Retrospective and prospective review of the generalized nonlinear strength theory for geomaterials

Shunchuan Wu; Jiaxin Wang; Shihuai Zhang; Shigui Huang; Lei Xia; Qianping Zhao

doi:10.1007/s12613-024-2929-1

Volume 31 Issue 8

Aug. 2024

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Article Navigation > International Journal of Minerals, Metallurgy and Materials > 2024 > 31(8): 1767-1787

Shunchuan Wu, Jiaxin Wang, Shihuai Zhang, Shigui Huang, Lei Xia, and Qianping Zhao, Retrospective and prospective review of the generalized nonlinear strength theory for geomaterials, Int. J. Miner. Metall. Mater., 31(2024), No. 8, pp. 1767-1787. https://doi.org/10.1007/s12613-024-2929-1

Cite this article as:

Shunchuan Wu, Jiaxin Wang, Shihuai Zhang, Shigui Huang, Lei Xia, and Qianping Zhao, Retrospective and prospective review of the generalized nonlinear strength theory for geomaterials, Int. J. Miner. Metall. Mater., 31(2024), No. 8, pp. 1767-1787. https://doi.org/10.1007/s12613-024-2929-1

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Review

Retrospective and prospective review of the generalized nonlinear strength theory for geomaterials

+ Author Affiliations

Corresponding authors:
Jiaxin Wang E-mail: wangjiaxin2727@163.com

Shihuai Zhang E-mail: zhangshihuai@ustc.edu.cn
Received: 23 November 2023; Revised: 8 May 2024; Accepted: 8 May 2024; Available online: 9 May 2024

Abstract

Abstract

Strength theory is the basic theory for calculating and designing the strength of engineering materials in civil, hydraulic, mechanical, aerospace, military, and other engineering disciplines. Therefore, the comprehensive study of the generalized nonlinear strength theory (GNST) of geomaterials has significance for the construction of engineering rock strength. This paper reviews the GNST of geomaterials to demonstrate the research status of nonlinear strength characteristics of geomaterials under complex stress paths. First, it systematically summarizes the research progress of GNST (classical and empirical criteria). Then, the latest research the authors conducted over the past five years on the GNST is introduced, and a generalized three-dimensional (3D) nonlinear Hoek‒Brown (HB) criterion (NGHB criterion) is proposed for practical applications. This criterion can be degenerated into the existing three modified HB criteria and has a better prediction performance. The strength prediction errors for six rocks and two in-situ rock masses are 2.0724%–3.5091% and 1.0144%–3.2321%, respectively. Finally, the development and outlook of the GNST are expounded, and a new topic about the building strength index of rock mass and determining the strength of in-situ engineering rock mass is proposed. The summarization of the GNST provides theoretical traceability and optimization for constructing in-situ engineering rock mass strength.
- rock mechanics;
- rock mass strength;
- strength theory;
- failure criterion;
- Hoek–Brown criterion;
- intermediate principal stress;
- deviatoric plane;
- smoothness and convexity

