1. Introduction
Functionally graded materials (FGMs) are a new type of composite materials with smooth and continuous variation in material properties in desired directions. This is achieved by gradually varying the volume fraction of the constituent materials. Such materials possess various advantages over conventional composite laminates, such as smaller stress concentration, higher fracture toughness and improved residual stress distribution. Recently, the FGMs have been used to build plate and shell components in various engineering applications, especially mechanical, aerospace, marine and civil engineering. In some cases, those FGM plates and shells are frequently subjected to dynamic loads, which leads to the vibration behaviours, which may cause fatigue damage and result in severe reduction in the strength and stability of the whole structures. Therefore, the vibration analysis of the FGM plates and shells is required and it is important to provide insight into dynamic behaviours and optimal design.
It is well known that vibration problems deal with two main concepts: plate and shell theories and computational approaches. The development of plate and shell theories has been subjected to significant research interest for many years, and many plate and shell theories have been proposed and developed. The main plate and shell theories can be classified into two categories: two-dimensional (2-D) plate and shell theories, including classic plate and shell theory (CPT) [1–4], the first-order shear deformation theory (FSDT) [5–16], and the higher order shear deformation theory (HSDT) [17–26], and three-dimensional (3-D) theory of elasticity [27–35]. However, all 2-D theories are approximate because they were developed based on certain kinematic assumptions that result in relatively simple expression and derivation of solutions. Actually, 3-D elasticity theory, which does not rely on any hypotheses about the distribution field of deformations and stress, not only provides realistic results but also allows for further physical insight. More attempts have been made for 3-D vibration analysis of plates and shells in the recent decades. Furthermore, many analytical, semi-analytical and numerical computational methods have also been developed, such as Ritz method, state-space method, differential quadrature method (DQM), Galerkin method, meshless method, finite element method (FEM) and discrete singular convolution (DSC) approach.
However, a close scrutiny of the literature in this field reveals that most investigations were carried out based on 2-D plate and shell theories, and a general 3-D solution for this subject seems to be limited. Moreover, the review also reveals that most of previous research efforts were restricted to vibration problems of FGM plates and shells with limited sets of classical boundary conditions. It is well recognized that there exist various possible boundary restraint cases for plates and shells in practical assembly and engineering applications. Consequently, it is necessary and of great significance to develop a unified, efficient and accurate method that is capable of universally dealing with FGM plates and shells with general boundary conditions.
In view of these apparent voids, the aim of this chapter is to develop an accurate semi-analysis method that is capable of dealing with vibrations of FGM plates and shells with general boundary conditions, including classical boundaries, elastic supports and their combinations and to provide a summary of known 3-D results of plates and shells with general boundary conditions, which may serve as benchmark solutions for future researches in this field.
In this chapter, 3-D vibration analysis of thick functionally graded plates and cylindrical shells with arbitrary elastic restraints is presented. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple power-law distributions in terms of volume fraction of constituents and are estimated by Voigt’s rule of mixture. By using the artificial spring boundary technique, the general boundary conditions can be obtained by setting proper spring stiffness. All displacements of the functionally graded plates and shells are expanded in the form of the linear superposition of standard 3-D cosine series and several supplementary functions, which are introduced to remove potential discontinuity problems with the original displacements along the edge. The RayleighRitz procedure is used to yield the accurate solutions.
2. Theoretical formulations
2.1. Preliminaries
A differential element of a shell with uniform thickness h is considered, as shown in Fig. 1. An orthogonal curvilinear coordinate system composed of coordinates α, β and z coordinates is located on the bottom surface. The u, v and w denote the displacement components of an arbitrary point in the α, β and z directions, respectively. Within the context of 3-D elasticity theory, the linear strain-displacement relations can be expressed as follows [36]:
where
In engineering applications, plates and shells are the basic structural elements. For the sake of brevity, this chapter will be confined to rectangular plates and cylindrical shells. According to Fig. 2, the coordinate systems and Lame coefficients are given as follows [36]: for rectangular plates,
(7) |
where
(8) |
where E(r) and μ(r) are the effective Young’s modulus and Poisson’s ratio of a FGMs, respectively. In this chapter, it is assumed that the FGMs are manufactured by ceramic and metallic constituents, and the effective material properties of FGMs can be expressed as follows [37]:
where E, μ, ρ and V are Young’s modulus, Poisson’s ratio, density and volume fraction, respectively. The subscripts c and m donate the ceramic and metallic constituents, respectively. The ceramic volume fraction follows simple power-law distribution:
where z is the thickness coordinate, and p is the power-law index that takes only positive values. The value of p equal to zero represents a fully ceramic plate, whereas infinite p indicates a fully metallic plate.
