We examine the atomistic scale dependence of material's resistance-to-failure by numerical simulations and analytical analysis in electrical analogs of brittle crystals. We show that fracture toughness depends on the lattice geometry in a way incompatible with Griffith's relationship between fracture and free surface energy. Its value finds its origin in the matching between the continuum displacement field at the engineering scale, and the discrete nature of solids at the atomic scale. The generic asymptotic form taken by this field near the crack tip provides a solution for this matching, and subsequently a way to predict toughness from the atomistic parameters with application to graphene.