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 adhoc cutoff radius. As the interatomic distance tends to zero, we prove that the suitably rescaled energy $\Gamma$converges under the assumption of wellseparateness 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 nonvanishing 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 wellseparateness.
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Macroscopic Plasticity as the $\Gamma$Limit of a Nonlinear Dislocation Energy with Mixed Growth
Research Group:
Janusz Ginster
Institution:
Department of Mathematical Sciences, Carnegie Mellon University
Location:
A133
Schedule:
Wednesday, June 21, 2017  14:00
Abstract:
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