Functionally Graded Material Core Research Articles

Thermal buckling analysis of a new type of rectangular FG sandwich plates using a new higher-order shear deformation theory

This article presents a novel higher-order shear deformation Theory to investigate the thermal buckling behavior of a new type of functionally graded material (FGM) sandwich structure. The study focuses on a specific configuration of rectangular FGM sandwich plate, consisting of FGM face layers and FGM core layer subjected to temperature gradients varying linearly and non-linearly across the thickness. By employing the principle of virtual work, the governing equations are derived, involving only four unknown functions. The Navier procedure is then utilized to obtain closed-form solutions. In this study, the material properties and thermal expansion coefficient of the two face layers of the sandwich plate are assumed to vary in the thickness direction according to a simple power-law distribution. In contrast, the core layer is assumed to have an exponential variation along the thickness direction. The critical thermal buckling is influenced by various factors, including the thickness, and aspect ratios of the sandwich plate, the gradient index, the type of loading, and the type of sandwich plate. The effects of these variables on the critical buckling temperature are thoroughly examined and analyzed.

Discrete shear projection method for mechanical buckling analysis of FGM sandwich plates

This paper deals with the mechanical buckling analysis of functionally graded material (FGM) sandwich plates using quadrilateral element Q4γs based on the discrete shear projection method (DSPM) to avoid shear locking. The element has four-node and five degrees of freedom per node. It allows modelling two types of FGM sandwich plates, i.e. type A with a homogeneous core and FGM face sheets, and type B with homogeneous top and bottom layers and an FGM core. Different thickness schemes have been considered leading to symmetrical or unsymmetrical geometrical stacking. The paper focuses on the buckling in simply supported square plates under uniaxial or biaxial loads. The linear classical buckling theory (LCBT) is retained to evaluate the critical loads. The results obtained by the Q4γs element are compared with those reported by Zenkour (2005) and using another in-house finite element based on DKMQ and DKQ elements. All results are in good agreement when the LCBT is used to define critical loads, but the paper includes selected non-linear analysis based on the general-purpose finite element software ABAQUS. The results show the limit of validity of the LCBT for unsymmetrical stacking in the sandwich of type A and all sandwich of type B.

Dynamic Instability of Hybrid Functionally Graded Porous Plate Under Arbitrary Pulsating Loads

In this paper, the dynamic instability of hybrid functionally graded porous (FGP) plates under arbitrary periodic load is studied. The FGP plate composes of ceramic, functionally graded material (FGM) and metal. Four FGM core layers discussed include a non-pore perfect FGM layer and three imperfect FGM layers with different porosity distribution patterns. FGM layer properties are described by constituent volume fraction, porosity volume fraction and porosity distribution pattern. The Galerkin method and eigenfunctions transforms are used to establish the Mathieu-type governing equations. The influences of the load parameter, layer thickness ratio, constituent volume fraction, porosity distribution pattern and porosity volume fraction on the dynamic instability of hybrid FGP plates are investigated and discussed.

Free vibration of functionally graded sandwich plates in thermal environments

AbstractThis paper presents an analytical solution for the free vibration of functionally graded material (FGM) sandwich plates in a thermal environment. An equivalent‐single‐layer (ESL) plate theory with four variables is used to obtain the solution. Two types of sandwich plates are examined in this study: one with FGM face sheets and a homogeneous core and the other with an FGM core and homogeneous face sheets. The governing equations of motion are derived based on Hamilton's principle and then solved using the Navier method. The results of natural frequencies of simply supported FGM sandwich plates are compared with the available solutions in the literature. The effects of volume fraction distribution, geometrical parameters, and temperature increments on the free vibration characteristics are discussed in detail.

