关联Drucker-Prager条件下等效塑性应变系数
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摘要
针对岩石类材料,试图在关联Drucker-Prager条件下建立更一般化的等效塑性应变定义方法,并研究等效塑性应变系数在塑性变形过程中的变化。首先列举了国内外常用的等效塑性应变定义方法,并分析了它们的不合理性。然后从等效应力和等效塑性应变的定义出发,推导出关联Drucker-Prager条件下等效塑性应变系数。结合室内致密砂岩三轴压缩试验,以塑性体应变为内变量,研究了致密砂岩峰后内摩擦角和粘聚力随内变量的变化,进而研究了等效塑性应变系数随塑性变形的变化。研究发现:等效塑性应变系数依赖于屈服准则的选取,等效塑性应变系数为(2/3)~(1/2)是Mises屈服准则下的特例;关联Drucker-Prager条件下等效塑性应变系数C<(2/3)~(1/2),并且随着塑性变形的发展而减小最终趋于稳定值;对于岩石类材料,很多有限元软件中不区分具体情况而直接选用C=(2/3)~(1/2)计算等效塑性应变存在较大的误差。
For geomaterials, the more general definition of the effective plastic strain based on associated flow of Drucker-Prager law is presented and the variation of the coefficient of the effective plastic strain is analyzed. Firstly, several common definitions of the effective plastic strain are listed and irrationalities of those definitions are pointed out. Then a more general definition of the effective plastic strain based on the associated flow of Drucker-Prager criterion is proposed from the basic definition of effective stress and effective plastic strain. By conducting triaxial compression test of tight sandstones in the laboratory and choosing the plastic volume strain as the plastic internal variable, the variations of internal friction angle and cohesion strength of tight sandstones are analyzed. And variation of the coefficient of effective plastic strain with the plastic deformation is also analyzed. Finally, some significant conclusions are reached. The coefficient of effective plastic strain is dependent on the yield criterion chosen. The coefficient of effective plastic strain, (2/3)~(1/2), is found to be a special case when Mises yield criterion is used. And what's more, the effective plastic strain coefficient based on the associated flow of Drucker-Prager law is no longer a constant but decreasing gradually with the plastic deformation and finally almost reaches a constant value, and its value is always smaller than (2/3)~(1/2). If the effective plastic strain based on associated flow of Drucker-Prager law is set to be (2/3)~(1/2) as conventionally adopted in almost all papers and FEM software, there will be a significant deviation from the effective plastic strain.
For geomaterials, the more general definition of the effective plastic strain based on associated flow of Drucker-Prager law is presented and the variation of the coefficient of the effective plastic strain is analyzed. Firstly, several common definitions of the effective plastic strain are listed and irrationalities of those definitions are pointed out. Then a more general definition of the effective plastic strain based on the associated flow of Drucker-Prager criterion is proposed from the basic definition of effective stress and effective plastic strain. By conducting triaxial compression test of tight sandstones in the laboratory and choosing the plastic volume strain as the plastic internal variable, the variations of internal friction angle and cohesion strength of tight sandstones are analyzed. And variation of the coefficient of effective plastic strain with the plastic deformation is also analyzed. Finally, some significant conclusions are reached. The coefficient of effective plastic strain is dependent on the yield criterion chosen. The coefficient of effective plastic strain, (2/3)~(1/2), is found to be a special case when Mises yield criterion is used. And what's more, the effective plastic strain coefficient based on the associated flow of Drucker-Prager law is no longer a constant but decreasing gradually with the plastic deformation and finally almost reaches a constant value, and its value is always smaller than (2/3)~(1/2). If the effective plastic strain based on associated flow of Drucker-Prager law is set to be (2/3)~(1/2) as conventionally adopted in almost all papers and FEM software, there will be a significant deviation from the effective plastic strain.
引文
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[13]Pourhosseini O,Shabanimashcool M.Development of an elasto-plastic constitutive model for intact rocks[J].International Journal of Rock Mechanics&Mining Sciences,2014,66(1):1-12.
