The Overview of Fracture Mechanics Models for Concrete

[1] Griffith, A. A. (1920). The phenomena of rupture and flow in solids. Philosophical Transactions, Series A, 222, 163–198. Griffith A. A. 1920 The phenomena of rupture and flow in solids Philosophical Transactions, Series A 222 163 198 Search in Google Scholar

[2] Irwin, G. R. (1948). Fracture Dynamics. Fracturing of Metals, 147–166. Irwin G. R. 1948 Fracture Dynamics. Fracturing of Metals 147 166 Search in Google Scholar

[3] Kurumatani, M., Terada, K., Kato, J., Kyoya, T., Kashiyama, K. (2016). An isotropic damage model based on fracture mechanics for concrete. Engineering Fracture Mechanics 155, 49–66. Kurumatani M. Terada K. Kato J. Kyoya T. Kashiyama K. 2016 An isotropic damage model based on fracture mechanics for concrete Engineering Fracture Mechanics 155 49 66 10.1016/j.engfracmech.2016.01.020 Search in Google Scholar

[4] Wang, J., Tao, M., Nie, X. (2017). Fracture energy-based model for average crack spacing of reinforced concrete considering size effect and concrete strength variation. Construction and Building Materials, Volume 148, 398–410. Wang J. Tao M. Nie X. 2017 Fracture energy-based model for average crack spacing of reinforced concrete considering size effect and concrete strength variation Construction and Building Materials, Volume 148 398 410 10.1016/j.conbuildmat.2017.05.082 Search in Google Scholar

[5] Xu, W., Waas, A. M. (2016). Modeling damage growth using the crack band model; effect of different strain measures. Engineering Fracture Mechanics 152, 126–138. Xu W. Waas A. M. 2016 Modeling damage growth using the crack band model; effect of different strain measures Engineering Fracture Mechanics 152 126 138 10.1016/j.engfracmech.2015.06.034 Search in Google Scholar

[6] Xue, J., Kirane, K. (2019). Strength size effect and post-peak softening in textile composites analyzed by cohesive zone and crack band models. Engineering Fracture Mechanics 212, 106–122. Xue J. Kirane K. 2019 Strength size effect and post-peak softening in textile composites analyzed by cohesive zone and crack band models Engineering Fracture Mechanics 212 106 122 10.1016/j.engfracmech.2019.03.025 Search in Google Scholar

[7] Zhao, L., Yan, T., Bai, X., Li, T., Cheng, J. (2013). Implementation of Fictitious Crack Model Using Contact Finite Element Method for the Crack Propagation in Concrete under Cyclic Load. Mathematical Problems in Engineering, vol. 2013, ID 726317. Zhao L. Yan T. Bai X. Li T. Cheng J. 2013 Implementation of Fictitious Crack Model Using Contact Finite Element Method for the Crack Propagation in Concrete under Cyclic Load Mathematical Problems in Engineering vol. 2013 ID 726317 10.1155/2013/726317 Search in Google Scholar

[8] Hilleborg, A., Modeer, M., Petersson, P. E. (1976). Analysis of Crack Formation and Crack Growth in Concrete by Means of Fract. Mech. and Finite Elements. Cement and ConcreteResearch, 6(6), 773–791. Hilleborg A. Modeer M. Petersson P. E. 1976 Analysis of Crack Formation and Crack Growth in Concrete by Means of Fract. Mech. and Finite Elements Cement and ConcreteResearch 6 6 773 791 10.1016/0008-8846(76)90007-7 Search in Google Scholar

[9] Słowik, M., Stroeven, P., Akram, A. (2020). Crack mechanisms in concrete – from micro to macro scale. Budownictwo i Architektura, 19(4), 53–66. Słowik M. Stroeven P. Akram A. 2020 Crack mechanisms in concrete – from micro to macro scale Budownictwo i Architektura 19 4 53 66 10.35784/bud-arch.2147 Search in Google Scholar

