Exact analytic solutions of the nonlinear partial differential equations governing rigid perfect plasticity problems

Abstract

We provide exact analytic solutions for the stress and velocity states in statically determinate rigid, perfectly plastic bodies under plane-strain conditions. The extracted solutions include more than one arbitrary function, a fact that permits us to use them for large classes of boundaries and boundary conditions. In addition, other solutions by making use of several ad hoc assumptions are constructed including one arbitrary function. For the stresses the solutions are extracted by the full decoupling of the system of the equilibrium equations and the appropriate von Mises-Hencky nonlinear condition, leading to a second order nonlinear partial differential equation (PDE) of the Monge type; for the velocities we use the Saint Venant-von Mises theory of plasticity PDEs. Several applications concerning the so-called direct problem are examined. The advantage of the proposed analytical solution methodology compared to the technique of characteristics is the general applicability delivered from the a priori construction of slip lines, as well as the demanded numerical solutions of the corresponding equations of characteristics.