A mixed-mode phase field fracture model in anisotropic rocks with consistent kinematics

Griffith, 1921, The phenomena of rupture and flow in solids, Phil. Trans. R. Soc. A, 221, 163 Rudnicki, 1980, Fracture mechanics applied to the Earth’s crust, Annu. Rev. Earth Planet. Sci., 8, 489, 10.1146/annurev.ea.08.050180.002421 Hutchinson, 1991, Mixed mode cracking in layered materials, Adv. Appl. Mech., 29, 63, 10.1016/S0065-2156(08)70164-9 Suits, 2009, Using high speed video imaging in the study of cracking processes in rock, Geotech. Test. J., 32, 164, 10.1520/GTJ101631 Bobet, 1998, Fracture coalescence in rock-type materials under uniaxial and biaxial compression, Int. J. Rock Mech. Min. Sci., 35, 863, 10.1016/S0148-9062(98)00005-9 Bobet, 2001, Numerical simulation of initiation of tensile and shear cracks Yang, 2012, An experimental study of the fracture coalescence behaviour of brittle sandstone specimens containing three fissures, Rock Mech. Rock Eng., 45, 563, 10.1007/s00603-011-0206-x Reyes, 1991, Failure mechanisms of fractured rock - a fracture coalescence model, 333 Vásárhelyi, 2000, Modeling of crack initiation, propagation and coalescence in uniaxial compression, Rock Mech. Rock Eng., 33, 119, 10.1007/s006030050038 Liebowitz, 1968 Nuismer, 1975, An energy release rate criterion for mixed mode fracture, Int. J. Fract., 11, 245, 10.1007/BF00038891 Shen, 1994, Modification of the G-criterion for crack propagation subjected to compression, Eng. Fract. Mech., 47, 177, 10.1016/0013-7944(94)90219-4 Shen, 1995, Coalescence of fractures under shear stresses in experiments, J. Geophys. Res. Solid Earth, 100, 5975, 10.1029/95JB00040 Zhang, 2017, A modification of the phase-field model for mixed mode crack propagation in rock-like materials, Comput. Methods Appl. Mech. Engrg., 322, 123, 10.1016/j.cma.2017.04.028 Bourdin, 2008, The variational approach to fracture, J. Elasticity, 91, 5, 10.1007/s10659-007-9107-3 Linder, 2009, Finite elements with embedded branching, Finite Elem. Anal. Des., 45, 280, 10.1016/j.finel.2008.10.012 Khoei, 2014 Moës, 2011, A level set based model for damage growth: The thick level set approach, Internat. J. Numer. Methods Engrg., 86, 358, 10.1002/nme.3069 Pandolfi, 2012, An eigenerosion approach to brittle fracture, Internat. J. Numer. Methods Engrg., 92, 694, 10.1002/nme.4352 Belytschko, 2013 Sun, 2017, Mixed Arlequin method for multiscale poromechanics problems, Internat. J. Numer. Methods Engrg., 111, 624, 10.1002/nme.5476 Sih, 1973, Some basic problems in fracture mechanics and new concepts, Eng. Fract. Mech., 5, 365, 10.1016/0013-7944(73)90027-1 Callari, 2002, Finite element methods for the analysis of strong discontinuities in coupled poro-plastic media, Comput. Methods Appl. Mech. Engrg., 191, 4371, 10.1016/S0045-7825(02)00374-2 Borja, 2002, Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a Petrov–Galerkin multiscale formulation, Comput. Methods Appl. Mech. Engrg., 191, 2949, 10.1016/S0045-7825(02)00218-9 Borja, 2008, Assumed enhanced strain and the extended finite element methods: A unification of concepts, Comput. Methods Appl. Mech. Engrg., 197, 2789, 10.1016/j.cma.2008.01.019 Sukumar, 2000, Extended finite element method for three-dimensional crack modelling, Internat. J. Numer. Methods Engrg., 48, 1549, 10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A Pandolfi, 2000, Three dimensional cohesive-element analysis and experiments of dynamic fracture in C300 steel, Int. J. Solids Struct., 37, 3733, 10.1016/S0020-7683(99)00155-9 Borja, 2013 Miehe, 2010, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Comput. Methods Appl. Mech. Engrg., 199, 2765, 10.1016/j.cma.2010.04.011 da Silva, 2013, Modeling of crack initiation, propagation and coalescence in rocks, Int. J. Fract., 182, 167, 10.1007/s10704-013-9866-8 Michael, 