XCII. A theoretical investigation of the compression of a ductile material between smooth flat dies

Abstract

A theory is suggested to describe the distribution of stress and velocity in a block of ductile material symmetrically indented on opposite sides by two smooth flat dies. The theory is two- dimensional and an ideal plastic-rigid material is assumed. Slip-line fields are proposed for all ratios of width of dies to height of block greater than one. It is found that the velocity field transforms into its corresponding slip-line field. Other plane plastic problems where this correspondence occurs are cited. Since the velocities are known along only one slip-line boundary, the usual numerical procedure for calculating the field has to be modified and the use of influence coefficients is introduced. Solutions are computed for width/height ratios between one and two, and it is observed that simple analytical relations hold between these ratios and the corresponding velocity discontinuities in the field. The calculated mean pressures on the dies are compared with upper and lower bound curves derived by Hill on the basis of extremum principles. The deformation after different finite indentations is illustrated by photographs of experiments using (a) plasticine models, and (b) aluminium alloy strip. Similar fields are proposed for extrusion or drawing through a wedge-shaped die with large reductions, but no solutions have been computed.