Nonlinear finite element vibration analysis of double-walled carbon nanotubes based on Timoshenko beam theory

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. The Von Karman type nonlinear strain-displacement relationships are employed where the ends of the nanotube are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. A superconvergent beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of these beam-modeled nanotubes. In this kind of beam element, the interpolating functions are obtained using the exact solution of the static analysis of the beam. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique to obtain the nonlinear vibration frequencies of double-walled carbon nanotubes with different boundary conditions. The effects of material constant of the surrounding elastic medium and the geometric parameters on the vibrational behavior 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. The present numerical results are validated by comparing the linear and nonlinear frequencies of double-walled carbon nanotubes with those available in the literature where possible. This comparison illustrates that the present scheme yields very accurate results in predicting the nonlinear frequencies.