Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory

Hall, Cameron L.; Chapman, S. Jonathan; Ockendon, J. R.

doi:10.1137/090778444

Cited by 18 publications

(37 citation statements)

References 7 publications

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“…Our study takes place in the context of a variety of recent mathematical results aiming to better understand surface effects in similar particle systems, notably [HCO10,GvMPS15] in the setting of dislocation pile-ups. While [HCO10] focuses on the case in which the interaction potential is homogeneous, and [GvMPS15,Hal11] studies boundary layers in a continuum model for the x(0) = 0…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

2017

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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 pile-ups 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.

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Section: Introductionmentioning

confidence: 99%

“…particle density, our contribution is the derivation of a discrete description of the boundary layer for a general class of interaction potentials. Two particular examples that we have in mind are pile-ups of dislocation walls [GvMPS15] and pile-ups of dislocation dipoles [HCO10].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

2017

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“…As in the present work, [9] and [10] involves approximating a sum by exploiting assumptions about the positions of the particles.…”

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“…In [10], it was found that the form of the continuum model depends strongly on the rate at which the inter-particle repulsion decays with distance. Since spherical magnets can be modelled as magnetic dipoles and the force of attraction between dipoles decays in inverse proportion to the fourth power of distance (see, for example, [12]), the results in [10] might lead us to expect local interactions within a chain of magnets to dominate over nonlocal interactions.…”

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confidence: 99%

The Mechanics of a Chain or Ring of Spherical Magnets

Hall¹,

Vella²,

Goriely³

2013

SIAM J. Appl. Math.

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Abstract. Strong magnets, such as neodymium-iron-boron magnets, are increasingly being manufactured as spheres. Because of their dipolar characters, these spheres can easily be arranged into long chains that exhibit mechanical properties reminiscent of elastic strings or rods. While simple formulations exist for the energy of a deformed elastic rod, it is not clear whether or not they are also appropriate for a chain of spherical magnets. In this paper, we use discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres. We find that the mechanical properties of a chain of magnets differ significantly from those of an elastic rod: while both magnetic chains and elastic rods support bending by change of local curvature, nonlocal interaction terms also appear in the energy formulation for a magnetic chain. This continuum model for the energy of a chain of magnets is used to analyse small deformations of a circular ring of magnets and hence obtain theoretical predictions for the vibrational modes of a circular ring of magnets. Surprisingly, despite the contribution of nonlocal energy terms, we find that the vibrations of a circular ring of magnets are governed by the same equation that governs the vibrations of a circular elastic ring.

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“…The relation between the distribution of dislocations and the decomposition of the strain into an elastic and a plastic part in geometrically nonlinear models was addressed by Reina and Conti [52]. Dislocation pile-ups have been studied by Focardi and Garroni [25] and by Hall et al [31] using formal asymptotics, and by Geers et al [30] using Γ -convergence.…”

Section: Introductionmentioning

confidence: 99%

The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations

Conti

Garroni

Ortiz

2015

Arch Rational Mech Anal

119

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We prove that the classical line-tension approximation for dislocations in crystals, that is, the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ -limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length ψ(b, t), which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length ψ 0 (b, t) obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its H 1 -elliptic envelope.

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Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory

Cited by 18 publications

References 7 publications

Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

The Mechanics of a Chain or Ring of Spherical Magnets

The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations

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