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.