Wave Analysis of Thick Rectangular Graphene Sheets: Thickness and Small-Scale Effects on Natural and Bifurcation Frequencies

Free vibration and wave analysis of thick rectangular graphene are studied by employing the wave propagation method. To consider small-scale effects and thickness of a plate in nanoscales, equations of motions are represented by the Eringen nonlocal theory coupled with the Mindlin plate theory of thick plates. To solve the governing equations of motion with the wave propagation technique, propagation and reflection matrices are derived. These matrices are combined to obtain exact natural frequencies of graphene sheets for all six possible boundary conditions. To check the accuracy and reliability of the method, natural frequencies are compared with the results of the literature, and excellent agreement is observed. Additionally, wave analysis of the graphene sheet is performed and different types of waves in the graphene sheet are captured. Deriving the dispersion relation of the graphene sheet, bifurcation frequencies (cut-off and escape frequencies) are analytically found. Finally, the effects of graphene sheet thickness and nonlocal parameter on the natural frequencies and bifurcation frequencies are investigated. It is observed that natural frequencies are highly dependent on the graphene sheet’s thickness and nonlocal parameter. More importantly, the number and order of bifurcation frequencies depend on these two parameters as well. Our findings are valuable for the sustainable design and fabrication of graphene-based sensors, in which structural health monitoring of embedded graphene sheets is of great importance.