Volume Fraction Index Research Articles

Buckling behavior of spiral stiffened sandwich FGM cylindrical shells with porous core under axial compression using the FSDT

The linear buckling behavior of functionally graded cylindrical shells with porous core stiffened by spiral stiffeners under axial compression using the first-order shear deformation theory is presented in this paper. The improved Lekhnitskii’s smeared stiffeners technique is applied for shear deformable spiral FGM stiffeners. Approximate analytical solutions are assumed to satisfy the simply supported boundary conditions and the adjacent equilibrium criterion is applied to obtain closed-form relations of buckling loads. Effects of the number of FGM stiffeners, stiffener angle, volume fraction index and porosity coefficient on the buckling behavior of shells are numerically investigated.

Buckling behavior of spiral stiffened sandwich FGM cylindrical shells with porous core under axial compression using the FSDT

The linear buckling behavior of functionally graded cylindrical shells with porous core stiffened by spiral stiffeners under axial compression using the first-order shear deformation theory is presented in this paper. The improved Lekhnitskii’s smeared stiffeners technique is applied for shear deformable spiral FGM stiffeners. Approximate analytical solutions are assumed to satisfy the simply supported boundary conditions and the adjacent equilibrium criterion is applied to obtain closed-form relations of buckling loads. Effects of the number of FGM stiffeners, stiffener angle, volume fraction index and porosity coefficient on the buckling behavior of shells are numerically investigated.

Static Analysis of Skew Functionally Graded Plate Using Novel Shear Deformation Theory.

In this article, the static response of a functionally graded material (FGM) plate is studied via hybrid higher-order shear deformation theory which uses hyperbolic and polynomial shape functions and includes the effect of thickness stretching. The composition of the plate comprises metallic and ceramic phases. The ceramic volume fraction varies gradually along with the thickness following the power law. The mechanical properties of the FGM plate are determined by the rule of mixtures and the Mori–Tanaka homogenization scheme. The displacement fields are defined to satisfy the requirement of traction-free boundary conditions at the bottom and top surfaces of the plate surface removing the need for determination of shear correction factor. A C0 continuity FE model is developed for the present mathematical model. Nine-node isoparametric elements with eight nodal unknowns at each node are developed. The present model comparison with existing literature is completed and found to be coherent. Inhouse MATLAB code is developed for the present work. Sinusoidal and uniformly distributed loading is analyzed in the present work. The parametric study is undertaken to explore the effect of the side-to-thickness ratio, aspect ratio, thickness, and volume fraction index on stresses and transverse displacements.

Effect of thermal environment on the asymmetric vibration of temperature-dependent two-dimensional functionally graded annular plate by Chebyshev polynomials

In the present paper, an accurate numerical technique is employed to analyze the effect of two-dimensional temperature variation on the free asymmetric vibrations of functionally graded (FG) annular plates using the first-order shear deformation theory. The temperature-dependent mechanical properties of the plate material are graded along the thickness and radial directions. The governing differential equations for asymmetric motion are derived from Hamilton’s principle and the exact solution for the two-dimensional heat conduction equation is obtained using the separation of variables method considering thermal boundary conditions. The Chebyshev collocation technique is adopted to compute the numerical values of fundamental frequency for different boundary conditions. The impact of volume fraction index, heterogeneity parameter, density parameter, nodal lines, radii ratio, and thermal boundary conditions on the frequencies of the plate is investigated. An excellent agreement between the results obtained by the present approach and other methods available in the literature validated the authors’ model and technique. Three-dimensional normalized mode shapes with different nodal lines are illustrated for the specified plates.

Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core

The functionally graded (FG) sandwich plate has attained much attention in recent years due to its potential for exhibiting the merits of sandwich construction and FGM. Accordingly, intensive studies have focused on FG sandwich plates to investigate their mechanical behaviors. However, these mechanical behaviors are still in need of further investigation, particularly with respect to the major parameters. In this context, this paper intends to parametrically investigate the free-vibration behavior of the FG sandwich plate with a homogeneous core by developing a reliable and effective numerical method. This numerical method was based on hierarchical models, developed from the spectral model accuracy, and the 2-D natural element method (NEM). The hierarchical models were derived from the 3-D elasticity and the NEM was characterized by high smooth interpolation functions. From the verification experiments, the proposed method shows a good agreement with the reference and a uniform convergence to the 3-D elasticity. The free vibration characteristics of FG sandwich plates with a homogeneous core were investigated using the proposed numerical method. It was found that the calibrated fundamental frequency was significantly influenced by the type of material composing the core, the volume fraction index, the relative thickness and position of core layer, and the plate aspect ratio.

