Skip to main content
SpringerLink
Log in

Computational Methods for Ductile Fracture Modeling at the Microscale

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

This paper is a state-of-the-art review of computational damage and fracture mechanics methods applied to model ductile fracture at the microscale. An emphasis is made on robust and stable methods that can handle heterogeneous structures, large deformations, and cracks initiation and coalescence. Ductile materials’ microstructures feature brittle and ductile components whose heterogeneous behavior can give raise to cracks initiation due to stress concentration. Due to large deformations, cracks initiated by brittle components failure transform into large voids. These major voids interact and coalesce by plastic localization within ductile components and lead to final failure. This process can involve minor voids nucleated directly within ductile components at sub-micron scales. State-of-the-art discontinuous approaches can be applied to discretize accurately brittle components and model their failure, given that large deformations can be handled. For ductile components, continuous approaches are discussed in this review as they can model the homogenized influence of minor voids, hence alleviating the burden and computational cost overhead that an explicit discretization of those voids would require. Close to final failure, when major voids are coalescing, and the influence of minor voids becomes comparable to that of major voids, the transition from a continuous damage process within ductile components to the initiation and propagation of discontinuous cracks within these components has to be modeled. This review ends with a discussion on computational methods that have successfully been applied to model the continuous-discontinuous transition, and that could be coupled to discontinuous approaches in order to model ductile fracture at the microscale in its full three-dimensional complexity.

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

Similar content being viewed by others

Notes

  1. Other variants such as dual phase steels and polycrystals with multiple ductile components are also considered in the present review.

  2. The word uncoupled means that in this work the softening effect was not modeled, as opposed to coupled damage models discussed in Sect. 3.1. A non local regularization was nevertheless judged necessary by Mediavilla et al. [83] to ”reduce the influence of local damage variations which are a result of the discretization”.

  3. Error estimators can also be based on mechanical variables as discussed in Sect. 4.4.1.

  4. Element enrichment approaches such as the X-FEM can also be used (Sect. 2.2).

  5. See Sect. 3.2 for comments on the relations between multiscale methods and micromechanical models.

  6. We refer the reader back to Sect. 2.4 for the definition of mesh adaption, as opposed to remeshing or full remeshing.

Abbreviations

2D:

Two-dimensional

3D:

Three-dimensional

CDM:

Continuum damage model

CDT:

Continuous-discontinuous transition

CZM:

Cohesive zone model

DNS:

Direct numerical simulation

FE:

Finite element

GFEM:

Generalized finite element method

GTN:

Gurson–Tvergaard–Needleman

LS:

Level-set

PF:

Phase-field

RVE:

Representative volume element

X-FEM:

eXtended finite element method

References

  1. Allen DH, Searcy CR (2001) A micromechanical model for a viscoelastic cohesive zone. Int J Fract 107(2):159–176

    Article  Google Scholar 

  2. Ambati M, De Lorenzis L (2016) Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements. Comput Methods Appl Mech Eng 312:351–373. https://doi.org/10.1016/j.cma.2016.02.017

    Article  MathSciNet  Google Scholar 

  3. Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Computat Mech 55(5):1017–1040. https://doi.org/10.1007/s00466-015-1151-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambati M, Kruse R, De Lorenzis L (2016) A phase-field model for ductile fracture at finite strains and its experimental verification. Computat Mech 57(1):149–167. https://doi.org/10.1007/s00466-015-1225-3

    Article  MathSciNet  MATH  Google Scholar 

  5. Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229

    Article  MATH  Google Scholar 

  6. Andrade Pires F, de Souza Neto E, Owen D (2004) On the finite element prediction of damage growth and fracture initiation in finitely deforming ductile materials. Comput Methods Appl Mech Eng 193(48–51):5223–5256. https://doi.org/10.1016/j.cma.2004.01.038

    Article  MATH  Google Scholar 

  7. Antretter T, Fischer F (1998) Particle cleavage and ductile crack growth in a two-phase composite on a microscale. Comput Mater Sci 13(1–3):1–7. https://doi.org/10.1016/S0927-0256(98)00039-1

    Article  Google Scholar 

  8. Aragón AM, Simone A (2017) The discontinuity-enriched finite element method. Int J Numer Methods Eng 112(11):1589–1613. https://doi.org/10.1002/nme.5570,nme.5570

    Article  MathSciNet  Google Scholar 

  9. Areias P, Dias-da Costa D, Alfaiate J, Júlio E (2009) Arbitrary bi-dimensional finite strain cohesive crack propagation. Comput Mech 45(1):61–75. https://doi.org/10.1007/s00466-009-0418-z

    Article  MathSciNet  MATH  Google Scholar 

  10. Areias P, Van Goethem N, Pires EB (2011) A damage model for ductile crack initiation and propagation. Comput Mech 47(6):641–656. https://doi.org/10.1007/s00466-010-0566-1

