Time-Dependent Propagation of 3-D Cracks in Rocks Under Hydromechanical Coupling

  • Jie Mei
  • Lei YangEmail author
  • Xiangchao Sheng
  • Guangxiao Song
  • Weimin Yang
  • Bo Zhang
Original Paper


Crack propagation and rock failure under hydromechanical coupling have typical time-dependent characteristics, and the subcritical crack propagation is one of the most important causes of rock instability. Rheological tests based on mortar specimens containing single internal 3-D cracks and the corresponding numerical simulations are carried out to investigate the time-dependent characteristics of the crack propagation, the failure mode of rocks, and the effects of water pressure and crack dip angle. The mortar specimen exhibits a tensile-shear failure mode under the hydromechanical coupling as observed in rheological tests, and a macro-fracture penetrates the upper and lower ends of the specimen. The propagation rate of the crack decreases first and then increases. The crack propagation can be divided into three stages according to the relationship between the propagation rate and fracture parameters. The water pressure significantly reduces the time required for rock failure due to its promoting effect on the crack propagation rate, which has negative effects on the long-term stability of rocks. Cracks with a dip angle of 45° are more likely to cause rock failure under hydromechanical coupling, while cracks with larger dip angles exhibit a better long-term stability.


Hydromechanical coupling Subcritical crack Stress intensity factor Water pressure Dip angle 

List of Symbols


Half-length of crack long-axis


Half-length of crack short-axis


Crack initial length


Crack critical length


Stress corrosion index


Water pressure


Crack propagation rate of the crack front


Propagation rate of the point i


Propagation rate of the point corresponding to the median propagation length

A, A′

Endpoints of crack long-axis

B, B′

Endpoints of crack short-axis


Crack propagation constant


Diameter of the specimen


Energy release rate


Height of the specimen


Mixed-mode fracture toughness


Equivalent stress intensity factor


Stress intensity factor for Mode I, II, and III fracture


The components of the tensile, slip, and tear stresses related to the resolved Mode I, II, and III stress intensity factors


Threshold value for subcritical propagation

M, N

Measuring points along the wing crack front


Stress intensity factor


Crack propagation time


Crack dip angle


Crack deflection angle


Crack kink angle


Weight coefficients corresponding to Mode I, II, and III stress intensity factors


Weight coefficients corresponding to the components of tensile, slip, and tear stresses


Position angle of the crack front


Crack initiation stress


Crack propagation length at point i


The specified median propagation length


Time required for each crack propagation step



This research work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51509146, 51879148 and 51879151).

