During the operation of structures, stress and deformation fields occur inside the materials used, which often ends in fatal damage of the entire structure. Therefore, the modelling of this damage, including the possible formation and growth of cracks, is at the forefront of numerical and applied mathematics. The finite element method (FEM) and its modification will allow us to predict the behaviour of these structural materials. Furthermore, some practical applications based on cohesive approach are tested. The main effort is devoted to composites with fibres and searching for procedures for their accurate modelling, mainly in the area where damage can be expected to occur. The use of the cohesive approach of elements that represent the physical nature of energy release in front of the crack front has proven to be promising not only in the direct use of cohesive elements, but also in combination with modified methods of standard finite elements.

Institute of Mathematics and Descriptive Geometry Faculty of Civil Engineering Brno University of Technology 602 00 Brno Czech Republic

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Yu W. A review of modeling of composite structures. Materials. 2024;17:446. doi: 10.3390/ma17020446. PubMed DOI PMC

Slatcher S., Evandt Ø. Practical application of the weakest link model to fracture toughness problems. Eng. Fract. Mech. 1986;24:495–508. doi: 10.1016/0013-7944(86)90223-7. DOI

Cui W. A state-of-the-art review on fatigue life prediction methods for metal structures. J. Mar. Sci. Technol. 2002;7:43–56. doi: 10.1007/s007730200012. DOI

Krejsa M., Seitl S., Brožovslý J., Lehner P. Fatigue damage prediction of short edge crack under various load: Direct optimized probabilistic calculation. Procedia Struct. Integr. 2017;5:1283–1290. doi: 10.1016/j.prostr.2017.07.107. DOI

Hun D.-A., Guilleminot J., Yvonnet J., Bornert M. Stochastic multiscale modeling of crack propagation in random heterogeneous media. Int. J. Numer. Methods Eng. 2019;119:1325–1344. doi: 10.1002/nme.6093. DOI

Kotrechko S., Kozák V., Zatsarna O., Zimina G., Stetsenko N., Dlouhý I. Incorporation of temperature and plastic strain effects into local approach to fracture. Materials. 2021;14:6224. doi: 10.3390/ma14206224. PubMed DOI PMC

Mieczkowski G., Szymczak T., Szpica D., Borawski A. Probabilistic modelling of fracture toughness of composites with discontinuous reinforcement. Materials. 2023;16:2962. doi: 10.3390/ma16082962. PubMed DOI PMC

Le B.D., Koval G., Chazallon C. Discrete element approach in brittle fracture mechanics. Eng. Comput. 2013;30:263–276.

Guan J., Zhang L., Li L., Yao X., He S., Niu L., Cao H. Three-dimensional discrete element model of crack evolution on the crack tip with consideration of random aggregate shape. Theor. Appl. Fract. Mech. 2023;127:104022. doi: 10.1016/j.tafmec.2023.104022. DOI

Xu G., Yue Q., Liu X. Deep learning algorithm for real-time automatic crack detection, segmentation, qualification. Eng. Appl. Artif. Intell. 2023;16:1070852. doi: 10.1016/j.engappai.2023.107085. DOI

Griffith A.A. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. A. 1920;221:163–198.

Sih G.C. Strain-energy density factor applied to mixed mode crack problems. Int. J. Fract. 1974;10:304–321. doi: 10.1007/BF00035493. DOI

Sun C.T., Jin Z.H. A comparison of cohesive zone modelling and classical fracture mechanics based on near tip stress field. Int. J. Solids Struct. 2006;43:1047–1060.

Rice J.R. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 1968;35:379–386. doi: 10.1115/1.3601206. DOI

Goutianos S. Derivation of path independent coupled mix mode cohesive laws from fracture resistance curves. Appl. Comp. Mater. 2017;24:983–997. doi: 10.1007/s10443-016-9568-2. DOI

Barenblatt G.I. The mathematical theory of equilibrium of cracks in brittle fracture. Adv. Appl. Mech. 1962;7:55–129.

Erdogan F., Sih G.C. On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 1963;85:519–527. doi: 10.1115/1.3656897. DOI

Enescu I. Some researches regarding stress intensity factors in crack closure problems. WSEAS Trans. Appl. Theor. Mech. 2018;13:187–192.

Tabiei A., Zhang W. Cohesive element approach for dynamic crack propagation: Artificial compliance and mesh dependency. Eng. Fract. Mech. 2017;180:23–42. doi: 10.1016/j.engfracmech.2017.05.009. DOI

Papenfuß C. Continuum Thermodynamics and Constitutive Theory. Springer; Berlin, Germany: 2020.

