FGM Plates Research Articles

Transient analysis of functionally graded axisymmetric plates via finite element method in the Laplace domain

The principal goal of this research article is to examine the forced vibration of Functionally Graded Material (FGM) axisymmetric plates with the aid of the finite element method in the Laplace space. The material properties are assumed to be isotropic, linear viscoelastic, or elastic and vary continuously in the direction of the thickness of the plate. The governing equations of motion are transferred to the Laplace domain and solved for a set of Laplace parameters. Then, a modified Durbin’s inverse Laplace method is employed to retransfer the obtained results back to the time domain. A convergence analysis is conducted to optimize the number of Laplace parameters and time steps. An 8-node quadratic quadrilateral element featuring two degrees of freedom per node is employed to generate the model of the axisymmetric plates. Different loads such as step load, square wave, and sawtooth wave loads are used to observe the impact of load type on the transient response of FGM plates. In viscoelastic analysis, the Kelvin damping model is preferred, and the effect of the damping ratio is investigated. The obtained results are validated with the aid of the available literature and the accuracy of the suggested model is confirmed. It is carried out that the material variation significantly affects the dynamic behavior of the plates, for instance as the material coefficient increases, displacement amplitudes decrease, and periods increase. The novelty of this paper is the employment of the finite element method together with Laplace transformation for addressing the present class of problems for the first time.

Rapid heating of FGM plates resting on elastic foundation

Rapid heating of FGM plates resting on elastic foundation

Mixed mode stress intensity factor analysis on edge cracked FGM plate with different material distribution models by XFEM

Mixed mode stress intensity factor analysis on edge cracked FGM plate with different material distribution models by XFEM

Numerical Optimization of Functionally Graded Ti-HAP Material for Tibial Bone Fixation System.

Functionally graded materials (FGMs) are heterogeneous composites characterized by outstanding properties. They are built from two or more components with a gradient distribution of chemical composition along a given direction. A promising graded material for biomedical engineering as an implant could be a FGM made of titanium (Ti) and hydroxyapatite (HAP). It would allow us to counteract the difference between the stiffness modulus of pure titanium and bone tissue. Moreover, it can be a good solution to the problem of stress shielding for bone fixation plates made of conventional titanium or steel. The presented paper aims to perform micromechanical modeling and optimization of a functionally Ti-HAP graded plate, followed by numerical analysis of a fractured tibia stabilization system under specific boundary conditions. Finite element analysis was performed using ANSYS Workbench 2021 software. The models of the FGM plate and tibial fixation system were made using the Space Claim tool. The ANSYS software allowed the optimization of the model considered and the selection of the appropriate structural parameters of the FGM Ti-HAP material. In general, the results proved that the osteosynthesis plate built of graded Ti-HAP material resulted in lower bone stress compared to titanium and steel plates. The results obtained confirmed the validity of the design and the possibility to use functionally graded Ti-HAP bone fixation plates.

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Evaluation of the mechanical and microstructural properties of an Al7075/B4C/Gr FGM plate fabricated using a stir casting process

Evaluation of the mechanical and microstructural properties of an Al7075/B4C/Gr FGM plate fabricated using a stir casting process

Frequency studies of nonlinear TSDT FGM plates in the linear shear corrections and thermal environment temperature

The effects of third-order shear deformation theory (TSDT) and varied linear shear correction coefficient on the vibration frequency of thick functionally graded material (FGM) plates with fully homogeneous equations under thermal environment and power law are investigated. The nonlinear coefficient c_1 term of displacement field of TSDT is included in the fully homogeneous equation under vibration of FGM plates. The determinant of the coefficient matrix in dynamic equilibrium differential equations under vibration can be represented in the fully fifth-order polynomial equation, thus the natural frequency can be found. The natural frequency with/without the nonlinear c_1 term of displacement fields is investigated. It is a significant novelty with the consideration of the nonlinear c_1 term in the frequency computation.

