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Investigation of Macroscopic Brittle Creep Failure Caused by Microcrack Growth Under Step Loading and Unloading in Rocks

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Abstract

The growth of subcritical cracks plays an important role in the creep of brittle rock. The stress path has a great influence on creep properties. A micromechanics-based model is presented to study the effect of the stress path on creep properties. The microcrack model of Ashby and Sammis, Charles’ Law, and a new micro–macro relation are employed in our model. This new micro–macro relation is proposed by using the correlation between the micromechanical and macroscopic definition of damage. A stress path function is also introduced by the relationship between stress and time. Theoretical expressions of the stress–strain relationship and creep behavior are derived. The effects of confining pressure on the stress–strain relationship are studied. Crack initiation stress and peak stress are achieved under different confining pressures. The applied constant stress that could cause creep behavior is predicted. Creep properties are studied under the step loading of axial stress or the unloading of confining pressure. Rationality of the micromechanics-based model is verified by the experimental results of Jinping marble. Furthermore, the effects of model parameters and the unloading rate of confining pressure on creep behavior are analyzed. The coupling effect of step axial stress and confining pressure on creep failure is also discussed. The results provide implications on the deformation behavior and time-delayed rockburst mechanism caused by microcrack growth on surrounding rocks during deep underground excavations.

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Abbreviations

a :

Radius of initial penny-shaped microcracks

F w :

Wedge force acting on initial microcrack

D o :

Initial damage

D :

Damage

l :

Wing crack length

l o :

Initial equilibrium crack length

K I :

Stress intensity factor for mode I crack

K IC :

Fracture toughness for mode I crack

m :

Material constant

n :

Stress erosion index

N V :

Number of pre-existing cracks per unit volume

t :

Time

v :

Characteristic crack velocity

α :

Cosine value of angle φ

β :

Constant

ε :

Axial strain

ε e :

Axial elastic strain

ε o :

Material constant

μ :

Friction coefficient between microcrack interface

σ 1 :

Axial stress

σ 1i :

Initial value for step loading of axial stress

σ 1c :

Critical stress of crack extension

σ 1peak :

Peak stress corresponding to short-term strength

σ 3 :

Confining pressure

σ 3i :

Initial value for step unloading of confining pressure

\(\sigma_{3}^{i}\) :

Internal stress between two wing crack tips

σ n :

Normal stress on microcrack plane

Δσ :

Step stress value

τ :

Shear stress on microcrack plane

φ :

Angle between each initial microcrack and maximum principal stress σ 1

References

  • Ashby MF, Hallam SD (1986) The failure of brittle solids containing small cracks under compressive stress states. Acta Metall 34(3):497–510

    Article  Google Scholar 

  • Ashby MF, Sammis CG (1990) The damage mechanics of brittle solids in compression. Pure Appl Geophys 133(3):489–521

    Article  Google Scholar 

  • Atkinson BK (1979) A fracture mechanics study of subcritical tensile cracking of quartz in wet environments. Pure Appl Geophys 117(5):1011–1024

    Article  Google Scholar 

  • Atkinson BK (1984) Subcritical crack growth in geological materials. J Geophys Res 89(B6):4077–4114

    Article  Google Scholar 

  • Aydan Ö, Ito T, Özbay U, Kwasniewski M, Shariar K, Okuno T, Özgenoğlu A, Malan DF, Okada T (2014) ISRM suggested methods for determining the creep characteristics of rock. Rock Mech Rock Eng 47(1):275–290

    Article  Google Scholar 

  • Baud P, Meredith PG (1997) Damage accumulation during triaxial creep of darley dale sandstone from pore volumometry and acoustic emission. Int J Rock Mech Min Sci 34(3–4):24.e1–24.e10

    Google Scholar 

  • Bellenger E, Bussy P (2001) Phenomenological modeling and numerical simulation of different modes of creep damage evolution. Int J Solids Struct 38(4):577–604

    Article  Google Scholar 

  • Bhat HS, Sammis CG, Rosakis AJ (2011) The micromechanics of Westerley granite at large compressive loads. Pure Appl Geophys 168(12):2181–2198

    Article  Google Scholar 

  • Boukharov GN, Chanda MW, Boukharov NG (1995) The three processes of brittle crystalline rock creep. Int J Rock Mech Min Geomech Abstr 32(4):325–335

    Article  Google Scholar 

  • Brantut N, Baud P, Heap MJ, Meredith PG (2012) Micromechanics of brittle creep in rocks. J Geophys Res 117:B08412

    Article  Google Scholar 

  • Bristow JR (1960) Microcracks, and the static and dynamic elastic constants of annealed and heavily cold-worked metals. Br J Appl Phys 11(2):81–85

    Article  Google Scholar 

  • Challamel N, Lanos C, Casandjian C (2005) Creep damage modelling for quasi-brittle materials. Eur J Mech A Solids 24(4):593–613

    Article  Google Scholar 

  • Chandler NA (2013) Quantifying long-term strength and rock damage properties from plots of shear strain versus volume strain. Int J Rock Mech Min Sci 59:105–110

