{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T00:29:00Z","timestamp":1768004940016,"version":"3.49.0"},"reference-count":110,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,15]],"date-time":"2023-01-15T00:00:00Z","timestamp":1673740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003093","name":"Ministry of Higher Education Malaysia","doi-asserted-by":"publisher","award":["FRGS\/1\/2022\/TK10\/UKM\/02\/30"],"award-info":[{"award-number":["FRGS\/1\/2022\/TK10\/UKM\/02\/30"]}],"id":[{"id":"10.13039\/501100003093","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>A review of nonlocal theories utilized in the fatigue and fracture modeling of solid structures is addressed in this paper. Numerous papers have been studied for this purpose, and various nonlocal theories such as the nonlocal continuum damage model, stress field intensity model, peridynamics model, elastic-plastic models, energy-based model, nonlocal multiscale model, microstructural sensitive model, nonlocal lattice particle model, nonlocal high cycle fatigue model, low cycle fatigue model, nonlocal and gradient fracture criteria, nonlocal coupled damage plasticity model and nonlocal fracture criterion have been reviewed and summarized in the case of fatigue and fracture of solid structures and materials.<\/jats:p>","DOI":"10.3390\/ma16020831","type":"journal-article","created":{"date-parts":[[2023,1,16]],"date-time":"2023-01-16T04:57:19Z","timestamp":1673845039000},"page":"831","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Review on Nonlocal Theories in Fatigue Assessment of Solids"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0047-9854","authenticated-orcid":false,"given":"Saeed H.","family":"Moghtaderi","sequence":"first","affiliation":[{"name":"Department of Mechanical and Manufacturing Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2106-0542","authenticated-orcid":false,"given":"Alias","family":"Jedi","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Manufacturing Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia"},{"name":"Centre for Automotive Research, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmad Kamal","family":"Ariffin","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Manufacturing Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.ijfatigue.2019.02.009","article-title":"Comparison of TCD and SED Methods in Fatigue Lifetime Assessment","volume":"123","author":"Hu","year":"2019","journal-title":"Int. J. Fatigue"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Orenes Moreno, B., Bessone, A., Solazzi, S., Vanti, F., Bagnera, F., Riva, A., and Botto, D. (2022). Linear Elastic Fracture Mechanics Assessment of a Gas Turbine Vane. Materials, 15.","DOI":"10.3390\/ma15134694"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chen, H., Sun, Z., Zhong, Z., and Huang, Y. (2022). Fatigue Factor Assessment and Life Prediction of Concrete Based on Bayesian Regularized BP Neural Network. Materials, 15.","DOI":"10.20944\/preprints202205.0232.v1"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"106516","DOI":"10.1016\/j.ijfatigue.2021.106516","article-title":"Fatigue Fracture and Fatigue Life Assessment of Railway Wheel Using Non-linear Model for Fatigue Crack Growth","volume":"153","author":"Berto","year":"2021","journal-title":"Int. J. Fatigue"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10704-021-00526-x","article-title":"Recent Advances on Size Effect in Metal Fatigue under Defects: A Review","volume":"234","author":"Zhu","year":"2022","journal-title":"Int. J. Fract."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"956473","DOI":"10.1155\/2014\/956473","article-title":"A State-of-the-Art Review on Fatigue Life Assessment of Steel Bridges","volume":"2014","author":"Ye","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107013","DOI":"10.1016\/j.engfailanal.2022.107013","article-title":"On the Need to Evaluate the Probabilistic of Fatigue Life Assessment of Random Strain Loading Considering Load Sequence Effects","volume":"145","author":"Chin","year":"2023","journal-title":"Eng. Fail. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Civalek, \u00d6., Uzun, B., and Yayl\u0131, M.\u00d6. (2022). Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-\/Micro-Rods with Deformable Boundary Conditions. Materials, 15.","DOI":"10.3390\/ma15196803"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Shi, F., Fantuzzi, N., Trovalusci, P., Li, Y., and Wei, Z. (2022). Stress Field Evaluation in Orthotropic Microstructured Composites with Holes as Cosserat Continuum. Materials, 15.","DOI":"10.3390\/ma15186196"},{"key":"ref_10","first-page":"1","article-title":"Nonlinear Vibrations of Gradient and Nonlocal Elastic Nano-Bars","volume":"15","author":"Moghtaderi","year":"2020","journal-title":"Mech. Based Des. Struct. Mach."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1007\/s42452-020-03217-9","article-title":"Applications of the Local Gradient Elasticity to the Description of the Size Effect of Shear Modulus","volume":"2","author":"Hrytsyna","year":"2020","journal-title":"SN Appl. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Tang, F., He, S., Shi, S., Xue, S., Dong, F., and Liu, S. (2022). Analysis of Size-Dependent Linear Static Bending, Buckling, and Free Vibration Based on a Modified Couple Stress Theory. Materials, 15.","DOI":"10.3390\/ma15217583"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Krysko, A.v., Papkova, I.v., Rezchikov, A.F., and Krysko, V.A. (2022). A New Mathematical Model of Functionally Graded Porous Euler\u2013Bernoulli Nanoscaled Beams Taking into Account Some Types of Nonlinearities. Materials, 15.","DOI":"10.3390\/ma15207186"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Eringen, A.C. (1984). Theory of Nonlocal Elasticity and Some Applications, Princeton University.","DOI":"10.21236\/ADA145201"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"035011","DOI":"10.1088\/2053-1591\/abe3c6","article-title":"Nonlinear Flexure Mechanics of Beams: Stress Gradient and Nonlocal Integral Theory","volume":"8","author":"Fazlali","year":"2021","journal-title":"Mater. Res. Express"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1016\/j.ijengsci.2010.09.020","article-title":"Nonlocal Nonlinear Formulations for Bending of Classical and Shear Deformation Theories of Beams and Plates","volume":"48","author":"Reddy","year":"2010","journal-title":"Int. J. Eng. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1016\/j.ijengsci.2007.04.004","article-title":"Nonlocal Theories for Bending, Buckling and Vibration of Beams","volume":"45","author":"Reddy","year":"2007","journal-title":"Int. J. Eng. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"023511","DOI":"10.1063\/1.2833431","article-title":"Nonlocal Continuum Theories of Beams for the Analysis of Carbon Nanotubes","volume":"103","author":"Reddy","year":"2008","journal-title":"J. Appl. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ni, Z., Xiong, T., Lei, J., Wang, L., Gao, T., Yu, J., and Mi, C. (2022). Life Prediction Method of Dissimilar Lightweight Materials Welded Joints with Precrack under Coupled Impact-Fatigue Loading. Materials, 15.","DOI":"10.3390\/ma15145077"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Choi, W.K., Ha, S., Kim, J.C., Park, J.C., Gong, A., and Kim, T.W. (2022). Oxidation Damage Evolution in Low-Cycle Fatigue Life of Niobium-Stabilized Austenitic Stainless Steel. Materials, 15.","DOI":"10.3390\/ma15124073"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Strohmann, T., Breitbarth, E., Besel, M., Zaunschirm, S., Witulski, T., and Requena, G. (2022). Damage Mechanisms and Anisotropy of an AA7010-T7452 Open-Die Forged Alloy: Fatigue Crack Propagation. Materials, 15.","DOI":"10.3390\/ma15113771"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sheng, J., Yu, Z., Dou, G., and Liu, H. (2022). Fatigue Damage Behaviors of TRC-Strengthened RC Beams. Materials, 15.","DOI":"10.3390\/ma15155113"},{"key":"ref_23","first-page":"1512","article-title":"Nonlocal Damage Theory","volume":"113","year":"1987","journal-title":"J. Eng. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1115\/1.3173674","article-title":"Nonlocal Continuum Damage, Localization Instability and Convergence","volume":"55","author":"Bazant","year":"1988","journal-title":"J. Appl. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4133","DOI":"10.1016\/S0020-7683(97)00306-5","article-title":"Nonlocal Models for Damage and Fracture: Comparison of Approaches","volume":"35","year":"1998","journal-title":"Int. J. Solids Struct."