Volume Fraction Exponent Research Articles

Energy harvesting properties of the functionally graded flexoelectric microbeam energy harvesters

Flexoelectricity may be an excellent candidate for piezoelectricity in micro and nano scaled energy harvesters due to its strong size dependency. In this paper, a size dependent analytical model of the cantilever-based functionally graded flexoelectric energy harvesters is developed. The Hamilton's principle is used to derive the governing equations. By means of the Galerkin's method, the approximated closed-form solutions of electrical output and energy conversion efficiency are obtained. For the functionally graded flexoelectric energy harvester, the effective flexoelectric response is controlled by not only the flexocoupling coefficient but also its first derivative. Numerical results of a 3 μm-thick Polyvinylidene Fluoride/Strontium Titanate composed functionally graded flexoelectric energy harvester demonstrate that when the volume fraction exponent varies from zero to infinity, the optimal working frequency gradually reduces from 41407 Hz to 7761 Hz and the optimal load resistance gradually increases from 0.99 MΩ to 83.91 MΩ. Meanwhile, shrinking the thickness from 3 μm to 0.3 μm will highly increase the normalized power density and the energy conversion efficiency about one and two orders, respectively. Moreover, when the strain gradient elastic coefficient becomes larger, the natural frequency will increase while the corresponding maximum electrical output will decrease.

A parametric study on thermo-mechanical vibration of axially functionally graded material pipe conveying fluid

In this article, we study the thermo-elastic vibration of axially functionally graded material (FGM) pipe conveying fluid considering temperature changes. The governing equation based on Euler–Bernoulli beam theory is solved by differential quadrature method. The FGM properties are defined by the property ratios and the volume fraction functions. Power volume fraction function and exponent volume fraction function are compared. We also use sigmoid volume fraction functions so that the exclusive influence of function distribution can be isolated from that of total material proportions. The property ratios’ effects of elasticity and thermo-elasticity gradient are also discussed. Based on the numerical results of first-order dimensionless frequencies and critical flow velocities, concerning thermo-elasticity gradient can theoretically change the stability of the pipe. And the influences of the pure distribution on the first-order critical flow velocities are much smaller than that of the varying total proportions of the component materials. These conclusions will hopefully be used as reference for FGM pipe designing and fabricating.

Nonlinear Vibration Analysis of Sigmoid Functionally Graded Sandwich Plate with Ceramic-FGM-Metal Layers

In the present study, free and forced nonlinear vibration characteristics of a sandwich functionally graded material (FGM) plate resting on Pasternak elastic foundation have been investigated. The formulation is based on non-polynomial higher-order shear deformation theory with inverse hyperbolic shape function. A new modified sigmoid law is presented to compute the effective material properties of sandwich FGM plate. The governing equilibrium equations have been formulated using Hamilton’s principle and the governing nonlinear coupled ordinary differential equations are derived using stress function method in conjunction with Galerkin approach. The resulting equations are then solved using fourth order Runge–Kutta time integration scheme for simply supported plate with immovable edges. Comprehensive parametric studies have been performed to analyze the influence of geometric configuration, volume fraction exponent, and elastic medium parameter on nonlinear time series analysis for different configurations of sandwich plate. A nonlinear complex behaviour of the plate is studied using time displacement response, phase-plane plot and Poincare map. For Pasternak foundation, system shows multi-loop periodic nature while it losses its periodicity and shows the quasi-periodic nature for Winkler foundation. In addition, it has been observed that the system shows quasi-periodic route to chaotic nature for thick plate to thin plate.

Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis

The study using numerical methods on porous functionally graded (FG) nanoplates is still somewhat limited. This paper focuses on porosity-dependent nonlinear transient analysis of FG nanoplates using isogeometric finite element approach. In order to capture the small size effects, the Eringen's nonlocal elasticity based on higher order shear deformation theory (HSDT) are used to model the porous FG nanoplates. Two distributions of porosities inside FG materials are incorporated and defined via a modified rule of mixture. The nonlinear transient nonlocal governing equations under transverse dynamic loads are formulated by using the von Kármán strains and are solved by Newmark time integration scheme to obtain geometrically nonlinear responses. It is indicated that nonlinear transient deflections of the porous FG nanoplate are significantly influenced by material composition, porosity, nonlocal parameters, volume fraction exponent, porosity distributions, geometrical parameters and dynamic load characteristics.

