{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,6]],"date-time":"2026-07-06T17:35:54Z","timestamp":1783359354650,"version":"3.54.6"},"reference-count":47,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,27]],"date-time":"2022-07-27T00:00:00Z","timestamp":1658880000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Sciences Foundation of China","doi-asserted-by":"publisher","award":["41172274"],"award-info":[{"award-number":["41172274"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The nonlinear mechanical behavior of rock significantly influences the design and construction of underground structures. Due to the complexity and diversity of the damage mechanisms of rock, the damage variable directly defined by partial-damage mechanisms is insufficient in reflecting the progressive-failure process of rock comprehensively. So, in this paper, a novel damage variable is introduced into the plastic-strain rate based on the theoretical framework of irreversible thermodynamics to overcome this defect. The general expression is derived according to the least energy dissipation rate principle. The proposed damage variable can represent the irreversible energy dissipation process and has a strictly theoretical basis in mechanics. Moreover, the granite and marble stress-strain curves are simulated and compared with the Lemaitre damage model, Mazars damage model, and statistical damage model. The results show that the form of the proposed damage variable is practical and straightforward and can better reflect the entire stress-strain relationship of rock. Furthermore, the initial value of the inelastic response strain can be given directly through the proposed damage variable. The model presented here can overcome the issue that the current models need to select the damage threshold indirectly or assume it in advance and ensures that the damage evolution characteristics follow the first principle entirely.<\/jats:p>","DOI":"10.3390\/e24081032","type":"journal-article","created":{"date-parts":[[2022,7,27]],"date-time":"2022-07-27T21:11:05Z","timestamp":1658956265000},"page":"1032","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Damage Evolution Modelling for Rock Materials Based on the Principle of Least Energy Dissipation Rate within Irreversible Thermodynamics"],"prefix":"10.3390","volume":"24","author":[{"given":"Xiaoming","family":"Li","sequence":"first","affiliation":[{"name":"School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei 230009, China"},{"name":"School of City and Architecture Engineering, Zaozhuang University, Zaozhuang 277160, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5680-5199","authenticated-orcid":false,"given":"Mingwu","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei 230009, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/0148-9062(65)90004-5","article-title":"The failure of rock","volume":"2","author":"Cook","year":"1965","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0020-7225(79)90058-2","article-title":"A continuum model for plastic-brittle behaviour of rock and concrete","volume":"17","author":"Dragon","year":"1979","journal-title":"Int. J. Eng. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4129","DOI":"10.1029\/JB094iB04p04129","article-title":"Micromechanics of the brittle to plastic transition in Carrara marble","volume":"94","author":"Fredrich","year":"1989","journal-title":"J. Geophys. Res."},{"key":"ref_4","unstructured":"Brady, B.H.G., and Brown, E.T. (1993). Rock Mechanics for Underground Mining, Chapman and Hall."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/S1365-1609(02)00036-9","article-title":"Application of a local degradation model to the analysis of brittle fracture of laboratory scale rock specimens under triaxial conditins","volume":"39","author":"Fang","year":"2002","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.ijrmms.2006.07.005","article-title":"Determination of residual strength parameters of jointed rock masses using the GSI system","volume":"44","author":"Cai","year":"2017","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4247","DOI":"10.1007\/s12665-013-2819-8","article-title":"Water absorption process effect on strength of Ayazini tuff, such as the uniaxial compressive strength (UCS), flflexural strength and freeze and thaw effect","volume":"71","author":"Celik","year":"2014","journal-title":"Environ. Earth Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s12665-013-2451-7","article-title":"Rock mass damage inducedby rockbursts occurring on tunnel floors: A case study of two tunnels at the Jinping II Hydropower Station","volume":"71","author":"Zhang","year":"2014","journal-title":"Enviro. Earth Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.ijrmms.2010.01.002","article-title":"An