节理岩体稳定分析的数值流形方法研究
|
摘要
节理岩体是岩土工程最普遍的施工对象之一,它的力学性质对岩土工程的安全和稳定起着决定性的作用,合理评价节理岩体的强度和变形特征,对于诸如边坡稳定、地下开挖、支护设计和锚固系统的场地选择、设计、施工等岩土工程问题具有十分重要的意义。而节理岩体的显著特点,是其中包含大量方向各异、物理性质各异的褶皱、断层、节理、片理等不连续面。在这些不连续面的切割下,岩体成为连续和不连续部分共存的材料介质,给数值建模分析带来了很大困难。
数值流形方法以两套网格的独特设置为手段,将数学分析中的有限覆盖技术引入数值计算,统一解决了连续与非连续变形计算的力学问题,成为应用于岩体工程领域的一种新方法。然而,现有数值流形方法囿于线弹性分析计算,且难以有效模拟岩土工程的施工过程,影响和限制了其推广应用。
基于上述考虑,本论文着眼于对节理岩体强度以及节理岩体工程的数值流形分析,从数值流形程序的开发入手,针对本文特定的研究目的,对数值流形方法本身提出了诸多改进和完善。在实现了预期的功能后,利用本文开发的数值流形程序,重点对节理岩体的不连续变形特征进行了分析研究。
取得的主要研究成果包括以下几个方面：
1、基于由规则矩形构成的数学网格,改进了数值流形覆盖系统的自动剖分算法；推导建立了二阶位移函数的单元矩阵。
2、提出了在数值流形方法中对不同约束条件的模拟方法；推导建立了高阶位移函数条件下求解动力问题的一系列相关矩阵；提出了在不连续面整个长度范围内设置分布接触弹簧的处理方法。
3、在数值流形方法中引入了连续介质的非线性弹塑性本构模型,实现了节理岩体弹塑性数值流形分析,据此评价节理岩体强度。
4、提出了开挖模拟在数值流形方法中的实现方法,为岩土工程施工过程模拟奠定了理论基础。
5、对两个具有典型不连续变形特征的岩土工程问题进行了数值流形分析,获得了对工程设计施工具有指导意义的分析结果。
As a major construction object in the geotechnical engineering, jointed rock mass often has a significant influence to the safety and stability of the project. Making a reasonable assessment of the deformation and strength characteristic of jointed rock mass is of great importance for the geotechnical engineering. Meanwhile, in jointed rock mass, it often contains large amount of discontinuities that are of different trend and physical properties, which causes big problems in modeling the issues by computer.
The numerical manifold method (NMM) employs two sets of meshes, combining the continuous and discontinuous deformation problems into a uniform frame. As a new method, it has been applied in geotechnical engineering. But the existing NMM is limited in elastic deformation analysis, and it becomes incapable when dealing with some construction simulations, such as the excavation process. That is a main obstacle for NMM's wide spread.
Based on the above consideration, a NMM program is developed in this paper first, and several improvements to the existing NMM theory are presented meanwhile. As the NMM program is capable of the nonlinear elasto-plastic analysis and has the ability to simulate the excavation process, it is used as an analytical tool for the stability analysis of the jointed rock mass.
The main achievements of this paper include:
1. Based on the rectangular mathematical mesh, several improvements are presented to the existing algorithm of automatically forming of cover system; establish the element matrix of 2-order displacement function.
2. Present the modeling method of different constraint conditions; deduce the inertia force matrix and velocity matrix based on high order displacement.function; present the distributed contact spring.
3. Add a nonlinear elasto-plastic model to NMM, and the strength analysis of jointed rock masses is carried out using the elasto-plastic NMM program.
4. An algorithm that can simulate the excavation problem in numerical manifold method is proposed, making it possible for NMM to be used in a wider range.
5. Two geotechnical projects that contain typical discontinuous feature are analyzed by the improved NMM, and some instructive results are obtained.
数值流形方法以两套网格的独特设置为手段,将数学分析中的有限覆盖技术引入数值计算,统一解决了连续与非连续变形计算的力学问题,成为应用于岩体工程领域的一种新方法。然而,现有数值流形方法囿于线弹性分析计算,且难以有效模拟岩土工程的施工过程,影响和限制了其推广应用。
基于上述考虑,本论文着眼于对节理岩体强度以及节理岩体工程的数值流形分析,从数值流形程序的开发入手,针对本文特定的研究目的,对数值流形方法本身提出了诸多改进和完善。在实现了预期的功能后,利用本文开发的数值流形程序,重点对节理岩体的不连续变形特征进行了分析研究。
取得的主要研究成果包括以下几个方面：
1、基于由规则矩形构成的数学网格,改进了数值流形覆盖系统的自动剖分算法；推导建立了二阶位移函数的单元矩阵。
2、提出了在数值流形方法中对不同约束条件的模拟方法；推导建立了高阶位移函数条件下求解动力问题的一系列相关矩阵；提出了在不连续面整个长度范围内设置分布接触弹簧的处理方法。
3、在数值流形方法中引入了连续介质的非线性弹塑性本构模型,实现了节理岩体弹塑性数值流形分析,据此评价节理岩体强度。
4、提出了开挖模拟在数值流形方法中的实现方法,为岩土工程施工过程模拟奠定了理论基础。
5、对两个具有典型不连续变形特征的岩土工程问题进行了数值流形分析,获得了对工程设计施工具有指导意义的分析结果。
As a major construction object in the geotechnical engineering, jointed rock mass often has a significant influence to the safety and stability of the project. Making a reasonable assessment of the deformation and strength characteristic of jointed rock mass is of great importance for the geotechnical engineering. Meanwhile, in jointed rock mass, it often contains large amount of discontinuities that are of different trend and physical properties, which causes big problems in modeling the issues by computer.
