在青藏高原、西伯利亚等高海拔、高纬度地区，冻结岩层广泛分布，冻结深度可达数百米。随着“一带一路”倡议的深入推进，在上述地区工程建设及采矿活动快速增加。冻结岩石既是上述工程活动的开挖对象，也是当地普遍使用的建筑材料。由于孔隙中水-冰-气三相共存，冻结岩石力学特性显著区别于常温岩石。但目前，对冻结岩石力学特征的研究主要关注力学参数的变化，其力学性质特殊性的细观机制尚未被揭示。因此，研究冻结岩石力学性质具有很强的理论价值和实践意义。

基于热力学原理分析了孔隙水的相变过程，发现了孔隙水的冻融滞回现象，提出了基于超声波参数的冻结状态判定方法。建立了冻结岩石力学特性随温度和饱和度变化的定量关系，并确定了变化的关键值。基于电阻率测试探究了冻结岩石受荷破坏过程中孔隙冰的压融效应，并阐明了其原理。分析了冻结岩石断裂韧度对温度和含水量变化的敏感性，基于声发射和DIC手段揭示了冻结岩石的断裂破坏特征。构建了冻结岩石孔隙结构模型，从微细观角度厘清了孔隙冰-未冻水-岩石骨架之间的耦合作用关系，阐明了冻结效应对岩石力学特性的改造机制。基于双胡克模型方法，建立了考虑孔隙冰压融效应与热融效应的冻结岩石本构模型，并进行了验证。主要得到以下结论：

（1）多孔岩石孔隙水的冻结过程具有显著的阶段性，存在典型的“过冷”、“热弛豫”及“冻融回滞”现象。①饱和岩石的冻结过程可分为“过冷-快速冻结-缓慢冻结”三个阶段，自由水和毛细水最先发生冻结，随后只有吸附水发生冻结。②初始饱和度的增加导致孔隙水的冻结过程由低饱和度（<10%）的“过冷-缓慢冻结”改变为高饱和度（>10%）的三阶段变化。③岩石冻融过程中总未冻水、自由水、毛细水以及吸附水均存在明显的回滞现象，饱和度低于30%时回滞由吸附水含量控制，高于30%时回滞由自由水和毛细水含量控制。④超声波参数随未冻水含量呈线性变化，可以用来判定岩石的冻结状态。

（2）冻结岩石的强度和变形特性受初始饱和度和冻结温度的显著影响。①岩石的单轴抗压强度和抗拉强度均随冻结温度的降低呈“三阶段递增”的变化趋势，其界限温度分别为-4℃，-12℃和-2℃，-12℃；破坏应变和弹性模量均增大。②冻结岩石的单轴抗压强度随初始饱和度的增大呈“减小-快速增大-缓慢增大”的三阶段变化，界限饱和度为20%和50%；抗拉强度呈“减小-缓慢增大-快速增大”的三阶段变化，界限饱和度为10%和50%；破坏应变增大，弹性模量降低。③冻结作用使得饱和岩石振铃计数的突增从峰值应力处转变至峰后阶段，而初始饱和度大于20%时也出现了这一转变；岩石的破坏模式由高冻结温度（>-4℃）的劈裂破坏转变至低冻结温度（<-4℃）的剪切破坏，而初始饱和度增大至40%时，其破坏模式也出现了相同的改变。④曲率预融和应力集中效应导致冻结岩石受压破坏过程中孔隙冰存在显著的压融效应，且主要发生在微裂纹压密阶段。

（3）冻结温度的降低和饱和度的增大导致岩石断裂韧度呈阶段性变化，且微裂纹不易发生扩展。①冻结岩石断裂韧度随初始饱和度的增大而呈“缓慢增大-快速增大-降低”三阶段变化，其界限饱和度为40%和90%，而随冻结温度的降低呈“快速增加-缓慢增加”的两阶段变化，其关键温度为-4℃。②初始饱和度越高，冻结岩石微裂纹扩展越缓慢，峰值时刻裂隙端部应变越小。冻结温度的降低也导致微裂纹扩展缓慢。③初始饱和度低于40%时，微裂纹扩展过程中声发射信号较小，高于40%后声发射信号显著提高，而冻结温度对声发射信号影响的临界温度是-2℃。④冻结温度的降低导致岩石的宏观裂纹扩展速率呈“快速增大-基本不变”的两阶段变化。

（4）不同含量的孔隙冰和未冻水（相组成分）赋存于孔隙和微裂隙中，决定了冰-水-岩石骨架之间的相互作用，继而改变了冻结岩石的力学特性。①冻结岩石中冰充填孔隙和微裂隙，未冻水膜赋存于冰和岩石骨架之间。②孔隙冰和未冻水对岩石骨架具有冰的支撑作用、充填作用和未冻结水膜的胶结作用三种强化效应以及冻胀损伤、压融水的润滑和未冻水的Rhebinder作用三种弱化作用。③高冻结温度中（>-4℃），孔隙冰的支撑作用和充填作用增强了岩石的力学特性，低冻结温度中（<-4℃），冰岩界面的胶结作用也导致了冻结岩石强度的增加。④在低饱和度冻结岩石中（<20%），压融水的润滑作用和未冻水的Rhebinder效应导致岩石强度的降低，而胶结作用和充填作用使得岩石断裂韧度增大。在高饱和度岩石中（>20%），孔隙冰和未冻水的强化效应高于弱化效应。

（5）基于双胡克模型方法，建立了考虑孔隙冰压融效应与热融效应的冻结岩石本构模型并进行了验证。①考虑微裂隙中冰的压融软化，将冻结岩石分为冰充填微裂隙的“软体”和冰充填孔隙的冻结岩石骨架“硬体”，分别采用真应变和广义应变计算其变形。②定义了冰充填微裂隙的压融损伤变量，考虑温度对孔隙冰界面预融和冰力学特性的影响，将冻结岩石骨架变形与温度建立联系。③基于Weibull分布定义了冻结岩石骨架损伤变量，得到考虑压融与热融效应的冻结岩石本构模型；④对模型验证发现其能较好地模拟冻结岩石的变形破坏过程。

