Cylindrical Bending Vibration of Multiple Graphene Sheet Systems Embedded in an Elastic Medium

Based on the Eringen nonlocal elasticity theory and multiple time scale method, an asymptotic nonlocal elasticity theory is developed for cylindrical bending vibration analysis of simply-supported, Nl-layered, and uniformly or nonuniformly-spaced, graphene sheet (GS) systems embedded in an elastic medium. Both the interactions between the top and bottom GSs and their surrounding medium and the interactions between each pair of adjacent GSs are modeled as one-parameter Winkler models with different stiffness coefficients. In the formulation, the small length scale effect is introduced to the nonlocal constitutive equations by using a nonlocal parameter. The nondimensionalization, asymptotic expansion, and successive integration mathematical processes are performed for a typical GS. After assembling the motion equations for each individual GS to form those of the multiple GS system, recurrent sets of motion equations can be obtained for various order problems. Nonlocal multiple classical plate theory (CPT) is derived as a first-order approximation of the current nonlocal plane strain problem, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal multiple CPT, although with different nonhomogeneous terms. Some nonlocal plane strain solutions for the natural frequency parameters of the multiple GS system with and without being embedded in the elastic medium and their corresponding mode shapes are presented to demonstrate the performance of the asymptotic nonlocal elasticity theory.