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Erschienen in: Geotechnical and Geological Engineering 1/2016

16.11.2015 | Original paper

Application of the Christensen Failure Criterion to Intact Rock

verfasst von: Hossein Rafiei Renani, C. Derek Martin, Evert Hoek

Erschienen in: Geotechnical and Geological Engineering | Ausgabe 1/2016

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Abstract

The Christensen criterion, originally introduced in materials science, has a simple mathematical form and uniaxial tensile and compressive strength as the only parameters, making it an attractive candidate for rock engineering purposes. In this study, the applicability of the criterion to rock materials is examined. Explicit equations for application of the criterion under biaxial, triaxial compression, triaxial extension, and polyaxial states of stresses are derived. A comprehensive strength data set including the results of tests on synthetic rock, chert dyke, Carrara marble and Westerly granite is utilized to examine the accuracy of the Christensen criterion to the failure of rock material. The two surprising findings about the Christensen criterion are the zero values of tensile strength and the very low slopes of the failure envelope obtained from fitting analyses for chert dyke and Westerly granite. It is shown that the two problems are interrelated and the values of tensile strength tend to zero to produce higher slopes. It is then mathematically proven that the maximum initial slope of the Christensen failure envelope is limited to 4 in triaxial compression and 2.5 in triaxial extension which is considerably lower than the slope of experimental data. The accuracy of the Christensen criterion was found to be significantly lower than the well-established Hoek–Brown criterion. The circular π-plane representations and brittle-to-ductile transition limits from the Christensen criterion are also inconsistent with the observed behavior of rocks.

