{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T10:03:17Z","timestamp":1780394597066,"version":"3.54.1"},"reference-count":93,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:00:00Z","timestamp":1780358400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:00:00Z","timestamp":1780358400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Postdoctoral Fellowship Program of CPSF","award":["GZC20241796"],"award-info":[{"award-number":["GZC20241796"]}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2025M771865"],"award-info":[{"award-number":["2025M771865"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Foreign Expert Project of China","award":["H20250282"],"award-info":[{"award-number":["H20250282"]}]},{"name":"the Fundamental Research Funds for the Central Universities","award":["2682025GH012"],"award-info":[{"award-number":["2682025GH012"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1007\/s00366-026-02348-8","type":"journal-article","created":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T08:22:07Z","timestamp":1780388527000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An area-based peridynamic theory for failure analysis"],"prefix":"10.1007","volume":"42","author":[{"given":"Mengjie","family":"Zheng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiongwu","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhanhui","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongsheng","family":"Mao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,2]]},"reference":[{"issue":"20\u201322","key":"2348_CR1","doi-asserted-by":"publisher","first-page":"2899","DOI":"10.1016\/0020-7683(95)00255-3","volume":"33","author":"GT Camacho","year":"1996","unstructured":"Camacho GT, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33(20\u201322):2899\u20132938","journal-title":"Int J Solids Struct"},{"issue":"9","key":"2348_CR2","doi-asserted-by":"publisher","first-page":"1397","DOI":"10.1016\/0022-5096(94)90003-5","volume":"42","author":"XP Xu","year":"1994","unstructured":"Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397\u20131434","journal-title":"J Mech Phys Solids"},{"issue":"15","key":"2348_CR3","doi-asserted-by":"publisher","first-page":"2221","DOI":"10.1002\/nme.849","volume":"57","author":"C Zi","year":"2003","unstructured":"Zi C, Belytschko T (2003) New crack-tip elements for xfem and applications to cohesive cracks. Int J Numer Methods Eng 57(15):2221\u20132240","journal-title":"Int J Numer Methods Eng"},{"issue":"5","key":"2348_CR4","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1002\/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S","volume":"45","author":"T Belytschko","year":"1999","unstructured":"Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5):601\u2013620","journal-title":"Int J Numer Methods Eng"},{"issue":"1","key":"2348_CR5","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1002\/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J","volume":"46","author":"N Mo\u00ebs","year":"1999","unstructured":"Mo\u00ebs N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131\u2013150","journal-title":"Int J Numer Methods Eng"},{"key":"2348_CR6","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1016\/j.enganabound.2019.03.028","volume":"104","author":"J Chen","year":"2019","unstructured":"Chen J, Zhou X (2019) The enhanced extended finite element method for the propagation of complex branched cracks. Eng Anal Bound Elem 104:46\u201362","journal-title":"Eng Anal Bound Elem"},{"key":"2348_CR7","doi-asserted-by":"publisher","first-page":"744","DOI":"10.1016\/j.cma.2016.11.029","volume":"315","author":"X Li","year":"2017","unstructured":"Li X, Chen J (2017) An extended cohesive damage model for simulating arbitrary damage propagation in engineering materials. Comput Methods Appl Mech Eng 315:744\u2013759","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"45\u201348","key":"2348_CR8","doi-asserted-by":"publisher","first-page":"2765","DOI":"10.1016\/j.cma.2010.04.011","volume":"199","author":"C Miehe","year":"2010","unstructured":"Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45\u201348):2765\u20132778","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2","key":"2348_CR9","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s00466-014-1109-y","volume":"55","author":"M