Curved double-walled carbon nanotubes (CDWCNTs) are crucial components in nanoelectronics, mechanical sensors, and composite materials due to their unique geometry and structural properties. Electron microscopy images have revealed that carbon nanotubes are rarely perfectly straight, often exhibiting curvature or waviness along their length due to inherent geometrical imperfections. The accurate mechanical modeling of these structures is essential, particularly to capture size-dependent effects that classical elasticity theories fail to account for. In this study, a novel analytical framework was introduced for combining the initial value method with the approximate transfer matrix approach to analyze the mechanical behavior of CDWCNTs under anti-plane loading within the framework of nonlocal elasticity theory. The proposed methodology provides an effective and computationally efficient alternative to traditional analytical approaches. By analyzing displacements, rotations, bending moments, and shear forces, substantial deviations were revealed between classical and nonlocal elasticity solutions, particularly as the dimensionless nonlocal parameter $$R/\gamma $$ R / γ decreased. The results show that nonlocal effects become dominant at smaller size parameters, especially in displacements, rotations, and bending moments, while shear forces remain unaffected. These findings emphasize the critical role of nonlocal effects in accurately predicting nanoscale mechanical responses and offer valuable insights for modeling advanced nanostructures in emerging technologies, such as microelectromechanical systems and nanotechnology. Convergence studies have confirmed the accuracy and stability of the proposed approach, thereby establishing this as a robust tool for modeling nanoscale structures.

Curved nanobeams are fundamental components of nanoscale systems, such as nano and microelectromechanical systems, where accurate modeling of small-scale effects is essential for achieving high precision. This study introduces a novel framework for analyzing the in-plane static behavior of curved nanobeams resting on an elastic foundation, particularly for small displacements. Nonlocal constitutive equations are derived based on nonlocal elasticity theory and solved using the initial value method and the approximate transfer matrix approach to address the challenges arising from the high degree of statical indeterminacy. A convergence analysis is conducted, showing that the proposed method provides a systematic and computationally efficient solution. A parametric analysis reveals that the nonlocal parameter, elastic foundation, and opening angle significantly influence the displacements in the tangential and normal directions, the rotation and bending moment in the binormal direction, the shear force in the normal direction, and the axial force in the tangential direction. These findings further elucidate the mechanics of curved nanostructures, contributing to the design and optimization of nanoscale devices.

In this study, a novel method is presented to analyze the buckling behavior of single‐walled carbon nanotubes (SWCNTs) using the initial value method (IVM) in conjunction with the approximate transfer matrix, within the framework of nonlocal elasticity theory. The study aims to accurately approximate critical buckling load parameters under various boundary conditions, without encountering high computational requirements. IVM enables the computation of displacements and stress resultants along the entire beam from given initial conditions. The approximate transfer matrix is employed to analyze system states at different points through successive integration of solutions, generating the principal matrix needed for IVM and ensuring systematic and precise results that optimize the accuracy of the analysis. A convergence study confirms the effectiveness and precision of the proposed method, revealing a decrease in the critical buckling load parameters as the nonlocal parameter increases, applicable across all boundary conditions studied (simply supported, clamped–clamped, clamped–simply supported, and clamped‐free). These results underscore the need to incorporate nonlocal effects for more accurate nanostructure mechanics predictions. The integration of IVM and the approximate transfer matrix provides a computationally efficient alternative to traditional numerical and semi‐analytical methods, aiding researchers and engineers working with SWCNTs and other nanomaterials.

Single-walled carbon nanotubes (SWCNTs) possess exceptional mechanical and dynamic properties, making them indispensable for diverse applications such as nanoelectromechanical systems, sensor technologies, energy storage devices and composite material design. However, a more accurate, efficient and systematic analysis of their vibrational behaviour under conditions involving elastic foundation stiffness and nanoscale effects remains essential for tailoring their applicability to specific purposes. This study introduces a novel methodological approach to analyse the vibrational behaviour of SWCNTs resting on an elastic foundation, leveraging nonlocal elasticity theory to incorporate small-scale effects. The proposed framework combines the initial value method with the approximate transfer matrix technique to efficiently compute natural frequencies. Vibrational analyses were performed under various boundary conditions, including simply supported, clamped–clamped and clamped–simply supported. Parametric studies revealed that increases in the nonlocal parameter significantly reduce natural frequencies, whereas a higher elastic foundation stiffness enhances the system rigidity and increases vibrational frequencies. This framework offers substantial computational efficiency and accuracy, presenting a robust tool for modelling SWCNT dynamics. The results provide valuable insights for designing effective SWCNT-based structures, contributing to their applications in nano-engineering and other engineering domains requiring precise vibrational analysis.