Free vibration of functionally graded carbon nanotube-reinforced conical panels integrated with piezoelectric layers subjected to elastically restrained boundary conditions

Abstract

This article presents the free vibration of piezoelectric functionally graded carbon nanotube-reinforced composite conical panels with elastically restrained boundary conditions. The material properties of carbon nanotube-reinforced composites are assumed to be temperature-dependent and are obtained using the extended rule of mixture. First-order shear deformation theory is adopted to obtain the kinematics of the hybrid panels, and the boundary spring technique is used to implement arbitrary boundary conditions. Meanwhile, two types of electrical boundary conditions, closed circuit and open circuit, are considered for the free surfaces of the piezoelectric layers. The complete sets of electro-mechanically coupled governing equations are obtained using the Rayleigh–Ritz procedure with the Chebyshev polynomial basis functions. The resultant eigenvalue problem is solved to obtain natural frequencies and mode shapes of the hybrid panels. Convergence and comparison studies have been conducted to verify the stability and accuracy of the proposed method. Several numerical examples are examined to reveal the influences of the carbon nanotube volume fractions, carbon nanotube distribution types, boundary conditions, geometrical parameters, and temperatures on the natural frequencies of the hybrid panel. Moreover, the mode shapes of the hybrid panels under various boundary conditions are also presented.

Introduction

As a novel class of material, carbon nanotubes (CNTs) have attracted increasing attention in recent years due to their remarkable mechanical and physical properties. For instance, they have a low density of 1.3 (g/cm³) and Young’s modulus with a value greater than 1 (TPa) which are superior to all carbon fibers. ¹ Due to such excellent and unique features, CNTs have been considered as a promising reinforcement for polymer composites, which will be widely applied in the areas where high strength and light weight are needed, such as aeronautics and astronautics. Extensive investigations about carbon nanotube-reinforced composite (CNTRC) materials have been conducted to study their material properties and constitutive modeling. Coleman et al. ² reported a review and comparison of mechanical properties of CNTRCs fabricated by different processing methods. Tensile test experiments of CNT composites have been conducted by Qian et al., ³ and their results demonstrated that reinforcement with only 1 wt% nanotubes resulted in 36%–42% increase in elastic modulus and 25% increase in break stress. Through molecular mechanic simulations and elasticity calculations, Liao and Li ⁴ studied the interfacial characteristics of a CNT-reinforced polystyrene composite system. They found that interfacial shear stress of the CNT–polystyrene system is approximately 160 MPa, which is significantly higher than most carbon fiber-reinforced polymer composite systems. By means of several micromechanics models, Li et al. ⁵ investigated reinforcing mechanisms of single-walled CNT-reinforced epoxy composites and found that the results of both Halpin–Tsai and Mori–Tanaka models are in close agreement with the experimental results. Classical molecular dynamic (MD) simulations of model polymer/CNT composites constructed by embedding a single-walled carbon nanotube (SWCNT) (10, 10) into two different amorphous polymer matrices are carried out by Han and Elliott. ⁶ The results showed that the MD results matched very well with those obtained from the rule of mixture.

Functionally graded materials (FGMs) are a new breed of composite materials with properties that vary spatially according to a certain non-uniform distribution of the reinforcement component. Inspired by the concept of FGMs, the pattern of functionally graded (FG) distribution of reinforcement has been successfully applied for CNTRC materials. FG-CNTRCs were first proposed by Shen. ⁷ Numerical results of Shen reveal that nonlinear bending behavior can be considerably improved using an FG distribution of CNTs in the matrix. Recently, Kwon et al. ⁸ have successfully achieved the linear distribution of CNTs in a matrix, rather than the uniform distribution of CNTs in a matrix using the power metallurgy fabrication process, which proves the feasibility of FG-CNTRC.

Generally, FG-CNTRC can be incorporated in beams, plates, and shells as structural components in practical engineering. Extensive studies have been devoted to analyzing static bending,^7,9 elastic buckling and postbuckling,^10–13 and linear and nonlinear free vibrations^14–17 of FG-CNTRC structures in recent years. Shen and colleagues^7,18–21 carried out a series of research on bending, vibration, and buckling problems of FG-CNTRC plates and cylindrical shells using the two-step perturbation techniques. Using the element-free kp-Ritz, Zhang et al.^22–24 dealt with the nonlinear deformation of triangular and quadrilateral FG-CNTRC plates. Alibeigloo and Liew ²⁵ investigated the bending behavior of simply-supported rectangular FG-CNTRC plates using the three-dimensional theory of elasticity under thermo-mechanical loads. Rafiee et al. ²⁶ proposed an exact closed-form solution for the thermal buckling and postbuckling of piezoelectric FG-CNTRC Timoshenko beam. More detailed description may be found in the review article given by Liew et al. ²⁷

Conical panels or shells are one of the most important structural components which have found a wide range of applications in marine, civil, mechanical, and aerospace engineering. Recently, a few studies have been done on the mechanical characteristics of FG-CNTRC conical shells. Heydarpour et al. ²⁸ examined the vibration of rotating FG-CNTRC truncated conical shells. Based on first-order shear deformation theory (FSDT), Jam and Kiani ²⁹ and Mirzaei and Kiani ³⁰ studied the linear mechanical and thermal buckling of FG-CNTRC conical shells, and hybrid Fourier-generalized differential quadrature (DQ) method was used to solve the partial differential equations. Adopting harmonic DQ (HDQ) method, Mehri et al. ³¹ dealt with the buckling and vibration responses of a FG-CNTRC conical shell based on Novozhilov nonlinear shell theory. Similarly, using the same method, Mehri et al. ³² analyzed the dynamic instability of a pressurized FG-CNTRC-truncated conical shell subjected to yawed supersonic airflow. Ansari et al. ³³ treated the buckling analysis of axially loaded FG-CNTRC conical panels using variational DQ (VDQ) method.

Although a few studies have been conducted on the mechanical behaviors of FG-CNTRC conical panels or shells, to the best of authors’ knowledge, there is no work on the free vibration of CNTRC conical panels. Meanwhile, in the above available literatures, only the classical boundary conditions are considered. However, in practical engineering applications, the panels may not always be classical in nature and there may be elasticity along the supports. Thus, understanding the vibrational characteristics of FG-CNTRC structures under elastically restrained boundary conditions is of great necessity, which will be the focus of this article.

Integrating the engineering structures with piezoelectric layers is practically of much importance. The piezoelectric layers, as distributed sensors and actuators embedded in structures, have wide application in structural vibration control, ³⁴ shape control, ³⁵ and structural health monitoring. ³⁶ Motivated by this, this article deals with the free vibrations of FG-CNTRC conical panels integrated with piezoelectric layers at the top and bottom surfaces and subjected to elastically restrained boundary conditions. Distribution of electric potential through the thickness of the piezoelectric layers is assumed to be linear. The material properties of the host panels are assumed to be temperature-dependent and graded in the thickness direction, and are estimated using an extended rule of mixtures which contains efficiency parameters. FSDT is adopted to obtain the kinematics of the panels, and the boundary spring technique is used to simulate arbitrary boundary conditions, including classical and elastic ones. Meanwhile, two types of electrical boundary conditions, namely, closed circuit and open circuit, are considered for the free surfaces of the piezoelectric layers. The complete sets of electro-mechanically coupled governing equations are obtained using the Rayleigh–Ritz procedure with the Chebyshev polynomial basis functions. Solving the resultant standard eigenvalue problem, the natural frequencies and the mode shapes of the panels are achieved. Several numerical examples will be presented to reveal the influences of the CNT volume fraction, CNT distribution type, geometrical parameters, and temperatures on the nature frequencies of the hybrid panel. Moreover, effects of stiffness coefficients of boundary springs and electrical boundary conditions are also investigated. Finally, the mode shapes of the hybrid conical panels with various boundary conditions are presented.

