A quantitative model for the Frank-Read dislocation source based on pinned mean curvature flow

TL;DR

The paper presents a minimal, quantitative model for the Frank--Read dislocation source by treating dislocation lines as undergoing a pinned, forced mean curvature flow driven by external shear, while ignoring long-range elastic interactions. Through nondimensionalisation, the dynamics depend on a single forcing parameter $f_s$, yielding a compact evolution equation for the dislocation curve. The authors develop a geometrically intrinsic finite-difference discretisation with collision handling and remeshing, and provide an open-source implementation. Validation against Ni–Fe experimental images shows qualitative agreement in source geometry, while the model predicts a quadratic growth law for total dislocation length, $\ell \propto t^2$, and a loop-generation rate that can be quantified from late-time data. Overall, the work offers a tractable benchmark for dislocation source dynamics and a basis for exploring more complex, anisotropic or elastic-interaction effects in future studies.

Abstract

This work introduces a simple quantitative model for the Frank--Read source, considered to be one of the most important micro-mechanical mechanisms of dislocation creation in crystalline materials. It has long been known that these sources create dislocations in a repetitive, oscillatory process, which is driven by an external shear force. Unlike the existing explanations in the literature, the model introduced in the present article is based on just a few simple physical principles, namely line tension and dislocation motion due to a single slip plane flow rule, together with a pinning constraint on the ends of the central dislocation line. A complete discretisation, including suitable re-meshing and ``topological cutting'' algorithms, is described and simulation results are discussed. Despite its conceptual simplicity, the model and discretisation described in the present work yield remarkably accurate predictions about the shape and properties of the Frank--Read source. In particular, it is shown that only one dimensionless parameter controls the dynamics of the Frank--Read source if one neglects crystal anisotropy. This allows to derive an emergent law about the length of dislocation line generated per shear energy.

Figures (8)

• Figure 1: Comparison between experimental image of a Frank--Read source in NiFe, adapted from Figure 1.2.28 of Tad24, and a simulated curve using our model. In (b), the dimensionless parameter is $f_s = 5 \times 10^{-3}$, and $t=0.61$.
• Figure 2: Log-log plot of the total length of the dislocation lines generated by a constant applied shear stress in a simulation (solid line); a fitted asymptotic quadratic rate is shown for reference (dotted line). The simulation used dimensionless parameter $f_s = 5 \times 10^{-3}$, and time step $\tau = 1.2 \times 10^{-3}$.
• Figure 3: Illustration of the geometric setup.
• Figure 4: An illustration of the numerical approximation procedure for finding the approximate normal at node $\boldsymbol{\varphi}^m_j$. The displacement vector between nodes adjacent nodes (indicated in blue) is normalised and rotated anticlockwise, giving $\boldsymbol{n}_j^m$.
• Figure 5: Illustration of merging, the interpolated mesh at time $m+t_\text{hit}$ replaces the mesh at time $m+1$ and the simulation continues with the split, interpolated mesh.