TWO NEW BASIC RHEOLOGICAL DISCRETE DYNAMIC SYSTEMS OF THE FRACTIONAL TYPE OF PIEZOELECTRIC PROPERTY: THE KELVIN-VOIGT-FARADAY OSCILLATOR, AND THE MAXWELL-FARADAY CRAWLERoscillator, and the Maxwell-Faraday crawler

Abstract

We introduce two new fundamental rheological elements: a fractional‑type Newtonian viscous element with a fractional‑order differential constitutive relation and inherent energy dissipation, and a Faraday ideally elastic piezoelectric element characterized by electrical polarization induced by mechanical deformation.
Using these elements, two new standard light rheological binder models are formulated: the standard light Kelvin-Voigt-Faraday fractional‑type model and the standard light Maxwell-Faraday fractional-type model. Fractional‑order differential constitutive relations are derived for both models. The former exhibits the property of delayed elasticity, while the latter demonstrates fractional stress relaxation.
Based on these models, two discrete rheological dynamical systems with fractional‑order behavior and piezoelectric properties are defined and their dynamics are investigated. According to their dynamic characteristics, the systems are termed the fractional Kelvin-Voigt-Faraday rheological oscillator and the fractional Maxwell-Faraday rheological crawler.
For the Kelvin-Voigt-Faraday oscillator, Laplace transforms and approximate analytical expressions for free and forced fractional‑order oscillatory modes with piezoelectric coupling are obtained. For the Maxwell-Faraday crawler, Laplace transforms the independent generalized coordinates-one external and two internal-are derived. The analytical results are illustrated by appropriate graphical representations. Two theorems are formulated and proved.
In the concluding remarks, several analogies between fractional‑type rheological mechanical systems and rheological electrical dynamical systems with piezoelectric properties are presented.