Abstract
Integral models in non-linear viscoelasticity
Summary
The multiparametric models of non-linear viscoelasticity, as typified by theBird-Carreau model, are ill-posed which leads to the practical impossibility of accurately determining the parameter values. This, in turn, makes it hazardous to infer the molecular behaviour of the material although the inaccurate parameters may correctly predict the bulk material behaviour. Instead, it is proposed that models which are defined directly in terms of the measured data be employed. One such model has been briefly described and this model is also well-posed, i.e. the error does not propagate uncontrollably in the calculation when one attempts to improve the accuracy or resolution of the model.
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Abbreviations
- d :
-
rate of deformation tensor
- F(s) :
-
spectral function inFredholm integral
- K(x, s) :
-
kernel function inFredholm integral
- n :
-
number of parameters or equations
- n 1 :
-
empirical constant
- P :
-
deviatoric stress tensor
- P 12 :
-
shear stress
- P 11 −P 22 :
-
first normal stress difference
- R 1 ,R 2 :
-
cumulative relative errors defined by eqs. [6] and [7]
- t :
-
time
- α 1 :
-
empirical constant
- \(\dot \gamma \) :
-
shear rate
- η(\(\eta (\dot \gamma ),\eta _p \) :
-
viscosity functions, parameters
- θ :
-
relaxation time ofWhite-Metzner model
- λ :
-
characteristic fluid relaxation time
- \(\overline{\overline \lambda } \) :
-
eigenvalues of the matrix, eq. [5]
- v :
-
defined by eq. [8]
- µ :
-
viscosity of non-linearMaxwell element
- φ(x) :
-
function of material property
References
Schwarzl, F. R. andL. C. E. Struick Advances in Molecular Relaxation Processes1, 201 (1967–68).
Schwarzl, F. J., Private communication to B. Hlaváček.
Sinevic, V. andB. Hlaváček Rheol. Acta8, 247 (1969).
Stanislav, J. andB. Hlaváček, Trans. Soc. Rheol.17, 331 (1973).
Hlaváček, B. andF. A. Seyer J. Appl. Poly. Sci.16, 423 (1972).
Markovitz, H., in: Rheology Vol. IV (New York, 1967).
Bernstein, B., E. Kearsley andL. Zapas Trans. Soc. Rheol.7, 391 (1963).
Bird, R. B. andP. J. Carreau Chem. Eng. Sci.23, 427 (1968).
Tanner, R. I. A. I. Ch. E. Journal15, 177 (1969).
Spriggs, T. W., J. D. Huppler andR. B. Bird Trans. Soc. Rheol.10, 191 (1966).
Huppler, J. D., L. A. Holmes andE. Ashare Trans. Soc. Rheol.11, 159 (1967).
Hlaváček, B., J. F. Stanislav andF. A. Seyer, Trans. Soc. Rheol.18, 27 (1974).
Babuška, I., M. Prager, andE. Vitásek, Numerical Processes in Differential Equations (London, 1966).
Hlaváček, B. andP. J. Carreau, to be published.
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Hlaváček, B., Stanislav, J. & Patterson, I. Integral parameter models in non-linear viscoelasticity. Rheol Acta 14, 812–815 (1975). https://doi.org/10.1007/BF01521410
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DOI: https://doi.org/10.1007/BF01521410