×

Comparative verification of discrete and smeared numerical approaches for the simulation of hydraulic fracturing. (English) Zbl 1453.86010

Summary: The numerical treatment of propagating fractures as embedded discontinuities is a challenging task for which an analyst has to select a suitable numerical method from a range of options. Since their inception in the mid-80s, smeared approaches for fracture simulation such as non-local damage, gradient damage or more lately phase-field modelling have steadily gained popularity. One of the appeals of a smeared implicit fracture representation, the ability to handle complex topologies with unknown crack paths in relatively coarse meshes as well as multiple-crack interaction and multiphysics, is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations which is technically more difficult to achieve with many other methods. However, in hydraulic fracturing simulations, not only the prediction of the fracture path but also the computation of fracture width and propagation pressure (frac pressure) is crucial for reliable and meaningful applications of the simulation tool; how to determine some of these quantities in smeared representations is not immediately obvious. In this study, two of the most popular smeared approaches of recent, namely non-local damage and phase-field models, and an approach in which the solution space is locally enriched to capture a strong discontinuity combined with a cohesive-zone model are verified against fundamental hydraulic fracture propagation problems in the toughness-dominated regime. The individual theoretical foundations of each approach are discussed and differences in the treatment of physical and numerical properties of the methods when applied to the same physical problems are highlighted through examples.

MSC:

86-08 Computational methods for problems pertaining to geophysics
86A20 Potentials, prospecting
74R99 Fracture and damage
74S05 Finite element methods applied to problems in solid mechanics
74L10 Soil and rock mechanics

Software:

OpenGeoSys; PETSc
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alessi, R., Marigo, J.J., Vidoli, S.: Gradient damage models coupled with plasticity: variational formulation and main properties. Mech. Mater. 80(PB), 351-367 (2015). https://doi.org/10.1016/j.mechmat.2013.12.005 · doi:10.1016/j.mechmat.2013.12.005
[2] Ambati, M., Gerasimov, T., De Lorenzis, L.: Phase-field modeling of ductile fracture. Comput. Mech. 55(5), 1017-1040 (2015). https://doi.org/10.1007/s00466-015-1151-4. arXiv:1011.1669v3 · Zbl 1329.74018 · doi:10.1007/s00466-015-1151-4
[3] Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999-1036 (1990). https://doi.org/10.1002/cpa.3160430805 · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[4] Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Unione Mat. Ital. 7, 105-123 (1992) · Zbl 0776.49029
[5] Amor, H., Marigo, J.J., Maurini, C.: Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J. Mech. Phys. Solids 57(8), 1209-1229 (2009). https://doi.org/10.1016/j.jmps.2009.04.011 · Zbl 1426.74257 · doi:10.1016/j.jmps.2009.04.011
[6] Balay, S.; Gropp, WD; McInnes, LC; Smith, BF; Arge, E. (ed.); Bruaset, AM (ed.); Langtangen, HP (ed.), Efficient management of parallelism in object oriented numerical software libraries, 163-202 (1997), Basel · Zbl 0882.65154 · doi:10.1007/978-1-4612-1986-6_8
[7] Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Technical Report ANL-95/11—Revision 3.8, Argonne National Laboratory (2017a). http://www.mcs.anl.gov/petsc
