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The present study proposes an analytical solution for the axisymmetric/asymmetric buckling analysis of thin circular/annular nanoplates under uniform radial compressive in-plane load. In order to consider small scale effects, nonlocal elasticity theory of Eringen is employed. To ensure the efficiency and stability of the present methodology, the results are compared with other presented in literature. Material properties including Young’s modulus, density, Poison’s ratio are assumed to be constant through the body of nanoplate. In addition, the effect of small scales on critical buckling loads for different parameters such as radius of the FG nanoplate, boundary condition, mode number and geometry parameters are investigated. In order to obtain the critical buckling load, the asymmetric modes as well as axisymmetric modes are considered. The thin nanoplate is modeled using Kirchhoff plate theory.

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In this paper, exact closed-form solutions in explicit forms are presented to investigate small scale effects on the buckling of Lévy-type rectangular nanoplates based on the Reddy’s nonlocal third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton’s principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on buckling load ratios and non-dimensional buckling loads of rectangular nanoplates with different combinations of boundary conditions are tabulated for various values of nonlocal parameters, aspect ratios and thickness to length ratios. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. Also, the present study may be useful for static and dynamic analysis of thicker nano scale plate-like structures, multi-layer graphene and graphite as composite or sandwich structures.

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In this paper, for the free vibration analysis of bilayer graphenes with interlayer shear effect the sandwich beam model is introduced. Because of the similarity between the bilayer graphene and the sandwich structures, in which at the top and the bottom of the bilayer graphene there is a single layer graphene and between them there is Vander walls bindings, the bilayer graphene is modeled as a sandwich beam and its free vibration is investigated for free-clamp end condition. To obtain the governing equations, each graphene layer is modeled based on the Euler-Bernoulli theory and in-plane displacements are also considered in addition to the transverse displacement. It is also assumed that the graphene layers do not have relative displacement during vibration. The effect of the Vander walls bindings is introduced in the governing equations as the shear modulus. The results obtained by the sandwich beam model, presented in this paper for the first time, include the first five natural frequencies of the bilayer graphenes with 7 to 20 nanometer lengths. These results are validated by the molecular dynamic and the Multi-Beam-Shear model results.

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In this paper, exact closed-form solutions in explicit forms are presented to investigate small scale effects on the transverse vibration behavior of Lévy-type rectangular nanoplates based on the Reddy’s nonlocal third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton’s principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on frequency ratios and non-dimensional fundamental natural frequencies of rectangular nanoplates with different combinations of boundary conditions are tabulated for various values of nonlocal parameters, aspect ratios and thickness to length ratios. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. Also, the present study may be useful for static and dynamic analysis of thicker nano scale plate-like structures, multi-layer graphene and graphite as composite or sandwich structures.

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In this study, by modeling van der Waals (vdWs) interactions based on the Lennard-Jones potential function interlayer tensile-compressive and shear moduli of bilayer graphene sheets are analytically calculated. To this end, by varying potential depth parameter which shows the strength of vdWs interactions a new model is presented for calculating interlayer in-plane and out-of-plane moduli for two different stacking patterns. In order to determine the interlayer vdWs moduli, a small flake of monolayer graphene is sliding on a large monolayer graphene substrate and accordingly variations of vdWs forces as well as the interlayer shear and normal strains are recorded. The relative displacements of layers cause linear strain and stress. In the model, bilayer graphene geometry (being armchair or zigzag, and stacking pattern) and potential depth parameter are two important parameters for determination of vdWs moduli. The accuracy of the method is verified by comparing the present results with those reported in literatures. Finally, close-form relations for interlayer tensile-compressive and shear moduli of vdWs interactions versus the depth potential parameter are presented for ABA and AAA stacking patterns as well as zigzag and armchair directions. It is observed that the interlayer moduli have linear relation with the potential depth parameter.

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In this paper, free axial vibration of nanorods is investigated by focusing on the inertia of the lateral motions effects. To this end, Rayleigh and nonlocal theories considering the inertia of the lateral motions and the small scale effects, respectively, are used. Then, by implementing the Hamilton’s principle nonlocal governing equation of motion and boundary conditions are derived. Since using nonlocal elasticity causes that the 2-order local governing equation is changed to the 4-order nonlocal governing equation while number of boundary condition remains constant (one boundary condition at each end of nanorod), the governing equation is solved using Rayleigh-Ritz method. In Rayleigh-Ritz method a suitable shape function for the problem should be selected. The shape function must at least satisfy the geometrical boundary conditions. In the present study, orthogonal polynomials are selected as shape functions then they are normalized by using the Gram-Schmidt process for more rapid convergence. Then, the first five axial natural frequencies of nanorod with clamped-clamped and clamped-free end conditions are obtained. In the next step, effects of various parameters like length of nanorod, diameter of nanorod and nonlocal parameter value on natural frequencies are investigated. Results of the present study can be useful in more accurate design of nano-electro-mechanical systems in which nanotubes are used.

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In the present paper, free axial vibration behavior of functionally graded nanorods is studied using the surface elasticity theory. For modelling of free axial vibration of nanorods, the Simple theory of rods is implemented. Besides using the Simple theory of rods, the surface elasticity theory is used for considering the surface energy parameters in the governing equations and boundary conditions. The surface energy parameters are the surface elasticity, the surface density, and the surface residual stress. The surface and bulk material properties of nanorod are considered to vary in the length direction according to the power law distribution. Then, the governing equation of motion and boundary conditions of nanorod are derived using the Hamilton’s principle. Due to considering the surface energy parameters, the obtained governing equation of motion becomes non-homogeneous. But in none of the previous researches, for example investigation of free transverse vibration of nanobeams and free torsional vibration of nanorods in presence of the surface energy, the surface energy parameters do not cause the non-homogeneity of the governing equation or the boundary conditions. To extract the natural frequencies of the nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using Galerkin method. In order to have a comprehensive research, effects of various parameters like the length and radius of nanorod on axial frequencies of functionally graded nanorod is investigated.