Three-dimensional numerical study on the failure characteristics of intermittent fissures under compressive-shear loads

  • Yun-Teng Wang
  • Xiao-Ping ZhouEmail author
  • Miao-Miao Kou
Research Paper


A 3-D conjugated bond-pair-based peridynamic model is developed to comprehensively investigate failure characteristics of rock-like materials with intermittent fissures in the compressive-shear loading tests. Rock-like specimens containing one single central fissure are first simulated. Numerical results indicate that the 3-D conjugated bond-pair-based peridynamic model can faithfully reproduce failure characteristics of rock-like materials under compressive-shear loads. Then, the failure characteristics of rock-like specimens containing two parallel central intermittent fissures are numerically investigated. Effects of fissure inclination angle, fissure ligament length and rock bridge angle on fracturing behaviors, such as crack coalescence patterns, are also studied as well as crack initiation stress and coalescence stress.


3-D numerical simulations Compressive-shear tests Conjugated bond-pair-based peridynamics Fracturing behaviors Intermittent fissures Rock-like materials 



This work is supported by the National Natural Science Foundation of China (Grant Nos. 51839009, 51679017), National Program on Key Basic Research 973 Project of China (Grant No. 2014CB046903), Fundamental Research Funds for the Central Universities (Grant No. 106112017CDJXSYY0001) and Natural Science Foundation Project of CQ-CSTC (Grant Nos. cstc2016jcyjys0005 and cstc2017jcyj-yszxX0013 and cstc2017jcyjA1250). Authors especially would like to thank the financial support from Mr. Hong-Wu Weng original research foundation in Peking University of China.


  1. 1.
    Amiri F, Anitescu C, Arroyo M, Bordas S, Rabczuk T (2014) XFEM interpolants, a seamless bridge between XFEM and enriched meshless methods. Comput Mech 53(1):45–57MathSciNetCrossRefGoogle Scholar
  2. 2.
    Areias P, Rabczuk T (2013) Finite strain fracture of plates and shells with configurational forces and edge rotations. Int J Numer Methods Eng 94(12):1099–1122MathSciNetCrossRefGoogle Scholar
  3. 3.
    Areias P, Rabczuk T (2017) Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elem Anal Des 132:27–41CrossRefGoogle Scholar
  4. 4.
    Areias P, Rabczuk T, Dias-da Costa D (2013) Element-wise fracture algorithm based on rotation of edges. Eng Fract Mech 110:113–137CrossRefGoogle Scholar
  5. 5.
    Areias P, Rabczuk T, Camanho P (2014) Finite strain fracture of 2d problems with injected anisotropic softening elements. Theoret Appl Fract Mech 72(1):50–63CrossRefGoogle Scholar
  6. 6.
    Areias P, Msekh M, Rabczuk T (2016) Damage and fracture algorithm using the screened poisson equation and local remeshing. Eng Fract Mech 158:116–143CrossRefGoogle Scholar
  7. 7.
    Areias P, Reinoso J, Camanho PP, Cesar de Sa J, Rabczuk T (2018) Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Eng Fract Mech 189:339–360CrossRefGoogle Scholar
  8. 8.
    Bi J, Zhou XP, Bi J, Qian QH (2016) The 3D numerical simulation for the propagation process of multiple pre-existing flaws in rock-like materials subjected to biaxial compressive loads. Rock Mech Rock Eng 49(5):1611–1627CrossRefGoogle Scholar
  9. 9.
    Bobet A (2000) The initiation of secondary cracks in compression. Eng Fract Mech 66:187–219CrossRefGoogle Scholar
  10. 10.
    Bobet A, Einstein HH (1998) Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int J Rock Mech Min Sci 35(7):863–888CrossRefGoogle Scholar
  11. 11.
    Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217:77–95MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cao P, Liu T, Pu C, Lin H (2015) Crack propagation and coalescence of brittle rock-like specimens with pre-existing cracks in compression. Eng Geol 187:113–121CrossRefGoogle Scholar
  13. 13.
