{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T07:45:07Z","timestamp":1777707907328,"version":"3.51.4"},"reference-count":192,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["Grant agreement No. 850437"],"award-info":[{"award-number":["Grant agreement No. 850437"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Damage Mechanics"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:p>This paper provides an extensive review of popular regularization methods utilized in numerical models to stabilize the structural response of materials exhibiting significant softening. The necessity for regularization is highlighted in cases of material softening, which is attributed to the loss of ellipticity in the governing differential equations. It discusses the advantages and disadvantages of the regularization methods most commonly employed in the scientific community. Furthermore, the paper highlights recent advancements, particularly in defining internal length within nonlocal models and characteristic element length in fracture energy regularization methods, as alternative solutions to address the limitations inherent in traditional approaches.<\/jats:p>","DOI":"10.1177\/10567895251329946","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T03:28:12Z","timestamp":1749439692000},"page":"72-118","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["State of art in regularization methods for numerical analysis of structures with softening"],"prefix":"10.1177","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-6575-1449","authenticated-orcid":false,"given":"Jiahui","family":"Shen","sequence":"first","affiliation":[{"name":"Mul2 Lab, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4140-2204","authenticated-orcid":false,"given":"M\u00e1rio Rui Tiago","family":"Arruda","sequence":"additional","affiliation":[{"name":"Department of Structures, Structural Behaviour Unit, LNEC \u2013 National Laboratory of Civil Engineering, Lisboa, Portugal"}]},{"given":"Alfonso","family":"Pagani","sequence":"additional","affiliation":[{"name":"Mul2 Lab, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy"}]}],"member":"179","published-online":{"date-parts":[[2025,6,9]]},"reference":[{"key":"e_1_3_2_2_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789508097541"},{"key":"e_1_3_2_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruct.2017.01.074"},{"key":"e_1_3_2_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2021.108129"},{"key":"e_1_3_2_5_1","doi-asserted-by":"publisher","DOI":"10.1002\/1097-0207(20001030)49:6<811::AID-NME985>3.0.CO;2-9"},{"key":"e_1_3_2_6_1","doi-asserted-by":"publisher","DOI":"10.1061\/JMCEA3.0002111"},{"key":"e_1_3_2_7_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1984)110:12(1693)"},{"key":"e_1_3_2_8_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1991)117:5(1070)"},{"key":"e_1_3_2_9_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1994)120:3(593)"},{"key":"e_1_3_2_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0013-7944(01)00084-4"},{"key":"e_1_3_2_11_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1984)110:12(1666)"},{"key":"e_1_3_2_12_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(2002)128:11(1119)"},{"key":"e_1_3_2_13_1","unstructured":"Ba\u017eant ZP Le JL Hoover CG (2010) Nonlocal boundary layer (NBL) model: Overcoming boundary condition problems in strength statistics and fracture analysis of quasi-brittle materials. In: Proceedings of the 7th international conference on fracture mechanics of concrete and concrete structures May 23\u201328 2010 Seoul South Korea: Korea Concrete Institute pp.135\u2013143."},{"key":"e_1_3_2_14_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1007486221395"},{"key":"e_1_3_2_15_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620260809"},{"key":"e_1_3_2_16_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9445(1988)114:11(2493)"},{"key":"e_1_3_2_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02486267"},{"key":"e_1_3_2_18_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1990)116:11(2485)"},{"key":"e_1_3_2_19_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.3173674"},{"key":"e_1_3_2_20_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1989)115:4(755)"},{"key":"e_1_3_2_21_1","doi-asserted-by":"publisher","DOI":"10.1201\/9780203756799"},{"key":"e_1_3_2_22_1","doi-asserted-by":"publisher","DOI":"10.1016\/0958-9465(94)00028-W"},{"key":"e_1_3_2_23_1","unstructured":"Bongers G (2011) A Stress-based Gradient-enhanced Damage Model. Delft Netherlands: Delft University of Technology."