Benchmark Solution for Free Vibration of Moderately Thick Functionally Graded Sandwich Sector Plates on Two-Parameter Elastic Foundation with General Boundary Conditions

Li, Haichao; Pang, Fuzhen; Wang, Xueren; Li, Shuo

doi:https://doi.org/10.1155/2017/4018629

Shock and Vibration

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Abstract Introduction Discussion Conclusions Appendix Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 4018629 | https://doi.org/10.1155/2017/4018629

Benchmark Solution for Free Vibration of Moderately Thick Functionally Graded Sandwich Sector Plates on Two-Parameter Elastic Foundation with General Boundary Conditions

Haichao Li,¹Fuzhen Pang,¹Xueren Wang,²and Shuo Li¹

Academic Editor: Lorenzo Dozio

Received02 Apr 2017

Revised04 Jun 2017

Accepted11 Jun 2017

Published12 Sept 2017

Abstract

The free vibration analysis of moderately thick functionally graded (FG) sector plates resting on two-parameter elastic foundation with general boundary conditions is presented via Fourier-Ritz method, which is composed of the modified Fourier series approach and the Ritz procedure. The material properties are assumed to vary continuously along the thickness according to the power-law distribution. The bilayered and single-layered functionally graded sector plates are obtained as the special cases of sandwich plates. The first-order shear deformation theory (FSDT) is adopted to construct the theoretical model. Under current framework, regardless of boundary conditions, each displacement and each rotation of plates is represented by the modified Fourier series consisting of a standard Fourier cosine series and several closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series representation. Then, the accurate solutions are obtained by using the Ritz procedure based on the energy function of sector plates. The present method shows good convergence, reliability, and accuracy by comprehensive investigation with some selected classical boundary conditions. Numerous new vibration results for moderately thick FG sandwich sector plates are provided. The effects of the elastic restraint parameters and so forth on free vibration characteristic of sector plates are presented.

1. Introduction

The sector plates have been widespread in many branches of engineering applications such as architectural structures, bridges, hydraulic structures, containers, airplane, missiles, ships, and instruments due to the excellent performance like light weight and an effective form with high load-carrying capacity, economy, and technological effectiveness. As is known to us all, the functionally graded material and the fiber-reinforced composite laminated structures possess outstanding characteristics such as the high specific strength and stiffness, good corrosion resistance, and long fatigue life. Thus, the work of combining the sector plate and composite material is a hot point for a lot of researchers.

In the last years, a large quantity of research efforts has been devoted to the vibration analysis of composite laminated and functionally graded sector plates [1–39] in the literature. In these researches, the scope of the boundary conditions and accuracy of solution for the vibration information of the plate rest on the plate theories and the numerical method. For the moment, the plate theories can be classified into three main categories, namely, the classical plate theory, moderately thick plate theory, and thick plate theory. As is well known, the classical plate theory is the simplest theory constructed from the Kirchhoff hypothesis and only has a few numbers of degrees of freedom in the calculation. Thus, the methods on the basis of the classical plate theory have a high computational efficiency. But, the applied range of the classical plate theory is only limited to the thin plate, and when the plate is relatively thick or when accurate solutions for higher modes of vibration are desired, the results may be unfaithful in practical applications since the classical plate theory neglects the effect of shear and normal deformation in the thickness direction. Thereafter, the moderately thick plate theory including the first-order shear deformation theory and higher-order shear deformation theory is developed which, respectively, introduce the shear correction factor and the assumption of high-order variations of in-plane displacements taking into account the transverse shear and normal deformation through the plate thickness. Compared with the classical plate theory and moderately thick plate theory, the thick plate theory does not rely on any hypotheses, deserves any numerical precision, and can solve the vibration problem of thick plates. But, it needs more computational resource to obtain the accurate results. Based on the reasonable plate theory, the computational methods of structures have played an important role in the understanding of the structural behaviors. There are many existing methods being applied to the numerical technique to solve the equation of motions and boundary conditions and then obtain the vibration results of the structures, such as differential quadrature method, generalized differential quadrature method, finite element method, and Galerkin’s method. In the above methods, when the boundary conditions change, the codes of the solutions need corresponding changes in the calculation process. However, the number of boundary condition in the practical engineering is very large; that is, the boundary conditions of the plate have more than two hundred. Besides, the existing methods are usually good at solving the classical boundary condition, but the boundary conditions of the structures are not always in certain classical case and contain a variety of possible boundary conditions such as elastic restraints in the practical engineering.

