Skip to main content
SpringerLink
Log in

Analytical Model of Layered Rock Considering Its Time-Dependent Behaviour

  • Technical Note
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Abbreviations

\(\Delta u, \, \tau , \, \gamma\) :

Shear deformation, shear stress and shear strain of the bedding planes

\(J(t)\), \(\overline{J(t)}\) :

Creep compliance of the bedding planes and layered rock mass

\(\varepsilon^{Tot}_{ij} , \, \varepsilon^{e}_{ij} , \, \varepsilon^{c}_{ij}\) :

Total strain, elastic strain, and creep strain term of the layered rock mass

\(\delta_{ij}\) :

Kronecker’s delta

\(C_{ij}\) :

Inverse of the stiffness matrix

\(F_{ij} , \, F_{ijkl}\) :

Second- and fourth-rank tensors of the bedding planes

\(k_{s} , \, k_{n}\) :

Non-dimensional parameters related to the normal and shear stiffnesses

\(K_{n} , \, K_{s}\) :

Normal and shear stiffnesses of the bedding planes

\(R_{0}\) :

Stiffness ratio of the normal stiffness to the shear stiffness

\(\theta , \, N, \, s^{c} , \, r^{c}\) :

Angle, number, spacing, and length of the bedding planes

\(G_{0} , \, G^{c}\) :

Shear modulus of the bedding planes

\(E_{m} , \, \nu_{m}\) :

Elastic modulus and Poisson’s ratio of the intact rock mass

\(\eta_{{1}}^{c} , \, \eta_{{2}}^{c}\) :

Coefficients of the shear viscosity of the bedding planes

\(E_{{0}} ,E,\eta_{{1}} , \, \eta_{{2}}\) :

Elastic and creep parameters of layered rock

\(\varepsilon_{11} , \, \varepsilon_{22} , \, \gamma_{12}\) :

Axial strain, lateral strain, and shear strain of the layered rock mass

\(\sigma_{11} , \, \sigma_{22} , \, \sigma_{12}\) :

Axial stress, lateral stress, and shear stress of the layered rock mass

\(\dot{\varepsilon }_{11} ,\;\dot{\varepsilon }_{s}\) :

Axial strain rate and steady-state creep rate of the layered rock mass

\(\lambda_{i} (i = 1,2,3)\) :

Non-dimensional parameters to be determined by the geometric parameters of the bedding planes

References

  • Adhikary DP, Guo H (2002) An orthotropic Cosserat elasto-plastic model for layered rocks. Rock Mech Rock Eng 35(3):161–170

    Article  Google Scholar 

  • Ali E, Guang W, Weixue J (2014) Assessments of strength anisotropy and deformation behavior of banded amphibolite rocks. Geotech Geol Eng 32(2):429–438

    Article  Google Scholar 

  • Amadei B, Goodman R.E (1981) A 3D constitutive relation for jointed rock mass. Proceedings of the International Symposium on the Mechanical Behavior of Structured Media Ottawa, Canada

  • Andargoli MBE, Shahriar K, Ramezanzadeh A, Goshtasbi K (2019) The analysis of dates obtained from long-term creep tests to determine creep coefficients of rock salt. Bull Eng Geol Environ 78(3):1617–1629

    Article  Google Scholar 

  • Bahaaddini M, Sharrock G, Hebblewhite BK (2013) Numerical investigation of the effect of joint geometrical parameters on the mechanical properties of a non-persistent jointed rock mass under uniaxial compression. Comput Geotech 49:206–225

    Article  Google Scholar 

  • Bandis S, Lumsden AC, Barton N et al (1981) Experimental studies of scale effects on the shear behaviour of rock layers. Int J Rock Mech Min Sci Geomech Abstr 18(1):1–21

    Article  Google Scholar 

  • Chenevert ME, Gatlin C (1965) Mechanical anisotropies of laminated sedimentary rocks. Soc Petrol Eng J 5(01):67–77

    Article  Google Scholar 

  • Cho JW, Kim H, Jeon S et al (2012) Deformation and strength anisotropy of Asan gneiss, Boryeong shale, and Yeoncheon schist. Int J Rock Mech Min Sci 50:158–169

