Vibration Of Nanoplates Research Articles

Size-dependent free vibration analysis of Mindlin nano-plates with curvilinear plan-forms by a high order curved hierarchical finite element

ABSTRACTIn the present study, Eringen nonlocal theory is employed in Mindlin plate theory to study the free vibration of nano-plates. The hierarchical finite element method based on a curved hierarchical quadrilateral element is developed and applied to free vibration analysis of arbitrary shaped Mindlin nano-plates. Equations of motion for free vibration of arbitrary nano-plate are derived according to the principle of virtual displacements. The convergence and accuracy of the proposed method are demonstrated by comparison with other methods. Finally the effect of aspect ratio, thickness ratio, mode number, and the boundary conditions on the natural frequencies have been studied.

A review of size-dependent elasticity for nanostructures

Nanotechnology is one of the pillars of human life in the future. This technology is growing fast and many scientists work in this field. The behavior of materials in nano size varies with that in macro dimension. Therefore scientists have presented various theories for examining the behavior of materials in nano-scale. Accordingly, mechanical behavior of nano-plates, nanotubes nano-beams and nano-rodes are being investigated by Non-classical elasticity theories. This review includes the last researches on bending, buckling, and vibration of nano-plates, nano-beams, nanorods, and nanotubes which were investigated by non-local elasticity theory and nonlocal strain gradient theory. Great scholars have written valuable reviews in the field of nanomechanics. Therefore, given a large number of researches and the prevention of repetition, the articles in the past year are reviewed.

Surface and Nonlocal Effects on Coupled In-Plane Shear Buckling and Vibration of Single-Layered Graphene Sheets Resting on Elastic Media and Thermal Environments using DQM

AbstractIn this article, surface and nonlocal effects are explored in the analysis of buckling and vibration in rectangular single-layered graphene sheets embedded in elastic media and subjected to coupled in-plane loadings and thermal conditions. The small-scale and surface effects are taken into account using the Eringen's nonlocal elasticity and Gurtin-Murdoch's theory, respectively. Using the principle of virtual work, the governing equations considering small-scale are derived for the nanoplate bulk and surface. The differential quadrature method (DQM) is utilized for the solution of the relevant problems and the results are validated against Navier's solutions. The impacts of the nonlocal parameter, Winkler and shear elastic moduli, temperature rise, boundary conditions, and the in-plane biaxial, uniaxial, and shear loadings on the surface effects of buckling and vibration are investigated. Numerical results show that increasing nonlocal parameter leads to enhanced surface effects on both buckling and vibration. This is in contrast to those reported elsewhere. Moreover, increasing in-plane loads are observed to enhance surface effects on vibration. On the other hand, the nonlocal parameter is observed to have more pronounced effects on shear buckling and vibration of plates subjected to coupled in-plane shear loads than those subjected to biaxial and uniaxial loads. This is while surface effects have greater impacts on biaxial buckling and vibration of nanoplates than on shear buckling and vibration.

Thermo-mechanical vibration, buckling, and bending of orthotropic graphene sheets based on nonlocal two-variable refined plate theory using finite difference method considering surface energy effects

In this article, the influence of temperature change on the vibration, buckling, and bending of orthotropic graphene sheets embedded in elastic media including surface energy and small-scale effects is investigated. To take into account the small-scale and surface energy effects, the nonlocal constitutive relations of Eringen and surface elasticity theory of Gurtin and Murdoch are used, respectively. Using Hamilton’s principle, the governing equations for bulk and surface of orthotropic nanoplate are derived using two-variable refined plate theory. Finite difference method is used to solve governing equations. The obtained results are verified with Navier’s method and validated results reported in the literature. The results demonstrated that for both isotropic and orthotropic material properties, by increasing the temperature changes, the degree of surface effects on the buckling and vibration of nanoplates could enhance at higher temperatures, while it would diminish at lower temperatures. In addition, the effects of surface and temperature changes on the buckling and vibration for isotropic material property are more noticeable than those of orthotropic. On the contrary, these results are totally reverse for bending problem.

Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load

In this paper, the size-dependent effects on the time-dependent bending and shearing responses of single-layer graphene sheets (SLGSs) induced by displacement of the concentrated moving load along the SLGSs are studied. Hence, the equations of motion are derived by applying two-variable refined plate theory in conjunction with the nonlocal elasticity theory via Hamilton’s principle. The two independent unknowns denoting the dynamic behaviour of plates are defined by the developing state-space method for this case, and Navier-solution method is employed to obtain the bending and shearing deflections due to the variation of velocity and time. The primary results for free and forced vibrations of nanoplates are approved by existing literature to illustrate the correctness of the present formulation and solution methods. Next, a solution including both the bending and shearing dynamic deflections of SLGSs under the moving load is derived for the first time in this research. Moreover, the roles of different parameters such as load velocity, length-to-thickness ratio, and small-scale effects on the dynamic deflections of SLGSs are addressed and discussed.

Large deformation analysis of single-crystalline nanoplates with cubic anisotropy

In this study, considering surface and bulk residual stresses, size- and orientation-dependent nonlinear bending, large amplitude vibration and postbuckling of single-crystalline nanoplates with cubic anisotropy are studied. Based on the first-order shear deformation plate theory, von-Karman geometrical nonlinearity and Gurtin-Murdoch theory, the model is formulated. In order to discretize the governing equations along with different combinations of boundary conditions, the generalized differential quadrature method is used and then a direct iterative method (Picard) is employed to solve the nonlinear system of equations. The presented results reveal that when thickness of the nanoplate increases, especially at high orientation and anisotropicity, ignoring shear deformation could cause some noticeable errors in the response of nonlinear bending, large amplitude vibration and postbuckling of nanoplates. Moreover, although orientation has a very direct effect on large amplitude vibration of nanoplates, it is observed that at a specific amplitude, the natural frequencies are independent from the orientation. The results of this study show that geometrical nonlinearity could be ignored in analyzing large amplitude vibration and postbuckling of highly anisotropic nanoplates in the [110] direction.

High resolution mass identification using nonlinear vibrations of nanoplates

Abstract This paper investigates nonlinear vibrations of the nanoplates for a high resolution mass identification. As background, the paper is furnished with a succinct review of the applications of the nanoplates, along with recent studies on nanoplate vibrations. In spite of a number of published works related to vibrations of nanoplates, their nonlinear behaviors have not been fully characterized. Accordingly, we present a mathematical model to examine nonlinear vibrations of nanoplates based on the nonlocal theory postulating nonlinear Winkler foundation. Besides, external force is taken into account to scrutinize nanoplate vibrations. The developed mathematical model consists of the assumption of a tiny mass stuck on the nanoplates. Utilizing the Galerkin’s method considering multiple scales method provides the primary mode of nanoplate oscillations. In view of that, parametric sensitivity analysis depicts that the frequency response of the nanoplates significantly changes by adding minuscule mass. Two different methods are developed for identification of the added mass. The first method works based on the jump phenomenon in nonlinear oscillators due to the added mass. In the second method, an adaptive parameter identifier estimates the added tiny mass based on a parametric dynamic model of the structure and its vibrational response to a periodic applied force. The effectiveness and straightforwardness of both developed methods are illustrated. Furthermore, a comparison with the molecular dynamics simulation is provided to verify the developed nonlocal model of nanoplate vibrations. It is concluded that the developed method can be considered as consequential and potential tools for high resolution mass sensing. The proposed approaches have the potential to be used in virus, enzyme and also humidity sensing.

Surface effect on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates with various edge supports

Surface stress and surface inertia effects may play a significant role in the mechanical characteristics of nanostructures with a high surface to volume ratio. The objective of this study is to present a comprehensive study on the surface stress and surface inertia effects on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates. To this end, the Gurtin–Murdoch theory, first-order shear deformation theory (FSDT) and Hamilton׳s principle are employed to develop a non-classical continuum plate model capable of taking the surface stress and surface inertia effects and also the rotary and in-plane inertias into account. To solve numerically the geometrically nonlinear forced vibration of nanoplates with different boundary conditions, the generalized differential quadrature (GDQ) method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are employed. The effects of parameters such as thickness, surface residual stress, surface elasticity, surface mass density, length-to-thickness ratio, width-to-thickness ratio and boundary conditions on the nonlinear forced vibration of rectangular nanoplates are fully investigated. The results demonstrate that surface effects on the nonlinear frequency response of aluminum (Al) nanoplate are more prominent in comparison with the silicon (Si) nanoplate.