Ambati","year":"2015","unstructured":"Ambati M, Gerasimov T, Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383\u2013405","journal-title":"Comput Mech"},{"key":"2348_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.tafmec.2018.09.015","volume":"102","author":"Y Zhang","year":"2019","unstructured":"Zhang Y, Zhuang X (2019) Cracking elements method for dynamic brittle fracture. Theoret Appl Fract Mech 102:1\u20139","journal-title":"Theoret Appl Fract Mech"},{"key":"2348_CR11","doi-asserted-by":"publisher","first-page":"2462","DOI":"10.1002\/nme.6315","volume":"121","author":"Y Zhang","year":"2020","unstructured":"Zhang Y, Mang HA (2020) Global cracking elements: a novel tool for Galerkin-based approaches simulating quasi-brittle fracture. Int J Numer Meth Eng 121:2462\u20132480","journal-title":"Int J Numer Meth Eng"},{"key":"2348_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2021.103573","volume":"195","author":"Y Zhang","year":"2021","unstructured":"Zhang Y, Huang J, Yuan Y, Mang HA (2021) Cracking elements method with a dissipation-based arc-length approach. Finite Elem Anal Des 195:103573","journal-title":"Finite Elem Anal Des"},{"key":"2348_CR13","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/j.mechmat.2014.12.010","volume":"83","author":"Y Yao","year":"2015","unstructured":"Yao Y, Liu L, Keer LM (2015) Pore pressure cohesive zone modeling of hydraulic fracture in quasi-brittle rocks. Mech Mater 83:17\u201329","journal-title":"Mech Mater"},{"key":"2348_CR14","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/j.piutam.2014.01.020","volume":"10","author":"A Needleman","year":"2014","unstructured":"Needleman A (2014) Some issues in cohesive surface modeling. Procedia IUTAM 10:221\u2013246","journal-title":"Procedia IUTAM"},{"issue":"3","key":"2348_CR15","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/S0045-7949(02)00431-5","volume":"81","author":"BL Karihaloo","year":"2003","unstructured":"Karihaloo BL, Xiao Q (2003) Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review. Comput Struct 81(3):119\u2013129","journal-title":"Comput Struct"},{"issue":"1","key":"2348_CR16","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0022-5096(99)00029-0","volume":"48","author":"SA Silling","year":"2000","unstructured":"Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175\u2013209","journal-title":"J Mech Phys Solids"},{"issue":"10","key":"2348_CR17","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/S0065-2156(10)44002-8","volume":"44","author":"SA Silling","year":"2010","unstructured":"Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44(10):73\u2013168","journal-title":"Adv Appl Mech"},{"issue":"17\u201318","key":"2348_CR18","doi-asserted-by":"publisher","first-page":"1526","DOI":"10.1016\/j.compstruc.2004.11.026","volume":"83","author":"SA Silling","year":"2005","unstructured":"Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17\u201318):1526\u20131535","journal-title":"Comput Struct"},{"key":"2348_CR19","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10659-007-9125-1","volume":"88","author":"SA Silling","year":"2007","unstructured":"Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151\u2013184","journal-title":"J Elast"},{"key":"2348_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.tust.2020.103289","volume":"97","author":"C Gao","year":"2020","unstructured":"Gao C, Zhou Z, Li Z, Li L, Cheng S (2020) Peridynamics simulation of surrounding rock damage characteristics during tunnel excavation. Tunn Undergr Space Technol 97:103289","journal-title":"Tunn Undergr Space Technol"},{"key":"2348_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2021.108088","volume":"258","author":"M Qin","year":"2021","unstructured":"Qin M, Yang D, Chen W, Yang S (2021) Hydraulic fracturing model of a layered rock mass based on peridynamics. Eng Fract Mech 258:108088","journal-title":"Eng Fract Mech"},{"issue":"1","key":"2348_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.compgeo.2021.104037","volume":"132","author":"Z Zhou","year":"2021","unstructured":"Zhou Z, Li Z, Gao C, Zhang D, Bai S (2021) Peridynamic micro-elastoplastic constitutive model and its application in the failure analysis of rock masses. Comput Geotech 132(1):104037","journal-title":"Comput