Axisymmetric Free Vibration of Functionally Graded Piezoelectric Circular Plates

Abstract

An analytical solution is presented for axisymmetric free vibration analysis of a functionally graded piezoelectric circular plate on the basis of the three-dimensional elastic theory of piezoelectric materials. The material properties are assumed to follow an exponential law distribution through the thickness of the circular plate. The state space equations for the free vibration behavior of the functionally graded piezoelectric circular plate are developed based on the state space method. The finite Hankel transform is utilized to obtain an ordinary differential equation with variable coefficients. By virtue of the proposed exponential law model, we have ordinary differential equations with constant coefficients. Then, the free vibration behaviors of the functionally graded piezoelectric circular plate with two kinds of boundary conditions are investigated. Some numerical examples are given to validate the accuracy and stability of the present model. The influences of the exponential factor and thickness-to-span ratio on the natural frequency of the functionally graded piezoelectric circular plate, constrained by different boundary conditions, are discussed in detail.

1. Introduction

Piezoelectric materials have been extensively used in smart systems owing to their electro-elastic coupling properties [1,2,3]. In the view of engineering, vibrational properties have become one of the most researched fields for piezoelectric materials. Shivashankar and Gopalakrishnan [4] presented a review paper about the application of piezoelectric materials for active vibration, flow, and noise control. On the basis of the nonlocal Mindlin plate theory, Liu et al. [5] studied the nonlinear vibration behavior of piezoelectric nanoplate. By using the Levinson plate theory, the closed-form solution for free vibration of the piezoelectric annular plate was obtained [6]. The nano-electric–mechanical systems have received increasing attention, and Zhao et al. [7] presented a literature review about the size-dependent vibrations and waves in piezoelectric nanostructures. Ke et al. [8] achieved the natural frequencies for the piezoelectric nanoplate with different boundary conditions and nonlocal parameters. Sun et al. [9] introduced a mathematical framework for analyzing the thickness–shear vibrations of a finite piezoelectric nanoplate. By utilizing the Kirchhoff thin plate theory, Wang and Li [10] studied the dependence of natural frequencies on the flexoelectric effects of piezoelectric nanoplate.

To achieve better performance of smart systems, piezoelectric materials are often constructed with other materials and serve as the multi-layer form in engineering. Such designs can bring various benefits but also cause some problems, such as interfacial stress concentration, interfacial crack, and even structural failure. These defects can lead to a decrease in the reliability and lifespan of the piezoelectric devices. To overcome the constraints of conventional laminated piezoelectric components, novel composite materials, named functionally graded piezoelectric materials (FGPMs), have been developed [11,12]. FGPMs represent a class of piezoelectric materials characterized by their spatially varying inhomogeneity, with mechanical properties that exhibit continuous variation across one or more spatial dimensions [13,14]. The continuity of the material properties of FGPMs can reduce the interfacial drawbacks, and they can be designed to possess special desired properties, which can be used as an active control, health monitoring, and so on [15].

Composite plate structures are widely used in engineering because of their superior stiffness-to-weight ratios. Understanding how the intrinsic material properties influence the dynamic behavior of FGPM plates is essential for designing FGPM devices. The growing demand for FGPMs has attracted many researchers. Wu et al. [16] presented a comprehensive review of diverse three-dimensional analytical methodologies for FGPM plates and shells. Sharma [17] generally presented an analysis of the vibration behavior of FGPM actuators subjected to shear excitation, employing the generalized differential quadrature method. Khalfi et al. [18] used a semi-analytical method to model the axisymmetric free vibration of the FGPM hollow cylinder resonator. Li and Pan [19] used the modified couple–stress theory to study the influences of the power-law index of material gradient and material length-scale parameters on the natural frequency of the FGPM microplate. By virtue of the first-order shear deformation theory, Shahdadi and Rahnama [20] derived an analytical solution for free vibration analysis of the FGPM annular sector plate. Li et al. [21] conducted a study on the active vibration suppression of an FGPM plate that was simply supported along its four edges. Zhong and Yu [22] derived the exact equations for determining the natural frequencies of free vibration of an FGPM rectangular plate.

From the literature review, the investigations that offer explicit closed-form solutions for the three-dimensional vibration analysis of plates, particularly circular and annular plates, are relatively few, primarily attributed to the inherent physical complexities and mathematical challenges involved. Based on the linear three-dimensional theory of elasticity coupled with the electric field, Ding et al. [23] studied the free axisymmetric vibration of the laminated piezoelectric circular plate. Using a direct method, Xu [24] developed a new state space formulation for the free vibration of circular/annular plates. By virtue of three-dimensional theory, Wang et al. [25] introduced a direct method for investigating the free axisymmetric vibration characteristics of transversely isotropic circular plates. To the authors’ knowledge, the state space method for homogeneous plates given by Ding et al. [23] has not been extended to present the three-dimensional exact solution for the axisymmetric free vibration of the FGPM circular plate with an exponential law distribution.

In this paper, the axisymmetric free vibration of an FGPM circular plate has been carried out utilizing the three-dimensional elastic theory of piezoelectric materials. Material properties of the FGPM circular plate are assumed to alter exponentially across the thickness direction [22]. Based on the state space method and the finite Hankel transform, the three-dimensional exact solution for the axisymmetric free vibration of the FGPM circular plate is presented. The natural frequencies are obtained for elastic simple support and rigid slipping support boundary conditions with different exponential factors and thickness-to-span ratios, and a comparison of the current model with those reported in the literature has been provided.

2. Theoretical Formulations

2.1. Geometrical Configuration and Material Properties

Consider an FGPM circular plate of radius a and thickness h consisting of N-ply in which the thickness of the j-th layer is h_j, as shown in Figure 1. A cylindrical coordinate system (r, θ, z) with origin O at the center of the top surface is utilized to describe the plate geometry and dimensions. The FGPM circular plate is modeled as the transversely isotropic piezoelectric materials in which the poling direction is along with the z-direction.