Theoretical formulation

A CNTRC conical panel integrated with two perfectly bounded piezoelectric layers at the top and bottom surfaces is considered in this study. As shown in Figure 1, the hybrid conical panel has a semi-vertex angle $\alpha$ , subtended angle ${\theta }_0$ , length $L_0$ , and small mean radius $R_0$ . The thicknesses of the host shell and the piezoelectric layers are denoted by h and $h_p$ , respectively. A coordinate system $(x,\theta ,z)$ with the origin located at the small end and on the middle plane of the hybrid conical panel is considered where the axes x and $\theta$ are pointed in the longitudinal and circumferential directions of the shell and z is in the direction of the outward normal to the middle surface. u, v, and w represent displacement components of an arbitrary point of the piezoelectric CNTRC shell along x, $\theta$ , and z axes, respectively. It is noted that the mean radius of the conical panel at any point along its length can be written as $r(x)=R_0+x\sin \alpha$ .

Figure 1.

Geometry of the CNTRC conical panel integrated with two piezoelectric layers at the top and bottom surfaces.

The host panel is made of a polymeric matrix reinforced with SWCNTs. Distribution of CNTs in a matrix may be FG or uniform along the direction of thickness. In this research, three types of FG and the uniformly distributed (UD) case are considered, as shown in Figure 2. FG-V, FG-X, and FG-O are assumed to be FG distributions of CNTs in the polymeric matrix. For FG-V type panel, the top surface of the host panel is CNT-rich. For FG-O type panel, the middle surface of the host panel is CNT-rich, and both top and bottom surfaces are CNT-rich for FG-X type panel. The CNT volume fraction $V_{\mathrm{CNT}}$ for each distribution type can be described as follows ⁷

$$V_{\mathrm{CNT}}(z)=\{\begin{array}{cc}{V}_{\mathrm{CNT}}^{*}\hfill & \mathrm{UD}\hfill \\ \multicolumn{1}{c}{(1+\frac{2z}{h})V_{\mathrm{CNT}}^{*}}& \mathrm{FG}-\mathrm{V}\hfill \\ \multicolumn{1}{c}{2(1-\frac{2\left|z\right|}{h})V_{\mathrm{CNT}}^{*}}& \mathrm{FG}-\mathrm{O}\hfill \\ \multicolumn{1}{c}{2\left(\frac{2\left|z\right|}{h}\right)V_{\mathrm{CNT}}^{*}}& \mathrm{FG}-\mathrm{X}\hfill \end{array}$$

(1)

where $V_{\mathrm{CNT}}^{*}$ is the total CNT volume fraction determined by

$$V_{\mathrm{CNT}}^{*}=\frac{w_{\mathrm{CNT}}}{w_{\mathrm{CNT}}+({\rho }^{\mathrm{CNT}}/{\rho }^m)-({\rho }^{\mathrm{CNT}}/{\rho }^m)w_{\mathrm{CNT}}}$$

(2)

in which $w_{\mathrm{CNT}}$ is the mass fraction of CNTs, and ${\rho }^{\mathrm{CNT}}$ and ${\rho }^m$ are the mass density of CNTs and matrix, respectively. It should be noted that all the distribution types of CNTs have the same value of CNT mass fraction.

Figure 2.

(a–d) Different CNT distributions in the thickness direction of the CNTRC conical panel with piezoelectric layers: (a) UD, (b) FG-V, (c) FG-X, and (d) FG-O.

The material properties of SWCNT and CNTRC were proved to be anisotropic by many researchers.^37,38 According to the rule of mixture and considering the CNT efficiency, the effective mechanical properties of FG-CNTRC host panel can be expressed as follows

$$\begin{array}{c}{E}_{11}^h={\eta }_1{V}_{\mathrm{CNT}}{E}_{11}^{\mathrm{CNT}}+V_m{E}^m,\hfill \\ \frac{{\eta }_2}{E_{22}^h}=\frac{V_{\mathrm{CNT}}}{E_{22}^{\mathrm{CNT}}}+\frac{V_m}{E^m},\frac{{\eta }_3}{G_{12}^h}=\frac{V_{\mathrm{CNT}}}{G_{12}^{\mathrm{CNT}}}+\frac{V_m}{G^m}\hfill \end{array}$$

(3)

where $E_{11}^{\mathrm{CNT}}$ , $E_{22}^{\mathrm{CNT}}$ , and $G_{12}^{\mathrm{CNT}}$ denote Young’s modulus and shear modulus of CNT, respectively, and $E^m$ and $G^m$ represent the corresponding properties of the isotropic matrix. CNT efficiency parameters ${\eta }_i(i=1,2,3)$ are introduced to account load transfer between the CNTs and polymeric phases (e.g. the surface effect, strain gradient effect, and intermolecular coupling effect) and other effects on the effective material properties of CNTRC, which are determined by matching Young’s moduli $E_{11}$ and $E_{22}$ of CNTRC obtained by rule of mixture to the MD results of Han and Elliott. ⁶ $V_m$ is the matrix volume fraction and satisfies the condition

$$V_{\mathrm{CNT}}+V_m=1$$

(4)

Similarly, Poisson’s ratio and the mass density of the host panel are expressed as

$$v_{12}^h=V_{\mathrm{CNT}}{v}_{12}^{\mathrm{CNT}}+V_m{v}^m;{\rho }^h=V_{\mathrm{CNT}}{\rho }^{\mathrm{CNT}}+V_m{\rho }^m$$

(5)

where $v_{12}^{\mathrm{CNT}}$ and $v^m$ are Poisson’s ratio of the CNTs and the matrix, respectively.

In order to consider the effects of through-the-thickness shear deformation and rotary inertia, FSDT is used to estimate the displacement components of the hybrid shell across the thickness. ³⁹ According to this theory, the displacement components at an arbitrary point in the hybrid panel can be expressed as follows

$$\begin{array}{c}u(x,\theta ,z,t)=u_0(x,\theta ,t)+z{\psi }_x(x,\theta ,t),\hfill \\ v(x,\theta ,z,t)=v_0(x,\theta ,t)+z{\psi }_{\theta }(x,\theta ,t)\hfill \\ w(x,\theta ,z,t)=w_0(x,\theta ,t)\hfill \end{array}$$

(6)

where $u_0$ , $v_0$ , and $w_0$ denote the displacement components of the corresponding point on the middle surface of the panel along the x, $\theta$ , and z axes, respectively. ${\psi }_x$ and ${\psi }_{\theta }$ are the transverse normal rotations about the $\theta$ and x axes, respectively. Based on FSDT, the strain components at arbitrary point in the hybrid panel are given by

$$\left\{\begin{array}{c}{\epsilon }_x\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\theta }}\\ \multicolumn{1}{c}{{\gamma }_{x\theta }}\\ \multicolumn{1}{c}{{\gamma }_{\mathrm{xz}}}\\ \multicolumn{1}{c}{{\gamma }_{\theta z}}\end{array}\right\}=\left\{\begin{array}{c}{\epsilon }_x^0\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\theta }^0}\\ \multicolumn{1}{c}{{\gamma }_{x\theta }^0}\\ \multicolumn{1}{c}{{\gamma }_{\mathrm{xz}}^0}\\ \multicolumn{1}{c}{{\gamma }_{\theta z}^0}\end{array}\right\}+z\left\{\begin{array}{c}{\kappa }_x\hfill \\ \multicolumn{1}{c}{{\kappa }_{\theta }}\\ \multicolumn{1}{c}{{\kappa }_{x\theta }}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right\}$$

(7)

where ${\epsilon }_x^0$ , ${\epsilon }_{\theta }^0$ , and ${\gamma }_{x\theta }^0$ are the in-plane strain components associated to the middle surface of the panel. ${\gamma }_{\mathrm{xz}}^0$ and ${\gamma }_{\theta z}^0$ are the through-thickness shear strain components. ${\kappa }_x$ , ${\kappa }_{\theta }$ , and ${\kappa }_{x\theta }$ are the components of change in curvature. These strain components above can be written in terms of the displacement and rotation components as follows

$$\begin{array}{c}{\epsilon }_x^0=\frac{\partial u_0}{\partial x},{\epsilon }_{\theta }^0=\frac{1}{r(x)}(\frac{\partial v_0}{\partial \theta }+u_0\sin \alpha +w_0\cos \alpha ),\hfill \\ {\gamma }_{x\theta }^0=\frac{\partial v_0}{\partial x}+\frac{\partial u_0}{r(x)\partial \theta }-\frac{v_0\sin \alpha }{r(x)}\hfill \\ {\gamma }_{\mathrm{xz}}^0=\frac{\partial w_0}{\partial x}+{\psi }_x,{\gamma }_{\theta z}^0=\frac{\partial w_0}{r(x)\partial \theta }-\frac{v_0\cos \alpha }{r(x)}+{\psi }_{\theta }\hfill \\ {\kappa }_x=\frac{\partial {\psi }_x}{\partial x},{\kappa }_{\theta }=\frac{1}{r(x)}(\frac{\partial {\psi }_{\theta }}{\partial \theta }+{\psi }_x\sin \alpha ),\hfill \\ {\kappa }_{x\theta }=\frac{\partial {\psi }_x}{r(x)\partial \theta }+\frac{\partial {\psi }_{\theta }}{\partial x}--\frac{{\psi }_{\theta }\sin \alpha }{r(x)}\hfill \end{array}$$

