Abstract
A multiphase-field approach for elasto-plastic and anisotropic brittle crack propagation in geological systems consisting of different regions of brittle and ductile materials is presented and employed to computationally study crack propagation. Plastic deformation in elasto-plastic materials such as frictional, granular or porous materials is modelled with the pressure-sensitive Drucker-Prager plasticity model. This plasticity model is combined with a multiphase-field model fulfilling the mechanical jump conditions in diffuse solid-solid interfaces. The validity of the plasticity model with phase-inherent stress and strain fields is shown, in comparison with sharp interface finite element solutions. The proposed model is capable of simulating crack formation in heterogeneous multiphase systems comprising both purely elastic and inelastic phases. We investigate the influence of different material parameters on the crack propagation with tensile tests in single- and two-phase materials. To show the applicability of the model, crack propagation in a multiphase domain with brittle and elasto-plastic components is performed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen, C.H., Chen, C.S., Wu, J.H.: Fracture toughness analysis on cracked ring disks of anisotropic rock. Rock Mech. Rock Eng. 41(4), 539–562 (2008)
Olson, J.E., Laubach, S.E., Lander, R.H.: Natural fracture characterization in tight gas sandstones: integrating mechanics and diagenesis. AAPG Bull. 93(11), 1535–1549 (2009)
Hancock, P.L.: Brittle microtectonics: principles and practice. J. Struct. Geol. 7(3-4), 437–457 (1985)
Pollard, D.D., Aydin, A.: Progress in understanding jointing over the past century. Geol. Soc. Am. Bull. 100(8), 1181–1204 (1988)
Anders, M.H., Laubach, S.E., Scholz, C.H.: Microfractures: a review. J. Struct. Geol. 69, 377–394 (2014)
Scholz, C.H.: Experimental study of the fracturing process in brittle rock. J. Geophys. Res. 73 (4), 1447–1454 (1968)
Henry, J.P., Paquet, J., Tancrez, J.P.: Experimental study of crack propagation in calcite rocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 14(2), 85–91 (1977)
Labuz, J.F., Shah, S.P., Dowding, C.H.: Experimental analysis of crack propagation in granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 22(2), 85–98 (1985)
Bobet, A., Einstein, H.H.: Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int. J. Rock Mech. Min. Sci. 35(7), 863–888 (1998)
Wong, R.H.C., Chau, K.T., Tang, C.A., Lin, P.: Analysis of crack coalescence in rock-like materials containing three flaws—part i: experimental approach. Int. J. Rock Mech. Min. Sci. 38(7), 909–924 (2001)
Sagong, M., Bobet, A.: Coalescence of multiple flaws in a rock-model material in uniaxial compression. Int. J. Rock Mech. Min. Sci. 39(2), 229–241 (2002)
Labuz, J.F., Biolzi, L.: Experiments with rock: remarks on strength and stability issues. Int. J. Rock Mech. Min. Sci. 44(4), 525–537 (2007)
Haeri, H., Shahriar, K., Marji, M.F., Moarefvand, P.: Experimental and numerical study of crack propagation and coalescence in pre-cracked rock-like disks. Int. J. Rock Mech. Min. Sci. 67, 20–28 (2014)
Bieniawski, Z.T.: Mechanism of brittle fracture of rock: part I—theory of the fracture process. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 4(4), 395–406 (1967)
Paterson, M.S., Wong, T.F.: Experimental rock deformation-the brittle field. Springer Science Business Media, Berlin (2005)
Evans, B., Fredrich, J.T., Wong, T.F.: The brittle-ductile transition in rocks: Recent experimental and theoretical progress. The Brittle-Ductile Transition in Rocks. Geophys. Monogr. Ser. 56, 1–20 (1990)
Wong, T.F., Baud, P.: The brittle-ductile transition in porous rock: a review. J. Struct. Geol. 44, 25–53 (2012)
