Nonlinear Vibration Analysis of Nano-Disks Based on Nonlocal Elasticity Theory Using Homotopy Perturbation Method

Abstract

This paper introduces a novel approach for small-scale effects on nonlinear free-field vibration of a nano-disk using nonlocal elasticity theory. The formulation of a nano-disk is based on the nonlinear model of von Kármán strain in polar coordinates and classical plate theory. To analyze the nonlinear geometric and small-scale effects, the differential equation based on nonlocal elasticity theory was extracted from Hamilton principle, while the inertial and shear-stress effects were neglected. The equation of motion was discretized using the Galerkin method on selecting an appropriate function based on the boundary condition used for the nano-disk. Due to presence of nonlinear terms, the homotopy method was used in conjunction with the perturbation method (HPM) to ease up the solution and completely solve the problem. For further comparison, the nonlinear equations were solved by the fourth-order Runge–Kutta method, the solution of which was compared with that of HPM. Excellent agreements in results were observed between the two methods, indicating that the latter method can simplify the solution, and hence, can be applied to nonlinear nano-disk problems to seek their solution with a high accuracy.

Computational Linear and Nonlinear Free Vibration Analyses of Micro/Nanoscale Composite Plate-Type Structures With/Without Considering Size Dependency Effect: A Comprehensive Review

“…Also, by reducing the ratios of geometrical factors the variation amplitude of the imperfect structures' has reduced. Moreover, Shishesaz et al [597] analyzed the NL FV of circular MPs based on NLET and exerted from HP by implementing the HPM. The model formulation was based on the NL model of VKT strain in polar coordinates combined with CPT.…”

Free vibration characteristics of viscoelastic nano-disks based on modified couple stress theory

“…According to the inefficiency of classical theories in determining the 1 behavior of these materials, non-classical theories of continuum mechanics have been developed. Among these theories, one can point out the first strain gradient theory (FSGT), 14 second strain gradient theory (SSGT), 15 micropolar theory, nonlocal elasticity theory, [16][17][18][19] and the modified couple stress theory (MCST). [20][21][22] MCST has been extensively employed for analyzing and determining the behavior of micro/ nanostructures.…”