Skip to main content
SpringerLink
Log in

A New Perspective on the Constant mi of the Hoek–Brown Failure Criterion and a New Model for Determining the Residual Strength of Rock

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

The constant mi is a fundamental parameter required in the Hoek–Brown (H–B) failure criterion for estimating rock strength. Triaxial tests for the calculation of this constant are time-consuming and expensive. In this paper, a new perspective is presented for the physical meaning of the parameter mi of the H–B model in consideration of the contact friction coefficient. A correlation between the contact friction coefficient and mi is established using published data. A practical approach is proposed to determine the value of the parameter mi. In addition, previously proposed methods of estimating the residual strength of rock are reviewed. A novel method based on the H–B failure criterion is established to predict the residual strength of rock. The single model parameter used in the new model controls the residual strength nonlinearity. The strengths predicted by the proposed method for limestone, granite, slate, and sandstone are in good agreement with those measured during laboratory tests. The corresponding errors are within a range of 15%. This method is applied to an actual rock mass around a tunnel in the Hanjiang to Weihe River Project of China.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

σ 1 :

Major minor principal stresses

σ 3 :

Minor principal stresses

UCS:

Unconfined compressive strength of intact rock

m i :

Material constant for intact rock

\(\kappa\) :

Coefficient for mixed mode fracture

\(\left| {\sigma_{t} } \right|\) :

Uniaxial tensile strength

F n :

Thrust force

F t :

Torque force

θ :

Contact friction angle between fractured rock faces

H :

Slope of Ft–Fn linear curve

a :

Rake angle of bit (geometric parameter of bit)

\(\varphi^{\prime}\) :

Friction angle between the intact rock and rock clastic

\(\varphi\) :

Internal friction angle

\(\mu\) :

Friction coefficient

\(\phi\) :

Crack surface friction angle

σ r :

Rock residual strength

σ cr :

Residual strength at the confining pressure of zero

b :

Dimensionless parameter

GSIr and mir :

Two model parameters in the GSI-softening model

λ, η and γ :

Dimensionless parameter

References

  • Arzúa J, Alejano LR (2013) Dilation in granite during servo-controlled triaxial strength tests. Int J Rock Mech Min Sci 61(1):43–56

    Google Scholar 

  • Arzúa J, Alejano LR, Walton G (2014) Strength and dilation of jointed granite specimens in servo- controlled triaxial tests. Int J Rock Mech Min Sci 69:93–104

    Google Scholar 

  • Basarir H, Ozsan A, Karakus M (2005) Analysis of support requirements for a shallow diversion tunnel at Guledar dam site, Turkey. Eng Geol 81:131–145

    Google Scholar 

  • Brown ET (1970) Strength of models of rock with intermittent joints. J Soil Mech Found Div 96(SM6):1935–1949

    Google Scholar 

  • Cai M, Kaiser PK, Tasaka Y, Minami M (2007) Determination of residual strength parameters of jointed rock masses using the GSI system. Int J Rock Mech Min Sci 44(2):247–265

    Google Scholar 

  • Cai M, Kaiser PK, Uno H, Tasaka Y, Minami M (2004) Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system. Int J Rock Mech Min Sci 41(1):3–19

    Google Scholar 

  • Crowder JJ, Bawden WF (2004) Review of post-peak parameters and behaviour of rock masses: current trends and research. RocNews Fall:13

    Google Scholar 

  • Cui L, Zheng J, Dong Y, Zhang B, Wang A (2017) Prediction of critical strains and critical support pressures for circular tunnel excavated in strain-softening rock mass. Eng Geol 224:43–61

    Google Scholar 

  • Cuss RJ, Rutter EH, Holloway RF (2003) The application of critical state soil mechanics to the mechanical behaviour of porous sandstones. Int J Rock Mech Min Sci 40(6):847–862

    Google Scholar 

  • Dagrain F (2001) Influence of the cutter geometry in rock cutting: an experimental approach. MS thesis. University of Minnesota

