Abstract
The establishment of mechanical model and brittleness index of frozen rocks can provide crucial guidance to the construction of rock mass engineering in permafrost area or artificial freezing engineering with water-rich soft rocks. The exponential function of strain difference is established for initial compacted stage and linear elastic stage in the stress–strain curve, thereby yielding the statistical damage model and damage evolution equation of frozen rock according to statistical damage theory, Hoek–Brown criterion and the strain difference function. The novel damage model is proposed to well simulate the whole failure process of frozen rocks, and based on that a novel brittleness index of frozen rocks is derived using energy method. Hence, a series of triaxial tests of saturated red sandstones under different low temperature are conducted. Confining pressure and low temperature have significant influences on the mechanical properties and brittleness of frozen rocks, and Hoek–Brown criterion is appropriate for expressing the nonlinear relation between principal stresses of frozen rocks. In addition, the rationality of the proposed damage model is proved by comparing with the test curve, and the superiority of the novel damage model compared with the existing ones is proved by two evaluation methods. Furthermore, the variation of damage evolution shows that confining pressure can enhance the rock ductility and low temperature can improve the rock brittleness, which is further verified by the novel brittleness index. The feasibility of the novel brittleness index is demonstrated by comparing that with the existing brittleness indexes. Eventually, the physical significance of parameters S0 and β for the damage model curves and brittleness index of frozen rocks are discussed. Consequently, the proposed damage model and novel brittleness index are suitable for the application of frozen rock engineering.
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References
Ai C, Zhang J, Li YW et al (2016) Estimation criteria for rock brittleness based on energy analysis during the rupturing process. Rock Mech Rock Eng 49(12):4681–4698
Altindag R (2002) The evaluation of rock brittleness concept on rotary blast hole drills. J S Afr Inst Min Metall 102(1):61–66
Bai Y, Shan RL, Ju YJ et al (2020) Study on the mechanical properties and damage constitutive model of frozen weakly cemented red sandstone. Cold Regions Sci Technol 171(C):102980–102980
Buller D, Hughes SN, Market J, et al (2010) Petrophysical evaluation for enhancing hydraulic stimulation in horizontal shale gas wells. In: SPE annual technical conference and exhibition held in Florence, Italy. SPE, pp 132990
Cao WG, Zhao H, Li X et al (2010) Statistical damage model with strain softening and hardening for rocks under the influence of voids and volume changes. Can Geotech J 47(8):857–871
Cao WG, Tan X, Zhang C et al (2018) Constitutive model to simulate full deformation and failure process for rocks considering initial compression and residual strength behaviors. Can Geotech J 56(5):649–661
Chen GQ, Jiang WZ, Sun X et al (2019) Quantitative evaluation of rock brittleness based on crack initiation stress and complete stress-strain curves. Bull Eng Geol Env 78(8):5919–5936
Chen K (2020) Constitutive model of rock triaxial damage based on the rock strength statistics. Int J Damage Mech 29(10):1487–1511
Chen YF, Lin H, Wang YX et al (2020) Damage statistical empirical model for fractured rock under freezing-thawing cycle and loading. Geofluids 2020:8842471
Chen YF, Lin H, Wang YX et al (2021) Statistical damage constitutive model based on the Hoek–Brown criterion. Arch Civ Mech Eng 21(3):117
Deng JA, Cu DS (2011) On a statistical damage constitutive model for rock materials. Comput Geosci 37(2):122–128
Fang W, Jiang N, Luo XD (2019) Establishment of damage statistical constitutive model of loaded rock and method for determining its parameters under freeze-thaw condition. Cold Reg Sci Technol 160:31–38
Gao F, Xiong X, Xu CS et al (2021) Mechanical property deterioration characteristics and a new constitutive model for rocks subjected to freeze-thaw weathering process. Int J Rock Mech Min Sci 140:104642
Gu QS, Ning JG, Tan YL et al (2018) Damage constitutive model of brittle rock considering the compaction of crack. Geomech Eng 15(5):1081–1089
