Research Article  Open Access
An Accurate Solution Method for the Static and Vibration Analysis of Functionally Graded ReissnerMindlin Rectangular Plate with General Boundary Conditions
Abstract
This paper presents an accurate solution method for the static and vibration analysis of functionally graded ReissnerMindlin plate with general boundary conditions on the basis of the improved Fourier series method. In the theoretical formulations, the governing equations and the general elastic boundary equations are obtained by using Hamilton’s principle. The components of admissible displacement functions are expanded as an improved Fourier series form which contains a 2D Fourier cosine series and auxiliary function in the form of 1D series. The major role of the auxiliary function is to remove the potential discontinuities of the displacement function and its derivatives at the edges and ensure and accelerate the convergence of the series representation. The characteristic equations are easily obtained via substituting admissible displacement functions into governing equations and the general elastic boundary equations. Several examples are made to show the excellent accuracy and convergence of the current solutions. The results of this paper may serve as benchmark data for future research in related field.
1. Introduction
The concept of functionally graded materials (FGMs) was firstly presented in 1987 by a group of material scientists in Sendai region of Japan during the first fiveyear project to study relaxation of thermal stress of materials in high speed aerospace vehicle [1, 2]. Since then, FGMs have received the major attention as heatshielding advanced structural materials in a variety of engineering applications and manufacturing industries, like aerospace, nuclear reactor automobile, aircraft, space vehicles, and biomedical and steel industries. The remarkable mechanical properties of the FGMs are achieved by gradually varying the volume fraction of the constituent materials whose properties vary from one interface to the other continuously.
As a most common infrastructure, the functionally graded rectangular plate has been widely used in practical engineering applications, i.e., aerospace, naval vessels, nuclear reactor, automobile, robot, and civil industries. On one hand, the research of static and vibration analysis of FG plate could provide the theoretical basis for practical engineering applications. On the other hand, the results of this paper could serve as the reference data in practical engineering applications in related field in future. The static and vibration analysis of the FG plate has been investigated by a large number of researches in the past decade. Among those available, Abrate [3] used the CPT to study the free vibration of FG thin rectangular plates with simply supported and clamped boundary conditions. Later, the free vibration, buckling and static deflections of FG square, and circular and skew plates with different combinations of boundary conditions were analyzed by Abrate [4] using the same method on the basis of the CPT, FSDT, and TSDT. Free and forced vibration analyses of both homogeneous and FG thick plates with classical boundary conditions were carried out by Qian et al. [5, 6] using the meshless local Petrov–Galerkin method based on the higherorder shear and normal deformable plate theory. Pradyumna and Bandyopadhyay [7] used the higherorder finite element method to investigate the free vibration of FG square plates with simply supported boundary condition. Ferreira et al. [8] employed the global collocation method and approximated the trial solution with multiquadric radial basis functions to study the free vibration of FG square plates with different classical boundary conditions on the basis of the FSDT and thirdorder shear deformation plate theory (TSDT). Praveen and Reddy [9] used the finite element method to obtain the static and dynamic thermoelastic behavior of the FG plates with some selected classical boundary conditions. Based on the thirdorder shear deformation theory and Reddy von Karmantype geometric nonlinearity theory, Reddy [10] carried out the static linear analysis and nonlinear static and dynamic analysis by using the finite element method. Croce and Venini [11] presented a new method of finite elements for analyzing ReissnerMindlin FG plates with classical boundary condition. Cheng and Batra [12, 13] applied the firstorder shear deformation theory and a thirdorder shear deformation theory to study the static, buckling, and steady state vibrations of FG polygonal plate with simply supported boundary conditions. Zhao et al. [14] extended the element free kpRitz method to study the free vibration analysis for FG square and skew plates with different boundary conditions on the basis of the FSDT. The nonlinear free vibration behavior of FG square thin plates was analyzed by Woo et al. [15] by presenting an analytical solution on the basis of the von Karman theory. Based on the CPT, Yang and Shen [16] presented an analytical method for free vibration and transient response of initially stressed FG rectangular thin plates resting on Pasternak elastic foundation having classical boundary condition subjected to impulsive lateral loads. Later, Yang and Shen [17] used a semianalytical approach to investigate the dynamic behavior of FG plates with impulsive lateral loads and complicated environment. Zhong and Yu [18] employed a statespace approach to analyze free and forced vibrations of an FG piezoelectric rectangular thick plate with simple support at its edges. Roque et al. [19] employed the