Functionally Graded Nanoplates Research Articles

Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory

This study is devoted to illustrate the mechanical buckling and free vibration analyses of double-porous functionally graded (FG) nanoplates embedded in an elastic foundation. A new quasi-3D refined plate theory is presented to model the displacement field. This theory contains only five unknown functions and considers the shear strain as well as thickness stretching. Based on the modified Mooney-type exponential relation, a new exponential law is presented to govern the materials variation and porosities distribution through the thickness of the nanoplates. The two porous nanoplates are bonded together by a set of parallel elastic springs and surrounded by Pasternak medium. The nonlocal strain gradient theory containing the nonlocal parameter and gradient coefficient is utilized to study the size-dependent effects. Based on Hamilton’s principle, the equations of motion are drawn including the material parameters, elastic foundation reaction and biaxial compressive forces. An analytical approach for simply-supported and clamped bilayer porous FG nanoplates is implemented. The obtained results are compared with those available in the literature. Additional numerical calculations are introduced to show the influences of the material length scale parameters, inhomogeneity parameter, porosity factor and other parameters on the critical buckling and frequencies of the double-porous FG nanoplates.

Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation

This work deals with the size-dependent wave propagation analysis of functionally graded (FG) anisotropic nanoplates based on a nonlocal strain gradient refined plate model. The present model incorporates two scale coefficients to examine wave dispersion relations more accurately. Material properties of FG anisotropic nanoplates are exponentially varying in the z-direction. In order to solve the governing equations for bulk waves, an analytical method is performed and wave frequencies and phase velocities are obtained as a function of wave number. The influences of several important parameters such as material graduation exponent, geometry, Winkler-Pasternak foundation parameters and wave number on the wave propagation of FG anisotropic nanoplates resting on the elastic foundation are investigated and discussed in detail. It is concluded that these parameters play significant roles on the wave propagation behavior of the nanoplates. From the best knowledge of authors, it is the first time that FG nanoplate made of anisotropic materials is investigated, so, presented numerical results can serve as benchmarks for future analysis of such structures.

The application of nonlocal elasticity to determine vibrational behavior of FG nanoplates

Nonlocal elasticity and Reddy plant theory are used to study the vibration response of functionally graded (FG) nanoplates resting on two parameters elastic medium called Pasternak foundation. Nonlocal higher order theory accounts for the effects of both scale and the effect of transverse shear deformation, which becomes significant where stocky and short nanoplates are concerned. It is assumed that the properties of FG nanoplate follow a power law through the thickness. In addition, Poisson\'s ratio is assumed to be constant in this model. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction of nanoplate with surrounding elastic medium. Using Hamilton\'s principle, size-dependent governing differential equations of motion and corresponding boundary conditions are derived. A differential quadrature approach is being utilized to discretize the model and obtain numerical solutions for various boundary conditions. The model is validated by comparing the results with other published results.

Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis

The study using numerical methods on porous functionally graded (FG) nanoplates is still somewhat limited. This paper focuses on porosity-dependent nonlinear transient analysis of FG nanoplates using isogeometric finite element approach. In order to capture the small size effects, the Eringen's nonlocal elasticity based on higher order shear deformation theory (HSDT) are used to model the porous FG nanoplates. Two distributions of porosities inside FG materials are incorporated and defined via a modified rule of mixture. The nonlinear transient nonlocal governing equations under transverse dynamic loads are formulated by using the von Kármán strains and are solved by Newmark time integration scheme to obtain geometrically nonlinear responses. It is indicated that nonlinear transient deflections of the porous FG nanoplate are significantly influenced by material composition, porosity, nonlocal parameters, volume fraction exponent, porosity distributions, geometrical parameters and dynamic load characteristics.

Transient response of porous FG nanoplates subjected to various pulse loads based on nonlocal stress-strain gradient theory

Based on nonlocal strain gradient theory (NSGT), transient behavior of a porous functionally graded (FG) nanoplate due to various impulse loads has been studied. The porous nanoplate has evenly and unevenly distributed pores inside its material structure. Impulse point loads are considered to be rectangular, triangular and sinusoidal types. These impulse loads lead to transient vibration of the nanoplate which is not studied before. NSGT introduces a nonlocal coefficient together with a strain gradient coefficient to characterize small size influences due to non-uniform stress and strain fields. Galerkin's approach has been performed to solve the governing equations and also inverse Laplace transform method is used to obtain transient response due to impulse loads. It is explained in this research that the transient response of a nanoplate is dependent on nonlocal coefficient, strain gradient parameter, pore dispersion, pore amount, type of impulse load and loading time.

Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals

In this paper, the buckling of rectangular functionally graded (FG) porous nanoplates based on three-dimensional elasticity is investigated. Since, similar researches have been done in two-dimensional analyses in which only large deflections with constant thickness were studied by using various plate theories; therefore, discussion of large deformations and change in thickness of plates after deflection in this study is examined. Moreover, porosity is assumed in two situations, even and uneven distributions considered in several conditions. Using nonlocal elasticity theory, nonlocal three-dimensional equations are obtained. Regarding difficulties in solving three-dimensional differential equations, simple analytical methods are assumed and proposed. The most important results show that even porosity makes the plate softer and results of uneven porosity are so close to the prefect material which leads to this considerable conclusion that porosity as an uneven distribution cannot be an important factor in static stability analyses of FG nanoplates.

Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis

This study aims to investigate the postbuckling response of functionally graded (FG) nanoplates by using the nonlocal elasticity theory of Eringen to capture the size effect. In addition, Reddy’s third-order shear deformation theory is adopted to describe the kinematic relations, while von Kámán’s assumptions are used to account for the geometrical nonlinearity. In order to calculate the effective material properties, the Mori-Tanaka scheme is adopted. Governing equations are derived based on the principle of virtual work. Isogeometric analysis (IGA) is employed as a discretization tool, which is able to satisfy the C1-continuity demand efficiently. The Newton-Raphson iterative technique with imperfections is employed to trace the postbuckling paths. Various numerical studies are carried out to examine the influences of gradient index, nonlocal effect, ratio of compressive loads, boundary condition, thickness ratio and aspect ratio on the postbuckling behaviour of FG nanoplates.

Nonlinear thermal vibration analysis of refined shear deformable FG nanoplates: two semi-analytical solutions

This paper deals with the semi-analytical nonlinear thermal vibration analysis of functionally graded (FG) nanoplates modeled by four-variable refined plate theory. The nanoplate is resting on a nonlinear hardening elastic medium and subjected to uniform and nonlinear temperature rises. Temperature-dependent gradient material properties of the nanoplate are described via simple power-law model. A closed-form expression for nonlinear frequency is presented using a novel Hamiltonian approach as well as He’s variational method. To the best of the author’s knowledge, the nonlinear governing equations and their solution for a refined four-variable nanoplate are very rare in the literature. It is seen that the nonlinear vibration frequency is affected by the uniform temperature rise, nonlinear temperature rise, scale parameter, maximum amplitude, material gradation, foundation parameters and temperature-dependency. The presented formulation and solution approaches can be used in future investigations on macro to nanoscale plates.

Three dimensional dynamic response of functionally graded nanoplates under a moving load

In this paper, reaction of functionally graded (FG) thick nanoplates resting on a viscoelastic foundation to a moving nanoparticle/load is investigated. Nanoplate is assumed to be thick by using second order shear deformation theory and small-scale effects are taken into account in the framework of Eringen\'s nonlocal theory. Material properties are varied through the thickness using FG models by having power-law, sigmoid and exponential functions for material changes. FG nanoplate is assumed to be on a viscoelastic medium which is modeled using Kelvin-Voight viscoelastic model. Galerkin, state space and fourth-order Runge-Kutta methods are employed to solve the governing equations. A comprehensive parametric study is presetned to show the influence of different parameters on mechanical behavior of the system. It is shown that material variation in conjunction with nonlocal term have a significant effect on the dynamic deformation of nanoplate which could be used in comprehending and designing more efficient nanostructures. Moreover, it is shown that having a viscoelastic medium could play an important role in decreasing these dynamic deformations. With respect to the fresh studies on moving atoms, molecules, cells, nanocars, nanotrims and point loads on different nanosctructures using scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding, predicting and controlling such kind of behaviors by showing the influence of the moving path, velocity etc. on dynamic reaction of the plate.