Hu QB, Liang HA, Yang T (2020) A new method for rock brittleness evaluation based on statistical damage constitutive relation. J Harbin Inst Technol 52(11):147–156
Huang SB, Liu QS, Cheng AP et al (2017) A statistical damage constitutive model under freeze-thaw and loading for rock and its engineering application. Cold Reg Sci Technol 145:142–150
Huang SB, Liu QS, Liu YZ et al (2018) Freezing strain model for estimating the unfrozen water content of saturated rock under low temperature. Int J Geomech 18(2):04017137
Hucka V, Das B (1974) Brittleness determination of rocks by different methods. Int J Rock Mech Mining Sci Geomech Abstr 1(10):389–392
Inada Y, Yokota K (1984) Some studies of low temperature rock strength. Int J Rock Mech Mining Sci Geomech Abstr 21(3):145–153
Jarvie DM, Hill RJ, Rublen TE et al (2007) Unconventional shale-gas systems: the Mississippian Barnett shale of north-central Texas as one model for thermogenic shale-gas assessment. AAPG Bull 91(4):475–499
Ji M, Chen K, Guo HJ (2018) Constitutive model of rock uniaxial damage based on rock strength statistics. Adv Civ Eng 2018:5047834
Jiang W, Lai Y, Yu F, Ma Q, Jiang H (2023) Mechanical properties investigation and damage constitutive models of red sandstone subjected to freeze-thaw cycles. Cold Reg Sci Technol. https://doi.org/10.1016/j.coldregions.2023.103776
Kivi IR, Ameri M, Molladavoodi M (2018) Shale brittleness evaluation based on energy balance analysis of stress-strain curves. J Petrol Sci Eng 167:1–19
Kodama J, Goto T, Fujii Y et al (2013) The effects of water content, temperature and loading rate on strength and failure process of frozen rocks. Int J Rock Mech Min Sci 62:1–13
Kuang ZC, Qiu SL, Li SH et al (2021) A new rock brittleness index based on the characteristics of complete stress-strain behaviors. Rock Mech Rock Eng 54:1109–1128
Lemaitre J (1984) How to use damage mechanics. Nucl Eng Des 80:233–245
Li HZ, Liao HJ, Xiong GD et al (2015) A three-dimensional statistical damage constitutive model for geomaterials. J Mech Sci Technol 29(1):71–77
Li LC, Zhai MY, Zhang LY et al (2019) Brittleness evaluation of glutenite based on energy balance and damage evolution. Energies 12(18):3421
Li N, Zou YS, Zhang SC et al (2019a) Rock brittleness evaluation based on energy dissipation under triaxial compression. J Petrol Sci Eng 183(C):106349–106349
Li YW, Jia D, Rui ZH et al (2017) Evaluation method of rock brittleness based on statistical constitutive relations for rock damage. J Petrol Sci Eng 153:123–132
Li YW, Long M, Zuo LH et al (2018) Brittleness evaluation of coal based on statistical damage and energy evolution theory. J Petrol Sci Eng 172:753–763
Li ZY, Yang GS (2021) Constitutive model of frozen red sandstone based on ice-solid binary medium. Arab J Geosci 14(16):1616
Liu B, Ma YJ, Sheng HL et al (2019) Experimental study on mechanical properties of Cretaceous red sandstone under different freezing temperatures and confining pressures. Chin J Rock Mechan Eng 38(3):455–466
Liu LL, Liu XY, Li Z et al (2019) Experimental analysis on the mechanical properties of saturated silty mudstone under frozen conditions. J Test Eval 47(1):20170080
Liu XB, Zhang ZY, Ge ZL et al (2021) Brittleness evaluation of saturated coal based on energy method from stress-strain curves of uniaxial compression. Rock Mech Rock Eng 54:3193–3207
Liu XS, Ning JG, Tan YL et al (2016) Damage constitutive model based on energy dissipation for intact rock subjected to cyclic loading. Int J Rock Mech Min Sci 85:27–32
Lu YN, Li XP, Chan AD (2019) Damage constitutive model of single flaw sandstone under freeze-thaw and load. Cold Reg Sci Technol 159:20–28
Lv ZT, Xia CC, Wang YS et al (2020) Frost heave and freezing processes of saturated rock with an open crack under different freezing conditions. Front Struct Civ Eng 14(4):947–960
Meng FZ, Wong LNY, Zhou H (2021) Rock brittleness indices and their applications to different fields of rock engineering: a review. J Rock Mech Geotech Eng 13(1):221–247
Munoz H, Taheri A, Chanda EK (2016) Fracture energy-based brittleness index development and brittleness quantification by pre-peak strength parameters in rock uniaxial compression. Rock Mech Rock Eng 49(12):4587–4606
Pan XH, Guo W, Wu SF (2020) An experimental approach for determination of the Weibull homogeneity index of rock or rock-like materials. Acta Geotech 15(2):375–391