In this work, the general boundary conditions can be described in terms of three groups of springs (
where the superscripts α1 and α2 denote the edges of α = 0 and α = L_{1}, respectively. For the rectangular plates, the similar conditions exist for the edges of β = constant. The classical boundary conditions and elastic restrains can be obtained by easily changing the values of boundary spring.
2.2. Energy functional
The energy functional of plates or shells can be expressed as follows:
where T is kinetic energy, U is elastic strain energy, and P denotes the potential energy stored in boundary springs.
The kinetic energy T can be written as follows:
where the over dot represents the differentiation with respect to time.
The strain energy U can be written in an integral form as follows:
Substituting Eqs. (1–16) into Eq. (15) together with Lame coefficients, one can obtain the explicit expressions of strain energy for rectangular plates and cylindrical shells.
The potential energy (P) stored in the boundary springs is given as follows:
where
2.3. Admissible functions
It is crucially important to construct the appropriate admissible displacement functions in the Rayleigh–Ritz method. Beam functions, orthogonal polynomials and Fourier series are often used as displacement functions of plates and shells. However, the use of beam function will lead to at least a very tedious solution process [38]. The problem with using a complete set of orthogonal polynomials is that the higher-order polynomials tend to become numerically unstable because of the computer round-off errors [38, 39]. These numerical difficulties can be avoided by the Fourier series because the Fourier series constitute a complete set and exhibit an excellent numerical stability. However, when the displacements are expressed in terms of conventional Fourier series, discontinuities potentially exist in the original displacements and their derivatives. In this chapter, a modified Fourier series defined as the linear superposition of a 3-D Fourier cosine series and some auxiliary polynomial functions is used to express the displacement components, which are given as follows [40–43]:
(17) |
(18) |
(19) |
where
It is easy to verify that
The similar conditions exist for the β- and z-related polynomials. It should be mentioned that as the circumferential symmetry of the cylindrical shells in the coordinate θ, the 3-D problem of the cylindrical shell can be transformed to 2-D analysis by using the Fourier series in circumferential direction.
2.4. Solution procedure
Substituting Eqs. (14–16) into Eq. (13) together with the displacement functions defined in Eqs. (17–19) and performing the Rayleigh–Ritz operation, a set of linear algebraic equation against the unknown coefficients can be obtained as follows:
where K is the total stiffness matrix for the structure and M is the total mass matrix. Both of them are symmetric matrices. X is the column matrix composed of unknown coefficients expressed in the following form:
where
The frequencies can be determined by solving Eq. (25) via the eigenfunction of MATLAB program. The mode shape corresponding to each frequency can be obtained by back substituting the eigenvector to the displacement functions in Eqs. (17–19).
3. Numerical examples and discussion
In this section, several vibration results of FGM plates and cylindrical shells with general boundary conditions are presented to illustrate the accuracy and reliability of the current formulation. To simplify presentation, C, S, F and E denote the clamped, simply supported, free and elastic restraints. Three types of elastic boundary conditions designated by symbols E_{1}, E_{2} and E_{3} are considered. E_{1}-type edge is considered to be elastic in normal direction; the support type E_{2} only allows elastically restrained displacement in both tangential directions; when all of three displacements along the edges are elastically restrained, the edge support is defined by E_{3}. The expressions of the different boundary conditions along the edge α = 0 are given as follows:
Free boundary condition (F):
Clamped boundary condition (C):
Simply supported boundary condition (S):
First type of elastic restraint (E_{1}):
Second type of elastic restraint (E_{1}):
Three type of elastic restraint (E_{1}):
A simple letter is used to describe the boundary conditions of structure. For example, SFCE denotes a plate having simply supported boundary condition at α = 0, free boundary condition at β = 0, clamped boundary condition at α = L_{α}, and elastic restraint at β = L_{β}; CS denotes a cylindrical shell having clamped boundary condition at α = 0 and simply supported boundary condition at α = L_{α}.
3.1. Rectangular plates
In this section, several numerical examples concerning the free vibration of FGM rectangular plates with different geometrical parameters and boundary conditions have been investigated to verify the convergence, accuracy and reliability of the present method. Some new vibration results of rectangular plates with elastic boundary conditions are given. Unless stated otherwise, the material properties for ceramic and metallic constituents of FGM plates are given as follows: E_{c} = 380 GPa, μ_{c} = 0.3 and ρ_{c} = 3800 kg/m^{3} and E_{m} = 70 GPa, μ_{m} = 0.3 and ρ_{m} = 2702 kg/m^{3}.
