# Study on the damage-softening constitutive model of rock and experimental verification

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## Abstract

A damage-softening model is presented to describe the stress–strain curve of rock. By comparing the Hoek–Brown (H–B) and Mohr–Coulomb (M–C) yield criterion, the equivalent M–C yield criterion is selected as the strength criterion in this model. To better characterize the rock damage and failure processes with considering the relationship between damage and deformation, the concept of yield stress ratio is introduced to describe the yield-strengthening deformation before rock peak stress. Damage events are described by two cumulative damage evolution laws. The evolution equations of tensile and shear damage are presented based on the equivalent plastic strains, and the maximum value between tensile and shear damage represents the total damage for rock. Considering that rock cannot bear tensile load after tensile failure but still has a certain shear strength, its tensile and shear strengths are small after shear failure. The elastic modulus is affected by tensile damage, whereas the angle of internal friction, the cohesion, and dilation angles are influenced by shear damage. The proposed damage-softening model describes the strain softening, brittle stress drop, and residual strength of rock after peak stress, and finally the model is implemented in FLAC3D. Comparing the test and the numerical calculation results, the damage-softening model better describes the total stress–strain curve of rock.

## Keywords

Rock deformation Damage evolution Softening Numerical calculation## List of symbols

*D*Damage variable

*D*^{t}Tensile damage

*D*^{s}Shear damage

*E*′Elastic modulus at yield strengthening stage

*E*,*E*_{s}Young’s modulus

*E*_{0}Initial value of elastic modulus

*K*,*G*Bulk and shear modulus

*k*_{1}Ratio of yield stress to peak strength

*k*_{2}Ratio of elastic modulus at yield strengthening stage to Young’s modulus

- H–B
Hoek–Brown yield criterion

- M–C
Mohr–Coulomb yield criterion

*m*_{i}Dimensionless empirical constant

*f*^{s}Shear yield plane

*g*^{t}Tensile yield plane

*h*Boundary plane between shear and tensile yield plane

- T
Tensile

- T–S
Tensile–shear

- C–S
Compressive–shear

*c*Cohesion

*c*′Equivalent cohesion

*c*_{p}Cohesion at peak strength point

*c*_{r}Cohesion at residual strength stage

*φ*Angle of internal friction

*φ*′Equivalent internal friction angle

*φ*_{p}Angle of internal friction at peak strength point

*φ*_{r}Angle of internal friction at residual strength stage

*ψ*Dilation angle

*σ*_{1},*σ*_{3}Major and minor principal stresses

*σ*_{c}Uniaxial compressive strength of rock

*σ*_{3max}The maximum confining pressure

*σ*_{t}Tensile strength

*σ*_{yield}Yield stress

*σ*_{peak}Peak stress

*σ*_{t}Tensile strength

*σ*_{t0}Initial value of tensile strength

*ε*^{ps}Equivalent shear plastic strain

*ε*^{pt}Equivalent tensile plastic strain

*ε*^{psL}Critical equivalent plastic strain of rock entering the residual deformation stage

*ω*Shear strength parameter [i.e., fraction angle (

*φ*), cohesion (*c*), and dilation angle (*ψ*)]*ω*^{p}Initial values of shear strength parameters

*ω*^{r}Residual values of shear strength parameters

**σ**_{I}^{e},**σ**_{N}^{e}Stress matrix of the unit before and after updating

- ∆
**ε**_{I} Strain matrix of the unit

**D**Stiffness matrix of the unit

- \(\Delta\varvec{\varepsilon}_{ij}^{\text{ps}}\)
Increment of shear plastic strain

- \(\Delta\varvec{\varepsilon}_{ij}^{\text{pt}}\)
Increment of tensile plastic strain

- Δ
*K*^{s} Increment of equivalent shear plastic strain

- Δ
*K*^{t} Increment of equivalent tensile plastic strain

## Notes

### Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grants 51734009 & 51179189), the Fifth “333” Project of Jiangsu Province (2016) and the China Postdoctoral Science Foundation (Grant 2018M642360). The authors would like to express their sincere gratitude to the editor and two anonymous reviewers for their valuable comments which have greatly improved this paper.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## References

