Flexural waves in multi-walled carbon nanotubes using gradient elasticity beam theory

The behavior of transverse waves propagating in carbon nanotubes (CNTs) in a free space and in an elastic matrix is investigated. The CNTs are modeled as Timoshenko beams of hybrid gradient elasticity theory and the elastic matrix is modeled as a bi-parameter Pasternak foundation. A governing equation with two scale factors is derived for Timoshenko beams where shear deformation and rotary inertia are taken into account. The dispersion relation of flexural waves in CNTs is given and confirmed by molecular dynamics simulations. A comparison of the phase velocity of single-walled CNTs is made when neglecting shear deformation and/or rotary inertia. The wave speed of acoustic branch is especially focused for multi-walled CNTs and the wave speed is dependent on van der Waals interaction. The effects of the surrounding medium and scale parameters on the velocity of bending waves are discussed, in particular for acoustic mode.