Analytical modeling of sound transmission through FGM sandwich cylindrical shell immersed in convected fluids

The main objective of this research work is focused on the acoustic response analysis of functionally graded material (FGM) sandwich cylindrical shell structure immersed in convected fluids. In the analysis, two common types of FGM sandwich cylindrical shells are considered, which include one with FGM face shell and homogeneous ceramic or metal core, and the other with FGM core and homogeneous face shell. Given the structure is immersed in a mean flowing fluid of velocity tangential to the acoustically deformed boundary and it is excited by an incidence oblique plane wave. Fluid–structure coupling is considered by imposing velocity continuity condition at fluid–structure interfaces. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, based on the developed theoretical mode, the influences of the gradient index, sandwich type, external mean flow, and skin-core-skin ratios on the structural sound transmission performance of FGM sandwich cylindrical shell have been investigated, wherein the first resonance dip shift noticeably to high frequency as the Mach number is increased, and case II has the largest amplitude fluctuation and the lowest STL curve magnitudes, in addition, when two branches, including positive refraction branch and the negative refraction branch, are found in the contour plots under opposite flow direction conditions, and the large skin-core-skin ratio is beneficial to improve the sound insulation characteristics of structures.

Nonlinear axisymmetric buckling analysis of the FGM sandwich shallow spherical shells under thermomechanical loads

Functionally graded material (FGM) sandwich shallow spherical shells are promising composite sandwich structures for modern engineering applications. In this paper, the axisymmetric snap-through buckling of spherical shells under uniform external pressure and non-uniform temperature rise along the shell thickness was studied. Based on the classical shell theory and the Sanders nonlinear kinematics equations, we established a two-point boundary value problem (BVP) posed by the displacement-type nonlinear ordinary differential equations governing the axisymmetric buckling of the FGM sandwich shallow spherical shells and the clamped boundary conditions. Then, the solutions of the above BVP were obtained using the numerical shooting method. The influence of initial term numbers of the infinite series defining the non-uniform temperature field on the accuracy of the solution was examined. Numerical calculations showed that a considerable initial term number of the series were needed to obtain the applicable solutions for engineering. Some valuable conclusions were drawn through detailed parameter studies. As the radius of the bottom circle increases, the lower critical load of shells decreases, the jump amplitude of buckling increases, and the upper critical load remains almost unchanged. The upper and lower critical loads and the jump amplitude increase as the circle radius of the middle surface of shells decreases. When the thickness of shells increases, their upper and lower critical loads increase, whereas the jump amplitude of buckling reduces. The gradient index has a significant effect on the steady-state response of pre-buckling spherical shells but little effect on that of post-buckling spherical shells. When the temperature rises, the upper critical load and the buckling jump amplitude of shells increase, but the lower critical load decreases. When the gradient index is greater (less) than about 0.35, the upper critical load decreases significantly (increases slightly) with the increase in the relative thickness of the FGM core. The present numerical results are well consistent with the finite element results and those in the available literature.

Effects of thermal environment and external mean flow on sound transmission loss of sandwich functionally graded magneto-electro-elastic cylindrical nanoshell

Similar to many new engineering ideas that are actually interdisciplinary subjects, magneto-electro-elastic (MEE) structures also need accurate ways of describing their structural behaviors. Based on nonlocal elasticity theory, the present article investigates the vibroacoustic behavior of a hollow multilayer cylindrical nanoshell where the core layer is made of isotropic functionally graded material (FGM) and other layers are made of MEE materials. The proposed system also is subjected to combined loads which contain a plane sound wave, the initial external electric and magnetic loads, and external mean airflow. The displacement field of structure is described using the third-order shear deformation assumption (TSDA). The derivation of vibroacoustic equations in the form of coupled relations is realized by implementing Hamilton's principle. The material properties of FGM core layer are supposed to vary along the in-plane and thickness directions based on the power-law model. The final objective is to analyze the sound transmission loss (STL) characteristics of the structure and inspect the accuracy of the developed method against existing data followed by comparing the results in terms of geometric and acoustic parameters. The obtained results and the described method can be used advantageously to better optimize such structures by choosing appropriate initial electric and magnetic potentials, flow Mach number, nonlocal parameter, material gradient index, incident angles.

