Analytical determination of strain components distribution and curvature in the direct extrusion process of symmetric sections based on Riemann mapping and Upper bound method

The aim of this paper is to analyze the strain components in the direct extrusion process to predict the curvature of the exit product. For this purpose, Riemann mapping theory is used to model the deformation zone and create a one-to-one correspondence between the input and output cross sections of the die. With the help of the Bezier curves, flow lines are created between these points and then an upper bound solution is obtained for the velocity field. The process pressure and the distribution of the strain components are determined for the square, hexagonal, and rectangular sections using the obtained velocity field. A theoretical method based on the elastic-plastic bending of beams is presented for calculating the curvature of the exit product for the eccentric dies. In this theoretical method, the distribution of stress components and the bending moments is calculated using the specified strain components. In fact, the amount of bending moments indicates the curvature of the exit product. Finally, the presented theoretical model is validated through comparison with the results of the finite element simulation and the previous studies. The results show that Riemann conformal mapping theory and upper bound method can be used to determine the distribution of strain components and predict the curvature of the output product, in addition to estimate the process pressure.