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straight edge-dislocations

European Journal of Applied Mathematics (1)

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1 results

Upscaling of dislocation walls in finite domains
P. VAN MEURS, A. MUNTEAN, M. A. PELETIER
Journal:

European Journal of Applied Mathematics / Volume 25 / Issue 6 / December 2014

Published online by Cambridge University Press:

28 August 2014, pp. 749-781
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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.