Free localized vibrations of a long double-walled carbon nanotube introduced into an inhomogeneous elastic medium

Abstract

Based on modified Flügge equations and nonlocal elasticity theory, free axisymmetric oscillations of a long double-walled carbon nanotube embedded into an inhomogeneous elastic medium is studied. The ambient medium is simulated by the Winkler foundation. Van der Waals forces are introduced in order to take into account the interaction between the nanotube walls. Using Tovstik’s asymptotic method, eigenmodes are constructed in the form of functions that decay far from the line on the surface of the outer wall, on which the modulus of subgrade reaction has a local minimum. Eigenmodes and eigenfrequencies corresponding to the coand counterdirected wall motions are found. It has been found that introducing a nonlocality parameter into the model results in eigenmodes that are not inherent in macroscale shells. In particular, an increase in the stretching force leads first to greater localization of vibrations and increase in the amplitudes of tangential atomic oscillations and, second, to reduction in the frequencies in the case when the tube lies in a sufficiently stiff medium.