开展冻融环境下岩石疲劳力学特性研究对于寒区岩体工程长期稳定性评价及其冻融灾害预测预报与治理具有重要意义。本文以单裂隙砂岩为研究对象，采用室内试验及理论分析相结合的方法开展研究工作。主要内容以及结论包括：

（1）完成了常温下单裂隙砂岩三（单）轴压缩试验。试验结果表明：单轴压缩下，裂隙倾角为90°的单裂隙砂岩强度最高，随着裂隙倾角的增大，强度降低；三轴压缩下，当围压一定时，随着裂隙倾角增大，强度呈递减趋势，当裂隙倾角一定时，随着围压的增大，单裂隙砂岩强度与变形均有提升，泊松比降低；单裂隙砂岩沿预制裂隙尖端处产生翼裂纹发生剪切破坏，围压增大，破坏模式由局部剪切破坏向整体剪切破坏过渡。

（2）开展了冻融循环前后单裂隙砂岩室内物理试验，得到岩样质量损失率与波速损失率。完成了不同冻融循环次数下单裂隙砂岩三轴压缩试验，同时进行了声发射监测试验。试验结果表明：当冻融循环次数与裂隙倾角一定时，随着围压的升高，单裂隙砂岩强度增大；当冻融循环次数与围压一定时，随着裂隙倾角的增大，强度呈递减趋势，裂隙倾角为90°时力学性能最弱；当裂隙倾角与围压一定时，随着冻融循环次数的增加，岩样内部在冻胀力作用下胶结程度减弱，加载时承载能力降低，峰值应变增大，弹性模量降低，泊松比升高；冻融单裂隙砂岩沿预制裂隙尖端处生成翼裂纹发生剪切破坏，随着冻融循环次数的增加，由局部剪切向整体剪切过渡，剪切破坏角度呈增大趋势；三轴压缩下的声发射信号特征表现为声发射振铃计数的不断累积，最终在破坏时达到极值。

（3）完成了冻融下不同裂隙倾角单裂隙砂岩三轴压缩疲劳加载试验，同时进行了声发射监测试验。试验结果表明：动载扰动下，单裂隙砂岩强度劣化加剧，随着裂隙倾角增大，强度降低；振幅增大，岩样所受扰动加剧，加载过程中所产生的疲劳应变增大，强度进一步劣化；破坏时沿主裂隙附近生成多条支裂隙，破坏形式复杂化；疲劳加载下声发射信号特征与静载下有所不同，在动载加载初期会出现第一个极值点，此阶段应变瞬间增大，微裂隙迅速扩展，声发射信号得到释放，随后回归平稳，并在破坏时达到最大值。

（4）完成了冻融前后、三轴压缩试验后以及疲劳加载试验后的单裂隙砂岩核磁共振细观试验，分析了单裂隙砂岩在不同裂隙倾角以及不同冻融循环次数因素下孔隙结构变化规律。试验结果表明：单裂隙砂岩T₂谱曲线存在三个主峰，前两个主峰占比高达99%，孔径分布主要在1~100μm范围内；不同裂隙倾角单裂隙砂岩在冻融循环后T₂谱曲线均有不同程度的左移，各个主峰的信号强度有所提升，说明岩样内部在冻胀力作用下萌生发育微小孔隙，随着循环次数的增多，微小孔隙不断扩展贯通，孔径变迁形式为微小孔-中孔-大孔；破坏后的大孔增幅明显，岩样内部孔隙在外荷载的作用下，扩展贯通形成宏观裂隙面，最终发生失稳破坏。

（5）给出了基于内变量理论的冻融单裂隙砂岩疲劳本构模型，理论模型拟合值与试验曲线基本吻合，建立的模型可以反映疲劳加载作用下冻融单裂隙砂岩疲劳力学特性。

The study of rock fatigue mechanical properties under freeze-thaw environment is of great significance for the long-term stability evaluation of rock engineering in cold regions and the prediction and treatment of freeze-thaw disasters. In this paper, single-fractured sandstone is taken as the research object, and the research work is carried out by combining laboratory test and theoretical analysis. The main contents and conclusions include:

(1) Triaxial (uniaxial) compression test of single-fractured sandstone at room temperature was completed. The experimental results show that: Under uniaxial compression, the strength of single-fractured sandstone with fracture dip angle of 90° is the highest, with the increase of fracture dip angle, the strength decreases; Under triaxial compression, when the confining pressure is constant, the strength decreases with the increase of fracture dip angle, when the fracture dip angle is constant, the strength and deformation of single-fractured sandstone increase with the increase of confining pressure, and the Poisson’s ratio decreases; The wing crack of single-fractured sandstone occurs shear failure along the tip of prefabricated crack, with the increase of confining pressure, the failure mode changes from local shear failure to overall shear failure.

(2) The indoor physical tests of single-fractured sandstone before and after freeze-thaw cycles were carried out, and the mass loss rate and wave velocity loss rate of rock samples were obtained. The triaxial compression test of single-fractured sandstone under different freeze-thaw cycles was completed, and the acoustic emission monitoring test was carried out. The experimental results show that: When the number of freeze-thaw cycles and fracture dip angle are constant, the strength of single-fractured sandstone increases with the increase of confining pressure; When the number of freeze-thaw cycles and confining pressure are constant, the strength decreases with the increase of fracture dip angle, the mechanical property is the weakest when the fracture dip angle is 90°; When the fracture dip angle and confining pressure are constant, with the increase of the number of freeze-thaw cycles, the cementation degree inside the rock sample decreases under the action of frost heaving force, and the bearing capacity decreases, the peak strain increases, the elastic modulus decreases, and the Poisson’s ratio increases under loading; The shear failure occurs along the wing crack generated at the tip of the prefabricated fracture in the freeze-thaw single-fractured sandstone. With the increase of the number of freeze-thaw cycles, the local shear transition to the overall shear, and the shear failure angle increases; The acoustic emission signal under triaxial compression is characterized by the continuous accumulation of acoustic emission ringing counts, and finally reaches the extreme value when the failure occurs.

(3) The triaxial compression fatigue loading test of single-fractured sandstone with different fracture dip angles under freezing-thawing was completed, and the acoustic emission monitoring test was carried out. The experimental results show that: Under dynamic load disturbance, the strength deterioration of single-fractured sandstone intensifies, and the strength decreases with the increase of fracture dip angle; With the increase of amplitude, the disturbance of rock sample increases, the fatigue strain generated during loading increases, and the strength further deteriorates; Several branch fractures are generated near the main fracture, and the failure mode is complicated; The characteristics of acoustic emission signals under fatigue loading are different from those under static loading, in the early stage of dynamic loading, the first extreme point will appear, at this stage, the strain increases instantaneously, the microcracks expand rapidly, the acoustic emission signals are released, then the regression is stable, and the maximum value is reached when the failure occurs.

(4) The NMR microscopic tests of single-fractured sandstone before and after freezing-thawing、after triaxial compression test and after fatigue loading test were completed, and the variation law of pore structure of single-fractured sandstone under different fracture inclination angles and different freeze-thaw cycles was analyzed. The experimental results show that: There are three main peaks in the T₂ spectrum curve of single fractured sandstone, the first two main peaks account for 99%, and the pore size distribution is mainly in the range of 1~100μm; The T₂ spectrum curves of single-fractured sandstone with different fracture angles have different degrees of left shift after freeze-thaw cycles, and the signal strength of each main peak is improved, it indicates that micro-pores are initiated and developed within the rock sample under the action of frost heaving force, with the increase of cycles, micro-pores continue to expand and penetrate, and the pore size changes in the form of micro-pores-mesopores-macropores; Macropores increase obviously after failure, under the action of external load, the pores in the rock sample expand and penetrate to form macroscopic fracture surface, and finally instability failure occurs.

