Shock and Vibration
Volume 2017 (2017), Article ID 4018629, 35 pages
https://doi.org/10.1155/2017/4018629

Research Article

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

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

¹College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
²Naval Academy of Armament, Beijing 100161, China

Correspondence should be addressed to Fuzhen Pang; pangfuzhen@hrbeu.edu.cn

Received 2 April 2017; Revised 4 June 2017; Accepted 11 June 2017; Published 12 September 2017

Academic Editor: Lorenzo Dozio

Copyright © 2017 Haichao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 .

(a)

(b)

Figure 1: Schematic diagram of FG sandwich annular sector plate: (a) geometry and dimensions of an annular sector plate; (b) partial boundary spring view of the annular sector plate.

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.

(a)

(b)

Figure 2: The material variation along the thickness of the FG sandwich annular sector plate: (a) FGM face sheet and homogeneous core; (b) homogeneous face sheet and FGM core.

(a)

(b)

(c)

(d)

Figure 3: Variation of volume fraction Vc along the thickness for different value of power-law exponent: (a) Type 1-1; (b) Type 1-2; (c) Type 2-1; (d) Type 2-2.

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 as

where 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 in

where

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 function

where 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.

(a)

(b)

Figure 4: An illustration of the possible discontinuities of the displacement at the end points and how they can be equivalently removed mathematically.

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 as

where , , 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.

Table 1: The corresponding elastic restraint parameters for various boundary conditions.

BC
= constant
= constant

F
0
0
0
0
0
0
0
0
0
0

C
10⁸ D
10⁸ D
10⁸ D
10⁸ D
10⁸ D
10⁸ D
10⁸ D
10⁸ D
10⁸ D
10⁸ D

S
0
10⁸ D
10⁸ D
0
10⁸ D
10⁸ D
0
10⁸ D
10⁸ D
0

SD
0
10⁸ D
10⁸ D
0
0
10⁸ D
0
10⁸ D
0
0

E¹
10² D
10² D
10² D
10⁸ D
10⁸ D
10² D
10² D
10² D
10⁸ D
10⁸ D

E²
10⁸ D
10⁸ D
10⁸ D
10² D
10² D
10⁸ D
10⁸ D
10⁸ D
10² D
10² D

E³
10² D
10² D
10² D
10² D
10² D
10² D
10² D
10² D
10² D
10² D

(a)

(b)

(c)

(d)

Figure 5: Variation of the frequency parameters ΔΩ versus the elastic restraint parameters for FG sandwich sector plate ( = 0.5, = 120°, = 0.2, and , 2-2-1): (a) Type 1-1; (b) Type 1-2; (c) Type 2-1; (d) Type 2-2.

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.

Table 2: Convergence of frequency parameters for clamped annular sector plate and circular sector plate (, = 120°, = 0.2, and ).

Mode number

1
2
3
4
8
10
12
16

Annular sector plate: Type 1-1 (2-2-1)

4 × 4
93.796
103.03
120.55
187.65
245.38
294.84
351.42
379.08

5 × 5
93.758
102.74
119.84
146.16
238.17
267.17
332.81
376.02

6 × 6
93.680
102.66
119.30
144.14
238.15
266.69
295.52
373.65

7 × 7
93.674
102.60
119.23
143.62
229.48
244.89
294.36
351.15

8 × 8
93.656
102.58
119.10
143.44
229.47
244.72
266.02
351.11

9 × 9
93.655
102.56
119.08
143.29
229.43
244.68
265.83
326.85

10 × 10
93.649
102.55
119.04
143.25
229.42
244.61
265.81
326.66

12 × 12
93.646
102.54
119.00
143.17
229.40
244.56
265.73
326.36

14 × 14
93.645
102.53
118.99
143.14
229.40
244.53
265.69
326.25

16 × 16
93.644
102.53
118.99
143.13
229.39
244.52
265.67
326.19

18 × 18
93.644
102.53
118.99
143.13
229.39
244.51
265.66
326.16

Circular sector plate: Type 1-1 (2-2-1)

4 × 4
41.679
68.054
91.458
100.58
184.45
221.56
243.92
302.38

5 × 5
41.617
67.665
91.386
99.181
171.33
201.68
221.52
257.70

6 × 6
41.585
67.575
91.114
98.546
169.74
184.82
214.70
249.33

7 × 7
41.574
67.496
91.101
98.424
168.53
184.81
212.39
243.85

8 × 8
41.565
67.477
91.040
98.282
167.86
184.81
208.42
243.85

9 × 9
41.562
67.452
91.037
98.255
167.61
184.81
207.67
243.85

10 × 10
41.559
67.447
91.017
98.208
167.40
184.80
207.22
243.85

12 × 12
41.557
67.435
91.007
98.179
167.24
184.80
206.88
243.84

14 × 14
41.556
67.430
91.003
98.167
167.18
184.80
206.75
243.84

16 × 16
41.555
67.427
91.001
98.160
167.15
184.80
206.69
243.84

18 × 18
41.555
67.426
91.001
98.159
167.15
184.80
206.66
243.84

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.