Temperature-dependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field

ABSTRACTThis study is focused on the wave propagation analysis of nanoplate made of temperature-dependent porous functionally graded (FG) materials rested on Winkler–Pasternak foundation under in-plane magnetic field. The material properties of FG nanoplate are supposed to vary through the thickness direction and described by power-law rule, in which the porosity distribution is considered as an even pattern. Hamilton’s principle is utilized to derive the governing equations on basis of second-order shear deformation theory in conjunction with nonlocal strain gradient theory. The influence of small-length parameters, thermal distribution, magnetic field, material composition, porosity, and Winkler–Pasternak foundation on wave dispersion is explored.

Nonlocal buckling analysis of functionally graded nano-plates subjected to biaxial linearly varying forces

The buckling problem of a thin, rectangular, functionally graded (FG) nano-plate with varying boundary conditions subjected to biaxial linearly varying forces is investigated utilizing the Eringen’s nonlocal theory. The plates are modeled using Kirchoff’s classical plate theory and the mechanical properties of the FG nano-plates vary continuously through the thickness of the plate following a power law model. The governing differential equations of motion of the nano-plates subjected to linearly varying biaxial loads is derived using the Hamilton’s principle, along with the constitutive relations. The Chebyshev spectral collocation method is then utilized to solve for the critical buckling loads. A parametric study is then performed to examine the impacts of varying the nonlocal parameter, power law index, aspect ratio, slopes of the axial loads, and varying boundary conditions on the nonlocal critical buckling loads of the nano-plates. It is shown that these parameters have significant influence on the stability characteristics of the FG nano-plates.

Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory

ABSTRACTIn this work, analytical solutions are presented for the wave propagation in functionally graded (FG) nanoplates using a nonlocal strain gradient theory and four-variable refined plate theory considering the magnetic field. The size effects are included using nonlocal strain gradient theory that has two length scale parameters, and the nanoplate is modeled as a plate using four-variable refined plate theory. From the knowledge of authors, it is the first time that the influences of magnetic field on the wave propagation in FG nanoplates are investigated based on present methodology.

Wave propagation in embedded inhomogeneous nanoscale plates incorporating thermal effects

In this article, an analytical approach is developed to study the effects of thermal loading on the wave propagation characteristics of an embedded functionally graded (FG) nanoplate based on refined four-variable plate theory. The heat conduction equation is solved to derive the nonlinear temperature distribution across the thickness. Temperature-dependent material properties of nanoplate are graded using Mori–Tanaka model. The nonlocal elasticity theory of Eringen is introduced to consider small-scale effects. The governing equations are derived by the means of Hamilton’s principle. Obtained frequencies are validated with those of previously published works. Effects of different parameters such as temperature distribution, foundation parameters, nonlocal parameter, and gradient index on the wave propagation response of size-dependent FG nanoplates have been investigated.

An analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position

ABSTRACTIn this article, an analytical method is presented for thermo-mechanical vibration analysis of functionally graded (FG) nanoplates with different boundary conditions under various thermal loadings including uniform, linear, and nonlinear temperature rise via a four-variable plate theory considering neutral surface position. The temperature-dependent material properties of FG nanoplate vary gradually along the thickness according to the Mori-Tanaka homogenization scheme. The exactness of solution is confirmed by comparing obtained results with those provided in the literature. A parametric study is performed investigating the effects of nonlocal parameter, temperature fields, gradient index, and boundary conditions on vibration behavior of FG nanoplates.

Size-dependent thermal stability analysis of embedded functionally graded annular nanoplates based on the nonlocal elasticity theory

In the present investigation, bifurcation-type buckling characteristics of heated functionally graded (FG) annular nanoplates resting on an elastic foundation and subjected to various types of thermal loading are carried out by presenting an exact analytical solution for the first time. Three kinds of frequently used thermal loading, i.e. uniform temperature rise, linear and nonlinear temperature distribution through the thickness direction are considered. Thermo-mechanical properties of FG nanoplate are supposed to vary smoothly and continuously throughout the thickness based on power law model whereas the Poisson's ratio is held constant. Eringen's nonlocal elasticity theory is exploited to describe the size dependency of nanoplate. Using the principle of virtual displacements, the equilibrium equations together with corresponding boundary conditions are obtained for the thermal buckling analysis of FG annular nanoplates including size effects. The pre-buckling analysis is performed and the proper boundary conditions are chosen to assure the existence of bifurcation-type buckling. Also, the nonlocal stability equations of FG annular nanoplate are derived by using the adjacent equilibrium criterion and they are solved by applying an exact asymmetrical solution. Moreover, in following a parametric study is accompanied to examine the effects of the several parameters such as nonlocal parameter, various temperature distributions, thickness to outer radius ratio, power law index, inner to outer radius ratios and elastic medium parameters on the critical buckling temperatures of the size-dependent FG nanoplates in detail. It is found that the small scale effects and thermal loading have a significant effect on thermal stability characteristics of FG annular nanoplates.

