Continuum Mechanics and Thermodynamics

, Volume 17, Issue 5, pp 387–408 | Cite as

Anisotropic damage mechanics for viscoelastic ice

  • A. Pralong
  • K. Hutter
  • M. FunkEmail author
Original Article


We present a formulation of continuum damage in glacier ice that incorporates the induced anisotropy of the damage effects but restricts these formally to orthotropy. Damage is modeled by a symmetric second rank tensor that structurally plays the role of an internal variable. It may be interpreted as a texture measure that quantifies the effective specific areas over which internal stresses can be transmitted. The evolution equation for the damage tensor is motivated in the reference configuration and pushed forward to the present configuration. A spatially objective constitutive form of the evolution equation for the damage tensor is obtained. The rheology of the damaged ice presumes no volume conservation. Its constitutive relations are derived from the free enthalpy and a dissipation potential, and extends the classical isotropic power law by elastic and damage tensor dependent terms. All constitutive relations are in conformity with the second law of thermodynamics.


Damage mechanics Induced anisotropy Constitutive relations Viscoelasticity Ice mechanics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hutter, K.: Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets. D. Reidel Publishing Company Tokyo (1983)Google Scholar
  2. 2.
    Sinha, N.-K.: Kinetics of microcracking and dilatation in polycrystalline ice. In: IUTAM/IAHR Symposium on Ice-Structure Interaction. 69–87 (1991)Google Scholar
  3. 3.
    Xiao, J., Jordaan, I.-J.: Application of damage mechanics to ice failure in compression. Cold Regions Science and Technology. 24(3), 305–322 (1996)CrossRefGoogle Scholar
  4. 4.
    Singh, S.-K., Jordaan, I.-J.: Constitutive behaviour of crushed ice. International Journal of Fracture. 97(1–4), 171–187 (1999)CrossRefGoogle Scholar
  5. 5.
    Szyszkowski, W., Glockner, P.-G.: On a multiaxial constitutive law for ice. Mechanics of Materials. 5(1), 49–71 (1986)CrossRefGoogle Scholar
  6. 6.
    Mahrenholtz, O., Wu, Z.: Determination of creep damage parameters for polycrystalline ice. Advances in Ice Technology. 181–192 (1992)Google Scholar
  7. 7.
    Kachanov, L.-M.: Rupture time under creep conditions (trans. from Russian, 1957). International Journal of Fracture. 97(1–4), xi–xviii (1999)Google Scholar
  8. 8.
    Schapery, R.-A.: Models for the deformation behavior of viscoelastic media with distributed damage and their applicability to ice. In: Sympsium on Ice-Structure Interaction. 191–230 (1991)Google Scholar
  9. 9.
    Melanson, P.-M., Jordaan, I.-J., Meglis, I.-L.: Modelling of damage in ice. In: Proceeding of the 14th International Symposium on Ice. vol. 2, 979–988 (1998)Google Scholar
  10. 10.
    Pralong, A., Funk, M., Lüthi, M.-P.: A description of crevasse formation using continuum damage mechanics. Annals of Glaciology. 37, 77–82 (2003)CrossRefGoogle Scholar
  11. 11.
    Weiss, J., Gay, M.: Fracturing of ice under compression creep as revealed by a multifractal analysis. Journal of Geophysical Research. 103(B10), 24005–24016 (1998)CrossRefGoogle Scholar
  12. 12.
    Lemaitre, J.: A Course on Damage Mechanics. Springer Berlin (1992)zbMATHGoogle Scholar
  13. 13.
    Wu, Z., Mahrenholtz, O.: Creep and creep damage of polycrystalline ice under multi-axial variable loading. In: Proceedings of the 12th International Conference on Offshore Mechanics and Arctic Engineering. vol. 4, 1–10 (1993)Google Scholar
  14. 14.
