Nonlinear Analysis of Torsional Vibration in Two-Phase Nanotubes using the concept of the local/nonlocal elasticity

Using the concept of the local/nonlocal elasticity theory instead of using its equivalent differential relation is the aim of this study to consider the nonlinear torsional vibration of two-phase nanotubes (TPhNTs) keeping in mind the ill-posedness of the pure nonlocal theory in some cases. To consider the geometrical nonlinearity effect, the von-Karman relations are utilized. Then, the local stress resultants are obtained and by implementing Hamilton’s principle the nonlinear governing equation of motion (NGEoM) are derived. Next, the NGEoM is presented in terms of displacement components using two relations: the relation between the nonlocal and local stress resultants; and the relation for defining the two-phase stress resultants versus the local and nonlocal ones. In continuation, the problem is solved using the multiple scales method and the frequency relation is obtained. Finally, after verification of the formulations, effects of length, diameter, mode number, vibration amplitude, nonlocal parameter, and local mixture parameter (K) on the natural frequencies are examined. Notably, the proposed formulation successfully yields results for the case of K = 0, a case that previous methods could not address. The results of this work can be a useful reference for the more accurate design and fabrication of nano-electro-mechanical systems.