基于应变能密度的非均质岩石损伤本构模型研究
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摘要
基于岩石细观单元弹性模量近似服从Weibull分布的假设,结合应变能密度理论,建立岩石损伤本构模型,借助声发射事件能量信号和岩石纵波波速提出岩石均质度系数m和弹性模量折减系数K_0确定方法,并对参数进行修正。利用建立的损伤本构模型进行单轴加载模拟,将模拟结果曲线与已有模型理论曲线以及单轴加载实验曲线进行对比,该模型能很好地描述试件应力、应变关系与声发射情况。在此基础上,利用该模型进行石膏试件反复加载模拟,分析其应力–应变及声发射特征,并与室内实验结果进行对比分析。研究结果表明:试件均质度越高,脆性破坏特征越明显;借助声发射信号和岩石纵波波速确定的均质度m以及弹性模量折减系数K_0的计算结果与试验结果吻合。基于应变能密度的岩石损伤本构模型的构建为综合考虑岩石的均质度及反复加载过程对岩石试件影响提供了新的理论依据。
Adopting the assumption that the elastic modulus of the rock mesoscopic unit approximately conforms to the Weibull distribution,a rock damage constitutive model was established based on the strain energy density theory. A method for determining the rock homogeneous degree coefficient m and the reduction factor of the elastic modulus K_0 was proposed using the AE(acoustic emission) energy signal and the rock longitudinal wave velocity,and the parameters were corrected as well. The uniaxial loading simulation was carried out by using the established damage constitutive model,and the simulation results were compared with the existing theoretical model and the uniaxial loading experimental results. It is shown that the proposed model can well describe the stress-strain relationship and acoustic emission conditions of specimens. The model was further used to simulate the repeated loading of plaster specimens and then to analyze stress,strain and acoustic emission characteristics,and comparison with the laboratory results was conducted. The research shows that the higher the homogeneous degree of the specimen is,the more obvious the characteristics of the brittle failure,and that the calculated results of m and K_0,which are determined based on the AE signal and the rock longitudinal wave velocity,agree with the experimental results. The established model provides a new theoretical basis for comprehensively considering the effects of rock homogeneous degree and repeated loading on rock specimens.
Adopting the assumption that the elastic modulus of the rock mesoscopic unit approximately conforms to the Weibull distribution,a rock damage constitutive model was established based on the strain energy density theory. A method for determining the rock homogeneous degree coefficient m and the reduction factor of the elastic modulus K_0 was proposed using the AE(acoustic emission) energy signal and the rock longitudinal wave velocity,and the parameters were corrected as well. The uniaxial loading simulation was carried out by using the established damage constitutive model,and the simulation results were compared with the existing theoretical model and the uniaxial loading experimental results. It is shown that the proposed model can well describe the stress-strain relationship and acoustic emission conditions of specimens. The model was further used to simulate the repeated loading of plaster specimens and then to analyze stress,strain and acoustic emission characteristics,and comparison with the laboratory results was conducted. The research shows that the higher the homogeneous degree of the specimen is,the more obvious the characteristics of the brittle failure,and that the calculated results of m and K_0,which are determined based on the AE signal and the rock longitudinal wave velocity,agree with the experimental results. The established model provides a new theoretical basis for comprehensively considering the effects of rock homogeneous degree and repeated loading on rock specimens.
引文
[1]谢和平,鞠杨,黎立云.基于能量耗散与释放原理的岩石强度与整体破坏准则[J].岩石力学与工程学报,2005,24(17):3 003-3 010.(XIE Heping,JU Yang,LI Liyun.Criteria for strength and structural failure of rocks based on energy dissipation and energy release principles[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(17):3 003-3 010.(in Chinese))
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[3]GAO F,ZHANG Z,LIU X.Research on rock burst proneness index based on energy evolution in rock[J].Disaster Advances,2012,5(4):1 367-1 371.
[4]WEN Z,WANG X,CHEN L,et al.Size effect on acoustic emission characteristics of coal-rock damage evolution[J].Advances in Materials Science and Engineering,2017.Doi:10.1155/2017/3472485.
