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Frontiers of Structural and Civil Engineering

, Volume 13, Issue 2, pp 273–287 | Cite as

Computational methods for fracture in rock: a review and recent advances

  • Ali JenabidehkordiEmail author
Review
  • 72 Downloads

Abstract

We present an overview of the most popular state-of-the-art computational methods available for modelling fracture in rock. The summarized numerical methods can be classified into three categories: Continuum Based Methods, Discrete Crack Approaches, and Block-Based Methods. We will not only provide an extensive review of those methods which can be found elsewhere but particularly address their potential in modelling fracture in rock mechanics and geotechnical engineering. In this context, we will discuss their key applications, assumptions, and limitations. Furthermore, we also address ‘general’ difficulties that may arise for simulating fracture in rock and fractured rock. This review will conclude with some final remarks and future challenges.

Keywords

numerical modelling method development rock mechanics fractured rock rock fracturing 

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References

  1. 1.
    Carpinteri A. Post-peak and post-bifurcation analysis of cohesive crack propagation. Engineering Fracture Mechanics, 1989, 32(2): 265–278CrossRefGoogle Scholar
  2. 2.
    Planas J, Elices M. Nonlinear fracture of cohesive materials. In: Current Trends in Concrete Fracture Research, Springer, 1991, 139–157CrossRefGoogle Scholar
  3. 3.
    Bažant Z P, Jirásek M. Nonlocal integral formulations of plasticity and damage: survey of progress. Journal of Engineering Mechanics, 2002, 128(11): 1119–1149CrossRefGoogle Scholar
  4. 4.
    Wheel M. A geometrically versatile finite volume formulation for plane elastostatic stress analysis. Journal of Strain Analysis for Engineering Design, 1996, 31(2): 111–116CrossRefGoogle Scholar
  5. 5.
    Selmin V. The node-centred finite volume approach: bridge between finite differences and finite elements. Computer Methods in Applied Mechanics and Engineering, 1993, 102(1): 107–138MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fallah N, Bailey C, Cross M, Taylor G. Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis. Applied Mathematical Modelling, 2000, 24(7): 439–455zbMATHCrossRefGoogle Scholar
  7. 7.
    Bailey C, Cross M. A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh. International Journal for Numerical Methods in Engineering, 1995, 38(10): 1757–1776zbMATHCrossRefGoogle Scholar
  8. 8.
    Mishev I D. Finite volume methods on voronoi meshes. Numerical Methods for Partial Differential Equations, 1998, 14(2): 193–212MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fryer Y, Bailey C, Cross M, Lai C H. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Applied Mathematical Modelling, 1991, 15(11–12): 639–645zbMATHCrossRefGoogle Scholar
  10. 10.
    Detournay C, Hart R. FLAC and numerical modelling in geomechanics. In: Proceedings of the International FLAC symposium on Numerical Modelling in Geomechanics, Minneapolis. Rotterdam: Balkema, 1999zbMATHGoogle Scholar
  11. 11.
    Fang Z. A local degradation approach to the numerical analysis of brittle fractures in heterogeneous rocks. Dissertation for PhD degree. Imperial College London (University of London), 2001Google Scholar
  12. 12.
    Martino S, Prestininzi A, Scarascia Mugnozza G. Mechanisms of deep seated gravitational deformations: parameters from laboratory testing for analogical and numerical modeling. In: Proc. Eurock, 2001, 137–142Google Scholar
  13. 13.
    Kourdey A, Alheib M, Piguet J, Korini T. Evaluation of slope stability by numerical methods. The 17th International Mining Congress and Exhibition of Turkey, 2001, 705–710Google Scholar
  14. 14.
    Marmo B A, Wilson C J L. A verification procedure for the use of FLAC to study glacial dynamics and the implementation of an anisotropic flow law. In: Särkkä P, Eloranta P, eds. Rock Mechanics—A Challenge for Society. Lisse: Swetz and Zeitlinger, 2001Google Scholar
  15. 15.
    Jing L, Hudson J. Numerical methods in rock mechanics. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(4): 409–427CrossRefGoogle Scholar
  16. 16.
    Wittke W, Sykes R. Rock Mechanics. Springer Berlin, 1990CrossRefGoogle Scholar
  17. 17.
    Peng W. The damage mechanics model for jointed rock mass and its nonlinear FEM analysis. Chinese Journal of Rock Mechanics and Engineering, 1988, 7(3): 193–202 (in Chinese)Google Scholar
  18. 18.
    Zheng Y R, Zhao S Y. Application of strength reduction FEM in soil and rock slope. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(19): 3381 (in Chinese)Google Scholar
  19. 19.
    Zhao S, Zheng Y, Deng W. Stability analysis on jointed rock slope by strength reduction FEM. Chinese Journal of Rock Mechanics and Engineering, 2003, 22(2): 254–260Google Scholar
  20. 20.
