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A Microplane-Based Anisotropic Damage Model for Deformation and Fracturing of Brittle Rocks

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Abstract

Anisotropy is an important property that is widely present in crustal rocks. Efforts have been devoted to providing a constitutive model that can describe both inherent and stress-induced anisotropy in rock. Different from classic models, that are based on stress invariants or strain tensors, we propose here an anisotropic damage microplane model to capture the characteristics of rock properties in different orientations (i.e., their anisotropy). The basic idea is to couple continuum damage mechanics with the classic microplane model. The stress tensor in the model is dependent on the integration of microplane stresses in all orientations. The damage state of any element in the model is determined by the microplane that satisfies the maximum tensile stress criterion or Mohr–Coulomb criterion. An ellipsoidal function was used to characterize the failure strength, where the orientation of the failure plane changes with the preferred orientation of defects in the rock. The proposed model is validated against laboratory experiments performed on brittle material with orientated cracks and granite under true triaxial compression. The fracture pattern and the effect of the intermediate principal stress are numerically predicted by our anisotropic damage model and we discuss relationships between the damage evolution and the anisotropy of the rock under true triaxial compression. The proposed numerical model, based on microplane theory, offers a new approach to analyzing the effect of crack orientation on the deformation and fracture of brittle rock.

Highlights

  • Microplane-based anisotropic damage model incorporating maximum tensile stress criterion and Mohr-Coulomb criterion is proposed.

  • Peak strength and elastic modulus vary with the preferred crack/damage angles.

  • Fracture pattern of brittle rock and the effect of intermediate principal stress in true triaxial compressive tests is numerically replicated.

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Data availability

The data that support the findings of this study are available on request from the corresponding author Tao Xu.

Abbreviations

\({\varepsilon }_{ij}, {\sigma }_{ij}\) :

Strain and stress tensor

\({\varepsilon }_{N}, {\varepsilon }_{T}\) :

Normal and tangential strain in microplane

\({\varepsilon }_{M}, {\varepsilon }_{L}\) :

Two components of the tangential strain in microplane

\({n}_{i}, {n}_{j}\) :

Unit normal vector, \(i, j=\mathrm{1,2},3.\)

\({l}_{i}, {l}_{j}, {m}_{i}, {m}_{j}\) :

Components of unit tangential vector, \(i, j=\mathrm{1,2},3.\)

\({E}_{N}, {E}_{T}\) :

Normal and tangential elastic modulus

ν :

Poisson’s ratio

Ω :

Surface of unit sphere

k, kth :

Microplane number

\({\omega }_{k}\) :

Integration weights of kth microplane

\({N}_{m}\) :

Total number of microplanes

u :

Scale parameter of elements

\({u}_{0}\) :

Average parameter of elements

w :

Heterogeneity index

\({\sigma }_{N}^{k}, {\sigma }_{T}^{k}\) :

Normal stress and tangential stress of kth microplane

\({\sigma }_{t0}, {\sigma }_{c0}\) :

Ultimate tensile strength and compression strength

\({S}_{0}\) :

Ultimate strength of element

\({\theta }_{f},{c}_{0}\) :

Friction angle and cohesion

\(E, E^{\prime}\) :

Undamaged and effective elastic modulus

D, D t , D c :

Damage variable, tensile damage, shear damage under compression

\({\varepsilon }_{t0}, {\varepsilon }_{c0}\) :

Ultimate strain corresponding to the \({\sigma }_{t0}\) and \({\sigma }_{c0}\)

η :

Residual strength coefficient (RSC)

AC, a, b, c :

Anisotropy coefficient

\({\theta }_{x}\) :

Angle between the preferred orientation and X-axis

References

  • Ashby MF, Sammis CG (1990) The damage mechanics of brittle solids in compression. Pure Appl Geophys 133:489–521

    Google Scholar 

  • Bai Q-s, Zhang C, Young RP (2022) Using true-triaxial stress path to simulate excavation-induced rock damage: a case study. International Journal of Coal Science & Technology 9:49

    Google Scholar 

  • Barton N, Quadros E (2014) Anisotropy is everywhere, to see, to measure, and to model. Rock Mech Rock Eng 48:1323–1339

    Google Scholar 

  • Batdorf SB, Budiansky B (1949) A mathematical theory of plasticity based on the concept of slip. Sov Appl Mech 12:1102–1112

    Google Scholar 

  • Baud P, Louis L, David C, Rawling G, Wong T-f (2005) Effects of bedding and foliation on mechanical anisotropy, damage evolution and failure mode. Geol Soc Lond Spec Publ 245:223–249

