裂隙岩体损伤本构模型研究及应用
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摘要
由细观力学与宏观唯象学理论相结合所建立的裂隙岩体损伤本构模型,既能反映材料内部的微观结构,又能说明其宏观力学性能。本文拟在细观力学Gibbs自由能函数的基础上,结合断裂力学、经典塑性力学和连续介质的热力学等,建立了弹性损伤本构模型和弹塑性损伤耦合本构模型。主要研究工作为:
①通过对栖霞山矿地质构造的研究,根据赤平极射投影原理,判断了X断裂,确定了地应力的方位和倾角。并据此在-625 m中段主巷道内进行取样工作,根据凯泽效应,进行岩石的声发射实验,得到了原岩地应力。另外,室内常规实验为后续的计算分析提供了所需的岩石力学参数。
②从Gibbs自由能函数出发,通过Legendre变换得到了应变空间表示的Helmholtz自由能函数,进而根据正交性原理,得到了三维弹性损伤本构方程。同时,运用断裂准则,分别确定了拉、压应力作用下平行分布裂纹和均匀分布裂纹的弹性损伤演化规律。
③将岩石峰前的应力-应变关系分为三个阶段分别考虑:1)在弹性初始损伤阶段,采用第三章的弹性损伤本构方程;2)在弹塑性初始损伤阶段,结合经典塑性力学和连续介质的热力学,得到了考虑损伤的弹塑性应变能和塑性乘子,并考虑等温过程,引入塑性流动法则,得到了弹塑性初始损伤本构方程。同时,根据裂纹的弯折扩展确定了损伤门槛值。3)在弹塑性损伤耦合阶段,将总的应力率和应变率分解为弹性、塑性和损伤三部分,引入塑性流动法则和共轭力表示的损伤演化方程,根据一致性条件,求得塑性乘子和损伤乘子,从而得到三维弹塑性损伤耦合阶段的非线性增量本构方程。
④退化三维模型为平变应变状态下的弹塑性损伤耦合本构模型,分别进行了单轴压缩和双轴压缩问题的验证,理论预测曲线与实验应力-应变曲线吻合较好。运用ANSYS 9.0,分别计算分析了在理想弹塑性模型和本文弹塑性损伤耦合本构模型下-625 m中段主巷道断面的的平面应变问题。结果表明,损伤模型较理想弹塑性模型更符合实际情况。
The damage constitutive model, which is established through the combination of micromechanics and macroscopical phenomenology, can both reflect the micro-structures of the material, and explain its macroscopic mechanical behaviors. Based on micromechanical Gibbs free energy function, as well as fracture mechanics, classic plastic mechanics and continuum thermodynamics, an elastic damage constitutive model and an elastic-plastic constitutive model coupled with damage are obtained. The specific work including:
①From research on geological structure of QiXiaShan mine by using equatorial horizon projection principle, the dips of ground stress are determined through the judgment of the X failure. Thus, specimen was picked up in the main roadway in 625 meters depth below ground level according to the determination. And the original ground stress was tested through Kaiser acoustic emissional experiments. In addition, indoor conventional experiments provided necessary parameters to numerical simulations.
②Helmholtz free energy function in the strain space is deduced through Legendre transformation from Gibbs energy function. Three dimensional elastic damage constitutive equations are acquired based upon orthogonality principle. Meanwhile, the elastic damage evolution laws are given for both parallel and uniformly distributed microcracks under both tension and compression respectively by using fracture criterion.
③The pre-peak stress-strain relationship is divided into three stages: Firstly, the elastic damage constitutive equations given in chapter three are adopted for elastic stage only with initial damage. Secondly, by taking classic plastic mechanics and continuum thermodynamics into account, the elastic-plastic strain energy including initial damage and plastic multiplicator are derived for elastic-plastic stage. Then, the elastic-plastic constitutive equations considering initial damage are gained by importing plastic flow rule and isothermal assumption. Threshold value is determined according to the kink expanding of microcracks. Lastly, the total stress rate and strain rate are resolved into three parts as elastic, plastic and damage ones. According to consistent condition, plastic and damage multiplicators are calculated by introducing plastic flow rule and damage evolution equation denoted by conjugated force. The three dimensional nonlinear incremental constitutive equations for the stage of elastic-plastic coupled with damage are developed.
