Computational Mechanics

, Volume 64, Issue 6, pp 1537–1556 | Cite as

Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method

  • Tao Wang
  • Xuan Ye
  • Zhanli LiuEmail author
  • Dongyang Chu
  • Zhuo ZhuangEmail author
Original Paper


The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials.


Phase field method Mohr–Coulomb failure criterion Compression-shear failure Explicit time integration Dynamic crack propagation 



This work is supported by National Natural Science Foundation of China, under Grant No. 11532008, the Special Research Grant for Doctor Discipline by Ministry of Education, China under Grant No. 20120002110075.


  1. 1.
    Krueger R (2004) Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 57(2):109–143. CrossRefGoogle Scholar
  2. 2.
    Elices M, Guinea GV, Gmez J, Planas J (2002) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69(2):137–163. CrossRefGoogle Scholar
  3. 3.
    Mos N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150.<131::AID-NME726>3.0.CO;2-J
  4. 4.
    Zhao J, Li Y, Liu WK (2015) Predicting band structure of 3d mechanical metamaterials with complex geometry via XFEM. Comput Mech 55(4):659–672. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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):632–670. MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Song J-H, Areias PMA, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Methods Eng 67(6):868–893. CrossRefzbMATHGoogle Scholar
  7. 7.
    Wang T, Liu Z, Zeng Q, Gao Y, Zhuang Z (2017) XFEM modeling of hydraulic fracture in porous rocks with natural fractures. Sci China Phys Mech Astron 60(8):84612. CrossRefGoogle Scholar
  8. 8.
    Miehe C, Mauthe S, Teichtmeister S (2015) Minimization principles for the coupled problem of Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture. J Mech Phys Solids 82:186–217. MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57(2):342–368. CrossRefzbMATHGoogle Scholar
  10. 10.
    Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Methods Eng 108(12):1451–1476. MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H (2010) A simple and robust three-dimensional cracking-particle method without enrichment. Comput Methods Appl Mech Eng 199(37):2437–2455. CrossRefzbMATHGoogle Scholar
  12. 12.
    Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87(4):045501. CrossRefGoogle Scholar
  13. 13.
    Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93(10):105504. CrossRefGoogle Scholar
  14. 14.
    Chu D, Li X, Liu Z (2017) Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling. Int J Fract 208(1):115–130. CrossRefGoogle Scholar
  15. 15.
    Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405. MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Molnr G, Gravouil A (2017) 2d and 3d Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture. Finite Elem Anal Des 130:27–38. CrossRefGoogle Scholar
  17. 17.
    Aldakheel F, Hudobivnik B, Hussein A, Wriggers P (2018) Phase-field modeling of brittle fracture using an efficient virtual element scheme. Comput Methods Appl Mech Eng 341:443–466. MathSciNetCrossRefGoogle Scholar
  18. 18.
    Aldakheel F, Wriggers P, Miehe C (2018) A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling. Comput Mech 62(4):815–833. MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826. MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5):577–685. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ambrosio L, Tortorelli VM (1990) Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun Pure Appl Math 43(8):999–1036. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Verhoosel CV, Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43–62. MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    McAuliffe C, Waisman H (2016) A coupled phase field shear band model for ductilebrittle transition in notched plate impacts. Comput Methods Appl Mech Eng 305:173–195. CrossRefzbMATHGoogle Scholar
  25. 25.
    Shen R, Waisman H, Guo L (2018) Fracture of viscoelastic solids modeled with a modified phase field method. Comput Methods Appl Mech Eng. CrossRefGoogle Scholar
  26. 26.
    Borden MJ, Hughes TJR, 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. MathSciNetCrossRefGoogle Scholar
  27. 27.
    Miehe C, Schnzel L-M, Ulmer H (2015) Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput Methods Appl Mech Eng 294:449–485. MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Miehe C, Hofacker M, Schnzel LM, Aldakheel F (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elasticplastic solids. Comput Methods Appl Mech Eng 294:486–522. CrossRefzbMATHGoogle Scholar
  29. 29.
    Miehe C, Mauthe S (2016) Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Comput Methods Appl Mech Eng 304:619–655. MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Geelen RJM, Liu Y, Hu T, Tupek MR, Dolbow JE (2018) A phase-field formulation for dynamic cohesive fracture. MathSciNetCrossRefGoogle Scholar
  31. 31.
    Spatschek R, Brener E, Karma A (2011) Phase field modeling of crack propagation. Philos Mag 91(1):75–95. CrossRefGoogle Scholar
  32. 32.
    Hofacker M, Miehe C (2013) A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int J Numer Methods Eng 93(3):276–301. MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ziaei-Rad V, Shen Y (2016) Massive parallelization of the phase field formulation for crack propagation with time adaptivity. Comput Methods Appl Mech Eng 312:224–253. MathSciNetCrossRefGoogle Scholar
  34. 34.
    Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95. MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Trabelsi H, Jamei M, Zenzri H, Olivella S (2012) Crack patterns in clayey soils: experiments and modeling. Int J Numer Anal Met 36(11):1410–1433. CrossRefGoogle Scholar
  36. 36.
    Cajuhi T, Sanavia L, De Lorenzis L (2018) Phase-field modeling of fracture in variably saturated porous media. Comput Mech 61(3):299–318. MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang X, Sloan SW, Vignes C, Sheng D (2017) A modification of the phase-field model for mixed mode crack propagation in rock-like materials. Comput Methods Appl Mech Eng 322:123–136. MathSciNetCrossRefGoogle Scholar
  38. 38.
    Bryant EC, Sun W (2018) A mixed-mode phase field fracture model in anisotropic rocks with consistent kinematics. Comput Methods Appl Mech Eng 342:561–584. MathSciNetCrossRefGoogle Scholar
  39. 39.
    Choo J, Sun W (2018) Coupled phase-field and plasticity modeling of geological materials: from brittle fracture to ductile flow. Comput Methods Appl Mech Eng 330:1–32. MathSciNetCrossRefGoogle Scholar
  40. 40.
    Labuz JF, Zang A (2012) Mohr–Coulomb failure criterion. Rock Mech Rock Eng 45(6):975–979. CrossRefGoogle Scholar
  41. 41.
    Remmers JJC, de Borst R, Needleman A (2008) The simulation of dynamic crack propagation using the cohesive segments method. J Mech Phys Solids 56(1):70–92. MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    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. MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics. Elsevier, Amsterdam [google-Books-ID: VvpU3zssDOwC]zbMATHGoogle Scholar
  44. 44.
    Kalthoff J, Winkler S (1987) Failure mode transition of high rates of shear loading. In: Chiem C, Kunze H, Meyer L (eds) Proceedings of the international conference on impact loading and dynamic behavior of materials, vol 1, pp 185–195Google Scholar
  45. 45.
    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):1873–1905. CrossRefzbMATHGoogle Scholar
  46. 46.
    Sharon E, Gross SP, Fineberg J (1995) Local crack branching as a mechanism for instability in dynamic fracture. Phys Rev Lett 74(25):5096–5099. CrossRefGoogle Scholar
  47. 47.
    Fliss S, Bhat HS, Dmowska R, Rice JR (2005) Fault branching and rupture directivity. J Geophys Res Solid Earth 110:B6. CrossRefGoogle Scholar
  48. 48.
    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):489–502. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Applied Mechanics Laboratory, School of Aerospace EngineeringTsinghua UniversityBeijingChina

Personalised recommendations