In high and cold regions such as the Qinghai-Tibet Plateau and Siberia, frozen rock formations are widely distributed, and freezing depths can reach several hundred meters. With the deepening of the “Belt and Road” Initiative, engineering construction and mining activities in these regions have rapidly increased. Frozen rocks are both the excavation object and the commonly building material used in the local construction activities. Due to the coexistence of “water-ice-gas” in the pores, the mechanical properties of frozen rock are significantly different from those of rocks at room temperatures. However, the study of mechanical characteristics of frozen rock mainly focuses on the change of mechanical parameters at present, and the microscopic mechanism of its special mechanical properties has not been revealed. Therefore, it is of great theoretical value and practical significance to study the mechanical properties of frozen rock.

Based on the principles of thermodynamics, the phase change process of pore water is analyzed, and the hysteresis phenomenon of pore water during freezing-thawing is discovered. A frozen state determination method based on ultrasonic parameters is proposed. The quantitative relationships between the mechanical properties of frozen rock and temperature and water content are established, and the key values of the changes are determined. The pressure-melting effect of pore ice in frozen rock under load is explored through electrical resistance testing, and the principle is elucidated. The sensitivity of the fracture toughness of frozen rock to changes in temperature and water content is explained, and the fracture characteristics of frozen rock are revealed based on acoustic emission (AE) and digital image correlation (DIC). A pore structure model of frozen rock is constructed to clarify the mutual effect between pore ice, unfrozen water, and rock skeleton, and the influence mechanism of freezing effects on rock mechanical properties is elucidated. Based on the “Two Part Hooke Model”, a constitutive model of frozen rock considering the pressure-melting effect and thawing of pore ice is established and verified. The main conclusions are as follows:

(1) The freezing process of pore water in porous rocks exhibits significant stages and typical phenomena such as “supercooling” “thermal relaxation” and “freezing-thawing hysteresis”. ①The freezing process of saturated rock can be divided into three stages of “supercooling-rapid freezing-slow freezing”. Free water and capillary water freeze first, followed by adsorbed water. ②The increase in initial saturation degree leads to the change from a “supercooling-slow freezing” process at low initial saturation degree (<10%) to a three-stage change at higher initial saturation degree (>10%). ③During the freezing-thawing process of rock, there is an obvious hysteresis in the total unfrozen water, free water, capillary water, and adsorbed water. When the initial saturation degree is less than 30%, hysteresis is controlled by the content of adsorbed water, while it is controlled by the content of free water and capillary water when it is higher than 30%. ④Ultrasonic parameters vary linearly with the content of unfrozen water, which can be used to determine the freezing state of rocks.

(2) The strength and deformation of frozen rock are significantly affected by the initial saturation degree and freezing temperature. ①The uniaxial compressive strength (UCS) and tensile strength (TS) of the rock both show a “three-stage increase” trend with decreasing freezing temperature, with critical temperatures of -4°C, -12°C for UCS, and -2°C, -12°C for TS. The failure strain and elastic modulus both increase. ②The UCS of frozen rocks shows a three-stage change of “decrease-rapid increase-slow increase” with the increase of initial saturation degree, and the critical water contents are 20% and 50%, respectively. The TS shows a “decrease-slow increase-rapid increase”, and the critical initial saturation degree are 10% and 50%, respectively. The failure strain increases, and the elastic modulus decreases. ③Freezing causes the sudden increase in AE counts of saturated rocks to shift from the peak stress to the post-peak stage, and this shift also occurs when the initial saturation degree is greater than 20%. The failure modes of rock change from splitting failure at high freezing temperatures (> -4℃) to shear failure at low freezing temperatures (< -4℃), and when the initial saturation degree increases to 40%, the same change in failure mode occurs.④Curvature induced premelting and stress concentration effects result in significant pressure melting of pore ice during the compressive failure of frozen rocks, mainly occurring in the microcrack compaction stage.

(3) The decrease in freezing temperature and increase in initial saturation degree significantly affect the fracture toughness and crack propagation characteristics of frozen rocks, and microcracks are not easily propagated. ①The fracture toughness of frozen rocks exhibits a three-stage change of “slowly increasing-rapidly increasing-decreasing” with an increase in initial saturation degree, and the critical water content is 40% and 90%, while it exhibits a two-stage change of “rapidly increasing-slowly increasing” with a decrease in freezing temperature, and the key temperature is -4°C. ②Higher initial saturation degree results in slower propagation of microcracks in frozen rocks and smaller strains at the crack tip under peak load. The decrease in freezing temperature also leads to slower propagation of microcracks. ③When the initial saturation degree is less than 40%, the AE counts during microcrack propagation are small, and the AE counts increase significantly when the water content is higher than 40%. The critical temperature of the freezing effect on AE counts being -2°C. ④The decrease in freezing temperature leads to a two-stage change of “rapid increase-basically unchanged” in the macroscopic crack propagation rate of frpzen rocks.

(4) The presence of different contents of porous ice and unfrozen water (phase compositions) in pores and microcracks determines the interactions between the ice-water-rock frameworks, thereby changing the mechanical properties of frozen rock. ①In frozen rock, ice fills pores and microcracks, while unfrozen water films exist between the ice and rock skeleton. ②Porous ice and unfrozen water have three strengthening effects on the rock skeleton: ice supporting effect, filling effect, and cementation effect of unfrozen water films, as well as three weakening effects: frost damage, lubrication of pressure melting water, and Rhebinder effect of unfrozen water. ③At high freezing temperatures (> -4℃), the supporting and filling effects of porous ice enhance the mechanical properties of the rock, while at low freezing temperatures (< -4℃), the cementation effect at the ice-rock interface also strengthen the frozen rock. ④In low water content frozen rock (< 20%), the lubrication effect of pressure melting water and the Rhebinder effect of unfrozen water lead to a decrease in rock strength, while the cementation and filling effects increase the rock's fracture toughness. In high water content rock (> 20%), the strengthening effects of porous ice and unfrozen water are greater than the weakening effects.

