Abstract
Crack propagation of the rock specimen containing a central closed flaw at different end friction coefficients (µ) under uniaxial compression and biaxial unloading confining pressure has been modeled under different loading conditions in this study employing a strength-based localized maximum stress (SLMS) criterion. The contact problem at the ends of specimens is solved via an augmented Lagrangian method. The effects of µ on tensile and shear crack propagation paths and lateral deformation of specimen ends are evaluated. Based on the variation of contact state at specimen ends under different loading and end conditions, the evolution of kinetic and static friction distribution at the specimen ends during crack propagation and the influence of µ on the kinetic and static friction distribution is investigated. The results show that the contact state at the specimen ends varies with different µ and can be classified into three types: complete sliding (µ = 0), sliding-sticking mixed (0 < µ < 0.3), and complete sticking (µ ≥ 0.3). The contact state of the end surfaces varies dynamically as the crack propagates when 0 < µ < 0.3. The friction on the end surfaces is kinetic-static mixed when 0 < µ < 0.3, while it is complete static friction when µ ≥ 0.3. For 0 < µ < 0.3, the static friction section in the end surfaces gradually increases with the crack propagation, which is more significant under biaxial unloading confining pressure due to the propagation of shear cracks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Chalabi M, Huang CL (1974) Stress distribution within circular cylinders in compression. Int J Rock Mech Min Sci Geomech Abstracts 11(2):45–56. https://doi.org/10.1016/0148-9062(74)92648-5
Aubertin M, Li L, Simon R (2000) A multiaxial stress criterion for short- and long-term strength of isotropic rock media. Int J Rock Mech Min Sci 37(8):1169–1193. https://doi.org/10.1016/S1365-1609(00)00047-2
Bai SW, Ling SS, Zhu WS (1982) Uniformity of stress distributions in rock cylinder specimen under uniaxial compression. Chin J Geotech Eng 4(4):193–203
Bathe K-J, Chaudhary A (1985) A solution method for planar and axisymmetric contact problems. Int J Numer Methods Eng 21(1):65–88. https://doi.org/10.1002/nme.1620210107
Bieniawski ZT, Bernede MJ (1979) Suggested methods for determining the uniaxial compressive strength and deformability of rock materials: part 1. Suggested method for determining deformability of rock materials in uniaxial compression. Int J Rock Mech Min Sci Geomech Abstracts 16(2):138–140. https://doi.org/10.1016/0148-9062(79)91451-7
Bobet A (2000) The initiation of secondary cracks in compression. Eng Fract Mech 66(2):187–219. https://doi.org/10.1016/S0013-7944(00)00009-6
Böhm J (1987) A comparison of different contact algorithms with applications. Comput Struct 26(1):207–221. https://doi.org/10.1016/0045-7949(87)90251-3
Brady BT (1971) Effects of inserts on the elastic behavior of cylindrical materials loaded between rough end-plates. Int J Rock Mech Min Sci Geomech Abstracts 8(4):357–369. https://doi.org/10.1016/0148-9062(71)90047-7
Brideau M-A, Yan M, Stead D (2009) The role of tectonic damage and brittle rock fracture in the development of large rock slope failures. Geomorphology 103(1):30–49. https://doi.org/10.1016/j.geomorph.2008.04.010
Brown E, Gonano L (1974) Improved compression test technique for soft rock. J Geotech GeoEnviron Eng 100:196–199
Carpenter NJ, Taylor RL, Katona MG (1991) Lagrange constraints for transient finite element surface contact. Int J Numer Methods Eng 32(1):103–128. https://doi.org/10.1002/nme.1620320107
Chan SK, Tuba IS (1971) A finite element method for contact problems of solid bodies—part I. Theory and validation. Int J Mech Sci 13(7):615–625. https://doi.org/10.1016/0020-7403(71)90032-4
Chang C, Haimson B (2000) True triaxial strength and deformability of the German Continental Deep Drilling Program (KTB) deep hole amphibolite. J Geophys Research: Solid Earth 105(B8):18999–19013. https://doi.org/10.1029/2000JB900184
