  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.