(8)

The constitutive relations describing the electrical and mechanical interactions for piezoelectric FG-CNTRC shell are given by

$$\begin{array}{cc}\multicolumn{1}{c}{\left\{\begin{array}{c}{\sigma }_x\hfill \\ \multicolumn{1}{c}{{\sigma }_{\theta }}\\ \multicolumn{1}{c}{{\tau }_{x\theta }}\\ \multicolumn{1}{c}{{\tau }_{\theta z}}\\ \multicolumn{1}{c}{{\tau }_{\mathrm{xz}}}\end{array}\right\}}& =\left[\begin{array}{ccccc}{Q}_{11}\hfill & Q_{12}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{Q_{12}}& Q_{22}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & Q_{66}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & Q_{44}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & Q_{55}\hfill \end{array}\right]\left\{\begin{array}{c}{\epsilon }_x\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\theta }}\\ \multicolumn{1}{c}{{\gamma }_{x\theta }}\\ \multicolumn{1}{c}{{\gamma }_{\theta z}}\\ \multicolumn{1}{c}{{\gamma }_{\mathrm{xz}}}\end{array}\right\}-\left[\begin{array}{ccc}0\hfill & 0\hfill & e_{31}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & e_{32}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& e_{24}\hfill & 0\hfill \\ \multicolumn{1}{c}{e_{15}}& 0\hfill & 0\hfill \end{array}\right]\left\{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{E_z}\end{array}\right\}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\multicolumn{1}{c}{\left\{\begin{array}{c}{D}_x\hfill \\ \multicolumn{1}{c}{D_{\theta }}\\ \multicolumn{1}{c}{D_z}\end{array}\right\}}& =\left[\begin{array}{ccccc}0\hfill & 0\hfill & 0\hfill & 0\hfill & e_{15}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & e_{24}\hfill & 0\hfill \\ \multicolumn{1}{c}{e_{31}}& e_{31}\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]\left\{\begin{array}{c}{\epsilon }_x\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\theta }}\\ \multicolumn{1}{c}{{\gamma }_{x\theta }}\\ \multicolumn{1}{c}{{\gamma }_{\theta z}}\\ \multicolumn{1}{c}{{\gamma }_{\mathrm{xz}}}\end{array}\right\}+\left[\begin{array}{ccc}{k}_{11}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& k_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & k_{33}\hfill \end{array}\right]\left\{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{E_z}\end{array}\right\}\hfill \end{array}$$

(10)

where $E_i$ and $D_i$ are the components of the electric field intensity and the electrical displacements, respectively. $Q_{\mathrm{ij}}(z)$ , $e_{\mathrm{ij}}(z)$ , and $k_{\mathrm{ij}}(z)$ , ( $i,j=1,2,3,4,5,6$ ), respectively, represent the reduced material stiffness coefficients compatible with the plane-stress conditions, piezoelectric stress constants, and dielectric permittivity coefficients for a constant elastic strain. The reduced elastic coefficients $Q_{\mathrm{ij}}^h$ and $Q_{\mathrm{ij}}^p$ of the CNTRC panel and the piezoelectric layers are, respectively, given by

$$\begin{array}{c}{Q}_{11}^h=\frac{E_{11}^h}{1-v_{12}^h{v}_{21}^h},Q_{22}^h=\frac{E_{22}^h}{1-v_{12}^h{v}_{21}^h},\hfill \\ Q_{12}^h=\frac{v_{12}^h{E}_{12}^h}{1-v_{12}^h{v}_{21}^h},Q_{66}^h=G_{12}^h,Q_{44}^h=1.2G_{12}^h,Q_{55}^h=G_{12}^h\hfill \\ Q_{11}^p=Q_{22}^p=\frac{E^p}{1-v^p{v}^p},Q_{12}^p=\frac{v^p{E}^p}{1-v^p{v}^p},\hfill \\ Q_{66}^p=Q_{44}^p=Q_{55}^p=G_{12}^p=\frac{E^p}{2(1+v^p)}\hfill \end{array}$$

(11)

in which $E^p$ and $v^p$ are Young’s modulus and Poisson’s ratio of the piezoelectric layers, respectively. $e_{\mathrm{ij}}^p$ and $k_{\mathrm{ij}}^p$ are, respectively, the piezoelectric stress constants and dielectric permittivity coefficients for a constant elastic strain for the piezoelectric layers. Since the piezoelectric layer is very thin, the electric potential is assumed to be linearly distributed in the thickness direction and the through-thickness component $E_z$ of the electric field is dominant. ⁴⁰ Therefore, for the top and bottom piezoelectric layers, the component $E_z$ of the electric field can written as given below ⁴¹

$$E_z^t=-\frac{{\varphi }_t(x,\theta ,t)}{h_p},E_z^b=-\frac{{\varphi }_b(x,\theta ,t)}{h_p}$$

(12)

where superscripts t and b are associated to the top and bottom piezoelectric layers, respectively. ${\varphi }^t(x,\theta ,t)$ and $-{\varphi }^b(x,\theta ,t)$ denote the electric potential at the free surface of the top and bottom piezoelectric layers, respectively.

Then, force, moment, and transverse shear force resultants per unit length can be expressed as

$$\begin{array}{c}\left[\begin{array}{c}{N}_x\hfill \\ \multicolumn{1}{c}{N_{\theta }}\\ \multicolumn{1}{c}{N_{x\theta }}\end{array}\right]=\underset{-H/2}{\overset{H/2}{\int }}\left[\begin{array}{c}{\sigma }_x\hfill \\ \multicolumn{1}{c}{{\sigma }_{\theta }}\\ \multicolumn{1}{c}{{\tau }_{x\theta }}\end{array}\right]\mathrm{dz},\hfill \\ \left[\begin{array}{c}{M}_x\hfill \\ \multicolumn{1}{c}{M_{\theta }}\\ \multicolumn{1}{c}{M_{x\theta }}\end{array}\right]=\underset{-H/2}{\overset{H/2}{\int }}z\left[\begin{array}{c}{\sigma }_x\hfill \\ \multicolumn{1}{c}{{\sigma }_{\theta }}\\ \multicolumn{1}{c}{{\tau }_{x\theta }}\end{array}\right]\mathrm{dz},\hfill \\ \left[\begin{array}{c}{Q}_{\theta z}\hfill \\ \multicolumn{1}{c}{Q_{\mathrm{xz}}}\end{array}\right]=\underset{-H/2}{\overset{H/2}{\int }}{k}_s\left[\begin{array}{c}{\tau }_{\theta z}\hfill \\ \multicolumn{1}{c}{{\tau }_{\mathrm{xz}}}\end{array}\right]\mathrm{dz}\hfill \end{array}$$

(13)

where $k_s$ is the shear correction factor and in this work is taken as $k_s=5/6$ . $H=h+2h_p$ is the total thickness of the hybrid panel.

