Skip to main content
SpringerLink
Log in

A nonlinear rheological damage model of hard rock

硬岩非线性流变损伤模型

  • Published:
Journal of Central South University Aims and scope Submit manuscript

Abstract

By adopting cyclic increment loading and unloading method, time-independent and time-dependent strains can be separated. It is more reasonable to describe the reversible and the irreversible deformations of sample separately during creep process. A nonlinear elastic-visco-plastic rheological model is presented to characterize the time-based deformational behavior of hard rock. Specifically, a spring element is used to describe reversible instantaneous elastic deformation. A reversible nonlinear visco-elastic (RNVE) model is developed to characterize recoverable visco-elastic response. A combined model, which contains a fractional derivative dashpot in series with another Hook’s body, and a St. Venant body in parallel with them, is proposed to describe irreversible visco-plastic deformation. Furthermore, a three-stage damage equation based on strain energy is developed in the visco-plastic portion and then nonlinear elastic-visco-plastic rheological damage model is established to explain the trimodal creep response of hard rock. Finally, the proposed model is validated by a laboratory triaxial rheological experiment. Comparing with theoretical and experimental results, this rheological damage model characterizes well the reversible and irreversible deformations of the sample, especially the tertiary creep behavior.

摘要

通过循环增量加–卸载的方法, 可以将岩石在恒定载荷下的时效变形进一步分离成瞬时弹性应 变、瞬时塑性应变、黏弹性应变和黏塑性应变。在蠕变过程中分别描述试样的可恢复和不可恢复的变 形是比较合理的。因此, 本文提出了一种可以描述硬岩时效变形的非线性流变模型。其中该模型采用 一个弹簧元件来描述可恢复的瞬时弹性变形; 一个改进的非线性黏弹性模型(RNVE)可以描述可恢 复的黏弹性变形; 一个分数阶黏壶和弹簧元件串联的组合模型与另一个塑性摩擦块并联形成一个新的 组合模型来描述不可恢复的瞬时塑性变形和黏塑性变形。此外, 在黏塑性元件中引入一个基于应变能 的含三阶段损伤模型来描述硬岩的加速蠕变变形。最后, 通过流变试验验证提出模型的合理性。对比 试验结果和理论结果, 本文提出的流变损伤模型可以很好地描述硬岩的可恢复和不可恢复变形, 尤其 是描述加速蠕变变形。

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.

Similar content being viewed by others

References

  1. YANG S Q, XU P, LI Y B, HUANG Y H. Experimental investigation on triaxial mechanical and permeability behavior of sandstone after exposure to different high temperature treatments [J]. Geothermics, 2017, 69: 93–109. DOI: https://doi.org/10.1016/j.geothermics. 2017.04.009.

    Article  Google Scholar 

  2. YANG S Q, RANJITH P G, JING H W, TIAN W L, JU Y. An experimental investigation on thermal damage and failure mechanical behavior of granite after exposure to different high temperature treatments [J]. Geothermics, 2017, 65: 180–197. DOI: https://doi.org/10.1016/j.geothermics.2016.09.008.

    Article  Google Scholar 

  3. YANG S Q, TIAN W L, RANJITH P G. Experimental investigation on deformation failure characteristics of crystalline marble under triaxial cyclic loading [J]. Rock Mechanics and Rock Engineering, 2017, 50: 2871–2889. DOI: https://doi.org/10.1007/s00603-017-1262-7.

    Article  Google Scholar 

  4. YANG S Q, TIAN W L, HUANG Y H. Failure mechanical behavior of pre-holed granite specimens after elevated temperature treatment by particle flow code [J]. Geothermics, 2018, 72: 124–137. DOI: https://doi.org/10.1016/j.geothermics.2017.10.018.

    Article  Google Scholar 

  5. BOUKHAROV G N, CHANDA M W, BOUKHAROV N G. The three processes of brittle crystalline rock creep [J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1995, 32(4): 325–335. DOI: https://doi.org/10.1016/0148-9062(94)00048-8.

    Article  Google Scholar 

  6. MARANINI E, BRIGNOLI M. Creep behaviour of a weak rock: experimental characterization [J]. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(1): 127–138. DOI: 10.1016/S1365-1609(98)00171-3.

    Article  Google Scholar 

  7. MALAN D F. Simulating the time-dependent behaviour of excavations in hard rock [J]. Rock Mechanics and Rock Engineering, 2002, 35(4): 225–254. DOI: 10.1007/s00603-002-0026-0.

