A Comprehensive Study on Heterogeneous Media with Spherical Cavity: The Higher-Order Fractional Triple-Phase-Lag Thermoelasticity with Local Kernels

Fractional calculus has recently been successfully integrated into thermoelastic theories. While many existing models rely on fractional operators with anomalous kernels, this study introduces a new approach that derives triple-phase-lag (TPL) thermoelastic thermal conductivity using higher-order fractional derivatives with local, nonanomalous kernels. By applying the Taylor series expansion of higher-order fractional derivatives, as proposed by Caputo and Fabrizio, this model offers enhanced approximations for heat flux, temperature gradients, and thermal displacement. Unlike previous models, it incorporates nonlocal effects, phase lags, and higher-order time derivatives, providing a more comprehensive understanding of the interactions between heat and mechanical forces on materials. The model’s effectiveness is demonstrated by its application to an infinite, heterogeneous thermoelastic medium with a spherical cavity, where it was analyzed under various parameters, including higher-order fractional time derivatives and the heterogeneity index. Results show that an increase in the heterogeneity index intensifies the temperature gradient, leading to greater temperature variations throughout the medium. This effect is attributed to the inhomogeneous properties of the material, which influence thermal conduction and heat transfer processes.