    Article  Google Scholar 

  • Colak K, Unlu T (2004) Effect of transverse anisotropy on the Hoek-Brown strength parameter for intact rocks. Int J Rock Mech Min Sci 6(41):1045–1052

    Article  Google Scholar 

  • Debecker B, Vervoort A (2009) Experimental observation of fracture patterns in layered slate. Int J Fract 159(1):51–62

    Article  Google Scholar 

  • Deng DP, Li L (2019) Failure modes and a calculation method for a stability analysis on a layered slope with a focus on interlayer sliding. Arab J Geosci 12(6):18

    Article  Google Scholar 

  • Dubey RK, Gairola VK (2008) Influence of structural anisotropy on creep of rocksalt from Simla Himalaya, India: an experimental approach. J Struct Geol 30(6):710–718

    Article  Google Scholar 

  • Fossum AF (1985) Effective elastic properties for a randomly jointed rock mass. Int J Rock Mech Min Sci Geomech Abstr 22:467–470

    Article  Google Scholar 

  • Goodman RE (1989) Introduction to rock mechanics, 2nd edn. John Wiley & Sons, USA

    Google Scholar 

  • Gutierrez M, Youn DJ (2015) Effects of fracture distribution and length scale on the equivalent continuum elastic compliance of fractured rock masses. J Rock Mech Geotech Eng 7(6):626–637

    Article  Google Scholar 

  • Khanlari G, Rafiei B, Abdilor Y (2015) An experimental investigation of the brazilian tensile strength and failure patterns of laminated sandstones. Rock Mech Rock Eng 48(2):843–852

    Article  Google Scholar 

  • Liang W, Yang C, Zhao Y, Dusseault MB, Liu J (2007) Experimental investigation of mechanical properties of bedded salt rock. Int J Rock Mech Min Sci 44(3):400–411

    Article  Google Scholar 

  • Lu H, Yao D, Hu Y et al (2017) Analysis of failure characteristics and stability of layered rock slope based on multi-fracture model of FLAC3D. Adv Sci Technol Water Res 37(4):36–41

    Google Scholar 

  • Menezes FF, Lempp C (2018) On the structural anisotropy of physical and mechanical properties of a Bunter Sandstone. J Struct Geol 114:196–205

    Article  Google Scholar 

  • Oda M (1982) Fabric tensor for discontinuous geological materials. Soils Found 22(4):96–108

    Article  Google Scholar 

  • Oda M (1986) An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock mass. Water Resour Res 22(13):1845–1856

    Article  Google Scholar 

  • Oda M (1988) An experimental study of the elasticity of mylonite rock with random cracks[C]. Int J Rock Mech Min Sci Geomech Abstr Pergamon 25(2):59–69

    Article  Google Scholar 

  • Oda M, Yamabe T, Kamemura K (1986) A crack tensor and its relation to wave velocity anisotropy in jointed rock mass. Int J Rock Mech Min Sci Geomech Abstr 23(6):387–397

    Article  Google Scholar 

  • Oda MA, Yamabe TA, Ishizuka YO, Kumasaka H, Kimura K (1993) Elastic stress and strain in jointed rock masses by means of crack tensor analysis. Rock Mech Rock Eng 26(2):89–112

    Article  Google Scholar 

  • Rulon JJ, Freeze RA (1985) Multiple seepage faces on layered slopes and their implications for slope-stability analysis. Can Geotech J 22(3):347–356

    Article  Google Scholar 

  • Salamon MDG (1968) Elastic moduli of a stratified rock mass. Int J Rock Mech Min Sci Geomech Abstr Pergamon 5(6):519–527

    Article  Google Scholar 

  • Shi X, Yang X, Meng Y, Li G (2016) An anisotropic strength model for layered rocks considering planes of weakness. Rock Mech Rock Eng 49(9):3783–3792

    Article  Google Scholar 

  • Sulem J, Panet M, Guenot A (1987) An analytical solution for time-dependent displacements in a circular tunnel. Int J Rock Mech Min Sci Geomech Abstr Pergamon 24(3):155–164

    Article  Google Scholar 

  • Tan X, Konietzky H, Thomas F, Dan DQ (2015) Brazilian tests on transversely isotropic rocks: laboratory testing and numerical simulations. Rock Mech and Rock Eng 48(4):1341–1351

