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The continuum limit of interacting dislocations on multiple slip systems

In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a $Γ$-convergence result on the interaction energy as the number of particles tends to infinity. As the leading application, we consider $n$ edge dislocations in multiple slip systems. Since the interaction potential of dislocations has a logarithmic singularity at zero with a sign that depends on the orientation of the slip systems, the interaction energy is unbounded from below. To make the minimization problem of this energy meaningful, we follow the common approach to regularise the interaction potential over a length-scale $δ> 0$. The novelty of our result is that we leave the \emph{type} of regularisation general, and that we consider the joint limit $n \to \infty$ and $δ\to 0$. Our result shows that the limit behaviour of the interaction energy is not affected by the type of the regularisation used, but that it may depend on how fast the \emph{size} (i.e., $δ$) decays as $n \to \infty$.