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Nanoscale circular plates are encountered in several nanotechnology-based devices such as nanoelectromechanical systems. Structures are size dependent at nanoscale due to surface energy effects. It is possible to capture such size dependence through special continuum models. In this paper, the Gurtin-Murdoch continuum theory is applied to develop a new continuum mechanics model for static deformation of thin and thick circular nanoplates. The relevant governing equations are established from basic principles. It is shown that the governing equations possess a closed-form analytical solution that makes the current approach suitable for device analysis and design. A series of closed-form analytical solutions is presented for static bending of thin and thick plates under common static loading (uniformly distributed and center point) and boundary conditions (simply supported and clamped edges). The analytical solution for a thin plate supported by a linear elastic substrate is also presented. Deflection profiles of selected silicon and aluminum plates are presented and compared with the classical plate theory results to examine the salient features of mechanical response and influence of surface elastic moduli, surface residual stress, and boundary conditions.