Compared with the composite laminated and functionally graded sector plate, the existing results of FG sandwich sector plates are too scarce for engineering applications and comparative studies. However, we can know the importance of the FG sandwich sector plates from the existing published literature about other FG sandwich structures [40–57]. Thus, an accurate frequency and mode shape determination are of considerable importance for the technical design of FG sandwich sector plates. The aim of this paper is to propose a benchmark solution for the vibration characteristics of the FG sandwich sector plates with general boundary conditions and resting on elastic foundation.

In this study, the theoretical model of the FG sandwich sector plate with general boundary conditions and resting on two-parameter elastic foundation is presented. Four common types of sandwich functionally graded sector plates are studied in the study. The material properties are assumed to vary continuously along the thickness according to the power-law distribution. The solution is obtained by using the Fourier-Ritz method, which leads to a generalized eigenvalue problem, and is applied to general boundary conditions of the sector plate. The mathematical fundamentals of the Fourier-Ritz method are discussed in detail in the published literature by Li [58, 59]. It is worthwhile to note that the interest of researches in this procedure is increasing due to its great simplicity and versatility. Therefore, this simple and direct procedure can be applied to a large number of structures, that is, beams [60–63], plates [5, 64–76], shells [77–83], coupled structures [84–88], and so on. In addition, compared with other methods, the present method can obtain accurate results and circumvent the difficulties of programming complex algorithms for the computer as well as the excessive use of storage and computer time.

2. Theoretical Formulations

2.1. Description of the Model

The basic configuration of the problem considered here is an annular sector plate as shown in Figure 1. The geometry and dimensions, namely, the uniform thickness h, inner radius , outer radius , width R () of plate in the radial direction, and the sector angle , are defined in a cylindrical coordinate system . It is noted that the circular sector plate can be viewed as a special case of an annular sector plate whose inner radius is .

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It is assumed that the FGM layers of sector plate are made of a mixture of metal and ceramic constituents. The effective material properties of layers vary continuously and smoothly in the thickness direction and are estimated by Voigt’s rule. Young’s modulus (), Poisson’s ratios (), and mass density () can be expressed as where the subscripts and represent the material properties of the ceramic and metal phase, respectively, and is the volume fraction of material constituent . In this paper, to simplify the work, the authors just choose four common types of the sandwich sector plate to investigate, and as shown in Figure 2, two types of moderately thick FG sandwich sector plates are considered in the present study: Type 1: FGM face sheet and homogeneous core; Type 2: homogeneous face sheet and FGM core. The volume fraction of the laminate FGM sector plate is defined as where the subscripts , , and are the power-law exponents used to determine the material profile in FGM. Type 1-1 and Type 1-2 moderately thick FG sandwich sector plates are the sandwich sector plates with FGM face sheets and isotropic core, while Type 2-1 and Type 2-2 moderately thick FG sandwich sector plates are the sandwich sector plates with isotropic face sheets and FGM core. The ratio of thickness of each layer from bottom to top is denoted by the combination of three numbers; for example, “2-2-1” denotes that : : = 2 : 2 : 1. It is noted that the single-layered and bilayered FG sector plates can be obtained by setting appropriate ratios of thickness of each layer. To clarify the behavior of (1a), (1b), (1c), and (2), the volume fraction in the thickness direction for the moderately thick FG sandwich sector plates with various values of power-law index is shown in Figure 3.

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2.2. Energy Expressions

The assumed displacement field for the FG sandwich sector plates based on the first-order shear deformation plate theory can be written as follows:where , , and denote the displacements of the corresponding point on the reference surface in r, , and directions, respectively. and are the rotations of the normal to the reference surface about the and direction, respectively, and is the time. Under the assumption of small deformation and linear strain-displacement relations, the strain components of moderately thick FG sandwich sector plates can be expressed aswhere the membrane strains, denoted by , , , , and , and the curvature changes, denoted by , , and , of the reference surface are given as

According to Hooke’s law, the corresponding stress-strain relations of the th layer can be written as where the elastic constants are functions of the thickness coordinate and are defined as

