Analytical determination of size-dependent natural frequencies of fully clamped rectangular microplates based on the modified couple stress theory

Abstract

This paper presents an analytical and size-dependent model for vibrational analysis of fully clamped rectangular microplates. Modified couple stress theory (MCST) and the Kirchhoff plate model are considered, and Hamilton's principle is employed to derive the sizedependent equation of motion that accounts for the effect of residual stresses. The natural frequencies of the microplate are extracted analytically by extended Kantorovich method. The present findings are validated with the available results in the literature, and an excellent agreement is observed between them. In addition, a parametric study is conducted to demonstrate the significant effects of couple stress components on the natural frequencies of fully clamped microplates. The ratio of MCST natural frequencies to those obtained with classical theory depends only on the Poisson's ratio of the plate and is independent of the aspect ratio of the plate for cases with no residual stresses.

The Influence of Couple Stress Components and Electrostatic Actuation on Free Vibration Characteristics of Thin Micro-Plates

“…It is worth noting that this important conclusion has also been reported numerically by Tsiatas [19] and analytically by Askari and Tahani [20] …”

“…[20]). It has been shown that continuing the present procedure would leads to the convergence of the results which is so rapid and usually occurred after the second iteration step [20].…”

A Levy solution for bending, buckling, and vibration of Mindlin micro plates with a modified couple stress theory

“…Jomezadeh et al [28] performed a vibration analysis of rectangular Kirchhoff micro plates using a Levytype solution method. Askari and Tahani [29] employed the extended Kantorovich method to analyze the vibration behavior of a Kirchhoff micro plate. They presented the results for a fully clamped micro plate.…”

“…While the effects of transverse shear and effects of boundary supports on the mechanical responses of plates are significant, it appears from the literature review that there is no analytical solution for micro plates having different boundary conditions and being modeled within shear deformation plate theories in conjugation with MCST. Even though a Levy-type solution [27,28,36], or the extended Kantorovich method [29] has been implemented to analyze Kirchhoff micro plates, the governing equations within the first order or higher order micro plate theories are solved using just the Navier method [32][33][34][35] or numerical methods [23,31,39]. That is, to the best of authors' knowledge, analytical solutions within shear deformation micro plate theories are available for just simply-supported ones.…”

An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates

“…The Modi ed Couple Stress Theory (MCST) bene ts from two main advantages of the inclusion of symmetric couple stress tensor and involvement of only one material length-scale parameter. Based on the modi ed couple stress theory, di erent studies have been conducted concerning vibrational behavior of micro-plates [17][18][19][20][21][22].…”