax Problem For Improving the Torsional Stability of Rectangular Plates”, J. Optim. Theory Appl., 177:1 (2018), 64–92  crossref  mathscinet  zmath  isi  scopus
    8. Ghosh A., Kozlov V.A., Nazarov S.A., Rule D., “A Two-Dimensional Model of the Thin Laminar Wall of a Curvilinear Flexible Pipe”, Q. J. Mech. Appl. Math., 71:3 (2018), 349–367  crossref  mathscinet  isi
    9. Berntsson F., Ghosh A., Kozlov V.A., Nazarov S.A., “A One Dimensional Model of Blood Flow Through a Curvilinear Artery”, Appl. Math. Model., 63 (2018), 633–643  crossref  mathscinet  isi  scopus
    10. Antunes P.R.S., Gazzola F., “Some Solutions of Minimaxmax Problems For the Torsional Displacements of Rectangular Plates”, ZAMM-Z. Angew. Math. Mech., 98:11 (2018), 1974–1991  crossref  mathscinet  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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