Calculus of Variations and Geometric Measure Theory

M. Ponsiglione

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

created by ponsiglio on 10 Apr 2006
modified on 02 Jul 2007


Accepted Paper

Inserted: 10 apr 2006
Last Updated: 2 jul 2007

Journal: SIAM J. Math. Anal.
Year: 2007


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. \par 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
$. \par 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 $\varepsilon$ tends to zero 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 $\varepsilon$ tends to zero a line tension model.

Keywords: calculus of variations, crystals, Analysis of microstructure, Stress concentration