Revisiting the empirical relation for the maximum shearing force using plasticity and ductile fracture mechanics

• Combination of plasticity with ductile fracture mechanics in the analysis of shearing.
• New level of understanding about the empirical relation for the maximum shearing force.
• Definition of the operating conditions for the maximum shearing force being determined by plastic instability in shear or by fracture before the plastic instability.

Combination of plasticity with ductile fracture mechanics in a simple plastic flow model for sheet metal cutting provides a new level of understanding of the empirical relation between the maximum shearing force Fmax and the ultimate tensile stress σUTS of the workpiece. The constant C in Fmax = C σUTStL, where t is the sheet thickness and L the total surface length of the cut contour, is shown to be determined either (a) by the load to cause plastic instability in shear with separation (cracking) occurring subsequently or (b) by the load to cause cracking when that occurs at a punch displacement smaller than that at plastic instability in which case no instability occurs. The usually encountered range of empirical values for C, viz.: 0.65 < C < 0.85, is shown to correspond with the load for instability and depends on the work hardening index, with less-ductile materials having C at the lower end of the range. Whether cracking can precede the instability depends on the toughness/strength ratio (R/k0) of the material and the workpiece thickness, where R is the fracture toughness and k0 the yield stress in shear. The thicker the sheet and the less ductile the material, i.e. the lower the (R/k0), promotes ductile fracture at a load smaller than that for plastic instability.