Functionally Graded Core Research Articles

Modeling and static analysis of porous functionally graded and FG-sandwich plates

The present work is emphasized towards the development of models for the porosity occurring in the functionally graded (FG) structures. The porosities are modeled across the thickness for the FG plate as well as for the FG core of the sandwich plate considering three different types of porosity distribution i.e. symmetric center enhanced, top enhanced, and bottom enhanced. The influence of the porosities on the material properties is examined and the properties are evaluated in the presence of porosities. Further, the influence of the porosities on the static behavior of FG and FG-sandwich plates is investigated quantitatively wherein the considered structures are modeled in the framework of an inverse hyperbolic shear deformation theory. The governing equations are obtained through principle of virtual work considering linear structural kinematics and generalized Hooke’s law. These equations are solved in the closed form for the simply supported boundary conditions and the response is obtained. The results are verified with the existing literature wherever possible and are in good agreement. The effect of various parameters such as porosity distribution, porosity parameter, core-thickness distribution, span-thickness ratio, power-law index, and loading conditions on the static response is examined and, on those basis, noteworthy conclusions are made. The results indicate that the plate with top enhanced porosity distribution possess lowest deflection while the plate with bottom enhanced distribution possess the highest deflection.

Random thermal-vibration mechanisms of sandwich ventral fin-type plate-shell systems with porous functionally graded core

The stochastic thermal-vibration mechanisms within a sandwich ventral fin-type plate-shell system, featuring a porous functionally graded (FG) core, are exhaustively analyzed under various random loading conditions employing an innovative node-based, meshless computational approach. The studied structure is decoupled into several plates and open cylindrical panel according to the geometric characteristics, and the mechanical relationships at the structural boundaries or connection interfaces are equivalently simulated by using penalty parameters. Following the general Hamilton's principle, the meshless approach combined with the first-order shear deformation theory (FSDT) incorporating thermal effects is employed to derive the vibration equations of the sandwich ventral fin-type plate-shell systems. Also, the pseudo excitation method (PEM) is introduced to calculate stationary and nonstationary random responses. In order to verify the accuracy of the meshless algorithm in this study, the convergence and correctness are studied comprehensively. And then, the effects of some parameters such as temperature variation, porosity parameters, power-law index and random excitations on the thermal vibration characteristics of the sandwich ventral fin-type plate-shell systems with porous FG core are presented. The results show that the power-law index and temperature can increase the frequency parameter of the structure. The smooth power spectral density (PSD) excitation only affects the amplitude of the response curve, and does not affect the frequency corresponding to the peak. In the analysis of non-stationary random vibration, the influence of modulation parameters on response is very significant.

On nonlinear dynamic analysis of hyperbolic tangent FG-GPL-reinforced shallow spherical cap under supersonic flow

Shallow spherical caps are commonly encountered in the design of spacecraft components, such as fuel tanks or structural elements. Understanding the dynamic behavior can contribute to the identification of potential failure modes. The modulus of elasticity of the augmented composite upper layer is obtained using the model Halpin–Tsai, and this layer is augmented with graphene platelets (GPLs). Also, GPLs’ volume fraction from the top layer is considered for different cases of GPLs. The bottom layer is assumed functionally graded (FG) with two kinds, type A: FG layers and homogeneous core, and type B: FG core and homogeneous layers. To approximate the displacement field, the new hyperbolic tangent shear deformation theory (HTSDT) has been used which is no need for shear modification coefficients. By adopting Hamilton’s principle, the equations of motion are obtained and are solved by the discrete differential quadrature method (DQM) as well as Newmark's time marching scheme by implementing the famous Newton–Raphson iterative technique. When the FG index is infinite, it is observed that the point of maximum displacement decreases by 12.5%, and critical load increases by about 23% compared to the case where it is zero.

Structural analysis of sandwich plates with variable stiffness composite skins and functionally graded cores using refined zig-zag theory

This work investigates the static flexural behavior of new hybrid Sandwich Plates (SP) with Functionally Graded (FG) cores and curvilinear fiber-reinforced Variable Stiffness (VS) skins. For simplicity and computational efficiency, the existing refined zig-zag theory formulation for curvilinear fiber-reinforced composites with three-node finite element (RZT3C) is enhanced to incorporate the thickness-wise material variation of the FG core. Benchmark problems of FG-SP under various boundary conditions are solved, with results compared against 3D Ansys and reference solutions. Contrary to costlier methods, the presented model accurately captures the complex flexural mechanics, paving the way for designing advanced structures.

