Skip to main content
SpringerLink
Log in

Critical state behaviors of a coarse granular soil under generalized stress conditions

  • Original Paper
  • Published:
Granular Matter Aims and scope Submit manuscript

Abstract

Critical state line (CSL) is the central concept in soil mechanics. A series of true triaxial compression tests under the constant-\({p}'\) and constant-b loading condition were carried out to investigate the CSL of a coarse granular soil. It was observed that the intermediate principal stress ratio (i.e., the b-value) greatly influenced the CSLs in both \(q{-}{p}'\) and \(e{-}{p}'\) spaces. The CSL slope in the \(q{-}{p}'\) space decreased with an increase in b-value. The intercept and gradient of the CSL in the \(e{-}{p}'\) space decreased with increasing b-value. CSLs incorporating the effects of the b-value in \(q{-}{p}'\) and \(e{-}{p}'\) spaces were extended to three-dimensional critical state surfaces (TCSSs) in \(q{-}{p}'{-}b\) and \(e{-}{p}'{-}b\) spaces. Two empirical equations were proposed for the two TCSSs in \(q{-}{p}'{-}b\) and \(e{-}{p}'{-}b\) spaces, respectively. The predictions by the two equations were in good agreement with the corresponding experimental data. The relationship between the excess friction angle (the difference between the peak state and critical state friction angles) and initial state parameter was influenced by the b-value. However, the relationship between the maximum dilatancy and initial state parameter was independent of the b-value.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Abbreviations

b :

Intermediate principal stress ratio

\(d_\mathrm{M}\) :

Maximum particle size (unit: mm)

\(C_\mathrm{u}\) :

Coefficient of uniformity

\(C_\mathrm{c}\) :

Coefficient of curvature

\(F_\mathrm{c}\) :

Fines content (unit: %)

\({\sigma }'_{1} ,{\sigma }'_{2}\) and \({\sigma }'_{3}\) :

Major, intermediate and minor effective principal stresses, respectively (unit: kPa)

\({p}'\) :

Mean effective stress (unit: kPa)

\({p}'_{\mathrm{cs}}\) :

Mean effective stress at the critical state (unit: kPa)

\(p_\mathrm{c}\) :

Confining pressure (unit: kPa)

\(p_\mathrm{a}\) :

Atmospheric pressure (unit: kPa)

q :

Deviatoric stress (unit: kPa)

\(q_{\mathrm{cs}}\) :

Critical state deviatoric stress (unit: kPa)

\(\eta \) :

Stress ratio

\(\varepsilon _\mathrm{a}\) :

Axial strain (unit: %)

\(\varepsilon _\mathrm{v}\) :

Volumetric strain (unit: %)

\(\hbox {d}\varepsilon _{v}^{p}\) :

Plastic volumetric strain increment (unit: %)

\(\hbox {d}\varepsilon _{s}^{p}\) :

Plastic shear strain increment (unit: %)

\(\hbox {d}\varepsilon _{v}\) :

Total volumetric strain increment (unit: %)

\(\hbox {d}\varepsilon _{s}\) :

Total shear strain increment (unit: %)

\(M_{\mathrm{cs}}\) :

Critical state stress ratio

\(M_{\mathrm{cs0}},\quad \chi _\mathrm{M}^\mathrm{b}\) and \(k_\mathrm{M}\) :

Material constants of CSL in the \(q{-}{p}'\) space

e :

Current void ratio

\(e_{0}\) :

Initial void ratio

\(e_{\mathrm{cs}}\) :

Critical state void ratio

\(e_{\mathrm{cs0}} ,\lambda _{\mathrm{cs}}\) and \(\xi \) :

Material constants of CSL in the \(e{-}{p}'\) space

\(e_{\mathrm{cs}0}^{0}\) and \(\chi _{\mathrm{e}}^{\mathrm{b}}\) :

Material constants in relation to \(e_{\mathrm{cs0}}\)

\(\lambda _{\mathrm{cs}}^{0}\) and \(\chi _{\lambda }^\mathrm{b}\) :

Material constants in relation to \(\lambda _{\mathrm{cs}}\)

d :

Dilatancy

\(d_{\max }\) :

Maximum dilatancy

\(\psi \) :

State parameter

\(\varphi _{\mathrm{cs}}\) :

Critical state friction angle (unit: \(^\circ \))

\(\varphi _{\mathrm{ps}}\) :

Peak state friction angle (unit: \(^\circ \))

\(\varphi _{\mathrm{ex}}\) :

Excess friction angle (unit: \(^\circ \))

