Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

Ilaria Lucardesi, [a] Marco Morandotti, [b] Riccardo Scala, [c] and Davide Zucco [b]

Confinement of dislocations inside a crystal with a prescribed external strain

Pages: 283-327
Accepted: 2 October 2018
Mathematics Subject Classification (2010): 74E15, (35J25, 74B05, 49J40, 31A05).
Keywords: Dislocations, core radius approach, harmonic functions, divergence-measure fields.
[a]: Institut Élie Cartan de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy, France
[b]: Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
[c]: Sapienza, Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy

Abstract: 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].

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