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Study on the micromechanical and crack characteristics of granite based on nanoindentation test and discrete element method

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Abstract

Exploring the mechanical properties and crack characteristics of granite at the grain scale is of greatly significant to understand brittle failures, such as spalling, slabbing, and rockburst of deep-buried hard rock under high geostress. The macroscopic engineering failure of a rock mass is often closely related to the microscopic mechanical properties and microstructure of the constituent minerals. This study derived the microscopic mechanical properties of granite minerals, including Young’s modulus, hardness, fracture toughness, and stiffness ratio based on nanoindentation tests. The relationship of the micromechanical parameters including Young’s modulus, hardness, and fracture toughness is presented in the following order: quartz > K-feldspar > plagioclase > biotite. A parameter calibration process that combines nanoindentation test and trial-and-error method was then proposed to reduce the randomness in the calibration process. This calibration process was adopted to the discrete element method simulation of granite, in which the microstructure of granite is specifically defined through a Voronoi tessellation. Finally, the microcrack evolution and crack characteristics of different minerals in granite were discussed based on the micro-X-ray computed tomography, scanning electron microscopy, and numerical results. The results reveal that the intragranular cracks play a crucial role in the failure process of brittle rocks and largely dominate the macroscopic properties of materials, in which the percentage of intragranular cracks increases from 61% to more than 80% when the compression test changes to the tension test.

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Data availability

Data will be made available on request.

Abbreviations

\(a\) :

Half-length of the precrack in the CSTBD specimen

\(A_{{\text{c}}}\) :

Contact area of the indenter on the sample

\(A_{\max }\) :

Maximum contact area of the indenter on the sample

\(B\) :

Thickness of the CSTBD specimen

\(d(p_{{\text{i}}} ,p_{{\text{j}}} )\) :

Euclidean distance

\(E\) :

Young’s modulus

\(E_{{\text{r}}}\) :

Reduced modulus

\(E^{*}\) :

Linear elastic modulus

\(\overline{E}^{*}\) :

Bond effective modulus

\(E_{{\text{i}}}\) :

Young’s modulus of the indenter

\(f_{{\text{I}}}\) :

Mode I normalized stress intensity factor

\(F_{\max }\) :

Maximum tensile force

\(G_{{\text{c}}}\) :

Critical energy release rate

\(G\) :

Shear modulus

\(h\) :

Indentation depth

\(h_{{\text{c}}}\) :

Indentation contact depth

\(h_{{\text{f}}}\) :

Residual indentation depth

\(h_{{\text{m}}}\) :

Maximum indentation depth

\(H\) :

Hardness

\(K\) :

Bulk modulus

\(K_{{{\text{{\rm I}C}}}}\) :

Tensile (mode I) fracture toughness

\(k_{{\text{n}}}\) :

Normal stiffness

\(k_{{\text{s}}}\) :

Shear stiffness

\(k_{{{\text{ratio}}}}\) :

Normal-to-shear stiffness ratio

\(\overline{k}_{{\text{n}}} /\overline{k}_{{\text{s}}}\) :

Bond normal-to-shear stiffness ratio

\(P\) :

Applied loading force of the indenter

\(P_{\max }\) :

Peak force of the indenter

\(R\) :

Radius of the CSTBD specimen

\(R_{\min }\) :

Minimum radius of the particle

\(R_{\max } /R_{\min }\) :

Maximum-to-minimum radius ratio

\(S\) :

Contact stiffness of the sample

\(t\) :

Thickness of the contact interface

\(U_{{{\text{crack}}}}\) :

Fracture energy

\(U_{{\text{e}}}\) :

Recoverable elastic energy

\(U_{{{\text{pp}}}}\) :

Pure plasticity energy

\(U_{{\text{t}}}\) :

Total energy

\(\nu\) :

Poisson’s ratio of the sample

\(\nu_{{\text{i}}}\) :

Poisson’s ratios of the indenter

\(\overline{c}\) :

