Strength behaviour of sandstone subjected to polyaxial state of stress
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摘要
The paper presents an experimental investigation on the strength behaviour of natural rock subjected to polyaxial state of stress. The polyaxial tests were conducted on cubical specimens of sandstone obtained from the Shivpuri district in Madhya Pradesh state of India, The specimens having nominal dimensions of100 mm x 100 mm x 100 mm were tested using a polyaxial testing machine. Twenty-five combinations of intermediate and minor principal stresses were applied and the specimens were loaded till failure occurs. It was observed that the intermediate principal stress has a substantial effect on the strength of the Shivpuri sandstone. A database of rock strength under various combinations of σ_2 and σ_3 was obtained for the Shivpuri sandstone. The database was used to study the predictability of five most commonly used strength criterion. Root mean square error(RMSE), average absolute relative error percentage(AAREP) and coefficient of accordance(COA) were used as indices for the measure of goodness of fit. It was observed that the least error in the prediction was shown by modified Mohr Coulomb criterion followed by modified Weibols and Cook criterion. A probability analysis of the error in prediction was also done.
The paper presents an experimental investigation on the strength behaviour of natural rock subjected to polyaxial state of stress. The polyaxial tests were conducted on cubical specimens of sandstone obtained from the Shivpuri district in Madhya Pradesh state of India, The specimens having nominal dimensions of100 mm x 100 mm x 100 mm were tested using a polyaxial testing machine. Twenty-five combinations of intermediate and minor principal stresses were applied and the specimens were loaded till failure occurs. It was observed that the intermediate principal stress has a substantial effect on the strength of the Shivpuri sandstone. A database of rock strength under various combinations of σ_2 and σ_3 was obtained for the Shivpuri sandstone. The database was used to study the predictability of five most commonly used strength criterion. Root mean square error(RMSE), average absolute relative error percentage(AAREP) and coefficient of accordance(COA) were used as indices for the measure of goodness of fit. It was observed that the least error in the prediction was shown by modified Mohr Coulomb criterion followed by modified Weibols and Cook criterion. A probability analysis of the error in prediction was also done.
The paper presents an experimental investigation on the strength behaviour of natural rock subjected to polyaxial state of stress. The polyaxial tests were conducted on cubical specimens of sandstone obtained from the Shivpuri district in Madhya Pradesh state of India, The specimens having nominal dimensions of100 mm x 100 mm x 100 mm were tested using a polyaxial testing machine. Twenty-five combinations of intermediate and minor principal stresses were applied and the specimens were loaded till failure occurs. It was observed that the intermediate principal stress has a substantial effect on the strength of the Shivpuri sandstone. A database of rock strength under various combinations of σ_2 and σ_3 was obtained for the Shivpuri sandstone. The database was used to study the predictability of five most commonly used strength criterion. Root mean square error(RMSE), average absolute relative error percentage(AAREP) and coefficient of accordance(COA) were used as indices for the measure of goodness of fit. It was observed that the least error in the prediction was shown by modified Mohr Coulomb criterion followed by modified Weibols and Cook criterion. A probability analysis of the error in prediction was also done.
引文
[1]Amadei B,Stephansson O.Rock stress and its measurement.London:Chapman and Hall;1997.
[2]Zang A,Stephansson 0.Stress field of the Earth's Crust.Dordrecht:Springer;2010.
[3]Murrell SAF.A criterion for brittle fracture of rocks and concrete under triaxial stress and the effect of pore pressure on the criterion.In:Proceedings of the5th symposium on rock mechan.Minneapolis:Pergamon Press;1963.
[4]Vonkármán T.Festigkeitsversuche unter allseitigem Druck.Z Ver Deut Ingr1911;55(2):1749-57.
[5]B(o|¨)ker R.Die Mechanik der bleibenden Formanderung in kristallinisch aufgebauten Korpern Verhandl Deut Ingr Mitt Forsch 1915;175:1-51.
[6]Murrell SAF.The effect of triaxial stress systems on the strength of rocks at atmospheric temperatures.Geophys J Int 1965;10(3):231-81.