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1002\/1097-0207(20001230)49:12<1547::AID-NME16>3.0.CO;2-D","article-title":"Gradient-Enhanced Damage Modelling of High-Cycle Fatigue","volume":"49","author":"Peerlings","year":"2000","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"de Borst, R., Geers, M.G.D., Peerlings, R.H.J., and Benallal, A. (1998). Some Remarks on Gradient and Nonlocal Damage Theories, Elsevier.","DOI":"10.1016\/S0922-5382(98)80044-5"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7723","DOI":"10.1016\/S0020-7683(01)00087-7","article-title":"A Critical Comparison of Nonlocal and Gradient-Enhanced Softening Continua","volume":"38","author":"Peerlings","year":"2001","journal-title":"Int. J. Solids Struct."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0997-7538(02)01211-1","article-title":"Localisation Issues in Local and Nonlocal Continuum Approaches to Fracture","volume":"21","author":"Peerlings","year":"2002","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3621","DOI":"10.1016\/S0020-7683(03)00144-6","article-title":"A Symmetric Nonlocal Damage Theory","volume":"40","author":"Borino","year":"2003","journal-title":"Int. J. 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Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/j.mechmat.2006.11.001","article-title":"Elasto-Viscoplastic Nonlocal Damage Modelling of Thermal Fatigue in Anisotropic Lead-Free Solder","volume":"39","author":"Ubachs","year":"2007","journal-title":"Mech. Mater."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1539","DOI":"10.1016\/j.engfracmech.2006.09.012","article-title":"Nonlocal Anisotropic Damage Model and Related Computational Aspects for Quasi-Brittle Materials","volume":"74","author":"Desmorat","year":"2007","journal-title":"Eng. Fract. Mech."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.engfracmech.2006.06.009","article-title":"An Implicit Gradient Application to Fatigue of Sharp Notches and Weldments","volume":"74","author":"Tovo","year":"2007","journal-title":"Eng. Fract. Mech."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s10704-008-9297-0","article-title":"Integral Equations and Nonlocal Damage Theory: A Numerical Implementation Using the BDEM","volume":"157","author":"Mallardo","year":"2009","journal-title":"Int. J. Fract."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/13632469.2012.746211","article-title":"Predicting Ductile Crack Initiation of Steel Bridge Structures Due to Extremely Low-Cycle Fatigue Using Local and Non-Local Models","volume":"17","author":"Kang","year":"2013","journal-title":"J. Earthq. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s10704-017-0225-z","article-title":"A Nonlocal Damage Model for Plain Concrete Consistent with Cohesive Fracture","volume":"207","author":"Lorentz","year":"2017","journal-title":"Int. J. Fract."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/978-981-10-7149-2_4","article-title":"Using a Non-Local Elastic Damage Model to Predict the Fatigue Life of Asphalt Pavement Structure","volume":"Volume PartF3","author":"Nguyen","year":"2018","journal-title":"Lecture Notes in Mechanical Engineering"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"113371","DOI":"10.1016\/j.cma.2020.113371","article-title":"Stochastic Nonlocal Damage Analysis by a Machine Learning Approach","volume":"372","author":"Feng","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1111\/ffe.13135","article-title":"Overview of the Geometrical Influence on the Fatigue Strength of Steel Butt Welds by a Nonlocal Approach","volume":"43","author":"Livieri","year":"2020","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"106000","DOI":"10.1016\/j.ijfatigue.2020.106000","article-title":"Fatigue Strength of Aluminium Welded Joints by a Non-Local Approach","volume":"143","author":"Livieri","year":"2021","journal-title":"Int. J. Fatigue"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"105863","DOI":"10.1016\/j.ijfatigue.2020.105863","article-title":"Efficient Higher-Order Cycle Jump Integration of a Continuum Fatigue Damage Model","volume":"141","author":"Titscher","year":"2020","journal-title":"Int. J. Fatigue"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1080\/14786435.2020.1713412","article-title":"A Non-Local Damage Model for the Fatigue Behaviour of Metallic Polycrystals","volume":"100","author":"Mareau","year":"2020","journal-title":"Philos. Mag."