(5) Based on the Two-part Hooke's model, a constitutive model of frozen rock considering the effects of pore ice pressure melting and thawing is established and validated. ①Considering the pressure melting softening of ice in microcracks, the frozen rock is divided into a “soft body” of ice filled microcracks and a “hard body” of rock skeleton that ice filled pores. The true strain and generalized strain are respectively used to calculate their deformation. ②The pressure melting damage variable of ice-filled microcracks is defined, and the relationship between the deformation of the frozen rock skeleton and temperature is considered by taking the influence of temperature on the pre-melting of pore ice interface and ice mechanical properties into account. ③The skeleton damage variables of frozen rock are defined based on Weibull distribution, and the constitutive model of frozen rock considering the pressure melting effect and thawing is obtained. ④ It is found that the model can simulate the deformation and failure process of frozen rock well.

[1] Wu Q, Zhang T. Recent permafrost warming on the Qinghai‐Tibetan Plateau[J]. Journal of Geophysical Research: Atmospheres, 2008, 113(D13).

[2] Sammis C, Biegel R. Mechanics of strengthening in crystalline rock at low temperatures: a preliminary assessment[C]//Proceedings of the 26th Seismic Research Review: Trends in Nuclear Explosion Monitoring. 2004: 475-484.

[3] Luo L, Ma W, Zhao W, et al. UAV-based spatiotemporal thermal patterns of permafrost slopes along the Qinghai–Tibet Engineering Corridor[J]. Landslides, 2018, 15: 2161-2172.

[4] 徐学祖, 王家澄, 张立新. 冻土物理学[M]. 科学出版社, 2001.

[5] Haeberli W. Construction, environmental problems and natural hazards in periglacial mountain belts[J]. Permafrost and periglacial processes, 1992, 3(2): 111-124.

[6] Huggel C, Salzmann N, Allen S, et al. Recent and future warm extreme events and high-mountain slope stability[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010, 368(1919): 2435-2459.

[7] Ravanel L, Allignol F, Deline P, et al. Rock falls in the Mont Blanc Massif in 2007 and 2008[J]. Landslides, 2010, 7: 493-501.

[8] 崔鹏, 陈容, 向灵芝, 等. 气候变暖背景下青藏高原山地灾害及其风险分析[J]. 气候变化研究进展, 2014, 10(02): 103-109.

[9] Gruber S, Haeberli W. Permafrost in steep bedrock slopes and its temperature‐related destabilization following climate change[J]. Journal of Geophysical Research: Earth Surface, 2007, 112(F2).

[10] Noetzli J, Hoelzle M, Haeberli W. Mountain permafrost and recent Alpine rock-fall events: a GIS-based approach to determine critical factors[C]//Proceedings of the 8th International Conference on Permafrost. Zürich: Swets & Zeitlinger Lisse, 2003, 2: 827-832.

[11] Dramis F, Govi M, Guglielmin M, et al. Mountain permafrost and slope instability in the Italian Alps: the Val Pola landslide[J]. Permafrost and Periglacial Processes, 1995, 6(1): 73-81.

[12] Freire-Lista D M, Fort R, Varas-Muriel M J. Freeze–thaw fracturing in building granites[J]. Cold Regions Science and Technology, 2015, 113: 40-51.

[13] Matsuoka N. Mechanisms of rock breakdown by frost action: an experimental approach[J]. Cold Regions Science and Technology, 1990, 17(3): 253-270.

[14] Yu S, Oguchi C T. Role of pore size distribution in salt uptake, damage, and predicting salt susceptibility of eight types of Japanese building stones[J]. Engineering Geology, 2010, 115(3-4): 226-236.

[15] Winkler E M. Frost damage to stone and concrete: geological considerations[J]. Engineering Geology, 1968, 2(5): 315-323.

[16] Hall K. Evidence for freeze–thaw events and their implications for rock weathering in northern Canada[J]. Earth Surface Processes and Landforms: The Journal of the British Geomorphological Research Group, 2004, 29(1): 43-57.

[17] 谭龙, 韦昌富, 田慧会, 等. 冻土未冻水含量的低场核磁共振试验研究[J]. 岩土力学, 2015, 36(06): 1566-1572.

[18] 孙颜顶, 刘波, 张静涛, 等. 冻融状态下粉质黏土电阻率、未冻水含量及温度关系[J]. 中国矿业大学学报, 2023, 52(01): 52-62.

[19] 罗豪良, 滕继东, 张升, 等. 冻土未冻水含量与电导率的关系研究[J]. 岩石力学与工程学报, 2021, 40(05): 1068-1079.

[20] 荣传新, 李承涛, 王彬, 等. 砂层冻结过程中未冻水含量核磁共振实验研究[J]. 重庆科技学院学报(自然科学版), 2022, 24(05): 90-96.

[21] 殷健超, 林键, 姚亚锋, 等. 基于NMR的冻融过程中砂土未冻水含量试验分析[J]. 兰州工业学院学报, 2021, 28(04): 1-4.

[22] Draebing D, Krautblatter M. P-wave velocity changes in freezing hard low-porosity rocks: a laboratory-based time-average model[J]. The Cryosphere, 2012, 6(5): 1163-1174.

[23] Sass O. Rock moisture fluctuations during freeze-thaw cycles: Preliminary results from electrical resistivity measurements[J]. Polar Geography, 2004, 28(1): 13-31.

[24] Timur A. Velocity of compressional waves in porous media at permafrost temperatures[J]. Geophysics, 1968, 33(4): 584-595.

[25] Kurfurst P J. Ultrasonic wave measurements on frozen soils at permafrost temperatures[J]. Canadian Journal of Earth Sciences, 1976, 13(11): 1571-1576.

