{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:11:41Z","timestamp":1760058701234,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,18]],"date-time":"2025-04-18T00:00:00Z","timestamp":1744934400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Steel structures subjected to earthquakes or extreme cyclic loadings may undergo extensive damage and fractures due to ultra-low-cycle fatigue (ULCF). Although assessments of damage initiation and evolution parameters have been carried out for some steels exposed to low-cycle fatigue, so far, these parameters for structural steels exposed to ULCF have neither been sufficiently studied nor quantified. Accordingly, this paper provides the results of finite element analysis (FEA) concerning the ULCF behaviors of S355 and S690 steel specimens. Calibration of the damage parameters is performed in SIMULIA Abaqus 6.11 FEA software using a direct cyclic algorithm and available experimental data. Kliman\u2019s model for the hysteresis energy of cyclic loading is used to analytically verify the damage parameters. In addition, available experimental data were obtained from cyclic axial strain tests on S355 and S690 steel specimens according to the ASTM International standard E606\/E606M-21. Finally, the non-linear Chaboche\u2013Lemaitre (C\u2013L) combined isotropic\u2013kinematic hardening model is used for the characterization of the ULCF behavior of S355 steel in a simple cylindrical bar. It is found that the two damage initiation parameters are 5.0 and \u22120.8, the first damage initiation parameter is dominant when modeling the number of cycles to failure, and the second damage initiation parameter is a material constant.<\/jats:p>","DOI":"10.3390\/sym17040615","type":"journal-article","created":{"date-parts":[[2025,4,18]],"date-time":"2025-04-18T06:23:56Z","timestamp":1744957436000},"page":"615","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finite Element Analysis-Based Assessment of Damage Parameters for Ultra-Low-Cycle Fatigue Behavior of Structural Steels"],"prefix":"10.3390","volume":"17","author":[{"given":"Ivan","family":"Milojevi\u0107","sequence":"first","affiliation":[{"name":"Faculty of Technical Sciences, University of Pri\u0161tina in Kosovska Mitrovica, Kneza Milo\u0161a St. 7, RS-38220 Kosovska Mitrovica, Serbia"},{"name":"Faculty of Architecture, Civil Engineering and Geodesy, University of Banja Luka, Vojvode Petra Bojovi\u0107a Blvd. 1A, BH-78000 Banja Luka, Bosnia and Herzegovina"}]},{"given":"Mirsad","family":"Tari\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Pri\u0161tina in Kosovska Mitrovica, Kneza Milo\u0161a St. 7, RS-38220 Kosovska Mitrovica, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0019-8371","authenticated-orcid":false,"given":"Dardan","family":"Klimenta","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Pri\u0161tina in Kosovska Mitrovica, Kneza Milo\u0161a St. 7, RS-38220 Kosovska Mitrovica, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-6926-6920","authenticated-orcid":false,"given":"Bojana","family":"Gruji\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Architecture, Civil Engineering and Geodesy, University of Banja Luka, Vojvode Petra Bojovi\u0107a Blvd. 1A, BH-78000 Banja Luka, Bosnia and Herzegovina"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9862-8917","authenticated-orcid":false,"given":"Darius","family":"Andriukaitis","sequence":"additional","affiliation":[{"name":"Department of Electronics Engineering, Faculty of Electrical and Electronics Engineering, Kaunas University of Technology, Studentu g. 50-438, LT-51368 Kaunas, Lithuania"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-9728-0695","authenticated-orcid":false,"given":"Sa\u0161a","family":"Jovanovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Pri\u0161tina in Kosovska Mitrovica, Kneza Milo\u0161a St. 7, RS-38220 Kosovska Mitrovica, Serbia"}]},{"given":"Milo\u0161","family":"\u010colovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Pri\u0161tina in Kosovska Mitrovica, Kneza Milo\u0161a St. 7, RS-38220 Kosovska Mitrovica, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,18]]},"reference":[{"key":"ref_1","unstructured":"El-Sayed, M.E.M. (2013, January 15\u201321). Transition from low cycle to high cycle in uniaxial fatigue. Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition, San Diego, CA, USA. IMECE2013-66202, V009T10A020."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kim, Y., and Hwang, W. (2019). High-cycle, low-cycle, extremely low-cycle fatigue and monotonic fracture behaviors of low-carbon steel and its welded joint. Materials, 12.","DOI":"10.3390\/ma12244111"},{"key":"ref_3","unstructured":"(2025, February 19). Siemens Digital Industries Software. Available online: https:\/\/community.sw.siemens.com\/s\/article\/what-is-a-sn-curve."