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

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 micro-to-meso-upscaling, and we obtain five expressions for the mesoscopic (continuum) internal stress. The upscaling method we illustrate here can be made mathematically rigorous, as we show in the companion paper (Geers et al., 2013. Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. Arch. Ration. Mech. Anal. 209, 495-539). The focus of the present paper is on the mechanical interpretation of the resulting internal stresses. In the continuum limit 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.