Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories

In the context of nonlocal continuum theory, seeking an analytical solution to the equations of motion of stocky double-walled carbon nanotubes (DWCNTs) with arbitrary boundary conditions is a very problematic task. Thereby, proposing efficient numerical techniques for frequency analysis and optimal design of such nanostructures is of great advantageous. Herein, free transverse vibration of an elastically supported stocky DWCNT embedded in an elastic matrix under initial axial force is of interest. To this end, the equivalent continuum structures (ECSs) associated with the innermost and outermost tubes are taken into account. The interaction of the DWCNT with its surrounding matrix is modeled using lateral and rotary continuous springs. Through consideration of interlayer van der Waals forces via an equivalent spring system, the two tubes are appropriately interacted. Using Hamiltonʹs principle, the dimensionless equations of motion of elastically supported DWCNTs are established using nonlocal Rayleigh, Timoshenko, and higher-order beam theories. The unknown fields of the equations of motion for each model are discretized in the spatial domain using reproducing kernel particle method. After tedious calculations, the set of eigenvalue equations pertinent to each model is extracted and numerically solved. The convergence checks of the proposed numerical models in predicting flexural frequencies of DWCNTs are carried out. The obtained results are also compared with those of other works and a reasonably good agreement is achieved. Through various numerical studies, the influences of slenderness ratio, ratio of the mean radius to the thickness of the ECSs, small-scale parameter, initial axial force, lateral and rotational stiffness of the surrounding matrix on the flexural frequencies of stocky DWCNTs are carefully examined for different boundary conditions. The capabilities of the proposed nonlocal models in capturing the flexural frequencies of stocky DWCNTs are discussed as well.