Forced vibration of nanoplate on viscoelastic substrate with consideration of structural damping: An analytical solution

Based on nonlocal theory, this article discusses forced vibration of visco-nanoplate resting on viscoelastic medium. Single layer viscoelastic graphene sheet (SLVGS) modeled as visco-nanoplate. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and viscoelastic Kelvin Voigt model, the governing equation is derived using Hamilton’s principle. The equation is solved analytically to obtain the response of VGS. The parametric study is thoroughly performed, concentrating on the series effects of nonlocal parameter, viscoelastic damping structure, aspect ratio, visco-Pasternak (VP) medium, mode number and quality factor (Q). In this study, both dimensional and nondimensional calculations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.

Nonlocal normal modes in nanoscale dynamical systems

This paper introduces the idea of nonlocal normal modes arising in the dynamic analysis of nanoscale structures. A nonlocal finite element approach is developed for the axial vibration of nanorods, bending vibration of nanobeams and transverse vibration of nanoplates. Explicit expressions of the element mass and stiffness matrices are derived in closed-form as functions of a length-scale parameter. In general the mass matrix can be expressed as a sum of the classical local mass matrix and a nonlocal part. The nonlocal part of the mass matrix is scale-dependent and vanishes for systems with larger lengths. Classical modal analysis and perturbation method are used to understand the dynamic behaviour of discrete nonlocal systems in the light of classical local systems. The conditions for the existence of classical normal modes for undamped and damped nonlocal systems are established. Closed-form approximate expressions of nonlocal natural frequencies, modes and frequency response functions are derived. Results derived in the paper are illustrated using examples of axial and bending vibration of nanotubes and transverse vibration of graphene sheets.

A two-variable first-order shear deformation theory coupled with surface and nonlocal effects for free vibration of nanoplates

A two-variable first-order shear deformation theory in combination with surface free energy and small scale (size-effect) is employed to present a simple and computationally efficient formulation for the free vibration of nanoplates with arbitrary boundary conditions. The free surfaces are modeled as two-dimensional membranes adhering to the underlying bulk material without slipping. To take into account the small scale effect, the nonlocal constitutive relations of Eringen are used. The equations of motion and the related boundary conditions are derived by employing Hamilton’s principle. An analytical solution for the free vibration of simply supported nanoplates is obtained and comparison studies with the results of other two-dimensional theories available in the open literature are performed to validate the proposed formulation. Then, by using the differential quadrature method, an approximate solution for nanoplates surrounded by an elastic media and with arbitrary boundary conditions is developed. Consequently, the effects of surface free energy, the small scale parameter and elastic constant of surrounding media together with geometrical parameters on the natural frequencies of nanoplates are investigated.

An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity

Eringen nonlocal theory is employed in Mindlin plate theory to consider small scale effects on free vibration of rectangular nano-plates. Introducing some auxiliary and potential functions, an exact analytical procedure is applied on the governing equations to decouple the displacement variables. It is believed that this method is new for solving vibration of nano-plates. The solution of natural frequencies is obtained for Levy-type boundary conditions (two opposite edges simply supported and the others arbitrary). In order to confirm the reliability of the method considered, the results are compared with several reported literature. The effect of nonlocal parameter is investigated on natural frequency of the nano-plate for different boundary conditions. Finally the influence of aspect ratio and thickness to length ratio on natural frequency is studied in detail. It is expected that results obtained in this paper serve as an accurate reference in future nano-structures issues.

Size dependent forced vibration of nanoplates with consideration of surface effects

In this article, an analytical method is presented to study the size dependent forced vibration of rectangular nanoplates under general external loading using a generalized form of Kirchhoff plate model. The effects of surface properties including surface elasticity, surface residual stresses and surface mass density are considered which are bases for size dependent behaviors due to increase in surface to volume ratios at smaller scales. At first, a complete discussion is given for size dependent natural frequencies which are then used in forced vibration analyses. It is shown that the surface properties compact the frequency spectrums of the nanoplates. Saving generality and using the superposition principle, closed form solution is derived for time response of nanoplates under general harmonic loads. As a result, some elliptic curves are obtained for which the surface properties will not change the time response of nanoplates when a point load is applied on any point of these curves. It is observed that, for actuations inside these ellipses, the surface effects reduce the vibration amplitude while increase it for actuations outside the curves. Sensitivity of the problem to the excitation frequencies is also studied and various examples are given to illustrate the trend of size dependencies.

