A numerical algorithm based on the element deletion method and rigid-viscoplastic finite element approach depending on Cockcroft-Latham and specific plastic work fracture criteria was applied to predict formation and evolution of possible cracking in a cold extrusion of aluminum and steel alloys. The Cockcroft-Latham fracture criterion induced an internal crack while an external crack occurred owing to the specific plastic work criterion in simulations. As a result, the Cockcroft-Latham criterion was found to be valid for predicting chevron cracking in comparison with the experimental observation available in the literature. Using the Cockcroft-Latham criterion, cracking was carefully investigated in terms of the size of the crack and gap distance between cracks depending on the number of elements and boundary condition at the punch interface. The critical damage values for the Cockcroft-Latham fracture criterion were also calculated based on the tensile instability and fracture conditions to investigate their effect on possible cracking. Finally, a processing map based on the Cockcroft-Latham fracture criterion for preventing chevron cracking in the cold extrusion of commercially available steel alloy was developed by considering processing parameters such as reduction in area and semicone angle. According to this investigation, the developed element deletion method with the Cockcroft-Latham fracture criterion was reasonably accurate for carrying out chevron cracking analyses in the cold extrusion with proper selection of a critical damage value.