Abstract
By applying the Griffith stress criterion of brittle failure, one can find that the uniaxial compressive strength (σc) of rocks is eight times the value of the uniaxial tensile strength (σt). The Griffith strength ratio is smaller than what is normally measured for rocks, even with the consideration of crack closure. The reason is that Griffith’s theories address only the initiation of failure. Under tensile conditions, the crack propagation is unstable so that the tensile crack propagation stress (σcd)t and the peak tensile strength σt are almost identical to the tensile crack initiation stress (σci)t. On the other hand, the crack growth after crack initiation is stable under a predominantly compressive condition. Additional loading is required in compression to bring the stress from the crack initiation stress σci to the peak strength σc. It is proposed to estimate the tensile strength of strong brittle rocks from the strength ratio of \( R = {\frac{{\sigma_{\text{c}} }}{{\left| {\sigma_{\text{t}} } \right|}}} = 8{\frac{{\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}}. \) The term \( {\frac{{\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}} \) accounts for the difference of crack growth or propagation in tension and compression in uniaxial compression tests. \( {\frac{{\sigma_{c} }}{{\sigma_{ci} }}} \) depends on rock heterogeneity and is larger for coarse grained rocks than for fine grained rocks. σci can be obtained from volumetric strain measurement or acoustic emission (AE) monitoring. With the strength ratio R determined, the tensile strength can be indirectly obtained from \( \left| {\sigma_{\text{t}} } \right| = {\frac{{\sigma_{\text{c}} }}{R}} = {\frac{{\sigma_{\text{ci}} }}{8}}. \) It is found that the predicted tensile strengths using this method are in good agreement with test data. Finally, a practical estimate of the Hoek–Brown strength parameter m i is presented and a bi-segmental or multi-segmental representation of the Hoek–Brown strength envelope is suggested for some brittle rocks. In this fashion, the rock strength parameters like σt and m i, which require specialty tests such as direct tensile (or Brazilian) and triaxial compression tests for their determination, can be reasonably estimated from uniaxial compression tests.
Similar content being viewed by others
References
Alber M, Heiland J (2001) Investigation of a limestone pillar failure part 1: geology, laboratory testing and numerical modeling. Rock Mech Rock Eng 34(3):167–186
Bell FG, Jermy CA (2000) The geotechnical character of some South African dolerites, especially their strength and durability. Q J Eng Geol Hydrogeol 33:59–76
Bell FG, Lindsay P (1999) The petrographic and geomechanical properties of some sandstones from the Newspaper Member of the Natal Group near Durban, South Africa. Eng Geol 53(1):57–81
Bieniawski ZT (1967) Mechanism of brittle fracture of rock, parts I, II and III. Int J Rock Mech Min Sci Geomech Abstr 4(4):395–430
Bordia SK (1972) Complete stress-volumetric strain equation for brittle rock up to strength failure. Int J Rock Mech Min Sci Geomech Abstr 9(1):17–24
Brace WF, Paulding B, Scholz C (1966) Dilatancy in the fracture of crystalline crocks. J Geophys Res 71(16):3939–3953
Brook N (1993) The measurement and estimation of basic rock strength. In: Comprehensive rock engineering, pp 41–66
Cai M, Kaiser PK (2007) Obtaining modeling parameters for engineering design by rock mass characterization. In: Proceedings of the 11th ISRM Congress, Talyor & Francis Group, London, pp 381–384
Cai M, Kaiser PK, Tasaka Y, Maejima T, Morioka H, Minami M (2004a) 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
Cai M, Kaiser PK, Uno H, Tasaka Y, Minami M (2004b) Estimation of rock mass strength and deformation modulus of jointed hard rock masses using the GSI system. Int J Rock Mech Min Sci 41(1):3–19
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
Carter BJ (1992) Size and stress gradient effects on fracture around cavities. Rock Mech Rock Eng 25:167–186
Carter BJ, Scott Duncan EJ, Lajtai EZ (1991) Fitting strength criteria to intact rock. Geotech Geol Eng 9:73–81
Chang SH, Seto M, Lee CI (2001) Damage and fracture characteristics of Kimachi sandstone in uniaxial compression. Geosystem Eng 4(1):18–26
Coviello A, Lagioia R, Nova R (2005) On the measurement of the tensile strength of soft rocks. Rock Mech Rock Eng 38(4):251–273
Diederichs MS (1999) Instability of hard rock masses: the role of tensile damage and relaxation. Ph.D. Thesis, p 566
