Functionally Graded Material Layer Research Articles

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

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

Nonlinear transient analysis of porous P-FGM and S-FGM sandwich plates and shell panels under blast loading and thermal environment

The present work is an attempt to develop a simple and accurate finite element formulation based on C0 continuity of transverse displacement for obtaining and comparing nonlinear transient response of porous functionally graded material (FGM) sandwich plates and shell panels. The volume fraction for the layers made of FGM is computed according to the power-law (P-FGM) or sigmoid (S-FGM) model. The FGM sandwich panel is subjected to blast loading and thermal environment considering the heat conduction in thickness direction and the material properties are assumed to be temperature-dependent. An eight-noded isoparametric element along with first-order shear deformation theory is used to develop a finite element model. The strain–displacement relation is obtained using Sander’s approximation incorporating von Karman type nonlinear strains. Two configurations, first having the top and the bottom layers made of pure ceramic and pure metal and the core of same composed of FGM, second having the top and the bottom layers made of FGM and the core composed of pure metal are considered for the present investigation. The FGM layers for the two configurations are porous and two types of porosity, viz., evenly spaced and unevenly spaced are considered for the analysis. The results obtained from the present finite element formulation are first validated with several benchmark examples available in the literature. Parametric studies are carried out to investigate the effect of volume fraction index, porosity model, temperature gradient, core to facesheet thickness ratio and blast load on nonlinear transient analysis of porous P-FGM and S-FGM sandwich plate and shell panels. It is observed that by selecting optimum parameters, the amplitude of the nonlinear transient response due to different blast loading is controlled.

Effect of an edge crack on stress concentration around hole surrounded by functionally graded material layer

The present work aims to investigate the effect of an edge crack on the stress concentration around the circular hole surrounded by Functionally Graded Material (FGM) in an infinite plate subjected to uniaxial tensile load. The numerical investigation has been carried out using Extended Finite Element Method (XFEM). Two cases have been analysed in this work, i.e. the whole plate made up of radial FGM and homogeneous material plate having radial FGM layer around the hole. Young’s modulus of FGM varies according to exponential and power law function. The relations of stress intensity factor (SIF) and stress concentration factor (SCF) with normalised crack length, Young’s modulus ratio, FGM layer thickness and power law index have been presented. It has been observed that the FGM layer case has low SCF around hole than FGM plate case in presence of an edge crack.

Computational analysis of a sandwich beam with FGM face sheets under flexural loading

Functionally graded materials (FGM) are nonhomogeneous materials at the microscopic level, which have a gradation in their structure and composition with the properties of the material varying along one or more directions. The applications of these materials are in the fields: Aerospace, Automotive, Biomedical-Medical, Optoelectronics-electronics, Machinery, Energy, Marine, etc. In the present work the problem of a sandwich beam with metal core and FGM face sheets and a sandwich beam with ceramic core and FGM face sheets under flexural loading is studied numerically via the Finite Element Method. The load is a distributed load on the upper face sheet of the beam. The elements that will be used are 8-node two-dimensional plane quadrilateral isoparametric elements. The elements used for the FGM layers, are graded finite elements where as for the core are homogeneous elements. The advantage of this study is the utilization of 8-node two-dimensional plane quadrilateral graded isoparametric elements in the FGM face sheets, for the study of a sandwich beam. The effect of the volume fraction index on the stress fields is examined.

Testing and Modeling of Functionally Graded Aluminum‐Doped Zinc Oxide Using Spark Plasma Sintering and Discrete Powder Layers of Varying Composition

Functionally graded material (FGM) ZnO is made via spark plasma sintering/field‐assisted sintering technique (SPS/FAST) by varying the Al dopant content along the z‐axis or pressing direction. A wide range of Al content (0–5 wt%) is used by adding Al powder to nanosized ZnO powder. Thermoelectric (TE) measurements are done on the FGM and individual layers made separately. X‐ray diffraction (XRD) shows two phases, ZnO and spinel phase (ZnAl2O4). High‐resolution Raman spectroscopy reveals doped ZnO and spinel phase (ZnAl2O4) spatially and shows segregation in the layer with highest Al content. Electron backscatter diffraction (EBSD) reveals noticeable grain growth with decreasing Al content, and there is a common, random preferred orientation in all layers. The tested properties are used to simulate efficiency curves for a discretely graded, five‐layer FGM as well as a homogenous material, where both graded structures provide an opportunity to widen the current density ranges and thus the temperature range of useful energy conversion.

