The continuum limit of interacting dislocations on multiple slip systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 102.

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 type of regularisation general, and that we consider the joint limit n →∞ and δ → 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 size (i.e., δ) decays as n →∞.

DOI : 10.1051/cocv/2020038
Classification : 82C22, 74Q05, 35A15, 82D35
Mots-clés : Particle system, many-particle limit, Γ-convergence, dislocations 
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     author = {van Meurs, Patrick},
     title = {The continuum limit of interacting dislocations on multiple slip systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2020038},
     mrnumber = {4185070},
     zbl = {1459.82198},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2020038/}
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van Meurs, Patrick. The continuum limit of interacting dislocations on multiple slip systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 102. doi : 10.1051/cocv/2020038. http://www.numdam.org/articles/10.1051/cocv/2020038/

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