Free localized vibrations of a long double-walled carbon nanotube embedded in nonhomogeneous elastic medium

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.