Geotech"},{"key":"2348_CR23","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1016\/j.engfracmech.2016.06.013","volume":"163","author":"Y Wang","year":"2016","unstructured":"Wang Y, Zhou X, Xu X (2016) Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng Fract Mech 163:248\u2013273","journal-title":"Eng Fract Mech"},{"issue":"2","key":"2348_CR24","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/j.ijnonlinmec.2004.08.004","volume":"40","author":"SA Silling","year":"2005","unstructured":"Silling SA, Bobaru F (2005) Peridynamic modeling of membranes and fibers. Int J Non-Linear Mech 40(2):395\u2013409","journal-title":"Int J Non-Linear Mech"},{"key":"2348_CR25","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.engfracmech.2016.11.004","volume":"169","author":"A Yaghoobi","year":"2017","unstructured":"Yaghoobi A, Chorzepa MG (2017) Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework. Eng Fract Mech 169:238\u2013250","journal-title":"Eng Fract Mech"},{"key":"2348_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.compgeo.2021.103998","volume":"133","author":"Y Jin","year":"2021","unstructured":"Jin Y, Li L, Jia Y, Shao J, Burlion N (2021) Numerical study of shrinkage and heating induced cracking in concrete materials and influence of inclusion stiffness with peridynamics method. Comput Geotech 133:103998","journal-title":"Comput Geotech"},{"key":"2348_CR27","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1002\/suco.202000113","volume":"22","author":"C Xu","year":"2021","unstructured":"Xu C, Yuan Y, Zhang Y, Xue Y (2021) Peridynamic modeling of prefabricated beams post-cast with steel-fiber reinforced high-strength concrete. Struct Concr 22:445\u2013456","journal-title":"Struct Concr"},{"key":"2348_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.engstruct.2021.112748","volume":"244","author":"N Zhang","year":"2021","unstructured":"Zhang N, Gu Q, Huang S, Xue X, Li S (2021) A practical bond-based peridynamic modeling of reinforced concrete structures. Eng Struct 244:112748","journal-title":"Eng Struct"},{"key":"2348_CR29","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1016\/j.compstruct.2018.09.034","volume":"207","author":"Y Gao","year":"2019","unstructured":"Gao Y, Oterkus S (2019) Fully coupled thermomechanical analysis of laminated composites by using ordinary state based peridynamic theory. Compos Struct 207:397\u2013424","journal-title":"Compos Struct"},{"key":"2348_CR30","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1016\/j.compstruct.2015.05.079","volume":"132","author":"YL Hu","year":"2015","unstructured":"Hu YL, Carvalho N, Madenci E (2015) Peridynamic modeling of delamination growth in composite laminates. Compos Struct 132:610\u2013620","journal-title":"Compos Struct"},{"key":"2348_CR31","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.cma.2012.01.016","volume":"217\u2013220","author":"W Hu","year":"2012","unstructured":"Hu W, Ha YD, Bobaru F (2012) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Methods Appl Mech Eng 217\u2013220:247\u2013261","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR32","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1016\/j.compstruct.2013.10.018","volume":"108","author":"Y Hu","year":"2014","unstructured":"Hu Y, Yu Y, Wang H (2014) Peridynamic analytical method for progressive damage in notched composite laminates. Compos Struct 108:801\u2013810","journal-title":"Compos Struct"},{"key":"2348_CR33","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.mechrescom.2016.06.006","volume":"76","author":"U Galvanetto","year":"2016","unstructured":"Galvanetto U, Mudric T, Shojaei A, Zaccariotto M (2016) An effective way to couple fem meshes and peridynamics grids for the solution of static equilibrium problems. Mech Res Commun 76:41\u201347","journal-title":"Mech Res Commun"},{"key":"2348_CR34","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1016\/j.cma.2018.11.028","volume":"346","author":"T Ni","year":"2019","unstructured":"Ni T, Zaccariotto M, Zhu QZ, Galvanetto U (2019) Static solution of crack propagation problems in peridynamics. Comput Methods Appl Mech Eng 346:126\u2013151","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR35","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1016\/j.euromechsol.2018.09.007","volume":"73","author":"Y Wang","year":"2019","unstructured":"Wang Y, Zhou X, Kou M (2019) An improved