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Phenomenological Constitutive Models for Polar Ice

  • Ryszard StaroszczykEmail author
Chapter
Part of the GeoPlanet: Earth and Planetary Sciences book series (GEPS)

Abstract

This chapter deals with the analysis of the behaviour of polar ice by applying a phenomenological approach, in which the macroscopic creep response of ice is determined solely in terms of the macroscopic stress, strain-rate, and deformation. The microscopic mechanism of re-orientation of individual crystals during the deformation of ice is also accounted for in order to model the evolution of the internal structure of the material. General forms of frame-indifferent constitutive flow laws, which express either the stress in terms of the strain-rate, or the strain-rate in terms of the stress, are derived on the assumption that the type of anisotropy which develops in polar ice sheets is close to orthotropy. The parameters in the derived constitutive models are determined by the correlation of model predictions with available experimental data. The phenomenological approach is also applied to model the mechanism of the dynamic recrystallization of polar ice. All constitutive models developed in this chapter are applied to simulate the evolution of the macroscopic viscous properties of polycrystalline ice with increasing shear and axial strains, and the results of these simulations are presented for various sets of material parameters defining the properties of ice.

References

  1. Budd WF, Jacka TH (1989) A review of ice rheology for ice sheet modelling. Cold Reg Sci Technol 16(2):107–144.  https://doi.org/10.1016/0165-232X(89)90014-1CrossRefGoogle Scholar
  2. Chadwick P (1999) Continuum mechanics: concise theory and problems, 2nd edn. Dover, Mineola, New YorkGoogle Scholar
  3. Ericksen JL, Rivlin RS (1954) Large elastic deformations of homogeneous anisotropic materials. J Ration Mech Anal 3:281–301Google Scholar
  4. Gagliardini O, Meyssonnier J (1999) Analytical derivations for the behavior and fabric evolution of a linear orthotropic ice polycrystal. J Geophys Res 104(B8):17797–17809.  https://doi.org/10.1029/1999JB900146CrossRefGoogle Scholar
  5. Hunter SC (1983) Mechanics of continuous media, 2nd edn. Ellis Horwood Ltd., ChichesterGoogle Scholar
  6. Jacka TH (1984) The time and strain required for development of minimum strain rates in ice. Cold Reg Sci Technol 8(3):261–268.  https://doi.org/10.1016/0165-232X(84)90057-0CrossRefGoogle Scholar
  7. Jacka TH, Maccagnan M (1984) Ice crystallographic and strain rate changes with strain in compression and extension. Cold Reg Sci Technol 8(3):269–286.  https://doi.org/10.1016/0165-232X(84)90058-2CrossRefGoogle Scholar
  8. Li J, Jacka TH, Budd WF (1996) Deformation rates in combined compression and shear for ice which is initially isotropic and after the development of strong anisotropy. Ann Glaciol 23:247–252CrossRefGoogle Scholar
  9. Liu IS (2002) Continuum mechanics. Springer, BerlinCrossRefGoogle Scholar
  10. Morland LW (1993) The flow of ice sheets and ice shelves. In: Hutter K (ed) Continuum mechanics in environmental sciences and geophysics. Springer, Wien, pp 403–466CrossRefGoogle Scholar
  11. Morland LW (2001) Influence of bed topography on steady plane ice sheet flow. In: Straughan B, Greve R, Ehrentraut H, Wang Y (eds) Continuum mechanics and applications in geophysics and the environment. Springer, Berlin, pp 276–304CrossRefGoogle Scholar
  12. Morland LW, Staroszczyk R (1998) Viscous response of polar ice with evolving fabric. Continuum Mech Thermodyn 10(3):135–152.  https://doi.org/10.1007/s001610050086CrossRefGoogle Scholar