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Anderson OL, Grew PC (1977) Stress corrosion theory of crack propagation with applications to geophysics. Rev Geophys Space Phys 15(1):77–104. CrossRefGoogle Scholar
  2. Atkinson BK (1979) A fracture mechanics study of subcritical tensile cracking of quartz in wet environments. Pure Appl Geophys 117(5):1011–1024. CrossRefGoogle Scholar
  3. Atkinson BK (1987) Fracture mechanics of rock. Academic Press, LondonGoogle Scholar
  4. Atkinson BK, Meredith PG (1987) Experimental fracture mechanics data for rocks and minerals. Fract Mech Rock 8(1):477–525. CrossRefGoogle Scholar
  5. Brace WF, Paulding BW, Scholz C (1963) A note on brittle crack propagation in compression. J Geophys Res 68(12):3709–3713. CrossRefGoogle Scholar
  6. Chang SH, Lee CI, Jeon S (2002) Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 66(1):79–97. CrossRefGoogle Scholar
  7. Charles RJ (1958) Static fatigue of glass. J Appl Phys 29(11):1549–1560. CrossRefGoogle Scholar
  8. Chen W, Konietzky H (2014) Simulation of heterogeneity, creep, damage and lifetime for loaded brittle rocks. Tectonophysics 633(1):164–175. CrossRefGoogle Scholar
  9. Chen W, Konietzky H, Abbas SM (2015) Numerical simulation of time-independent and -dependent fracturing in sandstone. Eng Geol 193:118–131. CrossRefGoogle Scholar
  10. Christensen RM (1979) A rate-dependent criterion for crack propagation. Int J Fract 15(1):3–21. CrossRefGoogle Scholar
  11. Dyskin AV, Jewell RJ, Joer H, Joer H, Sahouryeh E, Ustinov KB (1994) Experiments on 3-D crack propagation in uniaxial compression. Int J Fract 65(4):R77–R83. CrossRefGoogle Scholar
  12. Dyskin AV, Sahouryeh E, Jewell RJ, Joer H, Ustinov KB (2003) Influence of shape and locations of initial 3-D cracks on their propagation in uniaxial compression. Eng Fract Mech 70(15):2115–2136. CrossRefGoogle Scholar
  13. Farmer I (1983) Engineering behavior of rocks. Spring Netherl. CrossRefGoogle Scholar
  14. Fu JW, Liu SL, Zhu WS, Zhou H, Sun ZC (2018) Experiments on failure process of new rock-like specimens with two internal cracks under biaxial loading and the 3-D simulation. Acta Geotechnol 13(4):853–867. CrossRefGoogle Scholar
  15. Heap MJ, Baud P, Meredith PG, Bell AF, Main IG (2009) Time-dependent brittle creep in Darley Dale sandstone. J Geophys Res Sol Ea 114(B7):1–22. CrossRefGoogle Scholar
  16. Irwin GR (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech T ASME 24:351–369Google Scholar
  17. Kirby SH (1984) Introduction and digest to the special issue on chemical effect of water on the deformation and strengths of rocks. J Geophys Res 89(B6):3991–3995. CrossRefGoogle Scholar
  18. Knauss WG (1970) Delayed failure - the Griffith problem for linearly viscoelastic materials. Int J Fracture 6(1):7–20. CrossRefGoogle Scholar
  19. Konietzky H, Heftenberger A, Feige M (2009) Life-time prediction for rocks under static compressive and tensile loads: a new simulation approach. Acta Geotechnol 4(1):73–78. CrossRefGoogle Scholar
  20. Lajtai EZ, Schmidtke RH, Bielus LP (1987) The effect of water on the time-dependent deformation and fracture of a granite. Int J Rock Mech Min Sci Geomech Abstr 24(4):247–255. CrossRefGoogle Scholar
  21. Lee JS (2007) Time-dependent crack propagation in brittle rocks and field applications to geologic hazards. Dissertations and Theses, GradworksGoogle Scholar
  22. Li X, Konietzky H (2014) Simulation of time-dependent crack propagation in brittle rocks under constant loading conditions. Eng Fract Mech 119(3):53–65. CrossRefGoogle Scholar
  23. Li FZ, Shih CF, Needleman A (1985) A comparison of methods for calculating energy release rates. Eng Fract Mech 21(2):405–421. CrossRefGoogle Scholar
  24. Li JT, Cao P, Gu DS, Wu C (2008) Crack propagation time dependence analysis of granite under compressive-shear stresses state. J Coal Sci Eng Chin 14(1):34–37. CrossRefGoogle Scholar