Cohesion strength

\(\overline{\phi }\) :

Friction angle

\(\overline{\lambda }\) :

Radius multiplier

\(\sigma_{c}\) :

Uniaxial compressive strength

\(\sigma_{t}\) :

Brazilian tensile strength

\(\overline{\sigma }_{{\text{c}}}\) :

Tensile strength

\(\sigma_{c} /\sigma_{{\text{t}}}\) :

UCS-to-BTS ratio

\(\varepsilon_{{\text{p}}}\) :

Peak strain

\(\rho\) :

Density

\(\mu\) :

Friction coefficient

\({\mathbb{R}}^{2}\) :

Two-dimensional Euclidean plane

References

  1. Feng XT, Zhou YY, Jiang Q (2019) Rock mechanics contributions to recent hydroelectric developments in China. J Rock Mech Geotech Eng 11(3):511–526

    Article  Google Scholar 

  2. Xu DP, Huang X, Jiang Q, Li SJ, Zheng H, Qiu SL, Xu HS, Li YH, Li ZG, Ma XD (2021) Estimation of the three-dimensional in situ stress field around a large deep underground cavern group near a valley. J Rock Mech Geotech Eng 13(3):529–544

    Article  Google Scholar 

  3. Xia YL, Feng XT, Yang CX, Li SJ, Xu DP, Jiang Q, Kong R, Li YH, Li ZG (2023) Mechanism of excavation-induced cracking of the protective layer of a rock bench in a large underground powerhouse under high tectonic stress. Eng Geol 312:106951

    Article  Google Scholar 

  4. Li HB, Yang YG, Zhang XB, Zhou JW (2017) Deformation and failure analyses of large underground caverns during construction of the Houziyan Hydropower Station, Southwest China. Eng Fail Anal 88:164–185

    Article  Google Scholar 

  5. Zhou J, Lan HX, Zhang LQ, Yang DX, Song J, Wang S (2019) Novel grain-based model for simulation of brittle failure of Alxa porphyritic granite. Eng Geol 251(9):100–114

    Article  Google Scholar 

  6. Eberhardt E, Stimpson B, Stead D (1999) Effects of grain size on the initiation and propagation thresholds of stress-induced brittle fractures. Rock Mech Rock Eng 32(2):81–99

    Article  Google Scholar 

  7. Xia K, Nasseri MHB, Mohanty B, Lu F, Chen R, Luo SN (2008) Effects of microstructures on dynamic compression of Barre granite. Int J Rock Mech Min Sci 45(6):879–887

    Article  Google Scholar 

  8. Shao JF, Chau KT, Feng XT (2006) Modeling of anisotropic damage and creep deformation in brittle rocks. Int J Rock Mech Min Sci 43(4):582–592

    Article  Google Scholar 

  9. Bahrani N, Kaiser PK (2016) Numerical investigation of the influence of specimen size on the unconfined strength of defected rocks. Comput Geotech 77:56–67

    Article  Google Scholar 

  10. Hajiabdolmajid V, Kaiser PK, Martin CD (2002) Modelling brittle failure of rock. Int J Rock Mech Min Sci 39(6):731–741

    Article  Google Scholar 

  11. Martin CD, Chandler NA (1994) The progressive fracture of Lac du Bonnet granite. Int J Rock Mech Min Sci Geomech Abstr 31(6):643–659

    Article  Google Scholar 

  12. Peng J, Wong LNY, Teh CI (2017) Influence of grain size heterogeneity on strength and microcracking behavior of crystalline rocks. J Geophys Res Solid Earth 122(2):1054–1073

    Article  Google Scholar 

  13. Saadat M, Taheri A (2020) A cohesive grain based model to simulate shear behaviour of rock joints with asperity damage in polycrystalline rock. Comput Geotech 117:103254