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References

[1]	S.C. Wu, L.P. Li, and X.P. Zhang, Rock Mechanics, Higher Education Press, Beijing, 2021.
[2]	F.X. Ding, X. Wu, X.M. Zhang, et al., Reviews on research progress of strength theories for materials, J. Railway Sci. Eng., (2024). DOI: 10.19713/j.cnki.43-1423/u.T20240158
[3]	Q.F. Guo, X. Xi, S.T. Yang, and M.F. Cai, Technology strategies to achieve carbon peak and carbon neutrality for China’s metal mines, Int. J. Miner. Metall. Mater., 29(2022), No. 4, p. 626. doi: 10.1007/s12613-021-2374-3
[4]	M.H. Yu, Advances in strength theories for materials under complex stress state in the 20th century, Appl. Mech. Rev., 55(2002), No. 3, p. 169. doi: 10.1115/1.1472455
[5]	M.H. Yu, M. Yoshimine, H.F. Qiang et al., Advances and prospects for strength theory, Eng. Mech., 21(2004), No. 6, p. 1. doi: 10.3969/j.issn.1000-4750.2004.06.001
[6]	H.H. Zhu, Q. Zhang, and L.Y. Zhang, Review of research progresses and applications of Hoek‒Brown strength criterion, Chin. J. Rock Mech. Eng., 32(2013), No. 10, p. 1945. doi: 10.3969/j.issn.1000-6915.2013.10.001
[7]	H.T. Liu, Z. Han, Z.J. Han, et al., Nonlinear empirical failure criterion for rocks under triaxial compression, Int. J. Min. Sci. Technol., (2024). DOI: 10.1016/j.ijmst.2024.03.002
[8]	D. Zhang, E.L. Liu, X.Y. Liu, G. Zhang, and B.T. Song, A new strength criterion for frozen soils considering the influence of temperature and coarse-grained contents, Cold Reg. Sci. Technol., 143(2017), p. 1. doi: 10.1016/j.coldregions.2017.08.006
[9]	M. Asadi and M.H. Bagheripour, Modified criteria for sliding and non-sliding failure of anisotropic jointed rocks, Int. J. Rock Mech. Min. Sci., 73(2015), p. 95. doi: 10.1016/j.ijrmms.2014.10.006
[10]	J.Y. Pei, H.H. Einstein, and A.J. Whittle, The normal stress space and its application to constructing a new failure criterion for cross-anisotropic geomaterials, Int. J. Rock Mech. Min. Sci., 106(2018), p. 364. doi: 10.1016/j.ijrmms.2018.03.023
[11]	Y.N. Zheng, Q. Zhang, S. Zhang, C.J. Jia, and M.F. Lei, Yield criterion research on intact rock transverse isotropy based on Hoek–Brown criterion, Rock Soil Mech., 43(2022), No. 1, p. 139.
[12]	J.Y. Liang, C. Ma, Y.H. Su, D.C. Lu, and X.L. Du, A failure criterion incorporating the effect of depositional angle for transversely isotropic soils, Comput. Geotech., 148(2022), art. No. 104812. doi: 10.1016/j.compgeo.2022.104812
[13]	X.P. Lai, P.F. Shan, M.F. Cai, F.H. Ren, and W.H. Tan, Comprehensive evaluation of high-steep slope stability and optimal high-steep slope design by 3D physical modeling, Int. J. Miner. Metall. Mater., 22(2015), No. 1, p. 1. doi: 10.1007/s12613-015-1036-8
[14]	X.S. Li, Q.H. Li, Y.M. Wang, et al., Experimental study on instability mechanism and critical intensity of rainfall of high-steep rock slopes under unsaturated conditions, Int. J. Min. Sci. Technol., 33(2023), No. 10, p. 1243. doi: 10.1016/j.ijmst.2023.07.009
[15]	W. Liu, Q.H. Li, C.H. Yang, et al., The role of underground salt caverns for large-scale energy storage: A review and prospects, Energy Storage Mater., 63(2023), art. No. 103045. doi: 10.1016/j.ensm.2023.103045
[16]	X.L. Lü, M.S. Huang, and J.E. Andrade, Strength criterion for cross-anisotropic sand under general stress conditions, Acta Geotech., 11(2016), No. 6, p. 1339. doi: 10.1007/s11440-016-0479-z
[17]	J.X. Wang, S.C. Wu, H.Y. Cheng, J.L. Sun, X.L. Wang, and Y.X. Shen, A generalized nonlinear three-dimensional Hoek‒Brown failure criterion, J. Rock Mech. Geotech. Eng., (2024). DOI: 10.1016/j.jrmge.2023.10.022.
[18]	G.C. Nayak and O.C. Zienkiewicz, Convenient form of stress invariants for plasticity, J. Struct. Div., 98(1972), No. 4, p. 949. doi: 10.1061/JSDEAG.0003219
[19]	A.V. Hershey, The plasticity of an isotropic aggregate of anisotropic face-centered cubic crystals, J. Appl. Mech., 21(1954), No. 3, p. 241. doi: 10.1115/1.4010900
[20]	E.A. Davis, The Bailey flow rule and associated yield surface, J. Appl. Mech., 28(1961), No. 2, p. 310. doi: 10.1115/1.3641679
