An upper bound analysis of a process-induced side-surface defect in forgings: Part 1: The velocity fields and power terms

An upper bound analysis for an axisymmetric forging with a double ram action has been developed. The results of the analysis provide the geometric and processing conditions that can be used to avoid the development of unacceptable side-surface cracks. Kinematically admissible velocity fields that satisfy the incompressibility condition and the velocity boundary conditions have been determined and the corresponding upper bound solution for the flow pattern has been developed. To determine the conditions under which the process-induced surface cracks would occur, a kinematically admissible velocity field for defective flow was developed and analyzed by the upper bound approach. An extra power term that accounts for the power dissipated during crack formation has been added to the upper bound power for defective flow. This first paper in a two-part series presents the velocity fields and the individual power terms that were derived based upon these flow patterns, whilst the second paper shows the characteristics of the upper bound solution for this forging process. The second paper also presents the criteria for the prevention of the side-surface defects that were determined based upon the upper bound solution.