Analytical bending solutions for carbon nanotubes: paradoxes and their resolutions through doublet mechanics

This study focuses on the bending behavior of carbon nanotubes modeled as Timoshenko nano-beams under various boundary and loading conditions, which have not been previously examined within the framework of doublet mechanics. The sixth-order differential equation derived from equilibrium equations is solved analytically using the displacement field in doublet mechanics. The analyzes reveal paradoxes in some bending solutions, which have not been reported before. For example, the scale parameter effect disappears in clamped–clamped beams subjected to uniform load and in cantilever beams with a point load at the free end. Additionally, cantilever beams under uniform load exhibit unexpected stiffening behavior, which differs from that of clamped-pinned cases. To resolve these inconsistencies, a new analytical solution is developed based on variationally consistent boundary conditions, highlighting the originality of the developed approach. This approach reinstates the anticipated scale-dependent softening across all boundary conditions examined. Furthermore, a novel macro-stress expression has been identified and validated, thereby extending the previous formulations documented in the literature. The effects of the scale parameter, slenderness ratio, and boundary conditions on bending behavior are thoroughly examined. The results indicate that the influence of the scale parameter is most pronounced for the clamped–clamped, clamped-pinned, and clamped-free nano-beams, respectively. The proposed framework elucidates existing paradoxes and advances the application of doublet mechanics as a reliable analytical approach for nano-scale structural design.