Previous |  Up |  Next


computational mechanics; quasi-brittle fracture; nonlocal elasticity; smeared damage; extended finite element method
Computational analysis of quasi-brittle fracture in cement-based and similar composites, supplied by various types of rod, fibre, etc. reinforcement, is crucial for the prediction of their load bearing ability and durability, but rather difficult because of the risk of initiation of zones of microscopic defects, followed by formation and propagation of a large number of macroscopic cracks. A reasonable and complete deterministic description of relevant physical processes is rarely available. Thus, due to significance of such materials in the design and construction of buildings, semi-heuristic computational models must be taken into consideration. These models generate mathematical problems, whose solvability is not transparent frequently, which limits the credibility of all results of ad hoc designed numerical simulations. In this short paper such phenomena are demonstrated on a simple model problem, covering both micro- and macro-cracking, with references to needful generalizations and more realistic computational settings.
[1] Altan, S.: Existence in nonlocal elasticity. Arch. Mech. 47 (1989), 25–36.
[2] Bažant, Z.P.: Why continuum damage is nonlocal: micromechanics arguments. J. Eng. Mech. 117 (1991), 1070–1089.
[3] Bermúdez de Castro, A.: Continuum Thermomechanics. Birkhäuser, Basel, 2005. MR 2145925
[4] Bybordiani, M., Dias da Costa, D.: A consistent finite element approach for dynamic crack propagation with explicit time integration. Comput. Methods Appl. Mech. Eng. 376 (2021), 1–32, 113652. DOI 10.1016/j.cma.2020.113652 | MR 4200540
[5] de Vree, J.H.P., Brekelmans, W.A.M., van Gils, M.A.J.: Comparison of nonlocal approaches in continuum damage mechanics. Comput. Struct. 55 (1995), 581–588. DOI 10.1016/0045-7949(94)00501-S
[6] Drábek, P., Milota, I.: Methods of Nonlinear Analysis. Birkhäuser, Basel, 2013. MR 3025694
[7] Eringen, A.C.: Theory of Nonlocal Elasticity and Some Applications. Tech. report, Princeton University, Princeton, 1984.
[8] Evgrafov, A., Bellido, J.-C.: From nonlocal Eringen’s model to fractional elasticity. Math. Mech. Solids 24 (2019), 1935–1953. DOI 10.1177/1081286518810745 | MR 3954360
[9] Fasshauer, G.E., Ye, Q.: Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators. Numer. Math. 119 (2011), 585–611. DOI 10.1007/s00211-011-0391-2 | MR 2845629
[10] Fries, T.P., Belytschko, T.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Methods Eng. 68 (2006), 1358–1385. DOI 10.1002/nme.1761
[11] Giry, C., Dufour, F., Mazars, J.: Stress-based nonlocal damage model. Int. J. Solids Struct. 48 (2011), 3431–3443. DOI 10.1016/j.ijsolstr.2011.08.012
[12] Hashiguchi, K.: Elastoplasticity Theory. Springer Berlin, 2014. MR 3235845
[13] Havlásek, P., Grassl, P., Jirásek, M.: Analysis of size effect on strength of quasi-brittle materials using integral-type nonlocal models. Eng. Fract. Mech. 157 (2016), 72–85. DOI 10.1016/j.engfracmech.2016.02.029
[14] Ju, J.W.: Isotropic and anisotropic damage variables in continuum damage. J. Eng. Mech. 116 (1990), 2764–2770.
[15] Kamińska, I., Szwed, A.: A thermodynamically consistent model of quasibrittle elastic damaged materials based on a novel Helmholtz potential and dissipation function. MDPI Materials 14 (2021), 1–30, 6323.
