The three-dimensional free vibration analysis of simply supported, functionally graded piezoelectric material sandwich circular hollow cylinders with different surface conditions is presented. The material properties of each functionally graded piezoelectric material layer are regarded as heterogeneous through the thickness coordinate, and obey an exponent-law dependent on this. Pagano’s method is modified to be feasible for this study of functionally graded piezoelectric material sandwich cylinders, in which a displacement-based formulation is replaced by a mixed one; a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions using Euler’s formula; a successive approximation method is adopted to approximately transform each functionally graded piezoelectric material layer into homogeneous piezoelectric layers with an equal and small thickness for each layer in comparison with the mid-surface radius, and with the homogeneous material properties determined in an average thickness sense; and a transfer matrix method is developed so that the general solutions of system equations can be obtained layer-by-layer, which is significantly less time-consuming than usual. A parametric study of the influence of the mid-surface radius-to-thickness ratio, open- and closed-circuit surface conditions, the thickness ratio of each layer, and the material-property gradient index on the natural frequencies of functionally graded piezoelectric material sandwich cylinders is carried out.
As advanced materials have been rapidly developed in recent decades, multilayered homogeneous and functionally graded piezoelectric material (FGPM) structures have been widely used as sensors and actuators in active and passive vibration control and in noise suppression. Hence, a reliable model capable of providing satisfactory results for the analysis of these types of piezoelectric devices is of much importance in industrial applications. Some comprehensive literature surveys with regard to the theoretical methodologies and computational modeling for a variety of analyses of sandwiched elastic and piezoelectric plates and shells were carried out by Noor and Burton [1,2], Noor et al. [3,4], Burton and Noor [5], Saravanos and Heyliger [6], Gopinathan et al. [7], Carrera [8–10], Carrera and Ciuffreda [11], Carrera and Brischetto [12] and Wu et al. [13]. In addition, a well-known technique book related to sandwich structures was written by Vinson [14].
Among numerous modeling in the above-mentioned review articles [1–14], this literature survey focuses on the exact and approximate three-dimensional (3D) analyses of sandwiched (or multilayered) elastic and piezoelectric plates, doubly curved shells and circular hollow cylinders. Wu et al. [13] classified the exact 3D analytical methods into four different approaches, namely Pagano’s [15,16], state space [17–19], series expansion [20–22] and perturbation [23–26] approaches, in which the pioneers who initiated the applications of various approaches were mentioned, some comprehensive comparisons among the results obtained using the above-mentioned approaches were carried out, and extensions of these approaches to assorted 3D analyses of multilayered and functionally graded material (FGM) structures were collected and tabulated. Because Pagano’s method is simple and feasible for applications to the 3D analysis of laminated composite plates, it has been extensively used to that of laminated piezoelectric ones by Heyliger [27] and Heyliger and Brook [28], in which their formulation was based on the 3D piezoelectricity, and the generalized principle of virtual displacement (PVD) and Hamilton’s principle were used to derive the system equations for the static and free vibration problems of these plates, respectively. Based on the Stroh formalism, Lu et al. [29,30] presented the exact solutions of simply supported, FGPM plates under general and cylindrical bending types of mechanical loads. Some approximate 3D analyses of multilayered elastic, piezoelectric and magnetothermoelastic structures have also been presented using the PVD-based discrete layer theory combined with Ritz’s method by Ramirez et al. [31,32], asymptotic differential quadrature and asymptotic finite strip methods by Wu and Wu [33], Wu and Tsai [34], and Wu et al. [35], the PVD-based finite layer method (FLM) by Akhras and Li [36,37], Reisner’s mixed variational theorem (RMVT)-based FLM by Wu and Li [38,39], RMVT-based meshless collocation (MC) and element-free Galerkin (EFG) methods by Wu et al. [40], Wu and Chiu [41] and Wu and Yang [42,43].