In order to obtain the vibrational characteristics of the hybrid shell with the elastically restrained boundary conditions, the boundary spring technique is adopted. ⁴² Along each end of the shell, three groups of translational springs ( ${\varsigma }^u$ , ${\varsigma }^v$ , and ${\varsigma }^w$ ) and two group of rotational springs ( ${\varsigma }^{\eta }$ and ${\varsigma }^{\vartheta }$ ) which are distributed uniformly along the boundary are introduced to separately simulate the boundary forces and moments. Specifically, ${\varsigma }_{x0}^u$ , ${\varsigma }_{x0}^v$ , ${\varsigma }_{x0}^w$ , ${\varsigma }_{x0}^{\eta }$ , and ${\varsigma }_{x0}^{\vartheta }$ denote the sets of springs distributed along the edge $x=0$ . Similarly, by replacing the subscript $x0$ with xL, $\theta 0$ , and $\theta \theta$ , the other three sets of boundary springs at the corresponding ends can be designated, respectively. Therefore, arbitrary boundary conditions of the shell can be generated by assigning the translational and rotational springs at proper stiffnesses. For example, a clamped boundary can be generated by setting the stiffnesses of all the springs equal to infinite (which is represented by a very large number). Inversely, a free boundary is gained by setting the stiffnesses of the entire springs to zero. Thus, the general boundary conditions of an elastically restrained panel can be described as follows:

• At edge $x=0$

  $$\begin{array}{c}{\varsigma }_{x0}^u{u}_0=N_x,{\varsigma }_{x0}^v{v}_0=N_{x\theta },{\varsigma }_{x0}^w{w}_0=Q_{\mathrm{xz}},\hfill \\ {\varsigma }_{x0}^{\eta }{\psi }_x=M_x,{\varsigma }_{x0}^{\vartheta }{\psi }_{\theta }=M_{x\theta }\hfill \end{array}$$

  (14a)

• At edge $x=L$

  $$\begin{array}{c}{\varsigma }_{\mathrm{xL}}^u{u}_0=-N_x,{\varsigma }_{\mathrm{xL}}^v{v}_0=-N_{x\theta },\hfill \\ {\varsigma }_{\mathrm{xL}}^w{w}_0=-Q_{\mathrm{xz}},{\varsigma }_{\mathrm{xL}}^{\eta }{\psi }_x=-M_x,{\varsigma }_{\mathrm{xL}}^{\vartheta }{\psi }_{\theta }=-M_{x\theta }\hfill \end{array}$$

  (14b)

• At edge $\theta =0$

  $$\begin{array}{c}{\varsigma }_{\theta 0}^u{u}_0=N_{x\theta },{\varsigma }_{\theta 0}^v{v}_0=N_{\theta },{\varsigma }_{\theta 0}^w{w}_0=Q_{\theta z},\hfill \\ {\varsigma }_{\theta 0}^{\eta }{\psi }_x=M_{x\theta },{\varsigma }_{\theta 0}^{\vartheta }{\psi }_{\theta }=M_{\theta }\hfill \end{array}$$

  (14c)

• At edge $\theta ={\theta }_0$

  $$\begin{array}{c}{\varsigma }_{\theta \theta }^u{u}_0=-N_{x\theta },{\varsigma }_{\theta \theta }^v{v}_0=-N_{\theta },\hfill \\ {\varsigma }_{\theta \theta }^w{w}_0=-Q_{\theta z},{\varsigma }_{\theta \theta }^{\eta }{\psi }_x=-M_{x\theta },{\varsigma }_{\theta \theta }^{\vartheta }{\psi }_{\theta }=-M_{\theta }\hfill \end{array}$$

  (14d)

The energy-oriented Rayleigh–Ritz method is adopted to obtain the equations of motion of the shells in this study due to its good results and efficiency in modeling and solution procedure. To do this, the first step is to define the energy expressions of the hybrid panels. The sum of strain, electrostatic, and coupling energies of the piezoelectric FG-CNTRC panel can be written as

$$\begin{array}{c}\multicolumn{1}{c}{U=\frac{1}{2}\int \int \left\{\begin{array}{c}{N}_x{\epsilon }_x^0+N_{\theta }{\epsilon }_{\theta }^0+N_{x\theta }{\gamma }_{x\theta }^0+M_x{\kappa }_x+\hfill \\ \multicolumn{1}{c}{M_{\theta }{\kappa }_{\theta }+M_{x\theta }{\kappa }_{x\theta }+Q_{\mathrm{xz}}{\gamma }_{\mathrm{xz}}+Q_{\theta z}{\gamma }_{\theta z}}\end{array}\right\}r(x)\mathrm{dxd}\theta }\\ +\frac{1}{2}\int \int \int D_z{E}_{z}r(x)\mathrm{dxd}\theta \mathrm{dz}\hfill \end{array}$$

(15)

By substituting equations (7), (8), (10), and (12) into equation (15), U can be written in terms of displacement components and electrical potentials.

The strain energy stored in the boundary springs is expressed as

$$\begin{array}{c}\multicolumn{1}{c}{U_{\mathrm{SP}}=\frac{1}{2}\underset{0}{\overset{{\theta }_0}{\int }}\left\{\begin{array}{c}{[{\varsigma }_{x0}^u{u_0}^2+{\varsigma }_{x0}^v{v_0}^2+{\varsigma }_{x0}^w{w_0}^2+{\varsigma }_{x0}^{\eta }{{\psi }_1}^2+{\varsigma }_{x0}^{\vartheta }{{\psi }_2}^2]}_{x=0}{R}_0\hfill \\ \multicolumn{1}{c}{+{[{\varsigma }_{\mathrm{xL}}^u{u_0}^2+{\varsigma }_{\mathrm{xL}}^v{v_0}^2+{\varsigma }_{\mathrm{xL}}^2{w_0}^2+{\varsigma }_{\mathrm{xL}}^{\eta }{{\psi }_1}^2+{\varsigma }_{\mathrm{xL}}^{\vartheta }{{\psi }_2}^2]}_{x=L}{R}_1}\end{array}\right\}d\theta }\\ +\frac{1}{2}\underset{0}{\overset{L}{\int }}\left\{\begin{array}{c}{[{\varsigma }_{\theta 0}^u{u_0}^2+{\varsigma }_{\theta 0}^v{v_0}^2+{\varsigma }_{\theta 0}^w{w_0}^2+{\varsigma }_{\theta 0}^{\eta }{{\psi }_1}^2+{\varsigma }_{\theta 0}^{\vartheta }{{\psi }_2}^2]}_{\theta =0}\hfill \\ \multicolumn{1}{c}{+{[{\varsigma }_{\theta \theta }^u{u_0}^2+{\varsigma }_{\theta \theta }^v{v_0}^2+{\varsigma }_{\theta \theta }^2{w_0}^2+{\varsigma }_{\theta \theta }^{\eta }{{\psi }_1}^2+{\varsigma }_{\theta \theta }^{\vartheta }{{\psi }_2}^2]}_{\theta ={\theta }_0}}\end{array}\right\}r(x)\mathrm{dx}\hfill \end{array}$$

(16)

The kinetic energy of the panel is defined as

$$T=\frac{1}{2}\int \int \{{\rho }_0({{\stackrel{·}{u}}_0}^2+{{\stackrel{·}{v}}_0}^2+{{\stackrel{·}{w}}_0}^2)+2{\rho }_1({\stackrel{·}{u}}_0{\stackrel{·}{\psi }}_x+{\stackrel{·}{v}}_0{\stackrel{·}{\psi }}_{\theta })+{\rho }_2({{\stackrel{·}{\psi }}_x}^2+{{\stackrel{·}{\psi }}_{\theta }}^2)\}r(x)\mathrm{dxd}\theta$$

(17)

where the dot above the variables represents differentiation with respect to time. ${\rho }_0$ , ${\rho }_1$ , and ${\rho }_2$ are the inertial terms, defined by

$$[{\rho }_0,{\rho }_1,{\rho }_2]=\underset{-H/2}{\overset{H/2}{\int }}[\rho (z),z\rho (z),z^2\rho (z)]\mathrm{dz}$$

(18)

in which $\rho (z)$ is the mass density of the hybrid panel.

Thus, the Lagrangian energy function of the hybrid panel can be written as

$$L=T-U-U_{\mathrm{SP}}$$

(19)

Once the energy equations of the hybrid shell are established, the next step is to construct the proper admissible displacement functions and solve these functions.

