Deterministic and stochastic models of dislocation patterning

TL;DR

This paper introduces a continuum dislocation model that explains pattern formation through a energy minimization mechanism, validated by both hydrodynamic and cellular automaton simulations, resolving longstanding debates.

Contribution

It presents a unified physical interpretation of dislocation patterning and demonstrates robustness across different modeling approaches.

Findings

Patterns are consistent across models despite different assumptions.

Initial conditions influence early evolution but not final patterns.

Linear stability analysis predicts pattern emergence.

Abstract

We study a continuum model of dislocation transport in order to investigate the formation of heterogeneous dislocation patterns. We propose a physical mechanism which relates the formation of heterogeneous patterns to the dynamics of a driven system which tries to minimize an internal energy functional while subject to dynamic constraints and state dependent friction. This leads us to a novel interpretation which resolves the old 'energetic vs. dynamic' controversy regarding the physical origin of dislocation patterns. We demonstrate the robustness of the developed patterning scenario by considering the simplest possible case (plane strain, single slip) yet implementing the dynamics of the dislocation density evolution in two very different manners, namely (i) a hydrodynamic formulation which considers transport equations that are continuous in space and time while assuming a linear stress dependency of dislocation motion, and (ii) a stochastic cellular automaton implementation which assumes spatially and temporally discrete transport of discrete 'packets' of dislocation density which move according to an extremal dynamics. Despite the huge differences between both kinds of models, we find that the emergent patterns 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 which, however, do not affect the properties of the fully developed patterns.