In recent decades, an emerging class of FGPM structures, the material properties of which are heterogeneous and gradually and continuously vary through the thickness coordinate, has been developed to overcome some drawbacks resulting from sudden changes of the material properties, such as delamination and micro cracking, at the interfaces between adjacent layers in conventionally multilayered piezoelectric structures. The feature of continuous distributions of material properties through the thickness coordinate of FGPM structures, however, also increases the complexity and difficulty of analyzing such FGPM structures, and the literature with regard to the exact and approximate 3D analyses of FPGM structures is quite rare in comparison to that examining multilayered homogeneous ones.
The 3D analysis of FGPM plates and shells is an important treasure from the viewpoint of academic research because this may not only serve as a standard for assessing various approximate 2D theories of plates and shells, but also may provide a reference for making the appropriate kinetic and kinematic assumptions prior to developing more advanced 2D theories. Among the 3D approaches mentioned above, Pagano’s method is the most widely applied for multilayered plates and shells although it is not feasible for their FGM counterparts without further modifications. In order to achieve this, Wu et al. [44] and Wu and Lu [45] developed a modified Pagano’s method for exact 3D static and free vibration analyses, respectively, of simply supported, functionally graded magneto-electro-elastic plates, and it was further extended to the bending and thermoelastic analyses of FGPM sandwich cylinders by Wu and Tsai [46] and Wu and Jiang [47], respectively. Their modifications to the original Pagano method are as follows: (a) A mixed formulation rather than one that is displacement-based is used so that both the lateral boundary conditions on the outer surfaces and the continuity conditions at the interface between adjacent layers can be directly applied. (b) The sets of complex-valued solutions for system equations are transferred to the corresponding sets of real-valued solutions by means of Euler’s formula for the purpose of computational efficiency. (c) A successive approximation (SA) method, which was used by Soldatos and Hadjigeorgiou [48] for the analysis of homogeneous isotropic cylindrical shells using the state space approach, is adopted, and the functionally graded plate or shell is artificially divided into a certain number of individual layers with an equal and small thickness as compared to the in-plane dimension of the plate or the mid-surface radius of the shell for each layer. Using a refinement manipulation, one may reasonably approximate the variable material coefficients of each layer to the constant material coefficients in an average thickness sense so that the system of thickness-varying differential equations for each individual layer can be reduced to a system of thickness-invariant differential equations. (d) A transfer matrix method is developed, so that the general solutions of system equations can be obtained layer-by-layer. These modifications mean that Pagano’s method can be used in the 3D analysis of functionally graded plates and shells, and their earlier implementations have shown that the computation thus becomes less time-consuming than is usually the case and is independent of the total number of layers constituting the plates and shells.
Due to the benefits of the modified Pagano method, noted above, in this article it is extensively applied to the exact 3D free vibration analysis of simply supported, sandwiched (or multilayered) FGPM circular hollow cylinders with three different surface conditions, which include closed–closed, open–closed and open–open circuit conditions on the inner–outer lateral surfaces, in which the material properties of each FGPM layer are assumed to obey an exponent-law distribution through the thickness coordinate. Because the FGPM sandwich cylinder is transformed into a multilayered homogeneous piezoelectric one in this formulation using the SA method mentioned above, the analysis of multilayered (or sandwiched) hybrid elastic and piezoelectric cylinders can thus be included as a special case and be undertaken using the present formulation. A parametric study is thus carried out of the influence of the middle surface radius-to-thickness ratio, surface conditions, and material-property gradient index on the natural frequencies and their corresponding through-thickness distributions of modal elastic and electric field variables induced in the sandwiched FGPM cylinder.
Basic equations of 3D piezoelectricity
The free vibration of a simply supported, FGPM circular hollow sandwich cylinder with heterogeneous material properties through the thickness coordinate is considered. The cylindrical coordinate system and configuration of this cylinder are shown in Figure 1, in which a set of the cylindrical coordinates () is located on the center of the cylinder. The total thickness, length, radius to the middle surface, and the thickness coordinate of the cylinder are 2h, L, R and ζ, respectively, and .
The configuration and coordinates of an FGM sandwich cylinder.
