Abstract
In this paper, the small scale effects are incorporated into the free vibration analysis of single-layered graphene sheets (SLGSs) embedded in an elastic medium. To this end, Eringen’s nonlocal elasticity continuum are applied to the different types of plate theory namely as the classical plate theory (CLPT), first order shear deformation theory (FSDT), and higher order shear deformation theory (HSDT). Winkler and Pasternak foundation models used to simulate the surrounding elastic medium are compared with each other. Explicit expressions are derived to calculate the natural frequencies of square SLGSs corresponding to each type of nonlocal plate model. Selected numerical results are given to indicate the influence of the nonlocal parameter, Winkler and Pasternak elastic moduli, mode number, and the side length of SLGSs in detail. Also, comparison is made between the vibrational responses of SLGSs obtained through different nonlocal plate theories. It is found that the elastic foundation and value of nonlocal parameter have quite significant effects on the natural frequencies of SLGSs and these effects are influenced by mode number as well as side length.DOI: http://dx.doi.org/10.5755/j01.mech.23.5.14883
Highlights
Owing to their outstanding mechanical, electrical, and chemical properties, the family of carbon allotropes including carbon nanotubes, graphene sheets and fullerenes are becoming increasingly important in the emerging field of nanoscience and nanotechnology [1,2]
The plate theory is the first order shear deformation theory in which the effects of shear deformation and rotational inertia are taken into account, so the straight lines will no longer remain vertical to the mid-plane of the plate after deformation
Selected numerical results are presented to indicate the influence of the values of nonlocal parameter, Winkler modulus parameter, Pasternak modulus parameter, mode number, aspect ratio, and the type of nonlocal plate theory, in detail
Summary
To represent the behaviour of plates, there are different plate theories. The displacement components (u1, u2, u3) along the axes (x, y, z) can be written in a general form as: u1. The effects of shear deformation and rotational inertia are not considered in this type of plate theory. On the basis of Eq (1), the strain-displacement relations appropriate to CLPT can be obtained as: xx u1 x. Using the principle of virtual displacement, the equilibrium equation can be expressed for CLPT as:
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