Terzaghi, K.: Theoretical soil mechanics. Chapman and Hall, London (1951)
Drucker, D.C., Prager, W.: Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10(2), 157–165 (1952)
Zreid, I., Kaliske, M.: An implicit gradient formulation for microplane Drucker-Prager plasticity. Int. J. Plast. 83, 252–272 (2016)
Zreid, I., Kaliske, M.: A gradient enhanced plasticity–damage microplane model for concrete. Comput. Mech. 62(5), 1239–1257 (2018)
Griffith, A.A., Eng, M.: VI. The Phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. A 221(582-593), 163–198 (1921)
Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. (1957)
Zhang, X., Jeffrey, R.G.: Role of overpressurized fluid and fluid-driven fractures in forming fracture networks. J. Geochem. Explor. 144, 194–207 (2014)
Wu, K., Olson, J.E.: A simplified three-dimensional displacement discontinuity method for multiple fracture simulations. Int. J. Fract. 193(2), 191–204 (2015)
McClure, M.W., Babazadeh, M., Shiozawa, S., Huang, J.: Fully coupled hydromechanical simulation of hydraulic fracturing in 3D discrete-fracture networks. SPE J. 21(04), 1–302 (2016)
Ha, Y.D., Bobaru, F.: Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162(1-2), 229–244 (2010)
Ouchi, H., Katiyar, A., York, J., Foster, J.T., Sharma, M.M.: A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach. Comput. Mech. 55(3), 561–576 (2015)
Virgo, S., Abe, S., Urai, J.L.: Extension fracture propagation in rocks with veins: Insight into the crack-seal process using Discrete Element Method modeling. J. Geophys. Res. Solid Earth 118(10), 5236–5251 (2013)
Virgo, S., Abe, S., Urai, J.L.: The influence of loading conditions on fracture initiation, propagation, and interaction in rocks with veins: Results from a comparative Discrete Element Method study. J. Geophys. Res. Solid Earth 121(3), 1730–1738 (2016)
Fries, T.P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)
Wang, X., Shi, F., Liu, C., Lu, D., Liu, H., Wu, H.: Extended finite element simulation of fracture network propagation in formation containing frictional and cemented natural fractures. J. Nat. Gas Sci. Eng. 50, 309–324 (2018)
Mohammadnejad, M., Liu, H., Chan, A., Dehkhoda, S., Fukuda, D.: An overview on advances in computational fracture mechanics of rock Geosyst. Eng. 1–24 (2018)
Boettinger, W.J., Warren, J.A., Beckermann, C., Karma, A.: Phase-field simulation of solidification. Ann. Rev. Mater. Res. 32(1), 163–194 (2002)
Chen, L.Q.: Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32(1), 113–140 (2002)
Moelans, N., Blanpain, B., Wollants, P.: An introduction to phase-field modeling of microstructure evolution. Calphad 32(2), 268–294 (2008)
Qin, R.S., Bhadeshia, H.K.: Phase field method. Mater. Sci. Technol. 26(7), 803–811 (2010)
Nestler, B., Choudhury, A.: Phase-field modeling of multi-component systems. Curr. Opin. Solid State Mater. Sci. 15(3), 93–105 (2011)
Hoetzer, J., Kellner, M., Steinmetz, P., Nestler, B.: Applications of the phase-field method for the solidification of microstructures in multi-component systems. J. Indian Inst. Sci. 96(3), 235–256 (2016)
Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)
Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1-3), 5–148 (2008)
Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199 (45-48), 2765–2778 (2010)
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)
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, 77–95 (2012)
Hesch, C., Weinberg, K.: Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int. J. Numer. Methods Eng. 99(12), 906–924 (2014)
Ambati, M., Gerasimov, T., De Lorenzis, L.: Phase-field modeling of ductile fracture. Comput. Mech. 55(5), 1017–1040 (2015)
Ambati, M., Gerasimov, T., De Lorenzis, L.: A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 55(2), 383–405 (2015)