[8] Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc Web page (2017b). http://www.mcs.anl.gov/petsc, http://www.mcs.anl.gov/petsc
[9] Bazant, P.Z., Jirasek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128, 1119-1149 (2002) · doi:10.1061/(ASCE)0733-9399(2002)128:11(1119)
[10] Bažant, Z.P.: Why continuum damage is nonlocal: micromechanics arguments. J. Eng. Mech. 117(5), 1070-1087 (1991) · doi:10.1061/(ASCE)0733-9399(1991)117:5(1070)
[11] Belytschko, T., Moës, N., Usui, S., Parimi, C.: Arbitrary discontinuities in finite elements. Int. J. Numer. Methods Eng. 50(4), 993-1013 (2001) · Zbl 0981.74062 · doi:10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
[12] Belytschko, T., Chen, H., Xu, J., Zi, G.: Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int. J. Numer. Methods Eng. 58(12), 1873-1905 (2003) · Zbl 1032.74662 · doi:10.1002/nme.941
[13] Belytschko, T., Gracie, R., Ventura, G.: A review of extended/generalized finite element methods for material modeling. Model. Simul. Mater. Sci. Eng. 17(4), 043001 (2009) · doi:10.1088/0965-0393/17/4/043001
[14] Biot, M.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155-164 (1941) · JFM 67.0837.01 · doi:10.1063/1.1712886
[15] Böger, L., Keip, M.A., Miehe, C.: Minimization and saddle-point principles for the phase-field modeling of fracture in hydrogels. Comput. Mater. Sci. 138, 474-485 (2017) · doi:10.1016/j.commatsci.2017.06.010
[16] Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217-220, 77-95 (2012). https://doi.org/10.1016/j.cma.2012.01.008 · Zbl 1253.74089 · doi:10.1016/j.cma.2012.01.008
[17] Bouchard, P.O., Bay, F., Chastel, Y., Tovena, I.: Crack propagation modelling using an advanced remeshing technique. Comput. Methods Appl. Mech. Eng. 189(3), 723-742 (2000) · Zbl 0993.74060 · doi:10.1016/S0045-7825(99)00324-2
[18] Bourdin, B., Francfort, G., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797-826 (2000). https://doi.org/10.1016/S0022-5096(99)00028-9 · Zbl 0995.74057 · doi:10.1016/S0022-5096(99)00028-9
[19] Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91, 5-148 (2008) · Zbl 1176.74018 · doi:10.1007/s10659-007-9107-3
[20] Bourdin, B., Chukwudozie, C., Yoshioka, K.: A variational approach to the numerical simulation of hydraulic fracturing. In: Proceedings of the 2012 SPE Annual Technical Conference and Exhibition, vol. SPE 159154 (2012) · Zbl 1440.74121
[21] Bourdin, B., Marigo, J.J., Maurini, C., Sicsic, P.: Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys. Rev. Lett. 112, 014301 (2014). https://doi.org/10.1103/PhysRevLett.112.014301 · doi:10.1103/PhysRevLett.112.014301
[22] Brace, W., Paulding, B., Scholz, C.: Dilatancy in the fracture of crystalline rocks. J. Geophys. Res. 71(16), 3939-3953 (1966) · doi:10.1029/JZ071i016p03939
[23] Braides, A.: Approximation of Free-Discontinuity Problems. Springer, Berlin (1998) · Zbl 0909.49001 · doi:10.1007/BFb0097344
[24] Branco, R., Antunes, F., Costa, J.: A review on 3D-FE adaptive remeshing techniques for crack growth modelling. Eng. Fract. Mech. 141, 170-195 (2015) · doi:10.1016/j.engfracmech.2015.05.023
[25] Budyn, E., Zi, G., Moës, N., Belytschko, T.: A method for multiple crack growth in brittle materials without remeshing. Int. J. Numer. Methods Eng. 61(10), 1741-1770 (2004) · Zbl 1075.74638 · doi:10.1002/nme.1130
[26] Chessa, J., Belytschko, T.: An extended finite element method for two-phase fluids. J. Appl. Mech. 70(1), 10-17 (2003) · Zbl 1110.74391 · doi:10.1115/1.1526599