Table 3: Comparison of the fundamental frequency for Type 1-1 (1-0-0) annular sector plate with different boundary conditions and some values of , , , and .

Present
DQM [38]
Present
DQM [38]
Present
DQM [38]
Present
DQM [38]

SCSC

45
0.1
0.5
194.8160
194.9900
166.2082
166.3153
150.2243
150.2992
136.4071
136.4535

0.7
476.2485
476.6814
406.1966
406.4655
367.1207
367.3109
333.4284
333.5481

0.2
0.5
163.4673
163.6200
141.1496
141.2456
128.2058
128.2741
116.1314
116.1744

0.7
401.6972
402.0732
347.0488
347.2862
315.3959
315.5654
285.8517
285.9592

SSSS

120
0.1
0.5
84.6916
84.7699
71.8487
71.8978
64.7832
64.8181
58.8727
58.8950

0.7
218.2320
218.4339
185.1097
185.2357
166.8971
166.9869
151.6765
151.7341

0.2
0.5
80.1166
80.1911
68.2517
68.2982
61.6182
61.6513
55.9117
55.9327

0.7
170.8277
170.9907
153.3271
153.4349
142.0979
142.1771
128.3715
128.4221

SCSS

240
0.1
0.5
113.6641
113.7661
96.6646
96.7273
87.2537
87.2977
79.2726
79.3000

0.7
318.2095
318.4965
270.5833
270.7604
244.2267
244.3514
221.8914
221.9694

0.2
0.5
101.6405
101.7348
87.2198
87.2789
79.0084
79.0504
71.6453
71.6718

0.7
285.3312
285.5963
244.7812
244.9474
221.7106
221.8287
201.0608
201.1356

SCSF

360
0.1
0.5
25.4430
25.4653
21.5680
21.5816
19.4433
19.4527
17.6757
17.6815

0.7
72.2176
72.2815
61.2110
61.2499
55.1774
55.2045
50.1612
50.1778

0.2
0.5
24.7384
24.7610
21.0203
21.0343
18.9649
18.9748
17.2290
17.2352

0.7
70.4067
70.4710
59.7989
59.8388
53.9394
53.9677
49.0017
49.0195

Table 4: Comparison of frequency parameters ()1/2 for Type 1-1 (1-0-0) circular sector plate with different boundary conditions and some values of , , and = 0.005.

Source of result
Mode number

1
2
3
4
5
6

CCC

30
DQM [39]
161.6895
257.7845
358.6257
368.9981
496.3537
515.3727

Present
161.6245
257.6316
358.4922
368.7320
495.9707
515.0906

60
DQM [39]
64.9682
124.6827
127.8239
201.0241
209.1647
215.3604

Present
64.9699
124.6689
127.8671
200.9819
209.0478
215.2143

SSS

30
DQM [39]
85.1070
158.8688
239.4427
248.8392
355.3409
371.3462

Present
84.1225
157.9932
238.6164
247.5275
354.7307
370.1041

60
DQM [39]
34.6846
81.4720
84.4201
145.3033
152.9561
158.2822

Present
34.2925
81.1683
84.1225
144.8008
152.4740
157.8930

SFS

45
DQM [39]
19.2254
63.5700
70.7899
122.8581
152.5376
156.7239

Present
19.0321
63.2629
70.7192
122.3353
152.2428
156.3877

90
DQM [39]
4.7166
18.9667
30.4039
41.1333
63.3673
70.7733

Present
4.6654
18.9548
30.2890
41.0787
63.2629
70.7610

SCS

45
DQM [39]
60.1866
120.7868
144.6854
197.7130
242.6680
261.5216

Present
59.8481
120.3583
144.5406
197.1593
242.4294
261.2852

90
DQM [39]
30.0711
59.9173
72.8029
98.2194
120.4645
132.3217

Present
29.9584
59.8491
72.6442
98.1400
120.3593
132.0831

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.

Table 5: Fundamental frequency parameter Ω for Type 1-1 annular sector plate with various boundary conditions ( = 0.5, = 0.2, , and = 120°).