Kuhn, C., Noll, T., Müller, R.: On phase field modeling of ductile fracture. GAMM-Mitteilungen 39(1), 35–54 (2016)
Miehe, C., Aldakheel, F., Teichtmeister, S.: Phase-field modeling of ductile fracture at finite strains: a robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization. Int. J. Numer. Methods Eng. 111(9), 816–863 (2017)
Choo, J., Sun, W.: Coupled phase-field and plasticity modeling of geological materials: from brittle fracture to ductile flow. Comput. Methods Appl. Mech. Eng. 330, 1–32 (2018)
Kienle, D., Aldakheel, F., Keip, M.A.: A finite-strain phase-field approach to ductile failure of frictional materials. Int. J Solids Struct. (2019)
Spatschek, R., Müller-Gugenberger, C., Brener, E., Nestler, B.: Phase field modeling of fracture and stress-induced phase transitions. Phys. Rev. E 75(6), 066111 (2007)
Nestler, B., Schneider, D., Schoof, E., Huang, Y., Selzer, M.: Modeling of crack propagation on a mesoscopic length scale. GAMM-Mitteilungen 39(1), 78–91 (2016)
Abdollahi, A., Arias, I.: Numerical simulation of intergranular and transgranular crack propagation in ferroelectric polycrystals. Int. J. Fract. 174(1), 3–15 (2012)
Schneider, D., Schoof, E., Huang, Y., Selzer, M., Nestler, B.: Phase-field modeling of crack propagation in multiphase systems. Comput. Methods Appl. Mech. Eng. 312, 186–195 (2016)
Li, B., Peco, C., Millán, D., Arias, I., Arroyo, M.: Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy. Int. J. Numer. Methods Eng. 102(3-4), 711–727 (2015)
Teichtmeister, S., Kienle, D., Aldakheel, F., Keip, M.A.: Phase field modeling of fracture in anisotropic brittle solids. Int. J. Non-Linear Mech. 97, 1–21 (2017)
Nguyen, T.T., Rethore, J., Yvonnet, J., Baietto, M.C.: Multi-phase-field modeling of anisotropic crack propagation for polycrystalline materials. Comput. Mech. 60(2), 289–314 (2017)
Clayton, J.D., Knap, J.: Phase field modeling of directional fracture in anisotropic polycrystals. Comput. Mater. Sci. 98, 158–169 (2015)
Mikelić, A., Wheeler, M.F., Wick, T.: Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput. Geosci. 19(6), 1171–1195 (2015)
Wilson, Z.A., Landis, C.M.: Phase-field modeling of hydraulic fracture. J. Mech. Phys. Solids 96, 264–290 (2016)
Heider, Y., Markert, B.: A phase-field modeling approach of hydraulic fracture in saturated porous media. Mech. Res. Commun. 80, 38–46 (2017)
Chukwudozie, C., Bourdin, B., Yoshioka, K.: A variational phase-field model for hydraulic fracturing in porous media. Comput. Methods Appl. Mech. Eng. (2019)
Pham, K.H., Ravi-Chandar, K., Landis, C.M.: Experimental validation of a phase-field model for fracture. Int. J. Fract. 205(1), 83–101 (2017)
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, 80–99 (2018)
Wu, J.Y., Nguyen, V.P.: A length scale insensitive phase-field damage model for brittle fracture. J. Mech. Phys. Solids 119, 20–42 (2018)
Prajapati, N., Herrmann, C., Späth, M., Schneider, D., Selzer, M., Nestler, B.: Brittle anisotropic fracture propagation in quartz sandstone: insights from phase-field simulations. Comput. Geosci. 24, 1361–1376 (2020)
Herrmann, C., Schneider, D., Schoof, E., Schwab, F., Nestler, B.: Multiphase-Field Model for the Simulation of Brittle and Ductile Crack Propagation in Grey Cast Iron Microstructures, submitted (2020)
Herrmann, C., Schoof, E., Schneider, D., Schwab, F., Reiter, A., Selzer, M., Nestler, B.: Multiphase-field model of small strain elasto-plasticity according to the mechanical jump conditions. Comput. Mech. 1–14 (2018)
Schneider, D., Tschukin, O., Choudhury, A., Selzer, M., Böhlke, T., Nestler, B.: Phase-field elasticity model based on mechanical jump conditions. Comput. Mech. 55(5), 887–901 (2015)
