General Planar Ideal Flow Solutions with No Symmetry Axis

Abstract

Bulk ideal flows constitute a wide class of solutions in plasticity theory. Ideal flow solutions concern inverse problems. In particular, the solution determines part of the boundary of a region where it is valid. Bulk planar ideal flows exist in the case of (i) isotropic rigid/plastic material obeying an arbitrary pressure-independent yield criterion and its associated flow rule and (ii) the double sliding and rotation model based on the Mohr–Coulomb yield criterion. In the latter case, the intrinsic spin must vanish. Both models are perfectly plastic, and the complete equation systems are hyperbolic. All available specific solutions for both models describe flows with a symmetry axis. The present paper aims at general solutions for flows with no symmetry axis. The general structure of the solutions consists of two rigid regions connected by a plastic region. The characteristic lines between the plastic and rigid regions must be straight, which partly dictates the general structure of the characteristic nets. The solutions employ Riemann’s method in regions where the characteristics of both families are curvilinear. Special solutions that do not have such regions are considered separately. In any case, the solutions are practically analytical. A numerical technique is only necessary to evaluate ordinary integrals. The solutions found determine the tool shapes that produce ideal flows. In addition, the distribution of pressure over the tool’s surface is calculated, which is important for predicting the wear of tools.