Based on an equivalent continuum cylindrical shell model, we have predicted the intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube. It is found that the radial breathing and flexural vibration modes may appear intermittently, in certain circumstances, when the dominant parameters of the problem are in the instable region of the Mathieu stability diagram. The coupled nonlinear differential equations of the radial breathing and the flexural vibration modes are presented to a fourth-order approximation, which is then solved using a finite series expansion method. Intermittent transformation between radial breathing and flexural vibration modes is predicted, which may influence the physical properties of the carbon nanotube and the Raman spectroscope measurements.

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.

a b s t r a c tAtomic-scale finite element analyses show that 2:1 internal resonance mechanism exists in a range of single-walled carbon nanorings (10-60). When an initial radial breathing mode (RBM) vibration with sufficiently high velocity is imposed to a nanoring, circumferential flexural modes (CFMs) can be excited after a period of RBM-dominated vibration. Then, mode transformations between RBM and the excited CFMs can be observed in the subsequent vibration process. When single-walled carbon nanorings are assembled to make double-or triple-walled carbon nanorings, the 2:1 internal resonance may change to 1:1 internal resonance in a specific ring due to the strong interactions between these nanorings. Furthermore, mode transformations between RBM and the excited CFMs can become unstable in a specific ring if the excited CFMs in neighbouring layer rings are not symmetrically matching between each other. 2:1 internal resonance is also shown in selected armchair single-walled carbon nanotubes except in a special case (armchair (9, 9)), in which 1:1 internal resonance occurs.

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