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Literatur
Zurück zum Zitat Al-Ajmi AM, Zimmerman RW (2005) Relation between the Mogi and the Coulomb failure criteria. Int J Rock Mech Min Sci 42:431–439CrossRef Al-Ajmi AM, Zimmerman RW (2005) Relation between the Mogi and the Coulomb failure criteria. Int J Rock Mech Min Sci 42:431–439CrossRef
Zurück zum Zitat Benz T, Schwab R (2008) A quantitative comparison of six rock failure criteria. Int J Rock Mech Min Sci 45:1176–1186CrossRef Benz T, Schwab R (2008) A quantitative comparison of six rock failure criteria. Int J Rock Mech Min Sci 45:1176–1186CrossRef
Zurück zum Zitat Benz T, Schwab R, Kauther RA, Vermeer PAA (2008) Hoek–Brown criterion with intrinsic material strength factorization. Int J Rock Mech Min Sci 45(2):210–222CrossRef Benz T, Schwab R, Kauther RA, Vermeer PAA (2008) Hoek–Brown criterion with intrinsic material strength factorization. Int J Rock Mech Min Sci 45(2):210–222CrossRef
Zurück zum Zitat Christensen RM (1997) Yield functions/failure criteria for isotropic materials. Proceedings of mathematical, physical and engineering sciences, vol 453. The Royal Society, London, pp 1473–1491 Christensen RM (1997) Yield functions/failure criteria for isotropic materials. Proceedings of mathematical, physical and engineering sciences, vol 453. The Royal Society, London, pp 1473–1491
Zurück zum Zitat Christensen RM (2000) Yield functions, damage states and intrinsic strength. Math Mech Solids 5:285–300CrossRef Christensen RM (2000) Yield functions, damage states and intrinsic strength. Math Mech Solids 5:285–300CrossRef
Zurück zum Zitat Christensen RM (2004) A two property yield, failure (fracture) criterion for homogeneous, isotropic materials. J Eng Mater Technol 126:45–52CrossRef Christensen RM (2004) A two property yield, failure (fracture) criterion for homogeneous, isotropic materials. J Eng Mater Technol 126:45–52CrossRef
Zurück zum Zitat Christensen RM (2005) Exploration of ductile, brittle failure characteristics through a two parameter yield/failure criterion. Mater Sci Eng A 394:417–424CrossRef Christensen RM (2005) Exploration of ductile, brittle failure characteristics through a two parameter yield/failure criterion. Mater Sci Eng A 394:417–424CrossRef
Zurück zum Zitat Christensen RM (2006a) Yield functions and plastic potentials for BCC metals and possibly other materials. J Mech Mater Struct 1:183–200CrossRef Christensen RM (2006a) Yield functions and plastic potentials for BCC metals and possibly other materials. J Mech Mater Struct 1:183–200CrossRef
Zurück zum Zitat Christensen RM (2006b) A comparative evaluation of three isotropic, two property failure theories. J Appl Mech 73:852–859CrossRef Christensen RM (2006b) A comparative evaluation of three isotropic, two property failure theories. J Appl Mech 73:852–859CrossRef
Zurück zum Zitat Christensen RM (2007) A comprehensive theory of yielding and failure for isotropic materials. J Eng Mater Technol 129:173–181CrossRef Christensen RM (2007) A comprehensive theory of yielding and failure for isotropic materials. J Eng Mater Technol 129:173–181CrossRef
Zurück zum Zitat Colmenares LB, Zoback MD (2002) A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci 39:695–729CrossRef Colmenares LB, Zoback MD (2002) A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci 39:695–729CrossRef
Zurück zum Zitat Drucker D, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 10:157–165 Drucker D, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 10:157–165
Zurück zum Zitat 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 37:285–296CrossRef 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 37:285–296CrossRef
Zurück zum Zitat Hammah RE, Carvalho JL (2011) An introduction to the Christensen criterion—an answer to true, representative modelling of intact rock yielding/failure? In: CD-Rom proceedings of the international symposium on rock slope stability in open pit mining and civil engineering, Vancouver, BC Hammah RE, Carvalho JL (2011) An introduction to the Christensen criterion—an answer to true, representative modelling of intact rock yielding/failure? In: CD-Rom proceedings of the international symposium on rock slope stability in open pit mining and civil engineering, Vancouver, BC
Zurück zum Zitat Hoek E (1965) Rock fracture under static stress conditions. Ph.D. thesis, The Faculty of Engineering, University of Cape Town, South Africa Hoek E (1965) Rock fracture under static stress conditions. Ph.D. thesis, The Faculty of Engineering, University of Cape Town, South Africa
Zurück zum Zitat Hoek E (1983) Strength of jointed rock masses, 23rd Rankine Lecture. Géotechnique 33(3):187–223 Hoek E (1983) Strength of jointed rock masses, 23rd Rankine Lecture. Géotechnique 33(3):187–223
Zurück zum Zitat Hoek E (1990) Estimating Mohr-Coulomb friction and cohesion values from the HoekBrown failure criterion. Int J Rock Mech Min Sci Geomech Abst 12(3):227–229 Hoek E (1990) Estimating Mohr-Coulomb friction and cohesion values from the HoekBrown failure criterion. Int J Rock Mech Min Sci Geomech Abst 12(3):227–229
Zurück zum Zitat Hoek E, Brown ET (1980) Empirical strength criterion for rock masses. J Geotech Eng Div ASCE 106(GT9):1013–1035 Hoek E, Brown ET (1980) Empirical strength criterion for rock masses. J Geotech Eng Div ASCE 106(GT9):1013–1035
Zurück zum Zitat Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34:1165–1186CrossRef Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34:1165–1186CrossRef