④The elastic-plastic constitutive model coupled with damage under plane strain condition is obtained. Identifications, both on uniaxial and bioaxial compressions, have shown good consistency with the experimental curves of stress-strain. The main roadway in 625 meters depth below ground level under plane strain condition is studied through numerical simulation using ANSYS 9.0. Comparison is drawn between the elastic-plastic constitutive model coupled with damage and the ideal elastic-plastic one, and the former is more accord with actual situation.
①通过对栖霞山矿地质构造的研究,根据赤平极射投影原理,判断了X断裂,确定了地应力的方位和倾角。并据此在-625 m中段主巷道内进行取样工作,根据凯泽效应,进行岩石的声发射实验,得到了原岩地应力。另外,室内常规实验为后续的计算分析提供了所需的岩石力学参数。
②从Gibbs自由能函数出发,通过Legendre变换得到了应变空间表示的Helmholtz自由能函数,进而根据正交性原理,得到了三维弹性损伤本构方程。同时,运用断裂准则,分别确定了拉、压应力作用下平行分布裂纹和均匀分布裂纹的弹性损伤演化规律。
③将岩石峰前的应力-应变关系分为三个阶段分别考虑:1)在弹性初始损伤阶段,采用第三章的弹性损伤本构方程;2)在弹塑性初始损伤阶段,结合经典塑性力学和连续介质的热力学,得到了考虑损伤的弹塑性应变能和塑性乘子,并考虑等温过程,引入塑性流动法则,得到了弹塑性初始损伤本构方程。同时,根据裂纹的弯折扩展确定了损伤门槛值。3)在弹塑性损伤耦合阶段,将总的应力率和应变率分解为弹性、塑性和损伤三部分,引入塑性流动法则和共轭力表示的损伤演化方程,根据一致性条件,求得塑性乘子和损伤乘子,从而得到三维弹塑性损伤耦合阶段的非线性增量本构方程。
④退化三维模型为平变应变状态下的弹塑性损伤耦合本构模型,分别进行了单轴压缩和双轴压缩问题的验证,理论预测曲线与实验应力-应变曲线吻合较好。运用ANSYS 9.0,分别计算分析了在理想弹塑性模型和本文弹塑性损伤耦合本构模型下-625 m中段主巷道断面的的平面应变问题。结果表明,损伤模型较理想弹塑性模型更符合实际情况。
The damage constitutive model, which is established through the combination of micromechanics and macroscopical phenomenology, can both reflect the micro-structures of the material, and explain its macroscopic mechanical behaviors. Based on micromechanical Gibbs free energy function, as well as fracture mechanics, classic plastic mechanics and continuum thermodynamics, an elastic damage constitutive model and an elastic-plastic constitutive model coupled with damage are obtained. The specific work including:
①From research on geological structure of QiXiaShan mine by using equatorial horizon projection principle, the dips of ground stress are determined through the judgment of the X failure. Thus, specimen was picked up in the main roadway in 625 meters depth below ground level according to the determination. And the original ground stress was tested through Kaiser acoustic emissional experiments. In addition, indoor conventional experiments provided necessary parameters to numerical simulations.
②Helmholtz free energy function in the strain space is deduced through Legendre transformation from Gibbs energy function. Three dimensional elastic damage constitutive equations are acquired based upon orthogonality principle. Meanwhile, the elastic damage evolution laws are given for both parallel and uniformly distributed microcracks under both tension and compression respectively by using fracture criterion.
③The pre-peak stress-strain relationship is divided into three stages: Firstly, the elastic damage constitutive equations given in chapter three are adopted for elastic stage only with initial damage. Secondly, by taking classic plastic mechanics and continuum thermodynamics into account, the elastic-plastic strain energy including initial damage and plastic multiplicator are derived for elastic-plastic stage. Then, the elastic-plastic constitutive equations considering initial damage are gained by importing plastic flow rule and isothermal assumption. Threshold value is determined according to the kink expanding of microcracks. Lastly, the total stress rate and strain rate are resolved into three parts as elastic, plastic and damage ones. According to consistent condition, plastic and damage multiplicators are calculated by introducing plastic flow rule and damage evolution equation denoted by conjugated force. The three dimensional nonlinear incremental constitutive equations for the stage of elastic-plastic coupled with damage are developed.
④The elastic-plastic constitutive model coupled with damage under plane strain condition is obtained. Identifications, both on uniaxial and bioaxial compressions, have shown good consistency with the experimental curves of stress-strain. The main roadway in 625 meters depth below ground level under plane strain condition is studied through numerical simulation using ANSYS 9.0. Comparison is drawn between the elastic-plastic constitutive model coupled with damage and the ideal elastic-plastic one, and the former is more accord with actual situation.