It is known that the material properties of FGPMs can usually be seen as a function of coordinates (r, θ, z) and vary continuously in one or two directions [16]. In the present work, the material properties of the FGPM circular plate are presumed to change across the plate thickness with an exponential law distribution as follows [22]:

$$C_{ij}\left(z\right)=C_{ij}^0{\mathrm{e}}^{\eta z}, e_{ij}\left(z\right)=e_{ij}^0{\mathrm{e}}^{\eta z}, {\xi }_{ij}\left(z\right)= {\xi }_{ij}^0{\mathrm{e}}^{\eta z}, \rho \left(z\right)={\rho }^0{\mathrm{e}}^{\eta z},$$

(1)

where C_ij(z) denotes the elastic constants; e_ij(z) refers to the piezoelectric constants; ξ_ij(z) represents the dielectric constants; ρ(z) is the mass density; η stands for the exponential factor. Notice that $C_{ij}^0$, $e_{ij}^0$, ${\xi }_{ij}^0$, and ${\rho }_{ij}^0$ denote the corresponding material properties at the top surface of the FGPM circular plate.

2.2. Governing Equations for an FGPM Circular Plate

In the framework of axisymmetric free vibration of an FGPM circular plate, it is noted that all physical quantities are θ-independent, meaning that they do not vary with respect to the angular coordinate θ. Taking into account a harmonic solution, the equations of motion for each layer of the FGPM circular plate can be written as [26,27,28]:

$$\begin{array}{l}{\displaystyle \frac{\partial {\sigma }_{rr}}{\partial r}}+{\displaystyle \frac{\partial {\sigma }_{rz}}{\partial z}}+{\displaystyle \frac{{\sigma }_{rr}-{\sigma }_{\theta \theta }}{r}}=\rho {\displaystyle \frac{{\partial }^2{u}_r}{\partial t^2}},\\ {\displaystyle \frac{\partial {\sigma }_{rz}}{\partial r}}+{\displaystyle \frac{\partial {\sigma }_{zz}}{\partial z}}+{\displaystyle \frac{{\sigma }_{rz}}{r}}=\rho {\displaystyle \frac{{\partial }^2{u}_z}{\partial t^2}},\end{array}$$

(2)

where ${\sigma }_{ij}$ denotes the stress tensor; $u_r$ and $u_z$ are the displacements components in r and z directions, respectively; r is the radial coordinate (0 < r ≤ a); t is the time. The mass density ρ in Equation (2) obeys the exponential law distribution, namely, $\rho =\rho \left(z\right)={\rho }^0{\mathrm{e}}^{\eta z}$.

The Gaussian equation without electric charges can be expressed in the form of [26,27,28]:

$${\displaystyle \frac{\partial D_r}{\partial r}}+{\displaystyle \frac{\partial D_z}{\partial z}}+{\displaystyle \frac{D_r}{r}}=0,$$

(3)

where $D_r$ and $D_z$ represent the electric displacements in r and z directions, respectively.

The constitutive equations for the heterogeneous transversely isotropic piezoelectric materials, with respect to an axisymmetric problem in the cylindrical coordinate system, have the forms [26,27,28]:

$$\begin{array}{l}{\sigma }_{rr}=C_{11}{\epsilon }_{rr}+C_{12}{\epsilon }_{\theta \theta }+C_{13}{\epsilon }_{zz}-e_{31}{E}_z,\\ {\sigma }_{\theta \theta }=C_{12}{\epsilon }_{rr}+C_{11}{\epsilon }_{\theta \theta }+C_{13}{\epsilon }_{zz}-e_{31}{E}_z,\\ {\sigma }_{zz}=C_{13}{\epsilon }_{rr}+C_{13}{\epsilon }_{\theta \theta }+C_{33}{\epsilon }_{zz}-e_{33}{E}_z,\\ {\sigma }_{rz}=2C_{44}{\epsilon }_{zr}-e_{15}{E}_r,\\ D_r=2e_{15}{\epsilon }_{zr}+{\xi }_{11}{E}_r,\\ D_z=e_{31}{\epsilon }_{rr}+e_{31}{\epsilon }_{\theta \theta }+e_{33}{\epsilon }_{zz}+{\xi }_{33}{E}_z,\end{array}$$

(4)

in which ${\epsilon }_{rr}$, ${\epsilon }_{\theta \theta }$, ${\epsilon }_{zz}$, and ${\epsilon }_{zr}$ are strain components; $E_r$ and $E_z$ are the electric field intensities. The material constants C_ij, e_ij, and ξ_ij in Equation (4) follow the exponential law distribution, namely, $C_{ij}=C_{ij}\left(z\right)=C_{ij}^0{\mathrm{e}}^{\eta z}, e_{ij}=e_{ij}\left(z\right)=e_{ij}^0{\mathrm{e}}^{\eta z}, {\xi }_{ij}={\xi }_{ij}\left(z\right)= {\xi }_{ij}^0{\mathrm{e}}^{\eta z}.$

The general geometric equations are defined as

$$\begin{array}{l}{\epsilon }_{rr}={\displaystyle \frac{\partial u_r}{\partial r}}, {\epsilon }_{\theta \theta }={\displaystyle \frac{u_r}{r}}, {\epsilon }_{zz}={\displaystyle \frac{\partial u_z}{\partial z}}, {\epsilon }_{rz}=0.5\left({\displaystyle \frac{\partial u_z}{\partial r}}+{\displaystyle \frac{\partial u_r}{\partial z}}\right), \\ E_r=-{\displaystyle \frac{\partial \varphi }{\partial r}}, E_z=-{\displaystyle \frac{\partial \varphi }{\partial z}},\end{array}$$

(5)

where $\varphi$ is the electric potential.