(5) The fatigue constitutive model of freeze-thaw single-fractured sandstone based on internal variable theory is given, the fitting value of theoretical model is basically consistent with the experimental curve, the established model can reflect the fatigue mechanical properties of the freeze-thaw single-fractured sandstone under fatigue loading.

[1]李长洪, 肖永刚, 王宇, 等. 高海拔寒区岩质边坡变形破坏机制研究现状及趋势[J]. 工程科学学报, 2019, 41(11): 1374-1386.

[2]李宁, 程国栋, 谢定义. 西部大开发中的岩土力学问题[J]. 岩土工程学报, 2001， 23(3): 268-272.

[3]徐数祖, 王家澄, 张立新. 冻土物理学[M]. 北京: 科学出版社, 2001: 1-3.

[4]张海波. 动、静荷载作用下不同裂隙倾角裂隙岩体力学性能试验模拟研究[D]. 南京: 河海大学, 2007.

[5]彭建兵, 崔鹏, 庄建琦. 川藏铁路对工程地质提出的挑战[J]. 岩石力学与工程学报, 2020, 39(12): 2377-2389.

[6]INCE I, FENER M. A prediction model for uniaxial compressive stength of deteriorated pyroclastic rocks due to freeze–thaw cycle[J]. Journal of African Earth Sciences, 2016, 120:134–140.

[7]HAN T, SHI J, CAO X. Fracturing and damage to sandstone under coupling effects of chemical corrosion and freeze-thaw cycles[J]. Rock Mechanics and Rock Engineering, 2016, 49(11): 4245-4255.

[8]GHOBADI M H, BABAZADEH R. Experimental studies on the effects of cyclic freezing–thawing, salt crystallization, and thermal shock on the physical and mechanical characteristics of selected sandstones[J]. Rock Mechanics and Rock Engineering, 2015, 48(03): 1001-1016

[9]KOCK T D, BOONE M A, SCHRYVER T D, et al. A pore-scale study of fracture dynamics in rock using X-ray micro-CT under ambient freeze–thaw cycling[J]. Environmental science & technology, 2015, 49(05): 2867-2874.

[10]MUTLUTURK M, ALTINDAG R, TURK G. A decay function model for the integrity loss of rock when subjected to recurrent cycles of freezing-thawing and heating-cooling[J]. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(2): 237-244.

[11]MARTINS L, VASCONCELOS G, LOURENCO P B, et al. Influence of the freeze–thaw cycles on the physical and mechanical properties of granites[J]. Journal of Materials in Civil Engineering, 2016, 28(5).

[12]LUO X, JIANG N, ZUO C, et al. Damage characteristics of altered and unaltered diabases subjected to extremely cold freeze–thaw cycles[J]. Rock Mechanics and Rock Engineering, 2014, 47:1997-2004.

[13]FUKUDA M. Rock weathering by freeze–thaw cycles[J]. Low Temp Sci Series A. Phys. Sci, 1974, 32: 243-249.

[14]CHEN T C, YEUNG M R, MORI N. Effect of water saturation on deterioration of welded tuff due to freeze-thaw action[J]. Cold Regions Science and Technology, 2004, 38(2/3): 127-136.

[15]NICHOLSON D T, NICHOLSON F H. Physical deterioration of sedimentary rocks subjected to experimental freeze–thaw weathering[J]. Earth Surface Processes and Landforms, 2000, 25: 1295-1308.

[16]杨念哥, 周科平, 雷涛, 等. 冻融循环下砂岩动力特性及其破坏规律[J]. 中国有色金属学报, 2016, 26(10): 2181-2187.

[17]裴向军, 蒙明辉, 袁进科, 等. 干燥及饱水状态下裂隙岩石冻融特征研究[J]. 岩土力学, 2017, 38(07): 1999-2006.

[18]徐新木, 张耀平, 付玉华, 等. 冻融循环下含节理类岩石试样剪切破坏特性[J]. 吉林大学学报(地球科学版), 2021, 51(02): 483-494.