引文
[1] E J Sellers, P Klerck. Modeling of the effect of discontinuities on the extent of the fracture zone surrounding deep tunnels [J]. Tunneling and Underground Space Technology, 2000, 15(4): 463- 469.
[2] 何满潮, 谢和平, 彭苏萍等. 深部开采岩体力学研究[J]. 岩石力学与工程学报, 2005, 24(16): 2803- 2813.
[3] 何满潮. 深部的概念体系及工程评价指标[J]. 岩石力学与工程学报, 2005, 24(16): 2854- 2858.
[4] 杨光松. 损伤力学与复合材料损伤[M]. 北京, 国防工业出版社, 1995.
[5] 余寿文, 冯西桥. 损伤力学[M]. 北京, 清华大学出版社, 1997.
[6] 王军. 损伤力学的理论与应用[M]. 北京,科学出版社, 1997.
[7] 韦立德. 岩石力学损伤和流变本构模型研究[D]. 南京, 河海大学博士论文, 2003.
[8] L P Khoroshun, L V Nazarenk. A model of the short-term damageability of a transversally isotropic material [J]. International Applied Mechanics, 2001, 37(1): 66- 74.
[9] 杨友卿. 岩石强度的损伤力学分析[J]. 岩石力学与工程学报, 1999, 18(1): 23- 27.
[10] 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.
[11] 黄国明, 黄润秋. 岩石弹塑性损伤耦合本构模型[J]. 西安科技学院学报, 1996, 16(4): 328- 333.
[12] D F Malan. Manuel rocha medal recipient simulating the time-dependent behaviour of excavations in hard rock [J]. Rock Mechanics and rock Engineering, 2002, 35(4): 225- 254.
[13] Enrico Maranini, Tsutomu Yamaguchi. A non-associated viscoplastic model for the behaviour of granite in triaxial compression [J]. Mechanics of Materials, 2001, 33: 283- 293.
[14] Yang Chunhe, J J K Daemen, YmJianhua. Experimental investigation of creep behavior of salt rock [J]. Rock Mechanics and Rock Engineering, l998, 36(2): 233- 242.
[15] 柯孚久, 白以龙, 夏蒙棼. 固体中微裂纹系统统计演化的基本描述[J]. 力学学报, 1991, 23(3): 290- 298.
[16] 李广平, 陶振宇. 真三轴条件下的岩石细观损伤力学模型[J]. 岩土工程学报, 1995, 17(1): 24- 31.
[17] 杨更社. 岩石细观损伤力学特性及本构关系的 CT 识别[J]. 煤炭学报, 2000, 25(增): 102- 106.
[18] D Lydzba, J F Shao. Study of poroelasticity material coeffcients as response of microstructure[J]. Mechanics of Cohesive- frictional Materials, 2000, 5: 149- 171.
[19] G Duveau, J F Shao, J P Henry. Assessment of some failure criteria for strongly anisotropic geomaterials [J]. Mechanics of Cohesive- frictional Materials, 1998, 3: 1- 26.
[20] 葛修润, 任建喜, 蒲毅彬等. 煤岩三轴细观损伤演化规律的 CT 动态试验[J]. 岩石力学与工程学报, 1999, 18(5): 497- 502.
[21] Shao J F et al. Model of induced anisotropic damage in granites. International Journal of Rock Mechanics and Mining Sciences, 1999, 36: 1001- 1012.
[22] Yonghong Zhao. Crack Pattern evolution and a fractal damage constitutive model for rock [J]. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(3): 349- 366.
[23] 刘立, 邱贤德, 阎宗岭等. 复合岩石的微结构损伤破坏[J]. 矿山压力与顶板管理, 1999(2): 77- 80.
[24] 李广平. 考虑裂纹闭合效应的岩石损伤本构关系[J]. 应用力学学报, 1996, 13(1): 93- 97.
[25] D Zhao, G Swoboba, F Laabmayr. Damage mechanics and its application for the design of an underground theater [J]. Tunneling and Underground Space Technology, 2004, 19: 567- 575.
[26] 李浩, 陶振宇. 岩石在压应力作用下的裂纹扩展模型[J]. 武汉水利电力大学学报, 1999, 32(6): 10- 13.
[27] 赵永红. 岩石弹脆性分维损伤本构模型[J]. 地质科学, 1997, 32(4): 487- 494.
