Double-layered Nanoplates Research Articles

Effects of in-phase and anti-phase large amplitude nonlinear models for double-layer nanostructures

Double layer nano plates have magnificent applications in nanostructures such as, nano sensors, nano actuators and nano transistors. So, the main aim of the presented study is to propose an analytical model based on Reddy’s third-order shear deformation theory of plates which has a difficult procedure due to the great number of partial differential equations and nonlocal strain gradient theory. Due to the fact that numerical approaches aren’t good enough accurate and need more time to calculate, the multiple scales method which is a useful method in systems characterized by disparate time scales and in a variety of situations for extracting the slow time dependence of patterns or other systems is adopted. Based on von Karman equations as well as the theory of nonlocal strain gradient, the coupled nonlinear equations are derive. Compared to the other related published literature, acceptable agreement can be seen under different boundary conditions. Effect of size depended parameter on in-phase and anti-phase modes considering the SSSS and CCCC boundary condition is indescribable which means by increasing the nonlocal parameter nonlinear frequency and also the natural frequency of nanostructure reduces.

Comparing magnitudes of surface energy stress in synchronous and asynchronous bending/buckling analysis of slanting double-layer METE nanoplates

This article explores the magnitudes of surface energy stress in the synchronous and asynchronous bending/buckling analyses of skew double-layer magneto-electro-thermo-elastic (DLMETE) nanoplates, focusing on the effects of such main parameters as external electric potential, external magnetic potential, elastic foundations, skew angle, and temperature change on the rate of surface energy stress in the synchronous and asynchronous bending/buckling behaviors of skew DLMETE nanoplates. The two nanoplates examined are coupled via an elastic medium, termed van der Waals forces, while they are surrounded by Winkler and Pasternak foundations. Moreover, the two nanoplates are assumed to move in the same direction in the asynchronous mode but in opposite directions in the synchronous one. The refined plate, surface energy, and nonlocal hypotheses are manipulated to expand the governing equations, while the principle of virtual work is exploited to derive the relevant equilibrium equations. The equations thus obtained are solved using the Galerkin method and further validated via the Navier’s method. It is observed that the effects of surface energy stress on the different parameters of synchronous and asynchronous buckling behaviors of DLMETE nanoplates are radically different from those on their bending behavior.

Nonlocal nonlinear chaotic and homoclinic analysis of double layered forced viscoelastic nanoplates

The homoclinic phenomena and chaotic motions of the forced double layered nanoplates (DLNP) are investigated. By the nonlocal theory, the nonlinear equations of motion of DLNP subjected to transverse harmonic excitation are established. The buckled DLNP considered herein means that the parameter regime of the first mode are always unstable. It should be emphasized that the homoclinic Melnikov function is related to the nonlinear term which is affected by the boundary conditions. Hence two different boundary conditions, i.e. simply supported with movable and immovable edges are compared herein. The extended Melnikov method is employed to discuss the homoclinic phenomena and chaotic motions of the DLNP system. The criteria for the homoclinic motions of the four buckling cases are established. Then the results by the above global perturbation analyses are verified by the numerical integration evidences including Lyapunov exponential spectrums, Fourier spectrums and Poincaré sections. The influences of the structural parameters such as small scale effect and boundary conditions on the homoclinic behaviors are mainly discussed. From the results, the most remarkable fact can be seen that the rotary inertia term could break the symplectic symmetry of the unperturbed vibration system. The parametric regime where the chaotic motion may appear shrink with the increase of the nonlinear term r1, which means chaotic motion more likely appear for immovable edges than those with movable edges. Parametric regime where the transverse homoclinic phenomenon appears will decrease with the increase of the small scale parameter. And the homoclinic phenomena more likely appear in higher mode vibration.

