Variational gradient plasticity at finite strains. Part III: Local–global updates and regularization techniques in multiplicative plasticity for single crystals

The third part of our work on variational inelasticity with long-range effects outlines a formulation and finite element implementation of micromechanically-motivated multiplicative gradient plasticity for single crystals. In order to partially overcome the complexity of full multislip scenarios, we suggest a new viscous regularized formulation of rate-independent crystal plasticity, that exploits in a systematic manner the long- and short-range nature of the involved variables. To this end, we outline a multifield scenario, where the macro-deformation and the plastic slips on crystallographic systems are the primary fields. Related to these primary fields, we define as the long-range state the deformation gradient, the plastic slips and their gradients. We then introduce as the short-range plastic state the plastic deformation map, the dislocation density tensor and scalar hardening parameters associated with the slip systems. It is then shown that the evolution of the short range state is fully determined by the evolution of the long-range state. This separation into long- and short-range states is systematically exploited in the algorithmic treatment by a new update structure, where the short-range variables play the role of a local history base. The model problem under consideration accounts in a canonical format for basic effects related to statistically stored and geometrically necessary dislocation flow, yielding micro-force balances including non-convex cross-hardening, kinematic hardening and size effects. Further key ingredients of the proposed algorithmic formulation are geometrically exact updates of the short-range state and a distinct regularization of the rate-independent dissipation function that preserves the range of the elastic domain. The formulation is shown to be fully variational in nature, governed by rate-type continuous and incremental algorithmic variational principles. We demonstrate the modeling capabilities and algorithmic performance by means of representative numerical examples for multislip scenarios in fcc single crystals.