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[19]王水林,王威,吴振君.岩土材料峰值后区强度参数演化与应力-应变曲线关系研究[J].岩石力学与工程学报,2010,29(8):1524-1529.(Wang Shuilin,Wang Wei,Wu Zhenjun.Study of relationship between evolution of post-peak strength parameters and stress-strain curves of geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(8):1524-1529(in Chinese)).
[20]张帆,盛谦,朱泽奇,等.三峡花岗岩峰后力学特性及应变软化模型研究[J].岩石力学与工程学报,2008,27(增1):2651-2655.(Zhang Fan,Sheng Qian,Zhu Zeqi,et al.Study on post-peak mechanical behaviour and strain-softening model of three gorges granite[J].Chinese Journal of Rock Mechanics and Engineering,2008,27(S1):2651-2655(in Chinese)).
[21]Hajlabdolmajid V,Kaiser P K.Brittleness of rock and stability assessment in hard rock tunneling[J].Tunnelling and Underground Space Technology,2003,18(1):35-48.
[22]Franklin J A.Suggested methods for determining the strength of rock materials in triaxial compression:revised version[J].International Journal of Rack Mechanics and Mining Sciences&Geomechanics Abstracts,1983,20(6):285-290.
[23]王红才,赵卫华,孙东生,等.岩石塑性变形条件下的Mohr-Coulomb屈服准则[J].地球物理学报,2012,55(12):4231-4238.(Wang Hongcai,Zhao Weihua,Sun Dongsheng,et al.Mohr-Coulomb yield criterion in rock plastic mechanics[J].Chinese Journal of Geophsics,55(12):4231-4238(in Chinese)).
[24]Fjar E,Holt R M,Ragen A M,et al.Petroleum related rock mechanics[M].New York:Elsevier,2008.
[25]Bishop A.Shear strength parameters for undisturbed and remoulded soil specimens[C]//In:Proceedings of the Roscoe Memorial Symposium.Cambridge University,Cambridge:MA,1971:3-58.
[26]Hangzhou L,Xiong G,Zhao G.An elasto-plastic constitutive model for soft rock considering mobilization of strength[J].Transactions of Nonferrous Metals Society of China,2016,26(3):822-834.
[27]Schmertmann J H,Osterberg J O.An experimental study of the development of cohesion and friction with axial strain in saturated cohesive soils[C]//Research Conference on Shear Strength of Cohesive Soils.Colorado,USA:ASCE,1960:643-694.
[2]Chen W F,Salipu A F.Elasticity and plasticity[M].Beijing:China Architecture&Building Press,2005.
[3]Ba?ant Zdeněk P.Work inequalities for plastic fracturing materials[J].International Journal of Solids and Structures,1980,16(10):873-901.
[4]Han W,Reddy B D.Plasticity:mathematical theory and numerical analysis[M].New York:Springer,1999.
[5]Vermeer P A,Borst R D.Non-associated plasticity for soils,concrete and rock[J].Physics of Dry Granular Media,1984,29(3):1-62.
[6]Hill R.The mathematical theory of plasticity[M].Oxford:Oxford University Press,1950.
[7]Berg C A.Construction of the equivalent plastic strain increment[J].Studies in Applied Mathematics,1972,51(3):311-316.
[8]李纬民,刘助柏,贾薇.塑性变形时的应变等效系数[J].塑性工程学报,1997,4(1):47-50.(Li Weimin,Liu Zhubai,Jia Wei.On the strain equivalent coefficient in plastic deformation[J].Journal of Plasticity Engineering,1997,4(1):47-50(in Chinese)).
[9]Zhu Canjun,Xie Xiaoli,Lai Lian,et al.Study of two important plastic indexes of tresca yield criterion and mises yield criterion[J].Journal of Hunan University of Arts and Science:Natural Science Edition,2005,17(1):53-54.
[10]Chen W F,Zhang H.Structural plasticity:theory,problems,and CAEsoftware[M].New York:Springer-Verlag,1991.
[11]Alejano L R,Alonso E.Considerations of the dilatancy angle in rocks and rock masses[J].International Journal of Rock Mechanics&Mining Sciences,2005,42(4):481-507.
