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The self-interaction force of dislocation curves in metals depends on the local arrangement of the atoms and on the non-local interaction between dislocation curve segments. While these non-local 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 non-local interactions from the local contribution, various models depending on  			 		 $\varepsilon$ 		have been constructed to account for the non-local 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 discretisation error.

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