Thermomechanical vibration and buckling response of magneto-electro-elastic higher order laminated nanoplates

This study investigated the free vibration and buckling behavior of a magneto-electro-elastic (MEE) sandwich porous nanoplate under nonlinear thermal loads and electric and magnetic potentials. The equation of motion of the nanoplate consisting of barium-titanate and cobalt-ferrite components, with a functionally graded (FG) porous core and two face plates, was modeled by a sinusoidal higher-order shear (SHSDT) and the nonlocal strain gradient (NSGT) theories. Due to the significant effect of the temperature in the dynamics of the magneto-electro-elastic nanoplate, the temperature-dependent material properties of barium-titanate (BaTiO3) and cobalt ferrite (CoFe2O4) were defined and applied to the modeling for the first time in the literature. The laminated plate consists of upper and lower surface auxiliary plates and a main FG core plate. The core plate consists of functionally grading cobalt-ferrite and barium-titanate, while face plates can be in three different material compositions consisting of pure cobalt-ferrite, pure barium-titanate, or a combination of the two. It has been shown in this study that the dynamic behavior of the core plate, exposed to high temperatures, can be controlled through auxiliary surface plates. Especially in nanosensor applications, the disadvantage of thermal load can be reduced by the electric and magnetic field's strength applied to the surface plates. This study will make an essential contribution to the design and application of nanosensor or nanoelectromechanical systems, especially by controlling nanosensor responses in high-temperature applications.

On nonlinear free vibration of externally compressible fluid-loaded sandwich cylindrical shells: Curvature nonlinearity in bending and impermeability condition

A nonlinear fluid–structure interaction (FSI) model is presented for nonlinear vibration analysis of sandwich cylindrical shells subjected to an external compressible flow by considering the curvature nonlinearity in impermeability condition and bending. The sandwich shells are made of two face sheets and a central core of advanced materials including functionally graded (FG), metal foam, and anisogrid lattice composite. Based on the Kirchhoff–love hypotheses with the geometric nonlinearities in the normal strain and curvature of mid-surface, one decoupled nonlinear integral–differential equation is obtained for axisymmetric bending vibration of sandwich cylindrical shells. For the first time, the nonlinear impermeability condition is developed in order to take into account the large deformation of thin structures in FSI modeling. In both subsonic and supersonic regimes of steady irrotational compressible flow, a closed-form expression for nonlinear pressure distribution is presented according to the conservation of mass and generalized Bernoulli equation of extended potential theory. Using the Galerkin’s method in conjunction with the multiple-scale perturbation technique, the decoupled equation is solved to obtain nonlinear frequency and time–history response. Finally, parametric studies are carried out to investigate the effects of geometric ratios, boundary conditions, fluid specifications, and material variation on the responses of sandwich shells with different layups such as [Al; FG core; ZrO2], [Al; foam core; ZrO2], [Al; lattice core; Al]. Significant differences in pressure variation due to curvature nonlinearity are observed for short shells with larger vibration amplitude and larger number of half-waves in the axial direction. Therefore, the linear vibration analysis is inadequate for analyzing short sandwich cylindrical shells surrounded by an external fluid flow especially in the case of movable ends.

Buckling Response of Functionally Graded Porous Plates Due to a Quasi-3D Refined Theory

A quasi-3D refined theory is used to investigate the buckling response of functionally graded (FG) porous plates. The present theory takes into consideration the effect of thickness stretching. Three models of FG porous plates are presented: an isotropic FG porous plate, FG skins with a homogenous core, and an FG core with homogenous skins. The FG porous material properties vary along with the thickness of the FG layer based on modified polynomial law. By using the principle of total potential energy, the equilibrium equations are obtained. The buckling response is determined for simply supported FG porous plates. Analytical investigations are verified to present the accuracy of the current quasi-3D refined theory in predicting the buckling response of FG porous plates. The effect of thickness stretching and several parameters such as porosity coefficients, mechanical loadings, geometric parameters, gradient indexes, and layer thickness ratios are discussed. It is observed that the current theory shows more accurate results for the buckling response of FG plates compared with other shear deformation theories.

