Nonlinear vibration of a single-walled carbon nanotube embedded in a polymer matrix aroused by interfacial van der Waals forces

This paper studies the nonlinear vibration of a single-walled carbon nanotube (CNT) embedded in a polymer matrix aroused by van der Waals (vdW) forces using elastic beam models. The interfacial vdW forces are described by a nonlinear function in terms of the deflection of the CNT. According to different beam end conditions, the relation between deflection amplitudes and resonant frequencies of free vibrations of the CNT is derived through harmonic balance method. This relation is found to be sensitive to end conditions, diameters, and lengths of the embedded CNT. The axial load effect upon the vibrational behavior of the CNT and postbuckling of the embedded CNT are also discussed. Due to the influence of the surrounding polymer, the prediction on the critical buckling loads and resonant frequencies for embedded CNTs is quite different from that for free-standing CNTs. In addition, the applicability and accuracy of both Euler-Bernoulli and Timoshenko beam models are investigated. It is found that the Euler-Bernoulli beam model may provide comparable results as the Timoshenko beam model even for CNTs with smaller length-to-diameter ratios due to the constraint from the surrounding medium.