Upper-Bound Elemental Technique (UBET) with convex circular parallelepiped and convex spherical elements for three-dimensional forging analysis

In this study, two kinematically admissible velocity fields are derived for the proposed three-dimensional convex circular parallelepiped and convex spherical UBET elements. Those elements are applied to three-dimensional closed-die forging having convex curve surfaces; the capability of the proposed elements is then demonstrated. From the derived velocity fields, the upper-bound loads on the upper die and the velocity field are determined by minimizing the total energy consumption with respect to some chosen parameters. Also, experiments of two closed-die forgings are performed with commercial pure lead billets at the ambient temperature. The theoretical predictions of the forming load correlated well with the experimental results. The results in this study confirm that the elements proposed in this work can effectively be used to predict the forming load accurately in three-dimensional closed-die forging with convex curve surfaces.