By carrying the integration of the stresses over the cross-section, the force and moment resultants can be obtained:where , , and are the in-plane force resultants, , , and are moment resultants, and and are transverse shear force resultants. The shear correction factor is computed such that the strain energy due to transverse shear stresses in (9) equals the strain energy due to the true transverse stresses predicted by the three-dimensional elasticity theory. In application, the shear correction factor does not have to be the same in the and theta direction and depends on many factors such as the geometric and material parameters [89]. However, the aim of this paper is to study the vibration analysis of moderately thick functionally graded sector plates resting on two-parameter elastic foundation with general boundary conditions. Thus, in order to simplify this study and based on existing literature [5, 71], the shear correction factor is selected as a generic parameter in the next calculation. , , and (, = 1, 2, and 6) are the extensional, extensional-bending coupling, and bending stiffness, and they are, respectively, expressed as

The strain energy () of the FG sandwich sector plates during vibration can be defined as

Substituting (5a), (5b), (8), and (9) into (11), the strain energy expression of the structure can be written in terms of middle plane displacements and rotations.

The corresponding kinetic energy () function of the FG sandwich sector plate can be given as

Substituting , , and from (3a), (3b), and (3c) into (13) and performing the integration with respect to result inwhere

Since the main focus of this paper is to develop a unified solution for the vibration analysis of the moderately thick FG sandwich sector plate with general boundary conditions, thus, in order to satisfy the request, the artificial spring boundary technique [90] is adopted here. In this technique, five groups of boundary restraining springs are arranged at all sides of the sector plate to separately simulate the general boundary conditions. Assigning the stiffness of the boundary springs with various values is equivalent to imposing different boundary conditions of the sector plate. For example, the free boundary condition can be readily obtained by setting the spring coefficients to zeros, and the clamped boundary can be obtained by assigning the springs’ stiffness to infinity. Thus, the potential energy () stored in the boundary springs is given as

As mentioned previously, the main interests of this paper are focused on the moderately thick FG sandwich sector plate which rests on the elastic foundation. As illustrated in Figure 2, the elastic foundation is achieved by applying the two-parameter elastic foundation (Pasternak) mode with Winkler layer (stiffness ) and shear layer (stiffness ). Therefore, the total energy stored by foundation springs can be given by

2.3. Equations of Motion

By means of Hamilton’s principle, five equilibrium equations of motion of the considered moderately thick FG sandwich sector plate can be obtained, namely,

Considering (5a) and (5b), (18a), (18b), (18c), (18d), and (18e) show that each displacement and rotation component of the plates at most has second-order derivatives. In next subsection, a set of appropriately constructed admissible displacement functions of the FG sandwich sector plate elements are presented.

2.4. Admissible Displacement Functions and Solution Procedure

The admissible function of the displacement is essential to achieve an accurate and convergent solution in the Ritz procedure. Polynomials and traditional Fourier series are commonly used. However, for the polynomial [91, 92], the lower order polynomials cannot form a complete set, and the higher-order polynomials always tend to become numerically unstable due to the computer round-off errors. And for traditional Fourier series expression, it is only applicable for a few simple boundary conditions and can lead to unavoidable convergence problem for other boundary conditions, which limits the Fourier series to only a few ideal boundary conditions [58]. Take the beam problem, for example, the governing equations for free vibration of a general supported Euler beam are obtained as . From the equation, we can know that the displacement solution on a beam of length is required to have up to the fourth derivatives, that is, . In general, the displacement function defined over a domain can be expanded into a Fourier series inside the domain excluding the boundary points: . From the equation, we can see that the displacement function can be viewed as a part of an even function defined over , as shown in Figure 4(a). Thus, the Fourier cosine series is able to correctly converge to at any point over . However, its first-derivative is an odd function over leading to a jump at end locations. The corresponding Fourier expansion of continue on and can be differentiated term-by-term only if . Thus, its Fourier series expansion (sine series) will accordingly have a convergence problem due to the discontinuity at end points when is required to have up to the first-derivative continuity. Recently, Li [58, 59] has proposed an improved Fourier series technique to overcome this problem, and in this technique, a new function is considered in the displacement functionwhere the auxiliary function in (19) represents an arbitrary continuous function that, regardless of boundary conditions, is always chosen to satisfy the following equations: , , , . The actual values of the first and third derivatives (a sine series) at the boundaries need to be determined from the given boundary conditions. Essentially, represents a residual beam function which is continuous over and has zero slopes at both ends, as shown in Figure 4(b). Apparently, the cosine series representation of is able to converge correctly to the function itself and its first derivative at every point on the beam. Thus, based on the above analysis, can be understood as a continuous function which satisfies (19), and its form is not a concern but must be a closed form and sufficiently smooth over a domain of the beam to meet the requirements provided by the continuity conditions and boundary constraints. Moreover, it is noticeable that the auxiliary function can improve the convergent properties of the Fourier series.