[26] King M S. Acoustic velocities and electrical properties of frozen sandstones and shales[J]. Canadian Journal of Earth Sciences, 1977, 14(5): 1004-1013.

[27]Olhoeft G R. The electrical properties of permafrost[D]. University of Toronto, 1975.

[28] Pandit B I, King M S. A study of the effects of pore-water salinity on some physical properties of sedimentary rocks at permafrost temperatures[J]. Canadian Journal of Earth Sciences, 1979, 16(8): 1566-1580.

[29] Hunter K A. Direct disposal of liquified fossil fuel carbon dioxide in the ocean[J]. Marine and freshwater research, 1999, 50(8): 755-760.

[30] Zimmerman R W, King M S. The effect of the extent of freezing on seismic velocities in unconsolidated permafrost[J]. Geophysics, 1986, 51(6): 1285-1290.

[31] Herring T, Cey E, Pidlisecky A. Electrical resistivity of a partially saturated porous medium at subzero temperatures[J]. Vadose Zone Journal, 2019, 18(1): 1-11.

[32]梁珂. 冻土未冻水含量与比表面积定量关系的试验研究[D]. 北京建筑大学, 2021.

[33] Liu H, Yang G S, Shen Y J, et al. Analysis of water and ice content during rock freezing based on the three-valued segmentation of frozen rock CT images[C]//3rd ISRM Young Scholars Symposium on Rock Mechanics. OnePetro, 2014.

[34] Jia H, Ding S, Wang Y, et al. An NMR-based investigation of pore water freezing process in sandstone[J]. Cold Regions Science and Technology, 2019, 168: 102893.

[35] Weng L, Wu Z, Chu Z, et al. Evolution of the unfrozen water content for partially-saturated sandstones and the critical degree of saturation[J]. Rock Mechanics and Rock Engineering, 2023, 56(1): 1-18.

[36] 巫瑞锐. 结冰力学特性的理论与实验研究[D].南京航空航天大学, 2014.

[37] 赵耀. 多晶冰细观模拟方法及压力条件下的力学特性[J]. 海洋工程, 2002 (01): 15-19.

[38] 张明元, 孟广琳, 隋吉学. 辽东湾海冰物理力学特性[J]. 中国海洋平台, 1995 (02): 18-24+4.

[39] 王宝生. 原位高压冻结冰试样的制作工艺与单轴压缩力学特性[D]. 中国矿业大学, 2021.

[40] 张大长, 刘明源, 包涛. 淡水冰单轴受压力学特性的试验研究[J]. 工程力学, 2011, 28 (07): 238-244.

[41] Petrenko V. F., Whitworth R. W. Physics of ice[M]. New York Oxford University Press, 2002.

[42] Tammann G. Ueber die Grenzen des festen Zustandes[J]. Annalen der Physik, 1897, 298(10): 280-299.

[43] Gold, L.W. Some observations on the dependence of stain on stress for ice[J]. Can. J. Phys, 1958, 3610, 1265e1275.

[44] Sinha, N. K. Comparative study of ice strength data . In IAHR International Symposium on Ice, Quebec, Canada, July 27- 31, 1981. Proceedings. Vol. lI. Quebec, U niversite Laval, 1981, 581 – 592.

[45] Timco, G. W., and Frederking, R. M. W.. Flexural strength and fracture touglmess of sea ice. Cold Reg. Sci. and Technol. J., Amsterdam, The Netherlands, 1982, 8: 35-41.

[46] Sinha N K. Uniaxial compressive strength of first-year and multi-year sea ice[J]. Canadian Journal of Civil Engineering, 1984, 11(1): 82-91.

[47] Cole, D. M. and Durell, G. D. The cyclic loading of saline ice, Philos. Mag. A, 1995, 72: 209–229.

[48] Hawkes I, Mellor M. Deformation and fracture of ice under uniaxial stress[J]. Journal of glaciology, 1972, 11(61): 103-131.

[49] Mellor M. Mechanical properties of polycrystalline ice[C]//Physics and Mechanicsof Ice: Symposium Copenhagen, August 6–10, 1979, Technical University of Denmark. Springer Berlin Heidelberg, 1980: 217-245.

[50] Mellor M, Cole D M. Deformation and failure of ice under constant stress or constant strain-rate[J]. Cold Regions Science and Technology, 1982, 5(3): 201-219.

[51] Kuehn, G. A., Schulson, E. M., Jones, D. E., et al. The Compressive Strength of Ice Cubes of Different Sizes. ASME. J. Offshore Mech. Arct. Eng. May 1993; 115(2): 142–148.

[52] Schulson E M, Buck S E. The ductile-to-brittle transition and ductile failure envelopes of orthotropic ice under biaxial compression[J]. Acta Metallurgica et Materialia, 1995, 43 (10) :3661-3668.

[53] Jones S J. High strain-rate compression tests on ice[J]. The Journal of Physical Chemistry B, 1997, 101(32): 6099-6101.

[54] Michel B. Ice mechanics[M]. Quebec: Laval University Press, 1978.

[55] Tryde P. Physics and mechanics of ice[M]. Berlin, Heidelberg & New York: Springer-Verlag, 1980.

[56] Hobbs P. V. Ice physics[M]. USA: Oxford University Press, 2010.

[57] Kuhs W F. Physics and chemistry of ice[M]. Cambridge: The Royal Society of Chemistry, 2006.

[58] 李志军, 孟广琳, 隋吉学. 辽东湾海冰单轴压缩长期强度的初步分析[J]. 海洋学报(中文版), 1991(04): 571-575.

[59] 孟广琳, 张明元, 李志军, 等. 辽东湾北部三道沟平整冰单轴抗压强度特征[J]. 海洋通报, 1991, 10(3): 21-28

[60] 张明元, 孟广琳, 隋吉学, 等. 湘海湾海冰和黄河口河冰物理力学性质的测定和研究[J].海洋工程, 1993 (3) :39-45.

