Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Timoshenko beams

Variational formulation for multi-walled carbon nanotubes subject to buckling is derived by the semi-inverse method with governing equations based on the nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. The nonlocal theory improves the range and applicability of the physical model by modelling the nano-scale phenomenon more accurately. The natural and geometric boundary conditions are derived, which lead to a set of coupled boundary conditions for multi-walled nanotubes as opposed to uncoupled boundary conditions in the case of simply supported and clamped boundaries and also in the case of a local theory. The variational principle and the corresponding Rayleigh quotient facilitate the application of approximate and numerical methods of solution.