Multilayered Functionally Graded Material Research Articles

Static analysis of shear-deformable aircraft wings using a multilayered functionally graded material model

This study presents a comprehensive investigation of the static analysis of a shear-deformable aircraft wing made of functionally graded materials (FGMs). A finite element-based multilayered FGM model is employed for this purpose. In order to ensure the safe structural design of the aircraft wing, it is crucial to analyze the transverse displacement of the wing under various loading conditions. The presented model employs a power-law distribution based on the rule of mixture to derive the effective material properties of the FGM wing. The analysis focuses on three standard FGM aircraft wing profiles, namely NACA 0009, NACA 2424, and NACA 4415, which are representative of commonly used geometries in aircraft design. Additionally, the study explores the effects of volume fraction index, loading conditions, boundary conditions, and aspect ratio on the static analysis of the shear-deformable FGM aircraft wing. These parameters play a significant role in shaping the static behavior of the wings and offer valuable insights into the design of FGM aircraft wings.

3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

Bending response in smart functionally graded panels using a zigzag theory

Bending response in smart functionally graded material (FGM) panels is presented using a zigzag theory. The functionally graded panel has sensing and actuation capabilities due to the presence of two piezoelectric layers at its outer surfaces. The theory is capable of predicting structural responses in presence of material property continuity as well as discontinuity at the interfaces. Rule of mixtures is used in obtaining effective material properties across thickness of each layer. Governing equations and boundary conditions have been derived using variational principle. Responses in piezoelectric laminated FGM panels have been analyzed using two smart FGM panels of which one panel contains non-symmetric FGM substrate and the other contains symmetric FGM substrate. Effects of variation of the power law index have been studied in thick and moderately thick multilayered smart FGM panels.

A thermally-coupled elastic large-deformation model of a multilayered functionally graded material curved beam

A thermally-coupled elastic large-deformation model of a multilayered functionally graded material (FGM) curved beam with an arbitrary undeformed configuration is established based on two-variable geometrical relations that consider the effect of the curvature of an arbitrary undeformed configuration of a beam, thermally coupled constitutive relations, and balanced forces. By use of a reference strain and reference undeformed and deformed curvatures at the same reference layer of a multilayered FGM curved beam, the difficulty of determining the position of the neutral axis of a beam in its classical analysis is resolved. The differential quadrature method is used to solve the strongly nonlinear coupled model proposed in this work. As applications, thermally-coupled large-deformation analyses of a multilayered straight beam and FGM and multilayered FGM curved beams under thermal–mechanical loads are studied in detail. Numerical results show that there are significant differences between the current thermally coupled analysis and conventional thermoelastic analysis with a thermally-uncoupled constitutive relation. Various snap-through phenomena occur when the FGM curved beam is under thermal–mechanical loads.

Free Vibration Analysis of Single and Multilayered Sandwich FGM Plates-Assessment of Higher Order Refined Theories

Analytical formulations and solutions for natural frequency analysis of functionally graded material (FGM) plates based on two higher-order refined shear deformation theories with 9 and 12 degrees-of-freedom are presented. The displacement model with 12 degrees-of-freedom considers the effect of both transverse shear and normal strain/stress while the other considers only the effect of transverse shear deformation. In addition another higher-order model and the first-order model developed by other investigators and available in the literature are also presented for the evaluation purpose. For mathematical modeling purposes, the Poissons ratio of the material is considered as constant whereas Youngs modulus is assumed to vary through the thickness according to the power law function. The equations of motion are derived using Hamiltons principle. Solutions are obtained in closed-form using Naviers technique and solving the eigenvalue equation. The accuracy of the theoretical formulations and the solution method using the present two higher-order refined models is first established by comparing the results generated in the present investigation with the 3D elasticity solutions already reported in the literature. After establishing the accuracy of predictions, benchmark results for the natural frequencies using all the four models are presented for single layer FGM plate and multi layered FGM sandwich plate with varying edge ratios and side-to-thickness ratios.

