Vibration Of Nanoplates Research Articles

The role of nonlocality on low and high frequency behaviors of functionally graded sandwich nanoplates

AbstractThe role of nonlocality on low and high frequency behaviors and modes of the functionally graded (FG) sandwich nanoplates is investigated in this study for the first time using simple nonlocal higher‐order shear deformation theory (HSDT). The simple HSDT consists of only four unknowns in its equation of the displacement field. The nonlocal elasticity theory is used to consider the small‐scale effects on the behaviors of the nanoplates. The governing equations of motion are established by applying Hamilton's principle, then Navier's solution technique is employed to solve these equations to achieve the free vibration behaviors of the FG sandwich nanoplates. The vibrations of the nanoplates under both low and high frequency conditions are investigated, but the high frequency vibration of the nanoplates is discussed extensively. The influences of the geometrical dimensions, material gradient index, and nonlocal parameter on the high frequency vibration of the FG nanoplates are also considered and discussed comprehensively.

Mechanical responses of nanoplates resting on viscoelastic foundations in multi-physical environments

This paper presents an analysis of the static bending, thermal buckling, free vibrations, and forced vibrations of nanoplates using analytical approaches. This study introduces analytical formulations for all three aforementioned difficulties concerning nanoplates, taking into account the simultaneous impact of elements such as magnetic effects, temperature, and viscoelastic basis. The research at hand is characterized by its intricate nature and captivating subject matter, as it explores the phenomenon of the flexomagnetic impact on a plate. Furthermore, the plate is exposed to both mechanical and magnetic stresses concurrently inside a temperature-controlled environment. The plate is supported by a viscoelastic foundation, with the drag coefficient of the foundation being explicitly specified. The mathematical methods used in this study are derived from classical plate theory. An explicit expression is derived for the natural oscillation frequency, critical load resulting from thermal buckling, as well as the static and dynamic displacements of the plate, taking into account the drag coefficient offered by the foundation. The study further conducts a quantitative analysis to investigate the impact of several parameters, including temperature, viscoelastic foundation, and natural frequency of external stimulation force, on the mechanical behaviors of nanoplates. The findings of this study provide a scientific foundation for addressing practical challenges in the design, production, and use of nanoplates.

A Kirchhoff Plate Model for Longitudinal Vibration Analysis of Restrained Nanoplate Including Thermal Effects

This work attempts to apply the Kirchhoff plate theory to find out the vibrational analyses of a nanoplate incorporating thermal effects. The effects of thermal environments on the natural frequency of longitudinal vibration of restrained nanomaterials, especially for restrained nanoplates, have not been investigated, and most of the previous research has been carried out for unrestrained nanoplates. Therefore, it must be emphasized that the vibrations of restrained nanoplate, including thermal effects, are novel and applicable to the nanodevices, in which nanoplates act as the main structure of the nanocomposite. A novel motion and frequency equation are derived using the Kirchhoff plate model. The present study illustrates that a nanoplate’s longitudinal vibration characteristics strongly depend on the temperature change and stiffness coefficients. The numerical results clearly show that the longitudinal natural frequencies of the nanoplate are less than unity for both cases of low and high temperatures. This means that applying the Kirchhoff plate model for restrained nanoplate analysis would lead to an over-prediction of the frequency if the small thermal stress effect is neglected. Finally, the investigation of the restrained and thermal impact on longitudinal vibration of nanoplates may be used as a valuable reference for the application and the design of nanoelectronics and nano-drives devices, nano-oscillators, and nano-sensors, in which nanoplates act as essential elements.

Longitudinal Vibrations of Restrained Irregular Nanoplates

The main purpose of this work is to examine how surface irregularity affects the longitudinal vibration frequency of nanoplates. Based on the Kirchhoff plate model, the boundary conditions and governing equations are developed. The MATLAB R2013a Software is utilized to carry out the numerical solutions, and an accurate solution is shown. We derive a novel equation of motion as well as a new frequency equation. Examined is how several factors, such as thickness, stiffness, and surface irregularity, affect the longitudinal vibration of nanoplates. The studies revealed that from large to small irregularity parameters, the longitudinal natural frequencies of nanoplate increase. Additionally, it has been discovered that increasing the thickness of nanoplates is preferred for enhancing their vibration stability.