The numerical manifold method (NMM) employs two sets of meshes, combining the continuous and discontinuous deformation problems into a uniform frame. As a new method, it has been applied in geotechnical engineering. But the existing NMM is limited in elastic deformation analysis, and it becomes incapable when dealing with some construction simulations, such as the excavation process. That is a main obstacle for NMM's wide spread.
Based on the above consideration, a NMM program is developed in this paper first, and several improvements to the existing NMM theory are presented meanwhile. As the NMM program is capable of the nonlinear elasto-plastic analysis and has the ability to simulate the excavation process, it is used as an analytical tool for the stability analysis of the jointed rock mass.
The main achievements of this paper include:
1. Based on the rectangular mathematical mesh, several improvements are presented to the existing algorithm of automatically forming of cover system; establish the element matrix of 2-order displacement function.
2. Present the modeling method of different constraint conditions; deduce the inertia force matrix and velocity matrix based on high order displacement.function; present the distributed contact spring.
3. Add a nonlinear elasto-plastic model to NMM, and the strength analysis of jointed rock masses is carried out using the elasto-plastic NMM program.
4. An algorithm that can simulate the excavation problem in numerical manifold method is proposed, making it possible for NMM to be used in a wider range.
5. Two geotechnical projects that contain typical discontinuous feature are analyzed by the improved NMM, and some instructive results are obtained.
引文
[1]SHI Gen-hua.1996. Manifold method[C]//Proceeding of the 1st International Forum on Discontinuous Deformation Analysis and Simulation of Discontinuous Deformation Media.California:TSI Press,1996:52-204.
[2]沈明荣.岩体力学[M].上海：同济大学出版社.1999
[3]蔡美峰.岩石力学与工程[M].北京：科学出版社.2002
[4]凌贤长.蔡德所.岩体力学[M].哈尔滨：哈尔滨工业大学出版社.2002
[5]Jaeger, J.C., Cook, N.G.W.1969. Fundamentals of rock mechanics[M]. London. Chapman and Hall LTD
[6]高峰,谢和平等.节理岩体强度的分形统计分析[J].岩石力学与工程学报,23(21)：3608-3612
[7]李同录,罗世毅等.节理岩体力学参数的选取与应用[J].岩石力学与工程学报,2004,23(13)：2182-2186
[8]汪小刚,陈祖煌.应用蒙特卜洛法确定节理岩体的连通率和综合抗剪强度指标[J].岩石力学与工程学,1992,11(4)：345-355
[9]E.Hoek, E.T.Brown. Empirical strength criterion for rock masses [J]. Geotech Engng. Div. ASCE,1980,106:1013-1035
[10]Kalamaras G.S, Bieniawski Z.T A rock mass strength concept for coal seams incorporating the effect of time [R].8Th ISRM Congress. Tokyo.1995:295-302
[11]Kim M.K., Lade P.V. Modelling rock strength in three dimensions [J]. Int J Rock Mech Min Sci,1984,21:21-33.
[12]Arild Pahnstrom.Characterizing Rock Massesby the RMi for Use in Practical Rock Engineering,Partl:the develoment of the Rock Mass index[J]. Tunneling and Under Ground Space Technology,1996,11(2):175-188
[13]Ramamurthy T. Strength and modulus responses of anisotropic rocks[A]. In:Hudson JA. Editor. Comprehensive rock engineering,1994,1(13):313-29
[14]Bieniawski Z.T. Engineering classification of jointed rock masses[M]. John Wiley and Sons. New York.1989
[15]Serafim J.L., Pereira J.P. Considerations on the geomechanical classification of Bieniawski[A]. In:Proceedings of the Symposium on Engineering Geology and Underground Opening[C]. Portugal. Lisboa1983:1133-1144
[16]Barton N. Rock mechanics review, the shear strength of rock and rock joints[J]. Int. J. Rock Mech. Min. Sci. & Geomech,1976,13(4):255-279
[17]Hoek E., Brown E.T. Practical estimates of rock mass strength[J]. Int J Rock Mech Min Sci,1994,34:1165-86
[18]Palmstrom A., Singh R. The deformation modulus of rock masses comparisons between in situ tests and indirect estimates [J]. Tunneling and Underground Space Tech,2001,(16):115-131
[19]Barton N. Some new Q-value correlations to assist in site characteristics and tunnel design [J]. International Journal of Rock Mechanics & Mining Science,2002,39:185-216
[20]Kayabasi A., Gokceoglu C, Ercanoglu M. Estimating the deformation modulus of rock masses:a comparative study[J]. Int J Rock Mech Min Sci,2003,(40):55-63.
