Abstract
The Flügge and Donnell theories for thin cylindrical shells are used to obtain the approximate natural frequencies and mode shapes of single-wall and double-wall carbon nanotubes. Vibrations of carbon nanotubes are important in a number of nano-mechanical devices such as oscillators, charge detectors, clocks, field emission devices, and sensors (Gibson et al. 2007). Electron microscope observations of vibrating carbon nanotubes have been used indirectly and nondestructively to determine the effective elastic modulus and other aspects of mechanical behavior of carbon nanotubes.
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Notes
- 1.
This assumption leads to an inconsistency in the theory, since, having already assumed that \(e_\xi = 0\), it is straightforward to show that \(\sigma _\xi = \nu \left(\sigma _x + \sigma _\theta \right)\).
- 2.
A formula that relates the van der Waals interaction force between cylindrical surfaces and is an explicit function of the radii of the nanotubes can be found in (He et al. 2005).
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Magrab, E.B. (2012). Cylindrical Shells and Carbon Nanotube Approximations. In: Vibrations of Elastic Systems. Solid Mechanics and Its Applications, vol 184. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2672-7_7
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DOI: https://doi.org/10.1007/978-94-007-2672-7_7
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