FGM Plates Research Articles

Study on vibration behavior of functionally graded porous material plates immersed in liquid with general boundary conditions

This paper presents a novel energy method for vibration analysis of pre-loaded moving functionally graded material (FGM) plates in fluid. Porosity in plates, fluid–structure interactions and initial tensions are taken into consideration. The first-order shear deformation theory (FSDT) and linear potential flow theory are adopted to formulate the strain energy, kinetic energy and external work functions of the coupled system. The governing equations of the porous FGM plate immersed in liquid are deduced by the Hamilton’s principle and the unified solutions for general boundary conditions are obtained using the modified Ritz method. By comparing the obtained vibration results with those from the commercial software and literature, the accuracy of the proposed model is validated. The effects of axial speed, fluid density, material constituent, initial axial tension and porosity volume fraction on the vibration behaviors of the FGM plate are further discussed. Numerical cases show that the vibration behaviors of the FGM plate are significantly influenced by these key parameters.

Natural frequency prediction of functionally graded graphene-reinforced nanocomposite plates using ensemble learning and support vector machine models

Functionally graded materials (FGMs) are modern engineering materials with increasing application in various industrial fields. In this study, the free vibration behavior of graphene-reinforced FGM plate is investigated using finite element method and machine learning (ML) approaches. For this purpose, three advanced ML models including ensemble learning algorithms (bootstrap aggregation and gradient boosting) and Gaussian support vector machine are employed to predict the natural frequency of functionally graded graphene/epoxy nanocomposite plates. In this regard, first, hyperparameter optimization is carried out using Bayesian optimization algorithm. Then, regression analysis is performed using the aforementioned ML approaches. According to the obtained results, all ML models have a high coefficient of determination (more than 96%) with low mean squared error (MSE) values. However, the best performance is related to the gradient boosting method, followed by support vector machine and bootstrap aggregation, respectively. Finally, the significance degree of involved parameters on natural frequency is estimated using the Shapley values concept. The obtained results reveal that the most significant parameters affecting the natural frequency of graphene-reinforced FGM plates are clamp type, the volume fraction of graphene, followed by thickness ratio and distribution pattern, respectively.

Using the Equivalent Plate Concept to Calculate Functionally Graded Plates by Numerical Method

Functional graded materials are based on two materials, with very different properties, which vary continuously, according to a material law, on their thickness (a beam or a plate). The concept of equivalent plate involves the replacement of the given plate, a plate made of FGM, with a homogeneous and isotropic plate, with the same geometric characteristics (shape and dimensions), with the same load, which is attributed material properties that lead to the same behavior with the given plate. The aim of the paper is to provide a way to solve a complex problem (mathematical and structural), as in the case of FGM plates, with available means, with known professional calculation programs, with common finite elements, without intervention in the structure of its program. of finite element matrices, in other words without dedicated calculation programs. The finite element method is applied using the Ansys program and one of the best known finite elements, the 3D Solid185 element, both in the common version (structural) and in the multilayer finite element version.

Advanced frequency analysis of thick FGM plates using third-order shear deformation theory with a nonlinear shear correction coefficient

The effects of third-order shear deformation theory (TSDT) displacements and advanced nonlinear varied shear correction coefficient on the free vibration frequency of thick functionally graded material (FGM) plates under environment temperature are studied. The nonlinear coefficient term of TSDT displacements is included to derive the advanced equation of nonlinear varied shear correction coefficient for the thick FGM plates. The determinant of the coefficient matrix in dynamic equilibrium differential equations under free vibration can be represented into fully homogeneous equation and the natural frequency can be found. The parametric effects of nonlinear coefficient term of TSDT, environment temperature and FGM power law index on the natural frequency of thick FGM plates are investigated.