[28] 周光泉, 陈德华, 席道瑛. 岩石连续损伤本构方程[J]. 岩石力学与工程学报, 1995, 14(3): 229- 235.
[29] 殷有泉. 岩石的塑性、损伤及其本构表述[J]. 地质科学, 1995, 30(1): 63- 70.
[30] G Frantziskonis, C S Desai. Constitutive Model with Strain Softening [J]. International Journal of Solids and Structures, 1987, 23(6): 733- 768.
[31] 杨松岩, 俞茂宏. 一种基于混合物理论的非饱和岩土类材料的弹塑性损伤模型[J]. 岩土工程学报, 1998, 20(5): 58- 63.
[32] 韦立德, 徐卫亚, 杨春和等. 具有统计损伤的岩石弹塑性本构模型研究[J]. 岩石力学与工程学报, 2004, 23(12): 1971- 1975.
[33] 黄国明, 黄润秋. 岩石弹塑性损伤耦合本构模型[J]. 西安科技学院学报, 1996, 16(4): 328- 333.
[34] 曹文贵, 张升, 赵明华.基于新型损伤定义的岩石损伤统计本构模型探讨[J]. 岩土力学, 2006, 27(1): 41- 46.
[35] 周小平, 哈秋聆, 张永兴等. 峰前围压卸荷条件下岩石的应力-应变全过程分析和变形局部化研究[J]. 岩石力学与工程学报, 2005, 24(18): 3236- 3245.
[36] X P Zhou. Localization of deformation and stress-strain relation for mesoscopic heterogeneous brittle rock material under unloading [J].Theoretical and applied fracturemechanics, 2005, 44: 27- 43.
[37] 周小平, 张永兴, 朱可善. 中低围压下细观非均匀性岩石本构关系研究[J]. 岩土工程学报, 2003, 25(5): 606- 610.
[38] Xiao-ping Zhou, Yong-xing Zhang, Qiu-ling Ha, Ke-shan Zhu. Bounds on the complete stress-strain relation for a crack-weakened rock mass under compressive loads [J]. International Journal of Solids and Structures, 2004, 41: 6173- 6196.
[39] 周小平, 张永兴, 朱可善. 单轴拉伸条件下岩石本构理论研究[J]. 岩土力学, 2003, 24(增): 143- 147.
[40] M Cai, P K Kaiser, C D Martin. Quantification of rock mass damage in underground excavations from microseismic event monitoring [J]. International Journal of Rock Mechanics and Mining Sciences, 2001, 38: 1135- 1145.
[41] 周小平, 王建华, 张永兴等. 单轴拉伸条件下非均匀性岩石损伤局部化和应力应变关系分析[J]. 应用数学和力学, 2004, 25(9): 943- 950.
[42] Zhou Xiao-ping, Zhang Yong-xing, Ha Qiu-ling, Wang Jian-hua. Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous brittle rock subjected to compressive loads [J]. Applied Mathematics and Mechanics, 2004, 25(9): 1039- 1046.
[43] Zhou Xiao-ping, Wang Jian-hua, Zhang Yong-xing, Ha Qiu-ling. Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous rock under uniaxial tensile loading [J]. Applied Mathematics and Mechanics, 2004, 25(9): 1031- 1038.
[44] 周小平, 张永兴, 哈秋聆, 王建华. 单轴拉伸条件下细观非均匀性岩石变形局部化及其应力-应变全过程研究[J]. 岩石力学与工程学报, 2004, 23(1): 1- 6.
[45] 周小平, 张永兴, 朱可善. 压应力状态下断续节理岩体全过程应力-应变关系及其变形局部化分析[J]. 岩石力学与工程学报, 2005, 24(2): 217- 221.
[46] 周小平, 张永兴, 朱可善. 单轴拉伸条件下细观非均匀性岩石本构关系研究[J]. 土木工程学报, 2005, 38(3): 87- 93.
[47] J J Marigo. Modelling of brittle and fatigue damage for elastic material by growth of microvoids [J]. Engineering Fracture Mechanics, 1985, 21(4): 861- 874.
[48] J Lemaitre. A continuous damage mechanics model for ductile fracture [J]. Journal of Engineering Materials and Technology, 1985, 107(1): 83- 89.
[49] J C Simo, J W Ju. Strain-and stress-based continuum damage models, part I. formulation, part II. Computational aspects [J]. International Journal of Solids and Structures, 1988, 23(3): 821- 869.