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Jia, L.-J., and Ge, H. (2019). Ultra-Low-Cycle Fatigue Failure of Metal Structures Under Strong Earthquakes, Springer Nature. [1st ed.].","DOI":"10.1007\/978-981-13-2661-5"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hormozi, M.R., Biglari, F., and Nikbin, K.M. (2012, January 15\u201319). Study on sensitivity of damage parameters c1, c2, c3 and c4 on FB2 material under low cycle fatigue test. Proceedings of the 2012 ASME Pressure Vessels and Piping Division Conference, Toronto, ON, Canada.","DOI":"10.1115\/PVP2012-78810"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Xu, Y., Li, X., Zhang, Y., and Yang, J. (2023). Ultra-low cycle fatigue life prediction model\u2014A review. Metals, 13.","DOI":"10.3390\/met13061142"},{"key":"ref_7","first-page":"36","article-title":"A review on simulate fracture and ultra-low cycle fatigue in steel structures by ABAQUS","volume":"5","author":"Mirzaee","year":"2023","journal-title":"J. Civ. Eng. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2749","DOI":"10.1177\/13694332231199825","article-title":"Development of ultra-low cycle fatigue life prediction model for structural steel considering the effects of surface roughness, loading frequency, and loading amplitude","volume":"26","author":"Saleem","year":"2023","journal-title":"Adv. Struct. Eng."},{"key":"ref_9","first-page":"2338","article-title":"Study on ultra-low cycle fatigue fracture of thick-walled steel bridge piers","volume":"6","author":"Li","year":"2023","journal-title":"Proc. Civ. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107946","DOI":"10.1016\/j.ijfatigue.2023.107946","article-title":"A cyclic GTN model for ultra-low cycle fatigue analysis of structural steels","volume":"177","author":"Yin","year":"2023","journal-title":"Int. J. Fatigue"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liao, F., Yang, Z., Wang, J., Fang, P., Liu, X., and Li, X. (2024). Cyclic void growth model parameter calibration of Q460D steel and ER55-G welds after exposure to high temperatures. Buildings, 14.","DOI":"10.3390\/buildings14061622"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"108561","DOI":"10.1016\/j.jcsr.2024.108561","article-title":"Influence of corrosion on ultra-low cycle fatigue performance of steel butt-welded joints with various welding methods","volume":"215","author":"Zhang","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"108225","DOI":"10.1016\/j.jcsr.2023.108225","article-title":"Seismic damage behaviors of structural steels under different constitutive and damage models","volume":"212","author":"Hai","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108295","DOI":"10.1016\/j.jcsr.2023.108295","article-title":"Laboratory and ultra-low cycle fatigue evaluation of the seismic performance of BFP connection with different amounts of pre-tension in bolts","volume":"212","author":"Noferesti","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"108373","DOI":"10.1016\/j.jcsr.2023.108373","article-title":"Ultra-low cycle fatigue evaluation method for unstiffened steel piers using fiber model","volume":"213","author":"Yu","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108562","DOI":"10.1016\/j.jcsr.2024.108562","article-title":"Continuous damage model for structural steels and weld metals under ultra-low-cyclic loading","volume":"215","author":"Huang","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"108650","DOI":"10.1016\/j.jcsr.2024.108650","article-title":"New ultra-low cycle fatigue model for metal alloys","volume":"217","author":"Peng","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"108987","DOI":"10.1016\/j.jcsr.2024.108987","article-title":"Effect of loading sequence on the ultra-low-cycle fatigue performance of all-steel BRBs","volume":"222","author":"Liu","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109041","DOI":"10.1016\/j.jcsr.2024.109041","article-title":"Corrosion effect on ultra-low cycle fatigue capacity: A laboratory and numerical study","volume":"223","author":"Razmkhah","year":"2024","journal-title":"J. Constr. Steel Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109152","DOI":"10.1016\/j.jcsr.2024.109152","article-title":"Machine learning for ULCF life prediction of structural steels with synthetic data","volume":"224","author":"Yu","year":"2025","journal-title":"J. Constr. Steel Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"109268","DOI":"10.1016\/j.jcsr.2024.109268","article-title":"Comparative study on ultra-low-cycle-fatigue behaviour of three Indian structural steel grades","volume":"226","author":"Nambirajan","year":"2025","journal-title":"J. Constr. Steel Res."},{"key":"ref_22","unstructured":"(2025, February 19). SIMULIA Abaqus 6.11\u2014Abaqus Analysis User\u2019s Manual. Available online: http:\/\/orpheus.nchc.org.tw:2080\/v6.11\/books\/usb\/default.htm?startat=pt03ch06s02at06.html."