Abstract
We consider a model for dislocations in crystals introduced by Koslowski, Cuitiño and Ortiz, which includes elastic interactions via a singular kernel behaving as the H 1/2 norm of the slip. We obtain a sharp-interface limit of the model within the framework of Γ-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitté and Seppecher to which their rearrangement argument no longer applies. Instead, we show that the microstructure must be approximately one-dimensional on most length scales and we exploit this property to derive a sharp lower bound.
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Conti, S., Garroni, A. & Müller, S. Singular Kernels, Multiscale Decomposition of Microstructure, and Dislocation Models. Arch Rational Mech Anal 199, 779–819 (2011). https://doi.org/10.1007/s00205-010-0333-7
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DOI: https://doi.org/10.1007/s00205-010-0333-7