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Numerical determination of retardation and relaxation spectra optimalization of numerical process

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Summary

We have considered the optimal conditions for the determinations of relaxation spectra. The numerical process should be based on a well-posted method and should be free of artificial stability conditions. Such artificial conditions are those which restrict or prescribe the shape of the spectra. From the practical point of view we found that simple numerical processes which do not contain matrix inversion are convenient.

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Abbreviations

a ij :

elements of matrixt

b ij :

elements of matrixβ

k ij :

elements of matrix ϰ

m ij :

elements of matrixℳ

s :

variable in eq. [1]

v i :

elements of matrixV

x :

variable in eq. [1]

γ :

smoothing factor

λ :

matrix eigenvalue

v :

defined by eq. [6]

ϕ(x) :

experimental measured function, defined by eq. [1]

F(s) :

defined by eq. [1]

F * (s) :

defined by eq. [13]

G′(x) :

real part of dynamic modulus

H 3 (x) :

thirdSchwarzl-Staverman approximation of relaxation spectrum

J(x) :

creep compliance

K(x, s) :

kernel of integral in eq. [1]

\(\bar L(x)\) :

secondSchwarzl-Staverman approximation of retardation spectrum

R 1 :

cumulative relative error of approximation of relaxation spectrum; defined by eq. [3]

R 2 :

cumulative relative error of spectral relaxation function; defined by eq. [4]

References

  1. Alfrey, T. andP. Doty, J. Applied Phys.16, 700 (1945).

    Google Scholar 

  2. Andrews, R. D., Ind. Eng. Chem.44, 707 (1952).

    Google Scholar 

  3. Ferry, J. D. andM. L. Williams, J. Colloid Sci.14, 347 (1952).

    Google Scholar 

  4. Schwarzl, F. andA. J. Staverman, Appl. Sci. Res.A4, 127 (1953).

    Google Scholar 

  5. Okano, M., Busseiron Kenkyu3, 493 (1958).

    Google Scholar 

  6. Fujita, H., J. Applied Phys.29, 943 (1958).

    Google Scholar 

  7. Ninomiya, K. andJ. D. Ferry, J. Colloid Sci.14, 36 (1959).

    Google Scholar 

  8. Tschoegel, N. W., Rheol. Acta10, 582–600 (1971).

    Google Scholar 

  9. Tobolsky, A. V. andK. Murakami, J. Polym. Sci.40, 443 (1959).

    Google Scholar 

  10. Gradowczyk, M. H. andF. Moavenzadeh, Trans. Soc. Rheol.13, 173 (1969).

    Google Scholar 

  11. Clauser, J. F. andW. G. Knauss, Trans. Soc. Rheol.10, 191 (1966).

    Google Scholar 

  12. Twomey, S., J. Assn. Computing Machinery10 (January 1963).

  13. Phillips, D. L., J. Assn. Computing Machinery9 (January 1962).

  14. Tanner, R. I. andG. Williams, Trans. Soc. Rheol.14, 19 (1970).

    Google Scholar 

  15. Burger, H. C. andP. H. Van Cittert, Physik79, 722 (1932).

    Google Scholar 

  16. Hopkins, J. L., J. Polymer Sci.50, 59 (1961).

    Google Scholar 

  17. Roesler, F. C. andW. A. Twyman, Proc. Phys. Soc.B68, 97 (1955).

    Google Scholar 

  18. Hlaváček, B. andV. Kotrba, Rheol. Acta6, 288 (1967).

    Google Scholar 

  19. Hlaváček, B., Rheol. Acta,7, 225 (1968).

    Google Scholar 

  20. Hlaváček, B. andV. Sinevič, Rheol. Acta9, 312 (1970).

    Google Scholar 

  21. Hlaváček, B. andF. A. Seyer, J. Appl. Polym. Sci.16, 423 (1972).

    Google Scholar 

  22. Todd, J. Quart. J. Mech. Appl. Math.2, 469 (1949).

    Google Scholar 

  23. Von Neumann, J. andH. H. Goldstine, Bull. Amer. Math. Soc.53, 1021 (1947).

    Google Scholar 

  24. Stanislav, J. andB. Hlaváček (to be published).

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Authors and Affiliations

  1. Department of Chemical Engineering, The University of Calgary, Calgary, Alberta, Canada

    J. Stanislav & F. A. Seyer

  2. Department of Chemical and Petroleum Engineering, The University of Alberta, Edmonton, Alberta, Canada

    J. Stanislav & F. A. Seyer

  3. Department of Chemical Engineering, Ecole Polytechnique, Montréal 248, Canada

    J. Stanislav, F. A. Seyer & B. Hlaváček

Authors
  1. J. Stanislav
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  2. F. A. Seyer
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  3. B. Hlaváček
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With 2 figures and 3 tables

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Stanislav, J., Seyer, F.A. & Hlaváček, B. Numerical determination of retardation and relaxation spectra optimalization of numerical process. Rheol Acta 13, 602–607 (1974). https://doi.org/10.1007/BF01521762

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  • DOI: https://doi.org/10.1007/BF01521762

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