Free vibration of piezoelectric FGM sandwich beam with porous core embedded in thermal environment and elastic foundation

This paper aims to present thermo-electrical free vibration characteristics of functionally graded material (FGM) sandwich beam placed on the two-parameter elastic foundation. The beam is constructed of a foam core, two middle FGM layers, and two outer piezoelectric layers. It is assumed that the beam is subjected to a constant voltage and a uniform/linear temperature distribution. Physical properties of the core and two middle layers vary smoothly through the thickness according to the cosine and power-law forms, respectively. Lagrange equations in conjunction with the Reddy third-order beam theory is employed to derive the governing equations of motion. A simple polynomial trial function-based Ritz method is adopted for the approximation of the displacement field to obtain the vibration response. The correctness of the study is verified by comparisons with other authors’ published results. Influences of geometry parameters, material property distribution, applied voltage, elastic foundation, temperature distribution, temperature change, porosity coefficient, span-to-height ratio, and boundary conditions are investigated through parametric studies.

Level Set function-based Functionally Graded Material for the reduction of maximum stresses around a pair of inclined unequal circular holes

OBJECTIVE: The aim of this work is to present the methodology for grading the Functionally Graded Material (FGM) using Level Set (LS) sign distance function around the multiple holes and parametrically analyse the maximum stresses for a pair of inclined unequal circular holes surrounded by the FGM layer in an infinite plate subjected to uniaxial tensile load using the Extended Finite Element Method (XFEM). METHODS: The LS method has the ability to represent the multiple geometrical boundaries with a single sign distance function which can be effectively used for grading the FGM around the multiple discontinuities such as holes, inclusions, cracks, etc. When dealing with FGM material grading around multiple discontinuities, it is important to have smooth grading to minimise the stress concentration. The grading of the material with multiple functions may result in sharp changes in the material properties at the interference region which may lead to high stresses. The LS function-based FGM material grading eliminates such sharp changes as it uses a single function. RESULTS: The parametric analysis shows that applying the LS function-based power law FGM layer of Titanium – Titanium Mono Boride (Ti-TiB) around the pair of inclined unequal circular holes significantly reduces the values of maximum tensile as well as compressive hoop stresses when compared with the homogeneous material case.

Thermo-mechanical design optimization of symmetric and non-symmetric sandwich plates with ceramic-metal-ceramic functionally graded core to minimize stress, deformation and mass

This work explores the multiobjective design optimization of symmetric and non-symmetric sandwich plates with ceramic-metal-ceramic functionally graded (FG) core, along with ceramic faces (if included), under thermo-mechanical loadings. The FG core is made of two functionally graded material (FGM) layers, each with the volume fractions of the constituent phases defined by a power-law function through-thickness, with the interface between them metal-rich and the core outer surfaces ceramic-rich. In line with symmetric or non-symmetric sandwich plates, the design variables involve the thickness of each FGM layer of the core and the index of its power-law function, along with the thickness of each ceramic face. The thermal and mechanical problems are fully coupled using a layerwise mixed least-squares model with multi-field independent variables, including, most importantly, displacements, temperature and transverse stresses. The FG core z-continuous effective properties are also fully described by high-order z-expansions, as demonstrated adopting the rule of mixtures. The multiobjective optimization problem is solved by Direct MultiSearch derivative-free method, minimizing mass, transverse displacement and the stress field, measured by Tsai-Hill failure criteria, all together. Numerical results provide optimal symmetric and non-symmetric sandwich plates with ceramic-metal-ceramic FG core, considering distinct constituent materials and including three-dimensional approximate solutions for validation.

