Relaxation and retardation functions of the Maxwell model with fractional derivatives

152 of the function itself. This is correct for functions which vanish for negative time, as do the stress and strain to be considered in this paper. Comparison with experiments show that constitutive equations with fractional derivatives can be used in describing the rheological behavior of polymer melts and solutions. In contrast, the determination of the relaxation or retardation functions is more complicated. Their analytical determination via Laplace transform is hindered by the problems arising from the inverse Laplace transform [6]. The attempt to find the relaxation function in the time domain by solving the appropriate differential equations with fractional derivatives has not yet been made. Examples of solving these types of differential equations are contained in [10], where some solutions of diffusion equation are presented. Within this paper the analytic solution of the problem of relaxation and retardation function determination are given. These functions are calculated for a four-parameter Maxwell model with fractional derivatives of different orders for stress and strain. The obtained solutions are compared with other relaxation functions to demonstrate its behavior.
L Introduction Rheological constitutive equations with fractional derivatives have long played an important role in the description of the properties of polymer solutions and melts [ 1 - 6]. These equations are derived f r o m well known models (e.g., the Maxwell model) by substituting the ordinary derivatives of first, second and higher order by fractional derivatives of any noninteger order a. With this scheme, the order of the derivative relates to a material parameter which can be associated with degree of conversion as, for example, for a sol-gel transition [7, 8]. In other cases, it has been shown that constitutive equations employing fractional derivatives are linked to molecular theories [31 or are associated with system theory in general or with Cole-Cole behavior in particular [4, 9]. The arrangement of these models within the frame of linear theory of viscoelasticity is given in part by Tschoegl [9].
In [4] an operator f o r m of the constitutive equation containing fractional derivatives was derived and led to the complex modulus. In [6] a general three-dimensional constitutive equation applying fractional derivatives to linear viscoelasticity has been developed. The use of such an equation is limited because of its mathematical intractability in the time domain. For a five-parameter model with a fractional derivative of the stress and of the strain, the sinusoidal response and thermodynamic admissibility was investigated. Attempts to calculate the relaxation and retardation functions failed insofar as a less instructive numerical solution was given. Usually, the usefulness of these models with fractional derivatives will be tested with the help of dynamic experimental investigations. The complex modulus G* can be calculated easily because the Fourier transform of a fractional derivative of order a is the product of (ie)) a and the Fourier transform
Rheologica Acta
Rheol Acta 30:151-158 (1991)
Relaxation and retardation functions of the Maxwell model with fractional derivatives
Chr. Friedrich Institut ftir Makromolekulare Chemie, Albert-Ludwigs-Universit~tt Freiburg, FRG
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Rheologica Acta, Vol. 30, No. 2 (1991)
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Abstract: A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the RiemannLiouville definition, This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leftier function. The retardation function is determined via Laplace transformation and is solely a power law type. The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles. This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value. In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function. Key words: Fractional _derivative; _Maxwell _model; r_elaxation and _retardation functions; _Mittag-Leffler _functions; time c_onstants