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.     

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.     

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.     

Investigating the thermal environment effects on geometrically nonlinear vibration of smart functionally graded plates

An analytical solution for a sandwich circular FGM plate coupled with piezoelectric layers under one-dimensional heat conduction is presented. All materials of the device may be of any functional gradients in the direction of thickness. The solution exactly satisfies all the equilibrium conditions and continuity conditions for the stress, displacement and electric displacement as well as electric potential on the interfaces between adjacency layers. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the FG plate that is subjected to in-plane forces and applied actuator voltage in thermal environment in the case of simply supported boundary conditions. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectric coupled FGM plates are derived. The role of thermal environment as well as control effects on nonlinear static deflections and natural frequencies imposed by the piezoelectric actuators using high input voltages are investigated. Numerical examples are provided and simulation results are discussed. Numerical results for FGM plates with a mixture of metal and ceramic are presented in dimensionless forms. The good agreement between the results of this paper and those of the finite element (FE) analyses validated the presented approach. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage and thermal environment as well as gradient index of FG plate on the dynamics and control characteristics of the structure.     

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 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.     

Effect of loading and lamination parameters on the optimum design of laminated plates

This paper illustrates the effect of loading conditions and lamination parameters on the optimum design of the laminated composite plates. Optimum design procedure based on the flexibility criterion is presented in this paper. The objective is to determine the optimum thickness of the laminate layers and its optimum orientations without exhibiting any failure. The finite element method based on Mindlin plate theory is used in conjunction with an optimization method in order to determine the optimum design. Newmark implicit time integration scheme is used to discretize the time domain and calculate the transient response of the laminated plate. Exterior penalty method is exploited as a constrained minimization technique. For this purpose, four numerical examples are presented to figure out the effects of dynamic loading profile, boundary conditions, number of layers, and degree of orthotropy on the optimum weight design.     

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.     

Dynamic characteristics analysis of active constrained layer damping plate with various boundary conditions

Considering the direct and converse piezoelectric effect, expressions of piezoelectric membrane internal forces in the piezoelectric constrained layer were given. The control equations of the piezoelectric constrained layer and host plate were obtained in according with the thin plate theory. Based on the layer wised principle, the integrated first order differential equation of an active constrained layer damping (ACLD) plate was derived for the simply supported boundary condition. Then, this method was expanded to the ACLD plate with cantilever boundary condition by virtue of geometric analogy method. Employing the extended homogeneous capacity precision integration approach, a high precision semi-analytical method was proposed to analyze the dynamic characteristics of the ACLD plate with various boundary conditions. The comparison with the literature results has verified the accuracy and effectiveness of the present method.     

Thermoelastic analysis of functionally graded sandwich plates with a homogeneous core

This paper is concerned with the thermoelastic analysis of functionally graded (FG) sandwich plates with a homogeneous core by a numerical method. The core layer is homogeneous ceramic while two facesheets are inhomogeneous metal-ceramic FGMs having the power-law volume fractions. The metal-ceramic FG sandwich plates are characterized by the relative thicknesses of three layers, the width-thickness and aspect ratios of plate, and the volume fractions of metal and ceramic. Meanwhile, the problem is formulated using the hierarchical models exhibiting the spectral model accuracy and implemented by 2-D natural element method (NEM). The hierarchical models are based upon the 3-D elasticity and NEM is applied to the mid-surface of plate to approximate the triple-vectored in-plane displacement field. The accuracy of hierarchical models are examined with respect to the model order, from which the (3, 3, 4) hierarchical model is chosen for the thermoelastic analysis. The thermoelastic responses obtained by the present method are compared with the existing analytic solutions, and those are parametrically investigated with respect to the above-mentioned design parameters. It is found that the present method shows a reasonable accuracy and the thermoelastic responses of FG sandwich plates are remarkably influenced by the design parameters.

     

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.     

Bending and extension of a multilayered plate with body forces and thermal loading

A multilayered plate composed of thin layers of isotropic materials is analyzed. The problem for the multilayered plate with body forces is formulated by using the lamination theory in which displacement fields are expressed in terms of in-plane displacements on a main plane and transverse displacement. Placing the main plane at an appropriate distance from the lower surface of the plate, a set of equilibrium equations is shown to be written in uncoupled forms, which are identical to those for an uncoupled plate such as a single layer plate. It is proved that the complete solutions of the multilayered plates subject to the specified in-plane resultant tractions or in-plane displacements on its whole boundary can be obtained from the sum of solutions for uncoupled plates. Closed form solutions are obtained for a circular laminate clamped or simply supported on its the boundary as well as for a rotating disk with a constant angular velocity. The calculations of thermoelastic stresses and displacements in multilayered plates are also discussed. Closed form solutions are obtained for a circular laminate with distributed temperature varying in the radial direction and through the thickness.     

Semi-analytical solution of magneto-thermo-elastic stresses for functionally graded variable thickness rotating disks

In this paper, a semi-analytical solution for magneto-thermo-elastic problem in functionally graded (FG) hollow rotating disks with variable thickness placed in uniform magnetic and thermal fields is presented. Stresses and perturbation of magnetic field vector in FG rotating disks are determined using infinitesimal theory of magneto-thermo-elasticity under plane stress conditions. The material properties except Poisson’s ratio are modeled as power-law distribution of volume fraction. The profile of disk thickness is assumed to be a parabolic function of radius. The non-dimensional distribution of temperature, displacement, stresses and perturbation of magnetic field vector throughout radius are shown. Effects of material grading index, geometry of the disk and magnetic field on the stress and displacement fields are investigated. The results of stresses and radial displacements for two different boundary conditions with and without the effect of magnetic field are compared for a FG rotating disk with concave thickness profile. It has been found that imposing a magnetic field significantly decreases tensile circumferential stresses. Therefore the fatigue life of the disk will be significantly improved by applying the magnetic field. Results of this investigation could be applied for optimum design of FG hollow rotating disks with variable thickness.     

A novel approach to thick plate theory suggested by studies in foundation theory

A new approach to the statics of thick, elastic plates is presented. This approach to thick plate theory incorporates transverse normal stress and strain effects as well as those due to transverse shear deformation. The unusual boundary value problem resulting from the kinematic assumptions of this approach is obtained as a consequence of the principle of stationary potential energy. This boundary value problem is nonlinear so that the principle of superposition does not hold and it does not appear to be one which may be treated by standard methods. Therefore an iterative method of approximation is proposed. This iterative method is applied, analytically, to a problem of cylindrical plate bending for which an exact solution is known within the linear theory of elasticity. It is shown that for this particular problem the approximation method converges in two iterations to the exact elasticity solution. Since such simple results are not to be expected in general, there is some discussion of the need to develop a numerical algorithm to implement the proposed iterative method of approximation for arbitrary plate problems. Furthermore, certain theoretical difficulties with the new approach to thick plate theory are discussed.     

功能梯度压电材料壳体热残余应力

功能梯度压电材料结构成型冷却后会出现热残余现象,影响结构强度.借鉴复合材料层合结构的研究方法,将功能梯度压电材料球壳和圆柱壳沿厚度分为若干层,各层视为均匀材料,根据层间连续条件导出递推关系,得到显式的力—电—热多场耦合热残余解.统一了多层功能梯度压电材料壳体和连续功能梯度压电材料壳体热残余解.对于前者,其解为精确解;对于后者,其解为渐近解,随层数增加而收敛于精确解.其解也适用于功能梯度压电材料涂层.该方法对材料性能的变化方式(函数)没有要求,适应性强.并讨论影响热残余应力和界面强度的因素,球壳因双曲率的影响,热残余应力显著大于柱壳.     

