Nonlocal Mindlin Plate Research Articles

Transverse Free Vibration of Axisymmetric Functionally Graded Circular Nanoplates with Radial Loads

The functionally graded circular nanoplate is a commonly seen component in the nano-electromechanical system. It is indispensable to examine free vibration behaviors of such an axisymmetric nanostructure subjected to uniformly distributed loads in the radial direction. Although the vibration engineering and technology have been fully studied at the macro-scale, there are still many unsolved problems at the micro-scale. The present research aims to promote the theoretical characterization of vibration behaviors at micro/nano-scale and further provide a basis for the development of vibration testing technologies. Using the nonlocal strain gradient approach and Mindlin plate theory, we develop the theoretical model describing the transverse free vibration of the axisymmetric functionally graded circular nanoplate. First, by considering the nonlocal strain gradient constitutive relation, we derive the equation of motion via Hamilton’s principle in polar coordinate system. Subsequently, the differential quadrature method is employed to solve the equation of motion numerically. The natural frequencies of circular nanoplates decrease with an increase of a radial compression while increase with an increase of a radial tension. The first-mode natural frequency reduces to zero under a certain radial compression, bringing about the dynamical instability. The natural frequencies are sensitive to the radial compression, and the clamped boundary constraint is more resistant to external loads than the simply supported one. An increase in the nonlocal parameter results in lower natural frequencies, while an increase in the strain gradient characteristic parameter results in higher ones. It is demonstrated that the strain gradient characteristic parameter has a threshold in the present model for functionally graded circular nanoplates. In the circumstance of a lower nonlocal parameter than the strain gradient characteristic parameter, the circular nanoplate shows hardening behaviors. In the circumstance of a greater nonlocal parameter, the circular nanoplate shows softening behaviors. When the two internal characteristic parameters are equal, the stiffness of nanoplates remains unchanged and degenerates into its classical counterpart.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

Buckling and vibration analysis nanoplates with imperfections

In the present paper a coupling finite strip–finite element procedure is developed to investigate the buckling and vibration behaviour of imperfect nanoplates via nonlocal Mindlin plate theory. The imperfection can be either a thickness variation or a lack of planarity and it can be either localized or distributed on an entire edge of the nanoplate. The resulting nonlinear equations are solved exactly by applying the Kantorovich method. A finite element approach is proposed for coupling the in-plane and the out-of-plane buckling equations to describe properly the imperfections. Some numerical examples are carried out in order to show the sensitivity of the results to the nonlocal parameter and to the imperfection.

Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory

In the current paper, the sensitivity performance of functionally graded magneto-electro-elastic (FG-MEE) nanoplate with attached nanoparticles as a nanosensor is analyzed based on nonlocal Mindlin plate assumption. Power law distribution model is employed to display how the material properties of FG-MEE nanoplate vary across the thickness direction. It is supposed that FG-MEE nanoplate is under initial external electric and magnetic potentials. Boundary condition of each edge of FG-MEE nanoplate is assumed to be simply supported. Furthermore, a Pasternak substrate is applied for modelling the total reaction pressure between nanoplate and foundation. Partial differential equations and corresponding boundary conditions are first achieved using Hamilton’s variational principle and then analytically solved to determine the frequency shift utilizing Navier’s approach. Numerical examples are performed to elucidate the dependency of the sensitivity performance of FG-MEE nanosensor on the volume fraction exponent, nonlocal parameter, total attached mass and location of the nanoparticle, aspect ratio, mode number, initial external electric voltage, initial external magnetic potential, and Pasternak medium coefficients. It is clearly indicated that these factors have highly significant impacts on the variations of frequency shift.

Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories

This paper is concerned with the dynamic stability response of an embedded piezoelectric nanoplate made of polyvinylidene fluoride (PVDF). In order to present a realistic model, the material properties of nanoplate are assumed viscoelastic using Kelvin–Voigt model. The visco-nanoplate is surrounded by viscoelastic medium which is simulated by orthotropic visco-Pasternak foundation. The PVDF visco-nanoplate is subjected to an applied voltage in the thickness direction. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton’s principle. A novel numerical method namely as differential cubature method (DCM) in conjunction with the Bolotin’s method is applied to obtain the dynamic instability region (DIR) and parametric resonance responses of the visco-nanoplate. The effects of different parameters such as nonlocal parameter, external electric voltage, structural damping, boundary condition, dimension of nanoplate and viscoelastic medium are shown on DIR and parametric resonance frequency of structure. The accuracy of the proposed method is verified by comparing its numerical prediction with other theoretical and experimental published works as well as solution of system with differential quadrature method (DQM). Results indicate that the external electric voltage increases the frequency of the system especially in thick nanoplates.

Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method

In the present study a continuum model based on the nonlocal elasticity theory is developed for free vibration analysis of embedded orthotropic thick circular and elliptical nano-plates rested on an elastic foundation. The elastic foundation is considered to behave like a Pasternak type of foundations. Governing equations for vibrating nano-plate are derived according to the Mindlin plate theory in which the effects of shear deformations of nano-plate are also included. The Galerkin method is then employed to obtain the size dependent natural frequencies of nano-plate. The solution procedure considers the entire nano-plate as a single super-continuum element. Effect of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties, thickness and foundation on circular frequencies are investigated. It is seen that the nonlocal frequencies of the nano-plate are smaller in comparison to those from the classical theory and this is more pronounced for small lengths and higher vibration modes. It is also found that as the aspect ratio increases or the nanoplate becomes more elliptical, the small scale effect on natural frequencies increases. Further, it is observed that the elastic foundation decreases the influence of nonlocal parameter on the results. Since the effect of shear deformations plays an important role in vibration analysis and design of nano-plates, by predicting smaller values for fundamental frequencies, the study of these nano-structures using thick plate theories such as Mindlin plate theory is essential.

Buckling of FG circular/annular Mindlin nanoplates with an internal ring support using nonlocal elasticity

In this paper, the buckling analysis of FG circular/annular nanoplates under uniform in-plane radial compressive load with a concentric internal ring support and elastically restrained edges is studied using an exact analytical approach within the framework of nonlocal Mindlin plate theory. The material properties vary according to a power-law distribution of the volume fraction of the constituents whereas Poison's ratio is set to be constant. In solving this problem, the circular/annular FG nanoplate is first divided into an annular segment and a core circular/annular segment at the location of the internal ring support; accordingly solutions for two segments brought together by using the interfacial conditions. It is observed that an internal ring support can increase the buckling capacity, accordingly this capacity is maximized when the internal ring support is located at an optimal position. Furthermore, the effects of small scales on the maximum buckling load corresponding to the optimal radius of the internal ring support are investigated for various parameters such as radius and thickness of the FG nanoplate, boundary conditions, mode numbers and material properties.

Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory

In this article, non-uniform biaxial buckling analysis of orthotropic single-layered graphene sheet embedded in a Pasternak elastic medium is investigated using the nonlocal Mindlin plate theory. All edges of the graphene sheet are subjected to linearly varying normal stresses. The nanoplate equilibrium equations are derived in terms of generalized displacements based on first-order shear deformation theory (FSDT) of orthotropic nanoplates using the nonlocal differential constitutive relations of Eringen. Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. The accuracy of the present results is validated by comparing the solutions with those reported by the available literatures. Finally, influences of small scale effect, aspect ratio, polymer matrix properties, type of planar loading, mode numbers and boundary conditions are discussed in details.