Nonlinear approach on torsional buckling and postbuckling of functionally graded cylindrical shells reinforced by orthogonal and spiral stiffeners in thermal environment

A new nonlinear approach on the buckling and postbuckling of functionally graded orthogonal and/or spiral-stiffened circular cylindrical shells subjected to torsional loads is proposed in this paper. The shells skin are stiffened by eccentrically rings, stringers, and/or spiral stiffeners at the surface of shells assuming that the material distribution laws of shell skin and stiffeners are graded by two distribution models. Lekhnitskii’s smeared stiffeners technique is improved for spiral stiffeners with effect of thermal terms. This is the significant novelty and scientific contribution of this paper. Theoretical formulations were established by using the Donnell shell theory taking into account the geometrical nonlinearity of von Kármán. The obtained results investigated in numerical forms show effects of volume fraction exponent of shell skin and stiffeners, geometrical parameter and stiffeners on the torsional buckling, and postbuckling behavior of functionally graded cylindrical shells. Especially, very large effects of spiral stiffeners on torsional stability behavior are obtained in comparison with same quantity material of orthogonal stiffeners.

Application of sinusoidal shear deformation theory and physical neutral surface to analysis of functionally graded piezoelectric plate

Abstract The concept of neutral surface for a Functionally Graded Piezoelectric (FGP) plate is developed in this paper. The electro-elastic analysis of a FGP plate resting on Winkler-Pasternak foundation is performed in the theoretical framework provided by a two-variable sinusoidal shear deformation theory, including the aforementioned concept of neutral surface. First, the location of neutral surface is defined with respect to the position of the middle surface and then the governing bending equations are derived using the principle of virtual work. An analytical method is presented to investigate the variation of the displacement and stress components in terms of the main parameters of the model, which are volume fraction exponent of the constituents, the electric potential and foundation parameters. The numerical results are validated through the comparison with available references. The numerical results prove that the value of transverse bending deflection is greater than the corresponding shear deflection. In addition, it can be observed that the volume fraction exponent has peculiar influence on the distribution of displacements and stresses.

Differential Quadrature Free Vibration Analysis of Functionally Graded Thin Annular Sector Plates in Thermal Environments

In this paper, the free vibration behavior of functionally graded (FG) thin annular sector plates in thermal environment is studied using the differential quadrature method (DQM). The material properties of the FG plate are assumed to be temperature dependent and vary continuously through the thickness, according to the power-law distribution of the volume fraction of the constituents. The nonlinear temperature distribution along the thickness direction of the plate is considered. Based on the classical plate theory, the governing differential equations of motion of the plate are derived and solved numerically using DQM. The natural frequencies of thin FG annular sector plates in thermal environment under various combinations of clamped, free, and simply supported boundary conditions (BCs) are presented for the first time. To ensure the accuracy of the method, the natural frequencies of a pure metallic plate are calculated and compared with those existing in the literature for the homogeneous plate where the results are in good agreement. The effects of temperature field, BCs, volume fraction exponent, radius ratio, and the sector angle on the free vibrations of the FG-plate are examined.

A curved hierarchical finite element method for the nonlinear vibration analysis of functionally graded sandwich elliptic plates

ABSTRACTThis study of bi-FGM sandwich elliptic plates is carried out to investigate the vibration behavior using a curved hierarchical finite element. The plate is made up of two face-sheet layers with two different mixtures of FGM and one layer of homogeneous core. The blending function method is used to describe accurately the geometry of the elliptic plate. The accuracy of the present method is established by the comparisons made with the published results. Contour plots of frequency parameters as functions of the volume fraction exponents and thickness for bi-FGM sandwich plates are carried out.