Hashiguchi K. Elastoplasticity Theory. Springer; Berlin, Germany: 2014.

Morandotti M. Mathematical Analysis of Continuum Mechanics and Industrial Applications II: Proceedings of the International Conference CoMFoS16 16. Springer; Singapore: 2018. Structured deformation of continua: Theory and applications; pp. 125–136. Continuum Mechanics Focusing on Singularities.

Del Piero G., Owen D.R. Structured deformations of continua. Arch. Ration. Mech. Anal. 1993;124:99–155. doi: 10.1007/BF00375133. DOI

Ogawa K., Ichitsubo T., Ishioka S., Ahuja R. Irreversible thermodynamics of ideal plastic deformation. Cogent Phys. 2018;5:1496613. doi: 10.1080/23311940.2018.1496613. DOI

Taira S., Ohtani R., Kitamura T. Application of J-integral to high-temperature crack propagation, Part I—Creep crack propagation. J. Eng. Mater. Technol. 1979;101:154–161. doi: 10.1115/1.3443668. DOI

Landes J.D., Begley J.A. Mechanics of Crack Growth. ASTM International; West Conshohocken, PA, USA: 1976.

Riedel H. Fracture at High Temperatures. Springer; Berlin, Germany: 1987.

Kolednik O., Schöngrundner F., Fischer D. A new view on J-integrals in elastic–plastic materials. Int. J. Fract. 2014;187:77–107. doi: 10.1007/s10704-013-9920-6. DOI

Scheel J., Schlosser A., Ricoeur A. The J-integral for mixed-mode loaded cracks with cohesive zones. Int. J. Fract. 2021;227:79–94. doi: 10.1007/s10704-020-00496-6. DOI

Healy B., Gullerund A., Koppenhoefer K. WARP3D Release 18.3.6. User Manual, 3-D Dynamic Nonlinear Fracture Analysis of Solids. University of Illinois; Chicago, IL, USA: 2023.

Betegón C., Hancock J.W. Two-parameters characterization of elastic-plastic crack tip field. J. Appl. Mech. 1991;58:104–110. doi: 10.1115/1.2897135. DOI

Gupta M., Anderliesten R.C., Benedictus R. A review of T-stress and its effects in fracture mechanics. Eng. Fract. Mech. 2015;134:218–241. doi: 10.1016/j.engfracmech.2014.10.013. DOI

Cedolin L., Bažant Z.P. Effect of finite element choice in blunt crack band analysis. Comput. Methods Appl. Mech. Eng. 1980;24:205–316. doi: 10.1016/0045-7825(80)90067-5. DOI

Beissel S.R., Johnson G.R., Popelar C.H. An element-failure algorithm for dynamic crack propagation in general direction. Eng. Fract. Mech. 1998;61:407–425. doi: 10.1016/S0013-7944(98)00072-1. DOI

Hermosillo-Arteaga A., Romo M.P., Magaña R., Carrera J. Automatic remeshing algorithm of triangular elements during finite element analyses. Rev. Int. Metodos Numer. Calc. Diseno Ing. 2018;34:26.

Kachanov L.M. Introduction to Continuum Damage Mechanics. Martinus Nijhoff; Dordrecht, The Netherlands: 1986.

Gurson A.L. Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 1977;99:2–15. doi: 10.1115/1.3443401. DOI

Tvergaard V., Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 1984;32:157–169. doi: 10.1016/0001-6160(84)90213-X. DOI

Vlček L. Ph.D. Thesis. Brno University of Technology; Brno, Czech Republic: 2004. Numerical Analysis of the Bodies with Cracks.

Brocks W., Klingbeil D., Künecke G., Sun D.Z. Application of the Gurson model to ductile tearing resistance. In: Kirk M., Bakker A., editors. Constraint Effects in Fracture: Theory and Applications. ASTM; Dallas, TX, USA: 1995. pp. 232–252.

Zhan Z.L. A complete Gurson model. In: Alibadi M.H., editor. Nonlinear Fracture and Damage Mechanics. WIT Press; Southampton, UK: 2001. pp. 223–248.

Khoei A.R. Extended Finite Element Method: Theory and Applications. J. Wiley & Sons; Hoboken, NJ, USA: 2015.

