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Homogenization of vector-valued partition problems and dislocation cell structures in the plane

  • Sergio Conti
  • Adriana GarroniEmail author
  • Stefan Müller
Article
  • 126 Downloads

Abstract

We consider target-space homogenization for energies defined on partitions parametrized by a discrete lattice \(\mathcal {B}\subset \mathbb {R}^N\). For a small \(\sigma >0\), the variable is a piecewise constant function taking values in \(\sigma \mathcal {B}\), and the energy depends on the jumps and their orientation. In the limit as \(\sigma \rightarrow 0\) we obtain a homogenized functional defined on functions of bounded variation. This result is relevant in the study of dislocation structures in plastically deformed crystals. We review recent literature on the topic and propose our limiting effective energy as a continuum model for strain-gradient plasticity.

Keywords

Burger Vector Slip Plane Energy Density Dislocation Line Cell Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

SC and SM acknowledge financial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 “The mathematics of emergent effects”.

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Copyright information

© Unione Matematica Italiana 2016

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Dipartimento di MatematicaSapienza, Università di RomaRomeItaly

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