Evolution models of the strength parameters and shear dilation angle of rocks considering the plastic internal variable defined by a confining pressure function

Abstract

Based on the triaxial test results of 30 types of rocks, by analysing the confining pressure function and defining a new plastic internal variable, the evolution laws of the strength parameters and shear dilation angle with a defined plastic internal variable are studied, and the corresponding evolution models are established. First, the complete stress-strain curves of 30 types of rocks are collected from published literature; from these curves, the critical equivalent plastic strains under different confining pressures are extracted. With the confining pressure and critical equivalent plastic strain data of the 30 types of rocks, fitting is performed for 23 different functions. The results demonstrate that the three-parameter allometric power-type function is the best to serve as the confining pressure function to define the plastic internal variable. Second, the strength and plastic strain data of the 30 types of rocks are extracted and transformed into the strength and plastic internal variable data. By analysing the evolution laws of the strength parameters considering the plastic internal variable, the Gaussian function is adopted to uniformly characterise the variation in the strength parameters with the plastic internal variable. Third, the shear dilation angle, confining pressure and plastic strain data of the 30 types of rocks are extracted and transformed into shear dilation angle, confining pressure and plastic internal variable data. By analysing the evolution law of the shear dilation angle considering the confining pressure and the plastic internal variable, a negative exponential function is adopted to uniformly characterise the nonlinear evolution of the shear dilation angle. Finally, the proposed evolution models of the strength parameters and the shear dilation angle are integrated into ABAQUS. By comparing the simulated complete stress-strain curves with the experimental curves of the different rock types, it is verified that the proposed models can be used to correctly simulate the nonlinear deformation and failure of different rock types. This research overcomes the shortcomings of the existing models and has wide application prospective.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Abbreviations

κ :

Plastic internal variable

ε 1y and ε 3y :

Axial and lateral strains at the initial yield point

ε 1r and ε 3r :

Axial and lateral strains at the starting point of the residual stage

\( {\varepsilon}_{1\mathrm{r}}^{\mathrm{p}} \) and \( {\varepsilon}_{3\mathrm{r}}^{\mathrm{p}} \) :

Axial and lateral critical plastic strains

\( {\overline{e}}^{\mathrm{p}} \) and \( {\overline{e}}_{\mathrm{r}}^{\mathrm{p}} \) :

Equivalent plastic strain and critical equivalent plastic strain

f(σ 3/σ c):

Confining pressure function

σ 3 and σ c :

Confining pressure and unit stress

A1, A2 and A3 :

Parameters in the three-parameter allometric power-type confining pressure function

\( \Delta {\varepsilon}_1^{\mathrm{p}} \) and \( \Delta {\varepsilon}_{\mathrm{v}}^{\mathrm{p}} \) :

Axial and volumetric plastic strain increments

E :

Young’s modulus

ν :

Poisson’s ratio

c :

(Cohesion)

φ :

Internal friction angle

ψ :

Shear dilation angle

c max, c min, κ c and ξ c :

Parameters in the evolution function of cohesion (Eq. (7))

φ max, φ min, κ φ and ξ φ :

Parameters in the evolution function of the internal friction angle (Eq. (7))

α 1, α 2, β 1 and β 2 :

Parameters in the evolution function of the shear dilation angle (Eq. (9))

Δλ n :

Plastic multiplier determined by the plastic consistency condition

D :

Elastic matrix

σ :

Stress vector

\( {\boldsymbol{\sigma}}_{\mathrm{n}}^{\mathrm{trial}} \) :

Trial stress vector

Δε n :

Strain increment vector

e p and \( {\boldsymbol{e}}_{\mathrm{r}}^{\mathrm{p}} \) :

Deviatoric plastic strain vector and critical deviatoric plastic strain vector

References

  1. ABAQUS (2014) Analysis user’s manual. Dassault Systémes Simulia Corp, Providence

    Google Scholar 

  2. Alejano LR, Alonso E (2005) Considerations of the dilatancy angle in rocks and rock masses. Int J Rock Mech Min Sci 42:481–507

    Article  Google Scholar 

  3. Arzúa J, Alejano LR (2013) Dilation in granite during servo-controlled triaxial strength tests. Int J Rock Mech Min Sci 61:43–56

    Article  Google Scholar 

  4. Chen L, Wang CP, Liu JF et al (2015) Damage and plastic deformation modeling of Beishan granite under compressive stress conditions. Rock Mech Rock Eng 48(4):1623–1633

