Advertisement

A phenomenological modelling of rocks based on the influence of damage initiation

  • Heng Zhao
  • Shuai Zhou
  • Ling ZhangEmail author
Original Article
  • 76 Downloads

Abstract

It is recognized that the stress-induced damage impacts the progressive failure behavior of rocks. A phenomenological model for the compressive failure of rocks is thus presented in this study. The model addresses the progressive growth of damage that leads to the strength weakening on a macroscopic scale. Considering dramatic difference between uniaxial compression and tension strengths for rocks, the admitted Mises–Schleiche Drucker–Prager strength criterion is adopted to characterize the damage initiation. On this basis, a two-parameter Weibull-type probability function is used to define the strength distribution of representative volume elements, followed by the use of damage variable for addressing the accumulated probability of failure. The proposed damage variable essentially characterizes both the critical stress level of damage initiation and progressive damage evolution law. Detailed comparisons have been carried out between the predictions and experimental observations, and issues related to the damage evolution are particularly addressed. In addition, the results further validate the proposed model considering damage initiation.

Keywords

Rocks Damage initiation MSDP criterion Loading function Strain-softening 

List of symbols

\({\sigma _{\text{1}}},{\sigma _{\text{2}}},{\sigma _{\text{3}}}\)

Major, intermediate, and minor apparent principal stresses, respectively

\(\sigma _{1}^{*},\sigma _{2}^{*},\sigma _{3}^{*}\)

Major, intermediate, and minor net principal stresses, respectively

\({\sigma _{{\text{1}},{\text{p}}}}\)

Maximum principal stress at the peak point

\({\varepsilon _{\text{1}}},{\varepsilon _{\text{2}}},{\varepsilon _{\text{3}}}\)

Major, intermediate, and minor apparent principal strains, respectively

\({\varepsilon _{{\text{1}},{\text{p}}}}\)

Maximum principal axial strain at the peak point

\(A\)

An empirical constant

\({\sigma _{\text{c}}}\)

Uniaxial compression strength

\({\sigma _{{\text{cd}}}}\)

An empirical constant

\({\sigma _{\text{t}}}\)

Uniaxial tension strength

\({C_0}\)

Uniaxial compression strength

\(\left| {{T_0}} \right|\)

Uniaxial tension strength

\(D\)

Damage variable

\(E\)

Elasticity modulus of rocks

\({F^*}\)

A fictitious loading function

\(\nu\)

Poisson’s ratio of rocks

\(\varphi ,c\)

Cohesion and angles of internal friction of rocks, respectively

\({F_0},m\)

Statistical parameters for Weibull distribution

\({I_{\text{T}}}\)

Transition condition for the MSDP criterion

\({I_{\text{1}}}\)

First invariant of the stress tensor

\({J_{\text{2}}}\)

Second invariant of the deviatoric stress tensor

\({J_{\text{3}}}\)

Third invariant of the deviatoric stress tensor

\(a,k\)

Material constants for Drucker–Prager criterion

\(a\)

Fitting constant

\(b\)

Fitting constant

\({k_0}\)

Material constant

Subscripts

\({\text{p}}\)

Particular values at peak point of stress–strain curves

Superscript

*

Net stress

Notes

Acknowledgements

This work was fully supported by the National Natural Science Foundation of China under contract nos. 51608540 and 51678231, and the Basal Research Fund Support by Hunan University.

