基于摩尔-库仑准则的断续节理岩体复合损伤本构模型
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摘要
针对工程实际中断续节理裂隙岩体的损伤本构模型,假设岩石微元强度服从Weibull随机分布,以摩尔-库仑破坏准则作为描述微元强度的表示方法,推导出细观损伤变量。利用能量和断裂力学理论,综合考虑节理几何特征及力学特性,推导宏观损伤变量计算公式。基于Lemaitre应变等效假设,考虑宏细观缺陷耦合作用,推导出复合损伤变量,建立了基于摩尔-库仑准则的宏细观缺陷耦合作用的断续裂隙岩体损伤本构模型。研究结果表明:(1)采用摩尔-库仑准则作为描述微元强度的统计分布变量建立的损伤模型能够较好地反映岩石内部缺陷分布和变形特征,该模型真实地反映岩石微元强度受应力状态的影响。(2)该模型建立的理论曲线与断续节理岩体的试验曲线吻合较好。(3)节理裂隙岩体宏观损伤变量及峰值强度随节理倾角的变化规律与综合考虑宏细观耦合作用下的损伤变量及裂隙岩体峰值强度随节理倾角的变化规律基本一致。(4)宏细观耦合作用下的等效弹性模量与节理贯通率呈非线性负相关;在节理倾角一定的情况下,损伤变量与节理长度呈非线性正相关;在贯通率较小时,岩体的宏观损伤变量与内摩擦角的关系呈线性负相关变化,贯通率达到一定程度,线性关系变成非线性关系。
To investigate damage constitutive model of rock mass with intermittent joints in engineering practice with the assumption that the rock microscopic element strength obeys the Weibull random distribution function, the Mohr-Coulomb failure criterion is used to describe the strength of the microscopic element and the microscopic damage variable is deduced. A formula for macroscopic damage variable is deduced using energy and fracture mechanics theory and considering geometrical features and mechanical properties of the joint. Finally, based on the Lemaitre strain equivalence hypothesis, considering the macro-meso defect coupling effect, the composite damage variable is derived. A damage constitutive model of rock mass with intermittent joints under the action of macro-meso defect coupling is established based on the Mohr-Coulomb criterion. The results show that:(1) The damage model established by the Mohr-Coulomb criterion as the statistical distribution variable describing the strength of the micro-element can well reflect the characteristics of rock deformation and distribution of internal defects, which reflects that the micro-element strength of rock is influenced by the stress state.(2) The theoretical curve established by this model is in good agreement with the test curve of intermittent jointed rock mass.(3) The change of the macro damage variable and peak strength of jointed rock mass with the joint tilt angle and the change of damage variable considering the macro-meso coupling action and the peak strength of jointed rock mass with the joint tilt angle are generally consistent.(4) The relationship between the equivalent elastic moduli of macro-meso coupling and the crack connectivity rate exhibits nonlinear negative correlation. When the joint angle is constant, the relationship between the damage variable and joint length exhibits nonlinear positive correlation. When the connectivity rate is small, the relationship between the macroscopic damage variable and the internal friction angle of jointed rock mass shows a linear negative correlation, and the linear relations become nonlinear relations when the connectivity rate reaches a certain extent.