[10] Bažant, Z.P., Oh, B.H. (1983). Crack band theory for fracture of concrete. Matériaux et Construction, 16, 155–177. Bažant Z.P. Oh B.H. 1983 Crack band theory for fracture of concrete Matériaux et Construction 16 155 177 10.1007/BF02486267 Search in Google Scholar

[11] Bažant, Z.P. (editor) (1992). Fracture Mechanics of Concrete Structures. Elsevier Applied Science, London and New York, FraMCoS1. Bažant Z.P. 1992 Fracture Mechanics of Concrete Structures Elsevier Applied Science London and New York FraMCoS1 Search in Google Scholar

[12] Golewski, G. L. (2007). Analysis influence of Dmax on fracture mechanics parameters of concrete made of limestone aggregate at three point bending. Budownictwo i Architektura, 1, 5–16. Golewski G. L. 2007 Analysis influence of Dmax on fracture mechanics parameters of concrete made of limestone aggregate at three point bending Budownictwo i Architektura 1 5 16 10.35784/bud-arch.2298 Search in Google Scholar

[13] Mihashi, H., Nomura, N., Niiseki, S. (1991). Influence of aggregate size on fracture process zone of concrete detected with three dimensional acoustic emission technique. Cement and Concrete Research, 21(5), 737–744. Mihashi H. Nomura N. Niiseki S. 1991 Influence of aggregate size on fracture process zone of concrete detected with three dimensional acoustic emission technique Cement and Concrete Research 21 5 737 744 10.1016/0008-8846(91)90168-H Search in Google Scholar

[14] Woliński, S., Hordijk, D. A., Reinhardt, H. W., Cornelissen, H. A.W. (1987). Influence of aggregate size on fracture mechanics parameters of concrete. International Journal of Cement Composites and Lightweight Concrete, 9(2), 95–103. Woliński S. Hordijk D. A. Reinhardt H. W. Cornelissen H. A.W. 1987 Influence of aggregate size on fracture mechanics parameters of concrete International Journal of Cement Composites and Lightweight Concrete 9 2 95 103 10.1016/0262-5075(87)90025-X Search in Google Scholar

[15] Woliński, S. (1991). Tensile behaviour of concrete and their applications in nonlinear fracture mechanics of concrete. Scientific Works of Rzeszow University of Technology, 15. Woliński S. 1991 Tensile behaviour of concrete and their applications in nonlinear fracture mechanics of concrete Scientific Works of Rzeszow University of Technology 15 Search in Google Scholar

[16] Słowik, M., Błazik-Borowa, E. (2011). Numerical study of fracture process zone width in concrete members. Architecture Civil Engineering Environment, 4(2), 73–78. Słowik M. Błazik-Borowa E. 2011 Numerical study of fracture process zone width in concrete members Architecture Civil Engineering Environment 4 2 73 78 Search in Google Scholar

[17] Wang, X., Saifullah, H.A., Nishikawa, H., Nakarai, K. (2020). Effect of water-cement ratio, aggregate type, and curing temperature on the fracture energy of concrete. Construction and Building Materials, 259, 119646. Wang X. Saifullah H.A. Nishikawa H. Nakarai K. 2020 Effect of water-cement ratio, aggregate type, and curing temperature on the fracture energy of concrete Construction and Building Materials 259 119646 10.1016/j.conbuildmat.2020.119646 Search in Google Scholar

[18] Bažant, Z.P., Pfeiffer, P. A. (1987). Determination of Fracture Energy from Size Effect and Brittleness Number. ACI Materials Journal, 84(6), 463–480. Bažant Z.P. Pfeiffer P. A. 1987 Determination of Fracture Energy from Size Effect and Brittleness Number ACI Materials Journal 84 6 463 480 Search in Google Scholar

[19] Bažant, Z. P. (2001). Concrete fracture models: Testing and practice. Engineering Fracture Mechanics, 69(2), 165–205. Bažant Z. P. 2001 Concrete fracture models: Testing and practice Engineering Fracture Mechanics 69 2 165 205 10.1016/S0013-7944(01)00084-4 Search in Google Scholar

