Four-parameter Power-law Distributions Research Articles

A semi-analytic model for vibro-acoustic analysis of functionally graded shells of revolution

A semi-analytic model with wider applications is developed for vibro-acoustics of submerged functionally graded material (FGM) shells of revolution, and independent or coupled FGM shells are specific cases. For structural model, the shell is divided to many narrow segments treated as conical shells. Materials vary continuously in thickness direction according to the four-parameter power-law distributions about volume fractions of two constituents. Based on the first-order shear deformation theory (FSDT), differential equations are solved via expanding displacements as power series and Fourier series in meridional and circumferential directions, respectively. Continuity and boundary conditions are assembled to vibration governing equation. For acoustic model, Helmholtz integral equation is simplified to one line integral after expanding velocity and acoustic pressure as Fourier series in circumferential direction, and the pressure is further expressed as displacements of the shell. Then, structural and acoustic models are coupled as adding the acoustic pressure to continuity conditions. Through comparing results of developed model with the ones of literature and coupled finite element–boundary element method for some FGM shells, rapid convergence and high accuracy are demonstrated. Furthermore, discrepancies between vibro-acoustic responses of FGM, homogeneous and layered shells are slight, and effects of heavy fluid become more obvious as the volume fraction of the material with large stiffness decreases.

An analytical method for free vibrations of functionally graded cylindrical shells with arbitrary intermediate ring supports

An analytical method is presented to study free vibrations of functionally graded material (FGM) cylindrical shells with arbitrary intermediate ring supports. Material properties continuously vary in thickness direction in accordance with the four-parameter power-law distributions in terms of volume fractions of constituents. To establish the final governing equation, the shell is firstly divided into several shell segments according to locations of ring supports. Based on the first-order shear deformation theory (FSDT), differential equations about displacements are solved by expanding displacements as exponential functions in axial direction and Fourier series in circumferential direction. Correspondingly, five displacements and five forces at any cross section of segments are further expressed as 10 unknown coefficients for every circumferential mode number. The artificial spring technique is adopted to restrain displacements at boundaries and ring supports, and all segments are assembled to the overall shell through continuity conditions. To test the validity of the developed model, natural frequencies and mode shapes of several homogenous and FGM cylindrical shells with/without ring supports are compared with appropriate ones in the literature or calculated by finite element method (FEM), which demonstrates high accuracy and wide application of the proposed method. Furthermore, effects of ring supports and material parameters are analyzed. The results reveal that restraining radial and/or circumferential displacements at ring supports significantly affects natural frequencies, but mode shapes are mainly affected by radial displacements. In addition, natural frequencies of the FGM shell obviously vary with the power-law exponent, whereas the corresponding mode shapes almost keep unchanged.

Coupled free vibration analysis of functionally graded shaft-disk system by differential quadrature finite element method

The purpose of this paper is to investigate coupling vibration characteristics of the flexible functionally graded material (FGM) shaft-disk coupling system with variable thickness disk. The mathematical model of the shaft-disk system is based on the first-order shear deformation theory, combined with the Voigt model and the four-parameter power-law distribution. The energy expression is discretized by the differential quadrature finite element method, and furthermore the differential equation of the shaft-disk system is derived. In order to verify the convergence and calculation accuracy of the current model, a series of numerical examples are introduced. In addition, the influence of geometrical and material parameters on vibration results is discussed, while the boundary conditions and the cross-sectional shape of the disk are considered. It can be seen that the variation of the FGM parameters will affect the vibration characteristics within a certain range, while the frequency parameter with different disk cross-sectional shapes is different in sensitivity to parameter changes. The parametric analysis can provide reference for the determination of structure and material parameters.

Free Vibration Response of Four-Parameter Functionally Graded Thick Spherical Shell Formulation on FEA

The following study explored the free vibration characteristics of a rectangular spherical shell. An efficient formulation developed Higher-order shear deformation theory (HSDT), conjunction with the numerical procedure (FEM). To determine the desired volume fraction obtained through the Four-parameter power-law distribution considered on the top surface is ceramic-rich. In contrast, the base surface is metal-rich. Parameters distribution gives design flexibility and different kinds of material profiles exhibited. The efficiency of the model is verified by performing convergence studies and comparison tests. The numerical results compared with earlier available literature solutions to authenticate the stability and exactness of the current approach, and good agreement perceived. Illustrated numerical examples to find the influence on the appropriate selection of the four-parameters, the mechanical behavior of the composition and effects on geometrical parameters, skew angles, different boundary conditions on their non-dimensional frequency responses examined in detail.

