Thermomechanical Behavior of Functionally Graded Nanoscale Beams Under Fractional Heat Transfer Model with a Two-Parameter Mittag-Leffler Function

In this research, a new fractional framework for the non-Fourier thermal conductivity theory has been developed and analyzed in detail. Furthermore, a dual-phase-lag (DPL) model is presented to consider the discrete time intervals necessary for various microstructural processes. To overcome the singularity problem associated with traditional fractional derivatives, the Mittag-Leffler function is selected as a complementary kernel alongside the power-law function that defines the time-fractional derivative. The proposed model is based on a new fractional derivative that incorporates the Atangana and Baleanu operators with a two-parameter Mittag-Leffler kernel. The fractional DPL heat transfer model is employed to investigate transient heat transfer in functionally graded thermoelastic nanobeams. In addition, through the amalgamation of the Hamiltonian principle, non-local elasticity, and the Euler–Bernoulli beam theory, a set of mathematical equations has been formulated to delineate the characteristics of FG nanoscale beams. The Laplace transform method has been employed to ascertain the responses of thermo-mechanical fields in the transformed domain. To validate the proposed approach, a numerical illustration and a graphical depiction of pertinent numerical data have been included. The thermomechanical characteristics of nanobeams have been thoroughly investigated, encompassing their responsiveness to the power law index, nanoscale parameters, and fractional operators. Lastly, comparisons between different alternate kernels have revealed that both the fractional DPL models and the presented solutions effectively capture the heat transfer across the medium.