An upper-bound approach to plane-strain forging and extrusion—I

Abstract

To calculate the load required to perform certain kinds of plastic working operations and to account approximately for the modes of flow encountered, a description of a new upper-bound approach is first presented. To facilitate the calculation of good upper bounds, the concept of a unit region is introduced. After finding that the type of velocity field which consists of several rigid triangular parts is the most suitable for the present study, the lowest rate of energy dissipation and the related form of rigid-triangle velocity field are determined for each of the unit rectangular deforming regions having various height-width ratios and frictional boundary conditions. On analysing a particular working problem, the work-piece is supposed to be composed of several unit regions and the lowest upper bound for the working pressure, as well as the most suitable velocity field inside the material, is obtainable directly from those determined for unit regions.There follows from this method applied to several forging problems with open-dies—(a) Results on the compression of a plate between two flat parallel dies and on the indentation of a flat punch into a semi-infinite or a finite body and these are in close agreement with slip-linesolutions by other investigators. (b) The working pressures and the modes of deformation in open-die extrusion-forging are obtained and they explain some empirically known facts. (c) The working pressure and the mode of deformation in heading are estimated. Some of these results are compared with slip-line solutions and found to be in good agreement. (d) The indentation of a flat punch into a work-piece held in a container is analysed and the upper bound for punch pressure, mode of deformation and distortion of material are related to the dimensions of the working tool and the work-piece, as well as to the conditions of lubrication.From these results, the upper-bound approach method proposed here seems to offer a fairly accurate and simple means in analysing plane-strain forging problems.