The Study of Distorted Grid Patterns for Flow-Through Conical Converging Dies by the Multi-triangular Velocity Field

A variety of velocity fields may be used to analyze the intermediate and final distorted grids for the so-called “flow-through” metal forming processes such as wire drawing, rolling, extrusion, etc. In this paper the triangular velocity field describes the flow of homogeneous, perfectly plastic Mises’ material through a conical converging die. The traditional triangular velocity field was treated and the solution extended. The shape of the distorted grids was uniquely determined by the minimization of the power (drawing or extrusion stresses) required to cause its distortion for a given set of independent process parameters, i.e., process geometry-reduction in area and semi-cone angle, and friction. Actual power (forming stress requirements) was estimated by the upper-bound technique. For the unitriangular velocity field, the power was minimized with respect to the shape of the workpiece (the shape of the triangle). For the multitriangular velocity field, the power was minimized with respect to the shape and the number of triangles. Further, the number of triangles was treated as a real number. Thus, the accurate lower upper-bound was found and the reasonable solution in predicting real distortion grid patterns was then obtained. The analysis determines the severity of the distortion as a function of process geometry and friction.