Abstract
The creep model is the main form used to describe the creep behavior of weakly cemented soft rock (WCSR). To investigate the creep behavior of weakly cemented soft rock, multi-stage loading creep tests were performed by using GDS HPTAS creep triaxial apparatus. The creep curves, creep rates, and creep failure modes of weakly cemented soft rock under different water contents were obtained. A novel four-element fractional-order creep model to describe the three-stage creep behavior of weakly cemented soft rock was proposed using the Abel dashpot basing on the creep test results and fractional calculus theory. The formula of this model was developed according to the H-Fox special function. The trust-region method was used to obtain the model parameters, and the parameters were optimized using the ant colony optimization approach. The creep curves of weakly cemented soft rock under different water contents were calculated and compared with the experimental results, verifying the superiority of the four-element fractional-order creep model in describing the creep characteristics of weakly cemented soft rock. The calculated results were also in good agreement with the experimental results, which confirms that the four-element fractional order creep model can accurately reflect the complete creep behavior of weakly cemented soft rock.
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Abbreviations
- w :
-
Water content
- Γ :
-
Gamma function
- I :
-
Integral symbols
- α :
-
Fractional order
- σ s :
-
Stress threshold
- σ 1 :
-
Stress of the Hooke body
- σ 2 :
-
Stress of the Abel dashpot
- σ 3 :
-
Stress of the plastic
- σ c :
-
Uniaxial compressive strength
- ε a :
-
Axial strain
- Δt(Tomanovic):
-
Error of the creep model
- ρ :
-
Density
- δ :
-
Pheromone volatilization
- D :
-
Differential symbols
- ξ, ξ1, ξ2 :
-
Viscosity coefficient of the Abel dashpot
- E0, E1 :
-
Elastic moduli of the Hooke body
- η1, η2 :
-
Viscosity coefficients of the Newton body
- ε 1 :
-
Strain of the Hooke body
- ε 2 :
-
Strain of the Abel dashpot
- ε 3 :
-
Strain of the plastic element
- σ 0 :
-
Deviator stress
- ε r :
-
Radial strain
- β :
-
Basic parameter
References
Ajalloeian R, Lashkaripour GR (2000) Strength anisotropies in mudrock. Bull Eng Geol Environ 59:195–199. https://doi.org/10.1007/s100640000055
Arikoglu A (2014) A new fractional derivative model for linearly viscoelastic materials and parameter identification via genetic algorithms. Rheol Acta 53:219–233. https://doi.org/10.1007/s00397-014-0758-2
Bouras Y, Zorica D, Atanacković TM, Vrcelj Z (2018) A non-linear thermo-viscoelastic creep model based on fractional derivatives for high temperature creep in concrete. Appl Math Model 55:551–568. https://doi.org/10.1016/j.apm.2017.11.028
Cao S, Bian J, Li P (2002) Rheologic constitutive relationship of rock and a modifical model. Chin J Rock Mech Eng 21:632–634. https://doi.org/10.3321/j.issn:1000-6915.2002.05.004
Chen BR, Zhao XJ, Feng XT, Zhao HB, Wang SY (2014) Time-dependent damage constitutive model for the marble in the Jinping II hydropower station in China. Bull Eng Geol Environ 73:499–515. https://doi.org/10.1007/s10064-013-0542-z
Chen JR, Pu H, Xiao C (2015) Research on rheology model of broken mudstone based on the fractional theory. J China Univ Min Technol 44:996–1001. https://doi.org/10.13247/j.cnki.jcumt.000418
Chenevert EM (1970) Shale alteration by water adsorption. J Pet Technol 22:1141–1148. https://doi.org/10.2118/2401-pa
Deng HF, Zhou ML, Li JL, Sun XS, Huang YL (2016) Creep degradation mechanism by water–rock interaction in the red-layer soft rock. Arab J Geosci 9:601. https://doi.org/10.1007/s12517-016-2604-6
Duan XM, Yin DS, An LY, Zhang W (2013) The deformation study in viscoelastic materials based on fractional order calculus. Sci Sin Phys Mech Astron 43:971. https://doi.org/10.1360/132012-807
Feng WL, Qiao CS, Niu SJ (2020) Study on sandstone creep properties of Jushan Mine affected by degree of damage and confining pressure. Bull Eng Geol Environ 79(2):869–888. https://doi.org/10.1007/s10064-019-01615-x
Heymans N, Bauwens JC (1994) Fractal creep models and fractional differential equations for viscoelastic behavior. Rheol Acta 33:210–219. https://doi.org/10.1007/BF00437306