[21]Zhang L., Einstein H. Using RQD to estimate the deformation modulus of rock masses [J]. Int J Rock Mech Min Sci,2004,(41):337-341
[22]Singh B. Continuum characterization of jointed rock masses Part Ⅱ-significance of low shear modulus [J]. Int J Rock Mech Sci &Geomech Abstr,1973,10:337-49
[23]Amadei B., Goodman R.E. A 3-D constitutive relation for fractured rock masses[C]//Proc Int Symp on Mechanical Behavior of Structured media. Ottawa.1981:249-268.
[24]Kawamoto T.,. Ichikawa Y., Kyoya T. Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory [J]. Int J Numer Anal Methods Geomech,1988,12:1-30
[25]Cai M., Horii H. A constitutive model of highly jointed rock masses[J]. Mech Mater,1992,13:217-46.
[26]Goodman R.D., Taylor R., Brekke T.L. A model for the mechanics of jointed rock[J]. ASCE Journal of the soil mechanics and Foundations Division,1968,14(SM3):637-659
[27]Zienkiewicz O.C., Kelly D.W., Bettess P. The coupling of the finite element method and boundary solution procedures[J]. Int J Numer Methods Eng 1977,11:355-75
[28]Desai C.S., Zamman M.M., Lightner J.G., Siriwardane H.J. Thin layer element for interfaces and joint [J]. Int J Numer Anal Methods Geomech.1984,8:19-43
[29]Goodman R.E., Shi G Block theory and its application to rock engineering. Prentice-Hall, New Jersey
[30]Cundall P.A. A computer model for simulating progressive,large-scale movements in blocky rock systems. procecdings of the symposium of the International Society of Rock Mechanics.1971.Nancy.France.Vol(1):paper Ⅱ-8
[31]Shi G.H. Discontinuous deformation analysis-a new numerical model for the static and dynamics of block system. PHD Thesis [Dissertation]. Dept. of civil Engineering. University of Calofornia.Berkeley.1988
[32]Ke T.C. Artificial joint-based on manifold method. In:Working Forum on the Manifold of Material Analysis. Vol.1. Jenner, California, USA, WES, US Army Corps of Engineerings.1995-10-04-06.1995:21-37
[33]Chen G., Miki S., Ohnishi Y. Automatic creation of mathematical meshes in manifold method of material analysis. In:Working Forum on the Manifold of Material Analysis, Vol.1. Jenner, California, USA, WES, US Army Corps of Engineerings.1995-10-04-06. 1995:105-126
[34]Hideomi Ohtsubo, Katsuyuki Suzuki. Utilization of finite covers in the manifold method for accuracy control. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation,Kyoto,Japan,1997:317-322
[35]曹文贵,速宝玉.流形元覆盖系统自动形成方法之研究[J].岩土工程学报.2001,23(2)：1 87-190
[36]蔡永昌,朱合华,夏才初.流形方法覆盖系统自动生成算法[J].同济大学学报,2004,32(5)：585-590
[37]Wang C.Y., Sheng J., Chen M.H. et al. Approximation theories for the manifold method. In: Working Forum on the Manifold of Material Analysis,Vol.1,Jenner,California, USA, WES, US Army Corps of Engineerings.1995-10-04-06.1995:61-86
[38]田荣,栾茂田.高阶流形方法及其应用.工程力学,2001，18(2)：21-26
[39]王水林.四个物理覆盖构成一个单元的流形方法及应用[J].岩石力学与工程学报,1999,18(3)：312-316
[40]宋俊生,大西有三.高阶四边形单元的流形方法[J].岩石力学与工程学报,2003,22(6)：932-936
[41]彭自强,葛修润.数值流形方法在有限元三维二十结点单元上的实现[J].岩石力学与工程学报,2004,23(15)：2622-2627
[42]王芝银,王思敬,杨志法.岩石大变形的流形方法[J].岩石力学与工程学报,1997,16(5)：399-404.