Solution procedure

To adopt Rayleigh–Ritz procedure to obtain the discretized governing equations for the piezoelectric FG-CNTRC conical shell, the displacement components ( $u_0$ , $v_0$ , $w_0$ , ${\psi }_x$ , and ${\psi }_{\theta }$ ) and the electric potentials ( ${\varphi }_t$ and ${\varphi }_b$ ) involved in equation (19) should be expanded in terms of the admissible functions. In this study, the choice of the admissible displacement functions is considerably simplified, and any independent, complete basis functions may be employed to produce accurate results. The reason lies in the fact that the geometric boundary conditions of the panel are relaxed and enforced through boundary springs which can be seen as penalty parameters, and there is no need to explicitly satisfy the essential and natural conditions on these boundaries for the admissible functions. ⁴³ In this work, the Chebyshev polynomials of first kind are chosen as the basis functions to expand the displacement components. The displacement components can be expressed in the following forms

$$\begin{array}{c}{u}_0(x,\theta ,t)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{A}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t},\hfill \\ v_0(x,\theta ,t)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{B}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t}\hfill \\ w_0(x,\theta ,t)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{C}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t},\hfill \\ {\psi }_x(x,\theta ,t)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{D}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t}\hfill \\ {\psi }_{\theta }(x,\theta ,t)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{E}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t}\hfill \end{array}$$

(20)

where $\omega$ is the angular frequency of vibration; $A_{\mathrm{mn}}$ , $B_{\mathrm{mn}}$ , $C_{\mathrm{mn}}$ , $D_{\mathrm{mn}}$ , and $E_{\mathrm{mn}}$ are corresponding Chebyshev expanded coefficients; and $T_m(x)$ and $Q_n(\theta )$ are the mth and nth order Chebyshev polynomial for the displacement components in the x and $\theta$ directions, respectively. The Chebyshev polynomial series of first kind are defined as follows

$$T_0(x)=1,T_0(x)=1,T_m(x)=2x{T}_{m-1}(x)-T_{m-2}(x)(m>2)$$

(21a)

$$Q_0(\theta )=1,Q_1(\theta )=\theta ,Q_n(\theta )=2\theta Q_{n-1}(\theta )-Q_{n-2}(\theta ),(n>2)$$

(21b)

It should be noted that the Chebyshev polynomial series of first kind given in equation (21) are complete and orthogonal series are defined in the interval of $[-1,1]$ . Thus, a coordinate transformation from x and $\theta$ (for $x\in [0,L_0]$ and $\theta \in [0,{\theta }_0]$ ) to $\overline{x}$ and $\overline{\theta }$ ( $\overline{x},\overline{\theta }\in [-1,1]$ ) needs to be introduced, that is, $x=L_0(\overline{x}+1)/2$ and $\theta ={\theta }_0(\overline{\theta }+1)/2$ .

For the admissible electric potential functions, an auxiliary function is introduced to meet the electrical boundary condition. For electrical boundary condition, on each edge of the panel, either electric displacement or electric potential should be equal to zero. Here it is assumed that the piezoelectric layers are grounded all around, and therefore, on all four edges of the panel, the electric potentials are equal to zero. Electrical boundary conditions for the top and bottom piezoelectric layers may be expressed as

$$\begin{array}{c}x=0,L_0:{\varphi }_t={\varphi }_b=0\hfill \\ \theta =0,{\theta }_0:{\varphi }_t={\varphi }_b=0\hfill \end{array}$$

(22)

The electric potentials ( ${\varphi }_t$ and ${\varphi }_b$ ) may be expanded via Chebyshev polynomials and auxiliary functions as follows

$$\begin{array}{c}{\varphi }_t(x,\theta ,t)=R_t(x,\theta )\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{F}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t}\hfill \\ {\varphi }_b(x,\theta ,t)=R_b(x,\theta )\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{G}_{\mathrm{mn}}{T}_m(x)Q_n(\theta )e^{j\omega t}\hfill \end{array}$$

(23)

where $F_{\mathrm{mn}}$ and $G_{\mathrm{mn}}$ are the corresponding Chebyshev expanded coefficients. $R_t(x,\theta )$ and $R_b(x,\theta )$ are the auxiliary functions chosen according to the electrical boundary conditions. Consequently, in this study, these two auxiliary functions take the following form

$$R_t(x,\theta )=R_b(x,\theta )=x\theta (1-\frac{x}{L_0})(1-\frac{\theta }{{\theta }_o})$$

(24)

It should be noted that for an actual calculation, a limited number of polynomial terms must be used due to the limited computational cost and required numerical accuracy. Consequently, the numbers of polynomial terms truncated for displacements and electrical potentials are chosen as M and N to obtain the results with acceptable accuracy.

Once the admissible functions and energy expressions of the hybrid panel are established, the next task is to determine the coefficients in the admissible functions. By substituting equations (15)–(17) into equation (19) together with the admissible functions defined in equations (20) and (23) and minimizing the total expression of the Lagrangian energy function with respect to the undetermined coefficients, we obtain

$$\frac{\partial L}{\partial q}=0,q=A_{\mathrm{mn}},B_{\mathrm{mn}},C_{\mathrm{mn}},D_{\mathrm{mn}},E_{\mathrm{mn}},F_{\mathrm{mn}},G_{\mathrm{mn}}$$

(25)

a total of $7\times (M+1)\times (N+1)$ equations related to the unknown coefficients can be achieved and written in the form of matrix as ⁴²

$${\mathit{K}}_{\mathit{UU}}\mathit{U}+{\mathit{K}}_{\mathit{U}\mathbf{\Phi }}\mathbf{\Phi }-{\omega }^2{\mathit{M}}_{\mathit{UU}}\mathit{U}=0$$

(26a)

$${\mathit{K}}_{\mathbf{\Phi }\mathit{U}}\mathit{U}+{\mathit{K}}_{\mathbf{\Phi }\mathbf{\Phi }}\mathbf{\Phi }=0$$

(26b)

where ${\mathit{K}}_{\mathit{UU}}$ is the elastic matrix, ${\mathit{K}}_{\mathit{U}\mathbf{\Phi }}$ is the piezoelectric matrix, ${\mathit{M}}_{\mathit{UU}}$ is the mass matrix, ${\mathit{K}}_{\mathbf{\Phi }\mathbf{\Phi }}$ is the permittivity matrix, and ${\mathit{K}}_{\mathbf{\Phi }\mathit{U}}={\mathit{K}}_{\mathit{U}\Phi }^T$ . The superscript T represents the transposition operator of matrix. Besides, the mechanical displacement vector is denoted by U and the electric potential vector is denoted by $\mathbf{\Phi }$ , which are written as

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{U}=[A_{00},\dots ,A_{0n},\dots ,A_{1n},\dots ,A_{\mathrm{mn}},\dots ,A_{\mathrm{MN}},B_{00},\dots ,B_{0n},\dots ,B_{1n},\dots ,B_{\mathrm{mn}},\dots ,B_{\mathrm{MN}},}\\ C_{00},\dots ,C_{0n},\dots ,C_{1n},\dots ,C_{\mathrm{mn}},\dots ,C_{\mathrm{MN}},D_{00},\dots ,D_{0n},\dots ,D_{1n},\dots ,D_{\mathrm{mn}},\dots ,D_{\mathrm{MN}},\hfill \\ E_{00},\dots ,E_{0n},\dots ,E_{1n},\dots ,E_{\mathrm{mn}},\dots ,E_{\mathrm{MN}}{]}^T\hfill \\ \multicolumn{1}{c}{\mathbf{\Phi }=[F_{00},\dots ,F_{0n},\dots ,F_{1n},\dots ,F_{\mathrm{mn}},\dots ,F_{\mathrm{MN}},G_{00},\dots ,G_{0n},\dots ,G_{1n},\dots ,G_{\mathrm{mn}},\dots ,G_{\mathrm{MN}}{]}^T}\end{array}$$

(27)

Two cases of electrical boundary conditions, the closed circuit and open circuit, may be considered for the top and bottom surfaces of the piezoelectric layers. The former indicates that the top and bottom surfaces of the piezoelectric layers are grounded, and the latter implies that the electric potential on free surfaces of the piezoelectric layers is unknown.

