Scale effect on wave propagation of double-walled carbon nanotubes

Scale effect on transverse wave propagation in double-walled carbon nanotubes (DWNTs) is studied via nonlocal elastic continuous models. The nonlocal Euler–Bernoulli and Timoshenko beam models are proposed to study the small-scale effect on wave dispersion results for DWNTs with respect to the variation of DWNT’s wavenumbers and diameters by theoretical analyses and numerical simulations. The cut-off frequency, asymptotic phase velocity, and asymptotic frequency are also derived from the nonlocal continuum models. A rough estimation on the scale coefficient used in the nonlocal continuum models is proposed for the study of carbon nanotubes (CNTs) in the manuscript. The diameter-dependent dispersion relations for DWNTs via the nonlocal continuum models are observed as well. In addition, the applicability of the two beam models is explored by numerical simulations. The research work reveals the significance of the small-scale effect on wave propagation in multi-walled carbon nanotubes.