Nonlocal Anisotropic Shell Model of Linear Vibrations of Multi-walled Carbon Nanotubes

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 23, no. 1, 2020 (March)
Pages 14-26
Cited by J. of Mech. Eng., 2020, vol. 23, no. 1, pp. 14-26



Kostiantyn V. Avramov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-8740-693X

Balzhan N. Kabylbekova, M. Auezov South Kazakhstan State University (5, Tauke-khan Ave., Shymkent, 5160012, Kazakhstan), e-mail:, ORCID: 0000-0001-8461-8008

Kazira K. Seitkazenova, M. Auezov South Kazakhstan State University (5, Tauke-khan Ave., Shymkent, 5160012, Kazakhstan)

Darkhan S. Myrzaliyev, M. Auezov South Kazakhstan State University (5, Tauke-khan Ave., Shymkent, 5160012, Kazakhstan)

Vladimir N. Pecherskiy, M. Auezov South Kazakhstan State University (5, Tauke-khan Ave., Shymkent, 5160012, Kazakhstan)



A simply-supported multi-walled carbon nanotube (MWCNT) is considered. Its vibrations will be studied in a cylindrical coordinate system. The elastic constants in Hooke’s law depend on the CNT wall diameter, which is why each wall has its own elastic constants. CNT vibrations are described by the Sanders-Koiter shell theory. To derive partial differential equations (PDE) describing self-induced variations, a variational approach is used. The PDEs of vibrations are derived with respect to three projections of displacements. The model takes into account the Van der Waals forces between CNT walls. The three projections of displacements are expanded in basis functions. It was not possible to select the basis functions satisfying both geometric and natural boundary conditions. Therefore, selected are the basis functions that satisfy only geometric boundary conditions. To obtain a linear dynamic system with a finite number of degrees of freedom, the method of weighted residuals is used. To derive the basic relations of the method of weighted residuals, methods of variational calculus are used. The vibrational eigenfrequencies of single-walled (SW) CNTs are analyzed depending on the number of waves in the circumferential direction. With the number of waves in the circumferential direction from 2 to 4, the vibrational eigenfrequencies of CNTs are minimal. These numbers are smaller than those for the vibrational eigenfrequencies of engineering shells. Anisotropic models of tripple-walled (TW) CNTs were investigated. In their eigenforms, there is interaction between the basis functions and different numbers of waves in the longitudinal direction. This phenomenon was not observed in the isotropic CNT model. The appearance of such vibrations is a consequence of structural anisotropy.


Keywords: nanotube, Sanders-Koiter shell model, Van der Waals forces, nonlocal elasticity.


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Received 13 February 2020

Published 30 March 2020