Dynamic Green’s functions for lattice systems with frontiers possessing contrasting periodicity and material properties

Dynamic Green’s functions for lattice systems with frontiers possessing contrasting periodicity and material properties | 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 Dynamic Green’s functions for lattice systems with frontiers possessing contrasting periodicity and material properties Ege Aktunc, Michael Nieves, Gennady Mishuris This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8462342/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract We consider anti-plane waves propagating in a lattice, formed from periodically placed masses interconnected by springs, that possesses a bi-periodic frontier with contrasting inertial and elastic properties. In particular, we study the response of such a system to the application of a point force, corresponding to Green’s function. This problem is studied via the application of the Z-transform which yields Green’s function in an exact form in terms of a contour integral. We demonstrate how this approach also provides explicit dispersion relations for waves along the lattice frontier. We study the influence of the lattice frontier’s composition the pass band structure linked to the existence of these waves. A low frequency homogenisation model is also developed that incorporates the effects due to the elastic and inertial properties of the frontier. We also show how our technique can be extended to study the dissimilar lattice systems with a bi-periodic interface. All analytical results are accompanied by numerical illustrations. Lattice Green’s function Elastic lattice periodic mixed boundary conditions functional equations low frequency approximation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 05 Apr, 2026 Reviews received at journal 05 Apr, 2026 Reviews received at journal 23 Mar, 2026 Reviewers agreed at journal 09 Mar, 2026 Reviewers agreed at journal 09 Mar, 2026 Reviewers invited by journal 25 Feb, 2026 Editor assigned by journal 30 Dec, 2025 Submission checks completed at journal 28 Dec, 2025 First submitted to journal 27 Dec, 2025 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. 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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-8462342","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":596865017,"identity":"33f44a97-2ced-4f6d-bffe-6d22d9f17338","order_by":0,"name":"Ege Aktunc","email":"","orcid":"","institution":"Keele University","correspondingAuthor":false,"prefix":"","firstName":"Ege","middleName":"","lastName":"Aktunc","suffix":""},{"id":596865018,"identity":"ba2e1b92-4d1f-4e94-9eba-8b5edcc7d988","order_by":1,"name":"Michael Nieves","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2ElEQVRIie3RMQrCMBSA4VcEp9SucfIEQiTQqdqrJBS8gEsHh4JQl0DXegtBcE4J1CXiFfQGddJB1KqbQ8zokG94Q+CHFx6A4/wp+RoBasfx89CxS/qiHcw2eSPaNhlmfiXPeRRTraojS2EQZIgSUxLKHqtW+ZRv93lCmIZRKRFl5gQRhbRi4QGFmOfgrQFR+TO56UdMi+By5XeI7RJIpbf2RRd4BvyVmBdTiFQiTXipa4pZjZNSdWfm7+8Eba5kEgciOTXNPBoXy8UGm5LvG2DrQzqO4zgGT8X/R8HusERTAAAAAElFTkSuQmCC","orcid":"","institution":"Keele University","correspondingAuthor":true,"prefix":"","firstName":"Michael","middleName":"","lastName":"Nieves","suffix":""},{"id":596865019,"identity":"1651363d-0606-4b16-b156-0ac63b14b4e2","order_by":2,"name":"Gennady Mishuris","email":"","orcid":"","institution":"Aberystwyth University","correspondingAuthor":false,"prefix":"","firstName":"Gennady","middleName":"","lastName":"Mishuris","suffix":""}],"badges":[],"createdAt":"2025-12-27 16:53:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8462342/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8462342/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103632204,"identity":"db5f4b1b-18a1-41d9-9363-bae51ab9c4ad","added_by":"auto","created_at":"2026-02-28 04:06:39","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":8377745,"visible":true,"origin":"","legend":"","description":"","filename":"Latticeperiodicboundaryinterface.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8462342/v1_covered_0fdc8d27-460d-4569-9796-f724a305b772.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Dynamic Green’s functions for lattice systems with frontiers possessing contrasting periodicity and material properties","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"continuum-mechanics-and-thermodynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"cmat","sideBox":"Learn more about [Continuum Mechanics and Thermodynamics](http://link.springer.com/journal/161)","snPcode":"161","submissionUrl":"https://submission.nature.com/new-submission/161/3","title":"Continuum Mechanics and Thermodynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Lattice Green’s function, Elastic lattice, periodic mixed boundary conditions, functional equations, low frequency approximation","lastPublishedDoi":"10.21203/rs.3.rs-8462342/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8462342/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We consider anti-plane waves propagating in a lattice, formed from periodically placed masses interconnected by springs, that possesses a bi-periodic frontier with contrasting inertial and elastic properties. 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