### 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 language | English |
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Title of host publication | Analysis and Computation of Microstructure in Finite Plasticity |

Editors | Sergio Conti, Klaus Hackl |

Place of Publication | Switzerland |

Publisher | Springer |

Pages | 175-204 |

Number of pages | 30 |

ISBN (Electronic) | 9783319182421 |

ISBN (Print) | 9783319182414 |

DOIs | |

Publication status | Published - 31 May 2015 |

### Publication series

Name | Lecture Notes in Applied and Computational Mechanics |
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Volume | 78 |

### Fingerprint

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*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

}

TY - CHAP

T1 - Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations

AU - Müller, Stefan

AU - Scardia, Lucia

AU - Zeppieri, Caterina Ida

PY - 2015/5/31

Y1 - 2015/5/31

N2 - 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) ≤Cdist2(F,SO(2)) on the stored energy function.

AB - 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) ≤Cdist2(F,SO(2)) on the stored energy function.

UR - http://www.scopus.com/inward/record.url?scp=84928669504&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/978-3-319-18242-1_7

U2 - 10.1007/978-3-319-18242-1_7

DO - 10.1007/978-3-319-18242-1_7

M3 - Chapter

SN - 9783319182414

T3 - Lecture Notes in Applied and Computational Mechanics

SP - 175

EP - 204

BT - Analysis and Computation of Microstructure in Finite Plasticity

A2 - Conti, Sergio

A2 - Hackl , Klaus

PB - Springer

CY - Switzerland

ER -