Effect of natural boundary conditions on the upper-bound analysis of upset forging of ring and disks

The purpose of this investigation is to discuss the effect of natural boundary conditions derived using the method of variational calculus for upset forging of rings. In order to derive the natural boundary condition, an assumed admissible function with unknown constants is replaced by an arbitrary function. Rather than minimizing the unknown constants in the assumed function, the arbitrary function is found using variational calculus, and the natural boundary condition is derived as a consequence. The upper-bound solutions, with and without imposing the natural boundary condition, as well as finite element solutions were compared and discussed. The finite element solutions were determined using the commercial software package MARC. All solutions were experimentally verified using the experimental result of the calibration curve from ring tests. They were then discussed, by comparing with experimental bulged profiles as well as experimental forming load. 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 load. The variational upper-bound solution, which accounts for the natural boundary condition, has shown to present a significant improvement on the traditional upper-bound solution in general.