Scheme

Boundary conditions

CCCC
CSDCSD
CFCF
SSSS
CE¹E¹E¹
E²SE²S
E¹E¹E¹E¹
E²E²E²E²
E³E³E³E³

1-0-1
0.5
83.742
82.842
81.118
55.349
34.047
77.251
19.655
77.934
19.545

5
57.995
57.391
56.216
36.938
28.436
55.623
20.461
56.131
20.360

20
52.505
51.933
50.852
35.162
27.747
50.698
20.557
51.178
20.462

1-1-1
0.5
87.605
86.661
84.854
58.112
34.924
80.210
19.393
80.911
19.284

5
68.130
67.450
66.062
41.725
29.797
64.338
19.966
64.903
19.857

20
64.141
63.505
62.199
39.006
29.012
60.936
20.079
61.470
19.972

2-2-1
0.5
88.644
87.685
85.855
59.033
35.226
81.021
19.345
81.727
19.236

5
71.264
70.549
69.100
45.066
30.500
66.988
19.886
67.583
19.777

20
67.600
66.928
65.555
43.306
29.726
63.908
19.999
64.476
19.891

1-2-1
0.5
89.857
88.883
87.024
59.994
35.536
81.935
19.268
82.645
19.160

5
74.871
74.112
72.585
46.578
31.236
69.935
19.723
70.550
19.614

20
71.362
70.648
69.192
43.815
30.395
67.028
19.815
67.615
19.706

1-8-1
0.5
94.455
93.414
91.443
64.301
36.964
85.462
19.044
86.188
18.944

5
88.816
87.863
86.032
58.664
35.090
81.072
19.253
81.776
19.145

20
87.424
86.492
84.694
57.332
34.652
79.981
19.300
80.678
19.191

Table 6: Fundamental frequency parameter Ω for Type 1-2 annular sector plate with various boundary conditions ( = 0.5, = 0.2, , and = 60°).

Scheme

Boundary conditions

CCCC
CSDCSD
CFCF
SSSS
CE¹E¹E¹
E²SE²S
E¹E¹E¹E¹
E²E²E²E²
E³E³E³E³

1-0-1
0.5
102.850
95.998
87.258
78.574
38.733
91.343
22.240
95.758
22.163

5
76.806
71.963
65.038
60.777
34.563
70.427
23.721
74.004
23.636

20
64.244
60.045
54.466
49.650
31.736
59.240
23.789
62.580
23.703

1-1-1
0.5
94.916
88.841
80.407
74.595
37.235
85.537
22.951
89.569
22.871

5
71.671
67.123
60.701
56.482
33.443
65.929
23.804
69.388
23.718

20
62.000
57.896
52.590
47.469
31.205
57.188
23.750
60.497
23.664

2-2-1
0.5
92.094
86.160
78.038
72.068
36.881
83.145
23.083
87.211
23.002

5
70.286
65.781
59.549
55.071
33.105
64.653
23.805
68.127
23.720

20
61.477
57.391
52.155
46.926
31.077
56.702
23.737
60.009
23.652

1-2-1
0.5
89.217
83.617
75.535
70.882
36.331
81.015
23.298
84.838
23.216

5
68.690
64.293
58.193
53.831
32.770
63.268
23.818
66.672
23.732

20
60.784
56.729
51.574
46.262
30.914
56.071
23.722
59.363
23.636

1-8-1
0.5
74.201
69.559
62.817
58.962
34.055
68.220
23.787
71.684
23.702

5
62.252
58.139
52.800
47.727
31.266
57.420
23.756
60.731
23.670

20
58.386
54.429
49.575
47.727
30.333
53.861
23.649
57.119
23.564

Table 7: Fundamental frequency parameter for Type 2-1 annular sector plate with various boundary conditions ( = 0.5, = 0.2, , and = 240°).

