Calculus of Variations and Geometric Measure Theory

S. Conti - A. Garroni - S. Müller

Singular kernels, multiscale decomposition of microstructure, and dislocation models

created by garroni on 17 Feb 2010
modified on 18 Sep 2020


Submitted Paper

Inserted: 17 feb 2010
Last Updated: 18 sep 2020

Year: 2010

ArXiv: 1003.1917 PDF


We consider a model for dislocations in crystals introduced by Koslowski, Cuiti\~no 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 $\Gamma$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitt\'e 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 exploit this property to derive a sharp lower bound.

Keywords: non local energies, dislocations, microstructures