{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T05:50:19Z","timestamp":1775109019358,"version":"3.50.1"},"reference-count":54,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T00:00:00Z","timestamp":1774915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Buildings"],"abstract":"<jats:p>This study proposes a novel semi-discrete model of non-ordinary state-based peridynamics. It is used to simulate the tensile failure process of dog bone-shaped specimens of 3D-printed fiber-reinforced concrete with 0%, 1% and 2% fiber volume fractions. The results are compared with the literature laboratory results to verify the feasibility and reliability of the approach. In addition, it is utilized for a 3D-printable engineered cement-based composite (ECC) disk splitting simulation. Effects of different fiber lengths, printing interfaces, and fiber orientations on the failure process of disc specimens are investigated. It is found that ductile failure appears in the loading direction, while brittle failure appears in the other direction. Effect of fiber length on the bearing capacity is feeble. In addition, the non-ordinary state-based peridynamics semi-discrete model is used to simulate the crack propagation of three-point bending. The principal stress contours, damage diagrams, and displacement\u2013load curves of the concrete matrix at different time steps during the crack propagation process are obtained. The simulation is in great agreement with the experimental results. Finally, it is demonstrated that the novel non-ordinary state-based peridynamics approach proposed in this paper is accurate and efficient to simulate fracture behavior of 3D-printed ECC beams.<\/jats:p>","DOI":"10.3390\/buildings16071379","type":"journal-article","created":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T16:34:08Z","timestamp":1774974848000},"page":"1379","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Ordinary State-Based Peridynamics Simulation for Crack Propagation of 3D-Printed Fiber-Reinforced Concrete Beam Under Bending"],"prefix":"10.3390","volume":"16","author":[{"given":"Tao","family":"Zhu","sequence":"first","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2033-0520","authenticated-orcid":false,"given":"Yuching","family":"Wu","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3690-5018","authenticated-orcid":false,"given":"Peng","family":"Zhi","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8610-119X","authenticated-orcid":false,"given":"Peng","family":"Zhu","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1419-9463","authenticated-orcid":false,"given":"Meiyan","family":"Bai","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Guangxi University, Liuzhou 545026, China"}]},{"given":"Cheng","family":"Qi","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.engstruct.2015.02.020","article-title":"Behavior of steel fiber-reinforced high-strength concrete-filled FRP tube columns under axial compression","volume":"90","author":"Xie","year":"2015","journal-title":"Eng. Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.ijimpeng.2009.08.006","article-title":"Experimental and numerical investigations of low velocity impact behavior of high-performance fiber-reinforced cement based composite","volume":"37","author":"Farnam","year":"2010","journal-title":"Int. J. Impact Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2719","DOI":"10.1007\/BF00356823","article-title":"Experimental determination of the stress-crack opening relation in fibre cementitious composites with a crack-tip singularity","volume":"29","author":"Li","year":"1994","journal-title":"J. Mater. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4191","DOI":"10.1007\/BF00414198","article-title":"Properties of fibre reinforced concrete using recycled fibres from carpet industrial waste","volume":"29","author":"Wang","year":"1994","journal-title":"J. Mater. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0958-9465(96)00046-7","article-title":"Fiber-reinforced concrete: An overview after 30 years of development","volume":"19","author":"Zollo","year":"1997","journal-title":"Cem. Concr. Compos."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/j.conbuildmat.2004.04.027","article-title":"Mechanical properties of high-strength steel fiber-reinforced concrete","volume":"18","author":"Song","year":"2004","journal-title":"Constr. Build. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6517","DOI":"10.1007\/s10853-016-9917-4","article-title":"High-performance fiber-reinforced concrete: A review","volume":"51","author":"Afroughsabet","year":"2016","journal-title":"J. Mater. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103084","DOI":"10.1016\/j.autcon.2020.103084","article-title":"3D printing of a post-tensioned concrete girder designed by topology optimization","volume":"112","author":"Vantyghem","year":"2020","journal-title":"Autom. Constr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"112380","DOI":"10.1016\/j.engstruct.2021.112380","article-title":"Structural behaviour of 3D printed concrete beams with various reinforcement strategies","volume":"240","author":"Gebhard","year":"2021","journal-title":"Eng. Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/S0008-8846(01)00695-0","article-title":"Monotonic and fatigue performance in bending of fiber-reinforced engineered cementitious composite in overlay system","volume":"32","author":"Zhang","year":"2002","journal-title":"Cem. Concr. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"112081","DOI":"10.1016\/j.engstruct.2021.112081","article-title":"Repair of pre-damaged RC beams using hybrid fiber reinforced strain hardening cementitious composites","volume":"235","author":"Tinoco","year":"2021","journal-title":"Eng. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"04021226","DOI":"10.1061\/(ASCE)ST.1943-541X.0003164","article-title":"Shaking-table test on a two-story timber-framed masonry structure retrofitted with ultra-high ductile concrete","volume":"148","author":"Dong","year":"2022","journal-title":"J. Struct. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125824","DOI":"10.1016\/j.conbuildmat.2021.125824","article-title":"ECCs\/UHPFRCCs with and without FRP reinforcement for structural strengthening\/repairing: A state-of-the-art review","volume":"316","author":"Pan","year":"2022","journal-title":"Constr. Build. Mater."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.compositesb.2018.02.012","article-title":"Additive manufacturing (3D printing): A review of materials, methods, applications and challenges","volume":"143","author":"Ngo","year":"2018","journal-title":"Compos. Part B Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.conbuildmat.2019.01.008","article-title":"Mechanical anisotropy of aligned fiber reinforced composite for extrusion-based 3D printing","volume":"202","author":"Ma","year":"2019","journal-title":"Constr. Build. Mater."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.conbuildmat.2018.12.061","article-title":"Rheological and harden properties of the high-thixotropy 3D printing concrete","volume":"201","author":"Zhang","year":"2019","journal-title":"Constr. Build. Mater."},{"key":"ref_17","first-page":"131","article-title":"Static analysis of monoclinic plates via a three-dimensional model using differential quadrature method","volume":"72","author":"Bahrami","year":"2019","journal-title":"Struct. Eng. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1016\/S0022-460X(03)00625-4","article-title":"Natural frequencies of thick square plates made of orthotropic, trigonal, monoclinic, hexagonal and triclinic materials","volume":"270","author":"Batra","year":"2004","journal-title":"J. Sound Vib."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1016\/j.jsv.2008.06.034","article-title":"Natural frequencies of thick plates made of orthotropic, monoclinic, and hexagonal materials by a meshless method","volume":"319","author":"Ferreira","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"734","DOI":"10.1016\/j.jsv.2004.10.025","article-title":"Natural frequencies of orthotropic, monoclinic and hexagonal plates by a meshless method","volume":"285","author":"Ferreira","year":"2005","journal-title":"J. Sound Vib."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"111102","DOI":"10.1016\/j.ijsolstr.2021.111102","article-title":"Uncertainty quantification for the representative volume element of geometrically monoclinic 3D printed concrete","volume":"226","author":"Wu","year":"2021","journal-title":"Int. J. Solids Struct."},{"key":"ref_22","first-page":"101712","article-title":"Finite element analysis on the anisotropic behavior of 3D printed concrete under compression and flexure","volume":"39","author":"Xiao","year":"2021","journal-title":"Addit. Manuf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"112808","DOI":"10.1016\/j.compstruct.2020.112808","article-title":"Anisotropic behavior in bending of 3D printed concrete reinforced with fibers","volume":"254","author":"Ding","year":"2020","journal-title":"Compos. Struct."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"075036","DOI":"10.1088\/1361-665X\/aac789","article-title":"Mechanical characterization of 3D printed anisotropic cementitious material by the electromechanical transducer","volume":"27","author":"Ma","year":"2018","journal-title":"Smart Mater. Struct."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/BF02486267","article-title":"Crack band theory for fracture of concrete","volume":"16","author":"Oh","year":"1983","journal-title":"Mat\u00e9r. Constr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1617\/s11527-019-1384-3","article-title":"Experimental investigation on the reversibility of concrete creep under repeating loads","volume":"52","author":"Docevska","year":"2019","journal-title":"Mater. Struct."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1061\/(ASCE)0733-9399(1985)111:10(1227)","article-title":"Two parameter fracture model for concrete","volume":"111","author":"Jenq","year":"1985","journal-title":"J. Eng. Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1016\/0013-7944(85)90009-8","article-title":"A Fracture toughness criterion for concrete","volume":"21","author":"Jenq","year":"1985","journal-title":"Eng. Fract. Mech."},{"key":"ref_29","unstructured":"Swartz, S., and Refai, T. (1987, January 17\u201319). Influence of size effects on opening mode fracture parameters for precracked concrete beams in bending. Proceedings of the Fracture of Concrete and Rock: SEM-RILEM International Conference, Houston, TX, USA."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF00047063","article-title":"Determination of fracture energy, process zone longth and brittleness number from size effect, with application to rock and conerete","volume":"44","author":"Kazemi","year":"1990","journal-title":"Int. J. Fract."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1016\/S0013-7944(01)00128-X","article-title":"Extended finite element method for cohesive crack growth","volume":"69","author":"Belytschko","year":"2002","journal-title":"Eng. Fract. Mech."},{"key":"ref_32","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_33","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10659-007-9125-1","article-title":"Peridynamic states and constitutive modeling","volume":"88","author":"Silling","year":"2007","journal-title":"J. Elast."