FGM Plates Research Articles

Assessment of non‐polynomial shear deformation theories for the free vibration and transient analysis of plates with functionally‐graded materials supported on an elastic foundation

AbstractIn this article, various non‐polynomial higher‐order shear deformation theories are applied for the first time to analyze the free vibration and transient responses of plates with functionally graded material (FGM) supported on an elastic foundation. The shear deformation theories account for the non‐linear variation of the transverse shear strains with various warping functions, namely trigonometric, inverse hyperbolic, and inverse trigonometric ones. These models also inherently satisfy the traction‐free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plates and do not require any shear correction factor. A two‐parameter model, namely Winkler‐Pasternak's elastic foundation model, is utilized to develop the interaction between the FGM plates and the elastic medium. The governing equations of motion are obtained using Hamilton's principle and solved analytically using Navier's solution scheme. Furthermore, the transient responses of the plates are obtained using Newmark's average acceleration method. The applicability of the present theories is established by solving several numerical problems and validating the results with the solutions available in the literature. The effects of various parameters like span‐thickness ratios, aspect ratios, gradation coefficients, mechanical loads, and foundation stiffness on the fundamental frequencies and the transient responses of the plates are thoroughly investigated. The comparison of the results reveals the efficiency of the non‐polynomial functions, and the capability of efficient prediction of the structural responses of the FGM plates at a similar computational cost compared to established models in the literature. Furthermore, the results show that the stiffness of the elastic foundation can tweak the stiffness characteristics of the FGM plate resulting in significant changes in the natural frequencies and more controlled displacement‐time responses.

A high-order finite element continuation for buckling analysis of porous FGM plates

This work presents an effective approach to investigating the buckling and post-buckling behavior of porous FGM plates. The main objective of the present analysis is to use a High Order Continuation based on the Asymptotic Numerical Method with the Finite Element Method for nonlinear behaviors of a Porous Functionally Graded Material plate (PFGM) with different porosity distributions under various types of transverse loads. In our study, we will take into account the lack of uniform and non-uniform porosity using a Higher-order Shear Deformation Theory (HSDT). The adapted solver proves to be more effective for analyzing non-linear mechanical phenomena and more precisely the post-buckling analysis of the PFGM plates. Indeed, this numerical high order algorithm presents an adaptive step length to the local nonlinearity of the solution branch and to detect easily the bifurcation points and bifurcated branches. When compared to iterative techniques like Newton–Raphson, the proposed high-order algorithm has the benefit of speeding up the resolution of nonlinear problems. The mathematical formulation is based on the higher-order shear deformation plate theory taking into consideration geometrical nonlinearity and initial imperfections. Numerical results of a parametric study on the effect of various types of load, porosity distribution, grading index, porosity index and different span-to-thickness ratios will be investigated.

Vibration analysis of functionally graded material (FGM) double layered floating raft structure by the spectro-geometric method

This paper aims to analyze the vibration characteristics of functionally graded material (FGM) double layered floating raft structures under various boundary conditions. The artificial virtual spring technique is introduced to simulate the boundary conditions and coupling effect between the FGM plates by adjusting the spring stiffness. The spectro-geometric method is used to describe the displacement admissible function, which overcomes the discontinuity or jump phenomenon at the boundary for the conventional Fourier series. Based on the unified Rayleigh-Ritz energy framework, the dynamic equations of the FGM double layered floating raft structure are derived, and the corresponding vibration characteristics are obtained. The correctness of the presented method is verified by some numerical examples of comparisons with the calculation results obtained from the finite element method. After that, parametric analysis is carried out to investigate the effects of material parameters, geometric parameters, coupling spring stiffness, and coupling damping on the steady-state response of the FGM double layered floating raft with various boundary conditions.

NONLINEAR TRANSIENT RESPONSE OF A PERFORATED FGM PLATE UNDER THERMO-MECHANICAL CONDITIONS

The present study is aimed to investigate the nonlinear transient response of an aluminum-zirconia functionally graded material (FGM) plate with a central circular hole under thermo-mechanical loadings. The nonlinear formulation is based on the first-order shear deformation plate theory along with geometric nonlinearity incorporated in the von Karman sense. The finite element numerical technique is employed to solve approximately the governing equations of motion derived through Hamilton's principle. A MATLAB program, developed to carry out the investigation, is verified for accuracy by comparing the results with those reported in the literature-under mechanical and thermo-mechanical loadings. Numerical results are presented to show the effect of various parameters, viz. material gradation exponent, geometric variables, hole size, and boundary conditions, on the nonlinear transient response of the FGM plate. It is found that the material gradation exponent has a noticeable effect on the transverse deflection of vibration of the FGM plate with a hole, but the frequency of vibration is not affected much. Further, with the increase in the size of hole in the FGM plate, the amplitude of its vibration decreases while the frequency of vibration increases.