    Cao RH, Cao P, Lin H, Pu C, Ke O (2016) Mechanical behavior of brittle rock-like specimens with pre-existing fissures under uniaxial loading: experimental studies and particle mechanics approach. Rock Mech Rock Eng 49(3):763–783CrossRefGoogle Scholar
  14. 14.
    Cao RH, Cao P, Lin H, Ma G, Chen Y (2017) Failure characteristics of intermittent fissures under a compressive-shear test: experimental and numerical analyses. Theor Appl Fract Mech. CrossRefGoogle Scholar
  15. 15.
    Cen D, Huang D (2017) Direct shear tests of sandstone under constant normal tensile stress condition using a simple auxiliary device. Rock Mech Rock Eng 50(6):1425–1438CrossRefGoogle Scholar
  16. 16.
    Cheng H, Zhou X (2015) A multi-dimensional space method for dynamic cracks problems using implicit time scheme in the framework of the extended finite element method. Int J Damage Mech 24(6):859–890CrossRefGoogle Scholar
  17. 17.
    Cheng H, Zhou XP, Zhu J, Qian QH (2016) The effect of crack openings on crack initiation, propagation and coalescence behavior in rock-like materials under uniaxial compression. Rock Mech Rock Eng 49(4):3481–3494CrossRefGoogle Scholar
  18. 18.
    Colligon JS, Fischer G, Patel MH (1977) Fracture energy of plain and glass-reinforced gypsum plaster. J Mater Sci Lett 12(4):831–836CrossRefGoogle Scholar
  19. 19.
    Dey TN, Wang CY (1981) Some mechanisms of microcrack growth and interaction in compressive rock failure. Int J Rock Mech Min Sci Geomech Abstr 18(3):199–209CrossRefGoogle Scholar
  20. 20.
    Fu JW, Zhang XZ, Zhu WS, Chen K, Guan JF (2017) Simulating progressive failure in brittle jointed rock masses using a modified elastic-brittle model and the application. Eng Fract Mech 179:212–230CrossRefGoogle Scholar
  21. 21.
    Gehle C (2002) Breakage and shear behaviour of rock joints with intermittent material bridges. Ph.D. thesis, Ruhr-Universitat Bochum, GermanyGoogle Scholar
  22. 22.
    Gu J, Zhao ZY (2009) Considerations of the discontinuous deformation analysis on wave propagation. Int J Numer Anal Methods Geomech 33(12):1449–1465CrossRefGoogle Scholar
  23. 23.
    Ha YD, Lee J, Hong JW (2015) Fracturing patterns of rock-like materials in compression captured with peridynamics. Eng Fract Mech 144:176–193CrossRefGoogle Scholar
  24. 24.
    Huang D, Cen D, Ma G, Huang R (2015) Step-path failure of rock slopes with intermittent joints. Landslides 12(5):911–926CrossRefGoogle Scholar
  25. 25.
    Huang D, Gu D, Yang C, Huang R, Fu G (2016) Investigation on mechanical behaviors of sandstone with two preexisting flaws under triaxial compression. Rock Mech Rock Eng 49(2):375–399CrossRefGoogle Scholar
  26. 26.
    Huang YH, Yang SQ, Zhao J (2016) Three-dimensional numerical simulation on triaxial failure mechanical behavior of rock-like specimen containing two unparallel fissures. Rock Mech Rock Eng 49(12):4711–4729CrossRefGoogle Scholar
  27. 27.
    Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53(3):194–204CrossRefGoogle Scholar
  28. 28.
    Lee HW, Jeon SW (2011) An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression. Int J Solids Struct 48(6):979–999CrossRefGoogle Scholar
  29. 29.
    Lee J, Ha YD, Hong JW (2017) Crack coalescence morphology in rock-like material under compression. Int J Fract 203(1–2):211–236CrossRefGoogle Scholar
  30. 30.