    Cai M, Horii H. A constitutive model and FEM analysis of jointed rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1993, 30(4): 351–359CrossRefGoogle Scholar
  21. 21.
    Bazant Z P, Cedolin L. Fracture mechanics of reinforced concrete. Journal of the Engineering Mechanics Division, 1980, 106(6): 1287–1306Google Scholar
  22. 22.
    Goodman R E, Taylor R L, Brekke T L. A model for the mechanics of jointed rocks. Journal of Soil Mechanics & Foundations Division, 1968, 94(3): 637–660Google Scholar
  23. 23.
    Zienkiewicz O C, Best B, Dullage C, Stagg K G. Analysis of nonlinear problems in rock mechanics with particular reference to jointed rock systems. In: Proceedings of International Society of Rock Mechanics, 1970Google Scholar
  24. 24.
    Ghaboussi J, Wilson E, Isenberg J. Finite element for rock joints and interfaces. Journal of Soil Mechanics & Foundations Division, 1973, 99(10): 849–862Google Scholar
  25. 25.
    Desai C, Zaman M, Lightner J, Siriwardane H. Thin-layer element for interfaces and joints. International Journal for Numerical and Analytical Methods in Geomechanics, 1984, 8(1): 19–43CrossRefGoogle Scholar
  26. 26.
    Goodman R E. Methods of geological engineering in discontinuous rocks. New York: West Publishing, 1976Google Scholar
  27. 27.
    Katona M G. A simple contact–friction interface element with applications to buried culverts. International Journal for Numerical and Analytical Methods in Geomechanics, 1983, 7(3): 371–384zbMATHCrossRefGoogle Scholar
  28. 28.
    Bažant Z P. Why continuum damage is nonlocal: micromechanics arguments. Journal of Engineering Mechanics, 1991, 117(5): 1070–1087CrossRefGoogle Scholar
  29. 29.
    Peerlings R H J, de Borst R, Brekelmans W A M, de Vree J H P. Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering, 1996, 39(19): 3391–3403zbMATHCrossRefGoogle Scholar
  30. 30.
    Peerlings R H J, de Borst R, Brekelmans W, Geers M. Localisation issues in local and nonlocal continuum approaches to fracture. European Journal of Mechanics. A, Solids, 2002, 21(2): 175–189MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    de Borst R, Pamin J, Geers M G. On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. European Journal of Mechanics. A, Solids, 1999, 18(6): 939–962zbMATHCrossRefGoogle Scholar
  32. 32.
    Pasternak E, Dyskin A, Mühlhaus H B. Cracks of higher modes in Cosserat continua. International Journal of Fracture, 2006, 140(1–4): 189–199zbMATHCrossRefGoogle Scholar
  33. 33.
    Etse G, Willam K. Failure analysis of elastoviscoplastic material models. Journal of Engineering Mechanics, 1999, 125(1): 60–69CrossRefGoogle Scholar
  34. 34.
    Rabczuk T, Eibl J. Simulation of high velocity concrete fragmentation using SPH/MLSPH. International Journal for Numerical Methods in Engineering, 2003, 56(10): 1421–1444zbMATHCrossRefGoogle Scholar
  35. 35.
    Hillerborg A, Modéer M, Petersson P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 1976, 6(6): 773–781CrossRefGoogle Scholar
  36. 36.
    Miihlhaus H B, Triantafyllidis T. Surface waves in a layered halfspace with bending stiffness. Developments in Geotechnical Engineering, 1987, 44: 277–290CrossRefGoogle Scholar
  37. 37.
    Mühlhaus H B. Application of Cosserat theory in numerical solutions of limit load problems. Archive of Applied Mechanics, 1989, 59(2): 124–137Google Scholar
  38. 38.
    Vardoulakis I, Mühlhaus H. Local rock surface instabilities. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1986, 23: 379–383CrossRefGoogle Scholar
  39. 39.
    Rabczuk T. Computational methods for fracture in brittle and quasi-brittle solids: state of the art review and future perspectives. ISRN Applied Mathematics, 2013, 2013: 332–369MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    de Borst R. Fracture in quasi-brittle materials: a review of continuum damage-based approaches. Engineering Fracture Mechanics, 2002, 69(2): 95–112CrossRefGoogle Scholar
  41. 41.
    Jirásek M, Zimmermann T. Analysis of rotating crack model. Journal of Engineering Mechanics, 1998, 124(8): 842–851CrossRefGoogle Scholar
  42. 42.
    Jirásek M, Zimmermann T. Rotating crack model with transition to scalar damage. Journal of Engineering Mechanics, 1998, 124(3): 277–284CrossRefGoogle Scholar
  43. 43.
    Rabczuk T, Akkermann J, Eibl J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5–6): 1327–1354zbMATHCrossRefGoogle Scholar
  44. 44.
    Ohmenhäuser F, Weihe S, Kröplin B. Algorithmic implementation of a generalized cohesive crack model. Computational Materials Science, 1999, 16(1): 294–306CrossRefGoogle Scholar
  45. 45.