Substitution of equation (26b) into equation (26a) and elimination of $\mathbf{\Phi }$ lead to the electro-mechanically coupled system of equations as follows

$$({\mathit{K}}_{\mathit{UU}}-{\mathit{K}}_{\mathit{U}\Phi }{\mathit{K}}_{\mathbf{\Phi }\mathbf{\Phi }}^{-1}{\mathit{K}}_{\mathbf{\Phi }\mathit{U}}-{\omega }^2{\mathit{M}}_{\mathit{UU}})\mathit{U}=0$$

(28)

which is the equation of motion of the piezoelectric conical panels under open-circuit boundary conditions. For the closed-circuit conditions, the electric potential at the free surface of piezoelectric layers is identically zero, and thus, the piezoelectric coupled terms and electric potential vector $\mathbf{\Phi }$ disappear from equation (26). Therefore, for the closed-circuit condition, the system of equations (26) are simplified to the following form

$$({\mathit{K}}_{\mathit{UU}}-{\omega }^2{\mathit{M}}_{\mathit{UU}})\mathit{U}=0$$

(29)

The natural frequencies of the structure considered can be determined by solving the standard eigenvalue problem (equation (28) or (29)). Subsequently, the mode shapes of the hybrid panel can be obtained by substituting the corresponding coefficients into the displacement expressions. The solution procedure may be implemented in a MATLAB code.

Results and discussion

With the theoretical formulations presented in the previous sections, several examples for the free vibration analysis of FG-CNTRC conical panels integrated with piezoelectric layers with different boundary conditions and physical and geometrical parameters are conducted in this section to examine the accuracy, reliability, and the efficiency of the proposed method. In this section, convergence and comparison studies are presented first. Afterwards, parametric studies are carried out to examine the effects of involved parameters. In the rest of this article, a four-letter string is adopted to stand for the mechanical boundary condition, such as FCDS identify the panel with edges $x=0$ , $\theta =0$ , $x=L_0$ , and $\theta ={\theta }_0$ having free, clamped, shear-diaphragm, and simply-supported boundary conditions, respectively. Besides the above four classical boundaries, three types of elastic edge support conditions, designated by the symbols $E^1$ , $E^2$ , and $E^3$ , are considered here. The $E^1$ -type edge is considered to be axially elastic (i.e. $u_0\ne 0$ , $v_0=w_0={\psi }_x={\psi }_{\theta }$ ) and is characterized by a stiffness constant ${\varsigma }^u$ per unit length in the axial direction. The $E^2$ -type edge considers elastically restrained displacement in the circumferential direction (i.e. $v_0\ne 0$ , $u_0=w_0={\psi }_x={\psi }_{\theta }$ ) and is characterized by a stiffness constant ${\varsigma }^v$ per unit length in the circumferential direction. Moreover, for the $E^3$ -type edge, the transverse displacement is elastically restrained by a transverse boundary spring with stiffness constant ${\varsigma }^w$ per unit length (i.e. $w_0\ne 0$ , $u_0=v_0={\psi }_x={\psi }_{\theta }$ ). Since the boundary spring groups are introduced in this study, arbitrary boundary conditions of the panel can be easily generated by setting proper spring stiffness values. Taking edge $x=0$ , for example, the corresponding spring stiffness values for the above four types of classical boundaries and three elastically restrained boundaries are given as follows

$$\mathrm{Free}(\mathrm{F}):{\varsigma }_{x0}^u={\varsigma }_{x0}^v={\varsigma }_{x0}^w={\varsigma }_{x0}^{\eta }={\varsigma }_{x0}^{\vartheta }=0$$

(30a)

$$\mathrm{Clamped}(\mathrm{C}):{\varsigma }_{x0}^u={\varsigma }_{x0}^v={\varsigma }_{x0}^w={\varsigma }_{x0}^{\eta }={\varsigma }_{x0}^{\vartheta }={10}^8{D}^m$$

(30b)

$$\mathrm{Shear}\mathrm{-}\mathrm{diaphragm}(\mathrm{D}):{\varsigma }_{x0}^v={\varsigma }_{x0}^w={\varsigma }_{x0}^{\vartheta }={10}^8{D}^m,{\varsigma }_{x0}^v={\varsigma }_{x0}^{\eta }=0$$

(30c)

$$\mathrm{Simply}\mathrm{-}\mathrm{supported}(\mathrm{S}):{\varsigma }_{x0}^u={\varsigma }_{x0}^v={\varsigma }_{x0}^w={10}^8{D}^m,{\varsigma }_{x0}^{\eta }={\varsigma }_{x0}^{\vartheta }=0$$

(30d)

$$\begin{array}{c}\mathrm{Elastically\; restrained\; edge}(E^1):\hfill \\ {\varsigma }_{x0}^v={\varsigma }_{x0}^w={\varsigma }_{x0}^{\eta }={\varsigma }_{x0}^{\vartheta }={10}^8{D}^m,{\varsigma }_{x0}^u=K_0\hfill \end{array}$$

(30e)

$$\begin{array}{c}\mathrm{Elastically\; restrained\; edge}(E^2):\hfill \\ {\varsigma }_{x0}^u={\varsigma }_{x0}^w={\varsigma }_{x0}^{\eta }={\varsigma }_{x0}^{\vartheta }={10}^8{D}^m,{\varsigma }_{x0}^v=K_0\hfill \end{array}$$

(30f)

$$\begin{array}{c}\mathrm{Elastically\; restrained\; edge}(E^3):\hfill \\ {\varsigma }_{x0}^u={\varsigma }_{x0}^v={\varsigma }_{x0}^{\eta }={\varsigma }_{x0}^{\vartheta }={10}^8{D}^m,{\varsigma }_{x0}^w=K_0\hfill \end{array}$$

(30g)

where $D^m=E^m{h}^3/12[1-(v^m{)}^2]$ and $K_0$ is the given stiffness coefficient of the elastic edge. In our study, a considerably great stiffness of ${10}^8{D}^m$ is adopted to make sure the corresponding boundary displacement component is nearly equal to zero, which will be checked by several examples given in the following discussion.

Unless otherwise stated, poly(methyl methacrylate), referred to as PMMA, is chosen as the matrix with material properties $E^m=(3.52-0.0034T)\mathrm{GPa}$ , $v^m=0.34$ , and ${\rho }^m=1150\mathrm{kg}/m^3$ , where $T=T_0+\Delta T$ . Here, $\Delta T$ is the temperature change and $T_0$ is the reference temperature which is set equal to 300 K. (10, 10) armchair SWCNT is selected for the reinforcements. Young’s modulus and shear modulus of SWCNT are evaluated at three values of temperatures by Shen and Xiang and are given as follows

$$\begin{array}{c}T=300\mathrm{K},E_{11}^{\mathrm{CNT}}=5.6466\mathrm{TPa},E_{22}^{\mathrm{CNT}}=7.0800\mathrm{TPa},\hfill \\ G_{12}^{\mathrm{CNT}}=1.9445\mathrm{TPa}\hfill \\ T=500\mathrm{K},E_{11}^{\mathrm{CNT}}=5.5308\mathrm{TPa},E_{22}^{\mathrm{CNT}}=6.9348\mathrm{TPa},\hfill \\ G_{12}^{\mathrm{CNT}}=1.9643\mathrm{TPa}\hfill \\ T=700\mathrm{K},E_{11}^{\mathrm{CNT}}=5.4744\mathrm{TPa},E_{22}^{\mathrm{CNT}}=6.8641\mathrm{TPa},\hfill \\ G_{12}^{\mathrm{CNT}}=1.9644\mathrm{TPa}\hfill \end{array}$$

Poisson’s ratio and mass density of SWCNT are assumed to be temperature-independent and $v_{12}^{\mathrm{CNT}}=0.175$ and ${\rho }^{\mathrm{CNT}}=1400\mathrm{kg}/m^3$ . As stated earlier, three efficiency parameters are introduced in the rule of mixture to account for the size dependency of mechanical properties of the composite media. For three different volume fractions of CNTs, these parameters are as follows: ${\eta }_1=0.137$ and ${\eta }_2=1.022$ for $V_{\mathrm{CNT}}^{*}=0.12$ ; ${\eta }_1=0.142$ and ${\eta }_2=1.626$ for $V_{\mathrm{CNT}}^{*}=0.17$ ; and ${\eta }_1=0.141$ and ${\eta }_2=1.585$ for $V_{\mathrm{CNT}}^{*}=0.28$ . For each case, the efficiency parameter ${\eta }_3$ is equal to $0.7{\eta }_2$ . The shear modulus $G_{13}^{\mathrm{CNT}}$ is taken to be equal to $G_{12}^{\mathrm{CNT}}$ , whereas $G_{23}^{\mathrm{CNT}}$ is taken to be equal to $1.2G_{12}^{\mathrm{CNT}}$ .