A hybrid algorithm for modeling and studying of the effect of material and mechanical uncertainties on stability of sandwich FGM materials under thermal shock

Functionally graded material (FGM) structures are commonly used to withstand thermal stresses. These structures are very likely to experience thermal shocks. Given their nature, cylindrical shells are often subjected to a variety of dynamic loads so that the risk of instability (particularly dynamic instability) is very anticipated if they are subjected to a sudden thermal stress and a dynamic load, simultaneously. In fact, the possibility of uncertainty is very high in the fundamental parameters which determine thermal stresses. There is also uncertainty in the other properties of the structure including mechanical and thermal properties. Therefore, the present study addresses the dynamic stability of FGM cylindrical shells subjected to thermal and mechanical stresses based on Budiansky and Ruth’s criterion. The point estimation method and the artificial neural network-based algorithm were used to simulate the sources of uncertainty and to extract the statistical properties including the mean value, standard deviation, probability density functions (PDF), and cumulative distribution functions (CDF). The higher-order sheardeformationtheory (HSDT) was used to obtain the equilibrium equations, while the differential quadrature (DQ) method was used for their discretization. Functionally graded materials were assumed to change gradually along the thickness. The thermal shock distributions were based on a laboratory model (aerodynamic heating conditions) and two mathematical functions. The primary conditions, the type of thermal shock distribution, and the properties of the FGM core had the highest influence on the different stochastic properties of sandwich FGM cylindrical shells.

Free Vibrations of Flexoelectric FGM Conical Nanoshells with Piezoelectric Layers: Modeling and Analysis

Flexoelectric and piezoelectric effects have attracted the attention of researchers, owing to their applications in sensing systems and actuators. In this paper, the vibration of functionally graded material (FGM) conical nanoshell is studied, taking into account both piezoelectricity and flexoelectricity. The nanoshell has a sandwich-type structure with a FGM core and two layers of piezoelectric materials on its top and bottom. With the combination of the first order shear deformation and Eringen’s nonlocal theories, the vibration equation of the nanoshell is developed. In order to study the governing equations and the frequency of vibrations of nanoshell, the generalized differential quadrature method is implemented. Based on the developed numerical solution procedure, the effect of different parameters, such as flexoelectricity, piezoelectricity, nonlocal term and Pasternak foundation, are shown on the vibrations of conical nanoshell. The presented analysis provides a better insight into the behavior of conical nanoshells, which are highly applicable in bio-sensing and optical devices.

Vibration and Uncertainty Analysis of Functionally Graded Sandwich Plate Using Layerwise Theory

In the present study, free vibration analyses of porous functionally graded material (FGM) sandwich plates are carried out in thermal environment. Two types of sandwich plates such as sandwich with FGM face sheets and homogeneous core and sandwich with homogeneous face sheets and FGM core are considered for free vibration analysis with nonlinear temperature variation over the thickness direction. The metal and ceramic components are assumed to be temperature dependent. The higher-order layerwise theory is adopted with an eight-noded rectangular element to build a finite element model for free vibration analysis. A radial basis function network (RBFN) based surrogate model is developed to investigate the stochastic frequency response due to the uncertain material properties. The accuracy and the efficiency of the RBFN model are established by comparing the results with that of Monte Carlo simulation. Further, the developed surrogate model is implemented to perform various parametric studies and sensitivity analyses. The influence of the volume fraction index, types of porosity, and thermal gradient on the stochastic frequency response are analyzed. Though the influence of different stochastic input parameters is strongly dependent on the volume fraction index, the mass density of metal is found to be the most sensitive parameter for the sandwich plate with FGM face sheets and homogeneous core.

Thermomechanical Buckling Analysis of the E&P-FGM Beams Integrated by Nanocomposite Supports Immersed in a Hygrothermal Environment.

Due to the widespread use of sandwich structures in many industries and the importance of understanding their mechanical behavior, this paper studies the thermomechanical buckling behavior of sandwich beams with a functionally graded material (FGM) middle layer and two composite external layers. Both composite skins are made of Poly(methyl methacrylate) (PMMA) reinforced by carbon-nano-tubes (CNTs). The properties of the FGM core are predicted through an exponential-law and power-law theory (E&P), whereas an Eshelby–Mori–Tanaka (EMT) formulation is applied to capture the mechanical properties of the external layers. Moreover, different high-order displacement fields are combined with a virtual displacement approach to derive the governing equations of the problem, here solved analytically based on a Navier-type approximation. A parametric study is performed to check for the impact of different core materials and CNT concentrations inside the PMMA on the overall response of beams resting on a Pasternak substrate and subjected to a hygrothermal loading. This means that the sensitivity analysis accounts for different displacement fields, hygrothermal environments, and FGM theories, as a novel aspect of the present work. Our results could be replicated in a computational sense, and could be useful for design purposes in aerospace industries to increase the tolerance of target productions, such as aircraft bodies.