3.1.1. Convergence study
Theoretically, there are infinite terms in the modified Fourier series solution. However, the series is numerically truncated, and only finite terms are counted in actual calculations. The convergence of this method will be checked. Table 1 presents the first seven frequency parameters Ω of completely free FGM square plates. The frequency parameter Ω is defined as follows:
The geometrical parameters are given as follows: a/b = 1 and h/b = 0.1, 0.2 and 0.5. The power-law index p is taken to be p = 1. It is obvious that the results of this study show a monotonic trend, and the solutions converge quite rapidly as the truncated number increases. In the following examples, the truncated numbers of the displacement expressions will be uniformly selected as M × N × Q = 13 × 13 × 8.
As aforementioned, the boundary conditions can be easily obtained via changing the value of boundary springs. Therefore, the accuracy of the current method is strongly influenced by the values of springs’ stiffness. To determine the appropriate values of spring’s stiffness, the effects of elastic parameters on the frequencies of the FGM plate are investigated. The elastic parameter Γ is defined as ratios of corresponding spring’s stiffness to bending stiffness D_{c} = E_{c}h^{3}/12(1 – μ_{c}^{2}). The plates are free at y = constant and restrained by only one kind of spring whose stiffness parameter ranges from 10^{−1} to 10^{10} at x = constant. The first three frequency parameters of the FGM square plates with h/b = 0.2 and p = 1 are presented in Table 2. It is obvious that the increase of the elastic parameter leads to increase of the frequency parameters. When Γ ≥10^{7}, the influence of the elastic parameters on the frequencies of the plates can be neglected. The clamped boundary conditions can be simulated by assuming the elastic parameters equal to 10^{9}. The elastic boundary conditions can be obtained by assuming the elastic parameters equal to 100.
3.1.2. Plate with general boundary conditions
To illustrate the accuracy of the present method, the comparisons of the current results with those in the published literature are presented. Table 3 presents the first two frequency parameters of the FGM square plates with different boundary conditions. The results are compared with those presented by Huang et al. [32] using the Ritz method on the basis of 3-D elasticity theory. Table 4 presents the fundamental frequency parameters of the FGM square plates with SSSS boundary conditions. Numerical vibration results for the same problems have been reported by Hosseini-Hashemi et al. [18] and Matsunaga [20] using HSDTs, showing that excellent agreement of the results is achieved.
h/b | p = 0 | p = 0.5 | p = 1 | p = 4 | p = 10 | p = ∞ | |
0.1 | Ref. [18] | 0.0577 | 0.0490 | 0.0443 | 0.0381 | 0.0364 | 0.0293 |
Ref. [20] | 0.0577 | 0.0492 | 0.0442 | 0.0381 | 0.0364 | 0.0293 | |
Present | 0.0578 | 0.0491 | 0.0443 | 0.0381 | 0.0364 | 0.0294 | |
0.2 | Ref. [18] | 0.2113 | 0.1807 | 0.1631 | 0.1378 | 0.1301 | 0.1076 |
Ref. [20] | 0.2121 | 0.1819 | 0.1640 | 0.1383 | 0.1306 | 0.1077 | |
Present | 0.2122 | 0.1816 | 0.1640 | 0.1383 | 0.1306 | 0.1080 |
Several new numerical results for free vibration of FGM plates with general boundary conditions, including classical and elastic boundary conditions, are presented in Tables 5 and 6. The geometrical parameters are given as: a/b = 1 and 2, h/b = 0.1, 0.3 and 0.5. The different boundary conditions, including CSSSS, CCSS, CCCS, CFFF, CCFF, CCCF, E_{1}E_{1}E_{1}E_{1}, E_{2}E_{2}E_{2}E_{2} and E_{3}E_{3}E_{3}E_{3}, are studied. The power-law exponent is taken to be p = 0.6, 1, 5 and 10. From tables, it is obvious that the fundamental frequency of the plate strongly depends on the values of geometrical parameters, power-law index and boundary conditions. For the plates with classic boundary conditions, the foundational frequency parameters decrease with increase in the thickness-to-width ratio h/b. Except for plates with CFFF boundary conditions, the fundamental frequency parameters of the square plate (a/b = 1) are smaller than those of the plates with a/b = 2. The increase of the power-law index leads to decrease of the fundamental frequency parameter for all cases considered. For the plates with elastic restraints, the foundational frequency parameters increase as the length-to-width ratio a/b increases. Except for plates with h/b = 0.1 subjected to E_{2}E_{2}E_{2}E_{2} and E_{3}E_{3}E_{3}E_{3} boundary conditions, the foundational frequency parameters decrease with increase in the power-law index p. The effects of the thickness-to-width ratio h/b on the foundational frequency parameters are more complex. Some 3-D mode shapes for FGM plates with different boundary conditions are shown in Figs. 3 and 4.