The linear constitutive equations, valid for the nature of the symmetry class of the piezoelectric material considered, are given by
where and are the stress and strain components, respectively, Di and Ei (i = ) denote the electric displacement and electric field components, respectively, and (i, j = 1_6), (l = 1_3, j = 1_6) and (l, k = 1_3) are the elastic, piezoelectric and dielectric permeability coefficients, respectively, which are considered to be constants for a homogenous piezoelectric layer and to be variable through the thickness coordinate for an FGPM layer.
The strain-displacement relationships are
where ; are the displacement components.
The stress equilibrium equations without body forces are given by
where the commas stand for the partial differentiation with respect to the suffix variables, and ρ and t denote the mass density and time variable, respectively.
The equations of electrostatics for the piezoelectric material without the electric charge density are
The relations between the electric field and electric potential are
where Φ denotes the electric potential; , and .
Three different lateral surface conditions of the hollow cylinder are considered and are specified as follows:
Case 1. Closed–closed circuit conditions considered at both the inner–outer surfaces, respectively
Case 2. Open–closed circuit conditions considered at the innter–outer surfaces, respectively
Case 3. Open–open circuit conditions considered at both the inner–outer surfaces, respectively
The edge boundary conditions of the cylinder are considered as fully simple supports, suitably grounded, and are given as
There are 22 basic equations for 3D piezoelectricity, as listed in equations (1) to (8), and these are essentially a system of simultaneously partial differential equations with variable coefficients. In later discussion in this article, the modified Pagano method will be developed for this free vibration analysis of simply supported, FGPM sandwich circular hollow cylinders.
Nondimensionalization
In order to scale all the field variables within a close order of magnitude and prevent unexpected numerical instability in the computation process, we define a set of dimensionless coordinates and variables, as follows
where Q, e and stand for the reference elastic coefficients, piezoelectric coefficients and mass density, respectively, as well as the values of Q and e will be given as and in the following illustrative examples, in which the superscript 1 denotes the inner layer.
In this formulation, the elastic displacements (), the transverse shear and normal stresses () and the normal electric displacement and electric potential components (Dr,Φ) are selected as the primary field variables. The other field variables are secondary variables, and can be expressed in terms of these that are primary. Introducing the set of dimensionless coordinates and variables given in equations (13a–p) and using the method of direct elimination, we obtain one set of state space equations in terms of the primary field variables for the coupled analysis of FGPM sandwich cylinders, and they are given as follows
where
The relevant coefficients in the previous terms of are given in Appendix 1.
The in-surface stress and electric displacement components are dependent field variables, which can be calculated using the primary variables, which are determined, as follows
where
The dimensionless forms of the boundary conditions of the problem are specified as follows
At the edges, the following quantities are satisfied
The modified Pagano method
The double Fourier series expansion method
The double Fourier series expansion method is applied to reduce the system of partial differential equations (14) to (16) to a system of ordinary differential equations, such that by means of satisfying the edge boundary conditions, the primary variables are expressed in the following form
where , , and and are zeroes or positive integers; , in which ω is the circular frequency of the harmonic motion.
For brevity, the symbols of summation are omitted in the following derivation. Using the set of dimensionless coordinates and field variables, which are given in equation (13), and substituting equations (21) to (23) in equation (14), we have the resulting equations, as follows
where
and are given in Appendix 2, and the frequency parameter is included in the terms of , and .
Theories of the homogeneous linear systems
Equation (24), which is a system of eight simultaneously homogeneous ordinary differential equations in terms of eight primary variables, represents the state space equations for the free vibration responses of a simply supported, FGPM circular hollow cylinder, and the general solution of this is
where L is an 8 × 1 matrix of arbitrary constants; Ω is a fundamental matrix of equation (25) and is formed by eight linearly independent solutions in the form of , (i = 1,2, … , 8); and are the eigenvalues and their corresponding eigenvectors of the coefficient matrix in equation (24), respectively.