    Article  MathSciNet  MATH  Google Scholar 

  11. Areias P, Dias-da Costa D, Sargado JM, Rabczuk T (2013) Element-wise algorithm for modeling ductile fracture with the Rousselier yield function. Comput Mech 52(6):1429–1443. https://doi.org/10.1007/s00466-013-0885-0

    Article  MATH  Google Scholar 

  12. Areias P, Reinoso J, Camanho P, Rabczuk T (2015) A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Comput Mech 56(2):291–315. https://doi.org/10.1007/s00466-015-1172-z

    Article  MathSciNet  MATH  Google Scholar 

  13. Arriaga M, McAuliffe C, Waisman H (2015) Onset of shear band localization by a local generalized eigenvalue analysis. Comput Methods Appl Mech Eng 289:179–208

    Article  MathSciNet  MATH  Google Scholar 

  14. Babuska I, Melenk JM (1995) The partition of unity finite element method. Tech. rep., University of Maryland - Institute for Physical Science and Technology

  15. Banerjee A, Manivasagam R (2009) Triaxiality dependent cohesive zone model. Eng Fract Mech 76(12):1761–1770

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119)

    Article  Google Scholar 

  18. Béchet É, Minnebo H, Moës N, Burgardt B (2005) Improved implementation and robustness study of the x-fem for stress analysis around cracks. Int J Numer Methods Eng 64(8):1033–1056

    Article  MATH  Google Scholar 

  19. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5):601–620

    Article  MATH  Google Scholar 

  20. Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50(4):993–1013

    Article  MATH  Google Scholar 

  21. Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894

    Article  MathSciNet  MATH  Google Scholar 

  22. Belytschko T, Liu WK, Moran B, Khalil E (2013) Nonlinear finite elements for continua and structures, 2nd edn. Wiley, Hoboken

    MATH  Google Scholar 

  23. Besson J (2010) Continuum models of ductile fracture: a review. Int J Damage Mech 19(1):3–52. https://doi.org/10.1177/1056789509103482

    Article  Google Scholar 

  24. Böhm HJ (ed) (2004) Mechanics of microstructured materials. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2776-6

    Book  MATH  Google Scholar 

  25. Borden MJ, Hughes TJ, Landis CM, Anvari A, Lee IJ (2016) A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng 312:130–166. https://doi.org/10.1016/j.cma.2016.09.005

    Article  MathSciNet  Google Scholar 

  26. Borouchaki H, Laug P, Cherouat A, Saanouni K (2005) Adaptive remeshing in large plastic strain with damage. Int J Numer Methods Eng 63(1):1–36. https://doi.org/10.1002/nme.1274

    Article  MathSciNet  MATH  Google Scholar 

  27. Bosco E, Kouznetsova VG, Geers MGD (2015) Multi-scale computational homogenization-localization for propagating discontinuities using X-FEM. Int J Numer Methods Eng 102(3–4):496–527. https://doi.org/10.1002/nme.4838

    Article  MathSciNet  MATH  Google Scholar 

  28. Bouchard PO, Bay F, Chastel Y, Tovena I (2000) Crack propagation modelling using an advanced remeshing technique. Comput Methods Appl Mech Eng 189(3):723–742. https://doi.org/10.1016/S0045-7825(99)00324-2

    Article  MATH  Google Scholar 

  29. Bouchard PO, Bourgeon L, Fayolle S, Mocellin K (2011) An enhanced Lemaitre model formulation for materials processing damage computation. Int J Mater Form 4(3):299–315. https://doi.org/10.1007/s12289-010-0996-5

    Article  Google Scholar 

  30. Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826

    Article  MathSciNet  MATH  Google Scholar 

  31. Broumand P, Khoei A (2015) X-FEM modeling of dynamic ductile fracture problems with a nonlocal damage-viscoplasticity model. Finite Elements Anal Des 99:49–67. https://doi.org/10.1016/j.finel.2015.01.002

    Article  Google Scholar 

  32. Carter BJ, Wawrzynek PA, Ingraffea AR (2000) Automated 3-D crack growth simulation. International Journal for Numerical Methods in Engineering 47(1–3):229–253. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<229::AID-NME769>3.0.CO;2-2

    Article  MATH  Google Scholar 

  33. César de Sá J, Areias P, Zheng C (2006) Damage modelling in metal forming problems using an implicit non-local gradient model. Comput Methods Appl Mech Eng 195(48–49):6646–6660. https://doi.org/10.1016/j.cma.2005.02.037

    Article  MATH  Google Scholar 

  34. Chandra N, Li H, Shet C, Ghonem H (2002) Some issues in the application of cohesive zone models for metal-ceramic interfaces. Int J Solids Struct 39(10):2827–2855

    Article  MATH  Google Scholar 

  35. Chen J, Yuan H (2002) A micro-mechanical damage model based on gradient plasticity: algorithms and applications. Int J Numer Methods Eng 54(3):399–420. https://doi.org/10.1002/nme.431

