An Upper Bound Solution of Backward Tube Extrusion Process Through Curved Punches

Abstract

In this paper, an upper bound approach is used to analyze the process of backward tube extrusion through arbitrarily curved punches. Based on the assumptions of proportional angles and proportional distances from the mandrel surface in the deformation zone, two kinematically admissible velocity fields are proposed and those are used in upper bound solution. By using the developed upper bound solution, optimum punch lengths which minimize the extrusion forces are determined for a streamlined punch shape and also for a conical punch. The corresponding results are also determined by using a finite element code, ABAQUS, and by doing some experiments and compared with the analytical results. This comparisons show a good agreement. Keywords: Backward tube extrusion, Upper bound, Experiment IntroductionIn backward extrusion, there is no relative movement between the initial billet and the container and it is characterized by the absence of friction between the initial billet surface and the container. This process needs lower extrusion force and also it is suitable for producing partially extruded products because of the simplicity in ejecting of the extruded part from the container as compared with forward extrusion process. In this process, such as other metal forming processes, calculation and optimization of extrusion force are important. Among various analytical and numerical approximate methods of solution, the upper bound technique and the finite element method have been widely used for the analysis of the extrusion process. One of the limitations of most of the current FEM solution schemes for metal forming is that they do not provide parametric analysis. Hence, any parametric investigation is usually done manually by changing one FE model to another until a feasible solution is obtained. Establishment of analytical solutions for extrusion process facilitates parametric study and may help in understanding the mechanics behind the extrusion processes better. Even though the finite element gives detailed information, it takes considerable CPU time. Using the upper-bound technique has the merits of saving computer's CPU and it appears to be a useful tool for analyzing metal forming problems when the objective of such an analysis is limited to prediction of deformation load and/or to study metal flow during the process. A number of people have used the upper bound method to analyze the extrusion process. Avitzur [1-3] developed models for forward rod extrusion through conical dies using the upper bound approach. Chen and Ling [4] developed a velocity field for axisymmetric extrusions through cosine, elliptic and