Asymptotic analysis of a pile-up of regular edge dislocation walls

Hall, Cameron L.

doi:10.1016/j.msea.2011.09.065

Cited by 24 publications

(39 citation statements)

References 10 publications

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“…As a result, there may be interesting distinguished limits to consider depending on the relative sizes of and n −1 . In the present work, however, we assume that the ring contains a sufficiently large number of magnets (or that the deformation of the ring is sufficiently large), so that n −1 and we are justified in ignoring all of the higher-order corrections in powers of n. Hence, our inextensibility condition takes the form 9) and substituting (4.8) into (4.9) yields the result that…”

Section: Rings Of Magnetsmentioning

confidence: 99%

“…This is because the singularity in the summand causes conventional Euler-Maclaurin summation to yield series that are not asymptotic for the latter sums in (3.22). This problem indicates that we need to consider a discrete problem in boundary layer regions at the ends of the chain, analogous to the boundary layers described in [9,10].…”

Section: Continuum Formulationmentioning

confidence: 99%

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

mentioning

confidence: 99%

“…Another observation in [9,10] is that continuum dislocation models are inappropriate for describing the ends of a dislocation pile-up, and it is instead necessary to consider discrete problems in the boundary layers. Similarly, we find that the continuum model developed in this paper for a chain of magnets breaks down near the ends of the chain.…”

mentioning

confidence: 99%

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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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Section: Rings Of Magnetsmentioning

confidence: 99%

Section: Continuum Formulationmentioning

confidence: 99%

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

mentioning

confidence: 99%

mentioning

confidence: 99%

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The Mechanics of a Chain or Ring of Spherical Magnets

Hall¹,

Vella²,

Goriely³

2013

SIAM J. Appl. Math.

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“…A class of representative two-dimensional dislocation configurations widely studied in the literature are distributions (pile-ups) of dislocation walls consisting of straight and mutually parallel dislocations (e.g. [27,29,32,13,8,40,30,28]). In this scenario, dislocation-dislocation interactions take place in both directions that are in and normal to the dislocation slip planes.…”

Section: Introductionmentioning

confidence: 99%

A continuum model for distributions of dislocations incorporating short-range interactions

Niu

Zhu

Dai

et al. 2018

Communications in Mathematical Sciences

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Dislocations are the main carriers of the permanent deformation of crystals. For simulations of engineering applications, continuum models where material microstructures are represented by continuous density distributions of dislocations are preferred. It is challenging to capture in the continuum model the short-range dislocation interactions, which vanish after the standard averaging procedure from discrete dislocation models. In this study, we consider systems of parallel straight dislocation walls and develop continuum descriptions for the short-range interactions of dislocations by using asymptotic analysis. The obtained continuum short-range interaction formulas are incorporated in the continuum model for dislocation dynamics based on a pair of dislocation density potential functions that represent continuous distributions of dislocations. This derived continuum model is able to describe the anisotropic dislocation interaction and motion. Mathematically, these short-range interaction terms ensure strong stability property of the continuum model that is possessed by the discrete dislocation dynamics model. The derived continuum model is validated by comparisons with the discrete dislocation dynamical simulation results.

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The Equilibrium Measure for a Nonlocal Dislocation Energy

Mora

Rondi

Scardia³

2018

Comm Pure Appl Math

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In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well‐known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels. © 2018 Wiley Periodicals, Inc.

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Asymptotic analysis of a pile-up of regular edge dislocation walls

Cited by 24 publications

References 10 publications

The Mechanics of a Chain or Ring of Spherical Magnets

The Mechanics of a Chain or Ring of Spherical Magnets

A continuum model for distributions of dislocations incorporating short-range interactions

The Equilibrium Measure for a Nonlocal Dislocation Energy

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