Complete monotonicity of the relaxation moduli of distributed-order fractional Zener model

The fractional Zener constitutive law is extensively used as a model of solid-like viscoelastic behaviour. In this work the generalized distributed-order fractional Zener model is studied in the cases of discrete distribution and continuous distribution of power-law type. It is proven that the corresponding relaxation moduli are completely monotone functions under appropriate thermodynamic restrictions on the parameters. The asymptotic behaviour of the relaxation moduli is studied and integral representations are derived and used for numerical experiments.