Effects of a surrounding elastic medium on flexural waves propagating in carbon nanotubes via nonlocal elasticity

Abstract

The flexural wave behavior in carbon nanotubes embedded in an elastic medium is analyzed based on the classical and nonlocal theories of the Timoshenko beam. Emphasis is focused on the effects of small scale and the surrounding elastic medium on the phase velocity of the transverse wave. The system of basic equations for transverse deflection and rotation are derived, and further, a single fourth-order governing differential equation is reduced. The characteristic equation and dispersion relation are obtained for single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes (DWCNTs). The number of flexural wave branches depends only on the number of walls, but not on the surrounding elastic medium and the small scale parameter. A SWCNT has two phase velocities and a DWCNT has four phase velocities for extremely high frequencies. Critical or cutoff frequencies are independent of the small scale parameter. However, the lower critical frequencies depend on the surrounding elastic medium and the van der Waals force, and higher critical frequencies depend on the shear rigidity of tubes. Consideration of small scale decreases the corresponding wave speeds, and this effect is negligible for lower frequencies. A surrounding elastic medium affects the acoustic mode of phase velocity for lower frequencies, and hardly affects the optical mode. The classical/nonlocal Euler–Bernoulli and Rayleigh beam theories can be recovered as special cases of the present models. Moreover, the number of wave speeds for both SWCNTs and DWCNTs diminishes by half compared to that of the Timoshenko beam theory.