Variational Formulation of Vibrations of Multi-Walled Carbon Nanotubes With Geometric and Physical Nonlinearities

Abstract

Variational principles are derived for multi-walled carbon nanotubes (CNT) undergoing nonlinear vibrations. Two sources of nonlinearity are considered in the continuum modeling of CNTs with the Euler-Bernoulli beam model describing the dynamics of the CNTs. One source is the geometric nonlinearity which may arise as a result of large deflections. The second source is due to van der Waals forces between the nanotubes which can be modeled as a nonlinear force to improve the accuracy of the physical model. After deriving the applicable variational principle, Hamilton’s principle is given. Natural and geometric boundary conditions are derived using the variational formulation of the problem. Several approximate and computational methods of solution such as Rayleigh-Ritz and finite elements employ the variational formulation of the problem and as such these principles are instrumental in obtaining the solutions of vibration problems under complicated boundary conditions.