{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,10]],"date-time":"2026-09-10T05:09:54Z","timestamp":1789016994727,"version":"build-2803163510"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T00:00:00Z","timestamp":1788825600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Scientific Research Projects of Universities in Henan Province","award":["26B580002"],"award-info":[{"award-number":["26B580002"]}]},{"DOI":"10.13039\/501100004761","name":"Henan Provincial Natural Science Foundation","doi-asserted-by":"crossref","award":["262300421967"],"award-info":[{"award-number":["262300421967"]}],"id":[{"id":"10.13039\/501100004761","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Key Research and Development Projects of Henan Province","award":["262102240001"],"award-info":[{"award-number":["262102240001"]}]},{"name":"Key Research and Development Projects of Henan Province","award":["261111241000"],"award-info":[{"award-number":["261111241000"]}]},{"name":"Planned Project Supported by the Science and Technology Innovation Team of Universities and Colleges in Henan Province","award":["24IRTSTHN011"],"award-info":[{"award-number":["24IRTSTHN011"]}]},{"name":"Key Scientific and Technological Projects in Henan Province","award":["252102240031"],"award-info":[{"award-number":["252102240031"]}]},{"name":"Special Plan for Key Scientific Research Projects of Higher Education Institutions in Henan Province to Serve Industry Development","award":["25CY022"],"award-info":[{"award-number":["25CY022"]}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"crossref","award":["BK20250986"],"award-info":[{"award-number":["BK20250986"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Natural Science Foundation of the Jiangsu Higher Education Institutions of China","award":["25KJB580015"],"award-info":[{"award-number":["25KJB580015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Gels"],"abstract":"<jats:p>The service life of SBS-modified asphalt mixtures with physical gel structures is strongly influenced by the loading sequence\u2013interaction coupling effects. However, conventional linear models fail to accurately capture the influence of such coupling effects on fatigue life. This study, therefore, aims to develop a life prediction method for SBS-modified asphalt mixtures that explicitly accounts for the loading sequence\u2013interaction coupling effects. First, indirect tensile monotonic loading tests, as well as constant-amplitude and variable-amplitude indirect tensile repeated loading tests were conducted to determine the fatigue life of SBS-modified asphalt mixtures under different loading modes. Subsequently, a viscoelastic fatigue damage model was developed to analyze the accumulation of fatigue damage in SBS-modified asphalt mixtures. Subsequently, the effects of loading sequence and loading interaction on the fatigue damage accumulation process in SBS-modified asphalt mixtures were investigated. Based on these analyses, a nonlinear fatigue damage accumulation model considering the loading sequence\u2013interaction coupling effects was established. Finally, a nonlinear fatigue damage accumulation factor was introduced for developing a fatigue life prediction model for SBS-modified asphalt mixtures considering the loading sequence\u2013interaction coupling effects. The results indicate that the low\u2013high (\u03c3low\u2013\u03c3high) and high\u2013low (\u03c3high\u2013\u03c3low) loading sequences, respectively, retard and accelerate the fatigue damage accumulation process of SBS-modified asphalt mixtures. Consequently, the corresponding cumulative fatigue life fractions are greater than 1 and less than 1, respectively. Furthermore, the resulting fatigue life is positively correlated with the first-level fatigue life fraction under the \u03c3low\u2013\u03c3high sequence but negatively correlated under the \u03c3high\u2013\u03c3low sequence. The developed nonlinear fatigue damage accumulation model can accurately track the nonlinear fatigue damage accumulation of SBS-modified asphalt mixtures. The established nonlinear fatigue life prediction model accurately predicts the fatigue life of SBS-modified asphalt mixtures with physical gel structures under cyclic loading with variable stress amplitudes.<\/jats:p>","DOI":"10.3390\/gels12090825","type":"journal-article","created":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T12:51:48Z","timestamp":1788871908000},"page":"825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Variable-Amplitude Fatigue Life Prediction of SBS-Modified Asphalt Mixtures with Physical Gel Structures"],"prefix":"10.3390","volume":"12","author":[{"given":"Chenze","family":"Fang","sequence":"first","affiliation":[{"name":"School of Civil Engineering and Transportation, North China University of Water Resources and Electric Power, Zhengzhou 450045, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"given":"Jiahao","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Transportation, North China University of Water Resources and Electric Power, Zhengzhou 450045, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"given":"Menghao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215009, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0555-0824","authenticated-orcid":false,"given":"Hongbin","family":"Zhu","sequence":"additional","affiliation":[{"name":"College of Transportation Engineering, Dalian Maritime University, Dalian 116026, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-4362-1210","authenticated-orcid":false,"given":"Pingfan","family":"Hu","sequence":"additional","affiliation":[{"name":"Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122, USA"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1177\/1056789514553042","article-title":"Fatigue life prediction under variable amplitude loading using a nonlinear damage accumulation model","volume":"24","author":"Zuo","year":"2015","journal-title":"Int. J. Damage Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"137452","DOI":"10.1016\/j.conbuildmat.2024.137452","article-title":"Resurface of rubber modified asphalt mixture with stress absorbing membrane interlayer: From laboratory to field application","volume":"441","author":"Jin","year":"2024","journal-title":"Constr. Build. Mater."