Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented.

Automated summary

This article considers a class of generalized fractional Zener-type viscoelastic models with general fractional derivatives, and derives two integral representations for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform and using the Bernstein functions technique. The second integral representation is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail, and numerical examples are presented to illustrate the analytical results.