On Preserving Variational and Thermodynamic Consistency in Modeling Fracture of Zener-Type Viscoelastic Solids under Antiplane Loading

On Preserving Variational and Thermodynamic Consistency in Modeling Fracture of Zener-Type Viscoelastic Solids under Antiplane Loading | 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 On Preserving Variational and Thermodynamic Consistency in Modeling Fracture of Zener-Type Viscoelastic Solids under Antiplane Loading Sayahdin Alfat, Mardiana Napirah, Rahmat Rahmat, Ali Mulya Rende This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8830906/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 This work develops a variationally based and thermodynamically consistent phase-field model for fracture in Zener-type linear viscoelastic solids under antiplane loading. The formulation is cast as a rate-dependent dynamical system with internal variables, using a microforce balance structure to couple elastic, viscous, and fracture processes within a unified energetic framework. This guarantees that the governing equations and evolution laws satisfy the first and second laws of thermodynamics, yielding a dissipative and well-posed time evolution. The model captures the interaction between viscoelastic relaxation and crack growth in time-dependent materials. Numerical simulations based on an adaptive finite element method resolve the crack path and its temporal evolution efficiently. The results demonstrate the ability of the proposed framework to represent viscoelastic effects on fracture while preserving full variational and thermodynamic consistency. MSC Classification: 49S05 , 65M08 , 74A15 , 74A45 Phase-field fracture Zener-type viscoelasticity thermodynamically consistent modeling fracture evolution adaptive finite element method 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-8830906","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":588721304,"identity":"c228feb6-c0c3-40f9-b166-43d4d7ec8bc0","order_by":0,"name":"Sayahdin Alfat","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAklEQVRIiWNgGAWjYJACZgY2CRDV8IGBwYaBQYJ4LYyNMxgS0ojWwgDTcpiwFvn2M4afC8os8hjYDzY2/PxxPrF/dvPBBww1NtG4tBicyTGWnnFOopiBJ7GxsSfhduKMO8eSDRiOpeU24NLCkGMgzdsmkdjAkNj+gAeopeFGjpkEY8NhnFrk+98Y/wZr4X/Y2Pgn4VzifEJaGIAKILZIJDY28yQcSNxASIvBjWdl1jznJBLbJB42NsukJRtvvJGWbJCAxy/y/cmbb/OU1SX28ycfbHxjYyc770bywQcfamxwO4yBwwBMsUG5jmCVCTiVgwD7AxSuPV7Fo2AUjIJRMCIBACZlXsoijvJ8AAAAAElFTkSuQmCC","orcid":"","institution":"Haluoleo University","correspondingAuthor":true,"prefix":"","firstName":"Sayahdin","middleName":"","lastName":"Alfat","suffix":""},{"id":588721305,"identity":"68c97d91-9ebd-41a4-b038-9551ce98be0a","order_by":1,"name":"Mardiana Napirah","email":"","orcid":"","institution":"Haluoleo University","correspondingAuthor":false,"prefix":"","firstName":"Mardiana","middleName":"","lastName":"Napirah","suffix":""},{"id":588721306,"identity":"f8c0bacd-e60e-4f4b-a295-8caadf355310","order_by":2,"name":"Rahmat Rahmat","email":"","orcid":"","institution":"Haluoleo University","correspondingAuthor":false,"prefix":"","firstName":"Rahmat","middleName":"","lastName":"Rahmat","suffix":""},{"id":588721307,"identity":"380cb676-1446-4115-afa4-cf611b730ec0","order_by":3,"name":"Ali Mulya Rende","email":"","orcid":"","institution":"Haluoleo University","correspondingAuthor":false,"prefix":"","firstName":"Ali","middleName":"Mulya","lastName":"Rende","suffix":""}],"badges":[],"createdAt":"2026-02-09 13:23:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8830906/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8830906/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102423046,"identity":"b6789a2a-ac66-4af5-a7db-40910b90f5ff","added_by":"auto","created_at":"2026-02-11 13:59:08","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2370409,"visible":true,"origin":"","legend":"","description":"","filename":"Alfat2026.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8830906/v1_covered_d9a3e6ae-f673-42d6-b6fd-5fe47b8d9676.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On Preserving Variational and Thermodynamic Consistency in Modeling Fracture of Zener-Type Viscoelastic Solids under Antiplane Loading","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"Phase-field fracture, Zener-type viscoelasticity, thermodynamically consistent modeling, fracture evolution, adaptive finite element method","lastPublishedDoi":"10.21203/rs.3.rs-8830906/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8830906/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work develops a variationally based and thermodynamically consistent phase-field model for fracture in Zener-type linear viscoelastic solids under antiplane loading. The formulation is cast as a rate-dependent dynamical system with internal variables, using a microforce balance structure to couple elastic, viscous, and fracture processes within a unified energetic framework. This guarantees that the governing equations and evolution laws satisfy the first and second laws of thermodynamics, yielding a dissipative and well-posed time evolution. The model captures the interaction between viscoelastic relaxation and crack growth in time-dependent materials. Numerical simulations based on an adaptive finite element method resolve the crack path and its temporal evolution efficiently. 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