Reliable Determination of the Hoek–Brown ‘mi’ Parameter in Brittle Rocks using the Maximum Secant Modulus Criterion in Multistage Triaxial Test

Reliable Determination of the Hoek–Brown ‘mi’ Parameter in Brittle Rocks using the Maximum Secant Modulus Criterion in Multistage Triaxial Test | 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 Article Reliable Determination of the Hoek–Brown ‘mi’ Parameter in Brittle Rocks using the Maximum Secant Modulus Criterion in Multistage Triaxial Test Vahid Kordloo, Mehdi Talkhablou, Farhad Ahmadi Sheikhani This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7917253/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Feb, 2026 Read the published version in Scientific Reports → Version 1 posted 12 You are reading this latest preprint version Abstract Accurate estimation of the Hoek–Brown material constant ( mi ) is essential for reliable rock mass characterization, particularly in brittle, heterogeneous carbonate formations. This study introduces an optimized multi-stage triaxial compression test (ML-TCT) using a maximum secant modulus (Esec = max) criterion, designed to overcome common laboratory limitations like limited sample availability and the absence of automated control systems. Experimental tests were conducted on dolomitic limestone from the Shahbazan Formation using nine conventional single-stage tests (SL-TCT) and seven multi-stage tests under two protocols. The "loading–unloading" ML-TCT scheme proved superior, increasing data efficiency by generating 49 stress points from seven specimens compared to nine points from nine specimens in SL-TCT. Results show that ML-TCT yielded a higher and more representative mean mi value of 9.69, which is statistically significantly different from the SL-TCT results (p < 0.01). This higher mi is attributed to testing a single, persistent failure plane, which reduces inter-specimen variability. Conversely, SL-TCT produced unconfined compressive strength (σci) values that were 10–30 MPa higher, likely because cumulative micro-damage in the ML-TCT specimens weakens the rock's cohesive component. Numerical validation using RS2 software demonstrated that the parameters derived from ML-TCT result in predictions of a larger plastic zone and greater crown displacement, indicating that the proposed method offers a more conservative approach for characterizing brittle rocks. Ultimately, it offers a practical, cost-effective alternative for estimating the mechanical properties of brittle rocks under constrained laboratory conditions. Physical sciences/Engineering Physical sciences/Materials science Earth and environmental sciences/Solid earth sciences Hoek–Brown criterion mi​ parameter estimation Multi-stage triaxial test Maximum secant modulus Brittle carbonate rocks Dolomitic limestone Laboratory-to-field methods Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Feb, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 23 Dec, 2025 Reviewers agreed at journal 20 Dec, 2025 Reviews received at journal 18 Dec, 2025 Reviewers agreed at journal 07 Dec, 2025 Reviews received at journal 08 Nov, 2025 Reviewers agreed at journal 03 Nov, 2025 Reviewers agreed at journal 03 Nov, 2025 Reviewers invited by journal 02 Nov, 2025 Editor invited by journal 30 Oct, 2025 Editor assigned by journal 28 Oct, 2025 Submission checks completed at journal 28 Oct, 2025 First submitted to journal 21 Oct, 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-7917253","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":542080256,"identity":"24b7196d-93e8-4589-b0c4-0f26289a34ac","order_by":0,"name":"Vahid Kordloo","email":"","orcid":"","institution":"Tarbiat Modares University","correspondingAuthor":false,"prefix":"","firstName":"Vahid","middleName":"","lastName":"Kordloo","suffix":""},{"id":542080257,"identity":"e2bf4a58-b52c-492f-b2c5-b1001c9ca6e4","order_by":1,"name":"Mehdi 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Test","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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Hoek–Brown criterion, mi​ parameter estimation, Multi-stage triaxial test, Maximum secant modulus, Brittle carbonate rocks, Dolomitic limestone, Laboratory-to-field methods","lastPublishedDoi":"10.21203/rs.3.rs-7917253/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7917253/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate estimation of the Hoek\u0026ndash;Brown material constant (\u003cem\u003emi\u003c/em\u003e) is essential for reliable rock mass characterization, particularly in brittle, heterogeneous carbonate formations. This study introduces an optimized multi-stage triaxial compression test (ML-TCT) using a maximum secant modulus (Esec\u0026thinsp;=\u0026thinsp;max) criterion, designed to overcome common laboratory limitations like limited sample availability and the absence of automated control systems. Experimental tests were conducted on dolomitic limestone from the Shahbazan Formation using nine conventional single-stage tests (SL-TCT) and seven multi-stage tests under two protocols. The \"loading\u0026ndash;unloading\" ML-TCT scheme proved superior, increasing data efficiency by generating 49 stress points from seven specimens compared to nine points from nine specimens in SL-TCT. Results show that ML-TCT yielded a higher and more representative mean \u003cem\u003emi\u003c/em\u003e value of 9.69, which is statistically significantly different from the SL-TCT results (p\u0026thinsp;\u0026lt;\u0026thinsp;0.01). This higher \u003cem\u003emi\u003c/em\u003e is attributed to testing a single, persistent failure plane, which reduces inter-specimen variability. Conversely, SL-TCT produced unconfined compressive strength (σci) values that were 10\u0026ndash;30 MPa higher, likely because cumulative micro-damage in the ML-TCT specimens weakens the rock's cohesive component. Numerical validation using RS2 software demonstrated that the parameters derived from ML-TCT result in predictions of a larger plastic zone and greater crown displacement, indicating that the proposed method offers a more conservative approach for characterizing brittle rocks. Ultimately, it offers a practical, cost-effective alternative for estimating the mechanical properties of brittle rocks under constrained laboratory conditions.\u003c/p\u003e","manuscriptTitle":"Reliable Determination of the Hoek–Brown ‘mi’ Parameter in Brittle Rocks using the Maximum Secant Modulus Criterion in Multistage Triaxial Test","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-12 14:03:34","doi":"10.21203/rs.3.rs-7917253/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-12-23T13:28:57+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"308677934141029331104110748566929739476","date":"2025-12-20T06:01:29+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-12-18T09:35:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"239408012223961151671857007158112588502","date":"2025-12-08T04:02:30+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-09T03:52:35+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"182736081749094663450212288216322916260","date":"2025-11-03T13:53:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"163660577246956262735230935838607153972","date":"2025-11-03T06:12:06+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-11-03T01:44:25+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-10-30T12:21:44+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-10-28T05:07:47+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-10-28T05:06:35+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-10-21T14:09:24+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e0c0a2d2-efc2-4633-bbc3-03333b65efc6","owner":[],"postedDate":"November 12th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":57669868,"name":"Physical sciences/Engineering"},{"id":57669869,"name":"Physical sciences/Materials science"},{"id":57669870,"name":"Earth and environmental sciences/Solid earth sciences"}],"tags":[],"updatedAt":"2026-02-09T16:05:04+00:00","versionOfRecord":{"articleIdentity":"rs-7917253","link":"https://doi.org/10.1038/s41598-026-38702-5","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2026-02-06 15:59:34","publishedOnDateReadable":"February 6th, 2026"},"versionCreatedAt":"2025-11-12 14:03:34","video":"","vorDoi":"10.1038/s41598-026-38702-5","vorDoiUrl":"https://doi.org/10.1038/s41598-026-38702-5","workflowStages":[]},"version":"v1","identity":"rs-7917253","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7917253","identity":"rs-7917253","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}