[19]单仁亮, 杨昊, 郭志明, 等. 负温饱水红砂岩三轴压缩强度特性试验研究[J]. 岩石力学与工程学报, 2014, 33(S2): 3657-3664.

[20]陈国庆, 郭帆, 王剑超, 等. 冻融后石英砂岩三轴蠕变特性试验研究[J]. 岩土力学, 2017, 38(S1): 203-210.

[21]俞缙, 傅国锋, 陈旭, 等. 冻融循环后砂岩三轴卸围压力学特性试验研究[J]. 岩石力学与工程学报, 2015, 34(10): 2001-2009.

[22]王乐华, 姜照容, 李建林, 等. 冻融循环条件下层理砂岩卸荷力学特性试验研究[J]. 冰川冻土, 2016, 38(04): 1052-1058.

[23]刘杰, 雷岚, 王瑞红, 等.冻融循环中低应力水平加卸载作用下砂岩动力特性研究[J].岩土力学,2017,38(09): 2539-2550.

[24]吴冠男. 冻融循环条件下含弧状裂隙类岩石的裂纹扩展机理和力学特性研究[D]. 济南: 山东大学, 2021.

[25]张君岳, 田镇, 刘桓兑, 等. 冻融红砂岩物理力学性质损伤演化试验研究[J]. 矿业研究与开发, 2020, 40(10): 79-84.

[26]赵建军, 解明礼, 余建乐, 等. 冻融作用下含裂隙岩石力学特性及损伤演化规律试验研究[J]. 工程地质学报, 2019, 27(06): 1199-1207.

[27]陈招军, 王乐华, 王思敏, 等. 冻融循环条件下岩石加卸荷力学特性研究[J]. 长江科学院院报, 2017, 34(01): 98-103.

[28]张慧梅, 杨更社. 冻融岩石损伤劣化及力学特性试验研究[J]. 煤炭学报, 2013, 38(10): 1756-1762.

[29]路亚妮, 李新平, 肖家双. 单裂隙岩体冻融力学特性试验分析[J]. 地下空间与工程学报, 2014, 10(03): 593-598+649.

[30]任建喜, 蒋宇, 葛修润. 单轴压缩岩石疲劳寿命影响因素试验分析[J]. 岩土工程学报, 2005, 27(11): 1282-1285.

[31]宫凤强, 张乐, 李夕兵, 等. 不同预静载硬岩在动力扰动下断裂特性的试验研究[J]. 岩石力学与工程学报, 2017, 36(08): 1841-1854.

[32]冯春林, 吴献强, 丁德馨, 等. 周期荷载作用下白砂岩的疲劳特性研究[J]. 岩石力学与工程学报, 2009, 28(S1): 2749-2754.

[33]刘建锋, 谢和平, 徐进, 等. 循环荷载作用下岩石阻尼特性的试验研究[J]. 岩石力学与工程学报, 2008, 27(04): 712-717.

[34]葛修润, 蒋宇, 卢允德, 等. 周期荷载作用下岩石疲劳变形特性试验研究[J]. 岩石力学与工程学报, 2003, 22(10): 1581-1585.

[35]葛修润. 周期荷载下岩石大型三轴试件的变形和强度特性研究[J]. 岩土力学, 1987, 9(02): 11-19.

[36]杨永杰, 段会强, 邢鲁义. 周期荷载作用下煤的疲劳变形及能量演化特征[J]. 应用基础与工程科学学报, 2018, 26(01): 154-167.

[37]卢高明, 李元辉. 围压对黄砂岩疲劳破坏变形特性的影响[J]. 岩土力学, 2016, 37(07): 1847-1856.

[38]任松, 白月明, 姜德义, 等. 周期荷载作用下盐岩声发射特征试验研究[J]. 岩土力学, 2012,33(06): 1613-1618+1639.