Coupled effect of in-plane magnetic field and size effect on vibration properties of the completely free double-layered nanoplate system

The coupled effects of an in-plane magnetic field and the size effect on free vibration of a completely free magnetically affected orthotropic double-layered nanoplate system (DLNS) embedded in elastic media are examined by a Hamiltonian-based method combined with Eringen's nonlocal elasticity theory. The Lorentz magnetic force exerted on the DLNS is derived based on Maxwell's equations and Lorentz's formula. By introducing a full-state vector as the primary unknown, the classical governing equation is converted into its Hamiltonian form so that exact solutions of the DLNS are obtained by symplectic expansion and a superposition of boundaries. Analytical frequency equation and vibration mode functions are obtained simultaneously. Numerical results illustrate that, by changing the magnetic parameter, the natural frequencies of the DLNS show two opposite variation trends with increasing nonlocal parameter; and the vibration modes are re-arranged. These findings demonstrate that applying an in-plane magnetic field is an efficient way to control the natural frequencies and vibration modes of such magnetically affected DLNS.

Nonlinear resonant behaviors of embedded thick FG double layered nanoplates via nonlocal strain gradient theory

This research deals with the nonlinear forced vibration behavior of embedded functionally graded double layered nanoplates with all edges simply supported via nonlocal strain gradient elasticity theory based on the Mindlin plate theory along with von Karman geometric nonlinearity. The interaction of van der Waals forces between adjacent layers is included. For modeling surrounding elastic medium, the nonlinear Winkler–Pasternak foundation model is employed. The exact solution of the nonlinear forced vibration for primary and secondary resonance through the Harmonic Balance method is then established. For the double layered nanoplate, uniform distribution and sinusoidal distribution of loading are considered. It is assumed that the material properties of functionally graded double layered nanoplates are graded in the thickness direction and estimated through the rule of mixture. The Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations. The effects of different parameters such as length scale parameters, elastic foundation parameters, dimensionless transverse force and gradient index on the frequency responses of functionally graded double layered nanoplates are investigated. The results show that the length scale parameters give nonlinearity of the hardening type in nonlinear forced vibration and the increase in strain gradient parameter causes to increase in maximum response amplitude, whereas the increase in nonlocal parameter causes to decrease in maximum response amplitude.

On buckling of porous double-layered FG nanoplates in the Pasternak elastic foundation based on nonlocal strain gradient elasticity

In the present investigation, the buckling behaviours of porous double-layered functionally graded nanoplates in hygrothermal environment are presented for the first time. The nonlocal strain gradient theory with two material scale parameters is developed to examine buckling behaviour much accurately. Based on the new first order shear deformation theory the equations of equilibrium are obtained from the principle of minimum potential energy. To simplify the equations of equilibrium and removing the bending-extension coupling, the buckling behaviours of FG nanoplates are investigated based on physical neutral surface concept. The equations of equilibrium are solved for various boundary conditions using Galerkin's method. The obtained results are compared with the results available in the literature to valid the correctness of present solution method. The effects of nonlocal parameter, strain gradient parameter, porosity volume fraction, power-law index, temperature change, humidity change and boundary conditions on critical buckling load are presented.

Electro-thermal buckling of elastically supported double-layered piezoelectric nanoplates affected by an external electric voltage

PurposeThermal buckling of double-layered piezoelectric nanoplates has been analyzed by applying an external electric voltage on the nanoplates. The paper aims to discuss this issue.Design/methodology/approachDouble-layered nanoplates are connected to each other by considering linear van der Waals forces. Nanoplates are placed on a polymer matrix. A comprehensive thermal stress function is used for investigating thermal buckling. A linear electric function is used for taking external electric voltages into account. For considering the small-scale effect, the modified couple stress theory has been applied. An analytical solution has been used by taking various boundary conditions.FindingsEEV has a considerable impacted on the results of various half-waves in all boundary conditions. By increasing EEV, the reduction of critical buckling temperature in higher half-waves is remarkably slower than lower half-waves. By considering long lengths, the effect of EEV on the critical temperature will be markedly decreased.Originality/valueThis paper uses electro-thermal stability analysis. Double-layered piezoelectric nanoplates are analyzed. A comprehensive thermal stress function is applied for taking into account critical temperature.

