SpringerLink
Log in

Strain localization in planar shear of granular media: the role of porosity and boundary conditions

  • Regular Article - Flowing Matter
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract

Shear strain localization into shear bands is associated with velocity weakening instabilities and earthquakes. Here, we simulate steady-state plane-shear flow of numerical granular material (gouge), confined between parallel surfaces. Both constant shear stress and constant strain-rate boundary conditions are tested, and the two types of boundary conditions are found to yield distinct velocity profiles and friction laws. The inertial number, I, exerts the largest control on the layers’ behavior, but additional dependencies of friction on normal stress and thickness of the layer are observed under constant stress boundary condition. We find that shear-band localization, which is present in the quasistatic regime (\(I<10^{-3}\)) in rate-controlled shear, is absent under stress-controlled loading. In the latter case, flow ceases when macroscopic friction coefficient approaches the quasistatic friction value. The inertial regime that occurs at higher inertial numbers (\(I>10^{-3}\)) is associated with distributed shear, and friction and porosity that increase with shear rate (rate-strengthening regime). The finding that shear under constant stress boundary condition produces the inertial, distributed shear but never quasistatic, localized deformation is rationalized based on low fluctuations of shear forces in granular contacts for stress-controlled loading. By examining porosity within and outside a shear band, we also provide a mechanical reason why the transition between quasistatic and inertial shear coincides with the transition between localized and distributed strain.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Notes

  1. Porosity \(\phi \) should not be confused with solid (volume) fraction, equal to one minus porosity, which is unfortunately also often denoted by the same symbol.

  2. In DEM an asperity may be thought of as a grain configuration that requires above-average shear stress for deformation to proceed, either by dilatancy or by enhanced dissipation.

References

  1. C.H. Scholz, Geology 15(6), 493 (1987)

    Article  ADS  Google Scholar 

  2. F.M. Chester, J.P. Evans, R.L. Biegel, J. Geophys. Res. Solid Earth 98(B1), 771 (1993). https://doi.org/10.1029/92JB01866

    Article  Google Scholar 

  3. A. Billi, J. Struct. Geol. 27, 1823 (2005). https://doi.org/10.1016/j.jsg.2005.05.013

    Article  ADS  Google Scholar 

  4. M.L. Arboleya, T. Engelder, J. Struct. Geol. 17, 519 (1995). https://doi.org/10.1016/0191-8141(94)00079-F

    Article  ADS  Google Scholar 

  5. S. Cashman, K. Cashman, Geology 28, 111 (2000)

    Article  ADS  Google Scholar 

  6. N.W. Hayman, B.A. Housen, T.T. Cladouhos, K. Livi, J. Geophys. Res. Solid Earth 109, B5 (2004). https://doi.org/10.1029/2003JB002902

    Article  Google Scholar 

  7. A.M. Boullier, E.C. Yeh, S. Boutareaud, S.R. Song, C.H. Tsai, Geochem. Geophys. Geosyst. 10, Q03016 (2009). https://doi.org/10.1029/2008GC002252

    Article  ADS  Google Scholar 

  8. S. Siman-Tov, E. Aharonov, A. Sagy, S. Emmanuel, Geology 41(6), 703 (2013). https://doi.org/10.1130/G34087.1

    Article  ADS  Google Scholar 

  9. L. Smeraglia, A. Billi, E. Carminati, A. Cavallo, G. Di Toro, E. Spagnuolo, F. Zorzi, Sci. Rep. 7(1), 664 (2017). https://doi.org/10.1038/s41598-017-00717-4

    Article  ADS  Google Scholar 

  10. J.M. Logan, M. Friedman, N. Higgs, C. Dengo, T. Shimamoto, U.S. Geol. Surv. Open File Rep. 79–1239, 305 (1979)

  11. C. Marone, C.B. Raleigh, C.H. Scholz, J. Geophys. Res. 95, 7007 (1990). https://doi.org/10.1029/JB095iB05p07007

    Article  ADS  Google Scholar 

  12. J. Logan, C. Dengo, N. Higgs, Z. Wang, Fault mechanics and transport properties of rocks, in International Geophysics, vol. 51, ed. by B. Evans, T. Fong Wong (Academic Press, Cambridge, 1992), pp. 33–67