[20] Jenq, Y., Shah, S. P. (1985). Two parameter fracture model for concrete. Journal of Engineering Mechanics, 111(10), 1227–1241. https://doi.org/10.1061/(ASCE)0733-9399(1985)111:10(1227). Jenq Y. Shah S. P. 1985 Two parameter fracture model for concrete Journal of Engineering Mechanics 111 10 1227 1241 https://doi.org/10.1061/(ASCE)0733-9399(1985)111:10(1227) 10.1061/(ASCE)0733-9399(1985)111:10(1227) Search in Google Scholar

[21] Ince, R., Alyamaç, K. E. (2008). Determination of fracture parameters of concrete based on water-cement ratio. Indian Journal of Engineering & Materials Sciences, 15, 14–22. Ince R. Alyamaç K. E. 2008 Determination of fracture parameters of concrete based on water-cement ratio Indian Journal of Engineering & Materials Sciences 15 14 22 Search in Google Scholar

[22] Karamloo, M., Mazloom, M., Payganeh, G. (2016). Influences of water to cement ratio on brittleness and fracture parameters of self-compacting lightweight concrete. Engineering Fracture Mechanics, 168, part A, 227–241. Karamloo M. Mazloom M. Payganeh G. 2016 Influences of water to cement ratio on brittleness and fracture parameters of self-compacting lightweight concrete Engineering Fracture Mechanics 168 part A 227 241 10.1016/j.engfracmech.2016.09.011 Search in Google Scholar

[23] Bharatkumar, B. H., Raghuprasad, B. K., Ramachandramurthy, D. S., Narayanan, R., Gopalakrishnan, S. (2005). Effect of fly ash and slag on the fracture characteristics of high performance concrete. Materials and Structures volume, 38, 63–72. Bharatkumar B. H. Raghuprasad B. K. Ramachandramurthy D. S. Narayanan R. Gopalakrishnan S. 2005 Effect of fly ash and slag on the fracture characteristics of high performance concrete Materials and Structures volume 38 63 72 10.1007/BF02480576 Search in Google Scholar

[24] Sundara, K.T., Iyengar, R., Raviraj, S., VenkateswaraGupta, A. (1995). Graphical method to determine the parameters of the two-parameter fracture model for concrete. Engineering Fracture Mechanics, 51(5), 851–859. Sundara K.T. Iyengar R. Raviraj S. VenkateswaraGupta A. 1995 Graphical method to determine the parameters of the two-parameter fracture model for concrete Engineering Fracture Mechanics 51 5 851 859 10.1016/0013-7944(94)00299-W Search in Google Scholar

[25] Jansen, D. C., Weiss, W. J., Schleuchardt, S. H. F. (2000). Modified Testing Procedure for the Two Parameter Fracture Model for Concrete. The Proceedings of the 14th Engineering Mechanics Conference (EM2000): Austin, TX. Jansen D. C. Weiss W. J. Schleuchardt S. H. F. 2000 Modified Testing Procedure for the Two Parameter Fracture Model for Concrete The Proceedings of the 14th Engineering Mechanics Conference (EM2000) Austin, TX Search in Google Scholar

[26] Carpinteri, A., Berto, F., Fortese, G., Ronchei, C., Scorza, D., Vantadori, S. (2017). Modified two-parameter fracture model for bone. Engineering Fracture Mechanics, 174, 44–53. Carpinteri A. Berto F. Fortese G. Ronchei C. Scorza D. Vantadori S. 2017 Modified two-parameter fracture model for bone Engineering Fracture Mechanics 174 44 53 10.1016/j.engfracmech.2016.11.002 Search in Google Scholar

[27] Carpinteri, A., Fortese, G., Ronchei, C., Scorza, D., Vantadori, S. (2017). Mode I fracture toughness of fibre reinforced concrete. Theoretical and Applied Fracture Mechanics, 91, 66–75. Carpinteri A. Fortese G. Ronchei C. Scorza D. Vantadori S. 2017 Mode I fracture toughness of fibre reinforced concrete Theoretical and Applied Fracture Mechanics 91 66 75 10.1016/j.tafmec.2017.03.015 Search in Google Scholar

