Skip to main content
SpringerLink
Log in

On K-BKZ and other viscoelastic models as continuum generalizations of the classical spring-dashpot models

  • Original Contributions
  • Published:
Rheologica Acta Aims and scope Submit manuscript

Abstract

An alternate constitutive formulation for visco-elastic materials, with particular emphasis on macromolecular viscoelastic fluids, is presented by generalizing Maxwell's idealized separation of elastic and relaxation mechanisms. The notion ofrelative rate of change of elastic stress is identified, abstracted, and formulated with the help of the established theory of finitely elastic isotropic materials. This given a local rate-type constitutive relation for an elastic mechanism in a simple material.

For the simplest class of viscoelastic polymer melts, the notion of rate of change of elastic stress and its damped accumulation is identified and formulated. Under conditions of moderate strain rates, this scheme implies the reliable K-BKZ model for a class of polymer melts. An obvious extension generalizes the remaining classical spring-dashpot models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

I :

Set of second-order tensors.A ∈I is identified with a 3 × 3 matrix in a Cartesian co-ordinate system

I sym :

Set of symmetric second order tensors

Q :

Orthogonal tensor, i.e.Q T=Q −1.

\(^\aleph \mathop \mathcal{H}\limits_{\tau = t_0 }^t \left[ {F_{t_0 } \left( \tau \right)} \right]\) :

Symbol for the value of the functionalϰ H:X →I sym, whereX is the set of piecewise continuous and differentiable strain historiesF to : [t 0,t] →I Other functionals, unless otherwise specified, should be interpreted in a similar manner.

References

  1. Astarita G, Sarti GC (1975) Thermomechanics of Compressible Materials with Entropic Elasticity. In: Hutton JF, Pearson JRA, Walters K (eds) Theoretical Rheology, pp 123–137

  2. Berstein B, Kearsley EA, Zapas LJ (1963) Trans Soc Rheology 7:391–410

    Google Scholar 

  3. Bird RB, Armstrong RC, Hassager O (1977) Dynamics of Polymeric Liquids, vol 1. Wiley, New York

    Google Scholar 

  4. Bird RB, Curtiss CF (1984) Physics Today, Jan, pp 36–43

    Google Scholar 

  5. Chadwick P (1974) Phil Trans Roy Soc A276:371–403

    Google Scholar 

  6. Chadwick P, Creasy CFM (1984) J Mech Phys Solids 32:337–357

    Google Scholar 

  7. Chang H, Lodge AS (1972) Rheol Acta 11:127–129

    Google Scholar 

  8. Coleman BD, Gurtin ME, Herrera RI (1965) Arch Rational Mech Anal 19:1–19

    Google Scholar 

  9. Coleman BD, Noll W (1959) Arch Rational Mech Anal 3:289–303

    Google Scholar 

  10. Coleman BD, Noll W (1961) Rev Modern Phys 33:239–249

    Google Scholar 

  11. Crochet MJ, Davies AR, Walters K (1984) Numerical Simulation of Non-Newtonian Flows. Elsevier Science Publishers, Amsterdam

    Google Scholar 

  12. Curtiss CF, Bird RD (1981) J Chem Phys 74:2026–2033

    Google Scholar 

  13. Doi M, Edwards SF (1978) J Chem Soc, Faraday II:1818–1832

    Google Scholar 

  14. Fung YC (1969) A First Course in Continuum Mechanics. Prentice-Hall, Inc.

  15. Giesekus H (1982) Rheol Acta 21:366–375

    Google Scholar 

  16. Gurtin ME (1981) An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, vol 158. Academic Press, New York

    Google Scholar 

  17. Han CD (1976) Rheology in Polymer Processing. Academic Press, New York

    Google Scholar 

  18. Kaye A (1962) College of Aeronautics, Note 134. Cranfield, Bletchley, England

    Google Scholar 

  19. Larson RG, Monroe K (1984) Rheol Acta 23:10–13

    Google Scholar 

  20. Laun HM (1978) Rheol Acta 17:1–15

    Google Scholar 

  21. McKenna GB, Zapas LJ (1979) J Rheol 23:151–166

    Google Scholar 

  22. Narain A (1985) Impulse Dynamics of Viscoelastic Fluids and Some Solution by Integral Transforms (Accepted for publication in the Proceedings of IMSE 85, University of Texas, Arlington, Hemisphere Publishers)

  23. Narain A, Joseph DD (1982) Rheol Acta 21:228–250. See also Corrections, Rheol Acta 22:519 (1983)

    Google Scholar 

  24. Narain A, Joseph DD (1983) Rheol Acta 22:528–538

    Google Scholar 

  25. Noll W (1958) Arch Rational Mech Anal 2:197–226

    Google Scholar 

  26. Owen DR, Williams WO (1969) Arch Rational Mech Anal 33:288–306

    Google Scholar 

  27. Petrie CJS (1979) Elongational Flows: Aspects of the Behaviour of Model Elasticoviscous Fluids. Pitman, London

    Google Scholar 

  28. Renardy M (1982) Rheol Acta 21:251–254

    Google Scholar 

  29. Rivlin RS, Ericksen JL (1955) J Rational Mech Anal 4:323–425

    Google Scholar 

  30. Rivlin RS, Sawyers KN (1971) Ann Rev Fluid Mech 3:117–146

    Google Scholar 

  31. Saut JC, Joseph DD (1983) Arch Rational Mech Anal 81:53–95

    Google Scholar 

  32. Serrin J (1959) Mathematical Principles of Classical Fluids Mechanics. In: Flügge S, Truesdell C (eds) Handbuch der Physik, Bd VIII/1. Springer, Berlin

    Google Scholar 

  33. Tanner RI (1983) J Appl Mech 50:1181–1187

    Google Scholar 

  34. Truesdell C (1966) The Elements of Continuum Mechanics. Springer, Berlin

    Google Scholar 

  35. Truesdell C, Noll W (1965) The Non-Linear Field Theories of Mechanics. In: Flügge S (ed) Handbuch der Physik, Bd III/3. Springer, Berlin

    Google Scholar 

  36. Wagner MH (1976) Rheol Acta 15:136–142

    Google Scholar 

  37. Wagner MH (1977) Rheol Acta 16:43–50

    Google Scholar 

  38. Wagner MH, Stephenson SE (1979) J Rheology 23:489

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, 49931, Houghton, Mich., USA

    A. Narain

Authors
  1. A. Narain
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Narain, A. On K-BKZ and other viscoelastic models as continuum generalizations of the classical spring-dashpot models. Rheol Acta 25, 1–14 (1986). https://doi.org/10.1007/BF01369974

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01369974

Key words

Search

Navigation