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

3 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

Fingerprint

Dislocation
Homogenization
Plasticity
Gradient
Energy
Korn's Inequality
Gradient Plasticity
Continuum Limit
Energy Balance
Rescaling
Term
Energy Function
Estimate
Rigidity
Spacing
Invariance
Vector Field
Divergence
Refinement
Model

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). Switzerland : Springer. https://doi.org/10.1007/978-3-319-18242-1_7

Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations. / Müller, Stefan; Scardia, Lucia; Zeppieri, Caterina Ida.

Analysis and Computation of Microstructure in Finite Plasticity . ed. / Sergio Conti; Klaus Hackl . Switzerland : Springer, 2015. p. 175-204 ( Lecture Notes in Applied and Computational Mechanics; Vol. 78).

Research output: Chapter in Book/Report/Conference proceedingChapter

Müller, S, Scardia, L & Zeppieri, CI 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 . Lecture Notes in Applied and Computational Mechanics, vol. 78, Springer, Switzerland , pp. 175-204. https://doi.org/10.1007/978-3-319-18242-1_7
Müller S, Scardia L, Zeppieri CI. Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations. In Conti S, Hackl K, editors, Analysis and Computation of Microstructure in Finite Plasticity . Switzerland : Springer. 2015. p. 175-204. ( Lecture Notes in Applied and Computational Mechanics). https://doi.org/10.1007/978-3-319-18242-1_7
Müller, Stefan ; Scardia, Lucia ; Zeppieri, Caterina Ida. / Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations. Analysis and Computation of Microstructure in Finite Plasticity . editor / Sergio Conti ; Klaus Hackl . Switzerland : Springer, 2015. pp. 175-204 ( Lecture Notes in Applied and Computational Mechanics).
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