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 Γ-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 β : U ⊂ ℝ2 → ℝ2×2 with nonzero curl. We prove that the L2-distance of β from a single rotation matrix is bounded (up to a multiplicative constant) by the L2-distance of β from the group of rotations in the plane, modulo an error depending on the total mass of Curl β. This reduces to the classical rigidity estimate in the case Curl β = 0.