Geometric Rigidity for Incompatible Fields, and an Application to Strain-Gradient Plasticity

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 [9, Theorem 3.1] to the case of fields beta : U subset of R-2 -> R-2x2 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.