Thermal shock analysis of functionally graded sandwich curved beams using a new layerwise theory

AbstractThis work for the first time presents a simple, accurate and straightforward finite element formulation that requires a C0 continuity of transverse displacement for the transient stress analysis of curved beams under thermal shock using a new layerwise theory. The FGM sandwich curved beam is considered to be made of three layers in which the top, middle and the bottom layer is considered to be made of pure ceramic, functionally graded material (FGM) core and pure metal constituents, respectively. The proposed new layerwise theory assumes a higher‐order displacement field for FGM core and first‐order displacement field for both top and bottom layers. The material properties are assumed to be temperature‐dependent. The governing differential equation for the present investigation is formulated using Hamilton's principle. A wide range of comparison studies are presented to establish the accuracy of the present layerwise finite element formulation. A parametric study is conducted to show the simplicity and wider applicability of the proposed formulation. It is shown here that by selecting an appropriate value of volume fraction index, core to facesheet thickness ratio, magnitude of thermal shock, curvature and thickness ratio, the transient thermal stresses induced in FGM sandwich curved beams can be reduced substantially.

Dynamic analysis of functionally graded sandwich beams using a semi-analytic method named scaled boundary finite element method

In this research, free and forced vibrations of the functionally graded material (FGM) sandwich beams are investigated by using the scaled boundary finite element method (SBFEM). The innovation of this method is that the discretization only exists in the axial direction of the beam, which reduces the spatial dimension of problem by one, the time-consuming mesh generation processes are greatly reduced. The homogeneous second-order differential equations of the beams are derived by the method of virtual work principle with the radial coordinates as independent variables, which realizes the characteristics of the FGM changing along the radial direction. By using the state space method and the Padé series expansion method, the governing equations are further solved analytically, and the mass and stiffness matrices of each layer of the FGM sandwich beams are obtained. The combination between layers of the FGM sandwich beams is realized according to continuity conditions of displacement. Therefore, the free and forced vibrations using Newmark time integration method of the FGM sandwich beams are further realized. A special attention is given to two different types of the FGM sandwich beams, one is the beam with FGM face sheets and the other is the beam with FGM core. Three different types of dynamic loadings are considered in the analysis of forced vibration. The convergence of the proposed method for solving the dynamic characteristics of the beams is verified by increasing the order of the higher order spectral elements. In order to prove the numerical stability of the current method for dynamic analysis of the FGM sandwich beams, the frequency of the FGM sandwich beams under various parameters and the transient vertical displacement of the beam loading points under three different loadings are solved. The numerical results of the current method are compared with the results obtained by using different theories in the other literature, which shows a good agreement. In addition, on the basis of the research, the doped FGM core in the uniform core is discussed by using the current method, which provides some guidance for the optimal design of the dynamic characteristics of the beams.

Bending analysis of functionally graded sandwich plates using the refined finite strip method

In this study, static analysis of functionally graded (FG) sandwich plates is performed using the finite strip method based on the refined plate theory (RPT). Two types of common FG sandwich plates are considered. The first sandwich plate is composed of two FG material (FGM) face sheets and a homogeneous ceramic or metal core. The second one consists of two homogeneous fully metal and ceramic face sheets at the top and bottom, respectively, and a FGM core. Differential equations of FG sandwich plates are obtained using Hamilton's principle and stiffness and force matrices are formed using the finite strip method. The central deflection and the normal stress values are obtained for a sinusoidal loaded FG sandwich plate and the accuracy of the results are verified against those obtained from other theories such as the classical plate theory (CPT), the first-order shear deformation theory (FSDT), and the higher order shear deformation theory (HSDT). For the first time, this study presents a finite strip formulation in conjunction with the RPT to analyze FG Sandwich plates. While the proposed method is fast and simple, it is capable of modeling a variety of boundary conditions.

