Gradient theory for plasticity via homogenization of discrete dislocations

TL;DR

This paper derives a macroscopic strain gradient theory for plasticity from a model of discrete dislocations using homogenization, capturing the effects of dislocation patterns and crystalline structure.

Contribution

It introduces a new homogenized energy functional for plasticity derived from discrete dislocation models, explicitly characterizing the plastic energy density.

Findings

Gamma-limit of dislocation energy involves elastic and dislocation density terms

Plastic energy density is explicitly computed and depends on crystalline structure

Model accounts for dislocation pattern formations like walls

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.