If the coefficient matrix has a complex eigenvalue (i.e. ), then its complex conjugate (i.e. ) is also an eigenvalue of due to the fact that all of the coefficients of are real. In addition, are the corresponding eigenvectors of the complex conjugate pair, . Using Euler’s formula, we replace these complex-valued solutions with alternative two real-valued solutions for the consideration of computational efficiency, and these are given by
On the basis of the previous set of linearly independent real-valued solutions, a transfer matrix method can be developed for the analysis of multilayered piezoelectric hollow cylinders, and it can be extended to the analysis of FGPM sandwich cylinders using an SA method [53], where the FGPM sandwich cylinder is artificially divided into a finite number (Nl) of individual layers with equal and small thicknesses for each layer, compared with the mid-surface radius, as well as with constant material properties, determined in an average thickness sense. The exact solutions of the assorted field variables induced in the FGPM sandwich cylinder can thus be gradually approached by increasing the number of individual layers. It is noted that this solution process can be performed layer-by-layer, and the computational performance is independent of the total number of individual layers. Consequently, the implementation of the present approach is much less time-consuming than usual.
The successive approximation method
This article undertakes a 3D free vibration analysis of FGPM sandwich cylinders, which consists of an FGPM core bounded with two homogeneous piezoelectric face sheets, one of the widely utilized multilayered FGPM cylinders, in which the thickness of each layer is and is counted from the bottom layer), and , and the material properties ( ()) are assumed to be symmetric with respect to the mid-surface of the sandwich cylinder and obey an exponent-law distribution through the thickness coordinate, as follows
where and denote the material properties of the face sheets and the reference material properties at the mid-surface of the cylinder, respectively; denotes the material-property gradient index, which represents the degree of the material gradient along the thickness coordinate and it is noted that the material properties at the interfaces between adjacent layers are restricted to be continuous in this study, such that the material-property parameters, , and , are related to one another with the relationship, which is , or .
Because the material properties of the FGPM core in a FGPM sandwich cylinder vary along its thickness coordinate, resulting in a variant coefficient matrix in the system equation (i.e. equation (24)), the conventional Pagano method cannot be directly applied to this study of the FGPM sandwich cylinder. An SA method [48] is thus adopted to make the present approach feasible. In the SA method, the FGPM sandwich cylinder is artificially divided into an Nl-layered cylinder with an equal and small thickness compared with the mid-surface radius and with homogeneous material properties for each layer. For a typical mth layer in the upper half core layer of the cylinder, the material properties are regarded as constants and are determined in an average thickness sense, as follows
where and are the thickness coordinates, measured from the middle surface of the cylinder to the bottom and top surfaces of the mth layer in the upper half core layer, respectively, and where denotes the thickness of the mth layer, which is .
By means of equation (29), the modified Pagano method can be extensively applied to this analysis of FGPM sandwich cylinders. Increasing the number of artificial layers (Nl), we can approximate the exact solutions of this free vibration analysis of FGPM sandwich cylinders to any desired accuracy.
The transfer matrix method
As we noted above, the modified Pagano method can be applied to this study of FGPM sandwich cylinders using equation (29). The through-thickness distributions of material properties are modified as layerwise Heaviside functions, and the upper half of these are given by
where refers to the coefficients of of the mth layer in general; is the Heaviside function, and the material properties of the lower half of the cylinder are symmetric to these of the upper half with respect to the mid-surface of the cylinder, which were given above and thus are not repeated.
The solution procedure of the transfer matrix method for Nl-layered piezoelectric cylinders has been detailed described in the static counterpart of this article [46], and will not be repeated here. Manipulating the transfer matrix method, and imposing the boundary conditions prescribed on the lateral surfaces, we obtained a set of eight simultaneous algebraic equations as follows
where and denote the unknown variables on the outer and inner surfaces, respectively. In the surface conditions of Case 1,
and
in the surface conditions of Case 2,
and
in the surface conditions of Case 3,
and
According to equation (31), we have a set of homogeneous equations as
where is a 4 × 4 matrix, of which coefficients are related to the circular frequency ω.
A nontrivial solution of equation (32) exists if the determinant of the coefficient matrix vanishes. Hence, the natural frequencies of FGPM sandwich cylinders for a set of fixed values (, ) can be obtained by
Equation (33) is called the charateristic equation. Since the determinant of yields an implicit function of ω rather than an explicit function, a bisection method is used to determine the roots of equation (33).
