Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory

The deviation from the classical elastic characteristics induced by the free surface energy can be considerable for nanostructures due to the high surface to volume ratio. Consequently, this type of size dependency should be accounted for in the mechanical behaviors of nanoscale structures. In the current investigation, the influence of free surface energy on the nonlinear primary resonance of silicon nanoshells under soft harmonic external excitation is studied. In order to obtain more accurate results, the interaction between the first, third, and fifth symmetric vibration modes with the main oscillation mode is taken into consideration. Through the implementation of the Gurtin-Murdoch theory of elasticity into the classical shell theory, a size-dependent shell model is developed incorporating the effect of surface free energy. With the aid of the variational approach, the governing differential equations of motion including both of the cubic and quadratic nonlinearities are derived. Thereafter, the multi-time-scale method is used to achieve an analytical solution for the nonlinear size-dependent problem. The frequency-response and amplitude-response of the soft harmonic excited nanoshells are presented corresponding to different values of shell thickness and surface elastic constants as well as various vibration mode interactions. It is depicted that through consideration of the interaction between the higher symmetric vibration modes and the main oscillation mode, the hardening response of nanoshell changes to the softening one. This pattern is observed corresponding to both of the positive and negative values of the surface elastic constants and the surface residual stress.