coupled thermo-mechanic bond-based peridynamic model for cracking behaviors in brittle solids subjected to thermal shocks. Eur J Mech A Solids 73:282\u2013305","journal-title":"Eur J Mech A Solids"},{"issue":"1","key":"2348_CR36","doi-asserted-by":"publisher","first-page":"45","DOI":"10.2140\/jomms.2012.7.45","volume":"7","author":"E Oterkus","year":"2012","unstructured":"Oterkus E, Madenci E (2012) Peridynamic analysis of fiber reinforced composite materials. J Mech Mater Struct 7(1):45\u201384","journal-title":"J Mech Mater Struct"},{"issue":"12\u201313","key":"2348_CR37","doi-asserted-by":"publisher","first-page":"1250","DOI":"10.1016\/j.nucengdes.2006.10.002","volume":"237","author":"W Gerstle","year":"2007","unstructured":"Gerstle W, Sau N, Silling S (2007) Peridynamic modeling of concrete structures. Nucl Eng Des 237(12\u201313):1250\u20131258","journal-title":"Nucl Eng Des"},{"key":"2348_CR38","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.ijsolstr.2015.01.018","volume":"59","author":"SR Chowdhury","year":"2015","unstructured":"Chowdhury SR, Rahaman M, Roy D, Sundaram N (2015) A micropolar peridynamic theory in linear elasticity. Int J Solids Struct 59:171\u2013182","journal-title":"Int J Solids Struct"},{"issue":"1","key":"2348_CR39","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1016\/j.cma.2018.11.001","volume":"345","author":"H Chen","year":"2019","unstructured":"Chen H (2019) A comparison study on peridynamic models using irregular non-uniform spatial discretization. Comput Methods Appl Mech Eng 345(1):539\u2013554","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2019.106613","volume":"218","author":"E Madenci","year":"2019","unstructured":"Madenci E, Dorduncu M, Phan N, Gu X (2019) Weak form of bond-associated non-ordinary state-based peridynamics free of zero energy modes with uniform or non-uniform discretization. Eng Fract Mech 218:106613","journal-title":"Eng Fract Mech"},{"key":"2348_CR41","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2021.102930","volume":"113","author":"Y Zhang","year":"2021","unstructured":"Zhang Y, Yang X, Wang X, Zhuang X (2021) A micropolar peridynamic model with non-uniform horizon for static damage of solids considering different nonlocal enhancements. Theoret Appl Fract Mech 113:102930","journal-title":"Theoret Appl Fract Mech"},{"issue":"15","key":"2348_CR42","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.engfracmech.2018.02.006","volume":"197","author":"Y Hu","year":"2018","unstructured":"Hu Y, Chen H, Spencer BW, Madenci E (2018) Thermomechanical peridynamic analysis with irregular non-uniform domain discretization. Eng Fract Mech 197(15):92\u2013113","journal-title":"Eng Fract Mech"},{"key":"2348_CR43","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2020.102877","volume":"112","author":"TN Cong","year":"2021","unstructured":"Cong TN, Oterkus S (2021) Ordinary state-based peridynamics for geometrically nonlinear analysis of plates. Theoret Appl Fract Mech 112:102877","journal-title":"Theoret Appl Fract Mech"},{"key":"2348_CR44","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2019.106623","volume":"219","author":"TN Cong","year":"2019","unstructured":"Cong TN, Oterkus S (2019) Peridynamics for the thermomechanical behavior of shell structures. Eng Fract Mech 219:106623","journal-title":"Eng Fract Mech"},{"key":"2348_CR45","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1016\/j.ijsolstr.2015.04.040","volume":"69\u201370","author":"C Diyaroglu","year":"2015","unstructured":"Diyaroglu C, Oterkus E, Oterkus S, Madenci E (2015) Peridynamics for bending of beams and plates with transverse shear deformation. Int J Solids Struct 69\u201370:152\u2013168","journal-title":"Int J Solids Struct"},{"issue":"2","key":"2348_CR46","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1177\/1081286519873694","volume":"25","author":"U Yolum","year":"2020","unstructured":"Yolum U, G\u00fcler MA (2020) On the peridynamic formulation for an orthotropic Mindlin plate under bending. Math Mech Solids 25(2):263\u2013287","journal-title":"Math Mech Solids"},{"key":"2348_CR47","doi-asserted-by":"publisher","first-page":"416","DOI":"10.1016\/j.cma.2018.05.007","volume":"339","author":"G Hattori","year":"2018","unstructured":"Hattori G, Trevelyan J, Coombs WM (2018) A non-ordinary state-based