Variational principles for multi-walled carbon nanotubes undergoing non-linear vibrations by semi-inverse method

Variational principles are derived for multi-walled carbon nanotubes (CNT) undergoing non-linear vibrations. Two sources of non-linearity are considered in the continuum modelling of CNT with the Euler-Bernoulli beam model describing the dynamics of the CNT. One source is the geometric non-linearity, which may arise as a result of large deflections. The second source is owing to van der Waals forces between the nanotubes, which can be modelled as a non-linear force to improve the accuracy of the physical model. After deriving the applicable variational principle by the semi-inverse method, 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 therefore these principles are instrumental in obtaining the solutions of vibration problems under complicated boundary conditions.