Dynamic responses of a functionally graded pyroelectric hollow sphere for spherically symmetric problems

The spherically symmetric thermoelastic problem of a functionally graded pyroelectric hollow sphere is considered. The functionally graded material is restricted to the variations of density and other material constants as a certain power of the radial distance. By virtue of the introduction of a dependent variable as well as the separation of variables technique, a Volterra integral equation of the second kind related to a function with respect to time is derived, which is solved successfully by using an interpolation polynomial to approximate the unknown function. The corresponding recursive formulae are derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently and is suitable for a functionally graded pyroelectric hollow sphere with an arbitrary thickness and subjected to arbitrary spherically symmetric thermal loads. Numerical results are finally presented.     

Layer-wise finite element analysis of functionally graded cylindrical shell under dynamic load

In this paper, a layer-wise finite element formulation is developed for the analysis of a functionally graded material (FGM) cylindrical shell with finite length under dynamic load. For this purpose, FGM cylinder is divided into many sub-layers and then the general layerwise laminate theory is formulated by introducing piecewise continuous approximations through the thickness for each state In this model the radial displacement is approximated linearly through each “mathematical” layer. The properties are controlled by volume fraction that is an exponential function of radius. The governing equations are derived from virtual work statement and solved by finite element method. The main contribution of the present study is to develop a discrete layerwise finite element for a 2-dimensional thick FGM cylindrical shell. Results are obtained for the time history of the displacement and stress components with different exponent “n” of functionally graded material. In addition, natural frequency and mean velocity of the radial wave propagation for different exponent “n” of functionally graded material (FGM) are studied and compared with similar ones currently obtained for FGM cylindrical shell of infinite length.     

Vibration analysis of submerged thin FGM cylindrical shells

This study gives a brief work on vibration characteristics of cylindrical shells submerged in an incompressible fluid. The shell is presumed to be structured from functionally graded material. The effect of the fluid is introduced by using the acoustic wave equation. Love’s first order thin shell theory is utilized in the shell dynamical equations. The problem is framed by combining shell dynamical equations with the acoustic wave equation. Fluid-loaded terms are associated with Hankel function of second kind. Wave propagation approach is employed to solve the shell problem. Some comparisons of numerical results are performed for the natural frequencies of simply supported-simply supported, clamped-clamped and clamped-simply supported boundary conditions of isotropic as well as functionally graded cylindrical shells to check the validity of the present approach. The influence of fluid on the submerged functionally graded cylindrical shells is noticed to be very pronounced.     

Functionally graded piezoelectric strip with eccentric crack under anti-plane shear

In this paper, we examine the singular stresses and electric fields in a functionally graded piezoelectric ceramic strip containing an eccentric crack off the center line under anti-plane shear loading with the theory of linear piezoelectricity. It is assumed that the properties of the functionally graded piezoelectric ceramic strip vary continuously along the thickness. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate are obtained.     

球壳与柱壳之功能梯度压电涂层的热效应分析

针对以球壳和圆柱壳为基体的功能梯度压电涂层,由多场耦合控制方程和层间连续条件导出递推关系,建立了显式的力-电-热多场耦合解。对于多层功能梯度压电涂层,此解为精确解;对于连续功能梯度压电涂层,可将涂层分为若干层,只要层数取得足够大,所得的近似解将收敛于精确解。研究表明：压电效应对基体的应力影响可以忽略,但对涂层的应力影响非常显著;增加层数并不能有效降低最大应力,但增加层数可显著减小相邻层层间周向应力的突变幅度,减小层间应力集中;然而涂层与基体之间的界面径向应力却随层数的增大而使界面强度弱化。     

Dynamic propagation of a finite eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic propagation of an eccentric Griffith crack in a functionally graded piezoelectric ceramic strip under anti-plane shear is analyzed using the integral transform method. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a pair of dual integral equations, which is then expressed in a Fredholm integral equation of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties, crack moving velocity, and eccentricity. The dynamic stress intensity factors of a moving crack in functionally graded piezoelectric material increases when the crack moving velocity, eccentricity of crack location, material property gradient, and crack length increase. This paper was recommended for publication in revised form by Associate Editor Hyeon Gyu Beom Jeong Woo Shin received a B.S. and M.S. degree in Mechanical Engineering from Yonsei University in Seoul, Korea in 1998 and 2000, respectively. A major field of Mr. Shin is fracture mechanics. He is currently working on the KARI (Korea Aerospace Research Institute) as a senior researcher. He conducted load analysis of fixed wing aircraft and full scale airframe static test at the KARI. He is now developing landing gear in the KHP (Korea Helicopter Program) as a performance engineer.     

Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells

Based on the three-dimensional fundamental equations of anisotropic elasticity, a state equation with variable coefficients is derived in a unified matrix form. The free vibration of simply supported, fluid-filled cylindrically orthotropic functionally graded cylindrical shells with arbitrary thickness is then investigated. A laminate approximate model is employed which is suitable for an arbitrary variation of material constants along the radial direction. Numerical examples are presented and compared with existing results. The effects of related parameters on natural frequencies are discussed finally.     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.     

Dynamic characteristics of an eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic response of an eccentric Griffith crack in functionally graded piezoelectric ceramic strip under anti-plane shear impact loading is analysed using integral transform method. Laplace transform and Fourier transform are used to reduce the problem to two pairs of dual integral equations, which are then expressed to Fredholm integral equations of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties and electric loadings.     

