The upper bound theorem is a primary method for analyzing metal forming processes. A disadvantage of almost all upper bound solutions is that the constant friction law (Prandtl’s law) is adopted. This presentation employs an alternative formulation of the theorem that enables the inclusion of Coulomb friction in the boundary conditions of a boundary value problem. The general theorem is specialized to extrusion and drawing processes through conical dies. The general trial velocity field generalizes a typical kinematically admissible velocity field used in solutions based on the conventional formulation of the theorem. This trial velocity field is simplified, again using typical assumptions used in solutions based on the conventional formulation of the theorem. The calculation of the extrusion and drawing forces reduces to evaluating an ordinary integral numerically. The optimal die angle that minimizes the extrusion or drawing force is determined. The restrictions on the range of the validity of the solution are discussed. The general theorem is also specialized to the case of upsetting processes. The solutions obtained demonstrate that the alternative theorem is useful for metal-forming applications. It is expected that a typical trial velocity field required by this theorem might be slightly more complicated than the corresponding kinematically admissible velocity field. In particular, the trial velocity fields used in the presentation are more complicated than the kinematically admissible velocity field proposed in the literature. However, this additional complexity merely results in more cumbersome algebra.

Sergei Alexandrov is a professor at Beihang University (Beijing, China) and a research professor at the Institute for Problems of Mechanics of the Russian Academy of Science. He received his PhD in Physics and Mathematics in 1990 and DSc in Physics and Mathematics in 1994. He worked for ALCOA Technical Center (USA), GKSS Research Center (Germany), University of Aveiro (Portugal), and Seoul National University (South Korea). His research areas are plasticity theory, fracture mechanics, and their applications to metal forming and structural mechanics. He published three monographs and about 300 papers in journals included in SCIE. He is a member of the editorial boards of several international journals.