[61] 沈乐天, 赵士达, 卜锡年, 等. 天然淡水冰单轻压缩强度及其温度和应变率效应[J]. 冰川冻土, 1990 (2):141-146.

[62] 黄茂恒, 刘宗香. 黄河上游冰强度试验研究[C]. 中国地理学会冰川冻土学术会议论文选集,1982: 146-151.

[63] 严德成, 孟广琳, 张明远, 等. 黄河口低盐度海冰与黄河沙的抗压强度[J]. 海洋环境科学, 1994(3): 72-80.

[64] 于天来, 王金峰, 杜峰, 等. 呼玛河冰灾害试验研究[J]. 自然灾害学报, 2007 (4): 43-48.

[65] 王金峰, 于天来, 黄美兰. 河沙单辅无侧限抗压强度的试验研究[J]. 低温建筑技术, 2007(1): 11-13.

[66] Haynes F D. Temperature effect on the uni-axial strength of ice[C]. Proc. POAC 79,Trondheim, Norway, 1974: 667-681.

[67] Traetteberg A, Gold L W, Frederking R. The strain rate and temperature dependence of Young s modulus of ice[C]. Proceedings, IAHR Third International Symposium on Ice Problems, Hanover, New Hampshire, 1975:479-486.

[68] Richter J A, Cox G A. Preliminary examination of the effect of structure on the compressive strength of ice samples from nmulti-year pressure Iidges[C]. 1984:140-144.

[69] 张明元, 孟广琳, 隋吉学, 等. 渤海湾海冰和黄河口河冰物理力学性质的测定和研究[J].海洋工程, 1993, 11 (3):39-45

[70] 于天来, 袁正国, 黄美兰. 河冰力学性能试验研究 [J]. 辽宁工程技术大学学报(自然科学版), 2009, 28 (6): 937-940

[71] Glen J. W. The creep of polycrystalline ice[J]. 1955, 228: 519-538

[72] Timco G. W., Frederking R. M. W. Comparative strengths of fresh water ice[J]. Cold Regions Science and Technology, 1982, 6(1): 21-27

[73] Jones S J, Glen J W. The effect of dissolved impurities on the mechanical properties of ice crystals[J]. Philosophical Magazine, 1969, 19(157): 13-24

[74] Glen J W. The effect of hydrogen disorder on dislocation movement and plastic deformation of ice[J]. Phys kondens Materie, 1968, 7(1): 43-51

[75] Davarpanah S M, Török Á, Vásárhelyi B. Review on the mechanical properties of frozen rocks[J]. Rudarsko-geološko-naftni zbornik, 2022, 37(3).

[76] 杨更社, 奚家米, 李慧军, 等. 三向受力条件下冻结岩石力学特性试验研究[J]. 岩石力学与工程学报, 2009, 29(03): 459-464.

[77] 徐光苗, 刘泉声, 彭万巍, 等. 低温作用下岩石基本力学性质试验研究[J]. 岩石力学与工程学报, 2005, 25(12).

[78] Kurilko A S, Novopashin M D. Features of low temperature effect upon strength of enclosing rock and kimberlite in the “Udachnaya pipe[J]. Journal of Mining Science, 2005, 41: 119-122.

[79] Aoki K, Hibiya K, Yoshida T. Storage of refrigerated liquefied gases in rock caverns:characteristics of rock under very low temperatures[J]. Tunnelling and underground space technology, 1990, 5(4): 319-325.

[80] Török Á, Ficsor A, Davarpanah M, et al. Comparison of mechanical properties of dry, saturated and frozen porous rocks[C]//IAEG/AEG Annual Meeting Proceedings, San Francisco, California, 2018—Volume 6: Advances in Engineering Geology: Education, Soil and Rock Properties, Modeling. Springer International Publishing, 2019: 113-118.

[81]Mellor M. Strength and deformability of rocks at low temperatures[M]. Corps of Engineers, US Army, Cold Regions Research and Engineering Laboratory, 1971.

[82] Mellor M. Mechanical properties of rocks at low temperatures[C]//2nd International Conference on Permafrost, Yakutsk, International Permafrost Association. 1973: 334-344.

[83] Heins R W, Friz T O. The effect of low temperature on some physical properties of rock[C]//Drilling and Rock Mechanics Conference. OnePetro, 1967.

[84] 杨更社, 魏尧, 申艳军, 等. 冻结饱和砂岩三轴压缩力学特性及强度预测模型研究[J]. 岩石力学与工程学报, 2019 (4): 683-694.

[85] Dwivedi R D, Singh P K, Singh T N, et al. Compressive strength and tensile strength of rocks at sub-zero temperature[J]. 1998(5): 43-48.

[86] 杨更社. 冻结岩石力学的研究现状与展望分析[J]. 力学与实践, 2009(6): 9-16.

[87] Bai Y, Shan R, Ju Y, et al. Study on the mechanical properties and damage constitutive model of frozen weakly cemented red sandstone[J]. Cold Regions Science and Technology, 2020, 171: 102980.

[88] Ma M, Huang Y, Cao G, et al. Study on mechanical behavior of Jurassic frozen sandstone in western China based on NMR porosity[J]. Journal of Chemistry, 2020: 1-11.

[89] 朱杰, 徐颖, 李栋伟, 等. 泊江海子矿白垩纪地层冻结软岩力学特性试验[J]. 吉林大学学报(地球科学版), 2016, 46(3): 798-804.

[90] Liu L, Liu X, Li Z, et al. Experimental analysis on the mechanical properties of saturated silty mudstone under frozen conditions[J]. Journal of Testing and Evaluation, 2018, 47(1): 188-202.

[91] 刘波, 刘念, 李东阳, 等. 鄂尔多斯深部富水砂岩冻结强度的试验研究[J]. 矿业科学学报, 2017 (1): 25-32.