elasto-plastic damage constitutive model with double yield surfaces for saturated soft rock","volume":"47","author":"Zhou","year":"2010","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1016\/j.ijrmms.2015.09.012","article-title":"A damage plasticity model for different types of intact rock","volume":"80","author":"Unteregger","year":"2015","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.ijrmms.2015.11.011","article-title":"A coupled elastoplastic damage model for brittle rocks and its application in modelling underground excavation","volume":"84","author":"Zhang","year":"2016","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_12","first-page":"187","article-title":"A unified micromechancs-based damage model for instantaneous and time-dependent behaviors of brittle rocks","volume":"84","author":"Zhao","year":"2016","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijrmms.2016.11.005","article-title":"A micromechanics-based damage constitutive model of porous rocks","volume":"91","author":"Zhu","year":"2017","journal-title":"Int. J. Rock. Mech. Min. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1016\/0020-7683(82)90039-7","article-title":"Statistical aspects of the continuous damage theory","volume":"18","author":"Krajcinovic","year":"1982","journal-title":"Int. J. Solids Struct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1016\/j.ijsolstr.2008.09.033","article-title":"Is Weibull distribution the correct model for predicting probability of failure initiated by non-interacting flflaws?","volume":"46","author":"Todinov","year":"2009","journal-title":"Int. J. Solids Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.enggeo.2012.05.005","article-title":"A statistical damage constitutive model for softening behavior of rocks","volume":"143","author":"Li","year":"2012","journal-title":"Eng. Geol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"5905","DOI":"10.1007\/s12665-015-4228-7","article-title":"Progressive failure constitutive model for softening behavior of rocks based on maximum entropy theory","volume":"73","author":"Li","year":"2015","journal-title":"Environ. Earth Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"04016033","DOI":"10.1061\/(ASCE)GM.1943-5622.0000680","article-title":"Statistical damage constitutive model for rocks considering residual strength","volume":"17","author":"Zhao","year":"2017","journal-title":"Int. J. Geomech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5991","DOI":"10.1007\/s10064-019-01500-7","article-title":"Damage evolution behavior and constitutive model of s-andstone subjected to chemical corrosion","volume":"78","author":"Lin","year":"2019","journal-title":"Bull. Een. Geol. Environ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9035396","DOI":"10.1155\/2019\/9035396","article-title":"Statistical constitutive model of thermal damage fordeep rock considering initial compaction stage and residual strength. Math","volume":"2019","author":"Zhu","year":"2019","journal-title":"Probl. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1139\/cgj-2018-0178","article-title":"Constitutive model to simulate full deformation and failure process for rocks considering initial compression and residual strength behaviors","volume":"56","author":"Cao","year":"2019","journal-title":"Can. Geotech. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF01245574","article-title":"Determination of the post-failure behavior of brittle rock using a servo-controlled testing machine","volume":"2","author":"Rummel","year":"1970","journal-title":"Rock Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/S0020-7403(97)87627-8","article-title":"Constitutive and damage evolution equations of elastic-brittle materials based on irreversible thermodynamics","volume":"39","author":"Murakami","year":"1997","journal-title":"Int. J. Mech. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1016\/S0020-7683(98)00043-2","article-title":"Elastoplastic material with isotropic damage","volume":"36","author":"Cauvin","year":"1999","journal-title":"Int. J. Solids Structures."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1016\/S0020-7683(98)00044-4","article-title":"Damage mechanics: Basic variables in continuum theories","volume":"36","author":"Cauvin","year":"1999","journal-title":"Int. J. Solids Structures."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/j.euromechsol.2005.05.003","article-title":"Creep damage modelling for quasibrittle materials","volume":"24","author":"Challamel","year":"2005","journal-title":"Eur. J. Mech. A-Solids"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/j.ijmecsci.2005.01.002","article-title":"Strain-based anisotropic damage modelling and unilateral effects","volume":"47","author":"Challamel","year":"2005","journal-title":"Int. J. Mech. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1515\/JNETDY.2005.006","article-title":"Thermodynamic relationship between creep crack growth and creep deformation","volume":"30","author":"Yang","year":"2005","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1515\/JNETDY.2009.009","article-title":"Asymptotic Stability in Constrained Configuration Space for Solids","volume":"34","author":"Yang","year":"2009","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1177\/1056789514520801","article-title":"A discrete visco-plastic damage model for time-dependent behaviour of quasi-brittle rocks","volume":"24","author":"Wang","year":"2015","journal-title":"Int. J. Damage Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1016\/j.ijsolstr.2011.10.017","article-title":"A continuum damage mechanics framework for modeling micro-damage healing","volume":"49","author":"Darabi","year":"2011","journal-title":"Int. J. Solids Struct."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"111289","DOI":"10.1016\/j.ijsolstr.2021.111289","article-title":"On the constitutive modelling of elasto-plastic self-healing materials","volume":"234","author":"Subramanian","year":"2022","journal-title":"Int. J. Solids Struct."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1016\/j.apm.2020.12.032","article-title":"Anisotropic energy-based progressive damage model for laminated geomaterials","volume":"93","author":"Li","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2150045","DOI":"10.1142\/S0218348X21500456","article-title":"A thermodynamic consistent elastoplastic fractional time-dependent damage model for rock-like materials","volume":"29","author":"Qu","year":"2021","journal-title":"Fracaals"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/s00603-017-1322-z","article-title":"Experi mental and modelling investigations of the coupled elastoplastic damage of a quasi-brittle rock","volume":"51","author":"Zhang","year":"2018","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Chai, J.F. (2021). Research on dynamic response characteristics for basement structure of heavy haul railway tunnel with defects. Mathematics, 9.","DOI":"10.3390\/math9222893"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/s00603-021-02686-7","article-title":"Fatigue damage of wellbore cement sheath in gas storage salt cavern under alternating internal pressure","volume":"55","author":"He","year":"2011","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.apm.2020.05.002","article-title":"New damage evolution model of rock material","volume":"86","author":"Gao","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_39","unstructured":"Lemaitre, J., and Chaboche, J.L. (1988). M\u00e9canique des Mat\u00e9riaux Solides, Dunod."},{"key":"ref_40","unstructured":"Zhan, K.H., and Shen, X.F. (1982). Prigozin and the Theory of Dissipative Structures, Shaanxi Science and Technology Press."},{"key":"ref_41","first-page":"41","article-title":"Principle of least rate of energy consumption and some application of its solid mechanics","volume":"15","author":"Zhou","year":"1993","journal-title":"Nat. Sci. J. Xiangtan Univ."},{"key":"ref_42","unstructured":"Yang, G.T. (1980). Elastoplastic Mechanics, Beijing People\u2019s Education Press."},{"key":"ref_43","first-page":"57","article-title":"Prineiple of least Energy Consumption and its verification and application prospects","volume":"15","author":"Zhou","year":"1997","journal-title":"J. Chang. Railw. Univ."},{"key":"ref_44","unstructured":"Jaeger, J.C., Cook, N.G.W., and Zimmerman, R.W. (2007). Fundamentals of Rock Mechanics, Blackwell."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Lemaitre, J. (1992). A Course on Damage Mechanics, Springer.","DOI":"10.1007\/978-3-662-02761-5"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1016\/j.compgeo.2011.07.014","article-title":"A micromechanics-based elastoplastic damage model for quasi-brittle rocks","volume":"38","author":"Xie","year":"2011","journal-title":"Comput. Geotech."},{"key":"ref_47","unstructured":"Lemaitre, J., and Desmorat, R. (2005). Engineering Damage Mechanics, Springer."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1032\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:57:24Z","timestamp":1760140644000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1032"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,27]]},"references-count":47,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["e24081032"],"URL":"https:\/\/doi.org\/10.3390\/e24081032","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,27]]}}}