Nonlocal Plate Model Research Articles

Effect of the porous structure on the hygrothermal vibration analysis of functional graded nanoplates using nonlocal high-order continuum plate model

This study delves into the thermomechanical vibration behavior of functionally graded porous nanoplates under extreme thermal temperature and humidity conditions. The equation of motion of the nanoplate was derived using advanced theories in elasticity and deformation. The nanoplate consists of metal (SUS304) on the bottom surface and ceramic (Ni3S4) on the top surface, with the material distribution changing according to the power law across the plate thickness. The nanoplate was modeled with uniform and symmetric distributions of porosity reaching as high as 60%. Upon incorporating the thermal and moisture loads from the humid surroundings into the equations of motion derived from Hamilton's principle, the equations were solved using Navier's method and simplified to the eigenvalue equation. Analyzed within a broad framework are the thermomechanical vibration behavior of the nanoplate, temperature impact, humidity influence, porosity and its distribution, material grading parameter effects, and nonlocal integral elasticity effects. Observations indicate that variations in thermal temperature, humidity, and nonlocal parameters can lower the thermomechanical vibration frequency of the nanoplate, whereas porosity has the opposite effect. The effects mentioned are influenced by factors, such as the porosity ratio, porosity distribution, material ratios, and the size of the nonlocal parameter in the plate. The primary objective of this work is to uncover the nonlinear frequency response of nanoplates with high porosity in conditions characterized by high temperature and humidity.

Vibration analysis of higher-order nonlocal strain gradient plate via meshfree moving Kriging interpolation method

In this paper, a higher-order nonlocal strain gradient plate model combining nonlocal elasticity theory with strain gradient theory in the form of adding signs is constructed. The governing equation for the higher-order nonlocal strain gradient plate model is derived by Hamilton’s principle. Two higher-order parameters with nonlocal stress and strain gradient are introduced here to explain the size effect and dispersive behaviour. The meshfree moving Kriging interpolation method is utilized to address the frequency of higher-order nonlocal strain gradient plate under complicated boundary conditions. Nonlocal elasticity theory, strain gradient theory in the form of add signs and minus signs on the effect of frequency are compared by degenerating the higher-order nonlocal strain gradient plate model. The other goal of this paper is to compare the present model with higher-order nonlocal strain gradient theory in the form of minus signs on the effect of frequency. It is shown that these two items can describe the size effect well. The present method can also produce lower natural frequencies corresponding to ultrahigh-order mode shapes and applied in a discrete medium.

Formulation of non-local space-fractional plate model and validation for composite micro-plates

It is challenging to design nano/micro-devices because of their small size and tight engineering tolerances. Computer-aided design uses mathematical models of device parts, but contemporary models are not precise enough to capture all characteristics of the materials at the nano/micro-scale. In this work, we propose a novel mathematical model for composite nano/micro-plates in bending in the framework of a space-fractional continuum mechanics approach. This model yields closer mapping of experimental results compared to existing formulations. The developed theory is named the space-Fractional Kirchhoff–Love Plate (s-FKLP). We investigate how the parameters of the s-FKLP model, responsible for the scale effect description, affect the bending behavior of the micro-plates, considering different support conditions. In addition, we compare the results predicted by s-FKLP with the previously developed simpler (one-dimensional) space-Fractional Euler–Bernoulli Beam (s-FEBB) model to show that the choice of structural analysis strategy can influence design decisions. Despite its greater complexity, the s-FKLP is a more versatile approach to modeling micro-sized structures than the s-FEBB. The proposed s-FKLP model is empirically validated using real sandwich micro-plates. We conclude model alignment with the experimental data indicating that the model is suitable for representing sandwich micro-plates. We also find that the model’s length scale parameter corresponds to the microstructure’s grain size. The findings of this research may have implications for future work in the design and optimization of micro-sized devices.