[92] Yamabe T, Neaupane KM. Determination of some thermomechanical properties of Sirahama sandstone under subzero temperature condition. Int J Rock Mech Min Sci, 2001, 38(7): 1029–1034.

[93] Kodama J, Goto T, Fujii Y, et al. The effects of water content, temperature and loading rate on strength and failure process of frozen rocks[J]. International Journal of Rock Mechanics and Mining Sciences, 2013, 62: 1-13.

[94] Inada Y, Yokota K. Some studies of low temperature rock strength[C]//International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts. Pergamon, 1984, 21(3): 145-153.

[95] Jia H, Zi F, Yang G, et al. Influence of pore water (ice) content on the strength and deformability of frozen argillaceous siltstone[J]. Rock Mechanics and Rock Engineering, 2020, 53: 967-974.

[96] Dwivedi R D, Soni A K, Goel R K, et al. Fracture toughness of rocks under sub-zero temperature conditions[J]. International Journal of Rock Mechanics and Mining Sciences, 2000, 37(8): 1267-1275.

[97] Dong X, Yang G, Liu S. Experimental study on AE response and damage evolution characteristics of frozen sandstone under uniaxial compression[J]. Cold Regions Science and Technology, 2022, 193: 103424.

[98] Park J, Park H D. Investigation of frozen rock failure using thermal infrared image[J]. Tunnel and Underground Space, 2015, 25(2): 144-154.

[99] Yang G, Liang B, Liu H, et al. Mechanical Properties and Acoustic Emission Characteristics of Thawing Frozen Sandstone[J]. Advances in Materials Science and Engineering, 2022, 2022.

[100] Wang C, Li S, Zhang T, et al. Experimental study on mechanical characteristics and fracture patterns of unfrozen/freezing saturated coal and sandstone[J]. Materials, 2019, 12(6): 992.

[101] Liu B, Sun Y, Wang J, et al. Characteristic analysis of crack initiation and crack damage stress of sandstone and mudstone under low-temperature condition[J]. Journal of Cold Regions Engineering, 2020, 34(3): 04020020.

[102]Zhang G, Liu E, Chen S, et al. Effects of uniaxial and triaxial compression tests on the frozen sandstone combining with CT scanning[J]. International Journal of Physical Modelling in Geotechnics, 2019, 19(5): 261-274.

[103]Fan L F, Fan Y D, Xi Y, et al. Spatial failure mode analysis of frozen sandstone under uniaxial compression based on CT technology[J]. Rock Mechanics and Rock Engineering, 2022, 55(7): 4123-4138.

[104] Huang S, Liu Q, Cheng A, et al. A statistical damage constitutive model under freeze-thaw and loading for rock and its engineering application[J]. Cold Regions Science and Technology, 2018, 145: 142-150.

[105] Huang S, Liu Q, Cheng A, et al. A fully coupled thermo-hydro-mechanical model including the determination of coupling parameters for freezing rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2018, 103: 205-214.

[106] 杨涛, 霍树义, 金坎辉, 等. 冻融循环下砂岩损伤演化及本构模型[J]. 地质与勘探, 2020, 56(04): 826-831.

[107] 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.

[108] Shi Z, Li J, Wang J, et al. Experimental and numerical study on fracture characteristics and constitutive model of sandstone under freeze-thaw-fatigue[J]. International Journal of Fatigue, 2023, 166: 107236.

[109] Fang W, Jiang N, Luo X. Establishment of damage statistical constitutive model of loaded rock and method for determining its parameters under freeze-thaw condition[J]. Cold Regions Science and Technology, 2019, 160: 31-38.

[110] Liu B, Ma Y, Zhang G, et al. Acoustic emission investigation of hydraulic and mechanical characteristics of muddy sandstone experienced one freeze-thaw cycle[J]. Cold Regions Science and Technology, 2018, 151: 335-344.

[111] Jia H, Xiang W, Krautblatter M. Quantifying rock fatigue and decreasing compressive and tensile strength after repeated freeze‐thaw cycles[J]. Permafrost and Periglacial processes, 2015, 26(4): 368-377.

[112] 张慧梅, 杨更社. 冻融与荷载耦合作用下岩石损伤模型的研究[J]. 岩石力学与工程学报, 2010, 29(03): 471-476.

[113] 徐光苗, 刘泉声. 岩石冻融破坏机理分析及冻融力学试验研究[J]. 岩石力学与工程学报, 2005(17): 3076-3082.

[114] 肖鹏, 陈有亮, 杜曦, 等. 冻融循环作用下砂岩的力学特性及细观损伤本构模型研究[J]. 岩土工程学报, 2023, 45(04): 805-815.

[115] 王宇. 冻融单裂隙英安岩物理力学特性的相似材料试验及损伤本构模型研究[D]. 成都理工大学, 2019.

[116] 孟祥振. 冻融-荷载作用下含孔隙岩石的损伤本构模型[D]. 西安科技大学, 2018.

[117] Sammis C, Biegel R. Mechanics of strengthening in crystalline rock at low temperatures: a preliminary assessment[C]//Proceedings of the 26th Seismic Research Review: Trends in Nuclear Explosion Monitoring. 2004: 475-484.

[118] Bai Y, Shan R, Ju Y, et al. Study on the mechanical properties and damage constitutive model of frozen weakly cemented red sandstone[J]. Cold Regions Science and Technology, 2020, 171: 102980.

[119] Ming F, Zhang S, Niu F, et al. A study on crack damage stress and the damage constitutive model of frozen sandstone[J]. Bulletin of Engineering Geology and the Environment, 2021, 80: 6955-6970.

[120] Gong F, Wang Y, Ueda T, et al. Modeling and mesoscale simulation of ice-strengthened mechanical properties of concrete at low temperatures[J]. Journal of Engineering Mechanics, 2017, 143(6): 04017022.

[121] Zhao Y, Li S, Shi L, et al. Mechanical Damage Evolution and a Statistical Damage Model for Frozen Sandstone[J]. International Journal of Geomechanics, 2022, 22(10): 04022184.

