Nanomaterials have been attracted the attention of many researchers due to their superior mechanical, chemical and electronic properties. These desirable properties have led to its applications as components in nano electro-mechanical systems. Continuum modeling of nanostructures has received the great deal of attention of scientific community because controlled experiments in nanoscale are difficult and molecular dynamic simulations are highly computationally expensive. Since the Based on the nonlocal continuum mechanics, governing differential equations are derived. Numerical solutions for the buckling loads are obtained using Galerkin method. The vibration characteristic of variable thickness nanoplates embedded in an elastic medium is also investigated. The nonlocal governing equations of motion are derived taking into account the influences of the small scale based on the first order shear deformation theory (FSDT) of plates. Numerical solution for the vibration frequencies of nanoplates are obtained by employing the differential quadrature method (DQM) as a simple, efficient and accurate numerical tool for differential equations with variable coefficients. In another problem attempt is made to study the buckling behavior of orthotropic graphene sheets under various linearly varying in-plane normal forces. Based on the nonlocal elasticity theory, the small scale effects are introduced. Using the equilibrium equations of a differential element of a rectangular plate, the governing equations of single layered graphene sheet (SLGS) are derived. Differential quadrature method (DQM) is used to solve the governing equations for simply supported boundary conditions, clamped boundary conditions and various combinations of them. To verify the accuracy of the DQM solutions, the governing equation is also solved by the power series method (PSM) of Frobenius. In addition, the postbuckling of single-layered graphene sheet subjected to axial compression based on the nonlocal continuum mechanics is investigated. The geometrical nonlinearity is modeled with the use of von Karman’s assumptions. Galerkin method is applied to solve the governing nonlocal equations for postbuckling response. keyword : Buckling, Differential quadrature method, Nanoplate, Nonlocal elasticity