  25. Li SY, He TM, Yin XC (2010) Introduction of rock fracture mechanics. Press of University of Science and Technology of China, HefeiGoogle Scholar
  26. Li X, Konietzky H, Li XB (2016) Numerical study on time dependent and time independent fracturing processes for brittle rocks. Eng Fract Mech 163:89–107. CrossRefGoogle Scholar
  27. Liu JF, Wang L, Pei JL, Zheng L, Bian Y (2015) Experimental study on creep deformation and long-term strength of unloading-fractured marble. Eur J Environ Civ Eng 19(sup1):s97–s107. CrossRefGoogle Scholar
  28. Lockner DA (1993) Room temperature creep in saturated granite. J Geophys Res 98(B1):475–487. CrossRefGoogle Scholar
  29. Lu YL, Elsworth D, Wang LG (2013) Microcrack-based coupled damage and flow modeling of fracturing evolution in permeable brittle rocks. Comput Geotechnol 49:226–244. CrossRefGoogle Scholar
  30. Lu YL, Elsworth D, Wang LG (2014) A dual-scale approach to model time-dependent deformation, creep and fracturing of brittle rocks. Comput Geotechnol 60:61–76. CrossRefGoogle Scholar
  31. Martin CD, Christiansson R, Söderhäll J (2001) Rock stability considerations for siting and constructing a KBS-3 repository. University of Alberta, SKB TR-01-38, SwedenGoogle Scholar
  32. Masuda K (2001) Effects of water on rock strength in a brittle regime. J Struct Geol 23(11):1653–1657. CrossRefGoogle Scholar
  33. Nara Y, Takada M, Igarashi T, Hiroyoshi N, Kaneko K (2009) Subcritical crack propagation in rocks in an aqueous environment. Explor Geophys 62(1):163–171. CrossRefGoogle Scholar
  34. Nara Y, Yamanaka H, Oe Y, Kaneko K (2013) Influence of temperature and water on subcritical crack propagation parameters and long-term strength for igneous rocks. Geophys J Int 193(1):47–60. CrossRefGoogle Scholar
  35. Ojala IO (2003) Correlation of microseismic and chemical properties of brittle deformation in Locharbriggs sandstone. J Geophys Res 108(B5):2268. CrossRefGoogle Scholar
  36. Reiss CR, Taylor JS, Robert C, Nguyen BN, Onck PR, Giessen EV (2000) Crack-tip constraint effects on creep fracture. Eng Fract Mech 65(4):467–490. CrossRefGoogle Scholar
  37. Rinne M (2008) Fracture mechanics and subcritical crack propagation approach to model time-dependent failure in brittle rock (PhD dissertation). Helsinki University of TechnologyGoogle Scholar
  38. Swanson PL (1984) Subcritical crack propagation and other time- and environment- dependent behavior in crustal rocks. J Geophys Res Sol Earth 89(B6):4137–4152. CrossRefGoogle Scholar
  39. Tsang YW, Witherspoon PA (1981) Hydro-mechanical behavior of a deformable rock fracture subject to normal stress. J Geophys Res 86(B10):9187–9298. CrossRefGoogle Scholar
  40. Vitek V (1977) A theory of the initiation of creep crack propagation. Int J Fract 13(1):39–50. CrossRefGoogle Scholar
  41. Waza T, Kurita K, Mizutani H (1980) The effect of water on the subcritical crack propagation in silicate rocks. Tectonophys 67(1–2):25–34. CrossRefGoogle Scholar
  42. Yang SQ, Jing HW, Wang SY (2012) Experimental investigation on the strength, deformability, failure behavior and acoustic emission locations of red sandstone under triaxial compression. Rock Mech Rock Eng 45(4):583–606. CrossRefGoogle Scholar
  43. Yang L, Jiang YJ, Li SC, Li B (2013) Experimental and numerical research on 3D crack propagation in rocklike material subjected to uniaxial tension. J Geotech Geoenviron 139(10):1781–1788. CrossRefGoogle Scholar
  44. Zhao Y, Zhang L, Wang W, Wan W, Li SQ, Ma WH, Wang YX (2017) Creep behavior of intact and cracked limestone under multi-level loading and unloading cycles. Rock Mech Rock Eng 50(6):1409–1424. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  • Jie Mei
    • 1
  • Lei Yang
    • 1
    Email author
  • Xiangchao Sheng
    • 1
  • Guangxiao Song
    • 1
  • Weimin Yang
    • 1
  • Bo Zhang
    • 2
  1. 1.Geotechnical and Structural Engineering Research Center, Shandong UniversityJinanChina
  2. 2.School of Civil EngineeringShandong UniversityJinanChina

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