An efficient semi-analytical approach for the free vibration behavior of DWCNTs on elastic foundations based on nonlocal elasticity theory

In this study, a comprehensive analytical framework is developed to investigate the free vibration behavior of double-walled carbon nanotubes (DWCNTs) resting on an elastic foundation, based on Eringen's nonlocal elasticity theory. The DWCNTs are modeled as two coupled Euler–Bernoulli (EB) beams, explicitly incorporating intertube van der Waals (vdW) interactions and Winkler-type elastic foundation effects. The governing nonlocal differential equations are solved under various boundary conditions (BCs) using the method of initial values (IVM) in conjunction with the approximate transfer matrix (ATM) technique. This approach ensures computational efficiency while maintaining high accuracy in the determination of natural frequencies. A detailed parametric study is conducted to assess the influence of the nonlocal scale parameter, aspect ratio, and foundation stiffness on the dynamic response of DWCNTs. The results demonstrate that both the nonlocal effect and the elastic foundation significantly affect the natural frequencies. Validation is carried out by comparing the obtained results with existing benchmark solutions, showing excellent agreement. The proposed semi-analytical framework offers a robust and efficient alternative to conventional techniques and enhances the understanding of size-dependent dynamic behavior in nanostructures.