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Rock Mechanics and Rock Engineering

, Volume 50, Issue 10, pp 2677–2694 | Cite as

Validation of a New Elastoplastic Constitutive Model Dedicated to the Cyclic Behaviour of Brittle Rock Materials

  • B. Cerfontaine
  • R. Charlier
  • F. Collin
  • M. Taiebat
Original Paper

Abstract

Old mines or caverns may be used as reservoirs for fuel/gas storage or in the context of large-scale energy storage. In the first case, oil or gas is stored on annual basis. In the second case pressure due to water or compressed air varies on a daily basis or even faster. In both cases a cyclic loading on the cavern’s/mine’s walls must be considered for the design. The complexity of rockwork geometries or coupling with water flow requires finite element modelling and then a suitable constitutive law for the rock behaviour modelling. This paper presents and validates the formulation of a new constitutive law able to represent the inherently cyclic behaviour of rocks at low confinement. The main features of the behaviour evidenced by experiments in the literature depict a progressive degradation and strain of the material with the number of cycles. A constitutive law based on a boundary surface concept is developed. It represents the brittle failure of the material as well as its progressive degradation. Kinematic hardening of the yield surface allows the modelling of cycles. Isotropic softening on the cohesion variable leads to the progressive degradation of the rock strength. A limit surface is introduced and has a lower opening than the bounding surface. This surface describes the peak strength of the material and allows the modelling of a brittle behaviour. In addition a fatigue limit is introduced such that no cohesion degradation occurs if the stress state lies inside this surface. The model is validated against three different rock materials and types of experiments. Parameters of the constitutive laws are calibrated against uniaxial tests on Lorano marble, triaxial test on a sandstone and damage-controlled test on Lac du Bonnet granite. The model is shown to reproduce correctly experimental results, especially the evolution of strain with number of cycles.

Keywords

Fatigue Constitutive modelling Bounding surface model Cyclic loading Rock mechanics 

List of symbols

\(\alpha _0\)

Initial value of back-stress ratio of yield surface (–)

\(\alpha\)

Back-stress ratio of yield surface (–)

\(\alpha ^\mathrm{b}\)

Back-stress ratio of boundary surface (–)

\(\alpha ^\mathrm{pc}\)

Back-stress ratio of fatigue surface (–)

\(\delta _\mathrm{l}\)

Compression/extension opening ratio (–)

\(\epsilon _1\)

Vertical strain (–)

\(\epsilon _3\)

Lateral strain (–)

\(\epsilon _\mathrm{q}\)

Deviatoric strain (–)

\(\epsilon _\mathrm{v}\)

Volumetric strain (–)

\(\epsilon _\mathrm{q}^\mathrm{p}\)

Plastic deviatoric strain (–)

\(\epsilon _\mathrm{v}^\mathrm{p}\)

Plastic volumetric strain (–)

\(\lambda\)

Plastic multiplier (–)

\(\xi\)

Reduced deviatoric stress (–)

\(\nu\)

Poisson’s ratio (–)

\(\sigma _1\)

Vertical effective stress (MPa)

\(\sigma _3\)

Lateral effective stress (MPa)

\(\phi\)

Friction angle (–)

\(b_0\)

Hardening parameter (–)

c

Cohesion (MPa)

E

Young's modulus (MPa)

\({\mathbf {e}}\)

Strain vector (–)

\(f^\mathrm{y}\)

Yield surface (–)

\(f^\mathrm{l}\)

Limit surface (–)

\(h_1\)

Hardening direction of \(\alpha\) (–)

\(h_2\)

Hardening direction of \(p_\mathrm{c}\) (–)

\(n_{\alpha }\)

Parameter for hardening of \(\alpha\) (–)

\(n_\mathrm{pc}\)

Parameter for hardening of \(p_\mathrm{c}\) (–)

p

Mean stress (MPa)

\(p_\mathrm{c0}\)

Initial value of cohesion projection (MPa)

\(p_\mathrm{c}\)

Cohesion projection (MPa)

\(p_\mathrm{res}\)

Residual cohesion projection (MPa)

q

Deviatoric stress (MPa)