Hansbo A., Hansbo P. A finite element method for simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 2006;193:3523–3540. doi: 10.1016/j.cma.2003.12.041. DOI

Areias P.M.A., Belytschko T. Two-scale shear band evolution by local partition of unity. Int. J. Numer. Methods Eng. 2006;66:878–910. doi: 10.1002/nme.1589. DOI

Shen Y., Lew A. Stability and convergence proofs for a discontinuous Galerkin-based extended finite element method for fracture mechanics. Comput. Methods Appl. Mech. Eng. 2010;199:2360–2382. doi: 10.1016/j.cma.2010.03.008. DOI

Stolarska M., Chopp D.L., Moës N.N., Belytschko T. Modelling of crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 2001;51:943–960. doi: 10.1002/nme.201. DOI

Xiao Q.Z., Karihaloo B.L. Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery. Int. J. Numer. Methods Eng. 2006;66:1378–1410. doi: 10.1002/nme.1601. DOI

Moës N., Dolbow J., Belytchko T. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 1999;46:131–150. doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J. DOI

Cui C., Zhang G., Banerjee U., Babuška I. Stable generalized finite element method (SGFEM) for three-dimensional crack problems. Num. Math. 2022;152:475–509. doi: 10.1007/s00211-022-01312-0. DOI

Babuška I., Melenk J.M. The partition of unity method. Int. J. Numer. Methods Eng. 1997;40:727–758. doi: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N. DOI

Fries T.P., Belytschko T. The intrinsic XFEM: A method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Methods Eng. 2006;68:1358–1385. doi: 10.1002/nme.1761. DOI

Fries T.P., Belytschko T. The extended/generalized finite element method: An overview of the method and its applications. Int. J. Numer. Methods Eng. 2010;84:253–304. doi: 10.1002/nme.2914. DOI

Fries T.P., Baydoun M. Crack propagation with the extended finite element method and a hybrid explicit-implicit crack description. Int. J. Numer. Methods Eng. 2012;89:1527–1558. doi: 10.1002/nme.3299. DOI

Shi F., Wang D., Yang Q. An XFEM-based numerical strategy to model three-dimensional fracture propagation regarding crack front segmentation. Theor. Appl. Fract. Mech. 2022;118:103250. doi: 10.1016/j.tafmec.2022.103250. DOI

Panday V.B., Singh I.V., Mishra B.K. A new creep-fatigue interaction damage model and CDM-XFEM framework for creep-fatigue crack growth simulations. Theor. Appl. Fract. Mech. 2023;124:103740. doi: 10.1016/j.tafmec.2022.103740. DOI

Liu G., Guo J., Bao Y. Convergence investigation of XFEM enrichment schemes for modeling cohesive cracks. Mathematics. 2022;10:383. doi: 10.3390/math10030383. DOI

Xiao G., Wen L., Tian R. Arbitrary 3D crack propagation with improved XFEM: Accurate and efficient crack geometries. Comput. Methods Appl. Mech. Eng. 2021;377:113659. doi: 10.1016/j.cma.2020.113659. DOI

Jirásek M. Damage and smeared crack models. In: Hofstetter G., Meschke G., editors. Numerical Modelling of Concrete Cracking. Springer; Vienna, Austria: 2011. pp. 1–49.

Mazars J. A description of micro- and macroscale damage of concrete structures. Eng. Fract. Mech. 1986;25:729–737. doi: 10.1016/0013-7944(86)90036-6. DOI

Comi C. A non-local model with tension and compression damage mechanisms. Eur. J. Mech. A Solids. 2001;20:1–22. doi: 10.1016/S0997-7538(00)01111-6. DOI

Arruda M.R.T., Pacheco J., Castro L.M.S., Julio E. A modified Mazars damage model with energy regularization. Theor. Appl. Fract. Mech. 2023;124:108129. doi: 10.1016/j.engfracmech.2021.108129. DOI

Zhou X., Feng B. A smeared-crack-based field-enriched finite element method for simulating cracking in quasi-brittle materials. Theor. Appl. Fract. Mech. 2023;124:103817. doi: 10.1016/j.tafmec.2023.103817. DOI

Wu B., Li Z., Tang K. Numerical modeling on micro-to-macro evolution of crack network for concrete materials. Teor. Appl. Fract. Mech. 2020;107:102525. doi: 10.1016/j.tafmec.2020.102525. DOI

Giffin B.D., Zywicz E. A smeared crack modeling framework accommodating multi-directional fracture at finite strains. Int. J. Fract. 2023;239:87–109. doi: 10.1007/s10704-022-00665-9. DOI