Coupled effects of electrical polarization-strain gradient on vibration behavior of double-layered flexoelectric nanoplates

A vibrating double-layered nanoscale piezoelectric plate is developed accounting for the flexoelectricity and surface effects. The flexoelectricity is due to the coupling between electrical polarization and strain gradient. Applying Hamilton\'s principle, the governing equations and related boundary conditions are derived. Assuming suitable approximate functions, the governing equations are numerically solved for simply-supported and clamped boundary conditions. Obtained results indicate that both the flexoelectricity and surface effects possess notable impact on the vibration frequencies of the system. Only flexoelectricity yields a considerable difference between the present model and previous investigations on conventional piezoelectric nanoplates. Generally, a parametric study has been performed to examine the effects of surface elasticity, flexoelectricity, applied electric voltage, interlayer stiffness, geometrical parameters and boundary conditions on vibration frequencies of piezoelectric nanoplates.

Nonlocal stress-strain gradient vibration analysis of heterogeneous double-layered plates under hygro-thermal and linearly varying in-plane loads

This paper develops a nonlocal strain gradient plate model for vibration analysis of double-layered graded nanoplates under linearly variable in-plane mechanical loads in hygro-thermal environments. For more accurate analysis of nanoplates, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. The double-layered nanoplate is modeled via a four-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient nanoplate on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations. Effects of different factors such as in-plane loading, load factor, nonlocal parameter, length scale parameter, moisture percentage rise, temperature rise, elastic foundation, and boundary conditions on vibration characteristics of a double-layered nanoplate have been examined.

Investigating dynamic characteristics of porous double-layered FG nanoplates in elastic medium via generalized nonlocal strain gradient elasticity

For the first time, a vibrating porous double-nanoplate system under in-plane periodic loads is modeled via the generalized nonlocal strain gradient theory (NSGT). Based on the proposed theory, one can examine both stiffness-softening and stiffness-hardening effects for a more accurate analysis of nanoplates. Nanopores or nanovoids are incorporated to the model based on a modified rule of mixture. Modeling of porous double-layered nanoplate is conducted according to a refined four-variable plate theory with fewer field variables than first-order plate theory. The governing equations and related classical and nonclassical boundary conditions are derived based on Hamilton’s principle. These equations are solved for hinged nanoplates via Galerkin's method. It is shown that porosities, nonlocal parameter, strain gradient parameter, material gradation, interlayer stiffness, elastic foundation, side-to-thickness and aspect ratios have a notable impact on the vibration behavior of nanoporous materials.

Magneto-hygro-thermal vibration behavior of elastically coupled nanoplate systems incorporating nonlocal and strain gradient effects

Magneto-hygro-thermal vibration analysis of double-layered nanoplates made of functionally graded materials is presented based on higher order refined plate theory. For the first time, a double-layered nanoplate is modeled via nonlocal strain gradient theory in which both stiffness-softening and stiffness-hardening effects are incorporated. Another novelty of this paper is that the effects of magnetic and hygro-thermal fields on inhomogeneous double-layered nanoplates are considered to study their behavior under different physical fields. The gradation of material properties is considered using power-law model. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton’s principle. These equations are solved for hinged nanoplates via Galerkin’s method. It is indicated that type of vibration, moisture rise, temperature rise, nonlocal parameter, strain gradient parameter, material gradation, elastic foundation and side-to-thickness have a remarkable influence on vibration behavior of double-layered nanoscale plates.

Dynamic modeling and vibration analysis of double-layered multi-phase porous nanocrystalline silicon nanoplate systems

Abstract Nanocrystalline materials (NcMs) are multi-phase composites containing nanograins, nanovoids and interface. A softening mechanism due to the interface phase and nanovoids is observed in NcMs. For the first time, vibration behavior of double-layered nanocrystalline silicon nanoplates on elastic substrate is analyzed based on a two-variable refined plate model. Due to the experimentally observation of grains micro-rotation and strain gradients near interfaces, the strain gradient based couple stress theory is employed to describe the size-dependent behavior of the nanocrystalline nanoplate. A micromechanical model is employed to incorporate the effects of inclusions and their surface energies. Galerkin's method is implemented to obtain the natural frequencies of nanocrystalline nanoplate with different boundary conditions. One can see that the nanograins size, nanograins surface energy, nanovoids size, void percentage, interface region, scale parameter, foundation constants and boundary conditions have a great influence on the vibration behavior of a double-layered nanocrystalline nanoplate.

