Functionally Graded Core Research Articles

Vibro-acoustic response of sandwich plates with functionally graded core

This paper presents the vibro-acoustic modeling and analysis of sandwich plateswith metal–ceramic functionally graded (FG) core using a simplified first-order shear deformation theory and elemental radiator approach. A simply supported rectangular plate having functionally graded core, metal and ceramic facesheets is considered with aluminum as metal and alumina as ceramic. The material properties of the core are assumed to vary according to a power law distribution of the volume fraction of the constituents. The sound radiation due to point load and uniformly distributed load is computed by numerically solving the Rayleigh integral. The effective material properties of the sandwich plate are presented as a function of core thickness. The vibration parameters in terms of natural frequencies, plate displacement and velocity, and acoustic parameters such as radiated sound power level, radiated sound pressure level and radiation efficiency are computed for various values of the power law index. A comprehensive study of the influence of core thickness on vibro-acoustic performance is presented in terms of mean-squared velocity and overall sound power level. It is found that, for the plate being considered, the sound power level increases with increase in the power law index of the core at lower frequency segment. Increased vibro-acoustic response is observed in the high-frequency band for ceramic-rich FG core and in the low-frequency band for metal-rich FG core, respectively. A sandwich plate with metal-rich FG core configuration has shown improved flexural stiffness, compared to an FG plate with no significant rise in overall radiated sound. It is possible with this analysis to suitably tailor and optimize the sandwich FG plates for multifunctional performance and desired vibro-acoustic interaction.

Effects of shear stresses and rotary inertia on the stability and vibration of sandwich cylindrical shells with FGM core surrounded by elastic medium

ABSTRACTThe vibration and stability of axially loaded sandwich cylindrical shells with the functionally graded (FG) core with and without shear stresses and rotary inertia resting Pasternak foundation are investigated. The dynamic stability is derived based on the first order shear deformation theory (FSDT) including shear stresses. The axial load and dimensionless fundamental frequency for FG sandwich shell with shear stresses and rotary inertia and resting on the Pasternak foundation. Finally, the influences of variations of FG core, elastic foundations, shear stresses and rotary inertia on the fundamental frequencies and critical axial loads are investigated.

Temperature-dependent discrete layer-differential quadrature bending analysis of the multi-layered functionally graded annular plates rested on a two-parameter elastic foundation

ABSTRACTA discrete layer approach coupled with the differential quadrature method (DQM) is employed to temperature dependent analyze the laminated functionally graded (FG) annular plates under mechanical loading in a thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses and two-parameter elastic foundation. The material properties are assumed to be temperature-dependent and graded in the thickness direction. In order to accurately evaluate the effect of the thermal environment, the initial thermal stresses are obtained by solving the thermoelastic equilibrium equation. Comparison studies with the available solutions in the literature for FG plates are performed. Then, as an application, three common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous metal (soft) and ceramic (hard) core are analyzed. The influences of temperature rise, temperature-dependence of material properties, layers lay-out, foundation stiffness parameters, material graded index, and geometrical parameters on the solution are carried out. The new results can be used as benchmark solutions for future researches.

Bending behavior of sandwich structures with flexible functionally graded core based on high-order sandwich panel theory

In this study the problem of a sandwich beam with functionally graded (FG) core is attended with use of higher order sandwich panel theory forcing no pre-determined assumptions setting limits to the displacement and stress fields in the core. Face-sheets are analyzed using classical beam theory and the core is subjected to elasticity formulation. Governing equations and boundary conditions are derived using the principle of virtual displacements and governing equations based on five primary variables are solved using Navier’s solution. This method yields nonlinear patterns of displacements in the core and as a result it can manage to present localized effects of the core including the change in the core thickness under the load point which is an improvement on the previous classical and higher order theories. This present theory can handle any type of loadings distributed or concentrated and also any type of boundary conditions even when at the same edge, upper and lower face-sheets do not have the same boundary conditions. The solution is compared with present literature for a beam with conventional core. The results indicate positive effects of using FG core to reduce the possibility of delamination of the structure at interfaces between the upper and lower face-sheets and the core.