Scheme

Boundary conditions

CCCC
CSDCSD
CFCF
SSSS
CE¹E¹E¹
E²SE²S
E¹E¹E¹E¹
E²E²E²E²
E³E³E³E³

0-1-1
0.5
68.585
68.501
68.095
35.473
30.189
65.252
18.993
65.111
18.894

5
57.232
57.154
56.819
27.694
28.656
55.339
19.421
55.201
19.333

20
51.863
51.794
51.485
25.214
27.396
50.437
19.349
50.309
19.267

1-1-1
0.5
75.442
75.355
74.899
40.403
31.030
70.824
18.495
70.659
18.390

5
70.101
70.017
69.597
36.811
30.281
66.526
18.861
66.388
18.761

20
69.015
68.931
68.520
35.845
30.216
65.635
18.955
65.499
18.856

2-2-1
0.5
80.246
80.151
79.669
42.453
32.156
74.609
18.278
74.417
18.170

5
72.233
72.149
71.713
38.569
30.463
68.252
18.680
68.104
18.578

20
70.785
70.701
70.275
37.509
30.290
67.079
18.787
66.938
18.686

1-2-1
0.5
77.281
77.191
76.727
40.949
31.549
72.308
18.454
72.131
18.348

5
68.712
68.627
68.221
35.452
30.246
65.384
18.997
65.248
18.898

20
66.897
66.812
66.421
33.868
30.136
63.863
19.135
63.730
19.038

1-8-1
0.5
81.224
81.125
80.641
41.890
32.816
75.469
18.390
75.266
18.281

5
65.437
65.352
64.973
32.732
30.000
62.600
19.223
62.463
19.127

20
61.124
61.040
60.688
29.688
29.473
58.836
19.396
58.698
19.304

Table 8: Fundamental frequency parameter Ω for Type 2-2 annular sector plate with various boundary conditions (, = 0.2, , and = 350°).

Scheme

Boundary conditions

CCCC
CSDCSD
CFCF
SSSS
CE¹E¹E¹
E²SE²S
E¹E¹E¹E¹
E²E²E²E²
E³E³E³E³

0-1-1
0.5
88.492
88.459
88.180
43.936
34.434
80.910
17.573
80.937
17.470

5
75.956
75.929
75.677
39.875
30.844
71.202
18.046
71.227
17.952

20
73.509
73.483
73.235
38.734
30.337
69.250
18.167
69.275
18.075

1-1-1
0.5
75.223
75.197
74.946
39.359
30.759
70.633
18.110
70.659
18.017

5
69.903
69.878
69.642
35.863
30.034
66.362
18.485
66.388
18.396

20
68.819
68.794
68.563
34.922
29.974
65.474
18.580
65.499
18.492

2-2-1
0.5
72.482
72.456
72.214
37.588
30.375
68.454
18.311
68.479
18.220

5
66.472
66.446
66.231
32.806
29.890
63.505
18.780
63.531
18.694

20
64.791
64.765
64.561
31.458
29.771
62.068
18.888
62.093
18.804

1-2-1
0.5
77.052
77.025
76.773
39.890
31.273
72.105
18.066
72.131
17.971

5
68.515
68.489
68.262
34.539
30.006
65.222
18.622
65.248
18.534

20
66.702
66.676
66.460
32.995
29.904
63.704
18.763
63.730
18.677

1-8-1
0.5
80.972
80.942
80.685
40.804
32.530
75.240
17.995
75.266
17.897

5
65.240
65.214
65.008
31.888
29.774
62.438
18.854
62.463
18.769

20
60.927
60.902
60.716
28.921
29.262
58.674
19.039
58.698
18.957

Table 9: Fundamental frequency parameter for Type 1-1 circular sector plate with various boundary conditions ( = 0.2, , and = 120°).

Scheme

Boundary conditions

CCC
CSDC
CFC
SSS
CE¹E¹
E²SE²
E¹E¹E¹
E²E²E²
E³E³E³

1-0-1
0.5
44.469
35.027
11.934
27.121
20.578
33.940
16.508
41.593
16.347

5
30.024
23.414
7.849
17.890
18.659
23.195
16.508
29.195
16.363

20
28.149
22.255
7.637
17.323
18.372
22.108
16.451
27.491
16.315

1-1-1
0.5
46.635
36.771
12.549
28.509
20.755
35.468
16.338
43.277
16.178

5
34.280
26.468
8.724
19.955
19.113
26.042
16.396
32.954
16.240

20
32.115
24.754
8.139
18.622
18.801
24.429
16.355
31.040
16.202

2-2-1
0.5
47.319
37.352
12.773
29.005
20.836
35.982
16.311
43.815
16.152

5
36.114
27.952
9.251
21.140
19.373
27.426
16.425
34.560
16.266

20
34.053
26.301
8.677
19.836
19.097
25.883
16.412
32.764
16.256

1-2-1
0.5
48.050
37.956
12.996
29.503
20.905
36.508
16.261
44.370
16.102

5
38.089
29.524
9.790
22.372
19.591
28.880
16.376
36.236
16.217

20
35.966
27.787
9.166
20.966
19.301
27.267
16.361
34.412
16.203

1-8-1
0.5
51.195
40.668
14.066
31.858
21.294
38.879
16.129
46.770
15.972

5
47.136
37.121
12.641
28.733
20.742
35.754
16.232
43.632
16.073

20
46.164
36.284
12.313
28.009
20.613
35.008
16.253
42.864
16.093

Table 10: Fundamental frequency parameter for Type 1-2 circular sector plate with various boundary conditions ( = 0.2, , and = 60°).