[7]HandinJ,Heard HC,MagouirkJN.Effects of the intermediate principal stress on the failure of limestone,dolomite,and glass at different temperatures and strain rates.J Geophys Res 1967;72(2):611-40.
[8]Mogi K.Effect of the intermediate principal stress on rock failure.J Geophys Res 1967;72(20):5117-31.
[9]Mogi K.Fracture and flow of rocks under high triaxial compression.J Geophys Res 1971;76(5):1255-69.
[10]Beron AI,Chirkov SE.Investigation of strength of rocks under nonuniform triaxial compression conditions.Nauchnye soobshcheniya Instituta gornogo dela im A.A.Skochinskogo 1969;61:33-8.
[11]Chirkov SE.Results of studies of properties of rocks under nonuniform compression conditions.In:Proceedings of Issledovaniya Prochnosti Gornykh Porod.Moskva:Nauka;1973,p.113-34.
[12]Chirkov SE.Strength of rocks under unequal triaxial compression.FizikoTekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh 1976;1:11-7.
[13]Atlinson RH,Ko H.A fluid cushion,multiaxial cell for testing cubical rock specimens.Int J Rock Mech Min Sci Geomech Abstr 1973;10:351-61.
[14]Michelis P.A true triaxial cell for low and high pressure experiments.IntJ Rock Mech Min Sci Geomech Abstr 1985;22(3):183-8.
[15]Takahashi M,Koide H.Effect of the intermediate principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than2000 m.In:Proceedings of rock at great depth.Rotterdam:Balkema;1989.p.19-26.
[16]Kwasniewski M,Mogi K.Effect of intermediate principal stress on the failure of a foliated anisotropic rock In:Proceedings of mechanics of jointed and foul ted rock.Rotterdam:Balkema;1990.p.407-16.
[17]Smart BGD.A true Triaxial cell for testing cylindrical rock specimens,IntJ Rock Mech Min Sci Geomech Abstr 1995;32(3):269-75.
[18]Wawersik WR,Carlson LW,Holcomb DJ,Williams RJ.New method for truetriaxial rock testing,Int J Rock Mech Min Sci 1997:34.
[19]Haimson B,Chang C 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 2000;37(l):285-96.
[20]Chang C,Haimson B.True triaxial strength and deformability of the German Continental Deep Drilling Program(KTB)deep hole amphibolite.J Geophy Res:Solid Earth 2000;105(B8):18999-9013.
[21]Kwasniewslki M,Takahashi M,Li X.Volunme changes in sandstone under true triaxial compression conditions.In:Proceedings of technology roadmap for rock mechanics.South African:South African Institute of Mining and Metallurgy;2003.p.683-8.
[22]Chang C,Haimson B.Non-dilatant deformation and failure mechanism in two Long Valley Caldera rocks under true triaxial compression.IntJ Rock Mech Min Sci 2005;42(3):402-14.
[23]Tiwari RP,Rao KS.Physical modelling of a rock mass under a true Triaxial stress.Int J Rock Mech Min Sci 2004;41(3):1-6.
[24]Kwasniewski M.Mechanical behaviour of rocks under true triaxial compression conditions-volumetric strain and dilatancy.Arch Min Sci2007;52(3):409-35.
[25]Kwasniewski M,Takahashi M.Effect of confining pressure,intermediate principal stress and minimum principal stress on the mechanical behavior of a sandstone.In:Proceedings of the 11th congress,ISRM.Lisbon:Taylor&Francis Group,London;2007.p.237-47.
[26]Descamps F,Tshibangu JP.Modelling the limiting envelopes of rocks in the octahedral plane.Oil Gas Sci Technol 2007;62(5):683-94.
[27]Walsri C Poonprakon P,Thosuwan R,Fuenkajorn K.Compressive and tensile strengths of sandstones under true triaxial stresses.In:Proceedings of 2nd Thailand symposium on rock mechanics.Chonburi:published by Geomechanics Research Unit,.Suranaree University of Technology,Nakhon Ratchasima;2009.p.199-218,ISBN 9789745336247.