[5]XU Y H,CAI M.Influence of strain energy released from a test machine on rock failure process[J].Canadian Geotechnical Journal,2017,55(6):777-791.
[6]ZHOU X P,QIAN Q H,YANG H Q.Rock burst of deep circular tunnels surrounded by weakened rock mass with cracks[J].Theoretical and Applied Fracture Mechanics,2011,56(2):79-88.
[7]LEAL-GOMES M J A,DINIS-DA-GAMA C A J V.Limit equilibrium model for rock joints based on strain energies[J].International Journal of Geomechanics,2013,14(4):1-5.
[8]DAVIES I J.Empirical correction factor for the best estimate of Weibull modulus obtained using linear least squares analysis[J].Journal of Materials Science Letters,2001,20(11):997-999.
[9]ALIHA M R M,AYATOLLAHI M R.Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading-A statistical approach[J].Theoretical and Applied Fracture Mechanics,2014,69(2):17-25.
[10]ALIHA M R M,AMIRDEHI H R F.Fracture toughness prediction using Weibull statistical method for asphalt mixtures containing different air void contents[J].Fatigue and Fracture of Engineering Materials and Structures,2017,40(1):56-68.
[11]ALIHA M R M,SISTANINIA M,SMITH D J,et al.Geometry effects and statistical analysis of mode I fracture in guiting limestone[J].International Journal of Rock Mechanics and Mining Sciences,2012,51(4):128-135.
[12]SAKIN R,?RFAN A Y.Statistical analysis of bending fatigue life data using Weibull distribution in glass-fiber reinforced polyester composites[J].Materials and Design,2008,29(6):1 170-1 181.
[13]LAN H,MARTIN C D,HU B.Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading[J].Journal of Geophysical Research Solid Earth,2010,115(B1):Doi:10.1029/2009JB006496
[14]LIU H Y,ROQUETE M,KOU S Q,et al.Characterization of rock heterogeneity and numerical verification[J].Engineering Geology,2004,72(1/2):89-119.
[15]TANG C.Numerical simulation of progressive rock failure and associated seismicity[J].International Journal of Rock Mechanics and Mining Sciences,1997,34(2):249-261.
[16]TANG C A,LIU H,LEE P K K,et al.Numerical studies of the influence of microstructure on rock failure in uniaxial compressionPart I:effect of heterogeneity[J].International Journal of Rock Mechanics and Mining Sciences,2000,37(4):555-569.
[17]TANG C A,KAISER P K.Numerical simulation of cumulative damage and seismic energy release during brittle rock failure-Part I:fundamentals[J].International Journal of Rock Mechanics and Mining Sciences,1998,35(2):113-121.
[18]TANG C A,XU X H,KOU S Q,et al.Numerical investigation of particle breakage as applied to mechanical crushing-Part I:single-particle breakage[J].International Journal of Rock Mechanics and Mining Sciences,2001,38(8):1 147-1 162.
[19]GUY N,SEYEDI D M,HILD F.Characterizing fracturing of Clay-Rich lower watrous rock:from laboratory experiments to nonlocal damage-based simulations[J].Rock Mechanics and Rock Engineering,2018,51(5):1 777-1 787.
[20]PAKZAD R,WANG S Y,SLOAN S W.Numerical simulation of hydraulic fracturing in low-/high-permeability,Quasi-Brittle and heterogeneous rocks[J].Rock Mechanics and Rock Engineering,2018,51(4):1 153-1 171.
[21]WEDDFELT K,SAADATI M,LARSSON P L.On the load capacity and fracture mechanism of hard rocks at indentation loading[J].International Journal of Rock Mechanics and Mining Sciences,2017,100(1):170-176.
[22]CIL M B,BUSCARNERA G.DEM assessment of scaling laws capturing the grain size dependence of yielding in granular soils[J].Granular Matter,2016,18(3):36.
[23]KUNDU S,STROISZ A,PRADHAN S.A simple discrete-elementmodel of Brazilian test[J].European Physical Journal B,2016,89(5):1-7.
[24]STEFANOU I,SULEM J.Existence of a threshold for brittle grains crushing strength:two-versus three-parameter Weibull distribution fitting[J].Granular Matter,2016,18(2):1-10.
