Employing an analytical approach to study the thermo-mechanical vibration of a defective size-dependent graphene nanosheet

Abstract

In this study, an analytical method is developed to study the free vibration of a defected nano graphene sheet coupled with temperature change and embedded on foundation based on Reddy’s third-order shear deformation plate theory. The graphene sheet may be opposed to the structural defect during the production process. Therefore, it is important to analyze the vibration behavior of graphene sheet. Here, some of the defects are modelled as a hole. Reddy’ third-order shear deformation plate theory is employed because it satisfies the zero shear stress condition at the free surfaces and need not use any shear correction coefficient to obtain equations of motion of the defected nanosheet. The interaction of the defected graphene sheet with a viscoelastic medium is simulated as a visco-Pasternak foundation. The influence of the surrounding viscoelastic medium on the natural frequencies is analyzed. To get the equations of dynamic equilibrium and natural boundary conditions of the nanosheet, the Hamilton’s principle is implemented. The presented method is verified by comparing the results with their counterparts reported in the open literature and good agreement is observed. Effects of different boundary conditions such as C-C, S-S, C-F, C-S, inner-radius-outer-radius ratio, Winkler foundation parameter, damping modulus, shear modulus, nonlocal parameter and temperature change on the frequency of the defected graphene sheet are examined. Various natural frequencies in nondimensional form and mode shapes are developed. The results show that, by increasing the inner-radius-outer-radius ratio, the natural frequency has an increasing behavior for all kinds of boundary condition. It is observed that increasing the size of defect has a significant effect on the natural frequency. Moreover, it can be concluded that decreasing the nonlocal parameter as the small-scale effect makes the plate stiffer. Therefore, the natural frequency of the nanoplate increases.