Application of upper bound method to the powder-forging of cup-shaped axisymmetric preforms for estimating punch load

This paper is concerned with the analytic derivation of the punch load required to forge cup-shaped axisymmetric porous preforms. Splitting the two-dimensional preform domain into three rectangular regions, we assumed the kinematically admissible velocity elds to apply the upper bound method. According to the resulting formulas, we developed an incremental numerical algorithm for computing time-discretized step-wise punch loads and preform volumes. In order to illustrate the validity of the proposed estimate, we compared the numerical results obtained by the present analysis with those by the nite element simulation. From the comparison made through di erent preform dimensions, we observed that the present analysis expects the reliable upper bounds of process step-wise punch loads