Dynamic Instability Analysis of Embedded Multi-walled Carbon Nanotubes under Combined Static and Periodic Axial Loads using Floquet- Lyapunov Theory

The dynamic instability of single-walled carbon nanotubes (SWCNT), double-walled carbon nanotubes (DWCNT) and triple-walled carbon nanotubes (TWCNT) embedded in an elastic medium under combined static and periodic axial loads are investigated using Floquet–Lyapunov theory. An elastic multiple-beam model is utilized where the nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. Moreover, a radius-dependent vdW interaction coefficient accounting for the contribution of the vdW interactions between adjacent and non-adjacent layers is considered. The Galerkin’s approximate method on the basis of trigonometric mode shape functions is used to reduce the coupled governing partial differential equations to a system of extended Mathieu-Hill equations. Applying Floquet–Lyapunov theory, the effects of elastic medium, length, number of layers and exciting frequencies on the instability conditions of CNTs are investigated. Results show that elastic medium, length of CNTs, number of layer and exciting frequency have significant effect on instability conditions of multi-walled CNTs.