Effect of circular and elliptical cutouts on the free vibration analysis of sandwich FGM plate under the elastic foundations

The present article investigates the effect of circular and elliptical cutouts on vibrational characteristics of porous sandwich functionally graded material (SFGM) plates with FGM core resting on Pasternak elastic foundation. The structural kinematics of the plate are formulated based on nonpolynomial-based higher-order shear deformation theory (HSDT). Geometric discontinuities have been incorporated in terms of circular and elliptical cutouts in the SFGM plates. The sandwich structure is considered to be comprised of two homogeneous face sheets and a porous FGM core with various cutout shapes and sizes. The variation of material properties from the homogeneous face sheet to the FGM core has been carefully modeled using modified power law distribution. Further, a C0 continuous finite element formulation with a four-noded, isoparametric quadrilateral element with seven degrees of freedom (DOFs) per node has been employed to accomplish the results. The accuracy of the present results has been demonstrated through convergence and validation studies. A comprehensive study has been carried out to investigate the influence of volume fraction index, even and uneven porosity distribution, circular and elliptical cutouts, as well as H-elliptical and V-elliptical type cutout shapes on the frequency parameter of SFGM plates with the elastic foundation. The numerical results demonstrate the aforementioned parameters significantly impact the vibrational frequency of the SFGM plates.

A nested non-intrusive stochastic isogeometric method for nonlinear thermal vibration of FGM plates under random loading

This paper introduces an adaptive nested non-invasive dynamic SIGA method based on RBFNN for the nonlinear thermal vibration analysis of FGM plates under random loading. This method is robust and accurate in high-dimensional input spaces regardless of the sample sequence, addresses the low convergence rate of QMCS, and solves the stochastic dynamic response probability density function under random loading. Firstly, the multidimensional input space for random loading is constructed based on several mutually independent random variables via the stationary Gaussian stochastic process simulation technique. The nonlinear FGM plate model subjected to random loading in the thermal environment is modeled using HSDT with von Kármán strain-displacement relation within the IGA framework. Subsequently, a nested non-invasive dynamic response surface analysis method, SIGA-RBFNN, is proposed based on the RBFNN technique. Analysis of the FGM plate response under random loading demonstrates that SIGA-RBFNN, compared to tensor-product-based SIGA methods, is less affected by sample sequence types. It effectively addresses high-dimensional stochasticity and offers significant advantages in robustness, accuracy, and efficiency. Finally, SIGA-RBFNN compensates for the low convergence speed of QMCS and successfully solves the FGM plate displacement response statistics and probability density function under random loading.

New Analytical Solutions for Free Vibration of Cracked Functionally Graded Rectangular Plates

In aerospace engineering, the existence of cracks highly alters the mechanical behaviors of structural components. As a new class of key structural components, functionally graded material (FGM) rectangular plates with an interior/side crack are focused on in this study. New analytical free vibration solutions are put forward for such cracked FGM plates. First, a cracked plate is decomposed into several subplates with the necessary continuity conditions to address the physical discontinuity issue caused by the crack. The challenge then becomes deriving analytical free vibration solutions for each subplate. To accomplish this task, a symplectic superposition method (SSM) is developed to deal with FGM rectangular plates with various boundary conditions, including the complex continuity conditions. The SSM eliminates the need for a predefined solution form, and its solution procedure, including variable separation and eigen expansion, is mathematically rigorous. Afterward, the analytical solutions obtained via the SSM are integrated to serve as the final free vibration solution of the cracked FGM plate. Comprehensive results based on the obtained analytical solutions, including natural frequencies and vibration modes, are presented and validated. The solutions are further used to investigate the effects of some important parameters, such as crack length and crack location.

Thermal postbuckling and thermally induced postbuckled flutter of tri-directional functionally graded plates in yawed supersonic flow

The current work examines the thermal postbuckling, aero-elastic flutter and thermally induced postbuckled flutter about a static equilibrium state of tri-directional functionally graded (TFGM) rectangular and tapered plates in yawed supersonic flow. Based on the von Kármán nonlinear strains, the general higher-order shear deformation theory (GHSDT) and the first-order piston theory, the governing equations of motion for TFGM plates in yawed supersonic flow are established via the Hamilton's principle. The isogemetric analysis (IGA), the load continue strategy associated with the Newton Raphson iterative technique are exploited synthetically to capture the postbuckling paths, and then the postbuckled flutter behaviors are acquired with the static equilibrium state being obtained. Comparative and convergence studies are provided to improve reliabilities of the present material model, formulae and code implementations. Subsequently, detailed parametric investigations are carried out to evaluate the influences of material volume fractions, boundary conditions and airflow angles on the thermal postbuckling, aero-elastic flutter and postbuckled flutter behaviors of TFGM plates. The results show that it is the in-plane volume fractions, rather than the thickness volume fraction, that exert a greater influence on the thermal postbuckling and flutter behaviors of TFGM plates. Furthermore, TFGM plates exhibit more diverse postbuckling paths and more complex deformations due to the inhomogeneity of the in-plane material properties compared with z-directional FGM plates.