- 1.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. Mech. Mater.
**35**, 587–601 (2003)CrossRefGoogle Scholar - 2.Lyons, K.D., Honeygan, S., Moroz, T.: NETL extreme drilling laboratory studies high pressure high temperature drilling phenomena. J. Energy Resour. Technol.
**130**, 791–796 (2007)Google Scholar - 3.Barla, G.: Contributions to the understanding of time dependent behaviour in deep tunnels. Geomech. Tunn.
**4**, 255–265 (2011)CrossRefGoogle Scholar - 4.Shafiei, A., Dusseault, M.B.: Geomechanics of thermal viscous oil production in sandstones. J. Pet. Sci. Eng.
**103**, 121–139 (2013)CrossRefGoogle Scholar - 5.Holton, D., Myers, S., Carta, G., et al.: Application of a novel approach to assess the thermal evolution processes associated with the disposal of high-heat-generating waste in a geological disposal facility. Eng. Geol.
**211**, 102–119 (2016)CrossRefGoogle Scholar - 6.Frantziskonis, G., Desai, C.S.: Elastoplastic model with damage for strain softening geomaterials. Acta Mech.
**68**, 151–170 (1987)CrossRefGoogle Scholar - 7.Alonso, E., Alejano, L.R., Varas, F., et al.: Ground response curves for rock masses exhibiting strain-softening behaviour. Int. J. Numer. Anal. Methods Geomech.
**27**, 1153–1185 (2003)CrossRefzbMATHGoogle Scholar - 8.Lee, Y.K., Pietruszczak, S.: A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass. Tunn. Undergr. Space Technol.
**23**, 588–599 (2008)CrossRefGoogle Scholar - 9.Li, X., Cao, W.G., Su, Y.H.: A statistical damage constitutive model for softening behavior of rocks. Eng. Geol.
**143**, 1–17 (2012)CrossRefGoogle Scholar - 10.Krajcinovic, D., Fonseka, G.U.: The continuous damage theory of brittle materials, part 1: general theory. J. Appl. Mech. Trans. ASME
**48**, 809–815 (1981)CrossRefzbMATHGoogle Scholar - 11.Cao, W.G., Zhao, H., Li, X., et al.: Statistical damage model with strain softening and hardening for rocks under the influence of voids and volume changes. Can. Geotech. J.
**47**, 857–871 (2010)CrossRefGoogle Scholar - 12.Zhao, L.Y., Zhu, Q.Z., Xu, W.Y., et al.: A unified micromechanics-based damage model for instantaneous and time-dependent behaviors of brittle rocks. Int. J. Rock Mech. Min. Sci.
**84**, 187–196 (2016)CrossRefGoogle Scholar - 13.Atkinson, B.K.: Subcritical crack propagation in rocks: theory, experimental results and applications. J. Struct. Geol.
**4**, 41–56 (1982)CrossRefGoogle Scholar - 14.Fanella, D., Krajcinovic, D.: A micromechanical model for concrete in compression. Eng. Fract. Mech.
**29**, 49–66 (1988)CrossRefGoogle Scholar - 15.Xie, N., Zhu, Q.Z., Xu, L.H., et al.: A micromechanics-based elastoplastic damage model for quasi-brittle rocks. Comput. Geotech.
**38**, 970–977 (2011)CrossRefGoogle Scholar - 16.Huang, Y., Shao, J.F.: A micromechanical analysis of time-dependent behavior based on subcritical damage in claystone. Int. J. Damage Mech.
**22**, 773–790 (2012)CrossRefGoogle Scholar - 17.Zhou, X.P., Xia, E.M., Yang, H.Q., et al.: Different crack sizes analyzed for surrounding rock mass around underground caverns in Jinping I hydropower station. Theor. Appl. Fract. Mech.
**57**, 19–30 (2012)CrossRefGoogle Scholar - 18.Feng, X.Q., Yu, S.W.: Micromechanical modelling of tensile response of elastic-brittle materials. Int. J. Solids Struct.
**32**, 3359–3372 (1995)CrossRefzbMATHGoogle Scholar - 19.