Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations

Stefan Müller, Lucia Scardia, Caterina Ida Zeppieri

Research output: Chapter in Book/Report/Conference proceedingChapter

5 Citations (Scopus)

Abstract

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
PublisherSpringer
Pages175-204
Number of pages30
ISBN (Electronic)9783319182421
ISBN (Print)9783319182414
DOIs
Publication statusPublished - 31 May 2015

Publication series

Name Lecture Notes in Applied and Computational Mechanics
Volume78

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  • Cite this

    Müller, S., Scardia, L., & Zeppieri, C. I. (2015). Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations. In S. Conti, & K. Hackl (Eds.), Analysis and Computation of Microstructure in Finite Plasticity (pp. 175-204). ( Lecture Notes in Applied and Computational Mechanics; Vol. 78). Springer. https://doi.org/10.1007/978-3-319-18242-1_7