Relaxation and retardation functions of the Maxwell model with fractional derivatives

Abstract

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville 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-Leffler 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.