Numerical approaches to the modelling of quasi-brittle crack propagation

Vala, Jiří

Archivum mathematicum, Tome 59 (2023) no. 3, pp. 295-303

Abstract (VO)

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.

Détail

DOI : 10.5817/AM2023-3-295

Classification : 65M20, 65M60, 74A40, 74A45, 74H15
Keywords: computational mechanics; quasi-brittle fracture; nonlocal elasticity; smeared damage; extended finite element method

Bibliographie

[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

[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 | MR

[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

[6] Drábek, P., Milota, I.: Methods of Nonlinear Analysis. Birkhäuser, Basel, 2013. | MR

[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 | MR

[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 | MR

[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

[11] Giry, C., Dufour, F., Mazars, J.: Stress-based nonlocal damage model. Int. J. Solids Struct. 48 (2011), 3431–3443. | DOI

[12] Hashiguchi, K.: Elastoplasticity Theory. Springer Berlin, 2014. | MR

[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

[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

[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

[18] Mariani, S., Perego, U.: Extended finite element method for quasi-brittle fracture. Int. J. Numer. Meth. Engn. 58 (2003), 103–126. | DOI | MR

[19] Mielke, A., Roubíček, T.: Rate-Independent Systems. Springer, New York, 2015. | MR

[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

[21] Peerlings, R.H.J., R.de 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

[24] Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel, 2005. | MR

[25] Skala, V.: A practical use of radial basis functions interpolation and approximation. Investigación Operacional 37 (2016), 137–144. | MR

[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 | MR

[27] Sumi, Y.: Mathematical and Computational Analyses of Cracking Formation. Springer, Tokyo, 2014. | MR

[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

[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

[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

[33] Vala, J., Kozák, V.: Nonlocal damage modelling of quasi-brittle composites. Appl. Math. 66 (2021), 701–721. | DOI | MR

[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

[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

[36] Zlámal, M.: On the finite element method. Numer. Math. 12 (1968), 394–409. | DOI