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According to the above analysis, the displacement field of the moderately thick FG sandwich sector plate can be described by the improved Fourier series technique. The detailed expressions of the displacement and rotation components are given aswhere , , and , , , , and are the Fourier coefficients of two-dimensional Fourier series expansions for the displacement functions, respectively. , , , , , , , , , and are the supplemented coefficients of the auxiliary functions, where . , represent a set of closed-form sufficiently smooth functions defined over . In addition, the authors only choose the cosine series as the admissible function of the plate without containing the sine series because compared with the sine series, the cosine series can provide more convergence speed and better accuracy when the plate is with general boundary conditions. More information about the difference between the cosine series and sine series can be seen in [59].

According to the equilibrium equations (18a), (18b), (18c), (18d), and (18e), each of the displacement components at most has two derivatives; thus, it is required that at least two-order derivatives of the admissible functions exist and are continuous at any point on the plate. Therefore two auxiliary functions in r- and -direction are supplemented as demonstrated in (20a)–(20e). Such requirements can be readily satisfied by choosing simple auxiliary functions as follows:

It is easy to verify that

The Lagrangian energy function (L) of the moderately thick FG sandwich sector plate can be written as

Substituting (12), (13), (16), (17), (20a), (20b), (20c), (20d), and (20e) into (23) and performing the Ritz procedure with respect to each unknown coefficient, the equation of motion for plates can be yielded and is given in the matrix form: where

and (, = , , , , and ) are the submatrices of global stiffness and mass matrices. To clearly clarify the submatrices, the elements of typical matrices and are given in the Appendix. By solving (24), the frequencies (or eigenvalues) of moderately thick FG sandwich sector plate can be readily obtained and the mode shapes can be yielded by substituting the corresponding eigenvectors into series representations of displacement and rotation components.

3. Numerical Results and Discussion

In this section, the vibration information of the moderately thick FG sandwich sector plate can be easily obtained by using the MATLAB 7.11.0 to solve (24). Comparison studies and several numerical examples are presented for the verification of the accuracy and applicability of the present method. Take the moderately thick FG sandwich sector plate as the study object; its vibration characteristics under the general boundary conditions including the classical boundary condition, elastic boundary conditions, and their combinations are analyzed. In this paper, authors apply five groups of springs and set them proper stiffness to implement the corresponding boundary conditions as the innovation point. Thus, the first task is to study the boundary spring stiffness. For the sake of brevity, a symbolism is employed to represent the boundary condition of the annular sector plate and circular sector plate; for example, FCSE and CFE denote the annular sector plate with F (free), C (clamped), S (simply supported), and E (elastic restraint) boundary conditions at , , , and and the circular sector plate with C, F, and E boundary condition at , , and , respectively. Next, as a power method, the convergence, accuracy, and reliability of the present method should be investigated. Then, the vibration analysis and parameter studies of the moderately thick FG sandwich sector plate without the elastic foundation are presented. The parameter studies contain the power-law exponent, sector angle, material types, and thickness schemes. Lastly, the vibration analysis of the moderately thick FG sandwich sector plate resting on two-parameter foundations with different boundary conditions is conducted and the effects of the elastic foundation coefficients on the free vibration characteristic of the sector plate are presented. In addition, unless otherwise stated, the material constituents and are assumed to be alumina and aluminum, respectively. The material properties used in the following analyses are = 70 GPa, = 0.3, = 2707 kg/m³, = 380 GPa, = 0.3, and = 3800 kg/m³.

