A proof of Taylor scaling for curvature-driven dislocation motion through random arrays of obstacles

TL;DR

This paper proves that the critical shear stress needed for dislocation motion in a random obstacle array scales with the square root of obstacle density, confirming Taylor scaling in a new mathematical setting.

Contribution

It provides a rigorous mathematical proof of Taylor scaling for curvature-driven dislocation motion in the presence of random obstacles, extending previous theoretical understanding.

Findings

Critical shear stress scales with the square root of obstacle density.

Taylor scaling dominates at low obstacle densities.

Dislocation motion is characterized by a line-tension and curvature model.

Abstract

We prove Taylor scaling for dislocation lines characterized by line-tension and moving by curvature under the action of an applied shear stress in a plane containing a random array of obstacles. Specifically, we show--in the sense of optimal scaling--that the critical applied shear stress for yielding, or percolation-like unbounded motion of the dislocation, scales in proportion to the square root of the obstacle density. For sufficiently small obstacle densities, Taylor scaling dominates the linear-scaling that results from purely energetic considerations and, therefore, characterizes the dominant rate-limiting mechanism in that regime.