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

Based on the modified couple stress and first-order shear deformation plate theories, a Levy-type solution is presented for bending, buckling, and vibration analyses of rectangular isotropic micro plates with simple supports at opposite edges and different boundary conditions at the other two ones. The governing equations are derived using the Hamilton's principle, and then solved by a single Fourier series expansion and the state-space method, which its implementation has not been straightforward. The results are verified with the existing ones for a fully simply supported micro plate in the literature. Finally, the effect of geometric parameters and length scale parameter on bending, buckling, and vibration behaviors of micro plates is studied. Since, there are no analytical solutions for bending, buckling loads, and natural frequencies of Mindlin micro plates with different boundary conditions in the literature and the Navier method is the only available analytical solution for the Mindlin and higher order shear deformable micro plates within the modified couple stress theory, the results presented here can be used as a benchmark in future studies. In addition, it is shown that the difference between results of the Kirchhoff and the Mindlin plate models depends not only on the plate thickness but also on the length scale parameter to thickness ratio (l/h) as well as the boundary supports. This result emphasizes the significance of analytical solutions for shear deformation models of micro plates with different boundary supports.