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"142","DOI":"10.3221\/IGF-ESIS.35.17","article-title":"Damage and failure modeling of lotus-type porous material subjected to low-cycle fatigue","volume":"10","author":"Kramberger","year":"2015","journal-title":"Fractur. Struct. Integr."},{"key":"ref_24","unstructured":"Shrivastava, B. (2025, February 19). Mechanics of Solid Materials, Available online: https:\/\/catdir.loc.gov\/catdir\/samples\/cam034\/88022913.pdf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/s00707-020-02851-z","article-title":"Identification of Chaboche-Lemaitre combined isotropic-kinematic hardening model parameters assisted by the fuzzy logic analysis","volume":"232","author":"Skrzat","year":"2021","journal-title":"Acta Mech."},{"key":"ref_26","unstructured":"Versaillot, P.D. (2017). Effects of Cyclic Loading on the Mechanical Properties of Steel. [Master\u2019s Thesis, Politehnica University of Timi\u0219oara]."},{"key":"ref_27","unstructured":"(2021). Standard Test Method for Strain-Controlled Fatigue Testing (Standard No. E606\/E606M-21)."},{"key":"ref_28","unstructured":"Vucko, F., Bosch, C., and Delafosse, D. (2012, January 9\u201313). Experimental and numerical analysis of hydrogen interaction with plastic strain in a high strength steel. Proceedings of the European Corrosion Congress\u2014EUROCORR 2012, Istanbul, Turkey. Available online: https:\/\/hal-emse.ccsd.cnrs.fr\/emse-01063843."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Mazzolani, F., Ricles, J.M., and Sause, R. (2009). Numerical simulation of the cyclic inelastic behaviour of buckling restrained braces. Behaviour of Steel Structures in Seismic Areas, CRC Press Taylor & Francis Group. [1st ed.]. Session 15.","DOI":"10.1201\/9780203861592"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1016\/j.conbuildmat.2018.09.028","article-title":"Experimental cyclic behavior and constitutive modeling of high strength structural steels","volume":"189","author":"Hai","year":"2018","journal-title":"Constr. Build. Mater."},{"key":"ref_31","unstructured":"Vucko, F., Aoufi, A., Bosch, C., and Delafosse, D. (2013, January 1\u20135). Mobility and trapping of hydrogen in high-strength steel. Proceedings of the European Corrosion Congress\u2014EUROCORR 2013, Estoril, Portugal. Available online: https:\/\/hal-emse.ccsd.cnrs.fr\/file\/index\/docid\/993181\/filename\/Vucko_-_Mobility_and_trapping_of_hydrogen_in_high-strength_steel.pdf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/j.msea.2014.01.016","article-title":"Experimental investigations of internal and effective stresses during fatigue loading of high-strength steel","volume":"597","author":"Vucko","year":"2014","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"L7","DOI":"10.1016\/0025-5416(84)90083-1","article-title":"A comparison of two simple methods for measuring cyclic internal and effective stresses","volume":"64","author":"Dickson","year":"1984","journal-title":"Mater. Sci. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"61","DOI":"10.12913\/22998624\/175930","article-title":"Computational methods of the identification of Chaboche isotropic-kinematic hardening model parameters derived from the cyclic loading tests","volume":"18","author":"Gontarz","year":"2024","journal-title":"Adv. Sci. Technol. Res. J."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Song, W., Liu, X., Xu, J., Fan, Y., Shi, D., Yang, F., Xia, X., Berto, F., and Wan, D. (2021). Low-cycle fatigue life prediction of 10CrNi3MoV steel and undermatched welds by damage mechanics approach. Front. Mater., 8.","DOI":"10.3389\/fmats.2021.641145"},{"key":"ref_36","unstructured":"Ramberg, W., and Osgood, W.R. (2025, February 19). Description of Stress\u2013Strain Curves by Three Parameters, Technical Note No. 902, Available online: https:\/\/ntrs.nasa.gov\/api\/citations\/19930081614\/downloads\/19930081614.pdf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0025-5416(84)90238-6","article-title":"Fatigue life prediction for a material under programmable loading using the cyclic stress-strain properties","volume":"68","author":"Kliman","year":"1984","journal-title":"Mater. Sci. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0025-5416(84)90239-8","article-title":"Hysteresis energy of cyclic loading","volume":"68","author":"Kliman","year":"1984","journal-title":"Mater. Sci. Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/615\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:17:08Z","timestamp":1760030228000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/615"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,18]]},"references-count":38,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040615"],"URL":"https:\/\/doi.org\/10.3390\/sym17040615","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,4,18]]}}}