Studying the Vibrational Behavior of Single-Walled Carbon Nanotubes under Different Boundary Conditions using the Rayleigh-Ritz Technique

Abstract

The vibrational response of single-walled carbon nanotubes (SWCNTs) is studied in this paper. To take size effects into account, Eringen’s nonlocal elasticity equations are incorporated into the Donnell shell theory. The Rayleigh-Ritz method is employed in conjunction with the polynomial series as modal displacement functions to solve the problem. Four commonly used boundary conditions namely as simply supported-simply supported, clamped-clamped, clamped-simply supported, and clamped-free are considered. The fundamental frequencies of SWCNTs with various values of aspect ratios and nonlocal parameters are obtained. To propose the proper values of nonlocal parameter, the results of nonlocal shell model are matched with those of molecular dynamics (MD) simulations for armchair and zigzag SWCNTs through a nonlinear least square fitting procedure. The appropriate values of nonlocal parameter corresponding to each type of chirality and boundary condition are then derived. It is found that the results obtained via the presented nonlocal shell model with its proposed proper values of nonlocal parameter are in excellent agreement with those of MD simulations. 

Keywords

• carbon nanotube,nonlocal elasticity,free vibration,donnell shell theory,molecular dynamics simulation