Micro-mechanical Models for Polar Ice

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


This chapter is concerned with the behaviour of polar ice on geophysical time scales and its analysis by applying a micro-mechanical approach. Based on some assumptions regarding the anisotropic properties of an individual ice crystal and its microscopic deformation, frame-indifferent constitutive laws for creep response of the crystal are formulated. By applying homogenization methods, the microscopic laws are then used to derive the macroscopic constitutive relations for polycrystalline ice. These relations are employed to simulate the creep behaviour of ice in simple flow configurations in order to correlate parameters in the macroscopic flow laws with the observed anisotropic behaviour of polar ice. The chapter concludes with the analysis of the mechanism of dynamic (migration) recrystallization of polycrystalline ice. Three alternative dynamic recrystallization models are formulated, which are subsequently used in the simulations for simple flows to investigate the effect of the recrystallization process on the evolution of macroscopic viscosities of ice.


  1. Alley RB (1992) Flow-law hypotheses for ice-sheet modelling. J Glaciol 38(129):245–256CrossRefGoogle Scholar
  2. Arminjon M (1991) Limit distributions of the states and homogenization in random media. Acta Mech 88:27–59CrossRefGoogle Scholar
  3. Azuma N (1994) A flow law for anisotropic ice and its application to ice sheets. Earth Planet Sci Lett 128(3–4):601–614CrossRefGoogle Scholar
  4. Azuma N (1995) A flow law for anisotropic polycrystalline ice under uniaxial compressive deformation. Cold Reg Sci Technol 23:137–147CrossRefGoogle Scholar
  5. Bishop JFW, Hill R (1951) A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Phil Mag (7th Ser) 42(327):414–427Google Scholar
  6. Boehler JP (1987) Representations for isotropic and anisotropic non-polynomial tensor functions. In: Boehler JP (ed) Applications of tensor functions in solid mechanics. Springer, Wien, pp 31–53CrossRefGoogle Scholar
  7. Budd WF, Jacka TH (1989) A review of ice rheology for ice sheet modelling. Cold Reg Sci Technol 16(2):107–144. Scholar
  8. Castelnau O, Duval P, Lebensohn RA, Canova GR (1996) Viscoplastic modeling of texture development in polycrystalline ice with a self-consistent approach: comparison with bound estimates. J Geophys Res 101(B6):13851–13868. Scholar
  9. Chadwick P (1999) Continuum mechanics: concise theory and problems, 2nd edn. Dover, Mineola, New YorkGoogle Scholar
  10. De La Chapelle S, Castelnau O, Lipenkov V, Duval P (1998) Dynamic recrystallization and texture development in ice as revealed by the study of deep ice cores in Antarctica and Greenland. J Geophys Res 103(B3):5091–5105. Scholar
  11. Durand G, Svensson A, Persson A, Gagliardini O, Gillet-Chaulet F, Sjolte J, Montagnat M, Dahl-Jensen D (2009) Evolution of the texture along the EPICA Dome C ice core. In: Hondoh T (ed) Physics of ice core records II. Hokkaido University, Hokkaido, pp 91–105Google Scholar
  12. Duval P (1981) Creep and fabric of polycrystalline ice under shear and compression. J Glaciol 27(95):129–140CrossRefGoogle Scholar
  13. Duval P, Arnaud L, Brissaud O, Montagnat M, De La Chapelle S (2000) Deformation and recrystallization processes of ice from polar ice sheets. Ann Glaciol 30:83–87CrossRefGoogle Scholar
  14. Duval P, Castelnau O (1995) Dynamic recrystallization of ice in polar ice sheets. J Phys IV 5(C3):197–205Google Scholar
  15. Elvin AA (1996) Number of grains required to homogenize elastic properties of polycrystalline ice. Mech Mat 22(1):51–64CrossRefGoogle Scholar
  16. Faria SH (2006) Creep and recrystallization of large polycrystalline masses. III. Continuum theory of ice sheets. Proc R Soc Lond A 462(2073):2797–2816. Scholar
