Nonlinear finite element analysis for vibrations of double-walled carbon nanotubes

The large-amplitude free vibration analysis of double-walled carbon nanotubes embedded in an elastic medium is investigated by means of a finite element formulation. A double-beam model is utilized in which the governing equations of layers are coupled with each other via the van der Waals interlayer forces. Von-Karman type nonlinear strain-displacement relationships are employed where the ends of the nanotube are constrained to move axially. The amplitude-frequency response curves for large-amplitude free vibrations of single-walled and double-walled carbon nanotubes with arbitrary boundary conditions are graphically illustrated. The effects of material constant of the surrounding elastic medium and the geometric parameters on the vibration characteristics are investigated. For a double-walled carbon nanotube with different boundary conditions between inner and outer tubes, the nonlinear frequencies are obtained apparently for the first time. Comparison of the results with those from the open literature is made for the amplitude-frequency curves where possible. This comparison illustrates that the present scheme yields very accurate results in predicting the nonlinear frequencies.