Int. J. Mech. Sci. Vol.35, No. 3/4,pp. 209-229,1993 0020..-7403/93 $6.00 + .00 Printed in Great Britain. PergamonPressLtd CLOSED-FORM SOLUTION FOR WEDGE CUTTING FORCE THROUGH THIN METAL SHEETS T. WIERZBICKI and P. THOMAS Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Abstract--A simple kinematic model is developed which describes the main features of the process of the cutting of a plate by a rigid wedge. It is assumed in this model that the plate material curls up into two inclined cylinders as the wedge advances into the plate. This results in membrane stretching up to fracture of the material near the wedge tip, while the "flaps" in the wake of the cut undergo cylindrical bending. Self-consistent, single-term formulas for the indentation force and the energy absorption are arrived at by relating the "far-field" and "near-tip" deformation events through a single geometric parameter, the instantaneous rolling radius. Further analysis of this solution reveals a weak dependence on the wedge angle and a strong dependence on friction coefficient. The final equation for the approximate cutting force over a range of wedge semi- angles l0 ° < 0 < 30 ° and friction coefficients 0.1 </~ _< 0.4 is: F = 3.28ao(3t)°'21°'*t t'6/./°"4, which is identical in form and characteristics to the empirical results recently reported by Lu and Calladine lint. J. Mech. Sei. 32, 295-313 (1990)]. This analysis is believed to resolve a controversy recently developed in the literature over the interpretation of plate cutting experiments. NOTATION C empirical constant from Ref. [-15] /~b rate of bending work /~., rate of membrane work /~shcar rate of shear work F total cutting force Fp plastic resistance force Ff frictional resistance force g rate of curvature I cut length lp length of membrane stretch zone M 0 fully plastic bending moment M~a bending moment tensor N~p membrane force tensor n power exponent R rolling radius Ro essential work to fracture t plate thickness ti~ plate velocity vector V wedge velocity W work of cutting w near-tip lift function ct plate tilt angle 6t crack opening displacement parameter ~1 dimensionless crack opening displacement parameter t/ ratio of cutting force to withdrawal force from Ref. [15] strain rate e~ strain tensor 0 wedge semi-angle /~p rate of curvature tensor /~ friction coefficient ~(x) membrane deformation zone boundary a o material flow stress a=~ stress tensor rate of rotation 209