Rate dependent plastic microstructure evolution through strain gradient crystal plasticity

Plastic deformation and its possible combination with other loadings (thermal, irradiation etc.) induce various types of dislocation microstructure evolution, which eventually result in a spatially heterogeneous deformation field. Different type of dislocation microstructures exist in metallic materials at different length scales. Typical examples at macro scale are Lüders bands, Portevin-Le Chatelier (PLC) bands, while dislocation cell structures, labyrinth, mosaic, fence or carpet structures develop at meso scale, which are mainly due to self-organization of dislocations. The plastic localization induces macroscopic softening-hardening and stress-plateau type of responses, arising numerical issues in the solution procedure. The application of standard finite element methods yields mesh-dependent post-critical results due to loss of ellipticity of the incremental boundary value problem. In order to remedy these problems during the formation of microstructures several models have been proposed including, methods using calculus of variations in an incremental setting, non-local methods, viscous regularization techniques and Cosserat theories. A complete understanding of models which can simulate the patterning of dislocation slip or formation of dislocation substructures is not at hand. In order to contribute to this, inspired by the success of phase field models, an approach is proposed to illustrate the ability of non-convex field models to predict the emergence and evolution of dislocation slip microstructures in a rate dependent strain gradient crystal plasticity framework (see e.g. Yalcinkaya et al. [2011] and Yalcinkaya et al. [2012]). The framework studies the plastic slip patterning in a system with energetic hardening. Both the displacement and the plastic slip are considered as primary variables. These fields are determined on a global level by solving simultaneously the linear momentum balance and slip evolution equation which is derived from thermodynamical considerations. The slip law used in this context differs from the classical ones in the sense that it includes the non-convex free energy term leading to the patterning of this field. The non- convexity is treated as an intrinsic property of the free energy of the material. The numerical examples illustrate the microstructure evolution due to different types of non-convex contributions in a multi-slip 2D plane strain analysi