Calculus of Variations and Geometric Measure Theory

L. Scardia - C. I. Zeppieri

Line-tension model for plasticity as the $\Gamma$-limit of a nonlinear dislocation energy

created by zeppieri on 22 Dec 2010
modified on 11 Sep 2012


Published Paper

Inserted: 22 dec 2010
Last Updated: 11 sep 2012

Journal: SIAM J. Math. Anal.
Volume: 44
Pages: 2372--2400
Year: 2012


In this paper we rigorously derive a line-tension model for plasticity as the $\Gamma$-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The $\Gamma$-limit we obtain as the length of the Burgers vector tends to zero has the same form as the $\Gamma$-limit obtained by starting from a linear, semi-discrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration.

Keywords: $\Gamma$-convergence, nonlinear elasticity, dislocations, plasticity, rigidity estimate