Convergence of Interaction-Driven Evolutions of Dislocations with Wasserstein Dissipation and Slip-Plane Confinement

Mora, Maria Giovanna; Peletier, MA Mark; Scardia, Lucia

doi:10.1137/16m1096098

Cited by 30 publications

(36 citation statements)

References 32 publications

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“…(1) the phase-field model developed by Peierls and Nabarro [Pei40, Nab47, KCO02, GM06, MP12], (2) the removal of small balls around the dislocations from the elastic medium [CL05, GLP10,MPS17], (3) the smearing out of the dislocation core by a convolution kernel [ACH + 05, CAWB06, GLP10, CGO15], (4) a cut-off radius within which dislocations do not interact [HL82].…”

Section: This Brings Us To the Central Question Of This Papermentioning

confidence: 99%

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Garroni

Meurs

Peletier³

et al. 2019

Arch Rational Mech Anal

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We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals.Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale δn > 0, which converges to zero as the number of particles n tends to infinity.We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on both the theory of λ-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δn-regularised version of the Groma-Balogh equations, and a priori estimates for the Groma-Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n → ∞ we require δn to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δn converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations.These results show how the validity of the Groma-Balogh equations as the limit of manyparticle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

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Section: This Brings Us To the Central Question Of This Papermentioning

confidence: 99%

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Garroni

Meurs

Peletier³

et al. 2019

Arch Rational Mech Anal

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“…We establish the limsup-inequality (17b) by constructing a recovery sequence for µ in a dense subset of M ([0, 1]), which is similar to one used in [MPS14]. To construct this subset, we divide the domain of the dislocation walls in closed intervals I k with k = 1, .…”

Section: 2mentioning

confidence: 99%

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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“…In the case α = 1, the energy I 1 models interactions between edge dislocations of the same sign (see, e.g., [28,21]). The minimisers of I 1 were since long conjectured to be vertical walls of dislocations, and this has been confirmed only very recently, in [29], where the authors proved that the only minimiser of I 1 is the semi-circle law…”

Section: Introductionmentioning

confidence: 99%

The Ellipse Law: Kirchhoff Meets Dislocations

Carrillo

Mateu

Mora

et al. 2019

Commun. Math. Phys.

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In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α ∈ R. The case α = 0 corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α ∈ (0, 1) the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an ellipse. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses. Therefore we show a surprising connection between vortices and dislocations.

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Convergence of Interaction-Driven Evolutions of Dislocations with Wasserstein Dissipation and Slip-Plane Confinement

Cited by 30 publications

References 32 publications

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

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

The Ellipse Law: Kirchhoff Meets Dislocations

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