[50] J W Ju. On energy-based coupled elastoplastic damage theories:constitutive modelling andcomputational aspects [J]. International Journal of Solids and Structures, 1989, 25(7): 803- 833.
[51] D Halm, A Dragon. A model of anisotropic damage by mesocrack growth; unilateral effect [J]. International Journal of Damage Mechanics, 1996, 5(4): 384- 402.
[52] J F Shao. Poroelastic behaviour of brittle rock materials with anisotropic damage [J]. Mechanics of Materials, 1998, 30: 41- 53.
[53] J F Shao, J W Rudnicki. A microcracked-based continuous damage model for brittle geomaterials [J]. Mechanics of Materials, 2000, 32: 607- 619.
[54] 邵国建. 岩体初始地应力场的反演回归分析[J]. 水利水电科技进展, 2000, 20(5): 36- 38.
[55] 靳晓光, 李春晓, 沈军辉. 二郎山公路隧道围岩变形影响因素研究[J]. 地质灾害与环境保护, 2000, 11(1): 58- 62.
[56] 金艳丽, 刘汉东. 初始地应力场反演及回归分析方法研究[J]. 隧道建设, 2004, 24(2): 6- 8.
[57] 张彩昌. 利用 Kaiser 效应测定矿山原岩应力[J]. 云南冶金, 2002, 31(5): 15- 17.
[58] 黄志鹏, 朱可善, 郭映忠. 岩石 Kaiser 效应方向独立性试验研究[J]. 长江科学院院报, 1998, 15(2): 8- 9.
[59] 彭新明, 孙友宏, 李安宁. 岩石声发射技术的应用现状[J]. 世界地质, 2000, 19(3): 303- 306.
[60] 徐林生, 王兰生, 徐进等. 二郎山公路隧道岩体应力测试研究[J]. 华东公路, 2002, 134(1): 63- 64.
[61] 薛亚东, 高德利. 声发射地应力测量中凯塞点的确定[J]. 石油大学学报(自然科学版), 2000, 24(5): 1- 3.
[62] 北京有色冶金设计研究总院. 南京铅锌银矿扩建初步设计书, 1989.
[63] 南京栖霞山锌阳矿业有限公司. 35 万吨/年技改扩建工程可行性研究报告, 2001.
[64] 蔡美峰, 何满潮, 刘东燕. 岩石力学与工程[M]. 北京,科学出版社, 2002.
[65] 孙玉科, 古迅. 赤平极射投影在岩体工程地质力学中的应用[M]. 北京, 科学出版社, 1980.
[66] 周小平, 王建华. 测量地应力的新方法[J]. 岩土力学, 2002, 23(3): 316- 320.
[67] J F Shao, K T Chau, X T Feng. Modelling of anisotropic damage and creep deformation in brittle rocks [J]. International Journal of Rock Mechanics and Mining Sciences, 2006, 43: 582- 592.
[68] Q Yang, X Chen, L G. Tham. Relationship of crack fabric tensors of different orders [J]. Mechanics Research Communications, 2004, 31: 661- 666.
[69] Shou-Wen Yu, Xi-Qiao Feng. A micromechanics-based damage model formicrocrack-weakened brittle solids [J]. Mechanics of Materials, 1995, 20: 59- 76.
[70] 陈惠发著, 余天庆, 王勋文等译. 土木工程材料的本构方程(第二卷 塑性与建模)[M]. 武汉, 华中科技大学出版社, 2001.
[71] V A Lubarda, D Krajcinovic. Damage tensors and crack density distribution [J]. International Journal of Solids and Structures, 1993, 30(20): 2859- 2877.
[72] 张倬元, 王士天, 王兰生. 工程地质分析原理[M]. 北京, 地质出版社, 1994.
[73] 于学馥, 郑颖人, 刘怀恒等. 地下工程围岩稳定分析[M]. 北京, 煤炭工业出版社, 1980.
[74] 李华晔. 地下洞室围岩稳定性分析[M]. 北京, 中国水利水电出版社, 1999.
[75] 朱敬民. 岩体力学[M]. 北京, 中国建筑工业出版社, 1979.
[76] Zhou Xiaoping. Analysis of the localization of deformation and the complete stress-strain relation for mescoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading [J]. International Journal of Solids and Structures, 2004, 41(5-6): 1725- 1738.
[77] Xi-Qiao Feng, Shou-Wen Yu. Micromechanical modelling of tensile response of elastic-brittle materials [J]. International Journal of Solids and Structures, 1995, 32(22): 3359- 3372.
