Nonlinear complementarity functions for plasticity problems with frictional contact

In this paper, a unifying framework for the numerical treatment of frictional contact in combination with elastoplasticity with hardening is presented. Both kinds of problems lead to the mathematical structure of variational inequalities which are discretized by introducing additional variables for the contact stresses or the deviatoric stresses, leading to a mixed formulation. By defining a family of nonlinear complementarity functions which can be applied to both frictional and plastic yield conditions, these inequality constraints are reformulated as equalities. Combined with the equilibrium equations, a system of nonlinear equations is obtained that is solved in terms of a semismooth Newton method. To obtain a larger domain of convergence and better stability properties, the introduction of additional scaling parameters is motivated and their benefit to the convergence of the scheme is demonstrated. Due to the locality of the contact and plastic constraints, the resulting method can be interpreted as a primal-dual active set strategy. Furthermore, for linear hardening laws, our method is a generalization of the well-known radial return algorithm which is obtained for a special suboptimal choice of the plastic scaling parameters. The efficiency and the generality of the proposed quasi-Newton scheme is demonstrated by several numerical examples including the simulation of a three-dimensional contact problem with several elastoplastic bodies, exponential isotropic hardening and Coulomb friction.