Free vibration analysis of FGM plates and shallow shells by the R-functions method

Research of free vibration of porous functionally graded material (FGM) plates with complex shape is carried out. It is supposed that thickness of the plate changes in direction of one of the axes. Two types of porosity distributions through the thickness even and uneven are considered. Elastic foundation is two parameters (Winkler and Pasternak). To obtain the mathematical model of the given problem the first order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction is modelled by the power law. Variational Ritz’s method joined with the R-functions theory is used for solution of the formulated problem. The developed approach is tested on a lot of problems. Comparative analysis of the obtained results for rectangular plates confirms their verification. Demonstration of the efficiency of the proposed approach is fulfilled for FGM plate with complex shape and various boundary conditions. Effect of the different parameters on vibration characteristics such as porosity, volume exponent, elastic foundation, types of FGM, boundary conditions is studied.

Contribution to study the effect of (Reuss, LRVE, Tamura) models on the axial and shear stress of sandwich FGM plate (Ti–6A1–4V/ZrO<sub>2</sub>) subjected on linear and nonlinear thermal loads

<abstract> <p>The principal goal of the current work is to study the impact of three homogenization models (Reuss, LRVE, Tamura) on the axial and shear stress of sandwich functionally graded plate materials subjected on linear and nonlinear thermal loads with static and elastic behavior and it is simply supported using an integral higher shear deformation theory (HSDT). The governing partial differential equations are solved in the spatial coordinate by Navier solution. Those Numerous micromechanical models have been examined to attain the effective material properties of the two-phase FGM plate. The numerical results are compared with those given by other model existing in the literature to confirm the accuracy of the (HSDT). The present results are in good agreement with all models studied of homogenization for all values of the material index and all geometry configurations of the FG-sandwich plates.</p> </abstract>

Effect of porosity inclusions on the natural frequencies of the FGM plates using dynamic stiffness method

Effect of porosity inclusions on the natural frequencies of the FGM plates using dynamic stiffness method

Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load

Due to the high importance of viscoelastic materials in modern industrial applications, besides the intensive popularity of piezoelectric smart structures, analyzing their thermoelastic response in extreme temperature conditions inevitably becomes very important. Accordingly, this research explores the thermoviscoelastic response of sandwich plates made of a functionally-graded Boltzmann viscoelastic core and two surrounding piezoelectric face-layers subjected to electrothermal load in the platform of three-dimensional elasticity theory. The relaxation modulus of the FG viscoelastic layer across the thickness follows the power law model. the plate’s governing equations are expressed in the Laplace domain to handle mathematical complications corresponding to the sandwich plate with a viscoelastic core. Then, the state-space method, combined with Fourier expansion, is utilized to extract the plate response precisely. Finally, the obtained solution is converted to the time domain using the inverse Laplace technique. Verification of the present formulation is compared with those reported in the published papers. Finally, the influences of plate dimension, temperature gradient, and relaxation time constant on the bending response of the above-mentioned sandwich plate are examined. As an interesting finding, it is revealed that increasing the length-to-thickness ratio leads to a decrease in deflections and an increase in stresses.

Dynamic Analysis of Simply Supported Functionally Graded Plates

Functionally graded materials (FGMs) are considered as a special composite material that is synthesized for a specific purpose by gradual mixing of two or more different materials thus with varying material properties through their thickness. Due to smooth and continuous varying material properties from one face to another, FGMs are usually superior to the conventional composite materials in mechanical behavior. Hence the dynamic analysis of FGM plates is essential. In this paper, numerical solutions for free vibration analysis of moderately thick plates composed of functionally graded materials with simply supported boundary conditions using finite element analysis have been studied. First-order shear deformation plate theory (FSDT) has been applied in the research. Eight noded isoparametric serendipity plate bending elements have been used. Young’s modulus and density per unit volume are assumed to vary continuously through the plate thickness according to a power-law distribution. The results have been validated with the existing literature available from other analytical and numerical techniques. The effect of different plate parameters such as aspect ratios, edge constraints, thickness to length ratios, and gradient indices on the natural frequencies of FG rectangular plates are presented. It is found that the natural frequency depends considerably on the said parameters.