3.1. Determination of the Boundary Spring Stiffness

Figure 5 shows the variations of the lowest three frequency parameters of FG sandwich sector plate with respect to the restraint parameters , , , , and . The elastic restraint parameters are defined as ratios of the corresponding spring stiffness to the reference bending stiffness ; for example, , , , , and . Type 1-1, Type 1-2, Type 2-1, and Type 2-2 laminated FGM sector plates are considered in the analysis and the thickness scheme is 2-2-1. The geometric parameters and power-law exponents of the sector plate used are = 0.5, = 120°, = 0.2, and = = 5. The plates under consideration are completely clamped at the circumference boundary condition and free at the boundary , while at edge , the plates are elastically supported by only one group of spring components with stiffness varying from 10⁻⁴ to 10⁸. According to Figure 5, we can see that the change of the boundary elastic restraint parameter has little effect on the frequency parameter Ω when it is smaller than 10⁻². However, when it is increased in a certain range, the plate frequencies increase rapidly as the elastic parameters increase, and the frequency parameters approach their utmost and remain unchanged when approaches infinity. It is noted that the certain range is different with respect to different kinds of boundary elastic restraint parameters. The clamped boundary conditions of a plate can be realized by assigning all boundary spring stiffness to 10⁸D and the elastic edges can be obtained by setting the stiffness of springs to the proper value. As is well known, the boundary conditions contain a variety of elastic boundary conditions in the practical engineering applications. So, for fear of the overstaffing and unwieldiness, only three kinds of elastic boundary conditions are selected, and the relevant stiffness of the five types of boundary elastic restraint parameters is shown in Table 1.

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3.2. Convergence Study

In this subsection, the convergence of the moderately thick FG sandwich sector plate with different boundary conditions is studied. Table 2 shows the convergence of the frequency parameters for completely clamped sector plates. The material type and thickness ratios are taken to be Type 1-1 and 2-2-1, respectively. The geometrical parameters and power-law exponents for the sector plate used in the study are circular sector plate: = 120°, = 0.1; annular sector plate: = 0.5, = 120°, = 0.1; = = 5. For the fundamental mode, the maximum difference between the and is less than 0.162% for the worst case. However, for the higher modes (16th mode), the maximum difference is more than 16.225%. Thus, we can know that using the present method to predict the lower modes merely needs smaller truncation number. It means that the convergence of the present method in the lower modes is faster than that in the higher modes. From the table, we also can know that a highly desired convergence characteristic is observed in that (a) sufficiently accurate results can be obtained with only a small number of terms in the series expansions and (b) the solution is consistently refined as more terms are included in the expansions. However, this should not constitute a problem in practice because one can always verify the accuracy of the solution by increasing the truncation number until a desired numerical precision is achieved. As a matter of fact, this “quality control” scheme can be easily implemented automatically. In modal analysis, the natural frequencies for higher-order modes tend to converge slower (see Table 1). Thus, an adequate truncation number should be dictated by the desired accuracy of the largest natural frequencies of interest. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as in the following calculations. To further validate the accuracy and reliability of the current method, more numerical examples will be presented. In each case, the convergence study is performed and for brevity purpose, only the converged results are presented.

3.3. Sector Plate with Various Boundary Conditions

3.3.1. Validation and Some New Results

In this subsection, the present method is adopted to analyze the free vibration of moderately thick FG sandwich sector plate with different thickness ratios, power-law exponents, and boundary conditions. Firstly, the validity of the present method for the moderately thick FG sandwich circular sector plate is studied. The comparisons of the frequency parameters for the FG annular sector plate by the present method and other methods are shown in Table 3. The material type and thickness ratios are taken to be Type 1-1 and 1-0-0, respectively. The geometrical dimensions of the annular sector plates in the analysis are given as = 45°, 120°, 240°, and 360°, and 0.2, and = 0.5 and 0.7. The power-law exponents used are , 0.5, 1, and 2. Four types of boundary conditions are under consideration in this analysis, that is, SCSC, SSSS, SCSS, and SCSF. The reference results were reported by Saidi et al. [38] using the DQ method on the basis of FSDT. The comparison shows that the current results are in good agreement with those of Saidi et al. [38]. Next, the validity of the present method for the moderately thick FG sandwich circular sector plate is studied. Table 4 shows the first six frequency parameters of FG circular sector plate with different boundary conditions. The material type and thickness ratios are the same as those in Table 3. The geometrical parameters and power-law exponents of the FG circular sector plate are taken to be = 0.005 and . The results are compared with other published solutions reported by Mirtalaie et al. [39], which use the differential quadrature method based on the Kirchhoff hypothesis. From the table, we can see that there is a good agreement between the present results and the referential data. From Tables 3 and 4, we can learn that the present method has higher accuracy and reliability. In addition, the results of Tables 3 and 4 also show that it is appropriate to define the classical boundary conditions in terms of boundary spring rigidities as in Table 1. Having confidence in the current method, more numerical examples will be presented in the later examples.