Eberhardt E, Stead D, Stimpson B (1999) Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. Int J Rock Mech Min Sci 36(3):361–380
Griffith AA (1924) The theory of rupture. In: Proceedings of the 1st International Congress on Applied Mechanics, Delft, pp 54–63
Heard HC, Abey AE, Bonner BP, Schock RN (1974) Mechanical behaviour of dry Westerley Granite at high confining pressure. p 14
Hoek E (1965) Rock fracture under static stress conditions. National Mechanical Engineering Research Institute, Council for Scientific and Industrial Research, Pretoria, South Africa
Hoek E (2000) Practical rock engineering. http://www.rocscience.com, 313
Hoek E (2007) Practical rock engineering. http://www.rocscience.com. p 342
Hoek E, Bieniawski ZT (1984) Brittle fracture propagation in rock under compression. Int J Fracture 26(4):276–294
Hoek E, Brown ET (1980a) Empirical strength criterion for rock masses. J Geotech Eng Div ASCE 106(GT9):1013–1035
Hoek E, Brown ET (1980b) Underground excavation in rock. Institution of Mining and Metallurgy, London, p 527
Howarth DF, Rowlands JC (1987) Quantitative assessment of rock texture and correlation with drillability and strength properties. Rock Mech Rock Eng 20(1):57–85
Jaeger JC, Cook NGW (1979) Fundamentals of rock mechanics. Chapman-Hall and Science, London, p 585
Johnson JW, Friedman M, Hopkins TN (1987) Strength and microfracturing of Westerly granite extended wet and dry at temperature to 800°C to 200 MPa. In: Proceedings of the 28th US rock mechanics symposium. Tucson
Kemeny JM, Cook NGW (1987) Crack models for the failure of rocks in compression. In: Constitutive laws for engineering materials: theory and applications, Elsevier, Amsterdam, pp 879–887
Lajtai EZ (1998) Microscopic fracture processes in a granite. Rock Mech Rock Eng 31(4):237–250
Martin CD (1993) The strength of massive Lac du Bonnet granite around underground opening. Ph.D. thesis, p 278
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
Martin CD, Lanyon GW (2001) EDZ in clay shale: Mont Terri. GeoScience Ltd
Martin CD, Stimpson B (1994) The effect of sample disturbance on laboratory properties of Lac du Bonnet granite. Can Geotech J 31:692–702
McClintock FA, Walsh JB (1962) Friction on Griffith cracks in rocks under pressure. In: Proceedings of the 4th US National Congress on Applied Mechanics, New York, pp 1015–1021
Murrell SAF (1963) A criterion for brittle fracture of rocks and concrete under triaxial stress and the effect of pore pressure on the criterion. In: Rock Mechanics, 563-577
Murrell SAF (1964) The theory of the propagation of elliptical Griffith cracks under various conditions of plane strain or plane stress. Br J Appl Phys 15:1195–1223
Nemat-Nasser S, Horii H (1982) Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J Geophysical Research 87(B8):6805–6821
Olsson M, Niklasson B, Wilson L, Andersson C, Christiansson R (2004) Äspö HRL experiences of blasting of the TASQ tunnel. 77
Paterson MS, Wong TF (2005) Experimental rock deformation—The brittle field. Springer, Heidelberg
Read RS, Martin CD (1996) Technical summary of AECL’s Mine-by experiment, phase 1: excavation response. AECL, p 169
Schock RN, Heard HC, Stephens DR (1973) Stress–strain behaviour of granodiorite and two graywackes on compression to 20 kbar. J Geophys Res 78:5922–5941
Sheorey PR (1997) Empirical rock failure criteria. A.A. Balkema, Rotterdam, p 176
Tham LG, Liu H, Tang CA, Lee PKK, Tsui Y (2005) On tension failure of 2-D rock specimens and associated acoustic emission. Rock Mech Rock Eng 38(1):1–19
Vutukuri VS, Lama RD, Saluja SS (1974) Handbook on mechanical properties of rocks, vol I—testing techniques and results. Trans Tech Publications, p 280
Wong RHC, Tang CA, Chau KT, Lin P (2002) Splitting failure in brittle rocks containing pre-existing flaws under uniaxial compression. Eng Fract Mech 69(17):1853–1871
Acknowledgments
The author would like to thank Mr. S.J. Kim, Mr. G. Maybee, and Mr. D. Marr for conducting the laboratory tests at the Geomechanics Research Centre, MIRARCO-Mining Innovation, Laurentian University, Canada. The author also would like to thank Drs. D. McCreath and E. Hoek for reviewing the manuscript, and Drs. P. Kaiser and D. Martin for many helpful discussions on brittle failure of rocks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cai, M. Practical Estimates of Tensile Strength and Hoek–Brown Strength Parameter m i of Brittle Rocks. Rock Mech Rock Eng 43, 167–184 (2010). https://doi.org/10.1007/s00603-009-0053-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00603-009-0053-1