Some analytical and experimental studies of axi-symmetric cold forging and extrusion—I

Abstract

1. (i) In order to obtain upper-bound solutions for the axi-symmetric forging and extrusion of metals following the Mises yield criterion and the Levy-Mises stress-strain-rate relations, two types of admissible velocity fields are examined. The internal rate of energy dissipation associated with them is obtained without numerical integration if the material is assumed perfectly plastic. 2. (ii) To facilitate the calculation for a specific working problem, the conception of a unit cylindrical deforming region is introduced. The least rates of energy dissipation as determined from the two types of velocity field possible are graphed for individual unit regions having a wide range of dimensions and various frictional boundary conditions. 3. (iii) An upper-bound solution for problems with Coulomb friction is proposed. This is proved to be a better upper bound than Drucker's in certain cases. 4. (iv) A modification of upper-bound solutions as determined for a perfectly plastic material, for non-steady working processes of work-hardening material, is proposed. 5. (v) Several forging and extrusion problems with the tools having flat working surfaces are analysed by dividing the material into several unit regions and by using prepared diagrams of least rate of energy dissipation. These are the compression of a cylinder and an annulus, extrusion, piercing, extrusion-forging and symmetrically opposed extrusion-forging. 6. (vi) The results for the working pressure either agree with or improve on the analytical results of other investigators. They also agree with the experimental results obtained by others. The most suitable velocity fields well explain the internal deformations and extrusion defects observed hitherto.