Atomistic simulations using molecular dynamics (MD) method are conducted to check the conditions of the onset of fracture at the interface edges with a variety of angles. The simulations are facilitated with model bi-material systems interacting with Morse pair potentials. Three simulation models are considered, i.e. the interface edges with angles 45°, 90° and 135°, respectively. The simulation results show that, at the instant of crack initiation, the maximum stresses along the interfaces reach the ideal strength of the interface; also, the interface energies just decrease to below the value of the intrinsic cohesive energy of the interface. And the onset of fracture at the interface edges with different geometries is controlled by the maximum stresses or the cohesive interfacial energy.