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

Ali JENABIDEHKORDI

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Front. Struct. Civ. Eng. ›› 2019, Vol. 13 ›› Issue (2) : 273-287. DOI: 10.1007/s11709-018-0459-5
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REVIEW

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

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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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Ali JENABIDEHKORDI. Computational methods for fracture in rock: a review and recent advances. Front. Struct. Civ. Eng., 2019, 13(2): 273‒287 https://doi.org/10.1007/s11709-018-0459-5

References

[1]
Carpinteri A. Post-peak and post-bifurcation analysis of cohesive crack propagation. Engineering Fracture Mechanics, 1989, 32(2): 265–278
CrossRef Google scholar
[2]
Planas J, Elices M. Nonlinear fracture of cohesive materials. In: Current Trends in Concrete Fracture Research, Springer, 1991, 139–157
[3]
Bažant Z P, Jirásek M. Nonlocal integral formulations of plasticity and damage: survey of progress. Journal of Engineering Mechanics, 2002, 128(11): 1119–1149
CrossRef Google scholar
[4]
Wheel M. A geometrically versatile finite volume formulation for plane elastostatic stress analysis. Journal of Strain Analysis for Engineering Design, 1996, 31(2): 111–116
CrossRef Google scholar
[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): 107–138
CrossRef Google scholar
[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): 439–455
CrossRef Google scholar
[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): 1757–1776
CrossRef Google scholar
[8]
Mishev I D. Finite volume methods on voronoi meshes. Numerical Methods for Partial Differential Equations, 1998, 14(2): 193–212
CrossRef Google scholar
[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): 639–645
CrossRef Google scholar
[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, 1999
[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), 2001
[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, 137–142
[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, 705–710
[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, 2001
[15]
Jing L, Hudson J. Numerical methods in rock mechanics. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(4): 409–427
CrossRef Google scholar
[16]
Wittke W, Sykes R. Rock Mechanics. Springer Berlin, 1990
[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)
[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)
[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): 254–260
[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): 351–359
[21]
Bazănt Z P, Cedolin L. Fracture mechanics of reinforced concrete. Journal of the Engineering Mechanics Division, 1980, 106(6): 1287–1306
[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): 637–660
[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, 1970
[24]
Ghaboussi J, Wilson E, Isenberg J. Finite element for rock joints and interfaces. Journal of Soil Mechanics & Foundations Division, 1973, 99(10): 849–862
[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): 19–43
CrossRef Google scholar
[26]
Goodman R E. Methods of geological engineering in discontinuous rocks. New York: West Publishing, 1976
[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): 371–384
CrossRef Google scholar
[28]
Bažant Z P. Why continuum damage is nonlocal: micromechanics arguments. Journal of Engineering Mechanics, 1991, 117(5): 1070–1087
CrossRef Google scholar
[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): 3391–3403
CrossRef Google scholar
[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): 175–189
CrossRef Google scholar
[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): 939–962
CrossRef Google scholar
[32]
Pasternak E, Dyskin A, Mühlhaus H B. Cracks of higher modes in Cosserat continua. International Journal of Fracture, 2006, 140(1–4): 189–199
CrossRef Google scholar
[33]
Etse G, Willam K. Failure analysis of elastoviscoplastic material models. Journal of Engineering Mechanics, 1999, 125(1): 60–69
CrossRef Google scholar
[34]
Rabczuk T, Eibl J. Simulation of high velocity concrete fragmentation using SPH/MLSPH. International Journal for Numerical Methods in Engineering, 2003, 56(10): 1421–1444
CrossRef Google scholar
[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): 773–781
CrossRef Google scholar
[36]
Miihlhaus H B, Triantafyllidis T. Surface waves in a layered half-space with bending stiffness. Developments in Geotechnical Engineering, 1987, 44: 277–290
[37]
Mühlhaus H B. Application of Cosserat theory in numerical solutions of limit load problems. Archive of Applied Mechanics, 1989, 59(2): 124–137
[38]
Vardoulakis I, Mühlhaus H. Local rock surface instabilities. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1986, 23: 379–383
[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: 332–369
[40]
de Borst R. Fracture in quasi-brittle materials: a review of continuum damage-based approaches. Engineering Fracture Mechanics, 2002, 69(2): 95–112
CrossRef Google scholar
[41]
Jirásek M, Zimmermann T. Analysis of rotating crack model. Journal of Engineering Mechanics, 1998, 124(8): 842–851
CrossRef Google scholar
[42]
Jirásek M, Zimmermann T. Rotating crack model with transition to scalar damage. Journal of Engineering Mechanics, 1998, 124(3): 277–284
CrossRef Google scholar
[43]
Rabczuk T, Akkermann J, Eibl J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5–6): 1327–1354
CrossRef Google scholar
[44]
Ohmenhäuser F, Weihe S, Kröplin B. Algorithmic implementation of a generalized cohesive crack model. Computational Materials Science, 1999, 16(1): 294–306