[25]MOMBER A W.A transition function for the solid particle erosion of rocks[J].Wear,2015,328:348-355.
[26]PALUSZNY A,TANG X H,NEJATI M,et al.A direct fragmentation method with Weibull function distribution of sizes based on finite-and discrete element simulations[J].International Journal of Solids and Structures,2016,80:38-51.
[27]SALVIATO M,KIRANE K,BAZˇANT Z P.Statistical distribution and size effect of residual strength of quasibrittle materials after a period of constant load[J].Journal of the Mechanics and Physics of Solids,2014,64(2):440-454.
[28]孙倩,李树忱,冯现大,等.基于应变能密度理论的岩石破裂数值模拟方法研究[J].岩土力学,2011,32(5):1 575-1 582.(SUNQian,LI Shuchen,FENG Xianda,et al.Study of numerical simulation method of rock fracture based on strain energy density theory[J].Rock and Soil Mechanics,2011,32(5):1 575-1 582.(in Chinese))
[29]LI X,CAO W G,SU Y H.A statistical damage constitutive model for softening behavior of rocks[J].Engineering Geology,2012,143-144:1-17.
[30]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 Shengqi,XU Weiya,WEI Lide,et al.Statistical constitutive model for rock damage under uniaxial compression and its experimental study[J].Journal of Hehai University:Natural Science,2004,32(2):200-203.(in Chinese))
[31]曾晟,杨仕教,张新华,等.单轴压缩下石灰岩损伤统计本构模型与试验研究[J].南华大学学报:自然科学版,2005,19(1):69-72.(ZENG Sheng,YANG Shijiao,ZHANG Xinhua,et al.Statistical constitutive model for limestone rock damageunder uniaxial compression and its experimental study[J].Journal of University of South China:Science and Technology,2005,19(1):69-72.(in Chinese))
[32]杨明辉,赵明华,曹文贵.岩石损伤软化统计本构模型参数的确定方法[J].水利学报,2005,36(3):345-349.(YANG Minghui,ZHAOMinghua,CAO Wengui.Method for determining the parameters of statistical damage softening constitutive model for rock[J].Journal of Hydraulic Engineering,2005,36(3):345-349.(in Chinese))
[33]徐涛,唐春安,张哲,等.单轴压缩条件下脆性岩石变形破坏的理论、试验与数值模拟[J].东北大学学报:自然科学版,2003,24(1):87-90.(XU TAO,TANG Chun?an,ZHANG Zhe,et al.Theoretical,experimental and numerical studies on deformation and failure of brittle rock in uniaxial compression[J].Journal of Northeastern University:Natural Science,2003,24(1):87-90.(in Chinese))
[34]张晓君.岩石损伤统计本构模型参数及其临界敏感性分析[J].采矿与安全工程学报,2010,27(1):49-54.(ZHANG Xiaojun.Parameters of statistical damage constitutive model for rocks and its critical sensibility analysis[J].Journal of Mining and Safety Engineering,2010,27(1):49-54.(in Chinese))
[35]张富才.现场动力静力法弹性参数测定成果对比分析[J].水利水电技术,1990,(11):34-39.(ZHANG Fucai.Comparative analysis of results of elastic parameters measurement by dynamic hydrostatic method[J].Water Resources and Hydropower Engineering,1990,(11):34-39.(in Chinese))
[36]张培源,张晓敏,汪天庚.岩石弹性模量与弹性波速的关系[J].岩石力学与工程学报,2001,20(6):785-788.(ZHANG Peiyuan,ZHANG Xiaomin,WANG Tiangeng.Relationship between elastic moduli and wave velocities in rock[J].Chinese Journal of Rock Mechanics and Engineering,2001,20(6):785-788.(in Chinese))
[2]谢和平,鞠杨,黎立云,等.岩体变形破坏过程的能量机制[J].岩石力学与工程学报,2008,27(9):1 729-1 740.(XIE Heping,JUYang,LI Liyun,et al.Energy mechanism of deformation and failure of rock masses[J].Chinese Journal of Rock Mechanics and Engineering,2008,27(9):1 729-1 740.(in Chinese))
[3]GAO F,ZHANG Z,LIU X.Research on rock burst proneness index based on energy evolution in rock[J].Disaster Advances,2012,5(4):1 367-1 371.