    Google Scholar 

  • Bazant Z, Caner F, Carol I, Adley M, Akers S (2000) Microplane model M4 for concrete. I: formulation with work-conjugate deviatoric stress. J Eng Mech 126:944–953

    Google Scholar 

  • Bazant Z, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128:1129–1239

    Google Scholar 

  • Bazant Z, Kim J-K (1986) Creep of anisotropic clay: Microplane model. Journal of Geotechnical Engineering 112:458–475

    Google Scholar 

  • Bazant Z, Oh B (1985) Microplane model for progressive fracture of concrete and rock. J Eng Mech 111:559–582

    Google Scholar 

  • Bazant Z, Prat P (1987) Creep of anisotropic clay: New microplane model. J Eng Mech 113:1050–1064

    Google Scholar 

  • Bazant Z, Prat P (1988) Microplane model for brittle-plastic material: I. Theory Journal of Engineering Mechanics 114:1672–1688

    Google Scholar 

  • Bazant Z, Xiang Y, Prat P (1996) Microplane model for concrete. I: Stress-strain boundaries and finite strain. J Eng Mech 122:245–254

    Google Scholar 

  • Bazant Z, Zi G (2003) Microplane constitutive model for porous isotropic rocks. Int J Numer Anal Meth Geomech 27:25–47

    Google Scholar 

  • Bažant Z (1999) Size effect on structural strength: A review. Arch Appl Mech 69:703–725

    Google Scholar 

  • Bažant P, Oh B (1986) Efficient Numerical Integration on the Surface of a Sphere. Zeitschrift Angewandte Mathematik Und Mechanik 66:37–49

    Google Scholar 

  • Benson P, Schubnel A, Vinciguerra S, Trovato C, Meredith P, Young RP (2006) Modeling the permeability evolution of microcracked rocks from elastic wave velocity inversion at elevated isostatic pressure. Journal of Geophysical Research 111

  • Brocca M, Bazant Z (2000) Microplane constitutive model and metal plasticity. Appl Mech Rev 53:265–280

    Google Scholar 

  • Browning J, Meredith P, Stuart C, Harland S, Healy D, Mitchell T (2018) A directional crack damage memory effect in sandstone under true triaxial loading. Geophysical Research Letters 45

  • Browning J, Meredith P, Stuart C, Healy D, Harland S, Mitchell T (2017) Acoustic characterization of crack damage evolution in sandstone deformed under conventional and true triaxial loading: Crack damage evolution in sandstone. Journal of Geophysical Research: Solid Earth 122

  • COMSOL Multiphysics® v. 5.2. cn.comsol.com. Stockholm, Sweden, COMSOL AB

  • Caner F, Bazant Z (2013a) Microplane model M7 for plain concrete. I: Formulation. J Eng Mech 139:1714–1723

    Google Scholar 

  • Caner F, Bazant Z (2013b) Microplane model M7 for plain concrete. II: Calibration and verification. J Eng Mech 139:1724–1735

    Google Scholar 

  • Carol I, Bazant Z, Prat P (1991) Geometric damage tensor based on microplane model. J Eng Mech 117:2429–2448

    Google Scholar 

  • Cerfontaine B, Collin F (2018) Cyclic and fatigue behaviour of rock materials: review, interpretation and research perspectives. Rock Mech Rock Eng 51:391–414

    Google Scholar 

  • Chen X, Bazant Z (2014) Microplane damage model for jointed rock masses. Int J Numer Anal Meth Geomech 38:1431–1452

    Google Scholar 

  • Chow CL, Wang J (1987) An anisotropic theory of continuum damage mechanics for ductile fracture. Eng Fract Mech 27:547–558

    Google Scholar 

  • Daoud A, Browning J, Meredith P, Mitchell T (2020) Microstructural controls on thermal crack damage and the presence of a temperature-memory effect during cyclic thermal stressing of rocks. Geophysical Research Letters 47

  • David E, Brantut N, Hansen L, Mitchell T (2018) Absence of stress-induced anisotropy during brittle deformation in antigorite serpentinite. Journal of Geophysical Research: Solid Earth 123:10616–10644

    Google Scholar 

  • Douglass PM, Voight B (1969) Anisotropy of granites: a reflection of microscopic fabric. Geotechnique 19:376–398

    Google Scholar 

  • Feng X-T, Zhang X, Kong R, Wang G (2015) A novel mogi type true triaxial testing apparatus and its use to obtain complete stress–strain curves of hard rocks. Rock Mech Rock Eng 49:1649–1662