    Carpinteri A, Chiaia B, Cornetti P. A scale-invariant cohesive crack model for quasi-brittle materials. Engineering Fracture Mechanics, 2002, 69(2): 207–217CrossRefGoogle Scholar
  46. 46.
    François M, Royer-Carfagni G. Structured deformation of damaged continua with cohesive-frictional sliding rough fractures. European Journal of Mechanics. A, Solids, 2005, 24(4): 644–660MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    de Borst R, Remmers J J, Needleman A. Mesh-independent discrete numerical representations of cohesive-zone models. Engineering Fracture Mechanics, 2006, 73(2): 160–177CrossRefGoogle Scholar
  48. 48.
    Zhuang X, Huang R, Liang C, Rabczuk T. A coupled thermohydro-mechanical model of jointed hard rock for compressed air energy storage. Mathematical Problems in Engineering, 2014, 179169Google Scholar
  49. 49.
    Silani M, Talebi H, Hamouda A M, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23CrossRefGoogle Scholar
  50. 50.
    Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three-dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92CrossRefGoogle Scholar
  51. 51.
    Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38CrossRefGoogle Scholar
  52. 52.
    Talebi H, Silani M, Bordas S P, Kerfriden P, Rabczuk T. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071MathSciNetCrossRefGoogle Scholar
  53. 53.
    Budarapu P R, Gracie R, Bordas S P, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148CrossRefGoogle Scholar
  54. 54.
    Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143CrossRefGoogle Scholar
  55. 55.
    Talebi H, Silani M, Bordas S P, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541CrossRefGoogle Scholar
  56. 56.
    Belytschko T, Lin J I. A three-dimensional impact-penetration algorithm with erosion. Computers & Structures, 1987, 25(1): 95–104zbMATHCrossRefGoogle Scholar
  57. 57.
    Camacho G T, Ortiz M. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures, 1996, 33(20): 2899–2938zbMATHCrossRefGoogle Scholar
  58. 58.
    Xu X P, Needleman A. Void nucleation by inclusion debonding in a crystal matrix. Modelling and Simulation in Materials Science and Engineering, 1993, 1(2): 111–132CrossRefGoogle Scholar
  59. 59.
    Ortiz M, Leroy Y, Needleman A. A finite element method for localized failure analysis. Computer Methods in Applied Mechanics and Engineering, 1987, 61(2): 189–214zbMATHCrossRefGoogle Scholar
  60. 60.
    Pandolfi A, Krysl P, Ortiz M. Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture. International Journal of Fracture, 1999, 95(1–4): 279–297CrossRefGoogle Scholar
  61. 61.
    Pandolfi A, Guduru P, Ortiz M, Rosakis A. Three dimensional cohesive-element analysis and experiments of dynamic fracture in c300 steel. International Journal of Solids and Structures, 2000, 37 (27): 3733–3760CrossRefGoogle Scholar
  62. 62.
    Zhou F, Molinari J F. Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency. International Journal for Numerical Methods in Engineering, 2004, 59(1): 1–24zbMATHCrossRefGoogle Scholar
  63. 63.
    Falk M L, Needleman A, Rice J R. A critical evaluation of cohesive zone models of dynamic fracture. Journal de Physique. IV, 2001, 11(PR5): Pr5–43–Pr5–50CrossRefGoogle Scholar
  64. 64.
    Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Areias P, Rabczuk T, Camanho P. Finite strain fracture of 2d problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63CrossRefGoogle Scholar
  66. 66.
    Areias P, Rabczuk T, Dias-da Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137CrossRefGoogle Scholar
  67. 67.
    Areias P, Rabczuk T, Camanho P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947zbMATHCrossRefGoogle Scholar
  68. 68.
    Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41CrossRefGoogle Scholar
  70. 70.
    Areias P, Rabczuk T, de Sá J C. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Computational Mechanics, 2016, 58(6): 1003–1018MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350MathSciNetCrossRefGoogle Scholar
  72. 72.
    Areias P, Msekh M, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143CrossRefGoogle Scholar
  73. 73.
    Gens A, Carol I, Alonso E. Rock joints: FEM implementation and applications. Studies in Applied Mechanics, 1995, 42: 395–420CrossRefGoogle Scholar
  74. 74.
    Belytschko T, Fish J, Engelmann B E. A finite element with embedded localization zones. Computer Methods in Applied Mechanics and Engineering, 1988, 70(1): 59–89zbMATHCrossRefGoogle Scholar
  75. 75.
    Dvorkin E N, Cuitio A M, Gioia G. Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. International Journal for Numerical Methods in Engineering, 1990, 30(3): 541–564zbMATHCrossRefGoogle Scholar
  76. 76.
    Feist C, Hofstetter G. Three-dimensional fracture simulations based on the SDA. International Journal for Numerical and Analytical Methods in Geomechanics, 2007, 31(2): 189–212zbMATHCrossRefGoogle Scholar
  77. 77.
