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