Che Y, Song Y, Yang H, Guo X (2024) Creep properties and model of fractured sandstone under freezing Environment. Geomechanics for Energy and the Environmenthttps://doi.org/10.1016/j.gete.2024.100554. 100554
Darlington WJ, Ranjith PG, Choi SK (2011) The Effect of Specimen size on strength and other Properties in Laboratory Testing of Rock and Rock-Like Cementitious Brittle materials. Rock Mech Rock Eng 44(5):513–529. https://doi.org/10.1007/s00603-011-0161-6
Eftekhari M, Baghbanan A, Hashemolhosseini H (2016) Crack propagation in rock specimen under compressive loading using extended finite element method. Arab J Geosci 9(2):145. https://doi.org/10.1007/s12517-015-2196-6
Erdogan F, Sih G (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Engineering-ASME 85:519–527
Fairhurst C, Hudson J (1999) Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. Int J Rock Mech Min Sci 36:279–289
Feng X-T, Zhang X, Yang C, Kong R, Liu X, Peng S (2017) Evaluation and reduction of the end friction effect in true triaxial tests on hard rocks. Int J Rock Mech Min Sci 97:144–148. https://doi.org/10.1016/j.ijrmms.2017.04.002
Feng X-T, Haimson B, Li X, Chang C, Ma X, Zhang X, Suzuki K (2019) ISRM suggested Method: determining deformation and failure characteristics of rocks subjected to true Triaxial Compression. Rock Mech Rock Eng 52(6):2011–2020. https://doi.org/10.1007/s00603-019-01782-z
Ferrero AM, Migliazza MR (2009) Theoretical and numerical study on uniaxial compressive behaviour of marl. Mech Mater 41(5):561–572. https://doi.org/10.1016/j.mechmat.2009.01.011
Filon L (1902) On the elastic equilibrium of circular cylinders under certain practical systems of load. A198:147–233
Haimson B, Chang C (2000) A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. Int J Rock Mech Min Sci 37(1):285–296. https://doi.org/10.1016/S1365-1609(99)00106-9
Hao Y, Hao H, Li ZX (2013) Influence of end friction confinement on impact tests of concrete material at high strain rate. Int J Impact Eng 60:82–106. https://doi.org/10.1016/j.ijimpeng.2013.04.008
Hawkins AB (1998) Aspects of rock strength. Bull Eng Geol Environ 57(1):17–30. https://doi.org/10.1007/s100640050017
He M, Zhang Z, Zhu J, Li N, Li G, Chen Y (2021) Correlation between the rockburst proneness and friction characteristics of rock materials and a new method for rockburst proneness prediction: field demonstration. J Petrol Sci Eng 205:108997. https://doi.org/10.1016/j.petrol.2021.108997
Hoek E, Brown ET (1980) Underground excavations in rock. Spon, London
Horii H, Nemat-Nasser S (1985) Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J Geophys Research: Solid Earth 90(B4):3105–3125. https://doi.org/10.1029/JB090iB04p03105
Ingraham MD, Issen KA, Holcomb DJ (2013) Response of Castlegate sandstone to true triaxial states of stress. J Geophys Research: Solid Earth 118(2):536–552. https://doi.org/10.1002/jgrb.50084
International A (2014) ASTM D7012-14 - standard test methods for compressive strength and elastic moduli of intact rock core specimens under varying states of stress and temperatures. ASTM International, In. West Conshohocken, Pennsylvania
Jin L, Xu C, Han Y, Du X (2016) Effect of end friction on the dynamic compressive mechanical behavior of concrete under medium and low strain rates. Shock Vib 2016(6309073). https://doi.org/10.1155/2016/6309073
Labuz JF, Bridell JM (1993) Reducing frictional constraint in compression testing through lubrication. Int J Rock Mech Min Sci Geomech Abstracts 30(4):451–455. https://doi.org/10.1016/0148-9062(93)91726-Y
Lee H, Jeon S (2011) An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression. Int J Solids Struct 48(6):979–999. https://doi.org/10.1016/j.ijsolstr.2010.12.001
Li Y, Zhou H, Zhu W, Li S, Liu J (2016) Experimental and numerical investigations on the shear behavior of a jointed rock mass. Geosci J 20(3):371–379. https://doi.org/10.1007/s12303-015-0052-z
Li G, Feng S, Zhao J, Tang C (2024a) Modeling failure and instability of a single-jointed rock column based on finite deformation theory. Comput Geotech 165:105948. https://doi.org/10.1016/j.compgeo.2023.105948
