A variational upper-bound method for analysis of upset forging of rings

In this investigation, a variational upper-bound (VUB) method is employed. It is the method that determines an upper-bound solution using variational calculus. By using the method, the natural boundary condition, which was ignored in the traditional upper-bound (UB) method, can be theoretically derived for the stream function assumed to be arbitrary in prior. The VUB method has shown to be applicable to the problem of upset forgings of ring, and it presents an improvement on the UB method in general. From the result, we can clearly indicate that the natural boundary condition, which constrains plastic flow of the upsetting ring on the contact interface, significantly affects the upper-bound solution not only in predicting the bulge profile of the upset ring, but also in calculating the total forming energy rate. Theoretical predictions obtained using the VUB, the UB and the FEM methods were compared and discussed. Some experimental results were employed for verification of the theories as well.