Abstract
In this paper, a mesoscale model of concrete is presented, which considers particles, matrix material and the interfacial transition zone (ITZ) as separate constituents. Particles are represented as ellipsoides, generated according to a prescribed grading curve and placed randomly into the specimen. Algorithms are proposed to generate realistic particle configurations efficiently. The nonlinear behavior is simulated with a cohesive interface model for the ITZ. For the matrix material, different damage/plasticity models are investigated. The simulation of localization requires to regularize the solution, which is performed by using integral type nonlocal models with strain or displacement averaging. Due to the complexity of a mesoscale model for a realistic structure, a multiscale method to couple the homogeneous macroscale with the heterogeneous mesoscale model in a concurrent embedded approach is proposed. This allows an adaptive transition from a full macroscale model to a multiscale model, where only the relevant parts are resolved on a finer scale. Special emphasis is placed on the investigation of different coupling schemes between the different scales, such as the mortar method and the arlequin method, and a discussion of their advantages and disadvantages within the current context. The applicability of the proposed methodology is illustrated for a variety of examples in tension and compression.
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References
Scrivener KL (2004) Cem Concr Compos 26:935
Tasong WA, Lynsdale CJ, Cripps JC (1998) Cem Concr Res 28(10):1453
Lee KM, Buyukozturk O, Oumera A (1992) J Eng Mech 118(10):2031
Caliskan S, Karihaloo B, Barr B (2002) Mag Concr Res 54(6):449
Walz K (1976) Beton 3:95
Walz K (1976) Beton 4:135
Liao KY, Chang PK, Yaw-Nan P, Yang CC (2004) Cem Concr Res 34:977
Diamond S, Huang J (2001) Cem Concr Compos 23:179
Tasong WA, Lynsdale JC, Cripps JC (1999) Cem Concr Res 29:1019
Appa Rao G, Raghu Prasad B (2002) Cem Concr Res 32:253
Caliskan S, Karihaloo B, Barr B (2002) Mag Concr Res 54(6):463
Aquino MJ, Li Z, Shah SP (1995) Adv Cem Based Mater 2:211
Guinea G, El-Sayed K, Rocco C, Elices M, Planas J (2002) Cem Concr Res 32:1961
Feldkamp LA, Goldstein SA, Parfitt AM, Jesion G, Kleerekoper M (1989) J Bone Miner Res 4(1):3
Takano N, Kimura K, Zako M, Kubo F (2003) JSME Int J, Ser A, Solid Mech Mater Eng 46(4):527
Nagai G, Yamada T, Wada A (2000) In: Proceedings of the 8th ICCCBE, Stanford University, California, USA, pp 449–456
Hollister SJ, Kikuchi N (1994) Biotechnol Bioeng 43(7):586
Nagano Y, Ikeda Y, Kawamoto H (2004) In: Miyasaka C, Yokono Y, Bray D, Cho Y-H (eds) ASME/JSME pressure vessels and piping conference, San Diego, vol 484, pp 141–146
Zaitsev YB, Wittmann FH (1981) Mater Struct 14(5):357
Bažant ZP, Tabbara MR, Kazemi MT, Pijaudier-Cabot G (1990) J Eng Mech 116(8):1686
Schlangen E, van Mier JGM (1992) Mater Struct 25(9):534
Häfner S, Eckardt S, Könke C (2003) In: Proceedings of the 16th international conference on the application of computer science and mathematics in architecture and civil engineering, Weimar
Leite J, Slowik V, Mihashi H (2004) Cem Concr Res 34(6):1025
Leite JPB, Slowik V, Apel J (2007) Comput Struct 85(17–18):1293
Garboci E (2002) Cem Concr Res 32:1621
Häfner S, Eckardt S, Luther T, Könke C (2006) Comput Struct 84(7):450
Walraven JC (1980) PhD thesis, Delft University of Technology
Wriggers P, Moftah SO (2006) Finite Elem Anal Des 42(7):623 (Special issue)
Wang Z, Kwan A, Chan H (1999) Comput Struct 70(5):533