[28]Pobwandee T,Fuenkajorn K.Effect of intermediate principal stress on compressive strength of Phra Wihan sandstone.In:Proceedings of 3rd Thailand symposium on rock mechanics.Phetchaburi:published by Geomechanics Research Unit,Suranaree University of Technology,Nakhon Ratchasima;2011.p.55-62,ISBN 9789745336360.
[29]Sriapai T,Walsri C,Fuenkajorn K.True-triaxial compressive strength of Maha Sarakham salt.Int]Rock Mech Min Sci 2013;61:256-65.
[30]Wiebols GA,Cook NGW.An energy criterion for the strength of rock in polyaxial compression.Int J Rock Mech Min Sci Geomech Abstr 1968;5(6):529-49.
[31]Li D,Zhao F,Zheng M.Fractal characteristics of cracks and fragmeints generated in unloading rockburst tests.Int J Min Sci Technol 2014;24(6):819-23.
[32]He M,Nie W,Zhao Z,Cheng C Micro-and macro-fractures of coarse granite under true-triaxial unloading conditions.Int J Min Sci Technol 2011;21(3):389-94.
[33]Al-Ajmi AM.Wellbore stability analysis based on a new true-triaxial failure criterion.Land and Water Resources Engineering;2006.
[34]You M.True-Triaxial strength criteria for rock.IntJ Rock Mech Min Sci 2009;46(1):115-21.
[35]Rafiai H.New empirical polyaxial criterion for rock strength.IntJ Rock Mech Min Sci 2011;48(6):922-31.
[36]Drucker DC,Prager W.Soil mechanics and plastic analysis for limit design.Quart Appl Math 1952;10(2):157-65.
[37]Zhou S.A program to model the initial shape and extent of borehole breakout.Comput Geosci 1994;20(7-8):1143-60.
[38]Ewy RT.Wellbore-stability predictions by use of a modified Lade criterion.SPE Drill Completion 1999;14(2):85-91.
[39]Al-Ajmi AM,Zimmerman RW.Relation between the Mogi and the Coulomb failure criteria.Int J Rock Mech Min Sci 2005;42(3):431-9.
[40]Zhang L,Zhu H.Three-dimensional Hoek-Brown strength criterion for rocks.J Geotech Geoenvir Eng:ASCE 2007;133(9):1128-35.
[41]Singh M,Raj A,Singh B.Modified Mohr-Coulomb criterion for non-linear triaxial and polyaxial strength of intact rocks.Int J Rock Mech Min Sci 2011;48(4):546-55.
[42]Benz T,Schwab R A quantitative comparison of six rock failure criteria.Int J Rock Mech Min Sci 2008;45(7):1176-86.
[43]Colmenares LB,Zoback MD.A Statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks.Int J Rock Mech Min Sci 2002;39(6):695-729.
[44]Kwasniewski M.Mechanical behavior of rocks under true Triaxial compression conditions-a review.True triaxial testing of rocks.London:CRC Press;2012.
[45]Takahashi M,Narita T,Tomishima Y,Arai R Various loading systems for rock true Triaxial compression test.J Jpn Soc Eng Geol 2001;42(4):242-7.
[46]Mogi K.Experimental rock mechanics.Londan:Tylor and Francis;2007.
[47]Li X,Shi L,Bai B,Li Q,Xu D,Feng X.True-triaxial testing techniques for rocksState of the art and future perspectives.True triaxial testing of rocks.London:CRC Press;2012.
[48]Reches ZE,Dieterich JH.Faulting of rocks in three-dimensional strain fields 1.Failure of rocks in polyaxial,servo-control experiments.Tectonophysics1983;95(1):111-32.
[2]Zang A,Stephansson 0.Stress field of the Earth's Crust.Dordrecht:Springer;2010.
[3]Murrell SAF.A criterion for brittle fracture of rocks and concrete under triaxial stress and the effect of pore pressure on the criterion.In:Proceedings of the5th symposium on rock mechan.Minneapolis:Pergamon Press;1963.
[4]Vonkármán T.Festigkeitsversuche unter allseitigem Druck.Z Ver Deut Ingr1911;55(2):1749-57.
