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A delta-function method for the n-th approximation of relaxation or retardation time spectrum from dynamic data (1991)

Friedrich, Chr.

Springer

Rheologica acta 30 (1991), S. 7-13

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ISSN: 1435-1528

Keywords: Linear viscoelasticity ; complex modulus ; complex compliance ; relaxation and retardation spectrum ; approximative determination ; delta-function

Source: Springer Online Journal Archives 1860-2000

Topics: Chemistry and Pharmacology , Physics

Notes: Abstract A function series g(x; n, m) is presented that converges in the limiting case n→∞ and m = constant to the delta-function located at x = ωλ = 1. For every finite n, there exists 2n+1(−n≤m≤n) approximations of the delta-function δ(n)(x−x n,m ). x n,m is the argument where the function reaches its maximum. A formula for the calculation is given. The delta-function approximation is the starting point for the approximative determination of the logarithmic density function of the relaxation or retardation time spectrum. The n-th approximation of density functions based on components of the complex modulus (G*) or the complex compliance (J*) is given. It represents an easy differential operator of order n. This approach generalizes the results obtained by Schwarzl and Staverman, and Tschoegl. The symmetry properties of the approximations are explained by the symmetry properties of the function g(x; n, m). Therefore, the separate equations for each approximation given by Tschoegl can be subsumed in a single equation for G′ and G″, and in another for J′ and J″.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00366789

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Determination of relaxation time spectra by analytical inversion using a linear viscoelastic model with fractional derivatives (1995)

Friedrich, Chr. ; Braun, H. ; Weese, J.

Stamford, Conn. [u.a.] : Wiley-Blackwell

Polymer Engineering and Science 35 (1995), S. 1661-1669

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ISSN: 0032-3888

Keywords: Chemistry ; Chemical Engineering

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Chemistry and Pharmacology , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics

Notes: Recently, Friedrich proposed an empirical model for linear viscoelastic fluids corresponding to a constitutive equation with fractional derivatives [Phil. Mag. Lett., 66, 287 (1992)]. For this model, the relaxation modulus, the dynamic moduli, the relaxation time spectrum, and other material functions have been explicitly calculated as a function of the few parameters that characterize a viscoelastic fluid within this model. By fitting this model to experimental data, the model parameters can be determined and other material functions, in particular the relaxation time spectrum, can be calculated immediately. This paper reports to what extent this method, which may be called analytical inversion, is appropriate for the determination of relaxation time spectra. For that pupose, the spectra of a number of very different polymeric materials are determined with this method. The spectra calculated in this way are compared with the spectra obtained by nonlinear regularization. It turns out that the empirical model describes the linear viscoelastic properties of a variety of different materials with high accuracy. Keeping in mind that the determination of the relaxation time spectrum requires the solution of an ill-posed problem, the agreement between the relaxation time spectra obtained by analytical inversion and by regularization is satisfactory for these materials.

Additional Material: 9 Ill.