peridynamics framework for anisotropic materials. Comput Methods Appl Mech Eng 339:416\u2013442","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR48","doi-asserted-by":"crossref","unstructured":"Dipasquale D, Shojaei A, Yooyen S (2020) A novel stress tensor-based failure criterion for peridynamics. In: 1st IAAI - conference 2020, pp 1\u20134","DOI":"10.3390\/proceedings2019039023"},{"key":"2348_CR49","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijfatigue.2021.106638","volume":"156","author":"H Li","year":"2022","unstructured":"Li H, Hao Z, Li P, Li X, Zhang D (2022) A low cycle fatigue cracking simulation method of non-ordinary state-based peridynamics. Int J Fatigue 156:106638","journal-title":"Int J Fatigue"},{"issue":"5","key":"2348_CR50","doi-asserted-by":"publisher","first-page":"1186","DOI":"10.1016\/j.ijsolstr.2008.10.029","volume":"46","author":"TL Warren","year":"2009","unstructured":"Warren TL, Silling SA, Askari A, Weckner O, Epton MA, Xu J (2009) A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int J Solids Struct 46(5):1186\u20131195","journal-title":"Int J Solids Struct"},{"key":"2348_CR51","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1016\/j.tafmec.2019.03.006","volume":"101","author":"Y Shou","year":"2019","unstructured":"Shou Y, Zhou X, Berto F (2019) 3d numerical simulation of initiation, propagation and coalescence of cracks using the extended non-ordinary state-based peridynamics. Theor Appl Fract Mech 101:254\u2013268","journal-title":"Theor Appl Fract Mech"},{"key":"2348_CR52","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2022.103393","volume":"120","author":"X Cao","year":"2022","unstructured":"Cao X, Qin X, Li H, Shang S, Li S, Liu H (2022) Non-ordinary state-based peridynamic fatigue modelling of composite laminates with arbitrary fibre orientation. Theor Appl Fract Mech 120:103393","journal-title":"Theor Appl Fract Mech"},{"issue":"16","key":"2348_CR53","doi-asserted-by":"publisher","first-page":"4033","DOI":"10.1002\/nme.6691","volume":"122","author":"S Jin","year":"2021","unstructured":"Jin S, Hwang YK, Hong J (2021) Coupling of non-ordinary state-based peridynamics and finite element method with reduced boundary effect. Int J Numer Methods Eng 122(16):4033\u20134054","journal-title":"Int J Numer Methods Eng"},{"key":"2348_CR54","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/j.ijrmms.2016.09.010","volume":"89","author":"XP Zhou","year":"2016","unstructured":"Zhou XP, Wang YT (2016) Numerical simulation of crack propagation and coalescence in pre-cracked rock-like Brazilian disks using the non-ordinary state-based peridynamics. Int J Rock Mech Min Sci 89:235\u2013249","journal-title":"Int J Rock Mech Min Sci"},{"key":"2348_CR55","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1007\/s00366-022-01620-x","volume":"39","author":"S Yang","year":"2023","unstructured":"Yang S, Gu X, Xia X, Zhang Q (2023) Explosion damage analysis of concrete structure with bond-associated non-ordinary state-based peridynamics. Eng Comput 39:607\u2013624","journal-title":"Eng Comput"},{"issue":"6","key":"2348_CR56","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1007\/s00466-022-02148-z","volume":"69","author":"DL Tian","year":"2022","unstructured":"Tian DL, Zhou XP (2022) A viscoelastic model of geometry-constraint-based non-ordinary state-based peridynamics with progressive damage. Comput Mech 69(6):1413\u20131441","journal-title":"Comput Mech"},{"key":"2348_CR57","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1007\/s11012-019-00975-8","volume":"51","author":"M Asgari","year":"2019","unstructured":"Asgari M, Kouchakzadeh MA (2019) An equivalent Von Mises stress and corresponding equivalent plastic strain for elastic-plastic ordinary peridynamics. Meccanica 51:1001\u20131014","journal-title":"Meccanica"},{"issue":"11","key":"2348_CR58","doi-asserted-by":"publisher","first-page":"2809","DOI":"10.1007\/s11012-022-01600-x","volume":"57","author":"H Li","year":"2022","unstructured":"Li H, Hao Z, Li P, Li X, Zhang D (2022) Elastoplastic theory of finite deformation and its solution method for non-ordinary state-based peridynamics. Meccanica 57(11):2809\u20132820","journal-title":"Meccanica"},{"key":"2348_CR59","doi-asserted-by":"publisher","first-page":"1395","DOI":"10.1007\/s00366-021-01527-z","volume":"39","author":"T