# Gradient theory for plasticity via homogenization of discrete dislocations

Adriana Garroni; Giovanni Leoni; Marcello Ponsiglione

Journal of the European Mathematical Society (2010)

- Volume: 012, Issue: 5, page 1231-1266
- ISSN: 1435-9855

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topGarroni, Adriana, Leoni, Giovanni, and Ponsiglione, Marcello. "Gradient theory for plasticity via homogenization of discrete dislocations." Journal of the European Mathematical Society 012.5 (2010): 1231-1266. <http://eudml.org/doc/277518>.

@article{Garroni2010,

abstract = {We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations.
We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem.
The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the $\Gamma $-limit of this
energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form
$E=\int _\Omega (W(\beta ^e)+\varphi (\operatorname\{Curl\}\beta ^e))dx$,
where $\beta ^e$ represents the elastic part of the macroscopic strain, and $\operatorname\{Curl\}\beta ^e$ represents the geometrically necessary dislocation density. The plastic energy density $\varphi $ is defined explicitly through an
asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed
in crystals such as dislocation walls.},

author = {Garroni, Adriana, Leoni, Giovanni, Ponsiglione, Marcello},

journal = {Journal of the European Mathematical Society},

keywords = {variational models; energy minimization; relaxation; plasticity; strain gradient theories; stress concentration; dislocations; energy minimization; relaxation; stress concentration},

language = {eng},

number = {5},

pages = {1231-1266},

publisher = {European Mathematical Society Publishing House},

title = {Gradient theory for plasticity via homogenization of discrete dislocations},

url = {http://eudml.org/doc/277518},

volume = {012},

year = {2010},

}

TY - JOUR

AU - Garroni, Adriana

AU - Leoni, Giovanni

AU - Ponsiglione, Marcello

TI - Gradient theory for plasticity via homogenization of discrete dislocations

JO - Journal of the European Mathematical Society

PY - 2010

PB - European Mathematical Society Publishing House

VL - 012

IS - 5

SP - 1231

EP - 1266

AB - We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations.
We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem.
The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the $\Gamma $-limit of this
energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form
$E=\int _\Omega (W(\beta ^e)+\varphi (\operatorname{Curl}\beta ^e))dx$,
where $\beta ^e$ represents the elastic part of the macroscopic strain, and $\operatorname{Curl}\beta ^e$ represents the geometrically necessary dislocation density. The plastic energy density $\varphi $ is defined explicitly through an
asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed
in crystals such as dislocation walls.

LA - eng

KW - variational models; energy minimization; relaxation; plasticity; strain gradient theories; stress concentration; dislocations; energy minimization; relaxation; stress concentration

UR - http://eudml.org/doc/277518

ER -

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