Handle URI:
http://hdl.handle.net/10754/599934
Title:
The Mechanics of a Chain or Ring of Spherical Magnets
Authors:
Hall, Cameron L.; Vella, Dominic; Goriely, Alain
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 analyze 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. Copyright © by SIAM.
Citation:
Hall CL, Vella D, Goriely A (2013) The Mechanics of a Chain or Ring of Spherical Magnets. SIAM Journal on Applied Mathematics 73: 2029–2054. Available: http://dx.doi.org/10.1137/120897973.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Applied Mathematics
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
Jan-2013
DOI:
10.1137/120897973
Type:
Article
ISSN:
0036-1399; 1095-712X
Sponsors:
This work was supported by Award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).The first author's work was supported by grant EP/I017070/1 from the Engineering and Physical Sciences Research Council. The third author is a Wolfson Royal Society Merit Holder and was supported by a Reintegration grant under EC Framework VII.
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorHall, Cameron L.en
dc.contributor.authorVella, Dominicen
dc.contributor.authorGoriely, Alainen
dc.date.accessioned2016-02-28T06:32:46Zen
dc.date.available2016-02-28T06:32:46Zen
dc.date.issued2013-01en
dc.identifier.citationHall CL, Vella D, Goriely A (2013) The Mechanics of a Chain or Ring of Spherical Magnets. SIAM Journal on Applied Mathematics 73: 2029–2054. Available: http://dx.doi.org/10.1137/120897973.en
dc.identifier.issn0036-1399en
dc.identifier.issn1095-712Xen
dc.identifier.doi10.1137/120897973en
dc.identifier.urihttp://hdl.handle.net/10754/599934en
dc.description.abstractStrong 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 analyze 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. Copyright © by SIAM.en
dc.description.sponsorshipThis work was supported by Award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).The first author's work was supported by grant EP/I017070/1 from the Engineering and Physical Sciences Research Council. The third author is a Wolfson Royal Society Merit Holder and was supported by a Reintegration grant under EC Framework VII.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectApproximation of sumsen
dc.subjectAsymptotic analysisen
dc.subjectDiscrete-to-continuumen
dc.subjectMagnetismen
dc.titleThe Mechanics of a Chain or Ring of Spherical Magnetsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Applied Mathematicsen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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