Eckardt S, Könke C (2006) In: Gürlebeck K, Könke C (eds) Proceedings of the 17th international conference on the application of computer science and mathematics in architecture and civil engineering, Weimar
Devroye L (1986) Non-uniform random variate generation. Springer, New York
Wittmann FH, Roelfstra PE, Sadouki H (1985) Mater Sci Eng 68(2):239
Wang W, Wang J, Kim MS (2001) Comput Aided Geom Des 18(6):531
Barbeau EJ (2003) Polynomials. Springer, Berlin
Shewchuk JR (1996) In: Lin MC, Manocha D (eds) Applied computational geometry: towards geometric engineering. Lecture notes in computer science, vol 1148. Springer, Berlin, pp 203–222
Geuzaine C, Remacle JF (2008) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities (2008). http://geuz.org/gmsh/
Belytschko T, Black T (1999) Int J Numer Methods Eng 45(5):6001
Wells G, Sluys L (2001) Int J Numer Methods Eng 50:2667
Moës N, Belytschko T (2002) Eng Fract Mech 69:813
Sluys LJ (1992) PhD thesis, Delft University of Technology
Hofstetter G, Mang HA (1995) Computational mechanics of reinforced concrete structures. Vieweg, Weisbaden
Bažant ZP, Pijaudier-Cabot G (1989) J Eng Mech 115(4):755
Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, Cambridge
Kachanov L (1986) Introduction to continuum damage mechanics. Kluwer Academic, Dordrecht
Lemaitre J (1984) Nucl Eng Des 80(2):233
Rashid Y (1968) Nucl Eng Des 7:334
Bažant ZP, Gambarova P (1980) J Struct Div 106(4):819
Bažant ZP, Oh BH (1983) Mat Construct 16(93):155
Willam K, Pramono E, Sture S (1987) In: Shah S, Swartz S (eds) Proceedings of SEM/RILEM international conference on fracture of concrete and rock, pp 192–207
de Borst R, Nauta P (1985) Eng Comput 2(1):35
Cope RJ, Rao PV, Clark LA, Norris P (1980) In: Taylor C, Hinton E, Owen DRJ (eds) Numerical methods for nonlinear problems 1. Pineridge Press, Swansea, pp 457–470
Bažant ZP (1983) J Eng Mech 109(3):849
Gupta AK, Akbar H (1984) J Struct Eng 110(8):1735
Rots J (1988) PhD thesis, Delft University of Technology (1988)
Bažant ZP, Oh BH (1985) J Eng Mech 111(4):559
Bažant ZP, Prat PC (1988) J Eng Mech 114(10):1672
Carol I, Bažant ZP (1997) Int J Solids Struct 34(29):3807
Kuhl E, Ramm E (1998) Mech Cohes-Frict Mater 3:343
Bažant ZP, Caner FC, Carol I, Adley MD, Akers SA (2000) J Eng Mech 126(9):944
Patzak B, Jirásek M (2004) J Eng Mech 130:720
Taylor GI (1938) J Inst Metals Lond 62:307
Jirásek M, Bažant ZP (2001) Inelastic analysis of structures. Wiley, New York
Bažant ZP (1976) ASCE J Eng Mech Div 102(2):331
Pietruszczak S, Mróz Z (1981) Int J Numer Methods Eng 17(3):327
de Vree JHP, Brekelmans WAM, van Gils MAJ (1995) Comput Struct 55(4):581
Jirásek M, Grassl P (2008) Eng Fract Mech 75(8):1921
Mančevski D (1981) PhD thesis, Ruhr University Bochum
Oliver J (1989) Int J Numer Methods Eng 28(2):461
Jirásek M, Zimmermann T (1998) J Eng Mech 124:842
Eringen AC (1966) Int J Eng Sci 4(2):179
Kröner E (1967) Int J Solids Struct 3(5):731
Pijaudier-Cabot G, Bažant ZP (1987) J Eng Mech 113(10):1512
Bažant ZP, Pijaudier-Cabot G (1988) Journal of Applied Mechanics. Trans ASME 55(2):287. Cited By (since 1996): 192
Bažant ZP, Lin FB (1988) J Struct Eng 114(11):2493
Jirásek M (1998) Int J Solids Struct 35(31–32):4133
Bažant ZP (1991) J Eng Mech 117(5):1070
Bažant ZP (1994) J Eng Mech 120(3):593
Bažant ZP, Jirásek M (2002) J Eng Mech 128(11):1119
Jirásek M, Bažant ZP (1994) J Struct Eng ASCE 120(7):1521
Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP (1996) Int J Numer Methods Eng 39:3391
Peerlings RHJ (1999) PhD thesis, Technische Universiteit Eindhoven
Mazars J, Pijaudier-Cabot G (1996) Int J Solids Struct 33(20–22):3327