Zurück zum Zitat Hoek E, Wood D, Shah S (1992) A modified Hoek–Brown criterion for jointed rock masses. In: Hudson J (ed) Proceedings of rock characterization symposium, International Society for Rock Mechanics: Eurock ’92. British Geotechnical Society, London, pp 209–213 Hoek E, Wood D, Shah S (1992) A modified Hoek–Brown criterion for jointed rock masses. In: Hudson J (ed) Proceedings of rock characterization symposium, International Society for Rock Mechanics: Eurock ’92. British Geotechnical Society, London, pp 209–213
Zurück zum Zitat Hoek E, Carranza-Torres C, Corkum B (2002) Hoek–Brown failure criterion—2002 edition. In: Proceedings of NARMS-TAC Conference, Toronto, ON, pp 267–273 Hoek E, Carranza-Torres C, Corkum B (2002) Hoek–Brown failure criterion—2002 edition. In: Proceedings of NARMS-TAC Conference, Toronto, ON, pp 267–273
Zurück zum Zitat Howarth DF, Rowlands JC (1987) Quantitative assessment of rock texture and correlation with drillability and strength properties. Rock Mech Rock Eng 20:57–85CrossRef Howarth DF, Rowlands JC (1987) Quantitative assessment of rock texture and correlation with drillability and strength properties. Rock Mech Rock Eng 20:57–85CrossRef
Zurück zum Zitat Jaeger JC, Cook NGW (1979) Fundamentals of rock mechanics, 3rd edn. Chapman and Hal, London Jaeger JC, Cook NGW (1979) Fundamentals of rock mechanics, 3rd edn. Chapman and Hal, London
Zurück zum Zitat Johnson B, Friedman M, Hopkins TW (1987) Strength and microfracturing of Westerly granite extended wet and dry at temperatures to 800 °C and pressures to 200 MPa. In: Proceedings of the 28th U.S. symposium on rock mechanics (USRMS), Tucson, AZ, pp 399–412 Johnson B, Friedman M, Hopkins TW (1987) Strength and microfracturing of Westerly granite extended wet and dry at temperatures to 800 °C and pressures to 200 MPa. In: Proceedings of the 28th U.S. symposium on rock mechanics (USRMS), Tucson, AZ, pp 399–412
Zurück zum Zitat Kim MK, Lade PV (1984) Modelling rock strength in three dimensions. Int J Rock Mech Min Sci 21:21–33CrossRef Kim MK, Lade PV (1984) Modelling rock strength in three dimensions. Int J Rock Mech Min Sci 21:21–33CrossRef
Zurück zum Zitat Krech WW, Henderson FA, Hjelmstad KE (1974) A standard rock suite for rapid excavation research. Bureau of Mines Report of Investigations No. 7865, 29 p Krech WW, Henderson FA, Hjelmstad KE (1974) A standard rock suite for rapid excavation research. Bureau of Mines Report of Investigations No. 7865, 29 p
Zurück zum Zitat Mogi K (1971) Fracture and flow of rocks under high triaxial compression. J Geophys Res 76:1255–1269CrossRef Mogi K (1971) Fracture and flow of rocks under high triaxial compression. J Geophys Res 76:1255–1269CrossRef
Zurück zum Zitat Nguyen Si-H, Chemenda AI, Ambre J (2011) Influence of the loading conditions on the mechanical response of granular materials as constrained from experimental tests on synthetic rock analogue material. Int J Rock Mech Min Sci 48:103–115CrossRef Nguyen Si-H, Chemenda AI, Ambre J (2011) Influence of the loading conditions on the mechanical response of granular materials as constrained from experimental tests on synthetic rock analogue material. Int J Rock Mech Min Sci 48:103–115CrossRef
Zurück zum Zitat Pan XD, Hudson JA (1988) A simplified three dimensional Hoek–Brown yield criterion. In: Romana M (ed) Rock mechanics and power plants. Balkema, Rotterdam, pp 95–103 Pan XD, Hudson JA (1988) A simplified three dimensional Hoek–Brown yield criterion. In: Romana M (ed) Rock mechanics and power plants. Balkema, Rotterdam, pp 95–103
Zurück zum Zitat Priest SD (2005) Determination of shear strength and three-dimensional yield strength for the Hoek–Brown yield criterion. Rock Mech Rock Eng 38(4):299–327CrossRef Priest SD (2005) Determination of shear strength and three-dimensional yield strength for the Hoek–Brown yield criterion. Rock Mech Rock Eng 38(4):299–327CrossRef
Zurück zum Zitat Rafiei H (2011) New empirical polyaxial criterion for rock strength. Int J Rock Mech Min Sci 48:922–931CrossRef Rafiei H (2011) New empirical polyaxial criterion for rock strength. Int J Rock Mech Min Sci 48:922–931CrossRef
Zurück zum Zitat Ramsey JM, Chester FM (2004) Hybrid fracture and the transition from extension fracture to shear fracture. Nature 428:63–66CrossRef Ramsey JM, Chester FM (2004) Hybrid fracture and the transition from extension fracture to shear fracture. Nature 428:63–66CrossRef
Zurück zum Zitat Schwartz AE (1964) Failure of rock in the triaxial shear test. In: Proceedings of the 6th US rock mechanics symposium, Rolla, Missouri. pp 109–135 Schwartz AE (1964) Failure of rock in the triaxial shear test. In: Proceedings of the 6th US rock mechanics symposium, Rolla, Missouri. pp 109–135
Zurück zum Zitat Zhang L, Zhu H (2007) Three-dimensional Hoek–Brown strength criterion for rocks. J Geotech Geoenvir Eng ASCE 133(9):1128–1135CrossRef Zhang L, Zhu H (2007) Three-dimensional Hoek–Brown strength criterion for rocks. J Geotech Geoenvir Eng ASCE 133(9):1128–1135CrossRef
Metadaten
Titel
Application of the Christensen Failure Criterion to Intact Rock
verfasst von
Hossein Rafiei Renani
C. Derek Martin
Evert Hoek
Publikationsdatum
16.11.2015
Verlag
Springer International Publishing
Erschienen in
Geotechnical and Geological Engineering / Ausgabe 1/2016
Print ISSN: 0960-3182
Elektronische ISSN: 1573-1529
DOI
https://doi.org/10.1007/s10706-015-9945-x

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