Development of a more realistic upper bound solution for the three-dimensional problems in the forward extrusion process

• A new generalized formulation is presented for the extrusion of shaped sections.
• The velocity and strain distribution contours were determined theoretically.
• The results of the proposed method are comparable with FEM.
• Three-dimensional material flow was obtained at different stages.
• The upper bound results showed improvements over previous works data.

This paper provides a new generalized formulation based on the upper bound method for the three-dimensional analysis of the forward extrusion of non-symmetrical sections. The proposed method has been formulated in such a way as to present a more realistic definition of deforming region by eliminating internal discontinuities and also velocity discontinuities at the entry and exit surfaces which are defined by three-dimensional curved surfaces. Using this new formulation, extrusion of shapes such as L-shape, rectangle, hexagon and a spur gear was analyzed. The distribution of velocity and strain, three-dimensional material flow as well as extrusion load were obtained. Finite element analysis for the same problem was also carried out and the comparison of the results showed the capability of the present method. In fact, the results of the analytical velocity and strain distributions are in close agreement with those given by the simulation in terms of magnitude and distribution. Influence of the process parameters such as extrusion ratio and aspect ratio on the strain distributions was investigated. Comparison between theoretical and experimental data was carried out and close agreement was observed. Also, improvements of the results of the present method were observed comparing with previous works.