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.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

FSDPT based study for vibration analysis of piezoelectric coupled annular FGM plate

The vibration behavior of a piezoelectrically actuated thick functionally graded (FG) annular plate is studied based on first order shear deformation plate theory (FSDPT). A consistent formulation that satisfies the Maxwell static electricity equation is presented so that the full coupling effect of the piezoelectric layer on the dynamic characteristics of the annular FG plate can be estimated based on the free vibration results. The differential equations of motion are solved analytically for various boundary conditions of the plate. The analytical solutions are derived and validated by comparing the obtained resonant frequencies of the composite plate with those of an isotropic core plate. As a special case, assuming that the material composition of core plate varies continuously in the direction of the thickness according to a power law distribution, a comprehensive study is conducted to show the influence of functionally graded index on the vibration behavior of smart structure. Also, the good agreement between the results of this paper and those of the finite element (FE) analyses validates the presented approach. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Farzad Ebrahimi received his B.S. and M.S. degree in Mechanical Engineering from University of Tehran, Iran. He is currently working on his Ph.D. thesis under the title of “Vibration analysis of smart functionally graded plates” at Smart Materials and Structures Lab in Faculty of Mechanical Engineering of the University of Tehran. His research interests include vibration analysis of plates and shells, smart materials and structures and functionally graded materials.     

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.     

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.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Free vibration of cantilevered symmetrically laminated thick trapezoidal plates

To account for the effect of transverse shear deformation, the p-Ritz method incorporating Reddy’s third-order shear deformation theory has been developed for the vibration analysis of cantilevered, thick, laminated, trapezoidal plates. In the p-Ritz method, a set of uniquely defined polynomial functions, consisting of the product of a two-dimensional function and a basic function, are used as the admissible trial displacement and rotation functions in the Ritz minimization procedure. The energy integral is formulated based on Reddy’s third-order shear deformation theory. From the p-Ritz method, the governing eigenvalue equation is derived which is used to compute the vibration frequency parameters and mode shapes of the laminated plate. Convergence and comparison studies have been presented to demonstrate and verify the accuracy of the results.     

Free vibration analysis of multi-directional functionally graded circular and annular plates

This paper addresses the free vibration of multi-directional functionally graded circular and annular plates using a semianalytical/ numerical method, called state space-based differential quadrature method. Three-dimensional elasticity equations are derived for multi-directional functionally graded plates and a solution is given by the semi-analytical/numerical method. This method gives an analytical solution along the thickness direction, using a state space method and a numerical solution using differential quadrature method. Some numerical examples are presented to show the accuracy and convergence of the method. The most of simulations of the present study have been validated by the existing literature. The non-dimensional frequencies and corresponding displacements mode shapes are obtained. Then the influences of thickness ratio and graded indexes are demonstrated on the non-dimensional natural frequencies.     

Free vibration of transversely isotropic piezoelectric circular cylindrical panels

This paper presents an exact three-dimensional free vibration analysis of a transversely isotropic piezoelectric circular cylindrical panel. The general solution for coupled equations for piezoelectric media that was recently proposed by Ding et al. (Int. J. Solids Struct. 33 (1996) 2283) is employed. By using the variable separation method, three-dimensional exact solutions are obtained under several boundary conditions. Numerical results are finally presented and compared with available data in literature. The results show the non-dimensional frequencies of the piezoelectric panel are bigger than that of the non-piezoelectric one.     

Stability and free vibration analysis of thick piezoelectric composite plates using spline finite strip method

In the present study, a spline finite strip with higher-order shear deformation is formulated for stability and free vibration analysis of piezoelectric composite plates. At each knot, the electric potentials on the surfaces and middle plane of each piezoelectric layer are taken as nodal degrees of freedom. However, if a continuous electrode is installed on the surface of the layer, the electric potential on the electrode is changed to structural degree of freedom, so that the equipotential condition on the electrode is automatically satisfied. The analysis can be conducted based on Reddy's third-order shear deformation theory, Touratier's “Sine” model, Afaq's exponential model or Cho's higher-order zigzag laminate theory. Consequently, the shear correction coefficients are not required in the analysis, and an improved accuracy for thick plates over the first-order shear deformation theory is achieved at only little extra computational cost.The numerical results obtained based on different shear deformation theories are presented in comparison with the three-dimensional solutions. The effects of length-to-thickness ratio, fiber orientation, boundary conditions and electrical conditions on the natural frequency and critical buckling load of piezoelectric composite plates are investigated through numerical examples.     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

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.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

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.     

Free vibration of laminated composite plates using two variable refined plate theory

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.     

The free vibration analysis and optimal design of an adhesively bonded functionally graded single lap joint

In this study, three-dimensional free vibration and stress analyses of an adhesively bonded functionally graded single lap joint were carried. The effects of the adhesive material properties, such as modulus of elasticity, Poisson's ratio and density were found to be negligible on the first ten natural frequencies and mode shapes of the adhesive joint. Both the finite element method and the back-propagation artificial neural network (ANN) method were used to investigate the effects of the geometrical parameters, such as overlap length, plate thickness and adhesive thickness; and the material composition variation through the plate thickness on the natural frequencies, mode shapes and modal strain energy of the adhesive joint. The suitable ANN models were trained successfully using a series of free vibration and stress analyses for various random geometrical parameters and compositional gradient exponents. The ANN models showed that the support length, the plate thickness and the compositional gradient exponent played important role on the natural frequencies, mode shapes and modal strain energies of the adhesive joint whereas the adhesive thickness had a minor effect. In addition, the optimal joint dimensions and compositional gradient exponent were determined using genetic algorithm and ANN models so that the maximum natural frequency and the minimum modal strain energy conditions are satisfied for each natural frequency of the adhesively bonded functionally graded single lap joint.     

Free vibration analysis of moderately thick antisymmetric cross-ply laminated rectangular plates with elastic edge constraints

This study presents a simple formulation for studying the free vibration of shear-deformable antisymmetric cross-ply laminated rectangular plates having translational as well as rotational edge constraints. The aim is to fill the void in the available literature with respect to the free vibration results of antisymmetric cross-ply laminated rectangular plates. The spatial discretization of the resulting mathematical model in five field variables is carried out using the two-dimensional Differential Quadrature Method (DQM). Several combinations of translational and rotational elastic edge constraints are considered. Convergence study with respect to the number of nodes has been carried out and the results are compared with those from past investigations available only for simpler problems. Effects of stiffness parameters, geometrical features, moduli ratio and lamination schemes on the natural frequencies are studied.     

Analysis of nonlinear vibration response of a functionally graded truncated conical shell with piezoelectric layers

The nonlinear vibration response of a functionally graded materials (FGMs) truncated conical shell with piezoelectric layers is analyzed. The vibration amplitude is suppressed by the positive and inverse piezoelectric effects. And the bifurcation phenomenon is described to reveal the motion state of the conical shell. Firstly, a truncated conical shell composed of three layers is described. And the effective material properties of the FG layer are defined by the Voigt model and the power law distribution. Next, the electric potentials of piezoelectric layers are defined as cosine distribution along the thickness direction. Meanwhile, the constant gain negative velocity feedback algorithm is used to suppress the vibration amplitude by the electric potential produced by the sensor layer. Thereafter, considering the first-order shear deformation theory and the von Karman nonlinearity, the relationship between the strain and displacement is defined. And the corresponding energy of the conical shell is calculated. After that, the motion equations of the conical shell are derived based on the Hamilton principle. Again, the nonlinear single degree of freedom equation is derived by the Galerkin method and the static condensation method. In the end, the nonlinear vibration response of FGMs truncated conical shell with piezoelectric layers under the external excitation is analyzed via using the harmonic balance method and the Runge-Kutta method. The effects of various parameters, such as ceramic volume fraction exponent, external excitation’s amplitude, control gain and geometric parameters on the nonlinear vibration response of the system are evaluated by case studies. Results indicate that the control gain plays an important role on the suppression of the vibration amplitude. The ceramic volume fraction exponents are not sensitive to the nonlinear vibration response compared with other parameters. The bifurcation behavior is observed under different parameters. The FGMs truncated conical shell with piezoelectric layers has three types of motion state, such as periodic motion, multi-periodic motion, and chaos motion.