Weight optimisation of functionally graded beams using modified differential evolution

In this article, an efficient numerical approach for weight optimisation of functionally graded (FG) beams in the presence of frequency constraints is presented. For the analysis purpose, a finite element (FE) solution based on the first order shear deformation theory (FSDT) is established to analyse the free vibration behaviour of FG beams. A four-parameter power law distribution and a five-parameter trigonometric distribution are used to describe the volume fraction of material constituents in the thickness direction. The goal is to tailor the thickness and material distribution for minimising the weight of FG beams while constraining the fundamental frequency to be greater than a prescribed value. The constrained optimisation problem is effectively solved by a novel differential evolution (DE) algorithm. The validity and efficiency of the proposed approach is demonstrated through two numerical examples corresponding to the four-parameter distribution and the five-parameter distribution.Keywords: FGM beam; lightweight design; frequency constraint; differential evolution.

Free vibration analysis of laminated and FGM composite annular sector plates

In this article, the authors used two different numerical approaches for frequency response of annular sector and sector plates with functionally graded materials and laminated composite cases. First-order shear deformation (FSDT) and Love's conical shell theories are used for obtaining the annular sector plate equations via some suitable angles and geometric parameters. The method of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) have been used for numerical solution of the resulting governing equations for modal analysis. Simple power-law and four-parameter power law distributions in terms of the volume fractions of constituents have been used for FGM composites. Comparison and convergence study for the present numeral methods have been made via existing results available in the literature for sector and annular sector plates. Frequencies values for annular sector/sector plates have been obtained for different material and geometric parameters, boundary conditions and sector angles.

Vibration analysis of circular cylindrical panels with CNT reinforced and FGM composites

The aim of this study is to give a numerical solution for free vibration problem of functionally graded and carbon nanotube reinforced (CNTR) circular cylindrical panel. Based on the Love’s shell theory and first-order shear deformation theory (FSDT), related equations of motion have been obtained. Then, the methods of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) are used for numerical computations. Two different material properties are assumed to change continuously in the thickness direction according to the volume fraction power law and the general four-parameter power law distributions. Accuracy and convergence of the methods have been validated by comparing the obtained results with the existing results available in the open literature. Frequency values have been given for different material and geometric parameters, modes and boundary cases for circular cylindrical panels with FGM and CNTR cases. The effects of types of distributions of CNTR material are also discussed.

Vibration of laminated composite panels and curved plates with different types of FGM composite constituent

Abstract Free vibration analysis of truncated conical panels and annular sector plates with functionally graded materials (FGM) is carried out. Governing equations of motion are obtained based on two different shell theories such as Love's shell theory and first-order shear deformation theory (FSDT). The resulting governing differential equations have been solved using the differential quadrature (DQ) and discrete singular convolution (DSC) methods. As the FGM cases two different material properties of structures are assumed to change continuously in the thickness direction according to the volume fraction power law and the general four-parameter power law distributions in terms of the volume fractions of constituents. Accuracy, convergence and reliability of these two methods have been validated by comparing the obtained results with the existing results available in the open literature. Furthermore, the effects of the grid numbers on results for each method have been also investigated for different boundary conditions and mode numbers for conical panel vibrations. Then, using the DQ and DSC methods, the frequencies values have been calculated for different material and geometric parameters, modes and boundary cases for truncated conical panels with FGM. The effects of material power-law distribution are also discussed. The convergence, advantages and accuracy of the present two methodologies are examined in conjunctions with the vibration problem of truncated conical panels with functionally graded materials (FMG). Some results for annular sector plates and circular cylindrical panels have also been obtained via conical panel equations.

Nonlinear Dynamics Analysis of FGM Shell Structures with a Higher Order Shear Strain Enhanced Solid-Shell Element

In this paper, non-linear dynamics analysis of functionally graded material (FGM) shell structures is investigated using the higher order solid-shell element based on the Enhanced Assumed Strain (EAS). With this element, a quadratic distribution of the shear stress through the thickness is considered in an enhancing part. Material properties of the shell structure are varied continuously in the thickness direction according to the general four-parameter power-law distribution in terms of the volume fractions of the constituents. Performance and accuracy of the present higher order solid-shell element are confirmed by comparing the numerical results obtained from finite element analyses with results from the literature.

Frequencies of FGM shells and annular plates by the methods of discrete singular convolution and differential quadrature methods

Free vibration analysis of curved structural components such as truncated conical shells, circular cylindrical shells and annular plates has been investigated numerically in this paper. The method of discrete singular convolution (DSC) and the method differential quadrature (DQ) are used for numerical simulations, respectively. Related partial differential equations governed the motion of the structures obtained from higher-order shear deformation theory have been solved by using these two methods in the space domain. Different material properties have been considered such as isotropic, laminated and functionally graded material (FGM). Four-parameter power law and simple power law distributions have been used for ceramic volume fraction in FGM cases. The numerical results related to free vibration of conical shells have obtained by the present two techniques compare well with the results available in the literature. Results for circular cylindrical shells and annular plates have also been presented for different geometric and material parameters. The effects of grid number and types of the grid distribution have also been investigated for annular plate and shells.

Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions

This paper aims to present a unified vibration analysis approach for the four-parameter functionally graded moderately thick doubly-curved shells and panels of revolution with general boundary conditions. The first-order shear deformation theory is used in this formulation. The functionally graded panels structures consists of ceramic and metal which are set to vary continuously in the thickness direction according to the general four-parameter power-law distribution, and six types of power-law distributions are considered for the ceramic volume fraction. The admissible function of the FG panels and shells of revolution is obtained by the improved Fourier series with the help of the governing equations and the boundary conditions. The solution is obtained by using the variational operation in terms of the unknown expanded coefficients. By a great many numerical examples, the rapid convergence and good reliability and accuracy of the proposed approach are validated. A variety of new results for vibration problems of the FG doubly-curved shells and panels with different elastic restraints, geometric and material parameters are presented. The effects of the elastic restraint parameters, power-law exponent, circumference angle and power-law distributions on the free vibration characteristic of the panels are also presented, which can be served as benchmark data in the research and the actual production process.

Optimizing the natural frequencies of axially functionally graded beams and arches

In this paper a new design methodology is presented to optimize the natural frequencies of axially functionally graded beams and arches by tailoring appropriately their material distribution. Functionally Graded Materials (FGMs) are deemed to have an advantageous behavior over laminated composites due to the continuous variation of their material properties yet in all three dimensions which alleviate delamination, de-bonding and matrix cracking initiation issues. The design of FGM structures is ideally fitted to optimization techniques, as the optimum material composition is derived by varying the relative volume fraction of the constituent materials. In this study the Differential Evolution (DE) is employed for optimizing the natural frequencies of axially FG beams and arches. The evaluation of the objective function requires the solution of a free vibration problem of an arch with variable mass and stiffness properties. The arches are modeled using a generic curved beam model that includes both axial (tangential) and transverse (normal) deformation and the problem is solved using the analog equation method (AEM) for hyperbolic differential equations with variable coefficients. The Mori-Tanaka homogenization scheme is adopted in this investigation for the estimation of the effective material properties. Several beams and arches are analyzed, which illustrate the design method and demonstrate its efficiency. In this work, and without restricting the generality, the FGM is comprised of steel and aluminum. The volume fraction of steel is assumed to follow two alternative distributions, namely a four-parameter power law distribution (FGM-1) or a five-parameter trigonometric distribution (FGM-2). In all cases three model problems are examined. We seek the material distribution that the FGM structure vibrates with (i) the maximum fundamental frequency, (ii) the minimum mass and the fundamental frequency greater than a prescribed value and (iii) the minimum mass and frequencies which lie outside certain frequency bands.

Dynamic response of functionally graded material shells with a discrete double directors shell element

The purpose of the paper is to compute the dynamic behavior of functionally graded material (FGM) shell structures subjected to time-varying excitation using 3D-shell model based on a discrete double directors shell element. The third-order shear deformation theory is introduced in the present method to remove the shear correction factor and improve the accuracy of transverse shear stresses. Material properties of the shell are assumed to be graded in the thickness direction by varying the volume fraction of the ceramic and the metallic constituents using general four-parameter power-law distribution. The transient excitation is defined in the time domain and known at each time. The damping material is neglected and the time derivative is approximated by Newmark method. Numerical results for deflection and stresses are presented for plates and spherical caps. The effect of an imposed force on the response of the FGM shell is discussed. The numerical examples prove a good accuracy and reliability compared to the few results available in literature.

Vibration characteristics of fluid-conveying FGM cylindrical shells resting on Pasternak elastic foundation with an oblique edge

The flow-induced vibration characteristics (natural frequency and critical flow velocity) of FGM cylindrical shells partially resting on elastic foundation are investigated by an analytical method. These shells are completely filled with fluid or subjected to a flowing fluid. The cylindrical shells are partially surrounded by an elastic foundation with an oblique edge which is represented by the Pasternak model. The boundary edge of an elastic foundation lies in a plane that is oblique at an angle with the shell axis. The effective material properties of FGM models are predicted based on two homogenization methods such as Voigt and Mori-Tanaka models. Material properties vary continuously through the thickness according to a four-parameter power law distribution in terms of volume fraction of the constituents. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The fluid is described by the classical potential flow theory. The governing equation for eigenvalue problem is obtained using the Rayleigh-Ritz method. To validate the present method, numerical examples are presented and compared with the available existing results.

