The vibration of micro-circular ring of ceramic with viscothermoelastic properties under the classical Caputo and Caputo-Fabrizio of fractional-order derivative

This work introduces a novel mathematical framework for examining the thermal conduction characteristics of a viscothermoelastic, isotropic micro-circular ring. The foundation of the model is Kirchhoff's theory of love plates. The governing equations have been developed by using Lord and Shulman's generalized thermoelastic model. For a viscothermoelasticity material, Young's modulus incorporates an additional fractional derivative consideration such as the classical Caputo and Caputo-Fabrizio types, alongside the normal derivative. The outer bounding plane is thermally loaded by ramp-type heating. Laplace transform has been applied and its inverse has been obtained numerically. Graphical comparisons between the definitions of the ordinary derivative and the fractional derivatives were incorporated into the study. The objective was to study the impacts of the fractional derivative order on the vibration distribution of a ceramic micro-circular ring and obtain novel results. It is ascertained that the fractional derivative order and resonator thickness have no discernible effect on the distribution of thermal waves; nevertheless, the ramp heat parameter is identified as having a significant impact. The order of the fractional derivatives and the resonator's thickness, have a significant impact on the mechanical wave. It has been demonstrated that the ramp heat parameter effectively regulates the energy damping in ceramic resonators.