Investigation of Static Behaviour of Single-Walled Carbon Nanotubes with Helical Axes: Exact Analytical Solution

This study investigates the static behaviour of single-walled carbon nanotubes (SWCNTs) with helical axes using Eringen's nonlocal elasticity theory. The differential form of the nonlocal theory is employed to establish the relationship between local and nonlocal field variables within beam theory. These relationships are formulated in Frenet coordinates for a spatially curved beam model, yielding governing equations for helical SWCNTs with constant radius and uniform cross-section. Exact analytical solutions for the equations are obtained using the method of initial values, yielding closed-form solutions. Explicit expressions are provided for closed-coiled helical SWCNTs, representing the first exact analytical solution of Eringen's differential nonlocal elasticity theory applied to helical nanostructures. A comprehensive parametric study is conducted to systematically analyse the effects of helix geometry (pitch angle, aspect ratio, winding angle) and nonlocal parameters on the static response. The results reveal that pitch angle and helix geometry strongly influence the coupling between normal, binormal, and tangential displacements. For small pitch angles, binormal displacement dominates, while larger pitch angles substantially increase normal and tangential displacements. The parametric studies establish clear relationships between geometric configuration, nonlocal length scale, and mechanical response, providing essential design guidelines for helical nanostructures in engineering applications.