A concave circular parallelepiped element and its application to three-dimensional closed-die forging

In this study, a kinematically admissible velocity field is derived for a proposed three-dimensional concave circular parallelepiped UBET element. The element is applied to three-dimensional closed-die forging having concave curved surfaces: the capability of the proposed element is then demonstrated. From the derived velocity field, 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 into closed-die forging are performed with commercially pure lead billet at ambient temperature. The theoretical predictions of the forming load correlated well with the experimental results. The results in this study confirm that the element proposed in this work can be used effectively to predict the forming load accurately in three-dimensional closed-die forging having concave curved surfaces.