In this study, the change of the resistivity of carbon nanotubes and graphene sheets under strain was analyzed by applying a quantum chemical molecular dynamics analysis and the first principle calculation. Various combinations of double-walled carbon nanotube structures were modeled for the analysis. The change of the band structure was calculated by changing the amplitude of the applied strain. It was found in some cases that the band structure changes drastically from metallic band structure to semiconductive one, and this result clearly indicated that the electronic conductivity of the MWCNT decreased significantly in a three-dimensional strain field. It was also found that there is a critical strain at which the electronic band structure changes from metallic to semiconductive and vice versa. This result indicated that the metallic CNT changes a semiconductive CNT depending on the applied strain field. The effect of the diameter of the zigzag type CNT on the critical strain of buckling deformation was analyzed under uni-axial strain. In this analysis, the aspect ratio of each structure was fixed at 10. It was found that the critical strain decreased monotonically with the decrease of the diameter. This was because that the flexural rigidity of a cylinder decreased with the decrease of its diameter when the thickness of the wall of the cylinder was fixed. It was found that the critical strain decreased drastically from about 5% to 0.5% when the aspect ratio was changed from 10 to 30. Since the typical aspect ratio of CNTs often exceeds 1000, most CNTs should show buckling deformation when an axial compressive strain is applied to the CNTs. Finally, the shape of a six-membered ring of the CNT was found to be the dominant factor that determines the electronic band structure of a CNT. The change of the band structure of a grapheme sheet was analyzed by applying the abinitio calculation based on density functional theory. It was found that the fluctuation of the atomic distance among the six-membered ring is the most dominant factor of the electronic band structure. When the fluctuation exceeded about 10%, band gap appeared in the deformed six-membered ring, and thus, the electronic conductivity of the grapheme sheet change from metallic one to semiconductive one. It is therefore, possible to predict the change of the electronic conductivity of a CNT by considering the local shape of a six-membered ring in the deformed CNT.