For the convenience of deriving formulas, the following non-dimensional parameters are introduced as [29]:

$$\begin{array}{l}\overline{r}=r/a, \overline{z}=z/h, {\overline{h}}_j=h_j/h, s=h/a, {\overline{u}}_r=u_r/h, {\overline{u}}_z=u_z/h, \\ {\overline{\sigma }}_{rr}={\sigma }_{rr}/C_{11}^{(1)}, {\overline{\sigma }}_{\theta \theta }={\sigma }_{\theta \theta }/C_{11}^{(1)}, {\overline{\sigma }}_{zz}={\sigma }_{zz}/C_{11}^{(1)}, {\overline{\sigma }}_{rz}={\sigma }_{rz}/C_{11}^{(1)}, \\ {\overline{C}}_{ij}=C_{ij}/C_{11}^{(1)}, {\overline{\xi }}_{ij}={\xi }_{ij}/{\xi }_{33}^{(1)}, {\overline{e}}_{ij}=e_{ij}/\sqrt{C_{11}^{(1)}{\xi }_{33}^{(1)}}, \\ {\overline{D}}_i=D_i/\sqrt{C_{11}^{(1)}{\xi }_{33}^{(1)}}, \overline{\varphi }=\varphi \sqrt{{\xi }_{33}^{(1)}/C_{11}^{(1)}}/h, \overline{\rho }=\rho /{\rho }^{(1)}, \Omega =\omega h\sqrt{{\rho }^{(1)}/C_{11}^{(1)}},\end{array}$$

(6)

where s is the thickness-to-span ratio of the FGPM circular plate; ω refers to the natural frequency; $\Omega$ denotes the non-dimensional natural frequency; $C_{11}^{(1)}$, ${\xi }_{33}^{(1)}$, and ${\rho }^{(1)}$ are the elastic stiffness, dielectric coefficient, and mass density at the top surface of the FGPM circular plate, respectively. Therefore, the governing equations in Equations (2)–(4) can be rewritten as

$$\begin{array}{l}{\displaystyle \frac{\partial {\overline{\sigma }}_{rr}}{\partial \overline{r}}}+{\displaystyle \frac{1}{s}}{\displaystyle \frac{\partial {\overline{\sigma }}_{rz}}{\partial \overline{z}}}+{\displaystyle \frac{{\overline{\sigma }}_{rr}-{\overline{\sigma }}_{\theta \theta }}{\overline{r}}}=-{\displaystyle \frac{1}{s}}\overline{\rho }{\overline{u}}_r{\Omega }^2,\\ {\displaystyle \frac{\partial {\overline{\sigma }}_{rz}}{\partial \overline{r}}}+{\displaystyle \frac{1}{s}}{\displaystyle \frac{\partial {\overline{\sigma }}_{zz}}{\partial \overline{z}}}+{\displaystyle \frac{{\overline{\sigma }}_{rz}}{\overline{r}}}=-{\displaystyle \frac{1}{s}}\overline{\rho }{\overline{u}}_z{\Omega }^2,\\ {\displaystyle \frac{\partial {\overline{D}}_r}{\partial \overline{r}}}+{\displaystyle \frac{1}{s}}{\displaystyle \frac{\partial {\overline{D}}_z}{\partial \overline{z}}}+{\displaystyle \frac{{\overline{D}}_r}{\overline{r}}}=0,\end{array}$$

(7)

and

$$\begin{array}{l}{\overline{\sigma }}_{rr}=s{\overline{C}}_{11}{\displaystyle \frac{\partial {\overline{u}}_r}{\partial \overline{r}}}+s{\overline{C}}_{12}{\displaystyle \frac{{\overline{u}}_r}{\overline{r}}}+{\overline{C}}_{13}{\displaystyle \frac{\partial {\overline{u}}_z}{\partial \overline{z}}}+{\overline{e}}_{31}{\displaystyle \frac{\partial \overline{\varphi }}{\partial \overline{z}}},\\ {\overline{\sigma }}_{\theta \theta }=s{\overline{C}}_{12}{\displaystyle \frac{\partial {\overline{u}}_r}{\partial \overline{r}}}+s{\overline{C}}_{11}{\displaystyle \frac{{\overline{u}}_r}{\overline{r}}}+{\overline{C}}_{13}{\displaystyle \frac{\partial {\overline{u}}_z}{\partial \overline{z}}}+{\overline{e}}_{31}{\displaystyle \frac{\partial \overline{\varphi }}{\partial \overline{z}}},\\ {\overline{\sigma }}_{zz}=s{\overline{C}}_{13}{\displaystyle \frac{\partial {\overline{u}}_r}{\partial \overline{r}}}+s{\overline{C}}_{13}{\displaystyle \frac{{\overline{u}}_r}{\overline{r}}}+{\overline{C}}_{33}{\displaystyle \frac{\partial {\overline{u}}_z}{\partial \overline{z}}}+{\overline{e}}_{33}{\displaystyle \frac{\partial \overline{\varphi }}{\partial \overline{z}}},\\ {\overline{\sigma }}_{rz}={\overline{C}}_{44}\left(s{\displaystyle \frac{\partial {\overline{u}}_z}{\partial \overline{r}}}+{\displaystyle \frac{\partial {\overline{u}}_r}{\partial \overline{z}}}\right)+s{\overline{e}}_{15}{\displaystyle \frac{\partial \overline{\varphi }}{\partial \overline{r}}},\\ {\overline{D}}_r={\overline{e}}_{15}\left(s{\displaystyle \frac{\partial {\overline{u}}_z}{\partial \overline{r}}}+{\displaystyle \frac{\partial {\overline{u}}_r}{\partial \overline{z}}}\right)-s{\overline{\xi }}_{11}{\displaystyle \frac{\partial \overline{\varphi }}{\partial \overline{r}}},\\ {\overline{D}}_z=s{\overline{e}}_{31}{\displaystyle \frac{\partial {\overline{u}}_r}{\partial \overline{r}}}+s{\overline{e}}_{31}{\displaystyle \frac{{\overline{u}}_r}{\overline{r}}}+{\overline{e}}_{33}{\displaystyle \frac{\partial {\overline{u}}_z}{\partial \overline{z}}}-{\overline{\xi }}_{33}{\displaystyle \frac{\partial \overline{\varphi }}{\partial \overline{z}}}.\end{array}$$

(8)

2.3. State Space Equations for an FGPM Circular Plate

When an axisymmetric FGPM circular plate vibrates at its natural frequency ω, the following physical quantities can be assumed as

$$\begin{array}{l}{\overline{u}}_r={\overline{u}}_r(\overline{r}, \overline{z}){\mathrm{e}}^{i\omega t}, {\overline{\sigma }}_{zz}={\overline{\sigma }}_{zz}(\overline{r}, \overline{z}){\mathrm{e}}^{i\omega t}, {\overline{D}}_z={\overline{D}}_z(\overline{r}, \overline{z}){\mathrm{e}}^{i\omega t},\\ {\overline{\sigma }}_{rz}={\overline{\sigma }}_{rz}(\overline{r}, \overline{z}){\mathrm{e}}^{i\omega t}, {\overline{u}}_z={\overline{u}}_z(\overline{r}, \overline{z}){\mathrm{e}}^{i\omega t}, \overline{\varphi }=\overline{\varphi }(\overline{r}, \overline{z}){\mathrm{e}}^{i\omega t}.\end{array}$$