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105823","DOI":"10.1016\/j.ijfatigue.2020.105823","article-title":"An Improved Methodology Based on Continuum Damage Mechanics and Stress Triaxiality to Capture the Constraint Effect during Fatigue Crack Propagation","volume":"140","author":"Pandey","year":"2020","journal-title":"Int. J. Fatigue"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s40430-020-02777-6","article-title":"Current Challenges in Modelling Vibrational Fatigue and Fracture of Structures: A Review","volume":"43","author":"Kamei","year":"2021","journal-title":"J. Braz. Soc. Mech. Sci. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"113761","DOI":"10.1016\/j.compstruct.2021.113761","article-title":"A Practical Approach for the Non-Local Simulation of Progressive Damage in Quasi-Isotropic Fibre-Reinforced Composite Laminates","volume":"265","author":"Reiner","year":"2021","journal-title":"Compos. Struct."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"107976","DOI":"10.1016\/j.engfracmech.2021.107976","article-title":"An IGA Based Nonlocal Gradient-Enhanced Damage Model for Failure Analysis of Cortical Bone","volume":"255","author":"Soni","year":"2021","journal-title":"Eng. Fract. Mech."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Huang, J.S., Liew, J.X., and Liew, K.M. (2021). Data-Driven Machine Learning Approach for Exploring and Assessing Mechanical Properties of Carbon Nanotube-Reinforced Cement Composites. Compos. Struct., 267.","DOI":"10.1016\/j.compstruct.2021.113917"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/0142-1123(93)90182-P","article-title":"Stress Field Intensity Approach for Predicting Fatigue Life","volume":"15","author":"Weixing","year":"1993","journal-title":"Int. J. Fatigue"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/S0142-1123(99)00046-8","article-title":"Application of a New Model Proposal for Fatigue Life Prediction on Notches and Key-Seats","volume":"21","author":"Qylafku","year":"1999","journal-title":"Int. J. Fatigue"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1046\/j.1460-2695.2002.00527.x","article-title":"A Non-Local Theory Applied to High Cycle Multiaxial Fatigue","volume":"25","author":"Morel","year":"2002","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1214","DOI":"10.1111\/j.1460-2695.2007.01190.x","article-title":"Non-Local Stress Approach for Fatigue Assessment Based on Weakest-Link Theory and Statistics of Extremes","volume":"30","author":"Wormsen","year":"2007","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1016\/j.ijfatigue.2006.10.033","article-title":"A New Non-Local Criterion in High-Cycle Multiaxial Fatigue for Non-Proportional Loadings","volume":"29","author":"Chamat","year":"2007","journal-title":"Int. J. Fatigue"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1016\/j.ijfatigue.2008.01.007","article-title":"Non-Local Area Approach to Fatigue Life Evaluation under Combined Reversed Bending and Torsion","volume":"30","author":"Karolczuk","year":"2008","journal-title":"Int. J. Fatigue"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.commatsci.2008.04.005","article-title":"Non-Local Stress Gradient Approach for Multiaxial Fatigue of Defective Material","volume":"44","author":"Karolczuk","year":"2008","journal-title":"Comput. Mater. Sci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1376","DOI":"10.1016\/j.ijfatigue.2011.05.003","article-title":"Fatigue Life Estimation under Variable Amplitude Bending Using the Non-Local Damage Parameter and Multisurface Plasticity Model","volume":"33","author":"Karolczuk","year":"2011","journal-title":"Int. J. Fatigue"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1016\/j.ijfatigue.2009.03.003","article-title":"Multiaxial Fatigue Damage Modelling at Macro Scale of Ti-6Al-4V Alloy","volume":"31","author":"Marmi","year":"2009","journal-title":"Int. J. Fatigue"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.ijfatigue.2011.09.018","article-title":"Method of Effective Stress for Fatigue: Part i\u2014A General Theory","volume":"37","author":"Zhang","year":"2012","journal-title":"Int. J. Fatigue"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.ijfatigue.2013.03.005","article-title":"Stress Gradient Effect on the Crack Nucleation Process of a Ti-6Al-4V Titanium Alloy under Fretting Loading: Comparison between Non-Local Fatigue Approaches","volume":"54","author":"Fouvry","year":"2013","journal-title":"Int. J. Fatigue"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/s40194-015-0248-x","article-title":"Fatigue Assessment of Welded Joints Using Stress Averaging and Critical Distance Approaches","volume":"59","author":"Baumgartner","year":"2015","journal-title":"Weld. World"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"107150","DOI":"10.1016\/j.engfracmech.2020.107150","article-title":"Probabilistic Fatigue Assessment of Notched Components under Size Effect Using Critical Distance Theory","volume":"235","author":"He","year":"2020","journal-title":"Eng. Fract. Mech."