Three-dimensional static analysis of a viscoelastic rectangular functionally graded material plate embedded between piezoelectric sensor and actuator layers

The three-dimensional bending behavior of a viscoelastic functionally graded material (FGM) layer embedded between piezoelectric layers and subjected to an electric field as well as a uniform transverse pressure is studied. An analytical solution is computed for a simply supported viscoelastic smart FGM plate using the state space technique along the thickness direction and a Fourier expansion along the in-plane coordinates. The governing differential equations in the time domain are transformed to the Laplace domain, solved in the Laplace domain, and transformed back to the time domain using the inverse Laplace transform. In the present study, the relaxation modulus of the FGM layer is assumed in the form of a Prony series, and it varies according to the power law in the thickness direction. The validity of the proposed approach is assessed by comparison of the numerical results of the approach with those of published works in the literature. The effects of geometric dimensions, electromechanical loads, and relaxation time constant on the behavior of the smart plate are investigated.

Field Grading Composites Tailored by Electrophoresis—Part 2: Permittivity Gradient in Non-Uniform Electric Field

A series of three articles present an innovative way to build advanced functionally graded materials (FGM) based on polymer/ceramic composites tailored by electrophoresis from the process principle to their field grading application in power electronics. In this Part 2, it was studied the impact of a non-uniform electric field on the high-k SrTiO3 particle organization within an epoxy matrix. In that purpose, DBC substrates with sharp metallization tracks were used to generate a strong electric field divergence. This non-uniform field has been used to organize the particles around the field reinforcement regions during the electrophoresis process. It was discovered that the particles self-arrange into a conformal composite FGM layer that presents a local permittivity gradient: the highest ε being located around the electric field peak areas. This new process could be used to selectively `heal' the electric field-induced weaknesses of any high voltage electrical system by using its own design.

Nonlinear forced vibrations of FGM sandwich cylindrical shells with porosities on an elastic substrate

The nonlinear forced vibrations of functionally graded material (FGM) sandwich cylindrical shells with porosities on an elastic substrate are studied. A step function and a porosity volume fraction are introduced to describe the porosities in FGM layers of sandwich shells. Using the Donnell’s nonlinear shallow shell theory and Hamilton’s principle, an energy approach is employed to gain the nonlinear equations of motion. Afterwards, the multi-degree-of-freedom nonlinear ordinary differential equations are carried out by using Galerkin scheme, and subsequently the pseudo-arclength continuation method is utilized to perform the bifurcation analysis. Finally, the effects of the core-to-thickness ratio, porosity volume fraction, power-law exponent, and external excitation on nonlinear forced vibration characteristics of FGM sandwich shells with porosities are investigated in detail.

Zigzag Theory for Piezoelectric-Layer-Integrated Functionally Graded Material Plates

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

Effect of Silicon Carbide Weight Percentage and Number of Layers on Microstructural and Mechanical Properties of Al7075/SiC Functionally Graded Material

In this research, four different types of Al7075/SiC functionally graded materials (FGMs) are produced by varying the number of layers and weight percentages of silicon carbide(SiC). Each layer is fabricated with a predetermined configuration using the conventional powder metallurgy technique. The result of introducing different weight percentages of SiC particles in each layer and their effect on the functionally graded materials microstructure and mechanical properties are examined. The microstructural analysis clearly shows the uniform distribution of reinforcement material gradient from top to bottom and good interfacial bonding among the layers of FGMs. The functionally graded material’s density decreases with an increase in SiC weight percentage due to increased defects. The variation of hardness in each layer confirms the successful fabrication of Al7075/SiC functionally graded material with different layers. The mechanical properties of Al7075/SiC functionally graded materials mainly depend on the type of interfacial bonding between the layers, micro-cracks, pores, and SiC agglomerations. The prepared functionally graded materials exhibit extended mechanical and wear characteristics and are recommended in the manufacturing of gears, drive shafts, brake drums and bearings.

Centrifugally cast A356/SiC functionally graded composite: Fabrication and mechanical property assessment

Aluminum alloy with SiC particles reinforcement, functionally graded composite cylinder is developed via horizontal centrifugal casting technique. Cylindrical specimen is fabricated with 8 wt% of SiC particles reinforcement in A356 under mould rotational speed of 800 rpm. The comparative hardness at different locations on the cylinder is measured by taking five different samples along its longitudinal direction. Present investigation results shown about 204 percent improvement in hardness obtained for functionally graded cylinder at its outer region as compared to inner zone. The hardness value and its percentage of improvement shown a leaning trend from one end to other on the surface of cylinder liner. The outer layer of functionally graded material (FGM) shown about four times increase in hardness as compared to that of as cast A356 specimens. The inner layer of FGM also showed 1.3 times enhanced hardness value in comparison to A356 alloy.