(9)

By choosing ${\overline{u}}_r$, ${\overline{\sigma }}_{\mathit{zz}}, {\overline{D}}_z$, ${\overline{\sigma }}_{\mathit{rz}}$, ${\overline{u}}_z$, and $\overline{\varphi }$ as the state variables, the state space equations for layer j of the FGPM circular plate can be given as:

$${\displaystyle \frac{\partial {\overline{R}}_j(\overline{r}, \overline{z})}{\partial \overline{z}}}=\left[\begin{array}{cc}0& A_j\\ B_j& 0\end{array}\right]{\overline{R}}_j(\overline{r}, \overline{z}),$$

(10)

in which

$${\overline{R}}_j={\left[\begin{array}{cccccc}{\overline{u}}_r& {\overline{\sigma }}_{zz}& {\overline{D}}_z& {\overline{\sigma }}_{rz}& {\overline{u}}_z& \overline{\varphi }\end{array}\right]}^T,$$

(11)

$$A_j=\left[\begin{array}{ccc}{\alpha }_1& -s{\displaystyle \frac{\partial }{\partial \overline{r}}}& -{\alpha }_2{\displaystyle \frac{\partial }{\partial \overline{r}}}\\ -s({\displaystyle \frac{1}{\overline{r}}}+{\displaystyle \frac{\partial }{\partial \overline{r}}})& -\overline{\rho }{\Omega }^2& 0\\ -{\alpha }_2({\displaystyle \frac{1}{\overline{r}}}+{\displaystyle \frac{\partial }{\partial \overline{r}}})& 0& {\alpha }_3({\displaystyle \frac{1}{\overline{r}}}{\displaystyle \frac{\partial }{\partial \overline{r}}}+{\displaystyle \frac{{\partial }^2}{\partial {\overline{r}}^2}})\end{array}\right],$$

(12)

$$B_j=\left[\begin{array}{ccc}-\overline{\rho }{\Omega }^2+{\beta }_1({\displaystyle \frac{{\partial }^2}{\partial {\overline{r}}^2}}+{\displaystyle \frac{1}{\overline{r}}}{\displaystyle \frac{\partial }{\partial \overline{r}}}-{\displaystyle \frac{1}{{\overline{r}}^2}})& {\beta }_2{\displaystyle \frac{\partial }{\partial \overline{r}}}& {\beta }_3{\displaystyle \frac{\partial }{\partial \overline{r}}}\\ {\beta }_2({\displaystyle \frac{\partial }{\partial \overline{r}}}+{\displaystyle \frac{1}{\overline{r}}})& {\beta }_4& {\beta }_5\\ {\beta }_3({\displaystyle \frac{\partial }{\partial \overline{r}}}+{\displaystyle \frac{1}{\overline{r}}})& {\beta }_5& {\beta }_6\end{array}\right].$$

(13)

Meanwhile, the other derived variables can be expressed as

$$\begin{array}{l}{\overline{\sigma }}_{rr}(\overline{r},\overline{z})=-{\displaystyle \frac{1}{s}}\left[{\beta }_7{\displaystyle \frac{{\overline{u}}_r(\overline{r},\overline{z})}{\overline{r}}}+{\beta }_1{\displaystyle \frac{\partial {\overline{u}}_r(\overline{r},\overline{z})}{\partial \overline{r}}}+{\beta }_2{\overline{\sigma }}_{zz}(\overline{r},\overline{z})+{\beta }_3{\overline{D}}_z(\overline{r},\overline{z})\right],\\ {\overline{\sigma }}_{\theta \theta }(\overline{r},\overline{z})=-{\displaystyle \frac{1}{s}}\left[{\beta }_1{\displaystyle \frac{{\overline{u}}_r(\overline{r},\overline{z})}{\overline{r}}}+{\beta }_7{\displaystyle \frac{\partial {\overline{u}}_r(\overline{r},\overline{z})}{\partial \overline{r}}}+{\beta }_2{\overline{\sigma }}_{zz}(\overline{r},\overline{z})+{\beta }_3{\overline{D}}_z(\overline{r},\overline{z})\right],\\ {\overline{D}}_r(\overline{r},\overline{z})={\displaystyle \frac{1}{s}}\left[{\alpha }_2{\overline{\sigma }}_{rz}(\overline{r},\overline{z})-{\alpha }_3{\displaystyle \frac{\partial \overline{\varphi }(\overline{r},\overline{z})}{\partial \overline{r}}}\right],\end{array}$$

(14)

where the coefficients ${\alpha }_p$ (p = 1, 2, 3) and ${\beta }_q$ (q = 0, 2, …, 7) are

$$\begin{array}{l}{\alpha }_1=1/{\overline{C}}_{44}, {\alpha }_2=s{\overline{e}}_{15}/{\overline{C}}_{44}, {\alpha }_3=s^2\left({\overline{e}}_{15}^2+{\overline{C}}_{44}{\overline{\xi }}_{11}\right)/{\overline{C}}_{44},\\ {\beta }_0={\overline{e}}_{33}^2+{\overline{C}}_{33}{\overline{\xi }}_{33}, {\beta }_1=\left(-{\overline{C}}_{33}{\overline{e}}_{31}^2+2{\overline{C}}_{13}{\overline{e}}_{31}{\overline{e}}_{33}-{\overline{C}}_{11}{\overline{e}}_{33}^2+{\overline{C}}_{13}^2{\overline{\xi }}_{33}-{\overline{C}}_{11}{\overline{C}}_{33}{\overline{\xi }}_{33}\right)s^2/{\beta }_0,\\ {\beta }_2=\left(-{\overline{e}}_{31}{\overline{e}}_{33}-{\overline{C}}_{13}{\overline{\xi }}_{33}\right)s/{\beta }_0, {\beta }_3=\left({\overline{C}}_{33}{\overline{e}}_{31}-{\overline{C}}_{13}{\overline{e}}_{33}\right)s/{\beta }_0,\\ {\beta }_4={\overline{\xi }}_{33}/{\beta }_0, {\beta }_5={\overline{e}}_{33}/{\beta }_0, {\beta }_6=-{\overline{C}}_{33}/{\beta }_0,\\ {\beta }_7=\left(-{\overline{C}}_{33}{\overline{e}}_{31}^2+2{\overline{C}}_{13}{\overline{e}}_{31}{\overline{e}}_{33}-{\overline{C}}_{12}{\overline{e}}_{33}^2+{\overline{C}}_{13}^2{\overline{\xi }}_{33}-{\overline{C}}_{12}{\overline{C}}_{33}{\overline{\xi }}_{33}\right)s^2/{\beta }_0.\end{array}$$

