Skip to main content
SpringerLink
Log in

Modelling complex cracks with finite elements: a kinematically enriched constitutive model

  • Compmech
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

A continuum constitutive framework with embedded cohesive interface model is presented to describe the failure of quasi-brittle materials. Both cohesive behaviour for cracking inside the fracture process zone and elastic bulk behaviour are treated at integration points making implementation straightforward. In this sense, the proposed approach is simpler than existing ones that focus on element enrichments, such as the extended finite element method, while share similarities with smeared crack models, and offers the capability to correctly model quasi-brittle failure in post-peak regime at constitutive level. In this work, the formulation is introduced, numerical algorithms described and static and dynamic fracture simulations with complex crack patterns are conducted to demonstrate the capability and advantage of the proposed approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Notes

  1. One exception is the cohesive interface elements which are available in many FE packages such as Abaqus and LS-Dyna.

  2. Reasons for this choice will be given later in the paper.

  3. Einstein summation convention does not apply here.

  4. Exceptions include Abaqus of Dassault Systemes and Systus of ESI.

  5. The load-displacement curves were extracted from Nooru-Mohamed et al. (1993) using the software plotdigitizer which is freely available at http://plotdigitizer.sourceforge.net.

  6. In Nguyen et al. (2015b) we presented simulations with finite thin interfaces to indicate that our model can be used without interface elements.

  7. Periodicity was not considered simply because our microstructure generator is unable to generate periodic structures.

References

  • Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M (2014) Phase-field modeling of fracture in linear thin shells. Theor Appl Fract Mech 69:102–109

    Article  Google Scholar 

  • Armero F, Linder C (2009) Numerical simulation of dynamic fracture using finite elements with embedded discontinuities. Int J Fract 160(2):119–141

    Article  Google Scholar 

  • Ayachit U (2015) The paraview guide: a parallel visualization application. Kitware

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

    Article  Google Scholar 

  • Bažant Z (1991) Why continuum damage is nonlocal: micromechanics arguments. J Eng Mech 117(5):1070–1087

    Article  Google Scholar 

  • Bažant Z, Oh B (1983) Crack band theory for fracture in concrete. Mater Struct 16:155–177

    Google Scholar 

  • Belytschko T, Fish J, Engelmann BE (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59–89

    Article  Google Scholar 

  • Belytschko T, Liu WK, Moran B (2000a) Nonlinear finite elements for continua and structures. Wiley, Chichester

    Google Scholar 

  • Belytschko T, Organ D, Gerlach C (2000b) Element-free Galerkin methods for dynamic fracture in concrete. Comput Methods Appl Mech Eng 187(3–4):385–399

    Article  Google Scholar 

  • Bordas S, Nguyen PV, Dunant C, Guidoum A, Nguyen-Dang H (2007) An extended finite element library. Int J Numer Methods Eng 71(6):703–732

    Article  Google Scholar 

  • Borden MJ, Hughes TJ, Landis CM, Verhoosel CV (2014) A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng 273:100–118

    Article  Google Scholar 

  • Budarapu PR, Gracie R, Yang SW, Zhuang X, Rabczuk T (2014) Efficient coarse graining in multiscale modeling of fracture. Theor Appl Fract Mech 69:126–143

    Article  Google Scholar 

  • de Borst R, Guitérrez M, Wells G, Remmers J, Askes H (2004) Cohesive zone models, higher-order continuum theories and reliability methods for computational failure analysis. Int J Numer Methods Eng 60(1):289–315

    Article  Google Scholar 

  • de Borst R, Guitérrez M, Wells G, Remmers J, Askes H (2004) Cohesive zone models, higher-order continuum theories and reliability methods for computational failure analysis. Int J Numer Methods Eng 60(1):289–315

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Dias-da-Costa D, Alfaiate J, Sluys L, Júlio E (2010) A comparative study on the modelling of discontinuous fracture by means of enriched nodal and element techniques and interface elements. Int J Fract 161(1):97–119

    Article  Google Scholar 

  • Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104

    Article  Google Scholar 

  • Dvorkin EN, Cuitino AM, Gioia G (1990) Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int J Numer Methods Eng 30:541–564