[21]	W.F. Hosford, A generalized isotropic yield creterion, J. Appl. Mech., 39(1972), No. 2, p. 607. doi: 10.1115/1.3422732
[22]	F. Barlat and K. Lian, Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions, Int. J. Plast., 5(1989), No. 1, p. 51. doi: 10.1016/0749-6419(89)90019-3
[23]	J.J. Tan, The unified form for yield criteria of metallic materials, Chin. Sci. Bull., 36(1991), No. 9, p. 769.
[24]	A.P. Karafillis and M.C. Boyce, A general anisotropic yield criterion using bounds and a transformation weighting tensor, J. Mech. Phys. Solids, 41(1993), No. 12, p. 1859. doi: 10.1016/0022-5096(93)90073-O
[25]	D.R.J. Owen and D. Perić, Recent developments in the application of finite element methods to nonlinear problems, Finite Elem. Anal. Des., 18(1994), No. 1-3, p. 1. doi: 10.1016/0168-874X(94)90085-X
[26]	R.W. Bailey, The utilization of creep test data in engineering design, Proc. Inst. Mech. Eng., 131(1935), No. 1, p. 131.
[27]	F. Edelman and D.C. Drucker, Some extensions of elementary plasticity theory, J. Frankl. Inst., 251(1951), No. 6, p. 581. doi: 10.1016/0016-0032(51)90406-1
[28]	B. Dodd and K. Naruse, Limitations on isotropic yield criteria, Int. J. Mech. Sci., 31(1989), No. 7, p. 511. doi: 10.1016/0020-7403(89)90100-8
[29]	R. Hill, A user-friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci., 35(1993), No. 1, p. 19. doi: 10.1016/0020-7403(93)90061-X
[30]	F. Barlat, R.C. Becker, Y. Hayashida, et al., Yielding description for solution strengthened aluminum alloys, Int. J. Plast., 13(1997), No. 4, p. 385. doi: 10.1016/S0749-6419(97)80005-8
[31]	F. Barlat, Y. Maeda, K. Chung, et al., Yield function development for aluminum alloy sheets, J. Mech. Phys. Solids., 45(1997), No. 11-12, p. 1727. doi: 10.1016/S0022-5096(97)00034-3
[32]	O.C. Zienkiewicz, Some useful forms of isotropic yield surfaces for soil and rock mechanics, [in] G. Gudehus, ed., Finite Elements in Geomechanics, John Wiley & Sons Ltd, London, 1977, p. 179.
[33]	C.S. Desai, A general basis for yield, failure and potential functions in plasticity, Int. J. Numer. Anal. Methods Geomech., 4(1980), No. 4, p. 361. doi: 10.1002/nag.1610040406
[34]	R. de Boer, On plastic deformation of soils, Int. J. Plast., 4(1988), No. 4, p. 371.
[35]	Z.J. Shen, A stress–strain model for sands under complex loading, Adv. Constitutive Laws Eng. Mater., 1(1989), p. 303.
[36]	S. Krenk, Family of invariant stress surface, J. Engrg. Mech., 122(1996), No. 3, p. 201. doi: 10.1061/(ASCE)0733-9399(1996)122:3(201)
[37]	M.H. Yu, Unified Strength Theory and its Applications, Xi’an Jiaotong University Press, Xi’an, 2018.
[38]	M. Aubertin, L. Li, R. Simon, and S. Khalfi, Formulation and application of a short-term strength criterion for isotropic rocks, Can. Geotech. J., 36(1999), p. 947. doi: 10.1139/t99-056
[39]	Y.Q. Zhou, Q. Sheng, Z.Q. Zhu, and X.D. Fu, Subloading surface model for rock based on modified Drucker‒Prager criterion, Rock Soil Mech., 38(2017), No. 2, p. 400. doi: 10.16285/j.rsm.2017.02.013
[40]	F.X. Zhou and S.R. Li, Generalized Drucker–Prager strength criterion, Key Eng. Mater., 353-358(2007), p. 369. doi: 10.4028/www.scientific.net/KEM.353-358.369
[41]	D. Zhang, E.L. Liu, X.Y. Liu, and B.T. Song, Investigation on strength criterion for frozen silt soils, Rock Soil Mech., 39(2018), No. 9, p. 3237. doi: 10.16285/j.rsm.2016.2962
[42]	X.Y. Liu, E.L. Liu, D. Zhang, G. Zhang, and B.T. Song, Study on strength criterion for frozen soil, Cold Reg. Sci. Technol., 161(2019), p. 1. doi: 10.1016/j.coldregions.2019.02.009
[43]	Y.P. Yao, J. Hu, A.N. Zhou, T. Luo, and N.D. Wang, Unified strength criterion for soils, gravels, rocks, and concretes, Acta Geotech., 10(2015), No. 6, p. 749. doi: 10.1007/s11440-015-0404-x
[44]	X.L. Du, D.C. Lu, Q.M. Gong, and M. Zhao, Nonlinear unified strength criterion for concrete under three-dimensional stress states, J. Eng. Mech., 136(2010), No. 1, p. 51.
[45]	Y. Xiao, H.L. Liu, and R.Y. Liang, Modified Cam–Clay model incorporating unified nonlinear strength criterion, Sci. China Technol. Sci., 54(2011), No. 4, p. 805. doi: 10.1007/s11431-011-4313-4
