A well-defined local continuum damage model with softening via convexification and entropic regularization

A well-defined local continuum damage model with softening via convexification and entropic regularization | 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 A well-defined local continuum damage model with softening via convexification and entropic regularization Celine Lauff, André Schlichting, Thomas Böhlke, Matti Schneider This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9285926/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract We introduce a local continuum damage model at small strains which is able to represent material softening behavior and leads to well-posed mechanical boundary value problems at the same time. To be more precise, upon an implicit Euler discretization in time, the condensed incremental potentials are uniformly strongly convex, permitting standard optimization algorithms with convergence guarantees to be used. As no gradients or non-local averaging is required, the innovative material model may be integrated into material routines of standard finite-element codes. An isotropic damage model with rate limitation serves as the starting point. Then, the construction of the material model proceeds in two steps. The material model is convexified first based on an explicit expression of the convex envelope of the condensed incremental potentials, leading to the emergence of damage distributions encoded via probability measures. The resulting model is convex, but not strictly convex. Therefore, we augment the model by an entropic regularization term which is deliberately chosen to produce the desired characteristics. We discuss the mechanical and computational aspects of the model, i.e., thermomechanical consistency, how to efficiently solve the discrete evolution problem for the internal variables and the consistent algorithmic tangent. Last but not least, we present computational examples of the model both at material-point level and in a heterogeneous environment which demonstrate the significantly improved numerical stability due to the uniformly strongly convex condensed incremental energies. Continuum damage mechanics Softening material response Optimal transport with entropic regularization Generalized standard material Well-posed boundary value problem Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 26 Apr, 2026 Reviewers agreed at journal 03 Apr, 2026 Reviewers agreed at journal 02 Apr, 2026 Reviewers invited by journal 01 Apr, 2026 Editor assigned by journal 01 Apr, 2026 Submission checks completed at journal 01 Apr, 2026 First submitted to journal 31 Mar, 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-9285926","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":617183395,"identity":"d7fea5ea-639a-454a-b68e-dcb091a3b779","order_by":0,"name":"Celine Lauff","email":"","orcid":"","institution":"Karlsruhe Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Celine","middleName":"","lastName":"Lauff","suffix":""},{"id":617183396,"identity":"b848987b-05e1-4fc0-9c07-67df30d6be3f","order_by":1,"name":"André Schlichting","email":"","orcid":"","institution":"University of Ulm","correspondingAuthor":false,"prefix":"","firstName":"André","middleName":"","lastName":"Schlichting","suffix":""},{"id":617183397,"identity":"6b9ca682-7189-4607-baba-dd116fb6a971","order_by":2,"name":"Thomas Böhlke","email":"","orcid":"","institution":"Karlsruhe Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Thomas","middleName":"","lastName":"Böhlke","suffix":""},{"id":617183398,"identity":"1207646a-c4a9-4f7f-9f83-14ae2c8524fd","order_by":3,"name":"Matti Schneider","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVRIie2PsUrEQBCGJ6TNpj8OLk8gbBBiIz7LhsCl8bAQ5AoJCwd7r3BCfIe0dhsWTBO8dkHRhbQRzk4Q1N0zhcVGLS32g4FhmI9/BsDh+K8QXVgXDwBmAB6FwIz52L7v0e/K4d8UGBSzmdKhGVUO1ttGKSiio+mq5v3yMa/uVxSe2TGEDbUqSZuZw0R8U96SumzPF9VDTb1rNodJa49J+F7hBMtTLBAji0qm6zfEBGBJ7Mq2M0qhlbOdQO8kxzKlPmIfgJ+UXZH7FN+kgECUkEHhOsX+fiK7eEOwiCs5x7X+J77SilfeZcGkHTssVS+vyyLCMut2/SWJQpkr6C9OZmFjf/8LbJkFP+w7HA6H4xc+AeHBawcptfEXAAAAAElFTkSuQmCC","orcid":"","institution":"University of Duisburg-Essen","correspondingAuthor":true,"prefix":"","firstName":"Matti","middleName":"","lastName":"Schneider","suffix":""}],"badges":[],"createdAt":"2026-04-01 03:23:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9285926/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9285926/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106403681,"identity":"9e89564e-c876-42bd-b941-51382c974a8c","added_by":"auto","created_at":"2026-04-08 09:14:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1719321,"visible":true,"origin":"","legend":"","description":"","filename":"OTR.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9285926/v1_covered_7827a3cf-4fdd-4935-a5d9-4e5073fb18f5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eA well-defined local continuum damage model with softening via convexification and entropic regularization\u003c/p\u003e","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":"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":"Continuum damage mechanics, Softening material response, Optimal transport with entropic regularization, Generalized standard material, Well-posed boundary value problem","lastPublishedDoi":"10.21203/rs.3.rs-9285926/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9285926/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce a local continuum damage model at small strains which is able to represent material softening behavior and leads to well-posed mechanical boundary value problems at the same time. 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