Zhang","year":"2023","unstructured":"Zhang T, Zhou XP, Qian QH (2023) Drucker-Prager plasticity model in the framework of OSB-PD theory with shear deformation. Eng Comput 39:1395\u20131414","journal-title":"Eng Comput"},{"key":"2348_CR60","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.ijmecsci.2019.06.008","volume":"159","author":"H Wang","year":"2019","unstructured":"Wang H, Xu Y, Huang D (2019) A non-ordinary state-based peridynamic formulation for thermo-visco-plastic deformation and impact fracture. Int J Mech Sci 159:336\u2013344","journal-title":"Int J Mech Sci"},{"key":"2348_CR61","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/j.ijimpeng.2015.06.019","volume":"87","author":"J Amani","year":"2016","unstructured":"Amani J, Oterkus E, Areias P, Zi G, Nguyen T, Rabczuk T (2016) A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87:83\u201394","journal-title":"Int J Impact Eng"},{"issue":"1","key":"2348_CR62","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/j.cma.2018.05.002","volume":"339","author":"P Li","year":"2018","unstructured":"Li P, Hao ZM, Zhen WQ (2018) A stabilized non-ordinary state-based peridynamic model. Comput Methods Appl Mech Eng 339(1):262\u2013280","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"2348_CR63","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.cma.2017.03.043","volume":"322","author":"SA Silling","year":"2017","unstructured":"Silling SA (2017) Stability of peridynamic correspondence material models and their particle discretizations. Comput Methods Appl Mech Eng 322(1):42\u201357","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"15","key":"2348_CR64","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.cma.2014.01.002","volume":"272","author":"MS Breitenfeld","year":"2014","unstructured":"Breitenfeld MS, Geubelle PH, Weckner O, Silling SA (2014) Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput Methods Appl Mech Eng 272(15):233\u2013250","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR65","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2022.108557","volume":"270","author":"FS Vieira","year":"2022","unstructured":"Vieira FS, Ara\u00fajo AL (2022) On the role of bond-associated stabilization and discretization on deformation and fracture in non-ordinary state-based peridynamics. Eng Fract Mech 270:108557","journal-title":"Eng Fract Mech"},{"key":"2348_CR66","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.mechrescom.2018.04.004","volume":"90","author":"H Chen","year":"2018","unstructured":"Chen H (2018) Bond-associated deformation gradients for peridynamic correspondence model. Mech Res Commun 90:34\u201341","journal-title":"Mech Res Commun"},{"issue":"6","key":"2348_CR67","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1002\/nme.5973","volume":"117","author":"HL Chen","year":"2019","unstructured":"Chen HL, Spencer BW (2019) Peridynamic bond-associated correspondence model: stability and convergence properties. Int J Numer Methods Eng 117(6):713\u2013727","journal-title":"Int J Numer Methods Eng"},{"key":"2348_CR68","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/j.ijsolstr.2018.06.015","volume":"150","author":"J Luo","year":"2018","unstructured":"Luo J, Sundararaghavan V (2018) Stress-point method for stabilizing zero-energy modes in non-ordinary state-based peridynamics. Int J Solids Struct 150:197\u2013207","journal-title":"Int J Solids Struct"},{"key":"2348_CR69","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.enganabound.2020.03.016","volume":"117","author":"H Cui","year":"2020","unstructured":"Cui H, Li C, Zheng H (2020) A higher-order stress point method for non-ordinary state-based peridynamics. Eng Anal Boundary Elem 117:104\u2013118","journal-title":"Eng Anal Boundary Elem"},{"key":"2348_CR70","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2023.103980","volume":"126","author":"Y Bie","year":"2023","unstructured":"Bie Y, Ren H, Yan H, Chen J (2023) The unified nonlocal peridynamics-based phase-field damage theory. Theor Appl Fract Mech 126:103980","journal-title":"Theor Appl Fract Mech"},{"key":"2348_CR71","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116730","volume":"420","author":"Y Bie","year":"2024","unstructured":"Bie Y, Ren H, Rabczuk T, Quoc Bui T, Wei Y (2024) The fully coupled thermo-mechanical