A size-dependent FG micro-plate model incorporating higher-order shear and normal deformation effects based on a modified couple stress theory

Abstract

This paper presents a size-dependent functionally graded (FG) micro-plate model based on a modified couple stress theory requiring only one material length scale parameter. The model proposed accounts for both shear and normal deformation effects by a parabolic variation of all displacements across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the micro-plate. The effective material properties of the micro-plate are estimated via the homogenization method of power law distribution. Firstly, the equations of motion and related boundary conditions are derived from Hamilton׳s principle. Then based on these equations, closed-form solutions for static bending and free vibration analysis are obtained for a simply supported rectangular micro-plate. Furthermore, the validity of the derived formulation is established by comparing it with the ones available in literature. Finally, numerical examples are presented to investigate the influences of the power law index, material length scale parameter, plate׳s thickness, shear and normal deformation effects on the mechanical characteristics of the FG micro-plate. The results demonstrate that the inclusion of the size effects results in an increase in the micro-plate׳s stiffness, and consequently, leads to a reduction of deflections and an increase in natural frequencies, while the shear and normal deformation effects are just the opposite.