[27] Chukwudozie, C.: Application of the variational fracture model to hydraulic fracturing in poroelastic media. Dissertation, Louisiana State University (2016)
[28] Davila, C., Camanho, P., de Moura, M.: Mixed-mode decohesion elements for analyses of progressive delamination. In: 19th AIAA Applied Aerodynamics Conference, p. 1486 (2001)
[29] de Borst, R., Verhoosel, C.V.: Gradient damage vs phase-field approaches for fracture: similarities and differences. Comput. Methods Appl. Mech. Eng. 312, 78-94 (2016a) · Zbl 1439.74347 · doi:10.1016/j.cma.2016.05.015
[30] de Borst, R., Verhoosel, C.V.: Gradient damage vs phase-field approaches for fracture: similarities and differences. Comput. Methods Appl. Mech. Eng. 312, 78-94 (2016b) · Zbl 1439.74347 · doi:10.1016/j.cma.2016.05.015
[31] Dean, R.H., Schmidt, J.H.: Hydraulic-fracture predictions with a fully coupled geomechanical reservoir simulator. SPEJ (2009). https://doi.org/10.2118/116470-PA
[32] Desmorat, R., Gatuingt, F., Jirásek, M.: Nonlocal models with damage-dependent interactions motivated by internal time. Eng. Fract. Mech. 142, 255-275 (2015) · doi:10.1016/j.engfracmech.2015.06.015
[33] Detournay, E.: Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48, 311-339 (2016) · Zbl 1356.74181 · doi:10.1146/annurev-fluid-010814-014736
[34] Diederichs, M.: Manuel rocha medal recipient rock fracture and collapse under low confinement conditions. Rock Mech. Rock Eng. 36(5), 339-381 (2003) · doi:10.1007/s00603-003-0015-y
[35] Duarte, C.A., Reno, L., Simone, A.: A high-order generalized fem for through-the-thickness branched cracks. Int. J. Numer. Methods Eng. 72(3), 325-351 (2007) · Zbl 1194.74385 · doi:10.1002/nme.2012
[36] Duddu, R., Waisman, H.: A nonlocal continuum damage mechanics approach to simulation of creep fracture in ice sheets. Comput. Mech. 51(6), 961-974 (2013) · Zbl 1366.86005 · doi:10.1007/s00466-012-0778-7
[37] Economides, M.J., Nolte, E.K.G.: Reservoir Stimulation, vol. 2. Wiley, New York (2000)
[38] Elices, M., Guinea, G., Gomez, J., Planas, J.: The cohesive zone model: advantages, limitations and challenges. Eng. Fract. Mech. 69(2), 137-163 (2002) · doi:10.1016/S0013-7944(01)00083-2
[39] Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319-1342 (1998). https://doi.org/10.1016/S0022-5096(98)00034-9 · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[40] Freddi, F., Royer-Carfagni, G.: Regularized variational theories of fracture: a unified approach. J. Mech. Phys. Solids (2010). https://doi.org/10.1016/j.jmps.2010.02.010 · Zbl 1244.74114
[41] Fries, T.P., Belytschko, T.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Methods Eng. 68(13), 1358-1385 (2006) · Zbl 1129.74045 · doi:10.1002/nme.1761
[42] Fries, T.P., Schätzer, M., Weber, N.: XFEM-simulation of hydraulic fracturing in 3D with emphasis on stress intensity factors. In: Oñate, E. Oliver, J., Huerta, A. (eds.) 11th World Congress on Computational Mechanics (WCCM XI), 5th European Conference on Computational Mechanics (ECCM V), 6th European Conference on Computational Fluid Dynamics (ECFD VI) (2014)
[43] Garagash, D.I.: Plane-strain propagation of a fluid-driven fracture during injection and shut-in: asymptotics of large toughness. Eng. Fract. Mech. 73(4), 456-481 (2006). https://doi.org/10.1016/j.engfracmech.2005.07.012 · doi:10.1016/j.engfracmech.2005.07.012
[44] Gasser, T.C., Holzapfel, G.A.: Modeling 3D crack propagation in unreinforced concrete using PUFEM. Comput. Methods Appl. Mech. Eng. 194(25-26), 2859-2896 (2005) · Zbl 1176.74180 · doi:10.1016/j.cma.2004.07.025
