Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations

Stefan Müller, Lucia Scardia, Caterina Ida Zeppieri

Research output: Chapter or section in a book/report/conference proceedingChapter or section

9 Citations (SciVal)


This article gives a short description and a slight refinement of recentwork [MSZ15], [SZ12] on the derivation of gradient plasticity models fromdiscrete dislocations models.We focus on an array of parallel edge dislocations. This reduces the problem to a two-dimensional setting. As in the work Garroni, Leoni & Ponsiglione [GLP10] we show that in the regime where the number of dislocation Nε is of the order log 1/ε (where ε is the ratio of the lattice spacing and the macroscopic dimensions of the body) the contributions of the self-energy of the dislocations and their interaction energy balance. Upon suitable rescaling one obtains a continuum limit which contains an elastic energy term and a term which depends on the homogenized dislocation density. The main novelty is that our model allows for microscopic energies which are not quadratic and reflect the invariance under rotations. A key mathematical ingredient is a rigidity estimate in the presence of dislocations which combines the nonlinear Korn inequality of Friesecke, James & Müller [FJM02] and the linear Bourgain& Brezis estimate [BB07] for vector fields with controlled divergence. The main technical improvement of this article compared to [MSZ15] is the removal of the upper boundW(F) ≤Cdist<sup>2</sup>(F,SO(2)) on the stored energy function.

Original languageEnglish
Title of host publicationAnalysis and Computation of Microstructure in Finite Plasticity
EditorsSergio Conti, Klaus Hackl
Place of PublicationSwitzerland
Number of pages30
ISBN (Electronic)9783319182421
ISBN (Print)9783319182414
Publication statusPublished - 31 May 2015

Publication series

Name Lecture Notes in Applied and Computational Mechanics


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