    Article  Google Scholar 

  • Gasser T, Holzapfel G (2005) Modeling 3D crack propagation in unreinforced concrete using PUFEM. Comput Methods Appl Mech Eng 194(25–26):2859–2896

  • Geuzaine C, Remacle JF (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331

    Article  Google Scholar 

  • Giner E, Sukumar N, Tarancón J, Fuenmayor F (2009) An Abaqus implementation of the extended finite element method. Eng Fract Mech 76(3):347–368

    Article  Google Scholar 

  • Graça-e-Costa R, Alfaiate J, Dias-da-Costa D, Sluys L (2012) A non-iterative approach for the modelling of quasi-brittle materials. Int J Fract 178(1–2):281–298

    Article  Google Scholar 

  • Graça-e-Costa R, Alfaiate J, Dias-da-Costa D, Neto P, Sluys L (2013) Generalisation of non-iterative methods for the modelling of structures under non-proportional loading. Int J Fract 182(1):21–38

    Article  Google Scholar 

  • Henderson A (2007) ParaView guide, a parallel visualization application. Kitware Inc, Clifton Park

    Google Scholar 

  • Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 12:213–222

    Article  Google Scholar 

  • Hofacker M, Miehe C (2012) Continuum phase field modeling of dynamic fracture: variational principles and staggered fe implementation. Int J Fract 178(1–2):113–129

    Article  Google Scholar 

  • Jefferson A (2003) Craft–a plastic-damage-contact model for concrete. II. Model implementation with implicit return-mapping algorithm and consistent tangent matrix. Int J Solids Struct 40(22):6001–6022

    Article  Google Scholar 

  • Jirasek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Eng 188:307–330

    Article  Google Scholar 

  • John R, Shah S (1990) Mixed-mode fracture of concrete subjected to impact loading. J Struct Eng 116(3):585–602

    Article  Google Scholar 

  • Mandel J (1971) Plasticité Classique et Viscoplasticité. Springer, Udine

    Google Scholar 

  • Mergheim J, Kuhl E, Steinmann P (2004a) A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Commun Numer Methods Eng 20(7):511–519

    Article  Google Scholar 

  • Mergheim J, Kuhl E, Steinmann P (2004b) A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Commun Numer Methods Eng 20(7):511–519

    Article  Google Scholar 

  • Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Mosler J, Meschke G (2004) Embedded crack vs. smeared crack models: a comparison of elementwise discontinuous crack path approaches with emphasis on mesh bias. Comput Methods Appl Mech Eng 193(30–32):3351–3375

    Article  Google Scholar 

  • Needleman A (1988) Material rate dependence and mesh sensitivity in localization problems. Comput Methods Appl Mech Eng 67:69–85

    Article  Google Scholar 

  • Ngo D, Scordelis A (1967) Finite element analysis of reinforced concrete beams. ACI J 64(14):152–163

    Google Scholar 

  • Nguyen GD, Einav I, Korsunsky AM (2012) How to connect two scales of behaviour in constitutive modelling of geomaterials. Géotech Lett 2:129–134

    Article  Google Scholar 

  • Nguyen GD, El-Zein A, Bennett T (2014a) A conceptual approach to two-scale constitutive modelling for hydro-mechanical coupling. In: Proceedings, 7th international congress on environmental geotechnics ICEG2014, Melbourne

  • Nguyen GD, Korsunsky AM, Einav I (2014b) A constitutive modelling framework featuring two scales of behaviour: fundamentals and applications to quasi-brittle failure. Eng Fract Mech 115:221–240

    Article  Google Scholar 

  • Nguyen GD, Nguyen CT, Nguyen VP, Bui H (2015a) Constitutive modelling of compaction localisation in porous sandstones. Int J Rock Mech Min Sci 83:57–72

    Google Scholar 

  • Nguyen GD, Nguyen CT, Nguyen VP, Bui H, Shen L (2015b) A size-dependent constitutive modelling framework for localised failure analysis. Comput Mech. doi:10.1007/s00466-016-1293-z

  • Nguyen VP (2014a) Discontinuous Galerkin/extrinsic cohesive zone modeling: implementation caveats and applications in computational fracture mechanics. Eng Fract Mech 128:37–68

