Coupled size-dependent behavior of shear deformable microplates

Abstract

The aim of the present study is to investigate the nonlinear motion characteristics of a shear deformable microplate based on the modified couple stress theory. The microplate is modeled via the third-order shear deformation theory retaining in-plane displacements and inertia. Using the Lagrange equations together with an assumed-mode method, five sets of second-order nonlinear ordinary differential equations of motion with coupled terms are obtained. These five sets of equations (two for the in-plane motions, one for the out-of-plane motion, and two for rotations) are transformed into ten sets of first-order nonlinear ordinary differential equations. These resultant equations are then solved by means of a direct time integration technique and the pseudo-arclength continuation method in order to analyze the nonlinear response of the system. Apart from the nonlinear analysis, the linear natural frequencies of the system are obtained using an eigenvalue analysis. Results are shown through frequency–response and force–response curves. Points of interest in the parameter space in the form of time histories, phase-plane portraits, and fast Fourier transforms are also highlighted. Moreover, a comparison is made between the motion characteristics of the system based on the modified couple stress and classical continuum theories.