A size-dependent model for coupled 3D deformations of nonlinear microbridges

This paper presents a new nonlinear model for coupled 3D displacements of the doubly-clamp microbeams based on the modified couple stress theory. The microbeam is assumed to deflect in axial and lateral directions and the nonlinearities caused by the mid-plane stretching are considered. Utilizing modified couple stress theory and applying extended Hamilton's principle, nonlinear equations of the coupled three-dimensional motion of the microbridge are derived. As case studies, the static deflection and nonlinear free vibration of the doubly-clamp microbeams are investigated using the presented model. For vibration analysis, the method of multiple scales is utilized to calculate the frequency of the microbridge versus the amplitudes of vibration in two lateral directions. The effects of the applied loads and amplitudes of vibration in each direction are investigated on the static deflection and frequencies of vibration respectively. Comparing the results obtained by classical and nonclassical theories shows that the classical theory underestimates the stiffness and natural frequency of the structures especially for microbeams that have thicknesses in order of the length scale parameter. Furthermore, findings shows that the mid-plane stretching causes a coupling between lateral directions so that applying load in one direction leads to change in deflection of the other direction.