Once equation (33) is solved, the eigenvalues and their corresponding modal values of and can be determined from equations (31) and (32), respectively. Afterwards, these primary variables through the thickness coordinate of the hollow cylinder can be obtained by
and
where and .
Once the primary variables varing through the thickness of the cylinder are determined, the corresponding set of dependent variables in the elastic and electric fields can then be obtained using equations (15) and (16).
Illustrative examples
Multilayered orthotropic cylinders
The 3D exact solutions of free vibration of simply supported, orthotropic multilayered circular hollow cylinders presented by Noor and Rarig [49] are used here to validate the accuracy and convergence of the modified Pagano method in Table 1. The material properties of each layer are taken to be , , , and = 3, 10 and 40, where the subscripts L and T denote the directions perpendicular and parallel to the fiber direction; the geometric parameters are L/R = 1 and S = R/2h = 5, and the dimensionless frequency parameter is defined as .
Convergence study of the present modified Pagano method for the frequency parameters of simply supported, laminated orthotropic cylinders with different values of
Table 1 shows the solutions of least frequency parameters of [0°/90°] and [90°/0°/0°/90°] laminated cylinders with obtained using the modified Pagano method, in which the number of layers (Nl) is Nl = 2, 4, 8 and 16 for the [0°/90°] cylinders, and Nl = 4, 8, 16 and 32 for the [90°/0°/0°/90°] ones. It can be seen in Table 1 that these solutions are accurate and converge rapidly. When Nl = 8, these solutions are in excellent agreement with the 3D exact solutions obtained by Noor and Rarig [49], and the relative errors of these 8-layered solutions are below 0.05%, as compared with the available exact 3D solutions [49]. These solutions are also compared with the available results obtained from the equivalent single-layer theory with the PVD-based first- and second-order displacement models (ED1 and ED2), the PVD-based layerwise models (LD1 and LD2), and the RMVT-based layerwise models (LM1 and LM2), which were given by Carrera [50], as well as from the RMVT-based EFG method by Wu and Yang [43]. It is seen in Table 1 that the performance among these theories is LM>LD>ED on the basis of the same orders of field variables, in which ‘>’ means more accurate. In addition, the fundamental frequency parameters increase when the ratio of of the lamina becomes larger, which implies the cylinders with a high ratio of possess the high overall stiffness, thus increasing their corresponding frequency parameters, and the frequency parameters of the 4-layered symmetric cylinders ([90°/0°/0°/90°]) are higher than the two-layer anti-symmetric ones ([0°/90°]), which implies that the coupling extension-bending effect of anti-symmetric cylinders decreases their overall stiffness, thus decreasing their corresponding frequency parameters.
Tables 2 and 3 show the least frequency parameters of 2- and 10-layered anti-symmetric cylinders ([0°/90°] and [0°/90°]5) with ; and in Table 2, and and in Table 3, respectively, in which Nl is taken to be (8, 16, 32) and (10, 20, 40) for the [0°/90°] and [0°/90°]5 laminated cylinders, respectively. The material properties of each layer and dimensionless frequency parameter are identical to those used in Table 1, and the geometric parameters are L/R = 1 and S = R/2h = 20, 4 and 2.5. The present solutions are also compared with the available 3D exact [49], LM2 and LD2 [50], and RMVT-based EFG [43] solutions in Table 2. It is again shown that these convergent rapidly and are in excellent agreement with the 3D exact [49], LM2 [50] and RMVT-based EFG [43] solutions, and the relative errors of these 8-layered solutions are below 0.05%, as compared with the available exact 3D solutions [49]. It can also be seen in Tables 2 and 3 that when , the least frequency parameter first monotonically decreases with increases in the value of , and then it increases; and when , the least frequency parameter always monotonically increases with increasing of the value of . Moreover, it is also shown that the fundamental frequency parameters occur at = (1, 4) and (1, 2) for the [0°/90°] laminated cylinders with S = 20 (thin cylinders), and 4 and 2.5 (thick cylinders), respectively, and these occur at = (1, 3) and (1, 2) for laminated cylinders with S = 20, and 4 and 2.5, respectively.