[46]	M.C. Liu, Y.F. Gao, and H.L. Liu, A nonlinear Drucker–Prager and Matsuoka–Nakai unified failure criterion for geomaterials with separated stress invariants, Int. J. Rock Mech. Min. Sci., 50(2012), p. 1. doi: 10.1016/j.ijrmms.2012.01.002
[47]	Y.Q. Zhang, M. Bernhardt, G. Biscontin, R. Luo, and R.L. Lytton, A generalized Drucker–Prager viscoplastic yield surface model for asphalt concrete, Mater. Struct., 48(2015), No. 11, p. 3585. doi: 10.1617/s11527-014-0425-1
[48]	M.K. Darabi, R.K.A. Al-Rub, E.A. Masad, C.W. Huang, and D.N. Little, A modified viscoplastic model to predict the permanent deformation of asphaltic materials under cyclic-compression loading at high temperatures, Int. J. Plast., 35(2012), p. 100. doi: 10.1016/j.ijplas.2012.03.001
[49]	D. Lu, C.J. Ma, X.L. Du, L. Jin, and Q. Gong, Development of a new nonlinear unified strength theory for geomaterials based on the characteristic stress concept, Int. J. Geomech., 17(2017), art. No. 04016058. doi: 10.1061/(ASCE)GM.1943-5622.0000729
[50]	Z. Wan, R.D. Qiu, and J.X. Guo, A kind of strength and yield criterion for geomaterials and its transformation stress method, Chin. J. Theor. Appl. Mech., 49(2017), No. 3, p. 726. doi: 10.6052/0459-1879-16-297
[51]	Z. Wan, Y.Y. Liu, W. Cao, Y.J. Wang, L.Y. Xie, and Y.F. Fang, One kind of transverse isotropic strength criterion and the transformation stress space, Int. J. Numer. Anal. Meth. Geomech., 46(2022), No. 4, p. 798. doi: 10.1002/nag.3322
[52]	B.H. Liu, L.W. Kong, R.J. Shu, and T.G. Li, Mechanical properties and strength criterion of Zhanjiang structured clay in three-dimensional stress state, Rock Soil Mech., 42(2021), No. 11, p. 3090. doi: 10.16285/j.rsm.2021.0396
[53]	J.L. He, F.J. Niu, W.J. Su, and H.Q. Jiang, Nonlinear unified strength criterion for frozen soil based on homogenization theory, Mech. Adv. Mater. Struct., 30(2023), No. 19, p. 4002. doi: 10.1080/15376494.2022.2087126
[54]	S. Wang, Z. Zhong, B. Chen, X.R. Liu, and B.M. Wu, Developing a three dimensional (3D) elastoplastic constitutive model for soils based on unified nonlinear strength (UNS) criterion, Front. Earth Sci., 10(2022), art. No. 853962. doi: 10.3389/feart.2022.853962
[55]	G. Mortara, A yield criterion for isotropic and cross-anisotropic cohesive-frictional materials, Int. J. Numer. Anal. Methods Geomech., 34(2010), No. 9, p. 953. doi: 10.1002/nag.846
[56]	Y. Xiao, H.L. Liu, and J.G. Zhu, Failure criterion for granular soils, Chin. J. Geotech. Eng., 32(2010), No. 4, p. 586.
[57]	Z. Wan and Y.Y. Liu, A new generalized failure criterion and its plane strain strength characteristics, Arch. Appl. Mech., 93(2023), No. 4, p. 1699. doi: 10.1007/s00419-022-02353-5
[58]	X.T. Feng, R. Kong, C.X. Yang, et al., A three-dimensional failure criterion for hard rocks under true triaxial compression, Rock Mech. Rock Eng., 53(2020), No. 1, p. 103. doi: 10.1007/s00603-019-01903-8
[59]	X.T. Feng, C.X. Yang, R. Kong, et al., Excavation-induced deep hard rock fracturing: Methodology and applications, J. Rock Mech. Geotech. Eng., 14(2022), No. 1, p. 1. doi: 10.1016/j.jrmge.2021.12.003
[60]	D. Bigoni and A. Piccolroaz, Yield criteria for quasibrittle and frictional materials, Int. J. Solids Struct., 41(2004), No. 11-12, p. 2855. doi: 10.1016/j.ijsolstr.2003.12.024
[61]	G. Mortara, A new yield and failure criterion for geomaterials, Géotechnique., 58(2008), No. 2, p. 125.
[62]	P.V. Lade, Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces, Int. J. Solids Struct., 13(1977), No. 11, p. 1019. doi: 10.1016/0020-7683(77)90073-7
[63]	M.K. Kim and P.V. Lade, Modeling rock strength in three-dimensions, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 21(1984), No. 1, p. 21. doi: 10.1016/0148-9062(84)90006-8
[64]	G.T. Houlsby, A general failure criterion for frictional and cohesive materials, Soils Found., 26(1986), No. 2, p. 97. doi: 10.3208/sandf1972.26.2_97