},{"key":"e_1_3_2_24_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(03)00144-6"},{"key":"e_1_3_2_25_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.272"},{"key":"e_1_3_2_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2010.05.003"},{"key":"e_1_3_2_27_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-022-02143-4"},{"key":"e_1_3_2_28_1","doi-asserted-by":"publisher","DOI":"10.1002\/9781118536643"},{"key":"e_1_3_2_29_1","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1201\/b10546-34","article-title":"On the uniqueness of numerical solutions of shear failure of deep concrete beams: Comparison of smeared and discrete crack approaches","author":"Cervenka J","year":"2010","unstructured":"Cervenka J, Cervenka V (2010) On the uniqueness of numerical solutions of shear failure of deep concrete beams: Comparison of smeared and discrete crack approaches. In: Computational modelling of concrete structures: Proceedings of EURO-C 2010. Rohrmoos\/Schladming, Austria: CRC Press\/Balkema, pp.281\u2013290.","journal-title":"Computational modelling of concrete structures: Proceedings of EURO-C 2010"},{"key":"e_1_3_2_30_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2018.04.010"},{"key":"e_1_3_2_31_1","unstructured":"Cervenka V Pukl R Ozbolt J et al. (1995) Mesh sensitivity effects in smeared finite element analysis of concrete fracture. In: Wittmann FH (ed.) Fracture Mechanics of Concrete Structures Proceedings FRAMCOS-2. Freiburg Germany: AEDIFICATIO Publishers pp.1387\u20131396."},{"key":"e_1_3_2_32_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.518"},{"key":"e_1_3_2_33_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2010.04.005"},{"key":"e_1_3_2_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2010.06.013"},{"key":"e_1_3_2_35_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(00)00053-6"},{"key":"e_1_3_2_36_1","doi-asserted-by":"publisher","DOI":"10.1007\/s42102-024-00120-4"},{"key":"e_1_3_2_37_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compgeo.2022.105114"},{"key":"e_1_3_2_38_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0997-7538(00)01111-6"},{"key":"e_1_3_2_39_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(01)00066-X"},{"key":"e_1_3_2_40_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2021.02.010"},{"key":"e_1_3_2_41_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.3046"},{"key":"e_1_3_2_42_1","doi-asserted-by":"publisher","DOI":"10.1108\/eb023897"},{"key":"e_1_3_2_43_1","unstructured":"Desmorat R Gatuingt F (2007a) Introduction of an internal time in nonlocal integral theories. Technical Report 268 LMTCachan ENS Cachan\/CNRS\/Universit\u00e9 Paris 6\/PRES Universud Paris."},{"key":"e_1_3_2_44_1","unstructured":"Desmorat R Gatuingt F (2007b) Introduction of an internal time in nonlocal integral theories. Internal Report LMT-Cachan ENS Cachan CNRS Universit\u00e9 Paris 6 Cachan France."},{"key":"e_1_3_2_45_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2015.06.015"},{"key":"e_1_3_2_46_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2006.09.012"},{"key":"e_1_3_2_47_1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7949(94)00501-S"},{"key":"e_1_3_2_48_1","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.700"},{"key":"e_1_3_2_49_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2006.11.030"},{"key":"e_1_3_2_50_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2005.03.038"},{"key":"e_1_3_2_51_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1996)122:10(939)"},{"key":"e_1_3_2_52_1","unstructured":"Duvaut G Lions JL (1972) Les in\u00e9quations en m\u00e9canique et en physique. Paris: Dunod."},{"key":"e_1_3_2_53_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7225(66)90022-X"},{"key":"e_1_3_2_54_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(97)00119-4"},{"key":"e_1_3_2_55_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(2009)135:3(117)"},{"key":"e_1_3_2_56_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)1532-3641(2010)10:1(30)"},{"key":"e_1_3_2_57_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(98)80011-X"},{"key":"e_1_3_2_58_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01181824"},{"key":"e_1_3_2_59_1","doi-asserted-by":"crossref","unstructured":"Geers MGD Brekelmans WAM de Borst R (1994) Viscous regularization of strain-localisation for damaging materials. In: DIANA computational mechanics '94: Proceedings of the first international DIANA conference on computational mechanics October 24\u201325 1994 Dordrecht Netherlands: Springer pp.127\u2013138.","DOI":"10.1007\/978-94-011-1046-4_12"},{"key":"e_1_3_2_60_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2011.08.012"},{"key":"e_1_3_2_61_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.4047207"},{"key":"e_1_3_2_62_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620382105"},{"key":"e_1_3_2_63_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.479"},{"key":"e_1_3_2_64_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2014.05.023"},{"key":"e_1_3_2_65_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.2180"},{"key":"e_1_3_2_66_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0997-7538(98)80054-5"},{"key":"e_1_3_2_67_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijimpeng.2009.02.004"},{"key":"e_1_3_2_68_