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"105685","DOI":"10.1016\/j.ijmecsci.2020.105685","article-title":"Probabilistic Framework for Fatigue Life Assessment of Notched Components under Size Effects","volume":"181","author":"Liao","year":"2020","journal-title":"Int. J. Mech. Sci."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1007\/s10409-020-00957-0","article-title":"Review on Fatigue Life Prediction Models of Welded Joint","volume":"36","author":"Kang","year":"2020","journal-title":"Acta Mech. Sin."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"105531","DOI":"10.1016\/j.ijfatigue.2020.105531","article-title":"The Fatigue Limit Prediction of Notched Components\u2014A Critical Review and Modified Stress Gradient Based Approach","volume":"135","author":"Mei","year":"2020","journal-title":"Int. J. Fatigue"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"107005","DOI":"10.1016\/j.ijfatigue.2022.107005","article-title":"Probabilistic Fatigue Assessment of Notched Components under Size Effect Using Generalized Weakest-Link Model","volume":"162","author":"Zhu","year":"2022","journal-title":"Int. J. Fatigue"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0022-5096(99)00029-0","article-title":"Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces","volume":"48","author":"Silling","year":"2000","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1016\/j.compstruc.2004.11.026","article-title":"A Meshfree Method Based on the Peridynamic Model of Solid Mechanics","volume":"83","author":"Silling","year":"2005","journal-title":"Comput. Struct."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1115\/IMECE2011-64236","article-title":"A Nonlocal Approach to Modeling Crack Nucleation in AA 7075-T651","volume":"Volume 8","author":"Littlewood","year":"2011","journal-title":"Mechanics of Solids, Structures and Fluids, Vibration, Acoustics and Wave Propagation"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1063\/1.4864863","article-title":"Non-Local Modeling and Simulation of Wave Propagation and Crack Growth","volume":"Volume 1581","author":"Martowicz","year":"2014","journal-title":"AIP Conference Proceedings"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.engfracmech.2016.05.008","article-title":"Validation of a Peridynamic Model for Fatigue Cracking","volume":"162","author":"Zhang","year":"2016","journal-title":"Eng. Fract. Mech."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.ijsolstr.2016.01.019","article-title":"A Peridynamic Theory for Linear Elastic Shells","volume":"84","author":"Chowdhury","year":"2016","journal-title":"Int. J. Solids Struct."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.tafmec.2019.03.010","article-title":"A Nonlocal Peridynamics Modeling Approach for Corrosion Damage and Crack Propagation","volume":"101","author":"Rokkam","year":"2019","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"107373","DOI":"10.1016\/j.engfracmech.2020.107373","article-title":"An Energy-Based Peridynamic Model for Fatigue Cracking","volume":"241","author":"Nguyen","year":"2021","journal-title":"Eng. Fract. Mech."},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Nguyen, C.T., Oterkus, S., and Oterkus, E. (2020). A Peridynamic-Based Machine Learning Model for One-Dimensional and Two-Dimensional Structures. Contin. Mech. Thermodyn.","DOI":"10.1007\/s00161-020-00905-0"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"4655","DOI":"10.1007\/s11831-021-09549-y","article-title":"A Review on the Developments of Peridynamics for Reinforced Concrete Structures","volume":"Volume 28","author":"Hattori","year":"2021","journal-title":"Archives of Computational Methods in Engineering"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"108681","DOI":"10.1016\/j.engfracmech.2022.108681","article-title":"Data-Driven Bond-Based Peridynamics with Nonlocal Influence Function for Crack Propagation","volume":"272","author":"Ma","year":"2022","journal-title":"Eng. Fract. Mech."