On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory

In the present paper, a novel refined shear deformation beam theory is proposed and applied, for the first time, to investigate the bending behavior of functionally graded (FG) sandwich curved beam. The present theory is exploited to satisfy parabolic variation of shear stress distribution along the thickness direction thereby obviating the use of any shear correction factors. The material properties of FG sandwich beam change continuously from one surface to another according to a power-law function. Three common configurations of FG beams are considered for the study, namely: (a) single layer FG beam; (b) sandwich beam with FG face sheets and homogeneous core and (c) sandwich beams with homogeneous face sheets and FG core. The governing equations derived herein are solved by employing the finite element method using a two-noded beam element, developed for this purpose. A wide range of comparative studies is carried-out to establish the robustness and the performance of the present finite element model. A comparison study shows that the proposed model is: (a) accurate and comparable with the literature; b) of fast rate of convergence to the reference solution; c) excellent in terms of numerical stability and d) valid for both thin and thick FG sandwich curved beams. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of volume fraction index, radius of curvature, material distributions, length-to-thickness ratio, face-to-core- thickness ratio, loadings and boundary conditions on the static response of FG curved sandwich beam. New referential results for thin and thick FG sandwich curved beam are offered, in which can be used to establish benchmarks for future research.

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.

Analytical non-linear analysis of functionally graded sandwich solid/annular sector plates

Nonlinear behavior of functionally graded (FG) sandwich circular sector plates with simply supported radial edges under transverse loading is investigated using the first-order shear deformation theory with von Karman geometric nonlinearity. The nonlinear governing equations are reformulated and solved via single-parameter perturbation and Fourier series methods. Verification is performed by comparing numerical results with the available ones in the literature and the ones obtained via ABAQUS code. The effects of non-linearity, material constant, lay-up, and boundary conditions on bending of FG sandwich sector plates with FG core/homogenous face sheets and metallic core/FG face sheets are studied. It is shown that in sandwich plates with an FG core, the deflection and maximum tensile stress decrease by using thinner homogenous face sheets and in ones with a metallic core, increasing the thickness of FG face sheets reduces the maximum tensile and compressive stresses, while the total thickness is considered the same in all layups. Furthermore, in bending analysis of FG sandwich sector plates, linear theory is adequate for w/h<1 in plates with free circular edges, it is solely applicable for w/h<0.2 in plates with clamped circular edges and is inadequate for analysis of fully simply supported FG sandwich sector plate.

Vibration analysis of graphene nanoplatelets’ reinforced composite plates integrated by piezo-electromagnetic patches on the piezo-electromagnetic media

A new model of microplate is used to evaluate the vibrational response of a size-dependent three-layered sandwich structure, including functionally graded (FG) composite core by sinusoidal shear deformation plate theory. Also, this FG core is reinforced by graphene nanoplatelets (GPLs) and embedded between piezo-electromagnetic (PEM) face sheets. The matrix of the core is made of Epoxy and the extended rule of mixture (ERM) besides Halpin–Tsai model is chosen to predict the mechanical properties of the nanocomposite core. Moreover, BaTiO3-CoFe2O4 composition is considered as constituting material of PEM face sheets. Furthermore, the visco-Pasternak substrate is used to carry the whole structure. The aforementioned theories are employed efficiently to obtain motion equations. Finally, the vibrational response of the described model due to the different parameters’ variations is presented in tables and figures. Based on the results, temperature difference leads the structure to become stiffer and accordingly reduces the frequency. Moreover, using a square basis for a sandwich structure makes the lowest frequency and consequently stiffness.

Flutter phenomenon in composite sandwich beams with flexible core under follower force

The main purpose of the present work was to study the dynamic instability of a three-layered, thick composite sandwich beam with the functionally graded (FG) flexible core subjected to an axial compressive follower force. Flutter instability of a sandwich cantilever beam was analyzed using the high-order theory of sandwich beams, for the first time. The governing equations in general for sandwich beams with an FG core were extracted and could be used for all types of sandwich beams with any types of face sheets and cores. A polynomial function is considered for the vertical distribution of the displacement field in the core layer along the thickness, based on the results of the first Frosting's higher order model. The governing partial differential equations and the equations of boundary conditions of the dynamic system are derived using Hamilton's principle. By applying the boundary conditions and numerical solution methods of squares quadrature, the beam flutter phenomenon is studied. In addition, the effects of different geometrical and material parameters on the flutter threshold were investigated. The results showed that the responses of the dynamic instability of the system were influenced by the follower force, the coefficients of FGs and the geometrical parameters like the core thickness. Comparison of the present results with the published results in the literature for the special case confirmed the accuracy of the proposed theory. The results showed that the follower force of the flutter phenomenon threshold for long beams tends to the corresponding results in the Timoshenko beam.