Mathematical Approach for Mechanical Behaviour Analysis of FGM Plates on Elastic Foundation

This paper presents the flexural analysis of functionally graded plates resting on elastic foundations using new two-dimensional (2D) and quasi-three-dimensional (quasi-3D) higher order shear deformation theories. The main interesting feature of this theory is that it proposes a new displacement field with undetermined integral variables which involves only five unknown functions, unlike other shear and normal deformation theories, hence making it easier to use. A parabolic transverse shear deformation shape function satisfying the zero shear stress conditions on the plate outer surfaces is considered. The elastic foundation follows the Pasternak mathematical model. The material properties change continuously across the thickness of the FG plate using different distributions: power law, exponential, and Mori–Tanaka models. The governing equations of FG plates subjected to sinusoidal and uniformly distributed loads are established through the principle of virtual works and then solved via Navier’s procedure. In this work, a detailed discussion on the influence of material composition, geometric parameters, stretching effect, and foundation parameters on the deflection, axial displacements, and stresses is given, and the obtained results are compared with those published in previous works to demonstrate the accuracy and the simplicity of the present formulations. The different obtained results were found to be in good agreement with the available solutions of other higher-order theories. The proposed model is able to represent the cross section warping in the deformed shape and to demonstrate the validity and efficiency of the approach, the findings reported herein prove that this theory is capable of predicting displacements and stresses more accurately than other theories, as its results are closer when compared to numerical methods reported in other literatures.

Functionally graded materials in transport vehicles – overview, fabrication, application, modelling

This paper deals with functionally graded materials (FGM) as modern materials in transport vehicles. Basic methods of FGM production and areas of their application are presented, as well as an example of modelling an FGM plate. Bending of freely supported square FGM plates under the influence of sinusoidal transverse load was analyzed. The plates are made of metal (aluminium) and ceramic (alumina) in which material characteristics change as a function along the thickness of the plate. The static analysis was done by applying finite elements methods in the FEMAP software package for different mesh sizes. The results were compared with those obtained in the MATLAB software package using the classical plate theory, quasi 3D theory as well as high order shear deformation theories. The previously mentioned comparative results were analysed and certain observations and appropriate conclusions were made.

Advanced Dynamic Thermal Vibration of Laminated FGM Plates with Simply Homogeneous Equation by Using TSDT and Nonlinear Varied Shear Coefficient

The effects of advanced nonlinear varied shear coefficient and third-order shear deformation theory (TSDT) on the dynamic responses of thick functionally graded material (FGM) plates under thermal vibration are investigated. The nonlinear coefficient of the displacement field of TSDT is used to obtain the expression of advanced varied shear coefficient for the thick FGM plates. The dynamic displacements, shear rotations and stresses in numerical results under sinusoidal applied heat loads are obtained and investigated. Two parametric effects of environment temperature and FGM power law index on the dynamic responses of thermal stress and center deflection of thick FGM plates are also investigated. The transient responses of center deflection are found for the cases of simply homogeneous equation and fully homogeneous equation. Also, the transient responses of center deflection are found for cases of nonlinear and linear varied-modified coefficient of shear correction.

New quasi-three-, and two-dimensional trigonometric-cubic monomial HSDT for thermal buckling and thermo-mechanical bending analyses of FGM symmetrical/non-symmetrical sandwich plates with hard/soft core

This paper analyses the thermal buckling and thermomechanical bending responses of sandwich plates with FGM face layers, using new quasi-three-, and two-dimensional HSDTs in a unified theoretical formulation. The theory includes both the shear strain illustrated by a new shape function and the stretching effect.The theory makes use of the Hamilton principle to derive the plate governing equations. Then, the Navier procedure solved them for the simply supported boundary conditions. The theory assumed the thermal buckling loads to be uniform, linear, and non-linear distributed through the plate’s thickness, the FGM layers to be isotropically modelled according to a power law and the core layer to be isotropic and homogeneous.After validating the model's analytical solutions with those from the literature (to confirm the model's accuracy), various innovative studies extended the theory. They cover the effects of important parameter variations related to the plate dimensions, material, and the development of the proposed HSDT on the central deflection, stresses, and critical buckling temperature. The theory solutions also present the good performance of the current generalized formulation and the significant influence of the thermal effect. References in the literature can use these findings as foundational data for future research on FGM plates.