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"A159","DOI":"10.1115\/1.4009458","article-title":"Cumulative damage in fatigue","volume":"12","author":"Miner","year":"1945","journal-title":"J. Appl. Mech."},{"key":"ref_4","first-page":"68","article-title":"Investigating nonlinear fatigue damage accumulation of asphalt binders considering loading sequence effect","volume":"41","author":"Fang","year":"2024","journal-title":"Eng. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"138261","DOI":"10.1016\/j.conbuildmat.2024.138261","article-title":"Performance of rubber modified asphalt mixture with tire-derived aggregate subgrade","volume":"449","author":"Jin","year":"2024","journal-title":"Constr. Build. Mater."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1177\/1056789514524075","article-title":"A modified nonlinear fatigue damage accumulation model","volume":"24","author":"Lv","year":"2014","journal-title":"Int. J. Damage Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"127949","DOI":"10.1016\/j.jclepro.2021.127949","article-title":"Waste cathode-ray-tube glass powder modified asphalt materials: Preparation and characterization","volume":"314","author":"Jin","year":"2021","journal-title":"J. Clean. Prod."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"144963","DOI":"10.1016\/j.conbuildmat.2025.144963","article-title":"Property evolution and mechanism of rejuvenated bitumen during rejuvenation and secondary aging based on form and unsaturation of fatty acids","volume":"506","author":"Zhu","year":"2026","journal-title":"Constr. Build. Mater."},{"key":"ref_9","first-page":"627","article-title":"A concept of fatigue damage","volume":"76","author":"Marco","year":"1954","journal-title":"Trans. ASME"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1016\/S0142-1123(01)00027-5","article-title":"A new approach to low-cycle fatigue damage based on exhaustion of static toughness and dissipation of cyclic plastic strain energy during fatigue","volume":"23","author":"Ye","year":"2001","journal-title":"Int. J. Fatigue"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.ijfatigue.2004.08.005","article-title":"Sequential law in multiaxial fatigue, a new damage indicator","volume":"27","author":"Mesmacque","year":"2005","journal-title":"Int. J. Fatigue"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1177\/1056789512449638","article-title":"A fatigue driving stress approach to damage and life prediction under variable amplitude loading","volume":"22","author":"Kwofie","year":"2013","journal-title":"Int. J. Damage Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"106185","DOI":"10.1016\/j.ijfatigue.2021.106185","article-title":"A kinetics-based model of fatigue crack growth rate in bituminous material","volume":"148","author":"Li","year":"2021","journal-title":"Int. J. Fatigue"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107566","DOI":"10.1016\/j.engfracmech.2021.107566","article-title":"Stochastic fatigue damage in viscoelastic materials using probabilistic pseudo J-integral Paris\u2019 law","volume":"245","author":"Li","year":"2021","journal-title":"Eng. Fract. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"16974","DOI":"10.1021\/acs.langmuir.6c01856","article-title":"Atomistic prediction of the effects of structural differences in bio-oil fatty acids on thermodynamic properties and rejuvenation behavior of aged bitumen","volume":"42","author":"Zhu","year":"2026","journal-title":"Langmuir"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zhao, Z.Z., Wang, L.L., Wang, W.S., and Shangguan, X.H. (2022). Experimental investigation of the high-temperature rheological and aging resistance properties of activated crumb rubber powder\/SBS composite-modified asphalt. Polymers, 14.","DOI":"10.3390\/polym14091905"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.cocis.2006.09.001","article-title":"Relation between polymer architecture and nonlinear viscoelastic behavior of modified asphalts","volume":"11","author":"Polacco","year":"2006","journal-title":"Curr. Opin. Colloid Interface Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2226290","DOI":"10.1080\/10298436.2023.2226290","article-title":"Imperative role of SBS molecular structure on the performance properties of modified binders and asphalt mixes","volume":"24","author":"Islam","year":"2023","journal-title":"Int. J. Pavement Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"106417","DOI":"10.1016\/j.ijfatigue.2021.106417","article-title":"Energy dissipation and rheological property degradation of asphalt binder under repeated shearing with different oscillation amplitudes","volume":"152","author":"Zhang","year":"2021","journal-title":"Int. J. Fatigue"},{"key":"ref_20","first-page":"e04321","article-title":"Fatigue life prediction of asphalt binders under variable loading levels based on dissipated pseudo-strain energy","volume":"22","author":"Fang","year":"2025","journal-title":"Case Stud. Constr. Mater."