multiquadric radial basis function method and the HSDT to study free vibration of FG plates with different classical boundary conditions. The natural frequencies and buckling stresses of FG plates having simply supported edges were studied by Matsunaga [20] who utilized the method of power series expansion and twodimensional (2D) higherorder theory. HosseiniHashemi et al. [21, 22] presented a new exact closedform procedure for free vibration analysis of FG rectangular thick plates and ReissnerMindlin FG rectangular plates where the plate has two opposite edges of simply supported boundary conditions. Vel and Batra [23] presented an exact 3D elasticity solution to study the thermoelastic deformation, free and forced vibrations of FG simply supported plates. Based on the higherorder shear and normal deformation theory, Shiyekar et al. [24] presented the bidirectional flexure analysis of smart FG plate subjected to electromechanical loading by using Navier’s technique. Carrera et al. [25] evaluated the effect of thickness stretching in plate/shell structures made by materials which are functionally graded (FGM) in the thickness direction according to Carrera’s Unified Formulation. Based on Carrera’s Unified Formulation, Neves et al. [26] presented explicit governing equations to study static and free vibration analysis of FGM plates by using meshless technique based on collocation with radial basis functions. Zhang et al. [27] used the local Kriging meshless method to study the mechanical and thermal buckling behaviors of ceramic–metal functionally grade plates subject the classical boundary conditions. Belabed et al. [28] presented an efficient and simple higherorder shear and normal deformation theory for the bending and free vibration analysis of FG simply supported plates. Thai and Vo [29–34] extended a simple shear deformation theory to study the bending, buckling, and vibration of functionally graded plates with simply supported cases. Valizadeh et al. [35] applied isogeometric finite element method to study the static and dynamic characteristics of functionally graded material (FGM) plates with classical boundary condition. Based on isogeometric approach (IGA) and higherorder deformation plate theory (HSDT), Tran et al. [36] used the continuous element to investigate the static, dynamic, and buckling analysis of FG rectangular and circular plates with different classical boundary conditions. Mantari et al. [37] showed an analytical solution to static analysis of functionally graded plates (FGPs) based on a new trigonometric higherorder theory in which the stretching effect was considered. Xiang and Kang [38] conducted bending analysis for functionally graded plates according to a nthorder shear deformation theory and the meshless global collocation method based on the thin plate spline radial basis function and Reddy's thirdorder theory. Based on various plate theory, Tornabene et al. [39] applied the Generalized Differential Quadrature (GDQ) to study the bending, vibration, and dynamic analysis of FG plates and panels with different classical boundary conditions. Xuan et al. [40] presented a simple and effective approach which incorporated the isogeometric finite element analysis (IGA) with a refined plate theory (RPT) for static, free vibration, and buckling analysis of functionally graded material (FGM) plates with simply supported and clamped boundary conditions. Akavci [41] presented a closedform solution to study the free vibration analysis of functionally graded plate resting on Pasternak (twoparameter) model subject to classical boundary condition.
From the above review, we can know that most of existing literatures on the static and vibration analysis of functionally graded ReissnerMindlin plate are restricted to the classical boundary condition. However, the elastic edge restraints may be more common in the practical engineering, and the classical boundary condition is considered as a special case. Thus, to establish a unified, efficient and accurate formulation for static and vibration analysis of functionally graded ReissnerMindlin plate is necessary and significant.
To the best of author’s knowledge, the subject of static and vibration analysis of functionally graded ReissnerMindlin plate with general boundary conditions has not been performed in a specified work, yet. Therefore, the major focus of this paper is to present an accurate solution method for the static and vibration analysis of functionally graded ReissnerMindlin plate with general boundary conditions. The effective material properties of functionally graded materials are assumed to vary continuously in the thickness direction according to the powerlaw distribution in terms of the volume fraction of the constituents and are estimated by the Voigt model and Mori–Tanaka scheme. All the plate displacements are expanded as a modified Fourier series which is made up of a standard cosine Fourier series and some certain supplementary terms whatever the boundary conditions are. These supplementary terms contribute to removing the potential discontinuities at the edges and then ensure and accelerate the series convergence. The characteristic equations can be derived directly by solving the equation of motion and by combining the associated boundary equations and the modified Fourier series. The characteristic equations are easily obtained via substituting related modified Fourier series into governing equations and the general elastic boundary equations. A large number of numerical examples are made to show the convergence, reliability, and accuracy of the present method.
