Mesh variational r-adaptivity for sharp modeling of brittle fracture

Mesh variational r-adaptivity for sharp modeling of brittle fracture | 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 Mesh variational r-adaptivity for sharp modeling of brittle fracture Gatien Dony, Nicolas Moës, Jean-François Remacle This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8896760/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 7 You are reading this latest preprint version Abstract Purpose: Brittle fracture modeling is framed as a coupled evolution of spatial and material configurations. The displacement field between these configurations drives the fracture process and the crack path alters the geometry. This interplay has been expressed in the variational formalism of Francfort and Marigo. However, numerical models based on this formalism generally introduce a regularization to circumvent the need for geometry modification. Here, we propose to combine explicit sharp crack modeling with a variational formalism. Methods: Both mesh and displacement field are tuned to minimize the functional. Consequently, the crack path appears naturally. In this work, we first present the minimization of the fracture functional in a \((r)\) -adaptative scheme and the advantage it presents. Afterwards, we present a staggered optimization procedure to construct the spatial and material configurations of a body undergoing brittle fracture. Results: Afterwards, the method is demonstrated through several numerical examples. Namely, we recover theoretical behavior for a manufactured solution and present the crack path of single edge notch specimen. Conclusion: This method is a direct implementation of the variational formalism of fracture mechanics. Therefore, it provides a straightforward way to compute the energy release rate and crack path. Moreover, it does not require any regularization and the crack is directly represented in the geometry of the specimen. However, it is computationally expensive as the mesh needs to be optimized at each step of the loading. Fracture Mechanics Finite elements Mesh adaptation Solid Mechanics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 12 May, 2026 Reviews received at journal 10 May, 2026 Reviewers agreed at journal 19 Feb, 2026 Reviewers invited by journal 19 Feb, 2026 Editor assigned by journal 19 Feb, 2026 Submission checks completed at journal 17 Feb, 2026 First submitted to journal 16 Feb, 2026 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-8896760","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":594079162,"identity":"e9566650-1e26-4a8f-acf6-7ad7bf547746","order_by":0,"name":"Gatien Dony","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABC0lEQVRIiWNgGAWjYJCCA0DM2ADEDxgYLGAiB/BoYIZrYTZgYJAgTgsDVAubBEwLAz4t8u7nDx74uINBtl+6x6yap0ZCzuBG8sPDBQx35HBpMTyTzHBw5hkG45lzzpjd5jkmYWxwI83g8AyGZ8Y4tTQkMxzmbWNI3HAjx+w2b4NE4oYzBwwO8zAcTmzApaX/McPhv1AtxRAtxz+AtNTj0iIvAbSFEaqFGazleA/YlgRcDjOQeGxwsLdNwnjmjLRiyTlAv0ge7yk4zGPwzBCnLf2Jjz/8bLOR7ZdI3vjhTY2NHN9h9s2feSruyOO05QCYAkUHhwGyOC4NQFsQ1rM/wK1sFIyCUTAKRjQAAMnZXtE4OG+hAAAAAElFTkSuQmCC","orcid":"","institution":"Université Catholique de Louvain","correspondingAuthor":true,"prefix":"","firstName":"Gatien","middleName":"","lastName":"Dony","suffix":""},{"id":594079163,"identity":"dc29cea0-fb9f-48f0-9cb1-09ddfc86f952","order_by":1,"name":"Nicolas Moës","email":"","orcid":"","institution":"Université Catholique de Louvain","correspondingAuthor":false,"prefix":"","firstName":"Nicolas","middleName":"","lastName":"Moës","suffix":""},{"id":594079167,"identity":"cf19003c-5ec0-4ed1-99a6-e92df322e3ff","order_by":2,"name":"Jean-François Remacle","email":"","orcid":"","institution":"Université Catholique de Louvain","correspondingAuthor":false,"prefix":"","firstName":"Jean-François","middleName":"","lastName":"Remacle","suffix":""}],"badges":[],"createdAt":"2026-02-17 00:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8896760/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8896760/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103505970,"identity":"595c1ae5-63bb-4388-ae4f-af0aa5d7f86b","added_by":"auto","created_at":"2026-02-26 13:33:41","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3365163,"visible":true,"origin":"","legend":"","description":"","filename":"articleDony.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8896760/v1_covered_76b99830-fd25-4427-a37a-124e86435ad4.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Mesh variational r-adaptivity for sharp modeling of brittle fracture","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"international-journal-of-fracture","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"frac","sideBox":"Learn more about [International Journal of Fracture](http://link.springer.com/journal/10704)","snPcode":"10704","submissionUrl":"https://submission.nature.com/new-submission/10704/3","title":"International Journal of Fracture","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Fracture Mechanics, Finite elements, Mesh adaptation, Solid Mechanics","lastPublishedDoi":"10.21203/rs.3.rs-8896760/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8896760/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cb\u003ePurpose:\u003c/b\u003e\u003c/p\u003e\u003cp\u003eBrittle fracture modeling is framed as a coupled evolution of spatial and material configurations. The displacement field between these configurations drives the fracture process and the crack path alters the geometry. This interplay has been expressed in the variational formalism of Francfort and Marigo. However, numerical models based on this formalism generally introduce a regularization to circumvent the need for geometry modification. Here, we propose to combine explicit sharp crack modeling with a variational formalism.\u003c/p\u003e\u003cp\u003e\u003cb\u003eMethods:\u003c/b\u003e\u003c/p\u003e\u003cp\u003eBoth mesh and displacement field are tuned to minimize the functional. Consequently, the crack path appears naturally. In this work, we first present the minimization of the fracture functional in a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((r)\\)\u003c/span\u003e\u003c/span\u003e-adaptative scheme and the advantage it presents. Afterwards, we present a staggered optimization procedure to construct the spatial and material configurations of a body undergoing brittle fracture.\u003c/p\u003e\u003cp\u003e\u003cb\u003eResults:\u003c/b\u003e\u003c/p\u003e\u003cp\u003eAfterwards, the method is demonstrated through several numerical examples. Namely, we recover theoretical behavior for a manufactured solution and present the crack path of single edge notch specimen.\u003c/p\u003e\u003cp\u003e\u003cb\u003eConclusion:\u003c/b\u003e\u003c/p\u003e\u003cp\u003eThis method is a direct implementation of the variational formalism of fracture mechanics. Therefore, it provides a straightforward way to compute the energy release rate and crack path. Moreover, it does not require any regularization and the crack is directly represented in the geometry of the specimen. However, it is computationally expensive as the mesh needs to be optimized at each step of the loading.\u003c/p\u003e","manuscriptTitle":"Mesh variational r-adaptivity for sharp modeling of brittle fracture","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-24 06:07:39","doi":"10.21203/rs.3.rs-8896760/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-12T16:28:18+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T00:22:06+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"176437000899905815704317821493724115371","date":"2026-02-19T18:48:09+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-02-19T16:01:25+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-19T15:48:58+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-18T04:19:25+00:00","index":"","fulltext":""},{"type":"submitted","content":"International Journal of Fracture","date":"2026-02-17T00:12:34+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"international-journal-of-fracture","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"frac","sideBox":"Learn more about [International Journal of Fracture](http://link.springer.com/journal/10704)","snPcode":"10704","submissionUrl":"https://submission.nature.com/new-submission/10704/3","title":"International Journal of Fracture","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"bb0f1489-354a-41a2-9158-3a087b43562e","owner":[],"postedDate":"February 24th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-12T16:28:18+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T00:22:06+00:00","index":8,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-12T16:40:50+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-24 06:07:39","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8896760","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8896760","identity":"rs-8896760","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}