Small scale effect on linear vibration of buckled size-dependent FG nanobeams

The thermal stress due to the temperature rise in micro/nano-beams with immovable ends produces compressive axial force which can lead to buckling the beams if its value increases over the critical value. Hence, the investigation of dynamical behaviour of thermal buckled micro/nano-beams is an important topic. The present study is an attempt to present linear free vibration of buckled FG nano-beams. It is assumed that the material properties of FGMs are graded in the thickness direction. The partial differential equation of motion is derived based on Euler–Bernoulli beam theory, von-Karman geometric nonlinearity and Eringen’s nonlocal elasticity theory. The exact solution of the post-buckling configurations of FG nano-beams and polynomial-based differential quadrature method are employed to study the linear behaviour of vibrated nano-beams around their post-buckling configurations. The results show the important role of compressive axial force exerted on FG nano-beams in nonlocal behaviour of vibrating FG nano-beams