  Pristine monocrystalline graphene is claimed to be the strongest material
known with remarkable mechanical and electrical properties. However, graphene
made with scalable fabrication techniques is polycrystalline and contains
inherent nano-scale line and point defects - grain boundaries and
grain-boundary triple junctions - that lead to significant statistical
fluctuations in toughness and strength. These fluctuations become particularly
pronounced for nanocrystalline graphene where the density of defects is high.
Here we use large-scale simulation and continuum modeling to show that the
statistical variation in toughness and strength can be understood with
'weakest-link' statistics. We develop the first statistical theory of toughness
in polycrystalline graphene, and elucidate the nano-scale origins of the
grain-size dependence of its strength and toughness. Our results should lead to
more reliable graphene device design, and provide a framework to interpret
experimental results in a broad class of 2D materials.