    Sancho JM, Planas J, Fathy AM, Galvez J C, Cendon D A. Threedimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity. International Journal for Numerical and Analytical Methods in Geomechanics, 2007, 31(2): 173–187zbMATHCrossRefGoogle Scholar
  78. 78.
    Jirásek M. Comparative study on finite elements with embedded discontinuities. Computer Methods in Applied Mechanics and Engineering, 2000, 188(1): 307–330zbMATHCrossRefGoogle Scholar
  79. 79.
    Linder C, Zhang X. Three-dimensional finite elements with embedded strong discontinuities to model failure in electromechanical coupled materials. Computer Methods in Applied Mechanics and Engineering, 2014, 273: 143–160zbMATHCrossRefGoogle Scholar
  80. 80.
    Linder C, Armero F. Finite elements with embedded strong discontinuities for the modeling of failure in solids. International Journal for Numerical Methods in Engineering, 2007, 72(12): 1391–1433MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Oliver J, Huespe A, Blanco S, Linero D. Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach. Computer Methods in Applied Mechanics and Engineering, 2006, 195(52): 7093–7114zbMATHCrossRefGoogle Scholar
  82. 82.
    Foster C, Borja R, Regueiro R. Embedded strong discontinuity finite elements for fractured geomaterials with variable friction. International Journal for Numerical Methods in Engineering, 2007, 72(5): 549–581MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Nikolic M, Ibrahimbegovic A. Rock mechanics model capable of representing initial heterogeneities and full set of 3d failure mechanisms. Computer Methods in Applied Mechanics and Engineering, 2015, 290: 209–227MathSciNetCrossRefGoogle Scholar
  84. 84.
    Saksala T. Rate-dependent embedded discontinuity approach incorporating heterogeneity for numerical modeling of rock fracture. Rock Mechanics and Rock Engineering, 2015, 48(4): 1605–1622CrossRefGoogle Scholar
  85. 85.
    Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620zbMATHCrossRefGoogle Scholar
  86. 86.
    Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46(1): 131–150zbMATHCrossRefGoogle Scholar
  87. 87.
    Melenk J M, Babuška I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1): 289–314MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Rabinovich D, Givoli D, Vigdergauz S. Crack identification by arrival timeusing XFEM and a genetic algorithm. International Journal for Numerical Methods in Engineering, 2009, 77(3): 337–359MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Béchet E, Scherzer M, Kuna M. Application of the x-FEM to the fracture of piezoelectric materials. International Journal for Numerical Methods in Engineering, 2009, 77(11): 1535–1565MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Mayer U M, Gerstenberger A, Wall W A. Interface handling for three-dimensional higher-order XFEM-computations in fluid–structure interaction. International Journal for Numerical Methods in Engineering, 2009, 79(7): 846–869zbMATHCrossRefGoogle Scholar
  91. 91.
    Verhoosel C V, Remmers J J, Gutiérrez M A. A partition of unitybased multiscale approach for modelling fracture in piezoelectric ceramics. International Journal for Numerical Methods in Engineering, 2010, 82(8): 966–994zbMATHCrossRefGoogle Scholar
  92. 92.
    Nanthakumar S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176MathSciNetCrossRefGoogle Scholar
  93. 93.
    Zheng A X, Luo X Q, Shen H. Numerical simulation and analysis of deformation and failure of jointed rock slopes by extended finite element method. Rock and Soil Mechanics, 2013, 34(8): 2371–2376 (in Chinese)Google Scholar
  94. 94.
    Wan L L, Yü T T. Pre-processing of extended finite element method for discontinuous rock masses. Rock and Soil Mechanics, 2011, 32: 772–778 (in Chinese)Google Scholar
  95. 95.
    Goodarzi M, Mohammadi S, Jafari A. Numerical analysis of rock fracturing by gas pressure using the extended finite element method. Petroleum Science, 2015, 12(2): 304–315CrossRefGoogle Scholar
  96. 96.
    Zhuang X, Chun J, Zhu H. A comparative study on unfilled and filled crack propagation for rock-like brittle material. Theoretical and Applied Fracture Mechanics, 2014, 72: 110–120CrossRefGoogle Scholar
  97. 97.
    Réthoré J, Borst R, Abellan M A. A two-scale approach for fluid flow in fractured porous media. International Journal for Numerical Methods in Engineering, 2007, 71(7): 780–800MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Song C, Wolf J P. The scaled boundary finite-element methodalias consistent infinitesimal finite-element cell method for elastodynamics. Computer Methods in Applied Mechanics and Engineering, 1997, 147(3–4): 329–355MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Rabczuk T, Zi G, Gerstenberger A, Wall W A. A new crack tip element for the phantom-node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599zbMATHCrossRefGoogle Scholar
  100. 100.
    Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92: 242–256CrossRefGoogle Scholar
  101. 101.
    Hughes T J, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, nurbs, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39): 4135–4195MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146CrossRefGoogle Scholar
  103. 103.
