Except at the start and the end of the deformation, processes such as extrusion, drawing, and rolling are kinematically steady state. Steady-state solutions in these processes are needed for equipment design and die design and for controlling product properties. A variety of solutions for different conditions in extrusion and drawing have been obtained by applying the slip-line theory and the upper-bound theorems. Early applications of the finite-element method to the analysis of extrusion have been for the loading of a workpiece that fits the die and container, and for the extrusion of a small amount of it rather than extruding the workpiece until a steady state is reached. An exception is the work by Lee et al. for plane-strain extrusion with frictionless curved dies using the elastic-plastic finite-element method. In view of the computational efficiency, various numerical procedures particularly suited for the analysis of steady-state processes have been developed by several investigators. Shah and Kobayashi analyzed axisymmetric extrusion through frictionless conical dies by the rigid-plastic finite-element method. The technique involves construction of the flow lines from velocities and integration of strain-rates numerically along flow lines to determine the strain distributions. An improvement of the method was made by including friction at the die-workpiece interface. The steady-state deformation characteristics in extrusion and drawing were obtained as functions of material property, die-workpiece interface friction, die angle, and reduction. In kinematically transient or nonsteady-state forming problems, a mesh that requires continuous updating (Lagrangian) is used. In steady-state problems, a mesh fixed in space (Eulerian) is appropriate, since the process configuration does not change with time. For steady-state problems whose solutions depend on the loading history or strain history of the material, combined Eulerian-Lagrangian approaches are necessary. In deformation of rigid-plastic materials under the isothermal conditions, the solution obtained by the finite-element method is in terms of velocities and, hence, strain-rates. In the nonsteady-state processes, the effective strain-rates are added incrementally for each element to determine the effective strains after a certain amount of deformation.
Effect of the Forming Zone Length on Helical Rolling Processes for Manufacturing Steel Balls