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijimpeng.2012.08.002"},{"key":"e_1_3_2_69_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2016.03.022"},{"key":"e_1_3_2_70_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2016.02.029"},{"key":"e_1_3_2_71_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.09.010"},{"key":"e_1_3_2_72_1","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/2810108"},{"key":"e_1_3_2_73_1","doi-asserted-by":"publisher","DOI":"10.1016\/0008-8846(76)90007-7"},{"key":"e_1_3_2_74_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115559"},{"key":"e_1_3_2_75_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2024.105971"},{"key":"e_1_3_2_76_1","doi-asserted-by":"publisher","DOI":"10.1016\/0013-7944(90)90005-2"},{"key":"e_1_3_2_77_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(97)00306-5"},{"key":"e_1_3_2_78_1","doi-asserted-by":"publisher","DOI":"10.1080\/12795119.2004.9692625"},{"key":"e_1_3_2_79_1","doi-asserted-by":"publisher","DOI":"10.1080\/17747120.2007.9692973"},{"key":"e_1_3_2_80_1","doi-asserted-by":"publisher","DOI":"10.1080\/17747120.2007.9692974"},{"key":"e_1_3_2_81_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruc.2012.06.006"},{"key":"e_1_3_2_82_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2019.06.011"},{"key":"e_1_3_2_83_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2007.11.010"},{"key":"e_1_3_2_84_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1262"},{"key":"e_1_3_2_85_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7949(02)00078-0"},{"key":"e_1_3_2_86_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.364"},{"key":"e_1_3_2_87_1","first-page":"26","article-title":"Rupture time under creep conditions","volume":"8","author":"Kachanov L","year":"1958","unstructured":"Kachanov L (1958) Rupture time under creep conditions. Izvestia Akademii Nauk SSSR 8: 26\u201331.","journal-title":"Izvestia Akademii Nauk SSSR"},{"key":"e_1_3_2_88_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7683(87)90030-8"},{"key":"e_1_3_2_89_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruct.2017.08.040"},{"key":"e_1_3_2_90_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.4038610"},{"key":"e_1_3_2_91_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.3157739"},{"key":"e_1_3_2_92_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2009.07.007"},{"key":"e_1_3_2_93_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7683(67)90049-2"},{"key":"e_1_3_2_94_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2016.01.020"},{"key":"e_1_3_2_95_1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7949(92)90226-P"},{"key":"e_1_3_2_96_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(99)00214-5"},{"key":"e_1_3_2_97_1","doi-asserted-by":"publisher","DOI":"10.1061\/JENMDT.EMENG-7246"},{"key":"e_1_3_2_98_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00161-018-0669-z"},{"key":"e_1_3_2_99_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00161-022-01143-2"},{"key":"e_1_3_2_100_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compositesa.2007.01.017"},{"key":"e_1_3_2_101_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7683(88)90059-5"},{"key":"e_1_3_2_102_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0997-7538(02)01255-X"},{"key":"e_1_3_2_103_1","volume-title":"Mechanics of Solid Materials","author":"Lemaitre J","year":"1994","unstructured":"Lemaitre J, Chaboche JL (1994) Mechanics of Solid Materials. Cambridge, UK: Cambridge University Press."},{"key":"e_1_3_2_104_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0955-7997(01)00110-2"},{"key":"e_1_3_2_105_1","doi-asserted-by":"publisher","DOI":"10.3390\/su14052982"},{"key":"e_1_3_2_106_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcomc.2020.100006"},{"key":"e_1_3_2_107_1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(90)90073-U"},{"key":"e_1_3_2_108_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.conbuildmat.2022.126586"},{"key":"e_1_3_2_109_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compgeo.2024.106796"},{"key":"e_1_3_2_110_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2009.08.002"},{"key":"e_1_3_2_111_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmat.2007.03.006"},{"key":"e_1_3_2_112_1","unstructured":"Marconi F (2022) Damage-fracture transition by an eikonal-based gradient-type formulation for damage (-plastic) models. PhD Thesis Universit\u00e9 Paris-Saclay."},{"key":"e_1_3_2_113_1","unstructured":"Mazars J (1984) Application de la m\u00e9canique de l\u2019endommagement au comportement non lin\u00e9aire et \u00e0 la rupture du b\u00e9ton de structure. These de Docteur es Sciences Presentee a l\u2019Universite Pierre et Marie Curie-Paris 6."