Instability of dislocation fluxes in a single slip: Deterministic and stochastic models of dislocation patterning

We study a continuum model of dislocation transport in order to investigate the formation of heterogeneous dislocation patterns. We propose a physical mechanism that relates the formation of heterogeneous patterns with a well-defined wavelength to the stress-driven dynamics of dislocation densities that tries to minimize the internal energy while subject to dynamic constraints and a density-dependent, friction-like flow stress. This leads us to an interpretation that resolves the old energetic vs dynamic controversy regarding the physical origin of dislocation patterns and emphasizes the hydrodynamic nature of the instability that leads to dislocation patterning, which we identify as an instability of dislocation transport that is not dependent on processes such as dislocation multiplication or annihilation. We demonstrate the robustness of the developed patterning scenario by considering the simplest possible case (plane strain, single slip) in two model versions that consider the same driving stresses but implement the transport of dislocations that controls dislocation density evolution in two very different manners, namely (i) a hydrodynamic formulation that considers transport equations that are continuous in space and time, assuming that the dislocation velocity depends linearly on the local driving stress, and (ii) a stochastic cellular automaton implementation that assumes spatially and temporally discrete transport of discrete packets of dislocation density that move according to an extremal dynamics. Despite the differences, we find that the emergent patterns in both models are mutually consistent and in agreement with the prediction of a linear stability analysis of the continuum model. We also show how different types of initial conditions lead to different intermediate evolution scenarios that, however, do not affect the properties of the fully developed patterns.