\(\left( {{{\sigma }'_{1} }/{{\sigma }'_{3}}}\right) _{\mathrm{const}}\) :

Value of \(\left( {{{\sigma }'_{1}}/{{\sigma }'_{3}}} \right) \) at a constant volumetric strain

\(\left( {{{\sigma }'_{1}}/{{\sigma }'_{3}}}\right) _{\max }\) :

Maximum value of \(\left( {{{\sigma }'_{1}}/{{\sigma }'_{3}}}\right) \)

\(\chi _{\upvarphi }\) and \(\psi _\mathrm{a}\) :

Material constants in relation to \(\varphi _{\mathrm{ex}}\) (unit of \(\chi _{\upvarphi }\): \(^\circ \))

\(d_{0}\) and \(\chi _\mathrm{d}\) :

Material constants in relation to \(d_{\max }\)

\(\theta _{\max }\) :

Maximum dilatancy angle (unit: \(^\circ \))

\(\theta _{0}\) and \(\chi _{\uptheta }\) :

Material constants in relation to \(\theta _{\max }\) (unit: \(^\circ \))

References

  1. Roscoe, K.H., Schofield, A.N., Thurairajah, A.: Yielding of clays in states wetter than critical. Geotechnique 13, 211–240 (1963)

    Article  Google Scholar 

  2. Schofield, A.N., Wroth, C.P.: Critical State Soil Mechanics. McGraw-Hill, New York (1968)

    Google Scholar 

  3. Lade, P.V., Duncan, J.M.: Cubical triaxial tests on cohesionless soil. J. Soil Mech. Found. Div. ASCE 99, 793–812 (1973)

    Google Scholar 

  4. Yamada, Y., Ishihara, K.: Anisotropic deformation characteristics of sand under three dimensional stress conditions. Soils Found. 19, 79–94 (1979)

    Article  Google Scholar 

  5. Janardhanam, R., Desai, C.S.: Three-dimensional testing and modeling of ballast. J. Geotech. Eng. 109, 783–796 (1983)

    Article  Google Scholar 

  6. Ochiai, H., Lade, P.V.: Three Dimens. Behav. Sand Anisotropic Fabr. J. Geotech. Eng. 109, 1313–1328 (1983)

    Article  Google Scholar 

  7. Desai, C.S., Faruque, M.O.: Constitutive model for (geological) materials. J. Eng. Mech. 110, 1391–1408 (1984)

    Article  MATH  Google Scholar 

  8. Michelis, P.: Polyaxial yielding of granular rock. J. Eng. Mech. 111, 1049–1066 (1985)

    Article  Google Scholar 

  9. Chu, J., Lo, S.C.R., Lee, I.K.: Strain softening and shear band formation of sand in multi-axial testing. Geotechnique 46, 63–82 (1996)

    Article  Google Scholar 

  10. Shapiro, S., Yamamuro, J.A.: Effects of silt on three-dimensional stress–strain behavior of loose sand. J. Geotech. Geoenviron. Eng. 129, 1–11 (2002)

    Article  Google Scholar 

  11. Abelev, A.V., Lade, P.V.: Effects of cross anisotropy on three-dimensional behavior of sand. I: Stress–strain behavior and shear banding. J. Eng. Mech. 129, 160–166 (2003)

    Article  Google Scholar 

  12. Lade, P.V.: Analysis and prediction of shear banding under 3D conditions inn granular materials. Soils Found. 43, 161–172 (2003)

    Article  Google Scholar 

  13. Choi, C., Arduino, P., Harney, M.D.: Development of a true triaxial apparatus for sands and gravels. Geotech. Test. J. 31, 1–13 (2008)

  14. Xiao, Y., Liu, H., Chen, Y., Chu, J.: Influence of intermediate principal stress on the strength and dilatancy behavior of rockfill material. J. Geotech. Geoenviron. Eng. 140, 04014064 (2014)

    Article  Google Scholar 

  15. Xiao, Y., Liu, H.L., Zhu, J.G., Shi, W.C.: Dilatancy equation of rockfill material under the true triaxial stress condition. Sci. China Tech. Sci. 54, 175–184 (2011)

    Article  Google Scholar 

  16. Zhao, X., Evans, T.M.: Numerical analysis of critical state behaviors of granular soils under different loading conditions. Granul. Matter 13, 751–764 (2011)

    Article  Google Scholar 

  17. Barreto, D., O’Sullivan, C.: The influence of inter-particle friction and the intermediate stress ratio on soil response under generalised stress conditions. Granul. Matter 14, 505–521 (2012)