Scheme

Boundary conditions

CCC
CSDC
CFC
SSS
CE¹E¹
E²SE²
E¹E¹E¹
E²E²E²
E³E³E³

1-0-1
0.5
83.200
73.456
40.445
60.463
32.862
70.464
20.773
77.188
20.663

5
62.649
55.874
31.352
47.089
29.586
54.879
21.972
60.223
21.858

20
52.107
46.162
25.564
38.308
27.380
45.696
21.864
50.698
21.752

1-1-1
0.5
77.305
68.797
38.470
57.830
32.230
66.602
21.397
72.678
21.285

5
58.402
52.026
29.124
43.810
28.742
51.273
21.991
56.435
21.877

20
50.174
44.333
24.428
36.538
26.941
43.941
21.785
48.911
21.675

2-2-1
0.5
74.936
66.600
37.159
55.808
31.740
64.596
21.502
70.726
21.389

5
57.199
50.864
28.387
42.649
28.467
50.157
21.972
55.348
21.859

20
49.715
43.889
24.145
36.093
26.832
43.512
21.762
48.485
21.653

1-2-1
0.5
72.846
65.036
36.585
54.502
31.548
63.294
21.685
69.030
21.572

5
55.899
49.717
27.745
41.689
28.216
49.087
21.963
54.170
21.850

20
49.119
43.328
23.799
35.556
26.696
42.975
21.733
47.931
21.624

1-8-1
0.5
60.581
54.097
30.426
45.333
29.232
53.231
22.008
58.397
21.894

5
50.394
44.544
24.562
36.749
26.992
44.144
21.795
49.116
21.685

20
47.025
41.327
22.539
33.583
26.201
41.044
21.612
45.978
21.504

Table 11: Fundamental frequency parameter for Type 2-1 circular sector plate with various boundary conditions ( = 0.2, , and = 240°).

Scheme

Boundary conditions

CCC
CSDC
CFC
SSS
CE¹E¹
E²SE²
E¹E¹E¹
E²E²E²
E³E³E³

0-1-1
0.5
25.799
18.485
4.253
15.269
15.798
18.158
13.669
24.736
13.476

5
22.950
16.821
3.988
14.151
15.387
16.626
13.585
22.239
13.418

20
20.700
15.152
3.579
12.728
14.766
15.023
13.159
20.178
13.011

1-1-1
0.5
27.614
19.630
4.461
16.114
15.861
19.195
13.535
26.235
13.329

5
26.055
18.597
4.259
15.318
15.782
18.252
13.615
24.949
13.419

20
25.876
18.517
4.257
15.284
15.796
18.184
13.656
24.805
13.462

2-2-1
0.5
29.629
21.124
4.812
17.374
16.063
20.571
13.556
27.892
13.343

5
26.508
18.852
4.292
15.483
15.770
18.478
13.548
25.316
13.348

20
26.127
18.610
4.249
15.304
15.758
18.259
13.576
25.001
13.379

1-2-1
0.5
28.507
20.314
4.628
16.704
15.984
19.833
13.585
26.985
13.376

5
25.901
18.567
4.278
15.345
15.821
18.235
13.685
24.832
13.492

20
25.631
18.468
4.288
15.328
15.837
18.155
13.746
24.612
13.555

1-8-1
0.5
30.612
21.966
5.047
18.155
16.274
21.365
13.698
28.729
13.483

5
25.350
18.338
4.278
15.267
15.823
18.042
13.770
24.370
13.582

20
24.398
17.841
4.225
14.987
15.706
17.593
13.773
23.543
13.594

Table 12: Fundamental frequency parameter for Type 2-2 circular sector plate with various boundary conditions ( = 0.2, , and = 350°).