[4]WEN Z,WANG X,CHEN L,et al.Size effect on acoustic emission characteristics of coal-rock damage evolution[J].Advances in Materials Science and Engineering,2017.Doi:10.1155/2017/3472485.
[5]XU Y H,CAI M.Influence of strain energy released from a test machine on rock failure process[J].Canadian Geotechnical Journal,2017,55(6):777-791.
[6]ZHOU X P,QIAN Q H,YANG H Q.Rock burst of deep circular tunnels surrounded by weakened rock mass with cracks[J].Theoretical and Applied Fracture Mechanics,2011,56(2):79-88.
[7]LEAL-GOMES M J A,DINIS-DA-GAMA C A J V.Limit equilibrium model for rock joints based on strain energies[J].International Journal of Geomechanics,2013,14(4):1-5.
[8]DAVIES I J.Empirical correction factor for the best estimate of Weibull modulus obtained using linear least squares analysis[J].Journal of Materials Science Letters,2001,20(11):997-999.
[9]ALIHA M R M,AYATOLLAHI M R.Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading-A statistical approach[J].Theoretical and Applied Fracture Mechanics,2014,69(2):17-25.
[10]ALIHA M R M,AMIRDEHI H R F.Fracture toughness prediction using Weibull statistical method for asphalt mixtures containing different air void contents[J].Fatigue and Fracture of Engineering Materials and Structures,2017,40(1):56-68.
[11]ALIHA M R M,SISTANINIA M,SMITH D J,et al.Geometry effects and statistical analysis of mode I fracture in guiting limestone[J].International Journal of Rock Mechanics and Mining Sciences,2012,51(4):128-135.
[12]SAKIN R,?RFAN A Y.Statistical analysis of bending fatigue life data using Weibull distribution in glass-fiber reinforced polyester composites[J].Materials and Design,2008,29(6):1 170-1 181.
[13]LAN H,MARTIN C D,HU B.Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading[J].Journal of Geophysical Research Solid Earth,2010,115(B1):Doi:10.1029/2009JB006496
[14]LIU H Y,ROQUETE M,KOU S Q,et al.Characterization of rock heterogeneity and numerical verification[J].Engineering Geology,2004,72(1/2):89-119.
[15]TANG C.Numerical simulation of progressive rock failure and associated seismicity[J].International Journal of Rock Mechanics and Mining Sciences,1997,34(2):249-261.
[16]TANG C A,LIU H,LEE P K K,et al.Numerical studies of the influence of microstructure on rock failure in uniaxial compressionPart I:effect of heterogeneity[J].International Journal of Rock Mechanics and Mining Sciences,2000,37(4):555-569.
[17]TANG C A,KAISER P K.Numerical simulation of cumulative damage and seismic energy release during brittle rock failure-Part I:fundamentals[J].International Journal of Rock Mechanics and Mining Sciences,1998,35(2):113-121.
[18]TANG C A,XU X H,KOU S Q,et al.Numerical investigation of particle breakage as applied to mechanical crushing-Part I:single-particle breakage[J].International Journal of Rock Mechanics and Mining Sciences,2001,38(8):1 147-1 162.
[19]GUY N,SEYEDI D M,HILD F.Characterizing fracturing of Clay-Rich lower watrous rock:from laboratory experiments to nonlocal damage-based simulations[J].Rock Mechanics and Rock Engineering,2018,51(5):1 777-1 787.
[20]PAKZAD R,WANG S Y,SLOAN S W.Numerical simulation of hydraulic fracturing in low-/high-permeability,Quasi-Brittle and heterogeneous rocks[J].Rock Mechanics and Rock Engineering,2018,51(4):1 153-1 171.
[21]WEDDFELT K,SAADATI M,LARSSON P L.On the load capacity and fracture mechanism of hard rocks at indentation loading[J].International Journal of Rock Mechanics and Mining Sciences,2017,100(1):170-176.
[22]CIL M B,BUSCARNERA G.DEM assessment of scaling laws capturing the grain size dependence of yielding in granular soils[J].Granular Matter,2016,18(3):36.