[39]杨红伟, 许江, 吴鑫, 等. 周期水压力作用下砂岩变形试验与小波分析[J]. 重庆大学学报, 2011, 34(04): 6-12.

[40]许江, 鲜学福, 王鸿. 循环荷载作用下周期充水岩石变形规律的研究[J]. 地下空间与工程学报, 2006, 2(04): 556-560.

[41]周有成, 陈有亮, 邵伟. 含穿透裂纹岩石的高频疲劳特性研究[J]. 工程力学, 2011, 28(07): 98-102.

[42]VANEGHI R G, FERDOSI B, OKOTH A D, et al. Strength degradation of sandstone and granodiorite under uniaxial cyclic loading[J]. Journal of Rock Mechanics and Geotechnical Engineering, 2018, 10(01): 117-126.

[43]ZHOU X P, JIANG D C, ZHAO Z. Digital Evaluation of Micro-Pore Water Effects on Mechanical and Damage Characteristics of Sandstone Subjected to Uniaxial, Cyclic Loading–Unloading Compression by 3D Reconstruction Technique[J]. Rock Mechanics and Rock Engineering, 2021, 55(1): 147-167.

[44]Wang Y, Gao S H, Li C H, et al. Energy dissipation and damage evolution for dynamic fracture of marble subjected to freeze-thaw and multiple level compressive fatigue loading[J]. International Journal of Fatigue, 2020, 142: 105927.

[45]樊秀峰, 吴振祥, 简文彬. 循环荷载下砂岩疲劳损伤过程的声学特性分析[J]. 岩土力学, 2009, 30(S1): 58-62.

[46]赵百超, 陈四利, 侯芮. 冻融循环与疲劳荷载作用下水泥土力学特性试验研究[J]. 中外公路, 2021, 41(04): 362-365.

[47]张春会, 耿哲, 徐刚, 等. 液氮冻融循环作用下饱水煤样力学特性试验研究[J]. 煤炭科学技术, 2020, 48(10): 218-224.

[48]黄正均, 赵星光, 蔡美峰, 等. 三轴压缩加载频率对花岗岩疲劳特性影响试验[J]. 中国矿业, 2019, 28(04): 150-155+162.

[49]谢璨, 李树忱, 晏勤, 等. 不同尺寸裂隙岩石损伤破坏特性光弹性试验研究[J].岩土工程学报,2018,40(03): 568-575.

[50]宋勇军, 张磊涛, 任建喜, 等. 冻融后循环荷载作用下红砂岩力学特性试验研究[J]. 煤炭工程, 2019, 51(02): 112-117.

[51]申艳军, 杨更社, 王铭, 等. 冻融–周期荷载下单裂隙类砂岩损伤及断裂演化试验分析[J]. 岩石力学与工程学报, 2018, 37(03): 709-717.

[52]杨更社, 申艳军, 贾海梁, 等.冻融环境下岩体损伤力学特性多尺度研究及进展[J].岩石力学与工程学报,2018,37(03): 545-563.

[53]王琨, 周航宇, 赖杰, 等. 核磁共振技术在岩石物理与孔隙结构表征中的应用[J]. 仪器仪表学报, 2020, 41(02): 101-114.

[54]周科平, 李杰林, 许玉娟, 等. 冻融循环条件下岩石核磁共振特性的试验研究[J]. 岩石力学与工程学报, 2012, 31(04): 731-737.

[55]李杰林, 周科平, 柯波. 冻融后花岗岩孔隙发育特征与单轴抗压强度的关联分析[J]. 煤炭学报, 2015, 40(08): 1783-1789.

[56]白松涛, 程道解, 万金彬, 等. 砂岩岩石核磁共振T2谱定量表征[J].石油学报, 2016, 37(03): 382-391+414.

[57]宋勇军, 张磊涛, 任建喜, 等. 基于核磁共振技术的弱胶结砂岩干湿循环损伤特性研究[J]. 岩石力学与工程学报, 2019, 38(04): 825-831.