Influence of circular cutouts and elastic foundation on vibration characteristics of porous sandwich FGM plates under hygro-thermal environments

The present article investigates the effect of cutouts, porosity inclusion, and hygro-thermal environment on the vibrational characteristics of sandwich functionally graded material (FGM) plates resting on Pasternak elastic foundation. The governing equations for the sandwich FGM plates have been developed using the finite element method (FEM) with the conjunction of non-polynomial-based higher-order shear deformation theory (HSDT). The trigonometric shear-strain shape function has been used for the first time for sandwich FGM plates. The effective material properties with porosity inclusions have been computed using modified power–law distribution. The present results have been acquired by employing a C0 continuity isoparametric finite element (FE), using 4-noded elements with 7-DOF's per node. The accuracy of the current solution has been confirmed by convergence and validation studies. The influence of circular cutout, elastic foundations, hygro-thermal conditions, and geometric and boundary conditions on the vibrational frequency of porous sandwich FGM plates have been discussed in detail.

Porosity-dependent free vibration and transient responses of functionally graded composite plates employing higher order thickness stretching model

Tailoring the mechanical properties in hybrid materials, like functionally graded materials (FGMs), has become more pragmatic with the recent progress in material manufacturing and design. The porosities in FGMs, which develop during the manufacturing process, may become detrimental to armored plates or bulletproof designs. On the other hand, gradient-porosity distributions may have broad advantages in the design of lightweight and variable-stiffness aircraft components. The present study investigates the free vibration and transient responses of porosity-gradient FGM plates. A higher order theory, which considers the transverse shear and normal deformation, warping of the transverse cross-section, and higher order rotary inertia, is utilized for the first time to model the deformation of the porous FGM plate. The advantages of this model are (i) higher order kinematic terms for membrane and bending deformations due to the coupled membrane-bending behavior of FGMs in addition to the thickness-stretching effects, and (ii) the flexibility of applying external loads at different points along the thickness direction of the FGM plate. An analytical solution technique, popularly known as Navier’s approach, is adopted for the spatial solutions, and Newmark’s average acceleration method is utilized for the temporal solutions. The influence of the geometrical and porosity parameters on the structure’s vibration response is studied. Natural frequencies are affected significantly with uniform porosity distribution, while graded porosities can tweak vibration response in a small range but with better control. In the absence of the elasticity solutions, the present results may be considered the possible benchmark solutions for comparison of the other two-dimensional (2D) kinematic models.

Effect of thickness stretching and electro-mechanical loading on sandwich FGM plate with piezoelectric face sheet with different boundary conditions: A semi-analytical approach

Functionally graded material (FGM) belongs to the class of advanced composite materials with material properties that vary according to the volume fraction. FGM is the future material for smart structures, being used in place of conventional composite materials. The article aims at performing the static analysis of different types of sandwich FGM plates with a piezoelectric face sheet taking into consideration the porosity and thickness stretching effect. The problem is formulated using a variational approach and the bending solution is obtained for the first time using Galerkin Vlasov’s method. Three different types of porous sandwich plates are considered for the analysis when subjected to sinusoidally distributed mechanical and electrical loading. For the first time, a new uneven non-symmetric porosity model is put forward to capture the realistic bending behavior of the plate. The effect of boundary conditions, thickness stretching, and applied voltage on stress and deflection distribution along and across the porous plate’s surface is analyzed. It is observed that effect of the applied potential difference when assumed to be a function of position, the change in stresses and deflection are approximately 30%. This infers that voltage distribution plays a significant role in the distribution of stresses in the plate and cannot be neglected which is otherwise assumed as a constant by various researchers. To check the exactness and validity of the current formulation, various examples are discussed. The estimated results can be used as a validation study for the upcoming research on porous sandwich FGM plates with piezoelectric facesheets. Communicated by Francesco Tornabene

Accurate solution for free vibration behaviour of stepped FGM plates implementing the dynamic stiffness method

