Energy balance for fractional anti-Zener and Zener models in terms of relaxation modulus and creep compliance

TL;DR

This paper derives conditions ensuring the physical plausibility of fractional anti-Zener and Zener models by analyzing their relaxation modulus and creep compliance, including their monotonicity, oscillatory behavior, and asymptotic properties.

Contribution

It provides new restrictions on model parameters to guarantee thermodynamic consistency and analyzes the asymptotic and oscillatory behavior of these fractional models.

Findings

Relaxation modulus and creep compliance are shown to be completely monotone and Bernstein functions under certain parameter restrictions.

Models can exhibit oscillatory behavior with decreasing amplitude while remaining physically valid.

Asymptotic analysis near initial and large times reveals key dynamic properties of the models.

Abstract

Relaxation modulus and creep compliance corresponding to fractional anti-Zener and Zener models are calculated and restrictions on model parameters narrowing thermodynamical constraints are posed in order to ensure relaxation modulus and creep compliance to be completely monotone and Bernstein function respectively, that a priori guarantee the positivity of stored energy and dissipated power per unit volume, derived in time domain by considering the power per unit volume. Both relaxation modulus and creep compliance for model parameters obeying thermodynamical constraints, proved that can also be oscillatory functions with decreasing amplitude. Model used in numerical examples of relaxation modulus and creep compliance is also analyzed for the asymptotic behavior near the initial time instant and for large time.