3.2. Cylindrical shells
This section is concerned with the free vibration of FGM cylindrical shells with different boundary conditions. The convergence, accuracy and reliability of the present method are demonstrated by numerical examples and comparisons. New numerical results for the FGM cylindrical shells with the elastic boundary conditions are also presented. Unless stated otherwise the material properties for ceramic and metallic constituents of FGM cylindrical shells are given as follows: E_{c} = 168 GPa, μ_{c} = 0.3 and ρ_{c} = 5700 kg/m^{3} and E_{m} = 70 GPa, μ_{m} = 0.3 and ρ_{m} = 2707 kg/m^{3}.
3.2.1. Convergence study
The convergence studies of the first two frequencies for the completely free cylindrical shells with different circumferential wave numbers n are presented in Table 7. The different thickness-to-radius ratios (i.e., h/R_{0} = 0.1, 0.2 and 0.5) and circumferential wave numbers (i.e., n = 1, 2, 3 and 4) are considered. The power-law exponent is taken to be p = 1. From Table 7, it is evident that the present method has a good convergence, and the truncated numbers of the displacement expressions will be uniformly selected as M × Q = 13 × 13.
It is significant to investigate the effects of elastic parameters on the frequencies of the cylindrical shells. The cylindrical shells are restrained by only one kind of spring whose stiffness parameter ranges from 10^{−1} to 10^{10} at x = constant. The frequencies of the cylindrical shells with different circumferential wave numbers n are presented in Table 8. The geometrical parameters are used as R_{0} = 1 m, L/R_{0} = 2 and h/R_{0} = 0.2. The power-law index is taken to be p = 1. It is obvious that the increase of the elastic parameter leads to the increase of the frequency parameters. When Γ ≥10^{7}, the influence of the elastic parameters on the frequencies of the plates can be neglected. The clamped boundary conditions can be simulated by assuming the elastic parameters equal to 10^{9}. The elastic boundary conditions can be obtained by assuming the elastic parameters equal to 100.
3.2.2. Cylindrical shells with general boundary conditions
To illustrate the accuracy of the present method, the comparisons of the current results with those in published literature are presented. Table 9 presents the first three frequency parameters
Several new numerical results for free vibration of FGM cylindrical shells with general boundary conditions, including classical and elastic boundary conditions, are presented in Tables 11 and 12. The geometrical parameters are given as follows: R_{0} = 1 m, L/R_{0} = 2, h/R_{0} = 0.1, 0.3 and 0.5. The different boundary conditions, including CC, CS, SS, CF, SF, E_{1}E_{1}, E_{2}E_{2} and E_{3}E_{3}, are studied. The power-law exponent is taken to be p = 0, 0.6, 1, 2, 5, 10 and 20. It is observed from Table 11 that the boundary conditions have a significant effect on the frequencies of cylindrical shells. The higher constraints at edges may increase the flexural rigidity of the shell leading to higher frequency response. It is obvious that the increase of the thickness-to-radius ratio h/R_{0} leads to the increase of the frequency parameters. It is also seen that the fundamental frequencies decrease as the power-law index increases. From Table 12, the frequencies of shells also increase as thickness-to-radius ratio h/R_{0} increases. However, the effects of the power-law index on the frequencies of shells became more complex. Some 3-D mode shapes for FGM cylindrical shells with different boundary conditions are shown in Figs. 5 and 6.
