Vibration of Single Layer Graphene Sheet Based on Non local Elasticity and Higher Order Shear Deformation Theory

Higher order shear deformation theory (HSDT) is 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 all edges simply supported boundary conditions. Analytical solutions for the vibration of nanoplates such as graphene sheets are presented. Nonlocal elasticity theories are employed to bring out the size effect on the natural frequencies of graphene sheets. Effects of (i) nonlocal parameter, (ii) length (iii) thickness of the graphene sheets and (iv) higher order shear deformation theory on the vibration frequencies are investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories of graphene sheets.