[45]
Carpinteri A, Chiaia B, Cornetti P. A scale-invariant cohesive crack model for quasi-brittle materials. Engineering Fracture Mechanics, 2002, 69(2): 207–217
CrossRef Google scholar
[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): 644–660
CrossRef Google scholar
[47]
de Borst R, Remmers J J, Needleman A. Mesh-independent discrete numerical representations of cohesive-zone models. Engineering Fracture Mechanics, 2006, 73(2): 160–177
CrossRef Google scholar
[48]
Zhuang X, Huang R, Liang C, Rabczuk T. A coupled thermo-hydro-mechanical model of jointed hard rock for compressed air energy storage. Mathematical Problems in Engineering, 2014, 179169
[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: 18–23
CrossRef Google scholar
[50]
Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three-dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92
CrossRef Google scholar
[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: 30–38
CrossRef Google scholar
[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): 1047–1071
CrossRef Google scholar
[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): 1129–1148
CrossRef Google scholar
[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: 126–143
CrossRef Google scholar
[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): 527–541
[56]
Belytschko T, Lin J I. A three-dimensional impact-penetration algorithm with erosion. Computers & Structures, 1987, 25(1): 95–104
CrossRef Google scholar
[57]
Camacho G T, Ortiz M. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures, 1996, 33(20): 2899–2938
CrossRef Google scholar
[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): 111–132
CrossRef Google scholar
[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): 189–214
CrossRef Google scholar
[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): 279–297
CrossRef Google scholar
[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): 3733–3760
CrossRef Google scholar
[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): 1–24
CrossRef Google scholar
[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–Pr5-50
CrossRef Google scholar
[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): 291–315
CrossRef Google scholar
[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: 50–63
CrossRef Google scholar
[66]
Areias P, Rabczuk T, Dias-da Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137
CrossRef Google scholar
[67]
Areias P, Rabczuk T, Camanho P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947
CrossRef Google scholar
[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): 1099–1122
CrossRef Google scholar
[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: 27–41
CrossRef Google scholar
[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): 1003–1018
CrossRef Google scholar
[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: 322–350
CrossRef Google scholar
[72]
Areias P, Msekh M, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143
CrossRef Google scholar
[73]
Gens A, Carol I, Alonso E. Rock joints: FEM implementation and applications. Studies in Applied Mechanics, 1995, 42: 395–420
CrossRef Google scholar
[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): 59–89
CrossRef Google scholar
[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): 541–564
CrossRef Google scholar
[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): 189–212
CrossRef Google scholar
[77]
Sancho J M, Planas J, Fathy A M, Galvez J C, Cendon D A. Three-dimensional 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): 173–187
CrossRef Google scholar
[78]
Jirásek M. Comparative study on finite elements with embedded discontinuities. Computer Methods in Applied Mechanics and Engineering, 2000, 188(1): 307–330
CrossRef Google scholar
[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: 143–160
CrossRef Google scholar
[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): 1391–1433
CrossRef Google scholar
[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): 7093–7114
[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): 549–581
CrossRef Google scholar
[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: 209–227
CrossRef Google scholar
[84]
Saksala T. Rate-dependent embedded discontinuity approach incorporating heterogeneity for numerical modeling of rock fracture. Rock Mechanics and Rock Engineering, 2015, 48(4): 1605–1622
CrossRef Google scholar
[85]
Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620
CrossRef Google scholar
[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): 131–150
CrossRef Google scholar
[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): 289–314
CrossRef Google scholar
[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): 337–359
CrossRef Google scholar
[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): 1535–1565
CrossRef Google scholar
[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): 846–869
CrossRef Google scholar
[91]
Verhoosel C V, Remmers J J, Gutiérrez M A. A partition of unity-based multiscale approach for modelling fracture in piezoelectric ceramics. International Journal for Numerical Methods in Engineering, 2010, 82(8): 966–994
CrossRef Google scholar
[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): 153–176
CrossRef Google scholar
[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)
[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)
[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): 304–315
CrossRef Google scholar
[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: 110–120
CrossRef Google scholar
[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): 780–800