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. Geothermics
**72**, 124–137 (2018)CrossRefGoogle Scholar - 20.Yang, S.Q., Hu, B.: Creep and long-term permeability of a red sandstone subjected to cyclic loading after thermal treatments. Rock Mech. Rock Eng.
**51**, 2981–3004 (2018)CrossRefGoogle Scholar - 21.Khan, D., Singh, S., Needleman, A.: Finite deformation analysis of crack tip fields in plastically compressible hardening–softening–hardening solids. Acta. Mech. Sin.
**33**, 148–158 (2017)CrossRefGoogle Scholar - 22.Yang, S.Q., Huang, Y.H., Jing, H.W., et al.: Discrete element modeling on fracture coalescence behavior of red sandstone containing two unparallel fissures under uniaxial compression. Eng. Geol.
**178**, 28–48 (2014)CrossRefGoogle Scholar - 23.Zhou, X.P., Zhang, Y.X., Ha, Q.L., et al.: Micromechanical modelling of the complete stress–strain relationship for crack weakened rock subjected to compressive loading. Rock Mech. Rock Eng.
**41**, 747–769 (2008)CrossRefGoogle Scholar - 24.Swoboda, G., Yang, Q.: An energy-based damage model of geomaterials-I. Formulation and numerical results. Int. J. Solids Struct.
**36**, 1719–1734 (1999)CrossRefzbMATHGoogle Scholar - 25.Zhou, X.P., Yang, H.Q.: Micromechanical modeling of dynamic compressive responses of mesoscopic heterogeneous brittle rock. Theor. Appl. Fract. Mech.
**48**, 1–20 (2007)CrossRefGoogle Scholar - 26.Ren, Z., Peng, X., Wan, L.: A three-dimensional micromechanics model for the damage of brittle materials based on the growth and unilateral effect of elliptic microcracks. Eng. Fract. Mech.
**78**, 274–288 (2011)CrossRefGoogle Scholar - 27.Yu, M.H., Zan, Y.W., Zhao, J., et al.: A unified strength criterion for rock material. Int. J. Rock Mech. Min. Sci.
**39**, 975–989 (2002)CrossRefGoogle Scholar - 28.Zhou, X.P., Shou, Y.D., Qian, Q.H., et al.: Three-dimensional nonlinear strength criterion for rock-like materials based on the micromechanical method. Int. J. Rock Mech. Min. Sci.
**72**, 54–60 (2014)CrossRefGoogle Scholar - 29.Bi, J., Zhou, X.P., Qian, Q.H.: The 3D Numerical simulation for the propagation process of multiple pre-existing flaws in rock-like materials subjected to biaxial compressive loads. Rock Mech. Rock Eng.
**49**, 1611–1627 (2016)CrossRefGoogle Scholar - 30.Singh, M., Singh, B.: Modified Mohr-Coulomb criterion for non-linear triaxial and polyaxial strength of intact rocks. Int. J. Rock Mech. Min. Sci.
**48**, 546–555 (2011)CrossRefGoogle Scholar - 31.Nadai, A.: Theory of Flow and Fracture of Solids. McGraw Hill, New York (1950)Google Scholar
- 32.Jaeger, J.C., Cook, N.G.W.: Fundamentals of Rock Mechanics, 3rd edn. Chapman & Hall, London (1979)Google Scholar
- 33.Zhang, F., Sheng, Q., Zhu, Z.Q., et al.: Study on post-peak mechanical behaviour and strain-softening model of three gorges granite. Chin. J. Rock Mech. Eng.
**27**, 2651–2655 (2008). (in Chinese)Google Scholar - 34.Yang, S.Q., Xu, W.Y., Su, C.D.: Study on the deformation failure and energy properties of marble specimen under triaxial compression. Eng. Mech.
**24**, 136–142 (2007). (in Chinese)Google Scholar - 35.Zhang, C.H., Zhao, Q.S.: Triaxial tests of effects of varied saturations on strength and modulus for sandstone. Rock Soil Mech.
**35**, 951–958 (2014). (in Chinese)Google Scholar - 36.Yang, Y.J., Song, Y., Chen, S.J.: Test study of coal’s strength and deformation characteristics under triaxial compression. J. China Coal Soc.