[65]	S.L. Qiu, X.T. Feng, C.Q. Zhang, and S.L. Huang, Establishment of unified strain energy strength criterion of homogeneous and isotropic hard rocks and its validation, Chin. J. Rock Mech. Eng., 32(2013), No. 4, p. 714.
[66]	M. Aubertin, L. Li, and R. Simon, A multiaxial stress criterion for short- and long-term strength of isotropic rock media, Int. J. Rock Mech. Min. Sci., 37(2000), No. 8, p. 1169. doi: 10.1016/S1365-1609(00)00047-2
[67]	S.L. Huang, X.T. Feng, and C.Q. Zhang, A new generalized polyaxial strain energy strength criterion of brittle rock and polyaxial test validation, Chin. J. Rock Mech. Eng., 27(2008), No. 01, p. 124. doi: 10.3321/j.issn:1000-6915.2008.01.019
[68]	G.A. Wiebols and N.G.W. Cook, An energy criterion for the strength of rock in polyaxial compression, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 5(1968), No. 6, p. 529. doi: 10.1016/0148-9062(68)90040-5
[69]	V.A. Kolupaev, Generalized strength criteria as functions of the stress angle, J. Eng. Mech., 143(2017), No. 9, art. No. 04017095. doi: 10.1061/(ASCE)EM.1943-7889.0001322
[70]	P.L. Rosendahl, V.A. Kolupaev, and H. Altenbach, Extreme yield figures for universal strength criteria, [In] H. Altenbach and A. Öchsner, eds., State of the Art and Future Trends in Material Modeling, Springer, Cham, 2019, p. 259.
[71]	W. Ehlers, A single-surface yield function for geomaterials, Arch. Appl. Mech., 65(1995), No. 4, p. 246. doi: 10.1007/BF00805464
[72]	Z.Z. Li and X.W. Tang, Deduction and verification of a new strength criterion for soils, Rock Soil Mech., 28(2007), No. 6, p. 1247.
[73]	W.C. Shi, J.G. Zhu, and H.L. Liu, Influence of intermediate principal stress on deformation and strength of gravel, Chin. J. Geotech. Eng., 30(2008), No. 10, p. 1449.
[74]	J.Y. Liang and Y.M. Li, A failure criterion considering stress angle effect, Rock Mech. Rock Eng., 52(2019), No. 4, p. 1257. doi: 10.1007/s00603-018-1676-x
[75]	M.Z. Zheng and S.J. Li, A non-linear three-dimensional failure criterion based on stress tensor distance, Rock Mech. Rock Eng., 55(2022), p. 6741. doi: 10.1007/s00603-022-03034-z
[76]	X.S. Gao, M. Wang, C. Li, M.M. Zhang, and Z.H. Li, A new three-dimensional rock strength criterion based on shape function in deviatoric plane, Geomech. Geophys. Geo Energy Geo Resour., 10(2024), No. 1, art. No. 7. doi: 10.1007/s40948-023-00710-4
[77]	G. Mortara, A hierarchical single yield surface for frictional materials, Comput. Geotech., 36(2009), No. 6, p. 960. doi: 10.1016/j.compgeo.2009.03.007
[78]	J.Q. Jiang, R.Q. Xu, J.L. Yu, Z.J. Qiu, J.S. Qin, and X.B. Zhan, A practical constitutive theory based on egg-shaped function in elasto-plastic modeling for soft clay, J. Cent. South Univ., 27(2020), No. 8, p. 2424. doi: 10.1007/s11771-020-4459-y
[79]	J.C. Liu, X. Li, Y. Xu, and K.W. Xia, A three-dimensional nonlinear strength criterion for rocks considering both brittle and ductile domains, Rock Mech. Rock Eng., (2024). DOI: 10.1007/s00603-024-03823-8
[80]	X.D. Ma, J.W. Rudnicki, and B.C. Haimson, The application of a Matsuoka–Nakai–Lade–Duncan failure criterion to two porous sandstones, Int. J. Rock Mech. Min. Sci., 92(2017), p. 9. doi: 10.1016/j.ijrmms.2016.12.004
[81]	H.H. Chen, C.Y. Yang, J.P. Li, and D.A. Sun, A general method to incorporate three-dimensional cross-anisotropy to failure criterion of geomaterial, Int. J. Geomech., 21(2021), No. 12, art. No. 04021241. doi: 10.1061/(ASCE)GM.1943-5622.0002224
[82]	F. Zhou and H. Wu, A novel three-dimensional modified Griffith failure criterion for concrete, Eng. Fract. Mech., 284(2023), art. No. 109287. doi: 10.1016/j.engfracmech.2023.109287
[83]	H. Jiang, Simple three-dimensional Mohr–Coulomb criteria for intact rocks, Int. J. Rock Mech. Min. Sci., 105(2018), p. 145. doi: 10.1016/j.ijrmms.2018.01.036
[84]	H. Jiang, Failure criteria for cohesive-frictional materials based on Mohr–Coulomb failure function, Int. J. Numer. Anal. Meth. Geomech., 39(2015), No. 13, p. 1471. doi: 10.1002/nag.2366
[85]	H.Z. Li, J.T. Xu, Z.L. Zhang, and L. Song, A generalized unified strength theory for rocks, Rock Mech. Rock Eng., 56(2023), No. 11, p. 7759. doi: 10.1007/s00603-023-03471-4