Scheme

Boundary conditions

CCC
CSDC
CFC
SSS
CE¹E¹
E²SE²
E¹E¹E¹
E²E²E²
E³E³E³

0-1-1
0.5
29.907
20.840
4.640
19.172
15.254
20.389
12.621
27.916
12.377

5
24.600
16.933
3.734
15.548
14.713
16.692
12.413
23.527
12.190

20
23.758
16.341
3.604
15.003
14.621
16.128
12.371
22.804
12.155

1-1-1
0.5
24.421
16.819
3.712
15.444
14.711
16.585
12.427
23.377
12.206

5
23.029
15.914
3.530
14.622
14.616
15.729
12.446
22.189
12.238

20
22.861
15.832
3.520
14.553
14.619
15.654
12.476
22.046
12.270

2-2-1
0.5
23.685
16.337
3.613
15.005
14.669
16.130
12.443
22.752
12.228

5
22.589
15.745
3.523
14.493
14.632
15.577
12.552
21.814
12.352

20
22.348
15.646
3.516
14.419
14.613
15.486
12.581
21.601
12.385

1-2-1
0.5
25.197
17.390
3.843
15.972
14.819
17.130
12.496
24.043
12.270

5
22.877
15.866
3.532
14.588
14.637
15.688
12.503
22.062
12.298

20
22.622
15.759
3.525
14.503
14.633
15.589
12.548
21.843
12.347

1-8-1
0.5
27.025
18.761
4.166
17.247
15.066
18.437
12.648
25.592
12.414

5
22.364
15.631
3.505
14.397
14.606
15.468
12.560
21.611
12.363

20
21.505
15.171
3.431
14.012
14.456
15.032
12.535
20.841
12.348

Figure 6: Mode shapes of the FG sandwich annular sector plate with CE¹E¹E¹ boundary condition and different sector angles.

Figure 7: Mode shapes of the FG sandwich circular sector plate with E²SE² boundary condition and different sector angles.

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.

(a)

(b)

Figure 8: Variations of fundamental frequency parameter of Type 1 annular sector plate against different power-law exponents: (a) Type 1-1 (1-2-1); (b) Type 1-2 (1-2-1).

(a)

(b)

Figure 9: Variations of fundamental frequency parameter of Type 1 circular sector plate against different power-law exponents: (a) Type 1-1 (1-2-1); (b) Type 1-2 (1-2-1).

(a)

(b)

Figure 10: Variations of fundamental frequency parameter of Type 2 (2-2-1) sector plate against different power-law exponents: (a) annular sector plate; (b) circular sector plate.

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.

(a)

(b)

(c)

(d)

Figure 11: Variations of fundamental frequency parameter of annular sector plate against different sector angles ( = 0.5, = 120°, = 0.2, and = = 5): (a) CCCC; (b) CFCF; (c) CE¹E¹E¹; (d) E³E³E³E³.

(a)

(b)

(c)

(d)

Figure 12: Variations of fundamental frequency parameter Ω of circular sector plate against different sector angles ( = 0.2, = 120°, and = = 5, 2-2-1): (a) CCC (b) CFC; (c) CE¹E¹; (d) E³E³E³.

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.

(a)

(b)

Figure 13: Variations of fundamental frequency parameter Ω of annular sector plate with material types and different thickness schemes ( = 0.5, = 120°, = 0.2, and = = 5): (a) CCCC; (b) E³E³E³E³.

(a)

(b)

Figure 14: Variations of fundamental frequency parameter Ω of circular sector plate with material types and different thickness schemes ( = 120°, = 0.2, and = = 5): (a) CCC; (b) E³E³E³.

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.

Table 13: Comparison of frequency parameters for Type 1-1 (1-0-0) annular sector plates on Winkler foundations with various boundary conditions ( = 0.5, = 90°, = 0.2, and ).

(KW, KG)
Mode number
CCCC
FFFF
CSCS

Present
FEM
Error (%)
Present
FEM
Error (%)
Present
FEM
Error (%)

(10, 0)
1
101.04
101.61
0.56
27.05
27.24
0.70
98.73
99.29
0.57

2
119.56
120.17
0.51
37.04
37.25
0.56
113.70
114.26
0.49

3
148.92
149.68
0.51
50.79
50.81
0.03
128.81
129.06
0.19

4
154.35
154.56
0.14
58.99
59.17
0.31
141.92
142.58
0.47

5
184.85
185.88
0.55
77.94
78.16
0.28
152.79
153.00
0.14

6
196.96
198.36
0.71
87.03
87.05
0.03
178.77
179.68
0.51

(100, 0)
1
101.56
102.55
0.97
28.74
28.91
0.59
99.25
100.25
0.99

2
119.99
120.95
0.79
38.28
38.58
0.77
114.14
115.07
0.80

3
149.25
150.29
0.69
50.79
50.81
0.03
128.81
129.06
0.19

4
154.35
154.56
0.14
59.75
60.18
0.72
142.27
143.21
0.66

5
185.12
186.35
0.66
78.52
78.73
0.28
152.79
153.00
0.14

6
197.21
198.82
0.81
87.03
87.05
0.03
179.04
180.17
0.62

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.