    Google Scholar 

  • Griffiths L, Heap MJ, Xu T, Chen C, Baud P (2017) The influence of pore geometry and orientation on the strength and stiffness of porous rock. J Struct Geol 96:149–160

    Google Scholar 

  • Gupta I (1973) Seismic velocities in rock subjected to axial loading up to shear fracture. J Geophys Res 78:6936–6942

    Google Scholar 

  • Haimson B, Chang C (2000) A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. Int J Rock Mech Min Sci 37:285–296

    Google Scholar 

  • Healy D, Rizzo R, Cornwell D, Farrell N, Watkins H, Timms N, Gomez-Rivas E, Smith M (2017) FracPaQ: A MATLAB™ toolbox for the quantification of fracture patterns. J Struct Geol 95:1–16

    Google Scholar 

  • Heap MJ, Dg F, Meredith P, Vinciguerra S (2010) Elastic moduli evolution and accompanying stress changes with increasing crack damage: Implications for stress changes around fault zones and volcanoes during deformation. Geophys J Int 183:225–236

    Google Scholar 

  • Heap MJ, Vinciguerra S, Meredith P (2009) The evolution of elastic moduli with increasing crack damage during cyclic stressing of a basalt from Mt. Etna Volcano Tectonophysics 471:153–160

    Google Scholar 

  • Heap MJ, Meyer G, Noël C, Wadsworth F, Baud P, Violay M (2022) The permeability of porous volcanic rock through the brittle‐ductile transition. Journal of Geophysical Research: Solid Earth 127

  • Huang L, Li J, Tue NV, Němeček J, Püschel T (2016) Numerical aspects of microplane constitutive models for concrete. Appl Math Model 41:530–548

    Google Scholar 

  • Jin C, Salviato M, Li W, Cusatis G (2016) Elastic microplane formulation for transversely isotropic materials. Journal of Applied Mechanics 84

  • Kachanov M (1982a) A microcrack model of rock inelasticity part I: Frictional sliding on microcracks. Mech Mater 1:19–27

    Google Scholar 

  • Kachanov M (1982b) A microcrack model of rock inelasticity part II: Propagation of microcracks. Mech Mater 1:29–41

    Google Scholar 

  • Kachanov M (1982c) Microcrack model of rock inelasticity part III: Time-dependent growth of microcracks. Mech Mater 1:123–129

    Google Scholar 

  • Kawamoto T, Ichikawa Y, Kyoya T (1988) Deformation and fracture behaviour of discontinuous rock mass and damage mechanics theory. Int J Numer Anal Meth Geomech 12:1–30

    Google Scholar 

  • Lavrov A (2001) Kaiser effect observation in brittle rock cyclically loaded with different loading rates. Mech Mater 33:669–677

    Google Scholar 

  • Lavrov A (2003) The Kaiser effect in rocks: Principles and stress estimation techniques. Int J Rock Mech Min Sci 40:151–171

    Google Scholar 

  • Le H, Sun S, Kulatilake P, Wei J (2018) Effect of grout on mechanical properties and cracking behavior of rock-like specimens containing a single flaw under uniaxial compression. International Journal of Geomechanics 18

  • Li C, Bažant ZP, Xie H, Rahimi-Aghdam S (2019a) Anisotropic microplane constitutive model for coupling creep and damage in layered geomaterials such as gas or oil shale. Int J Rock Mech Min Sci 124:1365–1609

    Google Scholar 

  • Li C, Caner F, Chau V, Bazant Z (2017) Spherocylindrical microplane constitutive model for shale and other anisotropic rocks. J Mech Phys Solids 103:155–178

    Google Scholar 

  • Li S, Wang G (2004) On damage theory of a cohesive medium. Int J Eng Sci 42:861–885

    Google Scholar 

  • Li S, Zhang D, Bai X, Zhang X, Chu Y, Guo K (2019b) Experimental study on mechanical properties, acoustic emission energies and failure modes of pre-cracked rock materials under uniaxial compression. Pure Appl Geophys 176:4519–4532

    Google Scholar 

  • Li W, Rezakhani R, Jin C, Zhou X, Cusatis G (2016) A multiscale framework for the simulation of the anisotropic mechanical behavior of shale. International Journal for Numerical and Analytical Methods In Geomechanics 41

  • Liang Z, Tang CA, Tang SB, Xu T, Zuo YJ (2007) Characteristics of fractal and percolation of rocks subjected to uniaxial compression during their failure process. Yantu Gongcheng Xuebao/chinese Journal of Geotechnical Engineering 29:1386–1391