    Article  Google Scholar 

  14. Wong TF (1982) Micromechanics of faulting in westerly granite. Int J Rock Mech Min Sci Geomech Abstr 19(2):49–64

    Article  MathSciNet  Google Scholar 

  15. Cai M, Kaiser PK, Tasaka Y, Maejima T, Morioka H, Minami M (2004) Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. Int J Rock Mech Min Sci 41(5):833–847

    Article  Google Scholar 

  16. Jia LC, Chen M, Zhang W, Xu T, Zhou Y, Hou B, Jin Y (2013) Experimental study and numerical modeling of brittle fracture of carbonate rock under uniaxial compression. Mech Res Commun 50:58–62

    Article  Google Scholar 

  17. Lan HX, Martin CD, Hu B (2010) Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading. J Geophys Res Solid Earth 115:B01202

    Article  Google Scholar 

  18. Wang X, Cai M (2018) Modeling of brittle rock failure considering inter-and intra-grain contact failures. Comput Geotech 101:224–244

    Article  Google Scholar 

  19. Tang XH, Zhang YH, Xu JJ, Rutqvist J, Hu MS, Wang ZZ, Liu QS (2022) Determining Young’s modulus of granite using accurate grain-based modeling with microscale rock mechanical experiments. Int J Rock Mech Min Sci 157:105167

    Article  Google Scholar 

  20. Akono AT, Kabir P, Shi ZF, Fuchs S, Tsotsis TT, Jessen K, Werth CJ (2019) Modeling CO2–induced alterations in Mt. Simon sandstone via nanomechanics. Rock Mech Rock Eng 52:1353–1375

    Article  Google Scholar 

  21. Constantinides G, Chandran KSR, Ulm FJ, Van-Vliet KJ (2006) Grid indentation analysis of composite microstructure and mechanics: principles and validation. Mater Sci Eng A 430(1–2):189–202

    Article  Google Scholar 

  22. Liu XY, Xu DP, Li SJ, Qiu SL, Jiang Q (2023) An insight into the mechanical and fracture characterization of minerals and mineral interfaces in granite using nanoindentation and micro X-ray computed tomography. Rock Mech Rock Eng 56:3359–3375

    Article  Google Scholar 

  23. Luo SM, Lu YH, Wu YK, Song JL, DeGroot DJ, Jin Y, Zhang GP (2020) Cross-scale characterization of the elasticity of shales: Statistical nanoindentation and data analytics. J Mech Phys Solids 140:103945

    Article  Google Scholar 

  24. Wu YK, Li YC, Luo SM, Lu M, Zhou N, Wang DF, Zhang GP (2020) Multiscale elastic anisotropy of a shale characterized by cross-scale big data nanoindentation. Int J Rock Mech Min Sci 134:104458

    Article  Google Scholar 

  25. Xu JJ, Tang XH, Wang ZZ, Feng YF, Bian K (2020) Investigating the softening of weak interlayers during landslides using nanoindentation experiments and simulations. Eng Geol 277:105801

    Article  Google Scholar 

  26. Yang C, Xiong YQ, Wang JF, Li Y, Jiang WM (2020) Mechanical characterization of shale matrix minerals using phase–positioned nanoindentation and nano–dynamic mechanical analysis. Int J Coal Geol 229:103571

    Article  Google Scholar 

  27. Ganneau FP, Constantinides G, Ulm FJ (2006) Dual-indentation technique for the assessment of strength properties of cohesive-frictional materials. Int J Solids Struct 43(6):1727–1745

    Article  MATH  Google Scholar 

  28. Palkovic SD, Kupwade-Patil K, Yip S, Büyüköztürk O (2018) Random field finite element models with cohesive–frictional interactions of a hardened cement paste microstructure. J Mech Phys Solids 119:349–368

    Article  Google Scholar 

  29. Xu JJ, Zhang YH, Rutqvist J, Hu MS, Wang ZZ, Tang XH (2023) Thermally induced microcracks in granite and their effect on the macroscale mechanical behavior. J Geophys Res Solid Earth 128(1)

  30. Saadat M, Taheri A, Kawamura Y (2021) Investigating asperity damage of natural rock joints in polycrystalline rocks under confining pressure using grain-based model. Comput Geotech 135:104144

