Many coupled quantum mechanical/molecular mechanical (QM/MM) methods employ disjoint subdomains for the MM and QM regions together with link atoms to ameliorate the effects of severing covalent bonds that straddle the QM/MM interface. In the context of simulations of mechanical properties, this can be problematic because the interactions between the subdomains are then modeled by bonds involving link atoms and such bonds typically do not closely resemble those of the original system. In this paper we consider two coupling schemes that employ overlapping domains. The first is the ONlOM scheme of Morokuma et al. that includes an MM treatment of the entire system together with QM corrections for key subdomains. The second is a new approach that we will refer to as the overlapping domain link atom (ODLA) method. This method involves only a minimal overlap between the QM and MM subdomains. One important advantage of the ODLA scheme as compared to the ONIOM method is that, within the region that is treated entirely by QM methods, chemical interactions can be modeled for which reliable MM potentials are unavailable. Results of fracture studies of defected graphene sheets obtained with the ONIOM and ODLA methods are compared to benchmark results obtained by an entirely QM treatment. Both coupling methods perform well and the two coupling methods display very close agreement.