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Plasticity as the \({\Gamma}\)-Limit of a Two-dimensional Dislocation Energy: The Critical Regime Without the Assumption of Well-Separateness

  • Janusz GinsterEmail author
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Abstract

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work is that in this work the well-separateness of dislocations is not assumed. In order to prove meaningful lower bounds, the ball construction technique, which was developed in the context of Ginzburg–Landau by Jerrard and Sandier, is adapted and modified. To overcome the difficulty of loss of rigidity on thin annuli during the ball construction a combination of combinatorial arguments and local modifications of the occurring elastic strains is presented.

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Acknowledgements

The author is very grateful to Stefan Müller and Sergio Conti for bringing the problem to his attention and for many fruitful discussions.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Nonlinear AnalysisCarnegie Mellon UniversityPittsburghUSA

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