We study inextensional vibrations of zig-zag single-walled carbon nanotubes using nonlocal elasticity theories and molecular mechanics simulations employing MM3 potential. We find that the frequency expressions for the Rayleigh and Love modes of inextensional vibrations, predicted by the stress and strain gradient theories, differ from the classical continuum theory expressions by a multiplicative factor only. The factor is different for stress and strain gradient theories. We also observe that the strain gradient theory with positive sign exhibits a saturation-like behavior of the inextensional mode frequency with the circumferential wave number. Using this fact and molecular mechanics simulation data we derive an expression for the nonlocal parameter. © 2013 Elsevier Ltd. All rights reserved.