Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

Hall, Cameron L.; Hudson, Thomas E.; Meurs, Patrick

doi:10.1007/s10440-017-0119-0

Cited by 10 publications

(7 citation statements)

References 46 publications

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“…Especially in the case where D(U ) confines ϕ * , the graphs of ϕ * and ρ are close to each other. This observation is in line with the literature (see, e.g., [GvMPS16,GPPS13,HCO10,HHvM18]). In this paper we wish to finally quantify this observation.…”

Section: Introductionsupporting

confidence: 93%

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Quantitative estimate of the continuum approximations of interacting particle systems in one dimension

Kimura¹,

Meurs²

2020

Preprint

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We consider a large class of interacting particle systems in 1D described by an energy whose interaction potential is singular and non-local. This class covers Riesz gases (in particular, log gases) and applications to plasticity and numerical integration. While it is well established that the minimisers of such interaction energies converge to a certain particle density profile as the number of particles tends to infinity, any bound on the rate of this convergence is only known in special cases by means of quantitative estimates. The main result of this paper extends these quantitative estimates to a large class of interaction energies by a different proof. The proof relies on one-dimensional features such as the convexity of the interaction potential and the ordering of the particles. The main novelty of the proof is the treatment of the singularity of the interaction potential by means of a carefully chosen renormalisation.

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Section: Introductionsupporting

confidence: 93%

“…Plasticity The paper series started by [GPPS13,Hal11] and continued in [GvMPS16,HHvM18,vM18a,vM18b,vMMP14] studies the connection between models for plasticity of metals and an underlying microscopic model in a one-dimensional setting. This microscopic model is a minimisation problems of a certain E n of the form (1).…”

Section: Position In the Literaturementioning

confidence: 99%

Quantitative estimate of the continuum approximations of interacting particle systems in one dimension

Kimura¹,

Meurs²

2020

Preprint

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“…Finally, we observe small boundary-layer effects close to the barriers at x ∈ {0, 1}. We do not study these effects here, and refer to [HHvM16] for analysis and numerics of such boundary layers at equilibrium. (90)…”

Section: Using That Xk+lmentioning

confidence: 98%

Many-particle limits and non-convergence of dislocation wall pile-ups

Meurs

2017

Nonlinearity

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Abstract. The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repelling force on each other, resulting in a nontrivial equilibrium configuration. This class of particle systems covers the setting of pile-ups of dislocation walls, which is an idealised setup for studying the microscopic origin of several dislocation density models in the literature. Such density models are used to construct constitutive relations in plasticity models.Our aim is to pass to the many-particle limit. The main challenge is the combination of the nonlocal nature of the interactions, the singularity of the interaction potential between particles of the same type, the non-convexity of the the interaction potential between particles of the opposite type, and the interplay between the length-scale of the domain with the length-scale n of the decay of the interaction potential. Our main results are the Γ-convergence of the energy of the particle positions, the evolutionary convergence of the related gradient flows for n sufficiently large, and the non-convergence of the gradient flows for n sufficiently small.

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“…The mechanical behaviour of one-dimensional systems has been of interest for decades. Such systems serve as toy models for higher-dimensional theoretical investigations and are of interest with respect to one-dimensional structures, see, e.g., [7,8,10,11,19]. In order to understand the effective behaviour of materials, the systems are studied as the number of particles tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

Onset of fracture in random heterogeneous particle chains

Lauerbach¹,

Neukamm²,

Schäffner³

et al. 2021

Preprint

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In mechanical systems it is of interest to know the onset of fracture in dependence of the boundary conditions. Here we study a one-dimensional model which allows for an underlying heterogeneous structure in the discrete setting. Such models have recently been studied in the passage to the continuum by means of variational convergence (Γ-convergence). The Γ-limit results determine thresholds of the boundary condition, which mark a transition from purely elastic behaviour to the occurrence of a crack. In this article we provide a notion of fracture in the discrete setting and show that its continuum limit yields the same threshold as that obtained from the Γ-limit. Since the calculation of the fracture threshold is much easier with the new method, we see a good chance that this new approach will turn out useful in applications.

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Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

Cited by 10 publications

References 46 publications

Quantitative estimate of the continuum approximations of interacting particle systems in one dimension

Quantitative estimate of the continuum approximations of interacting particle systems in one dimension

Many-particle limits and non-convergence of dislocation wall pile-ups

Onset of fracture in random heterogeneous particle chains

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