Vibration and Buckling Characteristics of Sandwich FGM Plate with Conventional and Auxetic Honeycomb Core

The vibration and buckling characteristics of the sandwich plate with a honeycomb core and functionally graded material (FGM) face sheet have been evaluated in this paper. The honeycomb core, also known as conventional honeycomb and auxetic honeycomb, is modeled based on positive and negative Poisson’s ratio. Material properties of face sheets varied according to simple power-law functionally graded material (P-FGM). Hamilton’s principle has been employed to derive the equation of motion, and Navier’s method is used to solve the plate problem. Three different plate configurations are used to study the effect of the thickness layer. In addition, the outcome based on span-to-thickness ratio, aspect ratio, geometric parameters, and volume fraction exponent of honeycomb structure on frequency parameter and critical buckling load is examined and exhibited for three different plate configurations. The validation of the present formulation is ascertained by comparing it with other available results. Some novel results are presented for different angles of the unit cell that can be useful as a validation study for the forthcoming research on sandwich honeycomb rectangular plates. It is observed that the thickness of the honeycomb layer plays a significant role in affecting the behavior of the sandwich plate. Besides, the auxetic structure is highly sensitive to high excitation frequency applications compared to the conventional honeycomb structure.

Effect of Micromechanical Models on Mechanical Behavior of Functionally Graded Plates Based on a Quasi-3D Hyperbolic Shear Deformation Theory

Five alternative micromechanical models (Voigt, Reuss, Tamura, LRVE, Hashin) are used to predict the valid material properties of a simply-supported functionally graded materials (FGM) plate. The free vibration and static behavior are studied based on a quasi-3D hyperbolic shear deformation theory considering the stretching effect. The profile of an FGM plate is assumed by the power-law, Sigmoid and exponential functions. The numerical results of an FGM plate behavior reveal via Navier-type solution. The temperature effects of the micromechanical models on the fundamental frequency and transverse displacement for an FGM plate are also discussed. By analyzing the influence of different micro-models on the mechanical behavior of an FGM plate, researchers can objectively understand the establishment and differences of structural micromechanical models. And considering the actual high temperature situation, the deformation prediction ability of the structure is predicted. This study provides theoretical guidance for the analysis and modeling of FGM structures for the purpose of engineering practice.

EFG meshless-ANN approach for free vibration analysis of functionally graded material plates on elastic foundation in thermal environments

This study focuses on free vibration analysis of functionally graded material (FGM) plates supported by Winkler–Pasternak elastic foundation in thermal environment using element-free Galerkin (EFG) meshless method. Plate kinematics depend on first-order shear deformation theory. Uniform, linear, and nonlinear temperature variations through the thickness direction are considered, along with the temperature-dependent material properties. The numerical outcomes obtained from EFG method are compared with those available in the published literature to validate the proposed method’s accuracy. An artificial neural network (ANN) model that can easily predict the natural frequencies of the plate is constructed from the EFG method outcomes. Further, the effect of foundation parameters, power law index, thickness ratio, temperature variations, and different boundary conditions are investigated; results show that these significantly influence the vibration response of FGM plates supported by the elastic foundation. Increasing the temperature of FGM plates supported by the Winkler–Pasternack foundation causes a decrease in the dimensionless fundamental natural frequency, and the uniform temperature influence is greater than that of linear and nonlinear temperature variation. The proposed EFG-ANN prediction model saves approximately 98.80% computation time when predicting the natural frequency with an accuracy of approximately 98.76% compared to that by EFG meshless method alone.

Wave packet enriched finite element for accurate modelling of thermoelectroelastic wave propagation in functionally graded solids

A wave packet enriched finite element (FE) formulation featuring electrothermomechanical coupling is presented for solving two-dimensional wave propagation problems in functionally graded elastic and piezoelectric media. The FE equations for the Lord-Shulman and Green-Lindsay theories of generalized piezothermoelasticity are derived, where the standard Lagrangian interpolation functions of the temperature, electric potential, and displacement field variables are extrinsically enriched with element-domain sinusoidal wave-packet functions and are solved using the Newmark–β direct time integration. The effective properties of the functionally graded material (FGM) are computed using the volume-fraction-based micromechanics models, the Mori-Tanaka model and Voigt's rule of mixture (ROM) model. A variety of wave propagation problems, including the mechanically and electrically excited high-frequency Lamb wave propagation in FGM plates, impact waves in FGM bars, and thermal shock waves in functionally graded piezoelectric cylinders, are solved using the present formulation, and results are validated with available solutions. The efficacy of the current solution in comparison with the conventional FE is evaluated for all the studied problems. The effect of the inhomogeneity parameter on the transient wave behaviour is also presented for all the problems. The results of the Mori–Tanaka model are compared with those of the widely used ROM.