[28] Xu, Sh., Reinhardt, H. W. (1999). Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part I: Experimental investigation of crack propagation. International Journal of Fracture, 98, 111–149. Xu Sh. Reinhardt H. W. 1999 Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part I: Experimental investigation of crack propagation International Journal of Fracture 98 111 149 10.1023/A:1018668929989 Search in Google Scholar

[29] Neimitz, A. (1998). Fracture Mechanics. PWN, Warszawa. Neimitz A. 1998 Fracture Mechanics PWN Warszawa Search in Google Scholar

[30] Xu, S., Reinhardt, H. W. (1999). Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part II: Analytical evaluating and practical measuring methods for three-point bending notched beams. International Journal of Fracture, 98, 151–177. Xu S. Reinhardt H. W. 1999 Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part II: Analytical evaluating and practical measuring methods for three-point bending notched beams International Journal of Fracture 98 151 177 10.1023/A:1018740728458 Search in Google Scholar

[31] Xu, S., Reinhardt, H. W. (1999). Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part III: Compact tension specimens and wedge splitting specimens. International Journal of Fracture, 98, 179–193. Xu S. Reinhardt H. W. 1999 Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part III: Compact tension specimens and wedge splitting specimens International Journal of Fracture 98 179 193 10.1023/A:1018788611620 Search in Google Scholar

[32] Wang, B., Dai, J.G., Zhang, X.F., Xu, S.L. (2010). Experimental study on the double-k fracture parameters and brittleness of concrete with different strengths. Fracture Mechanics of Concrete and Concrete Structures – Assessment, Durability, Monitoring and Retrofitting of Concrete Structures -B. H. Oh, et al. (eds), Korea Concrete Institute, 703–708. Wang B. Dai J.G. Zhang X.F. Xu S.L. 2010 Experimental study on the double-k fracture parameters and brittleness of concrete with different strengths. Fracture Mechanics of Concrete and Concrete Structures – Assessment, Durability, Monitoring and Retrofitting of Concrete Structures Oh B. H. (eds), Korea Concrete Institute 703 708 Search in Google Scholar

[33] Kumar, S., Pandey, S. R., Srivastava, A.K.L. (2014). Determination of double-K fracture parameters of concrete using peak load method. Engineering Fracture Mechanics, 131, 471–484. Kumar S. Pandey S. R. Srivastava A.K.L. 2014 Determination of double-K fracture parameters of concrete using peak load method Engineering Fracture Mechanics 131 471 484 10.1016/j.engfracmech.2014.09.004 Search in Google Scholar

[34] Kumar, S., Pandey, S. R., Srivastava, A.K.L. (2013). Analytical methods for determination of double-K fracture parameters of concrete. Advances in Concrete Construction, 1(4), 319–340. DOI: http://dx.doi.org/10.12989/acc2013.1.4.319 Kumar S. Pandey S. R. Srivastava A.K.L. 2013 Analytical methods for determination of double-K fracture parameters of concrete Advances in Concrete Construction 1 4 319 340 DOI: http://dx.doi.org/10.12989/acc2013.1.4.319 10.12989/acc2013.1.4.319 Search in Google Scholar

[35] Qing, L. B., Nie, Y. T., Wang, J., Hu, Y. (2017). A simplified extreme method for determining double-K fracture parameters of concrete using experimental peak load. Fatigue & Fracture of Engineering Materials & Structures 40(2), 254–266. Qing L. B. Nie Y. T. Wang J. Hu Y. 2017 A simplified extreme method for determining double-K fracture parameters of concrete using experimental peak load Fatigue & Fracture of Engineering Materials & Structures 40 2 254 266 10.1111/ffe.12493 Search in Google Scholar

