Pre-critical deformation properties and dilatancy behaviors of Beishan granite under different stress paths
HAN Yang1, 2, 3, CHEN Liang4, ZHOU Zheng1, 2, PU Shikun1, 2, ZHANG Dengke1, 2, BAI Zhixiao1, 2, GUO Liwei2, 5, LI Erbing1, 2
(1. State Key Laboratory of Disaster Prevention and Mitigation of Explosion and Impact, Army Engineering University of PLA, Nanjing, Jiangsu 210007, China; 2. College of Defense Engineering, Army Engineering University of PLA, Nanjing, Jiangsu 210007, China; 3. Nanjing Urban Construction Management Group Co., Ltd., Nanjing, Jiangsu 210006, China; 4. Beijing Research Institute of Uranium Geology, Beijing 100029, China; 5. School of Civil Engineering, Southeast University, Nanjing, Jiangsu 211189, China)
Abstract： Accurately understanding the mechanical laws and mechanisms of micro-damage to the near-field surrounding rock in high-level radioactive waste geological disposal repository is of great significance for achieving the disposal encirclement function. Given that the mechanical behavior and response characteristics of rocks are related to stress paths, the mechanical tests were conducted on three typical stress paths, including loading axial pressure-fixing confining pressure, loading axial pressure-unloading confining pressure, and unloading axial pressure-unloading confining pressure. The Beishan granite in the pre-selected area for high-level radioactive waste geological disposal in China was taken as the research object, and the effects of stress path and initial confining pressure on the pre-critical deformation properties and dilatancy behavior of which under high stress levels were systematically studied. Firstly, taking the brittle stress-drop as the critical state, for the controllable and comparable stress path bifurcation stage, a unified quantitative comparison of macro deformation parameter evolution process and characteristics based on deviatoric stress ratio is proposed. It is found that when unloading confining pressure at high stress levels, axial pressure unloading actually exacerbates lateral deformation and dilatancy compared to loading; The secant Poisson?s ratio curves of secants between different paths are dispersed and clustered, with significant differences in evolutionary characteristics, but the confining pressure effect is not obvious; The relationship between the volumetric and radial strain increment follows a quadratic function, but there are significant differences in gradient and curvature of curves under different paths. Secondly, the plastic theory is used to describe the nonlinear deformation characteristics of rocks, and variables such as plastic strain compliance and radial axial plastic strain increment ratio (proportional coefficient) are defined. The results show that the greater the confining pressure, the more significant the inhibitory effect on lateral deformation and dilatancy, and the stronger the ability to withstand plastic deformation. Compared with axial compression loading, unloading delays the failure, making the development of plastic deformation more complete; Although initial confining pressure has a suppressive effect on rock deformation and failure, it has a greater promoting effect on plastic strain sensitivity and development, and has a greater impact on axial deformation. Finally, it is found that the relationship between the volumetric and axial plastic strain increment under each stress path follow an exponential distribution relationship. Based on this, a unified theoretical relationship between the apparent dilatancy angle and dilation angle before the critical point was constructed, and the influence of initial confining pressure was examined through the dilatancy index. The results show that the apparent dilatancy angle and the dilation angle of loading axial pressure-fixing confining pressure are the smallest, and the confining pressure effect of the dilatancy index is the most obvious; The increase gradient of dilation angle of loading axial pressure-unloading confining pressure is the largest, and the dilatancy trend is the most prominent; The apparent dilatancy angle and dilatation angle of unloading axial-unloading confining pressure are the largest, the dilatancy index is least affected by the initial confining pressure, and the dilatancy property is the most significant. The research results reveal the correlation between the pre-critical deformation evolution and dilatancy properties of Beishan granite under high stress level with stress path, indicating that the development of surrounding rock dilatancy micro-damage in multi-directional unloading area need be paid close attention to in the engineering practice and theoretical research of high-level radioactive waste geological disposal.
[1] WANG J，CHEN L，SU R，et al. The Beishan underground research laboratory for geological disposal of high-level radioactive waste in China：planning，site selection，site characterization and in situ tests[J]. Journal of Rock Mechanics and Geotechnical Engineering，2018，10(3)：411–435.