[86]	S.A.F. Murrell, The effect of triaxial stress system on the strength of rocks at atmospheric Temperatures, Geophys. J. Int., 10(1965), No. 3, p. 231. doi: 10.1111/j.1365-246X.1965.tb03155.x
[87]	Z.T. Bieniawski, Estimating the strength of rock materials, J. S. Afr. Inst. Min. Metall., 74(1974), No. 8, p. 312.
[88]	Y. Yudhbir, W. Lemanza, and F. Prinzl, An empirical failure criterion for rock masses, [in] The 5th ISRM Congress , Melbourne, Australia, 1983.
[89]	P.R. Sheorey, A.K. Biswas, and V.D. Choubey, An empirical failure criterion for rocks and jointed rock masses, Eng. Geol., 26(1989), No. 2, p. 141. doi: 10.1016/0013-7952(89)90003-3
[90]	D. Hobbs, The tensile strength of rocks, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 1(1964), p. 385. doi: 10.1016/0148-9062(64)90005-1
[91]	T. Ramamurthy and V.K. Arora, Strength predictions for jointed rock in confined and unconfined states, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 31(1994), No. 1, p. 9. doi: 10.1016/0148-9062(94)92311-6
[92]	J.A. Franklin, Triaxial strength of rock materials, Rock Mech., 3(1971), No. 2, p. 86. doi: 10.1007/BF01239628
[93]	E. Hoek and E.T. Brown, Empirical strength criterion for rock masses, J. Geotech. Engrg. Div., 106(1980), No. 9, p. 1013. doi: 10.1061/AJGEB6.0001029
[94]	N. Yoshida, N.R. Morgenstern, and D.H. Chan, A failure criterion for stiff soils and rocks exhibiting softening, Can. Geotech. J., 27(1990), No. 2, p. 195.
[95]	E.T. Brown and E. Hoek, Underground Excavations in Rock, CRC Press, London, 1980.
[96]	A.A. Griffith, Theory of rupture, [in] Proceedings of the 1st International Congress on Applied Mechanics, Delft, 1924, p. 55.
[97]	E. Hoek, Strength of rock and rock masses, ISRM News J., 2(1994), No. 2, p. 4.
[98]	E. Hoek, P.K. Kaiser, and W.F. Bawden, Support of Underground Excavations in Hard Rock, CRC Press, Florida, 2000.
[99]	E. Hoek and E.T. Brown, The Hoek–Brown failure criterion and GSI–2018 edition, J. Rock Mech. Geotech. Eng., 11(2019), No. 3, p. 445. doi: 10.1016/j.jrmge.2018.08.001
[100]	X.T. Feng, J.Y. Zhang, C.X. Yang, et al., A novel true triaxial test system for microwave-induced fracturing of hard rocks, J. Rock Mech. Geotech. Eng., 13(2021), No. 5, p. 961. doi: 10.1016/j.jrmge.2021.03.008
[101]	X.T. Feng, M. Tian, C.X. Yang, and B.G. He, A testing system to understand rock fracturing processes induced by different dynamic disturbances under true triaxial compression, J. Rock Mech. Geotech. Eng., 15(2023), No. 1, p. 102. doi: 10.1016/j.jrmge.2022.02.002
[102]	C. Zhu, M. Karakus, M.C. He, et al., Volumetric deformation and damage evolution of Tibet interbedded skarn under multistage constant–amplitude–cyclic loading, Int. J. Rock Mech. Min. Sci., 152(2022), art. No. 105066. doi: 10.1016/j.ijrmms.2022.105066
[103]	Y.K. Lee, S. Pietruszczak, and B.H. Choi, Failure criteria for rocks based on smooth approximations to Mohr–Coulomb and Hoek–Brown failure functions, Int. J. Rock Mech. Min. Sci., 56(2012), p. 146. doi: 10.1016/j.ijrmms.2012.07.032
[104]	Q. Zhang, H.H. Zhu, and L.Y. Zhang, Modification of a generalized three-dimensional Hoek–Brown strength criterion, Int. J. Rock Mech. Min. Sci., 59(2013), p. 80. doi: 10.1016/j.ijrmms.2012.12.009
[105]	Y.G. Yang, F. Gao, and Y.M. Lai, Modified Hoek–Brown criterion for nonlinear strength of frozen soil, Cold Reg. Sci. Technol., 86(2013), p. 98.
[106]	B.X. Li, Research on Failure Mechanism and 3-D Strength Criterion of Hard Rock in Deep Ground Engineering [Dissertation], Shandong University, Shandong, 2022.
[107]	X.D. Pan and J.A. Hudson, A simplified three dimensional Hoek–Brown yield criterion, [in] ISRM International Symposium, Madrid,1988.
[108]	L.Y. Zhang and H.H. Zhu, Three-dimensional Hoek–Brown strength criterion for rocks, J. Geotech. Geoenviron. Eng., 133(2007), No. 9, p. 1128. doi: 10.1061/(ASCE)1090-0241(2007)133:9(1128)
[109]	L. Zhang, A generalized three-dimensional Hoek–Brown strength criterion, Rock Mech. Rock Eng., 41(2008), No. 6, p. 893. doi: 10.1007/s00603-008-0169-8