    Article  MATH  Google Scholar 

  36. Crété JP, Longère P, Cadou JM (2014) Numerical modelling of crack propagation in ductile materials combining the gtn model and x-fem. Comput Methods Appl Mech Eng 275:204–233

    Article  MathSciNet  MATH  Google Scholar 

  37. Drabek T, Böhm H (2005) Damage models for studying ductile matrix failure in composites. Comput Mater Sci 32(3–4):329–336. https://doi.org/10.1016/j.commatsci.2004.09.035

    Article  Google Scholar 

  38. Duarte C, Hamzeh O, Liszka T, Tworzydlo W (2001) A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput Methods Appl Mech Eng 190(15):2227–2262

    Article  MATH  Google Scholar 

  39. El khaoulani R, Bouchard PO (2012) An anisotropic mesh adaptation strategy for damage and failure in ductile materials. Finite Elements Anal Des 59:1–10. https://doi.org/10.1016/j.finel.2012.04.006

    Article  MathSciNet  Google Scholar 

  40. El khaoulani R, Bouchard PO (2013) Efficient numerical integration of an elastic-plastic damage law within a mixed velocity-pressure formulation. Math Comput Simul 94:145–158. https://doi.org/10.1016/j.matcom.2013.06.004

    Article  MathSciNet  Google Scholar 

  41. Elguedj T, Gravouil A, Combescure A (2006) Appropriate extended functions for x-fem simulation of plastic fracture mechanics. Comput Methods Appl Mech Eng 195(7):501–515

    Article  MATH  Google Scholar 

  42. Feld-Payet S (2010) Amorçage et propagation de fissures dans les milieux ductiles non locaux. PhD thesis, Ecole Nationale Supérieure des Mines de Paris

  43. Feld-Payet S, Chiaruttini V, Besson J, Feyel F (2015) A new marching ridges algorithm for crack path tracking in regularized media. Int J Solids Struct 71:57–69. https://doi.org/10.1016/j.ijsolstr.2015.04.043

    Article  Google Scholar 

  44. Forest S (2009) Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J Eng Mech 135(3):117–131. https://doi.org/10.1061/(ASCE)0733-9399(2009)135:3(117)

    Article  Google Scholar 

  45. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

    Article  MathSciNet  MATH  Google Scholar 

  46. Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304

    MathSciNet  MATH  Google Scholar 

  47. Geers M, Kouznetsova V, Brekelmans W (2001) Gradient-enhanced computational homogenization for the micro-macro scale transition. Le Journal de Physique IV 11(Pr5):Pr5–145

    Google Scholar 

  48. Geers MG, Kouznetsova VG, Brekelmans W (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182

    Article  MATH  Google Scholar 

  49. Giang NA, Kuna M, Hütter G (2017) Influence of carbide particles on crack initiation and propagation with competing ductile-brittle transition in ferritic steel. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2017.05.015

    Article  Google Scholar 

  50. Gologanu M, Leblond JB, Devaux J (1993) Approximate models for ductile metals containing non-spherical voids-case of axisymmetric prolate ellipsoidal cavities. J Mech Phys Solids 41(11):1723–1754. https://doi.org/10.1016/0022-5096(93)90029-F

    Article  MATH  Google Scholar 

  51. Gravouil A, Moës N, Belytschko T (2002) Non-planar 3d crack growth by the extended finite element and level sets-part II: level set update. Int J Numer Methods Eng 53(11):2569–2586

    Article  MATH  Google Scholar 

  52. Gruau C, Coupez T (2005) 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric. Comput Methods Appl Mech Eng 194(48–49):4951–4976. https://doi.org/10.1016/j.cma.2004.11.020

    Article  MathSciNet  MATH  Google Scholar 

  53. Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: Part I-yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99(1):2. https://doi.org/10.1115/1.3443401

    Article  Google Scholar 

  54. Hirt C, Nichols B (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225. https://doi.org/10.1016/0021-9991(81)90145-5

    Article  MATH  Google Scholar 

  55. Hosokawa A, Wilkinson DS, Kang J, Kobayashi M, Toda H (2013) Void growth and coalescence in model materials investigated by high-resolution x-ray microtomography. Int J Fract 181(1):51–66

    Article  Google Scholar 

  56. Hu C, Ghosh S (2008) Locally enhanced Voronoi cell finite element model (LE-VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions. Int J Numer Methods Eng 76(12):1955–1992. https://doi.org/10.1002/nme.2400

    Article  MathSciNet  MATH  Google Scholar 

  57. Hutchinson JW (2012) Generalizing J 2 flow theory: fundamental issues in strain gradient plasticity. Acta Mech Sin 28(4):1078–1086. https://doi.org/10.1007/s10409-012-0089-4