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1186","DOI":"10.1016\/j.ijsolstr.2008.10.029","article-title":"A non-ordinary state-based peridynamic method to model solid material deformation and fracture","volume":"46","author":"Warren","year":"2009","journal-title":"Int. J. Solids Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1080\/17797179.2018.1547356","article-title":"Fast interaction functions for bond-based peridynamics","volume":"27","author":"Miranda","year":"2018","journal-title":"Eur. J. Comput. Mech."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.jmps.2016.05.017","article-title":"Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure","volume":"94","author":"Han","year":"2016","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.ijimpeng.2013.07.001","article-title":"Impact damage on a thin glass plate with a thin polycarbonate backing","volume":"62","author":"Hu","year":"2013","journal-title":"Int. J. Impact Eng."},{"key":"ref_38","first-page":"551","article-title":"Damage progression from impact in layered glass modeled with peridynamics","volume":"2","author":"Bobaru","year":"2012","journal-title":"Cent. Eur. J. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s10704-010-9442-4","article-title":"Studies of dynamic crack propagation and crack branching with peridynamics","volume":"162","author":"Ha","year":"2010","journal-title":"Int. J. Fract."},{"key":"ref_40","first-page":"523","article-title":"Impact damage assessment by using peridynamic theory","volume":"2","author":"Oterkus","year":"2012","journal-title":"Cent. Eur. J. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/j.jmps.2004.08.006","article-title":"The effect of long-range forces on the dynamics of a bar","volume":"53","author":"Weckner","year":"2005","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"113700","DOI":"10.1016\/j.cma.2021.113700","article-title":"Isogeometric analysis of cracks with peridynamics","volume":"377","author":"Xia","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.2140\/jomms.2007.2.1921","article-title":"An approach to modeling extreme loading of structures using peridynamics","volume":"2","author":"Demmie","year":"2007","journal-title":"J. Mech. Mater. Struct."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"04019049","DOI":"10.1061\/(ASCE)EM.1943-7889.0001628","article-title":"Simulating the fracture of notched mortar beams through extended finite-element method and peridynamics","volume":"145","author":"Das","year":"2019","journal-title":"J. Eng. Mech."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.ijimpeng.2017.08.008","article-title":"A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials","volume":"111","author":"Lai","year":"2018","journal-title":"Int. J. Impact Eng."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"103564","DOI":"10.1016\/j.ijnonlinmec.2020.103564","article-title":"Peridynamic correspondence model for finite elastic deformation and rupture in Neo-Hookean materials","volume":"126","author":"Behera","year":"2020","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_47","first-page":"103683","article-title":"Peridynamic buildability analysis of 3D-printed concrete including damage, plastic flow and collapse","volume":"73","author":"Zhu","year":"2023","journal-title":"Addit. Manuf."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.engfracmech.2016.11.004","article-title":"Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework","volume":"169","author":"Yaghoobi","year":"2017","journal-title":"Eng. Fract. Mech."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"115865","DOI":"10.1016\/j.engstruct.2023.115865","article-title":"Flexural performance of 3D-printed composite beams with ECC and recycled fine aggregate concrete: Experimental and numerical analysis","volume":"283","author":"Liu","year":"2023","journal-title":"Eng. Struct."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1177\/1081286513509811","article-title":"At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola","volume":"20","author":"Andreaus","year":"2015","journal-title":"Math. Mech. Solids"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.cemconres.2016.03.002","article-title":"Towards the design of an enriched concrete with enhanced dissipation performances","volume":"84","author":"Scerrato","year":"2016","journal-title":"Cem. Concr. Res."},{"key":"ref_52","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_53","unstructured":"Kong, Y. (2020). Experimental Study and Numerical Simulation of Mechanical Properties of ECC. [Ph.D. Thesis, Suzhou University of Science and Technology]."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"104310","DOI":"10.1016\/j.cemconcomp.2021.104310","article-title":"Mechanical anisotropy of ultra-high performance fibre-reinforced concrete for 3D printing","volume":"125","author":"Yang","year":"2022","journal-title":"Cem. Concr. Compos."}],"container-title":["Buildings"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-5309\/16\/7\/1379\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T04:27:44Z","timestamp":1775104064000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-5309\/16\/7\/1379"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,31]]},"references-count":54,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2026,4]]}},"alternative-id":["buildings16071379"],"URL":"https:\/\/doi.org\/10.3390\/buildings16071379","relation":{},"ISSN":["2075-5309"],"issn-type":[{"value":"2075-5309","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,31]]}}}