    Article  Google Scholar 

  31. Zhang T, Yu LY, Li J, Ma LJ, Su HJ, Zhang MW, Xu XL, Peng YX (2022) Numerical investigation of the effects of the micro-parameters of the transgranular contact on the mechanical response of granite. Theor Appl Fract Mech 118:103259

    Article  Google Scholar 

  32. Quey R, Dawson PR, Barbe F (2011) Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput Methods Appl Mech Eng 200(17):1729–1745

    Article  MATH  Google Scholar 

  33. Wang ZH, Yang SL, Li LH, Tang YS, Xu GL (2021) A 3D Voronoi clump based model for simulating failure behavior of brittle rock. Eng Fract Mech 248(1):107720

    Article  Google Scholar 

  34. Hu XJ, Gong XN, Hu HB, Guo PP, Ma JJ (2022) Cracking behavior and acoustic emission characteristics of heterogeneous granite with double pre-existing filled flaws and a circular hole under uniaxial compression: insights from grain-based discrete element method modeling. Bull Eng Geol Environ 81:162

    Article  Google Scholar 

  35. Chen S, Yue ZQ, Tham LG, Lee PKK (2004) Modeling of the indirect tensile test for inhomogeneous granite using a digital image-based numerical method. Int J Rock Mech Min Sci 41:447–447

    Article  Google Scholar 

  36. Chen S, Yue ZQ, Tham LG (2004) Digital image-based numerical modeling method for prediction of inhomogeneous rock failure. Int J Rock Mech Min Sci 41:939–957

    Article  Google Scholar 

  37. Yue ZQ, Chen S, Tham LG (2003) Finite element modeling of geomaterials using digital image processing. Comput Geotech 30:375–397

    Article  Google Scholar 

  38. Tan X, Konietzky H, Chen W (2016) Numerical simulation of heterogeneous rock using discrete element model based on digital image processing. Rock Mech Rock Eng 49:4957–4964

    Article  Google Scholar 

  39. Hu XJ, Xie N, Zhu QZ, Chen L, Li PC (2020) Modeling damage evolution in heterogeneous granite using digital image-based grain-based model. Rock Mech Rock Eng 53(11):4925–4945

    Article  Google Scholar 

  40. Li XF, Li HB, Zhang QB, Jiang JL, Zhao J (2018) Dynamic fragmentation of rock material: characteristic size, fragment distribution and pulverization law. Eng Fract Mech 199:739–759

    Article  Google Scholar 

  41. Li XF, Zhang QB, Li HB, Zhao J (2018) Grain-based discrete element method (GB-DEM) modelling of multiscale fracturing in rocks under dynamic loading. Rock Mech Rock Eng 51:3785–3817

    Article  Google Scholar 

  42. Zhang YH, Wong LNY, Chan KK (2019) An extended grain-based model accounting for microstructures in rock deformation. J Geophys Res Solid Earth 124:125–148

    Article  Google Scholar 

  43. Mahabadi OK, Randall NX, Zong Z, Grasselli G (2012) A novel approach for micro-scale characterization and modeling of geomaterials incorporating actual material heterogeneity. Geophys Res Lett 39(1):1303

  44. Wu ZJ, Ji XK, Liu QS, Fan LF (2020) Study of microstructure effect on the nonlinear mechanical behavior and failure process of rock using an image-based-FDEM model. Comput Geotech 121:103480

    Article  Google Scholar 

  45. Itasca Consulting Group Inc (2016) Particle flow code (PFC), version 5.0. Minneapolis

  46. Sun CL, Li GC, Gomah ME, Xu JH, Rong HY (2020) Meso–scale mechanical properties of mudstone investigated by nanoindentation. Eng Fract Mech 238:107245

    Article  Google Scholar 

  47. Oliver WC, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 7:1564–1583

    Article  Google Scholar 

  48. Manjunath GL, Jha B (2019) Geomechanical characterization of Gondwana shale across nano-micro-meso scales. Int J Rock Mech Min Sci 119:35–45

