Elastic vibrations of a cylindrical nanotube with the effect of surface stress and surface inertia

Vibration frequency analysis of nanostructures may be essential for study of their thermal conductivity and mechanical characterization. Given the high surface-to-volume ratio, the elastic vibrations of an infinitely long cylindrical nanotube have been studied by considering both the effects of surface stress and that of surface inertia within the framework of surface elasticity. The phonon dispersion and the resonant frequencies for the specific vibration modes have been calculated. Numerical results have indicated that the surface stress and the surface inertia have equally important effect on the vibration behavior of the nanotube that may depend on the vibration modes as well. Due to the surface effect, the vibration modes of lower order by the classical elasticity may be indeed the modes of higher order. The surface effect on the low-frequency Raman shift has also been found.