    Article  Google Scholar 

  8. MIURA K, OKUI Y, HORII H. Micromechanics-based prediction of creep failure of hard rock for long-term safety of high-level radioactive waste disposal system [J]. Mechanics and Materials, 2003, 35(3–6): 587–601. DOI: 10.1016/S0167-6636(02)00286-7.

    Article  Google Scholar 

  9. DEBERNARDI D, BARLA G. New viscoplastic model for design analysis of tunnels in squeezing conditions [J]. Rock Mechanics and Rock Engineering, 2009, 42(2): 259–288. DOI: 10.1007/s00603-009-0174-6.

    Article  Google Scholar 

  10. NOMIKOS P, RAHMANNEJAD R, SOFIANOS A. Supported axisymmetric tunnels within linear viscoelastic Burgers rocks [J]. Rock Mechanics and Rock Engineering, 2011, 44(5): 553–564. DOI: 10.1007/s00603-011-0159-0.

    Article  Google Scholar 

  11. HEAP M J, BAUD P, MEREDITH P G, VINCIGUERRA S, BELL A F, MAIN I G. Brittle creep in basalt and its application to time-dependent volcano deformation [J]. Earth and Planetary Science Letters, 2011, 307(1, 2): 71–82. DOI: 10.1016/j.epsl.2011.04.035.

    Article  Google Scholar 

  12. MALAN D F. Time-dependent behaviour of deep level tabular excavations [J]. Rock Mechanics and Rock Engineering, 1999, 32(2): 123–155. DOI: https://doi.org/10.1007/s006030050028.

    Article  Google Scholar 

  13. STERPI D, GIODA G. Visco-plastic behaviour around advancing tunnels in squeezing rock [J]. Rock Mechanics and Rock Engineering, 2009, 42(2): 319–339. DOI: 10.1007/s00603-007-0137-8.

    Article  Google Scholar 

  14. MARANINI E, YAMAGUCHI T. A non-associated viscoplastic model for the behaviour of granite in triaxial compression [J]. Mechanics and Materials, 2001, 33(5): 283–293. DOI: 10.1016/S0167-6636(01)00052-7.

    Article  Google Scholar 

  15. ZHAO Yan, WANG Yi, WANG Wei, WAN Wen, TANG Jin. Modeling of non-linear rheological behavior of hard rock using triaxial rheological experiment [J]. International Journal of Rock Mechanics and Mining Sciences, 2017, 93: 66–75. DOI: 10.1016/j.ijrmms.2017. 01.004.

    Article  Google Scholar 

  16. XU Wei, YANG Sheng, CHU Wei. Nonlinear viscoelasto-plastic rheological model (Hohai model) of rock and its engineering application [J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 25(3): 433–447. DOI: 10.3321/j.issn:1000-6915.2006.03.001. (in Chinese)

    Google Scholar 

  17. CAO G H, HU J X, ZHANG K. Coupling model for calculating prestress loss caused by relaxation loss, shrinkage, and creep of concrete [J]. Journal of Central South University, 2016, 23(2): 470–478. DOI: 10.1007/s11771-016-3092-2.

    Article  Google Scholar 

  18. BONINI M, DEBERNARDI D, BARLA M, BARLA G. The mechanical behaviour of clay shales and implications on the design of tunnels [J]. Rock Mechanics and Rock Engineering, 2009, 42: 361–388. DOI: https://doi.org/10.1007/s00603-007-0147-6. (in Chinese)

    Article  Google Scholar 

  19. YANG Wen, ZHANG Qiang, LI Shu, WANG Shu. Time-dependent behavior of diabase and a nonlinear creep model [J]. Rock Mechanics and Rock Engineering, 2014, 47(4): 1211–1224. DOI: https://doi.org/10.1007/s00603-013-0478-4.

    Article  Google Scholar 

  20. SCIUME G, BENBOUDJEMA F. A viscoelastic unitary crack-opening strain tensor for crack width assessment in fractured concrete structures [J]. Mechanics of Time-Dependent Materials, 2016, 21(2): 223–243. DOI: 10.1007/s11043-016-9327-7.

    Article  Google Scholar 

  21. SHAO J F, CHAU K T, FENG X T. Modeling of anisotropic damage and creep deformation in brittle rocks [J]. International Journal of Rock Mechanics and Mining Sciences, 2006, 43(4): 582–592. DOI: https://doi.org/10.1016/j.ijrmms.2005.10.004.