Navier solution for static and free vibration analysis of sandwich plate with auxetic honeycomb core resting on pasternak elastic foundation

This study investigates the bending behaviour and free vibration response of a rectangular sandwich plate with auxetic honeycomb core and functionally graded material (FGM) face sheets. Governing equations are established from Hamilton’s principle in the framework of the first-order deformation theory (FSDT). The Navier solution has been utilized to determine the deflection and the natural frequency of the simply supported sandwich plate. The reliability and accuracy of the proposed model have been validated via verification studies. The numerical investigations are conducted to investigate the influence of geometric parameters of auxetic material, volume fraction index of FGM, the thickness ratio of layers, and elastic foundation stiffness on deflection and fundamental frequency of the FG-Auxetic sandwich plate.

Thermal buckling loads of rectangular FG plates with temperature-dependent properties using Carrera Unified Formulation

In this article, a numerical analysis investigates thermal buckling loads of FG rectangular plates with temperature dependent properties. Constituent properties depend not only on thickness direction variable but also on temperature. Therefore calculation of the temperature distribution through the thickness is extremely significant. In this regard, an improved method is introduced for determining temperature distribution. Here, FEM is used to analyze the linear thermal buckling problem based on a higher-order plate theory called CUF. The virtual work principle is used to reach the governing equations for linear thermal buckling analysis. The equations are solved using the SLP method. The effects of some parameters like volume fraction index on the critical temperature differences of FG plates under different heat loads such as uniform, linear, and nonlinear have been examined. The influence of properties’ dependency on temperature is scrutinized too. The results prove the effectiveness of using CUF on critical temperature differences. It is also illustrated that if temperature-dependent properties are utilized in analyzing the thermal buckling loads, the results will be more accurate than using properties without depending on temperature.

Dynamics of imperfect inhomogeneous nanoplate with exponentially-varying properties resting on viscoelastic foundation

This article tries to investigate the dynamic deflection response of exponentially functionally graded material (E-FGM) nanoplate considering the role of porosities when embedded in a visco-elastic foundation and subjected to moving load, for the first time. The effective material properties are found using an exponential model of the rule of mixture. Next, the governing equations for the nanoplates while resting on a visco-Winkler foundation are found based on the third-order shear deformation theory in conjunction with Eringen nonlocal elasticity by developing Hamilton's principle. To solve the time-dependent governing motion equations, a state-space method is developed to find the response of the structure including simply-supported edges under moving load. Through some examples, the validation of the approach is provided before investigating the roles of nonlocality, volume fraction index (exponential parameter), porosity index, visco-elastic foundation coefficients, and velocity and time span of moving load on the forced vibration of embedded E-FGM nano-size plate under moving load.

Stochastic thermo-elastic vibration characteristics of functionally graded porous nano-beams using first-order perturbation-based nonlocal finite element model

This paper emphasized the size-dependent thermo-elastic vibration characteristics of uncertain functionally graded (FG) porous nano-beams by employing the first-order perturbation theory (FOPT) with the finite element method. An isoparametric quadratic beam element having three nodes is used for the finite element analysis of the nano-beams with variation in degree of uncertainty in material properties, geometric configuration, and thermal environment. The power-law model is employed to obtain the effective material properties of the considered beams. The governing equations are presented according to Eringen’s elasticity theory and Reddy’s beam theory using the minimum potential energy. The computed FOPT results for the vibration statistics are assessed with the help of Monte Carlo simulation (MCS) and a well agreement is found. Further, results are presented by analyzing the effects of degree of source uncertainty, size-dependent parameter, volume fraction indices, porosity distribution, porosity index, thermal environment, and aspect ratios on the fundamental frequency of the FGM nano-beams. The numerical results revealed the significant influence of thermal environment and porosity on the vibration statistics of the nano-beams and provided guidance for the precise and reliable design of nano-devices for the application in thermal environment.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.

Free vibration analysis of rotating FGP sandwich cylindrical shells with metal-foam core layer

This study offers a unique solution for the free vibration of rotating FGP sandwich cylindrical shells under different boundary conditions. A new type of sandwich structures with a metal-foam core and two coating FGM layers is investigated. Equations of motion governing the free vibration of rotating shells are derived by the first-order, shear-deformation shell theory. Single-term modal functions are introduced in Rayleigh-Ritz method to efficiently determine natural frequencies of the shell. After result validations, the influence of volume fraction index, distribution schemes, core-layer thickness, porosity coefficient, rotating velocity, geometrical parameters, and boundary conditions on the shell natural frequencies are studied.