  17. Faria SH, Kremer GM, Hutter K (2003) On the inclusion of recrystallization processes in the modeling of induced anisotropy in ice sheeets: a thermodynamicist’s point of view. Ann Glaciol 37:29–34CrossRefGoogle Scholar
  18. Faria SH, Ktitarev D, Hutter K (2002) Modelling evolution of anisotropy in fabric and texture of polar ice. Ann Glaciol 35:545–551CrossRefGoogle Scholar
  19. Faria SH, Weikusat I, Azuma N (2014) The microstructure of polar ice. Part I: highlights from ice core research. J Struct Geol 61:2–20. Scholar
  20. Gagliardini O, Arminjon M, Imbault D (2001) An inhomogeneous variational model applied to predict the behaviour of isotropic polycrystalline ice. Arch Mech 53(1):3–21Google Scholar
  21. 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. Scholar
  22. Gödert G, Hutter K (1998) Induced anisotropy in large ice shields: theory and its homogenization. Contin Mech Thermodyn 10(5):293–318CrossRefGoogle Scholar
  23. Gow AJ, Meese DA (2007) Physical properties, crystalline textures and c-axis fabrics of the Siple Dome (Antarctica) ice core. J Glaciol 53(183):573–584. Scholar
  24. Gow AJ, Meese DA, Alley RB, Fitzpatrick JJ, Anandakrishnan S, Woods GA, Elder BC (1997) Physical and structural properties of the Greenland Ice Sheet Project 2 ice core: a review. J Geophys Res 102(C12):26559–26575. Scholar
  25. Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2):127–140CrossRefGoogle Scholar
  26. Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc A 65(389):349–354CrossRefGoogle Scholar
  27. Hutchinson JW (1976) Bounds and self-consistent estimates for creep of polycrystalline materials. Proc R Soc Lond A 348(1652):101–127CrossRefGoogle Scholar
  28. 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. Scholar
  29. Jones SJ (1982) The confined compressive strength of polycrystalline ice. J Glaciol 28(98):171–177CrossRefGoogle Scholar
  30. Kamb WB (1961) The glide direction in ice. J Glaciol 3(30):1097–1106CrossRefGoogle Scholar
  31. Kennedy JH, Pettit EC, Di Prinzio CL (2013) The evolution of crystal fabric in ice sheets and its link to climate history. J Glaciol 59(214):357–373. Scholar
  32. Ktitarev D, Gödert G, Hutter K (2002) Cellular automaton model for recrystallization, fabric, and texture development in polar ice. J Geophys Res 107(B8):2165. Scholar
  33. Lile RC (1978) The effect of anisotropy on the creep of polycrystalline ice. J Glaciol 21(85):475–483CrossRefGoogle Scholar
  34. Liu IS (2002) Continuum mechanics. Springer, BerlinCrossRefGoogle Scholar
  35. Lliboutry L (1993) Anisotropic, transversely isotropic nonlinear viscosity of rock ice and rheological parameters inferred from homogenization. Int J Plast 9(5):619–632CrossRefGoogle Scholar
  36. Ma Y, Gagliardini O, Ritz C, Gillet-Chaulet F, Durand G, Montagnat M (2010) Enhancement factors for grounded ice and ice shelves inferred from an anisotropic ice-flow model. J Glaciol 56(199):805–812CrossRefGoogle Scholar
  37. Mangeney A, Califano F, Castelnau O (1996) Isothermal flow of an anisotropic ice sheet in the vicinity of an ice divide. J Geophys Res 101(B12):28189–28204CrossRefGoogle Scholar
  38. Mellor M, Cole DM (1982) Deformation and failure of ice under constant stress or constant strain-rate. Cold Reg Sci Technol 5(3):201–219CrossRefGoogle Scholar
  39. Meyssonnier J, Philip A (1996) A model for tangent viscous behaviour of anisotropic polar ice. Ann Glaciol 23:253–261CrossRefGoogle Scholar
  40. Meyssonnier J, Philip A (1999) Remarks on self-consistent modelling of polycrystalline ice. In: Hutter K, Wang Y, Beer H (eds) Advances in cold-region thermal engineering and sciences. Springer, Berlin, pp 225–236CrossRefGoogle Scholar