To investigate damage constitutive model of rock mass with intermittent joints in engineering practice with the assumption that the rock microscopic element strength obeys the Weibull random distribution function, the Mohr-Coulomb failure criterion is used to describe the strength of the microscopic element and the microscopic damage variable is deduced. A formula for macroscopic damage variable is deduced using energy and fracture mechanics theory and considering geometrical features and mechanical properties of the joint. Finally, based on the Lemaitre strain equivalence hypothesis, considering the macro-meso defect coupling effect, the composite damage variable is derived. A damage constitutive model of rock mass with intermittent joints under the action of macro-meso defect coupling is established based on the Mohr-Coulomb criterion. The results show that:(1) The damage model established by the Mohr-Coulomb criterion as the statistical distribution variable describing the strength of the micro-element can well reflect the characteristics of rock deformation and distribution of internal defects, which reflects that the micro-element strength of rock is influenced by the stress state.(2) The theoretical curve established by this model is in good agreement with the test curve of intermittent jointed rock mass.(3) The change of the macro damage variable and peak strength of jointed rock mass with the joint tilt angle and the change of damage variable considering the macro-meso coupling action and the peak strength of jointed rock mass with the joint tilt angle are generally consistent.(4) The relationship between the equivalent elastic moduli of macro-meso coupling and the crack connectivity rate exhibits nonlinear negative correlation. When the joint angle is constant, the relationship between the damage variable and joint length exhibits nonlinear positive correlation. When the connectivity rate is small, the relationship between the macroscopic damage variable and the internal friction angle of jointed rock mass shows a linear negative correlation, and the linear relations become nonlinear relations when the connectivity rate reaches a certain extent.
引文
[1]刘红岩,李俊峰.非贯通节理岩体损伤变量计算方法研究[J].岩土力学,2016,37(增刊1):95-100.LIU Hong-yan,LI Jun-feng.A method for calculating damage variable of rock mass with non-persistent joints[J].Rock and Soil Mechanics,2016,37(Supp.1):95-100.
[2]刘红岩,杨艳,李俊峰,等.基于TCK模型的非贯通节理岩体动态损伤本构模型[J].爆炸与冲击,2016,36(3):319-325.LIU Hong-yan,YANG Yan,LI Jun-feng,et al.Dynamic damage constitutive model for rock mass with non-persistent joints based on the TCK model[J].Explosion and Shock Waves,2016,36(3):319-325.
[3]GOODMAN R E,TAYLOR R E,BREKKE T.A model for the mechanics of jointed rock[J].Journal of the Soil Mechanics and Foundations Division,ASCE,1968,94(3):637-659.
[4]DESAI C S,ZAMAN M M,LIGHTNER J G,et al.Thin-layer element for interfaces and joints[J].International Journal for Numerical Methods in Geomechanics,1984,8(1):19-43.
[5]张力民,张慧,刘红岩.单轴压缩荷载下非贯通闭合节理岩体损伤本构模型[J].煤田地质与勘探,2016,44(1):79-84.ZHANG Li-min,ZHANG Hui,LIU Hong-yan.A damage constitutive model for rock mass with non-persistently closed joints under uniaxial compression load[J].Coal Geology and Exploration,2016,44(1):79-84.
[6]KAWAMOTO T,ICHIKAWA Y,KYOYA T.Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory[J].International Journal for Numerical Analysis Methodin Geomechanics,1988,12(1):1-30.
[7]SWOBODA G,SHEN X P,ROSAS L.Damage model for jointed rock mass and its application to tunneling[J].Computers and Geotechnics,1998,22(3/4):183-203.
[8]江权,冯夏庭,樊义林,等.高应力下硬岩卸荷破裂:白鹤滩水电站地下厂房玄武岩开裂观测实例分析[J].岩石力学与工程学报,2017,36(5):1076-1087.JIANG Quan,FENG Xia-ting,FAN Yi-lin,et al.Unloading break of hard rock under high-geo stress condition:Inner cracking observation for the basalt in the Baihetan's underground powerhouse[J].Chinese Journal of Rock Mechanics and Engineering,2017,36(5):1076-1087.
[9]SHAO J F,CHAUC K T,FENG X T.Modeling of anisotropic damage and creep deformation in brittle rocks[J].International Journal of Rock Mechanics and Mining Sciences,2006,43(4):582-592.
[10]唐春安.岩石破裂过程中的灾变[M].北京:煤炭工业出版社,1993.TANG Chun-an.Catastrophe in rock failure process[M].Beijing:China Coal Industry Publishing House,1993.
[11]杨圣奇,徐卫亚,韦立德,等.单轴压缩下岩石损伤统计本构模型与试验研究[J].河海大学学报(自然科学版),2004,32(2):200-203.YANG Sheng-qi,XU Wei-ya,WEI Li-de,et al.Rock damage constitutive model and experimental study under uniaxial compression[J].Journal of Hohai University(Natural Sciences),2004,32(2):200-203.
