Abstract
Uniaxial/triaxial compression and scanning/backscattered electron microscopy (SEM/BSEM) tests have been conducted in this study to explore the hydro-mechanical (H-M) response and failure mechanism of basalt from Baihetan hydropower station in China. The experimental findings show that the characteristic strengths, deformation behavior, and failure modes of basalt are greatly influenced by confining and pore pressures. Observations from SEM/BSEM tests reveal that the failure mode is due to the development and growth of intergranular/transgranular microcracks. Under H-M loading conditions, the permeability evolution of basalt exhibits nonlinear and distinct characteristics. Additionally, the permeability evolution and failure mode of basalt under the influence of both confining and pore pressures have been discussed. To further describe the H-M response of rocks under loading, continuum damage mechanics theory, in conjunction with strain equivalence and effective stress principles, has been used to establish an entropy distribution-based probabilistic damage constitutive model. To validate the model, the theoretical results have been compared with those obtained from experiments for basalt and other quasi-brittle rock-like materials. The outcome of this study has implications for the engineering design and safety evaluation for structures that lie within rock masses vulnerable to H-M coupling disasters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- A and L :
-
Cross-sectional area and length of rock samples, respectively
- c :
-
Cohesion
- E :
-
Elastic modulus
- \({{\tilde E}^\prime }\) :
-
Elastic modulus of the undamaged segment of materials
- F :
-
Mohr-Coulomb failure criterion or extent of meso-element strength
- k :
-
Permeability
- k 0 :
-
Initial permeability
- k 1 :
-
Minimum permeability
- k p :
-
Peak permeability
- m :
-
Highest order of moments
- N and N f :
-
Total and failed quantities of meso-element units, respectively
- p and q :
-
Rock material constants
- P(F):
-
Probability density function
- p p :
-
Pore pressure
- p(x) and P(x) or H(x):
-
Discrete and continuous probability density functions, respectively
- Q :
-
Fluid-flow rate
- S(x):
-
Information entropy
- u i (i=0, 1, 2,…, m):
-
Moment function
- υ :
-
Poisson’s ratio
- \({{\tilde v}^\prime }\) :
-
Poisson’s ratio of the undamaged segment of materials
- x :
-
Random variable
- (i, j, k):
-
(1, 2, 3) (2, 3, 1) or (3, 1, 2)
- β :
-
Biot’s effective stress coefficient
- ΔP :
-
Fluid pressure difference
- δ ij :
-
Kronecker product coefficient
- ε 1, ε 2, and ε 3 :
-
Axial, intermediate principal, and lateral strains, respectively
- ε 1t :
-
Measured differential strain
- ε 10 :
-
Initial strain
- ε cv :
-
Crack volumetric strain
- ε ev :
-
Elastic volumetric strain
- ε v :
-
Volumetric strain
- \(\tilde \varepsilon _i^\prime \,(i = 1,\,\,2,\,\,3)\) :
-
Effective strain of the undamaged segment of materials
- ϕ :
-
Damage variable
- η :
-
Dynamic water viscosity
- φ :
-
Internal friction angle
- λ i (i=0, 1, 2,…, m):
-
Lagrangian multipliers
- μi (i=0, 1, 2,…, m):
-
Moment constant
- σ 1, σ 2, σ 3 :
-
Axial, intermediate principal, confining stresses, respectively
- (σ 1–σ 3) or σ 1t :
-
Deviatoric stress
- σ cc :
-
Closure stress
- σ ci :
-
Initiation stress
- σ cd :
-
Damage stress
- σ p :
-
Peak stress
- \(\tilde \sigma _1^\prime \) :
-
Effective differential stress
- \(\tilde \sigma _3^\prime \) :
-
Effective confining pressure
- \(\tilde \sigma _c^\prime \) :
-
Uniaxial compression strength
- \(\sigma _i^\prime \,(i = 1,2,3)\) and \(\tilde \sigma _i^\prime \,(i = 1,2,3)\) :
-
Macroscopic stress and effective stress in the undamaged segment, respectively
- ξ and ψ :
-
Permeability fitting constants
References