[2] 王 驹，陈 亮，周志超，等. 我国高放废物地质处置新突破[J]. 原子能科学技术，2024，58(增2)：217–230.(WANG Ju，CHEN Liang，ZHOU Zhichao，et al. Geological disposal of high level radioactive waste in China：Progress and breakthrough during 2019-2024[J]. Atomic Energy Science and Technology，2024，58(Supp.2)：217–230.(in Chinese))
[3] 王 驹. 中国高放废物地质处置21世纪进展[J]. 原子能科学技术，2019，53(10)：2 072–2 082.(WANG Ju. Progress of geological disposal of high-level radioactive waste in China in the 21st Century[J]. Atomic Energy Science and Technology，2019，53(10)：2 072–2 082. (in Chinese))
[4] 康红普. 我国煤矿巷道围岩控制技术发展70年及展望[J]. 岩石力学与工程学报，2021，40(1)：1–30.(KANG Hongpu. Seventy years development and prospects of strata control technologies for coal mine roadways in China[J]. Chinese Journal of Rock Mechanics and Engineering，2021，40(1)：1–30.(in Chinese))
[5] 钱七虎. 地下工程建设安全面临的挑战与对策[J]. 岩石力学与工程学报，2012，31(10)：1 945–1 956.(QIAN Qihu. Challenges faced by underground projects construction safety and countermeasures[J]. Chinese Journal of Rock Mechanics and Engineering，2012，31(10)：1 945–1 956.(in Chinese))
[6] ALKAN H，CINAR Y，PUSCH G. Rock salt dilatancy boundary from combined acoustic emission and triaxial compression tests[J]. International Journal of Rock Mechanics and Mining Sciences，2007，44(1)：108–119.
[7] LAU J S O，CHANDLER N A. Innovative laboratory testing[J]. International Journal of Rock Mechanics and Mining Sciences，2004，41(8)：1 427–1 445.
[8] 黄润秋，黄 达. 卸荷条件下花岗岩力学特性试验研究[J]. 岩石力学与工程学报，2008，27(11)：2 205–2 213.(HUANG Runqiu，HUANG Da. Experimental research on mechanical properties of granites under unloading condition[J]. Chinese Journal of Rock Mechanics and Engineering，2008，27(11)：2 205–2 213.(in Chinese))
[9] ZHANG Z Z，DENG M，BAI J B，et al. Strain energy evolution and conversion under triaxial unloading confining pressure tests due to gob-side entry retained[J]. International Journal of Rock Mechanics and Mining Sciences，2020，126：104184.
[10] GUO Y T，YANG C H，MAO H J. Mechanical properties of Jintan mine rock salt under complex stress paths[J]. International Journal of Rock Mechanics and Mining Sciences，2012，56：54–61.
[11] XIE H Q，HE C H. Study of the unloading characteristics of a rock mass using the triaxial test and damage mechanics[J]. International Journal of Rock Mechanics and Mining Sciences，2004，41：74–80.
[12] 黄润秋，黄 达. 高地应力条件下卸荷速率对锦屏大理岩力学特性影响规律试验研究[J]. 岩石力学与工程学报，2010，29(1)：21–33.(HUANG Runqiu，HUANG Da. Experimental research on affection laws of unloading rates on mechanical properties of Jinping marble under high geostress[J]. Chinese Journal of Rock Mechanics and Engineering，2010，29(1)：21–33.(in Chinese))
[13] ZHANG L M，CONG Y，MENG F Z，et al. Energy evolution analysis and failure criteria for rock under different stress paths[J]. Acta Geotechnica，2021，16：569–580.
[14] FENG X T，XU H，YANG C X，et al. Influence of loading and unloading stress paths on the deformation and failure features of Jinping marble under true triaxial compression[J]. Rock Mechanics and Rock Engineering，2020，53：3 287–3 301.
[15] 李地元，孙 志，李夕兵，等. 不同应力路径下花岗岩三轴加卸载力学响应及其破坏特征[J]. 岩石力学与工程学报，2016，35(增2)：3 449–3 457.(LI Diyuan，SUN Zhi，LI Xibing，et al. Mechanical response and failure characteristics of granite under different stress paths in triaxial loading and unloading conditions[J]. Chinese Journal of Rock Mechanics and Engineering，2016，35(Supp.2)：3 449–3 457.(in Chinese))
[16] LI D Y，SUN Z，XIE T，et al. Energy evolution characteristics of hard rock during triaxial failure with different loading and unloading paths[J]. Engineering Geology，2017，228：270–281.