(15)

2.4. Finite Hankel Transform for an FGPM Circular Plate

The finite Hankel transform extended from the Hankel transform is commonly used to deal with the mechanical problems of structures with finite diameters [30]. In the present work, the finite Hankel transform for the axisymmetric free vibration analysis of an FGPM circular plate is given as [30,31]

$$J_{\mu }[f(\overline{r},\overline{z})]={\displaystyle {\int }_0^1\overline{r}f(\overline{r},\overline{z})J_{\mu }(k\overline{r})\mathrm{d}\overline{r}},$$

(16)

where $J_{\mu }(k\overline{r})$ is the first kind Bessel function of order μ. Therefore, we can express the state space vector in the Hankel transform domain as

$$R_j(k,\overline{z})={\left[\begin{array}{c}{U}_r(k,\overline{z})\\ S(k,\overline{z})\\ D(k,\overline{z})\\ T(k,\overline{z})\\ U_z(k,\overline{z})\\ F(k,\overline{z})\end{array}\right]}_j={\left[\begin{array}{c}{J}_1\left[{\overline{u}}_r(\overline{r},\overline{z})\right]\\ J_0\left[{\overline{\sigma }}_{zz}(\overline{r},\overline{z})\right]\\ J_0\left[{\overline{D}}_z(\overline{r},\overline{z})\right]\\ J_1\left[{\overline{\sigma }}_{rz}(\overline{r},\overline{z})\right]\\ J_0\left[{\overline{u}}_z(\overline{r},\overline{z})\right]\\ J_0\left[\overline{\varphi }(\overline{r},\overline{z})\right]\end{array}\right]}_j.$$

(17)

By using the finite Hankel transform to Equation (10), the state space equations can be rewritten as

$${\displaystyle \frac{\partial R_j(k,\overline{z})}{\partial \overline{z}}}=K_j(k)R_j(k,\overline{z})+Q_j(k,\overline{z}),$$

(18)

where the sub-matrices have the form of

$$K_j(k)=\left[\begin{array}{cccccc}0& 0& 0& {\alpha }_1& sk& {\alpha }_{2}k\\ 0& 0& 0& -sk& -\overline{\rho }{\Omega }^2& 0\\ 0& 0& 0& -{\alpha }_{2}k& 0& -{\alpha }_3{k}^2\\ -\overline{\rho }{\Omega }^2-{\beta }_1{k}^2& -{\beta }_{2}k& -{\beta }_{3}k& 0& 0& 0\\ {\beta }_{2}k& {\beta }_4& {\beta }_5& 0& 0& 0\\ {\beta }_{3}k& {\beta }_5& {\beta }_6& 0& 0& 0\end{array}\right],$$

(19)

and

$$Q_j(k,\overline{z})=\left[\begin{array}{c}-s{\overline{u}}_z(1,\overline{z})J_1(k)-{\alpha }_2\overline{\varphi }(1,\overline{z})J_1(k)\\ -s{\overline{\sigma }}_{rz}(1,\overline{z})J_0(k)\\ -{\alpha }_2{\overline{\sigma }}_{rz}(1,\overline{z})J_0(k)+{\alpha }_3{\displaystyle \frac{\partial \overline{\varphi }(1,\overline{z})}{\partial \overline{r}}}{J}_0(k)+{\alpha }_{3}k\overline{\varphi }(1,\overline{z})J_1(k)\\ {\beta }_1[{\displaystyle \frac{\partial {\overline{u}}_r(1,\overline{z})}{\partial \overline{r}}}{J}_1(k)-k{\overline{u}}_r(1,\overline{z})J_0(k)+{\overline{u}}_r(1,\overline{z})J_1(k)]+{\beta }_2{\overline{\sigma }}_{zz}(1,\overline{z})J_1(k)+{\beta }_3{\overline{D}}_z(1,\overline{z})J_1(k)\\ {\beta }_2{\overline{u}}_r(1,\overline{z})J_0(k)\\ {\beta }_3{\overline{u}}_r(1,\overline{z})J_0(k)\end{array}\right].$$

(20)

Let $\overline{r}=1$ in Equation (14) yields:

$$\begin{array}{l}s{\overline{\sigma }}_{rr}(1,\overline{z})=-{\beta }_7{\overline{u}}_r(1,\overline{z})-{\beta }_1{\displaystyle \frac{\partial {\overline{u}}_r(1,\overline{z})}{\partial \overline{r}}}-{\beta }_2{\overline{\sigma }}_{zz}(1,\overline{z})-{\beta }_3{\overline{D}}_z(1,\overline{z}),\\ s{\overline{D}}_r(1,\overline{z})={\alpha }_2{\overline{\sigma }}_{rz}(1,\overline{z})-{\alpha }_3{\displaystyle \frac{\partial \overline{\varphi }(1,\overline{z})}{\partial \overline{r}}}.\end{array}$$

(21)