    Google Scholar 

  • Charles RJ (1958) Static fatigue of glass. I. J Appl Phys 29(11):1549–1553

    Article  Google Scholar 

  • Chen ZH, Tang CA, Huang RQ (1997) A double rock sample model for rockbursts. Int J Rock Mech Min Sci 34(6):991–1000

    Article  Google Scholar 

  • Costin LS (1985) Damage mechanics in the post-failure regime. Mech Mat 4(2):149–160

    Article  Google Scholar 

  • Cruden DM (1970) A theory of brittle creep in rock under uniaxial compression. J Geophys Res 75(17):3431–3442

    Article  Google Scholar 

  • Damjanac B, Fairhurst C (2010) Evidence for a long-term strength threshold in crystalline rock. Rock Mech Rock Eng 43(5):513–531

    Article  Google Scholar 

  • Deshpande VS, Evans AG (2008) Inelastic deformation and energy dissipation in ceramics: a mechanism-based constitutive model. J Mech Phys Solids 56(10):3077–3100

    Article  Google Scholar 

  • Eslami J, Hoxha D, Grgic D (2012) Estimation of the damage of a porous limestone using continuous wave velocity measurements during uniaxial creep tests. Mech Mater 49:51–65

    Article  Google Scholar 

  • Evans AG (1972) A method for evaluating the time-dependent failure characteristics of brittle materials—and its application to polycrystalline alumina. J Mater Sci 7(10):1137–1146

    Article  Google Scholar 

  • Fabre G, Pellet F (2006) Creep and time-dependent damage in argillaceous rocks. Int J Rock Mech Min Sci 43(6):950–960

    Article  Google Scholar 

  • Grgic D, Amitrano D (2009) Creep of a porous rock and associated acoustic emission under different hydrous conditions. J Geophys Res 114:B10201

    Article  Google Scholar 

  • He MC, Miao JL, Feng JL (2010) Rock burst process of limestone and its acoustic emission characteristics under true-triaxial unloading conditions. Int J Rock Mech Min Sci 47(2):286–298

    Article  Google Scholar 

  • He MC, Nie W, Zhao ZY, Guo W (2012) Experimental investigation of bedding plane orientation on the rockburst behavior of sandstone. Rock Mech Rock Eng 45(3):311–326

    Article  Google Scholar 

  • Heap MJ, Baud P, Meredith PG, Bell AF, Main IG (2009) Time-dependent brittle creep in Darley Dale sandstone. J Geophys Res 114:B07203

    Article  Google Scholar 

  • Heap MJ, Baud P, Meredith PG, Vinciguerra S, Bell AF, Main IG (2011) Brittle creep in basalt and its application to time-dependent volcano deformation. Earth Planet Sci Lett 307(1–2):71–82

    Article  Google Scholar 

  • Jin F, Zhang CH, ASCE M, Wang G, Wang GL (2003) Creep modeling in excavation analysis of a high rock slope. J Geotech Geoenviron Eng 129(9):849–857

    Article  Google Scholar 

  • Johnson LR, Sammis CG (2001) Effects of rock damage on seismic waves generated by explosions. Pure Appl Geophys 158(11):1869–1908

    Article  Google Scholar 

  • Kachanov ML (1982) A microcrack model of rock inelasticity part II: propagation of microcracks. Mech Mater 1(1):29–41

    Article  Google Scholar 

  • Kachanov LM (1986) Introduction to continuum damage mechanics. Martinus Nijhoff, Dordrecht

    Book  Google Scholar 

  • Kostrov VV (1974) Seismic moment and energy of earthquakes, and seismic flow of rock. Izv Acad Sci USSR Phys Solid Earth 1:23–44

    Google Scholar 

  • Krajcinovic D (1983) Constitutive equations for damaging materials. J Appl Mech 50(2):355–360

    Article  Google Scholar 

  • Kranz RL (1979) Crack growth and development during creep of Barre granite. Int J Rock Mech Min Sci Geomech Abstr 16(1):23–35

    Article  Google Scholar 

  • Lacroix P, Amitrano D (2013) Long-term dynamics of rockslides and damage propagation inferred from mechanical modeling. J Geophys Res 118(4):2292–2307

    Article  Google Scholar 

  • Lemaitre J, Plumtree A (1979) Application of damage concepts to predict creep-fatigue failures. J Eng Mater Technol 101(3):284–292

    Article  Google Scholar 

  • Liu GT, Gao H, Chen FQ (2002) Microstudy on creep of concrete at early age under biaxial compression. Cem Concr Res 32(12):1865–1870

    Article  Google Scholar 

  • Lockner D, Byerlee J (1977) Acoustic emission and creep in rock at high confining pressure and differential stress. B Seismol Soc Am 67(2):247–258

    Google Scholar 

  • Ma L, Daemen JJK (2006) An experimental study on creep of welded tuff. Int J Rock Mech Min Sci 43(2):282–291

    Article  Google Scholar 

  • Mallet C, Fortin J, Guéguen Y, Bouyer F (2014) Evolution of the crack network in glass samples submitted to brittle creep conditions. Int J Fract 190(1–2):111–124