Buckling Analysis of Double-Layer Piezoelectric Nanoplates Surrounded by Elastic Foundations and Thermal Environments Considering Nonlocal and Surface Energy Models

AbstractThis study investigated the effects of considering surface and nonlocal energy parameters on the buckling analysis of double piezoelectric nanoplate (DPNP) embedded in elastic foundations and thermal environments. Both in-phase and out-of-phase modes of buckling and various boundary conditions are studied and compared with each other. The governing equations were derived by drawing on the principle of virtual work and then solved by employing the finite difference method. Finite difference solution was validated using Navier's method and journal references. A parametric study was also launched in order to investigate the effects of the external electric voltage, nonlocal parameters, different boundary conditions, elastic foundations and thermal environments on the surface effect of DPNP buckling. The obtained numerical results showed that the influence of surface stress on in-phase and out-of-phase modes of buckling of the DPNP was enhanced by augmenting the nonlocal parameters and external electric voltage; on the other hand, it was found to be decreased by increasing elastic foundations and temperature changes. In addition, the value of surface stress effects for the in-phase mode was higher than that of the out-of-phase one.

Hygro-thermal vibration analysis of graded double-refined-nanoplate systems using hybrid nonlocal stress-strain gradient theory

A new dynamic modeling and free vibrational analysis of double-layered nanoplates made of functionally graded (FG) materials in hygro-thermal environments is presented for the first time. A better description of size-dependent phenomena is presented using a nonlocal stress-strain gradient theory. The double-layered nanoplate is subjected to hygro-thermal loading and it is resting on elastic medium. The gradation of material properties is considered using power-law model. Modeling of double-layered nanoplate is conducted according to a refined four-variable plate theory with fewer field variables than first order plate theory. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton’s principle. These equations are solved for hinged nanoplates via Galerkin’s method. It is indicated that type of vibration, moisture rise, temperature rise, nonlocal parameter, strain gradient parameter, material gradation, elastic foundation and side-to-thickness have a remarkable influence on vibration behavior of double-layered nanoscale plates.

Nonlinear responses and stability analysis of viscoelastic nanoplate resting on elastic matrix under 3:1 internal resonances

The nonlinear responses and stability of double-layered nanoplate embedded in the elastic medium are investigated in the presence of 3:1 internal resonance. By the nonlocal theory, the method of multiple scales is employed to obtain the analytical nonlinear frequency-response relations. Two different external primary resonance conditions, i.e. the first and the second modes being directly excited, are considered. The influences of the small scale effect and viscous damping on the nonlinear vibration are explored in details. From the results, the frequency-response curves for the two primary resonance cases present complete different characteristics. It should be noted that the response curves are closed loops for the resonance of the second mode, which implies the steady-state response just exists in a finite frequency range. The regions of multi-values appear for both cases and the stability of the response is determined. When the first mode is directly excited, the impact of the viscidity of nanoplate and small scale effect on the frequency range of unstable response is rather significant. Furthermore, when the second mode is directly excited, a novel phenomenon, i.e. the frequency range for the closed loops of response diminished enormously as the increase of the viscidity of nanoplate, can be observed.

Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings

The in-phase and out-of-phase natural frequencies of double-layer piezoelectric nanoplates (DPNP) are studied under thermo-electro-mechanical loadings using the surface layer and nonlocal elasticity theories. For this purpose, two nanoplates are coupled via an elastic medium that induces van der Waals forces while the DPNP is surrounded by the Winkler and shear moduli of elastic foundations. Both in-phase and out-of-phase fundamental frequencies are investigated and compared under different boundary conditions. The Hamilton’s principle is employed to derive the governing equations and the finite difference method is subsequently used to solve them. The finite difference solution thus obtained is validated against the results obtained from the Navier’s method. A different aspect of the study involves the investigation of the effects of nonlocal parameters, thermo-electro-mechanical forces, various boundary conditions, and elastic foundations on the surface layer vibration of a DPNP. It is observed that the effects of surface energy layers on the in-phase and out-of-phase vibrations increase with increasing values of the nonlocal parameter and external electric voltage but that they reduce with rising mechanical and thermal loadings, elastic foundation, and stiffness for the boundary conditions defined. The results of the present work are expected to be used in designing NEMS/MEMS components using smart composite nanostructures.