Table 14: Fundamental frequency parameter Ω for annular sector plate with various boundary conditions and material types ( = 0.5, = 120°, = 0.2, and ).

(KW, KG)

Boundary conditions

CCCC
CSDCSD
CFCF
CSCS
CE¹CE¹
E²SE²S
E¹E¹E¹E¹
E²E²E²E²
E³E³E³E³

Type 1-1 (2-2-1)

(10, 10)
0.5
90.522
89.567
87.632
89.674
89.042
83.012
19.380
83.713
19.380

5
73.835
73.120
71.544
73.192
72.946
69.642
20.166
70.235
20.090

20
70.370
69.699
68.191
69.764
69.563
66.756
20.320
67.323
20.234

(10, 100)
0.5
105.518
104.589
101.772
104.683
103.066
98.766
19.380
99.425
19.380

5
93.191
92.491
89.841
92.546
91.158
89.502
20.173
90.074
20.173

20
90.876
90.217
87.584
90.265
88.880
87.711
20.366
88.263
20.365

(100, 100)
0.5
105.831
104.904
102.099
104.997
103.389
99.097
19.380
99.755
19.380

5
93.576
92.878
90.243
92.933
91.555
89.899
20.173
90.471
20.173

20
91.280
90.622
88.005
90.669
89.296
88.125
20.366
88.677
20.365

Type 1-2 (2-2-1)

(10, 10)
0.5
83.227
82.291
80.354
82.397
81.477
78.022
20.444
78.694
20.444

5
65.098
64.393
62.809
64.472
63.873
62.579
21.186
63.134
21.186

20
57.950
57.345
55.941
57.413
57.003
56.131
21.266
56.626
21.193

(10, 100)
0.5
101.277
100.373
97.371
100.460
98.366
96.747
20.444
97.387
20.444

5
85.865
85.396
74.553
85.411
75.644
85.318
21.187
85.839
21.187

20
77.059
76.696
66.901
76.701
68.098
76.697
21.305
77.056
21.305

(100, 100)
0.5
101.652
100.749
97.521
100.835
98.441
97.135
20.444
97.775
20.444

5
85.865
85.404
74.554
85.420
75.644
85.348
21.187
85.839
21.187

20
77.059
76.697
66.901
76.701
68.098
76.697
21.305
77.056
21.305

Type 2-1 (2-2-1)

(10, 10)
0.5
84.174
83.314
81.544
83.387
82.961
78.145
19.719
78.850
19.717

5
76.294
75.520
73.953
75.564
75.310
71.886
20.230
72.581
20.219

20
74.898
74.130
72.599
74.169
73.931
70.757
20.352
71.454
20.340

(10, 100)
0.5
100.681
99.821
97.138
99.874
98.445
95.280
19.722
95.986
19.721

5
95.109
94.256
91.774
94.272
93.031
91.124
20.247
91.929
20.243

20
94.226
93.337
90.916
93.346
92.149
90.445
20.372
91.302
20.368

(100, 100)
0.5
101.020
100.161
97.492
100.214
98.795
95.633
19.722
96.339
19.721

5
95.488
94.632
92.171
94.647
93.424
91.510
20.247
92.319
20.244

20
94.614
93.720
91.322
93.728
92.551
90.838
20.372
91.701
20.369

Type 2-2 (2-2-1)

(10, 10)
0.5
76.761
75.979
74.394
76.026
75.745
72.279
20.234
72.973
20.225

5
71.047
70.274
68.805
70.313
70.024
67.595
20.748
68.291
20.739

20
69.482
68.713
67.255
68.753
68.435
66.274
20.868
66.960
20.860

(10, 100)
0.5
95.506
94.654
92.140
94.673
93.389
91.466
20.248
92.258
20.246

5
92.119
90.905
85.329
90.907
87.243
88.700
20.771
89.851
20.767

20
91.249
89.712
82.231
89.730
83.971
87.816
20.891
89.202
20.888

(100, 100)
0.5
95.884
95.030
92.536
95.048
93.780
91.853
20.249
92.647
20.246

5
92.538
91.287
85.389
91.291
87.279
88.822
20.771
90.280
20.768

20
91.681
90.057
82.264
90.083
83.990
88.194
20.891
89.642
20.888

Table 15: Fundamental frequency parameter for circular sector plate with various boundary conditions and material types ( = 120°, = 0.2, and ).