[17] ZHANG Z T，ZHANG R，XIE H P，et al. Mining-induced coal permeability change under different mining layouts[J]. Rock Mechanics and Rock Engineering，2016，49：3 753–3 768.
[18] 王乐华，牛草原，张冰祎，等. 不同应力路径下深埋软岩力学特性试验研究[J]. 岩石力学与工程学报，2019，38(5)：973–981. (WANG Lehua，NIU Caoyuan，ZHANG Bingyi，et al. Experimental study on mechanical properties of deep-buried soft rock under different stress paths[J]. Chinese Journal of Rock Mechanics and Engineering，2019，38(5)：973–981.(in Chinese))
[19] 段淑倩，高 坡，江 权，等. 高地应力卸荷条件下错动带塑性变形规律与硬化特征[J]. 岩土力学，2022，43(1)：97–109.(DUAN Shuqian，GAO Po，JIANG Quan，et al. Plastic deformation and hardening characteristics of the staggered zone under high in-situ stress unloading conditions[J]. Rock and Soil Mechanics，2022，43(1)：97–109.(in Chinese))
[20] 苗胜军，段懿轩，尹紫微，等. 不同应力路径下辉绿岩能量演化与破坏机制研究[J]. 振动与冲击，2023，42(14)：154–161.(MIAO Shengjun，DUAN Yixuan，YIN Ziwei，et al. Energy evolution and failure mechanism of diabase under different stress paths[J]. Journal of Vibration and Shock，2023，42(14)：154–161.(in Chinese))
[21] 陈子全，李天斌，陈国庆，等. 不同应力路径下砂岩能耗变化规律试验研究[J]. 工程力学，2016，33(6)：120–128.(CHEN Ziquan，LI Tianbin，CHEN Guoqing，et al. Experimental study on energy evolution of sandstone under different stress paths[J]. Engineering Mechanics，2016，33(6)：120–128.(in Chinese))
[22] 何明明，陈蕴生，韩铁林，等. 不同应力路径下砂岩能耗特征的研究[J]. 岩石力学与工程学报，2015，34(增1)：2 632–2 638.(HE Mingming，CHEN Yunsheng，HAN Tielin，et al. Study of energy properties of sandstone under different loading paths[J]. Chinese Journal of Rock Mechanics and Engineering，2015，34(Supp.1)： 2 632–2 638.(in Chinese))
[23] 戴 兵，赵国彦，杨 晨，等. 不同应力路径下岩石峰前卸荷破坏能量特征分析[J]. 采矿与安全工程学报，2016，33(2)：367–374.(DAI Bin，ZHAO Guoyan，YANG Chen，et al. Energy evolution law of rocks in process of unloading failure under different paths[J]. Journal of Mining and Safety Engineering，2016，33(2)：367–374.(in Chinese))
[24] 由 爽，李虎振，侯晓旭，等. 采动应力路径下花岗岩变形破坏特性及能量演化机制[J]. 东北大学学报：自然科学版，2023，44(8)：1 177–1 187.(YOU Shuang，LI Huzhen，HOU Xiaoxu，et al. Deformation damage characteristics and energy evolution mechanism of granite under mining stress path[J]. Journal of Northeastern University：Natural Science，2023，44(8)：1 177–1 187.(in Chinese))
[25] 丛 宇，王在泉，郑颖人，等. 不同卸荷路径下大理岩破坏过程能量演化规律[J]. 中南大学学报：自然科学版，2016，47(9)：3 140–3 147. (CONG Yu，WANG Zaiquan，ZHENG Yingren，et al. Energy evolution principle of fracture propagation of marble with different unloading stress paths[J]. Journal of Central South University：Science and Technology，2016，47(9)：3 140–3 147.(in Chinese))
[26] SCHOCK R N. A constitutive relation describing dilatant behavior of Climax Stock granodiorate[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1976，13(7)：221–223.
[27] MAHMUTOGLU Y，VARDAR M. Effects of inelastic volume increase on fractured rock behaviour[J]. Bulletin of Engineering Geology and the Environment，2003，62：117–121.
[28] VAJDOVA V，BAUD P，WONG T F. Compaction，dilatancy，and failure in porous carbonate rocks[J]. Journal of Geophysical Research：Solid Earth，2004，109(B5)，https://doi.org/10.1029/2003JB002508.
[29] ZHANG Y，WANG Z F，XU D P，et al. True triaxial stresses mobilizing dilatant fracturing and engineering failure of hard rocks[J]. Engineering Failure Analysis，2023，154：107652.
