Applications of Mathematics, Vol. 66, No. 6, pp. 815-836, 2021


Non-local damage modelling of quasi-brittle composites

Jiří Vala, Vladislav Kozák

Received September 30, 2020.   Published online July 1, 2021.

Abstract:  Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering.
Keywords:  quasi-brittle composite; steel fibre concrete, micro- and macro-fracture, non-local viscoelasticity; cohesive interface; partial differential equations of evolution; method of discretization in time; extended finite element method
Classification MSC:  74R10, 74H15, 74S05, 74S20, 74E30


References:
[1] S. B. Altan: Existence in nonlocal elasticity. Arch. Mech. 41 (1989), 25-36. MR 1065652 | Zbl 0725.73022
[2] I. Babuška, J. M. Melenk: The partition of unity method. Int. J. Numer. Methods Eng. 40 (1997), 727-758. DOI 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N | MR 1429534 | Zbl 0949.65117
[3] T. Belytschko, T. Black: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45 (1999), 601-620. DOI 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S | Zbl 0943.74061
[4] A. Bermúdez de Castro: Continuum Thermomechanics. Progress in Mathematical Physics 43. Birkhäuser, Basel (2005). DOI 10.1007/3-7643-7383-0 | MR 2145925 | Zbl 1070.74001
[5] G. S. Bhatia, G. Arora: Radial basis function methods for solving partial differential equations: A review. Indian J. Sci. Technol. 9 (2016), Article ID 45, 18 pages. DOI 10.17485/ijst/2016/v9i45/105079
[6] L. Bouhala, A. Makradi, S. Belouettar, H. Kiefer-Kamal, P. Fréres: Modelling of failure in long fibres reinforced composites by X-FEM and cohesive zone model. Composites B, Eng. 55 (2013), 352-361. DOI 10.1016/j.compositesb.2012.12.013
[7] J. K. Bunkure: Lebesgue-Bochner spaces and evolution triples. Int. J. Math. Appl. 7 (2019), 41-52.
[8] A. Cianchi, V. Maz'ya: Sobolev inequalities in arbitrary domains. Adv. Math. 293 (2016), 644-696. DOI 10.1016/j.aim.2016.02.012 | MR 4074620 | Zbl 1346.46022
[9] D. S. Clark: Short proof of a discrete Gronwall inequality. Discrete Appl. Math. 16 (1987), 279-281. DOI 10.1016/0166-218X(87)90064-3 | MR 0878027 | Zbl 0612.39004
[10] G. Dal Maso, G. Lazzaroni: Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem. Discrete Contin. Dyn. Syst. 31 (2011), 1219-1231. DOI 10.3934/dcds.2011.31.1219 | MR 2836349 | Zbl 1335.74051
[11] G. Del Piero, D. R. Owen: Structured deformations of continua. Arch. Ration. Mech. Anal. 124 (1993), 99-155. DOI 10.1007/BF00375133 | MR 1237908 | Zbl 0795.73005
[12] L. Dlouhý, S. Pouillon: Application of the design code for steel-fibre-reinforced concrete into finite element software. Beton 116 (2020), 8-13.
[13] P. Drábek, J. Milota: Methods of Nonlinear Analysis: Applications to Differential Equations. Birkhäuser Advanced Texts. Basler Lehrbücher. Birkhäuser, Basel (2013). DOI 10.1007/978-3-0348-0387-8 | MR 3025694 | Zbl 1264.35001
[14] J. Eliáš, M. Vořechovský, J. Skoček, Z. P. Bažant: Stochastic discrete meso-scale simulations of concrete fracture: Comparison to experimental data. Eng. Fract. Mech. 135 (2015), 1-16. DOI 10.1016/j.engfracmech.2015.01.004
[15] E. Emmrich, D. Puhst: Measure-valued and weak solutions to the nonlinear peridynamic model in nonlocal elastodynamics. Nonlinearity 28 (2015), 285-307. DOI 10.1088/0951-7715/28/1/285 | MR 3297136 | Zbl 1312.35163
[16] A. C. Eringen: Theory of Nonlocal Elasticity and Some Applications. Technical Report 62. Princeton University Press, Princeton (1984).
[17] A. C. Eringen: Nonlocal Continuum Field Theories. Springer, New York (2002). DOI 10.1007/b97697 | MR 1918950 | Zbl 1023.74003
[18] A. Evgrafov, J. C. Bellido: From non-local Eringen's model to fractional elasticity. Math. Mech. Solids 24 (2019), 1935-1953. DOI 10.1177/1081286518810745 | MR 3954360 | Zbl 1425.74093
[19] T.-P. Fries, T. Belytschko: 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 | Zbl 1129.74045
