Continuum Mechanics and Thermodynamics

, Volume 22, Issue 3, pp 221–237

Continuum-mechanical, Anisotropic Flow model for polar ice masses, based on an anisotropic Flow Enhancement factor

  • Luca Placidi
  • Ralf Greve
  • Hakime Seddik
  • Sérgio H. Faria
Open Access
Original Article

Abstract

A complete theoretical presentation of the Continuum-mechanical, Anisotropic Flow model, based on an anisotropic Flow Enhancement factor (CAFFE model) is given. The CAFFE model is an application of the theory of mixtures with continuous diversity for the case of large polar ice masses in which induced anisotropy occurs. The anisotropic response of the polycrystalline ice is described by a generalization of Glen’s flow law, based on a scalar anisotropic enhancement factor. The enhancement factor depends on the orientation mass density, which is closely related to the orientation distribution function and describes the distribution of grain orientations (fabric). Fabric evolution is governed by the orientation mass balance, which depends on four distinct effects, interpreted as local rigid body rotation, grain rotation, rotation recrystallization (polygonization) and grain boundary migration (migration recrystallization), respectively. It is proven that the flow law of the CAFFE model is truly anisotropic despite the collinearity between the stress deviator and stretching tensors.

Keywords

Continuum mechanics Anisotropy Ice Mixtures Recrystallization 

References

  1. 1.
    Azuma N.: A flow law for anisotropic ice and its application to ice sheets. Earth Planet. Sci. Lett. 128(3–4), 601–614 (1994)CrossRefADSGoogle Scholar
  2. 2.
    Azuma N.: A flow law for anisotropic polycrystalline ice under uniaxial compressive deformation. Cold Reg. Sci. Technol. 23(2), 137–147 (1995)CrossRefGoogle Scholar
  3. 3.
    Azuma N., Higashi A.: Formation processes of ice fabric patterns in ice sheets. Ann. Glaciol. 6, 130–134 (1985)ADSGoogle Scholar
  4. 4.
    Blenk S., Ehrentraut H., Muschik W.: Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution. Int. J. Eng. Sci. 30, 1127–1143 (1992)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Boehler J.P.: Applications of Tensor Functions in Solid Mechanics. Springer, New York (1987)MATHGoogle Scholar
  6. 6.
    Budd W.F., Jacka T.H.: A review of ice rheology for ice sheet modelling. Cold Reg. Sci. Technol. 16(2), 107–144 (1989)CrossRefGoogle Scholar
  7. 7.
    Dafalias Y.F.: Orientation distribution function in non-affine rotations. J. Mech. Phys. Solids 49, 2493–2516 (2001)CrossRefADSMATHGoogle Scholar
  8. 8.
    Durand G., Persson A., Samyn D., Svensson A.: Relation between neighbouring grains in the upper part of the NorthGRIP ice core: implications for rotation recrystallization. Earth Planet. Sci. Lett. 265(3), 666–671 (2008). doi:10.1016/j.epsl.2007.11.002 CrossRefADSGoogle Scholar
  9. 9.
    EPICA Community Members: One-to-one coupling of glacial climate variability in Greenland and Antarctica. Nature 444(7116), 195–198 (2006). doi:10.1038/nature05301 CrossRefADSGoogle Scholar
  10. 10.
    Faria S.H.: Mixtures with continuous diversity: general theory and application to polymer solutions. Contin. Mech. Thermodyn. 13, 91–120 (2001)CrossRefADSMATHGoogle Scholar
  11. 11.
    Faria S.H.: Creep and recrystallization of large polycrystalline masses. I. General continuum theory. Proc. R. Soc. Lond. A 462(2069), 1493–1514 (2006a). doi:10.1098/rspa.2005.1610 CrossRefMathSciNetADSMATHGoogle Scholar
  12. 12.
    Faria S.H.: Creep and recrystallization of large polycrystalline masses. III. Continuum theory of ice sheets. Proc. R. Soc. Lond. A 462(2073), 2797–2816 (2006b). doi:10.1098/rspa.2006.1698 CrossRefMathSciNetADSMATHGoogle Scholar
  13. 13.
    Faria S.H.: The symmetry group of the CAFFE model. J. Glaciol. 54(187), 643–645 (2008)CrossRefADSGoogle Scholar
  14. 14.
    Faria S.H., Kremer G.M., Hutter K.: Creep and recrystallization of large polycrystalline masses. II. Constitutive theory for crystalline media with transversely isotropic grains. Proc. R. Soc. Lond. A 462(2070), 1699–1720 (2006). doi:10.1098/rspa.2005.1635 CrossRefMathSciNetADSMATHGoogle Scholar