    Article  Google Scholar 

  • Nguyen VP (2014b) An open source program to generate zero-thickness cohesive interface elements. Adv Eng Softw 74:27–39

    Article  Google Scholar 

  • Nguyen VP, Valls OL, Stroeven M, Sluys LJ (2010) On the existence of representative volumes for softening quasi-brittle materials-a failure zone averaging scheme. Comput Methods Appl Mech Eng 199(45–48):3028–3038

    Article  Google Scholar 

  • Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2011) Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks. Comput Methods Appl Mech Eng 200(9–12):1220–1236

    Article  Google Scholar 

  • Nguyen VP, Kerfriden P, Bordas S (2014c) Two- and three-dimensional isogeometric cohesive elements for composite delamination analysis. Compos Part B Eng 60:193–212

    Article  Google Scholar 

  • Nooru-Mohamed M, Schlangen E, van Mier JG (1993) Experimental and numerical study on the behavior of concrete subjected to biaxial tension and shear. Adv Cem Based Mater 1(1):22–37

    Article  Google Scholar 

  • Ortiz M, Leroy Y, Needleman A (1987) A finite element method for localized failure analysis. Comput Methods Appl Mech Eng 61(2):189–214

    Article  Google Scholar 

  • París F, Correa E, Mantic V (2006) Kinking of tranversal interface cracks between fiber and matrix. J Appl Mech 74(4):703–716

    Article  Google Scholar 

  • Park K, Paulino GH, Celes W, Espinha R (2012) Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture. Int J Numer Methods Eng 92(1):1–35

    Article  Google Scholar 

  • Peerlings RHJ, de Borst R, Brekelmans WAM, de Wree J (1996) Gradient enhanced damage for quasi brittle materials. Int J Numer Methods Eng 39:3391–3403

    Article  Google Scholar 

  • Pérez-Foguet A, Rodríguez-Ferran A, Huerta A (2001) Consistent tangent matrices for substepping schemes. Comput Methods Appl Mech Eng 190(35–36):4627–4647

    Article  Google Scholar 

  • Pietruszczak S, Mroz Z (2001) On failure criteria for anisotropic cohesive-frictional materials. Int J Numer Anal Methods Geomech 25(5):509–524

    Article  Google Scholar 

  • Pietruszczak S, Xu G (1995) Brittle response of concrete as a localization problem. Int J Solids Struc 32(11):1517–1533

    Article  Google Scholar 

  • Rabczuk T (2013) Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. ISRN Appl Math 2013. doi:10.1155/2013/849231

  • Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61(13):2316–2343

    Article  Google Scholar 

  • Rabczuk T, Areias PMA, Belytschko T (2007a) A meshfree thin shell method for non-linear dynamic fracture. Int J Numer Methods Eng 72(5):524–548

    Article  Google Scholar 

  • Rabczuk T, Bordas S, Zi G (2007b) A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Comput Mech 40(3):473–495

    Article  Google Scholar 

  • Radovitzky R, Seagraves A, Tupek M, Noels L (2011) A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method. Comput Methods Appl Mech Eng 200(1–4):326–344

    Article  Google Scholar 

  • Rashid Y (1968) Ultimate strength analysis of prestressed concrete pressure vessels. Nucl Eng Des 7(4):334–344

    Article  Google Scholar 

  • Rots J (1991) Smeared and discrete representations of localized fracture. In: Bažant Z (ed) Current trends in concrete fracture research. Springer, Netherlands, pp 45–59

    Chapter  Google Scholar 

  • Rots JG, Invernizzi S (2004) Regularized sequentially linear saw-tooth softening model. Int J Numer Anal Methods Geomech 28(7–8):821–856

    Article  Google Scholar 

  • Ruiz G, Pandolfi A, Ortiz M (2001) Three-dimensional cohesive modeling of dynamic mixed-mode fracture. Int J Numer Methods Eng 52(1–2):97–120

    Article  Google Scholar 

  • Sam CH, Papoulia KD, Vavasis SA (2005) Obtaining initially rigid cohesive finite element models that are temporally convergent. Eng Fract Mech 72(14):2247–2267

