A modified thermo piezoelectric model with spatiotemporal nonlocality and phase lags for solid spherical piezoelectric bodies under dynamic thermal loading

This study proposes a novel theoretical framework to analyze the dynamic response of a radially polarized piezoelectric solid sphere under time-dependent thermal loading in the presence of an applied magnetic field. The model incorporates spatiotemporal nonlocal elasticity to account for long-range interatomic interactions and intrinsic relaxation mechanisms. Additionally, it incorporates dual-phase-lag heat conduction to account for finite thermal propagation speeds. The governing equations for displacement, temperature, stress, electric potential, and electric displacement are formulated under spherical symmetry and are solved exactly in the Laplace transform domain, followed by numerical inversion to obtain time-domain solutions. The results demonstrate that both spatial and temporal nonlocal parameters significantly attenuate mechanical and electrical responses while smoothing thermal gradients. This process effectively eliminates the unphysical singularities predicted by classical theories. Moreover, a comparative analysis across multiple thermoelastic models confirms that conventional local approaches tend to overestimate stresses and electrical outputs. The proposed model provides a physically consistent description of size- and rate-dependent behavior in piezoelectric nanospheres, directly relevant to applications in nanoscale sensors, actuators, energy harvesters, and biomedical microdevices.