Both top and bottom piezoelectric layers are made from PZT-5A. Material properties of the piezoelectric materials are $E^p=63(1-0.0005\Delta T)\mathrm{GPa}$ , $v^p=0.35$ , ${\rho }^p=7600\mathrm{kg}/m^3$ , $e_{31}=e_{32}=7.209C/m^2$ , and $k_{33}=1.5\times {10}^{-8}F/m$ . It should be noted that, in all of the numerical results, except for Table 7, free vibrational analysis is performed at room temperature $T=300K$ .

Convergence and comparison studies

In this section, the convergence of the proposed method for free vibrational analysis of piezoelectric FG-CNTRC conical shell is examined to establish the number of the required polynomial terms used to achieve accurate results. Since no work has been reported on the free vibration of piezoelectric FG-CNTRC conical panel, some comparison studies are performed for the available results of FG-CNTRC cylindrical panels and plates. Some relevant comparison and convergence tests are provided in Table 1. The first six frequency parameters $\hat{\omega }=\omega L_0^2/h\sqrt{{\rho }^m/E^m}$ of an FG-X CNTRC cylindrical panel without piezoelectric layers under DDDD boundary conditions are evaluated and the results are compared with those of Mirzaei and Kiani. ¹⁵ From the table, we can find a very close agreement between present solutions and those of the available literature. It is obvious that by increasing M and N, the numerical results converge rapidly. In addition, the results are nearly unchanged with the growth of M and N when they reach a certain value, $M=N=11$ . Therefore, in the rest of this research, the number of the required polynomial terms is chosen as $M=N=11$ .

Table 1.

Comparison and convergence of first six frequency parameters $\hat{\omega }=\omega L^2/h\sqrt{{\rho }^m/E^m}$ for FG-CNTRC cylindrical panels with DDDD boundary conditions ( $L_0/R_0=1$ , $h/R_0=0.05$ , $\alpha =0\circ$ , and ${\theta }_0=180/\pi \circ$ ; $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern is considered).

$M=N$	${\hat{\omega }}_1$	${\hat{\omega }}_2$	${\hat{\omega }}_3$	${\hat{\omega }}_4$	${\hat{\omega }}_5$	${\hat{\omega }}_6$
3	25.1951	34.6858	44.7395	44.7399	72.1939	75.7101
5	24.7331	28.6228	44.4473	44.4477	53.5729	65.8275
7	24.7325	28.5133	43.4484	44.4469	44.4474	65.7198
9	24.7325	28.5128	43.1264	44.4469	44.4474	65.7194
11	24.7325	28.5128	43.1227	44.4469	44.4474	65.7194
13	24.7325	28.5128	43.1227	44.4469	44.4474	65.7194
Mirzaei and Kiani 15	24.8189	28.7742	43.5052	44.4268	44.4280	65.7565

A comparison study is presented in the next example for an FG-CNTRC cylindrical panel without piezoelectric layers under various boundary conditions and CNT distribution type. The first four frequency parameters $\hat{\omega }=\omega L_0^2/h\sqrt{{\rho }^m/E^m}$ are evaluated and compared with the available results in the open literature, as shown in Table 2. It is seen that the obtained frequency parameters are in good agreement with the available results in the open literature.

Table 2.

Comparison of first four frequency parameters $\hat{\omega }=\omega L^2/h\sqrt{{\rho }^m/E^m}$ for FG-CNTRC cylindrical panels with different boundary conditions ( $L_0/R_0=1$ , $h/R_0=0.05$ , $\alpha =0\circ$ , and ${\theta }_0=180/\pi \circ$ ; $V_{\mathrm{CNT}}^{*}=0.12$ ).

Type	Modes	DDDD		CCCC		SFSF		CFFF
		Present study	Mirzaei and Kiani 15	Present study	Mirzaei and Kiani 15	Present study	Mirzaei and Kiani 15	Present study	Mirzaei and Kiani 15
UD	${\hat{\omega }}_1$	17.8746	17.9355	30.7767	30.8185	15.8061	15.8241	6.3879	6.3903
	${\hat{\omega }}_2$	20.5992	20.7580	30.9693	31.0872	16.4883	16.4725	7.1181	7.0793
	${\hat{\omega }}_3$	31.4913	31.5916	43.4983	43.6652	18.4440	18.5829	9.9327	10.0725
	${\hat{\omega }}_4$	34.1604	34.1437	55.2504	55.2600	25.4701	25.7101	16.4973	16.5033
FG-X	${\hat{\omega }}_1$	19.7937	19.8544	32.2659	32.3134	18.0440	18.0629	7.3302	7.3335
	${\hat{\omega }}_2$	22.5095	22.6972	32.5585	32.6868	18.6422	18.6277	7.9392	7.9060
	${\hat{\omega }}_3$	33.2961	33.5735	45.2457	45.4421	20.3697	20.5093	10.6959	10.8363
	${\hat{\omega }}_4$	34.2671	34.2505	57.5399	57.5570	27.2760	27.5326	16.5477	16.5537
FG-O	${\hat{\omega }}_1$	15.1989	15.2680	28.1745	28.2171	12.4787	12.4981	5.1257	5.1275
	${\hat{\omega }}_2$	18.1273	18.3274	28.3902	28.5113	13.3485	13.3300	6.0657	6.0180
	${\hat{\omega }}_3$	29.5921	29.8442	41.3254	41.4832	15.8163	15.9688	9.0819	9.2289
	${\hat{\omega }}_4$	34.2669	34.2505	50.8472	50.8519	23.3692	23.6126	16.5448	16.5510
FG-V	${\hat{\omega }}_1$	16.5137	16.5635	29.3419	29.3736	15.3585	15.3792	5.6423	5.6401
	${\hat{\omega }}_2$	19.5987	19.7483	29.8658	29.9670	16.0414	16.0393	6.3917	6.3407
	${\hat{\omega }}_3$	31.0406	31.1231	42.8360	42.9944	18.1457	18.2548	9.7846	9.9216
	${\hat{\omega }}_4$	34.2670	34.2496	52.8232	52.8115	25.3466	25.5659	16.5576	16.5444

For the next example, an FG-CNTRC plate integrated with PZT-5A piezoelectric layers is considered. First four natural frequencies of such plate are compared with those obtained by Kiani, ⁴¹ as listed in Table 3. The material properties of CNT and piezoelectric layers are set to be equal to those in the previous sections, whereas those of the matrix are $E^m=3.522\mathrm{GPa}$ , $v^m=0.34$ , and ${\rho }^m=1150\mathrm{kg}/m^3$ . Two kinds of electrical boundary conditions are considered for the free piezoelectric surfaces, that is, open circuit and closed circuit. It is seen that results of this study match well with those obtained by Kiani. ⁴¹

Table 3.

Comparison of the first four natural frequencies (in Hz) for FG-CNTRC plates integrated with piezoelectric layers with closed-circuit electrical boundary conditions under various boundary conditions ( $R_0/h={10}^8$ , $L_0=0.4m$ , $h/L_0=0.05$ , $h_p/h=0.1$ , $\alpha =0$ , and ${\theta }_0=(360\times {10}^{-7}/\pi )\circ$ ; $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern is considered).

Boundary conditions	Electrical conditions	Source	$f_1$	$f_2$	$f_3$	$f_4$
CCCC	Closed	Kiani 41	1220.533	1877.468	2557.460	2969.362
		Present	1218.033	1885.627	2541.107	2980.809
	Open	Kiani 41	1249.769	1952.332	2586.436	3073.730
		Present	1246.860	1958.590	2570.399	3081.747
CFFF	Closed	Kiani 41	191.862	275.959	696.090	861.799
		Present	192.214	275.151	700.275	868.549
	Open	Kiani 41	194.315	277.499	729.130	861.907
		Present	196.909	275.627	728.753	869.689
SCSC	Closed	Kiani 41	843.448	1672.423	1881.033	2057.458
		Present	842.240	1676.078	1903.080	2074.104
	Open	Kiani 41	881.119	1757.160	1881.033	2098.108
		Present	880.210	1761.763	1904.004	2121.741
SSSS	Closed	Kiani 41	675.135	1287.031	1881.021	1881.021
		Present	672.201	1296.885	1900.141	1900.333
	Open	Kiani 41	703.692	1369.426	1881.021	1881.021
		Present	679.725	1380.216	1900.617	1901.680

The convergence and comparison studies show the excellent convergence and accuracy of the proposed formulations and solution method for the free vibration of FG-CNTRC conical panels with piezoelectric layers.

