On Green–Naghdi (II) thermoelasticity theory with time-fraction and memory-dependent derivatives

In this study, a novel thermomechanical model for elastic solids is developed within the framework of the fractional Green–Naghdi type II (GN-II) theory, enhanced by memory-dependent derivatives (MDD) and excluding energy dissipation. The model integrates generalized non-Fourier heat conduction to capture both time-delay and fractional-order effects in thermoelastic responses. Fundamental theoretical results are established, including a uniqueness theorem, a variational principle, and a reciprocity relation. The proposed framework is applied to a half-space subjected to a time-dependent thermal shock. A computational approach is used to perform inverse Laplace transforms, enabling the analysis of temperature, displacement, stress, and heat flux under different theoretical assumptions. Results show that the inclusion of a nonlinear memory kernel significantly affects the spatial and temporal distribution of field quantities. The findings confirm the model's capacity to accurately describe thermoelastic wave-diffusion phenomena. The proposed approach thus offers a more comprehensive and physically consistent alternative to classical models and lays the foundation for future studies on coupled thermal–mechanical behaviors in advanced materials.