Li J, Wang Q, Zan Y, Ju L, Jing C, Zhang Y (2024b) Research on wing crack propagation of closed crack under uniaxial compression based on peridynamics. Eng Anal Boundary Elem 158:121–138. https://doi.org/10.1016/j.enganabound.2023.10.015
Liang ZZ, Wu XK, Tang SB, Wnag WQ (2018) Numerical Silulatuon on End Effect of Rock specimens based on the anisotropic interface element Model. J Basic Sci Eng 26:526–537
Lin Q, Cao P, Wen G, Meng J, Cao R, Zhao Z (2021) Crack coalescence in rock-like specimens with two dissimilar layers and pre-existing double parallel joints under uniaxial compression. Int J Rock Mech Min Sci 139:104621. https://doi.org/10.1016/j.ijrmms.2021.104621
Liu J, Bai X, Elsworth D (2023) Evolution of pore systems in low-maturity oil shales during thermal upgrading —quantified by dynamic SEM and machine learning. Pet Sci. https://doi.org/10.1016/j.petsci.2023.12.021
Masoumi H, Roshan H, Hagan Paul C (2017) Size-dependent Hoek-Brown failure Criterion. Int J Geomech 17(2):04016048. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000706
Mijar AR, Arora JS (2000) Review of formulations for elastostatic frictional contact problems. Struct Multidisciplinary Optim 20(3):167–189. https://doi.org/10.1007/s001580050147
Mogi K (2007) Experimental rock mechanics. Taylor and Francis, London
Munoz H, Taheri A, Chanda EK (2016) Rock Mech Rock Eng 49(7):2541–2554. https://doi.org/10.1007/s00603-016-0935-y. Pre-Peak and Post-Peak Rock Strain Characteristics During Uniaxial Compression by 3D Digital Image Correlation
Nejati M, Bahrami B, Ayatollahi MR, Driesner T (2021) On the anisotropy of shear fracture toughness in rocks. Theoret Appl Fract Mech 113:102946. https://doi.org/10.1016/j.tafmec.2021.102946
Ning Y, Zhang G, Tang H, Shen W, Shen P (2019) Process analysis of toppling failure on Anti-dip Rock slopes under seismic load in Southwest China. Rock Mech Rock Eng 52(11):4439–4455. https://doi.org/10.1007/s00603-019-01855-z
Pan P-Z, Feng X-T, Hudson JA (2012) The influence of the intermediate principal stress on rock failure behaviour: a numerical study. Eng Geol 124:109–118. https://doi.org/10.1016/j.enggeo.2011.10.008
Pellegrino A, Sulem J, Barla G (1997) The effects of slenderness and lubrication on the uniaxial behavior of a soft limestone. Int J Rock Mech Min Sci 34(2):333–340. https://doi.org/10.1016/S0148-9062(96)00031-9
Peng SD (1971) Stresses within elastic circular cylinders loaded uniaxially and triaxially. Int J Rock Mech Min Sci Geomech Abstracts 8(5):399–432. https://doi.org/10.1016/1365-1609(71)90009-8
Rao Q, Sun Z, Stephansson O, Li C, Stillborg B (2003) Shear fracture (Mode II) of brittle rock. Int J Rock Mech Min Sci 40(3):355–375. https://doi.org/10.1016/S1365-1609(03)00003-0
Shen B, Stephansson O, Einstein HH, Ghahreman B (1995) Coalescence of fractures under shear stresses in experiments. J Geophys Research: Solid Earth 100(B4):5975–5990. https://doi.org/10.1029/95JB00040
Shi L, Li XC (2009) Analysis of end friction effect in true triaxial test. Rock Soil Mech 30:1159–1164
Simo JC, Laursen TA (1992) An augmented lagrangian treatment of contact problems involving friction. Comput Struct 42(1):97–116. https://doi.org/10.1016/0045-7949(92)90540-G
Szczepanik Z, Milne D, Hawkes C (2007) The confining effect of end roughness on unconfined compressive strength. Paper presented at the 1st Canada/United States Rock Mechanics Symposium, Vancouver, Canada
Tang CA, Tham LG, Lee PKK, Tsui Y, Liu H (2000) Numerical studies of the influence of microstructure on rock failure in uniaxial compression — part II: constraint, slenderness and size effect. Int J Rock Mech Min Sci 37(4):571–583. https://doi.org/10.1016/S1365-1609(99)00122-7
Tang S, Zhang L, Wang Q, Sun K, Li J, Ding S (2023) Numerical modeling of crack propagation from open and closed flaws in rock. Theoret Appl Fract Mech 128:104157. https://doi.org/10.1016/j.tafmec.2023.104157
Tang S, Wang J, Tang L, Ding S (2024) Automatic early warning of rockbursts from microseismic events by learning the feature-augmented point cloud representation. Tunn Undergr Space Technol 147:105692. https://doi.org/10.1016/j.tust.2024.105692