Barla G (1972) Rock anisotropy (L. Müller, Ed.), Springer Vienna, Vienna
Basu B, Tiwari D, Kundu D, Prasad R (2009) Is weibull distribution the most appropriate statistical strength distribution for brittle materials? Ceramics International 35(1):237–246, DOI: https://doi.org/10.1016/j.ceramint.2007.10.003
Brace WF, Paulding BW, Scholz C (1966) Dilatancy in the fracture of crystalline rocks. Journal of Geophysical Research 71(16):3939–3953, DOI: https://doi.org/10.1029/jz071i016p03939
Cao WG, Fang ZL, Tang X (1998) A study of statistical constitutive model for softening and damage rocks. Chinese Journal of Rock Mechanics and Engineering 17(6):628–633 (in Chinese)
Cao WG, Zhao H, Li X, Zhang YJ (2010) Statistical damage model with strain softening and hardening for rocks under the influence of voids and volume changes. Canadian Geotechnical Journal 47(8):857–871, DOI: https://doi.org/10.1139/T09-148
Carmeliet J (1995) Stochastic approaches for damage non-standard continua. International Journal of Solids and Structures 32(8–9):1149–1160, DOI: https://doi.org/10.1016/0020-7683(94)00182-V
Chaboche JL (1987) Continuum damage mechanics: Present state and future trends. Nuclear Engineering and Design 105(1):19–33, DOI: https://doi.org/10.1016/0029-5493(87)90225-1
Danzer R (2006) Some notes on the correlation between fracture and defect statistics: Are Weibull statistics valid for very small specimens? Journal of the European Ceramic Society 26(15):3043–3049, DOI: https://doi.org/10.1016/j.jeurceramsoc.2005.08.021
Danzer R, Supancic P, Pascual J, Lube T (2007) Fracture statistics of ceramics–Weibull statistics and deviations from Weibull statistics. Engineering Fracture Mechanics 74(18):2919–2932, DOI: https://doi.org/10.1016/j.engfracmech.2006.05.028
Deng J, Gu D (2011) On a statistical damage constitutive model for rock materials. Computers and Geosciences 37(2):122–128, DOI: https://doi.org/10.1016/j.cageo.2010.05.018
Fairhurst CE, Hudson JA (1999) Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. International Journal of Rock Mechanics and Mining Sciences 36(3):279–289, DOI: https://doi.org/10.1016/s0148-9062(99)00006-6
Frantziskonis G (1995) Heterogeneity and implicated surface effects: Statistical, fractal formulation and relevant analytical solution. Acta Mechanica 108(1–4):157–178, DOI: https://doi.org/10.1007/BF01177336
Grindrod PM, Heap MJ, Fortes AD, Meredith PG, Wood IG, Trippetta F, Sammonds PR (2010) Experimental investigation of the mechanical properties of synthetic magnesium sulfate hydrates: Implications for the strength of hydrated deposits on Mars. Journal of Geophysical Research 115(E6):1–15, DOI: https://doi.org/10.1029/2009JE003552
Han B, Xie SY, Shao JF (2016) Experimental investigation on mechanical behavior and permeability evolution of a porous limestone under compression. Rock Mechanics and Rock Engineering 49(9):3425–3435, DOI: https://doi.org/10.1007/s00603-016-1000-6
Hubbert K, Rubey W (1959) Role of fluid pressure in mechanics of overthrust faulting. Bulletin of the Geological Society of America 70(2):115–166, DOI: https://doi.org/10.1130/0016-7606(1959)70[115:ROFPIM]2.0.CO;2
Jia C, Zhang S, Xu W (2021) Experimental investigation and numerical modeling of coupled elastoplastic damage and permeability of saturated hard rock. Rock Mechanics and Rock Engineering 54(3):1151–1169, DOI: https://doi.org/10.1007/s00603-020-02319-5
Lee B, Rathnaweera TD (2016) Stress threshold identification of progressive fracturing in Bukit Timah granite under uniaxial and triaxial stress conditions. Geomechanics and Geophysics for Geo-Energy and Geo-Resources 2(4):301–330, DOI: https://doi.org/10.1007/s40948-016-0037-z
Lemaitre J, Chaboche J (1990) Mechanics of solid materials. Cambridge University Press, Cambridge, U.K.