    Article  Google Scholar 

  18. Sazzad, M.M., Suzuki, K.: Density dependent macro–micro behavior of granular materials in general triaxial loading for varying intermediate principal stress using DEM. Granul. Matter 15, 583–593 (2013)

    Article  Google Scholar 

  19. Huang, X., O’Sullivan, C., Hanley, K.J., Kwok, C.Y.: Discrete-element method analysis of the state parameter. Geotechnique 64, 954–965 (2014)

    Article  Google Scholar 

  20. Huang, X., O’Sullivan, C., Hanley, K.J., Kwok, C.Y.: DEM analysis of the influence of the intermediate stress ratio on the critical-state behaviour of granular materials. Granul. Matter 16, 641–655 (2014)

    Article  Google Scholar 

  21. Been, K., Jefferies, M.G.: A state parameter for sands. Geotechnique 35, 99–112 (1985)

    Article  Google Scholar 

  22. Been, K., Jefferiest, M.G., Hachey, J.: The critical state of sands. Geotechnique 41, 365–381 (1991)

    Article  Google Scholar 

  23. Coop, M.R.: The mechanics of uncemented carbonate sands. Geotechnique 40, 607–626 (1990)

    Article  Google Scholar 

  24. Coop, M.R., Sorensen, K.K., Bodas Freitas, T., Georgoutsos, G.: Particle breakage during shearing of a carbonate sand. Geotechnique 54, 157–163 (2004)

    Article  Google Scholar 

  25. Muir Wood, D., Belkheir, K., Liu, D.F.: Strain softening and state parameter for sand modelling. Geotechnique 44, 335–339 (1994)

  26. Muir Wood, D.: The magic of sands—the 20th Bjerrum Lecture presented in Oslo, 25 November 2005. Can. Geotech. J. 44, 1329–1350 (2007)

    Article  Google Scholar 

  27. Muir Wood, D., Maeda, K.: Changing grading of soil. Acta Geotech. 3, 3–14 (2008)

    Article  Google Scholar 

  28. Liu, M., Liu, H., Gao, Y.: New double yield surface model for coarse granular materials incorporating nonlinear unified failure criterion. J. Cent. South Univ. 19, 3236–3243 (2012)

    Article  Google Scholar 

  29. Zhao, J., Guo, N.: Unique critical state characteristics in granular media considering fabric anisotropy. Geotechnique 1, 695–704 (2013)

    Article  Google Scholar 

  30. ASTM: Standard test method for flat particles, elongated particles, or flat and elongated particles in coarse aggregate. West Conshohocken, PA (2010)

  31. CEN: Tests for mechanical and physical properties of aggregates Methods for the determination of resistance to fragmentation. Brussels, Belgium (2010)

  32. Xiao, Y., Liu, H., Chen, Y., Zhang, W.: Particle size effects in granular soils under true triaxial conditions. Geotechnique 64, 667–672 (2014)

    Article  Google Scholar 

  33. ASTM: Standard test method for consolidated undrained triaxial compression test for cohesive soils. West Conshohocken, PA (2002)

  34. Lin, S.Y., Lin, P.S., Luo, H.S., Juang, C.H.: Shear modulus and damping ratio characteristics of gravelly deposits. Can. Geotech. J. 37, 638–651 (2000)

    Article  Google Scholar 

  35. Indraratna, B., Salim, W.: Modelling of particle breakage of coarse aggregates incorporating strength and dilatancy. Geotech. Eng. 155, 243–252 (2002)

    Article  Google Scholar 

  36. Anderson, W.F., Fair, P.: Behavior of railroad ballast under monotonic and cyclic loading. J. Geotech. Geoenviron. Eng. 134, 316–327 (2008)

    Article  Google Scholar 

  37. Seif El Dine, B., Dupla, J.C., Frank, R., Canou, J., Kazan, Y.: Mechanical characterization of matrix coarse-grained soils with a large-sized triaxial device. Can. Geotech. J. 47, 425–438 (2010)

    Article  Google Scholar 

  38. Wang, J., Zhang, H., Tang, S., Liang, Y.: Effects of particle size distribution on shear strength of accumulation soil. J. Geotech. Geoenviron. Eng. 139, 1994–1997 (2013)