[23]KUNDU S,STROISZ A,PRADHAN S.A simple discrete-elementmodel of Brazilian test[J].European Physical Journal B,2016,89(5):1-7.
[24]STEFANOU I,SULEM J.Existence of a threshold for brittle grains crushing strength:two-versus three-parameter Weibull distribution fitting[J].Granular Matter,2016,18(2):1-10.
[25]MOMBER A W.A transition function for the solid particle erosion of rocks[J].Wear,2015,328:348-355.
[26]PALUSZNY A,TANG X H,NEJATI M,et al.A direct fragmentation method with Weibull function distribution of sizes based on finite-and discrete element simulations[J].International Journal of Solids and Structures,2016,80:38-51.
[27]SALVIATO M,KIRANE K,BAZˇANT Z P.Statistical distribution and size effect of residual strength of quasibrittle materials after a period of constant load[J].Journal of the Mechanics and Physics of Solids,2014,64(2):440-454.
[28]孙倩,李树忱,冯现大,等.基于应变能密度理论的岩石破裂数值模拟方法研究[J].岩土力学,2011,32(5):1 575-1 582.(SUNQian,LI Shuchen,FENG Xianda,et al.Study of numerical simulation method of rock fracture based on strain energy density theory[J].Rock and Soil Mechanics,2011,32(5):1 575-1 582.(in Chinese))
[29]LI X,CAO W G,SU Y H.A statistical damage constitutive model for softening behavior of rocks[J].Engineering Geology,2012,143-144:1-17.
[30]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 Shengqi,XU Weiya,WEI Lide,et al.Statistical constitutive model for rock damage under uniaxial compression and its experimental study[J].Journal of Hehai University:Natural Science,2004,32(2):200-203.(in Chinese))
[31]曾晟,杨仕教,张新华,等.单轴压缩下石灰岩损伤统计本构模型与试验研究[J].南华大学学报:自然科学版,2005,19(1):69-72.(ZENG Sheng,YANG Shijiao,ZHANG Xinhua,et al.Statistical constitutive model for limestone rock damageunder uniaxial compression and its experimental study[J].Journal of University of South China:Science and Technology,2005,19(1):69-72.(in Chinese))
[32]杨明辉,赵明华,曹文贵.岩石损伤软化统计本构模型参数的确定方法[J].水利学报,2005,36(3):345-349.(YANG Minghui,ZHAOMinghua,CAO Wengui.Method for determining the parameters of statistical damage softening constitutive model for rock[J].Journal of Hydraulic Engineering,2005,36(3):345-349.(in Chinese))
[33]徐涛,唐春安,张哲,等.单轴压缩条件下脆性岩石变形破坏的理论、试验与数值模拟[J].东北大学学报:自然科学版,2003,24(1):87-90.(XU TAO,TANG Chun?an,ZHANG Zhe,et al.Theoretical,experimental and numerical studies on deformation and failure of brittle rock in uniaxial compression[J].Journal of Northeastern University:Natural Science,2003,24(1):87-90.(in Chinese))
[34]张晓君.岩石损伤统计本构模型参数及其临界敏感性分析[J].采矿与安全工程学报,2010,27(1):49-54.(ZHANG Xiaojun.Parameters of statistical damage constitutive model for rocks and its critical sensibility analysis[J].Journal of Mining and Safety Engineering,2010,27(1):49-54.(in Chinese))
[35]张富才.现场动力静力法弹性参数测定成果对比分析[J].水利水电技术,1990,(11):34-39.(ZHANG Fucai.Comparative analysis of results of elastic parameters measurement by dynamic hydrostatic method[J].Water Resources and Hydropower Engineering,1990,(11):34-39.(in Chinese))
[36]张培源,张晓敏,汪天庚.岩石弹性模量与弹性波速的关系[J].岩石力学与工程学报,2001,20(6):785-788.(ZHANG Peiyuan,ZHANG Xiaomin,WANG Tiangeng.Relationship between elastic moduli and wave velocities in rock[J].Chinese Journal of Rock Mechanics and Engineering,2001,20(6):785-788.(in Chinese))