Analytical solution for dynamic behavior of multiwall carbon nanotubes subjected to mechanical shock loading

Dynamic behavior and frequency analysis of multiwall carbon nanotubes (MWCNTs) with finite length are investigated using an analytical method. By applying multiple elastic shells and a linearized model of van der Waals forces, a comprehensive continuum dynamic model of MWCNTs is developed. Also, by utilizing Laplace transform, time histories of MWCNTs subjected to shock lading are obtained. Then by using the fast Fourier transform (FFT), the time responses are transferred to the frequency domain and natural frequencies are found. The accuracy of results is verified by comparing the results of sudden loading with those obtained by numerical methods. Furthermore, an abrupt unloading after ramp loading is simulated, and the dependency of diameter and length on the axial and radial natural frequencies of MWCNTs is examined. Also, by characterizing the longitudinal displacement, the wave propagation velocities are obtained and an explicit expression is found for the axial natural frequencies of MWCNTs subjected to shock loading.