A higher-order MGT thermoelastic framework with Tempered-Caputo derivatives for power-law graded solids with spherical cavities

This work introduces a novel, thermodynamically consistent generalized thermoelastic framework that integrates the fourth-order Moore–Gibson–Thompson (4MGT) heat conduction law with Tempered-Caputo (TC) fractional derivatives. This marks the first application of the 4MGT model to power-law graded solids, representing a significant advancement over classical theories by concurrently capturing higher-order thermal inertia and fading memory effects without resulting in kernel singularities. The model is formulated for an infinite medium containing a spherical cavity, where the material properties transition continuously between copper and zirconia, following a strict power-law gradation. The cavity surface is subjected to full-wave rectified sinusoidal heating, simulating realistic high-frequency thermal cycling typical in various engineering systems. The coupled governing equations are solved analytically in the Laplace domain using a state-space formulation, and numerical inversion is accomplished via the Stehfest algorithm. Parametric studies reveal that key factors such as the grading index, fractional order, tempering parameter, and thermal excitation frequency critically influence the attenuation, dispersion, and localization of thermal and mechanical waves. The results demonstrate that the proposed 4MGT–TC framework effectively eliminates unphysical infinite propagation speeds and more accurately mitigates stress concentrations compared to classical or generalized models. These findings present actionable design guidelines for advanced components, such as thermal barrier coatings and microelectronic substrates, where achieving an optimal balance between thermal shielding and mechanical integrity under cyclic thermal loading is essential.