Axial vibration of embedded nanorods under transverse magnetic field effects via nonlocal elastic continuum theory

In this paper, the influence of a transverse magnetic field on the axial vibration of nanorods such as carbon nanotubes is theoretically modelled using nonlocal elasticity approach. Nonlocal elasticity handles the small-scale effects of vibrating nanorod. Nonlocal rod theory is utilised and detailed analytical solutions are obtained. The nanorod is assumed sensitive to magnetic field. Governing equations for nonlocal axial vibration of the nanorod under a transverse magnetic field are derived considering the Lorentz magnetic force obtained from Maxwell's relation. Nonlocal rod embedded in an elastic medium is also considered. Clamped–clamped and clamped–free boundary conditions are considered. A simple analytical expression for the natural frequencies is proposed. Results from the analytical model developed show that the transverse magnetic field exerted on the nanorod theoretically dampens the nonlocal effect of atom–atom interactions by increasing the natural frequencies. The variation of frequency with the increase of axial stiffness of elastic medium for an embedded nanorod in a magnetic field is more nonlinear with nonlocal effect than without nonlocal effects. This study provides the necessary physical insights for experimental studies on the dynamics of magnetically sensitive nanorods.