Kozák V., Vala J. Modelling of crack formation and growth using FEM for selected structural materials at static loading. WSEAS Trans. Appl. Theor. Mech. 2023;18:243–254. doi: 10.37394/232011.2023.18.23. DOI

Xie M., Gerstle W.H. Energy based cohesive crack propagation modelling. J. Eng. Mech. 1995;121:1349–1358. doi: 10.1061/(ASCE)0733-9399(1995)121:12(1349). DOI

Blal N., Daridon L., Monerie Y., Pagano S. Micromechanical-based criteria for the calibration of cohesive zone parameters. J. Comput. Appl. Math. 2013;246:206–214. doi: 10.1016/j.cam.2012.10.031. DOI

Sørensen B.F., Jacobsen T.K. Determination of cohesive laws by the J integral approach. Eng. Fract. Mech. 2003;70:1841–1858. doi: 10.1016/S0013-7944(03)00127-9. DOI

Jin Z.H., Sun C.T. Cohesive fracture model based on necking. Int. J. Fract. 2005;134:91–108. doi: 10.1007/s10704-005-7864-1. DOI

Cuvilliez S., Feyel F., Lorentz E., Michel-Ponnelle S. A finite element approach coupling a continuous gradient damage model and a cohesive zone model within the framework of quasi-brittle failure. Comput. Methods Appl. Mech. Eng. 2012;237:244–253. doi: 10.1016/j.cma.2012.04.019. DOI

Bouhala L., Makradi A., Belouettar S., Kiefer-Kamal H., Fréres P. Modelling of failure in long fibres reinforced composites by X-FEM and cohesive zone model. Compos. Part B. 2013;55:352–361. doi: 10.1016/j.compositesb.2012.12.013. DOI

Brighenti R., Scorza D. Numerical modelling of the fracture behaviour of brittle materials reinforced with unidirectional or randomly distributed fibres. Mech. Mater. 2012;52:12–27. doi: 10.1016/j.mechmat.2012.04.008. DOI

Afshar A., Daneshyar A., Mohammadi S. XFEM analysis of fiber bridging in mixed-mode crack propagation in composites. Compos. Struct. 2015;125:314–327. doi: 10.1016/j.compstruct.2015.02.002. DOI

Marfia S., Sacco E. Numerical techniques for the analysis of crack propagation in cohesive materials. Int. J. Numer. Methods Eng. 2003;57:1577–1602. doi: 10.1002/nme.732. DOI

Naghdinasab M., Farrokhabadi A., Madadi H. A numerical method to evaluate the material properties degradation in composite RVEs due to fiber-matrix debonding and induced matrix cracking. Finite Elem. Anal. Des. 2018;146:84–95. doi: 10.1016/j.finel.2018.04.008. DOI

Gong Y., Zhang H., Jiang L., Ding Z., Hu N. Determination of mixed-mode I/II fracture toughness and bridging law of composite laminates. Theor. Appl. Fract. Mech. 2023;127:104060. doi: 10.1016/j.tafmec.2023.104060. DOI

Cunha V.M.C.F., Barros J.A.O., Sena-Cruz J.M. An integrated approach for modelling the tensile behaviour of steel fibre reinforced self-compacting concrete. Cem. Concr. Res. 2011;41:64–76. doi: 10.1016/j.cemconres.2010.09.007. DOI

Kormaníková E., Kotrasová K. Mixed-mode delamination in laminate plate with crack. Adv. Mater. Proc. 2018;3:512–516. doi: 10.5185/amp.2018/1894. DOI

Lusis V., Krasnikovs A., Kononova O., Lapsa V.-A., Stonys C., Macanovskis A., Lukasnoks A. Effect of short fibers orientation on mechanical properties of composite material—Fiber reinforced concrete. J. Civ. Eng. Manag. 2017;23:1091–1099. doi: 10.3846/13923730.2017.1381643. DOI

Abadel A., Abbas H., Alrshoudi F., Altheeb A., Albidah A., Almusallam T. Experimental and analytical investigation of fiber alignment on fracture properties of concrete. Structures. 2020;28:2572–2581. doi: 10.1016/j.istruc.2020.10.077. DOI

Chen H., Zhang Y.X., Zhu L., Xiong F., Liu J., Gao W. A particle-based cohesive crack model for brittle fracture problems. Materials. 2020;13:3573. doi: 10.3390/ma13163573. PubMed DOI PMC

Vala J., Hobst L., Kozák V. Detection of metal fibres in cementitious composites based on signal and image processing approaches. WSEAS Trans. Appl. Theor. Mech. 2015;10:39–46.

