Effect of natural boundary condition and the neutral surface of nonlinear type on the upper-bound solution to upset forging of rings using a variational approach

The purpose of this work is to investigate the effect of natural boundary condition and the type of neutral surface on the upper-bound solutions using a variational approach. To this end, the neutral surface of the upset ring was considered a function of polynomials of various orders during deformation process, and then an upper-bound forming energy equation was formulated in terms of the function as well as the velocity field derived from the theory of stream function. Since the velocity field had been expressed in terms of an implicit stream function before the upper-bound forming energy equation J was established, a variational approach was fulfilled to determine an upper-bound solution by extremizing J. As a result, a set of boundary conditions was derived mathematically. To determine the upper-bound solution, the derived boundary conditions, which include the so-called natural boundary conditions and kinematically boundary conditions, were imposed in an optimization procedure. Such a method we have proposed to determine the solution is particularly termed the variational upper-bound (VUB) method in order to distinguish it from the traditional upper-bound (UB) method, which usually ignores the natural boundary condition perceptibly unobtainable. The natural boundary condition can be physically interpreted as to constrain plastic flow of the upset ring at frictional interfaces. However, in order to investigate the effect of the type of neutral surface on upper-bound solutions, a polynomial that may result in the neutral surface of nonlinear type was proposed. By comparing with some experimental results, it is found that the VUB method has considerably improved on the UB method in predicting the calibration curves, the bulged profiles of upset ring and disks, and the forming energy. The result presented in this work also demonstrates that the effect of the natural boundary condition is much more pronounced than that of the type of neutral surface used on improving the upper-bound solution. While keeping the same number of free parameters left for the purpose of minimization procedure, the VUB method is permitted to establish an approximate solution of higher order than the UB method, because it additionally satisfies the natural boundary condition derived in this work. The finite element solution determined using MARC, a commercial software package, is also presented in this work for discussion.