[20] Z. Gao, L. Zhang, W. Yu: A nonlocal continuum damage model for brittle fracture. Eng. Fract. Mech. 189 (2018), 481-500. DOI 10.1016/j.engfracmech.2017.10.019
[21] C. Giry, F. Dufour, J. Mazars: Stress-based nonlocal damage model. Int. J. Solids Struct. 48 (2011), 3431-3443. DOI 10.1016/j.ijsolstr.2011.08.012
[22] S. Grija, D. Shanthini, S. Abinaya: A review on fiber reinforced concrete. Int. J. Civil Eng. Technol. 7 (2016), 386-392.
[23] K. Hashiguchi: Elastoplasticity Theory. Lecture Notes in Applied and Computational Mechanics 69. Springer, Berlin (2014). DOI 10.1007/978-3-642-35849-4 | MR 3235845 | Zbl 1318.74001
[24] P. Havlásek, P. Grassl, M. Jirásek: 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
[25] A. Hoekstra: Design methodologies for steel-fibre-reinforced concrete and a new methodology for a real time quality control. Beton 116 (2020), 44-49.
[26] C. O. Horgan: Eigenvalue estimates and the trace theorem. J. Math. Anal. Appl. 69 (1979), 231-242. DOI 10.1016/0022-247X(79)90190-2 | MR 0535293 | Zbl 0412.35073
[27] A. Javili, R. Morasata, E. Oterkus, S. Oterkus: Peridynamics review. Math. Mech. Solids 24 (2019), 3714-3739. DOI 10.1177/1081286518803411 | MR 4000179 | Zbl 07273389
[28] M. Jirásek: Damage and smeared crack models. Numerical Modeling of Concrete Cracking CISM. Courses and Lectures 532. Springer, Wien (2011), 1-49. DOI 10.1007/978-3-7091-0897-0_1 | Zbl 1247.74055
[29] M. Kaliske, H. Dal, R. Fleischhauer, C. Jenkel, C. Netzker: Characterization of fracture processes by continuum and discrete modelling. Comput. Mech. 50 (2012), 303-320. DOI 10.1007/s00466-011-0578-5 | MR 2967876 | Zbl 1398.74347
[30] P. Kawde, A. Warudkar: Steel fibre reinforced concrete: A review. Int. J. Eng. Sci. Res. Technol. 6 (2017), 130-133. DOI 10.5281/zenodo.233321
[31] A. R. Khoei: Extended Finite Element Method: Theory and Applications. Wiley Series in Computational Mechanics. John Wiley & Sons, New York (2015). DOI 10.1002/9781118869673 | Zbl 1315.74001
[32] V. Kozák, Z. Chlup: Modelling of fibre-matrix interface of brittle matrix long fibre composite by application of cohesive zone method. Key Eng. Materials 465 (2011), 231-234. DOI 10.4028/www.scientific.net/KEM.465.231
[33] V. Kozák, Z. Chlup, P. Padělek, I. Dlouhý: Prediction of the traction separation law of ceramics using iterative finite element modelling. Solid State Phenomena 258 (2017), 186-189. DOI 10.4028/www.scientific.net/SSP.258.186
[34] M. Lazar, G. A. Maugin, E. C. Aifantis: On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43 (2006), 1404-1421. DOI 10.1016/j.ijsolstr.2005.04.027 | MR 2200992 | Zbl 1120.74342
[35] G. Lazzaroni: Quasistatic crack growth in finite elasticity with Lipschitz data. Ann. Mat. Pura Appl. (4) 190 (2011), 165-194. DOI 10.1007/s10231-010-0145-2 | MR 2747470 | Zbl 1215.35156
[36] X. Li, J. Chen: An extended cohesive damage model for simulating arbitrary damage propagation in engineering materials. Comput. Methods Appl. Mech. Eng. 315 (2017), 744-759. DOI 10.1016/j.cma.2016.11.029 | MR 3595276 | Zbl 1439.74334
[37] X. Li, W. Gao, W. Liu: A mesh objective continuum damage model for quasi-brittle crack modelling and finite element implementation. Int. J. Damage Mech. 28 (2019), 1299-1322. DOI 10.1177/1056789518823876
[38] R. W. Macek, S. A. Silling: Peridynamics via finite element analysis. Finite Elem. Anal. Des. 43 (2007), 1169-1178. DOI 10.1016/j.finel.2007.08.012 | MR 2393568
[39] Z. Majdisova, V. Skala: Radial basis function approximations: Comparison and applications. Appl. Math. Modelling 51 (2017), 728-743. DOI 10.1016/j.apm.2017.07.033 | MR 3694560 | Zbl 07166287
[40] A. Mielke, T. Roubíček: Rate-Independent Systems: Theory and Applications. Applied Mathematical Sciences 193. Springer, New York (2015). DOI 10.1007/978-1-4939-2706-7 | MR 3380972 | Zbl 1339.35006
