Abstract
The study refers the physical background and modelling of linear viscoelastic response functions and their reasonable relationships to the Caputo-Fabrizio fractional operator via the Prony (Dirichlet series) series decomposition. The problem of interconversion with power-law and exponential (single and multi-term functions) has been discussed. Special attentions have been paid on the Prony series decomposition approach, the related interconversion problems and the expression of the viscoelastic constitutive equations in terms of Caputo-Fabrizio fractional operator.
Highlights
This article addresses the physical background of modeling of dissipative phenomena, precisely response functions such as stress-strain relationships in the framework of the linear viscoelasticity and their reasonable relationships to the Caputo-Fabrizio fractional operator [1] via the Prony (Dirichlet series) series decomposition [2].The appearance of the new definition of fractional derivatives with non-singular kernels was provoked by needs to model dissipative transport processes in many new materials appearing in modern technologies [1]
As a matter of fact, we need a clear answer what really this new operator models and how it appears in the modeling of physical problems, despite the fact that some properties known from the classical fractional calculus, such as the index law [6], are not satisfied
The analysis done here and the principle task are oriented to modeling of viscoelastic constitutive relationships in terms of Caputo-Fabrizio operator naturally appearing through Prony series decomposition [2] of stress relaxation functions and not obeying power-law behaviors
Summary
This article addresses the physical background of modeling of dissipative phenomena, precisely response functions such as stress-strain relationships in the framework of the linear viscoelasticity and their reasonable relationships to the Caputo-Fabrizio fractional operator [1] via the Prony (Dirichlet series) series decomposition [2]. At the same time the Caputo-Fabrizio fractional operator was criticized [5,6,7] with points of view based on the classical fractional calculus with singular power-law memory kernel and with examples from the signal processing [6] and to some extent touching formal rheological relationship [5]. As a matter of fact, we need a clear answer what really this new operator models and how it appears in the modeling of physical problems, despite the fact that some properties known from the classical fractional calculus, such as the index law [6] , are not satisfied
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