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Analysis and Computation of Microstructure in Finite Plasticity pp 175–204Cite as

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

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Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 78))

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 N ε is of the order log \({1}\over{\varepsilon}\) (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 bound W(F) ≤ Cdist 2(F,SO(2)) on the stored energy function.

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Author information

Authors and Affiliations

  1. Hausdorff Center for Mathematics and Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Stefan Müller

  2. Department for Mathematical Sciences, University of Bath, Bath, UK

    Lucia Scardia

  3. Applied Mathematics, University of Münster, Münster, Germany

    Caterina Ida Zeppieri

Authors
  1. Stefan Müller
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  2. Lucia Scardia
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  3. Caterina Ida Zeppieri
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Correspondence to Stefan Müller .

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Editors and Affiliations

  1. Inst. für Angewandte Mathematik, Universität Bonn, Bonn, Germany

    Sergio Conti

  2. Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany

    Klaus Hackl

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Müller, S., Scardia, L., Zeppieri, C.I. (2015). Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations. In: Conti, S., Hackl, K. (eds) Analysis and Computation of Microstructure in Finite Plasticity. Lecture Notes in Applied and Computational Mechanics, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-319-18242-1_7

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  • DOI: https://doi.org/10.1007/978-3-319-18242-1_7

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18241-4

  • Online ISBN: 978-3-319-18242-1

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