\({\mathbf {s}}\)

Stress vector (–)

\({\mathbf {v}}\)

Hardening vector (–)

\(A_\mathrm{c1}\)

Parameter for cohesion evolution (–)

\(A_\mathrm{c2}\)

Parameter for cohesion evolution (–)

\(A_\mathrm{d}\)

Parameter for volumetric flow rule (–)

\(A_\mathrm{q}\)

Parameter for deviatoric flow rule (–)

\({\mathbf {E}}\)

Elastic tensor (MPa)

G

Shear modulus (MPa)

\(M^\mathrm{b}\)

Opening of the bounding surface (–)

\(M^\mathrm{l}\)

Opening of the limit surface (–)

\(M^\mathrm{pc}\)

Opening of the fatigue surface (–)

\(M^\mathrm{y}\)

Opening of the yield surface (–)

N

Number of cycles (–)

Notes

Acknowledgements

The authors gratefully acknowledge the financial support from Walloon Region (Belgium) and the SMARTWATER project. The first author would like to warmly thank the Pr. Derek Martin for his encouragement and data he provided.

References

  1. Alliche A (2004) Damage model for fatigue loading of concrete. Int J Fatigue 26(9):915–921. doi: 10.1016/j.ijfatigue.2004.02.006 CrossRefGoogle Scholar
  2. Attewell P, Farmer W (1973) Fatigue behaviour of rock. Int J Rock Mech Min Sci 10:1–9CrossRefGoogle Scholar
  3. Bastian T, Connelly B, Lazo Olivares C, Yfantidis N, Taheri A (2014) Progressive damage of Hawkesbury sandstone subjected to systematic cyclic loading. Min Educ Aust J Res Proj Rev 3:7–14Google Scholar
  4. Benz T, Schwab R (2008) A quantitative comparison of six rock failure criteria. Int J Rock Mech Min Sci 45:1176–1186CrossRefGoogle Scholar
  5. Bieniawski Z (1967a) Mechanism of brittle fracture of rock: part I—theory of the fracture process. Int J Rock Mech Min Sci 4:395–406CrossRefGoogle Scholar
  6. Bieniawski Z (1967b) Mechanism of brittle fracture of rock: part II—experimental studies. Int J Rock Mech Min Sci 4:407–423CrossRefGoogle Scholar
  7. Brantut N, Heap M, Meredith P, Baud P (2013) Time-dependent cracking and brittle creep in crustal rocks: a review. J Struct Geol 52:17–43. doi: 10.1016/j.jsg.2013.03.007 CrossRefGoogle Scholar
  8. Breccolotti M, Bonfigli M, D’Alessandro A, Materazzi A (2015) Constitutive modeling of plain concrete subjected to cyclic uniaxial compressive loading. Constr Build Mater 94:172–180. doi: 10.1016/j.conbuildmat.2015.06.067 CrossRefGoogle Scholar
  9. Burdine N (1963) Rock failure under dynamic loading conditions. Soc Pet Eng J 3(01):1–8CrossRefGoogle Scholar
  10. Cattaneo S, Labuz J (2001) Damage of marble from cyclic loading. J Mater Civ Eng 13:459–465CrossRefGoogle Scholar
  11. Cerfontaine B, Collin F, Charlier R (2016) Numerical modelling of transient cyclic vertical loading of suction caissons in sand. Geotechnique 66(2):121–136. doi: 10.1680/jgeot.15.P.061 CrossRefGoogle Scholar
  12. Chen H, Cong T, Yang W, Tan C, Li Y, Ding Y (2009) Progress in electrical energy storage system: a critical review. Prog Nat Sci 19(3):291–312. doi: 10.1016/j.pnsc.2008.07.014 CrossRefGoogle Scholar
  13. Chow TM, Meglis IL, Young RP (1995) Progressive microcrack development in tests on Lac du Bonnet granite—II. Ultrasonic tomographic imaging. Int J Rock Mech Min Sci Geomech Abstr 32(8):751–761. doi: 10.1016/0148-9062(95)00015-9 CrossRefGoogle Scholar
  14. Dafalias Y (1986) Bounding surface plasticity. i: mathematical foundation and hypoplasticity. J Eng Mech 112(9):966–987CrossRefGoogle Scholar
