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Upscaling of dislocation walls in finite domains

Published online by Cambridge University Press:  28 August 2014

P. VAN MEURS ,
A. MUNTEAN  and
M. A. PELETIER
P. VAN MEURS
Affiliation:
Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: p.j.p.v.meurs@tue.nl
A. MUNTEAN
Affiliation:
Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: p.j.p.v.meurs@tue.nl Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, The Netherlands
M. A. PELETIER
Affiliation:
Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: p.j.p.v.meurs@tue.nl Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, The Netherlands
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Abstract

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We wish to understand the macroscopic plastic behaviour of metals by upscaling the micro-mechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete model to be a one dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local. As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of Γ-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.

Keywords

PlasticityMultiscaleStraight edge-dislocationsDiscrete-to-continuum limitΓ-convergence74Q05, 74C05, 82B21, 49J45, 82D35
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 6 , December 2014 , pp. 749 - 781
Copyright
Copyright © Cambridge University Press 2014 

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