Schneider, D., Schwab, F., Schoof, E., Reiter, A., Herrmann, C., Selzer, M., Böhlke, T., Nestler, B.: On the stress calculation within phase-field approaches: a model for finite deformations. Comput. Mech. 60(2), 203–217 (2017)
Schneider, D., Schoof, E., Tschukin, O., Reiter, A., Herrmann, C., Schwab, F., Selzer, M., Nestler, B.: Small strain multiphase-field model accounting for configurational forces and mechanical jump conditions. Comput. Mech. 61(3), 277–295 (2018)
de Souza Neto, E.A., Peric, D., Owen, D.R.: Computational Methods for Plasticity: Theory and Applications. Wiley, New York (2011)
Schneider, D.: Phasenfeldmodellierung mechanisch getriebener Grenzflächenbewegungen in mehrphasigen Systemen (2016)
Cahn, J.W., Allen, S.M.: A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics. Le Journal de Physique Colloques 38(C7), C7–51 (1977)
Kuhn, C., Müller, R.: A continuum phase field model for fracture. Eng. Fract. Mech. 77(18), 3625–3634 (2010)
Silhavy, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997)
Simo, J.C., Hughes, T.J.: Computational Inelasticity. Springer, Berlin (1998)
Hötzer, J., Reiter, A., Hierl, H., Steinmetz, P., Selzer, M., Nestler, B.: The parallel multi-physics phase-field framework Pace3D. J. Comput. Sci. 26, 1–12 (2018)
Eiken, J.: The finite Phase-Field Method-A numerical diffuse interface approach for microstructure simulation with minimized discretization error. In: MRS Online Proceedings Library Archive, p 1369 (2011)
Yaşar, E.: Failure and failure theories for anisotropic rocks. In: 17th International Mining Congress and Exhibition of Turkey-IMCET, pp 417–424 (2001)
Atkinson, B.K.: Fracture toughness of Tennessee sandstone and Carrara marble using the double torsion testing method. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 16(1), 49–53 (1979)
Nara, Y., Yoneda, T., Kaneko, K.: Effect of humidity on fracture toughness and slow crack growth in sandstone. In: Rock Engineering in Difficult Ground Conditions-Soft Rocks and Karst, p 313 (2009)
Wang, S.R., Hagan, P., Li, Y.C., Zhang, C.G., Liu, X.L., Zou, Z.S.: Experimental study on deformation and strength characteristics of sandstone with different water contents. J. Eng. Sci. Technol. Rev. 10(4), 199–203 (2017)
Senseny, P.E., Pfeifle, T.W.: Fracture toughness of sandstones and shales. In: The 25th US Symposium on Rock Mechanics (USRMS). American Rock Mechanics Association (1984)
Heyliger, P., Ledbetter, H., Kim, S.: Elastic constants of natural quartz. J. Acoust. Soc. Am. 114(2), 644–650 (2003)
Kuna, M.: Numerische Beanspruchungsanalyse von Rissen. Vieweg+ Teubner 1, 408–410 (2008)
Ankit, K., Urai, J.L., Nestler, B.: Microstructural evolution in bitaxial crack-seal veins: a phase-field study. J. Geophys. Res. Solid Earth 120(5), 3096–3118 (2015)
Acknowledgements
Open Access funding enabled and organized by Projekt DEAL. The intense exchange with the project partners Janos Urai and Liene Spruzeniece is greatly acknowledged. The discussion on comparison of diffuse and sharp interface results has been facilitated through the Gottfried-Wilhelm Leibniz Price NE 822/31. The authors further appreciate the development of the elasto-plastic crack propagation model through Helmholtz programme “Renewable energies” and the topic “Geothermal energy systems”.
Funding
The research on crack propagation in combined brittle and ductile geological grain structures has been supported by the German Research foundation (DFG) through the project NE 822/34-1.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Späth, M., Herrmann, C., Prajapati, N. et al. Multiphase-field modelling of crack propagation in geological materials and porous media with Drucker-Prager plasticity. Comput Geosci 25, 325–343 (2021). https://doi.org/10.1007/s10596-020-10007-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-020-10007-0