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Continuum Mechanics and Thermodynamics

, Volume 19, Issue 7, pp 423–440 | Cite as

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material

I. On thermodynamically consistent evolution
  • Chung FangEmail author
  • Yongqi Wang
  • Kolumban Hutter
Original Article

Abstract

In the present study an evolution equation for the Cauchy stress tensor is proposed for an isotropic elasto-visco-plastic continuum. The proposed stress model takes effects of elasticity, viscosity and plasticity of the material simultaneously into account. It is ascribed with some scalar coefficient functions and, in particular, with an unspecified tensor-valued function N, which is handled as an independent constitutive quantity. It is demonstrated that by varying the values and the specific functional forms of these coefficients and N, different known models in non-Newtonian rheology can be reproduced. A thermodynamic analysis, based on the Müller–Liu entropy principle, is performed. The results show that these coefficients and N are not allowed to vary arbitrarily, but should satisfy certain restrictions. Simple postulates are made to further simplify the deduced general results of the thermodynamic analysis. They yield justification and thermodynamic consistency of the existing models for a class of materials embracing thermoelasticity, hypoelasticity and in particular hypoplasticity, of which the thermodynamic foundation is established successively for the first time in literature. The study points at the wide applicability and practical usefulness of the present model in different fields from non-Newtonian fluid to solid mechanics. In this paper the thermodynamic analysis of the proposed evolution-type stress model is discussed, its applications are reported later.