    Nguyen-Thanh N, Valizadeh N, Nguyen M, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff–love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Shi G H. Manifold method of material analysis. Tech. rep., DTIC Document, 1992Google Scholar
  105. 105.
    Shi G H. Modeling rock joints and blocks by manifold method. In: The 33th US Symposium on Rock Mechanics (USRMS), American Rock Mechanics Association, 1992Google Scholar
  106. 106.
    Ma G, An X, He L. The numerical manifold method: a review. International Journal of Computational Methods, 2010, 7(1): 1–32MathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    Li S, Cheng Y, Wu Y F. Numerical manifold method based on the method of weighted residuals. Computational Mechanics, 2005, 35 (6): 470–480zbMATHCrossRefGoogle Scholar
  108. 108.
    Lin J S. A mesh-based partition of unity method for discontinuity modeling. Computer Methods in Applied Mechanics and Engineering, 2003, 192(11): 1515–1532zbMATHCrossRefGoogle Scholar
  109. 109.
    Terada K, Asai M, Yamagishi M. Finite cover method for linear and non-linear analyses of heterogeneous solids. International Journal for Numerical Methods in Engineering, 2003, 58(9): 1321–1346zbMATHCrossRefGoogle Scholar
  110. 110.
    Terada K, Kurumatani M. Performance assessment of generalized elements in the finite cover method. Finite Elements in Analysis and Design, 2004, 41(2): 111–132CrossRefGoogle Scholar
  111. 111.
    Terada K, Ishii T, Kyoya T, Kishino Y. Finite cover method for progressive failure with cohesive zone fracture in heterogeneous solids and structures. Computational Mechanics, 2007, 39(2): 191–210zbMATHCrossRefGoogle Scholar
  112. 112.
    Zheng W, Zhuang X, Tannant D D, Cai Y, Nunoo S. Unified continuum/discontinuum modeling framework for slope stability assessment. Engineering Geology, 2014, 179: 90–101CrossRefGoogle Scholar
  113. 113.
    Kurumatani M, Terada K. Finite cover method with multi-cover layers for the analysis of evolving discontinuities in heterogeneous media. International Journal for Numerical Methods in Engineering, 2009, 79(1): 1–24zbMATHCrossRefGoogle Scholar
  114. 114.
    Gao H, Cheng Y. A complex variable meshless manifold method for fracture problems. International Journal of Computational Methods, 2010, 7(1): 55–81MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Zhang H, Li L, An X, Ma G. Numerical analysis of 2D crack propagation problems using the numerical manifold method. Engineering Analysis with Boundary Elements, 2010, 34(1): 41–50MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Chen G, Ohnishi Y, Ito T. Development of high-order manifold method. International Journal for Numerical Methods in Engineering, 1998, 43(4): 685–712zbMATHCrossRefGoogle Scholar
  117. 117.
    Jiang Q, Zhou C, Li D. A three-dimensional numerical manifold method based on tetrahedral meshes. Computers & Structures, 2009, 87(13): 880–889CrossRefGoogle Scholar
  118. 118.
    Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically nonlinear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75 (16): 4740–4758CrossRefGoogle Scholar
  120. 120.
    Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39 (6): 743–760zbMATHCrossRefGoogle Scholar
  121. 121.
    Rabczuk T, Areias P, Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548MathSciNetzbMATHCrossRefGoogle Scholar
  122. 122.
    Zi G, Rabczuk T, Wall W. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382zbMATHCrossRefGoogle Scholar
  123. 123.
    Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495zbMATHCrossRefGoogle Scholar
  124. 124.
    Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49zbMATHCrossRefGoogle Scholar
  125. 125.
    Rabczuk T, Areias P. A new approach for modelling slip lines in geological materials with cohesive models. International Journal for Numerical and Analytical Methods in Geomechanics, 2006, 30 (11): 1159–1172zbMATHCrossRefGoogle Scholar
  126. 126.
    Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23): 1391–1411CrossRefGoogle Scholar
  127. 127.
    Mossaiby F, Bazrpach M, Shojaei A. Extending the method of exponential basis functions to problems with singularities. Engineering Computations, 2015, 32(2): 406–423CrossRefGoogle Scholar
  128. 128.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37): 2437–2455zbMATHCrossRefGoogle Scholar
  129. 129.
    Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343zbMATHCrossRefGoogle Scholar
  130. 130.
    Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29): 2777–2799MathSciNetzbMATHCrossRefGoogle Scholar
  131. 131.
    Ai W, Augarde C E. An adaptive cracking particle method for 2D crack propagation. International Journal for Numerical Methods in Engineering, 2016, 108(13): 1626–1648MathSciNetCrossRefGoogle Scholar
  132. 132.
    Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid–structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71MathSciNetzbMATHGoogle Scholar
  133. 133.
