Mathematical modeling of functionally graded nanobeams via fractional heat Conduction model with non-singular kernels

Functionally gradient materials (FGM) in nanobeams are interesting issues in the theory of elasticity and thermoelasticity regarding thermal and mechanical stress. These advanced heat-resistant materials are used as structural components in contemporary technology. The thermoelastic interactions in functionally graded nanobeams (FGN) have been studied in this article. The basic equations that control the introduced model have been established based on the Euler–Bernoulli beam concept, Eringen’s theory, and the two phase-lag fractional heat conduction model. The heat equation has been modeled and fractionalized into a new formula that includes nonsingular and nonlocal differential operators. The physical properties of the nanobeam vary in graded according to its thickness. The FGN nanobeam is subject to a time-dependent and periodically varying heat flow. The differential equations are analyzed analytically in the Laplace transform field. The responses in the nanobeam are graphically depicted for various fractional-order values, the influence of the nonlocal parameter and the periodic frequency of the heat flux. The results show that the gap between classical and nonlocal theories widens with increasing nonlocal parameters and decreasing nanobeam length.