Dynamic responses analysis of P and S-FGM sandwich cylindrical shell panels using a new layerwise method

This research work presents a comparison of the dynamic response of the functionally graded sandwich cylindrical shell panels (FGSCS) using a new layerwise method. The layerwise method developed assumes a first-order shear deformation theory (FSDT) for top and bottom facesheets and a third-order shear deformation theory for the core. The strain-displacement relation for FGSCS panels is obtained using Sander's first approximation. Two different sandwich configurations are considered, one having a pure metallic core with top and bottom facesheets made of functionally graded material (FGM) and the other one having an FGM core with top and bottom facesheets made of pure ceramic and pure metal, respectively. Material properties of the FGM layers for the two configurations are varied along the thickness direction according to the power-law (P-FGM) and the sigmoid models (S-FGM) respectively. The newly developed layerwise finite element model in conjunction with Hamilton's principle is employed to obtain the governing differential equation. Subsequently, the Newmark-Beta time integration scheme is used to obtain the dynamic response of P and S functionally graded sandwich cylindrical shell (P and S-FGSCS) panels for two configurations. The results obtained are first compared with the exact analytical results available in the literature. Numerical results are presented to investigate the effect of volume fraction index, loading conditions, core-to-facesheet thickness ratio, curvature ratio and boundary conditions on the transient response of P and S-FGSCS panels. The analysis reveals by selecting optimum parameters and gradation model, the amplitude and frequency of dynamic response of P and S-FGSCS panels can be controlled substantially.

Buckling analysis of three-dimensional functionally graded sandwich plates using two-dimensional scaled boundary finite element method

For the stability analysis of functionally graded material (FGM) sandwich plates, a semi-analytical scaled boundary finite element method (SBFEM) is established within the frame work of layerwise theory in this paper. Two different configurations of FGM sandwich plates are studied, one of which consists of ceramic-rich core and FGM face sheets, and another involves metal-rich and ceramic-rich material face sheets as well as a FGM core, and the modulus of elasticity for each FGM layer is assumed to be changed continuously along thickness according to a power-law distribution. The layerwise theory performed in present work is based on the three-dimensional theory of elasticity for each individual layer satisfying the displacement continuity boundary conditions at layer interfaces, and the two-dimensional high order spectral element with three degrees of freedom per node is employed to discrete the reference surface of each individual layer. And then, based on the principle of virtual work, the SBFEM governing equation for each individual layer of FGM sandwich plates is derived, and a dual variable including the displacement and internal nodal force is used to reduce the governing equations to a system of first-order ordinary differential equation, which is solved analytically by a general approach. Finally, the critical buckling load can be obtained by solving the eigenvalue equation once the bending stiffness matrix and geometric stiffness matrix of FGM sandwich plates are determined. Fast rate of convergence of the proposed formulations is confirmed and comparison studies are presented to construct its high accuracy and excellent predictive capability. Furthermore, effects of gradient index, side-to-thickness ratio, layer configuration, and boundary conditions on dimensionless critical buckling loads of FGM sandwich plates are also studied. The proposed semi-analytical approach is simple in program implementation, accurate and computationally efficient.

Nonlinear generalized thermoelasticity of FGM finite domain based on Lord–Shulman theory

Current investigation deals with the nonlinear generalized thermoelasticity of a functionally graded material (FGM) layer. It is assumed that the elasticity modulus, thermal conductivity, thermal expansion coefficient, mass density and specific heat capacity follow the exponential type of property distribution while the thermal relaxation time is assumed to be constant for the FGM layer. The governing motion and energy equations for the layer are obtained under the assumptions of Lord and Shulman with a single relaxation time. The energy equation is kept in its nonlinear form and the assumption made in the previous investigations to linearize the energy equation is not performed in the current research. Two coupled equations are presented in a dimensionless presentation and solved by means of the generalized differential quadrature and Newmark time marching method. Numerical results of this study are validated for the case of homogeneous layer. After that, novel numerical results are provided to investigate the response of a layer subjected to thermal shock. As expected, the exponential index of the FGM layer, boundary conditions, coupling parameter, relaxation time and nonlinearity are all major factors on the response of the thermally affected layer. It is verified that under thermally nonlinear analysis the magnitudes of stress and displacement are underestimated. Also, temperature and displacement enhance as the exponential index of the FGM layer increases.

