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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 3, Pages 97–130 (Mi izv290)  

This article is cited in 17 scientific papers (total in 17 papers)

One-dimensional equations of deformation of thin slightly curved rods. Asymptotical analysis and justification

S. A. Nazarova, A. S. Slutskijb

a Saint-Petersburg State University
b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: We obtain asymptotics for the solution of the spatial problem of elasticity theory in a thin body (a rod) with a smoothly varying cross-section. Any anisotropy and any non-homogeneity of material is admitted. The ends of the a rod, which is under the action of volume forces, are rigidly fixed (clamped), and the lateral surface is under the action of forces. The small parameter $h$ is the ratio of the maximal diameter of the rod to its length. We suggest conditions on the differential properties and the structure of external load under which the solution of the one-dimensional equations yielded by asymptotical analysis provides an acceptable approximation to the three-dimensional displacement and stress fields. The error estimate is based on a special version of Korn's inequality, which is asymptotically sharp if suitable weight factors and powers of $h$ are introduced into the $L_2$-norms of displacements and their derivatives.


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English version:
Izvestiya: Mathematics, 2000, 64:3, 531–562

Bibliographic databases:

MSC: 73C02, 35J45, 35B45, 65P05, 35J55, 35J05, 73B40, 73K10, 73K12, 73V25, 73C50, 74K05
Received: 03.08.1998

Citation: S. A. Nazarov, A. S. Slutskij, “One-dimensional equations of deformation of thin slightly curved rods. Asymptotical analysis and justification”, Izv. RAN. Ser. Mat., 64:3 (2000), 97–130; Izv. Math., 64:3 (2000), 531–562

Citation in format AMSBIB
\by S.~A.~Nazarov, A.~S.~Slutskij
\paper One-dimensional equations of deformation of thin slightly curved rods. Asymptotical analysis and justification
\jour Izv. RAN. Ser. Mat.
\yr 2000
\vol 64
\issue 3
\pages 97--130
\jour Izv. Math.
\yr 2000
\vol 64
\issue 3
\pages 531--562

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    This publication is cited in the following articles:
    1. Nazarov S.A., Slutskii A.S., “Asymptotics of eigenfrequencies of a Pi-shaped elastic frame”, Doklady Mathematics, 64:2 (2001), 266–269  zmath  isi
    2. S. A. Nazarov, “Weighted anisotropic Korn's inequality for a junction of a plate and a rod”, Sb. Math., 195:4 (2004), 553–583  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Nazarov S.A., “On eigenoscillations of a solid with a blunted pick”, Doklady Physics, 52:10 (2007), 560–564  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. S. A. Nazarov, “The spectrum of the elasticity problem for a spiked body”, Siberian Math. J., 49:5 (2008), 874–893  mathnet  crossref  mathscinet  isi  elib  elib
    5. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Nazarov, SA, “Scenarios for the quasistatic growth of a slightly curved and kinked crack”, Pmm Journal of Applied Mathematics and Mechanics, 72:3 (2008), 347  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Zhikov, VV, “KORN INEQUALITIES ON THIN PERIODIC STRUCTURES”, Networks and Heterogeneous Media, 4:1 (2009), 153  crossref  mathscinet  zmath  isi  elib  scopus
    8. S. A. Nazarov, G. H. Sweers, A. S. Slutskij, “Homogenization of a thin plate reinforced with periodic families of rigid rods”, Sb. Math., 202:8 (2011), 1127–1168  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. S. A. Nazarov, “Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler”, Siberian Math. J., 53:2 (2012), 274–290  mathnet  crossref  mathscinet  isi
    10. S. A. Nazarov, “The Mandelstam Energy Radiation Conditions and the Umov–Poynting Vector in Elastic Waveguides”, J Math Sci, 2013  crossref  mathscinet  scopus
    11. Nazarov S.A., Slutskij A.S., Taskinen J., “Korn Inequality For a Thin Rod With Rounded Ends”, Math. Meth. Appl. Sci., 37:16 (2014), 2463–2483  crossref  mathscinet  zmath  isi  scopus
    12. Leugering G., Nazarov S.A., Slutskij A.S., “Asymptotic Analysis of 3-D Thin Piezoelectric Rods”, ZAMM-Z. Angew. Math. Mech., 94:6 (2014), 529–550  crossref  mathscinet  zmath  isi  scopus
    13. Kozlov V. Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944  mathscinet  zmath  isi  elib
    14. Kozlov V.A., Nazarov S.A., “Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies”, J. Elast., 132:1 (2018), 103–140  crossref  mathscinet  zmath  isi  scopus
    15. S. A. Nazarov, “The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section”, Sb. Math., 209:9 (2018), 1287–1336  mathnet  crossref  crossref  adsnasa  isi  elib
    16. Nazarov S.A., Slutskii A.S., “Asymptotics of Natural Oscillations of Elastic Junctions With Readily Movable Elements”, Mech. Sol., 53:1 (2018), 101–115  crossref  isi  scopus
    17. Leugering G., Nazarov S.A., Slutskij A.S., “The Asymptotic Analysis of a Junction of Two Elastic Beams”, ZAMM-Z. Angew. Math. Mech., 99:1 (2019), UNSP e201700192  crossref  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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