|Journal||Journal of Mechanical Engineering|
|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)|
|Cited by||J. of Mech. Eng., 2020, vol. 23, no. 1, pp. 14-26|
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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- Gibson, R. F., Ayorinde, E. O., & Wen, Y.-F. (2007). Vibrations of carbon nanotubes and their composites: A review. Composites Sci. and Technology, vol. 67, iss. 1, pp. 1–28. https://doi.org/10.1016/j.compscitech.2006.03.031.
- Sirtori, C. (2002). Applied physics: Bridge for the terahertz gap. Nature, no. 417, pp. 132–133. https://doi.org/10.1038/417132b.
- Jeon, T. & Kim, K. (2002). Terahertz conductivity of anisotropic single walled carbon nanotube films. Appl. Physics Letters, no. 80, pp. 3403–3405. https://doi.org/10.1063/1.1476713.
- Yoon, J., Ru, C. Q., & Mioduchowski, A. (2003). Sound wave propagation in multiwall carbon nanotubes. J. Appl. Physics, no. 93, pp. 4801–4806. https://doi.org/10.1063/1.1559932.
- Iijima, S., Brabec, C., Maiti, A., & Bernholc, J. (1996). Structural flexibility of carbon nanotubes. J. Chemical Physics, no. 104, pp. 2089–2092. https://doi.org/10.1063/1.470966.
- Yakobson, B. I., Campbell, M. P., Brabec, C. J., & Bernholc, J. (1997). High strain rate fracture and C-chain unraveling in carbon nanotubes. Computer Material Sci., vol. 8, iss. 4, pp. 241–248. https://doi.org/10.1016/S0927-0256(97)00047-5.
- Wang, C. Y. & Zhang, L. C. (2008). An elastic shell model for characterizing single-walled carbon nanotubes. Nanotechnology, no. 19. 195704. https://doi.org/10.1088/0957-4484/19/19/195704.
- Wang, Q. & Varadan, V. K. (2007). Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Material Structure, no. 16, pp. 178–190. https://doi.org/10.1088/0964-1726/16/1/022.
- Fu, Y. M., Hong, J. W., & Wang, X. Q. (2006). Analysis of nonlinear vibration for embedded carbon nanotubes. J. Sound and Vibration, vol. 296, iss. 4–5, pp. 746–756. https://doi.org/10.1016/j.jsv.2006.02.024.
- Ansari, R. & Hemmatnezhad, M. (2001). Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach. Mathematical and Computer Modeling, vol. 53, iss. 5–6, pp. 927–938. https://doi.org/10.1016/j.mcm.2010.10.029.
- Ansari, R. & Hemmatnezhad, M. (2012). Nonlinear finite element analysis for vibrations of double-walled carbon nanotubes. Nonlinear Dynamics, no. 67, pp. 373–383. https://doi.org/10.1007/s11071-011-9985-6.
- Hajnayeb, A. & Khadem, S. E. (2012). Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid. J. Sound and Vibration, vol. 331, iss. 10, pp. 2443–2456. https://doi.org/10.1016/j.jsv.2012.01.008.
- Avramov, K. V. (2018). Nonlinear vibrations characteristics of single-walled carbon nanotubes via nonlocal elasticity. Intern. J. Nonlinear Mech., vol. 107, pp. 149–160. https://doi.org/10.1016/j.ijnonlinmec.2018.08.017.
- Fazelzadeh, S. A. & Ghavanloo, E. (2012). Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality. Composite Structures, vol. 94, iss. 3, pp. 1016–1022. https://doi.org/10.1016/j.compstruct.2011.10.014.
- Ghavanloo, E. & Fazelzadeh, S. A. (2012). Vibration characteristics of single-walled carbon nanotubes based on an anisotropic elastic shell model including chirality effect. Appl. Math. Modelling, vol. 36, iss. 10, pp. 4988–5000. https://doi.org/10.1016/j.apm.2011.12.036.
- Chang, T. (2010). A molecular based anisotropic shell model for single-walled carbon nanotubes. J. Mech. and Physics Solids, vol. 58, iss. 9, pp. 1422–1433. https://doi.org/10.1016/j.jmps.2010.05.004.
- Chang, T., Geng, J., & Guo, X. (2006). Prediction of chirality- and size-dependent elastic properties of single-walled carbon nanotubes via a molecular mechanics model. Proc. Royal Society A, vol. 462, iss. 2072, pp. 2523–2540. https://doi.org/10.1098/rspa.2006.1682.
- He, X. Q., Kitipornchai, S., Wang, C. M., Xiang, Y., & Zhou, Q. (2010). A nonlinear Van Der Waals force model for multiwalled carbon nanotubes modeled by a nested system of cylindrical shells. ASME J. Appl. Mech., vol.77, iss. 6, 061006 (6 p.). https://doi.org/10.1115/1.4001859.
- Washizu, K. (1975). Variational methods in elasticity and plasticity. Oxford, United Kingdom: Pergamon Press, 420 p.
- Zienkiewicz, O. (1983). Finite elements and approximation. New York: John Wiley & Sons, 350 p.
- He, X. Q., Kitipornchai, S., & Liew, K. M. (2005). Buckling analysis of multi-walled carbon nanotubes: A continuum model accounting for Van der Waals interaction. J Mech. Phys. Solids, vol. 53, iss. 2, pp. 303–326. https://doi.org/10.1016/j.jmps.2004.08.003.
- Strozzi, M. & Pellicano, F. (2017). Linear vibrations of triple-walled carbon nanotubes. Mathematics and Mechanics of Solids, vol. 23, iss. 11, pp. 1456–1481. https://doi.org/10.1177/1081286517727331.
- Liew, K. M., He, X. Q., & Wong, C. H. (2004). On the study of elastic and plastic properties of multi-walled carbon nanotubes under axial tension using molecular dynamics simulation. Acta Materialia, vol. 52, iss. 9, pp. 2521–2527. https://doi.org/10.1016/j.actamat.2004.01.043.
- Lambin, Ph., Meunier, V., & Rubio, A. (2000). Electronic structure of polychiral carbon nanotubes. Physical review B, vol. 62, iss. 8, pp. 5129–5135. https://doi.org/10.1103/PhysRevB.62.5129.
Received 13 February 2020
Published 30 March 2020