The least frequency parameters of simply supported, laminated orthotropic cylinders with and
The free vibration of simply supported, FGPM sandwich circular hollow cylinders, which consist of two homogeneous piezoelectric face-sheets and an FGPM core, is studied. The thickness ratio of each layer of the sandwich cylinder considered is , in which and , and the material properties of each layer are given in equations (28a) to (28c).In this article, the material properties of a PZT-4 material are given as the reference material for the face-sheets, which are , and are listed in Table 4. The material properties can then be determined using once the value of is given, and the larger is, the softer and lighter the core-layer becomes. In addition, The dimensionless frequency parameter is defined as .
Elastic, piezoelectric and dielectric properties of the material PZT-4
Moduli
PZT-4
139.0 GPa
139.0 GPa
115.0 GPa
77.8 GPa
74.3 GPa
74.3 GPa
25.6 GPa
25.6 GPa
30.6 GPa
12.7 C/m2
12.7 C/m2
−5.2 C/m2
−5.2 C/m2
15.1 C/m2
6.46e–09 C2/Nm2
6.46e–09 C2/Nm2
5.62e–09 C2/Nm2
ρ
7600 kg/m3
The reference elastic and piezoelectric coefficients are given as and , respectively, in this study.
Table 5 presents the least frequency parameters of FGM sandwich cylinders for different vibration modes, in which , , = 5, , and = 1, 3 and 5. It can be seen in Table 5 that for a particular vibration mode, the natural frequency parameter decreases when the coupled elastic-electric effect is neglected in this formulation by letting , and this also means that this effect increases the overall stiffness of the FGM sandwich cylinder. In addition, the effect of surface conditions on the frequency parameter of the cylinder is minor, and the frequency parameter of the cylinder with closed–closed circuit conditions is slightly less than that of the cylinder with closed–open circuit conditions, which is further slightly less than that of the cylinder with open–open circuit conditions.
The least frequency parameters of FGPM sandwich cylinders for different vibration modes
()
Case 1
Case 2
Case 3
1
(1, 1)
0.03028
0.03005
0.03141
0.03005
0.03143
0.03005
(1, 2)
0.03612
0.03187
0.03648
0.03187
0.03666
0.03187
(1, 3)
0.08663
0.07562
0.08695
0.07562
0.08822
0.07562
(1, 4)
0.15264
0.13336
0.15339
0.13336
0.15675
0.13336
(2, 1)
0.07539
0.07468
0.07773
0.07468
0.07776
0.07468
(2, 2)
0.05915
0.05452
0.06058
0.05452
0.06068
0.05452
(2, 3)
0.09868
0.08657
0.09951
0.08657
0.10074
0.08657
(2, 4)
0.16222
0.14178
0.16335
0.14178
0.16676
0.14178
(3, 1)
0.11283
0.11090
0.11635
0.11090
0.11636
0.11090
(3, 2)
0.09236
0.08627
0.09484
0.08627
0.09496
0.08627
(3, 3)
0.12095
0.10714
0.12258
0.10714
0.12388
0.10714
(3, 4)
0.17921
0.15694
0.18098
0.15694
0.18455
0.15694
(4, 1)
0.14266
0.13826
0.14712
0.13826
0.14712
0.13826
(4, 2)
0.12814
0.11934
0.13149
0.11934
0.13175
0.11934
(4, 3)
0.15118
0.13487
0.15372
0.13487
0.15528
0.13487
(4, 4)
0.20347
0.17869
0.20605
0.17869
0.20995
0.17869
3
(1, 1)
0.03106
0.03013
