Asymptotic analysis of boundary layers in a repulsive particle system

Cameron L Hall (1) , Thomas Hudson (2) , Patrick Van Meurs (3)

1 MI - Mathematical Institute [Oxford]
2 MATHERIALS - MATHematics for MatERIALS
3 KU - Kanazawa University

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. Our setting covers pileups of dislocations, dislocation dipoles and dislocation walls. The main challenge is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via Γ–convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.

Mots clés: Boundary layer; Gamma convergence; Matched asymptotic expansions; Discrete to continuum limits

Domaines: Equations aux dérivées partielles [math.AP]

Cameron L Hall, Thomas Hudson, Patrick Van Meurs. Asymptotic analysis of boundary layers in a repulsive particle system. Acta Applicandae Mathematicae, 2018, ⟨10.1007/s10440-017-0119-0⟩. ⟨hal-01398449⟩