Static response of sandwich plates with FG core and piezoelectric faces under thermo-electro-mechanical loads and resting on elastic foundations

This study presents an analysis of the static bending of a sandwich plate composed of a functionally graded (FG) core and face sheets manufactured of piezoelectric material. The sandwich plate is resting on Pasternak's elastic foundations and subjected to sinusoidal thermo-electro-mechanical loads. Four-unknown shear deformation theory is utilized to define the displacement components. The equilibrium equations are established using the virtual work principle and solved by following Navier's solution method. The influences of applied voltage, thermal expansion coefficients, volume fraction exponent, elastic foundations parameters, side to-thickness ratio, and aspect ratio have been discussed. The current formulation is supported by comparisons with the published results.

Vibration Analysis of Different Types of Porous FG Conical Sandwich Shells in Various Thermal Surroundings

Vibration behavior of different types of porous functionally graded (FG) conical sandwich shells are investigated based on a modified high order sandwich shells theory for the first time. Sandwich shell includes FG face sheets covering a homogeneous core and the second one includes homogeneous face sheets and a FG core. Power law rule modified by considering two types of porosity distributions is used to model the functionally graded materials. All materials are temperature dependent and uniform, linear and nonlinear temperature distributions are used to model the effect of the temperature variation in the sandwiches. Governing equations are obtained by the Hamilton's energy principle and solved with Galerkin method. To verify the results, they are compared with ones achieved by finite element method obtained by Abaqus software for special cases with the results in literatures.

Thermoelastic Analysis of Functionally Graded Cylindrical Panels with Piezoelectric Layers

We propose a coupled thermoelastic approach based on the Lord-Shulman (L-S) and Maxwell’s formulations to study the wave propagation in functionally graded (FG) cylindrical panels with piezoelectric layers under a thermal shock loading. The material properties of the FG core layer feature a graded distribution throughout the thickness and vary according to a simple power law. A layerwise differential quadrature method (LW-DQM) is combined with a non-uniform rational B-spline (NURBS) multi-step time integration scheme to discretize the governing equations both in the spatial and time domains. The compatibility conditions of the physical quantities are enforced at the interfaces to describe their structural behavior in a closed form. A validation and comparative analysis with the available literature, together with a convergence study, show the efficiency and stability of the proposed method to handle thermoelastic problems. Numerical applications are herein performed systematically to check for the sensitivity of the thermoelastic response to the material graded index, piezoelectric layer thickness, external electrical voltage, opening angle, and shock thermal loading, which would be very helpful for practical engineering applications.

On the nonlinear bending and post-buckling behavior of laminated sandwich cylindrical shells with FG or isogrid lattice cores

The nonlinear governing equations of three shell theories (Donnell, Love, and Sanders) with first-order approximation and von Karman’s geometric nonlinearity for laminated sandwich cylindrical shells with isotropic, functionally graded (FG) or isogrid lattice layers are decoupled. This uncoupling makes it possible to present a semi-analytical solution for the nonlinear bending and post-buckling behavior of short and long doubly simply supported, doubly clamped, and cantilever laminated sandwich cylindrical shells subjected to various types of thermo-mechanical loadings. The results for deflection, stress, critical axial traction, and mode shapes in FG shells are verified with those obtained from ABAQUS code. Finally, the case studies are presented for FG shells and laminated sandwich shells with different layups such as $$[\hbox {Al; ZrO}_2]$$ , $$[\hbox {Al; FG core; ZrO}_2]$$ , $$[\hbox {Al; Gr; ZrO}_2]$$ , $$[\hbox {Al; Gr; FG core; ZrO}_2]$$ , $$[\hbox {Al; isogrid lattice core; Al}]$$ . The closed-form solutions presented here for the kinetic parameters and critical axial loading in a nonlinear analysis can be used in the conceptual design of laminated sandwich cylindrical shells with arbitrary layups and boundary conditions. Furthermore, introducing an equivalent Young’s modulus through the shell thickness, a simple formula is presented for the calculation of critical load in long shells with simple and clamped ends under axial loading with a maximum error of 10%. Moreover, findings show that the boundary-layer type behavior is seen only in long cylindrical shells in the pre-buckling region. Under thermal loading, snap-through buckling is observed in clamped shells. However, in simply supported shells by increasing the temperature, the transverse deflection increases, and while $$\Delta T-w/h$$ curves do not show any buckling phenomenon, the $$N^{0}/N_{\mathrm{cr}}^{*} -\Delta T$$ curves show such a behavior.

