Acta Geotechnica

, Volume 6, Issue 4, pp 219–229 | Cite as

A double slip non-coaxial flow rule for viscous-plastic Cosserat materials

  • Hans B. Mühlhaus
  • Jingyu Shi
  • Louise Olsen-Kettle
  • Louis Moresi
Research Paper


We propose a double slip non-coaxial plastic model within the framework of a Cosserat continuum theory. In a Cosserat continuum, a material point possesses the degrees of freedom of an infinitesimal rigid body: two translations and one rotation in 2D. We formulate the plastic model into viscous-plastic constitutive relationships and illustrate the viscous-plastic behaviour of the model by means of numerical solution of a simple shear problem.


Cosserat continuum Double slip Flow rule Non-coaxial 



We would like to acknowledge support from the ARC Discovery Grants DP0985662, DP110103024 and the ongoing support through Auscope/NCRIS. We are also grateful to Lutz Gross, Cihan Altinay and Vince Boros of ESSCC at the University of Queensland for various helps during the preparation of the manuscript.


  1. 1.
    Anand L (1983) Plane deformations of ideal granular materials. J Mech Phys Solids 31:105–122MATHCrossRefGoogle Scholar
  2. 2.
    Bercovici D, Schubert G, Zebib A (1988) Geoid and topography for infinite Prandtl number convection in a spherical shell. J Geophys Res 93:6430–6436CrossRefGoogle Scholar
  3. 3.
    Bunge HP, Richards MA, Baumgardner JR (1996) Effect of depth-dependent viscosity on the planform of mantle convection. Nature 379:436–438CrossRefGoogle Scholar
  4. 4.
    De Josselin de Jong G (1958) The undefiniteness in kinematics for friction materials. In: Proceedings of the conference on earth pressure problem, Brussels, vol 1, pp 55–70Google Scholar
  5. 5.
    De Josselin de Jong G (1971) The double sliding, free rotating model for granular assemblies. Geotechnique 21:155–162CrossRefGoogle Scholar
  6. 6.
    Gross L, Bourgouin L, Hale AJ, Mühlhaus HB (2007) Interface modeling in incompressible media using level sets in escript. Phys Earth Planet Inter 163:23–34CrossRefGoogle Scholar
  7. 7.
    Harris D (1995) A unified formulation for plasticity models of granular and other materials. Proc R Soc Lond A 450:37–49MATHCrossRefGoogle Scholar
  8. 8.
    Jiang MJ, Harris D, Yu HS (2005) Kinematic models for non-coaxial granular materials, part I. Int J Numer Anal Meth Geomech 29:643–661MATHCrossRefGoogle Scholar
  9. 9.
    Jiang MJ, Harris D, Yu HS (2005) Kinematic models for non-coaxial granular materials, part II. Int J Numer Anal Meth Geomech 29:663–689MATHCrossRefGoogle Scholar
  10. 10.
    Jiang MJ, Harris D, Yu HS (2005) A novel approach to examine double-shearing type models for granular materials. Granul Matter 7:157–168MATHCrossRefGoogle Scholar
  11. 11.
    Kreemer C, Holt WE, Haines AJ (2003) An integrated global model of present-day plate motions and plate boundary deformation. Geophys J Int 154:8–34CrossRefGoogle Scholar
  12. 12.
    Mehrabadi MM, Cowin SC (1978) Initial planar deformation of dilatant granular materials. J Mech Phys Solids 26:269–284MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Moresi L, Solomatov V (1998) Mantle convection with a brittle lithosphere: thoughts on the global tectonic style of the Earth and Venus. Geophys J 133:669–682CrossRefGoogle Scholar
  14. 14.
    Mühlhaus HB, Vardoulakis I (1987) The thickness of shear bands in granular materials. Geotechnique 37:271–283CrossRefGoogle Scholar
  15. 15.
    Mühlhaus H, Moresi L, Gross L, Grotowski J (2010) The influence of non-coaxiality on shear banding in viscous plastic materials. Granul Matter doi: 10.1007/s10035-010-0176-9
  16. 16.
    Mühlhaus HB, Shi J, Olsen-Kettle L, Moresi L (2011) Effects of a non-coaxial flow rule on shear bands in viscous-plastic materials. Granul Matter 13:205–210CrossRefGoogle Scholar
  17. 17.
    Prabhakar V, Reddy JN (2006) Spectral/hp penalty least-squares finite element formulation for the steady incompressible Navier–Stokes equations. J Comput Phys 215:274–297MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Spencer AJM (1964) A theory of the kinematics of ideal soils under plane strain conditions. J Mech Phys Solids 12:337–351MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Tackley PJ (1996) On the ability of phase transitions and viscosity layering to induce long-wavelength heterogeneity in the mantle. Geophys Res Lett 23:1985–1988CrossRefGoogle Scholar
  20. 20.
    Tackley PJ, Stevenson DJ, Glatzmaier GA, Schubert G (1993) Effects of an endothermic phase transition at 670 km depth in a spherical model of convection in the Earth’s mantle. Nature 361(6414):699–704CrossRefGoogle Scholar
  21. 21.
    Tejchman J (2000) Behaviour of granular bodies in induced shear zones. Granul Matter 2:77–96CrossRefGoogle Scholar
  22. 22.
    Tejchman J (2001) Shearing of an infinite narrow granular layer between two boundaries. In: Muhlhaus H-B (ed) Bifurcation and localisation theory in geomechanics. Swets and Zeitlinger, LisseGoogle Scholar
  23. 23.
    Tejchman J, Bauer E (2005) Fe-simulations of direct and a true simple shear test within a polar hypoplasticity. Comput Geotech 32:1–16CrossRefGoogle Scholar
  24. 24.
    Tejchman J, Gudehus G (2001) Shearing of a narrow granular layer with polar quantities. Int J Numer Anal Meth Geomech 25:1–28MATHCrossRefGoogle Scholar
  25. 25.
    Thornton C, Zhang L (2006) A numerical examination of shear banding and simple shear non-coaxial flow rules. Philos Mag 86:3425–3452CrossRefGoogle Scholar
  26. 26.
    van Heck H, Tackley PJ (2008) Planforms of self-consistently generated plate tectonics in 3-D spherical geometry. Geophys Res Lett 35:L19312. doi: 10.1029/2008GL035190 CrossRefGoogle Scholar
  27. 27.
    Zhong SJ, Zuber MT, Moresi L, Gurnis M (2000) Role of temperature-dependent viscosity and surface plates in spherical shell models of mantle convection. J Geophys Res 105:11063–11082CrossRefGoogle Scholar
  28. 28.
    Zhong SJ, Zhang N, Li ZX, Roberts J (2007) Supercontinent cycles, true polar wander, and very long-wavelength mantle convection. Earth Planet Sci Lett 261:551–564CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Hans B. Mühlhaus
    • 1
  • Jingyu Shi
    • 1
  • Louise Olsen-Kettle
    • 1
  • Louis Moresi
    • 2
  1. 1.School of Earth SciencesThe University of QueenslandBrisbaneAustralia
  2. 2.School of Geosciences, School of Mathematical SciencesMonash UniversityClaytonAustralia

Personalised recommendations