Nonlinear cylindrical bending analysis of shear deformable functionally graded plates under different loadings using analytical methods

An exact solution is presented for the nonlinear cylindrical bending and postbuckling of shear deformable functionally graded plates in this paper. A simple power law function and the Mori–Tanaka scheme are used to model the through-the-thickness continuous gradual variation of the material properties. The von Karman nonlinear strains are used and then the nonlinear equilibrium equations and the relevant boundary conditions are obtained using Hamilton's principle. The Navier equations are reduced to a linear ordinary differential equation for transverse deflection with nonlinear boundary conditions, which can be solved by exact methods. Finally, by solving some numeral examples for simply supported plates, the effects of volume fraction index and length-to-thickness ratio are studied. It is shown that there is no bifurcation point for simply supported functionally graded plates under compression. The behavior of near-boundary areas predicted by the shear deformation theory and the classical theory is remarkably different.     

Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity

This paper presents a new approach for analyzing transverse bending and vibration of circular cylindrical beams with radial nonhomogeneity. The radial nonhomogeneity may be continuous or piecewise-constant, corresponding a functionally graded circular cylinder or a multi-layered circular cylinder, respectively. Different from the Euler-Bernoulli and Timoshenko theories of beams, our analysis considers shear deformation, but does not need to introduce a shear correction factor. Using the shear-stress-free condition at the surface of the cylinder, coupled governing equations for deflection and rotation angle are derived, and then converted to a single governing equation. The influences of gradient index on the deflection and stress distribution for cantilever and simply-supported beams are studied. Natural frequencies of free vibration of a cylindrical beam with circular cross-section are calculated for different power-law gradients. In particular, a circular cylindrical shell may be taken as a special case of a bi-layered cylinder where the material properties of the inmost cylinder vanish. For this case, the natural frequencies for simply-supported and clamped-clamped cylindrical shells are evaluated and compared with those using three-dimensional theory. Our results coincide well with the previous.     

Effect of material inhomogeneity on the rotating functionally of a graded orthotropic hollow cylinder

The effect of the material inhomogeneity on the stress field of a rotating orthotropic infinite hollow cylinder made of functionally graded materials is investigated. The original functionally graded hollow cylinder is approximated by the laminate model, of which the solution will gradually approach the exact one with the increase in number of fictitious layers. The analysis is performed by means of the state space method. The validity of the solution is verified by utilizing the exact solution of a special non-homogeneous rotating hollow cylinder and earlier results. The distributions of the radial and tangential stresses of the internally pressurized rotating functionally graded hollow cylinder for various material inhomogeneity parameters are depicted graphically. The degree of the orthotropy on the stress fields is also discussed.     

Nonlinear analysis of functionally graded laminates considering piezoelectric effect

In this paper, static bending analysis of functionally graded plates with piezoelectric layers has been carried out considering geometrical nonlinearity in different sets of mechanical and electrical loadings. Only the geometrical nonlinearity has been taken into account. The governing equations are obtained using potential energy and Hamilton??s principle. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect by using higher order elements. The present finite element used displacement and electric potential as nodal degrees of freedom. Results are presented for two constituent FGM plate under different mechanical boundary conditions. Numerical results for FGM plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response of the plate are also studied.     

Transverse shear and rotary inertia effects on the stability analysis of functionally graded shells under combined static and periodic axial loadings

The effect of transverse shear and rotary inertias on the dynamic stability of functionally graded cylindrical shells subjected to combined static and periodic axial forces is investigated in this paper. Material properties of functionally graded cylindrical shells are considered temperature-dependent and are graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Numerical results for silicon nitride-nickel cylindrical shells are presented based on two different methods: the first-order shear deformation theory (FSDT) which considers the transverse shear strains and the rotary inertias, and the classical shell theory (CST). The results obtained show that the effect of transverse shear and rotary inertias on the dynamic stability of functionally graded cylindrical shells subjected to combined static and periodic axial forces is dependent on the shell’s material composition, environmental temperature, amplitude of static load, deformation mode, and the shell’s geometry parameters.     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

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.     

Thermo-electro-mechanical vibration and buckling analysis of a functionally graded piezoelectric porous cylindrical microshell

Journal of Mechanical Science and Technology - In this paper, the vibration and buckling behavior of a functionally graded piezoelectric porous cylindrical microshell under...     

Vibration of bilayered cylindrical shells with layers of different materials

In this analysis, a comparative study for natural frequencies of two-layered cylindrical shells was presented with one layer composed of functionally graded material and the other layer of isotropic material. Love’s thin shell theory was exploited for the strain-displacement and curvature-displacement relationships. For governing frequency equations, the Rayleigh-Ritz method was utilized to minimize the Lagrangian functional in the form of an eigenvalue problem. Frequency spectra were computed for long, short, thick, and thin cylindrical shells by varying the nondimensional geometrical parameters, length-to-radius and thickness-to-radius ratios for a simply supported end condition. Influence of different configurations of cylindrical shells on the shell frequencies was studied. For validity, the results obtained were compared with some results of isotropic and single-layered functionally graded cylindrical shells from the literature.     

Thermal performance of functionally graded parabolic annular fins having constant weight

The proposed work reports the performance of parabolic annular fins of constant weight made of functionally graded materials. The work involves computation of temperature gradient, efficiency and effectiveness of such fins and compares the performances for different functionally graded parabolic fin profiles obtained by varying grading parameters and profile parameters respectively keeping the weight of the fins constant. The functional grading of thermal conductivity is based on a power function of radial co-ordinate which consists of parameters, namely grading parameters, varying which different grading combinations are studied. A general second order ordinary differential equation has been derived for all the profiles and material grading. The efficiency and effectiveness of the annular fins of different profile and grading combinations have been calculated and plotted and the results reveal the dependence of fin performance on profile and grading parameter.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.