Internal resonance of vibrational modes in single-walled carbon nanotubes

We show, by molecular dynamics simulations, that 2 : 1 internal resonance may occur between a radial breathing mode (RBM) and a circumferential flexural mode (CFM) in single-walled carbon nanotubes (SWCNTs). When the RBM vibration amplitude is greater than a critical value, automatic transformations between the RBM and CFM with approximately half RBM-frequency are observed. This discovery in the discrete SWCNT atom assembly is similar to the 2 : 1 internal resonance mechanism observed in continuum shells. A non-local continuum shell model is employed to determine the critical conditions for the occurrence of observed 2 : 1 internal resonance between the RBM and CFMs based on two non-dimensional parameters and the Mathieu stability diagram.