    Google Scholar 

  13. N.M. Beeler, T.E. Tullis, M.L. Blanpied, J.D. Weeks, J. Geophys. Res. Solid Earth 101(B4), 8697 (1996). https://doi.org/10.1029/96JB00411

    Article  Google Scholar 

  14. A.R. Niemeijer, C.J. Spiers, J. Geophys. Res. 112, S78 (2007). https://doi.org/10.1016/j.ijggc.2012.09.018

    Article  Google Scholar 

  15. Z. Reches, D. Lockner, Nature 467, 452 (2010). https://doi.org/10.1038/nature09348

    Article  ADS  Google Scholar 

  16. B.P. Proctor, T.M. Mitchell, G. Hirth, D. Goldsby, F. Zorzi, J.D. Platt, G. Di Toro, J. Geophys. Res. Solid Earth 119(11), 8107 (2014). https://doi.org/10.1002/2014JB011057

    Article  ADS  Google Scholar 

  17. E.K. Mitchell, Y. Fialko, K.M. Brown, J. Geophys. Res. Solid Earth 121, 6932 (2016)

    Article  ADS  Google Scholar 

  18. P. Mora, D. Place, Geophys. Res. Lett. 26(1), 123 (1999). https://doi.org/10.1029/1998GL900231

    Article  ADS  Google Scholar 

  19. E. Aharonov, D. Sparks, Phys. Rev. E 65, 051302 (2002)

    Article  ADS  Google Scholar 

  20. J.K. Morgan, M.S. Boettcher, J. Geophys. Res. Solid Earth 104(B2), 2703 (1999). https://doi.org/10.1029/1998JB900056

    Article  Google Scholar 

  21. K. Mair, S. Abe, Earth Planet. Sci. Lett. 274(1), 72 (2008). https://doi.org/10.1016/j.epsl.2008.07.010

    Article  ADS  Google Scholar 

  22. K. Li, Y.F. Wang, Q.W. Lin, Q.G. Cheng, Y. Wu, Landslides 18, 1779 (2021). https://doi.org/10.1007/s10346-020-01607-z

    Article  Google Scholar 

  23. Y. Ben-Zion, C.G. Sammis, Pure Appl. Geophys. 160(3), 677 (2003)

    Article  ADS  Google Scholar 

  24. C. Marone, PAGEOPH 137, 409 (1991). https://doi.org/10.1007/BF00879042

    Article  Google Scholar 

  25. W.F. Brace, Tectonophysics 14, 189 (1972)

    Article  ADS  Google Scholar 

  26. M. Ikari, C. Marone, D. Saffer, Geology 39, 83 (2011). https://doi.org/10.1130/G31416.1

    Article  ADS  Google Scholar 

  27. T. Shimamoto, Science 231, 711 (1986)

    Article  ADS  Google Scholar 

  28. M. French, W. Zhu, J. Banker, Geophys. Res. Lett. 43, 4330 (2016). https://doi.org/10.1002/2016GL068893

    Article  ADS  Google Scholar 

  29. W. Wu, Y. Zou, X. Li, J. Zhao, Rev. Sci. Instrum. 85, 093902 (2014). https://doi.org/10.1063/1.4894207

    Article  ADS  Google Scholar 

  30. J. Samuelson, D. Elsworth, C. Marone, J. Geophys. Res. 114, B12404 (2009). https://doi.org/10.1029/2008JB006273

    Article  ADS  Google Scholar 

  31. D.R. Faulkner, C. Sanchez-Roa, C. Boulton, S.A.M. den Hartog, J. Geophys. Res. Solid Earth 123, 1 (2018). https://doi.org/10.1002/2017JB015130

