Implementation of an elastoplastic solver based on the Moreau–Yosida Theorem

We discuss a technique for solving elastoplastic problems with hardening. The one time-step elastoplastic problem can be formulated as a convex minimization problem with a continuous but non-smooth objective. We actually show that its objective structure satisfies conditions of the Moreau–Yosida Theorem known from convex analysis. Therefore, the substitution of the non-smooth plastic-strain p as a function of the total strain ɛ(u) yields an already smooth functional in the displacement u only. The second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. The numerical experiment states super-linear convergence of a Newton method or even quadratic convergence as long as the interface is detected sufficiently.