Parametric instability of shear deformable sandwich cylindrical shells containing an FGM core under static and time dependent periodic axial loads

The parametric instability of simply-supported sandwich cylindrical shell with a functionally graded (FG) core under static and time dependent periodic axial compressive loads is investigated in this study. The governing equations of sandwich cylindrical shell with an FG core are derived based on Donnell׳s shell theory using first order shear deformation theory (FSDT). Considering the time-dependent periodic axial compressive load, the governing equations are reduced the second order differential equation with the time-dependent periodic coefficient or Mathieu-type equation by using the Galerkin׳s method. This equation is solved by Bolotin׳s method and the closed form solutions for dimensionless excitation frequencies of parametric vibration of sandwich cylindrical shells with an FG core based on the FSDT are obtained. Numerical simulations are conducted to verify the analytical results. Finally, the effects of variations of volume fraction index of the FG core, static and dynamic load factors and shell characteristics on the dimensionless excitation frequencies of parametric vibration are studied numerically.

Stability and vibration of sandwich cylindrical shells containing a functionally graded material core with transverse shear stresses and rotary inertia effects

The dimensionless fundamental frequencies and critical axial loads of sandwich cylindrical thin shell with a functionally graded (FG) core are studied by taking into account the combined and separately influences of the shear stresses and rotary inertia. The governing equations of sandwich cylindrical shell with an FG core are derived based on Donnell’s shell theory using the shear deformation theory. The governing equations are reduced the sixth-order algebraic equation using the Galerkin’s method. Numerically solving this algebraic equation gives the magnitudes of the dimensionless fundamental frequency. In addition, the expressions for the dimensionless fundamental frequencies and critical axial loads of the sandwich cylindrical shell containing an FG core with and without the shear stresses are obtained in a special case. To validate the present method, the numerical example is presented and compared with the available existing results. Finally, the influences of variations of the FG core, shear stresses, rotary inertia and sandwich shell geometry parameters on the dimensionless fundamental frequencies and critical axial loads are analyzed numerically.

Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi-3D theories

This paper presents an analytical solution for vibration and buckling of functionally graded (FG) sandwich beams using various quasi-3D theories, which consider effects of both shear and normal deformation. Sandwich beams with FG skins–homogeneous core and homogeneous skins–FG core are considered. By using the Hamilton’s principle, governing equations of motion are derived. An analytical solution is presented, and the obtained results by various quasi-3D theories are compared with each other and with the available solutions in the literature. The effects of normal strain, power-law indexes, skin–core–skin thickness and slenderness ratios on vibration and buckling behaviour of sandwich beams are investigated.

Fonksiyonel Değişimli Çekirdeğe Sahip Üç Tabakalı Silindirik Kabukların Parametrik Titreşimi

In this study, the parametric vibration of three-layered cylindrical shells with functionally graded (FG) core is investigated. Firstly, analytical models of an FG core, which uses the intermediate member and outer sheath layers of the cylindrical shell formed from metal-rich and ceramic-rich represented as ternary systems. The governing equations of three-layered cylindrical shells with an FG core are derived and are reduced Mathieu-Hill equation by using Galerkin method. The expressions for non-dimensional frequency, critical axial load and boundaries of instability regions of three-layered cylindrical shell with an FG core are found. The expressions for non-dimensional frequency parameter, critical axial load and boundaries of instability regions are obtained for the monolayer FGM, pure metal and pure ceramic cylindrical shells, as a special case. Finally, the effects of variations of volume fractions, shell characteristics and variations of the thickness of a core on the values of critical parameters of three-layered cylindrical shells made of different types of FG core are studied numerically. Comparisons are made with the available studies in the open literature to validate this study

Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory

Static bending, buckling and free vibration behaviors of size-dependent functionally graded (FG) sandwich microbeams are examined in this paper based on the modified couple stress theory and Timoshenko beam theory. To avoid the use of a shear correction factor, equilibrium equations were used to compute the transverse shear force and shear stress. Two types of sandwich beams were considered: (1) homogeneous core and FG skins and (2) FG core and homogeneous skins. Numerical results were presented to illustrate the small scale effects on the behaviors of FG sandwich beams. The results reveals that the inclusion of the size effects results in an increase in the beam stiffness, and consequently, leads to a reduction of deflections and stresses and an increase in natural frequencies and critical buckling loads. Such effects are more pronounced when the beam depth was small, but they become negligible with the increase of the beam depth.

Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method

Free vibration of the functionally graded (FG) sandwich beams are studied by a meshfree boundary-domain integral equation method. Two sandwich beams, namely, FG core with homogeneous face sheets and homogeneous core with FG face sheets are considered in the study. Based on the two-dimensional elasticity theory, the boundary-domain integral equations are derived by applying the elastostatic fundamental solutions. Radial integration method is employed to transform the domain integrals related to material nonhomogeneity and inertia effect into boundary integrals, hence a meshfree scheme involving boundary integrals only is achieved. Each layer in the sandwich beam is modeled separately and the equilibrium and compatibility conditions are enforced at the interfaces to derive the equations for the whole sandwich beam. Extensive parametric study is presented to examine the influence of material composition, material gradient, layer thickness proportion, thickness to length ratio and boundary conditions on the free vibration of FG sandwich beams.

A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates

A new inverse trigonometric shear deformation theory is proposed for the static, buckling and free vibration analyses of isotropic and functionally graded (FG) sandwich plates. It accounts for a inverse trigonometric distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion obtained here are solved for three types of FG plates: FG plates, sandwich plates with FG core and sandwich plates with FG faces. Closed-form solutions are obtained to predict the deflections, stresses, critical buckling loads and natural frequencies of simply supported plates. A good agreement between the obtained predictions and the available solutions of existing shear deformation theories is found to demonstrate the accuracy of the proposed theory.

Free vibration of FGM layered beams by various theories and finite elements

The Carrera Unified Formulation (CUF) is used to perform free-vibrational analyses of functionally graded (FG) structures. CUF is a hierarchical formulation for obtaining refined structural theories that account for variable kinematic description. These theories can be obtained by expanding the unknown displacement variables over the beam section axes by adopting any kind of function. The number of the terms in the expansions is a free parameter of the analysis. For Taylor-like expansions, the linear case can result in classical beam theories. For the first time in the 1D CUF framework, the Finite Element method is used to solve the governing equations of functionally graded beams which are derived in a weak form by means of the Principle of Virtual Displacements. These equations are written in terms of f¨undamental nuclei.¨ Their forms do not depend on the expansions used. Several structures are considered, including a sandwich beam with FG core, laminated beams, thin- and thick-walled boxes as well as sandwich cylinders. The results are shown in terms of natural frequencies and compared with those available in existing literature.

Three-dimensional thermoelastic analysis of finite length laminated cylindrical panels with functionally graded layers

Based on the 3D thermoelasticity theory, the thermoelastic analysis of laminated cylindrical panels with finite length and functionally graded (FG) layers subjected to three-dimensional (3D) thermal loading are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The variations of the field variables across the panel thickness are accurately modeled by using a layerwise differential quadrature (DQ) approach. After validating the approach, as an important application, two common types of FG sandwich cylindrical panels, namely, the sandwich panels with FG face sheets and homogeneous core and the sandwich panels with homogeneous face sheets and FG core are analyzed. The effect of micromechanical modeling of the material properties on the thermoelastic behavior of the panels is studied by comparing the results obtained using the rule of mixture and Mori–Tanaka scheme. The comparison studies reveal that the difference between the results of the two micromechanical models is very small and can be neglected. Then, the effects of different geometrical parameters, material graded index and also the temperature dependence of the material properties on the thermoelastic behavior of the FG sandwich cylindrical panels are carried out.

Three-dimensional free vibration of laminated cylindrical panels with functionally graded layers

The three-dimensional (3D) free vibration of laminated cylindrical panels with finite length and functionally graded (FG) layers is presented. The cylindrical panels with two opposite axial edges simply supported and arbitrary boundary conditions at the curved edges can be analyzed via the present approach. The material properties vary continuously through the layers thickness. In order to accurately model the variation of the displacement components through the panel thickness, a layerwise-differential quadrature method (LW-DQM) is employed in this direction. Also, the in-plane variations of the displacement components are approximated using the trigonometric series in the circumferential direction and the DQM in the axial direction. The fast rate of convergence and accuracy of the method are demonstrated through different examples. As applications of the present approach, the free vibration of two common types of sandwich cylindrical panels, i.e. sandwich panels with FG face sheets and ceramic core and sandwich panels with FG core and ceramic/metal face sheets, and also bi-layered FG cylindrical panels are studied. The effects of geometrical and material parameters together with the boundary conditions on the frequency parameters of these types of panels are investigated.

