Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory

The equation of torsional motion is presented in this paper to investigate the free torsional vibration behaviors of tubes made of a bi-directional functionally graded (FG) material, which is composed of two different materials with continuously varying along the radius and length directions. To incorporate the size effect of long-range forces, the nonlocal elasticity theory is employed to derive the difference equation of torsional motion, which can be reduced to the classical governing equation by simply setting a zero nonlocal parameter. Suppose that the effective material properties of the nanotube vary in the length direction according to an exponential distribute function and in the radius direction according to a power-law function. The closed-form solutions of torsional frequencies and mode shapes are derived. It is shown that the torsional frequencies can be significantly affected by the through-radius and through-length gradings of the bi-directional FG nanotubes and hence can be prescribed by tailoring the bi-directional nano-structures of the FG material. The torsional frequencies can be increased with the decreasing nonlocal parameter, whereas the size-dependent behaviors on the mode shape cannot be observed.