RMVT- and PVD-based finite cylindrical layer methods for the three-dimensional buckling analysis of multilayered FGM cylinders under axial compression

The unified formulations of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) and the principle of virtual displacements (PVD) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under axial compression. The material properties of the FGM layer are assumed to obey the power-law distributions of the volume fraction of the constituents through the thickness coordinate. In these formulations, the cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the primary variables of each individual layer, respectively, as well as the related order of each primary variable can be freely chosen, such as the layerwise linear, quadratic or cubic function distribution through the thickness coordinate. The accuracy and convergence of the RMVT- and PVD-based FCLMs developed in this article are assessed by comparing their solutions with the exact 3D solutions available in the literature.

Exact solutions of free vibration of rotating multilayered FGM cylinders

A modified Pagano method is developed for the three-dimensional (3D) free vibration analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders with a constant rotational speed with respect to the meridional direction of the cylinders. The material properties of each FGM layer constituting the cylinders are regarded as heterogeneous through the thickness coordinate, and then specified to obey a power-law distribution of the volume fractions of the constituents, and the effects of centrifugal and Coriolis accelerations, as well as the initial hoop stress due to rotation, are considered. The Pagano method, which was developed for the static and dynamic analyses of multilayered composite plates, is modified in that a displacement-based formulation is replaced by a mixed formulation, the complex-valued solutions of the system equations are transferred to the real-valued solutions, a successive approximation method is adopted to extend its application to FGM cylinders, and a propagator matrix method is developed to reduce the time needed for its implementation. These modifications make the Pagano method feasible for multilayered FGM cylinders, and the computation in the implementation is independent of the total number of the layers, thus becoming less time-consuming than usual.

A semi-analytical element-free galerkin method for the 3D free vibration analysis of multilayered FGM circular hollow cylinders

A semi-analytical element-free Galerkin (EFG) method in conjunction with an earlier proposed differential reproducing kernel (DRK) interpolation is developed for the three-dimensional (3D) free vibration analysis of simply supported, multilayered composite and functionally graded material (FGM) circular hollow cylinders. Based on the Reissner’s mixed variational theorem (RMVT), the weak formulation of this 3D dynamic problem is derived, in which the material properties of each individual FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate of the layer. A parametric study of the influence of some geometric and material parameters, such as the radius-to-thickness and length-to-radius ratios, thickness ratio for each layer, and material-property gradient index, on the natural frequency parameters of multilayered FGM circular hollow cylinders is undertaken.

RMVT-based meshless collocation and element-free Galerkin methods for the approximate 3D analysis of multilayered composite and FGM circular hollow cylinders

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the approximate three-dimensional (3D) analysis of simply supported, multilayered composite and functionally graded material (FGM) circular hollow cylinders under mechanical loads. The strong and weak formulations of this 3D elasticity problem are derived on the basis of the Reissner mixed variational theorem (RMVT). The former consists of the Euler–Lagrange equations of this problem and its associated boundary conditions, while the latter represents a weighted residual integral in which the differentiation is equally distributed among the primary field variables and their variations. An earlier proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the boundary and continuity conditions related to the primary variables themselves can be directly applied. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation. In the illustrative examples, the accuracy and convergence rate of the present MC and EFG methods are examined, and some guidance for using these methods is suggested.

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates

Abstract A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) free vibration analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. Based on the Reissner Mixed Variational Theorem (RMVT), the strong and weak formulations of this problem are derived, in which the material properties of each individual FGM layer, constituting the plate, are assumed to obey the power-law distributions of the volume fractions of the constituents. The system motion equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation, in which the shape functions of the unknown functions satisfy the Kronecker delta properties, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. In the illustrative examples, the natural frequencies and their corresponding modal field variables varying along the thickness coordinate of the plate are studied. It is shown that the solutions obtained using these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates

Abstract A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. The strong and weak formulations of this 3D static problem are derived on the basis of the Reissner mixed variational theorem (RMVT) where the strong formulation consists of the Euler–Lagrange equations of the problem and its associated boundary conditions, and the weak formulation represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations. The early proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation. In the illustrative examples, it is shown that the solutions obtained from these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.

