Patterning in non convex strain gradient crystal plasticity

During forming processes most metals develop cellular dislocation structures due to dislocation slip patterning from moderate strains onwards. Typical examples of dislocation microstructures are dislocation cells and dislocation walls. Patterning typically refers to the self organization of dislocations with formation of regions of high dislocation density (dislocation walls) which envelop areas of low dislocation density (dislocation cell interiors), also to be regarded as domains of high plastic slip and low plastic slip activity. Due to the induced macroscopic anisotropic effects (the occurrence of dislocation microstructures and their evolution have been an interesting topic for the materials science community for decades. This paper present a non-convex rate dependent strain gradient plasticity framework for the description of plastic slip patterning in metal crystals. The non-convexity is treated as an intrinsic property of the free energy of the material. Departing from explicit expressions for the free energy and the dissipation potential, the non-convex strain gradient crystal plasticity model is derived in a thermodynamically consistent manner, including the accompanying slip law. For the numerical solution of the problem, the displacement and the plastic slip fields are considered as primary variables. These fields are determined on a global level by solving simultaneously the linear momentum balance and the resulting slip evolution equation. The slip law differs from classical ones in the sense that it naturally includes a contribution from the non-convex free energy term, which enables patterning of the deformation field. The formulation of the computational framework is partially dual to a Ginzburg Landau type of phase field modeling approach. The essential difference resides in the fact that a strong coupling exists between the deformation and the evolution of the plastic slip, whereas in the phase field type models the governing fields are only weakly coupled. The derivations and implementations are done in a transparent 1D setting [1], which allows for a thorough mechanistic understanding. The extension to 2D and multiple slip is discussed as well, whereby the non-convexity originates from the slip system interaction