{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T22:03:54Z","timestamp":1769810634287,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T00:00:00Z","timestamp":1681084800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2022XJLJ01"],"award-info":[{"award-number":["2022XJLJ01"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["41572270"],"award-info":[{"award-number":["41572270"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["52274148"],"award-info":[{"award-number":["52274148"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["202206003"],"award-info":[{"award-number":["202206003"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["202306040"],"award-info":[{"award-number":["202306040"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022XJLJ01"],"award-info":[{"award-number":["2022XJLJ01"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41572270"],"award-info":[{"award-number":["41572270"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52274148"],"award-info":[{"award-number":["52274148"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["202206003"],"award-info":[{"award-number":["202206003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["202306040"],"award-info":[{"award-number":["202306040"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"China University of Mining and Technology (Beijing) Undergraduate Innovation Training Program","award":["2022XJLJ01"],"award-info":[{"award-number":["2022XJLJ01"]}]},{"name":"China University of Mining and Technology (Beijing) Undergraduate Innovation Training Program","award":["41572270"],"award-info":[{"award-number":["41572270"]}]},{"name":"China University of Mining and Technology (Beijing) Undergraduate Innovation Training Program","award":["52274148"],"award-info":[{"award-number":["52274148"]}]},{"name":"China University of Mining and Technology (Beijing) Undergraduate Innovation Training Program","award":["202206003"],"award-info":[{"award-number":["202206003"]}]},{"name":"China University of Mining and Technology (Beijing) Undergraduate Innovation Training Program","award":["202306040"],"award-info":[{"award-number":["202306040"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Understanding the tensile strength properties and damage evolution mechanism in fissured rock is very important to fundamental research and engineering design. The effects of flaw dip angle on the tensile strength, macroscopic crack propagation and failure mode of symmetrical Brazilian discs of rock-like materials were investigated. A parallel bonding model was proposed to examine the damage of pre-flawed discs under splitting the load. The microscopic parameters of particles and bonds in the model that can characterize rock-like materials\u2019 mechanical and deformation properties were obtained by calibrating against the laboratory test results. The crack development, energy evolution and damage characteristics of Brazil discs containing a single pre-existing flaw were studied at the microscopic scale. The results show that the flaw significantly weakens the strength of the Brazilian disc, and both the peak load and the initial cracking load decrease with increasing flaw angle. The failure modes of the rock-like specimens are mainly divided into three types: wing crack penetration damage mode, tensile-shear penetration damage mode and radial penetration failure mode. Except for the flaw dip angle 0\u00b0, the wing cracks generally sprouted at the tip of the pre-flaw, and the wing cracks at both tips of the pre-flaw are centrosymmetric. Crack coalescence was concentrated in the post-peak stage. Based on the particle flow code (PFC) energy partitions, the damage variables characterized by dissipation energy were proposed. The disc specimen\u2019s pre-peak damage variables and peak damage variables decreased with increasing flaw angle, and the damage was concentrated in the post-peak phase.<\/jats:p>","DOI":"10.3390\/sym15040895","type":"journal-article","created":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T02:02:33Z","timestamp":1681178553000},"page":"895","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Strength Properties and Damage Evolution Mechanism of Single-Flawed Brazilian Discs: An Experimental Study and Particle Flow Simulation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9255-5390","authenticated-orcid":false,"given":"Yao","family":"Bai","sequence":"first","affiliation":[{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China"},{"name":"State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology (Beijing), Beijing 100083, China"}]},{"given":"Haoyu","family":"Dou","sequence":"additional","affiliation":[{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China"}]},{"given":"Peng","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China"}]},{"given":"Tiancheng","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China"}]},{"given":"Yujing","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China"}]},{"given":"Yuqin","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0148-9062(78)90003-7","article-title":"Suggested methods for determining tensile strength of rock materials","volume":"15","author":"Bieniawski","year":"1978","journal-title":"Int. J. Rock Mech. Min. Sci. Geomech. Abstr."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1016\/j.ijrmms.2004.04.001","article-title":"Consideration of fracture growth from an inclined slit and inclined initial fracture at the surface of rock and mortar in compression","volume":"41","author":"Fujii","year":"2004","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2059","DOI":"10.1007\/s11771-014-2155-5","article-title":"Mechanical behavior and failure analysis of brittle sandstone specimens containing combined flaws under uniaxial compression","volume":"21","author":"Yang","year":"2014","journal-title":"J. Cent. South Univ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1497","DOI":"10.1007\/s00603-015-0838-3","article-title":"An experimental and numerical study on cracking behavior of brittle sandstone containing two non-coplanar fissures under uniaxial compression","volume":"49","author":"Yang","year":"2016","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.compgeo.2017.03.015","article-title":"Strength failure behavior and crack evolution mechanism of granite containing pre-existing non-coplanar holes: Experimental study and particle flow modeling","volume":"88","author":"Huang","year":"2017","journal-title":"Comput. Geotech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.engfracmech.2018.10.009","article-title":"Fracture analysis of sandstone with a single filled flaw under uniaxial compression","volume":"204","author":"Miao","year":"2018","journal-title":"Eng. Fract. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.ijrmms.2008.03.006","article-title":"Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression","volume":"46","author":"Wong","year":"2009","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.ijrmms.2012.01.006","article-title":"Influence of the geometry of partially-spanning joints on the uniaxial compressive strength of rock","volume":"50","author":"Wasantha","year":"2012","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.coldregions.2019.03.013","article-title":"Strength and failure characteristics of rock-like material containing single crack under freeze-thaw and uniaxial compression","volume":"162","author":"Huang","year":"2019","journal-title":"Cold Reg. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"04019305","DOI":"10.1061\/(ASCE)MT.1943-5533.0002967","article-title":"Numerical simulations on cracking behavior of rock-like specimens with single flaws under conditions of uniaxial and biaxial compressions","volume":"31","author":"Zhao","year":"2019","journal-title":"J. Mater. Civ. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1007\/s11223-014-9525-y","article-title":"On the strength and crack propagation process of the pre-cracked rock-like specimens under uniaxial compression","volume":"46","author":"Haeri","year":"2014","journal-title":"Strength Mater."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.tafmec.2014.04.004","article-title":"A comparative study on unfilled and filled crack propagation for rock-like brittle material","volume":"72","author":"Zhuang","year":"2014","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1016\/j.jngse.2016.07.022","article-title":"Mechanical properties and failure modes of rock samples with grout-infilled flaws: A particle mechanics modeling","volume":"34","author":"Zhao","year":"2016","journal-title":"J. Nat. Gas. Sci. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1007\/s12517-020-05631-0","article-title":"Experimental study on the mechanical properties and damage evolution of red sandstone containing a single ice-filled flaw under triaxial compression","volume":"13","author":"Shan","year":"2020","journal-title":"Arab. J. Geosci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"103368","DOI":"10.1016\/j.tafmec.2022.103368","article-title":"Experimental investigation on mechanical properties and failure mechanism of rock-like specimens containing an arc-shaped ice-filled flaw under uniaxial compression","volume":"119","author":"Wang","year":"2022","journal-title":"Theoret. Appl. Fract. Mech."},{"key":"ref_16","first-page":"1","article-title":"Advances in discrete element method applied to soil, rock and concrete mechanics","volume":"8","author":"Richefeu","year":"2009","journal-title":"Electron. J. Geotech. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1016\/j.ijsolstr.2010.12.001","article-title":"An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression","volume":"48","author":"Lee","year":"2011","journal-title":"Int. J. Solids Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1007\/s00603-012-0323-1","article-title":"Crack initiation, propagation and coalescence in rock-like material containing two flaws: A numerical study based on bonded-particle model approach","volume":"46","author":"Zhang","year":"2013","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_19","first-page":"711","article-title":"Cracking processes in rock-like material containing a single flaw under uniaxial compression: A numerical study based on parallel bonded-particle model approach","volume":"45","author":"Zhang","year":"2012","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.geothermics.2017.10.018","article-title":"Failure mechanical behavior of pre-holed granite specimens after elevated temperature treatment by particle flow code","volume":"72","author":"Yang","year":"2018","journal-title":"Geothermics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1093\/jge\/gxz036","article-title":"PFC simulation of crack evolution and energy conversion during basalt failure process","volume":"16","author":"Zhang","year":"2019","journal-title":"J. Geophys. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"103266","DOI":"10.1016\/j.tafmec.2022.103266","article-title":"Experimental and DEM study on failure behavior and stress distribution of flawed sandstone specimens under uniaxial compression","volume":"118","author":"Huang","year":"2022","journal-title":"Theoret. Appl. Fract. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Shi, X., Zhang, C., and Zhou, X. (2020). The Statistical Damage Constitutive Model of the Mechanical Properties of Alkali-Resistant Glass Fiber Reinforced Concrete. Symmetry, 12.","DOI":"10.3390\/sym12071139"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2641","DOI":"10.1108\/EC-03-2019-0123","article-title":"Numerical Brazilian split test of pre-cracked granite with randomly distributed micro-components","volume":"37","author":"Liu","year":"2020","journal-title":"Eng. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.ijmst.2020.03.007","article-title":"Numerical investigation on the tensile fracturing behavior of rock-shotcrete interface based on discrete element method","volume":"30","author":"Qiu","year":"2020","journal-title":"Int. J. Min. Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"107993","DOI":"10.1016\/j.petrol.2020.107993","article-title":"Effect of natural filling fracture on the cracking process of shale Brazilian disc containing a central straight notched flaw","volume":"196","author":"Zhou","year":"2021","journal-title":"J. Petrol. Sci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.1007\/s00603-021-02746-y","article-title":"Analytical Solutions of Stress Intensity Factors for a Centrally Cracked Brazilian Disc Considering Tangential Friction Effects","volume":"55","author":"Liu","year":"2022","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1520\/GTJ20200054","article-title":"Development and application of a new triaxial testing system for subzero rocks","volume":"44","author":"Bai","year":"2021","journal-title":"Geotech. Test. J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1680\/geot.1979.29.1.47","article-title":"A discrete numerical model for granular assemblies","volume":"29","author":"Cundall","year":"1979","journal-title":"Geotechnique"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1016\/j.ijrmms.2007.02.002","article-title":"A clumped particle model for rock","volume":"44","author":"Cho","year":"2007","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.enggeo.2018.04.017","article-title":"Distinct element method simulations of rock-concrete interfaces under different boundary conditions","volume":"240","author":"Senent","year":"2018","journal-title":"Eng. Geol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"665955","DOI":"10.3389\/fmats.2021.665955","article-title":"Energy evolution law of marble failure process under different confining pressures based on particle discrete element method","volume":"8","author":"Yang","year":"2021","journal-title":"Front. Mater."},{"key":"ref_33","unstructured":"Itasca Consulting Group Inc. (2015). Users\u2019 Manual for Particle Flow Code in 2 Dimensions (PFC2D), Version 5.0, Itasca Consulting Group Inc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/895\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:13:30Z","timestamp":1760123610000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/895"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,10]]},"references-count":33,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040895"],"URL":"https:\/\/doi.org\/10.3390\/sym15040895","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,10]]}}}