Li X, Cao WG, Su YH (2012) A statistical damage constitutive model for softening behavior of rocks. Engineering Geology 143–144:1–17, DOI: https://doi.org/10.1016/j.enggeo.2012.05.005
Li Y, Jia D, Rui Z, Peng J, Fu C, Zhang J (2017) Evaluation method of rock brittleness based on statistical constitutive relations for rock damage. Journal of Petroleum Science and Engineering 153:123–132, DOI: https://doi.org/10.1016/j.petrol.2017.03.041
Li XF, Li HB, Liu LW, Liu YQ, Ju MH, Zhao J (2020) Investigating the crack initiation and propagation mechanism in brittle rocks using grain-based finite-discrete element method. International Journal of Rock Mechanics and Mining Sciences 127:1–20, DOI: https://doi.org/10.1016/j.ijrmms.2020.104219
Liu L, Li H, Li X (2022) A state-of-the-art review of mechanical characteristics and cracking processes of pre-cracked rocks under quasi-static compression. Journal of Rock Mechanics and Geotechnical Engineering 14(6):2034–2057, DOI: https://doi.org/10.1016/j.jrmge.2022.03.013
Liu Z, Wang H, Zhang C, Zhou B, Zhou H (2021) Size dependences of the mechanical behaviors of basalt rock blocks with hidden joints analyzed using a hybrid DFN-FDEM model. Engineering Fracture Mechanics 258:1–22, DOI: https://doi.org/10.1016/j.engfracmech.2021.108078
Liu Z, Zhang C, Zhang C, Gao Y, Zhou H, Chang Z (2019) Deformation and failure characteristics and fracture evolution of cryptocrystalline basalt. Journal of Rock Mechanics and Geotechnical Engineering 11(5):990–1003, DOI: https://doi.org/10.1016/j.jrmge.2019.04.005
Mead LR, Papanicolaou N (1984) Maximum entropy in the problem of moments Maximum entropy in the problem of moments. Journal of Mathematical Physics 25(8):2404–2417, DOI: https://doi.org/10.1063/1.526446
Mehranpour MH (2016) Comparison of six major intact rock failure criteria using a particle flow approach under true-triaxial stress condition. Geomechanics and Geophysics for Geo-Energy and Geo-Resources 2(4):203–229, DOI: https://doi.org/10.1007/s40948-016-0030-6
Murakami S (2012) Continuum damage mechanics - A continuum mechanics approach to the analysis of damage and fracture. Springer Dordrecht, London New York
Rutqvist J, Stephansson O (2003) The role of hydrochemical coupling in fractured rock engineering. Hydrogeology Journal 11(1):7–40, DOI: https://doi.org/10.1007/s10040-002-0241-5
Shannon CE (1948) A mathematical theory of communication. The Bell System Technical Journal 27:379–423, 623–656
Shi A, Li C, Hong W, Lu G, Zhou J, Li H (2022) Comparative analysis of deformation and failure mechanisms of underground powerhouses on the left and right banks of Baihetan hydropower station. Journal of Rock Mechanics and Geotechnical Engineering 14(3):731–745, DOI: https://doi.org/10.1016/j.jrmge.2021.09.012
Shi A, Wei Y, Wu J, Ren D, Tang M (2020) Study on the shear deformation of intralayer shear bands at the Baihetan hydropower station dam foundation. Bulletin of Engineering Geology and the Environment 79(7):3517–3532, DOI: https://doi.org/10.1007/s10064-020-01799-7
Tan T, Zhao Y, Zhao X, Chang L, Ren S (2022) Mechanical properties of sandstone under hydro-mechanical coupling. Applied Rheology 32:8–21, DOI: https://doi.org/10.1515/arh-2022-0120
Todinov MT (2009). Is Weibull distribution the correct model for predicting probability of failure initiated by non-interacting flaws? International Journal of Solids and Structures 46(3–4):887–901, DOI: https://doi.org/10.1016/j.ijsolstr.2008.09.033
Wan W, Li CC (2022) Microscopic and acoustic interpretations of the physics of rock burst and the difference in fracturing patterns in Class I and Class II rocks fast fourier transformation. Rock Mechanics and Rock Engineering 55(11):6841–6862, DOI: https://doi.org/10.1007/s00603-022-03015-2
Wang M, Li H, Han J, Xiao X, Zhou J (2019a) Large deformation evolution and failure mechanism analysis of the multi-freeface surrounding rock mass in the Baihetan underground powerhouse. Engineering Failure Analysis 100:214–226, DOI: https://doi.org/10.1016/j.engfailanal.2019.02.056
Wang HL, Xu WY (2013) Permeability evolution laws and equations during the course of deformation and failure of brittle rock. Journal of Engineering Mechanics 139(11):1621–1626, DOI: https://doi.org/10.1061/(ASCE)EM.1943-7889.0000608
Wang HL, Xu WY, Shao JF (2014) Experimental researches on hydro-mechanical properties of altered rock under confining pressures. Rock Mechanics and Rock Engineering 47(2):485–493, DOI: https://doi.org/10.1007/s00603-013-0439-y