Dynamic analysis of non-symmetric FG cylindrical shell under shock loading by using MLPG method

The Dynamic equations in the polar coordinates are drawn out using the MLPG method for the non-symmetric FG cylindrical shell. To simulate the mechanical properties of FGM, the nonlinear volume fractions for radial direction are used. The shape function applied in this paper is a form of the radial basis functions, by using this function all the requirements for an effective and suitable shape function are established. Hence in this study, the multiquadrics (MQ) radial basis functions are exploited as the shape function governing the problem. The MLPG method is combined with the the Newmark time approximation scheme to solve dynamic equations in the time domain. The obtained results by the MLPG method to be verified are compared with the analytical solution and the FEM. The obtained results through the MLPG method show a good agreement in comparison to other results and the MLPG method has high accuracy for dynamic analysis of the non-symmetric FG cylindrical shell. To demonstrate the capability of the present method to dynamic analysis of the non-symmetric FG cylindrical shell, it is analyzed dynamically with different volume fraction exponents under harmonic and rectangular shock loading. The present method shows high accuracy, efficiency and capability to dynamic analysis of the non-symmetric FG cylindrical shell with nonlinear grading patterns.

A new analysis of nonlinear free vibration behavior of bi-functionally graded sandwich plates using the p-version of the finite element method

ABSTRACTGeometrically nonlinear vibration of bi-functionally graded material (FGM) sandwich plates has been carried out by the p-version of the finite element method (FEM). The bi-FGM sandwich plate is made up of two face-sheet layers of two different FGM and one layer of homogeneous core. The nonlinear equations of motion of bi-FGM sandwich plates are establish using the harmonic balance method and solved iteratively by the linearized updated mode method. The effects of amplitude vibration, mechanical properties, geometrical parameters, thickness ratio of bi-FGM layers, and volume fraction exponent on the nonlinear vibration behavior of bi-FGM sandwich plates are plotted and investigated.

Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory

In the current paper, the sensitivity performance of functionally graded magneto-electro-elastic (FG-MEE) nanoplate with attached nanoparticles as a nanosensor is analyzed based on nonlocal Mindlin plate assumption. Power law distribution model is employed to display how the material properties of FG-MEE nanoplate vary across the thickness direction. It is supposed that FG-MEE nanoplate is under initial external electric and magnetic potentials. Boundary condition of each edge of FG-MEE nanoplate is assumed to be simply supported. Furthermore, a Pasternak substrate is applied for modelling the total reaction pressure between nanoplate and foundation. Partial differential equations and corresponding boundary conditions are first achieved using Hamilton’s variational principle and then analytically solved to determine the frequency shift utilizing Navier’s approach. Numerical examples are performed to elucidate the dependency of the sensitivity performance of FG-MEE nanosensor on the volume fraction exponent, nonlocal parameter, total attached mass and location of the nanoparticle, aspect ratio, mode number, initial external electric voltage, initial external magnetic potential, and Pasternak medium coefficients. It is clearly indicated that these factors have highly significant impacts on the variations of frequency shift.

R-functions theory applied to investigation of nonlinear free vibrations of functionally graded shallow shells

Nonlinear free vibration of functionally graded shallow shells with complex planform is investigated using the R-functions method and variational Ritz method. The proposed method is developed in the framework of the first-order shear deformation shallow shell theory. Effect of transverse shear strains and rotary inertia is taken into account. The properties of functionally graded materials are assumed to be varying continuously through the thickness according to a power law distribution. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. Admissible functions are constructed by the R-functions theory. To implement the proposed approach, the corresponding software has been developed. Comprehensive numerical results for three types of shallow shells with positive, zero and negative curvature with complex planform are presented in tabular and graphical forms. The convergence of the natural frequencies with increasing number of admissible functions has been checked out. Effect of volume fraction exponent, geometry of a shape and boundary conditions on the natural and nonlinear frequencies is brought out. For simply supported rectangular FG shallow shells, the results obtained are compared with those available in the literature. Comparison demonstrates a good accuracy of the approach proposed.