The RMVT- and PVD-based finite layer methods for the three-dimensional analysis of multilayered composite and FGM plates

Abstract The Reissner mixed variational theorem (RMVT)- and principle of virtual displacements (PVD)-based finite layer methods (FLMs) are developed for the three-dimensional (3D) analysis of simply-supported, multilayered composite and functionally graded material (FGM) plates subjected to mechanical loads. The material properties of the FGM layers are assumed to obey either an exponent-law exponentially varied with the thickness coordinate or the power-law distributions of the volume fractions of the constituents. In these formulations, the plate is divided into a number of finite layers, where the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the field variables of each individual layer, respectively. Because an h -refinement instead of a p -refinement process is adopted to yield the convergent solutions in this analysis, the layerwise either linear or parabolic function distribution through the thickness coordinate is assumed for the related field variables. The unified formulations of these two kinds of FLMs with freely-chosen orders for the in- and out-of-plane field variables are presented. The accuracy and convergence rate of a variety of RMVT- and PVD-based FLMs developed in this paper are assessed by comparing their solutions with the exact 3D solutions available in the literature.

Exact solution of the bending vibration problem of FGM beams with variation of material properties

In the theoretical part of this contribution we deal with deriving a fourth-order differential equation of the functionally graded material (FGM) beam deflection with variation of material properties. The variation of the effective elasticity modulus and effective mass density can be caused by variation of both the volume fraction and material properties of the FGM constituents in a one-layer beam or in the layers of a multilayered sandwich beam. Homogenization of the varying material properties of the beams is achieved by extended mixture rules and laminate theory. The linear beam theory has been used for establishing the equilibrium and kinematic equations of the FGM beam. The shear force deformation effect and the effect of consistent mass distribution and mass inertia moment have been taken into account too. Numerical experiments were performed to calculate the eigenfrequencies and corresponding eigenmodes of chosen one-layer beams and multilayered FGM sandwich beams. The solution results are discussed and compared with those obtained using a very fine mesh of two-dimensional solid elements of a commercial finite element model (FEM) code.

An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates

Abstract A Reissner mixed variational theorem (RMVT)-based third-order shear deformation theory (TSDT) is developed for the static analysis of simply-supported, multilayered functionally graded material (FGM) plates under mechanical loads. The material properties of the FGM layers are assumed to obey either the exponent-law distributions through the thickness coordinate or the power-law distributions of the volume fractions of the constituents. In this theory, Reddy’s third-order displacement model and the layerwise parabolic function distributions of transverse shear stresses are assumed in the kinematic and kinetic fields, respectively, a priori, where the effect of transverse normal stress is regarded as minor and thus ignored. The continuity conditions of both transverse shear stresses and elastic displacements at the interfaces between adjacent layers are then exactly satisfied in this RMVT-based TSDT. On the basis of RMVT, a set of Euler–Lagrange equations associated with the possible boundary conditions is derived. In conjunction with the method of variable separation and Fourier series expansion, this theory is successfully applied to the static analysis of simply-supported, multilayered FGM plates under mechanical loads. A parametric study of the effects of the material-property gradient index and the span-thickness ratio on the displacement and stress components induced in the plates is undertaken.

Optimization of Crack-Free Polytypoidally Joined Si3N4–Al2O3 Functionally Graded Materials (FGM) Using 3-Dimensional Modeling