In the Introduction, we mention that the results of the moderately thick FG sandwich sector plate are very scarce in the published literature. Therefore, in the next examples, the authors will give some new results for the moderately thick FG sandwich sector plate with various boundary conditions to support the predesign in practical engineering applications. Tables 5–8 show the fundamental frequency parameter for moderately thick FG sandwich annular sector plate with different boundary conditions including the classical case, elastic restraint, and their combination. Five kinds of thickness ratios, that is, 1-0-1, 1-1-1, 2-2-1, 1-2-1, and 1-8-1, are taken be in the analysis. The power-law exponents of the FG circular sector plate are used as , 5, and 20. The geometrical dimensions of the annular sector plates in the analysis are given as , , and the sector angle = 120°, 60°, 240°, and 350° corresponding to Tables 5–8, respectively. Also, Tables 9–12 show the fundamental frequency parameter for the moderately thick FG sandwich circular sector plate subject to various boundary conditions. The geometrical and material parameter are the same as Tables 5–8 expect for . From Tables 5–12, it is evident that the fundamental frequency parameters are quite sensitive to the change of the power-law exponents and boundary conditions. For the case of classical boundary conditions, the increase of the power-law exponent leads to the decrease of the fundamental frequency parameters of sector plates. However, when the sector plate is under the elastic boundary condition, the effects of the power-law exponent on the fundamental frequencies of the sector plate are different and a little more complex. For the sector plate with the conditions of E¹E¹E¹E¹ and E³E³E³E³, the fundamental frequencies increase as the power-law exponent increases. While the boundary condition of the sector plate is E²E²E²E², the variation of the frequencies for the sector plate is similar to the classical boundary conditions’. In addition, the thickness ratio and material type have significant influence on the fundamental frequencies of the sector plate regardless of the boundary conditions and power-law exponents. Some mode shapes for moderately thick FG sandwich sector plates with different boundary conditions and geometric and material parameters are depicted in Figures 6 and 7.

3.3.2. Parameter Studies

The effects of power-law exponents, sector angles, material types, and thickness schemes with different boundary conditions are investigated in this subsection. Figure 8 shows the variations of fundamental frequency parameter of Type 1 annular sector plate against different power-law exponents and . The geometric dimensions used for the analysis are , = 120°, and = 0.2. The scope of the power-law exponents and is from 0 to 10 and the interval of power-law exponents is equal to 0.5. The thickness ratio of the annular sector plate is taken to be 1-2-1. The classical boundary condition (CCCC), classical-elastic boundary condition (CE¹E¹E¹), and the elastic boundary condition (E³E³E³E³) are considered in this analysis. Also, the variations of the fundamental frequency parameter of Type 1 circular sector plate versus the power-law exponents and are shown in Figure 9. The geometric dimensions, thickness ratios, and material parameters are the same as those in Figure 8 expect for = 0. Three types of boundary conditions, that is, CCC, CE¹E¹, and E³E³E³, are selected in this analysis. From the figures, we can see that when the sector plate is under classical and classical-elastic boundary conditions, the frequency parameters decrease monotonically as the power-law exponents and increase irrespective of the shape and material type of plates. When all edges of the sector plate are under the E³ elastic boundary condition, the variations of the fundamental frequency parameters turn more complex. For Type 1 annular sector plate with the E³E³E³E³ boundary condition, the frequency parameters Ω increase monotonically as the power-law exponents and increase. For circular sector plate under E³E³E³, when the material type is Type 1-1, the variation rule of the fundamental frequency parameters are the same as the classical boundary condition. However, when the material type is Type 1-2, the variations of the fundamental frequency parameters firstly increase and then decrease as the power-law exponents and increase. Next, the variations of the fundamental frequencies of Type 2 sector plate with different boundary conditions and power-law exponent are shown in Figure 10. The geometric dimensions and material parameters are the same as those in Figures 8 and 9. The thickness ratio of the sector plate adopts 2-2-1 in this analysis. From the figure, we can see that the variation of fundamental frequencies of the sector plate decreases or increases as the power-law exponent increases regardless of the material type, when the sector plate is under the all clamped and elastic boundary conditions. For the annular sector plate with the condition CE¹E¹E¹, the fundamental frequencies decrease monotonically versus the increase of the power-law exponent irrespective of the material type. However, for the case of circular sector plate, the variations of fundamental frequencies between the materials of Type 2-1 and Type 2-2 are different. For Type 2-1, the fundamental frequencies monotonically decrease with the power-law exponent increasing. For Type 2-2, the fundamental frequencies firstly decrease and then slow up.

