Fractionalization of anti-Zener and Zener models via rheological analogy

Abstract

Fractional-order anti-Zener and Zener models are formulated using rheological schemes corresponding to the classical anti-Zener and Zener models and by considering fractional-order springs and dash-pots instead of the classical ones, with the fractionalization achieved by generalizing Hooke’s and Newton’s laws by the use of fractional integral and derivative, respectively. Following the aforementioned approach, four constitutive models, containing both fractional integrals and derivatives, are formulated and transformed into eight constitutive models having the asymmetric form in the sense that there is a different number of operators acting on stress and strain, as well as, inspired by the symmetric form of the classical anti-Zener and Zener models, into seven constitutive models having the symmetric form. Analysis of models’ thermodynamical consistency yielded that for a number of models there is no such combination of model parameters that implies energy dissipativity, while the energy dissipativity of the rest of the models is ensured by posing the restrictions on model parameters. A new type of constitutive equations containing fractional derivatives of order below one is obtained along, while some of the models containing fractional operators of order below two proved to have an operator of order obtained as the arithmetic mean of the orders of other two operators acting on the same physical quantity.