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The discrete dislocation dynamics of multiple dislocation loops

Stefania Patrizi, Mary Vaughan

TL;DR

This work derives a rigorous mesoscopic limit for a nonlocal, fractional Allen--Cahn equation arising from the Peierls--Nabarro model of slip-loop dislocations. By building global and local barriers from phase-transition layers and a carefully constructed corrector $\psi$, and employing the Barles–Da Lio–Souganidis front-propagation framework, the authors show that multiple dislocation fronts move independently by mean curvature as $\varepsilon\to0$, with inter-front interactions negligible at leading order. The result provides a formal and then rigorous passage from a microscopic dislocation model to discrete dislocation dynamics in dimensions $n\ge 2$, laying groundwork for extensions to many fronts, front collisions, and macroscopic elastoplastic limits. A key technical advance is the existence, sharp bounds, and asymptotics of the corrector $\psi$, which enables global barrier constructions despite the nonlocal nature of the dynamics. The methodology and estimates pave the way for analyzing ensembles of curved dislocations and potential macroscopic plastic-flow models derived from microscopic phase-field descriptions.

Abstract

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls-Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in $\mathbb{R}^n$, $n \geq 2$. After suitably rescaling the equation with a small phase parameter $\varepsilon>0$, the rescaled solution solves a fractional Allen-Cahn equation. We show that, as $\varepsilon \to 0$, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.

The discrete dislocation dynamics of multiple dislocation loops

TL;DR

This work derives a rigorous mesoscopic limit for a nonlocal, fractional Allen--Cahn equation arising from the Peierls--Nabarro model of slip-loop dislocations. By building global and local barriers from phase-transition layers and a carefully constructed corrector , and employing the Barles–Da Lio–Souganidis front-propagation framework, the authors show that multiple dislocation fronts move independently by mean curvature as , with inter-front interactions negligible at leading order. The result provides a formal and then rigorous passage from a microscopic dislocation model to discrete dislocation dynamics in dimensions , laying groundwork for extensions to many fronts, front collisions, and macroscopic elastoplastic limits. A key technical advance is the existence, sharp bounds, and asymptotics of the corrector , which enables global barrier constructions despite the nonlocal nature of the dynamics. The methodology and estimates pave the way for analyzing ensembles of curved dislocations and potential macroscopic plastic-flow models derived from microscopic phase-field descriptions.

Abstract

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls-Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in , . After suitably rescaling the equation with a small phase parameter , the rescaled solution solves a fractional Allen-Cahn equation. We show that, as , the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.
Paper Structure (44 sections, 26 theorems, 368 equations, 3 figures)

This paper contains 44 sections, 26 theorems, 368 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem and main result
  3. Strategies and significance
  4. The Peierls--Nabarro model for slip loop dislocations
  5. Organization of the paper
  6. Notations
  7. Motion by mean curvature
  8. The level set approach
  9. Generalized flows
  10. Extension of the signed distance function
  11. Preliminary results
  12. Fractional Laplacians
  13. Properties of solutions to \ref{['eq:pde']}
  14. The phase transition, the corrector, and the auxiliary functions
  15. The phase transition $\phi$
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $u^{\varepsilon} = u^{\varepsilon}(t,x)$ be the unique solution of the reaction-diffusion equation eq:pde with the initial datum $u^{\varepsilon}_0:\mathbb{R}^n \to (0,N)$ defined by where $\phi$ solves eq:standing wave, $d_i^0$ are given in eq:initial d_i, and $N\geq1$. Then, as $\varepsilon \to 0$, the solutions $u^{\varepsilon}$ satisfy where $( {^+}\Omega_t^i, \Gamma_t^i, {^-}\Omega_t^i)

Figures (3)

  • Figure 1: Initial configuration for $N=3$ in dimension $n=2$
  • Figure 2: Convergence result for $N=3$ in dimension $n=2$
  • Figure 3: Dislocation types in a slip loop dislocation with fixed Burgers' vector

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: Extension of the signed distance function
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3: Comparison principle in $\mathbb{R}^n$
  • Proposition 3.4: Comparison principle in bounded domains
  • Proposition 3.5: Existence and uniqueness
  • proof
  • Lemma 4.1
  • ...and 31 more