IUMJ

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

Authors: Lucia Scardia, Caterina Ida Zeppieri and Stafan Muller

Issue: Volume 63 (2014), Issue 5, 1365-1396

Abstract:

In this paper, we show that a strain-gradient plasticity model arises as the $\Gamma$-limit of a \emph{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 [G. Friesecke, R.D. James, and S. M\"uller, \textit{A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity}, Comm. Pure Appl. Math. \textbf{55} (2002), no. 11, 1461--1506, Theorem 3.1] to the case of fields $\beta:U\subset\mathbb{R}^2\to\mathbb{R}^{2\times2}$ 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 $\Curl\beta$. This reduces to the classical rigidity estimate in the case $\Curl\beta=0$.