    Lee J, Hong JW, Jung JW (2017) The mechanism of fracture coalescence in pre-cracked rock-type material with three flaws. Eng Geol 223:31–47CrossRefGoogle Scholar
  31. 31.
    Li G, Liang ZZ, Tang CA (2015) Morphologic interpretation of rock failure mechanisms under uniaxial compression based on 3D multiscale high-resolution numerical modeling. Rock Mech Rock Eng 48(6):2235–2262CrossRefGoogle Scholar
  32. 32.
    Liu WY, Hong JW (2012) Discretized peridynamics for linear elastic solids. Comput Mech 50(5):579–590MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ma GW, Wang XJ, Ren F (2011) Numerical simulation of compressive failure of heterogeneous rock-like materials using SPH method. Int J Rock Mech Min Sci 48(3):353–363CrossRefGoogle Scholar
  34. 34.
    Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, BostonCrossRefGoogle Scholar
  35. 35.
    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):1273–1311MathSciNetCrossRefGoogle Scholar
  36. 36.
    Moës N, Belytschko T (2002) Extendedfinite element method for cohesive crack growth. Eng Fract Mech 69(7):813–833CrossRefGoogle Scholar
  37. 37.
    Nanthakumar S, Lahmer T, Zhuang X, Zi G, Rabczuk T (2016) Detection of ma-terial interfaces using a regularized level set method in piezoelectric structures. Inverse Probl Sci Eng 24(1):153–176MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ni T, Zhu QZ, Zhao LY, Li PF (2018) Peridynamic simulation of fracture in quasi brittle solids using irregular finite element mesh. Eng Fract Mech 188:320–343CrossRefGoogle Scholar
  39. 39.
    Ning YJ, An XM, Ma GW (2011) Footwall slope stability analysis with the numerical manifold method. Int J Rock Mech Min Sci 48:964–975CrossRefGoogle Scholar
  40. 40.
    Oterkus S, Madenci E, Oterkus E (2017) Fully coupled poroelastic peridynamic formulation for fluid-filled fractures. Eng Geol 225:19–28CrossRefGoogle Scholar
  41. 41.
    Park CH, Bobet A (2009) Crack coalescence in specimens with open and closed flaws: a comparison. Int J Rock Mech Min Sci 46(5):819–829CrossRefGoogle Scholar
  42. 42.
    Prudencio M, Van Sint Jan M (2007) Strength and failure modes of rock mass models with non-persistent joints. Int J Rock Mech Min Sci 44(6):890–902CrossRefGoogle Scholar
  43. 43.
    Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61(13):2316–2343CrossRefGoogle Scholar
  44. 44.
    Rabczuk T, Belytschko T (2007) A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 196(29–30):2777–2799MathSciNetCrossRefGoogle Scholar
  45. 45.
    Rabczuk T, Ren H (2017) A peridynamics formulation for quasi-static fracture and contact in rock. Eng Geol 225:42–48CrossRefGoogle Scholar
  46. 46.
    Rabczuk T, Samaniego E (2008) Discontinuous modelling of shear bands using adaptive meshfree methods. Comput Methods Appl Mech Eng 197(6):641–658MathSciNetCrossRefGoogle Scholar
  47. 47.
    Rabczuk T, Zi G (2007) A meshfree method based on the local partition of unity for cohesive cracks. Comput Mech 39(6):743–760CrossRefGoogle Scholar
  48. 48.
    Rabczuk T, Bordas S, Zi G (2007) A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Comput Mech 40(3):473–495CrossRefGoogle Scholar
  49. 49.
    Rabczuk T, Areias P, Belytschko T (2007) A simplified mesh-free method for shear bands with cohesive surfaces. Int J Numer Methods Eng 69(5):993–1021CrossRefGoogle Scholar
  50. 50.
    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):2437–2455CrossRefGoogle Scholar
  51. 51.
    Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Meth Eng 108(12):1451–1476MathSciNetCrossRefGoogle Scholar
  52. 52.