[78] 刘东燕. 断续节理岩体的压剪断裂及其强度特性研究[D]. 重庆, 重庆建筑工程学院博士论文, 1993.
[2] 何满潮, 谢和平, 彭苏萍等. 深部开采岩体力学研究[J]. 岩石力学与工程学报, 2005, 24(16): 2803- 2813.
[3] 何满潮. 深部的概念体系及工程评价指标[J]. 岩石力学与工程学报, 2005, 24(16): 2854- 2858.
[4] 杨光松. 损伤力学与复合材料损伤[M]. 北京, 国防工业出版社, 1995.
[5] 余寿文, 冯西桥. 损伤力学[M]. 北京, 清华大学出版社, 1997.
[6] 王军. 损伤力学的理论与应用[M]. 北京,科学出版社, 1997.
[7] 韦立德. 岩石力学损伤和流变本构模型研究[D]. 南京, 河海大学博士论文, 2003.
[8] L P Khoroshun, L V Nazarenk. A model of the short-term damageability of a transversally isotropic material [J]. International Applied Mechanics, 2001, 37(1): 66- 74.
[9] 杨友卿. 岩石强度的损伤力学分析[J]. 岩石力学与工程学报, 1999, 18(1): 23- 27.
[10] 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.
[11] 黄国明, 黄润秋. 岩石弹塑性损伤耦合本构模型[J]. 西安科技学院学报, 1996, 16(4): 328- 333.
[12] D F Malan. Manuel rocha medal recipient simulating the time-dependent behaviour of excavations in hard rock [J]. Rock Mechanics and rock Engineering, 2002, 35(4): 225- 254.
[13] Enrico Maranini, Tsutomu Yamaguchi. A non-associated viscoplastic model for the behaviour of granite in triaxial compression [J]. Mechanics of Materials, 2001, 33: 283- 293.
[14] Yang Chunhe, J J K Daemen, YmJianhua. Experimental investigation of creep behavior of salt rock [J]. Rock Mechanics and Rock Engineering, l998, 36(2): 233- 242.
[15] 柯孚久, 白以龙, 夏蒙棼. 固体中微裂纹系统统计演化的基本描述[J]. 力学学报, 1991, 23(3): 290- 298.
[16] 李广平, 陶振宇. 真三轴条件下的岩石细观损伤力学模型[J]. 岩土工程学报, 1995, 17(1): 24- 31.
[17] 杨更社. 岩石细观损伤力学特性及本构关系的 CT 识别[J]. 煤炭学报, 2000, 25(增): 102- 106.
[18] D Lydzba, J F Shao. Study of poroelasticity material coeffcients as response of microstructure[J]. Mechanics of Cohesive- frictional Materials, 2000, 5: 149- 171.
[19] G Duveau, J F Shao, J P Henry. Assessment of some failure criteria for strongly anisotropic geomaterials [J]. Mechanics of Cohesive- frictional Materials, 1998, 3: 1- 26.
[20] 葛修润, 任建喜, 蒲毅彬等. 煤岩三轴细观损伤演化规律的 CT 动态试验[J]. 岩石力学与工程学报, 1999, 18(5): 497- 502.
[21] Shao J F et al. Model of induced anisotropic damage in granites. International Journal of Rock Mechanics and Mining Sciences, 1999, 36: 1001- 1012.
[22] Yonghong Zhao. Crack Pattern evolution and a fractal damage constitutive model for rock [J]. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(3): 349- 366.
[23] 刘立, 邱贤德, 阎宗岭等. 复合岩石的微结构损伤破坏[J]. 矿山压力与顶板管理, 1999(2): 77- 80.
[24] 李广平. 考虑裂纹闭合效应的岩石损伤本构关系[J]. 应用力学学报, 1996, 13(1): 93- 97.
[25] D Zhao, G Swoboba, F Laabmayr. Damage mechanics and its application for the design of an underground theater [J]. Tunneling and Underground Space Technology, 2004, 19: 567- 575.
[26] 李浩, 陶振宇. 岩石在压应力作用下的裂纹扩展模型[J]. 武汉水利电力大学学报, 1999, 32(6): 10- 13.
[27] 赵永红. 岩石弹脆性分维损伤本构模型[J]. 地质科学, 1997, 32(4): 487- 494.
[28] 周光泉, 陈德华, 席道瑛. 岩石连续损伤本构方程[J]. 岩石力学与工程学报, 1995, 14(3): 229- 235.