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"3714","DOI":"10.1177\/1081286518803411","article-title":"Peridynamics Review","volume":"24","author":"Javili","year":"2019","journal-title":"Math. Mech. Solids"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0997-7538(01)01200-1","article-title":"Thermodynamics and Continuum Fracture Mechanics for Nonlocal-Elastic Plastic Materials","volume":"21","author":"Polizzotto","year":"2002","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Belnoue, J.P., Nguyen, G.D., and Korsunsky, A.M. (2009). Consistent Tangent Stiffness for Local-Nonlocal Damage Modelling of Metals. Procedia Engineering, Elsevier Ltd.","DOI":"10.1016\/j.proeng.2009.06.041"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.engfracmech.2015.06.030","article-title":"Formulation of Nonlocal Damage Models Based on Spectral Methods for Application to Complex Microstructures","volume":"147","author":"Boeff","year":"2015","journal-title":"Eng. Fract. Mech."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.ijfatigue.2016.04.028","article-title":"A Non-Local Approach Based on the Hypothesis of Damage Dissipation Potential Equivalence to the Effect of Stress Gradient in Fretting Fatigue","volume":"90","author":"Shen","year":"2016","journal-title":"Int. J. Fatigue"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"04017091","DOI":"10.1061\/(ASCE)ST.1943-541X.0001827","article-title":"Uniaxial Nonlocal Formulation for Geometric Nonlinearity\u2013Induced Necking and Buckling Localization in a Steel Bar","volume":"143","author":"Kolwankar","year":"2017","journal-title":"J. Struct. Eng"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/S0142-1123(03)00048-3","article-title":"A Volumetric Energy Based High Cycle Multiaxial Fatigue Citerion","volume":"25","author":"Banvillet","year":"2003","journal-title":"Int. J. Fatigue"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.ijfatigue.2012.12.013","article-title":"Non-Local Energy Based Fatigue Life Calculation Method under Multiaxial Variable Amplitude Loadings","volume":"54","author":"Saintier","year":"2013","journal-title":"Int. J. Fatigue"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.ijfatigue.2013.12.004","article-title":"Fatigue Life Estimation of Notched Elements with Use of Non-Local Volumetric Method","volume":"61","author":"Krzyzak","year":"2014","journal-title":"Int. J. Fatigue"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1016\/j.ijfatigue.2016.12.008","article-title":"Non-Linear Incremental Fatigue Damage Calculation for Multiaxial Non-Proportional Histories","volume":"100","author":"Meggiolaro","year":"2017","journal-title":"Int. J. Fatigue"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.ijfatigue.2017.03.021","article-title":"Fatigue Crack Growth under Large Scale Yielding Condition: The Need of a Characteristic Length Scale","volume":"102","author":"Maurel","year":"2017","journal-title":"Int. J. Fatigue"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"952","DOI":"10.1016\/j.ijfatigue.2008.09.004","article-title":"An Engineering Model for Low Cycle Fatigue Life Based on a Partition of Energy and Micro-Crack Growth","volume":"31","author":"Maurel","year":"2009","journal-title":"Int. J. Fatigue"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.ijfatigue.2019.06.003","article-title":"A Mixed Strain Rate and Energy Based Fatigue Criterion for Short Fiber Reinforced Thermoplastics","volume":"127","author":"Raphael","year":"2019","journal-title":"Int. J. Fatigue"},{"key":"ref_92","doi-asserted-by":"crossref","unstructured":"Maurel, V., Chiaruttini, V., Koster, A., and Missoum-Benziane, D. (2022). Fatigue Crack Growth under Large Scale Yielding Condition: A Tool Based on Explicit Crack Growth. J. Theor. Comput. Appl. Mech., Available online: https:\/\/hal.science\/hal-03628416v2.","DOI":"10.46298\/jtcam.9296"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"4827","DOI":"10.1016\/S0045-7825(02)00401-2","article-title":"Computational Mechanics of Fatigue and Life Predictions for Composite Materials and Structures","volume":"191","author":"Fish","year":"2002","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1080\/15376490500259319","article-title":"A Nonlocal Multiscale Fatigue