Keywords

Stress tensor Hypoelasticity Hypoplasticity Müller–Liu entropy principle 

PACS

46.05.+b 46.35.+z 47.50.Cd 47.57.-s 

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References

  1. 1.
    Lubliner J. (1990). Plasticity Theory. Macmillian Publishing Company, New York zbMATHGoogle Scholar
  2. 2.
    Naghdi P.M. (1990). A critical review of the state of finite plasticity. J. Appl. Math. Phys. (ZAMP) 41: 315–394 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fung Y.C. (1994). A First Course of Continuum Mechanics, 3rd ed. Prentice Hall, Englewood Cliffs Google Scholar
  4. 4.
    Truesdell, C., Noll, W.: The Non-linear Theories of Mechanics. In: Flügge, S. (ed.) Handbuch der Physik, III/3. Springer, Berlin (1965)Google Scholar
  5. 5.
    Wang Y. (2006). Time-dependent Poiseuille flows of visco-elasto-plastic fluids. Acta Mech. 186: 187–201 zbMATHCrossRefGoogle Scholar
  6. 6.
    Svendsen B., Hutter K. and Laloui L. (1999). Constitutive models for granular materials including quasi-static frictional behaviour: toward a thermodynamic theory of plasticity. Continuum Mech. Thermodyn. 4: 263–275 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Tabor D. (1993). Gases, Liquids and Solids, 3rd ed. Cambridge University Press, Cambridge Google Scholar
  8. 8.
    Barnes H.B., Hutton J.F. and Walters K. (1989). An Introduction to Rheology. Elsevier, Amsterdam zbMATHGoogle Scholar
  9. 9.
    Tanner I. (1992). Engineering Rheology. Oxford University Press, New York Google Scholar
  10. 10.
    Kolymbas D. (1991). An outline of hypoplasticity. Arch. Appl. Mech. 61: 143–151 zbMATHGoogle Scholar
  11. 11.
    Kolymbas, D.: A generalized hypoplastic constitutive law. In: Proceedings of 11th international conference on soil mechanics and foundation engineering, Vol. 5. Balkma, 2626, 1985 (1988)Google Scholar
  12. 12.
    Kolymbas, D.: Eine konstitutive Theorie für Boden und andere körnige Stoffe. Publication Series of the Institute of Soil Mechanics and Rock Mechanics. Karlsruhe University, No. 109 (1988)Google Scholar
  13. 13.
    Wu, W.: Hypoplasticity as a mathematical model for the mechanical behaviour of granular materials. Publication Series of the Institute of Soil Mechanics and Rock Mechanics. Karlsruhe University, No. 129 (1992)Google Scholar
  14. 14.
    Houlsby G.T. and Puzrin A.M. (2000). An approach to plastical based on generalized thermodynamics. In: Kolymbas, D. (eds) Constitutive Modelling of Granular Materials, pp 319–331. Springer, Berlin Google Scholar
  15. 15.
    Houlsby, G.T.: Derivation of incremental stress–strain response for plasticity models based on thermodynamic functions. In: IUTAM symposium on mechanics of granular and porous materials. Cambridge, pp. 161–172. Kluwer, Dordrecht (1996)Google Scholar
  16. 16.
    Collins I.F. and Houlsby G.T. (1997). Application of thermomechanical principles to the modelling of geotechnical materials. Proc. R. Soc. Lond. A 453: 1975–2001 zbMATHCrossRefGoogle Scholar
  17. 17.
    Fang, C., Lee, C.H.: An unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material. II. Normal stress difference in a viscometric flow, and an unsteady flow with a moving boundary (in press)Google Scholar
  18. 18.
    Burth K. and Brocks W. (1992). Plastizität, Grundlagen und Anwendungen für Ingenieure. Vieweg Verlag, Braunschweig Google Scholar
  19. 19.
    Pitteri M. (1986). Continuum equations of balance in classical statistical mechanics. Arch. Ration. Mech. Anal. 94: 291–305 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Svendsen B. (1999). A statistical mechanical formulation of continuum fields and balance relations for granular and other materials with internal degree of freedom. In: Hutter, K. and Wilmanski, K. (eds) CISM Course on Kinetic and Continuum Mechanical Approaches to Granular and Porous Media, pp 245–308. Springer, Wien Google Scholar
  21. 21.
    Müller I. (1985). Thermodynamics. Pitmann, London zbMATHGoogle Scholar
  22. 22.
    Hutter K. (1977). The foundations of thermodynamics, its basic postulates and implications: a review of modern thermodynamics. Acta Mech. 27: 1–54 CrossRefMathSciNetGoogle Scholar
  23. 23.
    Green A.E. and Naghdi P.M. (1971). On thermodynamics, rate of work and energy. Arch. Ration. Mech. Anal. 40: 38–49 CrossRefMathSciNetGoogle Scholar
  24. 24.
    Green A.E. and Naghdi P.M. (1972). On continuum thermodynamics. Arch. Ration. Mech. Anal. 48: 352–278 zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Liu I. (1972). Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46: 131–148 zbMATHGoogle Scholar
  26. 26.
    Liu I. (1996). On the entropy flux-heat flux relation in thermodynamics with Lagrange multipliers. Continuum Mech. Thermodyn. 8: 247–256 zbMATHCrossRefGoogle Scholar
  27. 27.
    Hutter K. and Wang Y. (2003). Phenomenological thermodynamics and entropy principle. In: Greven, A., Keller, G. and Warnecke, G. (eds) Entropy, 1st edn, pp 57–77. Princeton University Press, New Jersey Google Scholar
  28. 28.
    Wang Y. and Hutter K. (1999). A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol. Acta 38: 214–223 CrossRefGoogle Scholar
  29. 29.
    Wang Y. and Hutter K. (1999). A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid–fluid mixtures. Granul. Matter 1: 163–181 CrossRefGoogle Scholar
  30. 30.
    Hutter K., Laloui L. and Vulliet L. (1999). Thermodynamically based mixture models of saturated and unsaturated soils. Mech. Cohes. Frict. Mater. 4: 295–338 CrossRefGoogle Scholar
  31. 31.
    Romano M.A. (1974). A continuum theory for granular media with a critical state. Arch. Mech. 20: 1011–1028 Google Scholar
  32. 32.
    Gudehus, G.: A comparison of some constitutive laws for soils under rapidally loading and unloading. In: Wittke, W. (ed.) Proceeding of the 3rd International Conference on Numerical Method in Geomechanics, pp. 1309–1323 (1979)Google Scholar
  33. 33.
    Wu W. and Kolymbas D. (2000). Hypoplasticity then and now. In: Kolymbas, D. (eds) Constitutive Modelling of Granular Materials, pp 57–105. Springer, Heidelberg Google Scholar
  34. 34.
    Luca I., Fang C. and Hutter K. (2004). A thermodynamic model of turbulent motions in a granular material. Continuum Mech. Thermodyn. 16: 363–390zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Civil EngineeringNational Cheng Kung UniversityTainan CityTaiwan
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany
  3. 3.ZürichSwitzerland

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