On the optimal die curvature in deep drawing processes

The paper presents an attempt to increase the limit drawing ratio of deep drawing processes by searching an optimal die curvature, which minimizes the drawing load. The search done here for an optimal die curvature is based on experimental observations and followed by a detailed upper bound analysis. The analysis takes into account the non-steady character of the process (from a 2D circular plane blank into a 3D axisymmetric cup). The plastic flow along the die curvature is expressed in a toroidal coordinate system which seemingly describes more naturally a smooth velocity field along the real toroidal profile of the die. The outcome provides more closely the relationship between the energy dissipation rate and the die curvature so that a preferred die curvature is obtainable by energy minimization. Circular sheet blanks, made from aluminum and copper, were drawn through dies with different radii of curvature (with at least five repetitions at each radius) to capture the features of the optimal dies whenever exists. The main result is that under certain circumstances an optimal die curvature does exist. It depends largely on the drawing ratio and the blank/die interfacial friction, m, but appears quite insensitive to the initial thickness of the blanks. The optimal die curvature is pronounced in the cases where the frictional resistance is relatively low, otherwise it is indistinctable and remains practically undeterminable by designers.