Abstract：The frost heave force is an important cause of rock mass failure in the cold region. The traditional analytical model cannot calculate the frost heave force after crack propagation. Considering the influence of crack propagation，it is of great significance to establish an analytical model of frost heave force in fractured rock mass. In this paper，an analytical model of frost heave force evolution in fractured rock mass is derived based on the theory of elastic mechanics，complex variable function and elastic-plastic fracture mechanics. The effects of crack size，freezing rate，water mobility and other factors on the evolution of frost heave force before fracture instability were analyzed. Combined with the estimation of frost heave force after fracture instability，the evolution mechanism and the whole process characteristics of frost heave force were studied. The analytical calculation results were compared with the experimental results. The results show that：(1) The analytical model of frost heave force proposed in this paper can calculate the frost heave force after crack propagation，rationally explain the mechanism of frost heave force fluctuation，and provide a basis for the evaluation of frost resistance of cracks to a certain extent. (2) Before fracture instability，the maximum frost heave force decreases with the increase of fracture length，and the time required to reach the maximum frost heave force increases with the increase of crack width. The maximum frost heave force may decrease with the increase of freezing rate. With the increase of water mobility，the maximum frost heaving force before fracture instability increases，and the time required to reach the maximum frost heave force also increases. (3) The variation of frost heave force can be caused by subcritical crack propagation and instability fracture，and the evolution of frost heave force in cracks is characterized by fluctuation. When there is no instability fracture in the freezing process，the frost heave force fluctuates and increases. When there is instability fracture in the freezing process，the frost heave force first increases and then decreases.
李 聪，张新宙，吴亮亮，谢 天，赵凯艺. 裂隙岩体冻胀力演化解析模型与裂纹亚临界扩展分析[J]. 岩石力学与工程学报, 2024, 43(S1): 3439-3449.
LI Cong，ZHANG Xinzhou，WU Liangliang，XIE Tian，ZHAO Kaiyi. Analytical model of frost heave force evolution in fractured rock mass and analysis of subcritical crack propagation. , 2024, 43(S1): 3439-3449.
[1] 刘泉声，黄诗冰，康永水，等. 低温冻结岩体单裂隙冻胀力与数值计算研究[J]. 岩土工程学报，2015，37(9)：1 572–1 580.(LIU Quansheng，HUANG Shibing，KANG Yongshui，et al. Numerical and theoretical studies on frost heaving pressure in a single fracture of frozen rock mass under low temperature[J]. Chinese Journal of Geotechnical Engineering，2015，37(9)：1 572–1 580.(in Chinese))
[2] 段思阳. 椭圆孔边裂纹的断裂力学行为[硕士学位论文][D]. 石家庄：石家庄铁道大学，2019.(DUAN Siyang. Fracture mechanical behavior of cracks emanating from an elliptical hole[M. S. Thesis][D]. Shijiazhuang：Shijiazhuang Tiedao University，2019.(in Chinese))
[3] 祝青钰. 复变函数方法在孔边裂纹问题研究中的应用[博士学位论文][D]. 北京：北京理工大学，2016.(ZHU Qingyu. Application of complex function method on the research of cracked-hole problem[Ph. D. Thesis][D]. Beijing：Beijing Institute of Technology，2016.(in Chinese))