2012, A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Engrg., 217, 77 Heister, 2015, A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach, Comput. Methods Appl. Mech. Engrg., 290, 466, 10.1016/j.cma.2015.03.009 Teichtmeister, 2017, Phase field modeling of fracture in anisotropic brittle solids, Int. J. Non-Linear Mech., 97, 1, 10.1016/j.ijnonlinmec.2017.06.018 Ambrosio, 1990, Approximation of functional depending on jumps by elliptic functional via t-convergence, Comm. Pure Appl. Math., 43, 999, 10.1002/cpa.3160430805 Lee, 2016, Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model, Comput. Methods Appl. Mech. Engrg., 305, 111, 10.1016/j.cma.2016.02.037 Borden, 2014, A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework, Comput. Methods Appl. Mech. Engrg., 273, 100, 10.1016/j.cma.2014.01.016 Khisamitov, 2018, Variational approach to interface element modeling of brittle fracture propagation, Comput. Methods Appl. Mech. Engrg., 328, 452, 10.1016/j.cma.2017.08.031 Na, 2018, Computational thermomechanics of crystalline rock, part I: a combined multi-phase-field/crystal plasticity approach for single crystal simulations, Comput. Methods Appl. Mech. Engrg., 338, 657, 10.1016/j.cma.2017.12.022 Gültekin, 2016, A phase-field approach to model fracture of arterial walls: theory and finite element analysis, Comput. Methods Appl. Mech. Engrg., 312, 542, 10.1016/j.cma.2016.04.007 Clayton, 2014, A geometrically nonlinear phase field theory of brittle fracture, Int. J. Fract., 189, 139, 10.1007/s10704-014-9965-1 Clayton, 2015, Phase field modeling of directional fracture in anisotropic polycrystals, Comput. Mater. Sci., 98, 158, 10.1016/j.commatsci.2014.11.009 Clayton, 2016, Phase field modeling and simulation of coupled fracture and twinning in single crystals and polycrystals, Comput. Methods Appl. Mech. Engrg., 312, 447, 10.1016/j.cma.2016.01.023 Choo, 2018, Coupled phase-field and plasticity modeling of geological materials: from brittle fracture to ductile flow, Comput. Methods Appl. Mech. Engrg., 330, 1, 10.1016/j.cma.2017.10.009 Choo, 2018, Cracking and damage from crystallization in pores: coupled chemo-hydro-mechanics and phase-field modeling, Comput. Methods Appl. Mech. Engrg., 335, 347, 10.1016/j.cma.2018.01.044 Shen, 1995, The mechanism of fracture coalescence in compression—experimental study and numerical simulation, Eng. Fract. Mech., 51, 73, 10.1016/0013-7944(94)00201-R Wilson, 2013, A phase-field model for fracture in piezoelectric ceramics, Int. J. Fract., 183, 135, 10.1007/s10704-013-9881-9 Wilson, 2016, Phase-field modeling of hydraulic fracture, J. Mech. Phys. Solids, 96, 264, 10.1016/j.jmps.2016.07.019 Kuhn, 2010, A continuum phase field model for fracture, Eng. Fract. Mech., 77, 3625, 10.1016/j.engfracmech.2010.08.009 Ambati, 2014, A review on phase-field models of brittle fracture and a new fast hybrid formulation, Comput. Mech., 55, 383, 10.1007/s00466-014-1109-y Wang, 2017, A unified variational eigen-erosion framework for interacting brittle fractures and compaction bands in fluid-infiltrating porous media, Comput. Methods Appl. Mech. Engrg., 318, 1, 10.1016/j.cma.2017.01.017 Gurtin, 1996, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92, 178, 10.1016/0167-2789(95)00173-5 Gültekin, 2018, Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: A rate-dependent anisotropic crack phase-field model, Comput. Methods Appl. Mech. Engrg., 331, 23, 10.1016/j.cma.2017.11.008 Miehe, 2015, Minimization principles for the coupled problem of Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture, J. Mech. Phys. Solids, 82, 186, 10.1016/j.jmps.2015.04.006 Miehe, 2015, Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids, Comput. Methods Appl. Mech. Engrg., 294, 449, 10.1016/j.cma.2014.11.016 Backers, 2012, ISRM suggested method for the