Calculus of Variations and Geometric Measure Theory

I. Lucardesi - M. Morandotti - R. Scala - D. Zucco

Confinement of dislocations inside a crystal with a prescribed external strain

created by morandott on 20 Oct 2016
modified on 14 Feb 2019


Published Paper

Inserted: 20 oct 2016
Last Updated: 14 feb 2019

Journal: Riv. Mat. Univ. Parma
Volume: 9
Number: 2
Pages: 283-327
Year: 2018

ArXiv: 1610.06852 PDF


A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied in the framework of linear elasticity. Imposing a suitable boundary condition for the strain, namely requesting the non-vanishing of its boundary integral, results in a confinement effect. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The result is obtained by formulating the problem via the core radius approach and by studying the asymptotics as the core size vanishes. An iterative scheme is devised to prove the main result. This work sets the basis for studying the upscaling problem, i.e., the limit as $n\to\infty$, which is treated in 17.

Keywords: dislocations, core radius approach, divergence-measure fields, harmonic functions