[45] Giovanardi, B., Scotti, A., Formaggia, L.: A hybrid XFEM-phase field (xfield) method for crack propagation in brittle elastic materials. Comput. Methods Appl. Mech. Eng. 320, 396-420 (2017) · Zbl 1439.74350 · doi:10.1016/j.cma.2017.03.039
[46] Gordeliy, E., Peirce, A.: Coupling schemes for modeling hydraulic fracture propagation using the XFEM. Comput. Methods Appl. Mech. Eng. 253, 305-322 (2013) · Zbl 1297.74104 · doi:10.1016/j.cma.2012.08.017
[47] Gupta, P., Duarte, C.A.: Particle shape effect on macro-and micro behaviours of monodisperse ellipsoids. Int. J. Numer. Anal. Methods Geomech. 38, 1397-1430 (2014). https://doi.org/10.1002/nag.732 · Zbl 1273.74055 · doi:10.1002/nag.732
[48] He, W., Wu, Y.F., Xu, Y., Fu, T.T.: A thermodynamically consistent nonlocal damage model for concrete materials with unilateral effects. Comput. Methods Appl. Mech. Eng. 297, 371-391 (2015) · Zbl 1423.74056 · doi:10.1016/j.cma.2015.09.010
[49] Heider, Y., Markert, B.: Simulation of hydraulic fracture of porous materials using the phase-field modeling approach. Pamm 16(1), 447-448 (2016). https://doi.org/10.1002/pamm.201610212 · doi:10.1002/pamm.201610212
[50] 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 Concr. Res. 6(6), 773-781 (1976) · doi:10.1016/0008-8846(76)90007-7
[51] Hoek, E., Martin, C.: Fracture initiation and propagation in intact rock—a review. J. Rock Mech. Geotech. Eng. 6(4), 287-300 (2014) · doi:10.1016/j.jrmge.2014.06.001
[52] Ji, J., Settari, A., Sullivan, R.: A novel hydraulic fracturing model fully coupled with geomechanics and reservoir simulation. SPE J. (2009). https://doi.org/10.2118/110845-PA
[53] Jiang, L., Sainoki, A., Mitri, H.S., Ma, N., Liu, H., Hao, Z.: Influence of fracture-induced weakening on coal mine gateroad stability. Int. J. Rock Mech. Min. Sci. 88, 307-317 (2016). https://doi.org/10.1016/j.ijrmms.2016.04.017 · doi:10.1016/j.ijrmms.2016.04.017
[54] Jirásek, M.: Comparison of nonlocal models for damage and fracture. LSC Report 98(02) (1998) · Zbl 0930.74054
[55] Johnson, L., Marschall, P., Zuidema, P., Gribi, P.: Effects of post-disposal gas generation in a repository for spent fuel, high-level waste and long-lived intermediate level waste sited in opalinus clay. Technical Report, National Cooperative for the Disposal of Radioactive Waste (NAGRA) (2004)
[56] Karma, A., Kessler, D.A., Levine, H.: Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87(4), 3-6 (2001). https://doi.org/10.1103/PhysRevLett.87.045501 · doi:10.1103/PhysRevLett.87.045501
[57] Khoei, A.R.: Extended Finite Element Method: Theory and Applications. Wiley, London (2014) · Zbl 1315.74001
[58] Khoei, A., Moslemi, H., Sharifi, M.: Three-dimensional cohesive fracture modeling of non-planar crack growth using adaptive FE technique. Int. J. Solids Struct. 49(17), 2334-2348 (2012) · doi:10.1016/j.ijsolstr.2012.04.036
[59] Klinsmann, M., Rosato, D., Kamlah, M., McMeeking, R.M.: An assessment of the phase field formulation for crack growth. Comput. Methods Appl. Mech. Eng. 294(Supplement C), 313-330 (2015). https://doi.org/10.1016/j.cma.2015.06.009 · Zbl 1423.74833 · doi:10.1016/j.cma.2015.06.009
[60] Kolditz, O., Bauer, S., Bilke, L., Böttcher, N., Delfs, J., Fischer, T., Görke, U., Kalbacher, T., Kosakowski, G., McDermott, C., et al.: OpenGeoSys: an open-source initiative for numerical simulation of thermo-hydro-mechanical/chemical (THM/C) processes in porous media. Environ. Earth Sci. 67(2), 589-599 (2012) · doi:10.1007/s12665-012-1546-x