[29] 殷有泉. 岩石的塑性、损伤及其本构表述[J]. 地质科学, 1995, 30(1): 63- 70.
[30] G Frantziskonis, C S Desai. Constitutive Model with Strain Softening [J]. International Journal of Solids and Structures, 1987, 23(6): 733- 768.
[31] 杨松岩, 俞茂宏. 一种基于混合物理论的非饱和岩土类材料的弹塑性损伤模型[J]. 岩土工程学报, 1998, 20(5): 58- 63.
[32] 韦立德, 徐卫亚, 杨春和等. 具有统计损伤的岩石弹塑性本构模型研究[J]. 岩石力学与工程学报, 2004, 23(12): 1971- 1975.
[33] 黄国明, 黄润秋. 岩石弹塑性损伤耦合本构模型[J]. 西安科技学院学报, 1996, 16(4): 328- 333.
[34] 曹文贵, 张升, 赵明华.基于新型损伤定义的岩石损伤统计本构模型探讨[J]. 岩土力学, 2006, 27(1): 41- 46.
[35] 周小平, 哈秋聆, 张永兴等. 峰前围压卸荷条件下岩石的应力-应变全过程分析和变形局部化研究[J]. 岩石力学与工程学报, 2005, 24(18): 3236- 3245.
[36] X P Zhou. Localization of deformation and stress-strain relation for mesoscopic heterogeneous brittle rock material under unloading [J].Theoretical and applied fracturemechanics, 2005, 44: 27- 43.
[37] 周小平, 张永兴, 朱可善. 中低围压下细观非均匀性岩石本构关系研究[J]. 岩土工程学报, 2003, 25(5): 606- 610.
[38] Xiao-ping Zhou, Yong-xing Zhang, Qiu-ling Ha, Ke-shan Zhu. Bounds on the complete stress-strain relation for a crack-weakened rock mass under compressive loads [J]. International Journal of Solids and Structures, 2004, 41: 6173- 6196.
[39] 周小平, 张永兴, 朱可善. 单轴拉伸条件下岩石本构理论研究[J]. 岩土力学, 2003, 24(增): 143- 147.
[40] M Cai, P K Kaiser, C D Martin. Quantification of rock mass damage in underground excavations from microseismic event monitoring [J]. International Journal of Rock Mechanics and Mining Sciences, 2001, 38: 1135- 1145.
[41] 周小平, 王建华, 张永兴等. 单轴拉伸条件下非均匀性岩石损伤局部化和应力应变关系分析[J]. 应用数学和力学, 2004, 25(9): 943- 950.
[42] Zhou Xiao-ping, Zhang Yong-xing, Ha Qiu-ling, Wang Jian-hua. Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous brittle rock subjected to compressive loads [J]. Applied Mathematics and Mechanics, 2004, 25(9): 1039- 1046.
[43] Zhou Xiao-ping, Wang Jian-hua, Zhang Yong-xing, Ha Qiu-ling. Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous rock under uniaxial tensile loading [J]. Applied Mathematics and Mechanics, 2004, 25(9): 1031- 1038.
[44] 周小平, 张永兴, 哈秋聆, 王建华. 单轴拉伸条件下细观非均匀性岩石变形局部化及其应力-应变全过程研究[J]. 岩石力学与工程学报, 2004, 23(1): 1- 6.
[45] 周小平, 张永兴, 朱可善. 压应力状态下断续节理岩体全过程应力-应变关系及其变形局部化分析[J]. 岩石力学与工程学报, 2005, 24(2): 217- 221.
[46] 周小平, 张永兴, 朱可善. 单轴拉伸条件下细观非均匀性岩石本构关系研究[J]. 土木工程学报, 2005, 38(3): 87- 93.
[47] J J Marigo. Modelling of brittle and fatigue damage for elastic material by growth of microvoids [J]. Engineering Fracture Mechanics, 1985, 21(4): 861- 874.
[48] J Lemaitre. A continuous damage mechanics model for ductile fracture [J]. Journal of Engineering Materials and Technology, 1985, 107(1): 83- 89.
[49] J C Simo, J W Ju. Strain-and stress-based continuum damage models, part I. formulation, part II. Computational aspects [J]. International Journal of Solids and Structures, 1988, 23(3): 821- 869.
[50] J W Ju. On energy-based coupled elastoplastic damage theories:constitutive modelling andcomputational aspects [J]. International Journal of Solids and Structures, 1989, 25(7): 803- 833.
