<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2108_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2108_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. After suitably rescaling the equation with a small phase parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2108_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the rescaled solution solves a fractional Allen–Cahn equation. We show that, as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2108_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.</p>

The Discrete Dislocation Dynamics of Multiple Dislocation Loops

期刊论文
发表日期：

2025年5月20日

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Stefania Patrizi,
Mary Vaughan

摘要

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 $\mathbb {R}^n$ $R^{n}$ , $n \ge 2$ $n \geq 2$ . After suitably rescaling the equation with a small phase parameter $\varepsilon >0$ $ε > 0$ , the rescaled solution solves a fractional Allen–Cahn equation. We show that, as $\varepsilon \rightarrow 0$ $ε \to 0$ , the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.