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From Tables 5–12, we can see that the variations of fundamental frequency parameter Ω of the sector plate subjected to classical case, elastic restraints, and classical-elastic restraints with different material types, power-law exponents, and sector angles are more complex. Thus, in the following examples, the authors will fully illustrate the effects of the boundary conditions and material type on vibration characteristics of the sector plates with different sector angles. Figures 11 and 12 show the variations of fundamental frequency parameter Ω of annular sector plate and circular sector plate with different sector angles, material types, and boundary conditions, respectively. The geometric dimensions are the same as Figure 10 and the power-law exponent is . From the figures, it is obvious that the frequency parameters Ω change monotonically as the sector angle increases regardless of the boundary condition and material type. For the circular sector plate, the frequency parameters Ω decrease monotonically as the sector angle increases irrespective of the type of the boundary condition and material. For the case of CCC, CFC, and CE¹E¹, when the sector angle is in the region of 5° to 70°, the frequency parameters Ω of plate rapidly decrease as the sector angle increases. While the sector angle is beyond this region, the frequency parameters firstly slow down and then remain unchanged when the sector angle reaches a critical value. But, for the E³E³E³ boundary condition, the frequency parameters always decrease with the increase of the sector angle in the whole region. Also, the variations of frequency parameters Ω for the annular sector plate with CCCC, CFCF, and CE¹E¹E¹ are similar to the circular sector plate with CCC, CFC, and CE¹E¹. However, for the case of E³E³E³E³ annular sector plate, the frequency parameters Ω firstly rapidly increase and then remain unchanged as the sector angle increases irrespective of the material type.

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Lastly, the authors will investigate the effect of material types and thickness schemes on vibration characteristics of the moderately thick FG sandwich sector plate. Figures 13 and 14, respectively, show the variations of fundamental frequency parameter Ω for the annular sector plate and circular sector plate with different material types and thickness schemes. The geometric dimensions and power-law exponent are the same as Figures 11 and 12. For Type 1-1 sector plate with compete clamped boundary conditions, the fundamental frequency parameters decrease as the thickness of layer increases except for 1-n-1 thickness schemes. On the contrary, the fundamental frequency parameters increase as the thickness of layer increases except for 1-n-1 thickness schemes while the boundary condition of sector plate is completely elastic boundary conditions (annular sector plate: E³E³E³E³; circular sector plate: E³E³E³). For Type 1-2 circular sector plates, the fundamental frequency parameters decrease as the thickness of layer increases irrespectively of the boundary condition. However, for the annular sector plate, the variation is different and a little more complex. For the case of the CCCC boundary condition with n-1-1 and 1-1-n thickness schemes, the fundamental frequency parameters increase as the thickness of layer increases. And for the plate with 1-n-1 thickness schemes, the fundamental frequency parameters decrease as the thickness of layer increases. As to the case of the E³E³E³E³ boundary condition, the increase of thickness of layer leads to the decrease of fundamental frequency parameters regardless of the thickness schemes. For Type 2 sector plate with completely clamped boundary condition, the fundamental frequency parameters decrease as the thickness of layer increases except for Type 2-1 n-1-1 and Type 2-2 1-1-n thickness schemes. The change rule of the fundamental frequency parameters on the sector plate with elastic boundary conditions reverses in contrast to the clamped boundary condition. In addition, it also can be easily seen that according to Figures 13 and 14, for Type 1 material distribution, the results of the 1-1-n and n-1-1 thickness schemes are coincident in the same material type. For Type 2 material distribution, the results of Type 2-1 and Type 2-2 are coincident in the same thickness schemes.

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Based on the above analysis, the vibration characteristics of the moderately thick FG sandwich sector plate strongly depend on the material type, thickness schemes, power-law exponent, and boundary conditions.