  15. 15.
    Gagliardini O., Gillet-Chaulet F., Montagnat M.: A review of anisotropic polar ice models: from crystal to ice-sheet flow models. In: Hondoh, T. (eds) Physics of Ice Core Records, vol. 2, Yoshioka Publishing, Kyoto, Japan (2009)Google Scholar
  16. 16.
    Gillet-Chaulet F., Gagliardini O., Meyssonnier J., Montagnat M., Castelnau O.: A user-friendly anisotropic flow law for ice-sheet modelling. J. Glaciol. 51(172), 3–14 (2005)CrossRefADSGoogle Scholar
  17. 17.
    Glen J.W.: The creep of polycrystalline ice. Proc. R. Soc. Lond. A 228, 519–538 (1955)CrossRefADSGoogle Scholar
  18. 18.
    Gödert G.: A mesoscopic approach for modelling texture evolution of polar ice including recrystallization phenomena. Ann. Glaciol. 37, 23–28 (2003)CrossRefADSGoogle Scholar
  19. 19.
    Gödert G., Hutter K.: Induced anisotropy in large ice shields: theory and its homogenization. Contin. Mech. Thermodyn. 10(5), 293–318 (1998)CrossRefMathSciNetADSMATHGoogle Scholar
  20. 20.
    Hutter K.: Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets. D Reidel Publishing Company, Dordrecht, The Netherlands (1983)Google Scholar
  21. 21.
    Jacka T.H., Budd W.F.: Isotropic and anisotropic flow relations for ice dynamics. Ann. Glaciol. 12, 81–84 (1989)ADSGoogle Scholar
  22. 22.
    Kamb B.: Experimental recrystallization of ice under stress. In: Heard, H.C., Borg, I.Y., Carter, N.L., Raileigh, C.B. (eds) Flow and Fracture of Rocks, pp. 211–241. American Geophysical Union, Washington, DC (1972)Google Scholar
  23. 23.
    Larson R.G.: Constitutive Equations for Polymer Melts and Solutions. Butterworths Series in Chemical Engineering. Butterworths, Boston (1988)Google Scholar
  24. 24.
    Larson R.G.: The Structure and Rheology of Complex Fluids. Oxford University press, Oxford (1999)Google Scholar
  25. 25.
    Liu I.-S.: Continuum Mechanics. Springer, Berlin (2002)MATHGoogle Scholar
  26. 26.
    Lliboutry L.: Anisotropic, transversely isotropic nonlinear viscosity of rock ice and rheological parameters inferred from homogenization. Int. J. Plast. 9, 619–632 (1993)CrossRefMATHGoogle Scholar
  27. 27.
    Mangeney A., Califano F., Castelnau O.: Isothermal flow of an anisotropic ice sheet in the vicinity of an ice divide. J. Geophys. Res. 101(B12), 28189–28204 (1996)CrossRefADSGoogle Scholar
  28. 28.
    Massart T.J., Peerlings R.H.J., Geers M.G.D.: Mesoscopic modeling of failure and damage-induced anisotropy in brick masonry. Eur. J. Mech. A Solids 23(5), 719–735 (2004). doi:10.1016/j.euromechsol.2004.05.003 CrossRefADSMATHGoogle Scholar
  29. 29.
    McConnel J.C.: On the plasticity of an ice crystal. Proc. R. Soc. Lond. 49, 323–343 (1891)Google Scholar
  30. 30.
    Miyamoto A.: Mechanical properties and crystal textures of Greenland deep ice cores. Doctoral thesis. Hokkaido University, Sapporo Japan (1999)Google Scholar
  31. 31.
    Morland L.W., Staroszczyk R.: Stress and strain-rate formulations for fabric evolution in polar ice. Contin. Mech. Thermodyn. 15(1), 55–71 (2003)CrossRefMathSciNetADSMATHGoogle Scholar
  32. 32.
    Motoyama H.: The second deep ice coring project at Dome Fuji, Antarctica. Sci. Drill. 5, 41–43 (2007). doi:10.2204/iodp.sd.5.05.2007 Google Scholar
  33. 33.
    Müller I.: Thermodynamics. Pitman Advanced Publishing Program, Boston (1985)MATHGoogle Scholar
  34. 34.
    Nye J.F.: The distribution of stress and velocity in glaciers and ice sheets. Proc. R. Soc. Lond. 239, 113–133 (1952)ADSGoogle Scholar
  35. 35.
    Papenfuss C.: Theory of liquid crystals as an example of mesoscopic continuum mechanics. Comput. Mater. Sci. 19, 45–52 (2000)CrossRefGoogle Scholar
  36. 36.
    Papenfuss C., Van P.: Scalar, vectorial, and tensorial damage parameters from the mesoscopic background. Proc. Est. Acad. Sci. 57(3), 132–141 (2008)CrossRefMATHGoogle Scholar
  37. 37.
    Paterson W.S.B.: The Physics of Glaciers. 3rd edn. Pergamon Press, Oxford (1994)Google Scholar
  38. 38.