    Article  Google Scholar 

  22. LI X Z, SHAO Z S. Investigation of macroscopic brittle creep failure caused by microcrack growth under step loading and unloading in rocks [J]. Rock Mechanics and Rock Engineering, 2016, 49: 2581–2593. DOI: https://doi.org/10.1007/s00603-016-0953-9.

    Article  Google Scholar 

  23. WEN J F, TU S T. A multiaxial creep-damage model for creep crack growth considering cavity growth and microcrack interaction [J]. Engineering Fracture Mechanics, 2014, 123: 197–210. DOI: 10.1016/j.engfracmech.2014. 03.001.

    Article  Google Scholar 

  24. KACHANOV L M. Time of the rupture process under creep conditions [J]. Izv Akad Nauk S S R Otd Tech Nauk, 1958, 8: 26–31.

    Google Scholar 

  25. MURAKAMI S. Notion of continuum damage mechanics and its application to anisotropic creep damage theory [J]. Journal of Engineering Materials and Technology, 1983, 105(2): 99–105. DOI:10.1115/1.3225633.

    Article  Google Scholar 

  26. KYO YA, ICHIKA WA, KAWAMO TO. A damage mechanics theory for discontinuous rock mass [J]. Rock Mechanics, 1985: 469–480.

    Google Scholar 

  27. LIU H Z, XIE H Q, HE J D, XIAO M L. Nonlinear creep damage constitutive model for soft rocks [J]. Mechanics of Time-Dependent Materials, 2017, 21(1): 73–96. DOI: 10.1007/s11043-016-9319-7.

    Article  Google Scholar 

  28. YANG Sheng, XU Peng, RANJITH P G, CHEN Guo, JING Hong. Evaluation of creep mechanical behavior of deep-buried marble under triaxial cyclic loading [J]. Arabian Journal of Geosciences, 2015, 8(9): 6567–6582. DOI: 10.1007/s12517-014-1708-0.

    Article  Google Scholar 

  29. HU B, WANG Z L, LIANG B, LI G, WANG J G, CHEN G. Experimental study of rock creep properties [J]. Journal of Experimental Mechanics, 2015, 30(04): 438–446. DOI: 10.7520/1001-4888-14-247.

    Google Scholar 

  30. BUI T A, WONG H, DELERUYELLE F, ZHOU A. Constitutive modelling of the time-dependent behaviour of partially saturated rocks [J]. Computers and Geotechnics, 2016, 78: 123–133. DOI: https://doi.org/10.1016/j.compgeo. 2016. 05.004.

    Article  Google Scholar 

  31. ROGERS L. Operators and fractional derivatives for viscoelastic constitutive equations [J]. Journal of Rheological, 1983, 27(4): 351. DOI: 10.1122/1.549710.

    Article  MATH  Google Scholar 

  32. BAGLEY R L, TORVIK P J. Fractional calculus-A different approach to the analysis of viscoelastically damped structures [J]. AIAA Journal, 1983, 21(5): 741–748. DOI: https://doi.org/10.2514/3.8142.

    Article  MATH  Google Scholar 

  33. KOELLER R C. Application of fractional calculus to the theory of viscoelasticity [J]. Journal of Applied Mechanics, 1984, 51(2): 299–307. DOI: 10.1115/1.3167616.

    Article  MathSciNet  MATH  Google Scholar 

  34. WELCH S W J, RORRER R A L, DUREN J R G. Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials [J]. Mechanics of Time-Dependent Materials, 1999, 3(3): 279–303. DOI: https://doi.org/10.1023/A:1009834317545.

    Article  Google Scholar 

  35. CELAURO C, FECAROTTI C, PIRROTTA a, COLLOP A C. Experimental validation of a fractional model for creep/recovery testing of asphalt mixtures [J]. Construction and Building Materials, 2012, 36(4): 458–466. DOI: 10.1016/j.conbuildmat.2012.04.028.

    Article  Google Scholar 

  36. ZHOU H W, WANG C P, HAN B B, DUAN Z Q. A creep constitutive model for salt rock based on fractional derivatives [J]. International Journal of Rock Mechanics and Mining Sciences, 2011, 48: 116–121. DOI: 10.1016/j.ijrmms. 2010.11.004.

    Article  Google Scholar 

  37. ZHOU H W, WANG C P, MISHNAEVSKY L Jr, DUAN Z Q, DING J Y. A fractional derivative approach to full creep regions in salt rock [J]. Mechanics of Time-Dependent Materials, 2013, 17: 413–425. DOI: 10.1007/s11043-012-9193-x.

