Single-patch surfaces for tool shape design and finite element analysis of hot extrusion

In this paper, a new mathematical representation of single-patch surfaces is reported. With this representation, complex tool shapes of metal forming can be described with a single surface patch which is prescribed by the boundary of the die at its entry and exit. For non-linearly converging die-surfaces, the die shapes can be effectively altered by changing the coefficients of the tangential boundary conditions. Using the obtained die-surfaces, the finite element analysis of the extrusion of hot metals from a round billet to a hexagon workpiece was carried out. A comparison of the extrusion forces and the grain sizes for the formed parts was made for the linearly and non-linearly converging dies. It was found that the non-linearly converging dies could reduce the deformation resistance remarkably, and produce more uniform grains than those using the linearly converging dies.