Dynamic analysis of finite periodic nanoplate structures with various boundaries

Dynamic responses of the finite periodic nanoplate structures with various boundary conditions are researched by using the wave method. The dynamics model of the nanoplate structure is established based on the nonlocal Mindlin plate theory. From the Fourier transform, the vibration responses of the single layer graphene sheets under various boundary conditions are obtained by the wave method. The dynamic responses of the finite periodic nanoplate can be determined by the continuous and boundary conditions. In the numerical calculation, the natural frequencies of single layer graphene sheets computed by the presented method agree well with those of molecular dynamics (MD) simulation, which verifies the reliability of the present method. It can be found that the dynamic responses of the finite periodic nanoplates are much smaller than those of the uniform nanoplates in some frequencies which are called the band gap frequencies. It is significant to design the periodic array structure for the vibration suppression of the nanoplates. Moreover, the influences of the nanoplate width and thickness and boundary conditions on the dynamic response of the finite periodic nanoplate structures are also studied.

Static analysis of nanoplates based on the nonlocal Kirchhoff and Mindlin plate theories using DQM

In this study, static analysis of the two-dimensional rectangular nanoplates are investigated by the Differential Quadrature Method (DQM). Numerical solution procedures are proposed for deflection of an embedded nanoplate under distributed nanoparticles based on the DQM within the framework of Kirchhoff and Mindlin plate theories. The governing equations and the related boundary conditions are derived by using nonlocal elasticity theory. The difference between the two models is discussed and bending properties of the nanoplate are illustrated. Consequently, the DQM has been successfully applied to analyze nanoplates with discontinuous loading and various boundary conditions for solving Kirchhoff and Mindlin plates with small-scale effect, which are not solvable directly. The results show that the above mentioned effects play an important role on the static behavior of the nanoplates.

Nonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via Mindlin plate theory

This paper is concerned with the vibration characteristics of an embedded nanoplate-based nanoelectromechanical sensor made of polyvinylidene fluoride (PVDF) carrying a nanoparticle with different masses at any position. The nanoplate is surrounded by elastic medium which is simulated as Pasternak foundation. The PVDF nanoplate is subjected to an applied voltage in the thickness direction. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton’s principle which are then solved by Galerkin method to obtain the natural frequency of the nanoplate. A detailed parametric study is conducted to elucidate the influences of the nonlocal parameter, external electric voltage, position and mass of nanoparticle, temperature changes and dimension of nanoplate and elastic medium. Results indicate that the frequency is increased as the nanoparticle comes closer to the center of the nanoplate; also increasing mass of the nanoparticle decreases the frequency of the system. This study might be useful for the design of PVDF nanoplate-based resonator as nanoelectromechanical sensor.

Modified Nonlocal Mindlin Plate Theory for Buckling Analysis of Nanoplates

AbstractThis article presents a modified nonlocal Mindlin plate theory for stability analysis of nanoplates subjected to both uniaxial and biaxial in-plane loadings. Closed-form solutions of buckling load are presented according to the nonlocal Kirchhoff, first-order and higher-order shear deformation plate theories for simply supported rectangular plates. It is shown that the nonlocal shear deformation plate theories cannot predict the critical buckling load correctly because the buckling load approaches zero as the mode numbers approach infinity. To find the critical buckling load by accounting for either the small scale or the shear deformation effects, a modified nonlocal first-order shear deformation plate theory is adapted. Finally, the critical buckling load and buckling mode numbers of nanoplates are obtained on the basis of the presented modified theory. The results show that variation of buckling load versus the mode number is physically acceptable.

Thermal buckling of a nanoplate with small-scale effects

In this paper, thermal buckling properties of a nanoplate with small-scale effects are studied. Based on the nonlocal continuum theory, critical temperatures for the nonlocal Kirchhoff and Mindlin plate theories are derived. The thermal buckling characteristics are presented with different models. The influences of the scale coefficients, half-wave numbers, width ratios, and the ratios of the width to the thickness are discussed. From this work, it can be observed that the small-scale effects are significant for the thermal buckling properties. Both the half-wave number and width ratio have influence. The nonlocal Kirchhoff plate theory is valid for the thin nanoplate, and the nonlocal Mindlin plate theory is more appropriate for simulating the mechanical behaviors of the thick nanoplate.

Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory

Abstract This study presents an analytical approach for buckling analysis and smart control of a single layer graphene sheet (SLGS) using a coupled polyvinylidene fluoride (PVDF) nanoplate. The SLGS and PVDF nanoplate are considered to be coupled by an enclosing elastic medium which is simulated by the Pasternak foundation. The PVDF nanoplate is subjected to an applied voltage in the thickness direction which operates in control of critical load of the SLGS. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. The exact analysis is performed for the case when all four ends are simply supported and free electrical boundary condition. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton's principle. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, stiffness of the internal elastic medium, graphene length, mode number and external electric voltage on the buckling smart control of the SLGS. The results depict that the imposed external voltage is an effective controlling parameter for buckling of the SLGS. This study might be useful for the design and smart control of nano-devices.

BUCKLING ANALYSIS OF GRAPHENE NANOSHEETS BASED ON NONLOCAL ELASTICITY THEORY

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.

Static bending behaviors of nanoplate embedded in elastic matrix with small scale effects

In this paper, the bending behaviors of the nanoplate with small scale effects are investigated by the nonlocal continuum theory. The governing equations for the nonlocal Mindlin and Kirchhoff plate models are derived. The expressions of the bending displacement are presented analytically. The difference between the two models is discussed and bending properties of the nanoplate are illustrated. It can be observed that the small scale effects are obvious for bending properties of the nanoplate. The half wave numbers, width ratios and elastic matrix properties also have significant influence on bending behaviors.

Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations

The potential applications of nanoplates in energy storage, chemical and biological sensors, solar cells, field emission, and transporting of nanocars have been attracted the attentions of the nanotechnology community to them during recent years. Herein, the later application of nanoplates from nonlocal elastodynamic point of view is of interest. To this end, dynamic response of a nanoplate subjected to a moving nanoparticle is examined within the context of nonlocal continuum theory of Eringen. The fully simply supported nanoplate is modeled based on the nonlocal Kirchhoff, Mindlin, and higher-order plate theories. The non-dimensional equations of motion of the nonlocal plate models are established. The effects of moving nanoparticle's weight and existing friction between the surfaces of the moving nanoparticle and nanoplate on the in-plane and out-of-plane vibrations of the nanoplate are incorporated into the formulations of the proposed models. The eigen function expansion and the Laplace transform methods are employed for discretization of the governing equations in the spatial and the time domains, respectively. The analytical expressions of the dynamic deformation field associated with each nonlocal plate theory are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight path (an opened path) as well as an ellipse path (a closed path). The dynamic in-plane forces and moments of each nonlocal plate model are also derived. Furthermore, the critical velocity and the critical angular velocity of the moving nanoparticle for the proposed models are expressed analytically for the aforementioned paths. Part II of this work consists in a comprehensive parametric study where the effects of influential parameters on dynamic response of the proposed nonlocal plate models are scrutinized in some detail.

Buckling analysis of a single-layer graphene sheet embedded in an elastic medium based on nonlocal Mindlin plate theory

The effect of length scale on buckling behavior of a single-layer graphene sheet embedded in a Pasternak elastic medium is investigated using a nonlocal Mindlin plate theory. An explicit solution is extracted for the buckling loads of graphene sheet and the influence of the nonlocal parameter and aspect ratio on dimensionless buckling loads is presented. It is found that the nonlocal assumptions exhibit larger buckling loads and stiffness of elastic medium in comparison to classical plate theory.

Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory

The present study proposes an analytical solution for the buckling analysis of rectangular nanoplates. In order to extract characteristic equations of the micro/nanoscale plate under in-plane load, the analysis procedure is based on the nonlocal Mindlin plate theory considering the small scale effects. The nonlocal Mindlin plate theory allows for small scale effects. The results show that buckling loads of biaxially compressed micro/nanoscale plate depend on the nonlocal parameter. In addition, the effects of small length scale on buckling loads are graphically presented for different geometrical parameters such as aspect ratios and loading factors. This study might be useful for the design of nanoelectronic devices such as atomic dust detectors and biological sensors.