},{"key":"ref_21","first-page":"e03286","article-title":"A new fatigue equation for asphalt mixtures considering loading sequence effects","volume":"20","author":"Fang","year":"2024","journal-title":"Case Stud. Constr. Mater."},{"key":"ref_22","first-page":"22080209","article-title":"Effects of loading sequence on fatigue damage accumulation of asphalt mixture","volume":"37","author":"Fang","year":"2023","journal-title":"Mater. Rep."},{"key":"ref_23","first-page":"267","article-title":"Research on cumulative fatigue damage of asphalt pavement based on multi-level amplitude loading","volume":"42","author":"Wang","year":"2017","journal-title":"Highw. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.conbuildmat.2016.05.006","article-title":"Developing an optimal multiple repeated load test to evaluate permanent deformation of asphalt mixtures based on axle load spectrum","volume":"122","author":"Jiang","year":"2016","journal-title":"Constr. Build. Mater."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2152027","DOI":"10.1080\/10298436.2022.2152027","article-title":"Effects of traffic load amplitude sequence on the cracking performance of asphalt pavement with a semi-rigid base","volume":"24","author":"Yao","year":"2022","journal-title":"Int. J. Pavement Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"04024142","DOI":"10.1061\/JMCEE7.MTENG-17329","article-title":"Investigating fatigue damage accumulation of asphalt binders considering amplitude sequence and loading interaction","volume":"36","author":"Fang","year":"2024","journal-title":"J. Mater. Civ. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.1460-2695.1988.tb01216.x","article-title":"A non-linear continuous fatigue damage model","volume":"11","author":"Chaboche","year":"1988","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.ijfatigue.2005.05.001","article-title":"Fatigue life prediction under variable loading based on a new nonlinear continuum damage mechanics model","volume":"28","author":"Dattoma","year":"2006","journal-title":"Int. J. Fatigue"},{"key":"ref_29","first-page":"151","article-title":"Strain energy-based fatigue life prediction under variable amplitude loadings","volume":"66","author":"Zhu","year":"2018","journal-title":"Struct. Eng. Mech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.ijfatigue.2018.04.006","article-title":"Strain energy gradient-based LCF life prediction of turbine discs using critical distance concept","volume":"113","author":"Zhu","year":"2018","journal-title":"Int. J. Fatigue"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.conbuildmat.2015.10.201","article-title":"Development and validation of nonlinear viscoelastic damage (NLVED) model for three-stage permanent deformation of asphalt concrete","volume":"102","author":"Zhang","year":"2016","journal-title":"Constr. Build. Mater."},{"key":"ref_32","first-page":"21","article-title":"Nonlinear fatigue damage model for asphalt mixtures","volume":"22","author":"Zheng","year":"2009","journal-title":"China J. Highw. Transp."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1016\/j.conbuildmat.2016.03.126","article-title":"A review of fatigue damage in bituminous mixtures: Understanding the phenomenon from a new perspective","volume":"113","year":"2016","journal-title":"Constr. Build. Mater."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1016\/j.matdes.2013.09.004","article-title":"Mean damage parameter for the characterization of fatigue cracking behavior in bituminous mixes","volume":"54","year":"2014","journal-title":"Mater. Des."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.measurement.2018.03.037","article-title":"Fatigue damage model of stone matrix asphalt with polymer modified binder based on tensile strain evolution and residual strength degradation using digital image correlation methods","volume":"123","author":"Jiang","year":"2018","journal-title":"Measurement"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhu, H., Guo, N., Li, Y., Ren, S., Wang, F., Zhang, J., Zheng, Z., Su, H., Zhao, Z., and Zhang, K. (2026). Towards Sustainable Utilization of Rejuvenated Bitumen: A Review with Emphasis on Secondary Aging, Aging Resistance, and Multiple Rejuvenation. Polymers, 18.","DOI":"10.3390\/polym18172103"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"127569","DOI":"10.1016\/j.conbuildmat.2022.127569","article-title":"Evaluation of lab performance of stamp sand and acrylonitrile styrene acrylate waste composites without asphalt as road surface materials","volume":"338","author":"Jin","year":"2022","journal-title":"Constr. Build. Mater."}],"container-title":["Gels"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2310-2861\/12\/9\/825\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,10]],"date-time":"2026-09-10T04:39:52Z","timestamp":1789015192000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2310-2861\/12\/9\/825"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,8]]},"references-count":37,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["gels12090825"],"URL":"https:\/\/doi.org\/10.3390\/gels12090825","relation":{},"ISSN":["2310-2861"],"issn-type":[{"value":"2310-2861","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,8]]}}}