[61] Kuhl, E., Ramm, E., de Borst, R.: An anisotropic gradient damage model for quasi-brittle materials. Comput. Methods Appl. Mech. Eng. 183(1), 87-103 (2000) · Zbl 0992.74060 · doi:10.1016/S0045-7825(99)00213-3
[62] Kuhn, C., Müller, R.: A continuum phase field model for fracture. Eng. Fract. Mech. 77(18), 3625-3634 (2010). https://doi.org/10.1016/j.engfracmech.2010.08.009. (computational Mechanics in Fracture and Damage: A Special Issue in Honor of Prof. Gross) · doi:10.1016/j.engfracmech.2010.08.009
[63] Kuhn, C., Lohkamp, R., Schneider, F., Aurich, J.C., Mueller, R.: Finite element computation of discrete configurational forces in crystal plasticity. Int. J. Solids Struct. 56, 62-77 (2015) · doi:10.1016/j.ijsolstr.2014.12.004
[64] Lee, S., Wheeler, M.F., Wick, T.: Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Comput. Methods Appl. Mech. Eng. 312, 509-541 (2016). https://doi.org/10.1016/j.cma.2016.02.037 · Zbl 1425.74419 · doi:10.1016/j.cma.2016.02.037
[65] Legarth, B., Huenges, E., Zimmermann, G.: Hydraulic fracturing in a sedimentary geothermal reservoir: results and implications. Int. J. Rock Mech. Min. Sci. 42(7-8), 1028-1041 (2005) · doi:10.1016/j.ijrmms.2005.05.014
[66] Lemaitre, J., Chaboche, J.L., Benallal, A., Desmorat, R.: Mécanique des matériaux solides-3eme édition. Dunod (2009)
[67] Li, T., Marigo, J.J., Guilbaud, D., Potapov, S.: Gradient damage modeling of brittle fracture in an explicit dynamic context. Int. J. Numer. Methods Eng. 00(March), 1-25 (2016). https://doi.org/10.1002/nme · doi:10.1002/nme
[68] Marigo, J.J., Maurini, C., Pham, K.: An overview of the modelling of fracture by gradient damage models. Meccanica 51(12), 3107-3128 (2016). https://doi.org/10.1007/s11012-016-0538-4 · Zbl 1374.74109 · doi:10.1007/s11012-016-0538-4
[69] Meschke, G., Leonhart, D.: A generalized finite element method for hydro-mechanically coupled analysis of hydraulic fracturing problems using space – time variant enrichment functions. Comput. Methods Appl. Mech. Eng. 290, 438-465 (2015) · Zbl 1423.74280 · doi:10.1016/j.cma.2015.03.005
[70] Meschke, G., Dumstorff, P., Fleming, W.: Variational extended finite element model for cohesive cracks: influence of integration and interface law. In: IUTAM Symposium on Discretization Methods for Evolving Discontinuities, pp. 283-301. Springer (2007) · Zbl 1209.74055
[71] Meyer, A., Rabold, F., Scherzer, M.: Efficient finite element simulation of crack propagation. Preprintreihe des Chemnitzer SFB 393 (2004) · Zbl 1095.74042
[72] Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations. Int. J. Numer. Methods Eng. 83(10), 1273-1311 (2010). https://doi.org/10.1002/nme.2861 · Zbl 1202.74014 · doi:10.1002/nme.2861
[73] Miehe, C., Mauthe, S., Teichtmeister, S.: Minimization Principles for the Coupled Problem of Darcy-Biot-Type Fluid Transport in Porous Media Linked to Phase Field Modeling of Fracture, vol. 82. Elsevier, Amsterdam (2015). https://doi.org/10.1016/j.jmps.2015.04.006
[74] Miehe, C., Aldakheel, F., Raina, A.: Phase Field Modeling of Ductile Fracture at Finite Strains: A Variational Gradient-extended Plasticity-damage Theory, vol 84. Elsevier, Amsterdam(2016). https://doi.org/10.1016/j.ijplas.2016.04.011
[75] Minkley, W., Brückner, D., Lüdeling, C.: Tightness of salt rocks and fluid percolation. In: 45. Geomechanik-Kolloqium, Freiberg, Germany (2016)
[76] Moës, N., Belytschko, T.: Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69(7), 813-833 (2002) · doi:10.1016/S0013-7944(01)00128-X
