Expansions for the linear-elastic contribution to the self-interaction force of dislocation curves

TL;DR

This paper develops asymptotic expansions for the self-interaction force of dislocation curves in metals, enabling quantitative comparison of models and improving numerical simulation accuracy.

Contribution

It provides the first asymptotic expansions of the self-interaction force beyond leading order, facilitating model comparison and error analysis.

Findings

Derived asymptotic expansions in  for self-interaction force

Developed numerical schemes based on these expansions

Bounded discretization errors in simulations

Abstract

The self-interaction force of dislocation curves in metals depends on the local arrangement of the atoms and on the nonlocal interaction between dislocation curve segments. While these nonlocal segment-segment interactions can be accurately described by linear elasticity when the segments are further apart than the atomic scale of size $\varepsilon$, this model breaks down and blows up when the segments are $O(\varepsilon)$ apart. To separate the nonlocal interactions from the local contribution, various models depending on $\varepsilon$ have been constructed to account for the nonlocal term. However, there are no quantitative comparisons available between these models. This paper makes such comparisons possible by expanding the self-interaction force in these models in $\varepsilon$ beyond the $O(1)$-term. Our derivation of these expansions relies on asymptotic analysis. The practical use of these expansions is demonstrated by developing numerical schemes for them, and by -- for the first time -- bounding the corresponding discretization error.