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 [10, Theorem 3.1] to the case of fields β : U ⊂ R → R2×2 with nonzero curl. We prove that the L-distance of β from a single rotation matrix is bounded (up to a multiplicative constant) by the L-distance of β from the group of rotations in the plane, modulo an error depending on the total mass of Curlβ. This reduces to the classical rigidity estimate in the case Curlβ = 0.