Vibration analysis of shell‐like curved carbon nanotubes using nonlocal strain gradient theory

The performance of nanotube‐based resonators is significantly affected by the variation of their natural frequencies. In reality, nanotubes may have some levels of initial imperfection, which definitely changes their vibrational characteristics. In the present article, the vibrational behavior of initially curved nanotubes is investigated on the basis of thin shell theory incorporating the nonlocal strain gradient theory. The three‐dimensional governing equations of motion are derived by adopting the curvilinear coordinate system through Hamilton's principle. A new formulation of Eringen's model containing the higher‐order strain gradients is employed to capture the effects of nonlocality and small scale parameter. Galerkin decomposition method is employed to acquire a bi‐cubic relation for the natural frequency of the considered nanoshell. The effects of different parameters like the ratio of radius of curvature to the cross section radius, small scale and nonlocality, and length‐to‐radius ratio on the variation of the natural frequency are presented. Excellent agreements are found in some comparative studies between the results of this work and those obtained by molecular dynamics and experimental findings in the literature. Finally, some interesting results are found by plotting the variation of natural frequency of the system versus the length‐to‐radius ratio. It is exhibited that by increasing the ratio L / r , the natural frequencies associated with different values of wavenumber m converge to the lower one and the convergence speed is highly dependent to the circumferential wavenumber n.