[58]孙中光, 姜德义, 谢凯楠, 等. 基于低场磁共振的北山花岗岩热损伤研究[J]. 煤炭学报, 2020, 45(03): 1081-1088.

[59]李克钢, 杨宝威, 秦庆词. 基于核磁共振技术的白云岩卸荷损伤与渗透特性试验研究[J]. 岩石力学与工程学报, 2019, 38(S2): 3493-3502.

[60]张娜, 王水兵, 严成钢, 等. 基于核磁共振技术的泥岩水化损伤孔隙结构演化试验[J]. 煤炭学报, 2019, 44(S1): 110-117.

[61]Yao Y, Liu D. Comparison of low-field NMR and mercury intrusion porosimetry in characterizing pore size distributions of coals[J]. Fuel, 2012, 95(1): 152-158.

[62]褚夫蛟, 刘敦文, 陶明, 等. 基于核磁共振的不同含水状态砂岩动态损伤规律[J]. 工程科学学报, 2018, 40(02): 144-151.

[63]李夕兵, 翁磊, 谢晓锋, 等. 动静载荷作用下含孔洞硬岩损伤演化的核磁共振特性试验研究[J]. 岩石力学与工程学报, 2015, 34(10): 1985-1993.

[64]吴志军, 卢槐, 翁磊, 等. 基于核磁共振实时成像技术的裂隙砂岩渗流特性研究[J]. 岩石力学与工程学报, 2021,40(02): 263-275.

[65]程桦, 陈汉青, 曹广勇, 等. 冻土毛细–薄膜水分迁移机制及其试验验证[J]. 岩土工程学报, 2020, 42(10): 1790-1799.

[66]张慧梅, 谢祥妙, 彭川, 等. 三向应力状态下冻融岩石损伤本构模型[J]. 岩土工程学报, 2017, 39(08): 1444-1452.

[67]彭述权, 王培宇, 樊玲, 等. 节理岩体弹塑黏性疲劳本构模型研究[J]. 岩土力学, 2021, 42(02): 379-389.

[68]郭建强, 黄质宏. 循环荷载作用下岩石疲劳本构模型初探[J]. 岩土工程学报, 2015, 37(09): 1698-1704.

[69]刘汉卿, 浦少云, 刘西金, 等. 周期荷载下岩石分数阶黏弹塑性本构模型研究[J]. 长江科学院院报, 2018, 35(09): 127-132+138.

[70]黄兴. 循环荷载作用下石膏岩疲劳损伤特性及其本构模型[D]. 西安: 长安大学, 2017.

[71]唐欣, 俞缙, 林立华, 等. 岩石疲劳应力等效化及非线性疲劳变形本构模型[J]. 岩土工程学报, 2021, 43(01): 102-111.

[72]张平阳, 夏才初, 周舒威, 等. 循环加-卸载岩石本构模型研究[J]. 岩土力学, 2015, 36(12): 3354-3359.

[73]Rice J R. Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity[J]. Journal of the Mechanics and Physics of Solids, 1971, 19(6): 433-455..

[74]葛修润, 蒋宇, 卢允德, 等. 周期载荷作用下岩石疲劳应变特性试验研究[J]. 岩石力学与工程学报, 2003, 22(10): 1581-1585.

[75]葛修润. 周期载荷作用下岩石大型三轴试验的应变和强度特性研究[J]. 岩土力学, 1987, 18(2): 11-19.

[76]乔丽苹, 王者超, 李术才, 等. 岩石内变量蠕变模型研究[J]. 岩土力学, 2012, 33(12): 3529-3537+3603.

[77]王者超, 赵建纲, 李术才, 等. 循环荷载作用下花岗岩疲劳力学性质及其本构模型[J]. 岩石力学与工程学报, 2012, 31(09): 1888-1900.

[78]宾婷婷. 三轴循环荷载作用下辉长岩力学特性及本构模型研究[D]. 成都: 成都理工大学, 2016.