determination of mode II fracture toughness, Rock Mech. Rock Eng., 45, 137, 10.1007/s00603-012-0271-9 Wu, 2015, On the equivalence between traction- and stress-based approaches for the modeling of localized failure in solids, J. Mech. Phys. Solids, 82, 137, 10.1016/j.jmps.2015.05.016 Niandou, 1997, Laboratory investigation of the mechanical behaviour of Tournemire shale, Int. J. Rock Mech. Min. Sci., 34, 3, 10.1016/S1365-1609(97)80029-9 Semnani, 2016, Thermoplasticity and strain localization in transversely isotropic materials based on anisotropic critical state plasticity, Int. J. Numer. Anal. Methods Geomech., 40, 2423, 10.1002/nag.2536 Na, 2017, Effects of spatial heterogeneity and material anisotropy on the fracture pattern and macroscopic effective toughness of Mancos shale in Brazilian tests, J. Geophys. Res. Solid Earth, 122, 6202, 10.1002/2016JB013374 Walpole, 1984, Fourth-rank tensors of the thirty-two crystal classes: multiplication tables, Proc. R. Soc. A Math. Phys. Eng. Sci., 391, 149 Schmidt, 2009, Eigenfracture: an eigendeformation approach to variational fracture, Multiscale Model. Simul., 7, 1237, 10.1137/080712568 Pandolfi, 2013, Modeling fracture by material-point erosion, Int. J. Fract., 184, 3, 10.1007/s10704-012-9788-x Liu, 2014, A regularized phenomenological multiscale damage model, Internat. J. Numer. Methods Engrg., 99, 867, 10.1002/nme.4705 Radovitzky, 2011, A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method, Comput. Methods Appl. Mech. Engrg., 200, 326, 10.1016/j.cma.2010.08.014 Shen, 2014, 181 Lyness, 1967, Numerical differentiation of analytic functions, SIAM J. Numer. Anal., 4, 202, 10.1137/0704019 Tanaka, 2014, Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis, Comput. Methods Appl. Mech. Engrg., 269, 454, 10.1016/j.cma.2013.11.005 Brothers, 2014, A comparison of different methods for calculating tangent-stiffness matrices in a massively parallel computational peridynamics code, Comput. Methods Appl. Mech. Engrg., 279, 247, 10.1016/j.cma.2014.06.034 Bangerth, 2007, deal.II –a general purpose object-oriented finite element library, ACM Trans. Math. Software, 33, 24/1, 10.1145/1268776.1268779 White, 2008, Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197, 4353, 10.1016/j.cma.2008.05.015 Choo, 2016, Hydromechanical modeling of unsaturated flow in double porosity media, Int. J. Geomech., D4016002, 10.1061/(ASCE)GM.1943-5622.0000558 Na, 2017, Computational thermo-hydro-mechanics for multiphase freezing and thawing porous media in the finite deformation range, Comput. Methods Appl. Mech. Engrg., 318, 667, 10.1016/j.cma.2017.01.028 Miehe, 2010, Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83, 1273, 10.1002/nme.2861 Wheeler, 2014, An augmented-Lagrangian method for the phase-field approach for pressurized fractures, Comput. Methods Appl. Mech. Engrg., 271, 69, 10.1016/j.cma.2013.12.005 Bobet, 1998, Numerical modeling of fracture coalescence in a model rock material, Int. J. Fract., 92, 221, 10.1023/A:1007460316400 Sargado, 2018, High-accuracy phase-field models for brittle fracture based on a new family of degradation functions, J. Mech. Phys. Solids, 111, 458, 10.1016/j.jmps.2017.10.015 Negri, 2008, Quasi-static crack propagation by Griffith’s criterion, Math. Models Methods Appl. Sci., 18, 1895, 10.1142/S0218202508003236 Lemaitre, 1986, Local approach of fracture, Eng. Fract. Mech., 25, 523, 10.1016/0013-7944(86)90021-4 Amor, 2009, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, J. Mech. Phys. Solids, 57, 1209, 10.1016/j.jmps.2009.04.011 Heister, 2015, Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 290, 466, 10.1016/j.cma.2015.03.009 May, 2015, A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-convergence and stress oscillations, Eur. J. Mech. A. Solids, 52, 72, 10.1016/j.euromechsol.2015.02.002