    Article  MathSciNet  MATH  Google Scholar 

  58. Huynh D, Belytschko T (2009) The extended finite element method for fracture in composite materials. Int J Numer Methods Eng 77(2):214–239

    Article  MathSciNet  MATH  Google Scholar 

  59. Ibijola E (2002) On some fundamental concepts of continuum damage mechanics. Comput Methods Appl Mech Eng 191(13–14):1505–1520. https://doi.org/10.1016/S0045-7825(99)90187-1

    Article  MATH  Google Scholar 

  60. Jackiewicz J, Kuna M (2003) Non-local regularization for FE simulation of damage in ductile materials. Comput Mater Sci 28(3–4):684–695. https://doi.org/10.1016/j.commatsci.2003.08.024

    Article  Google Scholar 

  61. Jain JR, Ghosh S (2009) Damage evolution in composites with a homogenization-based continuum damage mechanics model. Int J Damage Mech 18(6):533–568

    Article  Google Scholar 

  62. Jirásek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Eng 188(1):307–330

    Article  MATH  Google Scholar 

  63. Kachanov L (1958) Time of the rupture process under creep conditions. Bull SSR Acad Sci Division Tech Sci 8:26–31

    Google Scholar 

  64. Kouznetsova V, Geers M, Brekelmans W (2004) Size of a representative volume element in a second-order computational homogenization framework. Int J Multiscale Comput Eng 2(4):575–598

    Article  Google Scholar 

  65. Laborde P, Pommier J, Renard Y, Salaün M (2005) High-order extended finite element method for cracked domains. Int J Numer Methods Eng 64(3):354–381

    Article  MATH  Google Scholar 

  66. Lebensohn RA, Escobedo JP, Cerreta EK, Dennis-Koller D, Bronkhorst CA, Bingert JF (2013) Modeling void growth in polycrystalline materials. Acta Mater 61(18):6918–6932. https://doi.org/10.1016/j.actamat.2013.08.004

    Article  Google Scholar 

  67. Lemaitre J, Chaboche JL (1978) Phenomenological approach of damage rupture. J Mécanique Appl 2(3):317–365

    Google Scholar 

  68. Lemaitre J, Desmorat R, Sauzay M (2000) Anisotropic damage law of evolution. Eur J Mech A/Solids 19(2):187–208. https://doi.org/10.1016/S0997-7538(00)00161-3

    Article  MATH  Google Scholar 

  69. Li S, Ghosh S (2004) Debonding in composite microstructures with morphological variations. Int J Comput Methods 1(01):121–149

    Article  MATH  Google Scholar 

  70. Liang Y, Sofronis P (2003) Micromechanics and numerical modelling of the hydrogen-particle-matrix interactions in nickel-base alloys. Model Simul Mater Sci Eng 11(4):523–551. https://doi.org/10.1088/0965-0393/11/4/308

    Article  Google Scholar 

  71. Liu G, Zhou D, Bao Y, Ma J, Han Z (2017) Multiscale analysis of interaction between macro crack and microdefects by using multiscale projection method. Theor Appl Fract Mech 90:65–74

    Article  Google Scholar 

  72. Liu WK, Hao S, Belytschko T, Li S, Chang CT (1999) Multiple scale meshfree methods for damage fracture and localization. Comput Mater Sci 16(1–4):197–205. https://doi.org/10.1016/S0927-0256(99)00062-2

    Article  Google Scholar 

  73. Liu Z, Fleming M, Liu WK (2017) Microstructural material database for self-consistent clustering analysis of elastoplastic strain softening materials. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2017.11.005

    Article  Google Scholar 

  74. Loehnert S, Belytschko T (2007) A multiscale projection method for macro/microcrack simulations. Int J Numer Methods Eng 71(12):1466–1482

    Article  MathSciNet  MATH  Google Scholar 

  75. Lorentz E (2008) A mixed interface finite element for cohesive zone models. Comput Methods Appl Mech Eng 198(2):302–317

    Article  MATH  Google Scholar 

  76. Lorentz E, Besson J, Cano V (2008) Numerical simulation of ductile fracture with the Rousselier constitutive law. Comput Methods Appl Mech Eng 197(21–24):1965–1982. https://doi.org/10.1016/j.cma.2007.12.015

    Article  MATH  Google Scholar 

  77. Massart T, Peerlings R, Geers M (2007) An enhanced multi-scale approach for masonry wall computations with localization of damage. Int J Numer Methods Eng 69(5):1022–1059

    Article  MATH  Google Scholar 

  78. Mathur K, Needleman A, Tvergaard V (1994) Ductile failure analyses on massively parallel computers. Comput Methods Appl Mech Eng 119(3–4):283–309. https://doi.org/10.1016/0045-7825(94)90091-4

    Article  MATH  Google Scholar 

  79. Matouš K, Geers MGD, Kouznetsova VG, Gillman A (2017) A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys 330:192–220. https://doi.org/10.1016/j.jcp.2016.10.070

