Dislocation-induced stress in polycrystalline materials: mesoscopic simulations in the dislocation density formalism

TL;DR

This paper introduces a fast Fourier transform-based numerical method to evaluate dislocation-induced stress in polycrystalline materials, revealing complex stress behaviors influenced by dislocation density and grain size.

Contribution

The paper presents a novel, efficient numerical approach for calculating dislocation stress in materials with arbitrary dislocation orientations and densities, including analytical predictions of stress variance.

Findings

Stress distributions are highly nontrivial in polycrystalline layers with rapidly varying dislocation density.

The decay of dislocation density significantly affects stress features and their dependence on grain size.

Analytical predictions of stress variance agree with numerical simulations.

Abstract

In this paper we present a simple and effective numerical method which allows a fast Fourier transformation-based evaluation of stress generated by dislocations with arbitrary directions and Burgers vectors if the (site-dependent) dislocation density is known. Our method allows the evaluation of the dislocation stress using a rectangular grid with shape-anisotropic discretization cells without employing higher multipole moments of the dislocation interaction coefficients. Using the proposed method, we first simulate the stress created by relatively simple nonhomogeneous distributions of vertical edge and so-called 'mixed' dislocations in a disk-shaped sample, which is necessary to understand the dislocation behavior in more complicated systems. The main part of our research is devoted to the stress distribution in polycrystalline layers with the dislocation density rapidly varying with the distance to the layer bottom. Considering GaN as a typical example of such systems, we investigate dislocation-induced stress for edge and mixed dislocations, having random orientations of Burgers vectors among crystal grains. We show that the rapid decay of the dislocation density leads to many highly nontrivial features of the stress distributions in such layers and study in detail the dependence of these features on the average grain size. Finally we develop an analytical approach which allows us to predict the evolution of the stress variance with the grain size and compare analytical predictions with numerical results.