In this paper, a variational upper-bound (VUB) method is proposed. It is the method that determines an upper-bound solution using variational calculus. By using the method, a set of so-called ''naturally boundary conditions,'' which were all ignored in past investigations, can be theoretically derived for the rigid/plastic interface assumed to be arbitrary function in prior. This method is shown to be applicable to the cup ironing problem. From the result we can clearly indicate the role of these natural boundary conditions which reasonably account for the effect of friction on the pattern of metal flow. The VUB method not only presents an improvement on the traditional upper-bound method, but also favorably agrees with experimental result of Nielsen et al. (1963) in calculating upper-bound energy.
JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING-TRANSACTIONS OF THE ASME