With special physical structure and novel properties, single walled carbon nanotubes (SWCNTs) have attracted much attention in science and engineering. In order to develop a direct link between continuum analysis and atomistic simulation to investigate the mechanical properties of CNTs more effectively, an innovatory methodology to construct continuum model from the inner (inscribed surface) to the outer (covalent bond) is developed. Present study begins from studying Cauchy-Born rule. The differential geometry concept of inscription and the mean value theorem are adopted to modify the Cauchy-Born rule, and then this modified Cauchy-Born rule can build a hyper-elastic inscribed surface constitutive model Resorting to using Tersoff-Brenner potential, the Current theory to armchair CNTs is applied to evaluate the mechanical properties. The results are validated by the comparison with previously reported studies.