[43]朱以文,曾又林.岩石大变形分析的增量流形方法[J].岩石力学与工程学报,1999,18(1)：1-5
[44]王书法等.岩体弹塑性分析的数值流形方法[J].岩石力学与工程学报,2002,21(6)：900-904
[45]Qiu X.J. Manifold method without use of penalty springs[A]. In:Proc of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, California USA.1996,205-249
[46]Lin D.Z., Mo H.H. Manifold method of material analysis. In:Working Forum on the Manifold of Material Analysis, Vol.1. Jenner, California, USA, WES, US Army Corps of Engineerings.1995-10-04-06.1995:147-164
[47]张国新等.五强溪船闸裂缝的流形元模拟[J].水力学报,2006,24：37-42
[48]郑榕明,张勇慧.基于六面体覆盖的三维数值流形方法的理论探讨与应用[J].岩石力学与工程学报.2004,23(10)：1745-1754
[49]姜清辉,周创兵.四面体有限单元覆盖的三维数值流形方法[J].岩石力学与工程学报,2005,24(24)：4455-4460
[50]彭自强,葛修润.数值流形方法在有限元三维二十节点单元上的实现[J].岩石力学与工程学报,2004,23(15)：2622-2627
[51]骆少明,张湘伟,蔡永昌.非线性数值流形方法的变分原理与应用[J].应用数学与力学,2000,21(12)：1265-1270
[52]王书法,朱维申.考虑侧向影响的数值流形方法及其工程应用[J].岩石力学与工程学报,2001,20(3)：297-300
[53]王后裕,朱可善.高次数值流形方法及其在弹性地基板中的应用[J].力学与实践,2001,3：31-34
[54]栾茂田,张大林,杨庆等.有限覆盖无单元法在裂纹扩展数值分析问题中的应用[J].岩土工程学报,2003,25(5)：527-531
[55]Zhang Guoxin,Sugiura Yasuhito,Hasegawa Hiroo.Crack propagation and thermal fracture analysis by manifold method[A]. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation[C],Kyoto,Japan,1997.282-297
[56]王水林,葛修润.流形方法在模拟裂纹扩展中的应用[J].岩石力学与工程学报1997,16(5)：405-410
[57]Yaw Jeng Chiou, Ren Jow Tsay, Wailin Chuang. Crack propagation using manifold method[A]. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation[C],Kyoto, Japan,1997.298-308
[58]Tsay Ren Jow, Chiou Yaw Jeng, Chuang Wai lin. Crack growth prediction by manifold method[J]. Journal of Engineering Mechanics,1999,125(8):884-890
[59]Yaw Jeng Chiou, Yu Min Lee, Ren Jow Tsay. Mixed modefracture propagation by manifold method[J]. International Journal of Fracture,2002,114:327-347
[60]张大林,栾茂田,杨庆等.基于流形方法的动态应力强度因子数值算法[J].大连理工大学学报,2002,42(5)：590-593
[61]曹文贵,速宝玉.岩石锚固支护的数值流形方法模拟及其应用[J].岩土工程学报,2001,5：581-583
[62]董志宏，邬爱清,丁秀丽.数值流形方法中的锚固支护模拟及初步应用[J].岩石力学与工程学报,2005,24(10)：3754-3760
[63]姜清辉,王书法.锚固岩体的三维数值流形方法模拟[J].岩石力学与工程学报,2006,25(3)：528-532
[64]沈振中,郑磊.基于数值流形方法的水库岩体边坡稳定分析[J].水电能源科学,2006,24(1)：32-35
[65]彭华.岩石边坡稳定分析的数值流形法研究[博士论文].武汉：武汉大学,2001.
[66]KU C.Y. Modeling of jointed rock mass based on the numerical manifold method[Dissertation]. USA:University of Pittsburgh,2001.
[67]王芝银,李云鹏.岩体力学中的增量数值流形方法及其应用[J].西安矿业学院学报,1998,18(增)：23-27
[68]Ke Te-Chih, Tang Jyh-Haw. Modeling of solid-fluid interaction using the manifold method[A]. In:Proceedings of 2nd North American Rock Mechanics Symposium, [C] Montreal, Quebec, Canada,1996.1815-1822
[69]Yuzo Ohnishi, Makoto Tanaka, Tomofumi Koyama. Manifold method in saturated-unsaturated groundwater flow analysis[A]. In:3rd International Conference on Analysis of Discontinuous Deformation from Theory to Practice[C], Vail, Colorado,USA,1999.1213-1228
[70]石根华,裴觉民译.数值流形方法与非连续变形分析[M].北京：清华大学出版社,1997
[71]Lin J-S. A unified framework for discrete and continuum analysis [A].In:Cook BK, Jensen RP, editors. Discrete element methods-numerical modeling of discontinua. Geotechnical Special Publication No.117. New York:ASCE,2002,145-50
[72]张国新,金峰,王光纶.用基于流形元的子域奇异边界元法模拟重力坝的地震破坏[J].工程力学,2001,18(4)：18-27
[73]王俊奎,丁立祚.弹性固体力学[M].北京：中国铁道科学出版社,1990
[74]王芝银,李云鹏.数值流形方法中的几点改进[J].岩土工程学报.1998,20(6)：33-36
[75]林绍忠.单纯形积分的递推公式[J].长江科学院院报.2005,22(3)：32-34
[76]朱爱军.数值流形方法研究及其在岩土工程中的应用[博士论文].重庆：重庆大学,2005.