    Pralong, A., Funk, M.: Dynamic damage model of crevasse opening and application to glacier calving. Journal of Geophysical Research 110, 301–309 (doi: 1029/2004 33003104, 2005).Google Scholar
  15. 15.
    Holzapfel, G.-A.: Nonlinear Solid Mechanics, a Continuum Approach for Engineering. John Wiley & Sons, LTD, Chichester (2000)zbMATHGoogle Scholar
  16. 16.
    Liu, I.-S.: Method of Lagrange multipliers for exploitation of entropy principle. Archive for Rational Mechanics and Analysis. 46(2), 131–148 (1972)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Svendsen, B., Hutter, K., Laloui, L.: Constitutive models for granular materials including quasi-static frictional behaviour: Toward a thermodynamic theory of plasticity. Continuum Mechanics and Thermodynamics. 11(4), 263–275 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling : Continuum Mechanics, Dimensional Analysis, Turbulence. Springer, Berlin (2004)zbMATHGoogle Scholar
  19. 19.
    Liu, I.-S.: On entropy flux heat flux relation in thermodynamics with lagrange multipliers. Continuum Mechanics and Thermodynamics. 8(4), 247–256 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Betten, J.: Tensorrechnung für Ingenieure. B. G. Teubner, Stuttgart (1987)zbMATHGoogle Scholar
  21. 21.
    Lemaitre, J., Desmorat, R., Sauzay, M.: Anisotropic damage law of evolution. European Journal of Mechanics A-Solids. 19(2), 187–208 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Svendsen, B., Hutter, K.: A continuum approach for modelling induced anisotropy in glaciers and ice sheets. Annals of Glaciology. 23, 262–269 (1996)Google Scholar
  23. 23.
    Murakami, S., Ohno, N.: A continuum theory of creep and creep damage. In: Proc. of the 3rd IUTAM Symposium on Creep in Structures. 422–444 (1980)Google Scholar
  24. 24.
    Hayhurst, D.-R.: Creep-rupture under multi-axial states of stress. Journal of the Mechanics and Physics of Solids. 20(6):381–390 (1972)CrossRefGoogle Scholar
  25. 25.
    Gold, L.-W.: Time to formation of first cracks in ice. In: Physics of Snow and Ice, International Conference on low Temperature Science. vol. 1(Part 1), 359–370 (1967)Google Scholar
  26. 26.
    Gold, L.-W.: The failure process in columnar-grained ice. Technical Report 369, Division of Building Research, NRC Ottawa (1972)Google Scholar
  27. 27.
    Duval, P.: Creep and fabrics of polycrystalline ice under shear and compression. Journal of Glaciology. 27(95), 129–140 (1981)Google Scholar
  28. 28.
    Krausz, A.-S., Krausz, K.: Fracture Kinetics of Crack Growth. Kluwer Academic Publishers, Dordrecht (1988)zbMATHGoogle Scholar
  29. 29.
    Picasso, M., Rappaz, M.-J., Reist, A., Funk, M., Blatter, H.: Numerical simulation of the motion of a two dimensional glacier. International Journal for Numerical Methods in Engineering. 60(5), 995–1009 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Voight, B.: A method for prediction of volcanic-eruptions. Nature. 332(6160), 125–130 (1988)CrossRefGoogle Scholar
  31. 31.
    Röthlisberger, H.: Eislawinen und Ausbrüche von Gletscherseen. In: Gletscher und Klima - glaciers et climat, Jahrbuch der Schweizerischen Naturforschenden Gesellschaft, wissenschaftlicher Teil 1978. 170–212 (1981)Google Scholar
  32. 32.
    Lüthi, M.: Instability in glacial systems. In: Milestones in Physical Glaciology: From the Pioneers to a Modern Science. VAW, ETH Zürich. vol. 180, 63–70 (2003)Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Laboratory of Hydraulics, Hydrology and Glaciology,Swiss Federal Institute of TechnologyZurichSwitzerland
  2. 2.Department of MechanicsDarmstadt University of TechnologyDarmstadtGermany

Personalised recommendations