Gradient theory for plasticity via homogenization of discrete dislocations

Garroni, 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 -