Continuum Representation of Systems of Dislocation Lines: A General Method for Deriving Closed-Form Evolution Equations

TL;DR

This paper develops a continuum modeling approach for dislocation systems in plasticity, using the Maximum Information Entropy Principle to derive closed-form evolution equations that accurately replicate discrete dislocation dynamics in complex geometries.

Contribution

It introduces a novel methodology based on MIEP to derive closed-form evolution equations for dislocation densities, improving the representation of dislocation microstructure evolution.

Findings

Excellent agreement with discrete simulations using few dislocation density measures

Accurate reconstruction of dislocation orientation distributions

Effective modeling of complex, heterogeneous dislocation geometries

Abstract

Plasticity is governed by the evolution of, in general anisotropic, systems of dislocations. We seek to faithfully represent this evolution in terms of density-like variables which average over the discrete dislocation microstructure. Starting from T. Hochrainer's continuum theory of dislocations (CDD), we introduce a methodology based on the 'Maximum Information Entropy Principle' (MIEP) for deriving closed-form evolution equations for dislocation density measures of different order. These equations provide an optimum representation of the kinematic properties of systems of curved and connected dislocation lines with the information contained in a given set of density measures. The performance of the derived equations is benchmarked against other models proposed in the literature, using discrete dislocation dynamics simulations as a reference. As a benchmark problem we study dislocations moving in a highly heterogeneous, persistent slip-band-like geometry. We demonstrate that excellent agreement with discrete simulations can be obtained in terms of a very small number of averaged dislocation fields containing information about the edge and screw components of the total and excess (geometrically necessary) dislocation densities. From these the full dislocation orientation distribution which emerges as dislocations move through a channel-wall structure can be faithfully reconstructed.