[12]徐卫亚,韦立德.岩石损伤统计本构模型的研究[J].岩石力学与工程学报,2002,21(6):787-791.XU Wei-ya,WEI Li-de.Study on statistical damage constitutive model of rock[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(6):787-791.
[13]曹文贵,方祖烈,唐学军.岩石损伤软化统计本构模型之研究[J].岩石力学与工程学报,1998,17(6):628-633.CAO Wen-gui,FANG Zu-lie,TANG Xue-jun.A study of statistical constitutive model for soft and damage rock[J].Chinese Journal of Rock Mechanics and Engineering,1998,17(6):628-633.
[14]陈文玲,李宁.含非贯通裂隙岩体介质的损伤模型[J].岩土工程学报,2000,22(4):430-434.CHEN Wen-ling,LI Ning.Damage model of the rock mass medium with intermittent cracks[J].Chinese Journal of Geotechnical Engineering,2000,22(4):430-434.
[15]KRAJCINOVIC D,SILVA M A G.Statistical aspects of the continuous damage theory[J].International Journal of Solids and Structures,1982,18(7):551-562.
[16]曹文贵,张升.基于Mohr-Coulomb准则的岩石损伤统计分析方法研究[J].湖南大学学报(自然科学版),2005,32(1):43-47.CAO Wen-gui,ZHANG Sheng.Study on the statistical analysis of rock damage based on Mohr-Coulomb criterion[J].Journal of Hunan University(Natural Sciences),2005,32(1):43-47.
[17]杨更社,谢定义.岩体宏观细观损伤的耦合计算分析[C]//第六次全国岩石力学与工程学术大会论文集.武汉:[出版者不详],2000:327-329.YANG Geng-she,XIE Ding-yi.Coupling analysis on the macro-damage and meso-damage of rock masses[C]//Symposium on the Sixth National Rock Mechanics and Engineering Academic Conference.Wuhan:[s.n.],2000:327-329.
[18]张吉宏,刘红岩.综合考虑宏微观复合损伤的节理岩体本构模型[J].煤田地质与勘探,2013,41(6):49-52.ZHANG Ji-hong,LIU Hong-yan.Constitutive model of jointed rock mass by combining macroscopic and microcopic composite damage[J].Coal Geology&Exploration,2013,41(6):49-52.
[19]张力民,吕淑然,刘红岩.综合考虑宏细观缺陷的岩体动态损伤本构模型[J].爆炸与冲击,2015,35(3):428-436.ZHANG Li-min,LüShu-ran,LIU Hong-yan.A dynamic damage constitutive model of rock mass by comprehensively considering macroscopic and mesoscopic flaws[J].Explosion and Shock Waves,2015,35(3):428-436.
[20]赵怡晴,刘红岩,吕淑然,等.基于宏观和细观缺陷耦合的节理岩体损伤本构模型[J].中南大学学报(自然科学版),2015,46(4):1489-1496.ZHAO Yi-qing,LIU Hong-yan,LüShu-ran,et al.Damage constitutive model of jointed rock mass based on coupling macroscopic and mesoscopic flaws[J].Journal of Central South University(Science and Technology),2015,46(4):1489-1496.
[21]袁小清,刘红岩,刘京平.基于宏细观损伤耦合的非贯通裂隙岩体本构模型[J].岩土力学,2015,36(10):2804-2814.YUAN Xiao-qing,LIU Hong-yan,LIU Jing-ping.Constitutive model of rock mass with non-persistent joints based on coupling macroscopic and mesoscopic damages[J].Rock and Soil Mechanics,2015,36(10):2804-2814.
[22]LEMAITRE J.How to use damage mechanics[J].Nuclear Engineering and Design,1984,80(3):233-245.
[23]GRADY D E,KIPP M E.Continuum modeling of explosive fracture in oil shale[J].International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts,1987,17(3):147-157.
[24]NEMAT-NASSER S,OBATA N A.Microcrack model of dilatancy in brittle materials[J].Journal of Applied Mechanics,1988,55(1):24-35.
[25]宁建国,任会兰,方敏杰.基于椭圆形微裂纹演化与汇合的准脆性材料本构模型[J].科学通报,2012,57(21):1978-1986.NING Jian-guo,REN Hui-lan,FANG Min-jie.Constitutive model of quasi brittle materials based on Evolution and coalescence of elliptical microcracks[J].Chinese Science Bulletin,2012,57(21):1978-1986.
[26]PARK C H,BOBET A.Crack initiation,propagation and coalescence from frictional flaws in uniaxial compression[J].Engineering Fracture Mechanics,2010,14:2727-2748.
[27]吕建国,王志乔,刘红岩.岩石断裂与损伤[M].北京:地质出版社,2013:12-54.LüJian-guo,WANG Zhi-qiao,LIU Hong-yan.Rock fracture and damage[M].Beijing:Geological Publishing House,2013:141-146.
[28]HUANG C,SUBHASH G,VITTON S J.A dynamic damage growth model for uniaxial compressive response of rock aggregates[J].Mechanics of Materials,2002,34(5):267-277.
[29]HUANG C,SUBHASH G.Influence of lateral confinement on dynamic damage evolution during uniaxial compressive response of brittle solids[J].Journal of the Mechanics and Physics of Solids,2003,51(6):1089-1105.
[30]LEE S,RAVICHANDRAN G.Crack initiation in brittle solids under multiaxial compression[J].Engineering Fracture Mechanics,2003,70(13):1645-1658.
[31]PALIWAL B,RAMESH KT.An interacting micro-crack damage model for failure of brittle materials under compression[J].Journal of the Mechanics and Physics of Solids,2008,56(3):896-923.
[32]ISIDA M.On the tension of a strip with a central elliptical hole[J].Transactions of the Japan Society of Mechanical Engineering,1955,21(107):514-518.
[33]LI N,CHEN W,ZHANG P.The mechanical properties and a fatigue damage model for jointed rock mass subjected to dynamic cyclical loading[J].International Journal of Rock Mechanics and Mining Sciences,2001,38(7):1071-1079.
[34]陈新,廖志红,李德建.节理倾角及连通率对岩体强度、变形影响的单轴压缩试验研究[J].岩石力学与工程学报,2011,30(4):781-789.CHEN Xin,LIAO Zhi-hong,LI De-jian.Experimental study of effects of joint inclination angle and connectivity rate on strength and deformation properties of rock masses under uniaxial compression[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(4):781-789.
[2]刘红岩,杨艳,李俊峰,等.基于TCK模型的非贯通节理岩体动态损伤本构模型[J].爆炸与冲击,2016,36(3):319-325.LIU Hong-yan,YANG Yan,LI Jun-feng,et al.Dynamic damage constitutive model for rock mass with non-persistent joints based on the TCK model[J].Explosion and Shock Waves,2016,36(3):319-325.
[3]GOODMAN R E,TAYLOR R E,BREKKE T.A model for the mechanics of jointed rock[J].Journal of the Soil Mechanics and Foundations Division,ASCE,1968,94(3):637-659.
[4]DESAI C S,ZAMAN M M,LIGHTNER J G,et al.Thin-layer element for interfaces and joints[J].International Journal for Numerical Methods in Geomechanics,1984,8(1):19-43.
[5]张力民,张慧,刘红岩.单轴压缩荷载下非贯通闭合节理岩体损伤本构模型[J].煤田地质与勘探,2016,44(1):79-84.ZHANG Li-min,ZHANG Hui,LIU Hong-yan.A damage constitutive model for rock mass with non-persistently closed joints under uniaxial compression load[J].Coal Geology and Exploration,2016,44(1):79-84.
[6]KAWAMOTO T,ICHIKAWA Y,KYOYA T.Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory[J].International Journal for Numerical Analysis Methodin Geomechanics,1988,12(1):1-30.