Surface effects on nonlinear dynamics of NEMS consisting of double-layered viscoelastic nanoplates

The nonlinear flexural vibration behavior of double-layered viscoelastic nanoplates including surface effects is investigated based on nonlocal elasticity theory. Using nonlinear strain-displacement relations, the geometrical nonlinearity is modeled. To derive the governing equations, nonlocal plate theory and Hamilton's principle are employed and also to obtain the nonlinear eigenvalues, the differential quadrature method (DQM) is utilized. In particular, surface effects, including surface elasticity, residual surface stress and surface density, are considered. The detailed parametric study is conducted, focusing on the influences of nonlocal effect, aspect ratio of the plate, elastic foundation, Van der Walls (vdW) interaction, temperature and the viscidity of the plate. The influence of the viscoelastic coefficient is also discussed. Results are compared and validated with available studies, and a good agreement is seen. After validation of the present study, various plots for the nonlinear-to-linear frequencies against amplitude-to-thickness ratio and thickness for double visco-nanoplates are presented.

Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle

In this paper, dynamic behavior of double layered nanoplate systems (DLNPS) with respect to a moving nanoparticle is investigated. Both layers of DLNPS are assumed to be orthotropic and each layer is bearing a biaxial load while internal damping effects are also taken into account. Furthermore, coupling between layers are modeled using Kelvin-Voigt viscoelastic theory and moving nanoparticles path are assumed to be linear and circular with constant velocities. Governing equations of motion are derived by using D'Alembert's principle, Kirchhoff-Love plate and Eringen's nonlocal theory. Galerkin's and Laplace transform methods is used to solve the governing equations and analytical solution is presented for linear moving nanoparticle while close-form solution is obtained for circular moving nanoparticle. In order to clarify the influence of different parameters such as small scale effect, stiffness and damping in coupling, biaxial compression and tension of layers, path of the moving mass, etc. on dynamic behavior of each layer, parametric study is presented. Accordingly, with the brand new discussions in moving atoms, molecules, nanocars, nanotrims, point loads on different nanosctructures using scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding such kind of behaviors.

Rigorous vibration analysis of double-layered orthotropic nanoplate system

A rigorous analytical symplectic method is introduced into the free vibration of a rectangular double-layered orthotropic nanoplate system. Eringen's nonlocal elasticity theory is taken to capture the small size effect. Meanwhile, it leads to a high-order differential governing equation in the Lagrangian system, which can only be analytically solved by the semi-inverse method with pre-determined trail functions in previous studies. To overcome this drawback, a Hamiltonian system is established by introducing a new total unknown vector consisting of the displacement amplitude, rotation angle, shear force and bending moment. The governing equation is reduced to a set of one-order ordinary differential equations so that they can be systematically solved by the method of separation of variables and the expansion of eigenfunctions. Analytical frequency equations are derived for the Levy-type edges and vibration modes are expressed in terms of the total unknown vectors. Comparison is presented to verify the accuracy of the symplectic method and comprehensive numerical examples are given also.

Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric double-layered nanoplates

The homoclinic phenomena in double-layered nanoplates (DLNP) are investigated. Based on the nonlocal continuum theory, the nonlinear dynamical equations for DLNP subjected to in-plane excitation are derived by double-mode Galerkin truncation. The extended Melnikov method is utilized to discuss the homoclinic phenomena and chaotic motion for the buckled DLNP system. The criterions for the existence of transverse homoclinic orbits are established under different four buckling cases (i.e., the first- and second-type synchronous buckling as well as asynchronous buckling). And the results derived by the above-mentioned analysis are verified by molecular dynamics and Lyapunov exponent spectrum. Small-scale effect on the homoclinic motion is mainly inspected. From the result, it is rather novel that the transversality condition is independent of the nonlinear terms in the equations. This fact means that it is not necessary to distinguish whether the boundaries are movable or immovable for the in-plane parametric excitation DLNP. The parametric regime where homoclinic phenomena appear shrinks with the augment of nonlocal parameter for the first-type synchronous buckling case. However, this trend is just opposite for the other three buckling cases. Finally, it can be seen that homoclinic phenomena more likely occur on Mode I (i.e., lower mode) for the second-type synchronous and asynchronous buckling cases. However, the homoclinic motion for the first-type asynchronous buckling most likely takes place on Mode III.