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

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Abstract

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.

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References

  1. Krueger R (2004) Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 57(2):109–143. https://doi.org/10.1115/1.1595677

    Article  Google Scholar 

  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. https://doi.org/10.1016/S0013-7944(01)00083-2

    Article  Google Scholar 

  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. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

  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. https://doi.org/10.1007/s00466-015-1129-2

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1002/nme.4731

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1002/nme.1652

    Article  MATH  Google Scholar 

  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. https://doi.org/10.1007/s11433-017-9037-3

    Article  Google Scholar 

  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. https://doi.org/10.1016/j.jmps.2015.04.006

    Article  MathSciNet  Google Scholar 

  9. Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57(2):342–368. https://doi.org/10.1016/j.jmps.2008.10.012

    Article  MATH  Google Scholar 

  10. Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Methods Eng 108(12):1451–1476. https://doi.org/10.1002/nme.5257

    Article  MathSciNet  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2010.03.031

    Article  MATH  Google Scholar 

  12. Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87(4):045501. https://doi.org/10.1103/PhysRevLett.87.045501

    Article  Google Scholar 

  13. Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93(10):105504. https://doi.org/10.1103/PhysRevLett.93.105504

    Article  Google Scholar 

  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. https://doi.org/10.1007/s10704-017-0220-4

    Article  Google Scholar 

  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. https://doi.org/10.1007/s00466-014-1109-y

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1016/j.finel.2017.03.002

    Article  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2018.07.008

    Article  MathSciNet  Google Scholar 

  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. https://doi.org/10.1007/s00466-017-1530-0

    Article  MathSciNet  MATH  Google Scholar 

  19. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342. https://doi.org/10.1016/S0022-5096(98)00034-9

    Article  MathSciNet  MATH  Google Scholar 

  20. Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826. https://doi.org/10.1016/S0022-5096(99)00028-9

    Article  MathSciNet  MATH  Google Scholar 

  21. Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5):577–685. https://doi.org/10.1002/cpa.3160420503

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1002/cpa.3160430805

    Article  MathSciNet  MATH  Google Scholar 

  23. Verhoosel CV, Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43–62. https://doi.org/10.1002/nme.4553

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2016.02.018

    Article  MATH  Google Scholar 

  25. Shen R, Waisman H, Guo L (2018) Fracture of viscoelastic solids modeled with a modified phase field method. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2018.09.018

    Article  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2016.09.005

    Article  MathSciNet  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2014.11.016

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2014.11.017

    Article  MATH  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2015.09.021

    Article  MathSciNet  MATH  Google Scholar 

  30. Geelen RJM, Liu Y, Hu T, Tupek MR, Dolbow JE (2018) A phase-field formulation for dynamic cohesive fracture. https://doi.org/10.1016/j.cma.2019.01.026

    Article  MathSciNet  Google Scholar 

  31. Spatschek R, Brener E, Karma A (2011) Phase field modeling of crack propagation. Philos Mag 91(1):75–95. https://doi.org/10.1080/14786431003773015

    Article  Google Scholar 

  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. https://doi.org/10.1002/nme.4387

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2016.04.013

    Article  MathSciNet  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2012.01.008

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1002/nag.1060

    Article  Google Scholar 

  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. https://doi.org/10.1007/s00466-017-1459-3

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2017.04.028

    Article  MathSciNet  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2018.08.008

    Article  MathSciNet  Google Scholar 

  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. https://doi.org/10.1016/j.cma.2017.10.009

    Article  MathSciNet  Google Scholar 

  40. Labuz JF, Zang A (2012) Mohr–Coulomb failure criterion. Rock Mech Rock Eng 45(6):975–979. https://doi.org/10.1007/s00603-012-0281-7

    Article  Google Scholar 

  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. https://doi.org/10.1016/j.jmps.2007.08.003

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1002/nme.2861

    Article  MathSciNet  MATH  Google Scholar 

  43. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics. Elsevier, Amsterdam [google-Books-ID: VvpU3zssDOwC]

    MATH  Google Scholar 

  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 185–195

  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. https://doi.org/10.1002/nme.941

    Article  MATH  Google Scholar 

  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. https://doi.org/10.1103/PhysRevLett.74.5096

    Article  Google Scholar 

  47. Fliss S, Bhat HS, Dmowska R, Rice JR (2005) Fault branching and rupture directivity. J Geophys Res Solid Earth 110:B6. https://doi.org/10.1029/2004JB003368

    Article  Google Scholar 

  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. https://doi.org/10.1007/s00466-014-1001-9

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Applied Mechanics Laboratory, School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Tao Wang, Xuan Ye, Zhanli Liu, Dongyang Chu & Zhuo Zhuang

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  1. Tao Wang
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Appendix A Performance study

Appendix A Performance study

Fig. 27
figure 27

The domain division for different number of CPUs (8 and 12), each color represents a domain

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Fig. 28
figure 28

Performance study: Calculating wall time consumption under different numbers of CPUs and DOFs: a Computing time under different number of CPUs and b Computing time under different number of DOFs

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The phase-field method usually requires large-scale computation, and parallel computing is particularly important at this time. Explicit time integration scheme is very suitable for parallel computation. We use multi-CPU subregional computing to implement parallel computing. Here we take the example in Sect. 5.2 to study the efficiency of the parallel computing. To carry out parallel computation, we divide the whole model according to the number of CPUs used. Figure 27 gives the results of domain division of the model when using 8 CPUs and 12 CPUs, and then calculates each region with one CPU. The information on the common boundary of each domain is stored in the public variables, which are synchronized in multiple cpus via MPI function.

The wall time of the model with different number of CPUs is shown in Fig. 28a. The model has 7939500 DOFs and 53244 incremental steps. We can find that using multiple CPUs can greatly improve the efficiency of calculation, and the wall time is approximately inversely proportional to the number of CPUs used. Figure 28b shows the wall time consumed by the same model with the same number of CPUs (16) and different DOFs. It can be seen that the relationship between the wall time and the number of DOFs is basically linear, which is better than that of the implicit scheme.

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Wang, T., Ye, X., Liu, Z. et al. Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method. Comput Mech 64, 1537–1556 (2019). https://doi.org/10.1007/s00466-019-01733-z

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