},{"key":"e_1_3_2_114_1","doi-asserted-by":"publisher","DOI":"10.1016\/0013-7944(90)90145-7"},{"key":"e_1_3_2_115_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1989)115:2(345)"},{"key":"e_1_3_2_116_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruc.2007.04.004"},{"key":"e_1_3_2_117_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-874X(97)00007-3"},{"key":"e_1_3_2_118_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compositesb.2020.107921"},{"key":"e_1_3_2_119_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0141-0296(01)00125-0"},{"key":"e_1_3_2_120_1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(88)90069-2"},{"key":"e_1_3_2_121_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2019.04.011"},{"key":"e_1_3_2_122_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.6397"},{"key":"e_1_3_2_123_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2020.107511"},{"key":"e_1_3_2_124_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruct.2022.115677"},{"key":"e_1_3_2_125_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2011.02.002"},{"key":"e_1_3_2_126_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.09.019"},{"key":"e_1_3_2_127_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-012-0717-7"},{"key":"e_1_3_2_128_1","doi-asserted-by":"publisher","DOI":"10.5802\/crmeca.137"},{"issue":"1","key":"e_1_3_2_129_1","first-page":"303","article-title":"Numerical simulations of concrete specimens with the gradient-enhanced eikonal non-local damage model","volume":"41","author":"Nogueira BR","year":"2023","unstructured":"Nogueira BR, Rastiello G, Giry C, et al. (2023) Numerical simulations of concrete specimens with the gradient-enhanced eikonal non-local damage model. Academic Journal of Civil Engineering 41(1): 303\u2013312.","journal-title":"Academic Journal of Civil Engineering"},{"key":"e_1_3_2_130_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2023.109670"},{"key":"e_1_3_2_131_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620280214"},{"key":"e_1_3_2_132_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2012.06.004"},{"key":"e_1_3_2_133_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0013-7944(99)00111-3"},{"key":"e_1_3_2_134_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(03)00263-9"},{"key":"e_1_3_2_135_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2008.02.004"},{"key":"e_1_3_2_136_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0997-7538(02)01211-1"},{"key":"e_1_3_2_137_1","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D"},{"key":"e_1_3_2_138_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(01)00087-7"},{"key":"e_1_3_2_139_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijimpeng.2016.04.002"},{"key":"e_1_3_2_140_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0065-2156(08)70009-7"},{"key":"e_1_3_2_141_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1987)113:10(1512)"},{"key":"e_1_3_2_142_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7683(93)90232-V"},{"key":"e_1_3_2_143_1","doi-asserted-by":"publisher","DOI":"10.1080\/19648189.2010.9693260"},{"key":"e_1_3_2_144_1","doi-asserted-by":"publisher","DOI":"10.3934\/nhm.2014.9.575"},{"key":"e_1_3_2_145_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.367"},{"key":"e_1_3_2_146_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2009.01.003"},{"key":"e_1_3_2_147_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.5364"},{"issue":"25","key":"e_1_3_2_148_1","first-page":"4369","article-title":"Over-nonlocal gradient enhanced plastic-damage model for concrete","volume":"46","author":"Poh LH","unstructured":"Poh LH, Swaddiwudhipong S (2009b) Over-nonlocal gradient enhanced plastic-damage model for concrete. International Journal of Solids and Structures 46(25\u201326): 4369\u20134378.","journal-title":"International Journal of Solids and Structures"},{"key":"e_1_3_2_149_1","doi-asserted-by":"crossref","unstructured":"Rabotnov YN (1969) Creep rupture. In: Applied mechanics: Proceedings of the twelfth international congress of applied mechanics Stanford University August 26\u201331 1968 Berlin\/Heidelberg: Springer pp.342\u2013349.","DOI":"10.1007\/978-3-642-85640-2_26"},{"key":"e_1_3_2_150_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.12.006"},{"key":"e_1_3_2_151_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2003.11.015"},{"key":"e_1_3_2_152_1","doi-asserted-by":"publisher","DOI":"10.1002\/nag.422"},{"key":"e_1_3_2_153_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechrescom.2013.09.011"},{"key":"e_1_3_2_154_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijimpeng.2024.105209"},{"key":"e_1_3_2_155_1","unstructured":"Rots JG et al. (1988) Computational modeling of concrete fracture. PhD Thesis Delft University of Technology Delft Netherlands."