    Article  Google Scholar 

  39. Rowe, P.W.: The stress–dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. Lond. A 269, 500–527 (1962)

    Article  ADS  Google Scholar 

  40. Bolton, M.D.: The strength and dilatancy of sands. Geotechnique 36, 65–78 (1986)

    Article  Google Scholar 

  41. Vaid, Y.P., Sasitharan, S.: The strength and dilatancy of sand. Can. Geotech. J. 29, 522–526 (1992)

    Article  Google Scholar 

  42. Coop, M.R., Atkinson, J.H.: The mechanics of cemented carbonate sands. Geotechnique 43, 53–67 (1993)

    Article  Google Scholar 

  43. Indraratna, B.: Large-scale triaxial testing of greywacke rockfill. Geotechnique 43, 37–51 (1993)

    Article  Google Scholar 

  44. Jefferies, M.G.: Nor-Sand: a simple critical state model for sand. Geotechnique 43, 91–103 (1993)

    Article  Google Scholar 

  45. Schanz, T., Vermeer, P.A.: Angles of friction and dilatancy of sand. Geotechnique 46, 145–151 (1996)

    Article  Google Scholar 

  46. Manzari, M.T., Dafalias, Y.F.: A critical state two-surface plasticity model for sands. Geotechnique 47, 255–272 (1997)

    Article  Google Scholar 

  47. Li, X.S., Dafalias, Y.F.: Dilatancy for cohesionless soils. Geotechnique 50, 449–460 (2000)

    Article  Google Scholar 

  48. Salgado, R., Bandini, P., Karim, A.: Shear strength and stiffness of silty sand. J. Geotech. Geoenviron. Eng. 126, 451–462 (2000)

    Article  Google Scholar 

  49. Chu, J., Leong, W.K.: Pre-failure strain softening and pre-failure instability of sand: a comparative study. Geotechnique 51, 311–321 (2001)

  50. Dafalias, Y.F., Papadimitriou, A.G., Li, X.S.: Sand plasticity model accounting for inherent fabric anisotropy. J. Eng. Mech. 130, 1319–1333 (2004)

    Article  Google Scholar 

  51. Yang, J., Li, X.S.: State-dependent strength of sands from the perspective of unified modeling. J. Geotech. Geoenviron. Eng. 130, 186–198 (2004)

    Article  Google Scholar 

  52. Chu, J., Wanatowski, D.: Effect of loading mode on strain softening and instability behavior of sand in plane-strain tests. J. Geotech. Geoenviron. Eng. 135, 108–120 (2009)

    Article  Google Scholar 

  53. Guo, P., Su, X.: Shear strength, interparticle locking, and dilatancy of granular materials. Can. Geotech. J. 44, 579–591 (2007)

    Article  Google Scholar 

  54. Carrera, A., Coop, M., Lancellotta, R.: Influence of grading on the mechanical behaviour of Stava tailings. Geotechnique 61, 935–946 (2011)

  55. Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions. J. Eng. Mech. 140, 04014002 (2014)

  56. Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Bounding surface plasticity model incorporating the state pressure index for rockfill materials. J. Eng. Mech. 140, 04014087 (2014)

  57. Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Strength and deformation of fockfill material based on large-scale triaxial compression tests. I: Influences of density and pressure. J. Geotech. Geoenviron. Eng. 140, 04014070 (2014)

    Article  Google Scholar 

  58. Shi, W.C.: True Triaxial Tests on Coarse-Grained Soils and Study on Constitutive Model. Hohai University, Nanjing (2008)

    Google Scholar 

  59. Li, X.S., Wang, Y.: Linear representation of steady-state line for sand. J. Geotech. Geoenviron. Eng. 124, 1215–1217 (1998)

    Article  Google Scholar 

  60. Xiao, Y., Liu, H., Chen, Y., Jiang, J., Zhang, W.: Testing and modeling of the state-dependent behaviors of rockfill material. Comput. Geotech. 61, 153–165 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

The authors acknowledge the financial support from the 111 Project (Grant No. B13024), the National Natural Science Foundation of China (Grant No. 51509024) and the Fundamental Research Funds for the Central Universities (Grant No. 106112015CDJXY200008).

Author information

Authors and Affiliations

  1. State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing, 400030, China

    Yang Xiao

  2. Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, 400045, China

    Yang Xiao

  3. School of Civil Engineering, Chongqing University, Chongqing, 400450, China

    Yang Xiao, Hong Liu & Feng Yin

  4. College of Civil and Transportation Engineering, Hohai University, Nanjing, 210098, China

    Yang Xiao

  5. School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong City, NSW, 2522, Australia

    Yifei Sun

Authors
  1. Yang Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yifei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Feng Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Xiao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, Y., Sun, Y., Liu, H. et al. Critical state behaviors of a coarse granular soil under generalized stress conditions. Granular Matter 18, 17 (2016). https://doi.org/10.1007/s10035-016-0623-3

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s10035-016-0623-3

Keywords

Search

Navigation