Squeeze Film Effect in Surface Micromachined Nano Ultrasonic Sensor for Different Diaphragm Displacement Profiles

In the present paper, we have analytically explored the small variations of the local pressure in the trapped air film of both sides of the clamped circular capacitive micromachined ultrasonic transducer (CMUT), which consists of a thin movable membrane of silicon nitride (Si3N4). This time-independent pressure profile has been investigated thoroughly by solving the associated linear Reynold’s equation in the framework of three analytical models, viz. membrane model, plate model, and non-local plate model. The solution involves Bessel functions of the first kind. The Landau–Lifschitz fringing technique has been assimilated to engrave the edge effects in estimation of the capacitance of CMUT, which should be considered in the micrometer or lesser dimension. To divulge the dimension-based efficacy of the considered analytical models, various statistical methods have been employed. Our use of contour plots of absolute quadratic deviation revealed a very satisfactory solution in this direction. Though the analytical expression of the pressure profile is very cumbersome in various models, the analysis of these outputs exhibits that the pressure profile follows the displacement profile in all the cases indicating no viscous damping. A finite element model (FEM) has been used to validate the systematic analyses of displacement profiles for several radii and thicknesses of the CMUT’s diaphragm. The FEM result is further corroborated by published experimental results bearing excellent outcome.

Size-dependent free vibration and buckling analysis of magneto-electro-thermo-elastic nanoplates based on the third-order shear deformable nonlocal plate model

In this paper, a size-dependent nanoplate model is developed to describe the free vibration and buckling behaviors of magneto-electro-thermo-elastic (METE) rectangular nanoplates. It is assumed that the METE nanoplate is subjected to external electric voltage, external magnetic potential, and uniform temperature rise. The nonlocal elasticity theory along with the third-order shear deformation plate theory is employed for the size-dependent mathematical modeling of nanoplates. The presented model has two advantages over available models: (1) the need for the correction factor is bypassed and (2) it can be successfully applied to thick nanoplates. The governing equations and the corresponding boundary conditions are derived using the Hamilton’s principle which are then discretized on the space domain based on the generalized differential quadrature (GDQ) method. Afterward, an efficient numerical Galerkin procedure is adopted to reduce the discretized equations into Duffing-type ordinary differential equations. Numerical results are presented to examine the influences of nonlocal parameter, length-to-thickness ratio, temperature rise, external electric potential, external magnetic potential and type of boundary condition on the free vibration, and buckling behaviors of METE nanoplates. It is revealed that frequency and critical buckling load of nanoplates are dependent on magneto-electro-mechanical loadings, whereas they are less dependent on thermal loading.

Quasi-3D large deflection nonlinear analysis of isogeometric FGM microplates with variable thickness via nonlocal stress–strain gradient elasticity

Via the nonlocal stress–strain gradient continuum mechanics, the microscale-dependent linear and nonlinear large deflections of transversely loaded composite sector microplates with different thickness variation schemes are investigated. Microplates are assumed to be prepared from functionally graded materials (FGMs) the characteristics of which are changed along the thickness direction. A quasi-3D plate theory with a sinusoidal transverse shear function in conjunction with a trigonometric normal function was employed for the establishment of size-dependent modelling of FGM microplates with different thickness variation schemes. Then, to solve the nonlocal stress–strain gradient flexural problem, the non-uniform rational B-spline type of isogeometric solution methodology was applied for an accurate integration of geometric discerptions. It was found that the gap between load–deflection curves drawn for linear, concave and convex thickness variation patterns became greater by changing FGM composite microplate boundary conditions from clamped to simply supported. In addition, it was found that by considering only the nonlocal size effect, the plate deflection obtained by the nonlocal strain gradient quasi-3D plate model was greater than that extracted by the classical continuum elasticity because of the softening character of nonlocal size effect, while the strain gradient microstructural size dependency acted in opposite way and represented a stiffening character.

Static Stability Analysis of Mass Sensors Consisting of Hygro-Thermally Activated Graphene Sheets Using a Nonlocal Strain Gradient Theory

This paper develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygro-thermal environments with mass sensors. For a more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. The graphene sheet is modeled via a two-variable shear deformation plate theory that does not need shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on the elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors, such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, nanoparticle mass and geometrical parameters, on buckling characteristics of graphene sheets are examined and presented as dispersion graphs.

Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory

A nonlocal Kirchhoff plate model with fourth-order strain gradient theory is firstly proposed to study variations with band gap frequencies and vibration energy distribution. The temperature rise is supposed to vary linearly through the thickness of the periodic nanoplate structures. The dynamic equations of finite periodic nanoplate structures under thermal load with the small-scale effect and the nonlinear membrane strain taken into consideration are derived based on the finite element method. The structural intensity approach is developed to predict the vibration energy flow of the periodic nanoplates. Effects of the temperature rise on band gaps and the structural intensity are divided into two parts. One is nonlinear effects of temperature rise, and the other is effects of the thermal load. In the numerical calculation, the natural frequencies of single-layer graphene sheets computed by the nonlocal finite element method with fourth-order strain gradient agree well with analytical results, which validate the effectiveness of the present method. The influences of nonlocal parameters and thermal load on the band gap and structural intensity are considered, respectively. The boundary value has been achieved to determine the critical mechanical load or thermal load for analyzing in depth the effects of the mechanical load and thermal load on the structural intensity. The proposed method shows that the vibration energy flow pattern may be controlled by adjusting the magnitude of the mechanical or thermal load.

Geometrically nonlinear analysis of nonlocal plates using fractional calculus

This study presents the analytical formulation and the finite element solution of a geometrically nonlinear and nonlocal plate modeled via fractional-order mathematics. The finite nonlocal strains are obtained from a frame-invariant and thermodynamically consistent fractional-order continuum formulation. The fractional-order continuum model is used to capture softening nonlocal response via both Mindlin and Kirchhoff formulations. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal plates are obtained using variational principles. Further, a 2D nonlinear finite element model for the solution of this class of fractional-order systems is developed. The fractional finite element model is used to study the geometrically nonlinear response of nonlocal plates subject to various loading and boundary conditions. It is established that irrespective of the nature of boundary conditions, the fractional-order nonlocality leads to a reduction in the plate’s stiffness thereby increasing the magnitude of the transverse displacements. More specifically, the nonlocal plate model based on fractional-order strain-displacement relations is free from boundary effects that lead to mathematical ill-posedness and inaccurate predictions typical of classical strain-driven integral approaches to nonlocal elasticity under certain loading and boundary conditions. This consistency in the predictions of the fractional-order nonlocal model is a result of the well-posed nature of the fractional-order governing equations.

Nonlinear Vibration and Stability Analysis of Thermally Postbuckled Double-Layered Graphene Sheet under 1:1 and 3:1 Internal Resonance

In the present research, vibration behavior is presented for a thermally postbuckled double layered graphene sheet (DLGS). The DLGS is modeled as a nonlocal orthotropic plate and contains small-scale effects. The formulations are based on Kirchhoff’s plate theory and nonlinearity of von Karman type is considered in strain–displacement relations. The thermal effects, van der Waals forces between layers and chirality are also included, and some of the material properties are assumed to be temperature-dependent. A coupled system of equations is derived and a new semi-analytical solution is obtained using multiple time-scale methods. The effects of variation of small-scale parameter on the natural frequencies, deflections and response curve of DLGS are analyzed, and the numerical results are obtained from the nonlocal plate model; also, molecular dynamics (MD) simulations are used to investigate different vibration behaviors of DLGS and calibration of the small-scale coefficient. Numerical results are compared with those of similar researches. Effects of various parameters on the postbuckled vibration of DLGS in thermal environments such as scale parameter, environmental damping, length and thermal load are presented. The stability and occurrence of internal resonance between vibration modes around a stable buckled configuration are investigated.