    Article  MathSciNet  Google Scholar 

  80. Matsui K, Terada K, Yuge K (2004) Two-scale finite element analysis of heterogeneous solids with periodic microstructures. Comput Struct 82(7):593–606

    Article  Google Scholar 

  81. McAuliffe C, Waisman H (2016) A coupled phase field shear band model for ductile-brittle transition in notched plate impacts. Comput Methods Appl Mech Eng 305:173–195. https://doi.org/10.1016/j.cma.2016.02.018

    Article  MathSciNet  MATH  Google Scholar 

  82. McHugh P, Connolly P (2003) Micromechanical modelling of ductile crack growth in the binder phase of WC-Co. Comput Mater Sci 27(4):423–436. https://doi.org/10.1016/S0927-0256(03)00045-4

    Article  Google Scholar 

  83. Mediavilla J, Peerlings R, Geers M (2006) A robust and consistent remeshing-transfer operator for ductile fracture simulations. Comput Struct 84(8–9):604–623. https://doi.org/10.1016/j.compstruc.2005.10.007

    Article  Google Scholar 

  84. Mediavilla J, Peerlings RHJ, Geers MGD (2006) Discrete crack modelling of ductile fracture driven by non-local softening plasticity. Int J Numer Methods Eng 66(4):661–688. https://doi.org/10.1002/nme.1572

    Article  MATH  Google Scholar 

  85. Meng Q, Wang Z (2015) Prediction of interfacial strength and failure mechanisms in particle-reinforced metal-matrix composites based on a micromechanical model. Eng Fract Mech 142:170–183. https://doi.org/10.1016/j.engfracmech.2015.06.001

    Article  Google Scholar 

  86. 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):2765–2778

    Article  MathSciNet  MATH  Google Scholar 

  87. Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations. Int J Numer Methods Eng 83(10):1273–1311

    Article  MathSciNet  MATH  Google Scholar 

  88. 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  MathSciNet  MATH  Google Scholar 

  89. Moës N, Gravouil A, Belytschko T (2002) Non-planar 3d crack growth by the extended finite element and level sets-part i: mechanical model. Int J Numer Methods Eng 53(11):2549–2568

    Article  MATH  Google Scholar 

  90. Moës N, Stolz C, Bernard PE, Chevaugeon N (2011) A level set based model for damage growth: the thick level set approach. Int J Numer Methods Eng 86(3):358–380. https://doi.org/10.1002/nme.3069

    Article  MathSciNet  MATH  Google Scholar 

  91. Moorthy S, Ghosh S (1998) A Voronoi Cell finite element model for particle cracking in elastic-plastic composite materials. Comput Methods Appl Mech Eng 151(3–4):377–400. https://doi.org/10.1016/S0045-7825(97)00160-6

    Article  MATH  Google Scholar 

  92. Morgeneyer T, Helfen L, Sinclair I, Proudhon H, Xu F, Baumbach T (2011) Ductile crack initiation and propagation assessed via in situ synchrotron radiation-computed laminography. Scr Mater 65(11):1010–1013. https://doi.org/10.1016/j.scriptamat.2011.09.005

    Article  Google Scholar 

  93. Nahshon K, Hutchinson J (2008) Modification of the Gurson Model for shear failure. Eur J Mech A/Solids 27(1):1–17. https://doi.org/10.1016/j.euromechsol.2007.08.002

    Article  MATH  Google Scholar 

  94. Nègre P, Steglich D, Brocks W, Koçak M (2003) Numerical simulation of crack extension in aluminium welds. Comput Mater Sci 28(3–4):723–731. https://doi.org/10.1016/j.commatsci.2003.08.026

    Article  Google Scholar 

  95. Nguyen O, Repetto E, Ortiz M, Radovitzky R (2001) A cohesive model of fatigue crack growth. Int J Fract 110(4):351–369

    Article  Google Scholar 

  96. Nguyen VP, Stroeven M, Sluys LJ (2011) Multiscale continuous and discontinuous modeling of heterogeneous materials: a review on recent developments. J Multiscale Model 3(04):229–270

    Article  MathSciNet  Google Scholar 

  97. O’Keeffe SC, Tang S, Kopacz AM, Smith J, Rowenhorst DJ, Spanos G, Liu WK, Olson GB (2015) Multiscale ductile fracture integrating tomographic characterization and 3-D simulation. Acta Mater 82:503–510. https://doi.org/10.1016/j.actamat.2014.09.016

    Article  Google Scholar 

  98. Oliver J, Huespe A, Sanchez P (2006) A comparative study on finite elements for capturing strong discontinuities: E-fem vs x-fem. Comput Methods Appl Mech Eng 195(37):4732–4752

    Article  MathSciNet  MATH  Google Scholar 

  99. Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. International Journal for Numerical Methods in Engineering 44(9):1267–1282. https://doi.org/10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0.CO;2-7