[77]陈庭.球面数值流形方法及其在地壳运动中的应用研究.[博士论文].武汉：武汉大学，2005
[78]钟万勰,张洪武,吴承伟.参变量变分原理及其在工程中的应用[M].北京：科学出版 社.1997
[79]Jiro H., Takakoto S., Umetaro Y.1979. An elasto-plastic analysis applicable to the'strain softening' materials (in Japanese) [J]. Journal of the Mining and Metallurgical Institute of Japan 10:721-726.
[80]罗国煜,王培清,吴浩等.敷溪口松动变形边坡稳定性和破坏机制浅析[J].湖南水利,1981,(1)：12-16
[81]Jaeger J.C., Cook N. Pinching-off and discing of rocks. J Geophys Res.1963,68:1759-65
[82]彭光忠.单轴压应力下页岩岩块的结构面方向与其力学特性的关系[J].岩土工程学报,1983,5(2)：101-109
[83]张志刚.节理岩体强度确定方法及其各向异性特征研究[博士论文].北京：北京交通大学,2007
[84]Mana A.I., Clough G.W. Predicition of movement for braced cuts in clay[J]. J Geotechnical Engrg Division,1981,107:759-778
[85]郑宏,葛修润,谷先荣等.关于岩土工程有限元分析中的若干问题[J].岩土力学,1995,16(3)：7-11.
[86]王勇,殷宗泽.有限元计算深开挖挖方等效荷载的分析[J].河海大学学报,1998,26(5)：71-74
[87]章清.有限元分析中开挖释放荷载的正确计算[J].河海大学学报,1999,27(3)：112-115.
[88]高俊合,赵维柄,李兴文.深开挖有限元分析中释放荷载模拟[J].河海大学学报,1999,27(1)：47-52
[89]王敏强,许原.有限元分析中开挖释放荷载计算的讨论[J].武汉大学学报,2001,34(1)：56-59
[90]董志宏，邬爱清,丁秀丽.数值流形方法中的锚固支护模拟及初步应用[J].岩石力学与工程学报,2005,24(10)：3754-3760
[91]Jeen Shang Lin, Cheng Yu Ku. Two-scale modeling of jointed rock masses[J]. International Journal of Rock Mechanics & Mining Science,43(2006):426-436
[92]朱爱军.数值流形方法研究及其在岩土工程中的应用[博士论文].重庆：重庆大学,2005
[93]曹文贵,程晔,赵明华.公路路基岩溶顶板安全厚度确定的数值流形方法研究[J].岩土工程学报,2005,27(6)：621-625
[94]卢廷浩.岩土数值分析[M].北京：中国水利水电出版社,2008
[95]徐芝纶.弹性力学简明教程[M].北京：高等教育出版社,1980
[96]王水林.数值流形方法与裂纹扩展的模拟.[博士论文].武汉：中国科学院武汉岩土力学研究所,1998.
[97]Zhang G.X., Sugiura Y., Hasegawa H. Crack propagation and thermal fracture analysis by manifold method[A]. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation[C], Kyoto, Japan,1997.282-297
[98]张鲁新,马惠民.坡体病害地段隧道变形机理及其防治技术.铁道部科学研究院研究报告.1998
[99]马惠民.成昆线东荣河1#隧道变形与坡体位移规律.铁道部科学研究院研究报告.1998
[100]周德培.成昆线东荣河1#隧道变形与坡体病害相互作用机理德模型试验.铁道部科学研究院研究报告.1999
[101]Hoek E., Brown E.T.著,连志开等译.岩石地下工程[M].北京：冶金工业出版社,1980
[2]沈明荣.岩体力学[M].上海：同济大学出版社.1999
[3]蔡美峰.岩石力学与工程[M].北京：科学出版社.2002
[4]凌贤长.蔡德所.岩体力学[M].哈尔滨：哈尔滨工业大学出版社.2002
[5]Jaeger, J.C., Cook, N.G.W.1969. Fundamentals of rock mechanics[M]. London. Chapman and Hall LTD
[6]高峰,谢和平等.节理岩体强度的分形统计分析[J].岩石力学与工程学报,23(21)：3608-3612
[7]李同录,罗世毅等.节理岩体力学参数的选取与应用[J].岩石力学与工程学报,2004,23(13)：2182-2186
[8]汪小刚,陈祖煌.应用蒙特卜洛法确定节理岩体的连通率和综合抗剪强度指标[J].岩石力学与工程学,1992,11(4)：345-355
[9]E.Hoek, E.T.Brown. Empirical strength criterion for rock masses [J]. Geotech Engng. Div. ASCE,1980,106:1013-1035
[10]Kalamaras G.S, Bieniawski Z.T A rock mass strength concept for coal seams incorporating the effect of time [R].8Th ISRM Congress. Tokyo.1995:295-302
[11]Kim M.K., Lade P.V. Modelling rock strength in three dimensions [J]. Int J Rock Mech Min Sci,1984,21:21-33.
