Abstract Using the lattice Green function approach, we study the crack-dislocation effects in the fracture of crystalline materials. Firstly, we calculate the Green function for the defective lattice, with dislocation and crack, by solving the Dyson equation. After the lattice Green functions have been determined, the relaxation problem for the reconstituted bonds in the cohesive zone is solved. The external force F with tensile and shear components is applied, as a pair of forces, to the atoms at the center of the crack. In this calculation the dislocation emission is chosen to be on a cleavage plane as well as on a glide plane of the two-dimensional lattice (hexagonal and graphite lattices). We compare the cleavage and dislocation emission criteria of the atomistic simulation with those of linear elasticity theory.