    Article  MATH  Google Scholar 

  100. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49. https://doi.org/10.1016/0021-9991(88)90002-2

    Article  MathSciNet  MATH  Google Scholar 

  101. Östlund R, Golling S, Oldenburg M (2016) Microstructure based modeling of ductile fracture initiation in press-hardened sheet metal structures. Comput Methods Appl Mech Eng 302:90–108. https://doi.org/10.1016/j.cma.2015.11.035

    Article  MathSciNet  MATH  Google Scholar 

  102. Panchal JH, Kalidindi SR, McDowell DL (2013) Key computational modeling issues in Integrated Computational Materials Engineering. Comput-Aided Des 45(1):4–25. https://doi.org/10.1016/j.cad.2012.06.006

    Article  Google Scholar 

  103. Peerlings RHJ, De Borst R, Brekelmans WAM, De Vree JHP (1996) Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering 39(19):3391–3403. https://doi.org/10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D

    Article  MATH  Google Scholar 

  104. Perzyński K, Wrozyna A, Kuziak R, Legwand A, Madej L (2017) Development and validation of multi scale failure model for dual phase steels. Finite Elements Anal Des 124(October 2016):7–21. https://doi.org/10.1016/j.finel.2016.10.001

    Article  Google Scholar 

  105. Pineau A, Benzerga A, Pardoen T (2016) Failure of metals I: Brittle and ductile fracture. Acta Mater 107(January):424–483. https://doi.org/10.1016/j.actamat.2015.12.034

    Article  Google Scholar 

  106. Pourmodheji R, Mashayekhi M (2012) Improvement of the extended finite element method for ductile crack growth. Mater Sci Eng A 551:255–271

    Article  Google Scholar 

  107. Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct 88(23):1391–1411

    Article  Google Scholar 

  108. Ramazani A, Schwedt A, Aretz A, Prahl U, Bleck W (2013) Characterization and modelling of failure initiation in DP steel. Comput Mater Sci 75:35–44

    Article  Google Scholar 

  109. Reed WH, Hill T (1973) Triangular mesh methods for the neutron transport equation. Tech. rep., Los Alamos Scientific Lab., N. Mex. (USA)

  110. Ren B, Li S, Qian J, Zeng X (2011) Meshfree simulations of spall fracture. Comput Methods Appl Mech Eng 200(5–8):797–811. https://doi.org/10.1016/j.cma.2010.003

    Article  MathSciNet  MATH  Google Scholar 

  111. Rice J (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35:379–386

    Article  Google Scholar 

  112. Rice JR (1968) Mathematical analysis in the mechanics of fracture. Fract Adv Treatise 2:191–311

    Google Scholar 

  113. Rousselier G (1987) Ductile fracture models and their potential in local approach of fracture. Nuclear Eng Des 105(1):97–111. https://doi.org/10.1016/0029-5493(87)90234-2

    Article  Google Scholar 

  114. Roux E, Bernacki M, Bouchard PO (2013) A level-set and anisotropic adaptive remeshing strategy for the modeling of void growth under large plastic strain. Comput Mater Sci 68:32–46. https://doi.org/10.1016/j.commatsci.2012.10.004

    Article  Google Scholar 

  115. Roux E, Shakoor M, Bernacki M, Bouchard PO (2014) A new finite element approach for modelling ductile damage void nucleation and growth-analysis of loading path effect on damage mechanisms. Model Simul Mater Sci Eng 22(7):075,001. https://doi.org/10.1088/0965-0393/22/7/075001

    Article  Google Scholar 

  116. Salih S, Davey K, Zou Z (2016) Rate-dependent elastic and elasto-plastic cohesive zone models for dynamic crack propagation. Int J Solids Struct 90:95–115

    Article  Google Scholar 

  117. Samal M, Seidenfuss M, Roos E, Dutta B, Kushwaha H (2008) Finite element formulation of a new nonlocal damage model. Finite Elem Anal Des 44(6–7):358–371. https://doi.org/10.1016/j.finel.2007.12.002

    Article  Google Scholar 

  118. Scheider I (2009) Derivation of separation laws for cohesive models in the course of ductile fracture. Eng Fract Mech 76(10):1450–1459

    Article  Google Scholar 

  119. Scheyvaerts F, Onck P, Tekoglu C, Pardoen T (2011) The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension. J Mech Phys Solids 59(2):373–397. https://doi.org/10.1016/j.jmps.2010.003

    Article  MATH  Google Scholar 

  120. Seabra MR, de Sa JMC, Šuštarič P, Rodič T (2012) Some numerical issues on the use of xfem for ductile fracture. Comput Mech 50(5):611–629

    Article  MathSciNet  MATH  Google Scholar 

  121. Seabra MRR, Šuštarič P, Cesar de Sa JMA, Rodič T (2013) Damage driven crack initiation and propagation in ductile metals using XFEM. Comput Mech 52(1):161–179. https://doi.org/10.1007/s00466-012-0804-9