CrossRef Google scholar
[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): 329–355
CrossRef Google scholar
[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): 577–599
[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: 242–256
CrossRef Google scholar
[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): 4135–4195
CrossRef Google scholar
[102]
Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146
CrossRef Google scholar
[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: 265–291
CrossRef Google scholar
[104]
Shi G H. Manifold method of material analysis. Tech. rep., DTIC Document, 1992
[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, 1992
[106]
Ma G, An X, He L. The numerical manifold method: a review. International Journal of Computational Methods, 2010, 7(1): 1–32
CrossRef Google scholar
[107]
Li S, Cheng Y, Wu Y F. Numerical manifold method based on the method of weighted residuals. Computational Mechanics, 2005, 35(6): 470–480
CrossRef Google scholar
[108]
Lin J S. A mesh-based partition of unity method for discontinuity modeling. Computer Methods in Applied Mechanics and Engineering, 2003, 192(11): 1515–1532
CrossRef Google scholar
[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): 1321–1346
CrossRef Google scholar
[110]
Terada K, Kurumatani M. Performance assessment of generalized elements in the finite cover method. Finite Elements in Analysis and Design, 2004, 41(2): 111–132
CrossRef Google scholar
[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): 191–210
CrossRef Google scholar
[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: 90–101
CrossRef Google scholar
[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): 1–24
CrossRef Google scholar
[114]
Gao H, Cheng Y. A complex variable meshless manifold method for fracture problems. International Journal of Computational Methods, 2010, 7(1): 55–81
CrossRef Google scholar
[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): 41–50
CrossRef Google scholar
[116]
Chen G, Ohnishi Y, Ito T. Development of high-order manifold method. International Journal for Numerical Methods in Engineering, 1998, 43(4): 685–712
CrossRef Google scholar
[117]
Jiang Q, Zhou C, Li D. A three-dimensional numerical manifold method based on tetrahedral meshes. Computers & Structures, 2009, 87(13): 880–889
CrossRef Google scholar
[118]
Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256
CrossRef Google scholar
[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): 4740–4758
CrossRef Google scholar
[120]
Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760
CrossRef Google scholar
[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): 524–548
CrossRef Google scholar
[122]
Zi G, Rabczuk T, Wall W. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382
CrossRef Google scholar
[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): 473–495
CrossRef Google scholar
[124]
Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49
CrossRef Google scholar
[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): 1159–1172
CrossRef Google scholar
[126]
Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23): 1391–1411
CrossRef Google scholar
[127]
Mossaiby F, Bazrpach M, Shojaei A. Extending the method of exponential basis functions to problems with singularities. Engineering Computations, 2015, 32(2): 406–423
CrossRef Google scholar
[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): 2437–2455
CrossRef Google scholar
[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): 2316–2343
CrossRef Google scholar
[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): 2777–2799
CrossRef Google scholar
[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): 1626–1648
CrossRef Google scholar
[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): 48–71
[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): 209–228
CrossRef Google scholar
[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): 969–998
CrossRef Google scholar
[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)
[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): 175–209
CrossRef Google scholar
[137]
Silling S A, Epton M, Weckner O, Xu J, Askari E. Peridynamic states and constitutive modeling. Journal of Elasticity, 2007, 88(2): 151–184
CrossRef Google scholar
[138]
Ganzenmüller G C, Hiermaier S, May M. On the similarity of meshless discretizations of peridynamics and smooth-particle hydrodynamics. Computers & Structures, 2015, 150: 71–78
CrossRef Google scholar
[139]
Bessa M, Foster J, Belytschko T, Liu W K. A meshfree unification: reproducing kernel peridynamics. Computational Mechanics, 2014, 53(6): 1251–1264
CrossRef Google scholar
[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: 419–431
CrossRef Google scholar
[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): 852–877
CrossRef Google scholar
[142]
Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 108(12): 1451–1476
[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: 762–782
CrossRef Google scholar
[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: 176–193
CrossRef Google scholar
[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: 248–273
CrossRef Google scholar
[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: 235–249
CrossRef Google scholar
[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): 1650009
CrossRef Google scholar
[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, 1856–1870
[149]
Brebbia C A, Walker S. Boundary Element techniques in Engineering. Elsevier, 1980
[150]
Mi Y, Aliabadi M. Three-dimensional crack growth simulation using BEM. Computers & Structures, 1994, 52(5): 871–878