Substitution of Equation (21) into Equation (20) leads to

$$Q_j(k,\overline{z})=\left[\begin{array}{c}-s{\overline{u}}_z(1,\overline{z})J_1(k)-{\alpha }_2\overline{\varphi }(1,\overline{z})J_1(k)\\ -s{\overline{\sigma }}_{rz}(1,\overline{z})J_0(k)\\ -s{\overline{D}}_r(1,\overline{z})J_0(k)+{\alpha }_{3}k\overline{\varphi }(1,\overline{z})J_1(k)\\ ({\overline{C}}_{12}-{\overline{C}}_{11})s^2{\overline{u}}_r(1,\overline{z})J_1(k)-s{\overline{\sigma }}_{rr}(1,\overline{z})J_1(k)-{\beta }_{1}k{\overline{u}}_r(1,\overline{z})J_0(k)\\ {\beta }_2{\overline{u}}_r(1,\overline{z})J_0(k)\\ {\beta }_3{\overline{u}}_r(1,\overline{z})J_0(k)\end{array}\right].$$

(22)

To determine the parameter k, it is essential to consider the boundary conditions at the circumferential edge of the FGPM circular plate. It is easily found that the derived variables in Equation (14) have been utilized to achieve the matrix $Q_j$ in Equation (22). $Q_j=\left[0\right]$ can be satisfied for the following two boundary conditions; the first one is named an elastic simple support boundary condition and has the form of [31]:

$${\overline{u}}_z(1,\overline{z})=0, \overline{\varphi }(1,\overline{z})=0, [({\overline{C}}_{12}-{\overline{C}}_{11})s{\overline{u}}_r(1,\overline{z})-{\overline{\sigma }}_{rr}(1,\overline{z})]=0 \mathrm{and} J_0(k)=0.$$

(23)

The second one is called a rigid slipping support boundary condition and takes the form of [31]:

$${\overline{u}}_r(1,\overline{z})=0, {\overline{\sigma }}_{rz}(1,\overline{z})=0, {\overline{D}}_r(1,\overline{z})=0 \mathrm{and} J_1(k)=0.$$

(24)

By virtue of those two boundary conditions, Equation (18) can be reduced into

$${\displaystyle \frac{\partial R_j(k,\overline{z})}{\partial \overline{z}}}=K_j(k)R_j(k,\overline{z}).$$

(25)

2.5. Axisymmetric Free Vibration for an FGPM Circular Plate

It is apparent that Equation (25) is an ordinary differential equation with variable coefficients due to the non-homogeneous material properties. Therefore, it is difficult to obtain the solution of Equation (25). But luckily, utilizing the exponential law distribution of material properties shown in Equation (1), Equation (25) can be converted into an ordinary differential equation with constant coefficients, which is

$${\displaystyle \frac{\partial {\widehat{R}}_j(k,\overline{z})}{\partial \overline{z}}}={\widehat{K}}_j(k){\widehat{R}}_j(k,\overline{z}),$$

(26)

where

$${\widehat{R}}_j={\left[\begin{array}{cccccc}{U}_r(k,\overline{z})& \widehat{S}(k,\overline{z})& \widehat{D}(k,\overline{z})& \widehat{T}(k,\overline{z})& U_z(k,\overline{z})& F(k,\overline{z})\end{array}\right]}^T,$$

(27)

and $S(k,\overline{z})=\widehat{S}(k,\overline{z}){\mathrm{e}}^{\eta \overline{z}}, D(k,\overline{z})=\widehat{D}(k,\overline{z}){\mathrm{e}}^{\eta \overline{z}}, T(k,\overline{z})=\widehat{T}(k,\overline{z}){\mathrm{e}}^{\eta \overline{z}}$. The matrix ${\widehat{K}}_j(k)$ is

$${\widehat{K}}_j(k)=\left[\begin{array}{cccccc}0& 0& 0& {\alpha }_1^0& sk& {\alpha }_2^{0}k\\ 0& -\eta & 0& -sk& -{\overline{\rho }}^0{\Omega }^2& 0\\ 0& 0& -\eta & -{\alpha }_2^{0}k& 0& -{\alpha }_3^0{k}^2\\ -{\overline{\rho }}^0{\Omega }^2-{\beta }_1^0{k}^2& -{\beta }_2^{0}k& -{\beta }_3^{0}k& -\eta & 0& 0\\ {\beta }_2^{0}k& {\beta }_4^0& {\beta }_5^0& 0& 0& 0\\ {\beta }_3^{0}k& {\beta }_5^0& {\beta }_6^0& 0& 0& 0\end{array}\right],$$

(28)

and the parameters in Equation (28) are expressed as

$$\begin{array}{l}{\alpha }_1^0=1/{\overline{C}}_{44}^0, {\alpha }_2^0=s{\overline{e}}_{15}^0/{\overline{C}}_{44}^0, {\alpha }_3^0=s^2\left[{\left({\overline{e}}_{15}^0\right)}^2+{\overline{C}}_{44}^0{\overline{\xi }}_{11}^0\right]/{\overline{C}}_{44}^0,\\ {\beta }_0^0={\left({\overline{e}}_{33}^0\right)}^2+{\overline{C}}_{33}^0{\overline{\xi }}_{33}^0, {\beta }_1^0=\left[-{\overline{C}}_{33}^0{\left({\overline{e}}_{31}^0\right)}^2+2{\overline{C}}_{13}^0{\overline{e}}_{31}^0{\overline{e}}_{33}^0-{\overline{C}}_{11}^0{\left({\overline{e}}_{33}^0\right)}^2+{\left({\overline{C}}_{13}^0\right)}^2{\overline{\xi }}_{33}^0-{\overline{C}}_{11}^0{\overline{C}}_{33}^0{\overline{\xi }}_{33}^0\right]s^2/{\beta }_0^0,\\ {\beta }_2^0=\left(-{\overline{e}}_{31}^0{\overline{e}}_{33}^0-{\overline{C}}_{13}^0{\overline{\xi }}_{33}^0\right)s/{\beta }_0, {\beta }_3^0=\left({\overline{C}}_{33}^0{\overline{e}}_{31}^0-{\overline{C}}_{13}^0{\overline{e}}_{33}^0\right)s/{\beta }_0^0, \\ {\beta }_4^0={\overline{\xi }}_{33}^0/{\beta }_0^0, {\beta }_5^0={\overline{e}}_{33}^0/{\beta }_0^0, {\beta }_6^0=-{\overline{C}}_{33}^0/{\beta }_0^0.\end{array}$$

(29)

