Sigmoid Functionally Graded Material Research Articles

Symplectic superposition method for the free-vibrating problem of sigmoid functionally graded material rectangular thin plates clamped at four edges

The symplectic superposition method (SSM) is implemented to investigate the free vibration problem of sigmoid functionally graded material (FGM) rectangular thin plates clamped at four edges. Firstly, the vibration equation of the sigmoid FGM rectangular thin plate is transformed into Hamiltonian canonical equations. Then based on the analysis of boundary conditions of the plate, the vibration problem of the sigmoid FGM rectangular thin plate clamped at four edges is decomposed as two sub-problems simply supported on two opposite edges. And then based on the symplectic elasticity method (SEM), the series expansion solutions for the two sub-vibrating problems are obtained. Afterward, by superposing the series expansion solutions for the two sub-vibrating problems, a symplectic superposition solution of the freely vibrating sigmoid FGM rectangular thin plates clamped at four edges is obtained. Finally, we apply the obtained symplectic superposition solution to research the frequencies and vibration mode functions of some specific sigmoid FGM rectangular thin plates, and the impact of the aspect ratio a/b and material gradient index k on the vibration frequencies of the sigmoid FGM rectangular thin plates is studied.

Computational modeling of sigmoid functionally graded material (SFGM) plate

The first-order shear deformation theory (FSDT) was used to explore the natural frequency response of functionally graded piezoelectric plates subjected to static electrical and mechanical strain in this present study. A monomorph model for a functionally graded piezoelectric plate with material properties that change according to sigmoid law with respect to plate thickness has been considered. A three-dimensional finite element model with a free tetrahedral element mesh was created using COMSOL 4.2 Multiphysics® software, with each node having three degrees of freedom. Variations in the FGPM plate’s free vibration boundary conditions, composition, and geometry have all been investigated. In free vibration analysis, non-dimensional natural frequency of FGPM plate initially decreasing considerably and then remaining almost constant with the increase in volume fraction index when material property graded by power law. When material properties are varied by sigmoid law, with an increase in volume fraction index, the non-dimensional natural frequency of FGPM plates remains virtually constant. FGPM plates have a lower non-dimensional natural frequency if the thickness to width ratio is greater. Non-dimensional natural frequencies of Clamped-Clamped FGPM plates (C–C–C–C) are greater compared to Clamped-Free FGPM (C–F–C–F) and Simply Supported Free FGPM (S–F–S–F).

Sigmoid functionally graded plates embedded on Winkler-Pasternak foundation: Free vibration analysis by dynamic stiffness method

In this present paper, the dynamic stiffness method has been formulated to calculate the natural frequency of sigmoid functionally graded material (S-FGM) plate embedded on the Winkler-Pasternak elastic foundation. The material properties of S-FGM continuously vary along the transverse direction of the plate by using two power-law variations in terms of volume fraction of the constituent's material. Hamilton's principle is implemented to derive the governing partial differential equation of motion based on the classical plate theory considering the physical neutral surf ace of the FGM rectangular plate. The Wittrick -Williams algorithm is applied as a solution technique to solve the transcendental nature of the dynamic stiffness matrix and extract the natural frequencies of the FGM plate with the desired accuracy. The S-FGM plate parameters’ variation of natural frequencies with the change of parametric numerical values (aspect ratio, sigmoid volume fraction index, boundary conditions and elastic foundation parameters, density ratio, and modulus ratio) are also highlighted. The DSM results are compared and validated with the available published literature. A new set of natural frequency results for the S-FGM plate embedded on the Winkler-Pasternak elastic foundation are generated.