    Article  Google Scholar 

  5. Chen T, Deng JH, Sitar N et al (2017) Stability investigation and stabilization of a heavily fractured and loosened rock slope during construction of a strategic hydropower station in China. Eng Geol 221:70–81

    Article  Google Scholar 

  6. Chen ZQ, He C, Ma GY et al (2019) Energy damage evolution mechanism of rock and its application to brittleness evaluation. Rock Mech Rock Eng 52(4):1265–1274

    Article  Google Scholar 

  7. Clausen J, Damkilde L, Andersen L (2007) An efficient return algorithm for non-associated plasticity with linear yield criteria in principal stress space. Comput Struct 85(23–24):1795–1807

    Article  Google Scholar 

  8. Duan SQ, Feng XT, Jiang Q et al (2017) In situ observation of failure mechanisms controlled by rock masses with weak interlayer zones in large underground cavern excavations under high geostress. Rock Mech Rock Eng 50(9):2465–2493

    Article  Google Scholar 

  9. Feng XT, Guo HS, Yang CX et al (2018) In situ observation and evaluation of zonal disintegration affected by existing fractures in deep hard rock tunneling. Eng Geol 242:1–11

    Article  Google Scholar 

  10. Guan XR, Zhao YL, Wang JJ et al (2015) Numerical analysis of quasi-steady flow characteristics in large diameter pipes with low liquid loading under high pressure. J Nat Gas Sci Eng 26:907–920

    Article  Google Scholar 

  11. Guo SF, Qi SW, Zhan ZF et al (2017) Plastic-strain-dependent strength model to simulate the cracking process of brittle rocks with an existing non-persistent joint. Eng Geol 231:114–125

    Article  Google Scholar 

  12. Hajiabdolmajid V, Kaiser PK, Martin CD (2002) Modelling brittle failure of rock. Int J Rock Mech Min Sci 39:731–741

    Article  Google Scholar 

  13. Hu QJ, Shi RD, Zheng LN et al (2018) Progressive failure mechanism of a large bedding slope with a strain-softening interface. Bull Eng Geol Environ 77(1):69–85

    Article  Google Scholar 

  14. Huang JS, Griffiths DV (2008) Observations on return mapping algorithms for piecewise linear yield criteria. Int J Geomech 8(4):253–265

    Article  Google Scholar 

  15. Huang F, Zhu HH, Xu QW et al (2013) The effect of weak interlayer on the failure pattern of rock mass around tunnel – scaled model tests and numerical analysis. Tunn Undergr Space Tech 35:207–218

    Article  Google Scholar 

  16. Jia CJ, Xu WY, Wang SS et al (2019) Experimental analysis and modeling of the mechanical behavior of breccia lava in the dam foundation of the Baihetan Hydropower Project. Bull Eng Geol Environ 78(4):2681–2695

    Article  Google Scholar 

  17. Jiang Q, Zhong S, Cui J et al (2016) Statistical characterization of the mechanical parameters of intact rock under triaxial compression: an experimental proof of the Jinping marble. Rock Mech Rock Eng 49(12):4631–4646

    Article  Google Scholar 

  18. Jongpradist P, Tunsakul J, Kongkitkul W et al (2015) High internal pressure induced fracture patterns in rock masses surrounding caverns: experimental study using physical model tests. Eng Geol 197:158–171

    Article  Google Scholar 

  19. Kumar R, Sharma KG, Varadarajan A (2010) Post-peak response of some metamorphic rocks of India under high confining pressures. Int J Rock Mech Min Sci 47:1357–1362

    Article  Google Scholar 

  20. Liu J, Li JP (2011) Experimental research on sandstone pre-peak unloading process under high confining pressure. Chin J Rock Mech Eng 30(3):473–479 (in Chinese)

    Google Scholar 

  21. Liu ZB, Shao JF (2017) Strength behavior, creep failure and permeability change of a tight marble under triaxial compression. Rock Mech Rock Eng 50(3):529–541

    Article  Google Scholar 

  22. Lu YL, Wang LG, Yang F et al (2010) Post-peak strain softening mechanical properties of weak rock. Chin J Rock Mech Eng 29(3):640–648 (in Chinese)

    Google Scholar 

  23. Martin CD (1997) Seventeenth Canadian geotechnical colloquium: the effect of cohesion loss and stress path on brittle rock strength. Can Geotech J 34(5):698–725

    Article  Google Scholar 

  24. Mouzannar H, Bost M, Leroux M et al (2017) Experimental study of the shear strength of bonded concrete-rock interfaces: surface morphology and scale effect. Rock Mech Rock Eng 50(10):2601–2625