[77] Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131-150 (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[78] Morita, N., Black ,A.D., Guh, G.F.: Theory of Lost Circulation Pressure. SPE Annual Technical Conference and Exhibition, 23-26 September, New Orleans, Louisiana (1990). https://doi.org/10.2118/20409-MS
[79] Murakami, S.: Continuum Damage Mechanics: A Continuum Mechanics Approach to the Analysis of Damage and Fracture, vol. 185. Springer, Berlin (2012)
[80] Nagel, T., Görke, U.J., Moerman, K.M., Kolditz, O.: On advantages of the kelvin mapping in finite element implementations of deformation processes. Environ. Earth Sci. 75(11), 1-11 (2016). https://doi.org/10.1007/s12665-016-5429-4 · doi:10.1007/s12665-016-5429-4
[81] Nagel, T., Minkley, W., Böttcher, N., Naumov, D., Görke, U.J., Kolditz, O.: Implicit numerical integration and consistent linearization of inelastic constitutive models of rock salt. Comput. Struct. 182, 87-103 (2017) · doi:10.1016/j.compstruc.2016.11.010
[82] Nedjar, B.: On a concept of directional damage gradient in transversely isotropic materials. Int. J. Solids Struct. 88, 56-67 (2016) · doi:10.1016/j.ijsolstr.2016.03.026
[83] Needleman, A.: An analysis of decohesion along an imperfect interface. Int. J. Fract. 42(1), 21-40 (1990a) · doi:10.1007/BF00018611
[84] Needleman, A.: An analysis of tensile decohesion along an interface. J. Mech. Phys. Solids 38(3), 289-324 (1990b) · doi:10.1016/0022-5096(90)90001-K
[85] Nguyen, G.D., Houlsby, G.T.: Non-local damage modelling of concrete: a procedure for the determination of model parameters. Int. J. Numer. Anal. Methods Geomech. 31(7), 867-891 (2007) · Zbl 1196.74173 · doi:10.1002/nag.563
[86] Nguyen, O., Repetto, E., Ortiz, M., Radovitzky, R.: A cohesive model of fatigue crack growth. Int. J. Fract. 110(4), 351-369 (2001) · doi:10.1023/A:1010839522926
[87] Nguyen, G.D., Korsunsky, A.M., Belnoue, J.P.H.: A nonlocal coupled damage-plasticity model for the analysis of ductile failure. Int. J. Plast. 64, 56-75 (2015) · doi:10.1016/j.ijplas.2014.08.001
[88] Oliver, J.: On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations. Int. J. Solids Struct. 37(48-50), 7207-7229 (2000) · Zbl 0994.74004 · doi:10.1016/S0020-7683(00)00196-7
[89] Oliver, J., Huespe, A.E., Pulido, M., Chaves, E.: From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng. Fract. Mech. 69(2), 113-136 (2002) · doi:10.1016/S0013-7944(01)00060-1
[90] Parisio, F., Laloui, L.: Plastic-damage modeling of saturated quasi-brittle shales. Int. J. Rock Mech. Min. Sci. 93, 295-306 (2017) · doi:10.1016/j.ijrmms.2017.01.016
[91] Parisio, F., Samat, S., Laloui, L.: Constitutive analysis of shale: a coupled damage plasticity approach. Int. J. Solids Struct. 75, 88-98 (2015) · doi:10.1016/j.ijsolstr.2015.08.003
[92] Parisio, F., Tarokh, A., Makhnenko, R., Naumov, D., Miao, X.Y., Kolditz, O., Nagel, T.: Experimental characterization and numerical modelling of fracture processes in granite. Int. J. Solids Struct. (2018a, in press). https://doi.org/10.1016/j.ijsolstr.2018.12.019
[93] Parisio, F., Vilarrasa, V., Laloui, L.: Hydro-mechanical modeling of tunnel excavation in anisotropic shale with coupled damage-plasticity and micro-dilatant regularization. Rock Mech. Rock Eng. (2018b) https://doi.org/10.1007/s00603-018-1569-z
[94] Peerlings, R.H.J., De Borst, R., Brekelmans, W.A.M., De Vree, J.H.P.: Gradient enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39(19), 3391-3403 (1996) · Zbl 0882.73057 · doi:10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D
