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

Abstract 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.