Analysis of Flexural Vibrations of a Piezoelectric Semiconductor Nanoplate Driven by a Time-Harmonic Force.

The performance of devices fabricated from piezoelectric semiconductors, such as sensors and actuators in microelectromechanical systems, is superior; furthermore, plate structures are the core components of these smart devices. It is thus important to analyze the electromechanical coupling properties of piezoelectric semiconductor nanoplates. We established a nanoplate model for the piezoelectric semiconductor plate structure by extending the first-order shear deformation theory. The flexural vibrations of nanoplates subjected to a transversely time-harmonic force were investigated. The vibrational modes and natural frequencies were obtained by using the matrix eigenvalue solver in COMSOL Multiphysics 5.3a, and the convergence analysis was carried out to guarantee accurate results. In numerical cases, the tuning effect of the initial electron concentration on mechanics and electric properties is deeply discussed. The numerical results show that the initial electron concentration greatly affects the natural frequency and electromechanical fields of piezoelectric semiconductors, and a high initial electron concentration can reduce the electromechanical fields and the stiffness of piezoelectric semiconductors due to the electron screening effect. We analyzed the flexural vibration of typical piezoelectric semiconductor plate structures, which provide theoretical guidance for the development of new piezotronic devices.

Surface stress size dependency in nonlinear free oscillations of FGM quasi-3D nanoplates having arbitrary shapes with variable thickness using IGA

The present study aims to propose a computational package for analyzing the geometrically nonlinear large-amplitude vibrations of nanoplates having arbitrary shapes with variable thickness in the presence of surface stress type of size effect. Accordingly, isogeometric analysis (IGA) is employed to achieve exact geometrical description as well as higher-order efficient smoothness with no meshing difficulty. The associated size-dependent plate model is constructed for the first time via implementation of the Gurtin–Murdoch surface elasticity to a quasi-3D plate model having the capability to take the thickness stretching into consideration with only four variables. The nanoplates are assumed made of functionally graded materials (FGMs) through the variable thickness, the displacement along which is calculated independently via a trigonometric normal shape function. The variation of plate thickness obeys three different schemes including linear, concave, and convex ones. It is highlighted that by changing the pattern of the thickness variation from convex type to linear one, and then from linear type to concave one, in spite of the higher classical flexural stiffness, the surface elastic-based flexural stiffness of FGM composite nanoplates gets lower, which results in a lower nonlinear frequency corresponding to a very thin nanoplates. Moreover, it is portrayed that by increasing the value of material gradient index, the role of stiffer character of the surface stress in the nonlinear free oscillation behavior of FGM composite nanoplates is pronounced. This anticipation is the same for the both initial and deeper parts of the nonlinear frequency-deflection response.

A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation

This article proposes a finite element method (FEM) based on a quasi-3D nonlocal theory to study the free vibration of functionally graded material (FGM) nanoplates lying on the elastic foundation (EF) in the thermal environment. By applying Hamilton's principle, the governing equations of FGM nanoplates on the EF are obtained. Using the FEM helps solve many complicated problems that analytical solution (AS) cannot be performed yet, such as complex structures, asymmetric problems, variable thickness, etc. The numerical results of this work are compared with those of other published researches to verify accuracy and reliability. In addition, the effects of geometrical parameters, material properties such as the thickness, material exponents, nonlocal coefficients, elastic foundation stiffness, boundary conditions (BCs), and temperature on the free vibration of nanoplates are comprehensively investigated.

A rigorously analytical exploration of vibrations of arbitrarily shaped multi-layered nanomembranes from different materials

The chief aim of this paper is to present a novel analytical solution for vibrational scrutiny of multi-layered composite nanomembranes of any shape. Both the nonlocality and surface effects are incorporated into the governing equations based on the idea of the deflected nanomembrane’s curves with identical deflections. Irrespective of the regular equations of motion of nanomembranes whose stiffness terms are generally expressed as the partial differential of the unknown fields to the independent components of the coordinates system, the newly developed ones are ordinary differential equations, which are exclusively stated as a function of a single parameter. This is the main characteristic of the curves with equal deflection, and thereby, the established methodology is called isodeflection curve approach. Through presenting several examples, the characteristic equations associated with annular, elliptic, and parabolic multi-layered nanomembranes are derived and solved for the natural frequencies of synchronous vibration modes. With some efforts, the proposed analytical solution can also be extended to free transverse vibrations of nanoplates as well as other complex multi-layered nanocomposite structures.

Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

In this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

Two Differential Equations for Investigating the Vibration of Conductive Nanoplates in a Constant In-Plane Magnetic Field Based on the Energy Conservation Principle and the Local Equilibrium Equations

In this paper, we investigate the movement of nanoplates using two approaches: extracting its differential equation, and extracting an integral equation based on the energy conservation principle. To extract the differential equation describing the free vibration of nanoplates in constant in-plane magnetic fields, we first use the theories developed by Kirchhoff and Mendelian to investigate the deformation of the nanoplates. Then, we use Lorentz force to calculate the electromagnetic force, and we use the Eringen’s non-local theory to consider the non-local effects. The extracted equation has an exact solution for calculating the natural frequency of rectangle nanoplates with simply support boundary conditions. To extract the differential equation based on the energy conservation principle, we calculate the stresses based on local equilibrium equations. These stresses are then used to discover the relationship between inner moments and mid-plane deformation. After that, based on the energy conservation principle, an equation describing the vibration is obtained. Finally, based on the extracted equation, the curvatures are calculated so that the Eringen’s non-local theory is satisfied. These curvatures are used to calculate the elastic potential energy and rate of work done for the applied magnetic field. For a rectangular plate with simply support, the results indicate that the two equations are consistent with each other in predicting the frequency. However, as the power of the applied field increases, the existence of magnetic viscosity is predicted, and the difference between the results of these equations will become significant.

Applying nonlocal strain gradient theory to size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoplates

In this research paper, as initial endeavors, the vibrational responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates taking into account the effect of nonlocal parameter and strain gradient coefficient are investigated. The study aims at developing mathematical modeling via an analytical solution to FG-CNTRC nanoplate structure with allowance for the nonlocal strain gradient effect. The four types of CNT distribution are used and compared in the context of the vibration of nanoplate in the presence of the small length scale effects, namely the (a) UD, (b) FG-V, (c) FG-O, and (d) FG-X. Some theoretical equations based on the first-order shear deformation plate theory (FSDT) are presented to provide a lucid understanding of how the small length-scale influences the FG-CNTRC nanoplate. For the vibrational analysis of a nanoplate, which is simply supported boundary condition, Navier solutions are obtained. Also, in contrast to earlier studies, an analytical approach is used to establish the governing equations of the FG-CNTRC nanoplate. Some specific numerical examples are given and compared with the results presented in the literature. In the section of numerical results, the influence of the nonlocal parameter, strain gradient coefficient, geometric parameters and vibrational modes on the non-dimensional natural frequency are investigated and discussed in detail. These could be useful to analysts and designers to estimate the fundamental natural frequencies in each of the four CNT distributions that the FG-CNTRC nanoplate possesses.

On size-dependent mechanics of nanoplates

In this paper, a two-dimensional stress-driven nonlocal integral model is introduced for the bending and transverse vibration of rectangular nanoplates for the first time. An appropriate kernel function, which satisfies all essential properties, is proposed for two-dimensional problems in the Cartesian coordinate system. Using Leibniz integral rule and Hamilton's principle, the curvature-moment relations, classical and constitutive boundary conditions, as well as the equations of motion of rectangular small-scale plates are derived. Two differential quadrature techniques are utilised to implement both classical and non-classical boundary conditions and obtain an accurate numerical solution. The solution is used to simulate the bending and vibration of nanoplates. The Laplacian-based nonlocal strain gradient model of plates is also developed for the sake of comparison. It is found that the stress-driven integral model can better estimate the size-dependent mechanical characteristics of small-scale rectangular plates with various boundary conditions.

