##
**A comparative study on the modelling of discontinuous fracture by means of enriched nodal and element techniques and interface elements.**
*(English)*
Zbl 1273.74531

From the summary: In this paper, three different approaches used to model strong discontinuities are studied: a new strong embedded discontinuity technique, designated as the discrete strong embedded discontinuity approach \((DSDA)\), the generalized finite element method \((GFEM)\) and the use of interface elements. First, it is shown that all three descriptions are based on the same variational formulation. However, the main differences between these models lie in the way the discontinuity is represented in the finite element mesh, which is explained in the paper. Main focus is on the differences between the element enrichment technique, used in the \(DSDA\) and the nodal enrichment technique adopted in the \(GFEM\). In both cases, global enhanced degrees of freedom are adopted. Next, the numerical integration of the discretised equations in the three methods is addressed and some important differences are discussed. Two types of numerical tests are presented: first, simple academic examples are used to emphasize the differences found in the formulations and next, some benchmark tests are computed.

PDF
BibTeX
XML
Cite

\textit{D. Dias-da-Costa} et al., Int. J. Fract. 161, No. 1, 97--119 (2010; Zbl 1273.74531)

### References:

[1] | Alfaiate J, Sluys L (2005) Discontinuous numerical modelling of fracture using embedded discontinuities. In: Owen DRJ, Nate EO, Suárez B (eds) Computational plasticity, fundamentals and applications. COMPLAS VIII, Barcelona, Spain |

[2] | Alfaiate J, Sluys LJ (2004) On the use of embedded discontinuities in the framework of a discrete crack approach. In: Yao Z, Yuan WZM (eds) Sixth international congress on computational mechanics in conjunction with the second asian-pacific congress of computational mechanics. WCCMVI, Beijing, China |

[3] | Alfaiate J, Pires EB, Martins JAC (1992) A finite element model for the study of crack propagation. In: Aliabadi MH, Cartwright DJ, Nisitani H (eds) 2nd international conference on localised damage, Computational Mechanics Publications and Elsevier Applied Science. Southampton, United Kingdom, pp 261–282 |

[4] | Alfaiate J, Pires EB, Martins JAC (1997) A finite element analysis of non-prescribed crack propagation in concrete. Comput Struct 63(1): 17–26 · Zbl 0899.73512 |

[5] | Alfaiate J, Wells GN, Sluys LJ (2002) On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture. Eng Fract Mech 69(6): 661– 686 |

[6] | Alfaiate J, Simone A, Sluys LJ (2003) Non-homogeneous displacement jumps in strong embedded discontinuities. Int J Solids Struct 40(21): 5799–5817 · Zbl 1059.74548 |

[7] | Alfaiate J, Sluys LJ, Pires EB (2005) Mixed mode fracture in concrete and masonry. In: Ferro AC, Mai Y, Ritchie RG (eds) ICFXI, 11th international conference on fracture, Torino, Italy |

[8] | Alfaiate J, Moonen P, Sluys LJ (2007a) On the use of DSDA and X-FEM for the modelling of discontinuities in porous media. In: Oñate E, Owen DRJ (eds) IX International Conference on Computational Plasticity.. COMPLAS IX, Barcelona, Spain |

[9] | Alfaiate J, Moonen P, Sluys LJ, Carmeliet J (2007b) On the modelling of preferential moisture uptake in porous media. In: EPMESC XI, Kyoto, Japan · Zbl 1231.76303 |

[10] | Areias P, Belytschko T (2005) Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int J Numer Methods Eng 63: 760–788 · Zbl 1122.74498 |

[11] | Armero F, Garikipati K (1996) An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Int J Solids Struct 33(20–22): 2863–2885 · Zbl 0924.73084 |

[12] | Barenblatt G (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7: 55–129 |

[13] | Barpi F, Valente S (2000) Numerical simulation of prenotched gravity dam models. J Eng Mech 126(6): 611–619 |

[14] | Bazant ZP, Oh BH (1983) Crack band theory of concrete. Mater Struct 16: 155–177 |

