Coupled quantum mechanical/molecular mechanical (QM/MM) calculations were used to study the effects of large defects and cracks on the mechanical properties of carbon nanotubes and graphene sheets. The semi-empirical method PM3 was used to treat the QM subdomains and a Tersoff-Brenner potential was used for the molecular mechanics; some of the QM calculations were also done using density functional theory (DFT). Scaling of the Tersoff-Brenner potential so that the modulus and overall stress-strain behavior of the QM and MM models matched quite closely was essential for obtaining meaningful coupled calculations of the mechanical properties. The numerical results show that at the nanoscale, the weakening effects of holes, slits, and cracks vary only moderately with the shape of the defect, and instead depend primarily on the cross section of the defect perpendicular to the loading direction and the structure near the fracture initiation point. The fracture stresses for defective graphene sheets are in surprisingly good agreement with the Griffith formula for defects as small as $10\mathrm{\AA }$, which calls into question the notion of nanoscale flaw tolerance. The energy release rate at the point of crack extension in graphene was calculated by the $J$-integral method and exceeds twice the surface energy density by 10% for the QM(DFT)/MM results, which indicates a modest lattice trapping effect.

• Received 12 July 2006

DOI:https://doi.org/10.1103/PhysRevB.75.075412