Use of Upper Bound Solutions for Predicting the Pressure for the Plane Strain Extrusion of Materials

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Abstract
An upper bound solution, based on a kinematically admissible velocity field, is proposed for plane-strain symmetrical extrusion through wedge-shaped dies. The pattern chosen consists of a double-triangle arrangement of velocity discontinuities and the solution was optimized using a digital computer programme. Analysis of the results for a full range of die angles, reductions and frictional conditions provides useful predictions relating to the formation of dead zones. Such an analysis is given in detail for a 90° die. The results obtained compare favourably with the predictions of analytical slip-line field solutions in the regions where such solutions are available. The solution is also extended to accommodate the strain-hardening characteristics of the material. These results show that the alternative mean strain hypothesis is prone to errors in the presence of friction. This solution also yields an indication of the nature of the hardness distribution across the extruded product with particular reference to the effects of die geometry and reduction in area.

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