Transverse vibrations of double-walled carbon nanotubes under compressive axial load

Based on the Bernoulli-Euler beam theory, a double-elastic beam model is developed for transverse vibrations of double-walled carbon nanotubes under compressive axial load, which includes the interaction of van der Waals force between the inner and outer tubes and the effect of compressive axial load. Explicit expressions are derived for natural frequencies and associated amplitude ratios of the inner to the outer tubes for the case of simply supported double-walled carbon nanotubes. The influences of compressive axial load on the properties of vibrations are discussed. It is shown that the effects of compressive axial load on the natural frequencies of double-walled carbon nanotubes are sensitive to the vibration modes and aspect ratios. The natural frequencies are dependent on the axial load and decrease with increasing the axial load. However, the associated amplitude ratios of the inner to the outer tubes of double-walled carbon nanotubes are independent of the axial load. In addition, the critical axial buckling stress and strain for simply supported double-walled carbon nanotubes are obtained.