Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory

Abstract

The present investigation proposes a nonclassical mathematical model and an algorithm for the axisymmetrically nonlinear free vibration analysis of a circular microplate. Based on the modified couple stress theory and von Kármán geometrically nonlinear theory, the governing equations for microplates are established in variational form from Hamilton principle containing only one material length scale parameter which can capture the size-dependent behavior. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed simple harmonic function in time. The resulting nonlinear spatial boundary value problem is then solved numerically by shooting method, and the size-dependent characteristic relationships of nonlinear vibration frequency versus central amplitude of the microplates are obtained. The parametric studies are conducted for immovable clamped and simply supported edge conditions, some of the results in special cases are verified by comparing with those in the literature. The numerical results indicate that the microplates modeled by the modified couple stress theory cause more stiffness than modeled by the classical continuum plate theory, such that the differences between the results of these two theories are large when the thickness to material length scale ratio is small, whereas they are decreasing with the increase of the ratio.