An upper bound solution for deformation field analysis in differential velocity sideways extrusion using a unified stream function

An analytical model providing a detailed description of the material flow and deformation behaviour of extruded curved profiles produced by the novel differential velocity sideways extrusion (DVSE) process, has been developed on the basis of a unified stream function and the upper bound theorem. Plasticine experiments and finite element (FE) modelling were carried out to validate the proposed analytical model. The derived streamline equation contains a shape parameter n describing the degree of curvature of a flow line and the coordinate parameters x₀ and y₀ defining entering and leaving positions respectively of the flow line, from the plastic deformation zone (PDZ). The analytical model was able to closely model the material flow eccentricity ratio ξ (the relative amounts of work-piece material entering the deformation zone from two opposing directions), and flow lines obtained from experiments under different velocity ratios and extrusion ratios. The predicted value of ξ was found to be independent of n value and hardening of the material. The n value was found to increase from the corner near the die orifice to the corner around the dead material zone (DMZ). In addition, the n value increased with the increase of extrusion ratio and ratio of velocities of the two opposing extrusion rams, which enabled the representation of a decreased area of DMZ and more localised PDZ containing 1–99% accumulated effective strain. The predicted field distributions of the localised effective strain rate in the PDZ and inhomogeneous effective strain in the extrudates were consistent with FE modelling results.