Beta-Fractional Heat Flow and Rotational Effects in the Thermoelasticity Response of Microelongated Solids

This work conducts a comprehensive examination of the impact of rotational motion on the transmission of thermomechanical waves in microelongated thermoelastic materials. The analysis is conducted within the framework of fractional-order heat conduction by incorporating the β-fractional derivative into three established thermoelastic models: the Lord–Shulman (L–S) theory, the dual-phase-lag (DPL) model, and the refined dual-phase-lag (RDPL) model. The essential mathematical frameworks governing heat transfer, elastic deformation, and microelongation are methodically derived to precisely characterize the combined thermal and mechanical interactions. These equations explicitly incorporate limited thermal wave velocities, rotating influences, and the inherent microstructural elongation properties of the medium. To simplify the resulting mathematical formulation and facilitate analytical tractability, an appropriate nondimensionalization process is employed in conjunction with a suitable transform method. This methodology lowers the original set of highly coupled partial differential equations (PDEs) to a more manageable form, from which precise analytical formulas for the displacement components, microelongation field, temperature distribution, and stress tensors are derived. The derived solutions provide a comprehensive description of the spreading behavior of thermoelastic waves in microelongated materials subjected to rotational motion. To validate and illustrate the theoretical findings, numerical computations are carried out, allowing a detailed comparison of the responses predicted by the L–S, DPL, and RDPL models. These comparisons are performed for two distinct scenarios: one incorporating fractional-order effects and the other based on the classical nonfractional formulation. The findings of this work highlight the interaction of rotational motion, β-fractional heat conduction, and phase-lag effects in determining the dynamic thermoelastic response of microelongated materials. The proposed framework offers a more broad and physically accurate description of wave propagation events in such media, which could be useful for advanced material modeling and engineering applications.