Hong Z, Xia-Ting F, Xian-jie H (2015) A creep model for weakly consolidated porous sandstone including volumetric creep. Int J Rock Mech Min Sci 78:99–107. https://doi.org/10.1016/j.ijrmms.2015.04.021
Hou RB, Zhang K, Tao J, Xue XR, Chen YL (2019) A nonlinear creep damage coupled model for rock considering the effect of initial damage. Rock Mech Rock Eng:1275–1285. https://doi.org/10.1007/s00603-018-1626-7
Huang M, Liu XR (2011) Study on the relationship between parameters of deterioration creep model of rock under different modeling assumptions. Adv Mater Res 243-249:2571–2580. https://doi.org/10.4028/www.scientific.net/AMR.243-249.2571
Jiang HF, Liu DY, Huang W et al. (2014) Creep properties of rock under high confining pressure and different pore water pressures and a modified Nishihara model. Chin J Geotech Eng 36(3):443–451. https://doi.org/10.11779/CJGE201403006
Işık Y (2010) Influence of water content on the strength and deformability of gypsum. Int J Rock Mech Min Sci 47:342–347. https://doi.org/10.1016/j.ijrmms.2009.09.002
Ju N, Haifeng H, Da Z, Xin Z, Chengqiang Z (2016) Improved Burgers model for creep characteristics of red bed mudstone considering water content. Rock Soil Mech 67:–74. https://doi.org/10.16285/j.rsm.2016.S2.008
Li P, Liu J, Zhu JB, He HJ (2008) Research on effects of water content on shear creep behavior of weak structural plane of sandstone. Rock Soil Mech 29:1865–1871. https://doi.org/10.3901/JME.2008.10.294
Li RD, Yue JC, Zhu CZ, Sun ZY (2013) A nonlinear viscoelastic creep model of soft soil based on fractional order derivative. Appl Mech Mater 438-439:1056–1059. https://doi.org/10.4028/www.scientific.net/AMM.438-439.1056
Li DW, Chen JH, Zhou Y (2018) A study of coupled creep damaged constitutive model of artificial frozen soil. Adv Mater Sci Eng 2018:1–9. https://doi.org/10.1155/2018/7458696
Liao MK, Lai YM, Liu EL, Wan XS (2016) A fractional order creep constitutive model of warm frozen silt. Acta Geotech 12:1–13. https://doi.org/10.1007/s11440-016-0466-4
Liu X, Chao Y, Jing Y (2015) The influence of moisture content on the time-dependent characteristics of rock material and its application to the construction of a tunnel portal. Adv Mater Sci Eng:1–13. https://doi.org/10.1155/2015/725162
Liu HZ, Xie HQ, He JD, Xiao ML, Zhuo L (2016) Nonlinear creep damage constitutive model for soft rock. Mech Time Dependent Mater 21:1–24. https://doi.org/10.1007/s11043-016-9319-7
Matthess G (1988) Advances in modelling water-rock interaction in aquifers[M]// groundwater flow and quality modelling. Springer Netherlands. https://doi.org/10.1007/978-94-009-2889-3_23
Metzler R, Nonnenmacher TF (2003) Fractional relaxation processes and fractional creep models for the description of a class of viscoelastic materials. Int J Plast 19:941–959. https://doi.org/10.1016/s0749-6419(02)00087-6
Ofoegbu GI, Dasgupta B, Manepally C, Stothoff SA, Fedors R (2017) Modeling the mechanical behavior of unsaturated expansive soils based on bishop principle of effective stress. Environ Earth Sci 76:555. https://doi.org/10.1007/s12665-017-6884-2
Papoulia KD, Panoskaltsis VP, Kurup NV, Korovajchuk I (2010) Creep representation of fractional order viscoelastic material models. Rheol Acta 49:381–400. https://doi.org/10.1007/s00397-010-0436-y
Pu SY, Huang ZH, Yao JY, Mu R, Zheng HC, Wang TL, Liu XL, Li L, Wang YH (2018) Fractional-order Burgers constitutive model for rock under low dynamic stress. J Yangtze River Sci Res Inst. https://doi.org/10.11988/ckyyb.20170454
Roedder E , Bassett RL (1981) Problems in determination of the water content of rock-salt samples and its significance in nuclear-waste storage siting. Geol 9(11):92–104. https://doi.org/10.1130/0091-7613(1981)92.0.CO;2
Shi X, Cai W, Meng Y, Li G, Wen K, Zhang Y (2016) Weakening laws of rock uniaxial compressive strength with consideration of water content and rock porosity. Arab J Geosci 9:1–7. https://doi.org/10.1007/s12517-016-2426-6
Su T, Zhou HW, Zhao JW, Che J, Sun XT, Wang L (2019) A creep model of rock based on variable order fractional derivative. Chin J Rock Mech Eng 38(7):1355–1363. https://doi.org/10.13722/j.cnki.jrme.2018.1382