    Zhu H, Zhuang X, Cai Y, Ma G. High rock slope stability analysis using the enriched meshless shepard and least squares method. International Journal of Computational Methods, 2011, 8(2): 209–228MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    Zhuang X, Augarde C, Mathisen K. Fracture modeling using meshless methods and level sets in 3D: framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    Zhuang X, Huang F, Zhu H. Modelling 2D joint propagation in rock using the meshless methods and level sets. Chinese Journal of Rock Mechanics and Engineering, 2012, 31: 21872196 (in Chinese)Google Scholar
  136. 136.
    Silling S A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 2000, 48(1): 175–209MathSciNetzbMATHCrossRefGoogle Scholar
  137. 137.
    Silling S A, Epton M, Weckner O, Xu J, Askari E. Peridynamic states and constitutive modeling. Journal of Elasticity, 2007, 88(2): 151–184MathSciNetzbMATHCrossRefGoogle Scholar
  138. 138.
    Ganzenmüller G C, Hiermaier S, May M. On the similarity of meshless discretizations of peridynamics and smooth-particle hydrodynamics. Computers & Structures, 2015, 150: 71–78zbMATHCrossRefGoogle Scholar
  139. 139.
    Bessa M, Foster J, Belytschko T, LiuWK. A meshfree unification: reproducing kernel peridynamics. Computational Mechanics, 2014, 53(6): 1251–1264MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Shojaei A, Mudric T, Zaccariotto M, Galvanetto U. A coupled meshless finite point/peridynamic method for 2D dynamic fracture analysis. International Journal of Mechanical Sciences, 2016, 119: 419–431CrossRefGoogle Scholar
  141. 141.
    Bobaru F, Yang M, Alves L F, Silling S A, Askari E, Xu J. Convergence, adaptive refinement, and scaling in 1D peridynamics. International Journal for Numerical Methods in Engineering, 2009, 77(6): 852–877zbMATHCrossRefGoogle Scholar
  142. 142.
    Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 108 (12): 1451–1476Google Scholar
  143. 143.
    Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: a stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782MathSciNetCrossRefGoogle Scholar
  144. 144.
    Ha Y D, Lee J, Hong J W. Fracturing patterns of rock-like materials in compression captured with peridynamics. Engineering Fracture Mechanics, 2015, 144: 176–193CrossRefGoogle Scholar
  145. 145.
    Wang Y, Zhou X, Xu X. Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Engineering Fracture Mechanics, 2016, 163: 248–273CrossRefGoogle Scholar
  146. 146.
    Zhou X P, Wang Y T. Numerical simulation of crack propagation and coalescence in pre-cracked rock-like Brazilian disks using the non-ordinary state-based peridynamics. International Journal of Rock Mechanics and Mining Sciences, 2016, 89: 235–249CrossRefGoogle Scholar
  147. 147.
    Ren H, Zhuang X, Rabczuk T. A new peridynamic formulation with shear deformation for elastic solid. Journal of Micromechanics and Molecular Physics, 2016, 1(02): 1650009CrossRefGoogle Scholar
  148. 148.
    Mossaiby F, Shojaei A, Zaccariotto M, Galvanetto U. OpenCL implementation of a high performance 3D peridynamic model on graphics accelerators. Computers & Mathematics with Applications, 2017, 1856–1870Google Scholar
  149. 149.
    Brebbia C A, Walker S. Boundary Element Techniques in Engineering. Elsevier, 1980zbMATHGoogle Scholar
  150. 150.
    Mi Y, Aliabadi M. Three-dimensional crack growth simulation using BEM. Computers & Structures, 1994, 52(5): 871–878zbMATHCrossRefGoogle Scholar
  151. 151.
    Simpson R N, Bordas S P, Trevelyan J, Rabczuk T. A twodimensional isogeometric boundary element method for elastostatic analysis. Computer Methods in Applied Mechanics and Engineering, 2012, 209: 87–100MathSciNetzbMATHCrossRefGoogle Scholar
  152. 152.
    Lafhaj Z, Shahrour I. Use of the boundary element method for the analysis of permeability tests in boreholes. Engineering Analysis with Boundary Elements, 2000, 24(9): 695–698zbMATHCrossRefGoogle Scholar
  153. 153.
    Nguyen B, Tran H, Anitescu C, Zhuang X, Rabczuk T. An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 252–275MathSciNetCrossRefGoogle Scholar
  154. 154.
    Cundall P A. A computer model for simulating progressive largescale movements in blocky rock systems. In: Procedings of the Symposio of the International Society of Rock Mechanics, Nancy, 1971Google Scholar
  155. 155.
    Cundall P A. Rational design of tunnel supports: a computer model for rock mass behavior using interactive graphics for the input and output of geometrical data. Tech. rep., DTIC Document, 1974, 1–195Google Scholar
  156. 156.
    Cundall P A, Strack O D. A discrete numerical model for granular assemblies. Geotechnique, 1979, 29(1): 47–65CrossRefGoogle Scholar
  157. 157.
    Cundall P A. Formulation of a three-dimensional distinct element model part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1988, 25: 107–116Google Scholar
  158. 158.
