Kinetic and Dynamic Effects on the Upper-Bound Loads in Metal-Forming Processes

The regular upper-bound approach in metal-forming processes is extended to time-dependent processes. The ultimate goal is to estimate, in an approximate manner, time rate effects such as machine speed and material inertia, on the forming load of time independent materials. The admissible velocity field with associated jumps is used to generate an acceleration flow field and associated flow resistance. Two fundamental nondimensional numbers emerge from the analysis for all processes considered. One is related to the speed at which the deforming load is applied, ρu$02$/σ₀ (called the “kinetic head”) and the second is related to the acceleration of the deforming tool and its contact area with the flowing metal ρu˙₀R/σ₀ (called the “dynamic head”). The uniqueness of each specific process is characterized by appropriate functions representing the unsteady (or steady) pertinent geometry of the product and multiplying the foregoing numbers. The resulting expressions appear to be dominant only at relatively high speed and/or impact operations, and thus amplify the role of the time rate on the limit load. Three typical processes (forging, extrusion, and piercing) exemplify the approach with some experimental evidence.