Numerical analysis of single-layered graphene sheets by a mesh-free approach

The paper presents a numerical analysis of the behavior of single-layered graphene sheet (SLGS) using a mesh-free approach based on a nonlocal continuum plate model (NCPM). The adopted NCPM constructed by incorporating the nonlocal elasticity theory into the first order shear deformation elastic plate theory is able to capture small length scale effects. Through the NCPM, the SLGS is modeled as a continuous orthotropic nanoplate. the obtained nonlocal nonlinear partial differential equations completed by boundary conditions are solved numerically by a mesh-free approach associating the asymptotic numerical method with the mesh-free collocation method based on moving least square approximation. The effects of the small-scale parameter and aspect ratio on the nonlinear bending and post-buckling behaviors of SLGS are considered. Good agreement has been established between the obtained results and those of the literature.

Nonlinear Forced Vibration of Thermally Postbuckled Double-Layered Triangular Graphene Sheet with Clamped Boundary Conditions

In the present research, vibration behavior is presented for a thermally postbuckled double-layered triangular graphene sheet (DLTGS). The DLTGS is modeled as a nonlocal orthotropic plate and contains small-scale effects. The formulations are based on the Kirchhoff’s plate theory, and a nonlinearity of von Karman-type is considered in strain–displacement relations. The thermal effects and van der Waals forces between layers are also included and some of the material properties are assumed to be temperature-dependent. A semi-analytical solution is obtained using the multiple time scales method. The effects of variation of small-scale parameter to the natural frequencies, deflections and response curve of DLTGS are analyzed, and the numerical results are obtained from the nonlocal plate model. Numerical results are compared with those of similar researches. Effects of various parameters on the postbuckled vibration of DLTGS in thermal environments such as scale parameter and thermal load are presented. The stability and occurrence of the internal resonance between vibration modes around a stable buckled configuration is investigated.

Hygrothermal effects on static stability of embedded single-layer graphene sheets based on nonlocal strain gradient elasticity theory

This article develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygrothermal environments. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, elastic foundation and geometrical parameters on buckling characteristics a graphene sheets are examined.

Propagation of uncertainty in free vibration of graphene sheet rested on elastic foundation

In this study, a convex model method is used to predict the uncertainty propagation in frequencies of graphene sheets rested on elastic foundation, resulting from the unavoidable scatter in parameters. The nonlocal plate model is utilised to include nanoscale size-effects, and an expression for the natural frequencies is proposed in an explicit form. The predicted results for the nominal case are compared with those obtained from molecular dynamic simulations. Uncertainty analysis of vibrational frequencies of the graphene sheets by using a non-probabilistic method is the main novelty of the present study. By using this method, the lower and upper bounds of the natural frequencies are evaluated. Numerical results indicate that the uncertainties must be considered for designing nanodevices.

On the local and non-local plate models of single layer graphene

In the application of continuum plate theories for modeling the mechanics of single layer graphene, significant inconsistencies exist in the literature on how to deal with the discrepancy between the continuum assumptions and discrete nature of the graphene lattice, whether using the classical plate theories or the more recent non-local elasticity based plate models. Physically vague properties such as graphene thickness and non-local parameter are often used as fitting parameters for calibrating the numerical results against those obtained from atomistic models or experimental data. However, very often a set of parameters that work well for a particular case can lead to large errors in other cases. This work attempts to answer the following question: can the mechanical behavior of single layer graphene sheets be described accurately and consistently using a continuum plate model? By using the results from atomic lattice mechanics calculations as reference for error analysis, we demonstrate that, when the sources of error in the classical plate model are properly identified, mechanics of single layer graphene can be accurately described by a modified non-local plate model with an interatomic potential based constitutive relation and a corrected boundary configuration.