[5]B(o|¨)ker R.Die Mechanik der bleibenden Formanderung in kristallinisch aufgebauten Korpern Verhandl Deut Ingr Mitt Forsch 1915;175:1-51.
[6]Murrell SAF.The effect of triaxial stress systems on the strength of rocks at atmospheric temperatures.Geophys J Int 1965;10(3):231-81.
[7]HandinJ,Heard HC,MagouirkJN.Effects of the intermediate principal stress on the failure of limestone,dolomite,and glass at different temperatures and strain rates.J Geophys Res 1967;72(2):611-40.
[8]Mogi K.Effect of the intermediate principal stress on rock failure.J Geophys Res 1967;72(20):5117-31.
[9]Mogi K.Fracture and flow of rocks under high triaxial compression.J Geophys Res 1971;76(5):1255-69.
[10]Beron AI,Chirkov SE.Investigation of strength of rocks under nonuniform triaxial compression conditions.Nauchnye soobshcheniya Instituta gornogo dela im A.A.Skochinskogo 1969;61:33-8.
[11]Chirkov SE.Results of studies of properties of rocks under nonuniform compression conditions.In:Proceedings of Issledovaniya Prochnosti Gornykh Porod.Moskva:Nauka;1973,p.113-34.
[12]Chirkov SE.Strength of rocks under unequal triaxial compression.FizikoTekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh 1976;1:11-7.
[13]Atlinson RH,Ko H.A fluid cushion,multiaxial cell for testing cubical rock specimens.Int J Rock Mech Min Sci Geomech Abstr 1973;10:351-61.
[14]Michelis P.A true triaxial cell for low and high pressure experiments.IntJ Rock Mech Min Sci Geomech Abstr 1985;22(3):183-8.
[15]Takahashi M,Koide H.Effect of the intermediate principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than2000 m.In:Proceedings of rock at great depth.Rotterdam:Balkema;1989.p.19-26.
[16]Kwasniewski M,Mogi K.Effect of intermediate principal stress on the failure of a foliated anisotropic rock In:Proceedings of mechanics of jointed and foul ted rock.Rotterdam:Balkema;1990.p.407-16.
[17]Smart BGD.A true Triaxial cell for testing cylindrical rock specimens,IntJ Rock Mech Min Sci Geomech Abstr 1995;32(3):269-75.
[18]Wawersik WR,Carlson LW,Holcomb DJ,Williams RJ.New method for truetriaxial rock testing,Int J Rock Mech Min Sci 1997:34.
[19]Haimson B,Chang C 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 2000;37(l):285-96.
[20]Chang C,Haimson B.True triaxial strength and deformability of the German Continental Deep Drilling Program(KTB)deep hole amphibolite.J Geophy Res:Solid Earth 2000;105(B8):18999-9013.
[21]Kwasniewslki M,Takahashi M,Li X.Volunme changes in sandstone under true triaxial compression conditions.In:Proceedings of technology roadmap for rock mechanics.South African:South African Institute of Mining and Metallurgy;2003.p.683-8.
[22]Chang C,Haimson B.Non-dilatant deformation and failure mechanism in two Long Valley Caldera rocks under true triaxial compression.IntJ Rock Mech Min Sci 2005;42(3):402-14.
[23]Tiwari RP,Rao KS.Physical modelling of a rock mass under a true Triaxial stress.Int J Rock Mech Min Sci 2004;41(3):1-6.
[24]Kwasniewski M.Mechanical behaviour of rocks under true triaxial compression conditions-volumetric strain and dilatancy.Arch Min Sci2007;52(3):409-35.
[25]Kwasniewski M,Takahashi M.Effect of confining pressure,intermediate principal stress and minimum principal stress on the mechanical behavior of a sandstone.In:Proceedings of the 11th congress,ISRM.Lisbon:Taylor&Francis Group,London;2007.p.237-47.
[26]Descamps F,Tshibangu JP.Modelling the limiting envelopes of rocks in the octahedral plane.Oil Gas Sci Technol 2007;62(5):683-94.
