Spatial deformation of a ferromagnetic elastic rod

Spatial deformation of a ferromagnetic elastic rod | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Spatial deformation of a ferromagnetic elastic rod G R Krishna Chand Avatar, Vivekanand Dabade This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9425220/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Ferromagnetic elastic slender structures offer the potential for large actuation displacements under modest external magnetic fields, due to the magneto– mechanical coupling. This paper investigates the phase portraits of the Hamiltonian governing the three-dimensional deformation of inextensible ferromagnetic elastic rods subjected to combined terminal tension and twisting moment in the presence of a longitudinal magnetic field. The total energy functional is formulated by combining the Kirchhoff elastic strain energy with micromagnetic energy contributions appropriate to soft and hard ferromagnetic materials: magnetostatic (demagnetization) energy for the former, and exchange and Zeeman energies for the latter. Exploiting the circular cross-sectional symmetry and the integrable structure of the governing equations, conserved Casimir invariants are identified and the Hamiltonian is reduced to a single-degree-offreedom system in the Euler polar angle. Analysis of the resulting phase portraits reveals that purely elastic and hard ferromagnetic rods undergo a supercritical Hamiltonian Hopf pitchfork bifurcation, whereas soft ferromagnetic rods exhibit this bifurcation only within a restricted range of the magnetoelastic parameter, 0 < ˜KdM < 1/8. Both helical and localized post-buckling configurations are analyzed, and the corresponding load–deformation relationships are systematically characterized across a range of loading scenarios. Localized buckling modes, corresponding to homoclinic orbits in the Hamiltonian phase space, are constructed numerically. In contrast to the purely elastic case, the localized configurations of soft ferromagnetic rods exhibit non-collinear extended straight segments, a geometrically distinctive feature arising directly from the magnetoelastic coupling. Ferromagnetic rod Hamiltonian Spatial buckling Micromagnetics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9425220","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":627693532,"identity":"c4f431ec-9a6d-4753-bef1-6dc6c7cdfcb4","order_by":0,"name":"G R Krishna Chand Avatar","email":"","orcid":"","institution":"Indian Institute of Science Bangalore","correspondingAuthor":false,"prefix":"","firstName":"G","middleName":"R Krishna Chand","lastName":"Avatar","suffix":""},{"id":627693533,"identity":"d0fa8cd8-a298-4fda-a344-e2369f288a4b","order_by":1,"name":"Vivekanand Dabade","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvUlEQVRIiWNgGAWjYDCCAzwMDDwMNjxsUD4PsVrSSNdymAR38R0/e/DDm4rzMnz8hx8w/KhhkDEnpEXyTF6y5Jwzt3nYJNIMGHuOMfBYNhDQYnAgx0Catw2kBehC3gYGHoMDhLScf2P8m/ffOR42/jMMjH+J0nIjx0yat+EAMMRyGJiJskXyxrs0yznHksF+OSxzTIKwFr7zuYdvvKmxs5fvP/zw4ZsaG3uCWlAAULEEKepHwSgYBaNgFOACAHkfOeqRr4y1AAAAAElFTkSuQmCC","orcid":"","institution":"Indian Institute of Science Bangalore","correspondingAuthor":true,"prefix":"","firstName":"Vivekanand","middleName":"","lastName":"Dabade","suffix":""}],"badges":[],"createdAt":"2026-04-15 10:12:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9425220/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9425220/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107706572,"identity":"f9131b90-3285-4d3d-bf05-462405bf22bb","added_by":"auto","created_at":"2026-04-24 09:18:23","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4806991,"visible":true,"origin":"","legend":"","description":"","filename":"LateXdocuments.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9425220/v1_covered_88583764-b70a-4b33-840a-39d567326eb8.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Spatial deformation of a ferromagnetic elastic rod","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Ferromagnetic rod, Hamiltonian, Spatial buckling, Micromagnetics","lastPublishedDoi":"10.21203/rs.3.rs-9425220/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9425220/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFerromagnetic elastic slender structures offer the potential for large actuation displacements under modest external magnetic fields, due to the magneto– mechanical coupling. This paper investigates the phase portraits of the Hamiltonian governing the three-dimensional deformation of inextensible ferromagnetic elastic rods subjected to combined terminal tension and twisting moment in the presence of a longitudinal magnetic field. The total energy functional is formulated by combining the Kirchhoff elastic strain energy with micromagnetic energy contributions appropriate to soft and hard ferromagnetic materials: magnetostatic (demagnetization) energy for the former, and exchange and Zeeman energies for the latter. Exploiting the circular cross-sectional symmetry and the integrable structure of the governing equations, conserved Casimir invariants are identified and the Hamiltonian is reduced to a single-degree-offreedom system in the Euler polar angle. Analysis of the resulting phase portraits reveals that purely elastic and hard ferromagnetic rods undergo a supercritical Hamiltonian Hopf pitchfork bifurcation, whereas soft ferromagnetic rods exhibit this bifurcation only within a restricted range of the magnetoelastic parameter, 0 \u0026lt; ˜KdM \u0026lt; 1/8. Both helical and localized post-buckling configurations are analyzed, and the corresponding load–deformation relationships are systematically characterized across a range of loading scenarios. Localized buckling modes, corresponding to homoclinic orbits in the Hamiltonian phase space, are constructed numerically. In contrast to the purely elastic case, the localized configurations of soft ferromagnetic rods exhibit non-collinear extended straight segments, a geometrically distinctive feature arising directly from the magnetoelastic coupling.\u003c/p\u003e","manuscriptTitle":"Spatial deformation of a ferromagnetic elastic rod","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-24 04:20:05","doi":"10.21203/rs.3.rs-9425220/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"f1b60b88-3f4e-4e98-afa0-ff11173baf50","owner":[],"postedDate":"April 24th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-18T08:36:44+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-18T08:03:10+00:00","index":12,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-14T09:55:08+00:00","index":11,"fulltext":""},{"type":"reviewerAgreed","content":"58420377188972626438265611015791261363","date":"2026-05-04T06:29:50+00:00","index":10,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-18T08:40:03+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-24 04:20:05","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9425220","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9425220","identity":"rs-9425220","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}