The thermodynamics of three-dimensional, single-component elastic crystalline solids was developed by Slattery and Lagoudas (2005). Considering the in¯nitesimal deformations, the stress can be expressed as a function of the lattice vectors and density in the reference configuration and ¹(I;mn), which is defined as the derivative of specific Helmoholtz free energy with respect to the I(mn). Using the Cauchy - Born rule to connect the interatomic potential energy and the specific Helmholtz free energy, it is possible to calculate the elastic properties of both nano-scale materials such as carbon nanotubes and macro-scale materials such as diamond and silicon. In this study, we used Terso® (1988a) - Brenner (1990b) Potential, Terso® (1988b) potential and Finnis and Sinclair (1984) potential for carbon, silicon, and vanadium systems respectively. Using the interatomic potentials to describe the specific Helmholtz free energy, the elastic properties of graphite, diamond, silicon and vanadium were calculated. This method was also extended to the calculation of Young's modulus of single-walled carbon nanotubes (SWCNTs), which are composed of a two dimensional array of carbon atoms. For SWCNT, we get good agreement with the available experimental data. For diamond and silicon, C11 and C12 were consistent with both the superelastic model and the experimental data. The difference of C44 between the calculation and experimental data was due to accuracy of the potential functions.