An accurate numerical solution for the singular velocity field near the maximum friction surface in plane strain extrusion

In the case of rigid perfectly plastic material, the velocity field is singular in the vicinity of maximum friction surfaces (the equivalent strain rate approaches infinity in the vicinity of such surfaces). This causes significant difficulties with the convergence of finite element solutions. On the other hand, an accurate description of the velocity field near frictional interfaces is crucial for predicting the generation of a narrow fine grain layer that frequently appears near such interfaces. The present paper provides an accurate numerical solution for the singular velocity field in the vicinity of the maximum friction surface in plane strain extrusion. The singularity in the velocity field is represented by means of the strain rate intensity factor. The numerical approach is based on the method of characteristics. The strain rate intensity factor is determined by means of simple formulae once the solution for the radii of curvature of characteristic lines and velocities has been found. It is shown that the distribution of the strain rate intensity factor along the friction surface is discontinuous. Comparison with an analytic solution for material flow through an infinite wedge-shaped die is made and a high accuracy of the numerical solution is demonstrated. On the other hand, it is shown that the analytic solution is not accurate enough to find the strain rate intensity factor even for rather long dies.