[15] | Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5): 601–620 · Zbl 0943.74061 |

[16] | Bocca P, Carpinteri A, Valente S (1986) Mixed mode fracture of concrete. Int J Solids Struct 27: 1139–1153 |

[17] | Bolzon G (2001) Formulation of a triangular finite element with an embedded interface via isoparametric mapping. Comput Mech 27(6): 463–473 · Zbl 1052.74053 |

[18] | Borja RI (2008) Assumed enhanced strain and the extended finite element methods: a unification of concepts. Comput Methods Appl Mech Eng 197(33–40): 2789–2803 · Zbl 1194.74368 |

[19] | Carey GF, Ma M (1999) Joint elements, stress post-processing and superconvergent extraction with application to Mohr–Coulomb failure. Commun Numer Methods Eng 15(5): 335–347 · Zbl 0926.74102 |

[20] | Carol I, Prat P (1995) A multicrack model based on the theory of multisurface plasticity and two fracture energies. In: Owen DRJ, Oñate E, Hinton E (eds) Computational Plasticity, fundamentals and applications. Pineridge Press, Barcelona, Spain, pp 1583–1594 |

[21] | Cervera M, Chiumenti M (2006) Smeared crack approach: back to the original track. Int J Numer Anal Methods Geomech 30(12): 1173–1199 · Zbl 1196.74180 |

[22] | Dias-da-Costa D, Alfaiate J, Sluys LJ, Júlio E (2009) A discrete strong discontinuity approach. Eng Fract Mech 76(9): 1176–1201 |

[23] | Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48(12): 1741–1760 · Zbl 0989.74066 |

[24] | Duarte CAM, Oden JT (1995) H-p clouds-an h-p meshless method. Tech Rep 95–05 |

[25] | Duarte CAM, Babuška I, Oden JT (2000) Generalized finite element methods for three-dimensional structural mechanics problems. Comput Struct 77(2): 215–232 |

[26] | Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2): 100–104 |

[27] | Dvorkin EN, Cuitiño AM, Gioia G (1990) Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int J Numer Methods Eng 30(3): 541–564 · Zbl 0729.73209 |

[28] | Gasser TC, Holzapfel GA (2006) 3d crack propagation in unreinforced concrete: a two-step algorithm for tracking 3d crack paths. Comput Methods Appl Mech Eng 195(37–40): 5198–5219 · Zbl 1154.74376 |

[29] | Goodman RE, Taylor RL, Brekke TL (1968) A model for the mechanics of jointed rock. J Soil Mech Found Div 99: 637–659 |

[30] | Herrmann LR (1978) Finite element analysis of contact problems. ASCE J Eng Mech Div 104(5): 1043–1057 |

[31] | Hillerborg A, Modeer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6(6): 773–781 |

[32] | Ingraffea A (1989) Shear cracks. In: Elfgren L (eds) Fracture mechanics of concrete structures–from theory to applications, report of the Technical Committee 90-FMA Fracture Mechanics of Concrete-Applications. Chapman and Hall, London, United Kingdom, pp 231–233 |

[33] | Ingraffea A, Saouma V (1985) Numerical modelling of discrete crack propagation in reinforced and plain concrete. In: Sih GC, DiTommaso A(eds) Engineering Application of Fracture Mechanics. Martinus Nijhoff Publishers |

[34] | Jirásek M, Belytschko T (2002) Computational resolution of strong discontinuities. In: Mang HA, Rammerstorfer JEFG (ed) WCCM V, Fifth world congress on computational mechanics, Vienna, Austria |

[35] | Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies in Applied Mathematics 8, Philadelphia · Zbl 0685.73002 |

[36] | Klisinski M, Runesson K, Sture S (1991) Finite element with inner softening band. J Eng Mech 117(3): 575–587 |

[37] | Larsson R, Runesson K (1996) Element-embedded localization band based on regularized displacement discontinuity. J Eng Mech 122(5): 402–411 · Zbl 0919.73279 |

[38] | Linder C, Armero F (2007) Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int J Numer Methods Eng 72(12): 1391–1433 · Zbl 1194.74431 |

