After an intense earthquake, repair or replacement the damaged elements which absorb the seismic energy is very difficult. One of the most suitable and cost-effective elements for this purpose is steel slit dampers (SSDs). The design of these dampers should be as such to experience yielding and failure prior to the yielding of main structural elements. For this reason, in analyzing and designing structures equipped with dampers, determination the fracture time of dampers is very important. Currently, there is no proper criterion for estimating the failure time of an SSD. Hence, in this study, using the finite element method and ABAQUS software, most of possible criteria for estimating the failure time of the dampers are investigated. In this investigation, the fracture initiation of an SSD is estimated by monitoring the stress or strain level at the critical elements of an SSD. Results showed that among the various types of criteria, the use of the combination of the plastic equivalent strain and the fracture index, may lead to an acceptable estimation of the time of failure of the dampers.