    Zhu H, Wu W, Zhuang X, Cai Y, Rabczuk T. Method for estimating normal contact parameters in collision modeling using discontinuous deformation analysis. International Journal of Geomechanics, 2016, 17(5): E4016011CrossRefGoogle Scholar
  159. 159.
    Wriggers P. Computational Contact Mechanics. Springer Science & Business Media, 2006zbMATHCrossRefGoogle Scholar
  160. 160.
    Cundall P A, Hart R D. Numerical modelling of discontinue. Engineering Computations, 1992, 9(2): 101–113CrossRefGoogle Scholar
  161. 161.
    Curran J H, Ofoegbu G I. Modeling discontinuities in numerical analysis. Comprehensive Rock Engineering, 1993, 1: 443–468Google Scholar
  162. 162.
    Walton O R. Force models for particle-dynamics simulations of granular materials. In: Mobile Particulate Systems. Springer Netherlands, 1995, 287: 367–380Google Scholar
  163. 163.
    Luding S. About contact force-laws for cohesive frictional materials in 2D and 3D. In: Procedings of Behavior of granular media, 2006, 9: 137–147Google Scholar
  164. 164.
    Jing L. A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering. International Journal of Rock Mechanics and Mining Sciences, 2003, 40(3): 283–353CrossRefGoogle Scholar
  165. 165.
    Huang D, Wang J, Liu S. A comprehensive study on the smooth joint model in DEM simulation of jointed rock masses. Granular Matter, 2015, 17(6): 775–791CrossRefGoogle Scholar
  166. 166.
    Lorig L. A simple numerical representation of fully bonded passive rock reinforcement for hard rocks. Computers and Geotechnics, 1985, 1(2): 79–97CrossRefGoogle Scholar
  167. 167.
    Kochen R, Andrade J C O. Predicted behavior of a subway station in weathered rock. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(3): 160–e1–160.e13Google Scholar
  168. 168.
    Souley M, Hoxha D, Homand F. Distinct element modelling of an underground excavation using a continuum damage model. International Journal of Rock Mechanics and Mining Sciences, 1999, 35(4–5): 442–443Google Scholar
  169. 169.
    Rawlings C, Barton N, Bandis S, Addis M, Gutierrez M. Laboratory and numerical discontinuum modeling of wellbore stability. Journal of Petroleum Technology, 1993, 45(11): 1086–1092CrossRefGoogle Scholar
  170. 170.
    Gutierrez M, Makurat A. Coupled HTM modelling of cold water injection in fractured hydro-carbon reservoirs. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(3): 113.e1–113.e15Google Scholar
  171. 171.
    Jing L. Numerical modelling of jointed rock masses by distinct element method for two-and three-dimensional problems. Dissertation for PhD degree. Luleå University of Technology, SwedenGoogle Scholar
  172. 172.
    Harper T, Last N. Response of fractured rock subject to fluid injection part III. Practical application. Tectonophysics, 1990, 172 (1–2): 53–65CrossRefGoogle Scholar
  173. 173.
    Shi G H. Stereographic method for the stability analysis of the discontinuous rocks. Scientia Sinica, 1977, 3: 260–271Google Scholar
  174. 174.
    Warburton P M. Some modern developments in block theory for rock engineering. Analysis and Design Methods: Comprehensive Rock Engineering: Principles, Practice and Projects 2, 2013: 293–315Google Scholar
  175. 175.
    Goodman R E, Shi G H. Block Theory and Its Application to Rock Engineering. Prentice-Hall Englewood Cliffs, NJ, 1985Google Scholar
  176. 176.
    Shi G H, Goodman R E. The key blocks of unrolled joint traces in developed maps of tunnel walls. International Journal for Numerical and Analytical Methods in Geomechanics, 1989, 13 (2): 131–158CrossRefGoogle Scholar
  177. 177.
    Karaca M, Goodman R. The influence of water on the behaviour of a key block. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1993, 30: 1575–1578CrossRefGoogle Scholar
  178. 178.
    Jakubowski J, Tajdus A. The 3D Monte Carlo simulation of rigid block around a tunnel. In: Mechanics of Jointed and Faulted Rock. Rotterdam: Balkema, 1995, 551–6Google Scholar
  179. 179.
    Kuszmaul J, Goodman R. An analytical model for estimating key block sizes in excavations in jointed rock masses. In: Fractured and Jointed Rock Masses. Rotterdam: Balkema, 1995,19–26Google Scholar
  180. 180.
    Windsor C R. Block stability in jointed rock masses. In: Nedlands W A, ed. CSIRO Rock Mechanics Research Centre. Fractured and Jointed Rock Masses, Lake Tahoe, California. 1992, 65–72Google Scholar
  181. 181.
    Mauldon M, Chou K, Wu Y. Linear programming analysis of keyblock stability. In: Computer Methods and Advancements in Geomechanics, 1997, 1: 517–22Google Scholar
  182. 182.
