On the computational solution of vector-density based continuum dislocation dynamics models: a comparison of two plastic distortion and stress update algorithms

TL;DR

This paper compares two algorithms for updating plastic distortion in continuum dislocation dynamics models, demonstrating that a field dislocation mechanics-based scheme improves accuracy over traditional time integration methods.

Contribution

The authors introduce a new plastic distortion update scheme based on field dislocation mechanics that enhances numerical accuracy and consistency in dislocation dynamics simulations.

Findings

The new scheme reduces numerical errors in plastic distortion calculations.

Finite element implementation confirms improved accuracy of the new method.

Application to steel crystal under load demonstrates practical effectiveness.

Abstract

Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan's law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test is first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions.