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.     

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.     

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.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments

Nonlinear bending analysis is presented for a simply supported, functionally graded rectangular plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded plate are based on Reddy's higher-order shear deformation plate theory that includes thermal effects. Two cases of the in-plane boundary conditions are considered. A mixed Galerkin-perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded rectangular plates with two constituent materials. The influences played by temperature rise, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio and volume fraction distributions are studied.     

Thermally induced buckling of functionally graded hybrid composite plates

In this paper, thermal buckling analysis is performed on hybrid functionally graded plates (FGPs) with an arbitrary initial stress. The governing equations are derived using the average stress method, including the effect of transverse shear deformation. Then, an eigenvalue problem is formed to evaluate thermal buckling temperatures for simple supported initially stressed ceramic-FGM-metal plates. The effects of functionally graded material (FGM) layer thickness, volume fraction index, layer thickness ratio, thickness ratio, aspect ratio and initial stress on the thermal buckling temperature of hybrid FGPs are investigated. The results reveal that the volume fraction index, initial stresses and FGM layer thickness have significant influence on the thermal buckling of hybrid FGPs.     

An exact solution for functionally graded piezoelectric laminates in cylindrical bending

In the present paper, the exact solutions of a simply supported functionally graded piezoelectric plate/laminate under cylindrical bending are derived. With similar derivation procedure as that used for Stroh formalism, the eigenrelation and general solutions for the problems can be expressed in very concise forms, which are convenient for further treatments of both analytic and numerical studies. The exact solutions can be served as benchmarks to verify and improve various approximate theories and numerical methods. To show the influence of material gradients, numerical examples based on the exact solutions are given, and some properties of the mechanical and electric responses of the plates under mechanical and electrical forces are discussed.     

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.     

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.     

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.     

Thermal buckling of functionally graded rectangular plates subjected to partial heating

We present the thermal buckling analysis of functionally graded rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thickness. The plate is simply supported for out-of-plane deformation and perfectly clamped for in-plane deformation. It is assumed that the functionally graded material properties such as the coefficient of linear thermal expansion and Young's modulus are changed individually in the thickness direction of the plate with the power law, while Poisson's ratio is assumed to be constant. Analytical developments consist of two stages. First, the nonuniform in-plane resultant forces are determined by solving a plane thermoelastic problem. Then the critical buckling temperatures of the plates with the predetermined resultant forces are calculated as the generalized eigenvalue problem which is constructed by using the Galerkin method. Finally, the effects of material inhomogeneity, aspect ratio, and heated region on the critical buckling temperatures are examined.     

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.     

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 functionally graded plates with multiple circular and non-circular cutouts

Cutouts are inevitable in structures due to practical consideration.In order to investigate the free vibration of functionally graded plates with multiple circular and non-circular cutouts,finite eleme...     

Thermoelastic and vibration analysis of functionally graded cylindrical shells

The static response and free vibration of metal and ceramic functionally graded shells are analyzed using the element-free kp-Ritz method. The material properties are assumed to vary continuously along the depth direction. The displacement field is expressed in terms of a set of mesh-free kernel particle functions according to Sander's first-order shear deformation shell theory. The effects of the volume fraction, material property, boundary condition, and length-to-thickness ratio on the shell deflection, axial stress, and natural frequency are examined in detail. Convergence studies of node numbers are performed to verify the effectiveness of the proposed method. Comparisons reveal that the numerical results obtained from the proposed method agree well with those from the classical and finite element methods.     

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.     

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.     

Stochastic analysis of compositionally graded plates with system randomness under static loading

This paper presents an investigation of the stochastic bending response of moderately thick, compositionally graded plates with uncertainties of low variability and subjected to lateral load and uniform temperature change. System parameters such as the thermal and mechanical material properties of each constituent material, volume fraction index, and load intensity are taken as independent random variables. The basic formulations are based on Reddy's higher-order shear deformation plate theory and a semi-analytical method. A first-order perturbation technique is employed to obtain the second-order response statistics-mean and variance of the flexural deflection of plates with various boundary conditions. Typical results are presented for two types of plates containing functionally graded materials made of metallic phase Ni and ceramic phase Al₂O₃. It is found that the response sensitivity of the plate is very much dependent on the material composition. Variations in Young's modulus and lateral load have dominant effects on the stochastic characteristics compared to other random parameters. The deflection dispersion of compositionally graded plates shows the so-called “non-intermediate” characteristic even when thermal loading is absent.     

Geometrically nonlinear analysis of functionally graded shells

The nonlinear response of functionally graded ceramic-metal shell panels under mechanical and thermal loading is studied. The nonlinear formulation is based on a modified version of Sander's nonlinear shell theory, in which the geometric nonlinearity takes the form of von Kármán strains. It is assumed that the material properties vary through the thickness according to a power-law distribution of the volume fraction of the constituents. The displacement field is expressed in terms of a set of mesh-free kernel particle functions. The bending stiffness is evaluated using a stabilized conforming nodal integration technique, and the shear and membrane terms are computed using a direct nodal integration to eliminate shear and membrane locking. The arc-length method, combined with the modified Newton-Raphson approach, is employed to trace the full load-displacement path. The characteristic of the displacement and the axial stress in panels under thermal and mechanical loading is investigated, and the effects of the volume fraction exponent, boundary conditions, and material properties on the nonlinear response of shell panels are also examined.     

Exact solutions for the free in-plane vibrations of rectangular plates

All classical boundary conditions including two distinct types of simple support boundary conditions are formulated by using the Rayleigh quotient variational principle for rectangular plates undergoing in-plane free vibrations. The direct separation of variables is employed to obtain the exact solutions for all possible cases. It is shown that the exact solutions of natural frequencies and mode shapes can be obtained when at least two opposite plate edges have either type of the simply-supported conditions, and some of the exact solutions were not available before. The present results agree well with FEM results, which show that the present solutions are correct and the direct separation of variables is practical. The exact solutions can be taken as the benchmarks for the validation of approximate methods.