    Pimienta P., Duval P., Lipenkov V.Y.: Mechanical behaviour of anisotropic polar ice. In: Waddington, E.D., Walder, J.S. (eds) The Physical Basis of Ice Sheet Modelling, pp. 57–66. IAHS Publication IAHS Press, Wallingford, UK (1987)Google Scholar
  39. 39.
    Placidi, L.: Thermodynamically consistent formulation of induced anisotropy in polar ice accounting for grain-rotation, grain-size evolution and recrystallization. Doctoral thesis, Department of Mechanics, Darmstadt University of Technology, German (2004). Available at http://elib.tu-darmstadt.de/diss/000614/
  40. 40.
    Placidi L.: Microstructured continua treated by the theory of mixtures. Doctoral thesis. University of Rome, La Sapienza, Italy (2005)Google Scholar
  41. 41.
    Placidi L., Faria S.H., Hutter K.: On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets. Ann. Glaciol. 39, 49–52 (2004)CrossRefADSGoogle Scholar
  42. 42.
    Placidi, L., Hutter, K.: Characteristics of orientation and grain-size distributions. In: Proceedings of the 21st International Congress of Theoretical and Applied Mechanics. Warsaw, Poland (2004)Google Scholar
  43. 43.
    Placidi L., Hutter K.: An anisotropic flow law for incompressible polycrystalline materials. Z. angew. Math. Phys. 57, 160–181 (2006a). doi:10.1007/s00033-005-0008-7 CrossRefMathSciNetMATHGoogle Scholar
  44. 44.
    Placidi L., Hutter K.: Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity. Contin. Mech. Thermodyn. 17(6), 409–451 (2006b). doi:10.1007/s00161-005-0006-1 CrossRefMathSciNetADSMATHGoogle Scholar
  45. 45.
    Placidi L., Hutter K., Faria S.H.: A critical review of the mechanics of polycrystalline polar ice. GAMM-Mitt. 29(1), 80–117 (2006)MathSciNetMATHGoogle Scholar
  46. 46.
    Rashid M.M.: Texture evolution and plastic response of two-dimensional polycrystals. J. Mech. Phys. Solids 40, 1009–1029 (1992)CrossRefADSMATHGoogle Scholar
  47. 47.
    Russell-Head D.S., Budd W.F.: Ice sheet flow properties derived from borehole shear measurements combined with ice core studies. J. Glaciol. 24(90), 117–130 (1979)ADSGoogle Scholar
  48. 48.
    Seddik, H.: A full-Stokes finite-element model for the vicinity of Dome Fuji with flow-induced anisotropy and fabric evolution. Doctoral thesis, Graduate School of Environmental Science, Hokkaido University, Sapporo, Japan (2008). Available at http://hdl.handle.net/2115/34136
  49. 49.
    Seddik H., Greve R., Placidi L., Hamann I., Gagliardini O.: Application of a continuum-mechanical model for the flow of anisotropic polar ice to the EDML core, Antarctica. J. Glaciol. 54(187), 631–642 (2008)CrossRefADSGoogle Scholar
  50. 50.
    Seddik H., Greve R., Zwinger T., Placidi L.: A full-Stokes ice flow model for the vicinity of Dome Fuji, Antarctica, with induced anisotropy and fabric evolution. The Cryosphere Discuss. 3(1), 1–31 (2009)ADSGoogle Scholar
  51. 51.
    Staroszczyk R., Morland L.W.: Strengthening and weakening of induced anisotropy in polar ice. Proc. R. Soc. Lond. 457(2014), 2419–2440 (2001)CrossRefADSMATHGoogle Scholar
  52. 52.
    Svendsen B., Hutter K.: A continuum approach for modelling induced anisotropy in glaciers and ice sheets. Ann. Glaciol. 23, 262–269 (1996)ADSGoogle Scholar
  53. 53.
    Thorsteinsson T.: An analytical approach to deformation of anisotropic ice-crystal aggregates. J. Glaciol. 47(158), 507–516 (2001)CrossRefADSGoogle Scholar
  54. 54.
    Truesdell C.: Sulle basi della termomeccanica, Nota I. Rendiconnti Accademia dei Lincei 8/22, 33–38 (1957a)MathSciNetGoogle Scholar
  55. 55.
    Truesdell C.: Sulle basi della termomeccanica, Nota II. Rendiconnti Accademia dei Lincei 8/22, 158–166 (1957b)Google Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Luca Placidi
    • 1
    • 2
  • Ralf Greve
    • 3
  • Hakime Seddik
    • 3
  • Sérgio H. Faria
    • 4
  1. 1.Department of Structural and Geotechnical Engineering“Sapienza”, University of RomeRomeItaly
  2. 2.Smart Materials and Structures LaboratoryCisterna di LatinaItaly
  3. 3.Institute of Low Temperature ScienceHokkaido UniversitySapporoJapan
  4. 4.GZG, Department of CrystallographyUniversity of GöttingenGöttingenGermany

Personalised recommendations