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

Stefan Müller, Lucia Scardia, Caterina Ida Zeppieri

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)1365-1396
Number of pages32
JournalIndiana University Mathematics Journal
Volume63
Issue number5
DOIs
Publication statusPublished - 2014

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