[110]	H. Jiang, X.W. Wang, and Y.L. Xie, New strength criteria for rocks under polyaxial compression, Can. Geotech. J., 48(2011), No. 8, p. 1233. doi: 10.1139/t11-034
[111]	H. Jiang and Y.L. Xie, A new three-dimensional Hoek–Brown strength criterion, Acta Mech. Sin., 28(2012), No. 2, p. 393. doi: 10.1007/s10409-012-0054-2
[112]	H. Jiang and J.D. Zhao, A simple three-dimensional failure criterion for rocks based on the Hoek–Brown criterion, Rock Mech. Rock Eng., 48(2015), No. 5, p. 1807. doi: 10.1007/s00603-014-0691-9
[113]	H. Jiang, A failure criterion for rocks and concrete based on the Hoek–Brown criterion, Int. J. Rock Mech. Min. Sci., 95(2017), p. 62. doi: 10.1016/j.ijrmms.2017.04.003
[114]	H. Jiang, Three-dimensional failure criteria for rocks based on the Hoek–Brown criterion and a general lode dependence, Int. J. Geomech., 27(2017), No. 8, art. No. 04017023.
[115]	W.Q. Cai, H.H. Zhu, W.H. Liang, L.Y. Zhang, and W. Wu, A new version of the generalized Zhang–Zhu strength criterion and a discussion on its smoothness and convexity, Rock Mech. Rock Eng., 54(2021), No. 8, p. 4265. doi: 10.1007/s00603-021-02505-z
[116]	W.Q. Cai, H.H. Zhu, and W.H. Liang, Three-dimensional tunnel face extrusion and reinforcement effects of underground excavations in deep rock masses, Int. J. Rock Mech. Min. Sci., 150(2022), art. No. 104999. doi: 10.1016/j.ijrmms.2021.104999
[117]	W.Q. Cai, H.H. Zhu, and W.H. Liang, Three-dimensional stress rotation and control mechanism of deep tunneling incorporating generalized Zhang–Zhu strength-based forward analysis, Eng. Geol., 308(2022), art. No. 106806. doi: 10.1016/j.enggeo.2022.106806
[118]	W.Q. Cai, C.L. Su, H.H. Zhu, et al., Elastic-plastic response of a deep tunnel excavated in 3D Hoek–Brown rock mass considering different approaches for obtaining the out-of-plane stress, Int. J. Rock Mech. Min. Sci., 169(2023), art. No. 105425. doi: 10.1016/j.ijrmms.2023.105425
[119]	H.H. Chen, H.H. Zhu, and L.Y. Zhang, A unified constitutive model for rock based on newly modified GZZ criterion, Rock Mech. Rock Eng., 54(2021), No. 2, p. 921. doi: 10.1007/s00603-020-02293-y
[120]	H. Chen, H. Zhu, and L. Zhang, Further modification of a generalised 3D Hoek–Brown criterion: the GZZ criterion, Géotech. Lett., 12(2022), No. 4, p. 272.
[121]	B. Single, R.K. Goel, V.K. Mehrotra, S.K. Garg, and M.R. Allu, Effect of intermediate principal stress on strength of anisotropic rock mass, Tunn. Undergr. Space Technol., 13(1998), No. 1, p. 71. doi: 10.1016/S0886-7798(98)00023-6
[122]	S. Priest, Three-dimensional failure criteria based on the Hoek–Brown criterion, Rock Mech. Rock Eng., 45(2012), p. 989. doi: 10.1007/s00603-012-0277-3
[123]	H.Z. Li, T. Guo, Y.L. Nan, and B. Han, A simplified three-dimensional extension of Hoek–Brown strength criterion, J. Rock Mech. Geotech. Eng., 13(2021), No. 3, p. 568.
[124]	L.J. Ma, Z. Li, M.Y. Wang, J.W. Wu, and G. Li, Applicability of a new modified explicit three-dimensional Hoek–Brown failure criterion to eight rocks, Int. J. Rock Mech. Min. Sci., 133(2020), art. No. 104311. doi: 10.1016/j.ijrmms.2020.104311
[125]	X.C. Que, Z.D. Zhu, Z.H. Niu, S. Zhu, and L.X. Wang, A modified three-dimensional Hoek–Brown criterion for intact rocks and jointed rock masses, Geomech. Geophys. Geo Energy Geo Resour., 9(2023), No. 1, art. No. 7. doi: 10.1007/s40948-023-00560-0
[126]	F. Gao, Y.G. Yang, H.M. Cheng, and C.Z. Cai, Novel 3D failure criterion for rock materials, Int. J. Geomech., 19(2019), No. 6, art. No. 04019046. doi: 10.1061/(ASCE)GM.1943-5622.0001421
[127]	X.C. Shi, Q.L. Li, J.F. Liu, L.Y. Gao, and X. Zhou, An improved true triaxial Hoek–Brown strength criterion, Adv. Eng. Sci., 55(2023), No. 2, p. 214.
[128]	M.H. Yu, Y.W. Zan, J. Zhao, and M. Yoshimine, A unified strength criterion for rock material, Int. J. Rock Mech. Min. Sci., 39(2002), No. 8, p. 975. doi: 10.1016/S1365-1609(02)00097-7
[129]	S.D. Priest, Determination of shear strength and three-dimensional yield strength for the Hoek–Brown criterion, Rock Mech. Rock Eng., 38(2005), No. 4, p. 299. doi: 10.1007/s00603-005-0056-5