3.4. Sector Plate Resting on Two-Parameter Elastic Foundations

In the practical engineering, the structures are usually laid on a soil medium and the vibration analysis of these structures with such elastic foundation and general boundary conditions is necessary and of great significance. In this subsection, the first goal is to check the accuracy of the present method on the vibration analysis of the moderately thick FG sandwich sector plate resting on elastic foundations. The comparison of frequency parameters Ω for Type 1-1 (1-0-0) annular sector plates on Winkler foundations with various boundary conditions is shown in Table 13. The results obtained from the ABAQUS based on FEA method are tabulated in the table for comparison due to the lack of the reference data of the published literature. The geometry parameters of the annular sector plates are given as , = 90°, and = 0.2. For general purpose, the nondimensional foundation parameters are used in this analysis: and . Two kinds of Winkler foundation stiffness , that is, 10 and 100, are taken in this analysis. From the table, a consistent agreement of present results and referential data is clear. The discrepancy is very small and does not exceed 1% for the worst case. In addition, the table shows that the increasing of Winkler foundation stiffness contributes to the increase of frequencies of the sector plate.

What has been mentioned above is the results of vibration characteristics for the moderately thick FG sandwich sector plate resting on two-parameter elastic foundation, which is limited in the published literature. Thus, the author will present some new results of moderately thick FG sandwich sector plates resting on two-parameter elastic foundation with various boundary conditions, material types, and foundation coefficients using the present method in the next examples. These results can be as the benchmark solutions for future computational methods in this field. The fundamental frequency parameter Ω for the annular sector plate and circular sector plate with different boundary conditions, material types, and elastic foundation coefficients are shown in Tables 14 and 15, respectively. From the tables, it is clear that the variation of the coefficients has the significant effect on frequency parameters of the sector plate. In order to further study the influence of foundation coefficients on vibration characteristics of the moderately thick FG sandwich sector plate, the variations of the fundamental frequency parameters Ω versus the Winkler foundation stiffness and shearing layer stiffness for annular sector plate and circular sector plate are presented in Figures 15 and 16, respectively. It can be easily obtained that regardless of the boundary conditions, material type, and shape of the plate, there exists a certain range of the elastic foundation coefficients during which the frequency parameter increases and out of which the influence on frequency parameter Ω can be neglected.

(a)

(b)

(a)

(b)

4. Conclusions

In the present study, the vibration characteristics of the moderately thick FG sandwich sector plate are investigated. The annular sector plate and circular sector plate with four common types of material distributions, thickness schemes, classical and elastic boundary conditions, and various geometry constants are incorporated. The first-order shear deformation plate theory is employed to include the effects of rotary inertias and shear deformation. The energy expression of the plate is expressed as a function of five displacement components by using the kinematic and constitutive relations. Each generalized displacement of the sector plate is represented by the modified Fourier series consisting of a standard Fourier cosine series and several auxiliary closed-form functions introduced to remove any potential discontinuity of the original displacement and its derivatives and accelerate the convergence of the series representation. The general boundary conditions of the sector plate are realized by using the artificial spring boundary technique which contains three groups of liner springs and two groups of rotation springs. The convergence, accuracy, and reliability of the presented solutions are validated by numerical examples and comparison of the present results with those available in the literature and obtained by using FEA method and numerous new results for the moderately thick FG sandwich sector plate with various boundary conditions, material and geometric parameters, and elastic foundations are presented. The effects of elastic boundary restraint parameters, power-law exponents, sector angles, material types, thickness schemes, and foundation coefficients on the frequency values are examined and discussed in detail. The main innovation point and conclusions can be drawn:(1)Compared with other methods, the present method can easily obtain the vibration information of the moderately thick FG sandwich sector plate with general boundary conditions and enables rapid convergence, high reliability, and accuracy.(2)New results for moderately thick FG sandwich sector plates with various boundary conditions, material and geometric parameters, and elastic foundations are presented.(3)The vibration characteristics of the moderately thick FG sandwich sector plate strongly depend on boundary conditions, power-law exponents, sector angles, material types, thickness schemes, and foundation coefficients.

Appendix

The submatrices of global stiffness and mass matrices of the plate are written bywhere

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Funding

This study was funded by the National Key Research and Development Program of China (2016YFC0303406), Assembly Advanced Research Fund of China (6140210020105), Major Innovation Projects of High Technology Ship Funds of Ministry of Industry and Information of China, National Natural Science Foundation of China (no. 51209052), High Technology Ship Funds of Ministry of Industry and Information Technology of China, Fundamental Research Funds for the Central Universities (HEUCFD1515, HEUCFM170113), and China Postdoctoral Science Foundation (2014M552661).

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