(KW, KG)

Boundary conditions

CCC
CSDSD
CFC
CSC
CE¹E¹
E²CE²
E¹E¹E¹
E²E²E²
E³E³E³

Type 1-1 (2-2-1)

(10, 10)
0.5
49.067
35.093
15.397
39.729
21.501
47.290
16.598
45.630
16.459

5
38.629
27.661
12.888
31.189
20.261
37.845
16.857
37.116
16.730

20
36.774
26.473
12.541
29.772
20.058
36.126
16.902
35.522
16.779

(10, 100)
0.5
61.880
50.832
27.164
54.445
25.231
60.311
16.695
58.832
16.695

5
55.308
47.427
26.216
49.943
24.801
54.623
17.346
53.978
17.345

20
54.348
47.099
26.161
49.404
24.815
53.783
17.497
53.251
17.497

(100, 100)
0.5
62.411
51.467
28.366
55.040
26.536
60.855
16.695
59.388
16.695

5
55.956
48.171
27.583
50.652
26.194
55.279
17.346
54.642
17.345

20
55.023
47.866
27.563
50.135
25.911
54.465
17.497
53.939
17.497

Type 1-2 (2-2-1)

(10, 10)
0.5
48.273
36.215
16.399
40.329
22.268
46.823
17.555
45.444
17.418

5
38.187
28.868
14.075
32.353
21.065
37.571
17.809
36.981
17.690

20
33.685
25.898
12.888
28.601
20.241
33.299
17.566
32.930
17.463

(10, 100)
0.5
62.582
53.284
28.603
55.930
25.792
61.359
17.557
60.189
17.557

5
56.710
50.599
27.835
42.896
23.948
56.230
18.029
55.767
18.029

20
54.211
49.201
27.309
38.461
23.213
53.919
18.035
53.637
18.035

(100, 100)
0.5
63.184
53.975
29.902
56.093
25.792
61.972
17.557
60.813
17.557

5
57.452
51.410
29.325
42.897
23.948
56.978
18.029
56.520
18.029

20
55.007
50.058
28.869
38.461
23.213
54.719
18.035
54.440
18.035

Type 2-1 (2-2-1)

(10, 10)
0.5
45.085
32.207
14.401
36.416
21.073
43.762
16.779
42.542
16.642

5
40.914
29.476
13.586
33.169
20.750
40.008
17.073
39.180
16.942

20
40.378
29.203
13.540
32.813
20.753
39.528
17.161
38.750
17.031

(10, 100)
0.5
59.248
49.411
26.795
52.533
25.079
58.090
16.980
57.014
16.978

5
56.875
48.481
26.700
50.992
25.156
56.102
17.404
55.392
17.400

20
56.666
48.513
26.774
50.873
25.238
55.946
17.501
55.286
17.497

(100, 100)
0.5
59.822
50.085
28.061
53.168
26.444
58.675
16.980
57.610
16.978

5
57.508
49.246
28.050
51.662
26.029
56.744
17.404
56.042
17.400

20
57.311
49.205
28.141
51.536
25.828
56.599
17.501
55.947
17.497

Type 2-2 (2-2-1)

(10, 10)
0.5
41.276
29.759
13.690
33.489
20.813
40.343
17.091
39.489
16.959

5
39.492
29.102
13.702
32.507
20.919
38.751
17.486
38.066
17.359

20
39.075
29.014
13.745
32.337
20.949
38.375
17.578
37.723
17.454

(10, 100)
0.5
57.144
48.672
26.770
51.223
25.198
56.347
17.405
55.613
17.402

5
56.671
49.199
27.215
49.839
25.074
56.066
17.792
55.505
17.788

20
56.603
49.076
27.346
48.227
24.789
56.038
17.870
55.513
17.867

(100, 100)
0.5
57.775
49.395
28.117
51.893
26.033
56.987
17.405
56.262
17.402

5
57.354
49.956
28.631
50.040
25.076
56.756
17.793
56.202
17.789

20
57.300
49.096
28.780
48.295
24.791
56.742
17.871
56.224
17.867

(a)

(b)

Figure 15: Variation of the frequency parameters versus the elastic foundation coefficients for annular sector plates with different boundary conditions: (a) CCCC; (b) E²E²E²E².

(a)

(b)

Figure 16: Variation of the frequency parameters versus the elastic foundation coefficients for circular sector plates with different boundary conditions: (a) CCC; (b) E²E²E².

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 by

where

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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