Свободные локализованные колебания длинной двухстенной углеродной нанотрубки, внедренной в неоднородную упругую среду

Gennadiy I. Mikhasev; Marina G. Botogova

doi:10.21638/11701/spbu01.2016.117

Authors

Gennadiy I. Mikhasev Belarusian State University, pr. Nezavisimosti, 4, Minsk, 220030, Republic of Belarus;
Marina G. Botogova Belarusian State University, pr. Nezavisimosti, 4, Minsk, 220030, Republic of Belarus;

DOI:

https://doi.org/10.21638/11701/spbu01.2016.117

Abstract

On the basis of the modified Flu¨ugge equations for thin cylindrical shells and non-local theory of elasticity, free axisymmetric vibrations of a long double-walled carbon nanotube embedded in nonhomogeneous elastic medium is studied. The surrounding medium is modelled by the Winkler foundation. To take into account the interaction between the nanotube walls, the van-der-Waals forces are introduced into the governing equations. By using the asymptotic method of Tovstik, eigenmodes are constructed in the form of functions decaying far from the line on the outermost wall where the coefficient of soil reaction has a local minimum. Eigenmodes and natural frequencies corresponding to the like-directed and differently directed motions of walls have been found. It has been revealed that the introduction of a parameter of nonlocality into the model «generates» the eigenmodes which are nonrelevant for macro-sized shells. In particular, the increase of the tensile axil force results in: 1) more high rate of localization of vibrations and growing amplitudes of tangential oscillations of atoms; 2) increasing natural frequencies in the case when the tube is in the sufficiently stiffen medium. Refs 13. Figs 3. Tables 1.

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Литература

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9. Mikhasev G. On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium//ZAMM. 2014. Vol. 94 (1-2). P. 130-141.

10. Товстик П.Е. Двумерные задачи устойчивости и колебаний оболочек нулевой гауссовой кривизны//ДАН СССР. 1983. Т. 271 (1). С. 69-71.

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References

1. Bakshi S.R., Lahiri D., Agarwal A., “Carbon nanotube reinforced metal matrix composites — a review”, International Materials Reviews 55(1), 41–64 (2010).

2. Peng J. Wu J., Hwang K. C., Song J., Huang Y., “Can a single-wall carbon nanotube be modeled as a thin shell?”, J. Mech. Phys. Solids 56, 2213–2224 (2008).

3. Fazelzadeh S.A., Ghavanloo E., “Nonlocal anisotropic elastic shell model for vibrations of singlewalled carbon nanotubes with arbitrary chirality”, Composite Structures 94(3), 1016–1022 (2012).

4. Yoon J., Ru C.Q., Mioduchowski A., “Vibration of an embedded multiwalled carbon nanotube”, Compos. Sci. Technol. 63, 1533–1542 (2003).

5. Li R., Kardomateas G.A., “Vibration characteristics of multiwalled carbon nanotubes embedded in elastic media by a nonlocal elastic shell model”, J. of Appl. Mech. 74, 1087–1094 (2007).

6. Eringen A.C., Nonlocal continuum field theories (Springer, New York, 2002).

7. Mikhasev G. I., Tovstik P. E., Localized Vibrations and Waves in Thin Shells. Asymptotic Methods (Fizmatlit, Moscow, 2009, 290 p.) [in Russian].

8. Arani A.G., Barzoki A.M., Kolahchi R., Loghman A., “Pasternak foundation effect on the axial and torsional waves propagation in embedded DWCNTs using nonlocal elasticity cylindrical shell theory”, J. Mech. Technol. 25(9), 2385–2391 (2011).

9. Mikhasev G., “On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium”, ZAMM 94(1–2), 130–141 (2014).

10. Tovstik P. E., “Two-dimensional problems of buckling and vibrations of the shells of zero Gaussian curvature”, Soviet Phys. Dokl. 28(7), 593–594 (1983).

11. Mikhasev G. I., “Governing equations of a multi-walled carbon nanotube based on nonlocal theory of orthotropic shells”, Dokl. Acad. Sci. Belarus 55(6), 119–123 (2011) [in Russian].

12. Fl¨ugge W., Stresses in Shells (Heidelberg: Springer, Berlin, G¨ottingen, 1962).

13. Strozzi M., Manevitch L. I., Pellicano F., Smirnov V.V., Shepelev D. S., “Low-frequency linear vibrations of single-walled carbon nanotubes: Analytical and numerical models”, J. of Sound and Vibration 333, 2936–2957 (2014).