    Google Scholar 

  • Liang ZZ, Tang CA, Zhang YB, TANG S, MA T, (2008) Three-dimensional numerical study of direct tensile fracture of rock and associated fractal. Chin J Rock Mech Eng 27:1402–1410

    Google Scholar 

  • Liang Z, Tang C, Zhang Y, Tianhui MA, Zhang Y (2006) 3D numerical simulation of failure process of rock. Chin J Rock Mech Eng 25:931–936

    Google Scholar 

  • Louis L, Baud P, Wong T-f (2009) Microstructural inhomogeneity and mechanical anisotropy associated with bedding in Rothbach sandstone. Pure Appl Geophys 166:1063–1087

    Google Scholar 

  • Menéndez B, Zhu W, Wong T-f (1996) Micromechanics of brittle faulting and cataclastic flow in Berea sandstone. J Struct Geol 18:1–16

    Google Scholar 

  • Murakami S (1983) Notion of continuum damage mechanics and its application to anisotropic creep damage theory. J Eng Mater Technol 105:99–105

    Google Scholar 

  • Nara Y, Kaneko K (2006) Sub-critical crack growth in anisotropic rock. Int J Rock Mech Min Sci 43:437–453

    Google Scholar 

  • Nemat-Nasser S, Hori M (2013) Micromechanics: Overall properties of heterogeneous materials. Elsevier

    Google Scholar 

  • O’Connell R, Budiansky B (1974) Seismic Velocities in Dry and Saturated Cracked Solids. J Geophys Res 79:5412–5426

    Google Scholar 

  • Pan P, Feng X, Hudson JA (2012) The influence of the intermediate principal stress on rock failure behaviour: A numerical study. Eng Geol 124:109–118

    Google Scholar 

  • Pellet F, Hajdu A, Deleruyelle F, Besnus F (2005) A viscoplastic model including anisotropic damage for the time dependent behaviour of rock. Int J Numer Anal Meth Geomech 29:941–970

    Google Scholar 

  • Rizzo R, Healy D, Heap MJ, Farrell N (2018) Detecting the onset of strain localization using two-dimensional wavelet analysis on sandstone deformed at different effective pressures. Journal of Geophysical Research: Solid Earth 123:10460–10478

    Google Scholar 

  • Sadrnejad SA, Shakeri S (2017) Fabric assessment of damaged anisotropic geo-materials using the multi-laminate model. Int J Rock Mech Min Sci 91:90–103

    Google Scholar 

  • Sammis C, Ashby M (1986) The Failure of Brittle Porous Solids Under Compressive Stress States. Acta Metall 34:511–526

    Google Scholar 

  • Sayers CM, Munster JGV (1991) Microcrack-induced seismic anisotropy of sedimentary rocks. Journal of Geophysical Research: Solid Earth 96:16529–16533

    Google Scholar 

  • Sayers CM, Kachanov M (1995) Microcrack-induced elastic wave anisotropy of brittle rocks. Journal of Geophysical Research Solid Earth 100:4149–4156

    Google Scholar 

  • Sayers CM, Munster JGV, King MS (1990) Stress-induced ultrasonic anisotrophy in Berea sandstone. International Journal of Rock Mechanics & Mining Sciences & Geomechanics Abstracts 27:429–436

    Google Scholar 

  • Schubnel A, Benson P, Thompson B, Hazzard J, Young RP (2006) Quantifying damage, saturation and anisotropy in cracked rocks by inverting elastic wave velocities. Pure Appl Geophys 163:947–973

    Google Scholar 

  • Schubnel A, Guéguen Y (2003) Dispersion and anisotropy of elastic waves in cracked rocks. J Geophys Res: Solid Earth 108

  • Shao J, Chau K, Feng X-T (2006) Modeling of anisotropic damage and creep deformation in brittle rocks. Int J Rock Mech Min Sci 43:582–592

    Google Scholar 

  • Shirole D, Hedayat A, Ghazanfari E, Walton G (2020) Evaluation of an ultrasonic method for damage characterization of brittle rocks. Rock Mech Rock Eng 53:2077–2094

    Google Scholar 

  • Stevens J, Holcomb D (1980) Theoretical investigation of the sliding crack model of dilatancy. J Geophys Res 85:7091–7100

    Google Scholar 

  • Stuart CE, Meredith PG, Murrell SAF, Munster JGV (1993) Anisotropic crack damage and stress-memory effects in rocks under triaxial loading. International Journal on Rock Mechanics and Mining Sciences & Geomechanical Abstracts 30:937–941

