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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Janusz Ginster
Carnegie Melllon University
Title: Plasticity as the Gamma-Limit of a Nonlinear Dislocation Energy with Mixed Growth

Abstract: 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 includes an elastic energy with subquadratic growth for elastic large strains which allows us to treat dislocations without introducing an ad-hoc cut-off radius.

As the interatomic distance tends to zero, we prove that the suitably rescaled energy $\Gamma$-converges 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.

Recording: http://mm.math.cmu.edu/recordings/cna/JanuszGinster_small.mp4
Date: Tuesday, November 8, 2016
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  Ian Tice