Temperature-dependent shear correction factor with heat transfer based on micromechanical properties for FGM plates

For a development of aerospace structures in high-temperature regions, Functionally Graded Materials (FGMs) have been used rather than conventional composites. In this regard, present work considers the FGM models in thermal environments, and crucial evaluation to supplement the heat conduction is investigated. To model the thermal–structural interactions, the First-order Shear Deformation Theory is adopted with temperature-dependent characteristics including thermo-micromechanical properties. And the material distributions through the thickness direction are then examined in two types such as the Power- and the Sigmoid-laws. Furthermore, the Mori–Tanaka method for homogenization technique is employed for the precise micromechanical interactions of the mixture particles. Moreover, the un-symmetry of material properties in the structure is considered with physical neutral surface concept. To ensure the validity of this study, the previous literatures are used to compare with the present numerical results. And then, shear correction factors are fully discussed using homogenization methods and the temperature-dependent material properties for micromechanical modeling. Additionally, the effect of heat conduction is studied in detail according to the Poisson’s ratio and Young’s modulus of the mixtures.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Buckling Analysis of Functionally Graded Materials (FGM) Thin Plates with Various Circular Cutout Arrangements

In this paper, several analyses were conducted to investigate the buckling behavior of Functionally Graded Material (FGM) thin plates with various circular cutout arrangements. The computer model was simulated using the Finite Element (FE) software ABAQUS. The developed model was validated by the authors in previous research. A parametric analysis was employed to investigate the effect of plate thickness and circular cutout diameter on the buckling behavior of the FGM thin plates. The normalized buckling load was also calculated to compare the buckling performance of FGM plates with various dimensions. Moreover, von Mises stress analysis was examined to understand the yield capability of the FGM plates in addition to the buckling modes that show the stress distribution of the critical buckling stress. Hence, this research provides a comprehensive analysis to display the relation between the critical buckling load and the arrangement of the circular cutouts. The results show that the critical buckling load heavily depends on the dimension of the plate and the cutout size. For instance, an increase in the plate thickness and a decrease in the cutout diameter increase the critical buckling load. Moreover, the circular cutout in a horizontal arrangement exhibited the best buckling performance, and as the arrangement shifts to a vertical arrangement, the buckling performance deteriorates.

Analysis of FGM plates based on physical neutral surface using general third-order plate theory

We present a nonlinear finite element analysis to investigate the nonlinear behaviour of functionally graded materials (FGM) plates. The material properties are assumed to vary gradually across the thickness according to a power law distribution. The starting point of the investigation is the generalized third-order plate theory which is modified in the present analysis to include the position of the neutral surface and enhanced with additional terms to represent the distribution stresses better. The plates are subjected to bending and compression loads including buckling and post-buckling analysis.

Vibration Response Analysis of Tapered Porous FGM Plate Resting on Elastic Foundation

This paper attempts to demonstrate the vibration response of porous Functionally Graded Material (FGM) plate with variable thickness. For the first time, the porous tapered FGM plate being considered is mathematically modeled and assumed to be resting on a linear, parabolic, sinusoidal, and exponential varying Winkler’s elastic foundation. The even and uneven porosity distribution as a micro-defect is assumed in a tapered FGM plate that varies according to well-defined mathematical rules. The tapered FGM plate, across the edges, is supported with various boundary conditions. Simple Power (P-) and Sigmoid (S-) Law have been chosen for the homogenization of material properties that are tailored in the thickness direction. First-order shear deformation theory (FSDT) is applied to describe the displacement function for computing the strain field. The variational approach has been used to establish the formulation for free vibration response. The equation of motion has been derived using Hamilton’s principle and solved by implementing Galerkin Vlasov’s method. Parametric studies on elastic foundations have been done to explore the results and relevance to the real problems. It was observed that variable foundation has a significant effect on the tapered FGM plate rather than the tapered homogenous plate. In addition, it was witnessed that the effect of variable foundation effect diminishes as a constraint on the edges of the tapered FGM plate increases. Also, some benchmark results based on the porosity effect and the influence of variable elastic foundations are exhibited in this study for future reference.

Static bending, free vibration, and buckling analyses of two-layer FGM plates with shear connectors resting on elastic foundations

This paper aims to carry out comprehensively the static bending, free vibration, and buckling analyses of two-layer FGM plates with shear connectors resting on elastic foundations, where two-layer FGM plates resting on Winker's-Pasternak elastic foundations taking into account relative sliding movements are analyzed by using the combination of Navier's solution and finite element method based on nth-order shear deformation plate theory. The plate is made up of two layers of FG materials joined by metal shear connectors. Lagrange and Hermitian shape functions are used to establish stiffness matrices, mass matrices, geometric stiffness matrices, and element force vectors. The suggested method and mechanical model are accurate when compared to the computed data from this research and reputable literature. The computed data accurately describes the mechanical response of the structure. Then a series of parameter investigations are conducted to capture the effects of the structure's geometrical and material features on the mechanical behavior of two-layer FG plates. The numerical results of this study may be utilized as a basis for future research, applications, and design of similar structures in mechanical and civil engineering practice. This is also a critical foundation for studying the dynamic behavior of structures with shear connectors.