Wang L-F, Zhou X-P (2021) A field-enriched finite element method for simulating the failure process of rocks with different defects. Comput Struct 250:106539. https://doi.org/10.1016/j.compstruc.2021.106539
Wang Y, Zhou X, Xu X (2016) Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng Fract Mech 163:248–273. https://doi.org/10.1016/j.engfracmech.2016.06.013
Wang H, Dyskin A, Dight P, Pasternak E, Hsieh A (2020a) Review of unloading tests of dynamic rock failure in compression. Eng Fract Mech 225:106289. https://doi.org/10.1016/j.engfracmech.2018.12.022
Wang Y, Zhang H, Lin H, Zhao Y, Liu Y (2020b) Fracture behaviour of central-flawed rock plate under uniaxial compression. Theoret Appl Fract Mech 106:102503. https://doi.org/10.1016/j.tafmec.2020.102503
Wang H, Dyskin A, Pasternak E, Dight P (2022) Possible mechanism of spallation in rock samples under uniaxial compression. Eng Fract Mech 269:108577. https://doi.org/10.1016/j.engfracmech.2022.108577
Wu Z, Wong LNY (2012) Frictional crack initiation and propagation analysis using the numerical manifold method. Comput Geotech 39:38–53. https://doi.org/10.1016/j.compgeo.2011.08.011
Xie Y, Cao P, Liu J, Dong L (2016) Influence of crack surface friction on crack initiation and propagation: a numerical investigation based on extended finite element method. Comput Geotech 74:1–14. https://doi.org/10.1016/j.compgeo.2015.12.013
Xing Y, Kulatilake PHSW, Sandbak LA (2018) Effect of rock mass and discontinuity mechanical properties and delayed rock supporting on tunnel stability in an underground mine. Eng Geol 238:62–75. https://doi.org/10.1016/j.enggeo.2018.03.010
Xu YH, Cai M, Zhang XW, Feng XT (2017) Influence of end effect on rock strength in true triaxial compression test. Can Geotech J 54(6):862–880. https://doi.org/10.1139/cgj-2016-0393
Yang SQ, Su CD, Xu WY (2005) Experimental and theoretical study of size effect of rock material. Eng Mech 22(4):112–118
Yang H, Lin H, Chen Y, Wang Y, Zhao Y, Yong W, Gao F (2022) Influence of wing crack propagation on the failure process and strength of fractured specimens. Bull Eng Geol Environ 81(1):71. https://doi.org/10.1007/s10064-021-02550-6
Yu W, Jin L, Du X (2023) Experiment and meso-scale modelling on combined effects of strain rate and specimen size on uniaxial-compressive failures of concrete. Int J Damage Mech 32(5):683–714. https://doi.org/10.1177/10567895231160811
Yun X, Mitri HS, Yang X, Wang Y (2010) Experimental investigation into biaxial compressive strength of granite. Int J Rock Mech Min Sci 47(2):334–341. https://doi.org/10.1016/j.ijrmms.2009.11.004
Zhai H, Masoumi H, Zoorabadi M, Canbulat I (2020) Size-dependent behaviour of weak Intact rocks. Rock Mech Rock Eng 53(8):3563–3587. https://doi.org/10.1007/s00603-020-02117-z
Zhang L, Tang S, Li J, Sun K, Wang Q, Ding S (2024) Numerical study on the failure evolution of rock slopes containing multi-flaws by strength-based fracture method. Eng Fail Anal 157:107924. https://doi.org/10.1016/j.engfailanal.2023.107924
Zhao Y, Wang Y, Wang W, Tang L, Liu Q, Cheng G (2019) Modeling of rheological fracture behavior of rock cracks subjected to hydraulic pressure and far field stresses. Theoret Appl Fract Mech 101:59–66. https://doi.org/10.1016/j.tafmec.2019.01.026
Zhu Q, Li D, Han Z, Li X, Zhou Z (2019) Mechanical properties and fracture evolution of sandstone specimens containing different inclusions under uniaxial compression. Int J Rock Mech Min Sci 115:33–47. https://doi.org/10.1016/j.ijrmms.2019.01.010
Acknowledgements
This study was financially supported by the National Natural Science Foundation of China (Grant Nos. 51874065 and U1903112).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflicts of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, L., Tang, S. End friction and its effect on crack propagation in fractured rock specimens. Bull Eng Geol Environ 83, 223 (2024). https://doi.org/10.1007/s10064-024-03719-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10064-024-03719-5