0.03149
0.03013
0.03150
0.03013
(1, 2)
0.03714
0.03342
0.03758
0.03341
0.03781
0.03341
(1, 3)
0.08243
0.07483
0.08324
0.07483
0.08444
0.07483
(1, 4)
0.13515
0.12418
0.13681
0.12418
0.13938
0.12418
(2, 1)
0.07714
0.07498
0.07804
0.07498
0.07806
0.07498
(2, 2)
0.06003
0.05564
0.06104
0.05564
0.06119
0.05564
(2, 3)
0.09292
0.08479
0.09418
0.08479
0.09532
0.08479
(2, 4)
0.14246
0.13110
0.14446
0.13110
0.14704
0.13110
(3, 1)
0.11545
0.11160
0.11691
0.11160
0.11691
0.11160
(3, 2)
0.09254
0.08685
0.09412
0.08685
0.09431
0.08685
(3, 3)
0.11255
0.10360
0.11440
0.10361
0.11558
0.10361
(3, 4)
0.15570
0.14372
0.15821
0.14372
0.16085
0.14372
(4, 1)
0.14497
0.13901
0.14705
0.13901
0.14708
0.13901
(4, 2)
0.12597
0.11860
0.12817
0.11860
0.12852
0.11860
(4, 3)
0.13857
0.12854
0.14110
0.12854
0.14242
0.12854
(4, 4)
0.17461
0.16180
0.17776
0.16180
0.18054
0.16180
5
(1, 1)
0.03139
0.03014
0.03149
0.03014
0.03150
0.03014
(1, 2)
0.03022
0.02832
0.03054
0.02832
0.03059
0.02832
(1, 3)
0.05826
0.05545
0.05881
0.05543
0.05907
0.05543
(1, 4)
0.08995
0.08604
0.09078
0.08604
0.09134
0.08604
(2, 1)
0.07750
0.07478
0.07771
0.07478
0.07772
0.07478
(2, 2)
0.05287
0.05012
0.05331
0.05012
0.05333
0.05012
(2, 3)
0.06670
0.06352
0.06738
0.06352
0.06761
0.06352
(2, 4)
0.09491
0.09078
0.09585
0.09078
0.09639
0.09078
(3, 1)
0.11449
0.11017
0.11484
0.11017
0.11484
0.11017
(3, 2)
0.08269
0.07884
0.08325
0.07883
0.08327
0.07883
(3, 3)
0.08323
0.07941
0.08405
0.07941
0.08426
0.07941
(3, 4)
0.10464
0.10007
0.10572
0.10007
0.10625
0.10007
(4, 1)
0.13975
0.13403
0.14028
0.13403
0.14028
0.13403
(4, 2)
0.11069
0.10569
0.11132
0.10569
0.11136
0.10569
(4, 3)
0.10434
0.09970
0.10531
0.09970
0.10552
0.09970
(4, 4)
0.11890
0.11371
0.12016
0.11370
0.12069
0.11371
Table 6 shows the least frequency parameters of FGM sandwich cylinders with different middle surface radius-to-thickness ratios, material-property gradient indices, and thickness ratios for each layer, in which , = 5 and 10, , , and , 0.8h:0.4h:0.8h, 0.6h:0.8h:0.6h, 0.4h:1.2h:0.4h, and 0.2h:1.6h:0.2h. It can be seen in Table 6 that the least frequency parameters of FGM sandwich cylinders increase when the cylinder becomes thicker. As noted above, the greater the material-property gradient index is, the softer and lighter of the cylinder becomes. It can be seen in Table 6 that in the case of thick cylinder (R/2h = 5), the least frequency parameter decreases when the material-property gradient index becomes larger for most ratios of , except for , and this also means that the gross loss of the stiffness of the cylinder is more than that of the mass of the cylinder for most ratios of , and in the case of , the ratio of the gross loss stiffness-to-mass firstly decreases, and then it increases with increasing the value of . In addition, the above-mentioned variation of the least frequency parameter with for the cylinder with R/2h = 5 and is also observed in the cases of moderately thick FGM sandwich cylinders (R/2h = 10) with all ratios of .
