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

Abstract

Considering the linear constitutive model containing fractional integrals and Riemann-Liouville fractional derivatives, the power per unit volume is expressed in time domain in terms of stored energy and dissipated power per unit volume. 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. 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.