Advertisement

Rheologica Acta

, Volume 33, Issue 6, pp 517–529 | Cite as

Calculation of discrete retardation spectra from creep data — I. Method

  • J. Kaschta
  • R R. Schwarzl
Article

Abstract

A new method is proposed for the calculation of discrete retardation spectra from creep and recovery data. The calculation of the spectrum is not restricted to a special region of consistency, e.g., the terminal region. In a retardation time window which has to correspond to the time window of the original data set a spectrum can always be calculated. A linear regression technique is applied to the measured data in the iterative calculation of a spectrum with a logarithmically equidistant spacing of retardation times. In this way the number of retardation times is limited and problems with ill-posedness are avoided. In order to obtain only positive retardation strengths it is necessary to shift the set of prescribed logarithmically equidistant retardation times on the logarithmic time scale. It can be shown that there is a retardation time interval for this shift, in which the retardation times may be varied without obtaining negative retardation strengths. While varying the retardation times in this interval the relative error of description of the data passes through a distinct minimum. In this way a spectrum is obtained which best describes the input data. Generally, one retardation time per decade will be sufficient to describe the data within the limits of experimental error. In the case of noisy data, the method is shown to work just as well and leads to a smoothing of the original data set. The method may be used for the conversion of creep and recovery data to storage and loss compliance. The error connected with this procedure is discussed.

Key words

Creep compliance retardation spectrum relaxation spectrum linear viscoelastic theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baumgartel M, Winter HH (1989) Determination of discrete relaxation and retardation spectra from dynamic mechanical data. Rheol Acta 28:511–519Google Scholar
  2. Baumgartel M, Winter HH (1992) Interrelation between continuous and discrete relaxation time spectra. J NonNewt Fluid Mech 44:15 – 36Google Scholar
  3. Boltzmann L (1876) Sitzb Kgl Akad Wiss Wien 2 Abt 70:725Google Scholar
  4. Cost TL, Becker EB (1970) A multidata method of approximate Laplace transform inversion. Int J Num Methods in Engg 2:207Google Scholar
  5. Elster C, Honerkamp J, Weese J (1991) Using regularisation methods for the determination of relaxation and retardation spectra of polymeric liquids. Rheol Acta 31:161–174Google Scholar
  6. Elster C, Honerkamp J (1989) Modified maximum entropy method and its application to creep data. Macromolecules 24:310 – 314Google Scholar
  7. Emri I, Tschoegl NW (1992) Generating line spectra from experimental responses, Part 1: Relaxation modulus and creep compliance. Rheol Acta 32:311–321Google Scholar
  8. Ferry JD (1980) Viscoelastic properties of polymers, 3rd ed., J. Wiley, New YorkGoogle Scholar
  9. Ferry JD, Williams ML (1952) J Colloid Sci 6:347Google Scholar
  10. Friedrich Ch, Hoffmann B (1983) Nichtkorrekte Aufgaben in der Rheometrie. Rheol Acta 22:425–434Google Scholar
  11. Honerkamp J (1989) Ill-posed problems in rheology. Rheol Acta 363:371Google Scholar
  12. Kaschta J (1991) Zum Nachgiebigkeitsverhalten amorpher Polymere im Glas-Kautschuk-LJbergang. Thesis, ErlangenGoogle Scholar
  13. Kaschta J (1992) On the calculation of discrete retardation and relaxation spectra, in: Moldenaers P, Keunings R (eds) Theoretical and Applied Rheology, Vol 2, p 155, Elsevier, AmsterdamGoogle Scholar
  14. Kaschta J, Schwarzl FR (1994) Calculation of discrete retardation spectra from creep data — II. Analysis of measured creep functions. Submitted to Rheol ActaGoogle Scholar
  15. Malkin YA (1990) Some inverse problems in rheology leading to integral equations. Rheol Acta 29:511–518Google Scholar
  16. Schapery RA (1962) Approximation methods of transform inversion for viscoelastic stress analysis. Proc Fourth US Nat Congr Appl Mech 2:1075Google Scholar
  17. Tanner RI (1968) J Appl Polym Sci 12:1649Google Scholar
  18. Tschoegl NW (1971) A general method for the determination of approximations to the spectral distributions from dynamic response functions. Rheol Acta 10:582–594Google Scholar
  19. Tschoegl NW (1971) A general method for the determination of approximations to the spectral distributions from transient response functions. Rheol Acta 10:595 – 600Google Scholar
  20. Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behaviour. Springer, BerlinGoogle Scholar
  21. Schwarzl FR, Staverman A (1952) Higher approximations of relaxation spectra. Physica 18:791–799Google Scholar
  22. Schwarzl FR (1969) The numerical calculation of storage and loss compliance from creep data for linear viscoelastic materials. Rheol Acta 8:6–17Google Scholar
  23. Schwarzl FR (1970) On the interconversion between viscoelastic functions. Pure and Appl Chem 23:219–234Google Scholar
  24. Schwarzl FR (1975) Numerical calculation of stress relaxation modulus from dynamic data for linear viscoelastic materials. Rheol Acta 14:581–590Google Scholar
  25. Schwarzl FR (1989) Viscoelasticity, in: Mark HIT (ed) Ecyclopedia of polymer science and engineering 2nded, Vol 17, p 637. J. Wiley, New YorkGoogle Scholar
  26. Yasuda G, Ninomiya K (1966) Proc Japan Rubber Association 39:81Google Scholar

Copyright information

© Steinkopff-Verlag 1994

Authors and Affiliations

  • J. Kaschta
    • 1
  • R R. Schwarzl
    • 1
  1. 1.Institute for Materials ScienceUniversity of Erlangen-NürnbergErlangenGermany

Personalised recommendations