Surface elastic shell model for nonlinear primary resonant dynamics of FG porous nanoshells incorporating modal interactions

Abstract

The surface to volume ratio becomes significant at nanoscale, so the surface stress plays an essential role in the mechanical responses of nanostructures. The focus of the present study is on an accurate analytical nonlinear primary resonance solution of functionally graded (FG) porous nanoshells under an external soft harmonic excitation including surface stress effects. To this end, the constitutive differential equations are adopted within the framework of the Gurtin-Murdoch theory of elasticity. In order to capture more accurate results, the nonlinear modal interactions between the main oscillation mode and higher symmetric vibration modes are taken into account. In addition, the mechanical properties of the FG porous nanoshell are estimated on the basis of the closed-cell Gaussian-Random field scheme. Thereafter, with the aid of the multiple time-scales technique, analytical expressions are expressed for the nonlinear surface elastic-based frequency-responses and amplitude-responses associated with the nonlinear primary resonance of FG porous nanoshells corresponding to various nonlinear modal interactions. It is demonstrated that by taking the modal interaction between higher symmetric vibration modes with the main oscillation mode into account, the frequency-response of FG porous nanoshells changes from the hardening behavior to the softening one. As a consequence, the modal interaction causes to shift the peak of the nonlinear resonance from the excitation frequency range of Ω/ω > 1 to Ω/ω < 1.