  15. Dafalias Y, Manzari M (2004) Simple plasticity sand model accounting for fabric change effects. J Eng Mech 130(6):622–634CrossRefGoogle Scholar
  16. Dafalias Y, Papadimitriou A, Li X (2004) Sand plasticity model accounting for inherent fabric anisotropy. J Eng Mech 130(11):1319–1333CrossRefGoogle Scholar
  17. Dafalias Y, Taiebat M (2016) SANISAND-Z: zero elastic range sand plasticity model. Géotechnique 66(12):999–1013CrossRefGoogle Scholar
  18. Dal Pino R, Narducci P, Royer-Carfagni G (1999) A SEM investigation on fatigue damage of marble. J Mater Sci Lett 18:1619–1622CrossRefGoogle Scholar
  19. Eberhardt E, Stead D, Stimpson B (1999a) Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. Int J Rock Mech Min Sci 36(3):361–380. doi: 10.1016/S0148-9062(99)00019-4 CrossRefGoogle Scholar
  20. Eberhardt E, Stimpson B, Stead D (1999b) Effects of grain size on the initiation and propagation thresholds of stress-induced brittle fractures. Rock Mech Rock Eng 32(2):81–99. doi: 10.1007/s006030050026 CrossRefGoogle Scholar
  21. Erarslan N, Alehossein H, Williams DJ (2014) Tensile fracture strength of brisbane tuff by static and cyclic loading tests. Rock Mech Rock Eng 47(4):1135–1151. doi: 10.1007/s00603-013-0469-5 CrossRefGoogle Scholar
  22. Erarslan N, Williams DJ (2012) Investigating the effect of cyclic loading on the indirect tensile strength of rocks. Rock Mech Rock Eng 45(3):327–340. doi: 10.1007/s00603-011-0209-7 CrossRefGoogle Scholar
  23. Ferrero A, Migliazza M, Spagnoli A (2009) Theoretical modelling of bowing in cracked marble slabs under cyclic thermal loading. Constr Build Mater 23(6):2151–2159CrossRefGoogle Scholar
  24. Førsund R (2007) Hydropower economics, vol 112. Springer, New YorkGoogle Scholar
  25. Gatelier N, Pellet F, Loret B (2002) Mechanical damage of an anisotropic porous rock in cyclic triaxial tests. Int J Rock Mech Min Sci 39(3):335–354. doi: 10.1016/S1365-1609(02)00029-1 CrossRefGoogle Scholar
  26. Ghamgosar M, Erarslan N (2015) Experimental and numerical studies on development of fracture process zone (FPZ) in rocks under cyclic and static loadings. Rock Mech Rock Eng 49(3):893–908. doi: 10.1007/s00603-015-0793-z CrossRefGoogle Scholar
  27. Haimson BC, Kim CM (1971) Mechanical behaviour of rock under cyclic fatigue. Rock Mech 3:845–863Google Scholar
  28. Hajiabdolmajid V, Kaiser P (2002) Brittleness of rock and stability assessment in hard rock tunneling. Tunn Undergr Space Technol 18(1):35–48. doi: 10.1016/S0886-7798(02)00100-1 CrossRefGoogle Scholar
  29. Hashiguchi K (2009) Elastoplasticity theory. Springer-Verlag, NewYorkCrossRefGoogle Scholar
  30. Hoek E, Brown E (1980) Underground excavations in rock. The Institution of Mining and Metallurgy, LondonGoogle Scholar
  31. Hoek E, Brown E (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34(8):1165–1186. doi: 10.1016/S1365-1609(97)80069-X CrossRefGoogle Scholar
  32. Hueckel T (1991) Damping, cyclic strain buildup and fatigue of rocks a generalized Ramberg–Osgood approach. Comput Geotech 12(3):235–269CrossRefGoogle Scholar
  33. Khaledi K, Mahmoudi E, Datcheva M, Schanz T (2016) Stability and serviceability of underground energy storage caverns in rock salt subjected to mechanical cyclic loading. Int J Rock Mech Min Sci 86:115–131. doi: 10.1016/j.ijrmms.2016.04.010 Google Scholar
