doi: 10.1007/s12540-014-5005-y Met. Mater. Int., Vol. 20, No. 5 (2014), pp. 825~834 An Upper Bound Solution for Twist Extrusion Process M. Seyed Salehi 1 , N. Anjabin 2 , and H. S. Kim 3,* 1 K. N. Toosi University of Technology, Department of Materials Science and Engineering, P.O. Box: 15875-4416, Tehran, Iran 2 Shiraz University, School of Engineering, Department of Materials Science and Engineering, Zand Ave., Shiraz, Iran 3 Pohang University of Science and Technology, Department of Materials Science and Engineering, Pohang 790-784, Korea (received date: 14 August 2013 / accepted date: 27 December 2013) Twist extrusion, a promising severe plastic deformation technique for grain refinement down to ultrafine/ nanocrystalline microstructures, was introduced as an attempt to provide large plastic deformation condi- tions similar to those in high pressure torsion while allowing large workpiece dimensions for industrial applications. As a relatively new severe plastic deformation technique, twist extrusion requires in-depth investigation of its plastic deformation characteristics. In this study, the twist extrusion process with a square shape die cavity has been analyzed using an upper bound solution to estimate the required power, deformation pattern, and optimum process condition. The analysis has been performed based on two kine- matically admissible velocity fields while the effects of friction condition, die geometry, and mean equiva- lent strain have been considered. The results indicate that the die geometry and process parameters can dramatically change the deformation pattern and extrusion power. Keywords: severe plastic deformation, twist extrusion, upper bound solution, grain refinement 1. INTRODUCTION Ultrafine grained materials have been shown to have supe- rior mechanical properties in comparison with conventional coarse-grained polycrystalline materials [1-4]. Severe plastic deformation (SPD) processing has been employed as an effi- cient route for the production of bulk ultrafine- and even nano-grained structures in metallic materials [5,6]. In this regard, various methods and techniques are utilized to refine the grain structure by applying large shear strains associated with high hydrostatic pressure, such as equal channel angular pressing (ECAP) [7-12], accumulative roll bonding (ARB) [13- 15], high pressure torsion (HPT) [16-18], tubular channel angular pressing [19], caliber rolling [20], and twist extrusion (TE) [1,21-31]. In the TE process, a specimen with non-circular cross-sec- tion is extruded through a twisted die cavity while the original cross-section is preserved. In this method, the direction of material flow remains unchanged so extrusion die can be eas- ily installed in conventional industrial lines. Also, the amount of wasted material in this method is lower than the other routine SPDs, e.g. ECAP [22]. In this method shearing plastic strain is applied to billet on surface perpendicular to the extrusion axis. To estimate the required power and deformation pattern during SPD processes several numerical and analytical meth- ods such as the Finite Element Method (FE) [7,16,21,26] and Upper Bound (UB) theorem [27,32] have been employed. In some works, the results obtained from the FEM were used to validate the results of UB analysis of the SPD process [33,34]. Beygelzimer et al. [25] investigated the kinematics of metal flow during TE by reconstruction of experimental stream lines and by proposing a kinematically admissible velocity field which is closest to the experimental stream lines. Akbari Mousavi et al. [21] analyzed the TE of titanium alloys using an FE model which found the occurrence of maximum and minimum equivalent strains at the corner and the center of the extruded billet, respectively. Khoddam et al. [32] proposed an UB model to analyze the interaction between die and material and process pressure during axi-symmetric forward spiral extrusion. Seyed-Salehi and Serajzadeh [27] developed an UB solution in TE with elliptical die cross-section. They proposed a kinematically admissible velocity field which satisfies the velocity boundary conditions and mass conservation to investigate the flow pattern and the UB theorem used to cal- culate extrusion power for different die geometries and fric- *Corresponding author: hskim@postech.ac.kr KIM and Springer