*Published Paper*

**Inserted:** 25 jun 2008

**Last Updated:** 18 sep 2020

**Journal:** J. Eur. Math. Soc.

**Volume:** 12

**Number:** 5

**Pages:** 1231-1266

**Year:** 2010

**Abstract:**

In this paper, 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 in exam, 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 as the core radius tends to zero and the
number of dislocations tends to infinity of this energy (suitably rescaled),
takes the form $$ E= \int_{{\Om}} (W(\beta^{e)} + \f (\Curl \beta^{e))} dx, $$ where
$\beta^e$ represents the elastic part of the macroscopic strain, and $\Curl
\beta^e$ represents the geometrically necessary dislocation density. The
plastic energy density $\f$ 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.

**Keywords:**
relaxation, dislocations, strain gradient theories, plasticity

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