[79]黄兴建. 循环荷载作用下灰岩的力学特性及本构模型研究[D]. 成都: 成都理工大学, 2017.

[80]张开放, 赵翔, 李欣慰, 等. 基于应变强度理论的岩石疲劳损伤本构模型[J]. 江西建材, 2021(10): 40-42.

[81]罗吉安, 刘丰茂, 刘之喜, 等. 基于八面体理论的岩石循环加-卸载本构模型及修正[J]. 高压物理学报, 2020, 34(02): 82-90.

[82]刘之喜, 罗吉安. 基于畸变能理论的岩石损伤本构模型研究[J]. 长江科学院院报, 2020, 37(01): 137-141+155.

[83]王勇, 杜伟. 循环荷载下裂隙岩体的改进分数阶西原模型[J]. 粉煤灰综合利用, 2021, 35(02): 13-19+42.

[84]何聪. 静载荷与循环冲击组合作用下岩石损伤本构模型研究[D]. 赣州: 江西理工大学, 2016.

[85]李小峰, 王军保. 基于改进Harris分布的岩石损伤统计本构模型研究[J]. 地下空间与工程学报, 2012, 8(04): 767-771.

[86]张世殊, 刘恩龙, 张建海. 砂岩在低频循环荷载作用下的疲劳和损伤特性试验研究[J] .岩石力学与工程学报, 2014, 33(S1): 3212-3218.

[87]曹瑞琅, 贺少辉, 韦京, 等. 基于残余强度修正的岩石损伤软化统计本构模型研究[J]. 岩土力学, 2013,34(06): 1652-1660+1667.

[88]朱合华, 黄伯麒, 张琦, 等. 基于广义Hoek-Brown准则的弹塑性本构模型及其数值实现[J]. 工程力学, 2016, 33(02): 41-49.

[89]张宏博, 黄茂松, 宋修广. 循环荷载作用下粉细砂累积变形的等效黏塑性本构模型[J]. 水利学报, 2009, 40(06): 651-658.

[90]刘恩龙, 张建海, 何思明, 等. 循环荷载作用下岩石的二元介质模型[J]. 重庆理工大学学报(自然科学版), 2013, 27(09): 6-12+16.

[91]邓建, 肖明, 谢冰冰, 等. 循环荷载下岩体结构面本构关系与积分算法研究[J]. 岩土工程学报, 2017, 39(06): 1048-1057.

[92]孟祥振, 张慧梅, 康晓革. 含孔隙冻融岩石的损伤本构模型[J]. 西安科技大学学报, 2019, 39(04): 688-692.

[93]胡学龙. 基于统一强度理论的岩石动态损伤模型研究[D]. 北京: 北京科技大学, 2020.

[94]胡学龙, 李克庆, 璩世杰. 基于统一强度理论的岩石弹塑性本构模型及其数值实现[J]. 爆炸与冲击, 2019, 39(08): 130-138.

[95]胡学龙, 璩世杰, 李克庆. 基于统一强度理论的岩石弹塑性损伤模型研究[J]. 中国矿业大学学报, 2019, 48(02): 305-312.

[96]周永强, 盛谦, 罗红星, 等. 考虑率效应的岩石材料次加载面动态本构模型[J]. 岩土工程学报, 2018, 40(10): 1818-1826.

[97]周永强, 盛谦, 朱泽奇, 等. 基于修正Drucker-Prager准则的岩石次加载面模型[J]. 岩土力学, 2017, 38(02): 400-408+418.

[98]周永强, 盛谦, 冷先伦, 等. 基于循环加卸载的次加载面模型在岩石中的初步应用[J]. 岩石力学与工程学报, 2015, 34(10): 2073-2082.

[99]王春, 唐礼忠, 程露萍, 等. 三维高静载频繁动态扰动时岩石损伤特性及本构模型[J]. 岩土力学, 2017, 38(08): 2286-2296+2305.