Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects

In this article, the size-dependent free vibration of nanotubes with surface effects is investigated. An efficient shell-core-shell model is introduced to simulate the structure which includes the effect of additional surface elasticity and surface residual stresses. Love’s continuum model for longitudinal wave propagation is employed, which accounts for the effects of lateral contractions on the axial vibration of the structure. On the other hand, the Timoshenko beam model is modified to include the surface effects as well as shear deformations for transverse vibration of nanotubes under axial load. Temperature effects on material properties are also investigated to show the general trend of size dependencies. The generalized differential equation for each case is derived under general external loadings using Hamilton’s principle. The axial vibration is analyzed for different material types and boundary conditions. The linear stiffness modulus is assumed for the elastic bed in both directions with due dependence on the size of the nanotubes.