International Journal of Fracture

, Volume 215, Issue 1–2, pp 49–65 | Cite as

Validated simulations of dynamic crack propagation in single crystals using EFEM and XFEM

  • Q. ZengEmail author
  • M. H. Motamedi
  • A. F. T. Leong
  • N. P. Daphalapurkar
  • T. C. Hufnagel
  • K. T. Ramesh
Original Paper


Brittle and quasibrittle materials such as ceramics and geomaterials fail through dynamic crack propagation during impact events. Simulations of such events are important in a number of applications. This paper compares the effectiveness of the embedded finite element method (EFEM) and the extended finite element method (XFEM) in modeling dynamic crack propagation by validating each approach against an impact experiment performed on single crystal quartz together with in-situ imaging of the dynamic fracture using X-ray phase contrast imaging (XPCI). The experiment is conducted in a Kolsky bar (generating a strain rate on the order of \(10^3\,\text {s}^{-1}\)) that is operated at the synchrotron facilities at the advanced photon source (APS). The in situ XPCI technique can record the dynamic crack propagation with micron-scale spatial resolution and sub-microsecond temporal resolution, and the corresponding images are used to extract the time-resolved crack propagation path and velocity. A unified framework is first presented for the dynamic discretization formulations of EFEM and XFEM. This framework clarifies the differences between the two methods in enrichment techniques and numerical solution schemes. In both cases, a cohesive law is used to describe the fracture process after crack initiation. The simulations of the dynamic fracture experiment using the two simulation approaches are compared with the in situ experimental observations and measurements. The performance of each method is discussed with respect to capturing the early crack propagation process.


Dynamic fracture Embedded finite element method Extended finite element method In situ X-ray phase contrast imaging Dynamic fracture experiments 



This work was supported by the Defense Threat Reduction Agency, Basic Research Award # HDTRA1-15-1-0056, to Johns Hopkins University. The content, views, and conclusions contained in this document are those of the authors and should not be interpreted as representing the official positions or policies, either expressed or implied, of the Defense Threat Reduction Agency or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.