Wang S, Xu W, Yang L (2019b) Experimental and elastoplastic model investigation on brittle-ductile transition and hydro-mechanical behaviors of cement mortar. Construction and Building Materials 224:19–28, DOI: https://doi.org/10.1016/j.conbuildmat.2019.07.046
Wang H, Zhao K, Qu X, Xu J, Cai M (2020) Hydro-mechanical properties of rock-like specimens with pre-existing intermittent joints. European Journal of Environmental and Civil Engineering 27(7):2542–2551, DOI: https://doi.org/10.1080/19648189.2020.1763853
Wei Y, Xu M, Wang W, Shi A, Tang M, Ye Z (2011) Feasibility of columnar jointed basalt used for high-arch dam foundation 2 Formation of columnar jointed basalt. Journal of Rock Mechanics and Geotechnical Engineering 3:461–468, DOI: https://doi.org/10.3724/SP.J.1235.2011.00461
Weibull W (1951) A statistical distribution function of wide applicability. Journal of applied mechanics 18(3):293–297, DOI: https://doi.org/10.1115/1.4010337
Wen J, Zhou C, Huang L, Cheng Y, Huang L, You F (2011) Study on statistical damage softening constitutive model and determination of parameters for rock based on lognormal distribution. Applied Mechanics and Materials 69:33–38, DOI: https://doi.org/10.4028/www.scientific.net/AMM.69.33
Wu A, Fan L, Fu X, Zhang Y, Zhong Z, Yu M (2021a) Design and application of hydro-mechanical coupling test system for simulating rock masses in high dam reservoir operations. International Journal of Rock Mechanics and Mining Sciences 140:1–18, DOI: https://doi.org/10.1016/j.ijrmms.2021.104638
Wu H, Liu J, Wang X, Liu L, Tian Z (2021b) In situ monitoring and numerical experiments on vertical deformation profiles of large-scale underground caverns in giant hydropower stations. Geofluids 2021:1–12, DOI: https://doi.org/10.1155/2021/6673825
Xia Y, Zhang C, Zhou H, Chen J, Gao Y, Liu N, Chen P (2020) Structural characteristics of columnar jointed basalt in drainage tunnel of Baihetan hydropower station and its influence on the behavior of P-wave anisotropy. Engineering Geology 264:1–17, DOI: https://doi.org/10.1016/j.enggeo.2019.105304
Xiangjun L, Jiankun S, Lixi L, Lin H, Hong L (2011) Effects of pore pressure on rock strength properties. Chinese Journal of Rock Mechanics and Engneering 30(s2):3457–3463 (in Chinese)
Xiao YX, Feng XT, Chen BR, Feng GL, Yao ZB, Hu LX (2017) Excavation-induced microseismicity in the columnar jointed basalt of an underground hydropower station. International Journal of Rock Mechanics and Mining Sciences 97:99–109, DOI: https://doi.org/10.1016/j.ijrmms.2017.04.012
Xiao YX, Feng XT, Feng GL, Liu HJ, Jiang Q, Qiu SL (2016) Mechanism of evolution of stress-structure controlled collapse of surrounding rock in caverns: A case study from the Baihetan hydropower station in China. Tunnelling and Underground Space Technology 51:56–67, DOI: https://doi.org/10.1016/j.tust.2015.10.020
Xue L, Qin S, Sun Q, Wang Y, Lee LM, Li W (2014) A study on crack damage stress thresholds of different rock types based on uniaxial compression tests. Rock Mechanics and Rock Engineering 47(4):1183–1195, DOI: https://doi.org/10.1007/s00603-013-0479-3
Yang B (2005) Stress analysis in two-dimensional problems. Elsevier Inc.
Yuan SC, Harrison JP (2006) A review of the state of the art in modelling progressive mechanical breakdown and associated fluid flow in intact heterogeneous rocks. International Journal of Rock Mechanics and Mining Sciences 43(7):1001–1022, DOI: https://doi.org/10.1016/j.ijrmms.2006.03.004
Zhang C, Liu Z, Pan Y, Gao Y, Zhou H, Cui G (2020) Influence of amygdale on crack evolution and failure behavior of basalt. Engineering Fracture Mechanics 226:1–22, DOI: https://doi.org/10.1016/j.engfracmech.2019.106843
Zhao C, Zhang Z, Lei Q (2021a) Role of hydro-mechanical coupling in excavation-induced damage propagation, fracture deformation and microseismicity evolution in naturally fractured rocks. Engineering Geology 289:1–18, DOI: https://doi.org/10.1016/j.enggeo.2021.106169
Zhao Y, Zhang L, Wang Y, Lin H (2021b) Editorial: Hydro-mechanical coupling and creep behaviors of geomaterials. Frontiers in Earth Science 8(January):2020–2022, DOI: https://doi.org/10.3389/feart.2020.632135
Acknowledgments
The authors acknowledge the financial support provided by the Major Science and Technology Special Projects in Yunnan Province (Grant No. 202102AF080001).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abubakar, A.F., Wang, H. & Wu, Y. Investigation of the Hydro-mechanical Response and Entropy-based Probabilistic Damage Constitutive Model of Basalt under Compression. KSCE J Civ Eng (2024). https://doi.org/10.1007/s12205-024-1874-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12205-024-1874-x