The Theoretical Analysis of Metal-Forming Problems in Plane Strain

Abstract

Abstract The plastic flow in plane strain of an ideally plastic material subjected to large strains is considered. Elastic strains are negligible and a rigid-plastic type of analysis is adopted. The equations to be satisfied are detailed, and they include stress and velocity equations in the plastic regions, as well as consideration of the stress field in the rigid regions to check the validity of small strains there. Complete solutions satisfying these conditions require the determination of the rigid-plastic boundaries to delineate the regions in which the various conditions must be satisfied. The fallacy of static determinacy of such problems in terms of the stress equations only is emphasized. The study of complete solutions indicates errors in solutions commonly accepted in the literature which are based on the stress equations only. Examples are discussed. The general occurrence of velocities in the boundary conditions of forming problems is pointed out, and the difficulty of setting such problems in terms of boundary stresses only is illustrated by examples.