[12]Arild Pahnstrom.Characterizing Rock Massesby the RMi for Use in Practical Rock Engineering,Partl:the develoment of the Rock Mass index[J]. Tunneling and Under Ground Space Technology,1996,11(2):175-188
[13]Ramamurthy T. Strength and modulus responses of anisotropic rocks[A]. In:Hudson JA. Editor. Comprehensive rock engineering,1994,1(13):313-29
[14]Bieniawski Z.T. Engineering classification of jointed rock masses[M]. John Wiley and Sons. New York.1989
[15]Serafim J.L., Pereira J.P. Considerations on the geomechanical classification of Bieniawski[A]. In:Proceedings of the Symposium on Engineering Geology and Underground Opening[C]. Portugal. Lisboa1983:1133-1144
[16]Barton N. Rock mechanics review, the shear strength of rock and rock joints[J]. Int. J. Rock Mech. Min. Sci. & Geomech,1976,13(4):255-279
[17]Hoek E., Brown E.T. Practical estimates of rock mass strength[J]. Int J Rock Mech Min Sci,1994,34:1165-86
[18]Palmstrom A., Singh R. The deformation modulus of rock masses comparisons between in situ tests and indirect estimates [J]. Tunneling and Underground Space Tech,2001,(16):115-131
[19]Barton N. Some new Q-value correlations to assist in site characteristics and tunnel design [J]. International Journal of Rock Mechanics & Mining Science,2002,39:185-216
[20]Kayabasi A., Gokceoglu C, Ercanoglu M. Estimating the deformation modulus of rock masses:a comparative study[J]. Int J Rock Mech Min Sci,2003,(40):55-63.
[21]Zhang L., Einstein H. Using RQD to estimate the deformation modulus of rock masses [J]. Int J Rock Mech Min Sci,2004,(41):337-341
[22]Singh B. Continuum characterization of jointed rock masses Part Ⅱ-significance of low shear modulus [J]. Int J Rock Mech Sci &Geomech Abstr,1973,10:337-49
[23]Amadei B., Goodman R.E. A 3-D constitutive relation for fractured rock masses[C]//Proc Int Symp on Mechanical Behavior of Structured media. Ottawa.1981:249-268.
[24]Kawamoto T.,. Ichikawa Y., Kyoya T. Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory [J]. Int J Numer Anal Methods Geomech,1988,12:1-30
[25]Cai M., Horii H. A constitutive model of highly jointed rock masses[J]. Mech Mater,1992,13:217-46.
[26]Goodman R.D., Taylor R., Brekke T.L. A model for the mechanics of jointed rock[J]. ASCE Journal of the soil mechanics and Foundations Division,1968,14(SM3):637-659
[27]Zienkiewicz O.C., Kelly D.W., Bettess P. The coupling of the finite element method and boundary solution procedures[J]. Int J Numer Methods Eng 1977,11:355-75
[28]Desai C.S., Zamman M.M., Lightner J.G., Siriwardane H.J. Thin layer element for interfaces and joint [J]. Int J Numer Anal Methods Geomech.1984,8:19-43
[29]Goodman R.E., Shi G Block theory and its application to rock engineering. Prentice-Hall, New Jersey
[30]Cundall P.A. A computer model for simulating progressive,large-scale movements in blocky rock systems. procecdings of the symposium of the International Society of Rock Mechanics.1971.Nancy.France.Vol(1):paper Ⅱ-8
[31]Shi G.H. Discontinuous deformation analysis-a new numerical model for the static and dynamics of block system. PHD Thesis [Dissertation]. Dept. of civil Engineering. University of Calofornia.Berkeley.1988
[32]Ke T.C. Artificial joint-based on manifold method. In:Working Forum on the Manifold of Material Analysis. Vol.1. Jenner, California, USA, WES, US Army Corps of Engineerings.1995-10-04-06.1995:21-37
[33]Chen G., Miki S., Ohnishi Y. Automatic creation of mathematical meshes in manifold method of material analysis. In:Working Forum on the Manifold of Material Analysis, Vol.1. Jenner, California, USA, WES, US Army Corps of Engineerings.1995-10-04-06. 1995:105-126
[34]Hideomi Ohtsubo, Katsuyuki Suzuki. Utilization of finite covers in the manifold method for accuracy control. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation,Kyoto,Japan,1997:317-322
[35]曹文贵,速宝玉.流形元覆盖系统自动形成方法之研究[J].岩土工程学报.2001,23(2)：1 87-190
[36]蔡永昌,朱合华,夏才初.流形方法覆盖系统自动生成算法[J].同济大学学报,2004,32(5)：585-590
[37]Wang C.Y., Sheng J., Chen M.H. et al. Approximation theories for the manifold method. In: Working Forum on the Manifold of Material Analysis,Vol.1,Jenner,California, USA, WES, US Army Corps of Engineerings.1995-10-04-06.1995:61-86
[38]田荣,栾茂田.高阶流形方法及其应用.工程力学,2001，18(2)：21-26
[39]王水林.四个物理覆盖构成一个单元的流形方法及应用[J].岩石力学与工程学报,1999,18(3)：312-316
[40]宋俊生,大西有三.高阶四边形单元的流形方法[J].岩石力学与工程学报,2003,22(6)：932-936
[41]彭自强,葛修润.数值流形方法在有限元三维二十结点单元上的实现[J].岩石力学与工程学报,2004,23(15)：2622-2627
[42]王芝银,王思敬,杨志法.岩石大变形的流形方法[J].岩石力学与工程学报,1997,16(5)：399-404.