[4] 祝青钰，韩 峰，隋明丽. 无限大板孔边双裂纹应力强度因子和裂纹面张开位移[J]. 航空学报，2016，37(3)：883–893.(ZHU Qingyu，HAN Feng，SUI Mingli. Stress intensity factors and crack-surface opening displacements for two cracks emanating from a circular hole in an infinite plate[J]. Acta Aeronautica Et Astronautica Sinica，2016，37(3)：883–893.(in Chinese))
[5] 祝二浩. 基于复变函数方法的椭圆形隧洞锚杆力学分析[硕士学位论文][D]. 北京：华北电力大学(北京)，2019.(ZHU Erhao. Mechanical analysis of bolts in an elliptical tunnel based on complex variable method[M. S. Thesis][D]. Beijing：North China Electric Power University(Beijing)，2019.(in Chinese))
[6] 刘宝良，闫龙海，闫相桥. 内部压力作用下无限大板中椭圆孔及裂纹问题[J]. 辽宁工程技术大学学报：自然科学版，2012，31(4)：540–543.(LIU Baoliang，YAN Longhai，YAN Xiangqiao. An elliptical hole-crack problem in infinite plate under internal pressure[J]. Journal of Liaoning Technical University：Natural Science，2012，31(4)：540–543.(in Chinese))
[7] 赵新平，杨丽英，吴国荣，等. 部分边界受力情形下带裂纹的椭圆孔口问题的解析研究[J]. 内蒙古师范大学学报：自然科学版，2014，43(6)：665–668.(ZHAO Xinping，YANG Liying，WU Guorong，et al. Analysis study for the problem based on elliptic boundary with a crack under pressure on part of boundary[J]. Journal of Inner Mongolia Normal University：Natural Science，2014，43(6)：665–668.(in Chinese))
[8] 黄诗冰，刘泉声，刘艳章，等. 低温热力耦合下岩体椭圆孔(裂)隙中冻胀力与冻胀开裂特征研究[J]. 岩土工程学报，2018，40(3)：459–467.(HUANG Shibing，LIU Quansheng，LIU Yanzhang，et al. Frost heaving pressure and characteristics of frost cracking in elliptical cavity(crack) of rock mass under coupled thermal-mechanical condition at low temperature[J]. Chinese Journal of Geotechnical Engineering，2018，40(3)：459–467.(in Chinese))
[9] 黄诗冰，刘泉声，程爱平，等. 低温岩体裂隙冻胀力与冻胀扩展试验初探[J]. 岩土力学，2018，39(1)：78–84.(HUANG Shibing，LIU Quansheng，CHENG Aiping，et al. Preliminary experimental study of frost heaving pressure in crack and frost heaving propagation in rock mass under low temperature[J]. Rock and Soil Mechanics，2018，39(1)：78–84.(in Chinese))
[10] GUO J H，LU Z X，HAN H T，et al. The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid[J]. European Journal of Mechanics A：Solids，2010，29(4)：654–663.
[11] 郭俊宏，刘官厅. 带双裂纹的椭圆孔口问题的应力分析[J]. 力学学报，2007，39(5)：699–703.(GUO Junhong，LIU Guanting. Stress analysis for an elliptical hole with two straight cracks[J]. Chinese Journal of Theoretical and Applied Mechanics，2007，39(5)：699–703.(in Chinese))
[12] 范天佑. 断裂理论基础[M]. 北京：科学出版社，2003：20–112.(FAN Tianyou. Theoretical basis of fracture[M]. Beijing：Science Press，2003：20–112.(in Chinese))
[13] 李世愚，和泰名，尹祥础. 岩石断裂力学[M]. 北京：科学出版社，2015：100–124.(LI Shiyu，Hetaiming，YIN Xiangchu. Rock fracture mechanics[M]. Beijing：Science Press，2015：100–124.(in Chinese))
[14] WEISS J. Subcritical crack propagation as a mechanism of crevasse formation and iceberg calving[J]. Journal of Glaciology，2004，50：109–115.
[15] ATKINSON B K. A fracture mechanics study of subcritical tensile cracking of quartz in wet environments[J]. Pure and Applied Geophysics，1979，117(5)：1 011–1 024.
[16] 汪道兵，秦 浩，马海燕，等.干热岩亚临界裂纹扩展的实验研究[J]. 北京石油化工学院学报，2020，(4)：7–12.(WANG Daobing，QIN Hao，MA Haiyan，et al. Experimental study on subcritical crack growth in hot dry rocks[J]. Journal of Beijing Institute of Petrochemical Technology，2020，(4)：7–12.(in Chinese))
[17] XU S，REINHARDT H W. Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture[J]. International Journal of Fracture，1998，92(1)：71–99.