[7]SWOBODA G,SHEN X P,ROSAS L.Damage model for jointed rock mass and its application to tunneling[J].Computers and Geotechnics,1998,22(3/4):183-203.
[8]江权,冯夏庭,樊义林,等.高应力下硬岩卸荷破裂:白鹤滩水电站地下厂房玄武岩开裂观测实例分析[J].岩石力学与工程学报,2017,36(5):1076-1087.JIANG Quan,FENG Xia-ting,FAN Yi-lin,et al.Unloading break of hard rock under high-geo stress condition:Inner cracking observation for the basalt in the Baihetan's underground powerhouse[J].Chinese Journal of Rock Mechanics and Engineering,2017,36(5):1076-1087.
[9]SHAO J F,CHAUC K T,FENG X T.Modeling of anisotropic damage and creep deformation in brittle rocks[J].International Journal of Rock Mechanics and Mining Sciences,2006,43(4):582-592.
[10]唐春安.岩石破裂过程中的灾变[M].北京:煤炭工业出版社,1993.TANG Chun-an.Catastrophe in rock failure process[M].Beijing:China Coal Industry Publishing House,1993.
[11]杨圣奇,徐卫亚,韦立德,等.单轴压缩下岩石损伤统计本构模型与试验研究[J].河海大学学报(自然科学版),2004,32(2):200-203.YANG Sheng-qi,XU Wei-ya,WEI Li-de,et al.Rock damage constitutive model and experimental study under uniaxial compression[J].Journal of Hohai University(Natural Sciences),2004,32(2):200-203.
[12]徐卫亚,韦立德.岩石损伤统计本构模型的研究[J].岩石力学与工程学报,2002,21(6):787-791.XU Wei-ya,WEI Li-de.Study on statistical damage constitutive model of rock[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(6):787-791.
[13]曹文贵,方祖烈,唐学军.岩石损伤软化统计本构模型之研究[J].岩石力学与工程学报,1998,17(6):628-633.CAO Wen-gui,FANG Zu-lie,TANG Xue-jun.A study of statistical constitutive model for soft and damage rock[J].Chinese Journal of Rock Mechanics and Engineering,1998,17(6):628-633.
[14]陈文玲,李宁.含非贯通裂隙岩体介质的损伤模型[J].岩土工程学报,2000,22(4):430-434.CHEN Wen-ling,LI Ning.Damage model of the rock mass medium with intermittent cracks[J].Chinese Journal of Geotechnical Engineering,2000,22(4):430-434.
[15]KRAJCINOVIC D,SILVA M A G.Statistical aspects of the continuous damage theory[J].International Journal of Solids and Structures,1982,18(7):551-562.
[16]曹文贵,张升.基于Mohr-Coulomb准则的岩石损伤统计分析方法研究[J].湖南大学学报(自然科学版),2005,32(1):43-47.CAO Wen-gui,ZHANG Sheng.Study on the statistical analysis of rock damage based on Mohr-Coulomb criterion[J].Journal of Hunan University(Natural Sciences),2005,32(1):43-47.
[17]杨更社,谢定义.岩体宏观细观损伤的耦合计算分析[C]//第六次全国岩石力学与工程学术大会论文集.武汉:[出版者不详],2000:327-329.YANG Geng-she,XIE Ding-yi.Coupling analysis on the macro-damage and meso-damage of rock masses[C]//Symposium on the Sixth National Rock Mechanics and Engineering Academic Conference.Wuhan:[s.n.],2000:327-329.
[18]张吉宏,刘红岩.综合考虑宏微观复合损伤的节理岩体本构模型[J].煤田地质与勘探,2013,41(6):49-52.ZHANG Ji-hong,LIU Hong-yan.Constitutive model of jointed rock mass by combining macroscopic and microcopic composite damage[J].Coal Geology&Exploration,2013,41(6):49-52.