    Article  MathSciNet  MATH  Google Scholar 

  122. Shakoor M, Bernacki M, Bouchard PO (2015) A new body-fitted immersed volume method for the modeling of ductile fracture at the microscale: analysis of void clusters and stress state effects on coalescence. Eng Fract Mech 147:398–417. https://doi.org/10.1016/j.engfracmech.2015.06.057

    Article  Google Scholar 

  123. Shakoor M, Bernacki M, Bouchard PO (2017) Ductile fracture of a metal matrix composite studied using 3D numerical modeling of void nucleation and coalescence. Engi Fract Mech. https://doi.org/10.1016/j.engfracmech.2017.10.027

    Article  Google Scholar 

  124. Shakoor M, Bouchard PO, Bernacki M (2017) An adaptive level-set method with enhanced volume conservation for simulations in multiphase domains. Int J Numer Methods Eng 109(4):555–576. https://doi.org/10.1002/nme.5297

    Article  MathSciNet  Google Scholar 

  125. Shakoor M, Buljac A, Neggers J, Hild F, Morgeneyer TF, Helfen L, Bernacki M, Bouchard PO (2017) On the choice of boundary conditions for micromechanical simulations based on 3D imaging. Int J Solids Struct 112:83–96. https://doi.org/10.1016/j.ijsolstr.2017.02.018

    Article  Google Scholar 

  126. Shanthraj P, Sharma L, Svendsen B, Roters F, Raabe D (2016) A phase field model for damage in elasto-viscoplastic materials. Comput Methods Appl Mech Eng 312:167–185. https://doi.org/10.1016/j.cma.2016.05.006

    Article  MathSciNet  Google Scholar 

  127. Simkins DC, Li S (2006) Meshfree simulations of thermo-mechanical ductile fracture. Comput Mech 38(3):235–249. https://doi.org/10.1007/s00466-005-0744-8

    Article  MATH  Google Scholar 

  128. Simonsen BC, Li S (2004) Mesh-free simulation of ductile fracture. Int J Numer Methods Eng 60(8):1425–1450. https://doi.org/10.1002/nme.1009

    Article  MATH  Google Scholar 

  129. Singh I, Mishra B, Bhattacharya S (2011) Xfem simulation of cracks, holes and inclusions in functionally graded materials. Int J Mech Mater Des 7(3):199

    Article  Google Scholar 

  130. Soghrati S, Xiao F, Nagarajan A (2017) A conforming to interface structured adaptive mesh refinement technique for modeling fracture problems. Comput Mech 59(4):667–684

    Article  MathSciNet  Google Scholar 

  131. Steglich D, Siegmund T, Brocks W (1999) Micromechanical modeling of damage due to particle cracking in reinforced metals. Comput Mater Sci 16(1–4):404–413. https://doi.org/10.1016/S0927-0256(99)00083-X

    Article  Google Scholar 

  132. Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181(1):43–69

    Article  MathSciNet  MATH  Google Scholar 

  133. Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Meth Eng 48:1741–1760

    Article  MATH  Google Scholar 

  134. Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Methods Eng 48(11):1549–1570

    Article  MATH  Google Scholar 

  135. Sukumar N, Chopp D, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190(46–47):6183–6200. https://doi.org/10.1016/S0045-7825(01)00215-8

    Article  MathSciNet  MATH  Google Scholar 

  136. Sukumar N, Dolbow J, Moës N (2015) Extended finite element method in computational fracture mechanics: a retrospective examination. Int J Fract 196(1–2):189–206

    Article  Google Scholar 

  137. Svenning E, Larsson F, Fagerström M (2017) Two-scale modeling of fracturing solids using a smeared macro-to-micro discontinuity transition. Comput Mech 60:627–641. https://doi.org/10.1007/s00466-017-1426-z

    Article  MathSciNet  MATH  Google Scholar 

  138. Tekoǧlu C, Hutchinson JW, Pardoen T (2015) On localization and void coalescence as a precursor to ductile fracture. Doi, Philos Trans Ser A Math Phys Eng Sci. https://doi.org/10.1098/rsta.2014.0121

    Book  Google Scholar 

  139. Tian R, Chan S, Tang S, Kopacz AM, Wang JS, Jou HJ, Siad L, Lindgren LE, Olson GB, Liu WK (2010) A multiresolution continuum simulation of the ductile fracture process. J Mech Phys Solids 58(10):1681–1700

    Article  MATH  Google Scholar 

  140. Tomar V, Zhai J, Zhou M (2004) Bounds for element size in a variable stiffness cohesive finite element model. Int J Numer Methods Eng 61(11):1894–1920

    Article  MATH  Google Scholar 

  141. Toro S, Sánchez PJ, Podestá JM, Blanco PJ, Huespe AE, Feijóo RA (2016) Cohesive surface model for fracture based on a two-scale formulation: computational implementation aspects. Computat Mech 58(4):549–585. https://doi.org/10.1007/s00466-016-1306-y