Eshelby-Mori-Tanaka approach for vibrational behavior of functionally graded carbon nanotube-reinforced plate resting on elastic foundation

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of functionally graded (FG) nanocomposite plates reinforced by randomly-oriented straight single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation are considered. Material properties are graded in the thickness direction of the plate according to the volume fraction power law distribution. An embedded carbon nanotube (CNT) in a polymer matrix and its surrounding inter-phase which is perfectly bonded to surrounding resin is replaced with an equivalent fiber to predict the mechanical properties of the carbon nanotube/polymer composite. The Mori-Tanaka approach is employed to calculate the effective elastic moduli of the plate. The natural frequencies of the plate are obtained by means of the generalized differential quadrature (GDQ) method. Detailed parametric studies have been carried out to investigate the influences of the CNT volume fraction, Winkler foundation modulus, shear elastic foundation modulus and various geometrical parameters on the vibration behavior of the functionally graded carbon nanotube-reinforced (FG-CNTR) plates.     

Dynamic relaxation method for nonlinear buckling analysis of moderately thick FG cylindrical panels with various boundary conditions

Using new approach proposed by Dynamic relaxation (DR) method, buckling analysis of moderately thick Functionally graded (FG) cylindrical panels subjected to axial compression is investigated for various boundary conditions. The mechanical properties of FG panel are assumed to vary continuously along the thickness direction by the simple rule of mixture and Mori-Tanaka model. The incremental form of nonlinear formulations are derived based on First-order shear deformation theory (FSDT) and large deflection von Karman equations. The DR method combined with the finite difference discretization technique is employed to solve the incremental form of equilibrium equations. The critical mechanical buckling load is determined based on compressive load-displacement curve by adding the incremental displacements in each load step to the displacements obtained from the previous ones. A detailed parametric study is carried out to investigate the influences of the boundary conditions, rule of mixture, grading index, radius-to-thickness ratio, length-to-radius ratio and panel angle on the mechanical buckling load. The results reveal that with increase of grading index the effect of radius-to-thickness ratio on the buckling load decreases. It is also observed that effect of distribution rules on the buckling load is dependent to the type of boundary conditions.

     

NUMERICAL ANALYSIS AND EXPERI- MENT OF UNSTEADY THERMAL FIELD OF ROTOR PLATE FOR EDDY CURRENT RETARDER

The physical model based on heat transfer theory and virtual boundary method for analyzing unsteady thermal field of rotor plate for eddy current retarder used in automobile is established and boundary conditions are also defined. The finite element governing equation is derived by Galerkin method. The time differential item is discrete based on Galerkin format that is stable at any condition. And a new style of varying time step method is used in iteration process. The thermal field on the rotor plate at the radial and axle directions is analyzed and varying temperature at appointed points on two side-surfaces is measured. The testing and analytical data are uniform approximately. Finite element method can be used for estimating thermal field of the rotor plate at initial design stage of eddy current retarder.     

Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures

A spline finite strip capability is described for predicting the buckling stresses and natural frequencies of vibration of prismatic plate structures which may be of composite laminated construction with arbitrary lay-ups. The plate structures may have general boundary conditions. The capability embraces analyses based on the use of first-order shear deformation plate theory and of classical plate theory, and utilizes substructuring procedures which include the use of superstrips. The theoretical development is not detailed since the present paper reports a very direct extension of a theoretical study developed for the analysis of single plates in an earlier paper in this Journal. A considerable range of buckling and vibration applications is documented and comparison of spline finite strip numerical values of buckling stresses and frequencies is made with results generated using the semi-analytical finite strip method and, in some cases, the finite element method. Buckled and vibrational mode shapes are presented for some applications.     

Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory

In this paper, for the first time, the modified strain gradient theory is used as a new size-dependent Kirchhoff micro-plate model to study the effect of interlayer van der Waals (vdW) force for the vibration analysis of multilayered graphene sheets (MLGSs). The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. After obtaining the governing equations based on modified strain gradient theory via principle of minimum potential energy, as only infinitesimal vibration is considered, the net pressure due to the vdW interaction is assumed to be linearly proportional to the deflection between two layers. To solve the governing equation subjected to the boundary conditions, the Fourier series is assumed for w=w(x, y). To show the accuracy of the formulations, present results in specific cases are compared with available results in literature and a good agreement can be seen. The results indicate that the present model can predict prominent natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.