Non-material Poincaré-Cosserat equations: application to slender sliding structures and fluid-structure interactions

Non-material Poincaré-Cosserat equations: application to slender sliding structures and fluid-structure interactions | 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 Non-material Poincaré-Cosserat equations: application to slender sliding structures and fluid-structure interactions Frédéric Boyer, Shucheng Zhang, Vincent Lebastard, Johann Hérault, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8885529/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 The counterpart of Lagrange's equations on a noncommutative Lie group of congurations is called Poincaré's equations. Applied to the Lie group SE(3), these equations give the well-known Newton-Euler dynamic model of a rigid body. Beyond rigid bodies, Poincaré's approach can be extended to a Cosserat medium, i.e., a continuous set of rigid microstructures glued together along material dimensions. The resulting set of equations is called the Poincaré-Cosserat material equations. Applied to a Cosserat rod, it provides the Simo-Reissner model of geometrically exact rods. Recently, these equations have been extended to the case of elongated structures sliding uniformly along a non-material control domain. In this article, we extend this initial result to a new set of equations that we call non-material Poincaré-Cosserat equations. Derived from a non-material version of the principle of least action, these equations can be applied to complex situations involving several media sliding along each other with a non uniform velocity. To illustrate their use in practical situations, we will apply them to three emblematic examples taken from solid and fluid mechanics. The first is a sliding rod extending and retracting in three dimensions from a rigid sleeve. The second is the garden hose, modeled as an ideal fluid flowing inside an elastic pipe clamped in a wall. The third is a model of fish swimming developed by J. Lighthill, called LAEBT (Large Amplitude Elongated Body Theory), which has been used in recent years by fluid mechanics, biomechanics, and robotics to explain the performance of swimming sh and control bio-inspired robots resembling fish. Sliding rods Lagrangian mechanics nite deformations geometrically exact beam theory garden hose Lighthill's swimming theory. Full Text Additional Declarations No competing interests reported. Supplementary Files VideononmaterialPoincareCosseratEquations.mp4 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-8885529","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":592958445,"identity":"adc0877d-45e3-46f9-8f48-1c3e18c5099b","order_by":0,"name":"Frédéric 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hose, Lighthill's swimming theory.","lastPublishedDoi":"10.21203/rs.3.rs-8885529/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8885529/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The counterpart of Lagrange's equations on a noncommutative Lie group of congurations is called Poincaré's equations. Applied to the Lie group SE(3), these equations give the well-known Newton-Euler dynamic model of a rigid body. Beyond rigid bodies, Poincaré's approach can be extended to a Cosserat medium, i.e., a continuous set of rigid microstructures glued together along material dimensions. The resulting set of equations is called the Poincaré-Cosserat material equations. Applied to a Cosserat rod, it provides the Simo-Reissner model of geometrically exact rods. Recently, these equations have been extended to the case of elongated structures sliding uniformly along a non-material control domain. In this article, we extend this initial result to a new set of equations that we call non-material Poincaré-Cosserat equations. Derived from a non-material version of the principle of least action, these equations can be applied to complex situations involving several media sliding along each other with a non uniform velocity. To illustrate their use in practical situations, we will apply them to three emblematic examples taken from solid and fluid mechanics. The first is a sliding rod extending and retracting in three dimensions from a rigid sleeve. The second is the garden hose, modeled as an ideal fluid flowing inside an elastic pipe clamped in a wall. The third is a model of fish swimming developed by J. 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