    Article  Google Scholar 

  49. Cheng YT, Li ZY, Cheng CM (2002) Scaling relationships for indentation measurements. Philos Mag A 82:1821–1829

    Article  Google Scholar 

  50. Zhang GP, Wei ZX, Ferrell RE (2009) Elastic modulus and hardness of muscovite and rectorite determined by nanoindentation. Appl Clay Sci 43:271–281

    Article  Google Scholar 

  51. Liu KQ, Ostadhassan M (2017) Microstructural and geomechanical analysis of Bakken shale at nanoscale. J Pet Sci Eng 153:133–144

    Article  Google Scholar 

  52. Pöhl F (2019) Pop-in behavior and elastic-to-plastic transition of polycrystalline pure iron during sharp nanoindentation. Sci Rep 9:1564–1583

    Article  Google Scholar 

  53. Gao FQ, Stead D, Elmo D (2016) Numerical simulation of microstructure of brittle rock using a grain-breakable distinct element grain-based model. Comput Geotech 78:203–217

    Article  Google Scholar 

  54. Li XF, Li HB, Liu LW, Liu YQ, Ju MH, Zhao J (2020) Investigating the crack initiation and propagation mechanism in brittle rocks using grain-based finite-discrete element method. Int J Rock Mech Min Sci 127:104219

    Article  Google Scholar 

  55. Peng J, Wong LNY, Teh CI, Li Z (2018) Modeling micro-cracking behavior of Bukit Timah granite using grain-based model. Rock Mech Rock Eng 51(1):135–154

    Article  Google Scholar 

  56. Yılmaz NG, Goktan RM, Kibici Y (2011) Relations between some quantitative petrographic characteristics and mechanical strength properties of granitic building stones. Int J Rock Mech Min Sci 48(3):506–513

    Article  Google Scholar 

  57. Potyondy DO (2010) A grain-based model for rock: approaching the true microstructure, In: Proceedings of the rock mechanics in the Nordic Countries. Kongsberg, vol 9–12, pp 225–234

  58. Liu G, Cai M (2020) Modeling time-dependent deformation behavior of brittle rock using grain-based stress corrosion method. Comput Geotech 118:103323

    Article  Google Scholar 

  59. Farahmand K, Diederichs MS (2021) Calibration of coupled hydro-mechanical properties of grain-based model for simulating fracture process and associated pore pressure evolution in excavation damage zone around deep tunnels. J Rock Mech Geotech Eng 13(1):60–83

    Article  Google Scholar 

  60. Hofmann H, Babadagli T, Yoon JS, Zang A (2015) A grain based modeling study of mineralogical factors affecting strength, elastic behavior and micro fracture development during compression tests in granites. Eng Fract Mech 147:261–275

    Article  Google Scholar 

  61. Mayer JM, Stead D (2017) Exploration into the causes of uncertainty in UDEC grain boundary models. Comput Geotech 82:110–123

    Article  Google Scholar 

  62. Liang K, Xie LZ, He B, Zhao P, Zhang Y, Hu WZ (2021) Effects of grain size distributions on the macro-mechanical behavior of rock salt using micro-based multiscale methods. Int J Rock Mech Min Sci 138:104592

    Article  Google Scholar 

  63. Mavko G, Mukerji T, Dvorkin J (2009) The rock physics handbook. Cambridge University Press, Cambridge

    Book  Google Scholar 

  64. Yang SQ, Huang YH, Jing HW, Liu XR (2014) Discrete element modeling on fracture coalescence behavior of red sandstone containing two unparallel fissures under uniaxial compression. Eng Geol 178:28–48

    Article  Google Scholar 

  65. Liu WJ, Zhu XH, Shi CS (2020) On the failure mechanism of brittle granite in 2-D rock indentation. Geotech Test J 43(5):1–9

