Nonlinear finite element analysis of vibration of multi-walled carbon nanotubes with geometric imperfection resting on elastic foundations in a thermal–magnetic environment

Abstract

This paper presents nonlinear finite element analysis of the systems of partial differential equations (PDEs) that govern the vibration dynamics and stability responses of slightly curved Multi-walled carbon nanotubes (MWCNTs) by means of nonlocal elasticity principle. The MWCNTs under investigation rest on Pasternak and Winkler foundations and operate in a magneto–thermal environment. With the aid of the nonlinear finite element analysis (FEA) of the developed governing equations, the effects of primary geometric imperfection, nonlocal parameter, magnetic field, temperature, elastic foundation parameters, number of walls, and boundary conditions on the nonlinear vibration of the CNTs are investigated. The simulated and presented results show how frequency ratio reduces with an increase in the number of elastic foundation parameters, strength of magnetic field term, temperature, tube wall numbers and ratio of curvature radius to length of the slightly curved nanotubes under investigation. Similar trend of results is recorded for the different boundary conditions considered. However, frequency ratio is highest for MWCNTs with clamped-pinned boundary conditions and lowest for the case of clamped supports at both ends. Furthermore, it is established that the vibration of quadruple MWCNTs approximates that of linear vibration even as the foundations parameters are augmented. Such result can be used to restrains chaos in vibration of CNTs. It is hoped that the results from this study will help in the design of MWCNTs for various applications.

Keywords

• Nonlinear finite element method
• Nonlocal elasticity theory
• Multi-walled carbon nanotubes
• Small-scale effects