    Article  Google Scholar 

  32. L. Goren, E. Aharonov, Earth Planet. Sci. Lett. 277(3), 365 (2009). https://doi.org/10.1016/j.epsl.2008.11.002

    Article  ADS  Google Scholar 

  33. F.M. Chester, J. Geophys. Res. Solid Earth 99, 7247 (1994)

    Article  Google Scholar 

  34. E. Aharonov, C.H. Scholz, J. Geophys. Res. Solid Earth 123(2), 1591 (2018). https://doi.org/10.1002/2016JB013829

    Article  ADS  Google Scholar 

  35. K.M. Frye, C. Marone, J. Geophys. Res. Solid Earth 107(B11), ETG11 (2002)

    Article  Google Scholar 

  36. J.W. Rudnicki, J.R. Rice, J. Mech. Phys. Solids 23, 371 (1975)

    Article  ADS  Google Scholar 

  37. J.R. Rice, The localization of plastic deformation, in Theoretical and applied mechanics, vol. 1, ed. by W. T. Koiter (North-Holland Publishing Co., Delft, 1976), pp. 207–220

  38. I. Vardoulakis, Mech. Res. Commun. 3(3), 209 (1976). https://doi.org/10.1016/0093-6413(76)90014-8

    Article  Google Scholar 

  39. I. Vardoulakis, Int. J. Numer. Anal. Methods Geomech. 4(2), 103 (1980). https://doi.org/10.1002/nag.1610040202

    Article  Google Scholar 

  40. H.B. Muhlhaus, I. Vardoulakis, Géotechnique 37(3), 271 (1987). https://doi.org/10.1680/geot.1987.37.3.271

    Article  Google Scholar 

  41. J. Sulem, I. Vardoulakis, Acta Mech. 83, 195 (1990). https://doi.org/10.1007/BF01172981

    Article  Google Scholar 

  42. R. Larsson, K. Runesson, K. Axelsson, Int. J. Numer. Anal. Methods Geomech. 20(11), 771 (1996)

  43. G. Weir, R. Young, Int. J. Numer. Anal. Methods Geomech. 27(15), 1299 (2003). https://doi.org/10.1002/nag.322

    Article  Google Scholar 

  44. I. Einav, M. Randolph, Géotechnique 56(7), 501 (2006). https://doi.org/10.1680/geot.2006.56.7.501

    Article  Google Scholar 

  45. I. Vardoulakis, J. Sulem, Bifurcation Analysis in Geomechanics (Taylor & Francis, Oxon, England, 1995)

    Google Scholar 

  46. J.R. Rice, J. Geophys. Res. 111, B05311 (2006). https://doi.org/10.1029/2005JB004006

    Article  ADS  Google Scholar 

  47. J. Sulem, V. Famin, J. Geophys. Res. Solid Earth 114(B3), B03309 (2009). https://doi.org/10.1029/2008JB006004

    Article  ADS  Google Scholar 

  48. J. Sulem, I. Stefanou, E. Veveakis, Granul. Matter 13, 261 (2011). https://doi.org/10.1007/s10035-010-0244-1

    Article  Google Scholar 

  49. N. Brantut, J. Sulem, J. Appl. Mech. 79, 3 (2012). https://doi.org/10.1115/1.4005880

    Article  Google Scholar 

  50. M. Veveakis, I. Stefanou, J. Sulem, Geotech. Lett. 3, 31 (2013). https://doi.org/10.1680/geolett.12.00063

    Article  Google Scholar 

  51. J.R. Rice, J.W. Rudnicki, J.D. Platt, J. Geophys. Res. Solid Earth 119(5), 4311 (2014). https://doi.org/10.1002/2013JB010710

    Article  ADS  Google Scholar 

  52. J. Desrues, R. Chambon, M. Mokni, F. Mazerolle, Géotechnique 46(3), 529 (1996). https://doi.org/10.1680/geot.1996.46.3.529

    Article  Google Scholar 

  53. T.W. Lambe, R.V. Whitman, Soil Mechanics (John Wiley, New York, 1969)

    Google Scholar 

  54. G.D.R. MiDi, Eur. Phys. J. E 14, 341 (2004)

    Article  Google Scholar 

  55. Y. Forterre, O. Pouliquen, Annu. Rev. Fluid Mech. 40, 1 (2008)

    Article  ADS  Google Scholar 

  56. F. da Cruz, S. Emam, M. Prochnow, J.N. Roux, F. Chevoir, Phys. Rev. E 72, 021309 (2005)

    Article  ADS  Google Scholar 

  57. A. Singh, V. Magnanimo, K. Saitoh, S. Luding, New J. Phys. 17, 043028 (2015). https://doi.org/10.1088/1367-2630/17/4/043028