    Google Scholar 

  • Tang C (1997) Numerical simulation of progressive rock failure and associated seismicity. Int J Rock Mech Min Sci 34:249–261

    Google Scholar 

  • Wu X, Baud P, Wong T-f (2000) Micromechanics of compressive failure and spatial evolution of anisotropic damage in Darley Dale sandstone. Int J Rock Mech Min Sci 37:143–160

    Google Scholar 

  • Xu L, Gong F, Luo S (2021a) Effects of pre-existing single crack angle on mechanical behaviors and energy storage characteristics of red sandstone under uniaxial compression. Theoret Appl Fract Mech 113:102933

    Google Scholar 

  • Xu T, P.G R, Wasantha PLP, Zhao J, Tang CA, Zhu W, (2013) Influence of the geometry of partially-spanning joints on mechanical properties of rock in uniaxial compression. Eng Geol 167:134–147

    Google Scholar 

  • Xu T, Tang CA, Zhao J, Li LC, Heap MJ (2012) Modelling the time-dependent rheological behaviour of heterogeneous brittle rocks. Geophys J Int 189:1781–1796

    Google Scholar 

  • Xu T, Yuan Y, Heap MJ, Zhou G-L, Perera MSA, Ranjith PG (2021b) Microwave-assisted damage and fracturing of hard rocks and its implications for effective mineral resources recovery. Minerals Engineering 160

  • Xue Y, Xu T, Zhu W, Heap MJ, Heng Z, Wang X (2021) Full-field quantification of time-dependent and -independent deformation and fracturing of double-notch flawed rock using digital image correlation. Geomechanics and Geophysics for Geo-Energy and Geo-Resources 7

  • Yang S-Q, Jing H-W (2011) Strength failure and crack coalescence behavior of brittle sandstone samples containing a single fissure under uniaxial compression. Int J Fract 168:227–250

    Google Scholar 

  • Yuan Y, Xu T, Heap MJ, Meredith PG, Yang T, Zhou G (2021) A three-dimensional mesoscale model for progressive time-dependent deformation and fracturing of brittle rock with application to slope stability. Computers and Geotechnics 135

  • Zhao XG, Wang J, Cai M, Su G (2021) Influence of intermediate principal stress on the strainburst characteristics of beishan granite with consideration of end effect. Rock Mech Rock Eng 54:4771–4791

    Google Scholar 

  • Zhenlong S, Yin G, P.G R, Li M, Huang J, Liu C, (2019) Influence of the intermediate principal stress on sandstone failure. Rock Mech Rock Eng 52:3033–3046

    Google Scholar 

  • Zhou GL, Xu T, Heap MJ, Meredith PG, Mitchell TM, Sesnic ASY, Yuan Y (2020) A three-dimensional numerical meso-approach to modeling time-independent deformation and fracturing of brittle rocks. Comput Geotech 117:103274

    Google Scholar 

  • Zienkiewicz O, Pande G (1977) Time dependent multilaminate model of rocks—A numerical study of deformation and failure of rock masses. Int J Numer Anal Meth Geomech 1:219–247

    Google Scholar 

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Acknowledgements

The work was jointly supported by National Natural Science Foundation of China (42172312, 51974062, and 52211540395) and a Royal Society-Newton Mobility Grant (IEC\NSFC\170625). M.J. Heap acknowledges support from the Institut Universitaire de France (IUF).

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Authors and Affiliations

  1. Center for Rock Instability and Seismicity Research, Northeastern University, Shenyang, 110819, China

    Yang Yuan & Tao Xu

  2. Department of Earth Sciences, University College London, London, WC1E 6BT, UK

    Philip G. Meredith, Thomas M. Mitchell & Ashley Stanton-Yonge Sesnic

  3. Institut Terre et Environnement de Strasbourg, UMR 7063, Université de Strasbourg, CNRS, 5 Rue René Descartes, 67084, Strasbourg, France

    Michael J. Heap

  4. Institut Universitaire de France (IUF), 1 Rue Descartes, 75231, Paris, France

    Michael J. Heap

  5. College of Energy and Mining Engineering, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Guanglei Zhou

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Yuan, Y., Xu, T., Meredith, P.G. et al. A Microplane-Based Anisotropic Damage Model for Deformation and Fracturing of Brittle Rocks. Rock Mech Rock Eng 56, 6219–6235 (2023). https://doi.org/10.1007/s00603-023-03363-7

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