Thermal Vibration of Thick FGM Conical Shells by Using Third-Order Shear Deformation Theory.

A time-dependent third-order shear deformation theory (TSDT) approach on the displacements of thick functionally graded material (FGM) conical shells under dynamic thermal vibration is studied. Dynamic equations of motion with TSDT for thick FGM conical shells are applied directly with the partial derivative of variable R*θ in the curve coordinates (x, θ, z) instead of y in the Cartesian coordinates (x, y, z) for thick FGM plates, where R* is the middle-surface radius at any point on conical shells. The generalized differential quadrature (GDQ) numerical method is used to solve the dynamic differential equations in equilibrium matrix forms under thermal loads. It is the novelty of the current study to identify the parametric effects of shear correction coefficient, environment temperature, TSDT model, and FGM power law index on the displacements and stresses in the thick conical shells only subjected to sinusoidal heating loads. The physical parts with values on the length-to-thickness ratio equals 5, and 10 FGMs can be used in an area of an airplane engine that usually operates near more than 1000 K of temperatures when the thermal stress is considered and affected. The important findings of the presented study are listed as follows. The values of normal stress are in decreasing tendencies with time in cases when the coefficient c1 equals 0.925925/mm2 in TSDT and length-to-thickness ratio equals 5. The shear stress values in x plane z direction on the minor middle-surface radius (r) equals the major middle-surface radius (R) over 8 and length-to-thickness ratio equals to 5 can withstand T = 1000 K of pressure.

Hygrothermal buckling analysis of thin rectangular FGM plate with variable thickness

PurposeThis paper aims to contribute novel insights into the analysis of thin functionally graded material (FGM) plates with variable thickness, considering both temperature-dependent and independent material properties, focusing on critical linear buckling temperature rise and the effect of critical linear moisture for various moisture concentrations.Design/methodology/approachThe study derives stability and equilibrium equations for thin rectangular FGM plates under hygrothermal loading, employing classical plate theory (CPT). Buckling behavior is examined using Galerkin’s method to obtain pre-buckling force resultants.FindingsThe findings highlight significant increases in critical buckling temperature with aspect ratio, distinct temperature sensitivity between materials and increasing moisture susceptibility with larger aspect ratios. These insights inform material selection and design optimization for FGM plates under hygrothermal loading, enhancing engineering applications.Research limitations/implicationsThis research primarily focuses on hypothetical scenarios and mathematical model development and analysis.Originality/valueThis paper presents original contributions in the field by addressing the hygrothermal buckling analysis of thin FGM rectangular plates with variable thickness, utilizing CPT, thereby enriching the understanding of structural behavior in varying environmental conditions.

Effect of bi-directional material gradation on thermo-mechanical bending response of metal-ceramic FGM sandwich plates using inverse trigonometric shear deformation theory

PurposeThe purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO2) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.Design/methodology/approachIn this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.FindingsThe bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.Originality/valueFor the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

Bending of Sandwich FGM Plates with a Homogeneous Core Either Hard or Soft Via a Refined Hyperbolic Shear Deformation Plate Theory

Bending of Sandwich FGM Plates with a Homogeneous Core Either Hard or Soft Via a Refined Hyperbolic Shear Deformation Plate Theory

Free vibration of cracked FGM Mindlin plate in fluid

The free vibration of cracked functionally graded material (FGM) plate in fluid is investigated in this paper. The material components are supposed following the power law and formulated by the Voigt model. The Mindlin plate theory (MPT) including the shear deformation effect is established and the physical neutral plane is adopted to simplify the derivation due to the in-plane displacements of this plane assumed as zero. A massless linear rotational spring model (LRSM) is established to simulate the through-width surface crack. The hydrodynamic pressure at the fluid-plate interface is derived by the potential flow theory, and is regarded as the added mass of cracked FGM plate during the analyses of fluid-plate coupled vibration. The derivation and discretization of vibrational equations for cracked FGM plates in fluid are respectively achieved though the Hamilton’s principle and differential quadrature (DQ) method. The frequencies and mode shapes are iteratively calculated with these values of the vacuum case. The influences of immersed depth, fluid density, gradient index, crack depth, crack location, length-to-width ratio, length-to-thickness ratio, and boundaries on vibration characteristics are conducted in the section of numerical results. The present vibration analyses of cracked structures can be applicable to avoid structural resonance and failure.