Comparison of nano-plate bending behaviour by Eringen nonlocal plate, Hencky bar-net and continualised nonlocal plate models

This paper is concerned with the bending behaviour of small-scale simply supported plates as predicted by using the Eringen nonlocal plate model (ENM), the Hencky bar-net model (HBM) and the continualised nonlocal plate model (CNM). HBM comprises rigid beam segments connected by rotational and torsional springs. CNM is a nonlocal model derived by using a continualisation approach that does away with the unknown scale coefficient $$e_{0}$$ in ENM. The exact bending solutions for simply supported rectangular nano-plates are derived by using ENM, HBM and CNM. By making the segment length $$\ell $$ of HBM equal to the scale length of continualised and Eringen’s nonlocal plate model and noting the phenomenological similarities between ENM, HBM and CNM, the Eringen’s length scale value $$e_0 $$ is found to be dependent on the aspect ratio of the simply supported plate and independent of the applied transverse loading. For a very small scale length $$\ell $$ , $$e_0$$ of ENM converges to values ranging from $$1/\sqrt{8}$$ to $$1/\sqrt{6}$$ for square plate to longish rectangular plate when calibrated by either HBM or CNM.

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.

Hygro-thermal vibration analysis of bilayer graphene sheet system via nonlocal strain gradient plate theory

In this research a nonlocal strain gradient plate model is introduced for hygro-thermal vibration analysis of double-layer graphene sheets resting on elastic medium. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. In fact, frequency increment due to the strain gradient effect is neglected in all previous papers related to double-layer graphene sheets. Governing equations of a nonlocal strain gradient double-layer graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as hygro-thermal loading, nonlocal parameter, length-scale parameter, elastic foundation, interlayer stiffness and boundary conditions on vibration characteristics a double-layer graphene sheet are studied.

Lattice and continualized models for the buckling study of nonlocal rectangular thick plates including shear effects

The present study investigates the exact buckling behavior of a new lattice plate theory, called microstructured thick plate model, that accounts for the shear effect. Three different kinds of continuous nonlocal elasticity plate theories for capturing the small length scale effect of this bending-shear lattice plate have been compared: a fourth and a sixth order continualized models based on the derivation of continuous equations from discrete ones and a phenomenological nonlocal Uflyand–Mindlin plate model. The phenomenological nonlocal model is the stress gradient model of Eringen (1983) applied at the plate scale for the bending part and eventually the shear part of the constitutive law. Each nonlocal plate model is calibrated with respect to the lattice plate model. By definition, the length scale coefficients are constant for the two proposed continualized models. For the phenomenological model, these coefficients are structural dependent and vary with different parameters, such as the aspect ratio or the buckling mode. More especially, it is shown that the calibrated small length scale coefficient is influenced by the shear effect. The comparison of the nondimensional buckling loads calculated with the optimal value of the small length scale coefficient, which differ for each nonlocal model, shows that the continualized models constitute a much better, more consistent and more accurate approximation than the phenomenological nonlocal approach.

Nonlocal Approaches for the Vibration of Lattice Plates Including Both Shear and Bending Interactions

The present study investigates the dynamical behavior of lattice plates, including both bending and shear interactions. The exact natural frequencies of this lattice plate are calculated for simply supported boundary conditions. These exact solutions are compared with some continuous nonlocal plate solutions that account for some scale effects due to the lattice spacing. Two continualized and one phenomenological nonlocal UflyandMindlin plate models that take into account both the rotary inertia and the shear effects are developed for capturing the small length scale effect of microstructured (or lattice) thick plates by associating the small length scale coefficient introduced in the nonlocal approach to some length scale coefficients given in a Taylor or a rational series expansion. The nonlocal phenomenological model constitutes the stress gradient Eringen’s model applied at the plate scale. The continualization process constructs continuous equation from the one of the discrete lattice models. The governing partial differential equations are solved in displacement for each nonlocal plate model. An exact analytical vibration solution is obtained for the natural frequencies of the simply supported rectangular nonlocal plate. As expected, it is found that the continualized models lead to a constant small length scale coefficient, whereas for the phenomenological nonlocal approaches, the coefficient, calibrated with respect to the element size of the microstructured plate, is structure-dependent. Moreover, comparing the natural frequencies of the continuous models with the exact discrete one, it is concluded that the continualized models provide much more accurate results than the nonlocal Uflyand–Mindlin plate models.