Coupled quantum mechanical/molecular mechanical ͑QM/MM͒ 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 ͑DFT͒. 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 Å, 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 QM͑DFT͒/MM results, which indicates a modest lattice trapping effect.

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Carbon nanotubes ͑CNTs͒ can undergo collapse from their customary cylindrical configurations to ribbons. The energy minima corresponding to these two states are identified using either atomistic molecular mechanics or the theory of finite crystal elasticity with reduced dimensionality. The minimum energy path between these two minima is found using the nudged elastic band reaction-pathway sampling scheme. The energetics of CNT collapse is explored for both single-and multi-walled CNTs as well as small bundles. The process has a strong diameter dependence, with collapse being more favorable for the larger diameter tubes, but is nearly independent of chirality. The saddle point always lies close to the collapsed state, and the absolute barrier energieseven for fairly short tube lengths-are sufficiently high, even when the reaction is highly exothermic, that thermal activation cannot initiate collapse via this pathway. The hydrostatic pressure required to buckle and collapse CNTs of various diameters is discussed.

SUMMARYWe present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain-decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least-square approximation which includes both extensional and angle-bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single-layer atomic sheets and nanotubes are given.

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