New Numerical Solver for Elastoplastic Problems based on the Moreau-Yosida Theorem

We discuss a new solution algorithm 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 functional dependening on unknown displacement smoothly and on the plastic strain non-smoothly. It is shown that the functional structure allows the application of the Moreau-Yosida Theorem known in convex analysis. It guarantees that the substitution of the non-smooth plastic-strain as a function of the linear strain which depends on the displacement only yields an already smooth functional in the displacement only. Moreover, the second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. This allows the efficient implementation of the Newton-Ralphson method. For easy implementation most essential Matlab c ○ functions are provided. Numerical experiments in two dimensions state quadratic convergence of a Newton-Ralpshon method as long as the elastoplastic interface is detected sufficiently precisely.