dual-horizon peridynamic correspondence damage model for homogeneous and heterogeneous materials. Comput Methods Appl Mech Eng 420:116730","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR72","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.117225","volume":"430","author":"Y Bie","year":"2024","unstructured":"Bie Y, Ren H, Bui TQ, Madenci E, Rabczuk T, Wei Y (2024) Dual-horizon peridynamics modeling of coupled chemo-mechanical-damage for interface oxidation-induced cracking in thermal barrier coatings. Comput Methods Appl Mech Eng 430:117225","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR73","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1016\/j.apm.2022.05.025","volume":"109","author":"D Tian","year":"2022","unstructured":"Tian D, Zhou XP (2022) A novel kinematic-constraint-inspired non-ordinary state-based peridynamics. Appl Math Model 109:709\u2013740","journal-title":"Appl Math Model"},{"key":"2348_CR74","doi-asserted-by":"publisher","first-page":"4906","DOI":"10.2514\/1.J061453","volume":"60","author":"I Javaheri","year":"2022","unstructured":"Javaheri I, Luo J, Lakshmanan A, Sundararaghavan V (2022) Higher-order approximations for stabilizing zero-energy modes in non-ordinary state-based peridynamics models. AIAA J 60:4906\u20134922","journal-title":"AIAA J"},{"key":"2348_CR75","doi-asserted-by":"publisher","DOI":"10.1016\/j.soildyn.2022.107250","volume":"157","author":"L Wang","year":"2022","unstructured":"Wang L, Huang S, Gu Q, Sun B, Li S, Lin Z (2022) Simulation of highly nonlinear materials based on a stabilized non-ordinary state-based peridynamic model. Soil Dyn Earthq Eng 157:107250","journal-title":"Soil Dyn Earthq Eng"},{"key":"2348_CR76","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2021.107767","volume":"249","author":"M Braun","year":"2021","unstructured":"Braun M, Iv\u00e1\u00f1ez I, Ariza MP (2021) A numerical study of progressive damage in unidirectional composite materials using a 2d lattice model. Eng Fract Mech 249:107767","journal-title":"Eng Fract Mech"},{"key":"2348_CR77","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2022.111561","volume":"243","author":"C Meng","year":"2022","unstructured":"Meng C, Liu L (2022) Damage-augmented nonlocal lattice particle method for fracture simulation of solids. Int J Solids Struct 243:111561","journal-title":"Int J Solids Struct"},{"key":"2348_CR78","doi-asserted-by":"publisher","DOI":"10.1016\/j.compscitech.2020.108335","volume":"200","author":"M Braun","year":"2020","unstructured":"Braun M, Ariza M (2020) A progressive damage based lattice model for dynamic fracture of composite materials. Compos Sci Technol 200:108335","journal-title":"Compos Sci Technol"},{"key":"2348_CR79","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1016\/j.cma.2014.04.002","volume":"276","author":"M Ghajari","year":"2014","unstructured":"Ghajari M, Iannucci L, Curtis P (2014) A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Methods Appl Mech Eng 276:431\u2013452","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR80","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2021.106413","volume":"199","author":"DL Tian","year":"2021","unstructured":"Tian DL, Zhou XP (2021) A continuum-kinematics-inspired peridynamic model of anisotropic continua: elasticity, damage, and fracture. Int J Mech Sci 199:106413","journal-title":"Int J Mech Sci"},{"key":"2348_CR81","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1016\/j.ijmecsci.2017.05.019","volume":"128\u2013129","author":"Y Wang","year":"2017","unstructured":"Wang Y, Zhou X, Shou Y (2017) The modeling of crack propagation and coalescence in rocks under uniaxial compression using the novel conjugated bond-based peridynamics. Int J Mech Sci 128\u2013129:614\u2013643","journal-title":"Int J Mech Sci"},{"key":"2348_CR82","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.ijsolstr.2017.10.022","volume":"134","author":"Y Wang","year":"2018","unstructured":"Wang Y, Zhou X, Wang Y, Shou Y (2018) A 3-d conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids. Int J Solids Struct 134:89\u2013115","journal-title":"Int J Solids Struct"},{"key":"2348_CR83","doi-asserted-by":"publisher","first-page":"1161","DOI":"10.1007\/s11440-018-0709-7","volume":"14","author":"YT Wang","year":"2019","unstructured":"Wang YT, Zhou