Archive of Applied Mechanics

, Volume 88, Issue 1–2, pp 287–316 | Cite as

Material point method for crack propagation in anisotropic media: a phase field approach



A novel phase field formulation implemented within a material point method setting is developed to address brittle fracture simulation in anisotropic media. The case of strong anisotropy in the crack surface energy is treated by considering an appropriate variational, i.e. phase field approach. Material point method is utilized to efficiently treat the resulting coupled governing equations. The brittle fracture governing equations are defined at a set of Lagrangian material points and subsequently interpolated at the nodes of a fixed Eulerian mesh where solution is performed. As a result, the quality of the solution does not depend on the quality of the underlying finite element mesh and is relieved from mesh distortion errors. The efficiency and validity of the proposed method are assessed through a set of benchmark problems.


Brittle fracture Anisotropy Phase field Material point method 



The research described in this paper has been financed by the University of Nottingham through the Dean of Engineering Prize, a scheme for pump priming support for early-career academic staff. The authors are grateful to the University of Nottingham for access to its high-performance computing facility.


  1. 1.
    Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57(1), 149–167 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bandara, S., Soga, K.: Coupling of soil deformation and pore fluid flow using material point method. Comput. Geotech. 63, 199–214 (2015)CrossRefGoogle Scholar
  4. 4.
    Bardenhagen, S., Kober, E.: The generalized interpolation material point method. Comput. Model. Eng. Sci. 5(6), 477–495 (2004)Google Scholar
  5. 5.
    Bardenhagen, S.G., Nairn, J.A., Lu, H.: Simulation of dynamic fracture with the material point method using a mixed J-integral and cohesive law approach. Int. J. Fract. 170(1), 49–66 (2011)CrossRefMATHGoogle Scholar
  6. 6.
    Bathe, K.J.: Finite Element Procedures. Prentice Hall, Upper Saddle River, NJ (2007)MATHGoogle Scholar
  7. 7.
    Batra, R., Zhang, G.: Search algorithm, and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method. Comput. Mech. 40(3), 531–546 (2007)CrossRefMATHGoogle Scholar
  8. 8.
    Bobaru, F., Hu, W.: The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176(2), 215–222 (2012)CrossRefGoogle Scholar
  9. 9.
    Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217–220, 77–95 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Borden, M.J., Hughes, T.J., Landis, C.M., Verhoosel, C.V.: 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 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1–3), 5–148 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Clayton, J., Knap, J.: Phase field modeling and simulation of coupled fracture and twinning in single crystals and polycrystals. Comput. Methods Appl. Mech. Eng. 312, 447–467 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Daphalapurkar, N.P., Lu, H., Coker, D., Komanduri, R.: Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method. Int. J. Fract. 143(1), 79–102 (2007)CrossRefMATHGoogle Scholar
  14. 14.
    Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. A 221, 163–198 (1921)CrossRefGoogle Scholar
  16. 16.
    Gültekin, O., Dal, H., Holzapfel, G.A.: A phase-field approach to model fracture of arterial walls: theory and finite element analysis. Comput. Methods Appl. Mech. Eng. 312, 542–566 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Homel, M.A., Herbold, E.B.: Field-gradient partitioning for fracture and frictional contact in the material point method. Int. J. Numer. Methods Eng. 109(7), 1013–1044 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hughes, T., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5–8), 301–313 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jassim, I., Stolle, D., Vermeer, P.: Two-phase dynamic analysis by material point method. Int. J. Numer. Anal. Methods Geomech. 37(15), 2502–2522 (2013)CrossRefGoogle Scholar
  20. 20.
    Kakouris, E.G., Triantafyllou, S.P.: Phase-field material point method for brittle fracture. Int. J. Numer. Methods Eng. (2017). doi: 10.1002/nme.5580 Google Scholar
  21. 21.
    Li, B., Peco, C., Milln, D., Arias, I., Arroyo, M.: Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy. Int. J. Numer. Methods Eng. 102(3–4), 711–727 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, T., Marigo, J.J., Guilbaud, D., Potapov, S.: Gradient damage modeling of brittle fracture in an explicit dynamics context. Int. J. Numer. Methods Eng. 108(11), 1381–1405 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Miehe, C., Hofacker, M., Welschinger, F.: 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 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83(10), 1273–1311 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Miehe, C., Aldakheel, F., Raina, A.: Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int. J. Plast. 84, 1–32 (2016)CrossRefGoogle Scholar
  26. 26.
    Nairn, J.A.: Material point method calculations with explicit cracks. Comput. Model. Eng. Sci. 4(6), 649–664 (2003)MATHGoogle Scholar
  27. 27.
    Nairn, J.A., Hammerquist, C., Aimene, Y.E.: Numerical implementation of anisotropic damage mechanics. Int. J. Numer. Methods Eng. (2017). doi: 10.1002/nme.5585 Google Scholar
  28. 28.
    Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Sadeghirad, A., Brannon, R.M., Burghardt, J.: A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. Int. J. Numer. Methods Eng. 86(12), 1435–1456 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Sadeghirad, A., Brannon, R., Guilkey, J.: Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces. Int. J. Numer. Methods Eng. 95(11), 928–952 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sanchez, J., Schreyer, H., Sulsky, D., Wallstedt, P.: Solving quasi-static equations with the material-point method. Int. J. Numer. Methods Eng. 103(1), 60–78 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Scholtès, L., Donzé, F.V.: Modelling progressive failure in fractured rock masses using a 3D discrete element method. Int. J. Rock Mech. Min. Sci. 52, 18–30 (2012)CrossRefGoogle Scholar
  33. 33.
    Schreyer, H., Sulsky, D., Zhou, S.J.: Modeling delamination as a strong discontinuity with the material point method. Comput. Methods Appl. Mech. Eng. 191(23–24), 2483–2507 (2002)CrossRefMATHGoogle Scholar
  34. 34.
    Shanthraj, P., Svendsen, B., Sharma, L., Roters, F., Raabe, D.: Elasto-viscoplastic phase field modelling of anisotropic cleavage fracture. J. Mech. Phys. Solids 99, 19–34 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Steffen, M., Kirby, R.M., Berzins, M.: Analysis and reduction of quadrature errors in the material point method (MPM). Int. J. Numer. Methods Eng. 76(6), 922–948 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Steffen, M., Wallstedt, P.M., Guilkey, J., Kirby, R., Berzins, M.: Examination and analysis of implementation choices within the material point method (MPM). Comput. Model. Eng. Sci. 31(2), 107–127 (2008)Google Scholar
  37. 37.
    Sulsky, D., Chen, Z., Schreyer, H.L.: A particle method for history-dependent materials. Comput. Methods Appl. Mech. Eng. 118(1–2), 179–196 (1994)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Sulsky, D., Kaul, A.: Implicit dynamics in the material-point method. Comput. Methods Appl. Mech. Eng. 193(1214), 1137–1170 (2004)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sulsky, D., Schreyer, L.: MPM simulation of dynamic material failure with a decohesion constitutive model. Eur. J. Mech. A/Solids 23(3), 423–445 (2004)CrossRefMATHGoogle Scholar
  40. 40.
    Ting, T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York (1996)MATHGoogle Scholar
  41. 41.
    Yang, P., Gan, Y., Zhang, X., Chen, Z., Qi, W., Liu, P.: Improved decohesion modeling with the material point method for simulating crack evolution. Int. J. Fract. 186(1), 177–184 (2014)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Centre for Structural Engineering and InformaticsThe University of NottinghamNottinghamUK

Personalised recommendations