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Due to the difference of nanotube diameters, the single-walled carbon nanotubes (SWCNTs) show the different energy and mechanical properties. In order to take the effect of the curvature of nanotubes into account in the modeling of those structures, the present paper proposes an atomistic based continuum model with using a type of modified Cauchy-Born to link the continuum strain energy to the interatomic potential. This modified Cauchy-Born is developed by incorporating the concept of differential mean value theorem into the standard Cauchy-Born rule. The present model not only can bridge the microscopic and macroscopic length scales, but also can investigate the curvature effect of a single layer film on the continuum level. Application of the current model to armchair carbon nanotubes and graphite shows an excellent prediction of the size dependent strain energy which are compared in a good agreement with the existing experimental and theoretical results.