Active cancellation of unsteady stress oscillation in a functionally graded piezoelectric thin plate subjected to impact loading

Abstract When a functionally graded material plate is subjected to impact pressure, an unsteady oscillation of dynamic stress with tensile stress is induced and thus may cause serious damage to the plate. In this paper, active cancellation of the unsteady stress oscillation through piezoelectric actuation is investigated. First, a one-dimensional electro-elastodynamic problem in a functionally graded piezoelectric thin plate subjected to impact pressure is mathematically analyzed under the assumption that material properties vary exponentially in the thickness direction. The analytical results lead to the oscillation cancelling condition that the dynamic component disappears from the mathematical expression of stress. Secondly, employing the method of characteristics, a similar electro-elastodynamic problem is numerically analyzed when material properties are assumed to vary in the thickness direction according to a power low distribution. The actuation voltages which cancel the unsteady stress oscillations for various values of the volume fraction exponent are found to be obtained by utilizing the oscillation cancelling condition derived from the mathematical analysis of the first problem. Applying the obtained results, active cancellation of the unsteady stress oscillation may be possible even in the case of another functionally graded piezoelectric thin plate.

Size-dependent vibration and stability of multi-span viscoelastic functionally graded material nanopipes conveying fluid using a hybrid method

Functionally graded materials (FGMs) and fluid-conveying nanopipes may find significant applications in nanotechnology. In this paper, the size-dependent free vibration and stability of multi-span viscoelastic FGM nanopipes conveying fluid are investigated by nonlocal elasticity theory. And a hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to determine the natural frequencies. Present analysis is verified by comparing results with those in existing literature. Then, the effects of fluid velocity, nonlocal parameter, volume fraction exponent and internal damping on stability of multi-span FGM nanopipes conveying fluid are discussed. It is found that the stability decreases with the increase of nonlocal parameter and increases with increasing volume fraction exponent. It is also found that distributions of natural frequencies of FGM nanopipes can be modulated by designing the volume fraction exponent.

Size-dependent vibration analysis of multi-span functionally graded material micropipes conveying fluid using a hybrid method

Microscale fluid-conveying pipes and functionally graded materials (FGMs) have many potential applications in engineering fields. In this paper, the free vibration and stability of multi-span FGM micropipes conveying fluid are investigated. The material properties of FGM micropipes are assumed to change continuously through thickness direction according to a power law. Based on modified couple stress theory, the governing equation and boundary conditions are derived by applying Hamilton’s principle. Subsequently, a hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to determine the natural frequencies, and the results determined by present method are compared with those in the existing literature. Then, the effects of material length scale parameter, volume fraction exponent, location and number of supports on dynamic characteristics of multi-span FGM micropipes conveying fluid are discussed. The results show that the size effect is significant when the diameter of micropipe is comparable to the length scale parameter, and the natural frequencies determined by modified couple stress theory are larger than those obtained by classical beam theory. It is also found that natural frequencies and critical velocities increase rapidly with the increase of volume fraction exponent when it is less than 10, and the intermediate supports could improve the stability of pipes conveying fluid significantly.

Free vibration analysis of functionally graded double-tapered beam rotating in thermal environment considering geometric nonlinearity, shear deformability, and Coriolis effect

The present work investigates the free vibration behavior of double-tapered functionally graded beams rotating in thermal environment, using an improved mathematical model. The functional gradation for ceramic–metal compositions, following power-law, is considered to be symmetric with respect to the mid-plane, leading to metal-rich core and ceramic-rich outer surfaces of the beam. The temperature dependence of the material properties are considered using Touloukian model. The nonlinearity in strain–displacement relationships for both the axial and transverse shear strains are considered. Firstly, the governing equations for deformed beam configuration under time-independent centrifugal loading are obtained using minimum total potential energy principle, and the solution is obtained following Ritz method. Then the free vibration problem of the centrifugally deformed beam is formulated employing Lagrange’s principle and considering tangent stiffness of the deformed beam configuration. Coriolis effect is considered in the mathematical model, and the governing equations are transformed to the state-space for obtaining an eigenvalue problem. The results for the first two modes of both chord-wise and flap-wise vibrations are presented in nondimensional plane to show the effects of taperness parameter, root-offset parameter, volume fraction exponent, operating temperature, and functionally graded material composition. The results in comparative form are presented for both temperature-dependent and temperature-independent material properties.

