A new definition of shape complexity for metal extrusion

According to a basic definition, complexity index for a given die profile is the ratio of the pressure required to extrude the profile relative to the extrusion pressure for a round cross-section of the same area. However, both intuitively and by definition, complexity index is a function of geometry only. Various researchers and experimenters have coined different definitions for shape complexity of extrusion dies, based solely on geometrical parameters. The simplest definition depicts complexity index as the ratio of profile perimeter to profile cross-sectional area. Other definitions are in terms of ratios of perimeter to weight, circumscribing circle diameter to minimum wall thickness, profile perimeter and equivalent circular perimeter, etc. Unfortunately, these definitions do not yield a consistent ranking of die profiles in terms of shape complexity. Moreover, the extrusion pressure predicted by these definitions often gives widely fluctuating values. The current paper aims at a new and more consistent definition of complexity for extrusion profiles. All extrusion runs for the study have been carried out in collaboration with a local commercial extrusion facility (hot extrusion of structural aluminum). A total of 27 different dies are used, profile complexity ranging from simple solids and hollows to quite complex ones. Die material (heat treated and surface hardened H-13 steel) and billet material (Al-6063) is the same for all experiments. Various runs at a ram speed of 3.8 ±0.1 mm/s have been carried out for 15 different die profiles, while another set of 12 profiles are extruded at 3.2 ±0.1 mm/s speed. After running a variety of regression runs, using various exponential, logarithmic and power law formats, the best-suited model for die complexity has been determined to be of the form C = α + β (P_s/P₀)^γ, where P_s and P ₀ are the perimeters of the actual profile section and an equivalent circular section of equal area, respectively. The constants α, β and γ are determined statistically by regression of experimental hot extrusion data obtained for a variety of die complexities and somewhat different operating conditions. In comparison with the existing definitions, the curve that follows the reference complexity curve (ratio of actual to circular-area pressure) most closely is that for the new definition.