FE Analysis of FGM Plates on Arbitrarily Orthotropic Pasternak Foundations for Membrane Effects

In this study, the finite element analysis of sigmoid functionally graded material (S-FGM) plates resting on orthotropic Pasternak elastic foundations with different material angles is presented. For modelling, SAP2000 software package is used in which the required adjustments are made to obtain the expected behaviour of the plate-foundation system. The plate is modelled both using solid elements and layered shell elements by defining a number of solid elements and layers in the thickness direction having elastic properties equivalent to the properties of the S-FGM plate. The interaction between the plate and the foundation is treated by equalizing the vertical displacements of the plate and foundation nodal points. The orthotropic Pasternak foundation is modelled using plane strain elements with some adjustments to the elastic properties. The membrane effects of the simply supported S-FGM plate on Pasternak foundation are considered by assigning the edge boundaries of the system as pinned supports and these effects are excluded by converting all boundary nodes into roller supports except one of the corner nodes of the plate and the foundation due to the stability requirement. A number of verification examples are performed to demonstrate the convenience and robustness of the proposed model. This work can be easily extended to static and dynamic analyses of FGM plates with various geometries resting on arbitrarily orthotropic Pasternak elastic foundations for further studies.

Free vibration of sigmoid functionally graded plates using the dynamic stiffness method and the Wittrick-Williams algorithm

In this paper, the free vibration characteristics of Sigmoid Functionally Graded Material (S-FGM) Levy-type plates are investigated by developing the Dynamic Stiffness Method (DSM) through the application of the Wittrick-Williams algorithm, as solution technique. Kirchoff-Love Plate Theory (KLPT) and Hamilton principle are utilised to derive the governing equation of motion and associated natural boundary conditions. Based on two power-law distribution functions, the material properties are gradually varied along the thickness direction. Using the proposed theory, a substantial number of numerical examples showing the natural vibration characteristics of plates made of sigmoid functionally graded material are illustrated to demonstrate the accuracy of the method. Some numerical results are compared with published results and found to be in excellent agreement. An extensive investigation is carried out and the results are examined and discussed in detail. The variations of material properties such as the Young’s modulus ratio and density ratio are seen to affect the natural frequencies of S-FGM plates significantly. The proposed method is not only accurate but also, quite simple and straightforward to compute the natural frequencies and mode shapes of S-FGM plates. The results presented can be used as benchmark solution for further investigation of FGM plates.

Analyses of thermal buckling and secondary instability of post-buckled S-FGM plates with porosities based on a meshfree method

The present research deals with thermal buckling and postbuckling of sigmoid functionally graded material (S-FGM) plates with porosities resting on elastic foundation based on the popular third-order shear deformation theory and the well-known von Kármán large deflection assumption. Special attention is paid to scrutinize the secondary instability that occurs during postbuckling regime. In addition, the transitions of deflection shapes of the plates under various boundary conditions between primary and secondary postbuckling paths are investigated. A displacement control strategy and a meshfree radial point interpolation method (RPIM) which employs a radial basis function without any adaptive parameters are utilized synthetically to carry out thermal buckling loads and postbuckling paths of the plates numerically. Numerical comparisons are performed to verify the convergence of the modified RPIM and the accuracy of iteration method in postbuckling path analysis. The influences of functionally graded index, porosity coefficient and parameters of elastic foundation to behaviors of the thermal buckling and especially the secondary instability occurs during postbuckling regime are investigated. It is found that the secondary instability is sensitive to the boundary conditions.

An analytical framework for rectangular FGM tapered plate resting on the elastic foundation

This article presented the free vibration analysis of tapered rectangular functionally graded material (FGM) plates resting on the two-parameter (Winkler and Pasternak) elastic foundation model. Higher-order shear deformation theory (HSDT) to satisfy the traction free boundary conditions on the top and bottom surface of the sigmoid FGM (S-FGM) plate. The material properties are varied according to the two simple power-law distributions of volume fractions of the constituents. The thickness of the plate linearly varies in the y–direction to both sides of the middle plane. The variational Hamilton principle establishes the equation of motions, and Galerkin-Vlasov’s method is used to solve the equation of motion. The free vibration is evaluated with the inclusion of an elastic foundation with different taper ratios. A parametric study has been investigated for the different span ratio, aspect ratio, volume exponent index, taper ratio, and combination of elastic foundation.

