Inserted: 27 aug 2018
Last Updated: 28 aug 2018
The upscaling of a system of screw dislocations in a material subject to an external strain is studied. The $\Gamma$-limit of a suitable rescaling of the renormalized energy is characterized in the space of probability measures. This corresponds to a discrete-to-continuum limit of the dislocations, which, as a byproduct, provides information on their distribution when the circulation of the tangential component of the external strain becomes larger and larger. In particular, dislocations are shown to concentrate at the boundary of the material and to distribute as the limiting external strain.
Keywords: $\Gamma$-convergence, dislocations, core radius approach, divergence-measure fields, discrete-to-continuum limit, Ginzburg-Landau vortices