    Article  ADS  Google Scholar 

  58. S. Parez, E. Aharonov, Front. Phys. 3, 80 (2015). https://doi.org/10.3389/fphy.2015.00080

    Article  Google Scholar 

  59. S. Parez, E. Aharonov, R. Toussaint, Phys. Rev. E 93, 042902 (2016). https://doi.org/10.1103/PhysRevE.93.042902

    Article  ADS  Google Scholar 

  60. Z. Shojaaee, J.N. Roux, F. Chevoir, D.E. Wolf, Phys. Rev. E 86, 011301 (2012)

    Article  ADS  Google Scholar 

  61. E. DeGiuli, M. Wyart, Proc. Natl. Acad. Sci. 114(35), 9284 (2017). https://doi.org/10.1073/pnas.1706105114

    Article  ADS  MathSciNet  Google Scholar 

  62. J.A. Dijksman, G.H. Wortel, L.T.H. van Dellen, O. Dauchot, M. van Hecke, Phys. Rev. Lett. 107, 108303 (2011). https://doi.org/10.1103/PhysRevLett.107.108303

    Article  ADS  Google Scholar 

  63. O. Kuwano, R. Ando, T. Hatano, Geophys. Res. Lett. 40(7), 1295 (2013). https://doi.org/10.1002/grl.50311

    Article  ADS  Google Scholar 

  64. B. Andreotti, Y. Forterre, O. Pouliquen, Granular Media: Between Fluid and Solid (Cambridge University Press, Cambridge, England, 2013)

    Book  Google Scholar 

  65. H.J. Melosh, J. Geophys. Res. Solid Earth 84(B13), 7513 (1979). https://doi.org/10.1029/JB084iB13p07513

    Article  Google Scholar 

  66. T. Barker, J.M.N.T. Gray, J. Fluid Mech. 828, 5–32 (2017). https://doi.org/10.1017/jfm.2017.428

    Article  ADS  MathSciNet  Google Scholar 

  67. A.H. Clark, J.D. Thompson, M.D. Shattuck, N.T. Ouellette, C.S. O’Hern, Phys. Rev. E 97, 062901 (2018). https://doi.org/10.1103/PhysRevE.97.062901

  68. F.C. Frank, Rev. Geophys. 3(4), 485 (1965). https://doi.org/10.1029/RG003i004p00485

    Article  ADS  Google Scholar 

  69. O. Reynolds, Phylos. Mag. Ser. 5(20), 469 (1885)

    Article  Google Scholar 

  70. P.A. Cundall, O.D. Strack, Géotechnique 29, 47 (1979)

    Article  Google Scholar 

  71. D. Frenkel, B. Smit, Understanding Molecular Simulations (Academic Press, San Diego, 2002)

    MATH  Google Scholar 

  72. Z. Shojaaee, L. Brendel, J. Torok, D.E. Wolf, Phys. Rev. E 86, 011302 (2012)

    Article  ADS  Google Scholar 

  73. L.E. Silbert, D. Ertas, G.S. Grest, T.C. Halsey, D. Levine, S.J. Plimpton, Phys. Rev. E 64, 051302 (2001)

  74. A. Favier de Coulomb, M. Bouzid, P. Claudin, E. Clément, B. Andreotti, Phys. Rev. Fluids 2, 102301 (2017). https://doi.org/10.1103/PhysRevFluids.2.102301