Kozák V., Vala J. Crack growth modelling in cementitious composites using XFEM. Procedia Struct. Integr. 2023;43:47–52. doi: 10.1016/j.prostr.2022.12.233. DOI

Shanmugasundaram N., Praveenkumar S. Mechanical properties of engineered cementitious composites (ECC) incorporating different mineral admixtures and fibre: A review. J. Build. Pathol. Rehabil. 2022;7:40. doi: 10.1007/s41024-022-00182-1. DOI

Wen C., Zhang P., Wang J., Hu S. Influence of fibers on the mechanical properties and durability of ultra-high-performance concrete: A review. J. Build. Eng. 2022;52:104370. doi: 10.1016/j.jobe.2022.104370. DOI

Yan Y., Tian L., Zhao W., Aires Master Lazaro S., Li X., Tang S. Dielectric and mechanical properties of cement pastes incorporated with magnetically aligned reduced graphene oxide. Dev. Built Environ. 2024;18:100471. doi: 10.1016/j.dibe.2024.100471. DOI

Rabinowitch O. Debonding analysis of fiber-reinforced-polymer strengthened beams: Cohesive zone modelling versus a linear elastic fracture mechanics approach. Eng. Fract. Mech. 2008;75:2842–2859. doi: 10.1016/j.engfracmech.2008.01.003. DOI

Barani Q.R., Khoei A.R., Mofid M. Modelling of cohesive crack growth in partially saturated porous media: A study on the permeability of cohesive fracture. Int. J. Fract. 2011;167:15–31. doi: 10.1007/s10704-010-9513-6. DOI

Hillerborg A., Modéer M., Peterson P.-E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 1976;6:773–781. doi: 10.1016/0008-8846(76)90007-7. DOI

Kozák V. Ductile crack growth modelling using cohesive zone approach. In: Kompiš V., editor. Composites with Micro- and Nano-Structure. Springer; Berlin, Germany: 2008. pp. 191–208.

Kozák V., Chlup Z., Padělek P., Dlouhý I. Prediction of traction separation law of ceramics using iterative finite element method. Solid State Phenom. 2017;258:186–189. doi: 10.4028/www.scientific.net/SSP.258.186. DOI

Moës N., Belytschko T. Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 2002;69:813–833. doi: 10.1016/S0013-7944(01)00128-X. DOI

Airoldi A., Davila C.G. Identification of material parameters for modelling delamination in the presence of fibre bridging. Comp. Struct. 2012;94:3240–3249. doi: 10.1016/j.compstruct.2012.05.014. DOI

Coq A., Diani J., Brach S. Comparison of the phase-field approach and cohesive element modelling to analyse the double cleavage drilled compression fracture test of an elastoplastic material. Int. J. Fract. 2024;245:1–14. doi: 10.1007/s10704-023-00755-2. DOI

Yuan Z., Fish J. Are the cohesive zone models necessary for delamination analysis? Comput. Methods Appl. Mech. Eng. 2016;310:567–604. doi: 10.1016/j.cma.2016.06.023. DOI

Aliabadi M.H., Saleh A.L. Fracture mechanics analysis of cracking in plain and reinforced concrete using the boundary element method. Eng. Fract. Mech. 2002;69:267–280. doi: 10.1016/S0013-7944(01)00089-3. DOI

Belytschko T., Gracie R., Ventura G. A review of extended/generalized finite element methods for material modelling. Modeling Simul. Mater. Sci. Eng. 2009;17:043001. doi: 10.1088/0965-0393/17/4/043001. DOI

Yu T.T., Gong Z.W. Numerical simulation of temperature field in heterogeneous material with the XFEM. Arch. Civ. Mech. Eng. 2013;13:199–208. doi: 10.1016/j.acme.2013.02.004. DOI

Park K., Paulino G.H., Roesler J.R. Cohesive fracture model for functionally graded fibre reinforced concrete. Cem. Concr. Res. 2010;40:956–965. doi: 10.1016/j.cemconres.2010.02.004. DOI

Ye C., Shi J., Cheng G.J. An extended finite element method (XFEM) study on the effect of reinforcing particles on the crack propagation behaviour in a metal-matrix composite. Int. J. Fatigue. 2012;44:151–156. doi: 10.1016/j.ijfatigue.2012.05.004. DOI

Eringen C.A. Nonlocal Continuum Field Theories. Springer; New York, NY, USA: 2002.