[122] Zhao Y, Li S, Zhao J, et al. A temperature-dependent elastoplastic damage model for frozen sandstone[J]. Cold Regions Science and Technology, 2023, 208: 103792.

[123] Liu S, Yang G, Dong X, et al. Energy characteristics and damage constitutive model of frozen sandstone under triaxial compression[J]. Journal of Cold Regions Engineering, 2022, 36(1): 04021021.

[124] 伊明, 赵涛, 马飞飞, 等. 基于Weibull分布的冻结砂岩损伤本构模型研究[J].煤田地质与勘探, 2022, 50(08): 116-124.

[125] 杨林. 人工冻结砂岩力学特性及本构关系研究[D]. 安徽理工大学, 2019.

[126] Atkins P, Paula J d. Physical Chemistry Thermodynamics, Structure, and Change. WH Freeman and Company New York, 2014.

[127] Kittel, C., and H. Kroemer. Thermal physics. London: Macmillan, 1980.

[128] Scherer G W . Freezing gels[J]. Journal of Non-Crystalline Solids, 1993, 155(1):1-25.

[129] Sun Z, Scherer G W. Pore size and shape in mortar by thermoporometry[J]. Cement and Concrete Research, 2010, 40(5): 740-751.

[130] Sun Z, Scherer G W. Effect of air voids on salt scaling and internal freezing[J]. Cement and Concrete Research, 2010, 40(2): 260-270.

[131] Yang R, Lemarchand E, Fen-Chong T, et al. A micromechanics model for partial freezing in porous media[J]. International Journal of Solids and Structures, 2015, 75: 109-121.

[132] Zeng Q, Li K, Fen-Chong T. Effect of supercooling on the instantaneous freezing dilation of cement-based porous materials[J]. Journal of Building Physics, 2016, 40(2): 101-124.

[133] Scherer G W, Valenza J J. Mechanisms of frost damage[J]. Materials science of concrete, 2005, 7(60): 209-246.

[134] Fagerlund G. Determination of pore-size distribution from freezing-point depression[J]. Matériaux et construction, 1973, 6: 215-225.

[135] Gong F, Zhi D, Zhou Y, et al. Empirical modeling of pore size distribution for rock materials with its impact on pore water freezing[J]. Cold Regions Science and Technology, 2022, 201: 103619.

[136] Brun M, Lallemand A, Quinson J F, et al. A new method for the simultaneous determination of the size and shape of pores: the thermoporometry[J]. Thermochimica acta, 1977, 21(1): 59-88.

[137] Zhao Y, Sun Y, Liu S, et al. Pore structure characterization of coal by NMR cryoporometry[J]. Fuel, 2017, 190: 359-369.

[138] Tice A R, Anderson D M, Sterrett K F. Unfrozen water contents of submarine permafrost determined by nuclear magnetic resonance[M]//Developments in Geotechnical Engineering. Elsevier, 1982, 28: 135-146.

[139] Watanabe K, Wake T. Measurement of unfrozen water content and relative permittivity of frozen unsaturated soil using NMR and TDR[J]. Cold Regions Science and Technology, 2009, 59(1): 34-41.

[140] Coates G R, Marschall D, Mardon D, et al. A New Characterization Of Bulk-Volume Irreducible Using Magnetic Resonance[J]. Log Analyst, 1997, 39(1):51-63.

[141] Dastidar R. Nuclear magnetic resonance (NMR) study of freezing and freezing of saturated porous media and application to shale and pore volume compressibility estimation[D], 2007.

[142] Cowan B. Nuclear Magnetic Resonance and Relaxation: Introduction[M]. Cambridge University Press, 1997.

[143] Godefroy S, Korb J P, Fleury M, et al. Surface nuclear magnetic relaxation and dynamics of water and oil in macroporous media[J]. Physical Review E, 2001, 64(2): 021605.

[144] Matteson A, Tomanic J P, Herron M M, et al. NMR relaxation of clay-brine mixtures, SPE-49008[C]//1998 SPE Annual Technical Conference and Exhibition Proceedings, v. omega, Formation evaluation and reservoir geology: Society of Petroleum Engineers. 1998: 205-212.

[145] Martinez G A, Davis L A. Petrophysical measurements on shales using NMR[C]//SPE/AAPG Western Regional Meeting. OnePetro, 2000.

[146] Carr H Y, Purcell E M. Effects of diffusion on free precession in nuclear magnetic resonance experiments[J]. Physical review, 1954, 94(3): 630.

[147] Meiboom S, Gill D. Modified spin‐echo method for measuring nuclear relaxation times[J]. Review of scientific instruments, 1958, 29(8): 688-691.

[148] 路建国, 张明义, 张熙胤, 等. 冻融过程中未冻水含量及冻结温度的试验研究[J].岩石力学与工程学报, 2017, 36(7): 1803-1812.

[149] Lee I M, Han S I, Kim H J, et al. Evaluation of rock bolt integrity using Fourier and wavelet transforms[J]. Tunnelling and underground space technology, 2012, 28: 304-314.

[150] Santos C A, Urdaneta V, Jaimes G, et al. Ultrasonic spectral and complexity measurements on brine and oil saturated rocks[J]. Rock mechanics and rock engineering, 2010, 43: 351-359.

[151] Grossmann A, Morlet J. Decomposition of Hardy functions into square integrable wavelets of constant shape[J]. SIAM journal on mathematical analysis, 1984, 15(4): 723-736.

[152] Mavko G M, Nur A. Wave attenuation in partially saturated rocks[J]. Geophysics, 1979, 44(2): 161-178.

[153] Pan B, Li K. A fast digital image correlation method for deformation measurement[J]. Optics and Lasers in Engineering, 2011, 49(7): 841-847.

[154] Pan B, Xie H, Wang Z. Equivalence of digital image correlation criteria for pattern matching[J]. Applied optics, 2010, 49(28): 5501-5509.