    Wibowo J L. Consideration of secondary blocks in key-block analysis. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(3): 333.e1–333.e2Google Scholar
  183. 183.
    Song J J, Lee C I, Seto M. Stability analysis of rock blocks around a tunnel using a statistical joint modeling technique. Tunnelling and Underground Space Technology, 2001, 16(4): 341–351CrossRefGoogle Scholar
  184. 184.
    Lee I M, Park J K. Stability analysis of tunnel key-block: a case study. Tunnelling and Underground Space Technology, 2000, 15 (4): 453–462CrossRefGoogle Scholar
  185. 185.
    Warburton P. Vector stability analysis of an arbitrary polyhedral rock block with any number of free faces. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1981, 18: 415–427CrossRefGoogle Scholar
  186. 186.
    Shi G H, Goodman R E. Two-dimensional discontinuous deformation analysis. International Journal for Numerical and Analytical Methods in Geomechanics, 1985, 9(6): 541–556zbMATHCrossRefGoogle Scholar
  187. 187.
    Shi G H. Three-dimensional discontinuous deformation analyses. In: DC Rocks 2001, The 38th US Symposium on Rock Mechanics (USRMS), American Rock Mechanics Association, 2001Google Scholar
  188. 188.
    Zhang X, Lu M. Block-interfaces model for non-linear numerical simulations of rock structures. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(7): 983–990CrossRefGoogle Scholar
  189. 189.
    Shyu K. Nodal-based discontinuous deformation analysis. Dissertation for PhD degree. University of California, Berkeley, 1993Google Scholar
  190. 190.
    Jing L. Formulation of discontinuous deformation analysis (DDA) an implicit discrete element model for block systems. Engineering Geology, 1998, 49(3): 371–381CrossRefGoogle Scholar
  191. 191.
    Lin C T, Amadei B, Jung J, Dwyer J. Extensions of discontinuous deformation analysis for jointed rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1996, 33: 671–694CrossRefGoogle Scholar
  192. 192.
    Kim Y I, Amadei B, Pan E. Modeling the effect of water, excavation sequence and rock reinforcement with discontinuous deformation analysis. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(7): 949–970CrossRefGoogle Scholar
  193. 193.
    Jiang Q, Yeung M. A model of point-to-face contact for threedimensional discontinuous deformation analysis. Rock Mechanics and Rock Engineering, 2004, 37(2): 95–116CrossRefGoogle Scholar
  194. 194.
    Hsiung S M. Discontinuous deformation analysis (DDA) with nth order polynomial displacement functions. In: DC Rocks 2001, The 38th US Symposium on Rock Mechanics (USRMS), American Rock Mechanics Association, 2001Google Scholar
  195. 195.
    Koo C, Chern J. The development of DDA with third order displacement function. In: Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, CA. TSI Press: Albuquerque, 1996, 12–14Google Scholar
  196. 196.
    Tonon F. Analysis of single rock blocks for general failure modes under conservative and non-conservative forces. International Journal for Numerical and Analytical Methods in Geomechanics, 2007, 31(14): 1567–1608zbMATHCrossRefGoogle Scholar
  197. 197.
    Tonon F, Asadollahi P. Validation of general single rock block stability analysis (bs3d) for wedge failure. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(4): 627–637CrossRefGoogle Scholar
  198. 198.
    Nezami E G, Hashash Y M, Zhao D, Ghaboussi J. A fast contact detection algorithm for 3-D discrete element method. Computers and Geotechnics, 2004, 31(7): 575–587CrossRefGoogle Scholar
  199. 199.
    Shi G H. Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures. Engineering Computations, 1992, 9(2): 157–168MathSciNetCrossRefGoogle Scholar
  200. 200.
    Wu W, Zhu H, Zhuang X, Ma G, Cai Y. A multi-shell cover algorithm for contact detection in the three-dimensional discontinuous deformation analysis. Theoretical and Applied Fracture Mechanics, 2014, 72: 136–149CrossRefGoogle Scholar
  201. 201.
    Li H, Bai Y, Xia M, Ke F, Yin X. Damage localization as a possible precursor of earthquake rupture. Pure and Applied Geophysics, 2000, 157: 1929–1943CrossRefGoogle Scholar
  202. 202.
    Mühlhaus H, Sakaguchi H, Wei Y. Particle based modelling of dynamic fracture in jointed rock. In: Proceedings of the 9th international conference of the international association of computer methods and advances in geomechanics–IACMAG, 1997, 97: 207–216Google Scholar
  203. 203.
    Napier J, Dede T. A comparison between random mesh schemes and explicit growth rules for rock fracture simulation. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(3): 217.e1–217.e3Google Scholar
  204. 204.
    Place D, Mora P. Numerical simulation of localisation phenomena in a fault zone. Pure and Applied Geophysics, 2000, 157, 11–12: 1821–1845CrossRefGoogle Scholar

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© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Structural MechanicsBauhaus Universität-WeimarWeimarGermany

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