Free vibration analysis of FG nanoplates embedded in elastic medium based on second-order shear deformation plate theory and nonlocal elasticity

In this paper, the second-order shear deformation plate theory is developed for a study of free vibration analysis of the functionally graded (FG) nanoplates embedded in an elastic medium based on nonlocal elasticity of Eringen. The material distributions of FG nanoplates are considered as power law, sigmoid and exponential models. The complete governing equations of FG nanoplates are derived by using the Hamilton’s principle. The analytical solution for natural frequencies and corresponding mode shapes of simply supported FG nanoplates are established. Furthermore, the expressions of the elements of stiffness and mass matrices are completely manifested. The influences of the power index, exponential index, nonlocal parameter, the stiffness of an elastic medium and the environmental temperature on free vibration responses are investigated. The obtained analytical results show an excellent agreement with other available solutions. The formulation and these analytical results of the proposed method could serve as a benchmark for an evaluation of future research.

On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model

In this paper, a new nonlocal hyperbolic refined plate model is presented for free vibration properties of functionally graded (FG) plates. This nonlocal nano-plate model incorporates the length scale parameter which can capture the small scale effect. The displacement field of the present theory is chosen based on a hyperbolic variation in the in-plane displacements through the thickness of the nano-plate. By dividing the transverse displacement into the bending and shear parts, the number of unknowns and equations of motion of the present theory is reduced, significantly facilitating structural analysis. The material properties are assumed to vary only in the thickness direction and the effective properties for the FG nano-plate are computed using Mori-Tanaka homogenization scheme. The governing equations of motion are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined four variable plate theory via Hamilton\'s principle. Analytical solution for the simply supported FG nano-plates is obtained to verify the theory by comparing its results with other available solutions in the open literature. The effects of nonlocal parameter, the plate thickness, the plate aspect ratio, and various material compositions on the dynamic response of the FG nano-plate are discussed.

Axisymmetric/asymmetric buckling of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity

This paper provides the axisymmetric/asymmetric buckling analysis of moderately thick circular and annular functionally graded (FG) nanoplates under uniform compressive in-plane loads. The material properties change continuously through the thickness of the FG nanoplate, which can vary according to a power-law distribution of the volume fraction of the constituents whereas Poison’s ratio is set to be constant. To reflect the small scale effects on the critical buckling loads of the Mindlin model, the nonlocal elasticity of Eringen is employed. An exact closed-form solution is used to solve the nonlocal governing equations. The effect of small scales on buckling loads for different parameters such as diameter of the FG nanoplate, boundary conditions, material properties and mode numbers are investigated in details. It is found that the modes corresponding to the lowest critical buckling loads for circular and annular FG nanoplates may be axisymmetric or asymmetric. Therefore, the asymmetric modes as well as axisymmetric ones have been considered here.

Size-dependent free flexural vibration behavior of functionally graded nanoplates

In this paper, size dependent linear free flexural vibration behavior of functionally graded (FG) nanoplates are investigated using the iso-geometric based finite element method. The field variables are approximated by non-uniform rational B-splines. The nonlocal constitutive relation is based on Eringen’s differential form of nonlocal elasticity theory. The material properties are assumed to vary only in the thickness direction and the effective properties for the FG plate are computed using Mori–Tanaka homogenization scheme. The accuracy of the present formulation is demonstrated considering the problems for which solutions are available. A detailed numerical study is carried out to examine the effect of material gradient index, the characteristic internal length, the plate thickness, the plate aspect ratio and the boundary conditions on the global response of the FG nanoplate. From the detailed numerical study it is seen that the fundamental frequency decreases with increasing gradient index and characteristic internal length.