Application of nonlocal elastic shell theory in wave propagation analysis of carbonnanotubes

Abstract

Wave propagation in carbon nanotubes (CNTs) is studied based on the proposed nonlocalelastic shell theory. Both theoretical analyses and numerical simulations have explicitlyrevealed the small-scale effect on wave dispersion relations for different CNT wavenumbersin the longitudinal and circumferential directions and for different wavelengths in thecircumferential direction. The applicability of the proposed nonlocal elastic shell theory isespecially explored and analyzed based on the differences between the wave solutionsfrom local and nonlocal theories in numerical simulations. It is found that thenewly proposed nonlocal shell theory is indispensable in predicting CNT phonondispersion relations at larger longitudinal and circumferential wavenumbers andsmaller wavelength in the circumferential direction when the small-scale effectbecomes dominant and hence noteworthy. In addition, the asymptotic frequency,phase velocities and cut-off frequencies are also derived from the nonlocal shelltheory. Moreover, an estimation of the scale coefficient is provided based on thederived asymptotic frequency. The research findings not only demonstrate greatpotential of the proposed nonlocal shell theory in studying vibration and phonondispersion relations of CNTs but also signify limitations of local continuum mechanicsin analysis of small-scale effects, and thus are of significance in promoting thedevelopment of nonlocal continuum mechanics in the design of nanostructures.