**31**, 150–153 (2006). (in Chinese)Google Scholar - 37.Zhang, L.M., Wang, Z.Q., Li, H.F., et al.: Theoretical and experimental study on siltstone brittle stress drop in post-failure region. J. Exp. Mech.
**23**, 234–240 (2008). (in Chinese)Google Scholar - 38.Yu, H.C., Li, Y.L.: Research on conventional triaxial compression test and constitutive model of silty mudstone. Yangtze River
**42**, 56–60 (2011). (in Chinese)Google Scholar - 39.Detournay, E.: Elastoplastic model of a deep tunnel for a rock with variable dilatancy. Rock Mech. Rock Eng.
**19**, 99–108 (1986)CrossRefGoogle Scholar - 40.Hoek, E., Brown, E.T.: Practical estimates of rock mass strength. Int. J. Rock Mech. Min. Sci.
**34**, 1165–1186 (1997)CrossRefGoogle Scholar - 41.Vermeer, P.A., Borst, R.: Non-associated plasticity for soils, concrete and rock. Heron
**29**, 1–64 (1984)Google Scholar - 42.Kachanov, L.M.: Time of the rupture process under creep conditions. Izv Akad Nauk SSSR Otdelenie Tekniches
**8**, 26–31 (1958)Google Scholar - 43.Lemaitre, J.: Evaluation of dissipation and damage in metals submitted to dynamic loading. In: Proceedings ICM-1, Kyoto, Japan (1971)Google Scholar
- 44.Kyoya, T., Ichikawa, Y., Kawamoto, T.: Damage mechanics theory for discontinuous rock mass. In: Proceedings of 5th International Conference on Numerical Methods in Geomechanics, Nagoya, Apr 1–5 (1985)Google Scholar
- 45.Dragon, A., Mroz, Z.: A continuum model for plastic–brittle behaviour of rock and concrete. Int. J. Eng. Sci.
**17**, 121–137 (1979)CrossRefzbMATHGoogle Scholar - 46.Mazars, J., Pijaudier-Cabot, G.: Continuum damage theory-application to concrete. J. Eng. Mech.
**115**, 345–365 (1989)CrossRefGoogle Scholar - 47.Hoek, E., Brown, E.T.: Underground Excavations in Rock. Institution of Mining and Metallurgy, London (1980)Google Scholar
- 48.Hoek, E., Brown, E.T.: Empirical strength criterion for rock masses. J. Geotech. Geoenviron. Eng.
**106**, 1013–1035 (1980)Google Scholar - 49.Hoek, E., Martin, C.D.: Fracture initiation and propagation in intact rock—a review. J. Rock Mech. Geotech. Eng.
**6**, 287–300 (2014)CrossRefGoogle Scholar - 50.Brace, W.F.: Brittle Fracture of Rocks. State of Stress in the Earth’s Crust, 111–174. American Elsevier, New York (1964)Google Scholar
- 51.Evert, H.: Strength of jointed rock masses. Geotechnique
**23**, 187–223 (1983)Google Scholar - 52.Bobich, J.K.: Experimental analysis of the extension to shear fracture transition in Berea sandstone. M.S. thesis. Texas A and M University, Texas (2005)Google Scholar
- 53.Hoek, E., Bieniawski, Z.T.: Brittle fracture propagation in rock under compression. Int. J. Fract.
**1**, 137–155 (1965)CrossRefGoogle Scholar - 54.Hoek, E., Carranza-Torres C., Corkum. B.: Hoek–Brown failure criterion-2002 edition. In: Proceedings of the Fifth North American Rock Mechanics Symposium, vol.
**1**, pp. 18–22 (2002)Google Scholar - 55.Itasca Consulting Group, Inc. Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 3.0, and User’s Manual. Itasca Consulting Group, Inc., Minneapolis (2005)Google Scholar
- 56.Xu, P., Yang, S.Q., Cheng, G.F.: Modified Burgers model of rocks and its experimental verification. J. China Coal Soc.
**39**, 1993–2000 (2014). (in Chinese)Google Scholar - 57.Yang, S.Q., Jing, H.W., Li, Y.S., et al.: Experimental investigation on mechanical behavior of coarse marble under six different loading paths. Exp. Mech.
**51**, 315–334 (2011)CrossRefGoogle Scholar