A quasi-3D trigonometric shear deformation theory for wave propagation analysis of FGM sandwich plates with porosities resting on viscoelastic foundation

In the present work, a quasi three-dimensional trigonometric shear deformation plate theory (quasi-3D TSDPT) is proposed for the wave propagation analysis of porous functionally graded material (FGM) sandwich plates resting on a viscoelastic foundation. The present plate theory accounts for the transverse shear and normal deformations by dividing the transverse displacement into bending, shear, and stretching components. Different types of FGM sandwich plates are taken into account. The porosities in the FGM layers of the sandwich structures are described by introducing the porosity volume fraction and the step function. The equations of motion governing the wave propagation behavior of porous FGM sandwich plates are derived by employing Hamilton’s principle. The analytical solutions to the wave dispersion relations are presented. Additionally, the parametric research is conducted to highlight the effects of the wave number, the porosity volume fraction, the viscoelastic foundation, the power-law exponent and the core-to-thickness ratio on the wave propagation. Results manifest that the influences of these parameters are significant on the wave propagation characteristics of porous FGM sandwich plates.

Impacts of laser cladding residual stress and material properties of functionally graded layers on titanium alloy sheet

Laser cladding induces high tensile residual stress (RS), which can compromise the quality of a specimen. Therefore, it is critical to accurately predict the RS distribution in cladding and understand its formation mechanism. In this study, functionally graded material (FGM) layers were successfully deposited on the surface of a titanium alloy Ti6Al4V sheet by laser cladding technology. A corresponding thermo-mechanical coupling simulation model of the laser cladding process was developed to investigate the formation mechanism of RS in the laser cladding FGM layers. The results show that high tensile RS forms in cladding components. Subsequent cladding can effectively alleviate the RS in cladding components although the position of maximum RS remains unchanged. The measurement results of the longitudinal RS on the top and bottom surfaces of cladding components by the X-ray diffraction (XRD) method agreed with the simulation results, thereby proving the accuracy of the simulation. In addition, the formation mechanism of RS in the laser cladding FGM layers was revealed by discussing the individual impact of each material property on RS. It was indicated that the RS distribution in the laser cladding FGM layers was significantly affected by material properties (in particular, coefficient of thermal expansion and Young’s modulus), except for the temperature gradient induced by the laser cladding process.

Design optimization of functionally graded plates under thermo-mechanical loadings to minimize stress, deformation and mass

This work addresses the multiobjective design optimization of metal-ceramic functionally graded (FG) plates, which are composed of a main functionally graded material (FGM) layer and may include metal and/or ceramic faces, under thermo-mechanical loadings. The design variables are the thickness of the FGM layer, the index of its power-law distribution of metal-ceramic volume fractions, and if included, the thickness of the metal and/or ceramic faces. The three objectives focus on mass, maximum transverse displacement and maximum value of the Tsai-Hill failure criteria to measure the stress field, aiming to minimize all together. Both thermal and mechanical problems are solved simultaneously using a layerwise mixed model based on least-squares formulation with multi-field independent variables, namely, displacements, temperature, transverse stresses, transverse heat flux, in-plane strains and in-plane components of the thermal gradient. The FGM layer z-continuous effective properties are fully described via high-order z-expansions, similarly to finite element approximations. The multiobjective optimization problem is solved by Direct MultiSearch (DMS) derivative-free method, which uses the notion of Pareto dominance to retain a list of feasible non-dominated solutions. Numerical results provide optimal designs of FG plates under thermo-mechanical loadings, exploring distinct metal-ceramic constituent materials and different side-to-thickness ratios, including three-dimensional approximate solutions for validation.