[39] | Lofti HR, Shing PB (1995) Embedded representation of fracture in concrete with mixed finite elements. Int J Numer Methods Eng 38(8): 1307–1325 · Zbl 0824.73070 |

[40] | Lourenço P, Rots JG (1997) A multi-surface interface model for the analysis of masonry structures. ASCE J Eng Mech 123(7): 660–668 |

[41] | Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall International, Englewood Cliffs, New Jersey |

[42] | Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1–4): 289–314 · Zbl 0881.65099 |

[43] | Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1): 131–150 · Zbl 0955.74066 |

[44] | Moës N, Sukumar N, Moran B, Belytschko T (2000) An extended finite element method (x-fem) for two- and three-dimensional crack modeling. In: ECCOMAS 2000, Barcelona · Zbl 0963.74067 |

[45] | Nooru-Mohamed MB (1992) Mixed-mode fracture of concrete: an experimental approach. PhD thesis, Delft University of Technology |

[46] | Ohlsson U, Olofsson T (1997) Mixed-mode fracture and anchor bolts in concrete analysis with inner softening bands. J Eng Mech 123(10): 1027–1033 |

[47] | Oliver J (1996) Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. part 1: fundamentals. Int J Numer Methods Eng 39(21): 3575–3600 · Zbl 0888.73018 |

[48] | Oliver J, Huespe AE, Pulido MDG, Chaves E (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 69: 113–136 |

[49] | Oliver J, Huespe AE, Sanchez PJ (2006) A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM. Comput Methods Appl Mech Eng 195(37–40): 4732–4752 · Zbl 1144.74043 |

[50] | Pivonka P, Ozbolt J, Lackner R, Mang HA (2004) Comparative studies of 3d-constitutive models for concrete: application to mixed-mode fracture. Int J Numer Methods Eng 60(2): 549–570 · Zbl 1098.74674 |

[51] | Remmers JJC (2006) Discontinuities in materials and structures: a unifying computational approach. PhD thesis, Delft University of Technology |

[52] | Rots JG (1988) Computational modeling of concrete fracture. PhD thesis, Delft University of Technology |

[53] | Sancho J, Planas J, Gálvez J, Cendón D, Fathy A (2005) Three-dimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity. In: Owen DRJ, nate EO, Suárez B (eds) Computational plasticity, fundamentals and applications. COMPLAS VIII, Barcelona, Spain · Zbl 1158.74042 |

[54] | Schlangen E (1993) Experimental and numerical analysis of fracture processes in concrete. PhD thesis, Delft University of Technology |

[55] | Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8): 1595–1638 · Zbl 0724.73222 |

[56] | Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12: 277–296 · Zbl 0783.73024 |

[57] | Simone A (2004) Partition of unity-based discontinuous elements for interface phenomena: computational issues. Commun Numer Methods Eng 20(6): 465–478 · Zbl 1058.74082 |

[58] | Simone A, Wells GN, Sluys LJ (2003) From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput Methods Appl Mech Eng 192(41–42): 4581–4607 · Zbl 1054.74719 |

[59] | Simone A, Duarte CAM, Van der Giessen E (2006) A generalized finite element method for polycrystals with discontinuous grain boundaries. Int J Numer Methods Eng 67(8): 1122–1145 · Zbl 1113.74076 |

[60] | Tijssens MGA, Sluys LJ, Van der Giessen E (2000) Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces. Eur J Mech, A/Solids 19(5): 761–779 · Zbl 0993.74073 |

[61] | Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method. Int J Numer Methods Eng 66(5): 761–795 · Zbl 1110.74858 |

[62] | Wells GN, Sluys LJ (2000) Application of embedded discontinuities for softening solids. Eng Fract Mech 65(2–3): 263–281 |

[63] | Wells GN, Sluys LJ (2001a) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12): 2667–2682 · Zbl 1013.74074 |

[64] | Wells GN, Sluys LJ (2001b) Three-dimensional embedded discontinuity model for brittle fracture. Int J Solids Struct 38(5): 897–913 · Zbl 1004.74065 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.