Solving Equation (26) gives the following solution:

$${\widehat{R}}_j(k, \overline{z})={\widehat{T}}_j(k, \overline{z}){\widehat{R}}_j(k, 0),$$

(30)

where

$${\widehat{T}}_j(k, \overline{z})=\exp [\widehat{K}(k)\overline{z}].$$

(31)

Based on the solution obtained in Equation (30), we can rewrite the state space vector in Equation (17) as

$$R_j(k, \overline{z})=T_j(k, \overline{z})R_j(k, 0),$$

(32)

where $T_j\left(k, \overline{z}\right)=M{\widehat{T}}_j\left(k, \overline{z}\right)$, and the matrix $M$ is

$$M=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& {\mathrm{e}}^{\eta \overline{z}}& 0& 0& 0& 0\\ 0& 0& {\mathrm{e}}^{\eta \overline{z}}& 0& 0& 0\\ 0& 0& 0& {\mathrm{e}}^{\eta \overline{z}}& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right].$$

(33)

It is assumed that the perfect interface is adopted between layer j and layer j + 1, the state space vector $R_N(k, 1)$ can be derived as

$$R_N(k, 1)=P(k)R_1(k, 0),$$

(34)

in which

$$P(k)={\displaystyle \prod _{j=1}^N{T}_j(k, {\overline{h}}_j)}.$$

(35)

The boundary conditions at the circumferential edge of the FGPM circular plate are presented in Equations (23) and (24). As for the free vibration analysis problem, the boundary condition at the top and bottom surfaces of the FGPM circular plate is

$${\overline{\sigma }}_{zz}={\overline{\sigma }}_{rz}=0, \mathrm{and} {\overline{D}}_z=0, \mathrm{at} \overline{z}=0, 1.$$

(36)

With Equations (17), (27), and (36), Equation (32) can be transformed into

$$\left[\begin{array}{c}{U}_r(k,1)\\ 0\\ 0\\ 0\\ U_z(k,1)\\ F(k,1)\end{array}\right]=\left[\begin{array}{cccccc}{P}_{11}& P_{12}& P_{13}& P_{14}& P_{15}& P_{16}\\ P_{21}& P_{22}& P_{23}& P_{24}& P_{25}& P_{26}\\ P_{31}& P_{32}& P_{33}& P_{34}& P_{35}& P_{36}\\ P_{41}& P_{42}& P_{43}& P_{44}& P_{45}& P_{46}\\ P_{51}& P_{52}& P_{53}& P_{54}& P_{55}& P_{56}\\ P_{61}& P_{62}& P_{63}& P_{64}& P_{65}& P_{66}\end{array}\right]\left[\begin{array}{c}{U}_r(k,0)\\ 0\\ 0\\ 0\\ U_z(k,0)\\ F(k,0)\end{array}\right],$$

(37)

and then we have

$$\left[\begin{array}{ccc}{P}_{21}& P_{25}& P_{26}\\ P_{31}& P_{35}& P_{36}\\ P_{41}& P_{45}& P_{46}\end{array}\right]\left[\begin{array}{c}{U}_r(k,0)\\ U_z(k,0)\\ F(k,0)\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right].$$

(38)

It is obvious that Equation (38) is a homogeneous equation, and the natural frequencies for the axisymmetric free vibration of the FGPM circular plate can be derived by setting the determinant of the coefficients matrix in Equation (38) to zero. Based on Equations (23) and (24), we can derive an infinite number of k_i > 0, so a range of natural frequency values can be derived from Equation (38) corresponding to k_i, which cannot be achieved by classical theories of the plate.

3. Numerical Examples

In this section, the axisymmetric free vibration behavior of the FGPM circular plate is investigated. In the following section, we will first validate the solutions presented. Secondly, the influences of the exponential factor, thickness-to-span ratio, and boundary condition on the natural frequency of the FGPM circular plate are presented.

3.1. Verification of Results

Consider a circular plate with top and bottom surfaces being traction-free. The circular plate made of homogeneous transversely isotropic piezoelectric materials was studied by Ding et al. [23], and the results were obtained using the numerical method and finite element method. The circular plate made of functionally graded materials obeying the exponential law was investigated by Wang et al. [25] and Roshanbakhsh et al. [32], who presented the results based on the direct displacement method and displacement potential functions, respectively. The material properties utilized in the previous paper are tabulated in

Table 1. Material properties at the top surface of the circular plate.

	${\mathit{C}}_{\mathit{i}\mathit{j}}^0(\mathbf{G}\mathbf{P}\mathbf{a})$	${\mathit{e}}_{\mathit{i}\mathit{j}}^0/(\mathbf{C}/{\mathbf{m}}^2)$	${\mathit{\xi }}_{\mathit{i}\mathit{j}}^0({\mathbf{C}}^2\times {\mathbf{N}}^{-1}{\mathbf{m}}^{-2})$	${\mathit{\rho }}^0(\mathbf{K}\mathbf{g}/{\mathbf{m}}^3)$
Ref. [23]	$\begin{array}{l}{C}_{11}^0=139 C_{12}^0=77.8 C_{13}^0=74.3\\ C_{33}^0=115 C_{44}^0=25.6\end{array}$	$\begin{array}{l}{e}_{31}^0=-5.2 e_{33}^0=15.1\\ e_{15}^0=12.7\end{array}$	$\begin{array}{l}{\xi }_{11}^0=6.46\times {10}^{-9} \\ {\xi }_{33}^0=5.62\times {10}^{-9}\end{array}$	7500
Ref. [25]	$\begin{array}{l}{C}_{11}^0=139 C_{12}^0=77.9 C_{13}^0=74.3\\ C_{33}^0=115 C_{44}^0=25.6\end{array}$	—	—	7500

.

Table 2. Comparison of the first non-dimensional natural frequencies Ω of the transversely isotropic piezoelectric circular plate with two boundary conditions (η = 0, k_i = k₁).

Boundary Conditions	Source of Results	s = 0.1	s = 0.2	s = 0.3	s = 0.4	s = 0.5
Elastic simple support	Ding et al. [23]	0.0154	0.0600	0.1297	0.2195	0.3246
	Present results	0.0154	0.0600	0.1297	0.2195	0.3246
Rigid slipping support	Ding et al. [23]	0.0385	0.1451	0.3003	0.4865	0.6913
	Present results	0.0386	0.1451	0.3004	0.4865	0.6913

presents the comparison results of the first dimensionless natural frequencies of the homogeneous circular plate with different thickness-to-span ratios and boundary conditions. It is clear that the present model (η = 0) predicts exactly the same natural frequencies as in Ding et al. [23].