References

  1. Aubertin M, Li L (2004) A porosity-dependent inelastic criterion for engineering materials. Int J Plast 20(12):2179–2208CrossRefGoogle Scholar
  2. Aubertin M, Simon R (1997) A damage initiation criterion for low porosity rocks. Int J Rock Mech Min Sci 34(3–4):17Google Scholar
  3. Aubertin M, Li L, Simon R, Khalfi S (1999) Formulation and application of a short-term strength criterion for isotropic rocks. Can Geotech J 36(5):947–960CrossRefGoogle Scholar
  4. Aubertin M, Li L, Simon R (2000) A multiaxial criterion for short term and long term strength of rock media. Int J Rock Mech Min Sci 37:1169–1193CrossRefGoogle Scholar
  5. Bui QV (2010) Initiation of damage with implicit gradient-enhanced damage models. Int J Solids Struct 47:2425–2435CrossRefGoogle Scholar
  6. Cai M, Kaiser PK, Tasaka Y, Maejima T, Morioka, Minami M H (2004) Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. Int J Rock Mech Min Sci 41:833–847CrossRefGoogle Scholar
  7. Cao WG, Zhang S (2005) Study on random statistical method of damage for softening hardening constitutive model of rock. In: Proceedings of the 2nd China–Japan geotechnical symposium, Shanghai, China, OctoberGoogle Scholar
  8. Cao WG, Fang ZL, Tang XJ (1998) A study of statistical constitutive model for softening and damage rocks. Chin J Rock Mechan Eng 17(6):628–633Google Scholar
  9. Cao WG, Zhao H, Li X, Zhang YJ (2010) Statistical damage model with strain softening and hardening for rocks under the influence of voids and volume changes. Can Geotech J 47(8):857–871CrossRefGoogle Scholar
  10. Castro LAM, Grabinsky MW, McCreath DR (1997) Damage initiation through extension fracturing in a moderately jointed brittle rock mass. Int J Rock Mech Min Sci 34:3–4CrossRefGoogle Scholar
  11. Diederichs MS, Kaiser PK, Eberhardt E (2004) Damage initiation and propagation in hard rock during tunneling and the influence of near-face stress rotation. Int J Rock Mech Min Sci 41:785–812CrossRefGoogle Scholar
  12. Huang RQ, Wu LZ, He Q, Li JH (2017) Stress intensity factor analysis and the stability of overhanging rock. Rock Mech Rock Eng 50(8):2135–2142CrossRefGoogle Scholar
  13. Krajcinovic D (1996) Damage mechanics. North-Holland Press, AmsterdamGoogle Scholar
  14. Krajcinovic D, Silva MAD (1982) Statistical aspects of the continuous damage theory. Int J Solids Struct 18(7):551–562CrossRefGoogle Scholar
  15. Li G, Tang CA (2015) A statistical meso-damage mechanical method for modeling trans-scale progressive failure process of rock. Int J Rock Mech Min Sci 74:133–150CrossRefGoogle Scholar
  16. Li L, Aubertin M, Simon R, Bussière B (2005) Formulation and application of a general inelastic locus for geomaterials with variable porosity. Can Geotech J 42(2):601–623CrossRefGoogle Scholar
  17. Li X, Cao WG, Su YH (2012) A statistical damage constitutive model for softening behavior of rocks. Eng Geol 143/144:1–17CrossRefGoogle Scholar
  18. 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
  19. Martin CD, Chandler NA (1994) The progressive fracture of Lac du Bonnet granite. Int J Rock Mech Min Geomech 31(6):643–659CrossRefGoogle Scholar
  20. Millard A, Massmann J, Rejeb A, Uehara S (2009) Study of the initiation and propagation of excavation damaged zones around openings in argillaceous rock. Environ Geol 57(6):1325–1335CrossRefGoogle Scholar
  21. Mohammsdi M, Taxakoli H (2015) Comparing the generalized Hoek–Brown and Mohr–Coulomb failure criteria for stress analysis on the rocks failure plane. Geomech Eng 9(1):115–124CrossRefGoogle Scholar
  22. Okubo S, Fukui K (1996) Complete stress–strain curves for various rock types in uniaxial tension. Int J Rock Mech Min Geomech 33(6):549–556CrossRefGoogle Scholar
  23. Pestman BJ, van Munster JG (1996) An acoustic emission study of damage development and stress memory effects in sandstone. Int J Rock Mech Min Geomech 33(6):585–593CrossRefGoogle Scholar
  24. Scholz CH (1968) Microfracturing and the inelastic deformation of rock in compression. J Geophys Res 73(4):1417–1432CrossRefGoogle Scholar
  25. Simone A, Askes H, Sluys LJ (2004) Incorrect initiation and propagation of failure in nonlocal and gradient-enhanced media. Int J Solids Struct 41:351–363CrossRefGoogle Scholar
  26. Wu LZ, Li B, Huang RQ, Wang QZ (2016) Study on Mode I–II hybrid fracture criteria for the stability analysis of sliding overhanging rock. Eng Geol 209:187–195CrossRefGoogle Scholar
  27. Wu LZ, Li B, Huang RQ, Sun P (2017) Experimental study and modeling of shear rheology in sandstone with non-persistent joints. Eng Geol 222:201–211CrossRefGoogle Scholar
  28. Wu LZ, Shao GQ, Huang RQ, He Q (2018) Overhanging rock: theoretical, physical and numerical modeling. Rock Mech Rock Eng 51:3585–3597CrossRefGoogle Scholar
  29. Yang SQ (2015) An experimental study on fracture coalescence characteristics of brittle sandstone specimens combined various flaws. Geomech Eng 8(4):541–557CrossRefGoogle Scholar
  30. Yang SQ, Jiang YZ, Xu WY, Chen XQ (2008) Experimental investigation on strength and failure behavior of pre-cracked marble under conventional triaxial compression. Int J Solids Struct 45:4796–4819CrossRefGoogle Scholar
  31. Yumlu M, Ozbay MU (1995) Study of the behaviour of brittle rocks under plane strain and triaxial loading conditions. Int J Rock Mech Min Geomech 32(7):725–733CrossRefGoogle Scholar
  32. Zhang P, Li N, Li XB, Nordlund E (2009) Compressive failure model for brittle rocks by shear faulting and its evolution of strength components. Int J Rock Mech Min Sci 46:830–841CrossRefGoogle Scholar
  33. Zhao H, Zhang C, Cao W, Zhao M (2016) Statistical meso-damage model for quasi-brittle rocks to account for damage tolerance principle. Environ Earth Sci 75:862CrossRefGoogle Scholar
  34. Zhao H, Shi C, Zhao M, Li X (2017) Statistical damage constitutive model for rocks considering residual strength. Int J Geomech 17(1):04016033CrossRefGoogle Scholar
  35. Zhao H, Zhou S, Zhao MH, Shi CJ (2018) Statistical micromechanics-based modeling for low-porosity rocks under conventional triaxial compression. Int J Geomech 18(5):04018019CrossRefGoogle Scholar
  36. Zhu J, Cheng H, Yao Y (2013) Statistical damage softening model of fractured rock based on SMP failure criterion and its application. Chin J Rock Mech Eng 32(Supp. 2):3160–3168Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Geotechnical EngineeringHunan UniversityChangshaPeople’s Republic of China

Personalised recommendations