Nonlinear oscillations of sigmoid functionally graded material plates moving in longitudinal direction

Geometrically nonlinear oscillations are investigated on sigmoid functionally graded material (S-FGM) plates with a longitudinal speed. The material properties of the plates obey a sigmoid distribution rule along the thickness direction. Based on the D’Alembert’s principle, a nonlinear equation of motion is derived for the moving S-FGM plates, where the von K´arm´an nonlinear plate theory is adopted. Utilizing the Galerkin method, the equation of motion is discretized and solved via the method of harmonic bal-ance. The approximate analytical solutions are validated through the adaptive step-size fourth-order Runge-Kutta method. Besides, the stability of the steady-state solutions is examined. The results reveal that the mode interaction behavior can happen between the first two modes of the moving S-FGM plates, leading to a complex nonlinear frequency response. It is further found that the power-law index, the longitudinal speed, the exci-tation amplitude, and the in-plane pretension force can significantly affect the nonlinear frequency-response characteristics of longitudinally traveling S-FGM plates.

Large-amplitude vibration of sigmoid functionally graded thin plates with porosities

This research focuses on the large-amplitude vibration of sigmoid functionally graded material (S-FGM) thin plates with porosities. Porosities in S-FGM plates can happen due to technical issues during the preparation of S-FGMs. Two types of porosity distribution, i.e., even and uneven distribution, are taken into account. The material properties of S-FGM plates with porosities change smoothly along the thickness direction based on the sigmoid distribution law, which is described by modified piecewise functions. The geometrical nonlinearity is considered by applying the von Kármán non-linear plate theory. The nonlinear governing equation of S-FGM plates with porosities is derived using the D′Alembert's principle. By applying the Galerkin method with the first three modes, the governing equation is discretized to three ordinary differential equations. Then, the method of harmonic balance is used to solve these discretized equations. Analytical results are verified numerically with the adaptive step-size fourth-order Runge-Kutta method. The stability of the steady-state response is examined by means of the perturbation technique. Furthermore, the maximum amplitudes of each mode during the vibration period are obtained and shown in the neighborhood of the fundamental mode. Study demonstrates that the S-FGM plates with porosities possess hardening spring characteristics in nonlinear frequency response. Moreover, a complex multi-solution phenomenon occurs in the present dynamic system which is rooted from the nonlinear mode interaction. Finally, investigation is made on the effects of porosity along with other key parameters on large-amplitude vibration response of porous S-FGM plates.

Vibration characteristics of moving sigmoid functionally graded plates containing porosities

This study investigates vibration characteristics of longitudinally moving sigmoid functionally graded material (S-FGM) plates containing porosities. Two types of porosity distribution, i.e., the even and uneven distributions, are taken into account. In accordance with the sigmoid distribution rule, the material properties of porous S-FGM plates vary smoothly along the plate thickness direction. The nonlinear geometrical relations are adopted by using the von Karman non-linear plate theory. Based on the d’Alembert’s principle, the nonlinear governing equation of the system is derived. Then, the governing equation is discretized to a set of ordinary differential equations via the Galerkin method. These discretized equations are subsequently solved by using the method of harmonic balance. Analytical solutions are verified with the aid of the adaptive step-size fourth-order Runge–Kutta method. By using the perturbation technique, the stability of the steady-state response is highlighted. Finally, both natural frequencies and nonlinear forced responses of moving porous S-FGM plates are examined. Results demonstrate that the moving porous S-FGM plates exhibit hardening spring characteristics in the nonlinear frequency response. Moreover, it is shown that the type of porosity distribution, moving speed, porosity volume fraction, constituent volume fraction and in-plane pretension all have significant influence on the nonlinear forced responses of moving porous S-FGM plates.

Thermo-Micromechanical Vibration Behaviors of FGM Structures with Neutral Surface

This work presents the micro-mechanical models in thermal environment for the vibration behavior of Functionally Graded Materials (FGMs) plate using First-order Shear Deformation Theory (FSDT). In the formulation, the heat transfer effects and the temperature-dependent material properties are considered. Relative estimation of micromechanical behaviors of Mori-Tanaka Method (MTM) is used. And, neutral surface concept is adopted as the reference plane due to the asymmetry in the thickness direction of the model. In the numerical analysis, Finite Element Method is applied for various volume fractions and temperature rising conditions. Also Power-law and Sigmoid FGMs are discussed in thermo-elastic vibration characteristics.

Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory

The modified couple stress-based third-order shear deformation theory is presented for sigmoid functionally graded materials (S-FGM) plates. The advantage of the modified couple stress theory is the involvement of only one material length scale parameter which causes to create symmetric couple stress tensor and to use it more easily. Analytical solution for dynamic instability analysis of S-FGM plates on elastic medium is investigated. The present models contain two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton\'s energy principle. The governing equations are then written in the form of Mathieu-Hill equations and then Bolotin\'s method is employed to determine the instability regions. The boundaries of the instability regions are represented in the dynamic load and excitation frequency plane. It is assumed that the elastic medium is modeled as Pasternak elastic medium. The effects of static and dynamic load, power law index, material length scale parameter, side-to-thickness ratio, and elastic medium parameter have been discussed. The width of the instability region for an S-FGM plate decreases with the decrease of material length scale parameter. The study is relevant to the dynamic simulation of micro structures embedded in elastic medium subjected to intense compression and tension.

Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory

The hygro-thermo-mechanical bending behavior of sigmoid functionally graded material (S-FGM) plate resting on variable two-parameter elastic foundations is discussed using a four-variable refined plate theory. The material characteristics are distributed within the thickness direction according to the two power law variation in terms of volume fractions of the constituents of the material. By employing a four variable refined plate model, both a trigonometric distribution of the transverse shear strains within the thickness and the zero traction boundary conditions on the top and bottom surfaces of the plate are respected without utilizing shear correction factors. The number of independent variables of the current formulation is four, as against five in other shear deformation models. The governing equations are deduced based on the four-variable refined plate theory incorporating the external load and hygro-thermal influences. The results of this work are compared with those of other shear deformation models. Various numerical examples introducing the influence of power-law index, plate aspect ratio, temperature difference, elastic foundation parameters, and side-to-thickness ratio on the static behavior of S-FGM plates are investigated.

Effects of variable thickness and imperfection on nonlinear buckling of sigmoid-functionally graded cylindrical panels

The use of variable thickness can help the designers and researches reduce the weight of the functionally graded (FG) panel structures. For cases where reduction of weight is of high importance, such as space structures, ocean engineering, this type of panel is the best choice. Hence, this paper analyzes the effect of the variable thickness on nonlinear buckling of imperfect cylindrical panels made of sigmoid-functionally graded material (S-FGM) under combined axial compression and external pressure. The governing equations are in nonlinear form based on the classical shell theory with the von Karman assumption. By applying Galerkin procedure and the Airy stress function, the resulting equations are solved to obtain closed form expressions for critical buckling load and load–deflection curves. In numerical results, effect of variable thickness, the volume fraction index, imperfection size, loading as well as the geometric parameters on the load–dimensionless deflection curves are discussed in details.

A new approach for nonlinear dynamic buckling of S-FGM toroidal shell segments with axial and circumferential stiffeners

This paper presents an analytical approach to study the nonlinear dynamic buckling of sigmoid functionally graded material (S-FGM) toroidal shell segments surrounded by elastic foundations subjected to axial compression. Shells are reinforced by closely spaced rings and stringers. The nonlinear dynamic buckling analysis is based on the classical shell theory with von Kármán–Donnell-type of geometrical nonlinearity. By using the Galerkin method, the closed-form expression of the deflection–time curves is obtained. Numerical solutions are presented in tabular and graphical forms to investigate the deflection–time curves of FGM shells surrounded by elastic foundations. Finally, the influences of geometrical parameters, volume fraction index, elastic foundations, and the effectiveness of stiffeners on the nonlinear dynamic buckling of the S-FGM toroidal shell segments are discussed. The present results are compared with the available data for a special case.

Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation

A refined higher-order shear deformation theory is developed for the analysis of free and forced vibration of sigmoid functionally graded materials (S-FGM) plates. The theory, proposed in this paper, considers the parabolic distribution of the transverse shear stress, and satisfies the condition that requires the transverse shear stress to be zero on the upper and lower surfaces of the plate, without the shear correction factor. Unlike the conventional higher-order shear deformation theory, the refined higher-order shear deformation theory, even though it uses only four unknown variables, shares strong similarities with classical plate theory (CPT) in many aspects such as boundary conditions, equation of motion, and stress-resultant expressions. The material properties of the plate are assumed to vary according to the two power law distributions of the volume fractions of the constituents. The equations of motion are derived from Hamilton׳s principle. The solutions for a simply supported plate are derived, and a comparative analysis is carried out by comparing the results obtained with first-order shear deformation theory and another, higher-order shear deformation theory. The results of the comparative analysis with the proposed theory provide accurate and relevant results for free-vibration problems of Functionally Graded Materials (FGM) plates. Analytical solutions for the forced-vibration problems are presented so as to reveal the effects of the power law index, length, aspect ratio, loading time interval, elastic medium parameters and side-to-thickness ratio of the plate on the dynamic response.

Characteristics of Power Law and Sigmoid FGMs Models in Thermal Effect

In thermal environment, vibration behavior of Functionally Graded Materials (FGMs) plates is investigated, and the materials are developed with mixing ceramic and metal. Present study is based on the first-order shear deformation theory of plate. Then, mixture methods such as Power law (P-) and Sigmoid (S-) models are chosen. According to a volume fraction, the material properties are assumed to vary continuously through the thickness direction and to be temperature dependent properties. Further, thermal effects are considered as uniform temperature rise and one dimensional heat transfer. For the structure analysis, FEM is used to obtain the natural frequencies based on the virtual work principle.

Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium

With the aim of reducing the weight of structures, functionally graded materials (FGM) plates with variable thickness have been widely used in various engineering applications such as aeronautical, mechanical and ocean structures. However so far, the analytical approaches for analyzing the instability behaviors of FGM plates with variable thickness are still somehow limited in the literature. The paper hence presents an analytical approach to investigate the influences of variable thickness on buckling and postbuckling behavior of imperfect sigmoid FGM (S-FGM) plates resting on elastic medium subjected to compressive loading. The material properties of S-FGM plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Governing equations are based on the classical plate theory with von Karman-type geometric nonlinearity. The initial geometrical imperfections of plates are also accounted. By using the Galerkin procedure and the Airy stress function, the resulting equations are solved to obtain the closed form expressions for nonlinear equilibrium paths. The effects of power-law indices, coefficients of foundation, initial geometrical imperfections and geometrical parameters on nonlinear stability of plates are comprehensively investigated. The results reveal that the variable thickness of plate has a significant effect on the buckling behavior of S-FGM plates under compression loading.

Structural Stability and Dynamics of FGM Plates Using an Improved 8-ANS Finite Element

I investigate the vibration and buckling analysis of functionally graded material (FGM) structures, using a modified 8-node shell element. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function. The modified 8-ANS shell element has been employed to study the effect of power law index on dynamic analysis of FGM plates with various boundary conditions and buckling analysis under combined loads, and interaction curves of FGM plates are carried out. To overcome shear and membrane locking problems, the assumed natural strain method is employed. In order to validate and compare the finite element numerical solutions, the reference results of plates based on Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are compared. Results of the present study show good agreement with the reference results. The solutions of vibration and buckling analysis are numerically illustrated in a number of tables and figures to show the influence of power law index, side-to-thickness ratio, aspect ratio, types of loads, and boundary conditions in FGM structures. This work is relevant to the simulation of wing surfaces, aircrafts, and box structures under various boundary conditions and loadings.

Nonlinear Response of Imperfect Eccentrically Stiffened Ceramic-Metal-Ceramic Sigmoid Functionally Graded Material (S-FGM) Thin Circular Cylindrical Shells Surrounded on Elastic Foundations Under Uniform Radial Load

This article researches nonlinear response of imperfect eccentrically stiffened symmetric FGM thin circular cylindrical shells with ceramic-metal-ceramic layers, which are symmetric through the middle surface by Sigmoid-law distribution (S-FGM) and have stiffeners surrounded on elastic foundations under uniform radial load. The Donnell classical shell theory, stress function, and Galerkin method are used for investigation of the nonlinear stability of the S-FGM shell. The obtained results show the effects of the stiffeners, elastic foundations, mechanical load, and material parameters on the nonlinear buckling response of symmetric S-FGM circular cylindrical shells.