},{"key":"e_1_3_2_156_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2022.108552"},{"key":"e_1_3_2_157_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.dib.2019.104533"},{"key":"e_1_3_2_158_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-020-01908-z"},{"key":"e_1_3_2_159_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2019.04.001"},{"key":"e_1_3_2_160_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-012-0769-8"},{"key":"e_1_3_2_161_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.3171860"},{"key":"e_1_3_2_162_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2021.106410"},{"key":"e_1_3_2_163_1","doi-asserted-by":"crossref","unstructured":"Shen J Arruda M Pagani A et al. (2024) Mesh objective characteristic element length for higher-order finite beam elements. Advances in Engineering Software 195: 103709.","DOI":"10.1016\/j.advengsoft.2024.103709"},{"key":"e_1_3_2_164_1","doi-asserted-by":"publisher","DOI":"10.1080\/15376494.2022.2098430"},{"key":"e_1_3_2_165_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tws.2022.109094"},{"key":"e_1_3_2_166_1","first-page":"105678952312155","article-title":"A consistent crack bandwidth for higher-order beam theories: Application to concrete","volume":"33","author":"Shen J","unstructured":"Shen J, Tiago Arruda MR, Pagani A (2023b) A consistent crack bandwidth for higher-order beam theories: Application to concrete. International Journal of Damage Mechanics 33: 10567895231215557.","journal-title":"International Journal of Damage Mechanics"},{"key":"e_1_3_2_167_1","doi-asserted-by":"publisher","DOI":"10.1080\/17747120.2007.9692975"},{"key":"e_1_3_2_168_1","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.597"},{"key":"e_1_3_2_169_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2003.09.020"},{"key":"e_1_3_2_170_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.673"},{"key":"e_1_3_2_171_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2013.06.005"},{"key":"e_1_3_2_172_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7683(92)90151-I"},{"key":"e_1_3_2_173_1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(96)00997-8"},{"key":"e_1_3_2_174_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116189"},{"key":"e_1_3_2_175_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)GM.1943-5622.0000852"},{"key":"e_1_3_2_176_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compgeo.2016.10.016"},{"key":"e_1_3_2_177_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijimpeng.2021.103965"},{"key":"e_1_3_2_178_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.02.031"},{"key":"e_1_3_2_179_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechrescom.2020.103620"},{"key":"e_1_3_2_180_1","first-page":"1","article-title":"A nonlocal gradient damage model with energy limiter for dynamic brittle fracture","volume":"73","author":"Tran HT","year":"2023","unstructured":"Tran HT, Bui TQ (2023) A nonlocal gradient damage model with energy limiter for dynamic brittle fracture. Computational Mechanics 73: 1\u201326.","journal-title":"Computational Mechanics"},{"key":"e_1_3_2_181_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116123"},{"key":"e_1_3_2_182_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compositesb.2023.111120"},{"key":"e_1_3_2_183_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.12.027"},{"key":"e_1_3_2_184_1","unstructured":"Vermeer P Brinkgreve R (1994) A new effective non-local strain measure for softening plasticity. In: Chambon R Desrues J and Vardoulakis I (eds) Localisation and Bifurcation Theory for Soils and Rocks. Netherlands: A.A. Belkema Rotterdam pp.89\u2013100."},{"key":"e_1_3_2_185_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2022.108843"},{"key":"e_1_3_2_186_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2022.109032"},{"key":"e_1_3_2_187_1","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19971030)40:20<3839::AID-NME245>3.0.CO;2-6"},{"key":"e_1_3_2_188_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.06.029"},{"key":"e_1_3_2_189_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2015.06.034"},{"key":"e_1_3_2_190_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2024.04.053"},{"key":"e_1_3_2_191_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.4056324"},{"key":"e_1_3_2_192_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789520983872"},{"key":"e_1_3_2_193_1","doi-asserted-by":"publisher","DOI":"10.1177\/10567895221109622"}],"container-title":["International Journal of Damage Mechanics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10567895251329946","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/10567895251329946","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10567895251329946","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:51:16Z","timestamp":1777456276000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/10567895251329946"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,9]]},"references-count":192,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["10.1177\/10567895251329946"],"URL":"https:\/\/doi.org\/10.1177\/10567895251329946","relation":{},"ISSN":["1056-7895","1530-7921"],"issn-type":[{"value":"1056-7895","type":"print"},{"value":"1530-7921","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,9]]}}}