  34. Khaledi K, Mahmoudi E, Datcheva M, Schanz T (2016) Stability and serviceability of underground energy storage caverns in rock salt subjected to mechanical cyclic loading. Int J Rock Mech Min Sci 86(15):115–131. doi: 10.1016/j.ijrmms.2016.04.010 Google Scholar
  35. Li N, Zhang P, Chen Y, Swoboda G (2003) Fatigue properties of cracked, saturated and frozen sandstone samples under cyclic loading. Int J Rock Mech Min Sci 40(1):145–150. doi: 10.1016/S1365-1609(02)00111-9 CrossRefGoogle Scholar
  36. Li X, Dafalias Y (2012) Anisotropic critical state theory: role of fabric. J Eng Mech 138(3):263–275CrossRefGoogle Scholar
  37. Liu E, He S (2012) Effects of cyclic dynamic loading on the mechanical properties of intact rock samples under confining pressure conditions. Eng Geol 125:81–91. doi: 10.1016/j.enggeo.2011.11.007 CrossRefGoogle Scholar
  38. Liu J, Xie H, Hou Z, Yang C, Chen L (2014) Damage evolution of rock salt under cyclic loading in unixial tests. Acta Geotech 9(1):153–160. doi: 10.1007/s11440-013-0236-5 CrossRefGoogle Scholar
  39. Mahmoudi E, Khaledi K, Miro S, König D, Schanz T (2016) Probabilistic analysis of a rock salt cavern with application to energy storage systems. Rock Mech Rock Eng. doi: 10.1007/s00603-016-1105-y Google Scholar
  40. Manzari M, Dafalias Y (1997) A critical state two-surface plasticity model for sands. Geotechnique 47(2):255–272CrossRefGoogle Scholar
  41. Martin C, Chandler N (1994) The progressive fracture of Lac du Bonnet granite. Int J Rock Mech Min Sci Geomech Abstr 31(6):643–659CrossRefGoogle Scholar
  42. Martin CD (1997) Seventeenth Canadian geotechnical colloquium: the effect of cohesion loss and stress path on brittle rock strength. Can Geotech J 34(5):698–725CrossRefGoogle Scholar
  43. Mazars J, Hamon F, Grange S (2015) A new 3D damage model for concrete under monotonic, cyclic and dynamic loadings. Mater Struct. doi: 10.1617/s11527-014-0439-8 Google Scholar
  44. Mira P, Tonni L, Pastor M, Fernandez-Merodo J (2009) A generalized midpoint algorithm for the integration of a generalized plasticity model for sands. Int J Numer Method Geomech 77:1201–1223. doi: 10.1002/nme CrossRefGoogle Scholar
  45. Papamichos E, Papanicolopulos S, Larsen I, Alnæs L, Rescic S (2004) Method for in situ, quasi non-destructive diagnosis of mechanical properties and damage of natural building stones. In Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), American Rock Mechanics AssociationGoogle Scholar
  46. Peng X, Meyer C (2000) A continuum damage mechanics model for concrete reinforced with randomly distributed short fibers. Comput Struct 78(4):505–515. doi: 10.1016/S0045-7949(00)00045-6 CrossRefGoogle Scholar
  47. Pouya A, Zhu C, Arson C (2016) Micro-macro approach of salt viscous fatigue under cyclic loading. Mech Mater 93:13–31. doi: 10.1016/j.mechmat.2015.10.009 CrossRefGoogle Scholar
  48. Prevost J-H (1985) A simple plasticity theory for frictional cohesionless soils. Soil Dyn Earthq Eng 4(1):9–17Google Scholar
  49. Prost G (1988) Jointing at rock contacts in cyclic loading. Int J Rock Mech Min Sci Geomech Abstr 25(5):263–272. doi: 10.1016/0148-9062(88)90003-4 CrossRefGoogle Scholar