  41. Molinari A, Canova GR, Ahzy S (1987) A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metallurgica 35(12):2983–2994CrossRefGoogle Scholar
  42. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21(5):571–574CrossRefGoogle Scholar
  43. Morland LW (2002) Influence of lattice distortion on fabric evolution in polar ice. Continuum Mech Thermodyn 14(1):9–24. Scholar
  44. 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. Scholar
  45. Paterson WSB (1994) The physics of glaciers, 3rd edn. Butterworth-Heinemann, OxfordCrossRefGoogle Scholar
  46. Pimienta P, Duval P, Lipenkov VY (1987) Mechanical behavior of anisotropic polar ice. In: International association of hydrological sciences publication, no 170, pp 57–66. (Symp. Physical Basis of Ice Sheet Modelling, Vancouver 1987)Google Scholar
  47. Placidi L, Greve R, Seddik H, Faria SH (2010) Continuum-mechanical, anisotropic flow model for polar ice masses, based on an anisotropic flow enhancement factor. Continuum Mech Thermodyn 22(3):221–237. Scholar
  48. Placidi L, Hutter K, Faria SH (2006) A critical review of the mechanics of polycrystalline polar ice. GAMM-Mitt 29(1):80–117CrossRefGoogle Scholar
  49. Rigsby GP (1958) Effect of hydrostatic pressure on velocity of shear deformation of single ice crystals. J Glaciol 3(24):273–278CrossRefGoogle Scholar
  50. Spencer AJM (1980) Continuum mechanics. Longman, HarlowGoogle Scholar
  51. Staroszczyk R (2001) A uniform stress, discrete-grain model for induced anisotropy of ice. In: Szmidt K (ed) Applications of mechanics in civil- and hydro-engineering. IBW PAN Publishing House, Gdańsk, pp 295–314Google Scholar
  52. Staroszczyk R (2002) A uniform strain, discrete-grain model for evolving anisotropy of polycrystalline ice. Arch Mech 54(2):103–126Google Scholar
  53. Staroszczyk R (2004) Constitutive modelling of creep induced anisotropy of ice. IBW PAN Publishing House, GdańskGoogle Scholar
  54. Staroszczyk R (2009) A multi-grain model for migration recrystallization in polar ice. Arch Mech 61(3–4):259–282Google Scholar
  55. Staroszczyk R (2011) A uniform stress, multi-grain model for migration recrystallization in polar ice. Acta Geophys 59(5):833–857. Scholar
  56. Staroszczyk R, Morland LW (2000) Plane ice-sheet flow with evolving orthotropic fabric. Ann Glaciol 30:93–101CrossRefGoogle Scholar
  57. Staroszczyk R, Morland LW (2001) Strengthening and weakening of induced anisotropy in polar ice. Proc R Soc Lond A 457(2014):2419–2440. Scholar
  58. Svendsen B, Hutter K (1996) A continuum approach for modelling induced anisotropy in glaciers and ice sheets. Ann Glaciol 23:262–269CrossRefGoogle Scholar
  59. Thorsteinsson T (2002) Fabric development with nearest-neighbor interaction and dynamic recrystallization. J Geophys Res 107(B1). Scholar
  60. Thorsteinsson T, Kipfstuhl J, Miller H (1997) Textures and fabrics in the GRIP ice core. J Geophys Res 102(C12):26583–26599. Scholar
  61. Thorsteinsson T, Waddington ED, Taylor KC, Alley RB, Blankenship DD (1999) Strain-rate enhancement at Dye 3, Greenland. J Glaciol 45(150):338–345CrossRefGoogle Scholar
  62. 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. Scholar
  63. Truesdell C, Noll W (2004) The non-linear field theories of mechanics, 3rd edn. Springer, BerlinCrossRefGoogle Scholar
  64. Van der Veen CJ, Whillans IM (1994) Development of fabric in ice. Cold Reg Sci Technol 22(2):171–195. Scholar
  65. Wenk HR, Canova GR, Molinari A, Kocks UF (1989) Viscoplastic modelling of texture development in quartzite. J Geophys Res 94(B12):17895–17906CrossRefGoogle Scholar
  66. Zhang Y, Jenkins JT (1993) The evolution of the anisotropy of a polycrystalline aggregate. J Mech Phys Solids 41(7):1213–1243CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations