Ad hoc exact solutions for the stress and velocity fields in rigid perfectly plastic materials under plane-stress conditions

Abstract

Making use of convenient and original ad hoc assumptions we construct closed-form solutions for the problem of stress and velocity states in statically determinate rigid, perfectly plastic materials under plane-stress conditions. The obtained solutions are expressed implicitly including arbitrary functions. For the stresses they are extracted by means of the two equilibrium linear PDEs and the appropriate strongly non-linear von Mises–Hencky condition; for the velocities we use the Saint Venant–von Mises theory of plasticity PDEs. Finally, an application concerning the analytical derivation of the stress state, the lines of discontinuity and the limiting load for an obtuse wedge under lateral normal pressure on one side is developed. The advantage of the proposed analytical solution methodology compared to the technique of characteristics is the general applicability delivering from the a priori construction of the slip-lines, as well as the demanded numerical solutions of the corresponding equations of characteristics.