[18] 徐世烺. 混凝土断裂力学[M]. 北京：科学出版社，2011：312–317.(XU Shilang. Concrete fracture mechanics[M]. Beijing：Science Press，2011：312–317.(in Chinese))
[19] 董 伟，何化南，吴智敏，等. 基于裂缝扩展准则的K_R阻力曲线研究[J]. 工程力学，2011，28(7)：13–19.(DONG Wei，HE Huanan，WU Zhimin，et al. Study on K_R-curves based on crack propagation criterion in concrete[J]. Engineering Mechanics，2011，28(7)：13–19.(in Chinese))
[20] COOK R F，CLARKE B R. Fracture stability，R-curve and strenghth variability[J]. Acta Metall，1988，36(3)：555–562.
[21] AROSIO D，LONGONI L，MAZZA F，et al. Freeze-thaw cycle and rockfall monitoring[C]// World Landslide Forum. [S.l.]：[s.n.]，2013：385–390.
[22] 田 镇，李银平，王贵宾，等. 饱水红砂岩裂隙冻胀力与变形试验研究[J]. 岩石力学与工程学报，2022，41(增1)：2 857–2 868.(TIAN Zhen，LI Yinping，WANG Guibin，et al. Experimental study on frost heaving force and deformation of water saturated red sandstone fractures[J]. Chinese Journal of Rock Mechanics and Engineering，2022，41(Supp.1)：2 857–2 868.(in Chinese))
[23] 乔 趁，王 宇，宋正阳，等. 饱水裂隙花岗岩周期冻胀力演化特性试验研究[J]. 岩土力学，2021，42(8)：2 141–2 150.(QIAO Chen，WANG Yu，SONG Zhengyang，et al. Experimental study on the evolution characteristics of cyclic frost heaving pressure of saturated fractured granite[J]. Rock and Soil Mechanics，2021，42(8)：2 141–2 150.(in Chinese))
[24] 刘 昊，王 宇，王华建，等. 冻融循环作用下岩石含冰裂隙冻胀力演化试验研究[J]. 工程地质学报，2022，30(4)：1 122–1 131.(LIU Hao，WANG Yu，WANG Huajian，et al. Experimental study on frost heaving pressure evolution of rock ice cracks under freezing-thawing cycles[J]. Journal of Engineering Geology，2022，30(4)：1 122–1 131. (in Chinese))
[25] 刘艳章，王 瑾，尹 东，等. 冻融作用下单裂隙类砂岩冻胀力与强度变化规律研究[J]. 矿冶工程，2021，41(6)：115–119.(LIU Yanzhang，WANG Jin，YIN Dong，et al. Experimental study on frost heaving pressure and strength variation of single crack quasi?sandstone under freeze?thaw process[J]. Mining and Metallurgical Engineering，2021，41(6)：115–119.(in Chinese))
[26] 贾海梁，赵思琪，丁 顺，等. 含水裂隙冻融过程中冻胀力演化及影响因素研究[J]. 岩石力学与工程学报，2022，41(9)：1 832–1 845. (JIA Hailiang，ZHAO Siqi，DING Shun，et al. Study on the evolution and influencing factors of frost heaving force of waterbearing cracks during freezing-thawing process[J]. Chinese Journal of Rock Mechanics and Engineering，2022，41(9)：1 832–1 845.(in Chinese))
[27] 单仁亮，白 瑶，孙鹏飞，等. 裂隙红砂岩冻胀力特性试验研究[J].煤炭学报，2019，44(6)：1 742–1 752.(SHAN Renliang，BAI Yao，SUN Pengfei，et al. Experimental study on frost heaving pressure properties in fractured red sandstone[J]. Journal of China Coal Society，2019，44(6)：1 742–1 752.(in Chinese))
[28] 郑新雨，徐 飞，孙浩凯，等. 寒区富水隧道冻结圈围岩冻胀力演化规律研究[J]. 中国公路学报，2022，35(9)：340–351.(ZHENG Xinyu，XU Fei，SUN Haokai，et al. Evolution law of frost-heaving pressure of frozen circles in surrounding rock around water-rich tunnels in cold regions[J]. China Journal of Highway and Transport，2022，35(9)：340–351.(in Chinese))
[29] LI C，ZHANG R T，ZHU J B，et al. Mechanism of progressive failure of a slope with a steep joint under the action of freezing and thawing：model test[J]. Journal of Mountain Science，2022，19(10)：2 999–3 012.
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