[130]	N. Melkoumian, S.D. Priest, and S.P. Hunt, Further development of the three-dimensional Hoek–Brown yield criterion, Rock Mech. Rock Eng., 42(2009), No. 6, p. 835. doi: 10.1007/s00603-008-0022-0
[131]	T. Benz, R. Schwab, R.A. Kauther, and P.A. Vermeer, A Hoek–Brown criterion with intrinsic material strength factorization, Int. J. Rock Mech. Min. Sci., 45(2008), No. 2, p. 210. doi: 10.1016/j.ijrmms.2007.05.003
[132]	T. Benz and R. Schwab, A quantitative comparison of six rock failure criteria, Int. J. Rock Mech. Min. Sci., 45(2008), No. 7, p. 1176. doi: 10.1016/j.ijrmms.2008.01.007
[133]	J.Q. Huang, M. Zhao, X.L. Du, F. Dai, C. Ma, and J.B. Liu, An elasto-plastic damage model for rocks based on a new nonlinear strength criterion, Rock Mech. Rock Eng., 51(2018), No. 5, p. 1413. doi: 10.1007/s00603-018-1417-1
[134]	M.V. da Silva and A.N. Antão, A new Hoek–Brown–Matsuoka-Nakai failure criterion for rocks, Int. J. Rock Mech. Min. Sci., 172(2023), art. No. 105602. doi: 10.1016/j.ijrmms.2023.105602
[135]	A.K. Schwartzkopff, A. Sainoki, T. Bruning, and M. Karakus, A conceptual three-dimensional frictional model to predict the effect of the intermediate principal stress based on the Mohr–Coulomb and Hoek–Brown failure criteria, Int. J. Rock Mech. Min. Sci., 172(2023), art. No. 105605. doi: 10.1016/j.ijrmms.2023.105605
[136]	J.P. Zuo, H.T. Li, H.P. Xie, Y. Ju, and S.P. Peng, A nonlinear strength criterion for rock-like materials based on fracture mechanics, Int. J. Rock Mech. Min. Sci., 45(2008), No. 4, p. 594. doi: 10.1016/j.ijrmms.2007.05.010
[137]	J.P. Zuo, H.H. Liu, and H.T. Li, A theoretical derivation of the Hoek–Brown failure criterion for rock materials, J. Rock Mech. Geotech. Eng., 7(2015), No. 4, p. 361. doi: 10.1016/j.jrmge.2015.03.008
[138]	Z.F. Wang, P.Z. Pan, J.P. Zuo, and Y.H. Gao, A generalized nonlinear three-dimensional failure criterion based on fracture mechanics, J. Rock Mech. Geotech. Eng., 15(2023), No. 3, p. 630. doi: 10.1016/j.jrmge.2022.05.006
[139]	X.P. Zhou, Y.D. Shou, Q.H. Qian, and M.H. Yu, Three-dimensional nonlinear strength criterion for rock-like materials based on the micromechanical method, Int. J. Rock Mech. Min. Sci., 72(2014), p. 54. doi: 10.1016/j.ijrmms.2014.08.013
[140]	H. Saroglou and G. Tsiambaos, A modified Hoek–Brown failure criterion for anisotropic intact rock, Int. J. Rock Mech. Min. Sci., 45(2008), No. 2, p. 223. doi: 10.1016/j.ijrmms.2007.05.004
[141]	Q.G. Zhang, B.W. Yao, X.Y. Fan, et al., A modified Hoek–Brown failure criterion for unsaturated intact shale considering the effects of anisotropy and hydration, Eng. Fract. Mech., 241(2021), art. No. 107369. doi: 10.1016/j.engfracmech.2020.107369
[142]	J. Peng, G. Rong, M. Cai, X.J. Wang, and C.B. Zhou, An empirical failure criterion for intact rocks, Rock Mech. Rock Eng., 47(2014), No. 2, p. 347. doi: 10.1007/s00603-012-0355-6
[143]	J. Peng and M. Cai, A cohesion loss model for determining residual strength of intact rocks, Int. J. Rock Mech. Min. Sci., 119(2019), p. 131.
[144]	S.C. Wu, S.H. Zhang, C. Guo, and L.F. Xiong, A generalized nonlinear failure criterion for frictional materials, Acta Geotech., 12(2017), No. 6, p. 1353.
[145]	S.H. Zhang, Study on Strength and Deformability of Hard Brittle Sandstone [Dissertation], University of Science and Technology Beijing, Beijing, 2019.
[146]	J.X. Wang, S.C. Wu, X.K. Chang, H.Y. Cheng, Z.H. Zhou, and Z.J. Ren, A novel three-dimensional nonlinear unified failure criterion for rock materials, Acta Geotech., 19(2024), p. 3337.
[147]	S.C. Wu, S.H. Zhang, and G. Zhang, Three-dimensional strength estimation of intact rocks using a modified Hoek–Brown criterion based on a new deviatoric function, Int. J. Rock Mech. Min. Sci., 107(2018), p. 181.
[148]	S.H. Zhang, S.C. Wu, and G. Zhang, Strength and deformability of a low-porosity sandstone under true triaxial compression conditions, Int. J. Rock Mech. Min. Sci., 127(2020), art. No. 104204. doi: 10.1016/j.ijrmms.2019.104204