Calculus of Variations and Geometric Measure Theory

S. Müller - L. Scardia - C. I. Zeppieri

Geometric rigidity for incompatible fields and an application to strain-gradient plasticity

created by zeppieri on 02 Feb 2013
modified by zeppieri1 on 08 Dec 2014


Published Paper

Inserted: 2 feb 2013
Last Updated: 8 dec 2014

Journal: Indiana univ. Math. J.
Volume: 63
Pages: 1365-1396
Year: 2014


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