*Published Paper*

**Inserted:** 2 feb 2013

**Last Updated:** 8 dec 2014

**Journal:** Indiana univ. Math. J.

**Volume:** 63

**Pages:** 1365-1396

**Year:** 2014

**Abstract:**

In this paper we show that a strain-gradient plasticity model arises as the $\Gamma$-limit of a *nonlinear* semi-discrete dislocation energy.
We restrict our analysis to the case of plane elasticity, so that edge dislocations can be modelled as point singularities of the strain field.

A key ingredient in the derivation is the extension of the Rigidity Estimate to the case of fields $\beta: U\subset \mathbb{R}^2\to \mathbb{R}^{2\times 2}$ with nonzero curl. We prove that the $L^2$-distance of $\beta$ from a single rotation matrix is bounded (up to a multiplicative constant) by the $L^2$-distance of $\beta$ from the group of rotations in the plane, modulo an error depending on the total mass of ${\rm Curl}\,\beta$. This reduces to the classical Rigidity Estimate in the case ${\rm Curl}\,\beta = 0$.

**Keywords:**
$\Gamma$-convergence, rigidity estimate, nonlinear plane elasticity, edge dislocations, strain-gradient plasticity

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