    Article  Google Scholar 

  25. Nian TF, Li P, Mao Y et al (2018) Connections between chemical composition and rheology of aged base asphalt binders during repeated freeze-thaw cycles. Constr Build Mater 159:338–350

    Article  Google Scholar 

  26. Peng J, Cai M, Rong G et al (2017) Determination of confinement and plastic strain dependent post-peak strength of intact rocks. Eng Geol 218:187–196

    Article  Google Scholar 

  27. Pourhosseini O, Shabanimashcool M (2014) Development of an elasto-plastic constitutive model for intact rocks. Int J Rock Mech Min Sci 66:1–12

    Article  Google Scholar 

  28. Rafiei Renani H, Martin CD (2018a) Cohesion degradation and friction mobilization in brittle failure of rocks. Int J Rock Mech Min Sci 106:1–13

    Article  Google Scholar 

  29. Rafiei Renani H, Martin CD (2018b) Modeling the progressive failure of hard rock pillars. Tunn Undergr Space Tech 74:71–81

    Article  Google Scholar 

  30. Sun C, Zhang SG, Jia BX et al (2015) Physical and numerical model tests on post-peak mechanical properties of granite. Chinese J Geo Eng 37(5):847–852 (in Chinese)

    Google Scholar 

  31. Tian HM, Chen WZ, Yang DS et al (2016) Numerical analysis on the interaction of shotcrete liner with rock for yielding supports. Tunn Undergr Space Tech 54:20–28

    Article  Google Scholar 

  32. Walton G, Diederichs MS (2015) A new model for the dilation of brittle rocks based on laboratory compression test data with separate treatment of dilatancy mobilization and decay. Geotech Geol Eng 33(3):661–679

    Article  Google Scholar 

  33. Walton G, Arzúa J, Alejano LR et al (2015) A laboratory-testing-based study on the strength, deformability, and dilatancy of carbonate rocks at low confinement. Rock Mech Rock Eng 48(3):941–958

    Article  Google Scholar 

  34. Walton G, Hedayat A, Kim E et al (2017) Post-yield strength and dilatancy evolution across the brittle–ductile transition in Indiana limestone. Rock Mech Rock Eng 50(7):1691–1710

    Article  Google Scholar 

  35. Wang JC, Wang ZH, Yang SL (2017) A coupled macro- and meso-mechanical model for heterogeneous coal. Int J Rock Mech Min Sci 94:64–81

    Article  Google Scholar 

  36. Wang Q, Jiang B, Pan R et al (2018) Failure mechanism of surrounding rock with high stress and confined concrete support system. Int J Rock Mech Min Sci 102:89–100

    Article  Google Scholar 

  37. Wang M, Zheng DJ, Niu SJ et al (2019) Large deformation of tunnels in longwall coal mines. Environ Earth Sci 78:45

    Article  Google Scholar 

  38. Xia YJ, Li LC, Tang CA et al (2017) A new method to evaluate rock mass brittleness based on stress-strain curves of class I. Rock Mech Rock Eng 50(5):1123–1139

    Article  Google Scholar 

  39. Xie N, Zhu QZ, Xu LH et al (2011) A micromechanics-based elastoplastic damage model for quasi-brittle rocks. Comput Geotech 38:970–977

    Article  Google Scholar 

  40. Yang SQ (2016) Experimental study on deformation, peak strength and crack damage behavior of hollow sandstone under conventional triaxial compression. Eng Geol 213:11–24

    Article  Google Scholar 

  41. Yang SQ, Jing HW, Wang SY (2012) Experimental investigation on the strength, deformability, failure behavior and acoustic emission locations of red sandstone under triaxial compression. Rock Mech Rock Eng 45(4):583–606

    Article  Google Scholar 

  42. Yang FJ, Zhou H, Zhang CQ et al (2018) An elastoplastic coupling mechanical model for hard and brittle marble with consideration of the first stress invariant effect. Eur J Environ Civ Eng 22(4):405–428

    Article  Google Scholar 

  43. Yang SQ, Tian WL, Jing HW et al (2019) Deformation and damage failure behavior of mudstone specimens under single-stage and multi-stage triaxial compression. Rock Mech Rock Eng 52(3):673–689

    Article  Google Scholar 

  44. Yao MD, Rong G, Zhou CB et al (2016) Effects of thermal damage and confining pressure on the mechanical properties of coarse marble. Rock Mech Rock Eng 49(6):2043–2054

