Mechanics of dislocation pile-ups: A unification of scaling regimes

TL;DR

This paper unifies various models of dislocation pile-ups by identifying five scaling regimes through a dimensionless parameter, providing rigorous continuum stress expressions and connecting discrete dislocation arrangements to continuum theories.

Contribution

It introduces a unifying framework for dislocation pile-up models by rigorously deriving continuum stress expressions across five scaling regimes based on a key dimensionless parameter.

Findings

Identifies five distinct scaling regimes for dislocation pile-ups.

Derives explicit continuum internal stress expressions for each regime.

Connects discrete dislocation configurations to continuum models rigorously.

Abstract

This paper unravels the problem of an idealised pile-up of n infinite, equi-spaced walls of edge dislocations at equilibrium. We define a dimensionless parameter that depends on the geometric, constitutive and loading parameters of the problem, and we identify five different scaling regimes corresponding to different values of that parameter for large n. For each of the cases we perform a rigorous micro-to-meso upscaling, and we obtain five expressions for the mesoscopic (continuum) internal stress. We recover some expressions for the internal stress that are already in use in the mechanical community, as well as some new models. The results in this paper offer a unifying approach to such models, since they can be viewed as the outcome of the same discrete dislocation setup, for different values of the dimensionless parameter (i.e., for different local dislocations arrangements). In addition, the rigorous nature of the upscaling removes the need for ad hoc assumptions.