Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with surface energy

This paper deals with transverse vibration of axially functionally graded tapered nano-scaled beams acted upon by a longitudinal temperature gradient. Using surface elasticity theory of Gurtin–Murdoch, the equations of motion of the nanostructure are displayed based on the hypotheses of the Rayleigh, Timoshenko, and higher-order beam theory. Due to the variation of the material and the cross-section along the nanobeam, seeking an analytical solution to the resulting governing equations is a very cumbersome job. To conquer this difficulty, reproducing kernel particle method is proposed, and the natural frequencies of the thermally affected nanostructure are numerically calculated. Subsequently, the roles of the slenderness ratio, temperature gradient, diameter of the nanobeam, and variation of both the cross section and the material property along the length of the nanobeam on its free dynamic response are investigated. In each parametric study, the effects of both surface energy and shear deformation on the natural frequencies are addressed and explained.