Analysis of vibration characteristics of FGM sandwich joined conical–conical shells surrounded by elastic foundations

The main objective of this paper is focused on the vibration behavior analysis of functionally graded material (FGM) sandwich joined conical–conical shell surround by elastic foundations. In the analysis, four kinds of sandwich distributions, including FGM core and isotropic skins, and isotropic core and FGM skins are considered and the power law distribution is employed to estimate the volume of the constituents. The governing equation of conical shell surround by elastic foundations is established by the first-order shear deformation shell theory and Hamilton’s principle. Then generalized differential quadrature (GDQ) method is used to discretize the obtained equations. By using the continuity conditions of displacement, rotation, force and moment between two adjacent shells and the boundary conditions at the end of the connected shell, a set of homogeneous equations are obtained, which control the free vibration of the two joined shell. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of sandwich distribution types, gradient indexes, skin–core–skin ratio and boundary conditions on the vibration behavior of FGM sandwich joined conical–conical shell have been investigated. It has found that for different sandwich distribution types, the influences of gradient index on the natural frequency are obvious when the gradient index is minor, but when high values of the gradient index are taken, the gradient index has no significant effect on the natural frequency.

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.

Study on dynamic instability characteristics of functionally graded material sandwich conical shells with arbitrary boundary conditions

This paper presents analytical studies on the dynamic instability of functionally graded material (FGM) sandwich conical shell subjected to time dependent periodic parametric axial and lateral load. In the analysis, four kinds of sandwich distributions, including FGM core and isotropic skins, and isotropic core and FGM skins are considered and the power law distribution is employed to estimate the volume of the constituents. The arbitrary boundary conditions of the FGM sandwich conical shell are achieved by using the artificial spring boundary. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton’s principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin’s method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of sandwich distribution types, gradient indexes, skin-core-skin ratio and load forms on the dynamic stability of FGM sandwich conical shell have been investigated.

Vibro-acoustic characteristics of eccentrically stiffened functionally graded material sandwich cylindrical shell under external mean fluid

The main objective of this research work is focused on acoustic radiation analysis of functionally graded material (FGM) sandwich cylindrical shell structure reinforced by periodically eccentrically ring stiffeners. In the analysis, two common types of FGM sandwich cylindrical shell are considered, which include one with FGM face shell and homogeneous ceramic or metal core, and the other with FGM core and homogeneous face shell. Given the structure is immersed in a mean flowing fluid of velocity tangential to the accoustically deformed boundary and it is excited by a radial point force. The ring stiffeners are assumed to be identical and uniformly spaced, and the in-plane flexural and out-of-plane torsional motions of the ring stiffeners are employed to accurately describe the force moment coupling between the stiffeners and the shell. Fluid-structure coupling is considered by imposing velocity continuity condition at fluid-structure interfaces. By applying the Poisson summation formula and the Fourier transformation technique for periodic ring stiffeners FGM shell structures, the sound pressure level (SPL) about structural acoustic radiation performance index is expressed in a superposition form of space harmonics for a given wave number. The validity of the proposed theoretical model is verified by comparisons with previously published numerical results. Based on the developed theoretical mode, the influences of the FGM properties, gradient index, sandwich type, the size of ring stiffener, external mean flow and skin-core-skin ratios type on the structural acoustic radiation performance have been investigated. In addition, the results show that under the same skin-core-skin ratio and thickness, the acoustic radiation performances of type C seem better, and the larger the section size of ring stiffened, the greater the influence of structure acoustic radiation characteristics, whereas the magnitudes of the SPL curves increase with the increasing values of Mach number.

A Finite Element Formulation and Nonlocal Theory for the Static and Free Vibration Analysis of the Sandwich Functionally Graded Nanoplates Resting on Elastic Foundation

This article presents a finite element method (FEM) integrated with the nonlocal theory for analysis of the static bending and free vibration of the sandwich functionally graded (FG) nanoplates resting on the elastic foundation (EF). Material properties of nanoplates are assumed to vary through thickness following two types (Type A with homogeneous core and FG material for upper and lower layers and Type B with FG material core and homogeneous materials for upper and lower layers). In this study, the formulation of the four-node quadrilateral element based on the mixed interpolation of tensorial components (MITC4) is used to avoid “the shear-locking” problem. On the basis of Hamilton’s principle and the nonlocal theory, the governing equations for the sandwich FG nanoplates are derived. The results of the proposed model are compared with published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and material properties on the static and free vibration behaviors of nanoplates are investigated in detail.