In this article, free vibration response of stepped functionally graded material (FGM) plate is studied through the implementation of dynamic stiffness method (DSM). The material model of the plate is assumed to have the power-law (P-FGM) property variation through the thickness. The kinematic variables within the Kirchhoff’s plate theory are defined by considering the concept of the physical neutral surface to avoid the bending-stretching coupling of the P-FGM plate. The governing equation of motion pertaining to a single plate element are obtained through the application of Hamilton’s principle. Levy type (closed form) solution to this equation of motion is sought to formulate the DS matrix through the implementation of displacement and force edge conditions at the two opposite ends of the plate. The dynamic stiffness (DS) matrix is first formulated for a single FGM plate element and later these DS matrices are suitably assembled to obtain the global DS matrix of the stepped FGM plate. The assembled global DS matrix is finally solved using the Wittrick-William’s algorithm to compute the accurate natural frequencies of the stepped FGM plates. Through different comparisons, it has been demonstrated that the frequency results obtained from the DSM are very accurate for the stepped P-FGM plates. New set of results are also presented by varying different geometrical and material parameter of the stepped plate. The results reported from this parametric study will be beneficial in the design phase of various stepped P-FGM plates.

Thermo‐mechanical nonlinear stability analysis of geometrically imperfect functionally graded plates with microstructural defects using logarithmic structure kinematics: An unified expression

AbstractIn the present paper, thermo‐mechanical stability analysis of geometrically imperfect porous functionally graded plates (FGP) with geometric nonlinearity is presented. The equilibrium, stability, and compatibility equations are derived using Logarithmic structural kinematics in conjunction with the von‐Karman type of geometric nonlinearity. A logarithmic‐based shear‐strain function has been used for the first time for the buckling response of functionally graded material (FGM) plates. Generic imperfection function has been implemented to incorporate the various imperfection modes like sin‐type or global type in the formulation. The effective materials properties of the plates with porosity inclusion have been computed using modified power law. The plate is subjected to uniaxial and biaxial compression as well as combined compression and tension along with thermal loading. An exact expression for the critical buckling load and critical buckling thermal load of geometrically imperfect porous FGM plate for each loading case has been developed. After confirming the excellent accuracy of the current exact solutions, the effect of geometric imperfection, porosity inclusion, and geometric configurations on the nonlinear stability of the FGM plate have been discussed extensively. The results presented in this paper will be used as a benchmark for future research.

Advanced dynamic thermal vibration of thick FGM plates-cylindrical shells

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates-cylindrical shells under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient c1 term of TSDT displacement field is used to obtain the equations of motion and the advanced nonlinear shear correction coefficient expression for thermal vibration of thick FGM plates and cylindrical shells, respectively. At the intersection of FGM plates-cylindrical shells, the displacements, stress resultants and moment resultants could be considered and reasonably assumed in the continuity condition for the differential volume in the plates and cylindrical shells, respectively. The effects of advanced nonlinear shear correction, FGM power law index and environment temperature on the thermal stress and center displacement of FGM plates-cylindrical shells are investigated.

Semi-analytical solutions of static and dynamic degenerate, nondegenerate and functionally graded electro-elastic multilayered plates

In this paper mathematical modelling and semi-analytical solutions are elaborated based on the extended Stroh-like formalism for 3-D static problems of electro-elastic mono and multilayered plates. For the dynamic analysis of such structures under various excitation types, a time trigonometric series was coupled with the Stroh-like formalism. The unknown coefficient vectors are solution of the resulted eigenvalue problems and the solution is obtained in the associated basis. Mechanical and electrical forces are applied on the top and/or bottom layer and may take arbitrary forms. A flexible matrix formulation is elaborated to easily account for arbitrary mechanical and electrical surface loads and excitations. The methodological approach focuses on various aspects including the electro-elastic effects, the degeneracy of elastic isotropic materials as well as the effect of the graded material’s factor. Analytical solutions are obtained for the static behavior of degenerate and non-degenerate Elastic FGM plates, where the eigenvalues and eigenvectors are explicitly given. Mainly, semi-analytical solutions are presented for the dynamic behavior of piezoelectric sandwich plates by coupling Stroh-like formalism and trigonometric series. The presented methodological approach is applied to the dynamical analysis of sandwich plates under various excitation forms, namely sinusoidal, rectangular and sawtooth excitation pulses. The obtained numerical results demonstrate the effect of the graded factor, degeneracy and the applied surface load as well as of the excitation on mechanical and electrical responses for various laminated plates.