Two Dimensional Minimax Theory on Shear Stress and a General Classification Model for Nonlinear Constitutive Relations

  • Jinro Ukita
Conference paper
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 94)


A theoretical framework that provides a general representation of two-dimensional nonlinear constitutive relations is proposed. The theory is based upon minimization of maximum shear stress that links functional forms of energy dissipation and constitutive relation. Its application to a polynomial energy dissipation function leads to a power-law relationship between strain rate and stress magnitudes. On the basis of this relationship, a general categorization model for constitutive relations is developed and used to analyze forms commonly encountered in ice studies.


Constitutive Relation Maximum Shear Stress Dissipation Function Constitutive Form Principal Strain Rate 
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  1. Alley, R. B., 1992. Flow-law hypotheses for ice-sheet modeling. J. Glaciol. 38: 245–256.ADSGoogle Scholar
  2. Bagnold, R. A., 1954. Experiments on a gravity free dispersion of large solid sphere in a Newtonian fluid under shear. Proc. R. Soc. London, Ser. A 225: 49–63.ADSCrossRefGoogle Scholar
  3. Glen, J. W., 1958. The flow law of ice. A discussion of the assumptions made in glacier theory, their experimental foundations and consequences. IASH 47: 171–183.Google Scholar
  4. Hibler, W. D. 1979. A dynamic thermodynamic model of sea ice. J. Phys. Oceanogr. 9: 815–846.ADSCrossRefGoogle Scholar
  5. Jaeger, H. M. and S. R. Nagel, 1992. Physics of granular flow. Science 255: 1523–1531.ADSCrossRefGoogle Scholar
  6. Landau, L. D. and E. M. Lifshitz, 1959. Fluid Mechanics. Trans. by J. B. Sykes and W. H. Reid, London, Pergamon Press; Reading, Mass., Addison-Wesley Pub. Co.Google Scholar
  7. Moritz, R. E. and J. Ukita, 2000. Geometry and the deformation of pack ice, Part I: A simple kinematic model. Ann. Glaciol. 31: 313–322.ADSCrossRefGoogle Scholar
  8. Overland, J. E. and J. Ukita, 2000. Arctic sea ice dynamics workshop. EOS Trans. AGU 81: 309, 314.ADSCrossRefGoogle Scholar
  9. Parmerter, R. R. and M. D. Coon, 1972. Model of pressure ridge formation in sea ice. J. Geophys. Res. 77: 6565–6575.ADSCrossRefGoogle Scholar
  10. Paterson, W. S. B., 1981. The Physics of Glaciers. Oxford, Pergamon Press (2nd Ed.).Google Scholar
  11. Rothrock, D. A., 1975. The energetics of the plastic deformation of pack ice by ridging. J. Geophys. Res. 80: 4514–4519.ADSCrossRefGoogle Scholar
  12. Shen, H. H., W. D. Hibler and M. Lepparanta, 1987. The role of floe collisions in sea ice rheology. J. Geophys. Res. 92: 7085–7096.ADSCrossRefGoogle Scholar
  13. Stern, H. L., D. A. Rothrock and R. Kwok, 1995. Open water production in Arctic sea ice: satellite measurements and model parameterizations. J. Geophys. Res. 100: 20,601–20,612.CrossRefGoogle Scholar
  14. Thorndike, A. S., D. A. Rothrock, G. A. Maykut and R. Colony, 1975. The thickness distribution of sea ice. J. Geophys. Res. 80: 4501–4513.ADSCrossRefGoogle Scholar
  15. Ukita, J. and R. E. Moritz, 1995. Yield curves and flow rules of pack ice. J. Geophys. Res. 95: 4545–4557.ADSCrossRefGoogle Scholar
  16. Ukita, J. and R. E. Moritz, 2000. Geometry and the deformation of pack ice, Part II: Simulation with a random isotropic model and implication in sea ice rheology. Ann. Glaciol. 31: 323–326.ADSCrossRefGoogle Scholar
  17. Ziegler, H and W. Wehrli, 1987. The derivation of constitutive relations from the free energy and the dissipation function. Adv. App. Mech. 25: 183–238.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Jinro Ukita
    • 1
  1. 1.Goddard Earth Science and Technology CenterUniversity of Maryland Baltimore County and NASA Goddard Space Flight CenterGreenbeltUSA

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