Sun J, Wang S (2000) Rock mechanics and rock engineering in China: developments and current state-of-the-art. 37:447–465. https://doi.org/10.1016/S1365-1609(99)00072-6
Tang H, Wang DP, Huang RQ, Pei XJ, Chen WL (2017) A new rock creep model based on variable-order fractional derivatives and continuum damage mechanics. Bull Eng Geol Environ:1–9. https://doi.org/10.1007/s10064-016-0992-1
Tang H, Duan Z, Wang DP, Dang Q (2020) Experimental investigation of creep behavior of loess under different moisture contents. Bull Eng Geol Environ 79(1):411–422. https://doi.org/10.1007/s10064-019-01545-8
Tomanovic Z (2006) Creep model of soft rock creep based on the tests on marl mechanics of time-dependent materials. 10:135–154. https://doi.org/10.1007/s11043-006-9005-2
Wang JB, Liu XR, Song ZP et al (2018) A whole process creeping model of salt rock under uniaxial compression based on inverse S function. Chin J Rock Mech Eng 37(11):27–40
Wei HZ, Xu WJ, Wei CF, Meng QS (2018) Influence of water content and shear rate on the mechanical behavior of soil–rock mixtures. Sci China Technol Sci 61:1127–1136. https://doi.org/10.1007/s11431-017-9277-5
Wu F, Xie HP, Liu JF, Bian Y, Pei JL (2014) Experimental study of fractional viscoelastic-plastic creep model. Chin J Rock Mech Eng 33:964–970. https://doi.org/10.13722/j.cnki.jrme.2014.05.012
Wu F, Liu JF, Wang J (2015) An improved Maxwell creep model for rock based on variable-order fractional derivatives. Environ Earth Sci 73:6965–6971. https://doi.org/10.1007/s12665-015-4137-9
Xu Z, Chen W (2013) A fractional-order model on new experiments of linear viscoelastic creep of Hami Melon. Comput Math Appl 66:677–681. https://doi.org/10.1016/j.camwa.2013.01.033
Xu HY, Jiang XY (2017) Creep constitutive models for viscoelastic materials based on fractional derivatives. Comput Math Appl 73:1377–1384. https://doi.org/10.1016/j.camwa.2016.05.002
Yang CH, Wang YY, Jian-Guang LI, Gao F (2007) Testing study about the effect of different water content on rock creep law. J China Coal Soc 32:695–699. https://doi.org/10.1007/s10870-007-9222-9
Yao Z, Zhang Q, Niu L (2016) Analysis of fractional order derivative Nishihara model of frozen remolded clay based on ant colony algorithm. J Yangtze River Sci Res Inst 33:81–86. https://doi.org/10.11988/ckyyc.20150419
Yuan WW (2014) Research on fractional damage model of mudstone dynamic creep. Northeast Petroleum Univ:29–39
Zhang Q, Li SC, Li LP, Zhang QQ (2013) Experimental study on creep of highly-weathered breccia. Appl Mech Mater 477-478:588–591. https://doi.org/10.4028/www.scientific.net/amm.477-478.588
Zhang Y, Xu WY, Wang W, Shao JF, Gu JJ (2014) Study of creep tests and its parameters identification of soft rock in fractured belt. Chin J Rock Mech Eng 33:3412–3420. https://doi.org/10.13722/j.cnki.jrme.2014.s2.004
Zhou HW, Wang CP, Han BB, Duan ZQ (2011) A creep constitutive model for salt rock based on fractional derivatives. Int J Rock Mech Min Sci 48:116–121. https://doi.org/10.1016/j.ijrmms.2010.11.004
Zhou HW, Wang CP, Mishnaevsky L, Duan ZQ, Ding JY (2013) A fractional derivative approach to full creep regions in salt rock. Mech Time Dependent Mater 17:413–425. https://doi.org/10.1007/s11043-012-9193-x7
Zhuang Y, Bai ZL, Xu YF (2011) Research on parameters of support vector machine based on ant colony algorithm computer simulation. 28:216–219. https://doi.org/10.1016/j.cageo.2010.07.006
Acknowledgments
This work was supported by the State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology (grant no. SKLGDUEK1914), the State Key Program of National Natural Science of China (grant no. 51734009), the National Natural Science Foundation of China (grant no. 51704144), and Liaoning Natural Science Foundation Guidance Project (grant no. 20180551162). The authors gratefully acknowledge the helpful comments made by the reviewers.
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Liu, J., Jing, H., Meng, B. et al. A four-element fractional creep model of weakly cemented soft rock. Bull Eng Geol Environ 79, 5569–5584 (2020). https://doi.org/10.1007/s10064-020-01869-w
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DOI: https://doi.org/10.1007/s10064-020-01869-w