[19]张力民,吕淑然,刘红岩.综合考虑宏细观缺陷的岩体动态损伤本构模型[J].爆炸与冲击,2015,35(3):428-436.ZHANG Li-min,LüShu-ran,LIU Hong-yan.A dynamic damage constitutive model of rock mass by comprehensively considering macroscopic and mesoscopic flaws[J].Explosion and Shock Waves,2015,35(3):428-436.
[20]赵怡晴,刘红岩,吕淑然,等.基于宏观和细观缺陷耦合的节理岩体损伤本构模型[J].中南大学学报(自然科学版),2015,46(4):1489-1496.ZHAO Yi-qing,LIU Hong-yan,LüShu-ran,et al.Damage constitutive model of jointed rock mass based on coupling macroscopic and mesoscopic flaws[J].Journal of Central South University(Science and Technology),2015,46(4):1489-1496.
[21]袁小清,刘红岩,刘京平.基于宏细观损伤耦合的非贯通裂隙岩体本构模型[J].岩土力学,2015,36(10):2804-2814.YUAN Xiao-qing,LIU Hong-yan,LIU Jing-ping.Constitutive model of rock mass with non-persistent joints based on coupling macroscopic and mesoscopic damages[J].Rock and Soil Mechanics,2015,36(10):2804-2814.
[22]LEMAITRE J.How to use damage mechanics[J].Nuclear Engineering and Design,1984,80(3):233-245.
[23]GRADY D E,KIPP M E.Continuum modeling of explosive fracture in oil shale[J].International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts,1987,17(3):147-157.
[24]NEMAT-NASSER S,OBATA N A.Microcrack model of dilatancy in brittle materials[J].Journal of Applied Mechanics,1988,55(1):24-35.
[25]宁建国,任会兰,方敏杰.基于椭圆形微裂纹演化与汇合的准脆性材料本构模型[J].科学通报,2012,57(21):1978-1986.NING Jian-guo,REN Hui-lan,FANG Min-jie.Constitutive model of quasi brittle materials based on Evolution and coalescence of elliptical microcracks[J].Chinese Science Bulletin,2012,57(21):1978-1986.
[26]PARK C H,BOBET A.Crack initiation,propagation and coalescence from frictional flaws in uniaxial compression[J].Engineering Fracture Mechanics,2010,14:2727-2748.
[27]吕建国,王志乔,刘红岩.岩石断裂与损伤[M].北京:地质出版社,2013:12-54.LüJian-guo,WANG Zhi-qiao,LIU Hong-yan.Rock fracture and damage[M].Beijing:Geological Publishing House,2013:141-146.
[28]HUANG C,SUBHASH G,VITTON S J.A dynamic damage growth model for uniaxial compressive response of rock aggregates[J].Mechanics of Materials,2002,34(5):267-277.
[29]HUANG C,SUBHASH G.Influence of lateral confinement on dynamic damage evolution during uniaxial compressive response of brittle solids[J].Journal of the Mechanics and Physics of Solids,2003,51(6):1089-1105.
[30]LEE S,RAVICHANDRAN G.Crack initiation in brittle solids under multiaxial compression[J].Engineering Fracture Mechanics,2003,70(13):1645-1658.
[31]PALIWAL B,RAMESH KT.An interacting micro-crack damage model for failure of brittle materials under compression[J].Journal of the Mechanics and Physics of Solids,2008,56(3):896-923.
[32]ISIDA M.On the tension of a strip with a central elliptical hole[J].Transactions of the Japan Society of Mechanical Engineering,1955,21(107):514-518.
[33]LI N,CHEN W,ZHANG P.The mechanical properties and a fatigue damage model for jointed rock mass subjected to dynamic cyclical loading[J].International Journal of Rock Mechanics and Mining Sciences,2001,38(7):1071-1079.
[34]陈新,廖志红,李德建.节理倾角及连通率对岩体强度、变形影响的单轴压缩试验研究[J].岩石力学与工程学报,2011,30(4):781-789.CHEN Xin,LIAO Zhi-hong,LI De-jian.Experimental study of effects of joint inclination angle and connectivity rate on strength and deformation properties of rock masses under uniaxial compression[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(4):781-789.