2. Theoretical and Numerical Formulations
2.1. Geometrical Configuration
Consider a flat and moderately thick FG plate with a with uniform thickness , length , and width of plate in the , , and direction, as shown in Figure 1. In addition, the coordinate system () is also shown in Figure 1, which will be used in the analysis. The arbitrary boundary technique [43–47] is introduced to implement the general boundary condition in which one group of liner spring (, , , and denote the location of the spring; i.e., represents the location of the edge ) and two groups of rotation springs and are introduced to simulate the related boundary forces in each boundary of a plate, as shown in Figure 1. The general boundary condition is easily obtained by assigning the stiffness of the boundary springs with various values. Take the clamped boundary condition as an example; when the spring stiffness is set larger enough than the bending rigidity of plate, it can be obtained essentially. In addition, a common uniform pressure is acted on the rectangular area [], and when the acting area is narrowed down to a point, a special case named the point load will be obtained, as illustrated in Figure 1.
2.2. Material Properties
It is supposed that the FGM of plate is made of two material constituents. In this study, the top surface () of the plate is ceramicrich () whereas the bottom surface () is metalrich (). Both the Voigt model and Mori–Tanaka scheme are adopted to evaluate the effective material properties [48]. In the Voigt model, it is assumed that the material properties including Young’s modulus , density , and Poisson’s ratio are proportional to the volume fraction according toin which the subscripts and represent the metallic and ceramic constituents, respectively.
Based on the Mori–Tanaka scheme, the effective local bulk modulus and the shear modulus of the FGM plate can be expressed aswhere and . The effective mass density which is defined by (1) is also utilized in the Mori–Tanaka scheme. The effective Young’s modulus and Poisson’s ratio are
In the above FG scheme, the common volume fraction may be given bywhere is the thickness coordinate and denotes the powerlaw exponent and only takes the positive values. The material parameters , , and determine the material variation profile through the functionally graded plate thickness. It should be noted that, in order to make the results become more universal in the future, the authors just choose the Voigt model and material parameters of the volume fraction are given as and . The variations of volume fraction for different values of the parameters , , , and are depicted in Figure 2.
2.3. Governing Equations and Boundary Conditions
Based on the Mindlin plate theory, the displacement components of the plate are assumed to bewhere is the displacements of the middle surface in directions and and are the rotation functions. Following assumptions of the small deformation and the linear strain–displacement relation, the strain components of FG plates can be written as
According to Hook’s law, the stress components of FG plates can be expressed aswhere the elastic constants are the functions of thickness coordinate :
By carrying the integration of the stresses over the crosssection, the force and moment resultants are written as follows:where and are the extensional, bending stiffness, and they are, respectively, expressed as
The strain energy and kinetic energy of the FG plate can be described as
In this paper, to develop a unified solution of the FG plate subjected to general elastic restrains, the static and vibration analysis is focused. So, the strain energy stored in the boundary springs during vibration can be defined as
Also, the external force on the plate during vibration can be defined as
To determine the static deflection, the load function should be expressed as a Fourier cosine series and the detailed information can be seen in [49].
The Lagrangian energy functional () of the plate is written as
Within arbitrary length of time, 0 to , Hamilton’s principle can be stated as follows:
By substituting (11)–(14) into (16), the variation equation can be rewritten as
Integrating by parts to relieve the virtual displacements , , and , we have
Since the virtual displacements , , and are arbitrary, (18) can be satisfied only if the coefficients of the virtual displacements are zero. Thus, the governing equation of motion and general boundary conditions for the static and vibration analysis of a FG plate can be derived as
Further, substituting (6a), (6b), and (9) into (19) and (20), the governing equation of motion can be expressed in matrix form:where
Similarly, the general boundary conditions can be written aswhere
2.4. Admissible Displacement Functions
In the structure vibration problem, the scope of boundary condition and the accuracy of the solution strongly depend on the choice of the admissible function of structures. Generally, for the commonly used polynomial expression, their convergence is uncertain. In other words, the lowerorder polynomials cannot form a complete set, and, on the contrary, it may lead to be numerically unstable owing to the computer roundoff errors when the higherorder polynomials are applied. To avoid the above weakness, the admissible functions can be expanded as the form of Fourier series due to the excellent numerical stability of the Fourier series. However, the conventional Fourier series just adapt to a few of simple boundary conditions due to the convergence problem along the boundary conditions. Recently, a modified Fourier series technique proposed by Li [50] has been widely applied in the vibration problems of plates and shells subject to different boundary conditions by the Ritz method, e.g., [42, 51–60]. In this technique, each displacement of the structure under study is written in the form of a conventional cosine Fourier series and several supplementary terms. The detailed principle and merit of the improved Fourier series can be seen in the related book [61] (entitled “Structural Vibration A Uniform Accurate Solution for Laminated Beams, Plates and Shells with General Boundary Conditions”). On the basis of the modified Fourier series technique, Jin et al. [48, 62] present an exact series solution to study the free vibration of functionally graded sandwich beams, composite laminated deep curved beams, and so on. Compared with most of the existing methods, the exact series solution not only owns the excellent convergence and accuracy but also can be applied to general boundary conditions. Therefore, in this formulation, the modified Fourier series technique is adopted and extended to conduct the static and vibration analysis of functionally graded ReissnerMindlin plate with general boundary conditions.