[95] Pham, K., Amor, H., Marigo, J.J., Maurini, C.: Gradient damage models and their use to approximate brittle fracture. Int. J. Damage Mech. 20(4), 618-652 (2011). https://doi.org/10.1177/1056789510386852 · doi:10.1177/1056789510386852
[96] Rice, J.: The Mechanics of Earthquake Rupture. Division of Engineering, Brown University, Providence (1979)
[97] Roth, S.N., Léger, P., Soulaïmani, A.: Coupled hydro-mechanical cracking of concrete using XFEM in 3D. In: Saouma, V, Bolander, J., Landis, E. (eds.) 9th International Conference on Fracture Mechanics of Concrete and Concrete Structures FraMCoS-9 (2016) · Zbl 1506.74437
[98] Santillán, D., Juanes, R., Cueto-Felgueroso, L.: Phase field model of fluid-driven fracture in elastic media: immersed-fracture formulation and validation with analytical solutions. J. Geophys. Res. Solid Earth 122(4), 2565-2589 (2017). https://doi.org/10.1002/2016JB013572 · doi:10.1002/2016JB013572
[99] Silani, M., Talebi, H., Hamouda, A.M., Rabczuk, T.: Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. J. Comput. Sci. 15, 18-23 (2016) · doi:10.1016/j.jocs.2015.11.007
[100] Sneddon, I., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. The SIAM Series in Applied Mathematics. Wiley, London (1969) · Zbl 0201.26702
[101] Tanné, E., Li, T., Bourdin, B., Marigo, J.J., Maurini, C.: Crack nucleation in variational phase-field models of brittle fracture. J. Mech. Phys. Solids 110(Supplement C), 80-99 (2018). https://doi.org/10.1016/j.jmps.2017.09.006 · doi:10.1016/j.jmps.2017.09.006
[102] Turon, A., Davila, C.G., Camanho, P.P., Costa, J.: An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng. Fract. Mech. 74(10), 1665-1682 (2007) · doi:10.1016/j.engfracmech.2006.08.025
[103] Vtorushin, E.: Application of mixed finite elements to spatially non-local model of inelastic deformations. GEM-Int. J. Geomath. 7(2), 183-201 (2016) · Zbl 1348.74299 · doi:10.1007/s13137-016-0083-2
[104] Watanabe, N., Wang, W., Taron, J., Görke, U., Kolditz, O.: Lower-dimensional interface elements with local enrichment: application to coupled hydro-mechanical problems in discretely fractured porous media. Int. J. Numer. Methods Eng. 90(8), 1010-1034 (2012). https://doi.org/10.1002/nme.3353/full · Zbl 1242.74172 · doi:10.1002/nme.3353/full
[105] Wheeler, M., Wick, T., Wollner, W.: An augmented-lagrangian method for the phase-field approach for pressurized fractures. Comput. Methods Appl. Mech. Eng. 271(Supplement C), 69-85 (2014). https://doi.org/10.1016/j.cma.2013.12.005 · Zbl 1296.65170 · doi:10.1016/j.cma.2013.12.005
[106] Wilson, Z.A., Landis, C.M.: Phase-field modeling of hydraulic fracture. J. Mech. Phys. Solids 96, 264-290 (2016). https://doi.org/10.1016/j.jmps.2016.07.019 · Zbl 1482.74020 · doi:10.1016/j.jmps.2016.07.019
[107] Yoshioka, K., Bourdin, B.: A variational hydraulic fracturing model coupled to a reservoir simulator. Int. J. Rock Mech. Min. Sci. 88(Supplement C), 137-150 (2016). https://doi.org/10.1016/j.ijrmms.2016.07.020 · doi:10.1016/j.ijrmms.2016.07.020
[108] Zhang, Z., Guazzato, M., Sornsuwan, T., Scherrer, S.S., Rungsiyakull, C., Li, W., Swain, M.V., Li, Q.: Thermally induced fracture for core-veneered dental ceramic structures. Acta Biomater. 9(9), 8394-8402 (2013) · doi:10.1016/j.actbio.2013.05.009
[109] Zhang, X., Vignes, C., Sloan, S.W., Sheng, D.: Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale. Comput. Mech. 59(5), 737-752 (2017). https://doi.org/10.1007/s00466-017-1373-8 · doi:10.1007/s00466-017-1373-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.