    Article  Google Scholar 

  • Tien YM, Kuo MC, Juang CH (2006) An experimental investigation of the failure mechanism of simulated transversely isotropic rocks. Int J Rock Mech Min Sci 43(8):1163–1181

    Article  Google Scholar 

  • Vervoort A, Min KB, Konietzky H, Cho JW, Debecker B, Dinh QD, Thomas F, Abbass T (2014) Failure of transversely isotropic rock under brazilian test conditions. Int J Rock Mech Min Sci 70(70):343–352

    Article  Google Scholar 

  • Wang Y, Li JL (2011) Investigation on shear creep mechanical properties of soft rock. Adv Mater Res 261:1024–1028

    Google Scholar 

  • Wu C, Chen Q, Basack S, Shi Z (2016) Biaxial creep test study on the influence of structural anisotropy on rheological behavior of hard rock. J Mate Civ Eng 28(10):04016104

    Article  Google Scholar 

  • Wu CZ, Chen QS, Basack S, Sudip B, Shivakumar K (2018) Laboratory investigation on rheological properties of greenschist considering anisotropy under multi-stage compressive creep condition. J Struct Geol 114:111–120

    Article  Google Scholar 

  • Xiong LX, Yang LD (2009) Creep model for rock mass considering normal creep of rock joint plane. J Cent South Univ (Sci and Technol) 40(3):814–821 ((in Chinese))

    Google Scholar 

  • Xu DP, Feng XT, Chen DF, Zhang CQ, Fan QX (2017) Constitutive representation and damage degree index for the layered rock mass excavation response in underground openings. Tunn Undergr Space Technol 64:133–145

    Article  Google Scholar 

  • Xu G, He C, Yan J, Ma G (2019) A new transversely isotropic nonlinear creep model for layered phyllite and its application. Bull Eng Geol Environ 78:5387–5408

    Article  Google Scholar 

  • Xu T, Xu Q, Tang CA, Ranjith PG (2013) The evolution of rock failure with discontinuities due to shear creep. Acta Geotech 8(6):567–581

    Article  Google Scholar 

  • Zhou YY, Feng XT, Xu DP, Fan QX (2016) Experimental investigation of the mechanical behavior of bedded rocks and its implication for high sidewall caverns. Rock Mech and Rock Eng 49(9):3643–3669

    Article  Google Scholar 

  • Zhou YY, Feng XT, Xu DP, Fan QX (2017) An enhanced equivalent continuum model for layered rock mass incorporating bedding structure and stress dependence. Int J Rock Mech Min Sci 97:75–98

    Article  Google Scholar 

  • Zhou XP, Zhu BZ, Juang CH, Wong LNY (2018) A stability analysis of a layered-soil slope based on random field. Bull Eng Geol Environ 78(4):2611–2625

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of China Geological Survey (Grant No.DD20190309), Key Research Program of the Chinese Academy of Sciences (Grant No.KFZD-SW-423), National Natural Science Foundation of China (Grant No. 51991392, U1806226), Hubei Provincial Natural Science Foundation of China (No. 2018ACA134), Hubei Province Innovative Research Groups (2018CFA012), and Science & Technology Research and Development Program of China Railway (No. P2018G045).

Author information

Authors and Affiliations

  1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Beijing, China

    Y. Tian, W. Z. Chen, H. M. Tian, J. P. Yang, Z. Y. Zhang & X. Y. Shu

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Y. Tian & Z. Y. Zhang

  3. Research Centre of Geotechnical and Structural Engineering, Shandong University, Jinan, 250061, Shandong, China

    W. Z. Chen

  4. School of Civil and Communications, Hohai University, Nanjing, 40041, Jiangsu, China

    X. Y. Shu

Authors
  1. Y. Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Z. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. M. Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. P. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Z. Y. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. X. Y. Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to W. Z. Chen or H. M. Tian.

Ethics declarations

Conflict of Interest

The authors declare that they have 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, Y., Chen, W.Z., Tian, H.M. et al. Analytical Model of Layered Rock Considering Its Time-Dependent Behaviour. Rock Mech Rock Eng 54, 5937–5944 (2021). https://doi.org/10.1007/s00603-021-02421-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-021-02421-2

Keywords

Search

Navigation