The least frequency parameters of simply supported, FGPM sandwich cylinders with different thickness ratios for each layer and material-property gradient indices
R/2h
Surface conditions
h:0: h
0.8 h:0.4 h:0.8 h
0.6 h: 0.8 h: 0.6 h
0.4 h:1.2 h:0 4 h
0.2 h:1.6 h:0.2 h
5
1
Case 1
0.03324
0.03405
0.03488
0.03563
0.03612
Case 2
0.03355
0.03437
0.03521
0.03598
0.03648
Case 3
0.03362
0.03446
0.03534
0.03613
0.03666
3
Case 1
0.03324
0.03337
0.03414
0.03546
0.03714
Case 2
0.03355
0.03368
0.03448
0.03585
0.03758
Case 3
0.03362
0.03376
0.03459
0.03601
0.03781
5
Case 1
0.03324
0.02870
0.02766
0.02818
0.03022
Case 2
0.03355
0.02893
0.02788
0.02844
0.03054
Case 3
0.03362
0.02894
0.02789
0.02846
0.03059
10
1
Case 1
0.00996
0.01019
0.01042
0.01061
0.01072
Case 2
0.01017
0.01039
0.01061
0.01080
0.01092
Case 3
0.01018
0.01040
0.01062
0.01081
0.01093
3
Case 1
0.00996
0.01038
0.01080
0.01125
0.01166
Case 2
0.01017
0.01051
0.01091
0.01136
0.01177
Case 3
0.01018
0.01051
0.01092
0.01137
0.01179
5
Case 1
0.00996
0.01006
0.01025
0.01065
0.01125
Case 2
0.01017
0.01013
0.01032
0.01072
0.01132
Case 3
0.01018
0.01013
0.01032
0.01072
0.01133
In order to have a clear picture with regard to the distributions of modal field variables induced in the FGM sandwich cylinder, we show the through-thickness distributions of a variety of modal in- and out-of-surface elastic and electric variables of the cylinder with surface conditions of Cases 1–3, in which , = 5, , , and = 0.2h:1.6h:0.2h. As above noted, when = 0, the cylinder is reduced to be a single-layered homogeneous piezoelectric cylinder with the material properties given in Table 4. It is seen in Figures 2 to 4 that when = 0, the through-thickness distributions of modal displacement and in-surface stress, transverse stress, and electric potential and displacement components appear to be linear, parabolic and higher order polynomial functions, respectively, while when , these distributions appear to be piecewise higher order polynomial functions. The through-thickness distributions of modal field variables induced in a homogeneous piezoelectric cylinder are much different from those induced in an FGM cylinder. In addition, Figures 2 to 4 also show that these solutions obtained using the modified Pagano method exactly satisfy the surface conditions at the lateral surfaces of the cylinder, as well as the continuity conditions of elastic displacement, transverse stress, electric potential and electric displacement components at the face sheets/core interfaces of the cylinder.
The through-thickness distributions of various modal elastic and electric variables induced in the FGPM sandwich cylinders with the surface condition case 1.
The through-thickness distributions of various modal elastic and electric variables induced in the FGPM sandwich cylinders with the surface condition case 2.
The through-thickness distributions of various modal elastic and electric variables induced in the FGPM sandwich cylinders with the surface condition case 3.
Concluding remarks
In this article, we have developed a modified Pagano method for the quasi-3D free vibration analysis of simply supported, FGPM sandwich cylinders with closed–closed, open–closed, and open–open circuit surface conditions. The accuracy and convergence of the solutions obtained using this method are evaluated in comparison with the available exact 3D solutions, with which the present solutions are shown to converge rapidly and be in excellent agreement. A parametric study of the influences of the middle surface radius-to-thickness ratio, surface condition, thickness ratio of each layer, and material-property gradient index on the frequency parameters and their corresponding through-thickness distributions of assorted modal elastic and electric variables are undertaken. The present solutions may serve as benchmarks for assessing the accuracy and convergence of various approximate 2D theories of FGPM cylinders, and they also can provide a reference for making suitable kinetic and kinematic assumptions prior to developing more advanced 2D theories of FGPM cylinders.
Footnotes
Funding
This work was supported by the National Science Council of the Republic of China through Grant NSC 100-2221-E-006-180-MY3.
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