Model","volume":"12","author":"Fish","year":"2005","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"121","DOI":"10.21278\/brod72407","article-title":"On Multiscale Damage Modelling of Heterogeneous Materials Using Nonlocal Continuum Theory","volume":"72","author":"Putar","year":"2021","journal-title":"Brodogradnja"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1016\/j.ijfatigue.2010.02.003","article-title":"Probabilistic Framework for a Microstructure-Sensitive Fatigue Notch Factor","volume":"32","author":"Owolabi","year":"2010","journal-title":"Int. J. Fatigue"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1016\/j.ijfatigue.2010.01.003","article-title":"Microstructure-Sensitive Computational Modeling of Fatigue Crack Formation","volume":"32","author":"McDowell","year":"2010","journal-title":"Int. J. Fatigue"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/j.ijsolstr.2015.12.026","article-title":"A Non-Local 3D Lattice Particle Framework for Elastic Solids","volume":"81","author":"Chen","year":"2016","journal-title":"Int. J. Solids Struct."},{"key":"ref_99","doi-asserted-by":"crossref","unstructured":"Gao, Y., Yao, H., Wei, H., and Liu, Y. (2020). Physics-Based Deep Learning for Probabilistic Fracture Analysis of Composite Materials. AIAA Scitech 2020 Forum, American Institute of Aeronautics and Astronautics Inc. (AIAA).","DOI":"10.2514\/6.2020-1860"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1111\/ffe.12329","article-title":"Non-Local High Cycle Fatigue Strength Criterion for Metallic Materials with Corrosion Defects","volume":"38","author":"Saintier","year":"2015","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.engfracmech.2018.05.041","article-title":"A Non-Local Approach for Probabilistic Assessment of LCF Life Based on Optimized Effective-Damage-Parameter","volume":"199","author":"Li","year":"2018","journal-title":"Eng. Fract. Mech."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1134\/S1029959919060079","article-title":"Nonlocal and Gradient Fracture Criteria for Quasi-Brittle Materials under Compression","volume":"22","author":"Suknev","year":"2019","journal-title":"Phys. Mesomech."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.ijplas.2014.08.001","article-title":"A Nonlocal Coupled Damage-Plasticity Model for the Analysis of Ductile Failure","volume":"64","author":"Nguyen","year":"2015","journal-title":"Int. J. Plast."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"3129","DOI":"10.1007\/s11012-016-0560-6","article-title":"GraFEA: A Graph-based Finite Element approach for the Study of Damage and Fracture in Brittle Materials","volume":"51","author":"Khodabakhshi","year":"2016","journal-title":"Meccanica"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"3593","DOI":"10.1007\/s00707-019-02479-8","article-title":"A Nonlocal Fracture Criterion and Its Effect on the Mesh Dependency of GraFEA","volume":"230","author":"Khodabakhshi","year":"2019","journal-title":"Acta Mech."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"115727","DOI":"10.1016\/j.cma.2022.115727","article-title":"A Robust Moore\u2013Penrose Pseudoinverse-Based Static Finite-Element Solver for Simulating Non-Local Fracture in Solids","volume":"403","author":"Alebrahim","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"105097","DOI":"10.1016\/j.jmps.2022.105097","article-title":"Modeling Impact Fracture in a Quasi-Brittle Solids Using a 3D Nonlocal Graph-Based Finite Element Analysis: Theory, Finite Element Simulations, and Experimental Verification","volume":"170","author":"Shin","year":"2023","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"101618","DOI":"10.1016\/j.eml.2022.101618","article-title":"On Simulating Impact Fracture in High-Strength Concrete Using GraFEA","volume":"52","author":"Shin","year":"2022","journal-title":"Extrem. Mech. Lett."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1016\/j.cma.2019.05.032","article-title":"Fracture of Viscoelastic Materials: FEM Implementation of a Non-local & Rate Form-based Finite-deformation Constitutive Theory","volume":"354","author":"Thamburaja","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1080\/15376494.2020.1716414","article-title":"Numerical Simulations of Damage and Fracture in Viscoelastic Solids Using a Nonlocal Fracture Criterion","volume":"27","author":"Sarah","year":"2020","journal-title":"Mech. Adv. Mater. 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