Nonlinear Vibration of Single-Walled Carbon Nanotubes Under Magnetic Field by Stochastic Finite Element Method

Abstract

In the present study, we investigate the statistical nonlinear dynamic behaviors of a single-walled carbon nanotube (SWCNT) subjected to a longitudinal magnetic field by considering the effect of geometric nonlinearity. We consider both the Young’s modulus of elasticity and mass density of the SWCNT as stochastic with respect to the position to actually characterize the random material properties of the SWCNT. In addition, we use the theory of nonlocal elasticity to investigate the small scale effect on the nonlinear vibration of the SWCNT. By using the Hamilton’s principle, the nonlinear governing equations of the SWCNT subjected to a longitudinal magnetic field are derived. We utilize the stochastic finite element method along with the perturbation technique to compute the statistical response of the SWCNT. Some statistical dynamic response of the SWCNT, such as the mean values and standard deviations of the midpoint deflections, are computed and checked by the Monte Carlo simulation, besides, the effects of the small scale coefficients, magnetic field and the elastic stiffness of matrix on the statistical dynamic response of the SWCNT are studied and discussed.