[12]Zhao X G,Cai M.A mobilized dilation angle model for rocks[J].International Journal of Rock Mechanics&Mining Sciences,2010,47(3):368-384.
[13]Pourhosseini O,Shabanimashcool M.Development of an elasto-plastic constitutive model for intact rocks[J].International Journal of Rock Mechanics&Mining Sciences,2014,66(1):1-12.
[14]殷有泉.岩石力学与岩石工程的稳定性[M].北京:北京大学出版社,2011.(Yin Youquan.Rock mechanics and stability of rock engineering[M].Beijing:Peking University Press,2011(in Chinese)).
[15]殷有泉.岩石类材料塑性力学[M].北京:北京大学出版社,2014.(Yin Youquan.Plasticity of geomaterials[M].Beijing:Peking University Press,2014(in Chinese)).
[16]Chen W F,Liu X L.Limit analysis in soil mechanics[M].Amsterdam:Elsevier,1990.
[17]李文婷,李树忱,冯现大,等.基于莫尔-库仑准则的岩石峰后应变软化力学行为研究[J].岩石力学与工程学报,2011,30(7):1460-1466.(Li Wenting,Li Shuchen,Feng Xianda,et al.Study of post-peak strain softening mechanical properties of rock based on Mohr-Coulomb criterion[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(7):1460-1466(in Chinese)).
[18]韩建新,李术才,李树忱,等.基于强度参数演化行为的岩石峰后应力-应变关系研究[J].岩土力学,2013,34(2):342-346.(Han Jianxin,Li Shucai,Li Shuchen,et al.Study of post-peak stress-strain relationship of rock material based on evolution of strength parameters[J].Rock and Soil Mechanics,2013,34(2):342-346(in Chinese)).
[19]王水林,王威,吴振君.岩土材料峰值后区强度参数演化与应力-应变曲线关系研究[J].岩石力学与工程学报,2010,29(8):1524-1529.(Wang Shuilin,Wang Wei,Wu Zhenjun.Study of relationship between evolution of post-peak strength parameters and stress-strain curves of geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(8):1524-1529(in Chinese)).
[20]张帆,盛谦,朱泽奇,等.三峡花岗岩峰后力学特性及应变软化模型研究[J].岩石力学与工程学报,2008,27(增1):2651-2655.(Zhang Fan,Sheng Qian,Zhu Zeqi,et al.Study on post-peak mechanical behaviour and strain-softening model of three gorges granite[J].Chinese Journal of Rock Mechanics and Engineering,2008,27(S1):2651-2655(in Chinese)).
[21]Hajlabdolmajid V,Kaiser P K.Brittleness of rock and stability assessment in hard rock tunneling[J].Tunnelling and Underground Space Technology,2003,18(1):35-48.
[22]Franklin J A.Suggested methods for determining the strength of rock materials in triaxial compression:revised version[J].International Journal of Rack Mechanics and Mining Sciences&Geomechanics Abstracts,1983,20(6):285-290.
[23]王红才,赵卫华,孙东生,等.岩石塑性变形条件下的Mohr-Coulomb屈服准则[J].地球物理学报,2012,55(12):4231-4238.(Wang Hongcai,Zhao Weihua,Sun Dongsheng,et al.Mohr-Coulomb yield criterion in rock plastic mechanics[J].Chinese Journal of Geophsics,55(12):4231-4238(in Chinese)).
[24]Fjar E,Holt R M,Ragen A M,et al.Petroleum related rock mechanics[M].New York:Elsevier,2008.
[25]Bishop A.Shear strength parameters for undisturbed and remoulded soil specimens[C]//In:Proceedings of the Roscoe Memorial Symposium.Cambridge University,Cambridge:MA,1971:3-58.
[26]Hangzhou L,Xiong G,Zhao G.An elasto-plastic constitutive model for soft rock considering mobilization of strength[J].Transactions of Nonferrous Metals Society of China,2016,26(3):822-834.
[27]Schmertmann J H,Osterberg J O.An experimental study of the development of cohesion and friction with axial strain in saturated cohesive soils[C]//Research Conference on Shear Strength of Cohesive Soils.Colorado,USA:ASCE,1960:643-694.