Nanotechnology has the potential of material control and utilization of material properties, instruments and modern systems in nano scale. This technology is utilized in different fields like medicine, agriculture, traortation, water and air purification, electronics. Nano materials have engrossed great deal of attention of scientific community due to their novel mechanical, chemical, thermal, electrical and electronic properties. Graphene sheet is one of the most important productions of nanomaterials. As controlled experiments in nanoscale are difficult and molecular dynamic simulations are highly computationally expensive, theoretical analysis of nanostructures becomes an important issue concerning application of nanostructures. Grapene sheets are used widely in electronic devices.There are three ways for mechanical analysis of these sheets, including experiments in nano scale, simulation by software, and theoretical analysis. Continuum modeling of nanostructures has thus received increasing deal of attention. However there is a need to upgrade theclassical continuum theories.In this research by using nonlocal elasticity theory of eringen. Then we investigated natural frequencies of rectangular nanoplates embedded in winkler elastic foundation by using differential quadratic method. Influence of wrinkle and sheer elastic factors on natural frequencies are studied. By using third order sheer deformation theory and it’s assumptions displacement and strain relations are derived. Its governing equations are derived by third order sheer deformation theory. Navier’s solution is applied to solve governing equation, and by using that natural frequencies of nanoplate are obtained. The refined plate theory is expanded for nanoplates, and natural frequencies are obtained by refined plate theory for isotropic and orthotropic rectangular nanoplates. Then lateral vibration of rectangular nanoplate under in-plane load is investigated. Vibration equations of circular nanoplates are derived by using nonlocal elasticity theory And exact solution is given for clamped boundary condition. In the end frequency equation for annular nanoplate is derived for clamped boundary condition on outer and inner radius. Keywords: Nonlocal elasticity, Rectangular nanoplate, Circular nanoplate, Small scale effect.
