In this paper, the higher-order elasticity constants are evaluated in the theoretical scheme of higher-order continuum. A single-walled carbon nanotube is treated as a higher-order continuum cylindrical tube with a thin wall, and the representative cell is chosen as a triangle unit that contains four carbon atoms. The Brenner potential is employed to describe the C-C atomic interaction, and the higher-order constitutive relationship is derived by virtue of the higher-order Cauchy-Born rule. The higher-order elasticity constants of carbon nanotubes are evaluated based on the derived higher-order constitutive model, which can provide a foundation for the further analysis of the mechanical properties of carbon nanotubes in the theoretical scheme of higher-order continuum.