A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution

In this paper, we proposed a simple mathematical procedure to solve the differential equations governing the buckling and bending analysis of FG thick rectangular plates resting on two-parametric foundation based on Mindlin assumption. All edges are set on the simply supported conditions. Young modulus of the FG plate was assumed to vary according to a simple four-parameter power law across the thickness direction. For bending analysis, the plate was subjected to two kinds of loading: sinusoidal and uniform. For bucking analysis, two kinds of in-plane loading were applied to the plate: uniaxial and biaxial. Variations of FG material variation profile, thickness ratio, and foundation parameters on buckling critical load and out-plane displacement were examined. The distribution of axial and shear stress across the thickness, when the plate is exposed to uniform transverse loading, was further studied.

A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions

The vibrations of functionally graded circular plates, annular plates, and annular, circular sectorial plates have been traditionally treated as different boundary value problems, which results in numerous specific solution algorithms and procedures. It is the problem itself that has been an overwhelming task for a new researcher or application engineer to comprehend. Furthermore each type of plate usually needs treating separately when different boundary conditions are involved. In this paper, a unified method is presented for the vibration analysis of the plates mentioned above with general boundary conditions based on the first-order shear deformation theory and Ritz procedure. The material properties are assumed to vary continuously through the thickness according to the general four-parameter power-law distribution. Regardless of the shapes of the plates and the types of boundary conditions, the displacements of the plates are described as an improved Fourier series expansion which is composed of a double Fourier cosine series and several auxiliary functions. As an innovative point of this work, the auxiliary functions are introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and to accelerate the convergence of series representations. The accuracy, reliability and versatility of the current solution are fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions. Some new results of functionally graded circular, annular and sector plates with various boundary conditions are presented, which may serve as datum solutions for future computational methods. In addition, the influence of boundary conditions, the material and geometric parameters on the vibration characteristics of the plates are also reported.

An isogeometric finite element approach for three-dimensional static and dynamic analysis of functionally graded material plate structures

This article proposes an efficiently computational tool based on an isogeometric finite element formulation of three-dimensional (3-D) elasticity for static and dynamic response analysis of functionally graded material (FGM) plates. The material properties of FGM plate structures are continuously assumed to vary according to the four-parameter power-law distributions in the thickness direction. The weak form is discretized and approximated by NURBS-based isogeometric approach. The method allows us to preserve exact geometry and is flexible to employ higher-order approximations for the desired accuracy. Through numerical tests, a quartic NURBS element is strongly recommended for eliminating naturally shear locking. Moreover, this element can produce an ultra-accurate solution with a low number of degrees of freedom. Numerical validation shows excellent performance of the present method in comparison with other available methods.

Free vibration analysis of FGM shell structures with a discrete double directors shell element

Free vibration response of functionally graded material (FGM) shell structures is studied by using an efficient 3d-shell model based on a discrete double directors shell element. With this element, the vanishing of transverse shear strains on top and bottom faces is considered in a discrete form. The mechanical properties of the shell structure are assumed to vary continuously in the thickness direction according to the general four-parameter power-law distribution in terms of the volume fractions of the constituents. The fundamental equations for the FGM shell structures are derived using principle of virtual work. Performance and accuracy of the present discrete double directors shell model (DDDSM) are confirmed by comparing the present solutions obtained from finite element analyses with the existing results in the literature.

Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge

The free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge are investigated by an analytical method. The cylindrical shell is partially surrounded by an elastic foundation which is represented by the Pasternak model. An edge of an elastic foundation lies in a plane that is oblique at an angle with the shell axis. The motion of shell is represented based on the first order shear deformation theory (FSDT) to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shell is composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a four-parameter power law distribution in terms of volume fraction of the constituents. The equation of motion for eigenvalue problem is obtained using Rayleigh–Ritz method and variational approach. To validate the present method, the numerical example is presented and compared with the available existing results.

Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories

The theoretical framework of the present manuscript covers the dynamic analysis of doubly-curved shell structures using the generalized displacement field of the Carrera Unified Formulation (CUF), including the Zig-Zag (ZZ) effect given by the Murakami’s function. The partial differential system of equations is solved by using the Generalized Differential Quadrature (GDQ) method. This numerical approach has been proven to be accurate, reliable and stable in several engineering applications. The current paper focuses on Functionally Graded (FG) doubly-curved shells and panels using various higher-order equivalent single layer theories, introduced and applied for the first time by the authors to completely doubly-curved shell structures, and different through-the-thickness volume fraction distributions, such as four-parameter power law, Weibull and exponential distributions. Moreover, the classic theory of mixtures is compared to the Mori–Tanaka scheme for the calculation of the mechanical properties of the materials. In particular, the numerical applications presented in this work are related to particular FG configurations in which it is possible to model a soft-core structure using a continuous variation of the mechanical properties of the materials at hand. The natural frequencies and mode shapes of several structures are presented and compared to numerical solutions taken from the literature.