Calculus of Variations and Geometric Measure Theory

A. Garroni - G. Leoni - M. Ponsiglione

Gradient theory for plasticity via homogenization of discrete dislocations

created by ponsiglio on 25 Jun 2008
modified by garroni on 18 Sep 2020


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

ArXiv: 0808.2361 PDF


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 $$ E= \int{\Om} (W(\betae) + \f (\Curl \betae)) dx, $$ 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