Application of nonlocal beam models for carbon nanotubes

Investigations of wave and vibration properties of single- or multi-walled carbon nanotubes based on nonlocal beam models have been reported recently. However, there are numerous inconsistencies in the handling of the governing equations, applied forces, and boundary conditions based on some of the reported nonlocal beam models. In this paper, the consistent equations of motion for the nonlocal Euler and Timoshenko beam models are provided, and some issues on the nonlocal beam theories are discussed. The models are then applied to the studies of wave properties of single- and double-walled nanotubes. The wave and vibration properties of the nanotubes based on the presented nonlocal beam equations are studied, and scale effects are discussed.