CrossRef Google scholar
[151]
Simpson R N, Bordas S P, Trevelyan J, Rabczuk T. A two-dimensional isogeometric boundary element method for elastostatic analysis. Computer Methods in Applied Mechanics and Engineering, 2012, 209: 87–100
CrossRef Google scholar
[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): 695–698
CrossRef Google scholar
[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: 252–275
CrossRef Google scholar
[154]
Cundall P A. A computer model for simulating progressive large-scale movements in blocky rock systems. In: Procedings of the Symposio of the International Society of Rock Mechanics, Nancy, 1971
[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, 1–195
[156]
Cundall P A, Strack O D. A discrete numerical model for granular assemblies. Geotechnique, 1979, 29(1): 47–65
[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: 107–116
[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): E4016011
[159]
Wriggers P. Computational Contact Mechanics. Springer Science & Business Media, 2006
[160]
Cundall P A, Hart R D. Numerical modelling of discontinue. Engineering Computations, 1992, 9(2): 101–113
CrossRef Google scholar
[161]
Curran J H, Ofoegbu G I. Modeling discontinuities in numerical analysis. Comprehensive Rock Engineering, 1993, 1: 443–468
[162]
Walton O R. Force models for particle-dynamics simulations of granular materials. In: Mobile Particulate Systems. Springer Netherlands, 1995, 287: 367–380
[163]
Luding S. About contact force-laws for cohesive frictional materials in 2D and 3D. In: Procedings of Behavior of granular media, 2006, 9: 137–147
[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): 283–353
CrossRef Google scholar
[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): 775–791
CrossRef Google scholar
[166]
Lorig L. A simple numerical representation of fully bonded passive rock reinforcement for hard rocks. Computers and Geotechnics, 1985, 1(2): 79–97
CrossRef Google scholar
[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.e13
[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): 442–443
[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): 1086–1092
CrossRef Google scholar
[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.e15
[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, Sweden
[172]
Harper T, Last N. Response of fractured rock subject to fluid injection part III. Practical application. Tectonophysics, 1990, 172(1–2): 53–65
CrossRef Google scholar
[173]
Shi G H. Stereographic method for the stability analysis of the discontinuous rocks. Scientia Sinica, 1977, 3: 260–271
[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: 293–315
[175]
Goodman R E, Shi G H. Block Theory and Its Application to Rock Engineering. Prentice-Hall Englewood Cliffs, NJ, 1985
[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): 131–158
CrossRef Google scholar
[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: 1575–1578
[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, 551–6
[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, 1995,19–26
[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, 65–72
[181]
Mauldon M, Chou K, Wu Y. Linear programming analysis of key-block stability. In: Computer Methods and Advancements in Geomechanics, 1997, 1: 517–22
[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.e2
[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): 341–351
CrossRef Google scholar
[184]
Lee I M, Park J K. Stability analysis of tunnel key-block: a case study. Tunnelling and Underground Space Technology, 2000, 15(4): 453–462
CrossRef Google scholar
[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: 415–427
[186]
Shi G H, Goodman R E. Two-dimensional discontinuous deformation analysis. International Journal for Numerical and Analytical Methods in Geomechanics, 1985, 9(6): 541–556
CrossRef Google scholar
[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, 2001
[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): 983–990
CrossRef Google scholar
[189]
Shyu K. Nodal-based discontinuous deformation analysis. Dissertation for PhD degree. University of California, Berkeley, 1993
[190]
Jing L. Formulation of discontinuous deformation analysis (DDA) an implicit discrete element model for block systems. Engineering Geology, 1998, 49(3): 371–381
CrossRef Google scholar
[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: 671–694
[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): 949–970
CrossRef Google scholar
[193]
Jiang Q, Yeung M. A model of point-to-face contact for three-dimensional discontinuous deformation analysis. Rock Mechanics and Rock Engineering, 2004, 37(2): 95–116
CrossRef Google scholar
[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, 2001
[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, 12–14
[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): 1567–1608
CrossRef Google scholar
[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): 627–637
CrossRef Google scholar
[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): 575–587
CrossRef Google scholar
[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): 157–168
CrossRef Google scholar
[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: 136–149
CrossRef Google scholar
[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: 1929–1943
[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: 207–216
[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.e3
[204]
Place D, Mora P. Numerical simulation of localisation phenomena in a fault zone. Pure and Applied Geophysics, 2000, 157, 11–12: 1821–1845

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