Macroscopic Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

Janusz Ginster
Department of Mathematical Sciences, Carnegie Mellon University
Wednesday, June 21, 2017 - 14:00

Crystal Plasticity is the effect of a crystal undergoing a permanent change of shape in response to applied forces. At the atomic scale, dislocations -- local defects of the crystalline lattice concentrated on lines -- are considered to play a main role in this effect. In this talk, we present a rotationally invariant model for straight parallel edge dislocations in an infinite cylindrical domain. In contrast to existing literature, our model features an elastic energy with subquadratic growth at infinity which allows us to treat dislocations without the need to introduce an ad-hoc cut-off radius. As the interatomic distance tends to zero, we prove that the suitably rescaled energy $\Gamma$-converges under the assumption of well-separateness of dislocations to a macroscopic plasticity model. The main mathematical tool to control the rotational invariance of the energy and obtain compactness is a geometric rigidity result for fields with non-vanishing curl. We discuss the proof of this result and show how fine estimates for measures, whose divergence is bounded in certain critical negative Sobolev spaces, enter. Moreover, we give a brief idea of how to overcome the technical assumption of well-separateness.

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