  50. Pujades E, Willems T, Bodeux S, Orban P, Dassargues A (2016) Underground pumped storage hydroelectricity using abandoned works (deep mines or open pits) and the impact on groundwater flow. Hydrogeol J 24(6):1531–1546. doi: 10.1007/s10040-016-1413-z CrossRefGoogle Scholar
  51. Raju M, Kumar Khaitan S (2012) Modeling and simulation of compressed air storage in caverns: a case study of the Huntorf plant. Appl Energy 89(1):474–481. doi: 10.1016/j.apenergy.2011.08.019 CrossRefGoogle Scholar
  52. Royer-Carfagni G, Salvatore W (2000) The characterization of marble by cyclic compression loading: experimental results. Mech Cohesive Frict Mater 5(7):535–563CrossRefGoogle Scholar
  53. Simo J, Hughes T (1998) Computational inelasticity. Springer-Verlag, NewYorkGoogle Scholar
  54. Sloan SW, Abbo AJ, Sheng D (2001) Refined explicit integration of elastoplastic models with automatic error control. Eng Comput 18(1/2):121–194. doi: 10.1108/02644400110365842 CrossRefGoogle Scholar
  55. Stavropoulou M, Liolios P, Exadaktylos G (2004) Calibration of the triaxial hyperbolic Mohr–Coulomb elastoplastic model parameters on laboratory rock mechanics tests. Int J Geomech 12:618–631CrossRefGoogle Scholar
  56. Steffen B (2012) Prospects for pumped-hydro storage in Germany. Energy Policy 45:420–429CrossRefGoogle Scholar
  57. Suaris W, Ouyang C, Fernando V (1990) Damage model for cyclic loading of concrete. J Eng Mech 116(5):1020–1035CrossRefGoogle Scholar
  58. Taheri A, Royle A, Yang Z, Zhao Y (2016) Study on variations of peak strength of a sandstone during cyclic loading. Geomech Geophy Geo-Energy Geo-Resour 2(1):1–10. doi: 10.1007/s40948-015-0017-8 CrossRefGoogle Scholar
  59. Taiebat M, Dafalias Y (2008) SANISAND: simple anisotropic sand plasticity model. Int J Numer Anal Methods Geomech 32:915–948CrossRefGoogle Scholar
  60. Taiebat M, Dafalias Y (2010) Simple yield surface expressions appropriate for soil plasticity. Int J Geomech 10(4):161–169CrossRefGoogle Scholar
  61. Wang Z, Li S, Qiao L, Zhang Q (2015) Finite element analysis of the hydro-mechanical behavior of an underground crude oil storage facility in granite subject to cyclic loading during operation. Int J Rock Mech Min Sci 73:70–81. doi: 10.1016/j.ijrmms.2014.09.018 CrossRefGoogle Scholar
  62. Wang Z, Li S, Qiao L, Zhao J (2013) Fatigue behavior of granite subjected to cyclic loading under triaxial compression condition. Rock Mech Rock Eng 46(6):1603–1615. doi: 10.1007/s00603-013-0387-6 CrossRefGoogle Scholar
  63. Wu J, Li J, Faria R (2006) An energy release rate-based plastic-damage model for concrete. Int J Solids and Struct 43(3–4):583–612. doi: 10.1016/j.ijsolstr.2005.05.038 CrossRefGoogle Scholar
  64. Xiao J, Ding D, Jiang F, Xu G (2010) Fatigue damage variable and evolution of rock subjected to cyclic loading. Int J Rock Mech Min Sci 47(3):461–468. doi: 10.1016/j.ijrmms.2009.11.003 CrossRefGoogle Scholar
  65. Xiao J, Ding D, Xu G, Jiang F (2009) Inverted S-shaped model for nonlinear fatigue damage of rock. Int J Rock Mech Min Sci 46(3):643–648. doi: 10.1016/j.ijrmms.2008.11.002 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  • B. Cerfontaine
    • 1
  • R. Charlier
    • 1
  • F. Collin
    • 1
  • M. Taiebat
    • 2
  1. 1.Urban and Environmental EngineeringUniversity of LiegeLiegeBelgium
  2. 2.Department of Civil EngineeringUniversity of British ColumbiaVancouverCanada

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