    Article  Google Scholar 

  • Mallet C, Fortin J, Guéguen Y, Bouyer F (2015) Brittle creep and subcritical crack propagation in glass submitted to triaxial conditions. J Geophys Res 120(2):879–893

    Article  Google Scholar 

  • Martin CD, Chandler NA (1994) The progressive fracture of Lac du Bonnet granite. Int J Rock Mech Min Sci Geomech Abstr 31(6):643–659

    Article  Google Scholar 

  • Miura K, Okui Y, Horii H (2003) Micromechanics-based prediction of creep failure of hard rock for long-term safety of high-level radioactive waste disposal system. Mech Mater 35(3):587–601

    Article  Google Scholar 

  • Munson DE (1997) Constitutive model of creep in rock salt applied to underground room closure. Int J Rock Mech Min Sci 34(2):233–247

    Article  Google Scholar 

  • Nara Y, Takada M, Mori D, Owada H, Yoneda T, Kaneko K (2010) Subcritical crack growth and long-term strength in rock and cementitious material. Int J Fract 164(1):57–71

    Article  Google Scholar 

  • Nara Y, Yamanaka H, Oe Y, Kaneko K (2013) Influence of temperature and water on subcritical crack growth parameters and long-term strength for igneous rocks. Geophys J Int 193(1):47–60

    Article  Google Scholar 

  • Ohnaka M (1983) Acoustic emission during creep of brittle rock. Int J Rock Mech Min Sci Geomech Abstr 20(3):121–134

    Article  Google Scholar 

  • Ranaivomanana N, Multon S, Turatsinze A (2013) Tensile, compressive and flexural basic creep of concrete at different stress levels. Cem Concr Res 52:1–10

    Article  Google Scholar 

  • Rice JR (1975) Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. Constitutive equations in plasticity. Massachusetts Institute of Technology Press, Cambridge, pp 23–79

    Google Scholar 

  • Scholz CH (1968) Mechanism of creep in brittle rock. J Geophys Res 73(10):3295–3302

    Article  Google Scholar 

  • She CX, Cui X (2010) Influence of high pore water pressure on creep properties of rock. Chin J Rock Mech Eng 29(8):1603–1609

    Google Scholar 

  • Singh DP (1975) A study of creep of rocks. Int J Rock Mech Min Sci Geomech Abstr 12(9):271–276

    Article  Google Scholar 

  • Walsh JB (1965a) The effect of cracks on the compressibility of rock. J Geophys Res 70(2):381–389

    Article  Google Scholar 

  • Walsh JB (1965b) The effect of cracks in rocks on Poisson’s ratio. J Geophys Res 70(20):5249–5257

    Article  Google Scholar 

  • Wan LH, Cao P, Huang YH, Wang YX (2010) Study on subcritical crack growth of rocks and threshold values in different environments. Chin J Rock Soil Mech 31(9):2737–2742

    Google Scholar 

  • Wang TJ, Lou ZW (1990) A continuum damage model for weld heat affected zone under low cycle fatigue loading. Eng Fract Mech 37(4):825–829

    Article  Google Scholar 

  • Wang B, Zhu JB, Wu AQ, Hu JM, Xiong ZM (2008) Experimental study on mechanical properties of jinping marble under loading and unloading stress paths. Chin J Rock Mech Eng 27(10):2138–2145

    Google Scholar 

  • Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18(2):293–297

    Google Scholar 

  • Xia CC, Yan ZJ, Wang XD, Zhang CS, Zhao X (2009) Research on elasto-viscoplastic constitutive relation of marble under unloading condition. Chin J Rock Mech Eng 28(3):459–466

    Google Scholar 

  • Yang C, Daemen JJK (1997) Temperature effects on creep of tuff and its time-dependent damage analysis. Int J Rock Mech Min Sci Geomech Abstr 34(3):383–384

    Article  Google Scholar 

  • Yang W, Zhang Q, Li S, Wang S (2014) Time-dependent behavior of diabase and a nonlinear creep model. Rock Mech Rock Eng 47(4):1211–1224

    Article  Google Scholar 

  • Zhang M, Li ZK, Su X (2005) Probabilistic volume element modeling in elastic damage analysis of quasi-brittle materials. Chin J Rock Mech Eng 24(23):4282–4288

    Google Scholar 

  • Zhang ZL, Xu WY, Wang W, Wang RB (2012) Triaxial creep tests of rock from the compressive zone of dam foundation in Xiangjiaba Hydropower Station. Int J Rock Mech Min Sci 50:133–139

    Article  Google Scholar 

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  1. School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Xiaozhao Li & Zhushan Shao

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  1. Xiaozhao Li
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  2. Zhushan Shao
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Li, X., Shao, Z. Investigation of Macroscopic Brittle Creep Failure Caused by Microcrack Growth Under Step Loading and Unloading in Rocks. Rock Mech Rock Eng 49, 2581–2593 (2016). https://doi.org/10.1007/s00603-016-0953-9

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