The nanostructures under rotation have high promising future to be used in nano-machines, nano-motors and nano-turbines. They are also one of the topics of interests and it is new in designing of rotating nano-systems. In this paper, the scale-dependent vibration analysis of a nanoplate with consideration of the axial force due to the rotation has been investigated. The governing equation and boundary conditions are derived using the Hamilton’s principle based on Nonlocal elasticity theory. The boundary conditions of the nanoplate are considered as free-free in y direction and two clamped-free (cantilever plate) and clamped-simply (propped cantilever) in x direction. The equations have been solved using differential quadrature method to determine natural frequencies of the rotating nanoplate. For validation, in special cases, it has been shown that the obtained results coincide with literatures. The effects of the Nonlocal parameter, aspect ratio, hub radius, angular velocity and different boundary conditions on the first three frequencies have been investigated. Results show that vibration behavior of the rotating nanoplate with cantilever boundary condition is different from other boundary conditions.

This paper proposed analytical solutions for the buckling analysis of rectangular single-layered graphene sheets under in-plane loading on all edges simply is supported. The characteristic equations of the graphene sheets are derived and the analysis formula is based on the Nonlocal Mindlin plate. This theory is considering both the small length scale effects and transverse shear deformation effect. Nonlocal elasticity theory takes into account the small length scale effects as examining nanostructures such as nanoplates. It is presented graphically that the small scale or Nonlocal effects on the nondimensional buckling loads in the presence of aspect ratio and buckling modes.

In the current study, a finite element method is developed using the principle of total potential energy based on Nonlocal integral elasticity theory to investigate the free vibration behavior of nano-scaled plates. The classical plate theory is considered for deriving the formulations of the plate. The eigenvalue problem is extracted by using the variational principle and corresponding natural frequencies of free vibration are obtained. Different boundary conditions and various geometries can now be appropriately analyzed by using the Nonlocal finite element method proposed in the current article. The results of the current study are compared with those available in the literature. Then the effects of Nonlocal parameters, geometrical parameters, various boundary conditions and surface effects on the free vibration behavior of nano-scaled plates are investigated.

In this article، analytical solutions of low velocity transverse impact on a nanobeam are presented using the Nonlocal theory to bring out the effect of the Nonlocal behavior on dynamic deflections. Impact of a projectile (mass) on simply supported and clamped nanobeams are investigated using Nonlocal Timoshenko beam theory. In order to obtain an analytical result for this problem، an approximate method has been developed wherein the applied impulse is replaced by a suitable boundary condition and initial momentum of projectile and nanobeam. A number of numerical examples with analytical solutions for Nonlocal nanobeam and classical beam (steel and aluminum) have been presented and discussed. When the value of the striker mass is increased، the frequencies are decreased and the maximum dynamic deflection at the center of the beam is increased for both of the simply supported and the clamped-clamped nanobeams. The inclusion of the Nonlocal effect increases the magnitudes of dynamic deflections and decreases frequencies. Furthermore، the mass and the velocity of the nanoparticle (striker) have significant effects on the dynamic behavior of nanobeam.

Increasing the electrical resistance and reducing the conductivity of nanotubes due to the impact forces have attracted a lot of their use as power sensors. In this paper, the impact of a nanoparticle on simply supported and clamped double-walled nanotubes has been investigated by using elasticity Nonlocal theory. The spring constant for each nanobeam, which is influenced by the geometry of the structure and its shear and bending deformations, has been obtained using mass-spring system. The results of this analytical method have been compared with the results in the background research. According to the results, by increasing the speed, the dynamic deformation in the middle of the double-walled nanobeam increases. The dynamic deformation in the double-walled nanotube is smaller than single-walled nanotube and the maximum dynamic deformation of the double-walled nanotubes is increased by increasing the mass of nanoparticle. Also by considering the van der Waals force between the layers of double-walled nanotube, maximum dynamic deformation in the middle of nanobeam reduces.

