Twisting statics of functionally graded nanotubes using Eringen's nonlocal integral model

In this contribution, a nonlocal integral model is formulated to investigate the twisting static behaviors of through-radius functionally graded (FG) nanotubes based on Eringen's nonlocal integral elasticity. In comparison to the widely-used nonlocal differential model in the literature, the nonlocal integral model developed herein is self-consistent and well-posed. Closed-form solutions of rotational angle are derived for both clamped-clamped and clamped-free FG nanotubes. It is shown that the nonlocal parameter has a stiffness-softening effect on both clamped-clamped and clamped-free tubes, and the scaling effect is more sensitive to the former than the latter. The serious unresolved paradox that the torsional angle of a clamped-free FG nanotube with a concentrated torque at free end, when being modeled using the nonlocal differential model, is found to be identical with the classical result can now be well resolved based on the developed nonlocal integral model. The illustrative examples show that the maximum torsional angle can be prescribed by tailoring the through-radius microstructures of the nanotubes, and the increment of the nonlocal parameter and the applied torques can increase the torsional angle, while the increment of the effective shear rigidity can decrease the torsional angle.