On vibration and stability analysis of porous plates reinforced by graphene platelets under aerodynamical loading

The vibration and stability analyses of functionally graded (FG) reinforced porous plates with piezoelectric layers under supersonic flow are investigated. The plate has a FG core reinforced by nanocomposite graphene platelets (GPLs) which surrounded by two piezoelectric layers. The GPLs are distributed thorough the thickness direction both uniformly and non-uniformly. The first-order shear deformation plate theory (FSDT) and first-order piston theory are applied to analyze the stability of FG porous plates. Using Hamilton's principle and Maxwell's equation, the governing equations of motion as well as electrical and mechanical boundary conditions are obtained. Applying the Galerkin approach, the partial differential governing equations are converted to the ordinary differential equations. The numerical results show that the flutter aerodynamic pressure and natural frequencies decrease as the porosity coefficient increases. Besides, the symmetric porosity distribution together with GPL pattern A predicts the highest flutter aerodynamic pressure and natural frequencies. Also, the best efficient way to increase the stability region is considering GPL pattern A, with dispersing more GPLs fillers near the top and bottom surfaces of FG porous plate along with symmetric porosity distribution. Furthermore, FG porous plate enclosed by piezoelectric layers in open circuit condition predicts higher flutter aerodynamic pressure and natural frequencies than similar plate in closed circuit condition.

Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory

This paper is devoted to examining the nonlinear vibrational behaviors of functionally graded (FG) sandwich nanobeams in the presence of initial geometric imperfection. Based on the nonlocal strain gradient theory, the governing equation of the FG sandwich nanobeam with consideration of the Von-Karman nonlinearity and initial geometric imperfection is derived. The nonlinear oscillator frequency is obtained with the aid of He’s variational principle. Three types of nanobeams, i.e., FG nanobeam (Type A), sandwich nanobeam with homogeneous core and FG skins (Type B), and sandwich nanobeam with FG core and homogeneous skins (Type C) are taken into account. A cosine function similar to the mode shape form is employed to describe the geometric imperfection mode. Firstly, the present theoretical model is verified by comparing with previous perfect FG sandwich beams. Then, several key parameters such as the power-law exponent, the amplitudes of the nonlinear oscillator and the geometric imperfection, as well as the nonlocal and material characteristic parameters are investigated in detail. Finally, apart from the structural types, the influence of thickness distribution scheme is also thoroughly elucidated. The results obtained in this paper are helpful for exploring the FG sandwich design to enhance the mechanical performance of nano-devices.

Vibration and stability analysis of functionally graded sandwich beams by a multi-layer finite element

Abstract This paper presents a finite element model based on the first-order shear deformation theory for free vibration and buckling analyses of functionally graded (FG) sandwich beams. The present element has 3 N + 7 degrees-of-freedom for an N-layer beam. Lagrange's equations are employed for derivation of the equations of motion. Two types of FG sandwich beams are considered: (a) Type A with FG faces and homogeneous ceramic core, and (b) Type B with homogeneous ceramic and metal faces and FG core. Natural frequencies and buckling loads are calculated numerically for different boundary conditions, power-law indices, and span-to-height ratios. Accuracy of the present element is demonstrated by comparisons with the results available, and discussions are made on the results given in graphs and tables for the sandwich beams considered.

Electro-elastic analysis of a sandwich thick plate considering FG core and composite piezoelectric layers on Pasternak foundation using TSDT

Third order shear deformation theory is used to evaluate electro-elastic solution of a sandwich plate with considering functionally graded (FG) core and composite face sheets made of piezoelectric layers. The plate is resting on the Pasternak foundation and subjected to normal pressure. Short circuited condition is applied on the top and bottom of piezoelectric layers. The governing differential equations of the system can be derived using Hamilton\'s principle and Maxwell\'s equation. The Navier\'s type solution for a sandwich rectangular thick plate with all edges simply supported is used. The numerical results are presented in terms of varying the parameters of the problem such as two elastic foundation parameters, thickness ratio (hp/2h), and power law index on the dimensionless deflection, critical buckling load, electric potential function, and the natural frequency of sandwich rectangular thick plate. The results show that the dimensionless natural frequency and critical buckling load diminish with an increase in the power law index, and vice versa for dimensionless deflection and electrical potential function, because of the sandwich thick plate with considering FG core becomes more flexible; while these results are reverse for thickness ratio.