[36] Qing, L. B., Nie, Y. T. (2020). The relationship between double-K parameters of concrete based on fracture extreme theory. Journal of Theoretical and Applied Mechanics, 58(1), 59–71. Qing L. B. Nie Y. T. 2020 The relationship between double-K parameters of concrete based on fracture extreme theory Journal of Theoretical and Applied Mechanics 58 1 59 71 10.15632/jtam-pl/115240 Search in Google Scholar

[37] Kumar, S., Barai, S.V. (2012). Size-effect of fracture parameters for crack propagation in concrete: a comparative study. Computers and Concrete, 9(1), 1–19. Kumar S. Barai S.V. 2012 Size-effect of fracture parameters for crack propagation in concrete: a comparative study Computers and Concrete 9 1 1 19 10.12989/cac.2012.9.1.001 Search in Google Scholar

[38] Walraven, J.C. (2007). Fracture mechanics of concrete and its role in explaining structural behaviour. Fracture Mechanics of Concrete and Concrete Structures (FRAMCOS 6), 1265–1275. Walraven J.C. 2007 Fracture mechanics of concrete and its role in explaining structural behaviour Fracture Mechanics of Concrete and Concrete Structures (FRAMCOS 6) 1265 1275 Search in Google Scholar

[39] Tang, T., Ouyang, C., Libardi, W., Shah, S.P. (1995). Determination of K Ics and CTOD c from peak loads and relationship between two-parameter fracture model and size effect model. Fracture Mechanics of Concrete Structures, 3 (Edited by F.H. Wittmann), 135–144. Tang T. Ouyang C. Libardi W. Shah S.P. 1995 Determination of K Ics and CTOD c from peak loads and relationship between two-parameter fracture model and size effect model Fracture Mechanics of Concrete Structures 3 (Edited by Wittmann F.H. 135 144 Search in Google Scholar

[40] Bhowmik, S., Ray, S. (2019). An experimental approach for characterization of fracture process zone in concrete. Engineering Fracture Mechanics, 211, 401–419. Bhowmik S. Ray S. 2019 An experimental approach for characterization of fracture process zone in concrete Engineering Fracture Mechanics 211 401 419 10.1016/j.engfracmech.2019.02.026 Search in Google Scholar

[41] Bažant, Z.P., Kazemi, M.T. (1991). Size dependence of concrete fracture energy determined by RILEM work-of-fracture method. Int. J.Fract., 51, 121–138. Bažant Z.P. Kazemi M.T. 1991 Size dependence of concrete fracture energy determined by RILEM work-of-fracture method Int. J.Fract. 51 121 138 10.1007/978-94-011-3638-9_9 Search in Google Scholar

[42] Bažant, Z.P., Pfeiffer, P. A. (1987). Determination of Fracture Energy from Size Effect and Brittleness Number. ACI Materials Journal, 84(6), 463–480. Bažant Z.P. Pfeiffer P. A. 1987 Determination of Fracture Energy from Size Effect and Brittleness Number ACI Materials Journal 84 6 463 480 Search in Google Scholar

[43] Hoover, C.G., Bažant, Z.P. (2014). Cohesive crack, size effect, crack band and work-of-fracture models compared to comprehensive concrete fracture tests. Int J Fract 187, 133–143. https://doi.org/10.1007/s10704-013-9926-0 Hoover C.G. Bažant Z.P. 2014 Cohesive crack, size effect, crack band and work-of-fracture models compared to comprehensive concrete fracture tests Int J Fract 187 133 143 https://doi.org/10.1007/s10704-013-9926-0 10.1007/s10704-013-9926-0 Search in Google Scholar

[44] Carpinteri, A. (1992). Applications of Fracture Mechanics to Reinforced Concrete. CRC Press. Carpinteri A. 1992 Applications of Fracture Mechanics to Reinforced Concrete CRC Press Search in Google Scholar