An isogeometric approach for size-dependent buckling analysis of FGM nanoplates

In this paper, a suitable and simple computational formulation based on Isogeometric Analysis (IGA) integrated with higher-order shear deformation theory (HSDT) is introduced for size-dependent buckling analysis of functionally graded material (FGM) nanoplates. The material properties of FGM based on the Mori–Tanaka schemes and the rule of mixture are used. The differential nonlocal equations are utilized to take into account size effects. The nonlocal governing equations are approximated according to IGA based on HSDT, which satisfies naturally the higher-order derivatives continuity requirement in weak form of FGM nanoplates. The effect of nonlocal approach on the behaviors of the FGM nanoplates with several volume fraction exponents is investigated to show the reliability of the proposed method.

Stability analysis of multi-span viscoelastic functionally graded material pipes conveying fluid using a hybrid method

Abstract In this paper, a hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to investigate the stability of multi-span viscoelastic functionally graded material (FGM) pipes conveying fluid. The material properties of FGM pipes are considered as graded distribution along thickness direction according to a power-law. A parametric study is conducted to verify the effectiveness of present method and investigate effects of volume fraction exponent, fluid velocity, internal pressure and internal damping on stability of the FGM pipes conveying fluid. The numerical results demonstrate that the present method provides accurate results by using only a small number of elements and the viscoelastic FGM pipes exhibit some special dynamic behaviors. Moreover, the results also reveal that the natural frequencies of FGM piping system could be adjusted by devising the volume fraction exponent. This particular feature of FGM pipes can be tailored to fulfill the special applications in engineering.

Investigation of the mechanical properties on the large amplitude free vibrations of the functionally graded material sandwich plates

In this paper, the geometrically nonlinear formulation based on von Karman’s assumptions is employed to study the large amplitude free vibrations of functionally graded materials sandwich plates. The functionally graded material sandwich plate is made up of two layers of power-law functionally graded material face sheet and one layer of ceramic homogeneous core. A hierarchical finite element is employed to define the model, taking into account the effects of the transverse shear deformation and the rotatory inertia. The equations of motion for the nonlinear vibration of the functionally graded material sandwich plates are obtained using Lagrange’s equations. Employing the harmonic balance method, the equations of motion are converted from time domain to frequency domain and then solved iteratively using the linearized updated mode method. Results for linear and nonlinear frequency parameters of the simply supported functionally graded material sandwich plates are computed and compared with the published values, and an excellent agreement was found. The influence of the mechanical properties of the functionally graded material, thickness ratio of FGM layers, and volume fraction exponent on the backbone curves and on the nonlinear frequency parameters are investigated. The effects of the material properties of two different types of ceramics on the large amplitude vibration behaviors of the functionally graded material sandwich plates is also presented and discussed for the first time.

An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates

Abstract In the present investigation, a suitable and simple computational formulation based on Isogeometric Analysis (IGA) integrated with higher-order shear deformation theory (HSDT) is introduced for size-dependent geometrically nonlinear transient analysis of functionally graded material (FGM) nanoplates. The material properties of FGM based on the Mori–Tanaka schemes and the rule of mixture are used. The nonlinear transient nonlocal governing equations approximated according to IGA based on HSDT, which satisfies naturally the higher-order derivatives continuity requirement in weak form of the FGM nanoplates, are formed using the von Karman strains and solved by Newmark time integration scheme. The effect of nonlocal approach on the behaviors of the FGM nanoplates with several volume fraction exponents is investigated. Several numerical results are presented to demonstrate the reliability of the proposed method.