Pike M.G., Oskay C. XFEM modelling of short microfibre reinforced composites with cohesive interfaces. Finite Elem. Anal. Des. 2005;106:16–31. doi: 10.1016/j.finel.2015.07.007. DOI

Li X., Chen J. An extensive cohesive damage model for simulation arbitrary damage propagation in engineering materials. Comput. Methods Appl. Mech. Eng. 2017;315:744–759. doi: 10.1016/j.cma.2016.11.029. DOI

Vala J., Kozák V. Computational analysis of quasi-brittle fracture in fibre reinforced cementitious composites. Theor. Appl. Fract. Mech. 2020;107:102486. doi: 10.1016/j.tafmec.2020.102486. DOI

Ebrahimi S.H. Singularity analysis of cracks in hybrid CNT reinforced carbon fiber composites using finite element asymptotic expansion and XFEM. Int. J. Solids Struct. 2021;14–15:1–17. doi: 10.1016/j.ijsolstr.2021.01.001. DOI

Vala J. Numerical approaches to the modelling of quasi-brittle crack propagation. Arch. Math. 2023;59:295–303. doi: 10.5817/AM2023-3-295. DOI

Langenfeld K., Kurzeja P., Mosler J. How regularization concepts interfere with (quasi-)brittle damage: A comparison based on a unified variational framework. Contin. Mech. Thermodyn. 2022;34:1517–1544. doi: 10.1007/s00161-022-01143-2. DOI

Lu X., Guo X.M., Tan V.B.C., Tay T.E. From diffuse damage to discrete crack: A coupled failure model for multi-stage progressive damage of composites. Comput. Methods Appl. Mech. Eng. 2021;379:113760. doi: 10.1016/j.cma.2021.113760. DOI

Vilppo J., Kouhia R., Hartikainen J., Kolari K., Fedoroff A., Calonius K. Anisotropic damage model for concrete and other quasi-brittle materials. Int. J. Solids Struct. 2021;225:111048. doi: 10.1016/j.ijsolstr.2021.111048. DOI

Ottosen N. A failure criterion for concrete. J. Eng. Mech. 1977;103:527–535. doi: 10.1061/JMCEA3.0002248. DOI

Kachanov I. Effective elastic properties of cracked solids: Critical review of some basic concepts. Appl. Mech. Rev. 1992;45:304–335. doi: 10.1115/1.3119761. DOI

Frémond M., Nejdar B. Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 1996;33:1083–1103. doi: 10.1016/0020-7683(95)00074-7. DOI

Akagi G., Schimperna G. Local well-posedness for Frémond’s model of complete damage in elastic solids. Eur. J. Appl. Math. 2020;33:309–327. doi: 10.1017/S0956792521000024. DOI

Bui T.Q., Tran H.T., Hu X., Wu C.-T. Simulation of dynamic brittle and quasi-brittle fracture: A revisited local damage approach. Int. J. Fract. 2022;236:59–85. doi: 10.1007/s10704-022-00635-1. DOI

Zhu X. J-integral resistance curve testing and evaluation. J. Zhejiang Univ. Sci. A. 2009;10:1541–1560. doi: 10.1631/jzus.A0930004. DOI

Kumar M., Mukhopadhyay S. Efficient modelling of progressive damage due to quasi-static indentation on multidirectional laminates by a mesh orientation independent kinematically enriched continuum damage model. Compos. Part A. 2024;178:108002. doi: 10.1016/j.compositesa.2023.108002. DOI

Wang Y. A 3D stochastic damage model for concrete under monotonic and cyclic loadings. Cem. Concr. Res. 2023;171:107208. doi: 10.1016/j.cemconres.2023.107208. DOI

Michael N.E., Bansal R.C., Ismail A.A.A., Elnady A., Hasan S. A cohesive structure of Bi-directional long-short-term memory (BiLSTM)-GRU for predicting hourly solar radiation. Renew. Energy. 2024;222:119943. doi: 10.1016/j.renene.2024.119943. DOI

Oldfield M., Dini D., Giordano G., Rodriguez y Baena F. Detailed finite element modelling of deep needle insertions into a soft tissue phantom using a cohesive approach. Comput. Methods Biomech. Biomed. Eng. 2013;16:530–543. doi: 10.1080/10255842.2011.628448. PubMed DOI

Vellwock A.E., Libonati F. XFEM for composites, biological, and bioinspired materials: A review. Materials. 2024;17:745. doi: 10.3390/ma17030745. PubMed DOI PMC

Some Peculiarities of Using the Extended Finite Element Method in Modelling the Damage Behaviour of Fibre-Reinforced Composites