Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories

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In the present study, a vibration analysis of functionally graded rectangular nano-/microplates was considered based on modified nonlinear coupled stress exponential and trigonometric shear deformation plate theories. Modified coupled stress theory is a non-classical continuum mechanics theory. In this theory, a material-length scale parameter is applied to account for the effect of nanostructure size that earlier classical plate theories are not able to explain. The material properties of the plate were assumed to vary according to a power-law form in the thickness direction. The governing equation of the motion of functionally graded, rectangular nano-/microplates with different boundary conditions were obtained based on the Rayleigh-Ritz method using complete algebraic polynomial displacement and rotation functions. The advantage of the present Rayleigh-Ritz method is that it can easily handle the different conditions at the boundaries of moderately thick rectangular plates (e.g., clamped, simply supported, and free). A comparison of the results with those available in the literature has been made. Finally, the effect of various parameters, such as the power-law index, thickness-to-length scale parameter ratio h/l, and aspect ratio a/b, on the natural frequency of nano/micro-plates are presented and discussed in detail.