Continuum Model and Numerical Method for Dislocation Structure and Energy of Grain Boundaries

TL;DR

This paper introduces a continuum model for predicting the dislocation structure and energy of low angle grain boundaries in three dimensions, validated against atomistic simulations.

Contribution

It develops a novel continuum approach using dislocation density potential functions and constrained energy minimization for accurate grain boundary modeling.

Findings

Accurately predicts dislocation densities and energies of grain boundaries.

Validates model predictions against atomistic simulation results.

Effective for both planar and curved low angle grain boundaries.

Abstract

We present a continuum model to determine the dislocation structure and energy of low angle grain boundaries in three dimensions. The equilibrium dislocation structure is obtained by minimizing the grain boundary energy that is associated with the constituent dislocations subject to the constraint of Frank's formula. The orientation-dependent continuous distributions of dislocation lines on grain boundaries are described conveniently using the dislocation density potential functions, whose contour lines on the grain boundaries represent the dislocations. The energy of a grain boundary is the total energy of the constituent dislocations derived from discrete dislocation dynamics model, incorporating both the dislocation line energy and reactions of dislocations. The constrained energy minimization problem is solved by the augmented Lagrangian method and projection method. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of the energy and dislocation densities of both planar and curved low angle grain boundaries.