Leak-before-break (LBB) assessment of nuclear piping involves ductile fracture analysis of pipes or elbows with postulated through-wall cracks. Due to the fact that the crown part of an elbow is one of the positions that crack initiation occurs in most frequently, the calculation of J-integrals to investigate fracture behavior are important research topics. This paper proposes a 3-D finite element model of an elbow embedded with an axial through-wall crack to estimate the J-integral parameters under bending moment. The J-integral values can be calculated by using ABAQUS and taking into account the effects of geometrical and model of material in non-linear analysis. The results show that the non-linear deformation and contact condition of crack surfaces play important roles for the J-integral values. In addition, the J values estimated by the proposed model are more conservative and realistic than previous studies.