[27]Walsri C Poonprakon P,Thosuwan R,Fuenkajorn K.Compressive and tensile strengths of sandstones under true triaxial stresses.In:Proceedings of 2nd Thailand symposium on rock mechanics.Chonburi:published by Geomechanics Research Unit,.Suranaree University of Technology,Nakhon Ratchasima;2009.p.199-218,ISBN 9789745336247.
[28]Pobwandee T,Fuenkajorn K.Effect of intermediate principal stress on compressive strength of Phra Wihan sandstone.In:Proceedings of 3rd Thailand symposium on rock mechanics.Phetchaburi:published by Geomechanics Research Unit,Suranaree University of Technology,Nakhon Ratchasima;2011.p.55-62,ISBN 9789745336360.
[29]Sriapai T,Walsri C,Fuenkajorn K.True-triaxial compressive strength of Maha Sarakham salt.Int]Rock Mech Min Sci 2013;61:256-65.
[30]Wiebols GA,Cook NGW.An energy criterion for the strength of rock in polyaxial compression.Int J Rock Mech Min Sci Geomech Abstr 1968;5(6):529-49.
[31]Li D,Zhao F,Zheng M.Fractal characteristics of cracks and fragmeints generated in unloading rockburst tests.Int J Min Sci Technol 2014;24(6):819-23.
[32]He M,Nie W,Zhao Z,Cheng C Micro-and macro-fractures of coarse granite under true-triaxial unloading conditions.Int J Min Sci Technol 2011;21(3):389-94.
[33]Al-Ajmi AM.Wellbore stability analysis based on a new true-triaxial failure criterion.Land and Water Resources Engineering;2006.
[34]You M.True-Triaxial strength criteria for rock.IntJ Rock Mech Min Sci 2009;46(1):115-21.
[35]Rafiai H.New empirical polyaxial criterion for rock strength.IntJ Rock Mech Min Sci 2011;48(6):922-31.
[36]Drucker DC,Prager W.Soil mechanics and plastic analysis for limit design.Quart Appl Math 1952;10(2):157-65.
[37]Zhou S.A program to model the initial shape and extent of borehole breakout.Comput Geosci 1994;20(7-8):1143-60.
[38]Ewy RT.Wellbore-stability predictions by use of a modified Lade criterion.SPE Drill Completion 1999;14(2):85-91.
[39]Al-Ajmi AM,Zimmerman RW.Relation between the Mogi and the Coulomb failure criteria.Int J Rock Mech Min Sci 2005;42(3):431-9.
[40]Zhang L,Zhu H.Three-dimensional Hoek-Brown strength criterion for rocks.J Geotech Geoenvir Eng:ASCE 2007;133(9):1128-35.
[41]Singh M,Raj A,Singh B.Modified Mohr-Coulomb criterion for non-linear triaxial and polyaxial strength of intact rocks.Int J Rock Mech Min Sci 2011;48(4):546-55.
[42]Benz T,Schwab R A quantitative comparison of six rock failure criteria.Int J Rock Mech Min Sci 2008;45(7):1176-86.
[43]Colmenares LB,Zoback MD.A Statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks.Int J Rock Mech Min Sci 2002;39(6):695-729.
[44]Kwasniewski M.Mechanical behavior of rocks under true Triaxial compression conditions-a review.True triaxial testing of rocks.London:CRC Press;2012.
[45]Takahashi M,Narita T,Tomishima Y,Arai R Various loading systems for rock true Triaxial compression test.J Jpn Soc Eng Geol 2001;42(4):242-7.
[46]Mogi K.Experimental rock mechanics.Londan:Tylor and Francis;2007.
[47]Li X,Shi L,Bai B,Li Q,Xu D,Feng X.True-triaxial testing techniques for rocksState of the art and future perspectives.True triaxial testing of rocks.London:CRC Press;2012.
[48]Reches ZE,Dieterich JH.Faulting of rocks in three-dimensional strain fields 1.Failure of rocks in polyaxial,servo-control experiments.Tectonophysics1983;95(1):111-32.