  • Detournay E, Defourny P (1992) A phenomenological model for the drilling action of drag bits. Int J Rock Mech Min Sci 29:13–23

    Google Scholar 

  • Du XZ (2015) Hanjiang-to-Weihe river water diversion project construction management practices. CHN Water Res 14:42–45 (in Chinese)

    Google Scholar 

  • Gao FQ, Kang HP (2016) Effects of pre-existing discontinuities on the residual strength of rock mass-insight from a discrete element method simulation. J Struct Geol 85:40–50

    Google Scholar 

  • Gerbaud L, Menand S, Sellami H (2006) PDC bits: all comes from the cutter rock interaction. In: Proceedings of the IADC/SPE drilling conference. Society of Petroleum Engineers; no. SPE-98988-MS

  • Griffith AA (1924) Theory of rupture. In: Proceedings of the 1st international congress on applied mechanics, Delft, The Netherlands, pp 55–63

  • Gowd TN, Rummel F (1980) Effect of confining pressure on the fracture behaviour of a porous rock. Int J Rock Mech Min Sci 17(4):225–229

    Google Scholar 

  • Hajiabdolmajid V, Kaiser PK, Martin CD (2002) Modelling brittle failure of rock. Int J Rock Mech Min Sci 39:431–441

    Google Scholar 

  • He MM, Ning L, Zhang ZQ, Yao XC, Chen YS, Zhu CH (2019a) An empirical method for determining the mechanical properties of jointed rock mass using drilling energy. Int J Rock Mech Min Sci 116:64–74

    Google Scholar 

  • He MM, Zhang ZQ, Ren J, Huan JY, Li GF, Chen YS, Li N (2019b) Deep convolutional neural network for fast determination of the rock strength parameters using drilling data. Int J Rock Mech Min Sci 123:104084. https://doi.org/10.1016/j.ijrmms.2019.104084

    Article  Google Scholar 

  • He MM, Li N, Zhu J, Chen YS (2020) Advanced prediction for field strength parameters of rock using drilling operational data from impregnated diamond bit. J Petrol Sci Eng 187:106847

    Google Scholar 

  • Hoek E (1965) Rock fracture under static stress conditions. CSIR report MEG, Pretoria, South Africa, p 383

  • Hoek E, Brown ET (1980a) Underground excavations in rock. Institution of Mining and Metallurgy, London

    Google Scholar 

  • Hoek E, Brown ET (1980b) Empirical strength criterion for rock masses. J Geotech Eng Div 106(GT9):1013–1035

    Google Scholar 

  • Hoek E (1983) 23rd Rankine-lecture—strength of jointed rock masses. Geotechnique 33:185–223

    Google Scholar 

  • Hoek E, Brown ET (2019) The Hoek–Brown failure criterion and GSI 2018 edition. J Rock Mech Geotech Eng 11:445–463

    Google Scholar 

  • Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34(8):1165–1186

    Google Scholar 

  • Hudson JA, Crouch SL, Fairhurst C (1972) Soft, stiff and servo-controlled testing machines: a review with reference to rock failure. Eng Geol 6(3):155–189

    Google Scholar 

  • Jiang XY, Cui P, Liu CZ (2016) A chart-based seismic stability analysis method for rock slopes using Hoek–Brown failure criterion. Eng Geol 209(15):196–208

    Google Scholar 

  • Joseph TG (2000) Estimation of the post-failure stiffness of rock. Ph.D. thesis, University of Alberta, Edmonton, Alberta, p 540

  • Kalantari S, Hashemolhosseini H, Baghbanan A (2018) Estimating rock strength parameters using drilling data. Int J Rock Mech Min Sci 104:45–52

    Google Scholar 

  • Kerisel J (1975) Cours de mécanique des sols. Ecole National Des Ponts Et Chaussées, Paris (in French)