Table 3. Comparison of the first non-dimensional natural frequencies Ω of the FGM transversely isotropic circular plate with two boundary conditions (s = 0.5, k_i = k₁).

Boundary Conditions	Source of Results	η = 0	η = 1	η = 2	η = 10	η = 100
Elastic simple support	Wang et al. [25]	0.2700	0.2645	0.2495	—	—
	Roshanbakhsh et al. [32]	0.2691	0.2635	0.2485	0.1066	0.0116
	Present results	0.2700	0.2645	0.2495	0.1099	0.0117
Rigid slipping support	Wang et al. [25]	0.5574	0.5470	0.5193	—	—
	Roshanbakhsh et al. [32]	0.5563	0.5459	0.5182	0.2581	0.0296
	Present results	0.5574	0.5470	0.5193	0.2586	0.0296

shows the influences of the exponential factor and boundary condition on the natural frequencies obtained from the present degenerated model and the previous models [25,32], which also validates the accuracy of the presented solution. Furthermore, the results with η = 100 calculated from the present model also show the numerical stability of the obtained solution.

3.2. Parametric Studies and Discussion

The numerical results of the free vibration analysis of a single-layer FGPM circular plate with various exponential factors, thickness-to-span ratios, and boundary conditions are presented in this subsection. The material properties at the top surface of the FGPM circular plate are the same as those given by Ding et al. [23], which are tabulated in Table 1. The exponential factors are chosen as 0, 1, 2, 5, and 10, and the thickness-to-span ratios are selected as 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. Two different boundary conditions at the circumferential edge of the FGPM circular plate are considered, so we have k₁ = 2.40483 for the elastic simple support circular plate and k₁ = 3.83171 for the rigid slipping support circular plate.

Table 4. First non-dimensional natural frequencies Ω of the FGM circular plate with elastic simple support boundary condition.

s = h/a	η = 0	η = 1	η = 2	η = 5	η = 10
0.1	0.0154	0.0150	0.0140	0.0097	0.0054
0.2	0.0600	0.0585	0.0546	0.0381	0.0212
0.3	0.1297	0.1265	0.1180	0.0829	0.0470
0.4	0.2195	0.2141	0.1996	0.1414	0.0820
0.5	0.3246	0.3164	0.2950	0.2108	0.1251

illustrates the dependence of the first non-dimensional natural frequencies on the exponential factor and thickness-to-span ratio of the FGPM circular plate with elastic simple support boundary conditions.

Table 5. First non-dimensional natural frequencies Ω of the FGM circular plate with rigid slipping support boundary condition.

s = h/a	η = 0	η = 1	η = 2	η = 5	η = 10
0.1	0.0386	0.0376	0.0351	0.0244	0.0135
0.2	0.1451	0.1415	0.1320	0.0929	0.0529
0.3	0.3004	0.2928	0.2730	0.1947	0.1150
0.4	0.4865	0.4741	0.4419	0.3197	0.1961
0.5	0.6913	0.6732	0.6273	0.4604	0.2927

shows the influence of the exponential factor and thickness-to-span ratio on the first non-dimensional natural frequencies of the FGPM circular plate with rigid slipping support boundary conditions. It is easily found from Table 4 that the first non-dimensional natural frequencies Ω increase with increasing the thickness-to-span ratio s, which can be related to increasing flexural rigidity. Another observation from Table 4 is that the increment of exponential factor η leads to the decrease in the first non-dimensional natural frequencies Ω; this is because of the change in material stiffness. The same trends for Ω related to the thickness-to-span ratio and exponential factor are also observed in Table 5. Examining the data presented in Table 4 and Table 5, we found that the first non-dimensional natural frequencies Ω for the rigid slipping support circular plate were larger than those for the elastic simple support circular plate. This occurrence is attributed to the use of a less restrictive boundary condition on the circumferential edge of the FGPM circular plates, which reduces the natural frequencies.

Figure 2 shows the variations in the first non-dimensional natural frequencies versus the exponential factor for different boundary conditions. Figure 2 indicates that the natural frequencies reduce by increasing the exponential factor η. Furthermore, the severity of this decrease is more pronounced for a larger thickness-to-span ratio. The results also indicate that the natural frequencies for the FGPM circular plate are lower than those of the homogeneous circular plate. Comparing the vibration behaviors of different boundary conditions, it can be observed that the more the FGPM circular plate is constrained at the boundary, the larger natural frequencies are induced.

4. Conclusions

FGPMs combine the characteristics of both FGPM and piezoelectric materials, which can be widely used to manufacture smart devices. In order to effectively utilize FGPMs and realize their designability, the dynamic characteristics of smart FGPM structures demand detailed investigation. Therefore, this paper proposes a mathematical model to study the axisymmetric free vibration of FGPM circular plates with exponent-law-dependent material properties. An exact solution is derived for the free vibration of an axisymmetric FGPM circular plate with exponent-law-dependent material properties. The free vibration behavior of an FGPM circular plate is studied using the state space method as well as the Hankel finite transform. The natural frequencies for the FGPM circular plate are obtained by solving the state space equation. The present results are compared with those in the literature and show excellent agreement. The influences of the exponential factor, thickness-to-span ratio, and boundary condition on the natural frequency are studied. According to the numerical results, some vital conclusions can be drawn as follows: (i) the natural frequencies for the homogeneous circular plate are larger than those for the FGPM circular plate, and they decrease with increasing the exponential factor, and the severity of this decreasing is more significant for larger thickness-to-span ratio; (ii) the natural frequencies increase with increasing the thickness-to-span ratio due to the increasing flexural rigidity; (iii) the natural frequencies for the rigid slipping support circular plate are larger than those for the elastic simple supported circular plate since a less constraining boundary is adopted for the elastic simple supported circular plate. The obtained solution can serve as a benchmark result for evaluating other analytical and numerical methods for FGPM circular plates.