BC | n | λ _{1} | λ _{2} | λ _{3} | ||||||
LW-DQ [31] | FEM [31] | Present | LW-DQ [31] | FEM [31] | Present | LW-DQ [31] | FEM [31] | Present | ||
CC | 1 | 1.7860 | 1.7972 | 1.7905 | 2.6043 | 2.6222 | 2.6050 | 3.4148 | 3.4192 | 3.4245 |
2 | 1.7452 | 1.7573 | 1.7500 | 3.2942 | 3.3114 | 3.2949 | 3.4921 | 3.5150 | 3.5012 | |
3 | 1.8867 | 1.8862 | 1.8912 | 3.6024 | 3.6320 | 3.6099 | 3.9416 | 3.9257 | 3.9447 | |
4 | 2.1966 | 2.2072 | 2.2004 | 3.8126 | 3.8228 | 3.8193 | 4.2757 | 4.3215 | 4.2783 | |
5 | 2.6385 | 2.6617 | 2.6415 | 4.1302 | 4.1327 | 4.1364 | 4.7010 | 4.7322 | 4.7031 | |
CF | 1 | 0.7514 | 0.7546 | 0.7516 | 1.7563 | 1.7692 | 1.7568 | 1.8800 | 1.8996 | 1.8812 |
2 | 0.6620 | 0.6713 | 0.6622 | 1.8962 | 1.9256 | 1.8980 | 2.1305 | 2.1557 | 2.1324 | |
3 | 0.9246 | 0.9301 | 0.9247 | 2.0610 | 2.0668 | 2.0630 | 2.5165 | 2.5482 | 2.5179 | |
4 | 1.4021 | 1.4282 | 1.4021 | 2.4030 | 2.4646 | 2.4049 | 2.9919 | 3.0342 | 2.9930 | |
5 | 1.9814 | 2.0228 | 1.9814 | 2.8666 | 2.8571 | 2.8684 | 3.5251 | 3.5628 | 3.5258 | |
FF | 1 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0001 | 0.0003 | 1.0710 | 1.0734 | 1.0709 |
2 | 0.2576 | 0.2608 | 0.2576 | 0.3800 | 0.3831 | 0.3799 | 1.3533 | 1.3594 | 1.3532 | |
3 | 0.6884 | 0.6890 | 0.6884 | 0.9253 | 0.9377 | 0.9252 | 1.8689 | 1.8794 | 1.8689 | |
4 | 1.2302 | 1.2525 | 1.2302 | 1.5160 | 1.5307 | 1.5158 | 2.4754 | 2.4917 | 2.4753 | |
5 | 1.8427 | 1.8694 | 1.8426 | 2.1343 | 2.1532 | 2.1341 | 3.1169 | 3.1417 | 3.1169 |
p = 0 | p = 0.6 | p = 1 | p = 5 | |||||
Ref. [13] | Present | Ref. [13] | Present | Ref. [13] | Present | Ref. [13] | Present | |
f_{1} | 152.93 | 152.13 | 150.03 | 148.67 | 149.29 | 147.76 | 148.75 | 147.10 |
f_{2} | 152.93 | 152.13 | 150.03 | 148.67 | 149.29 | 147.76 | 148.75 | 147.10 |
f_{3} | 220.06 | 219.31 | 212.94 | 211.89 | 212.22 | 211.00 | 219.49 | 218.00 |
f_{4} | 220.06 | 219.31 | 212.94 | 211.89 | 212.22 | 211.00 | 219.49 | 218.00 |
f_{5} | 253.78 | 254.30 | 250.74 | 250.36 | 249.31 | 248.68 | 243.43 | 242.86 |
f_{6} | 253.78 | 254.30 | 250.74 | 250.36 | 249.31 | 248.68 | 243.43 | 242.86 |
f_{7} | 383.55 | 384.04 | 370.63 | 370.69 | 369.46 | 369.21 | 383.71 | 382.79 |
f_{8} | 383.55 | 384.04 | 370.63 | 370.69 | 369.46 | 369.21 | 383.71 | 382.79 |
f_{9} | 420.51 | 420.86 | 415.47 | 414.68 | 412.97 | 411.88 | 402.56 | 401.57 |
f_{10} | 431.45 | 428.75 | 420.39 | 416.91 | 418.46 | 414.66 | 423.57 | 419.16 |
4. Conclusions
A new 3-D exact solution for free vibration analysis of thick functionally graded plates and cylindrical shells with arbitrary boundary conditions is presented in this chapter. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple power-law distributions in terms of volume fraction of constituents and are estimated by Voigt’s rule of mixture. By using the artificial spring boundary technique, the general boundary conditions can be obtained by setting proper spring stiffness. All displacements of the functionally graded plates and shells are expanded in the form of the linear superposition of standard 3-D cosine series and several supplementary functions, which are introduced to remove potential discontinuity problems with the original displacements along the edge. The Rayleigh-Ritz procedure is used to yield the accurate solutions. The convergence, accuracy and reliability of this formulation are verified by numerical examples and by comparng the current results with those in published literature. The influence of the geometrical parameters and elastic foundation on the frequencies of rectangular plates and cylindrical shells is investigated.