In this article, classical plate theory (CPT) is reformulated using the Nonlocal differential constitutive relations of Eringen to develop an equivalent continuum model for orthotropic triangular nanoplates. The equations of motion are derived and the Galerkin’s approach in conjunction with the area coordinates is used as a basis for the solution. Nonlocal theories are employed to bring out the effect of the small scale on natural frequencies of nano scaled plates. Effect of Nonlocal parameter, lengths of the nanoplate, aspect ratio, mode number, material properties, boundary condition and in-plane loads on the natural frequencies are investigated. It is shown that the natural frequencies depend highly on the non-locality of the nanoplate, especially at the very small dimensions, higher mode numbers and stiffer edge condition.

Conventional Ritz and Galerkin methods based on local theory of elasticity employ polynomials as their approximating functions, however these methods are not convenient to use in three-dimensional Nonlocal analysis. In the present study, to conquer this difficulty, a type of weighted residual approach with a set of trigonometric approximating functions was developed. By using appropriate trigonometric approximating functions it is possible to consider the effect of various edges boundary conditions on frequency behavior of nanoplate. Validation of present formulation is carried out by comparing numerical result with the published results. It is concluded that the effect of Nonlocal parameter on natural frequencies is significant, especially in higher modes due to the lower wavelength of the mode. The research shows that in Nonlocal elasticity there are distinct discrepancies between behaviors of two and three-dimensional results. In addition, the difference between the two- and three-dimensional results in local elasticity is not as noticeable as in Nonlocal elasticity. Furthermore, the effects of length to thickness ratio, aspect ratio, Nonlocal parameter and different boundary conditions on fundamental natural frequency of nanoplates were studied. This benchmark solution can be used to assess the accuracy of conventional two-dimensional theories.

In this paper, an analytical method is presented to study thermo-elastic behavior of nanoscale spherical shell subjected to thermal shock based on Nonlocal elasticity theory. The shell is considered as elastic, homogeneous and isotropic solid. The Nonlocal differential equation of motion is derived in terms of radial displacement. The analytical solution of equation of motion is obtained by Laplace transform and differential transform method (DTM). Mechanical boundary conditions are used to obtain unknown parameters that get in recurrence equation in Laplace domain. The results in Laplace domain is transferred to time domain by employing the fast inverse Laplace transform method (FLIT). Accuracy of obtained results is evaluated by wellknown similar articles. The results have a good agreement in comparison with published data in pervious literatures. Also, the effects of Nonlocal parameter and wall thickness of shell on the dynamic characteristics of nanoscale spherical shell are studied in various points across the thickness of shell under thermal shock. The present analytical method provides an appropriate field for an alysis of times histories of radial and hoop stresses in a nanoscale shells subjected to various time dependent thermo-mechanical loads.

In the present research, vibration behavior is presented for a thermally postbuckled two side clamped monolayer graphene nanoribbon. The monolayer graphene nanoribbon is modeled as      a Nonlocal orthotropic plate strip which contains small scale effects. The formulations are based on   the Kirchhoff’s plate theory, and von Karman-type nonlinearity is considered in strain-displacement relations. The thermal effects are also included and the material properties are assumed to be temperature-dependent. The initial deflection caused by thermal postbuckling and internal loads are taken into account. A coupled system of equations is derived and a new semi analytical solution is obtained. The effects of variation of small scale parameter e0 a to the natural frequencies, deflections and mode shapes of graphene nanoribbon are analyzed and the numerical results are obtained from the Nonlocal plate model; also, molecular dynamics simulations are used to investigate different properties of graphene nanoribbon including both buckling and vibrational behaviors. The small scale coefficient is calibrated using molecular dynamics simulations. Numerical results are compared with those of similar researches. Effects of various parameters on the postbuckled vibration of graphene nanoribbon in thermal environments such as scale parameter, length and thermal load are presented. Stability and occurrence probability of internal resonance between vibration modes around a buckled configuration is investigated.