    Article  MathSciNet  MATH  Google Scholar 

  142. Turon A, Davila CG, Camanho PP, Costa J (2007) An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng Fract Mech 74(10):1665–1682

    Article  Google Scholar 

  143. Turon A, Camanho P, Costa J, Renart J (2010) Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: definition of interlaminar strengths and elastic stiffness. Compos Struct 92(8):1857–1864

    Article  Google Scholar 

  144. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32(1):157–169. https://doi.org/10.1016/0001-6160(84)90213-X

    Article  Google Scholar 

  145. Vajragupta N, Uthaisangsuk V, Schmaling B, Münstermann S, Hartmaier A, Bleck W (2012) A micromechanical damage simulation of dual phase steels using XFEM. Comput Mater Sci 54(1):271–279. https://doi.org/10.1016/j.commatsci.2011.10.035

    Article  Google Scholar 

  146. Vaz M, Owen DRJ (2001) Aspects of ductile fracture and adaptive mesh refinement in damaged elasto-plastic materials. International Journal for Numerical Methods in Engineering 50(1):29–54. https://doi.org/10.1002/1097-0207(20010110)50:1<29::AID-NME18>3.0.CO;2-G

    Article  MATH  Google Scholar 

  147. Vocialta M, Richart N, Molinari JF (2017) 3d dynamic fragmentation with parallel dynamic insertion of cohesive elements. Int J Numer Methods Eng 109(12):1655–1678

    Article  MathSciNet  Google Scholar 

  148. Wang Z, Yu T, Bui TQ, Trinh NA, Luong NTH, Duc ND, Doan DH (2016) Numerical modeling of 3-d inclusions and voids by a novel adaptive xfem. Adv Eng Softw 102:105–122

    Article  Google Scholar 

  149. Wolf J, Longère P, Cadou JM, Crété JP (2017) Numerical modeling of strain localization in engineering ductile materials combining cohesive models and X-FEM. Int J Mech Mater Des. https://doi.org/10.1007/s10999-017-9370-9

    Article  Google Scholar 

  150. Wolff C, Richart N, Molinari JF (2015) A non-local continuum damage approach to model dynamic crack branching. Int J Numer Methods Eng 101(12):933–949

    Article  MathSciNet  MATH  Google Scholar 

  151. Wulf J, Steinkopff T, Fischmeister H (1996) Fe-simulation of crack paths in the real microstructure of an Al(6061)/SiC composite. Acta Mater 44(5):1765–1779. https://doi.org/10.1016/1359-6454(95)00328-2

    Article  Google Scholar 

  152. Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434

    Article  MATH  Google Scholar 

  153. Xue L (2008) Constitutive modeling of void shearing effect in ductile fracture of porous materials. Eng Fract Mech 75(11):3343–3366. https://doi.org/10.1016/j.engfracmech.2007.07.022

    Article  Google Scholar 

  154. Ye C, Shi J, Cheng GJ (2012) An extended finite element method (xfem) study on the effect of reinforcing particles on the crack propagation behavior in a metal-matrix composite. Int J Fatigue 44:151–156

    Article  Google Scholar 

  155. Yuan Z, Fish J (2008) Toward realization of computational homogenization in practice. Int J Numer Methods Eng 73(3):361–380

    Article  MathSciNet  MATH  Google Scholar 

  156. Zhang Z, Naga A (2005) A new finite element gradient recovery method: superconvergence property. SIAM J Sci Comput 26(4):1192–1213. https://doi.org/10.1137/S1064827503402837

    Article  MathSciNet  MATH  Google Scholar 

  157. Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineerng analysis. Int J Numer Methods Eng 24(2):337–357. https://doi.org/10.1002/nme.1620240206

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The supports of the Carnot M.I.N.E.S Institute and the French Agence Nationale de la Recherche (ANR) through the COMINSIDE project are gratefully acknowledged.

Author information

Authors and Affiliations

  1. MINES ParisTech, PSL Research University, CEMEF-Centre de mise en forme des matériaux, CNRS UMR 7635, CS 10207 rue Claude Daunesse, 06904, Sophia Antipolis Cedex, France

    Modesar Shakoor, Victor Manuel Trejo Navas, Daniel Pino Munõz, Marc Bernacki & Pierre-Olivier Bouchard

Authors
  1. Modesar Shakoor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor Manuel Trejo Navas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Pino Munõz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marc Bernacki
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Pierre-Olivier Bouchard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Modesar Shakoor.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shakoor, M., Trejo Navas, V.M., Pino Munõz, D. et al. Computational Methods for Ductile Fracture Modeling at the Microscale. Arch Computat Methods Eng 26, 1153–1192 (2019). https://doi.org/10.1007/s11831-018-9276-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-018-9276-1

Keywords

Search

Navigation