[16] Kozák, V., Chlup, Z., Padělek, P., Dlouhá, I.: Prediction of the traction separation law of ceramics using iterative finite element modelling. Solid State Phenomena 258 (2017), 186–189. DOI 10.4028/
[17] Li, H., Li, J., Yuan, H.: A review of the extended finite element method on macrocrack and microcrack growth simulations. Theor. Appl. Fract. Mech. 97 (2018), 236–249. DOI 10.1016/j.tafmec.2018.08.008
[18] Mariani, S., Perego, U.: Extended finite element method for quasi-brittle fracture. Int. J. Numer. Meth. Engn. 58 (2003), 103–126. DOI 10.1002/nme.761 | MR 1999981
[19] Mielke, A., Roubíček, T.: Rate-Independent Systems. Springer, New York, 2015. MR 3380972
[20] Mousavi, S.M.: Dislocation-based fracture mechanics within nonlocal and gradient elasticity of bi-Helmholtz type. Int. J. Solids Struct. 87 (2016), 92–93, 105–120. DOI 10.1016/j.ijsolstr.2015.10.033
[21] Peerlings, R.H.J., Borst, , Brekelmans, W.A.M., Geers, M.: Gradient enhanced damage modelling of concrete fracture. Int. J. Numer. Anal. Methods Geomech. 3 (1998), 323–342.
[22] Pijaudier-Cabot, G., Mazars, J.: Damage models for concrete. Handbook of Materials Behavior Models (Lemaitre, J., ed.), Academic Press, Cambridge (Massachusetts, USA), 2001, pp. 500–512.
[23] Pike, M.G., Oskay, C.: XFEM modeling of short microfiber reinforced composites with cohesive interfaces. Finite Elem. Anal. Des. 106 (2015), 16–31. DOI 10.1016/j.finel.2015.07.007
[24] Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel, 2005. MR 2176645
[25] Skala, V.: A practical use of radial basis functions interpolation and approximation. Investigación Operacional 37 (2016), 137–144. MR 3479842
[26] Štekbauer, H., Němec, I., Lang, R., Burkart, D., ValaSte22, J.: On a new computational algorithm for impacts of elastic bodies. Appl. Math. 67 (2022), 28 pp., in print. DOI 10.21136/AM.2022.0129-21 | MR 4505704
[27] Sumi, Y.: Mathematical and Computational Analyses of Cracking Formation. Springer, Tokyo, 2014. MR 3234571
[28] Sun, Y., Edwards, M.G., Chen, B., Li, C.: A state-of-the-art review of crack branching. Eng. Fract. Mech. 257 (2021), 1–33, 108036.
[29] Szabó, B., Babuška, I.: Finite Element Analysis: Method, Verification and Validation. J. Wiley & Sons, Hoboken, 2021. MR 1164869
[30] Turner, M.J., Clough, R.W., Martin, H.C., Top, L.J.: Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences 23 (1956), 805–823.
[31] Vala, J.: On a computational smeared damage approach to the analysis of strength of quasi-brittle materials. WSEAS Trans. Appl. Theor. Mech. 16 (2021), 283–292. DOI 10.37394/232011.2021.16.31
[32] Vala, J., Kozák, V.: Computational analysis of quasi-brittle fracture in fibre reinforced cementitious composites. Theor. Appl. Fract. Mech.. 107 (2020), 1–8, 102486. DOI 10.1016/j.tafmec.2020.102486
[33] Vala, J., Kozák, V.: Nonlocal damage modelling of quasi-brittle composites. Appl. Math. 66 (2021), 701–721. DOI 10.21136/AM.2021.0281-20 | MR 4342610
[34] Vala, J., Kozák, V., Jedlička, M.: Scale bridging in computational modelling of quasi-brittle fracture of cementitious composites. Solid State Phenomena 325 (2021), 56–64. DOI 10.4028/
[35] 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. 225 (2021), 1–13, 111048. DOI 10.1016/j.ijsolstr.2021.111048
[36] Zlámal, M.: On the finite element method. Numer. Math. 12 (1968), 394–409. DOI 10.1007/BF02161362
Partner of
EuDML logo