[43]朱以文,曾又林.岩石大变形分析的增量流形方法[J].岩石力学与工程学报,1999,18(1)：1-5
[44]王书法等.岩体弹塑性分析的数值流形方法[J].岩石力学与工程学报,2002,21(6)：900-904
[45]Qiu X.J. Manifold method without use of penalty springs[A]. In:Proc of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, California USA.1996,205-249
[46]Lin D.Z., Mo H.H. Manifold method of material analysis. In:Working Forum on the Manifold of Material Analysis, Vol.1. Jenner, California, USA, WES, US Army Corps of Engineerings.1995-10-04-06.1995:147-164
[47]张国新等.五强溪船闸裂缝的流形元模拟[J].水力学报,2006,24：37-42
[48]郑榕明,张勇慧.基于六面体覆盖的三维数值流形方法的理论探讨与应用[J].岩石力学与工程学报.2004,23(10)：1745-1754
[49]姜清辉,周创兵.四面体有限单元覆盖的三维数值流形方法[J].岩石力学与工程学报,2005,24(24)：4455-4460
[50]彭自强,葛修润.数值流形方法在有限元三维二十节点单元上的实现[J].岩石力学与工程学报,2004,23(15)：2622-2627
[51]骆少明,张湘伟,蔡永昌.非线性数值流形方法的变分原理与应用[J].应用数学与力学,2000,21(12)：1265-1270
[52]王书法,朱维申.考虑侧向影响的数值流形方法及其工程应用[J].岩石力学与工程学报,2001,20(3)：297-300
[53]王后裕,朱可善.高次数值流形方法及其在弹性地基板中的应用[J].力学与实践,2001,3：31-34
[54]栾茂田,张大林,杨庆等.有限覆盖无单元法在裂纹扩展数值分析问题中的应用[J].岩土工程学报,2003,25(5)：527-531
[55]Zhang Guoxin,Sugiura Yasuhito,Hasegawa Hiroo.Crack propagation and thermal fracture analysis by manifold method[A]. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation[C],Kyoto,Japan,1997.282-297
[56]王水林,葛修润.流形方法在模拟裂纹扩展中的应用[J].岩石力学与工程学报1997,16(5)：405-410
[57]Yaw Jeng Chiou, Ren Jow Tsay, Wailin Chuang. Crack propagation using manifold method[A]. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation[C],Kyoto, Japan,1997.298-308
[58]Tsay Ren Jow, Chiou Yaw Jeng, Chuang Wai lin. Crack growth prediction by manifold method[J]. Journal of Engineering Mechanics,1999,125(8):884-890
[59]Yaw Jeng Chiou, Yu Min Lee, Ren Jow Tsay. Mixed modefracture propagation by manifold method[J]. International Journal of Fracture,2002,114:327-347
[60]张大林,栾茂田,杨庆等.基于流形方法的动态应力强度因子数值算法[J].大连理工大学学报,2002,42(5)：590-593
[61]曹文贵,速宝玉.岩石锚固支护的数值流形方法模拟及其应用[J].岩土工程学报,2001,5：581-583
[62]董志宏，邬爱清,丁秀丽.数值流形方法中的锚固支护模拟及初步应用[J].岩石力学与工程学报,2005,24(10)：3754-3760
[63]姜清辉,王书法.锚固岩体的三维数值流形方法模拟[J].岩石力学与工程学报,2006,25(3)：528-532
[64]沈振中,郑磊.基于数值流形方法的水库岩体边坡稳定分析[J].水电能源科学,2006,24(1)：32-35
[65]彭华.岩石边坡稳定分析的数值流形法研究[博士论文].武汉：武汉大学,2001.
[66]KU C.Y. Modeling of jointed rock mass based on the numerical manifold method[Dissertation]. USA:University of Pittsburgh,2001.