  1. Abedi R, Haber RB, Clarke PL (2017) Effect of random defects on dynamic fracture in quasi-brittle materials. Int J Fract 208(1):241–268. Google Scholar
  2. Areias P, Belytschko T (2005) Non-linear analysis of shells with arbitrary evolving cracks using XFEM. Int J Numer Methods Eng 62(3):384–415Google Scholar
  3. 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):2863–2885.
  4. Armero F, Linder C (2009) Numerical simulation of dynamic fracture using finite elements with embedded discontinuities. Int J Fract 160(2):119–141Google Scholar
  5. Bažant ZP (2002) Concrete fracture models: testing and practice. Eng Fract Mech 69(2):165–205.
  6. Bechmann R (1958) Elastic and piezoelectric constants of alpha-quartz. Phys Rev 110:1060–1061. Google Scholar
  7. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5):601–620Google Scholar
  8. Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58(12):1873–1905Google Scholar
  9. Belytschko T, Liu WK, Moran B, Elkhodary K (2013) Nonlinear finite elements for continua and structures. Wiley, LondonGoogle Scholar
  10. Benedetti I, Aliabadi M (2013) A three-dimensional cohesive–frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials. Comput Methods Appl Mech Eng 265:36–62.
  11. Borja RI (2000) A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation. Comput Methods Appl Mech Eng 190(11):1529–1549.
  12. 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.
  13. Camacho G, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33(20–22):2899–2938.
  14. Carol I, Prat P, López C (1997) Normal/shear cracking model: application to discrete crack analysis. J Eng Mech 123(8):765–773. 123:8(765)Google Scholar
  15. de Borst R, Remmers JJ, Needleman A (2006) Mesh-independent discrete numerical representations of cohesive-zone models. Eng Fract Mech 73(2):160–177.
  16. Dias-da Costa D, Alfaiate J, Sluys LJ, Júlio E (2009) 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. Google Scholar
  17. Dolbow J, Moës N, Belytschko T (2001) An extended finite element method for modeling crack growth with frictional contact. Comput Methods Appl Mech Eng 190(51):6825–6846Google Scholar
  18. Espinosa HD, Zavattieri PD (2003) A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials. Part I: Theory and numerical implementation. Mech Mater 35(3–6):333–364. Google Scholar
  19. Fineberg J, Gross SP, Marder M, Swinney HL (1991) Instability in dynamic fracture. Phys Rev Lett 67(4):457Google Scholar
  20. Follansbee P, Frantz C (1983) Wave propagation in the split Hopkinson pressure bar. J Eng Mater Technol 105(1):61–66Google Scholar
  21. Gálvez J, Cervenka J, Cendon D, Saouma V (2002) A discrete crack approach to normal/shear cracking of concrete. Cem Concr Res 32(10):1567–1585.
  22. Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(33):3523–3540.
  23. Huespe A, Oliver J, Sanchez P, Blanco S, Sonzogni V (2006) Strong discontinuity approach in dynamic fracture simulations. Mecán Comput 25:1997–2018Google Scholar
  24. Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood CliffsGoogle Scholar
  25. Ida Y (1972) Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy. J Geophys Res 77(20):3796–3805. Google Scholar
  26. IEEE (1988) IEEE standard on piezoelectricity. ANSI/IEEE Std 176-1987.
  27. Iwasa M, Bradt R (1987) Cleavage of natural and synthetic single crystal quartz. Mater Res Bull 22(9):1241–1248.
  28. Iwasa M, Ueno T, Bradt RC (1981) Fracture toughness of quartz and sapphire single crystals at room temperature. J Soc Mater Sci Jpn 30(337):1001–1004. Google Scholar
  29. Jirásek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Eng 188(1):307–330.
  30. Kalthoff J, Winkler S (1988) Failure mode transition at high rates of shear loading. Int Conf Impact Load Dyn Behav Mater 1:185–195Google Scholar
  31. Leon S, Spring D, Paulino G (2014) Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements. Int J Numer Methods Eng 100(8):555–576Google Scholar
  32. Leong AFT, Robinson AK, Fezzaa K, Sun T, Sinclair N, Casem DT, Lambert PK, Hustedt CJ, Daphalapurkar NP, Ramesh KT, Hufnagel TC (2018) Quantitative in situ studies of dynamic fracture in brittle solids using dynamic X-ray phase contrast imaging. Exp Mech
  33. Melenk J, Babus̆ka I, (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1):289–314.
  34. Menouillard T, Réthoré J, Combescure A, Bung H (2006) Efficient explicit time stepping for the extended finite element method (X-FEM). Int J Numer Methods Eng 68(9):911–939Google Scholar
  35. 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. 10.1002/(SICI)1097-0207(19990910)46:1\(<\)131::AID-NME726\(>\)3.0.CO;2-JGoogle Scholar