Thermoelastic wave propagation due to local thermal shock on the functionally graded media

The present study is based on investigating the effect of locally applied thermal shock on the ceramic side of the functionally graded (FG) domain using nonlinear Lord-Shulman generalized thermoelastic theory. The constituent materials and therefore their properties in the two-dimensional FG domain are considered to be graded as a function of horizontal or vertical coordinates according to the power law. The finite element (FE) form of the nonlinear coupled differential equation is solved using the Picard method at each time-step in the framework of the Newmark time-integration scheme. The gradation of the material properties due to FGM is incorporated using graded finite elements in the FE formulation. The numerical results concerned with studying the effect of volume fraction index and orientation of the gradation of material properties across the FG domain are presented. It is observed from the study that both the gradation of material properties, as well as the direction of orientation of material gradation, have a substantial effect on the propagation of the thermal and elastic waves in the domain. Moreover, the thermal waves appear to be more dominant than the elastic waves at the time of application of thermal shock, which may also alter as time progresses.

Application of Haar wavelet discretization and differential quadrature methods for free vibration of functionally graded micro-beam with porosity using modified couple stress theory

The present investigation is aimed at the implementation of Haar Wavelet Discretization Method (HWDM) and Differential Quadrature Method (DQM) on the free vibration of a Functionally Graded (FG) micro-beam with uniformly distributed porosity along the thickness. As per the power-law exponent model, the material properties such as Young's modulus, and mass density are varied along the thickness of the FG micro-beam and the beam is made up of Aluminum (Al) as metal constituent and Alumina (Al2O3) as ceramic constituent. Modified couple stress theory is employed to capture the small scale effect and pointwise convergence studies for HWDM as well as DQM have also been carried out to exhibit the effectiveness of the methods with respect to the undertaken problem. The results obtained by both methods are compared to demonstrate the accuracy of the present model, revealing excellent accuracy. The effect of power-law exponent, porosity volume fraction index, and thickness to material length scale parameter on the natural frequencies is thoroughly investigated with proper physical explanations for Hinged-Hinged (H-H), Clamped-Hinged (C-H), Clamped-Clamped (C-C), and Clamped-Free (C-F) boundary conditions. Further, mode shapes are also plotted for qualitatively assessing the dynamics of the structural component.

Influence of microstructural defects on free flexural vibration of cracked functionally graded plates in thermal medium using XFEM

This paper presents the influence of microstructural defects on the vibration characteristics of the cracked functionally graded plate in a thermal environment. Extended finite element formulation is implemented to model the cracked functionally graded plate based on higher-order shear deformation theory (HSDT). The level set function is implemented to track the crack in the functionally graded plate domain. The crack in the functionally graded plate domain is addressed by enriching the primary variable with additional functions using the partition of unity technique. The power-law distribution is implemented for the gradation of the material properties along the thickness direction to form functionally graded material (FGM). The microstructural defects are assumed as porosities in the plate. The porosity index determines the porosity in the FGM plate. The convergence and comparative study have been performed to verify the current formulation’s efficacy, accuracy, and reliability. Further, various numerical examples are presented. The effects of different influential parameters like porosity, volume fraction index, crack length, thickness ratio, and boundary condition on the natural frequencies have been discussed in detail. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of the cracked FGM porous plates, which can provide a reference for future research by researchers.

New finite modelling of the nonlinear static bending analysis of piezoelectric FG sandwich plates resting on nonlinear elastic foundations

This paper presents a new finite modelling of the nonlinear static bending analysis of piezoelectric functionally graded (FG) sandwich plates resting on nonlinear elastic foundations. Finite element formulations are derived by using the first-order shear deformation theory (FSDT) of Mindlin and the finite element method. The proposed theory and mathematical model of this work are verified by comparing the results with those of other methods, and they are in good agreement. A parameter study is conducted to investigate the effects of geometrical and physical properties such as nonlinear factors, volume fraction index, nonlinear foundation parameters, applied voltages, boundary conditions, etc., on the nonlinear mechanical behaviors of piezoelectric FG sandwich plates. The novel numerical results of this work are very important, and can be used as a good reference to examine related structures as well as of use in engineering practice.

Thermal stability analysis of functionally graded non-uniform asymmetric circular and annular nano discs: Size-dependent regularity and boundary conditions

In this article, the size-dependent thermal buckling analysis of nonuniform functionally graded asymmetric circular and annular nanodiscs is presented on the basis of Kirchhoff's plate theory, Eringen's nonlocal elasticity theory, and physical neutral plane. For the first time, the nonlocal regularity conditions and boundary conditions are obtained for the asymmetric discs. The thickness of the nanodiscs is assumed to be varying linearly and parabolically in the radial direction. The Power-law model is adopted to compute the temperature-independent effective mechanical properties of the functionally graded materials (FGMs). The size-dependent stability equation is obtained from Euler-Lagrange's equation which is derived from Hamilton's principle. This equation and corresponding boundary conditions are discretized by the differential quadrature method (DQM) and provide an eigenvalue problem. The numerical value of the lowest eigenvalue is reported as a critical temperature difference on the surfaces of the nanodiscs. The effects of various parameters such as nonlocal parameter, volume fraction index, nodal lines, and taper parameters for thickness variations are studied.

