Nonlocal elasticity theory for vibration of nanoplates

Classical plate theory (CLPT) and first-order shear deformation theory (FSDT) of plates are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. Navier's approach has been used to solve the governing equations for simply supported boundary conditions. Analytical solutions for vibration of the nanoplates such as graphene sheets are presented. Nonlocal theories are employed to bring out the effect of the nonlocal parameter on natural frequencies of the nanoplates. The developed theory has been extended to the analysis of double layered nanoplates. Effect of (i) nonlocal parameter, (ii) length, (iii) height, (iv) elastic modulus and (v) stiffness of Winkler foundation of the plate on nondimensional vibration frequencies are investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories of nanoplates and nanoshells.