Free vibration of nanoplates based on a nonlocal two-variable refined plate theory

The application of a two-variable refined plate theory is extended to the free vibration of nanoplates. The theory takes into account the effects of small scale and the quadratic variation of the transverse shear strains through the thickness of the nanoplate and hence, it does not require the use of shear correction factors. The equations of motion and the related boundary conditions are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined plate theory via Hamilton’s principle. The obtained equations are only dynamically coupled. Exact solution for the simply supported nanoplates is obtained to verify the theory by comparing its results with other available solutions in the open literature. Then, using the differential quadrature method (DQM) as an accurate and computationally efficient numerical method, the effect of small scale parameter on the frequency parameters of the nanoplates with some different types of boundary conditions are studied.

Forced transverse vibration of nanoplates using nonlocal elasticity

In the present study, forced vibration of nano-plates is studied using the nonlocal elasticity theory. The Navier type solution method is used for simply supported nano-plates. Results are given for different nonlocality parameter, different length of plates and different loading conditions. The results show that the nonlocality effects should be considered for nano-scale plates. The nonlocality effects increase with increasing mode number. The present result can be used for single layer graphene sheets.

Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory

This article deals with the small-scale effect on the nonlinear free vibration of orthotropic single-layered graphene sheets using the nonlocal elasticity plate theory. The formulations are based on the Mindlin plate theory, and von Karman-type nonlinearity is considered in strain displacement relations. Virtual work principle is used to derive the nonlinear nonlocal plate equations in which the effects of rotary inertia and transverse shear are included. The differential quadrature method is employed to reduce the governing nonlinear partial differential equations to a system of nonlinear algebraic eigenvalue equations. The efficiency and accuracy of the method are demonstrated by comparing the developed result with those available in literature. The methodology is capable of studying large-amplitude vibration characteristics of nanoplates with different sets of boundary conditions. The effects of various parameters on the nonlinear vibrations of nanoplates are presented.

Mechanisms of nonlocal effect on the vibration of nanoplates

This letter presents a study of the mechanisms of nonlocal effect on the transverse vibration of two-dimensional (2D) nanoplates, e.g., monolayer layer graphene and boron-nitride sheets. It is found that such a nonlocal effect stems from a distributed transverse force due to (1) the curvature change in the nanoplates and (2) the surface stress due to the nonlocal atom-atom interaction. A single equivalent vibration wavelength is defined to measure the nonlocal effect on the vibration of 2D nanoplates. The critical equivalent wavelength of order 0.55 to 2.23 nm is obtained for significant nonlocal effect on monolayer graphene.

Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory

In the present study, buckling and vibration of nanoplates are studied using nonlocal elasticity theory. Navier type solution is used for simply supported plates and Levy type method is used for plates with two opposite edge simply supported and remaining ones arbitrary. Results are given for different nonlocality parameter, different length of plates and different boundary conditions. The results show that nonlocality effects should be considered for nanoscale plates. Clamped boundary conditions are more sensitive to nonlocality effects. In the vibration problem nonlocality effects increase with increase in the mode number. Present result can be used for single layer graphene sheets.

Size dependent dynamic analysis of nanoplates

Determination of accurate behavior of nanostructures is important in nanoelectromechanical systems in view of the shortcoming of classical theories of mechanics. In this study size-dependent transverse vibration of nanoplates is investigated. Surface properties which include surface elasticity and residual stresses are taken into account. Elasticity modulus of the bulk part is considered dependent on size as well as temperature. Kirchhoff theory of laminated plates is used to derive the governing differential equation of nanoplate structure.

Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model

In the present paper, small-scale effects on the free in-plane vibration (FIV) of nanoplates are investigated employing nonlocal continuum mechanics. Equations of motion of the nonlocal plate model for the aforementioned study are derived and presented. Explicit relations for natural frequencies are obtained through direct separation of variables. It has been shown that nonlocal effects are quite significant in in-plane vibration studies and need to be included in the continuum model of nanoplates such as in graphene sheets.