Upscaling of dislocation walls in finite domains

TL;DR

This paper investigates how the size of finite domains influences the macroscopic plastic behavior of metals by modeling dislocation walls as a one-dimensional particle system and deriving effective equations through b3-convergence.

Contribution

It introduces a simplified dislocation network model in finite domains and derives effective macroscopic equations considering size effects, advancing understanding of realistic geometries.

Findings

Size of the domain significantly affects the macroscopic dislocation models.

Effective equations for dislocation density are derived via b3-convergence.

Classification of macroscopic models based on domain size.

Abstract

We wish to understand the macroscopic plastic behaviour of metals by upscaling the micro-mechanics of dislocations. We consider a highly simplified dislocation network, which allows our microscopic model to be a one dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local. As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of \Gamma-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.