A nonlocal shell model for mode transformation in single-walled carbon nanotubes

Abstract

A second-order strain gradient nonlocal shell model is established to study the modetransformation in single-walled carbon nanotubes (SWCNTs). Nonlocal length iscalibrated carefully for SWCNTs in reference to molecular dynamics (MD) simulationsthrough analysis of nonlocal length effects on the frequencies of the radial breathingmode (RBM) and circumferential flexural modes (CFMs) and its effects on modetransformation. All analyses show that only a negative second-order nonlocal shell model isappropriate to SWCNTs. Nonlocal length is evidently related to vibration modes andthe radius-to-thickness ratio. It is found that a nonlocal length is approximately0.1 nm in an average sense when RBM frequency is concerned. A nonlocal lengthof 0.122–0.259 nm is indicated for the mode transformation in a selected groupof armchair SWCNTs. 2:1 and 1:1 internal resonances are found for the sameSWCNT based on different models, which implies that the internal resonancemechanism depends on the model employed. Furthermore, it is shown that aneffective thickness of approximately 0.1 nm is more appropriate to SWCNTs than0.066 nm.