[30] 程建超，贾 震，侯孟冬，等. 砂岩三轴循环加卸载变形特性分析及扩容逾渗模型研究[J]. 岩石力学与工程学报，2024，43(11)：2 687–2 699.(CHEN Jianchao，JIA Zheng，HOU Mengdong，et al. Percolation modelling of dilation deformation evolution of sandstone under tri-axial cyclic loading-unloading[J]. Chinese Journal of Rock Mechanics and Engineering，2024，43(11)：2 687–2 699.(in Chinese))
[31] 赵星光，李鹏飞，马利科，等. 循环加、卸载条件下北山深部花岗岩损伤与扩容特性[J]. 岩石力学与工程学报，2014，33(9)：1 740–1 748.(ZHAO Xingguang，LI Pengfei，MA Like，et al. Damage and dilation characteristics of deep granite at Beishan under cyclic loading-unloading conditions[J]. Chinese Journal of Rock Mechanics and Engineering，2014，33(9)：1 740–1 748.(in Chinese))
[32] 李鹏飞，赵星光，郭 政，等. 北山花岗岩在三轴压缩条件下的强度参数演化[J]. 岩石力学与工程学报，2017，36(7)：1 599–1 610. (LI Pengfei，ZHAO Xingguang，GUO Zheng，et al. Variation of strength parameters of Beishan granite under triaxial compression[J]. Chinese Journal of Rock Mechanics and Engineering，2017，36(7)：1 599–1 610.(in Chinese))
[33] 陈学章，何江达，肖明砾，等. 三轴卸荷条件下大理岩扩容与能量特征分析[J]. 岩土工程学报，2014，36(6)：1 106–1 112.(CHEN Xuezhang，HE Jiangda，XIAO Mingli，et al. Dilatancy and energy properties of marble under triaxial unloading condition[J]. Chinese Journal of Geotechnical Engineering，2014，36(6)：1 106–1 112.(in Chinese))
[34] 刘新荣，刘 俊，李栋梁，等. 不同初始卸荷水平对深埋砂岩力学特性影响规律试验研究[J]. 岩土力学，2017，38(11)：3 081–3 088. (LIU Xinrong，LIU Jun，LI Dongliang，et al. Experimental research on the effect of different initial unloading levels on mechanical properties of deep-buried sandstone[J]. Rock and Soil Mechanics，2017，38(11)：3 081–3 088.(in Chinese))
[35] 尹光志，鲁 俊，李 星，等. 中间主应力对砂岩扩容及强度特性影响[J]. 煤炭学报，2017，42(4)：879–885.(YIN Guangzhi，LU Jun，LI Xing，et al. Influence of intermediate principal stress on dilation and strength characteristics of sandstone[J]. Journal of China Coal Society，2017，42(4)：879–885.(in Chinese))
[36] 赵国彦，戴 兵，董陇军，等. 不同应力路径下岩石三轴卸荷力学特性与强度准则研究[J]. 岩土力学，2015，36(11)：3 121–3 127. (ZHAO Guoyan，DAI Bing，DONG Longjun，et al. Experimental research on mechanical characteristics and strength criterion of rock triaxial unloading tests under different stress paths[J]. Rock and Soil Mechanics，2015，36(11)：3 121–3 127.(in Chinese))
[37] WALTON G，GAINES S. Evaluation of stress path and load rate effects on rock strength using compression testing data for Stanstead Granite[J]. International Journal of Rock Mechanics and Mining Sciences，2023，169：105455.
[38] CHEN L，ZHAO X G，LIU J，et al. Progress on rock mechanics research of Beishan granite for geological disposal of high-level radioactive waste in China[J]. Rock Mechanics Bulletin，2023，2(3)：100046.
[39] ULUSAY R，HUDSON J A. 1974-2006 The complete ISRM suggested methods for rock characterization，testing and monitoring[S]. Ankara，Turkey：ISRM Turkish National Group，2007.
[40] 中华人民共和国行业标准编写组. SL/T 264—2020 水利水电工程岩石试验规程[S]. 北京：中国水利水电出版社，2020.(The Professional Standards Compilation Group of the People′s Republic of China. SL/T 264—2020 Specifications for rock tests in water conservancy and hydroelectric engineering[S]. Beijing：China Water Power Press，2020.(in Chinese))
[41] LIU J F，QIU X S，YANG J X，et al. Failure transition of shear-to-dilation band of rock salt under triaxial stresses[J]. Journal of Rock Mechanics and Geotechnical Engineering，2024，16(1)：56–64.
