*Published Paper*

**Inserted:** 22 dec 2010

**Last Updated:** 11 sep 2012

**Journal:** SIAM J. Math. Anal.

**Volume:** 44

**Pages:** 2372--2400

**Year:** 2012

**Abstract:**

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

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