Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations

Abstract

Abstract. We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary.We study the behaviour of the energy as the number n of walls tends to infinity, and characterise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter βn, corresponding to βn 1/n, 1/n βn 1, and βn 1, and the two critical regimes βn ∼ 1/n and βn ∼ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes.The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter βn.

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

“…[3,14]. Finally, our results are valid in the presence of certain non-local interactions, that may arise in applications such as surface tension [34], defects in metal crystals (dislocations) [20,26], pedestrian dynamics [8,18] and swarming [5].…”

“…The transition to a particle system takes place by substitution ofμ (20). Moreover, inρ t and q we replace μ t byμ (5)- (6) is used.…”

“…The additional terms follow from similar steps as the ones leading to (20). We omit further details.…”

“…Such formulation incorporates both the limit and the approximating sequence. Hence, we focus on the measure-formulations (20) and (23), without the specific choice μ 0 =μ n 0 . Our convergence proof is applicable to a class of approximating measures that is much broader than just sums of Dirac deltas.…”

“…We use −η · u, which is a simplified version of the usual viscous term in SPH that (also) involves W h * u; see [27]. We assign the value θ = 0 to the formulation in (23), and θ = 1 to (20). Both equations are now simultaneously written as:¨…”