    Article  Google Scholar 

  45. Yi K, Kang HP, Ju WJ et al (2020) Synergistic effect of strain softening and dilatancy in deep tunnel analysis. Tunn Undergr Space Tech 97:103280

    Article  Google Scholar 

  46. Yu HC, Zhang XS, Liu HD et al (2016) Stress relaxation behavior of silty mudstone considering the effect of confining pressure. Environ Earth Sci 75:1001

    Article  Google Scholar 

  47. Zhang JC (2018) Experimental and modelling investigations of the coupled elastoplastic damage of a quasi-brittle rock. Rock Mech Rock Eng 51(2):465–478

    Article  Google Scholar 

  48. Zhang HJ, Li CC (2019) Effects of confining stress on the post-peak behaviour and fracture angle of Fauske marble and Iddefjord granite. Rock Mech Rock Eng 52(5):1377–1385

    Article  Google Scholar 

  49. Zhang K, Zhou H, Shao JF (2013a) An experimental investigation and an elastoplastic constitutive model for a porous rock. Rock Mech Rock Eng 46(6):1499–1511

    Article  Google Scholar 

  50. Zhang L, Gao S, Wang Z et al (2013) Analysis of marble failure energy evolution under loading and unloading conditions. Chin J Rock Mech Eng 32(8):1572–1578 (in Chinese)

    Google Scholar 

  51. Zhang CQ, Feng XT, Zhou H et al (2014) Rock mass damage induced by rockbursts occurring on tunnel floors: a case study of two tunnels at the Jinping II Hydropower Station. Environ Earth Sci 71(1):441

    Article  Google Scholar 

  52. Zhang Q, Jiang BS, Lv HJ (2016) Analytical solution for a circular opening in a rock mass obeying a three-stage stress-strain curve. Int J Rock Mech Min Sci 86:16–22

    Article  Google Scholar 

  53. Zhang J, Ai C, Li YW et al (2018) Energy-based brittleness index and acoustic emission characteristics of anisotropic coal under triaxial stress condition. Rock Mech Rock Eng 51(11):3343–3360

    Article  Google Scholar 

  54. Zhang SG, Liu JQ, Chen PP et al (2015) Experimental research on mechanical properties of granite under water-rock coupling. Chin J Rock Mech Eng 34(3):520–527 (in Chinese)

  55. Zhao XG, Cai M (2010) A mobilized dilation angle model for rocks. Int J Rock Mech Min Sci 47:368–384

    Article  Google Scholar 

  56. Zhou YD, Deng A, Wang C (2013) Finite-difference model for one-dimensional electro-osmotic consolidation. Comput Geotech 54:152–165

    Article  Google Scholar 

  57. Zong YJ, Han LJ, Wei JJ et al (2016) Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol 26(4):601–607

    Article  Google Scholar 

  58. Zong YJ, Han LJ, Meng QB et al (2020) Strength properties and evolution laws of cracked sandstone samples in re-loading tests. Int J Min Sci Technol 30(2):251–258

    Article  Google Scholar 

Download references

Acknowledgements

No conflict of interest exits in the submission of this manuscript that is approved by all authors for publication. We declare that the work described is an original research that has not been published previously. All the authors listed have approved the manuscript that is enclosed.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Chengxue She.

Appendix

Appendix

In Fig. 13, the values of the strength parameters for different plastic internal variables of rocks - are plotted, and the fitting curves are also drawn via Eq. (7) with the parameter values listed in Table 8.

Fig. 13
figure13

Strength parameter values for different κ values (open squares represent the cohesion, solid circles represent the internal friction angle) of rocks - , and the curves fitted by Eq. (7). ⑮

Table 8 Parameter values of Eq. (7) fitted by the strength parameter data of rocks -

In Fig. 14, the shear dilation angle values assumed for plastic internal variables under different confining pressures of rocks - are displayed, and the fitting curves by Eq. (9) are also given with the parameter values listed in Table 9.

Fig. 14
figure14

Calculated shear dilation angles (symbols) of rocks - , and the curves fitted by Eq. (9)

Table 9 Parameter values of Eq. (9) fitted by the shear dilation angle data of rocks -

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jin, J., She, C. & Shang, P. Evolution models of the strength parameters and shear dilation angle of rocks considering the plastic internal variable defined by a confining pressure function. Bull Eng Geol Environ (2021). https://doi.org/10.1007/s10064-020-02040-1

Download citation

Keywords

  • Plastic internal variable
  • Confining pressure function
  • Strength parameters
  • Shear dilation angle
  • Evolution model