Three Dimensional Analysis of the Extrusion Process Using the Slip Line Field Solution

Many metal forming problems have been solved using the slip line field method but all of them have either been two-dimensional (plane strain or plain stress) or axisymmetric problems. In this paper a procedure has been proposed by which the slip line field solution to three dimensional problems of metal forming becomes possible. For this purpose the extrusion of shaped sections has been taken as a case study. For this problem the geometry of the deforming zone has been defined by streamlines and stream surfaces. Clearly for the three dimensional extrusion of shaped sections the stream surfaces are not plane surfaces, and on the other hand the slip line field formulation could only be applied to plane surfaces, therefore an approximation has been made to accommodate this difficulty. In fact the three dimensional surface has been approximated to a plane surface so that for each three dimensional stream surface there exists an approximate plane surface. Unlike the case for the axisymmetric problem where there was only one plane surface on which the formulation was defined and the revolved 360 degrees to complete the deforming zone here there are many plane stream surfaces that by summing them up together the deforming zone is defined. The slip line field formulation was then applied to each and every one of these surfaces and the extrusion pressure on each surface was calculated separately and by adding up all the components of the pressure on each surface the total extrusion pressure was obtained.
To account for the error evolved from the approximations made in the formulations, error functions were developed which showed how much error was developed due to the approximations. To verify the results comparison were made to the results obtained by upper bound and experimental methods. These comparisons showed very good agreements.