A computational method for evaluating the resistance of fracture capacity of austenitic stainless steel (SUS304L) under ambient and cryogenic temperatures was proposed in the present study. To do this, a unified elasto-viscoplastic-damage model which can take into consideration the material nonlinearity under cryogenic temperature was adopted and implicitly formulated. In addition, the developed model is implemented into ABAQUS as a type of user-defined material subroutine UMAT. In order to verify the proposed method, the simulation results of resistance of fracture capacity are compared to a series of crack tests such as double-edge-cracked and center-cracked tension tests of SUS304L plates under ambient and cryogenic temperatures. Namely, the relationships between force and displacement as well as allowable crack length and allowable stress regarding to room and cryogenic temperatures are evaluate/predicted using the proposed method.