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

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.

DOI: https://doi.org/10.4213/im290

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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

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
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• http://mi.mathnet.ru/eng/izv290
• https://doi.org/10.4213/im290
• http://mi.mathnet.ru/eng/izv/v64/i3/p97

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. Nazarov S.A., “On eigenoscillations of a solid with a blunted pick”, Doklady Physics, 52:10 (2007), 560–564
4. S. A. Nazarov, “The spectrum of the elasticity problem for a spiked body”, Siberian Math. J., 49:5 (2008), 874–893
5. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107
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
7. Zhikov, VV, “KORN INEQUALITIES ON THIN PERIODIC STRUCTURES”, Networks and Heterogeneous Media, 4:1 (2009), 153
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
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
10. S. A. Nazarov, “The Mandelstam Energy Radiation Conditions and the Umov–Poynting Vector in Elastic Waveguides”, J Math Sci, 2013
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
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
13. Kozlov V. Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944
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
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
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
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
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