[155] Zhang L, Yang C, Wang D, et al. Freezing point depression of soil water depending on its non-uniform nature in pore water pressure[J]. Geoderma, 2022, 412: 115724.

[156] Hansen-Goos H, Wettlaufer J S. Theory of ice premelting in porous media[J]. Physical Review E, 2010, 81(3): 031604.

[157] Hertzberg R W, Vinci R P, Hertzberg J L. Deformation and fracture mechanics of engineering materials[M]. John Wiley & Sons, 2020.

[158] Meng W, Guo Y. Experimental study on mechanical properties of ice. Paper presented at Proceedings of the AASRI International Conference on Industrial Electronics and Applications, London, UK. 2015.

[159] Yang X, Jiang A, Li M. Experimental investigation of the time-dependent behavior of quartz sandstone and quartzite under the combined effects of chemical erosion and freeze–thaw cycles[J]. Cold Regions Science and Technology, 2019, 161: 51-62.

[160] Kuruppu M D, Obara Y, Ayatollahi M R, et al. ISRM-suggested method for determining the mode I static fracture toughness using semi-circular bend specimen[J]. Rock Mechanics and Rock Engineering, 2014, 47: 267-274.

[161] Peng K, Lv H, Zou Q, et al. Evolutionary characteristics of mode-I fracture toughness and fracture energy in granite from different burial depths under high-temperature effect[J]. Engineering Fracture Mechanics, 2020, 239: 107306.

[162] Miao S, Pan P Z, Yu P, et al. Fracture analysis of Beishan granite after high-temperature treatment using digital image correlation[J]. Engineering Fracture Mechanics, 2020, 225: 106847.

[163] Wu Z M, Rong H, Zheng J J, et al. An experimental investigation on the FPZ properties in concrete using digital image correlation technique[J]. Engineering Fracture Mechanics, 2011, 78(17): 2978-2990.

[164] Lin Q, Wan B, Wang S, et al. Visual detection of a cohesionless crack in rock under three-point bending[J]. Engineering Fracture Mechanics, 2019, 211: 17-31.

[165] Gutenberg B, Richter C F. Frequency of earthquakes in California[J]. Bulletin of the Seismological society of America, 1944, 34(4): 185-188.

[166] W. Goebel T H, Schorlemmer D, Becker T W, et al. Acoustic emissions document stress changes over many seismic cycles in stick‐slip experiments[J]. Geophysical Research Letters, 2013, 40(10): 2049-2054.

[167] Colombo I S, Main I G, Forde M C. Assessing damage of reinforced concrete beam using “b-value” analysis of acoustic emission signals[J]. Journal of materials in civil engineering, 2003, 15(3): 280-286.

[168] Jiang L, Cheng Y, Zhang Y, et al. Numerical investigation on the freezing evolution process of rock mass at sub-zero temperature[J]. Journal of Applied Science and Engineering, 2019, 22(2): 337-348.

[169] Dash J G, Rempel A W, Wettlaufer J S. The physics of premelted ice and its geophysical consequences[J]. Reviews of modern physics, 2006, 78(3): 695.

[170] Wilen L A, Wettlaufer J S, Elbaum M, et al. Dispersion-force effects in interfacial premelting of ice[J]. Physical Review B, 1995, 52(16): 12426.

[171] Guerin F, Laforte C, Farinas M I, et al. Analytical model based on experimental data of centrifuge ice adhesion tests with different substrates[J]. Cold Regions Science and Technology, 2016, 121: 93-99.

[172] Chen H, Wu Y, Xia H, et al. Review of ice-pavement adhesion study and development of hydrophobic surface in pavement deicing[J]. Journal of traffic and transportation engineering (English edition), 2018, 5(3): 224-238.

[173] Jellinek H H G. Adhesive properties of ice[J]. Journal of colloid science, 1959, 14(3): 268-280.

[174] Petrovic J J. Review mechanical properties of ice and snow[J]. Journal of materials science, 2003, 38: 1-6.

[175] Jia H, Xiang W, Krautblatter M. Quantifying rock fatigue and decreasing compressive and tensile strength after repeated freeze‐thaw cycles[J]. Permafrost and Periglacial processes, 2015, 26(4): 368-377.

[176] Prick A. Critical degree of saturation as a threshold moisture level in frost weathering of limestones[J]. Permafrost and Periglacial Processes, 1997, 8(1): 91-99.

[177] Orowan E. The fatigue of glass under stress[J]. Nature, 1944, 154(3906): 341-343.

[178] Liu H H, Rutqvist J, Berryman J G. On the relationship between stress and elastic strain for porous and fractured rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2009, 46(2): 289-296.

[179] Freed AD. Natural strain. J Eng Mater Technol 1995; 117379–85.

[180] Jaeger J C, Cook N G W, Zimmerman R. Fundamentals of rock mechanics[M]. John Wiley & Sons, 2009.

[181] Li X, Cao W G, Su Y H. A statistical damage constitutive model for softening behavior of rocks[J]. Engineering Geology, 2012, 143: 1-17.

[182] 徐芝纶. 弹性力学[M]. 北京 高等教育出版社, 2006.

[183] 曹文贵, 张超, 贺敏, 等. 考虑空隙压密阶段特征的岩石应变软化统计损伤模拟方法[J]. 岩土工程学报, 2016, 38(10): 1754-1761.

[184] Kleinberg R L, Kenyon W E, Mitra P P. Mechanism of NMR relaxation of fluids in rock[J]. Journal of magnetic resonance, Series A, 1994, 108(2): 206-214.

[185] Ausbrooks R, Hurley N F, May A, et al. Pore-size distributions in vuggy carbonates from core images, NMR, and capillary pressure[C]//SPE annual technical conference and exhibition. OnePetro, 1999.

[186] 刘波, 马永君, 盛海龙, 等. 不同围压与冻结温度下白垩系红砂岩力学性质试验研究[J]. 岩石力学与工程学报, 2019, 38(03): 455-466.