XP, Kou MM (2019) Three-dimensional numerical study on the failure characteristics of intermittent fissures under compressive-shear loads. Acta Geotech 14:1161\u20131193","journal-title":"Acta Geotech"},{"issue":"1","key":"2348_CR84","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruct.2020.112760","volume":"253","author":"YL Hu","year":"2020","unstructured":"Hu YL, Yu Y, Madenci E (2020) Peridynamic modeling of composite laminates with material coupling and transverse shear deformation. Compos Struct 253(1):112760","journal-title":"Compos Struct"},{"key":"2348_CR85","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruct.2022.116182","volume":"301","author":"YL Hu","year":"2022","unstructured":"Hu YL, Wang JY, Madenci E, Zhou M, Yu Y (2022) Peridynamic micromechanical model for damage mechanisms in composites. Compos Struct 301:116182","journal-title":"Compos Struct"},{"key":"2348_CR86","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-8465-3","volume-title":"Peridynamic theory and its applications","author":"E Madenci","year":"2014","unstructured":"Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, New York, NY"},{"key":"2348_CR87","first-page":"245","volume":"1","author":"P Underwood","year":"1983","unstructured":"Underwood P (1983) Dynamic relaxation, computational methods for transient analysis. Am Soc Mech Eng 1:245\u2013265","journal-title":"Am Soc Mech Eng"},{"issue":"3","key":"2348_CR88","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/j.tafmec.2010.08.001","volume":"53","author":"B Kilic","year":"2010","unstructured":"Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53(3):194\u2013204","journal-title":"Theor Appl Fract Mech"},{"issue":"4","key":"2348_CR89","first-page":"504","volume":"28","author":"ZN Zhang","year":"2006","unstructured":"Zhang ZN, Ge XR, Li YH (2006) Application of.virtual multi-dimensional internal bond (vmib) in uniaxial failure of rocklike materials. Chin J Geotech Eng 28(4):504\u2013509","journal-title":"Chin J Geotech Eng"},{"key":"2348_CR90","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.ijrmms.2012.06.001","volume":"55","author":"HQ Yang","year":"2012","unstructured":"Yang HQ, Zhou XP (2012) Multiscale numerical modeling of propagation and coalescence of multiple cracks in rock masses. Int J Rock Mech Min Sci 55:15\u201327","journal-title":"Int J Rock Mech Min Sci"},{"issue":"11","key":"2348_CR91","doi-asserted-by":"publisher","first-page":"1563","DOI":"10.1016\/j.engfracmech.2009.02.016","volume":"76","author":"MR Ayatollahi","year":"2009","unstructured":"Ayatollahi MR, Aliha MRM (2009) Analysis of a new specimen for mixed mode fracture tests on brittle materials. Eng Fract Mech 76(11):1563\u20131573","journal-title":"Eng Fract Mech"},{"key":"2348_CR92","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113398","volume":"372","author":"Y Bie","year":"2020","unstructured":"Bie Y, Liu Z, Yang H, Cui X (2020) Abaqus implementation of dual peridynamics for brittle fracture. Comput Methods Appl Mech Eng 372:113398","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2348_CR93","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2025.105298","volume":"141","author":"Y Bie","year":"2026","unstructured":"Bie Y, Ren H, Duan Y, Bui TQ, Zhuang X, Madenci E, Rabczuk T, Wei Y (2026) Abaqus implementation of coupled thermo-mechanical dual-horizon bond-based peridynamics for thermally induced fractures in thermal barrier coatings. Theor Appl Fract Mech 141:105298","journal-title":"Theor Appl Fract Mech"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-026-02348-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-026-02348-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-026-02348-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T09:03:00Z","timestamp":1780390980000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-026-02348-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,2]]},"references-count":93,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["2348"],"URL":"https:\/\/doi.org\/10.1007\/s00366-026-02348-8","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,2]]},"assertion":[{"value":"5 December 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 May 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 June 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"114"}}