    Google Scholar 

  • Kumar R, Sharma KG, Varadarajan A (2010) Post-peak response of some metamorphic rocks of India under high confining pressures. Int J Rock Mech Min Sci 47(8):1357–1362

    Google Scholar 

  • Labrie D (2017) Frictional properties of rocks as a function of rock type, specimen size and confining pressure. In: The 51st US rock mechanics symposium. American Rock Mechanics Association

  • Labuz JF, Zang A (2012) Mohr–Coulomb failure criterion. In: Ulusay R (ed) The ISRM suggested methods for rock characterization, testing and monitoring: 2007–2014. Springer, Berlin, pp 227–231

    Google Scholar 

  • Ma L, Xu H, Tong Q, Dong L, Zhang N, Li J (2014) Post-yield plastic frictional parameters of a rock salt using the concept of mobilized strength. Eng Geol 177:25–31

    Google Scholar 

  • Martin CD (1997) Seventeenth Canadian geotechnical colloquium: the effect of cohesion loss and stress path on brittle rock strength. Can Geotech J 34(5):698–725

    Google Scholar 

  • Masoumi H (2013) Investigation into the mechanical behaviour of intact rock at different sizes. In: Ph.D. thesis, University of New South Wales, Sydney, Australia

  • Mclintock FA, Walsh JB (1962) Friction on Griffith cracks in rocks under pressure. In: Proceedings of the 4th US national congress of applied mechanics. American Society of Mechanical Engineers, Berkeley, USA. New York, pp 1015–1021

  • Mehmet S (2012) An improved method of fitting experimental data to the Hoek–Brown failure criterion. Eng Geol 127:27–35

    Google Scholar 

  • Munoz H, Taheri A, Chanda EK (2016) Rock drilling performance evaluation by an energy dissipation based rock brittleness index. Rock Mech Rock Eng 49(8):3343–3355

    Google Scholar 

  • Pariseau WG (2007) Fitting failure criteria to laboratory strength tests. Int J Rock Mech Min Sci 44:637–646

    Google Scholar 

  • Peng J, Cai M, Rong G, Yao MD, Jiang QH, Zhou CB (2017) Determination of confinement and plastic strain dependent post-peak strength of intact rocks. Eng Geol 218:187–196

    Google Scholar 

  • Peng J, Rong G, Cai M, Zhou CB, Du W (2013) Simulating brittle failure of rocks by a new strain-softening model. In: Paper 715 Proceedings of the 47th US rock mechanics/geomechanics symposium, San Francisco, CA, USA

  • Peng J, Rong G, Cai M, Wang XJ, Zhou CB (2014) An empirical failure criterion for intact rocks. Rock Mech Rock Eng 47(2):347–356

    Google Scholar 

  • Peng J, Cai M (2019) A cohesion loss model for determining residual strength of intact rocks. Int J Rock Mech Min Sci 119:131–139

    Google Scholar 

  • Renani HR, Martin CD (2018) Cohesion degradation and friction mobilization in brittle failure of rocks. Int J Rock Mech Min Sci 106:1–13

    Google Scholar 

  • Richard T, Dagrain F, Poyol E, Detournay E (2012) Rock strength determination from scratch tests. Eng Geol 147–148:91–100

    Google Scholar 

  • Sari M (2012) An improved method of fitting experimental data to the Hoek–Brown failure criterion. Eng Geol 127:27–35

    Google Scholar 

  • Sun CW, Chai JR, Xu ZG, Qin Y, Chen XZ (2016) Stability charts for rock mass slopes based on the Hoek–Brown strength reduction technique. Eng Geol 214(30):94–106

    Google Scholar 

  • Taheri A, Qao Q, Chanda E (2016) Drilling penetration rate estimation using rock drillability characterization index. J Inst Eng Ser D 97(2):159–170

    Google Scholar 

  • Teale R (1965) The concept of specific energy in rock drilling. Int J Rock Mech Min Sci 2:57–73