[67]王芝银,李云鹏.岩体力学中的增量数值流形方法及其应用[J].西安矿业学院学报,1998,18(增)：23-27
[68]Ke Te-Chih, Tang Jyh-Haw. Modeling of solid-fluid interaction using the manifold method[A]. In:Proceedings of 2nd North American Rock Mechanics Symposium, [C] Montreal, Quebec, Canada,1996.1815-1822
[69]Yuzo Ohnishi, Makoto Tanaka, Tomofumi Koyama. Manifold method in saturated-unsaturated groundwater flow analysis[A]. In:3rd International Conference on Analysis of Discontinuous Deformation from Theory to Practice[C], Vail, Colorado,USA,1999.1213-1228
[70]石根华,裴觉民译.数值流形方法与非连续变形分析[M].北京：清华大学出版社,1997
[71]Lin J-S. A unified framework for discrete and continuum analysis [A].In:Cook BK, Jensen RP, editors. Discrete element methods-numerical modeling of discontinua. Geotechnical Special Publication No.117. New York:ASCE,2002,145-50
[72]张国新,金峰,王光纶.用基于流形元的子域奇异边界元法模拟重力坝的地震破坏[J].工程力学,2001,18(4)：18-27
[73]王俊奎,丁立祚.弹性固体力学[M].北京：中国铁道科学出版社,1990
[74]王芝银,李云鹏.数值流形方法中的几点改进[J].岩土工程学报.1998,20(6)：33-36
[75]林绍忠.单纯形积分的递推公式[J].长江科学院院报.2005,22(3)：32-34
[76]朱爱军.数值流形方法研究及其在岩土工程中的应用[博士论文].重庆：重庆大学,2005.
[77]陈庭.球面数值流形方法及其在地壳运动中的应用研究.[博士论文].武汉：武汉大学，2005
[78]钟万勰,张洪武,吴承伟.参变量变分原理及其在工程中的应用[M].北京：科学出版 社.1997
[79]Jiro H., Takakoto S., Umetaro Y.1979. An elasto-plastic analysis applicable to the'strain softening' materials (in Japanese) [J]. Journal of the Mining and Metallurgical Institute of Japan 10:721-726.
[80]罗国煜,王培清,吴浩等.敷溪口松动变形边坡稳定性和破坏机制浅析[J].湖南水利,1981,(1)：12-16
[81]Jaeger J.C., Cook N. Pinching-off and discing of rocks. J Geophys Res.1963,68:1759-65
[82]彭光忠.单轴压应力下页岩岩块的结构面方向与其力学特性的关系[J].岩土工程学报,1983,5(2)：101-109
[83]张志刚.节理岩体强度确定方法及其各向异性特征研究[博士论文].北京：北京交通大学,2007
[84]Mana A.I., Clough G.W. Predicition of movement for braced cuts in clay[J]. J Geotechnical Engrg Division,1981,107:759-778
[85]郑宏,葛修润,谷先荣等.关于岩土工程有限元分析中的若干问题[J].岩土力学,1995,16(3)：7-11.
[86]王勇,殷宗泽.有限元计算深开挖挖方等效荷载的分析[J].河海大学学报,1998,26(5)：71-74
[87]章清.有限元分析中开挖释放荷载的正确计算[J].河海大学学报,1999,27(3)：112-115.
[88]高俊合,赵维柄,李兴文.深开挖有限元分析中释放荷载模拟[J].河海大学学报,1999,27(1)：47-52
[89]王敏强,许原.有限元分析中开挖释放荷载计算的讨论[J].武汉大学学报,2001,34(1)：56-59
[90]董志宏，邬爱清,丁秀丽.数值流形方法中的锚固支护模拟及初步应用[J].岩石力学与工程学报,2005,24(10)：3754-3760
[91]Jeen Shang Lin, Cheng Yu Ku. Two-scale modeling of jointed rock masses[J]. International Journal of Rock Mechanics & Mining Science,43(2006):426-436
[92]朱爱军.数值流形方法研究及其在岩土工程中的应用[博士论文].重庆：重庆大学,2005
[93]曹文贵,程晔,赵明华.公路路基岩溶顶板安全厚度确定的数值流形方法研究[J].岩土工程学报,2005,27(6)：621-625
[94]卢廷浩.岩土数值分析[M].北京：中国水利水电出版社,2008
[95]徐芝纶.弹性力学简明教程[M].北京：高等教育出版社,1980
[96]王水林.数值流形方法与裂纹扩展的模拟.[博士论文].武汉：中国科学院武汉岩土力学研究所,1998.
[97]Zhang G.X., Sugiura Y., Hasegawa H. Crack propagation and thermal fracture analysis by manifold method[A]. In:Proc of the Second International Conference on Analysis of Discontinuous Deformation[C], Kyoto, Japan,1997.282-297
[98]张鲁新,马惠民.坡体病害地段隧道变形机理及其防治技术.铁道部科学研究院研究报告.1998
[99]马惠民.成昆线东荣河1#隧道变形与坡体位移规律.铁道部科学研究院研究报告.1998
[100]周德培.成昆线东荣河1#隧道变形与坡体病害相互作用机理德模型试验.铁道部科学研究院研究报告.1999
[101]Hoek E., Brown E.T.著,连志开等译.岩石地下工程[M].北京：冶金工业出版社,1980