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Power series approximations of dynamic moduli and relaxation spectrum
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10.1122/1.4789787
/content/sor/journal/jor2/57/2/10.1122/1.4789787
http://aip.metastore.ingenta.com/content/sor/journal/jor2/57/2/10.1122/1.4789787

Figures

Image of FIG. 1.
FIG. 1.

Effect of the order of polynomial for the regression of dynamic moduli. (a) Modulus data (symbols) and moduli obtained from regression (line). Moduli data are generated by the spectrum of Eq. (43) without statistical errors. (b) The calculated spectra for various orders of polynomials. The open symbols in (b) are the spectra calculated from the regression of loss modulus and the closed symbols are the spectra from the regression of storage modulus. The line behind the symbols is Eq. (43) .

Image of FIG. 2.
FIG. 2.

Test for the regression using data partitioning. (a) Moduli generated from the given spectrum Eq. (45) (line) and regression results. The different symbols denote the regressions for different partitions of data. All regressions were done by without the use of an orthogonal polynomial. (b) Comparison of the given spectrum (line) and spectra calculated from storage modulus data (symbols), which are obtained from partitioned data. (c) Comparison of the given spectrum (line) and spectra calculated from loss modulus.

Image of FIG. 3.
FIG. 3.

Spectrum calculated using an orthogonal polynomial and data partitioning. Modulus data used are the same as those in Fig. 2 . Regression for loss modulus is mad by use of the Chebyshev polynomial of the first kind. (a) The 12th order polynomial is used for the regression of loss modulus. (b) The 13th order polynomial is used for the regression of loss modulus.

Image of FIG. 4.
FIG. 4.

Effect of regression of loss modulus by use of Eq. (1) for the entire data. The Chebyshev polynomial is used. (a) Loss modulus data of Fig. 2 and the regression result. The order of polynomial is 17. (b) Spectra calculated by (symbols) and the given spectrum (line).

Image of FIG. 5.
FIG. 5.

Effect of errors. The errors are generated from the Gaussian distribution with the mean of zero and the standard deviation of unity and added to the modulus data of Fig. 1 . (a) The error level is 5% and the regression order is 8. (b) The error level is 5% and the regression order is 10. (c) The error level is 3% and the regression order is 8. The dotted lines indicate the range of modulus data. The error bar is the standard deviation calculated from 11 spectra, which are calculated from independently noised data. When the error level is fixed, a lower order of regression gives better stability of spectrum, and when the order of regression is fixed, a lower level of error gives a more stable spectrum.

Image of FIG. 6.
FIG. 6.

Precision of regression ( ) and spectrum ( ) as functions of the regression order. These measures of precision were calculated from the data in Fig. 1 with or without error. (a) Effect of error on the precision of regression. (b) Effect of error on the precision of spectrum. The insertion is the semilogarithm plot of the main plot. Although decreases monotonically as N increases, shows local minimum.

Image of FIG. 7.
FIG. 7.

Application to experimental data. The modulus data were measured by Schausberger et al. (1985) for nearly monodisperse polystyrene (PS6). (a) Regression of the whole data of loss modulus with the order of 10. (b) Regression of the whole data of loss modulus with the order of 12. (c) Comparison of spectra calculated by and . These regression orders show local minimum of for .

Image of FIG. 8.
FIG. 8.

Application of data partitioning for the data in Fig. 7 . The partitioning was done by for the partition I, for the partition II, and for the partition III. The regression orders are 5 for the partitions I and II and 11 for the partition III. (a) Spectra from data partitioning (symbols) and regression spectrum (line). The regression spectrum was obtained using Eq. (47) for the data shown as symbols in (a). (b) Modulus data and calculated moduli from the regression spectrum [line in (a)].

Tables

Generic image for table
TABLE I.

Constants for the power series expansion.

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/content/sor/journal/jor2/57/2/10.1122/1.4789787
2013-02-04
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Power series approximations of dynamic moduli and relaxation spectrum
http://aip.metastore.ingenta.com/content/sor/journal/jor2/57/2/10.1122/1.4789787
10.1122/1.4789787
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