The Upper Bound Approach to Plane Strain Problems Using Linear and Rotational Velocity Fields—Part I: Basic Concepts

This paper investigates an upper bound approach to plane strain deformation of a rigid, perfectly plastic material. In this approach the deformation region is divided into a finite number of rigid triangular bodies that slide with respect to one another. Neighboring rigid body zones are analyzed in specific cases where the zones are (1) both in rotational motion, (2) one in linear, the other in rotational motion and (3) both in linear motion. Specific equations are presented that describe surfaces of velocity discontinuity (shear boundaries) between the moving bodies, and the velocity discontinuities and shear power losses for each of the three cases. The shape of the surface of velocity discontinuity is uniquely determined by the velocity ratios of neighboring bodies, their relative directions of motion and, where applicable, the positions of their centers of rotation. Where one or both neighboring bodies exhibit rotational motion, the surface of velocity discontinuity is found to be a cylindrical surface. In the case of two neighboring bodies, each with linear motion, the surface of velocity discontinuity is found to be planar. The velocity discontinuity is found to be constant along the entire surface of velocity discontinuity. The characteristics of the surfaces of velocity discontinuity in plane strain deformation are investigated. The upper-bound approach to plane strain problems can be successfully adapted to real metal forming processes, including sheet and strip drawing, extrusion, forging, rolling, leveling, ironing, and machining.