A four node quadrilateral shell element for free vibration response of functionally graded spherical shell panels

Abstract The present work involves the free vibration analysis of functionally graded spherical shells using a four-node quadrilateral shell element. This four-node, shell element has seven DOF (Degrees of Freedom) per node namely; three displacements, two rotations of mid-plane, and two transverse shear strain com-ponents. The element is developed by modifying the discrete Kirchhoff quadri-lateral shell element based on Reddy’s third-order theory developed earlier by the authors for the cylindrical shell. By transforming all the degrees of freedom from local to global coordinates using a transformation matrix converts the plate bending element into a flat shell element, resulting in the present formulation being suitable to give results for thin shells as well as for thick shells. In the present work functionally graded spherical shell panels with simply supported as well as clamped boundary condition, with various radius to span ratios and with different volume fraction indices are analyzed for free vibration response. The power law property variation through the thickness is considered in this study. To assess the performance of the developed element, the results, of non-dimen-sionalised frequencies are compared with the results presented in the literature. Comparison of results shows that the developed element yields quite accurate results even with the coarser mesh, which indicates the computational efficiency. It is also seen that the percentage difference between the present results and the other results available in the literature is less than 2 in most of the cases.

Transient vibration and sound radiation analysis of simply supported functionally graded sandwich plates

Vibroacoustic response of simply supported functionally graded sandwich plates under transient loading is reported in this work. Simply supported square sandwich plate configurations of different thickness ratios and material property gradation in the core and face sheet are analyzed. The effective material properties of the sandwich plate are derived by considering an approximate laminate model where the Poisson’s ratio of the constituents of the sandwich are assumed to be constant and the plate kinematics is modelled using the first-order shear deformation theory. The structural system matrices of the functionally graded sandwich plate and plate deformations are computed using the finite element method. The time history of the structural response is obtained using the Newmark Beta time marching scheme. The radiated acoustic pressure in the far-field is determined using the time domain Rayleigh integral that is solved numerically by an elementary radiator approach. A detailed parametric study investigates the influence of the volume fraction index and thickness ratio on the transient vibroacoustic response of several sandwich configurations.

Sigmoid functionally graded plates embedded on Winkler-Pasternak foundation: Free vibration analysis by dynamic stiffness method

In this present paper, the dynamic stiffness method has been formulated to calculate the natural frequency of sigmoid functionally graded material (S-FGM) plate embedded on the Winkler-Pasternak elastic foundation. The material properties of S-FGM continuously vary along the transverse direction of the plate by using two power-law variations in terms of volume fraction of the constituent's material. Hamilton's principle is implemented to derive the governing partial differential equation of motion based on the classical plate theory considering the physical neutral surf ace of the FGM rectangular plate. The Wittrick -Williams algorithm is applied as a solution technique to solve the transcendental nature of the dynamic stiffness matrix and extract the natural frequencies of the FGM plate with the desired accuracy. The S-FGM plate parameters’ variation of natural frequencies with the change of parametric numerical values (aspect ratio, sigmoid volume fraction index, boundary conditions and elastic foundation parameters, density ratio, and modulus ratio) are also highlighted. The DSM results are compared and validated with the available published literature. A new set of natural frequency results for the S-FGM plate embedded on the Winkler-Pasternak elastic foundation are generated.

Instability Analysis of Fluid-Conveying Functionally Graded Thin-Walled Pipes on Random Elastic Foundations

The aims in this work are to study the effects of random variables and physical parameters on the critical fluid velocity of functionally graded material thin-walled pipes resting on random elastic foundations which carry the axial flow. The variation of the mean of the critical flow velocity with the spatial correlation length are monotonically increased or not monotonic which is conditional on the values of the coefficient of variation. The length of the intervals for purely ceramic or fully metal is shorter than these for other values of volume fraction index in which the mean of critical flow velocity is higher than that of the deterministic foundation stiffness. As the volume fraction index and temperature gradient of FGMs increase, the curves of the mean of critical flow velocity vary greatly for the fixed values of the coefficient of variation. The numerical results have been of practical interest for the design of the functionally graded pipes.