A model for an adaptable axisymmetric extrusion die with a bearing length

The adaptable design method uses upper bound models to determine die shapes that meet specified criteria, such as minimizing distortion in the product. This method has been developed for the axisymmetric extrusion process. In order to extend the methodology to the three-dimensional extrusion of a non-axisymmetric shape, the deformation zone in the upper bound model would need to be extended into the bearing region of the extrusion die. The necessary equations and conditions needed to make such an extension are presented in this paper. Finite element modeling (FEM) is used to compare the results of extrusion through dies designed by the adaptable die design method, for a model with the deformation zone extended into the bearing region, to a model without such an extension. The results indicate that the upper bound model incorporating a bearing length provides a realistic flow field. The results also demonstrate that the upper bound model can be used to analyze a multi-sectioned die, so long as: (1) the die surface and first derivative of the surface are continuous between sections, and (2) calculation of the internal power of deformation is made for each section separately. The results provide further support for the findings that the average effective strain and the volumetric effective strain rate deviation are robust criteria, which can be used to determine optimal adaptable die shapes with an upper bound model.