Qualitative and quantitative properties of the dynamics of screw dislocations

TL;DR

This paper investigates screw dislocation dynamics in elastic media, analyzing boundary interactions, collision events, and confinement effects using a semi-discrete model, with analytical results on forces, collision times, and energetic stability.

Contribution

It provides new analytical insights into boundary attraction, collision bounds, and confinement stability of screw dislocations under external stress.

Findings

Boundaries attract dislocations and influence collision times.

Dislocations tend to stay confined under external stress.

Analytical expressions for Peach--Koehler force near boundaries.

Abstract

This note collects some results on the behaviour of screw dislocation in an elastic medium. By using a semi-discrete model, we are able to investigate two specific aspects of the dynamics, namely (i) the interaction with free boundaries and collision events and (ii) the confinement inside the domain when a suitable Dirichlet-type boundary condition is imposed. In the first case, we analytically prove that free boundaries attract dislocations and we provide an expression for the Peach--Koehler force on a dislocation near the boundary. Moreover, we use this to prove an upper bound on the collision time of a dislocation with the boundary, provided certain geometric conditions are satisfied. An upper bound on the collision time for two dislocations with opposite Burgers vectors hitting each other is also obtained. In the second case, we turn to domains whose boundaries are subject to an external stress. In this situation, we prove that dislocations find it energetically favourable to stay confined inside the material instead of getting closer to the boundary. The result first proved for a single dislocation in the material is extended to a system of many dislocations, for which the analysis requires the careful treatments of the interaction terms.