A Strong Form Meshless Method for the Solution of FGM Plates

Laminated composite structures suffer from failure because of concentrations of gradient fields on interfaces due to discontinuity of material properties. The rapid development of material science enables designers to replace classical laminated plate elements in aerospace structures with more advanced ones made of functionally graded materials (FGM), which are microscopic composite materials with continuous variation of material coefficients according to the contents of their micro-constituents. Utilization of FGM eliminates the inconvenience of laminated structures but gives rise to substantial changes in structural design This paper deals with the presentation of a strong formulation meshless method for the solution of FGM composite plates. Recall that the fourth-order derivatives of deflections are involved in the governing equations for plate structures. However, the high-order derivatives of field variables in partial differential equations (PDE) lead to increasing inaccuracy of approximations. For that reason, the decomposition of the high-order governing equations into the second-order PDE is proposed. For the spatial approximation of field variables, the meshless moving least square (MLS) approximation technique is employed. The reliability (numerical stability, convergence, and accuracy) as well as computational efficiency of the developed method is illustrated by several numerical investigations of the response of FGM plates with the transversal gradation of material coefficients under stationary and/or transient mechanical and thermal loadings.

Carry-over Factors of Levy-type rectangular FGM plates subjected to edge moment or deflection

The purpose of this study is to carry out the rotational stiffness, shear stiffness, and carry-over-factors (COFs) of Levy-type rectangular FGM plates on the basis of the classical elastic theory. Three fundamental problems of Levy-type rectangular FGM plates subjected to edge moment or vertical deflection are under consideration. The Poisson's ratios of these FGM plates are assumed to be constant, while their Young's moduli vary continuously throughout the thickness direction with the volume fractions of the constituents obeying sigmoid functions. By using the technique of one-direction Fourier series expanding, the general solutions of FGM rectangular plate with Levy boundary conditions are derived. Then the COFs of these Levy-type fundamental problems are obtained. In addition, by superposing the fundamental solutions, the series-solutions of the S-FGM and S-FGM-coated plates subjected to constant edge moment are also developed and investigated. Results reveal that the obtained COFs of three fundamental problems are independent of Young's modulus, but depend on the aspect ratio. And, although the COFs, rotational and shear stiffnesses of the fundamental problems remain constant as the aspect ratio is small than one, the COFs sharply decay to zero while the rotational and shear stiffnesses rapidly increase as the aspect ratio is approximately greater than one. This event is attributed to that the simply-supported edges significantly affect the COFs for big aspect ratio. Notably, the obtained COFs and stiffnesses reported for the first time are applicable to the technique of moment distribution in solving the continuous FGM plates, and the obtained stiffnesses can serve as entries of the stiffness matrix of FGM plates in numerical calculation.

Static and Dynamic Deflection Characteristics of Cracked Porous FG Panels

The deflection responses of the cracked porous graded structure under the variable loading conditions (static and dynamic) are investigated numerically with the help of a commercial tool (ABAQUS). The model includes the grading in two different directions (along the thickness and length by setting the relevant volume fraction exponents [Formula: see text] and [Formula: see text], i.e., of the functionally graded material (FGM) for the analysis purpose. Additionally, different material gradings, i.e., power-law (P-FGM), sigmoid (S-FGM), and exponential (E-FGM), are adopted to achieve the distribution patterns within the FG panel. Further, the porosity and the damage (crack) have been introduced in the graded structural panel. The python scripting language is used to model the FGM plate via the batch input technique, i.e., ABAQUS kernel, instead of the standard GUI interface. The accuracy has been confirmed by validating the results with the published available results, and the applicability is stated through a series of numerical examples by changing input parameters.

Analytical Solution for the Three-Dimensional Vibration of a Rectangular Functionally Graded Material Plate with Two Simply Supported Opposite Faces

Analytical solutions based on three-dimensional (3D) elasticity for the vibrations of functionally graded material (FGM) plates are valuable for assessing the validity and accuracy of various plate theories and numerical approaches. Few benchmark 3D analytical solutions for the vibrations of FGM plates are available in the literature. In this study, analytical solutions based on Fourier series and 3D elasticity were developed for the first time for the vibrations of FGM rectangular plates with two simply supported opposite edge faces. The distributions of the properties of FGMs through the thickness follow a simple power law. The proposed solutions were validated by conducting comprehensive convergence studies on the vibration frequencies of square plates with different thickness-to-side ratios and boundary conditions as well as comparisons with published results. The benchmark nondimensional frequencies were tabulated for plates with free boundary conditions on the top and bottom faces and six combinations of boundary conditions on the other two faces. Moreover, the effects of aspect ratio and gradient index on the vibration frequencies of FGM plates were investigated. The influence of the thickness ratio of the FGM layer to the homogenous layer on the vibration frequencies of sandwich plates with FGM face sheets and a homogeneous core was also studied.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

Static Analysis of Skew Functionally Graded Plate Using Novel Shear Deformation Theory.

In this article, the static response of a functionally graded material (FGM) plate is studied via hybrid higher-order shear deformation theory which uses hyperbolic and polynomial shape functions and includes the effect of thickness stretching. The composition of the plate comprises metallic and ceramic phases. The ceramic volume fraction varies gradually along with the thickness following the power law. The mechanical properties of the FGM plate are determined by the rule of mixtures and the Mori–Tanaka homogenization scheme. The displacement fields are defined to satisfy the requirement of traction-free boundary conditions at the bottom and top surfaces of the plate surface removing the need for determination of shear correction factor. A C0 continuity FE model is developed for the present mathematical model. Nine-node isoparametric elements with eight nodal unknowns at each node are developed. The present model comparison with existing literature is completed and found to be coherent. Inhouse MATLAB code is developed for the present work. Sinusoidal and uniformly distributed loading is analyzed in the present work. The parametric study is undertaken to explore the effect of the side-to-thickness ratio, aspect ratio, thickness, and volume fraction index on stresses and transverse displacements.

Radial basis collocation method for free vibration analysis of elastically supported porous bi-directional FGM plate under various types of porosity distribution

ABSTRACT In this paper, the radial basis collocation method (RBCM) for the free vibration response of elastically supported porous bi-directional functionally graded (BDPFG) plate is investigated in a framework of higher-order shear deformation theory (HSDT). The material properties of BDPFG plates are assumed to be varying continuously in the length and thickness direction. The material properties are evaluated based on Voigt’s micro-mechanical model considering power law distribution method with arbitrary power index. Governing differential equations (GDEs) of the porous BFGM plate area achieved utilising Hamilton’s principle and multiquadric radial basis function (MQRBF) is used for discretizing the GDEs. To incorporate porosity in the BDPFG plate, five types of porosity distribution models are carried out and analysed. The present results are validated with the available published results in the literature to certify the efficacy and accuracy. The influence of grading index in the length and thickness direction, porosity distribution, porosity index, span to thickness ratio, and elastic foundation over natural frequency of bi-directional BDPFG plate is investigated which is the main contribution.

A new higher-order shear deformation theory for frequency analysis of functionally graded porous plates

The current investigation aims to account for the influence of porosity modeling while studying the dynamic response of functionally graded material (FGM) plates within the framework of a new, higher-order shear deformation theory for the first time. In the newly developed porosity modeling, the reciprocal effect of mass density and Young’s modulus is considered. Furthermore, silicon carbide in the ceramic phase and nickel in the metallic phase are selected as the constituent materials of the functionally graded plate. Based upon the new higher-order theory, the kinetic and kinematic relations are derived. This theory does not need any shear correction factor. The dynamic form of the principle of virtual work is employed to achieve governing equations of structure. Afterward, Galerkin’s method is implemented to solve obtained governing equations of porous, functionally graded plates. In order to verify the accuracy of the method, the obtained results were compared with the results reported in the literature, and good agreement was found. At last, the influences of some parameters such as porosity coefficient, length to thickness ratio, and aspect ratio of the porous, FGM plate on the dimensionless frequency are studied and the results are illustrated in detail.