Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory

Abstract

This paper is devoted to examining the nonlinear vibrational behaviors of functionally graded (FG) sandwich nanobeams in the presence of initial geometric imperfection. Based on the nonlocal strain gradient theory, the governing equation of the FG sandwich nanobeam with consideration of the Von-Karman nonlinearity and initial geometric imperfection is derived. The nonlinear oscillator frequency is obtained with the aid of He’s variational principle. Three types of nanobeams, i.e., FG nanobeam (Type A), sandwich nanobeam with homogeneous core and FG skins (Type B), and sandwich nanobeam with FG core and homogeneous skins (Type C) are taken into account. A cosine function similar to the mode shape form is employed to describe the geometric imperfection mode. Firstly, the present theoretical model is verified by comparing with previous perfect FG sandwich beams. Then, several key parameters such as the power-law exponent, the amplitudes of the nonlinear oscillator and the geometric imperfection, as well as the nonlocal and material characteristic parameters are investigated in detail. Finally, apart from the structural types, the influence of thickness distribution scheme is also thoroughly elucidated. The results obtained in this paper are helpful for exploring the FG sandwich design to enhance the mechanical performance of nano-devices.