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MiS Preprint
37/2006

Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuum

Marcello Ponsiglione

Abstract

This paper deals with the passage from discrete to continuum in modeling the static elastic properties, in the setting of anti-planar linear elasticity, of vertical screw dislocations in a cylindrical crystal.

We study, in the framework of $\Gamma$-convergence, the asymptotic behavior of the elastic stored energy induced by dislocations as the atomic scale $\varepsilon$ tends to zero, in the regime of dilute dislocations, i.e., rescaling the energy functionals by $1/\varepsilon^2|\log \varepsilon|$.

First we consider a continuum model, where the atomic scale is introduced as an internal scale, usually called core radius. Then we focus on a purely discrete model. In both cases, we prove that the asymptotic elastic energy as $\varepsilon\to 0$ is essentially given by the number of dislocations present in the crystal. More precisely the energy per unit volume is proportional to the length of the dislocation lines, so that our result recovers in the limit as $\varepsilon\to 0$ a line tension model.

Received:
Apr 10, 2006
Published:
Apr 10, 2006
MSC Codes:
74N05, 74N15, 74G70, 74G65, 74C15, 74B15, 74B10
Keywords:
Crystals, Analysis of microstructure, Stress concentration, calculus of variations

Related publications

inJournal
2007 Repository Open Access
Marcello Ponsiglione

Elastic energy stored in a crystal induced by screw dislocations: From discrete to continuous

In: SIAM journal on mathematical analysis, 39 (2007) 2, pp. 449-469