    Google Scholar 

  • Vassilis M, Trevor GC (2018) Maintaining geological reality in application of GSI for design of engineering structures in rock. Eng Geol 239:282–297

    Google Scholar 

  • Walton G, Labrie D, Alejano LR (2019) On the residual strength of rocks and rockmasses. Rock Eng Rock Mech. https://doi.org/10.1007/s00603-019-01879-5

    Article  Google Scholar 

  • Walton G (2017) Scale effects observed in compression testing of Stanstead granite including post-peak strength and dilatancy. Geotech Geol Eng 36:1091–1111

    Google Scholar 

  • Walton G, Hedayat A, Kim E, Labrie D (2017) Post-yield strength and dilatancy evolution across the brittle-ductile transition in Indiana limestone. Rock Mech Rock Eng 50(7):1691–1710

    Google Scholar 

  • Walton G, Arzua J, Alejano LR, Diederichs MS (2015) A laboratorytesting-based study on the strength, deformability, and dilatancy of carbonate rocks at low confinement. Rock Mech Rock Eng 48(3):941–958

    Google Scholar 

  • Wang WJ, Shen JY (2017) Comparison of existing methods and a new tensile strength based model in estimating the Hoek–Brown constant mi for intact rocks. Eng Geol 224(22):87–96

    Google Scholar 

  • Yang SQ, Jing HW, Wang SY (2012) Experimental investigation on the strength, deformability, failure behavior and acoustic emission locations of red sandstone under triaxial compression. Rock Mech Rock Eng 45(4):583–606

    Google Scholar 

  • Yang SQ, Jiang YZ, Xu WY, Chen XQ (2008) Experimental investigation on strength and failure behavior of pre-cracked marble under conventional triaxial compression. Int J Solids Struct 45(17):4796–4819

    Google Scholar 

  • Yumlu M, Ozbay MU (1995) A study of the behaviour of brittle rocks under plane strain and triaxial loading conditions. Int J Rock Mech Min Sci 32(7):725–733

    Google Scholar 

  • Zhou J, Lan H, Zhang L, Yang D, Song J, Wang S (2019) Novel grain-based model for simulation of brittle failure of Alxa porphyritic granite. Eng Geol 251:100–114

    Google Scholar 

  • Zuo JP, Li HT, Xie HP, Ju Y, Peng SP (2008) A nonlinear strength criterion for rocklike materials based on fracture mechanics. Int J Rock Mech Min Sci 45(4):594–599

    Google Scholar 

  • Zuo JP, Liu H, Li H (2015) A theoretical derivation of the Hoek–Brown failure criterion for rock materials. J Rock Mech Geotech Eng 7(4):361–366

    Google Scholar 

Download references

Acknowledgements

This study is sponsored by the National Natural Science Foundation of China (Grants nos. 11902249, 11872301 and 51779207) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant nos. 2019JQ395 and 17JS091). The financial support provided by this sponsor is greatly appreciated.

Author information

Authors and Affiliations

  1. State Key Laboratory of Eco-Hydraulics in Northwest Arid Region, Xi’an University of Technology, Xi’an, 710048, China

    Mingming He, Zhiqiang Zhang & Ning Li

  2. Shaanxi Key Laboratory of Loess Mechanics and Engineering, Xi’an University of Technology, Xi’an, 710048, China

    Mingming He, Zhiqiang Zhang & Ning Li

  3. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, China

    Jun Zheng

  4. Xi’an University of Science and Technology, Shaanxi, 710054, China

    Fangfang Chen

Authors
  1. Mingming He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiqiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fangfang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ning Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingming He.

Ethics declarations

Conflict of Interest

The authors declare 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, M., Zhang, Z., Zheng, J. et al. A New Perspective on the Constant mi of the Hoek–Brown Failure Criterion and a New Model for Determining the Residual Strength of Rock. Rock Mech Rock Eng 53, 3953–3967 (2020). https://doi.org/10.1007/s00603-020-02164-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-020-02164-6

Keywords

Search

Navigation