Phillips M, Haberkorn A, Rhyner H (2017) Snowpack characteristics on steep frozen rock slopes. Cold Reg Sci Technol 141:54–65
Shen YJ, Wang YZ, Zhao XD et al (2018) The influence of temperature and moisture content on sandstone thermal conductivity from a case using the artificial ground freezing (AGF) method. Cold Reg Sci Technol 155:149–160
Tang GP, Zhao LH, Li L et al (2017) Combined influence of nonlinearity and dilation on slope stability evaluated by upper-bound limit analysis. J Central South Univ 24(7):1602–1611
Tarasov B, Yves P (2013) Universal criteria for rock brittleness estimation under triaxial compression. Int J Rock Mech Min Sci 59:57–69
Ulusay R (2015) The ISRM suggested methods for rock characterization, testing and monitoring 2007–2014. Springer International Publishing, London
Wang JB, Song ZP, Zhao BY et al (2018) A study on the mechanical behavior and statistical damage constitutive model of sandstone. Arab J Sci Eng 43(10):5179–5192
Wang ZL, Li YC, Wang JG (2007) A damage-softening statistical constitutive model considering rock residual strength. Comput Geosci 33(1):1–9
Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297
Xia YJ, Li LC, Tang CA et al (2017) A new method to evaluate rock mass brittleness based on stress-strain curves of class I. Rock Mech Rock Eng 50(5):1123–1139
Xu G, Wu W, Qi J (2016) Modeling the viscous behavior of frozen soil with hypoplasticity. Int J Numer Anal Methods Geomech 40(15):2061–2075. https://doi.org/10.1002/nag.2516
Xu G, Wu W, Kong L, Qi J (2018) Hypoplastic modeling for the mechanical behavior of frozen soil in stress path testing. Int J Geomech 18(6). https://doi.org/10.1061/(ASCE)GM.1943-5622.0001117
Yagiz S (2008) Assessment of brittleness using rock strength and density with punch penetration test. Tunn Undergr Space Technol 24(1):66–74
Yamabe T, Neaupane KM (2001) Determination of some thermo-mechanical properties of Sirahama sandstone under subzero temperature condition. Int J Rock Mech Min Sci 38(7):1029–1034
Yang GS, Wei Y, Shen YJ et al (2019) Mechanical behavior and strength forecast model of frozen saturated sandstone under triaxial compression. Chin J Rock Mechan Eng 38(04):683–694
Yao X, Zhang M, Xu G, Yu F (2020) Influence of mechanical properties on the coefficient of lateral earth pressure at rest of frozen soils. Cold Reg Sci Technol. https://doi.org/10.1016/j.coldregions.2020.103057
Zhang HM, Meng XZ, Yang GS (2020) A study on mechanical properties and damage model of rock subjected to freeze-thaw cycles and confining pressure. Cold Reg Sci Technol 174:103056–10305654
Zhang HM, Yuan C, Yang GS et al (2019) A novel constitutive modelling approach measured under simulated freeze-thaw cycles for the rock failure. Eng Comput 37:1–14
Zhang J, Ai C, Li YW et al (2018) Energy-based brittleness index and acoustic emission characteristics of anisotropic coal under triaxial stress condition. Rock Mech Rock Eng 51(11):3343–3360
Zhang LM, Cong Y, Meng FZ et al (2021) Energy evolution analysis and failure criteria for rock under different stress paths. Acta Geotech 16:569–580
Zhao H, Shi CJ, Zhao MH et al (2016) Statistical damage constitutive model for rocks considering residual strength. Int J Geomech 17(1):04016033–04016033
Zhao Y, Yang TH, Zhang PH et al (2017) The analysis of rock damage process based on the microseismic monitoring and numerical simulations. Tunn Undergr Space Technol 69:1–17
Zhao YC, Li SY, Shi LH et al (2022) Mechanical damage evolution and a statistical damage model for frozen sandstone. Int J Geomech 22(10):04022184
Zhu CQ, Ma HY, Zhao PT et al (2021) Strength characteristics of frozen coal-rock interface for rock crosscut coal uncovering. Energy Explor Exploit 40(1):460–472
Acknowledgements
This paper was supported by the National Natural Science Foundation of China (Grant No. U2268215), National Key Research and Development Program of China (Grant No. 2018YFC0809605) and Key Research Program of Frontier Sciences of Chinese Academy of Sciences (Grant No. QYZDY-SSW-DQC015).
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Jiang, W., Lai, Y., Ma, Q. et al. Mechanical damage model and brittleness index of frozen rocks based on statistical damage theory. Acta Geotech. 18, 4687–4713 (2023). https://doi.org/10.1007/s11440-023-01861-0
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DOI: https://doi.org/10.1007/s11440-023-01861-0