Numerical analysis of three-dimensional extrusion of arbitrarily shaped sections by the method of weighted residuals

Abstract

By employing a method of numerical analysis based on the method of weighted residuals, the stresses and strains in three-dimensional cold extrusion of arbitrarily shaped sections from round billets through continuous dies are calculated with sufficient accuracy. The previous method using global flow functions lacks generality for its application. In the present work, instead of using global flow functions the velocity components and stress functions are taken in the form of global functions expressed in polynomial terms for the generalized three-dimensional plastic flow. The velocity boundary condition that the velocity component normal to the die surface and the velocity component normal to the plane of symmetry should be zero is considered by additional error equations. The stress and boundary conditions are also considered simultaneously. The errors defined for the governing equations and the boundary conditions are minimized by introducing the method of weighted residuals. The work-hardening effect is considered by integrating the effective strain rate following a streamline. As computational examples for arbitrarily shaped products, ellipse, clover, rounded square and trocoidal gear section are chosen for the extruded sections. Experiments were carried out for an aluminum alloy at room temperature. The computed results are compared with the experimental results as well as with those by the finite element method. The extrusion load and the flow pattern computed by the present method are in good agreement with the experimental results and those of the finite element method.