[42] 李建贺，盛 谦，朱泽奇，等. Mine-by试验洞开挖过程中围岩应力路径与破坏模式分析[J]. 岩石力学与工程学报，2017，36(4)：821–830.(LI Jianhe，SHENG Qian，ZHU Zeqi，et al. Analysis of stress path and failure mode of surrounding rock during mining of Mine-by experimental cave[J]. Chinese Journal of Rock Mechanics and Engineering，2017，36(4)：821–830.(in Chinese))
[43] 朱合华，蔡武强，梁文灏. GZZ岩体强度三维分析理论与深埋隧道应力控制设计分析方法[J]. 岩石力学与工程学报，2023，42(1)：1–27.(ZHU Hehua，CAI Wuqiang，LIANG Wenhao. GZZ strength-based three-dimensional analysis theory and stress-controlled design method in deep tunneling[J]. Chinese Journal of Rock Mechanics and Engineering，2023，42(1)：1–27.(in Chinese))
[44] 刘 宁，张春生，陈祥荣，等. 深埋隧洞开挖围岩应力演化过程监测及特征研究[J]. 岩石力学与工程学报，2011，30(9)：1 729–1 737. (LIU Ning，ZHANG Chunsheng，CHEN Xiangrong，et al. Monitoring and characteristics study of stress evolution of surrounding rock during deep tunnel excavation[J]. Chinese Journal of Rock Mechanics and Engineering，2011，30(9)：1 729–1 737.(in Chinese))
[45] ZHOU X P. Localization of deformation and stress–strain relation for mesoscopic heterogeneous brittle rock materials under unloading[J]. Theoretical and Applied Fracture Mechanics，2005，44(1)：27–43.
[46] 王传乐，杜广印，李二兵，等. 北山深部花岗岩常规三轴压缩条件下的强度参数演化及能量耗散[J]. 岩石力学与工程学报，2021，40(11)：2 238–2 248.(WANG Chuanle，DU Guangyin，LI Erbing，et al. Evolution of strength parameters and energy dissipation of Beishan deep granite under conventional triaxial compression[J]. Chinese Journal of Rock Mechanics and Engineering，2021，40(11)：2 238–2 248.(in Chinese))
[47] ZHAO X G，WANG J，CAI M，et al. Influence of intermediate principal stress on the strainburst characteristics of Beishan granite with consideration of end effect[J]. Rock Mechanics and Rock Engineering，2021，54(9)：4 771–4 791.
[48] 赵星光，王 驹，秦向辉，等. 中国高放废物地质处置地下实验室场址深部岩体地应力测量及工程应用[J]. 中南大学学报：自然科学版，2021，52(8)：2 634–2 645.(ZHAO Xingguang，WANG Ju，QIN Xianghui，et al. In-situ stress measurements at depth and engineering application at China?s underground research laboratory site for high-level radioactive waste disposal[J]. Journal of Central South University：Science and Technology，2021，52(8)：2 634–2 645. (in Chinese))
[49] 朱泽奇，盛 谦，张占荣. 脆性岩石侧向变形特征及损伤机理研究[J]. 岩土力学，2008，29(8)：2 137–2 143.(ZHU Zeqi，SHENG Qian，ZHANG Zhanrong. Study of lateral deformation characteristics and damage mechanism of brittle rock[J]. Rock and Soil Mechanics，2008，29(8)：2 137–2 143.(in Chinese))
[50] 陈 亮，刘建锋，王春萍，等. 北山深部花岗岩不同应力状态下声发射特征研究[J]. 岩石力学与工程学报，2012，31(增2)：3 618–3 624.(CHEN Liang，LIU Jianfeng，WANG Chunping，et al. Study of acoustic emission characteristics of Beishan deep granite under different stress conditions[J]. Chinese Journal of Rock Mechanics and Engineering，2012，31(Supp.2)：3 618–3 624.(in Chinese))
[51] XU X T，HUANG R Q，LI H，et al. Determination of Poisson's ratio of rock material by changing axial stress and unloading lateral stress test[J]. Rock Mechanics and Rock Engineering，2015，48(6)：853–857.