    Article  Google Scholar 

  • Schellekens J, Borst RD (1993) A non-linear finite element approach for the analysis of mode-i free edge delamination in composites. Int J Solids Struct 30(9):1239–1253

    Article  Google Scholar 

  • Schreyer H, Sulsky D, Munday L, Coon M, Kwok R (2006) Elastic-decohesion constitutive model for sea ice. J Geophys Res 111:C11S26

  • Silani M, Talebi H, Ziaei-Rad S, Hamouda AM, Zi G, Rabczuk T (2014) A three dimensional extended Arlequin method for dynamic fracture. Comput Mater Sci 96(Part B):425–431

  • Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209

    Article  Google Scholar 

  • Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12(5):277–296

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Simone A, Duarte CA, 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

    Article  Google Scholar 

  • Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190(32–33):4081–4193

    Article  Google Scholar 

  • Tijssens MGA, Sluys LJ, van der Giessen E (2001) Simulation of fracture of cementitious composites with explicit modeling of microstructural features. Eng Fract Mech 68(11):1245–1263

    Article  Google Scholar 

  • Turon A, Camanho P, Costa J, Dávila C (2006) A damage model for the simulation of delamination in advanced composites under variable-mode loading. Mech Mater 38(11):1072–1089

    Article  Google Scholar 

  • Wells GN, Sluys LJ (2001) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12):2667–2682

    Article  Google Scholar 

  • Wyart E, Duflot M, Coulon D, Martiny P, Pardoen T, Remacle JF, Lani F (2008) Substructuring FE-XFE approaches applied to three-dimensional crack propagation. J Comput Appl Math 215(2):626–638

    Article  Google Scholar 

  • Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42:1397–1434

    Article  Google Scholar 

  • Yang SW, Budarapu PR, Mahapatra DR, Bordas SP, Zi G, Rabczuk T (2015) A meshless adaptive multiscale method for fracture. Comput Mater Sci 96(Part B):382–395

    Article  Google Scholar 

  • Zhou F, Molinari JF (2004) Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency. Int J Numer Methods Eng 59(1):1–24

    Article  Google Scholar 

  • Zhuang X, Augarde C, Mathisen K (2012) Fracture modeling using meshless methods and level sets in 3D: framework and modeling. Int J Numer Methods Eng 92(11):969–998

    Article  Google Scholar 

  • Zi G, Chen H, Xu J, Belytschko T (2005) The extended finite element method for dynamic fractures. J Sound Vib 12:9–23

    Google Scholar 

Download references

Acknowledgments

Funding support from the Australian Research Council via projects DP140100945 (Luming Shen, Giang D. Nguyen, Vinh Phu Nguyen, Abbas El-Zein, Federico Maggi), FT140100408 (Giang D. Nguyen) and DE 150101703 (Daniel Dias-da-Costa) is gratefully acknowledged. Daniel Dias-da-Costa would like to extend his acknowledgements to the Portuguese Foundation for Science and Technology through Project No. FCOMP-01-0124-FEDER-020275-PTDC/ECM/119214/2010. VP Nguyen would like to express the gratitude towards Drs. Erik Jan Lingen and Martijn Stroeven at the Dynaflow Research Group, Houtsingel 95, 2719 EB Zoetermeer, The Netherlands for providing us the numerical toolkit jem/jive.

Author information

Author notes

  1. Vinh Phu Nguyen & Chi Thanh Nguyen

    Present address: Department of Civil Engineering, Monash University, Clayton, VIC, 3800, Australia

Authors and Affiliations

  1. School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia

    Vinh Phu Nguyen, Giang Dinh Nguyen & Chi Thanh Nguyen

  2. School of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

    Luming Shen, Daniel Dias-da-Costa, Abbas El-Zein & Federico Maggi

Authors
  1. Vinh Phu Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giang Dinh Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chi Thanh Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Luming Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Daniel Dias-da-Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Abbas El-Zein
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Federico Maggi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vinh Phu Nguyen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, V.P., Nguyen, G.D., Nguyen, C.T. et al. Modelling complex cracks with finite elements: a kinematically enriched constitutive model. Int J Fract 203, 21–39 (2017). https://doi.org/10.1007/s10704-016-0114-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-016-0114-x

Keywords

Search

Navigation