[41] M. Moradi, A. R. Bagherieh, M. R. Esfahani: Constitutive modeling of steel fiber-reinforced concrete. Int. J. Damage Mech. 29 (2020), 388-412. DOI 10.1177/1056789519851159
[42] M. Morandotti: Structured deformation of continua: Theory and applications. Mathematical Analysis of Continuum Mechanics and Industrial Applications II. Springer, Singapore (2018), 125-136. DOI 0.1007/978-981-10-6283-4_11
[43] N. Nakamura: Extended Rayleigh damping model. Front. Built Environ. 2 (2016), Article ID 14, 13 pages. DOI 10.3389/fbuil.2016.00014
[44] R. H. J. Peerlings, R. de Borst, W, A. M. Brekelmans, M. Geers: Gradient enhanced damage modelling of concrete fracture. Mech. Cohesive-frictional Mater. 3 (1998), 323-342. DOI 10.1002/(SICI)1099-1484(1998100)3:4<323::AID-CFM51>3.0.CO;2-Z
[45] G. Pijaudier-Cabot, J. Mazars: Damage models for concrete. Section 6.13. Handbook of Materials Behavior Models. Volume II (J. Lemaitre, ed.). Academic Press, London (2001), 500-512. DOI 10.1016/B978-012443341-0/50056-9
[46] M. G. Pike, C. Oskay: XFEM modeling of short microfiber reinforced composites with cohesive interfaces. Finite Elem. Anal. Des. 106 (2005), 16-31. DOI 10.1016/j.finel.2015.07.007
[47] Yu. Z. Povstenko: The nonlocal theory of elasticity and its application to the description of defects in solid bodies. J. Math. Sci. 97 (1999), 3840-3845. DOI 10.1007/BF02364923
[48] P. Ray: Statistical physics perspective of fracture in brittle and quasi-brittle materials. Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 377 (2019), Article ID 20170396, 13 pages. DOI 10.1098/rsta.2017.0396
[49] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications 4. D. Reidel, Dordrecht (1982). MR 0689712 | Zbl 0505.65029
[50] T. Roubíček: Nonlinear Partial Differential Equations with Applications. ISNM. International Series of Numerical Mathematics 153. Birkhäuser, Basel (2005). MR 2176645 | Zbl 1087.35002
[51] T. Roubíček: Thermodynamics of rate-independent processes in viscous solids at small strains. SIAM J. Math. Anal. 42 (2010), 256-297. DOI 10.1137/080729992 | MR 2596554 | Zbl 1213.35279
[52] M. Šilhavý: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Complex Syst. 5 (2017), 191-215. DOI 10.2140/memocs.2017.5.191 | MR 3669123 | Zbl 1447.49026
[53] X. T. Su, Z. J. Yang, G. H. Liu: Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials: A 3D study. Int. J. Solids Struct. 47 (2010), 2336-2345. DOI 10.1016/j.ijsolstr.2010.04.031 | Zbl 1194.74313
[54] Y. Sumi: Mathematical and Computational Analyses of Cracking Formation. Mathematics for Industry (Tokyo) 2. Springer, Tokyo (2014). DOI 10.1007/978-4-431-54935-2 | MR 3234571 | Zbl 1395.74001
[55] E. Svenning, F. Larsson, M. Fagerström: A two-scale modeling framework for strain localization in solids: XFEM procedures and computational aspects. Comput. Struct. 211 (2019), 43-54. DOI 10.1016/j.compstruc.2018.08.003
[56] R. F. Swati, L. H. Wen, H. Elahi, A. A. Khan, S. Shad: Extended finite element method (XFEM) analysis of fiber reinforced composites for prediction of micro-crack propagation and delaminations in progressive damage: A review. Microsyst. Technol. 25 (2019), 747-763. DOI 10.1007/s00542-018-4021-0
[57] J. Vala: Structure identification of metal fibre reinforced cementitious composites. Algoritmy: 20th Conference on Scientific Computing. STU Bratislava, Bratislava (2016), 244-253.
[58] J. Vala, V. Kozák: Computational analysis of quasi-brittle fracture in fibre reinforced cementitious composites. Theor. Appl. Fract. Mech. 107 (2020), Article ID 102486, 8 pages. DOI 10.1016/j.tafmec.2020.102486

Affiliations:   Jiří Vala (corresponding author), Vladislav Kozák, Brno University of Technology, Faculty of Civil Engineering, Institute of Mathematics and Descriptive Geometry, Žižkova 17, 602 00 Brno, Czech Republic, e-mail: vala.j@fce.vutbr.cz, kozak.v@fce.vutbr.cz


 
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