    Article  ADS  Google Scholar 

  75. G. Koval, J.N. Roux, A. Corfdir, F. Chevoir, Phys. Rev. E 79, 021306 (2009)

    Article  ADS  Google Scholar 

  76. L. Bocquet, A. Colin, A. Ajdari, Phys. Rev. Lett. 103, 036001 (2009). https://doi.org/10.1103/PhysRevLett.103.036001

    Article  ADS  Google Scholar 

  77. K. Kamrin, G. Koval, Phys. Rev. Lett. 108, 178301 (2012)

    Article  ADS  Google Scholar 

  78. K. Kamrin, D.L. Henann, Soft Matter 11, 179 (2015)

    Article  ADS  Google Scholar 

  79. O. Pouliquen, Y. Forterre, Phil. Trans. R. Soc. A 367, 5091 (2009). https://doi.org/10.1098/rsta.2009.0171

    Article  ADS  Google Scholar 

  80. P. Jop, Y. Forterre, O. Pouliquen, Nature 441, 727 (2006)

    Article  ADS  Google Scholar 

  81. P.W. Rowe, G.I. Taylor, Proceedings of the royal society of London. Ser. A Math. Phys. Sci. 269(1339), 500 (1962). https://doi.org/10.1098/rspa.1962.0193

    Article  Google Scholar 

  82. N. Makedonska, D.W. Sparks, E. Aharonov, L. Goren, J. Geophys. Res. 116, B09302 (2011)

    ADS  Google Scholar 

  83. Z. Lyu, J. Rivière, Q. Yang, C. Marone, Tectonophysics 763, 86 (2019). https://doi.org/10.1016/j.tecto.2019.04.010

    Article  ADS  Google Scholar 

  84. K.J. Hsu, Albert Heim: Observations on landslides and relevance to modern interpretations, in Rockslides and avalanches, vol. 14, ed. by B. Voight (Elsevier, Amsterdam, 1978), pp. 71–93

  85. F. Guillard, P. Golshan, L. Shen, J.R. Valdes, I. Einav, Nat. Phys. 11(10), 835 (2015). https://doi.org/10.1038/nphys3424

    Article  Google Scholar 

  86. S. Ben-Zeev, E. Aharonov, R. Toussaint, S. Parez, L. Goren, Phys. Rev. Fluids 5, 054301 (2020). https://doi.org/10.1103/PhysRevFluids.5.054301

    Article  ADS  Google Scholar 

  87. S. Ben-Zeev, L. Goren, S. Parez, R. Toussaint, C. Clément, E. Aharonov, The combined effect of buoyancy and excess pore pressure in facilitating soil liquefaction, in Poromechanics VI: Proceedings of the Sixth Biot Conference on Poromechanics, vol. 1, ed. by M. Vandamme, P. Dangla, J. M. Pereira, S. Ghabezloo (American Society of Civil Engineers, Reston, VA, 2017). https://doi.org/10.1061/9780784480779.013

Download references

Acknowledgements

S.P., T.T. and M.S. are grateful for the support of Grant No. 19-21114Y from the Czech Science Foundation (GA CR). E.A. acknowledges the support of ISF Grant No. 910/17. Computational resources were supplied by the project “e-Infrastruktura CZ” (e-INFRA LM2018140) provided within the program Projects of Large Research, Development and Innovations Infrastructures.

Author information

Authors and Affiliations

  1. Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Prague, Czech Republic

    Stanislav Parez, Tereza Travnickova & Martin Svoboda

  2. Faculty of Science, Jan Evangelista Purkyně University in Ústí nad Labem, Ústí nad Labem, Czech Republic

    Stanislav Parez

  3. Institute of Earth Sciences, Hebrew University of Jerusalem, Jerusalem, Israel

    Einat Aharonov

Authors
  1. Stanislav Parez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tereza Travnickova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Svoboda
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Einat Aharonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

S.P. designed the test of boundary conditions, contributed to the data analysis, interpreted the results and wrote the article. T.T. and M.S. conducted the numerical simulations and contributed to the data analysis. E.A. designed the theoretical model for porosity control over strain localization and contributed to writing of the article.

Corresponding author

Correspondence to Stanislav Parez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parez, S., Travnickova, T., Svoboda, M. et al. Strain localization in planar shear of granular media: the role of porosity and boundary conditions. Eur. Phys. J. E 44, 134 (2021). https://doi.org/10.1140/epje/s10189-021-00138-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/s10189-021-00138-2

Search

Navigation