Investigation of Material Nonlinearities in Surface Effects and Bulk on the Vibration Characteristics of Nanobeams

The surface effects play an important role in the vibrational properties of nanostructures. Investigating the nonlinear vibrations of nanobeams, the nonlinear material effects on surface effects have not been considered. This study aims to do research on the effect of material nonlinear coming from the stress-strain nonlinear equation on surface effects in order to increase the calculation accuracy. The effect of materially nonlinear behaviors of the bulk and surface effects in the presence of nonlinear Von Kármán strains are considered simultaneously. The governing equation based on the Hamilton principle and then the governing nonlinear differential equation based on applying the Galerkin’s method have been extracted. The obtained nonlinear differential equation possesses cubic and quantic nonlinearities, which are due to the geometric and the materially nonlinear behaviors, respectively. The quantic nonlinearity is only due to the materially nonlinear behavior of bulk and surface effects. The temporal response and nonlinear frequency of nanobeams are obtained by solving the nonlinear differential equation based on the modified Lindstedt–Poincaré method. The results represent the simultaneous effects of the materially nonlinear behaviors of the bulk and surface layers on the temporal response and nonlinear frequency of the nanobeam. To examine the validity of the results, the obtained natural frequencies are compared with other studies in the absence of the materially nonlinear term.