    Article  Google Scholar 

  38. KANG Jian, ZHOU Fu, LIU Chun, LIU Ying. A fractional non-linear creep model for coal considering damage effect and experimental validation [J]. International Journal of Non-Linear Mechanics, 2015, 76: 20–28. DOI: 10.1016/j.ijnonlinmec.2015.05.004.

    Article  Google Scholar 

  39. XIA Cai, ZHONG Shi. Experimental data processing method in consideration of influence of loading history on rock specimen deformation [J]. Journal of Central South Institute of Mining and Metallurgy, 1989, 20(1): 18–24. (in Chinese)

    MathSciNet  Google Scholar 

  40. ZHAO Yan, ZHANG Lian, WANG Wei, WAN Wen, LI Shu, MA Wen, WANG Yi, Creep behavior of intact and cracked limestone under multi-level loading and unloading cycles [J]. Rock Mechanics and Rock Engineering, 2017, 50(6): 1409–1424. DOI: https://doi.org/10.1007/s00603-017-1187-1.

    Article  Google Scholar 

  41. YU Tian, WANG Xun, LIU Zai. Elasticity and plasticity [M]. Beijing: China Architecture and Building Press, 2004. (in Chinese)

    Google Scholar 

  42. SUN Jun. Rheological behavior of geomaterials and its engineering applications [M]. Beijing: China Architecture and Building Press, 1999. (in Chinese)

    Google Scholar 

  43. YANG Sheng, CHENG Long. Non-stationary and nonlinear visco-elastic shear creep model for shale [J]. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(6): 1011–1020. DOI: 10.1016/j.ijrmms. 2011.06.007.

    Article  Google Scholar 

  44. YIN De, WU Hao, CHENG Chen, CHEN Yang. Fractional order constitutive model of geomaterials under the condition of triaxial test [J]. International Journal for Numerical Analytical Methods in Geomechanics, 2013, 37(8): 961–972. DOI: 10.1002/nag.2139.

    Article  Google Scholar 

  45. KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations [M]. Amsterdam: Elsevier, 2006.

    MATH  Google Scholar 

  46. XIE He, PENG Rui, JU Yang. Energy dissipation of rock deformation and fracture [J]. Chinese Journal Rock Mechanics and Engineering, 2004, 23(21): 3565–3570. DOI: 10.3321/j.issn:1000-6915.2004.21.001. (in Chinese)

    Google Scholar 

  47. HEAP M J, BAUD P, MEREDITH P G. Influence of temperature on brittle creep in sandstones [J]. Geophysical Research Letters, 2009, 36(19): L19305. DOI: 10.1029/2009GL039373.

    Article  Google Scholar 

  48. HEAP M J, BAUD P, MEREDITH P G, BELL A F, MAIN I G. Time-dependent brittle creep in Darley Dale sandstone [J]. Geophysical Research Letters, 2009, 114: B07203. DOI: 10.1029/2008JB006212.

    Google Scholar 

  49. LIN Q X, LIU Y M, THAM L G, TANG C A, LEE P K K, WANG J. Time-dependent strength degradation of granite [J]. International Journal of Rock Mechanics and Mining Sciences, 2009, 46(7): 1103–1114. DOI: 10.1016/j.ijrmms. 2009.07.005.

    Article  Google Scholar 

  50. YANG Sheng, XU Peng, RANJITH P G. Damage model of coal under creep and triaxial compression [J]. International Journal of Rock Mechanics and Mining Sciences, 2015, 80(8): 337–345. DOI: 10.1016/j.ijrmms. 2015.10.006.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Bo Hu  (胡波), Sheng-qi Yang  (杨圣奇) & Peng Xu  (徐鹏)

  2. School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Sheng-qi Yang  (杨圣奇)

Authors
  1. Bo Hu  (胡波)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sheng-qi Yang  (杨圣奇)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peng Xu  (徐鹏)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sheng-qi Yang  (杨圣奇).

Additional information

Foundation item: Project(BK20150005) supported by the Natural Science Foundation of Jiangsu Province for Distinguished Young Scholars, China; Project(2015XKZD05) supported by the Fundamental Research Funds for the Central Universities, China

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, B., Yang, Sq. & Xu, P. A nonlinear rheological damage model of hard rock. J. Cent. South Univ. 25, 1665–1677 (2018). https://doi.org/10.1007/s11771-018-3858-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11771-018-3858-9

Key words

关键词

Search

Navigation