    Ren H, Zhuang X, Rabczuk T (2017) Dual-horizon peridynamics: a stable solution to varying horizons. Comput Methods Appl Mech Eng 318:762–782MathSciNetCrossRefGoogle Scholar
  53. 53.
    Savilahti T, Nordlund E, Stephansson O (1990) Shear box testing and modeling of joint bridge, In: Proceedings of international symposium on shear box testing and modeling of joint bridge Rock Joints. Norway, pp 295–300Google Scholar
  54. 54.
    Shen B (1995) The mechanism of fracture coalescence in compression experimental study and numerical simulation. Eng Fract Mech 51(1):73–85CrossRefGoogle Scholar
  55. 55.
    Shi GH (1991) Manifold method of material analysis. In: Transactions of the 9th army conference on applied mathematics and computing. Minneapolis, USA, pp 57–76Google Scholar
  56. 56.
    Silling SA (2000) Reformulation of elasticity theory for discontinuities and long range forces. J Mech Phys Solids 48(1):175–209MathSciNetCrossRefGoogle Scholar
  57. 57.
    Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17):1526–1535CrossRefGoogle Scholar
  58. 58.
    Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184MathSciNetCrossRefGoogle Scholar
  59. 59.
    Stillinger F, Weber T (1985) Computer simulation of local order in condensed phase of silicon. Phys Rev B 31(8):5262–5271CrossRefGoogle Scholar
  60. 60.
    Tang CA (1997) Numerical simulation of progressive rock failure and associated seismicity. Int J Rock Mech Min Sci 34(2):249–262CrossRefGoogle Scholar
  61. 61.
    Vesga LF, Vallejo LE, Lobo-Guerrero S (2008) DEM analysis of the crack propagation in brittle clays under uniaxial compression tests. Int J Numer Anal Methods Geomech 32(11):1405–1415CrossRefGoogle Scholar
  62. 62.
    Wang YT, Zhou XP, Xu X (2016) Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng Fract Mech 163:248–273CrossRefGoogle Scholar
  63. 63.
    Wang YT, Zhou XP, Shou YD (2017) The modeling of crack propagation and coalescence in rock under uniaxial compression using the novel conjugated bond-based peridynamics. Int J Mech Sci 128–129:614–643CrossRefGoogle Scholar
  64. 64.
    Wang YT, Zhou XP, Wang Y, Shou YD (2018) A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids. Int J Solids Struct 134:89–115CrossRefGoogle Scholar
  65. 65.
    Wang YT, Zhou XP, Kou MM (2018) A coupled thermo-mechanical bond-based peridynamics for simulating thermal cracking in rocks. Int J Fract 211(1–2):13–42CrossRefGoogle Scholar
  66. 66.
    Wang YT, Zhou XP, Kou MM (2018) Peridynamic investigation on thermal fracturing behavior of ceramic nuclear fuel pellets under power cycles. Ceram Int 44(10):11512–11542CrossRefGoogle Scholar
  67. 67.
    Wong RHC, Chau KT (1998) Crack coalescence in a rock-like material containing two cracks. Int J Rock Mech Min Sci 35(2):147–164CrossRefGoogle Scholar
  68. 68.
    Wong LNY, Einstein HH (2009) Crack coalescence in molded gypsum and Carrara marble: part 1. Macroscopic observations and interpretation. Rock Mech Rock Eng 42(3):475–511CrossRefGoogle Scholar
  69. 69.
    Wong LNY, Li HQ (2013) Numerical study on coalescence of two pre-existing coplanar flaws in rock. Int J Solids Struct 50(22–23):3685–3706CrossRefGoogle Scholar
  70. 70.
    Wu ZJ, Wong LNY (2012) Frictional crack initiation and propagation analysis using the numerical manifold method. Comput Geotech 39:38–53CrossRefGoogle Scholar
  71. 71.
    Wu W, Zhuang X, Zhu H, Liu X, Ma G (2017) Centroid sliding pyramid method for removability and stability analysis of fractured hard rock. Acta Geotech 12(3):627–644CrossRefGoogle Scholar
  72. 72.