A 3D Discrete Layer-Differential Quadrature Free Vibration of Multi-Layered FG Annular Plates in Thermal Environment

A three-dimensional (3D) discrete layer approach coupled with the differential quadrature method (DQM) is developed for the free vibration analysis of the laminated functionally graded (FG) annular plates subjected to a thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses. After demonstrating the convergence behavior and accuracy of the method, two common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous core, are analyzed. The influences of different parameters on the solution are carried out.

Transient response of rotating laminated functionally graded cylindrical shells in thermal environment

Based on the elasticity theory, the transient analysis of dynamically pressurized rotating multi-layered functionally graded (FG) cylindrical shells in thermal environment is presented. The variations of the field variables across the shell thickness are accurately modeled by dividing the shell into a set of co-axial mathematical layers in the radial direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The differential quadrature method and Newmark's time integration scheme are employed to discretize the obtained governing equations of each mathematical layer. After performing the convergence and comparison studies, parametric studies for two common types of FG sandwich shells, namely, the shell with homogeneous inner/outer layers and FG core and the shell with FG inner/outer layers and homogeneous core are carried out. The influences of the temperature dependence of material properties, material graded index, the convective heat transfer coefficient, the angular velocity, the boundary condition and the geometrical parameters (length and thickness to outer radius ratios) on the dynamic response of the FG shells are investigated.

Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method

Transient responses and natural frequencies of sandwich beams with inhomogeneous functionally graded (FG) core are investigated. To serve this purpose, we propose a novel truly meshfree method in which the displacement field is approximated by the radial point interpolation method (RPIM) regardless of predefined mesh, and the domain integrals are evaluated by the so-called Cartesian transformation method (CTM) to obviate the need for a background cell. The effective properties of the FG core are obtained either by the rule of mixture or by the Mori–Tanaka micromechanics scheme, while the penalty technique is adopted to treat the material discontinuities at the interface between the core and the two face sheets. The accuracy and the efficiency of the present formulation are demonstrated by examining a series of numerical examples. The results are compared to those obtained by alternative methods, and excellent agreements are obtained.

Behavior of sandwich beams with functionally graded rubber core in three point bending

AbstractThe three‐point bending behavior of sandwich beams made up of jute epoxy skins and piecewise linear functionally graded (FG) rubber core reinforced with fly ash filler is investigated. This work studies the influence of the parameters such as weight fraction of fly ash, core to thickness ratio, and orientation of jute on specific bending modulus and strength. The load displacement response of the sandwich is traced to evaluate the specific modulus and strength. FG core samples are prepared by using conventional casting technique and sandwich by hand layup. Presence of gradation is quantified experimentally. Results of bending test indicate that specific modulus and strength are primarily governed by filler content and core to sandwich thickness ratio. FG sandwiches with different gradation configurations (uniform, linear, and piecewise linear) are modeled using finite element analysis (ANSYS 5.4) to evaluate specific strength which is subsequently compared with the experimental results and the best gradation configuration is presented. POLYM. COMPOS., 2011. © 2011 Society of Plastics Engineers

Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers

In this paper, a free vibration analysis of moderately thick circular functionally graded (FG) plate integrated with two thin piezoelectric (PZT4) layers is presented based on Mindlin plate theory. The material properties of the FG core plate are assumed to be graded in the thickness direction, while the distribution of electric potential field along the thickness of piezoelectric layers is simulated by sinusoidal function. The differential equations of motion are solved analytically for two boundary conditions of the plate: clamped edge and simply supported edge. The analytical solution is validated by comparing the obtained resonant frequencies with those of an isotropic host plate. The emphasis is placed on investigating the effect of varying the gradient index of FG plate on the free vibration characteristics of the structure. Good agreement between the results of this paper and those of the finite element analyses validated the presented approach.