Chang-Whan Lee

Finite element analysis of steady-state three-dimensional extrusion of sections through curved dies

Abstract A finite element analysis is made for steady-state three-dimensional extrusion of non-circular sections such as elliptic and clover-leaf sections through curved dies. In treating the incompressibility, the penalty constraint method, based on the rigid-plastic material model, is used. The initial guess for the initial velocity field is based on that of general upper-bound solutions for extrusion of arbitrarily shaped sections from round billets. Finite element computations are carried out for extrusion of elliptic and clover-leaf sections, but the present analysis can be further applied to general non-circular sections. The work-hardening effect is considered by integrating the effective strain rate along each stream line through interpolation by the least squares method. Experiments are carried out at room temperature and the experimental results are compared with numerical results in flow pattern and strain distribution. It is thus shown that the proposed numerical method is effective for detailed and reliable analysis of steady-state three-dimensional extrusion.

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.

Analysis of axisymmetric extrusion of rods by the method of weighted residuals using body-fitted coordinate transformation

Abstract The method of weighted residuals with the use of the finite difference method based on a coordinate transformation to nonorthogonal curvilinear coordinate system to fit the boundary of an arbitrary shape is proposed to determine the stresses and strains in axisymmetric extrusion of rods through continuous dies. By using the body-fitted coordinate transformation, the finite difference expressions at and adjacent to the boundary do not need any interpolation between boundaries and interior grid points. The errors defined for governing equations and boundary conditions are minimized by introducing the method of weighted residuals. The work-hardening effect is incorporated by integrating the effective strain rate along the flow line. As computational examples, four types of dies with different reductions of area are chosen. Experiments are carried out for steel and aluminum alloys 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 excellent agreement with the experimental results and those by the finite element method.

Analysis of three-dimensional extrusion of arbitrarily shaped tubes

Abstract In cold lubricated tube extrusion of arbitrarily shaped sections, the material properties and surface quality of the extruded products are influenced by the die profile. In the present analysis the axial velocity component on the intermediate cross-section in the plastically deforming region is expressed as a general function of the coordinates and this allows the description of complete three-dimensional distortion or flow rendering more realistic flow patterns. From the derived kinematically admissible velocity field, the corresponding upper-bound extrusion pressure is then obtained by optimizing the pressure with respect to the given parameters. The grid distortion is computed by tracing the metal flow along the streamlines. The effective strains are computed in order to study the work-hardening pattern of the extruded products. Experiments have been carried using A12024 as the working material. In the computation, three product shapes namely clover, trocoidal gear and ellipse are chosen. The effects of area reduction, die length, product shape complexity and frictional condition on the extrusion pressure, the metal flow and distribution of the final effective strain have been studied extensively. The theoretical predictions both in extrusion load and metal flow are in good agreement with the experimental results.

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

Abstract A method of numerical analysis based on the method of weighted residuals is proposed to determine the stresses and strains in three-dimensional extrusion of elliptic sections. In the proposed method, two flow functions and three stress functions are appropriately assumed for the three-dimensional plastic flow. The errors defined for the governing equations and boundary conditions are minimized by introducing the method of weighted residuals in the solution procedure. The work-hardening effect is considered by integrating the effective strain rate following a streamline to a spatially fixed point. Experiments are 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 theoretical predictions by the present method in both extrusion load and deformation patterns are in excellent agreement with the experimental results and with those by the finite element method.