Fractional Cattaneo Heat Equation in a Multilayer Elliptic Ring Membrane and its Thermal Stresses

A fractional Cattaneo model from the generalized Cattaneo model with two fractional derivatives of different orders is considered for studying the thermoelastic response for a multilayer elliptic ring membrane with source function. The solution is obtained by applying an integral transform technique analogous to Vodicka's approach considering series expansion functions in terms of an eigenfunction to the generalized fractional Cattaneo-type heat conduction equation within an elliptic coordinates system. The analytical expressions of displacement and stress components employing Airy's stress function approach are investigated. The results are obtained as a series solution in terms of Mathieu functions and hold convergence test. The effects of fractional parameters on the temperature fields and their thermal stresses are also discussed. The findings are depicted graphically for different kinds of surface temperature gradients, and it is distinguished that the higher the fractional-order parameter, the higher the thermal response. Lastly, the generalized theory of thermoelasticity predicts an instantaneous response, but the fractional theory, which is currently under consideration, predicts a delayed response to physical stimuli, which is something that can be seen occurring in nature. This delayed response can be explained by the fact that fractional theories are currently being considered. This gives credibility to the motivation behind this topic of study in the research.