Krayani A, Pijaudier-Cabot G, Dufour F (2009) Eng Fract Mech 76(14):2217
Borino G, Failla B, Parrinello F (2003) Int J Solids Struct 40(13–14):3621
Askes H, Sluys LJ (2000) Eur J Mech A, Solids 19(3):447
Bažant ZP, Ozbolt J (1990) J Eng Mech 116(11):2485
Jirásek M (1998) Comparison of nonlocal models for damage and fracture. LSC Internal report 98/02, Swiss Federal Institute of Technology (EPFL), Laboratory of Structural and Continuum Mechanics
Jirásek M, Marfia S (2005) Int J Numer Methods Eng 63(1):77
Carol I, Rizzi E, Willam K (2001) Int J Solids Struct 38(4):519
Hansen E, Willam K, Carol I (2001) In: de Borst R, Mazars J, Pijaudier-Cabot G, van Mier J (eds) Fracture mechanics of concrete structures, pp 549–556
Grassl P, Jirásek M (2006) Int J Numer Anal Methods Geomech 30(1):71
Ju J (1989) Int J Solids Struct 25(7):803
Jason L, Huerta A, Pijaudier-Cabot G, Ghavamian S (2006) Comput Methods Appl Mech Eng 195(52):7077
Lubliner J, Oliver J, Oller S, Oñate E (1989) Int J Solids Struct 25(3):299
Ananiev S, Ožbolt J (2004) In: Li V, Leung C, Willam K, Billington S (eds) Fracture mechanics of concrete structures, pp 271–278
Unger JF, Eckardt S, Könke C (2011) Comput. Concr. 8(4):401
Simo J, Hughes T (1997) Computational inelasticity. Springer, Berlin
Grassl P, Jirásek M (2010) Int J Solids Struct 47(7–8):957
Hillerborg A, Modéer M, Petersson P (1976) Cem Concr Res 6:773
Dugdale D (1960) J Mech Phys Solids 8:100
Barenblatt G (1962) Adv Appl Mech 7:55
Tvergaard V (2003) Eng Fract Mech 70:1859
Ortiz M, Pandolfi A (1999) Int J Numer Methods Eng 44(9):1267
Kessler-Kramer C (2002) PhD thesis, Universität Karlsruhe (TH), Germany
Reuss A (1929) Z Angew Math Mech 9(1):49
van Vliet MRA, van Mier JGM (1995) In: Wittman F (ed) Fracture mechanics of concrete structures (FraMCoS-2, Zürich), pp 383–396
Unger JF (2009) PhD thesis, Bauhaus University Weimar
Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials, 2nd edn. Amsterdam, Elsevier
Hill R (1963) J Mech Phys Solids 11(5):357
Hashin Z (1983) Journal of Applied Mechanics. Trans ASME 50(3):481
Gitman I (2006) PhD thesis, Delft University of Technology
Gitman IM, Askes H, Sluys LJ (2007) Eng Fract Mech 74(16):2518
Belytschko T, Song JH (2010) Int J Numer Methods Eng 81(5):537
Kröner E (1958) Z Phys 151:504
Voigt W (1889) Ann Phys 274(12):573
Hill R (1952) Proc Phys Soc A 65(5):349
Hashin Z, Shtrikman S (1963) J Mech Phys Solids 11(2):127
Mura T (1982) Micromechanics of defects in solids. Nijhoff, Dordrecht
Aboudi J (1991) Mechanics of composite materials—a unified micromechanical approach. Amsterdam, Elsevier
Zohdi T, Feucht M, Gross D, Wriggers P (1998) Int J Numer Methods Eng 43(3):493
Suquet RM (1987) In: Sanchez-Palencia E, Zaoui A (eds) Homogenization techniques for composite media: lectures delivered at the CISM international center for mechanical sciences, Udine, Italy, July 1–5, 1985. Lecture notes in physics, vol 272. Springer, Berlin, pp 193–278
Döbert C (2001) Dissertation, Universität Hannover
Reese S (2003) Int J Solids Struct 40(4):951
Ghaboussi J, Garret JH, Wu X (1990) In: Proceedings of the international conference on numerical methods in engineering: theory and applications, Swansea, UK, pp 701–717
Ghaboussi J, Garret J, Wu X (1991) J Eng Mech Div ASCE 117(1):132
Waszczyszyn Z, Ziemianski L (2001) Comput Struct 79(22–25):2261
Furukawa T, Yagawa G (1998) Int J Numer Methods Eng 43(2):195
Lefik M, Schrefler BA (2001) Comput Struct 80(22):1699
Sidarta DE, Ghaboussi J (1998) Comput Geotech 22(1):53
Fu Q, Hashash YMA, Jung S, Ghaboussi J (2007) Comput Geotech 34(5):330
