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

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.