Upscaling of dislocation walls in finite domains

P.J.P. Meurs, van, A. Muntean, M.A. Peletier

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Abstract

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 G-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. Key words: Plasticity; Multiscale; Straight edge-dislocations; Discrete-to-continuum limit; G-convergence
Original languageEnglish
Pages (from-to)749-781
JournalEuropean Journal of Applied Mathematics
Volume25
Issue number6
DOIs
Publication statusPublished - 2014

Keywords

  • Γ-convergence
  • 74Q05, 74C05, 82B21, 49J45, 82D35
  • Discrete-to-continuum limit
  • Multiscale
  • Plasticity
  • Straight edge-dislocations

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