[52] 郭建强，刘新荣，赵 青. 岩石卸荷力学特性的理论研究[J]. 岩土力学，2017，38(增2)：123–130.(GUO Jianqiang，LIU Xinrong，ZHAO Qing. Theoretical research on rock unloading mechanical characteristics[J]. Rock and Soil Mechanics，2017，38(Supp.2)：123–130.(in Chinese))
[53] 裴建良，刘建锋，徐 进. 层状大理岩卸荷力学特性试验研究[J]. 岩石力学与工程学报，2009，28(12)：2 496–2 502.(PEI Jianliang，LIU Jianfeng，XU Jin. Experimental study of mechanical properties of layered marble under unloading condition[J]. Chinese Journal of Rock Mechanics and Engineering，2009，28(12)：2 496–2 502.(in Chinese))
[54] 高春玉，徐 进，何 鹏，等. 大理岩加卸载力学特性的研究[J]. 岩石力学与工程学报，2005，24(3)：456–460.(GAO Chunyu，XU Jin，HE Peng，et al. Study on mechanical properties of marble under loading and unloading conditions[J]. Chinese Journal of Rock Mechanics and Engineering，2005，24(3)：456–460.(in Chinese))
[55] CAI W Q，ZHU H H，LIANG W H，et al. A post-peak dilatancy model for soft rock and its application in deep tunnel excavation[J]. Journal of Rock Mechanics and Geotechnical Engineering，2023，15(3)：683–701.
[56] 杨以荣，谢红强，肖明砾，等. 卸荷条件下横观各向同性岩体扩容与能量特性分析[J]. 岩土力学，2017，38(6)：1 589–1 599.(YANG Yirong，XIE Hongqiang，XIAO Mingli，et al. Dilatancy and energy characteristics analysis of transverse-isotropic rock mass under triaxial unloading condition[J]. Rock and Soil Mechanics，2017，38(6)：1 589–1 599.(in Chinese))
[57] 邱士利，冯夏庭，张传庆，等. 不同卸围压速率下深埋大理岩卸荷力学特性试验研究[J]. 岩石力学与工程学报，2010，29(9)：1 807–1 817.(QIU Shili，FENG Xiating，ZHANG Chuanqing，et al. Experimental research on mechanical properties of deep-buried marble under different unloading rates of confining pressures[J]. Chinese Journal of Rock Mechanics and Engineering，2010，29(9)：1 807–1 817.(in Chinese))
[58] 邱士利，冯夏庭，张传庆，等. 不同初始损伤和卸荷路径下深埋大理岩卸荷力学特性试验研究[J]. 岩石力学与工程学报，2012，31(8)：1 686–1 697.(QIU Shili，FENG Xiating，ZHANG Chuanqing，et al. Experimental research on mechanical properties of deep marble under different initial damage levels and unloading paths[J]. Chinese Journal of Rock Mechanics and Engineering，2012，31(8)：1 686–1 697.(in Chinese))
[59] 黄 兴，刘泉声，刘恺德，等. 深部软弱地层TBM掘进围岩变形破坏特性室内试验研究[J]. 岩石力学与工程学报，2015，34(1)：76–92.(HUANG Xing，LIU Quansheng，LIU Kaide，et al. Laboratory study of deformation and failure of soft rock for deep ground tunnelling with TBM[J]. Chinese Journal of Rock Mechanics and Engineering，2015，34(1)：76–92.(in Chinese))
[60] YUAN S C，HARRISON J P. An empirical dilatancy index for the dilatant deformation of rock[J]. International Journal of Rock Mechanics and Mining Sciences，2004，41(4)：679–686.
[61] 俞 缙，刘泽瀚，林立华，等. 变幅循环加卸载作用下大理岩扩容特性试验研究[J]. 岩土力学，2021，42(11)：2 934–2 942.(YU Jin，LIU Zehan，LIN Lihua，et al. Characteristics of dilatancy of marble under variable amplitude cyclic loading and unloading[J]. Rock and Soil Mechanics，2021，42(11)：2 934–2 942.(in Chinese))
[62] VERMEER P A. Non-associated plasticity for soils，concrete and rock[M]. Dordrecht：Springer Netherlands，1998：163–196.
[63] ZHAO X G，CAI M. A mobilized dilation angle model for rocks[J]. International Journal of Rock Mechanics and Mining Sciences，2010，47(3)：368–384.
2025, Vol. 44 