[51] D Halm, A Dragon. A model of anisotropic damage by mesocrack growth; unilateral effect [J]. International Journal of Damage Mechanics, 1996, 5(4): 384- 402.
[52] J F Shao. Poroelastic behaviour of brittle rock materials with anisotropic damage [J]. Mechanics of Materials, 1998, 30: 41- 53.
[53] J F Shao, J W Rudnicki. A microcracked-based continuous damage model for brittle geomaterials [J]. Mechanics of Materials, 2000, 32: 607- 619.
[54] 邵国建. 岩体初始地应力场的反演回归分析[J]. 水利水电科技进展, 2000, 20(5): 36- 38.
[55] 靳晓光, 李春晓, 沈军辉. 二郎山公路隧道围岩变形影响因素研究[J]. 地质灾害与环境保护, 2000, 11(1): 58- 62.
[56] 金艳丽, 刘汉东. 初始地应力场反演及回归分析方法研究[J]. 隧道建设, 2004, 24(2): 6- 8.
[57] 张彩昌. 利用 Kaiser 效应测定矿山原岩应力[J]. 云南冶金, 2002, 31(5): 15- 17.
[58] 黄志鹏, 朱可善, 郭映忠. 岩石 Kaiser 效应方向独立性试验研究[J]. 长江科学院院报, 1998, 15(2): 8- 9.
[59] 彭新明, 孙友宏, 李安宁. 岩石声发射技术的应用现状[J]. 世界地质, 2000, 19(3): 303- 306.
[60] 徐林生, 王兰生, 徐进等. 二郎山公路隧道岩体应力测试研究[J]. 华东公路, 2002, 134(1): 63- 64.
[61] 薛亚东, 高德利. 声发射地应力测量中凯塞点的确定[J]. 石油大学学报(自然科学版), 2000, 24(5): 1- 3.
[62] 北京有色冶金设计研究总院. 南京铅锌银矿扩建初步设计书, 1989.
[63] 南京栖霞山锌阳矿业有限公司. 35 万吨/年技改扩建工程可行性研究报告, 2001.
[64] 蔡美峰, 何满潮, 刘东燕. 岩石力学与工程[M]. 北京,科学出版社, 2002.
[65] 孙玉科, 古迅. 赤平极射投影在岩体工程地质力学中的应用[M]. 北京, 科学出版社, 1980.
[66] 周小平, 王建华. 测量地应力的新方法[J]. 岩土力学, 2002, 23(3): 316- 320.
[67] J F Shao, K T Chau, X T Feng. Modelling of anisotropic damage and creep deformation in brittle rocks [J]. International Journal of Rock Mechanics and Mining Sciences, 2006, 43: 582- 592.
[68] Q Yang, X Chen, L G. Tham. Relationship of crack fabric tensors of different orders [J]. Mechanics Research Communications, 2004, 31: 661- 666.
[69] Shou-Wen Yu, Xi-Qiao Feng. A micromechanics-based damage model formicrocrack-weakened brittle solids [J]. Mechanics of Materials, 1995, 20: 59- 76.
[70] 陈惠发著, 余天庆, 王勋文等译. 土木工程材料的本构方程(第二卷 塑性与建模)[M]. 武汉, 华中科技大学出版社, 2001.
[71] V A Lubarda, D Krajcinovic. Damage tensors and crack density distribution [J]. International Journal of Solids and Structures, 1993, 30(20): 2859- 2877.
[72] 张倬元, 王士天, 王兰生. 工程地质分析原理[M]. 北京, 地质出版社, 1994.
[73] 于学馥, 郑颖人, 刘怀恒等. 地下工程围岩稳定分析[M]. 北京, 煤炭工业出版社, 1980.
[74] 李华晔. 地下洞室围岩稳定性分析[M]. 北京, 中国水利水电出版社, 1999.
[75] 朱敬民. 岩体力学[M]. 北京, 中国建筑工业出版社, 1979.
[76] Zhou Xiaoping. Analysis of the localization of deformation and the complete stress-strain relation for mescoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading [J]. International Journal of Solids and Structures, 2004, 41(5-6): 1725- 1738.
[77] Xi-Qiao Feng, Shou-Wen Yu. Micromechanical modelling of tensile response of elastic-brittle materials [J]. International Journal of Solids and Structures, 1995, 32(22): 3359- 3372.
[78] 刘东燕. 断续节理岩体的压剪断裂及其强度特性研究[D]. 重庆, 重庆建筑工程学院博士论文, 1993.