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.

我们引入了位错密度张量，并从三维晶体变形的相场描述中推导出其运动学演化规律。相场晶体 (PFC) 模型用于定义晶格畸变，包括拓扑奇点和相关的构型应力。我们推导出由相场演化确定的位错线速度的精确表达式，并表明 PFC 中的位错运动是由 Peach-Koehler 力驱动的。从早期的 PFC 模型研究中众所周知，一般场配置的配置应力不是无散度的。因此，我们还提出了一种方法 (PFCMEq)，通过添加独立且可积分的变形将扩散动力学约束到机械平衡，从而使总产生的应力无发散。在 PFCMEq 模型中，远场应力与连续弹性的预测非常吻合，而位错核心周围的近场应力则通过相场的平滑性质进行规则化。我们应用该框架来研究在其滑行平面中播种的位错环的收缩率。