Ghaboussi J, Pecknold D, Zhang M, Haj-Ali R (1998) Struct Eng Mech 42(1):105
Haj-Ali R, Pecknold DA, Ghaboussi J, Voyiadjis GZ (2001) J Eng Mech 127(7):730
Hashash Y, Jung S, Ghaboussi J (2004) Int J Numer Methods Eng 59:989
Unger JF, Könke C (2008) Comput Struct 86(21–22):1994
Smit RJM, Brekelmans WAM, Meijer HEH (1998) Comput Methods Appl Mech Eng 155(1–2):181
Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) Comput Mech 27(1):37
Miehe C, Koch A (2002) Arch Appl Mech 72(4–5):300
Bayreuther C (2004) Dissertation, Universität Stuttgart
Wilson EL (1974) Int J Numer Methods Eng 8(1):198
Feyel F, Chaboche JL (2000) Comput Methods Appl Mech Eng 183(3–4):309
Feyel F (2003) Comput Methods Appl Mech Eng 192(28–30):3233
Ghosh S, Lee K, Raghavan P (2001) Int J Solids Struct 38(14):2335
Kouznetsova V (2002) PhD thesis, Technische Universiteit Eindhoven
Kouznetsova VG, Geers MGD, Brekelmans WAM (2002) Int J Numer Methods Eng 54(8):1235
Kouznetsova VG, Geers MGD, Brekelmans WAM (2004) Comput Methods Appl Mech Eng 193(48–51):5525
Gitman IM, Askes H, Sluys LJ (2008) Eur J Mech A, Solids 27(3):302
Belytschko T, Loehnert S, Song JH (2008) Int J Numer Methods Eng 73(6):869
Song JH, Belytschko T (2009) Compos, Part B, Eng 40(6):417
Noor AK, Kamel HA, Fulton RE (1978) Comput Struct 8(5):621
Hirai I, Wang BP, Pilkey WD (1984) Int J Numer Methods Eng 20(9):1671
Mao KM, Sun CT (1991) Int J Numer Methods Eng 32(1):29
Whitcomb JD (1991) Comput Struct 40(4):1027
Mote CD (1971) Int J Numer Methods Eng 3(4):565
Noor AK, Peters JM (1980) Int J Numer Methods Eng 15(9):1363
Belytschko T, Fish J, Bayliss A (1990) Comput Methods Appl Mech Eng 81(1):71
Hughes TJ (1995) Comput Methods Appl Mech Eng 127(1–4)
Hughes TJR, Feijó GR, Mazzei L, Quincy J (1998) Comput Methods Appl Mech Eng 166(1–2):3
Fish J (1992) Comput Struct 43(3):539
Rank E, Krause R (1997) Comput Struct 64(1–4):139
Krause R, Rank E (2003) Comput Methods Appl Mech Eng 192(35–36):3959
Bernardi C, Maday Y, Patera AT (1994) In: Brezis H, Lions JL (eds) Nonlinear partial differential equations and their applications. Collège de France Seminar, vol XI, Paris, 1989–1991. Pitman research notes in mathematics series, vol 299. Longman Scientific & Technical, Harlow, pp 13–51
Wohlmuth BI (2001) Lectures Notes in Computational Science and Engineering, vol 17
Lamichhane BP, Wohlmuth BI (2004) Computing 72:333
Wohlmuth BI (1999) SIAM J Numer Anal 36(5):1636
Belgacem FB (1999) Numer Math 84(2):173
Ben Dhia H (1998) C R Acad Sci IIB, Mech Phys Astron 326(12):899
Ben-Dhia H, Rateau G (2005) Int J Numer Methods Eng 62(11):1442
Hu H, Belouettar S, Potier-Ferry M, Daya EM (2008) Finite Elem Anal Des 45(1):37
Eckardt S (2009) PhD thesis, Bauhaus-University Weimar
Bauman PT, Dhia HB, Elkhodja N, Oden JT, Prudhomme S (2008) Comput Mech 42(4):511
Xiao SP, Belytschko T (2004) Comput Methods Appl Mech Eng 193(17–20):1645
Eckardt S, Könke C (2008) J Algorithms Comput Technol 2(2):275
Winkler B (2001) PhD thesis, University of Innsbruck
van Vliet MRA, van Mier JGM (2000) Eng Fract Mech 65(2–3):165
van Vliet MRA, van Mier JGM (1998) In: de Borst R, van der Giessen E (eds) Material instabilities in solids. Wiley, New York
Jirásek M, Rolshoven S, Grassl P (2004) Int J Numer Anal Methods Geomech 28(7–8):653
Vořechovský M, Sadílek V (2008) Int J Fract 154(1–2):27
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Unger, J.F., Eckardt, S. Multiscale Modeling of Concrete. Arch Computat Methods Eng 18, 341–393 (2011). https://doi.org/10.1007/s11831-011-9063-8
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DOI: https://doi.org/10.1007/s11831-011-9063-8