A phase field crystal theory of the kinematics of dislocation lines

TL;DR

This paper develops a phase field crystal model to describe dislocation line kinematics, deriving an exact velocity expression and a method to ensure divergence-free stress, improving understanding of dislocation dynamics in crystals.

Contribution

It introduces a dislocation density tensor and a kinematic evolution law derived from a phase field crystal model, including a new method to enforce mechanical equilibrium.

Findings

Dislocation velocity is driven by Peach-Koehler force.

The PFCMEq model ensures divergence-free stress fields.

Near-field stress is regularized by the phase field.

Abstract

We introduce a dislocation density tensor and derive its kinematic evolution law from a phase field description of crystal deformations in three dimensions. The phase field crystal (PFC) model is used to define the lattice distortion, including topological singularities, and the associated configurational stresses. We derive an exact expression for the velocity of dislocation line determined by the phase field evolution, and show that dislocation motion in the PFC is driven by a Peach-Koehler force. As is well known from earlier PFC model studies, the configurational stress is not divergence free for a general field configuration. Therefore, we also present a method (PFCMEq) to constrain the diffusive dynamics to mechanical equilibrium by adding an independent and integrable distortion so that the total resulting stress is divergence free. In the PFCMEq model, the far-field stress agrees very well with the predictions from continuum elasticity, while the near-field stress around the dislocation core is regularized by the smooth nature of the phase-field. We apply this framework to study the rate of shrinkage of an dislocation loop seeded in its glide plane.