    Xie Y, Cao P, Liu J, Dong L (2016) Influence of crack surface friction on crack initiation and propagation: a numerical investigation based on extended finite element method. Comput Geotech 74:1–14CrossRefGoogle Scholar
  73. 73.
    Yang SQ, Jiang YZ, Xu WY, Chen XQ (2008) Experimental investigation on strength and failure behavior of pre-cracked marble under conventional triaxial compression. Int J Solids Struct 45(17):4796–4819CrossRefGoogle Scholar
  74. 74.
    Zhang Z, Ge X (2005) A new quasi-continuum constitutive model for crack growth in an isotropic solid. Eur J Mech A Solid 24(2):243–252CrossRefGoogle Scholar
  75. 75.
    Zhang XP, Wong LNY (2012) Cracking processes in rock-like material containing a single flaw under uniaxial compression: a numerical study based on parallel bonded-particle model approach. Rock Mech Rock Eng 45(5):711–737Google Scholar
  76. 76.
    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 322:123–136MathSciNetCrossRefGoogle Scholar
  77. 77.
    Zhao G, Fang J, Zhao J (2011) A 3D distinct lattice spring model for elasticity and dynamic failure. Int J Numer Anal Methods Geomech 35(8):859–885CrossRefGoogle Scholar
  78. 78.
    Zhou XP, Wang YT (2016) Numerical simulation of crack propagation and coalescence in pre-cracked rock-like Brazilian disks using the non-ordinary state-based peridynamics. Int J Rock Mech Min Sci 89:235–249CrossRefGoogle Scholar
  79. 79.
    Zhou XP, Cheng H, Feng YF (2014) An experimental study of crack coalescence behavior in rock-like materials containing multiple flaws under uniaxial compression. Rock Mech Rock Eng 47(6):1961–1986CrossRefGoogle Scholar
  80. 80.
    Zhou XP, Bi J, Qian QH (2015) Numerical simulation of crack growth and coalescence in rock-like materials containing multiple pre-existing flaws. Rock Mech Rock Eng 48(3):1097–1114CrossRefGoogle Scholar
  81. 81.
    Zhou XP, Wang YT, Xu X (2016) Numerical simulation of initiation, propagation and coalescence of cracks using the non-ordinary state-based peridynamics. Int J Fract 201(2):213–234CrossRefGoogle Scholar
  82. 82.
    Zhou XP, Wang YT, Qian QH (2016) Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended non-ordinary state-based peridynamics. Eur J Mech A Solid 60:277–299MathSciNetCrossRefGoogle Scholar
  83. 83.
    Zhou XP, Wang YT, Shou YD, Kou MM (2018) A novel conjugated bond linear elastic model in bond-based peridynamics for fracture problems under dynamic loads. Eng Fract Mech 188:151–183CrossRefGoogle Scholar
  84. 84.
    Zhou S, Zhuang X, Zhu H, Rabczuk T (2018) Phasefield modelling of crack propagation, branching and coalescence in rocks. Theor Appl Fract Mech 96:174–192CrossRefGoogle Scholar
  85. 85.
    Zhu QZ, Ni T (2017) Peridynamic formulations enriched with bond rotation effects. Int J Eng Sci 121:118–129MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Yun-Teng Wang
    • 1
    • 2
    • 3
  • Xiao-Ping Zhou
    • 1
    • 2
    • 3
    • 4
    Email author
  • Miao-Miao Kou
    • 2
    • 3
  1. 1.School of Civil EngineeringWuhan UniversityWuhanPeople’s Republic of China
  2. 2.School of Civil EngineeringChongqing UniversityChongqingPeople’s Republic of China
  3. 3.Key Laboratory of New Technology for Construction of Cities in Mountain AreaChongqing UniversityChongqingPeople’s Republic of China
  4. 4.State Key Laboratory of Coal Mine Disaster Dynamics and ControlChongqing UniversityChongqingPeople’s Republic of China

Personalised recommendations