A general upper-bound formulation of stream function of upset forging of ring using a variational approach

Based on an upper-bound energy dissipation equation established as a functional, a general formulation is derived for stream function of upset forging of ring using a variational approach. The general formulation consists of an equilibrium equation of the stream function along with a set of statically boundary conditions which were usually ignored in past investigation. In order to demonstrate the effect of the statically boundary condition on upper-bound solution and in order to verify the validity of the upper-bound solution as well, the experimental results also carried out in this investigation have been compared with the upper-bound solution for discussion. Both VUB and corresponding UB solutions were determined by optimizing a chosen stream function under consideration of the same degree of freedom. From the result we can clearly indicate that the VUB solution does present an improvement on the corresponding UB solution in predicting bulged profiles of upset forging of ring and disk. Such an improvement is only due to the fact that it is additionally imposed on the VUB solution the available statically boundary condition, which constrains plastic flow of the upsetting ring or disk on the frictional interface during upsetting process.