An integral nonlocal model for the free vibration analysis of Mindlin nanoplates using the VDQ method

Eringen’s nonlocal theory (ENT) provides an efficient tool to consider the size dependency for the structural analysis of small-scale structures. Recently, it has been indicated that in some cases, inconsistent results may be obtained using the Eringen differential model (EDM) and Eringen integral model (EIM) should be employed. In this regard, the vibration of nanoplates resting on the elastic medium is investigated on the basis of the EIM using variational differential quadrature (VDQ) method. On the basis of the first-order shear deformation theory (FSDT), the EIM and EDM formulations are given. An effective numerical technique is applied in the framework of the energy method to find the natural frequencies. Comprehensive results are reported to compare the EIM and EDM. The results show that using the proposed integral model, the paradox related to cantilever nanoplates is resolved.

Buckling and vibration analysis of FG-CNT-reinforced composite rectangular thick nanoplates resting on Kerr foundation based on nonlocal strain gradient theory

This paper investigates the buckling and free vibration analysis of functionally graded carbon nanotube-reinforced composite thick rectangular nanoplates resting on a Kerr foundation under different boundary conditions. Quasi-three-dimensional hyperbolic shear deformation theory is employed to study the effects of transverse shear deformation and thickness stretching. To capture the small-size effects of nanoscale dimensions, the nonlocal strain gradient theory is used, which includes nonlocal parameters and length scale of the material. In this study, rectangular nanocomposite plates are reinforced by carbon nanotubes which are assumed to be graded through the thickness direction with four types of distributions, namely, uniformly, FG-O, FG-V, and FG-X. The governing equations and boundary conditions are extracted within Hamilton’s principle. They are discretized and numerically solved by utilizing a generalized differential quadrature method. The critical buckling loads and natural frequencies are determined by solving the eigenvalue problem. The accuracy of present results is validated with those available in the literature. Also, the effect of various factors, such as aspect ratio, length-to-thickness ratio, in-plane loading factor, length scale parameter, nonlocal parameter, volume fraction and dispersion profile of carbon nanotubes, elastic foundation coefficients, and different boundary conditions, on the buckling behavior and free vibration of nanoplates is investigated.

Natural vibration of stepped nanoplate with crack on an elastic foundation

The small scale effect on the vibrational characteristic of isotropic, rectangular nanoplate embedded in an elastic medium is investigated. The formulation is based on the plate theory on aggregate with the nonlocal elasticity theory. The solution procedure is derived using the governing differential equations of physical phase that are converted into set of linear algebraic equations. Latter these are solved by computer code. The effects of aspect ratio, step, crack and rotatory inertia on the different modal vibrations of nanoplate are explored. The results show the significant effect of different physical and geometrical parameters on the vibration of nanoplate.

A Novel Refined Plate Theory for Free Vibration Analyses of Single-Layered Graphene Sheets Lying on Winkler-Pasternak Elastic Foundations

In this article, the analyses of free vibration of nanoplates, such as single-layered graphene sheets (SLGS), lying on an elastic medium is evaluated and analyzed via a novel refined plate theory mathematical model including small-scale effects. The noteworthy feature of theory is that the displacement field is modelled with only four unknowns, which is even less than the other shear deformation theories. The present one has a new displacement field which introduces undetermined integral variables, the shear stress free condition on the top and bottom surfaces of the plate is respected and consequently, it is unnecessary to use shear correction factors. The theory involves four unknown variables, as against five in case of other higher order theories and first-order shear deformation theory. By using Hamilton’s principle, the nonlocal governing equations are obtained and they are solved via Navier solution method. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all examined. From this work, it can be observed that the small-scale effects and elastic foundation parameters are significant for the natural frequency.

Buckling and Free Vibrations of Nanoplates—Comparison of Nonlocal Strain and Stress Approaches

The buckling and free vibrations of rectangular nanoplates are considered in the present paper. The refined continuum transverse shear deformation theory (third and first order) is introduced to formulate the fundamental equations of the nanoplate. Besides, the analysis involve the nonlocal strain and stress theories of elasticity to take into account the small-scale effects encountered in nanostructures/nanocomposites. Hamilton’s principle is used to establish the governing equations of the nanoplate. The Rayleigh-Ritz method is proposed to solve eigenvalue problems dealing with the buckling and free vibration analysis of the nanoplates considered. Some examples are presented to investigate and illustrate the effects of various formulations.

Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory

In this paper, a new model for studying the effects of small-scale parameters simultaneously, on large amplitude vibrations of sandwich plates is developed using the higher-order nonlocal strain gradient theory. Considering the higher-order theories for capturing the size effects of nanostructures results in a set of nonlinear partial differential (PD) equations, including bi-nonlocal terms. By employing Hamilton's principle, the equations of motion for symmetric and anti-symmetric sandwich plates are derived based on the classical plate theory. The partial nonlinear differential equations of motion are reduced to an ordinary differential equation for transverse vibrations of nanoplates using the Galerkin's method. An analytical solution procedure is employed to obtain the closed-form frequency equation as a function of the vibration amplitude, small-scale parameters and sandwich layers elasticity, density and thickness coefficients. Numerical results are presented in order to investigate the sandwich layers coefficients on nonlinear vibrational behavior of nanoplates as same as small-scale parameters and the amplitude of vibrations. It is found that the vibration amplitude plays the main role in nonlinear vibrational behavior of nanoplates in which, nonlinear frequency and its ratio to linear frequency will be increased by increasing it. Moreover, there are non-uniform behaviors by increasing the sandwich layers coefficients and small-scale parameters. In addition, in the case of large amplitude vibrations, effects of sandwich layers’ coefficients and small-scale parameters on the nonlinear frequency and its ratio to linear frequency will become more noticeable. In order to validate the present solution procedure, the results are compared with those obtained from molecular dynamics simulations, the higher-order nonlocal strain gradient theory and the higher-order shear deformation plate theory.

How does porosity affect the free vibration of single-layered graphene sheets?

This paper aims to investigate the influence of porosity and length size on the free vibration of single-layered graphene sheets (SLGSs). Frequency analysis is performed using a finite element based molecular structural mechanics (MSM) approach mimicking the SLGSs as frame-like structures constructed out of the beam elements. Defining a porous unit cell, 320 SLGSs with different arrangements and values of porosities and various length sizes ranging from 4 to 32 nm are considered. Results reveal that increasing porosity as well as length size both decrease the natural frequencies of SLGSs, significantly. To improve the applicability of the results, a nonlocal small scale parameter introduced by the analytical solutions for vibration of nanoplates in the literature is calibrated in such a way that the obtained frequencies by MSM match the analytical solutions based on the nonlocal theory of elasticity. Both neural network and genetic programming processes are successfully implemented for the calibration. The proposed calibrated parameter can be easily applied to evaluate the natural frequencies of SLGSs for certain values of porosities and length sizes.

Nonlinear size-dependent vibration behavior of graphene nanoplate considering surfaces effects using a multiple-scale technique

ABSTRACTIn this article, a multiple-scale perturbation method is employed to analyze nonlinear free vibration of nanoplate incorporating surface effects. Eringen's nonlocal theory as well as surface elasticity theory of Gurtin and Murdoch is used to consider small scale effect by presenting the nonlocal parameter and the surface effects at the top and bottom of the bulk part of the nanoplate as a membrane with different mechanical properties, respectively. A multiple-scale perturbation method is suggested to consider the expansion representing the response to be a function of multiple independent variables, or scales, instead of a single variable. It also should be mentioned that graphene sheets are considered suitable candidates in nanoplates to increase the mechanical properties of the nanostructures. Employing the Hamilton's principle, the three coupled nonlinear equations of motion are obtained based on Hooke's law and the von Kármán nonlinear strain–displacement relationship. To show the accuracy of the presented method results are compared with literature and a good agreement is observed. Effects of two kinds of boundary conditions SSSS1 and SSSS2, as well as nonlocal parameter, surface residual stress and surface elastic modulus parameter on nonlinear frequency of nanoplate are analyzed. The results illustrate that by increasing the nonlocal parameter the nonlinear frequency shows a decreasing behavior while by increasing the surface residual stress and surface elastic modulus parameter, nanoplate's stiffness increases therefore the nonlinear frequencies increase. Also, it can be mentioned that the surface effects have very small effects on nonlinear frequencies and can be ignored. Therefore, they don't play an important role on the nonlinear fundamental frequencies of nanoplate.