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.
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References
Azuma N.: A flow law for anisotropic ice and its application to ice sheets. Earth Planet. Sci. Lett. 128(3–4), 601–614 (1994)
Azuma N.: A flow law for anisotropic polycrystalline ice under uniaxial compressive deformation. Cold Reg. Sci. Technol. 23(2), 137–147 (1995)
Azuma N., Higashi A.: Formation processes of ice fabric patterns in ice sheets. Ann. Glaciol. 6, 130–134 (1985)
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)
Boehler J.P.: Applications of Tensor Functions in Solid Mechanics. Springer, New York (1987)
Budd W.F., Jacka T.H.: A review of ice rheology for ice sheet modelling. Cold Reg. Sci. Technol. 16(2), 107–144 (1989)
Dafalias Y.F.: Orientation distribution function in non-affine rotations. J. Mech. Phys. Solids 49, 2493–2516 (2001)
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
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
Faria S.H.: Mixtures with continuous diversity: general theory and application to polymer solutions. Contin. Mech. Thermodyn. 13, 91–120 (2001)
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
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
Faria S.H.: The symmetry group of the CAFFE model. J. Glaciol. 54(187), 643–645 (2008)
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
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)
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)
Glen J.W.: The creep of polycrystalline ice. Proc. R. Soc. Lond. A 228, 519–538 (1955)
Gödert G.: A mesoscopic approach for modelling texture evolution of polar ice including recrystallization phenomena. Ann. Glaciol. 37, 23–28 (2003)
Gödert G., Hutter K.: Induced anisotropy in large ice shields: theory and its homogenization. Contin. Mech. Thermodyn. 10(5), 293–318 (1998)
Hutter K.: Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets. D Reidel Publishing Company, Dordrecht, The Netherlands (1983)
Jacka T.H., Budd W.F.: Isotropic and anisotropic flow relations for ice dynamics. Ann. Glaciol. 12, 81–84 (1989)
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)
Larson R.G.: Constitutive Equations for Polymer Melts and Solutions. Butterworths Series in Chemical Engineering. Butterworths, Boston (1988)
Larson R.G.: The Structure and Rheology of Complex Fluids. Oxford University press, Oxford (1999)
Liu I.-S.: Continuum Mechanics. Springer, Berlin (2002)
Lliboutry L.: Anisotropic, transversely isotropic nonlinear viscosity of rock ice and rheological parameters inferred from homogenization. Int. J. Plast. 9, 619–632 (1993)
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)
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
McConnel J.C.: On the plasticity of an ice crystal. Proc. R. Soc. Lond. 49, 323–343 (1891)
Miyamoto A.: Mechanical properties and crystal textures of Greenland deep ice cores. Doctoral thesis. Hokkaido University, Sapporo Japan (1999)
Morland L.W., Staroszczyk R.: Stress and strain-rate formulations for fabric evolution in polar ice. Contin. Mech. Thermodyn. 15(1), 55–71 (2003)
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
Müller I.: Thermodynamics. Pitman Advanced Publishing Program, Boston (1985)
Nye J.F.: The distribution of stress and velocity in glaciers and ice sheets. Proc. R. Soc. Lond. 239, 113–133 (1952)
Papenfuss C.: Theory of liquid crystals as an example of mesoscopic continuum mechanics. Comput. Mater. Sci. 19, 45–52 (2000)
Papenfuss C., Van P.: Scalar, vectorial, and tensorial damage parameters from the mesoscopic background. Proc. Est. Acad. Sci. 57(3), 132–141 (2008)
Paterson W.S.B.: The Physics of Glaciers. 3rd edn. Pergamon Press, Oxford (1994)
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)
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/
Placidi L.: Microstructured continua treated by the theory of mixtures. Doctoral thesis. University of Rome, La Sapienza, Italy (2005)
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)
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)
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
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
Placidi L., Hutter K., Faria S.H.: A critical review of the mechanics of polycrystalline polar ice. GAMM-Mitt. 29(1), 80–117 (2006)
Rashid M.M.: Texture evolution and plastic response of two-dimensional polycrystals. J. Mech. Phys. Solids 40, 1009–1029 (1992)
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)
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
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)
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)
Staroszczyk R., Morland L.W.: Strengthening and weakening of induced anisotropy in polar ice. Proc. R. Soc. Lond. 457(2014), 2419–2440 (2001)
Svendsen B., Hutter K.: A continuum approach for modelling induced anisotropy in glaciers and ice sheets. Ann. Glaciol. 23, 262–269 (1996)
Thorsteinsson T.: An analytical approach to deformation of anisotropic ice-crystal aggregates. J. Glaciol. 47(158), 507–516 (2001)
Truesdell C.: Sulle basi della termomeccanica, Nota I. Rendiconnti Accademia dei Lincei 8/22, 33–38 (1957a)
Truesdell C.: Sulle basi della termomeccanica, Nota II. Rendiconnti Accademia dei Lincei 8/22, 158–166 (1957b)
Acknowledgements
The authors would like to thank Kolumban Hutter and Leslie W. Morland for many productive discussions. Comments of the scientific editor Wolfgang Müller and an anonymous reviewer helped considerably to improve the structure and clarity of the manuscript. This study was supported by a Grant-in-Aid for Creative Scientific Research (No. 14GS0202) from the Japanese Ministry of Education, Culture, Sports, Science and Technology, by a Grant-in-Aid for Scientific Research (No. 18340135) from the Japan Society for the Promotion of Science, and by a grant (Nr. FA 840/1-1) from the Priority Program SPP-1158 of the Deutsche Forschungsgemeinschaft (DFG).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Placidi, L., Greve, R., Seddik, H. et al. Continuum-mechanical, Anisotropic Flow model for polar ice masses, based on an anisotropic Flow Enhancement factor. Continuum Mech. Thermodyn. 22, 221–237 (2010). https://doi.org/10.1007/s00161-009-0126-0
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DOI: https://doi.org/10.1007/s00161-009-0126-0