Torsion of a functionally graded material

Within the framework of the nonlocal strain gradient theory, a size-dependent shaft model, which can account for the through-radius power-law variation of two-constituent functionally graded (FG) materials, is derived to investigate the small-scaled effects on the static and dynamic torsion behaviors. The equations of torsional motion and corresponding boundary conditions of the size-dependent FG shaft are derived in terms of the Hamilton's principle. The shaft models can account for the small-scaled effects of the inter-atomic long-range force and the microstructure deformation mechanism by introducing material length scale and nonlocal parameters. An analysis on the harmonic propagation with time torsional waves in a nonlocal strain gradient FG shaft is carried out. In the case of clamped-clamped boundary conditions, analytical solutions are obtained for the free vibration and static torsion problems of nonlocal strain gradient FG shafts. The effects of small-scaled parameters and the through-radius power-law variation of a two-constituent FG material on wave propagation, free vibration and static torsion are investigated in details.