Upscaling of dislocation walls in finite domains

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 Γ-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.

References in corpus (5)

• Asymptotic behaviour of a pile-up of infinite walls of edge dislocations
• Statistical approach to dislocation dynamics: From dislocation correlations to a multiple-slip continuum plasticity theory
• Homogenization of dislocation dynamics
• Wasserstein gradient flows from large deviations of thermodynamic limits
• Gradient theory for plasticity via homogenization of discrete dislocations

Cited by in corpus (4)

• Upscaling of the dynamics of dislocation walls
• Convergence and non-convergence of many-particle evolutions with multiple signs
• Many-particle limits and non-convergence of dislocation wall pile-ups
• Quantitative estimate of the continuum approximations of interacting particle systems in one dimension