Combining (6a), (6b), (9), and (19), it can be known that each displacement/rotation component of a FG plate is required to have up to the second derivative. Therefore, no matter what the boundary conditions are, each displacement/rotation component of the plate is assumed to be a twodimensional modified Fourier series aswhere , , and , ,and are the Fourier coefficients of twodimensional Fourier series expansions for the displacements functions, respectively. and are the truncation numbers. and , , represent the auxiliary functions defined over whose major role is to eliminate all the discontinuities potentially associated with the firstorder derivatives at the boundary and then ensure and accelerate the convergence of the series expansion of the plate displacement. Here, it should be noted that the auxiliary functions just satisfy , and , and ; however, the concrete form is not the focus of attention. , , , , , and are the corresponding supplemented coefficients of the auxiliary functions, where , 2. Those auxiliary functions are defined as follows:
Further, the modified Fourier series expressions presented in (25) can be rewritten in the matrix form aswhere
2.5. Governing Eigenvalue Equations
Substituting (27) into (21) results inwhere
In the same way, substituting (27) into (23), the general boundary conditions of the plate can be rewritten aswhere
To derive the constraint equations of the unknown Fourier coefficients, all the sine terms, the auxiliary polynomial functions, and their derivatives in (29) and (31) will be expanded into Fourier cosine series, letting
Multiplying (29) with in the left side and integrating it from 0 to and 0 to separately with respect to and obtains where
Similarly, multiplying (31) with in the left side and then integrating it from 0 to with respect to along the edges and and multiplying (31) with in the left side and then integrating it from 0 to with respect to at the edges and , we havewhere
Thus, (36) can be rewritten as
Finally, combining (34) and (37) results inwhere
In (40), K is the stiffness matrix for the plate and the M is the mass matrix. is the load vector. In the static analysis, the Fourier coefficients, , will be firstly solved from (39) by setting and then the remaining Fourier coefficients will be calculated by using (38). The actual displacement function can then be easily determined from (37). While the load vector is equal to zero (), the vibration behavior which consists of the natural frequencies (or eigenvalues) and associated mode shapes (or eigenvectors) of FG plates can be readily obtained.
3. Numerical Results and Discussion
In order to demonstrate the present method, the static and free vibration analysis of FG plates with different boundary conditions will be considered in the following examples. Four types of material properties will be used in the following examples as seen in Table 1. For the purpose of describing the boundary condition sequence of the FG plate, a simple letter string is employed to simplify this study, as shown in Figure 1. The corresponding stiffness for the restraining springs is specified in Table 2. In addition, the new function of the shear correction factor will be introduced in the FG plate which can adapt to the actual model, and the detailed expression is defined aswhere is the thicknesstolength ratio and and are the corresponding constant coefficients, values of which are listed in Table 3. If you need more detailed information about the principle and reason of the new shear correction factor model, you can read [63].



3.1. Static Deflections
The first example considers the deformations of the isotropic plates under the action of a uniform pressure Pa with various boundary conditions, i.e., CCCC, CFCF, CCCF, CSCS, SSSS and CSFF, and different action regions as ], [], [], [], and [], and the results are shown in Table 4. Also, the deformations of the isotropic plates with central point loading N are also considered in the Table 4. The locations of the maximum deflections obviously depend on the boundary condition, while the maximum deflection obviously occurs at the center of the clamped plate, but its locations will not be so clear for other boundary conditions. For comparison, the maximum deflections predicted by the ABAQUS based on the finite element method (FEM) are also given in Table 4 due to the lack of the reference data. Besides, the deflection fields of isotropic plate for the CCCC, CCCF, CFCF, and CSCF by means of the present method and finite element method are presented in Figure 3. The element type and mesh sizes of the ABAQUS model are the S4R and 0.01 m × 0.01 m, respectively. The geometrical parameters of the plate are defined as , , and m, and the material parameter is chosen as the steel. From Table 4 and Figure 3, it is obvious that the present solution has excellent prediction accuracy for the static deflection.