[45] Cifuentes, H., Alcalde, M., Medina, F. (2013). Comparison of the Size-Independent Fracture Energy of Concrete obtained by Two Test Methods. In van Mier et al. (Eds). Proceedings of 8th International Conference on Fractur Mechanics of Concrete and ConcreteStructures (FraMCoS-8), 1–8. Cifuentes H. Alcalde M. Medina F. 2013 Comparison of the Size-Independent Fracture Energy of Concrete obtained by Two Test Methods. In Mier van (Eds). Proceedings of 8th International Conference on Fractur Mechanics of Concrete and ConcreteStructures (FraMCoS-8) 1 8 Search in Google Scholar

[46] Gustafsson, P. J., Hillerborg, A. (1988). Sensitivity in Shear Strength of Longitudinally Reinforced Concrete Beams to Fracture Energy of Concrete. Structural Journal, 85(3), 286–294. Gustafsson P. J. Hillerborg A. 1988 Sensitivity in Shear Strength of Longitudinally Reinforced Concrete Beams to Fracture Energy of Concrete Structural Journal 85 3 286 294 Search in Google Scholar

[47] Weibull, W. (1951). A Statistical Distribution Function of Wide Applicability. ASME Journal of Applied Mechanics, 18, 293–297. Weibull W. 1951 A Statistical Distribution Function of Wide Applicability ASME Journal of Applied Mechanics 18 293 297 10.1115/1.4010337 Search in Google Scholar

[48] Duan, K., Hu, X. Z., Wittmann, F. H. (2006). Scaling of quasi-brittle fracture: Boundary and size effect. Mech. Mater., 38, 128–141. Duan K. Hu X. Z. Wittmann F. H. 2006 Scaling of quasi-brittle fracture: Boundary and size effect Mech. Mater. 38 128 141 10.1016/j.mechmat.2005.05.016 Search in Google Scholar

[49] Hu, X. Z. (2002). An asymptotic approach to size effect on fracture toughness and fracture energy of composites. Engineering Fracture Mechanics, 69(5), 555–564. Hu X. Z. 2002 An asymptotic approach to size effect on fracture toughness and fracture energy of composites Engineering Fracture Mechanics 69 5 555 564 10.1016/S0013-7944(01)00102-3 Search in Google Scholar

[50] Hoover, C. G., Bažant, Z. P. (2014). Universal Size-Shape Effect Law Based on Comprehensive Concrete Fracture Tests. Journal of Engineering Mechanics, 140(3), 473–479. Hoover C. G. Bažant Z. P. 2014 Universal Size-Shape Effect Law Based on Comprehensive Concrete Fracture Tests Journal of Engineering Mechanics 140 3 473 479 10.1061/(ASCE)EM.1943-7889.0000627 Search in Google Scholar

[51] Carpinteri, A., Chiaia, B., Ferro, G. (1995). Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Materials and Structures, 28, 311–317. Carpinteri A. Chiaia B. Ferro G. 1995 Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder Materials and Structures 28 311 317 10.1007/BF02473145 Search in Google Scholar

[52] Carpinteri, A., Chiaia, B. (1997). Multifractal scaling laws in the breaking behaviour of disordered materials. Chaos, Solitons & Fractals, 8(2), 135–150. Carpinteri A. Chiaia B. 1997 Multifractal scaling laws in the breaking behaviour of disordered materials Chaos, Solitons & Fractals 8 2 135 150 10.1016/S0960-0779(96)00088-4 Search in Google Scholar

[53] Duan, K., Hu, X., Wittmann, F.H. (2003). Boundary effect on concrete fracture and non-constant fracture energy distribution. Engineering Fracture Mechanics, 70(16), 2257–2268. Duan K. Hu X. Wittmann F.H. 2003 Boundary effect on concrete fracture and non-constant fracture energy distribution Engineering Fracture Mechanics 70 16 2257 2268 10.1016/S0013-7944(02)00223-0 Search in Google Scholar

[54] Duan, K., Hu, X. (2004). Asymptotic analysis of boundary effect on strength of concrete. Conference: FraMCos-5, vol. 1. Duan K. Hu X. 2004 Asymptotic analysis of boundary effect on strength of concrete Conference: FraMCos-5 vol. 1 Search in Google Scholar