Small ligaments connecting the crack surfaces just behind the moving crack front are assumed to exist during fast cleavage crack growth. The production and conduction of heat during the process of tearing these ligaments are studied. The straining is computed by assuming an elastic visco-plastic material model. The produced heat during plastic work is calculated. Heat conduction is considered. The results reveal that the temperature increases several hundred centigrades for a low yield stress. For a high yield stress the temperature increases to > 1000 deg C. The appropriateness of the selected constitutive relationship is discussed in view of the result (for structural steels).