A Modified Couple Stress-based Model for the Nonlinear Vibrational Analysis of Nano-disks using Multiple Scales Method

Abstract

In this article, the nonlinear vibrational behavior of a nano-disk was analyzed using the multiple scales method (MSM). The modified couple stress theory was used to consider the small-scale effect via the application of nonlocal parameter. Employing Hamilton's principle, two coupled nonlinear differential equations were derived based on the nonlinear von-Karman strain-displacement relation and the classical plate theory. The Galerkin-based procedure was utilized to obtain a Duffing-type nonlinear ordinary differential equation with a cubic nonlinear term and solved by the application of MSM. The effects of nonlocal parameter, aspect ratio, different boundary conditions, and the nonlinear shift frequencies, were obtained on the overall behavior of the nano-disk. Results indicate that increasing the central dimensionless amplitude of the nano-disk, the nonlinear frequency, and the shift index exhibit an increasing behavior, while the increase in the non-dimensional nonlocal parameter, causes a decrease in the nonlinear frequency ratios and the shift index. Additionally, the increase in h/r increases the effect of dimensionless central amplitude on the nonlinear frequencies ratios. Additionally, comparison of the current results with those previously published in the literature shows good agreements. This indicates that the MSM can ease up the solution, and hence, can be applied to the solution of nonlinear nano-disks with high accuracy.