  36. Motamedi M, Weed D, Foster C (2016) Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach. Int J Solids Struct 85–86:44–56.
  37. 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. 10.1002/(SICI)1097-0207(19961115)39:21\(<\)3575:: AID-NME65\(>\)3.0.CO;2-EGoogle Scholar
  38. Oliver J, Huespe A, Pulido M, Chaves E (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 69(2):113–136.
  39. Oliver J, Huespe A, Sánchez P (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.
  40. Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44(9):1267–1282. 10.1002/(SICI)1097-0207(19990330)44:9\(<\)1267::AID-NME486\(>\)3.0.CO;2-7Google Scholar
  41. Rabczuk T (2013) Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. ISRN Appl Math 2013:38Google Scholar
  42. Ramulu M, Kobayashi AS (1985) Mechanics of crack curving and branching—a dynamic fracture analysis. Int J Fract 27(3):187–201. Google Scholar
  43. Rangarajan R, Chiaramonte MM, Hunsweck MJ, Shen Y, Lew AJ (2015) Simulating curvilinear crack propagation in two dimensions with universal meshes. Int J Numer Methods Eng 102(3–4):632–670Google Scholar
  44. Rinehart AJ, Bishop JE, Dewers T (2015) Fracture propagation in Indiana limestone interpreted via linear softening cohesive fracture model. J Geophys Res Solid Earth 120(4):2292–2308.
  45. Saksala T, Brancherie D, Harari I, Ibrahimbegovic A (2015) Combined continuum damage-embedded discontinuity model for explicit dynamic fracture analyses of quasi-brittle materials. Int J Numer Methods Eng 101(3):230–250Google Scholar
  46. Sancho JM, Planas J, Galves JC, Reyes E, Cendon DA (2006) An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*. Fatigue Fract Eng Mater Struct 29(12):1056–1065. Google Scholar
  47. Scheider I, Brocks W (2003) Simulation of cup-cone fracture using the cohesive model. Eng Fract Mech 70(14):1943–1961.
  48. Sharon E, Fineberg J (1996) Microbranching instability and the dynamic fracture of brittle materials. Phys Rev B 54:7128–7139. Google Scholar
  49. 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. Google Scholar
  50. 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. Google Scholar
  51. Simulia (2014) Abaqus analysis user’s guide, version 6.14. SimuliaGoogle Scholar
  52. Song JH, Belytschko T (2009) Dynamic fracture of shells subjected to impulsive loads. J Appl Mech 76(5):051301Google Scholar
  53. Song JH, Areias P, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Methods Eng 67(6):868–893Google Scholar
  54. Song JH, Wang H, Belytschko T (2008) A comparative study on finite element methods for dynamic fracture. Comput Mech 42(2):239–250. Google Scholar
  55. 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):6183–6200.
  56. Ward RW (1984) The constants of alpha quartz. In: 38th annual symposium on frequency control, pp 22–31.
  57. Weed DA, Foster CD, Motamedi MH (2017) A robust numerical framework for simulating localized failure and fracture propagation in frictional materials. Acta Geotech 12(2):253–275. Google Scholar
  58. Wells G, Sluys L (2000) Application of embedded discontinuities for softening solids. Eng Fract Mech 65(2–3):263–281.
  59. Wells GN, Sluys L (2001) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12):2667–2682Google Scholar
  60. Xu D, Liu Z, Liu X, Zeng Q, Zhuang Z (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Comput Mech 54(2):489–502Google Scholar
  61. Yoffe EH (1951) The moving Griffith crack. Lond Edinb Dublin Philos Mag J Sci 42(330):739–750. Google Scholar
  62. Zeng Q, Liu Z, Wang T, Gao Y, Zhuang Z (2018) Fully coupled simulation of multiple hydraulic fractures to propagate simultaneously from a perforated horizontal wellbore. Comput Mech 61(1):137–155. Google Scholar
  63. Zhang ZJ, Paulino GH, Celes W (2007) Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials. Int J Numer Methods Eng 72(8):893–923. Google Scholar
  64. Zhou F, Molinari JF, Ramesh KT (2006) Effects of material properties on the fragmentation of brittle materials. Int J Fract 139(2):169–196. Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Q. Zeng
    • 1
    Email author
  • M. H. Motamedi
    • 1
  • A. F. T. Leong
    • 1
  • N. P. Daphalapurkar
    • 1
    • 2
    • 4
  • T. C. Hufnagel
    • 1
    • 3
  • K. T. Ramesh
    • 1
    • 4
  1. 1.Hopkins Extreme Materials InstituteJohns Hopkins UniversityBaltimoreUSA
  2. 2.T3 Group, Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Department of Materials Science and EngineeringJohns Hopkins UniversityBaltimoreUSA
  4. 4.Department of Mechanical EngineeringJohns Hopkins UniversityBaltimoreUSA

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