  13. Morland LW, Staroszczyk R (2003a) Strain-rate formulation of ice fabric evolution. Ann Glaciol 37:35–39CrossRefGoogle Scholar
  14. Morland LW, Staroszczyk R (2003b) Stress and strain-rate formulations for fabric evolution in polar ice. Contin Mech Thermodyn 15(1):55–71.  https://doi.org/10.1007/s00161-002-0104-2CrossRefGoogle Scholar
  15. Morland LW, Staroszczyk R (2009) Ice viscosity enhancement in simple shear and uni-axial compression due to crystal rotation. Int J Eng Sci 47(11–12):1297–1304.  https://doi.org/10.1016/j.ijengsci.2008.09.011CrossRefGoogle Scholar
  16. Rivlin RS (1955) Further remarks on the stress-deformation relations for isotropic materials. J Ration Mech Anal 4:681–701Google Scholar
  17. Rivlin RS, Ericksen JL (1955) Stress-deformation relations for isotropic materials. J Ration Mech Anal 4:323–425Google Scholar
  18. Smith GD, Morland LW (1981) Viscous relations for the steady creep of polycrystalline ice. Cold Reg Sci Technol 5(2):141–150CrossRefGoogle Scholar
  19. Smith GF (1994) Constitutive equations for anisotropic and isotropic materials. North-Holland, AmsterdamGoogle Scholar
  20. Smith GF, Rivlin RS (1957) The anisotropic tensors. Q Appl Math 15:309–314CrossRefGoogle Scholar
  21. Spencer AJM (1980) Continuum mechanics. Longman, HarlowGoogle Scholar
  22. Spencer AJM (1987a) Anisotropic invariants and additional results for invariant and tensor representations. In: Boehler JP (ed) Applications of tensor functions in solid mechanics. Springer, Wien, pp 171–186CrossRefGoogle Scholar
  23. Spencer AJM (1987b) Isotropic polynomial invariants and tensor functions. In: Boehler JP (ed) Applications of tensor functions in solid mechanics. Springer, Wien, pp 141–169CrossRefGoogle Scholar
  24. Staroszczyk R (2001) An orthotropic constitutive model for secondary creep of ice. Arch Mech 53(1):65–85Google Scholar
  25. Staroszczyk R (2003) Plane ice sheet flow with evolving and recrystallising fabric. Ann Glaciol 37(1):247–251.  https://doi.org/10.3189/172756403781815834CrossRefGoogle Scholar
  26. Staroszczyk R (2004) Constitutive modelling of creep induced anisotropy of ice. IBW PAN Publishing House, GdańskGoogle Scholar
  27. Staroszczyk R, Gagliardini O (1999) Two orthotropic models for the strain-induced anisotropy of polar ice. J Glaciol 45(151):485–494CrossRefGoogle Scholar
  28. Staroszczyk R, Morland LW (1999) Orthotropic viscous model for ice. In: Hutter K, Wang Y, Beer H (eds) Advances in cold-region thermal engineering and sciences. Springer, Berlin, pp 249–258.  https://doi.org/10.1007/BFb0104187
  29. Staroszczyk R, Morland LW (2000a) Orthotropic viscous response of polar ice. J Eng Math 37(1–3):191–209.  https://doi.org/10.1023/A:1004738429373CrossRefGoogle Scholar
  30. Staroszczyk R, Morland LW (2000b) Plane ice-sheet flow with evolving orthotropic fabric. Ann Glaciol 30:93–101CrossRefGoogle Scholar
  31. Staroszczyk R, Morland LW (2001) Strengthening and weakening of induced anisotropy in polar ice. Proc R Soc Lond A 457(2014):2419–2440.  https://doi.org/10.1098/rspa.2001.0817CrossRefGoogle Scholar
  32. Treverrow A, Budd WF, Jacka TH, Warner RC (2012) The tertiary creep of polycrystalline ice: experimental evidence for stress-dependent levels of strain-rate enhancement. J Glaciol 58(208):301–314.  https://doi.org/10.3189/2012JoG11J149CrossRefGoogle Scholar
  33. Truesdell C, Noll W (2004) The non-linear field theories of mechanics, 3rd edn. Springer, BerlinCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Hydro-EngineeringPolish Academy of SciencesGdańskPoland

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