Joining Si3N4 and Al2O3 using 15 layers has been achieved by a unique approach that introduces SiAlON polytypoids as a functionally graded material (FGM) bonding layer. Previously, the hot press sintering of multilayered FGM with 20 layers, each 500 µm thick, has been achieved successfully. In the present study, the number of layers for FGM was reduced from 20 to 15 to increase optimization. Samples were fabricated by hot pressing at 48 MPa during the temperature ramp to 1650°C and cooling at 2°C/min to minimize residual stresses from sintering. Moreover, a finite element method (FEM) program based on the maximum principal stress theory and the maximum tensile stress theory was applied to design optimized and reduced FGM layers that produced a crack-free joint. The sample had a 3-dimensional cylindrical shape that was transformed to a 2-dimensional axisymmetric mode. By determining the expected thermal stress from the calculated elastic modulus and coefficient of thermal expansion, we were able to predict and prevent damage due to thermal stresses. These analyses are especially useful for FGM samples where it is very difficult to measure the residual stresses experimentally. Finally, oriented Vickers indentation testing was used to qualitatively characterize the strengths of the joint and the various interfaces. The indentation cracks were deflected at the SiAlON layers, implying weak interfaces. In other areas, cracks were not deflected, implying strong interfaces.

Reduction of Functionally Graded Material Layers for Si<SUB>3</SUB>N<SUB>4</SUB>-Al<SUB>2</SUB>O<SUB>3</SUB> System Using Three-Dimensional Finite Element Modeling

Numerical analysis method was used to reduce the number of functionally graded material (FGM) layers for joining Si3N4-Al2O3 using polytypoid interlayer by estimating the position of crack. In the past, hot press sintering of multi-layered FGMs with 20 layers of thickness 500 mm each have been fabricated successfully. In this paper, thermal residual stresses were calculated using finite element method (FEM) to find the optimized number of layers and its thicknesses of FGM joint. The number of layers for FGM was reduced to 15 layers from 20 layers. Thicknesses were varied to minimize residual stresses within the layers while reducing the number of FGM layers. The damage caused by thermal residual stress was estimated using maximum principal stress theory and maximum tensile stress theory. The calculated maximum stress was found to be axial stress of 430 MPa around 90% 12H/10% Al2O3 area. For each case, calculated strength of each FGM layer by linear rule of mixture was compared with computed thermal residual stresses. Thermal analysis results correctly predicted the position of crack, and this position agreed well with fabricated joints. Therefore, this numerical analysis method can be applied to reduced FGM layers of crack free joint. Finally, new composition profile of crack free joint was proposed using FGM method. [doi:10.2320/matertrans.MRA2007319]

Three Dimensional Analysis of Thermal Stress and Prediction of Failure of Polytypoidally Joined SI<SUB>3</SUB>N<SUB>4</SUB>-AL<SUB>2</SUB>O<SUB>3</SUB> Functionally Graded Material (FGM)

Three-dimensional analysis methods to calculate residual stress for functionally graded material (FGM) sample using sialon polytypoids to join silicon nitride and Alumina are introduced in this paper. The various multilayered FGM samples with 3, 9, and 20 layers were sintered to fabricate a crack-free joining of heterogeneous ceramics. To calculate three-dimensional thermal stresses of those fabricated FGM samples, a finite element analysis tool, ALGOR, was used. The Von Mises failure criterion and the maximum stress criterion were applied to predict failures in the FGM samples. For each case, calculated strength of each FGM layer by rule of mixture was compared with predicted thermal residual stresses. The Von Mises failure criterion predicted the locations of cracks more precisely than the maximum stress criterion. Such analyses are especially useful for graded FGM samples where the residual stresses are very difficult to measure experimentally. [doi:10.2320/matertrans.MRA2007092]

Effect of functionally graded material (FGM) layers on the residual stress of polytypoidally joined Si 3N 4–Al 2O 3

A unique approach introducing sialon polytypoids as a functionally graded material (FGM) bonding has been used to join silicon nitride and alumina. The various multilayered FGM samples ranging from 3 to 20 layers were sintered to fabricate a crack-free joining of heterogeneous ceramics. To calculate thermal stresses for the various multilayered FGM samples, the finite element analysis program (FEAP) was used. These analyses results matched experimental results and showed why some samples had large residual stresses that resulted in fracture. Moreover, the electron probe X-ray microanalysis (EPMA) from a crack-free FGM sample had a smooth concentration profile, which verifies the interface diffusion during sintering at each graded layer and confirms a successful joining.