Based on the verification, some new results of the static deflection of the FG plate with different boundary conditions and FGs type will been shown in the Tables 5–7. Table 5 shows the maximum deflection for Al/Al_{2}O_{3} and Al/ZrO_{2} plate having CCCC, CSCS, CFCF, and SSSS boundary cases under a central concentrated force N. Also, the maximum deflections of Al/Al_{2}O_{3} and Al/ZrO_{2} plate having different boundary conditions with the uniform pressure acting on entire rectangular area and part rectangular area [] are, respectively, performed in Tables 6 and 7. The geometrical parameters of the above Tables are used as follows: , , and m. From the above tables, we can see that the maximum deflection increases with the increase of the powerlaw exponent , regardless of the boundary conditions and load functions. In order to complete this study and to further enhance the understanding of this phenomenon, the relations between the maximum deflection and powerlaw exponent are shown in Figures 4 and 5. The geometrical parameters of those are the same as those of Tables 5–7. The variation of the static deflection versus the powerlaw exponent for FG plate with central concentrated force subject to different boundary condition is presented in Figure 4. Two FG materials Al/Al_{2}O_{3} and Al/ZrO_{2} are considered in Figure 4. For the Al/Al_{2}O_{3} plate, no matter what the boundary condition is, the maximum deflection keeps increasing when the powerlaw exponent increases. However, for the Al/ZrO_{2} plate, the maximum deflection trace climbs up, then declines, and reaches its crest around in the critical value. Figure 5 shows the variations of the static deflection versus the powerlaw exponent for Al/Al_{2}O_{3} plate having the uniform pressure with different acting regions which are the same as those of Tables 6 and 7. Four kinds boundary condition, i.e., CCCC, CSCS, CFCF, and SSSS, are considered. We can see clearly that the maximum deflection always climbs up versus the increase of the powerlaw exponent regardless of the boundary condition and load type. Finally, those results in Tables 5–7 and Figures 45 may serve as the benchmark data of FG plate for the future works in this filed.



(a)
(b)
(a)
(b)
3.2. Free Vibration Analysis
In dynamic analysis like the steady state response and transient response, the free vibration plays an important role. Thus, in this subsection, our attention will be focused on the modal results. To start with, a verification study is given to display the accuracy and reliability of the current method. In Table 8, the present method is verified by comparing the evaluation of fundamental frequency parameters (/)^{1/2} for a simply supported Al/ZrO2 square plate with those of the finite element HSDT method [7], finite element FSDT method [7], twodimensional higherorder theory [20], an analytical FSDT solution [63], an exact closedform procedure on basis FSDT [22], and threedimensional theory by employing the power series method [23]. From Table 8, it is not hard to see that the results obtained by the present method are in close agreement with those obtained by other methods. For the sake of completeness, the comparison for the fundamental frequency (/)^{1/2} of Al/Al_{2}O_{3} and Al/ZrO_{2} plates under six combinations of boundary conditions, i.e., SSSC, SCSC, SSSF, SCSF, and SFSF, is given in Tables 9 and 10, respectively. The results reported by Hashemi et al. [63] on the basis of the FSDT are included in the comparison. Although different solution approaches are used in the literature, it is still clearly seen that the present results and referential data agree well with each other.



After the verification, some new vibration results of FG plate with a variety of boundary conditions will be obtained in Tables 11 and 12 by using the present solution. In cases of Table 11, the FG plates are fabricated from aluminum (metal) and alumina (ceramic). In cases of Table 12, the FG plates are fabricated from aluminum (metal) and zirconia (ceramic). The geometrical parameters and powerlaw exponents of the FG plate are taken to be , , 0.1, and 0.2, and , 0.5, 1, 5, and 10. Some selected mode shapes of the Al/Al_{2}O_{3} plate are as shown in Figure 6. From Tables 1112, the powerlaw exponent significantly affects the fundamental frequency of the FG plate. To have a more intuitive understanding, the variation of the fundamental frequencies of FG plate with powerlaw exponents and different boundary conditions is depicted in Figures 78. From Figure 7, we can see that the fundamental frequencies decrease monotonously while increasing the powerlaw index . Moreover, the fundamental frequencies rapidly decrease, then increase, and finally decrease with the powerlaw index increasing, when the FG plate is fabricated from aluminum (metal) and zirconia (ceramic).