    Article  Google Scholar 

  66. Cho N, Martin CD, Sego DC (2007) A clumped particle model for rock. Int J Rock Mech Min Sci 44(7):997–1010

    Article  Google Scholar 

  67. Zhou J, Zhang LQ, Yang DX, Braun A, Han ZH (2017) Investigation of the quasi-brittle failure of Alashan granite viewed from laboratory experiments and grain-based discrete element modeling. Mater 10:835

    Article  Google Scholar 

  68. Bahrani N, Kaiser PK, Valley B (2014) Distinct element method simulation of an analogue for a highly interlocked, non-persistently jointed rockmass. Int J Rock Mech Min Sci 71:117–130

    Article  Google Scholar 

  69. Moore DE, Lockner DA (1995) The role of microcracking in shear-fracture propagation in granite. J Struct Geol 17(1):95–111

    Article  Google Scholar 

  70. Azar HF, Choupani N, Afshin H, Moghadam RH (2015) Effect of mineral admixtures on the mixed-mode (I/II) fracture characterization of cement mortar: CTS, CSTBD and SCB specimens. Eng Fract Mech 134:20–34

    Article  Google Scholar 

  71. Zhou L, Gao WT, Yu LY, Zhu ZM, Chen JX, Wang XK (2022) Thermal effects on fracture toughness of cracked straight-through Brazilian disk green sandstone and granite. J Rock Mech Geotech Eng 14(5):1447–1460

    Article  Google Scholar 

  72. Brace WF, Paulding BW, Scholz C (1966) Dilatancy in the fracture of crystalline rocks. J Geophys Res 71(16):3939–3953

    Article  Google Scholar 

  73. Martin CD (1993) The strength of massive Lac du Bonnet granite around underground openings. Ph.D. thesis, University of Manitoba, Manitoba, Canada

  74. Kotwaliwale N, Weckler PR, Brusewitz GH (2006) X-ray attenuation coefficients using polychromatic X-ray imaging of pecan components. Biosyst Eng 94(2):199–206

    Article  Google Scholar 

  75. Fujii Y, Takemura T, Takahashi M, Lin WR (2007) Surface features of uniaxial tensile fractures and their relation to rock anisotropy in Inada granite. Int J Rock Mech Min Sci 44:98–107

    Article  Google Scholar 

  76. Mahabadi OK, Tatone BSA, Grasselli G (2014) Influence of microscale heterogeneity and microstructure on the tensile behavior of crystalline rocks. J Geophys Res-Solid Earth 119:5324–5341

    Article  Google Scholar 

  77. Nasseri MHB, Rezanezhad F, Young RP (2011) Analysis of fracture damage zone in anisotropic granitic rock using 3D X-ray CT scanning techniques. Int J Fract 168:1–13

    Article  Google Scholar 

  78. Wong TF, Biegel R (1985) Effects of pressure on the micromechanics of faulting in San Marcos gabbro. J Struct Geol 7(6):737–749

    Article  Google Scholar 

  79. Tapponnier P, Brace WF (1976) Development of stress-induced microcracks in Westerly granite. Int J Rock Mech Min Sci Geomech Abstr 13(4):103–112

    Article  Google Scholar 

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Acknowledgements

This research as supported by the National Natural Science Foundation of China under Grant Nos. 51979268, 52279117, and 52279114.

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Authors and Affiliations

  1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, Hubei, China

    Xiu-yang Liu, Ding-ping Xu, Huai-sheng Xu, Guang-liang Feng, Shi-li Qiu & Quan Jiang

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiu-yang Liu & Huai-sheng Xu

  3. School of Hydraulic and Civil Engineering, Zhengzhou University, Zhengzhou, 450001, Henan, China

    Shu-qian Duan

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Liu, Xy., Xu, Dp., Duan, Sq. et al. Study on the micromechanical and crack characteristics of granite based on nanoindentation test and discrete element method. Comp. Part. Mech. (2023). https://doi.org/10.1007/s40571-023-00664-0

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