Parametric studies

In this subsection, parametric studies are carried out to examine the effects of the electrical and mechanical boundary conditions, distribution type of CNTs, volume fraction of CNTs, and geometric parameters on the natural frequencies of the FG-CNTRC conical panel integrated with piezoelectric layers.

Table 4 presents the fundamental natural frequency parameters $\hat{\omega }=\omega L_0^2/h\sqrt{{\rho }^m/E^m}$ of CNTRC conical panel integrated with two identical piezoelectric layers at the top and bottom surfaces. In this case, three different volume fractions of CNTs, four different types of CNT dispersion and two different types of electrical boundary conditions are taken into consideration. Table 4 is associated to eight types of boundary conditions, including classical and elastic boundary conditions. It is seen from the table that the fundamental frequency parameter of the hybrid panel increases as the electrical boundary conditions change from the closed circuit to open circuit. The reason lies in the fact that the closed-circuit piezoelectric layer discharges electric potential during vibration, while the open circuit one converts it to mechanical energy. Consequently, piezoelectric effect is more prominent in panels bonded with open-circuit piezoelectric layers. It is found that the increase in the volume fraction of CNTs increases the fundamental frequency parameter of the hybrid panel. In addition, among the four possible graded types of the CNTs, FG-X type of CNT distribution results in higher frequency parameter and also FG-O type has the lowest one. This is because the reinforcements distributed close to the top and bottom surfaces are more efficient than those distributed near the mid-surface for increasing the stiffness of CNTRC panel. As expected, among all the cases of mechanical boundary conditions, CCCC panels have the highest natural frequencies compared to plates with other boundary conditions. Comparing the cases with CFCF, CFE¹F, CFE²F, CFE³F, and CFFF boundary conditions, it is found that CFCF panel has the highest fundamental natural frequency parameters, followed by CFE²F, CFE¹F, CFE³F, and CFFF. In other words, the elastic boundary conditions have a remarkable effect on the fundamental natural frequency parameters of the hybrid panels.

Table 4.

Fundamental natural frequency parameter $\omega L^2/h\sqrt{{\rho }^m/E^m}$ for FG-CNTRC conical panels integrated with piezoelectric layers with the open-circuit or closed-circuit electrical boundary conditions with different volume fractions and distribution types under various boundary conditions ( $R_0=1m$ , $L_0=2m$ , $h/R_0=0.05$ , $h_p/h=0.1$ , $\alpha =30\circ$ , and ${\theta }_0=90\circ$ ; $K_0={10}^3{D}_m$ ).

$V_{\mathrm{CNT}}^{*}$	Type	Electrical conditions	Boundary conditions
			DDDD	CCCC	SSSS	CFFF	CFCF	CFE1F	CFE2F	CFE3F
0.12	UD	Closed	41.042	62.648	49.242	7.243	36.474	35.546	36.029	32.521
		Open	43.262	65.667	51.433	7.322	37.032	36.119	36.641	32.951
	FG-X	Closed	41.821	64.685	49.910	7.614	39.869	39.010	39.453	34.431
		Open	44.003	67.584	52.066	7.687	40.358	39.510	39.991	34.824
	FG-O	Closed	40.243	60.407	48.571	6.835	32.535	31.502	32.041	30.005
		Open	42.504	63.566	50.802	6.922	33.187	32.177	32.757	30.506
	FG-V	Closed	41.583	61.894	49.218	7.030	34.053	33.081	33.515	31.104
		Open	43.744	64.908	51.383	7.111	34.633	33.681	34.127	31.592
0.17	UD	Closed	42.826	65.920	51.370	7.937	40.269	39.396	39.814	35.060
		Open	44.955	68.791	53.466	8.009	40.758	39.896	40.352	35.451
	FG-X	Closed	43.950	68.688	52.342	8.428	44.595	43.792	44.167	37.237
		Open	46.027	71.410	54.392	8.495	45.013	44.217	44.628	37.605
	FG-O	Closed	41.713	62.892	50.463	7.393	35.107	34.114	34.595	32.006
		Open	43.894	65.938	52.607	7.473	35.701	34.726	35.248	32.462
	FG-V	Closed	43.299	64.750	51.381	7.635	37.049	36.120	36.493	33.320
		Open	45.368	67.634	53.442	7.709	37.570	36.656	37.040	33.769
0.28	UD	Closed	44.360	69.295	52.658	8.873	45.758	45.041	45.308	37.884
		Open	46.398	71.970	54.678	8.938	46.156	45.445	45.750	38.252
	FG-X	Closed	46.196	73.345	54.311	9.576	51.423	50.760	48.689	40.248
		Open	48.154	75.823	56.260	9.635	51.749	51.090	48.696	40.618
	FG-O	Closed	42.703	65.013	51.387	8.106	38.756	37.915	38.248	34.382
		Open	44.817	67.927	53.471	8.179	39.271	38.441	38.817	34.787
	FG-V	Closed	44.609	67.467	52.840	8.431	41.421	40.637	40.871	35.945
		Open	46.590	70.188	54.804	8.498	41.862	41.086	41.333	36.356

Table 5 examines the influence of semi-vertex angle $\alpha$ and thickness–radius ratio $h/R_0$ on the fundamental natural frequency of FG-CNTRC conical panel integrated with closed-circuit piezoelectric layers. Three different ratios of $h/R_0$ and five different semi-vertex angles are considered. Volume fraction of CNT is chosen as $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern of CNTs is considered. This table covers the fundamental frequencies with various boundary conditions, including classical and elastic ones. It is worth noting that when thickness–radius ratio varies from 0.05 to 0.15, the fundamental frequency increases, whereas the fundamental frequency decreases as semi-vertex angles change from 15° to 75°. A similar effect of mechanical boundary conditions of the panel on the fundamental frequency can also be obtained in this case.

Table 5.

Fundamental natural frequencies (in Hz) for FG-CNTRC conical panels integrated with piezoelectric layers with closed-circuit electrical boundary conditions with different semi-vertex angles and thickness under various boundary conditions ( $R_0=1m$ , $L=2m$ , $h_p/h=0.1$ , and ${\theta }_0=90\circ$ ; $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern is considered; $K_0={10}^3{D}_m$ ).

$\alpha$	$h/R_0$	Boundary conditions
		DDDD	CCCC	SSSS	CFFF	CFCF	CFE1F	CFE2F	CFE3F
15°	0.01	72.860	97.887	88.995	13.686	45.182	42.642	44.915	44.509
	0.05	152.545	235.266	180.378	29.551	134.623	131.934	133.615	117.494
	0.10	188.772	355.575	268.403	47.811	213.168	211.172	205.301	175.787
30°	0.01	61.829	82.532	75.824	10.320	40.991	38.887	40.759	40.197
	0.05	128.916	201.479	153.533	24.722	130.810	128.453	129.555	109.226
	0.10	173.745	301.436	226.572	40.304	209.743	207.964	196.393	168.026
45°	0.01	51.636	68.940	62.120	8.137	36.970	35.243	36.720	35.961
	0.05	111.908	178.238	129.543	21.188	126.358	124.684	125.206	100.261
	0.10	154.889	262.453	197.134	35.792	206.355	205.114	192.097	161.260
60°	0.01	40.902	55.625	47.565	6.466	32.911	31.629	32.598	31.488
	0.05	92.895	156.657	107.491	18.407	121.967	121.073	121.223	91.611
	0.10	135.460	233.837	156.034	32.864	203.346	202.711	189.982	155.691
75°	0.01	28.061	42.410	32.913	4.946	28.569	28.035	28.371	26.172
	0.05	70.072	131.446	79.335	16.415	118.555	118.301	117.754	84.620
	0.10	120.858	215.520	125.925	30.175	199.703	199.331	189.001	150.139

Table 6 presents the effect of piezoelectric thickness on the first four natural frequencies of FG-CNTRC conical panel integrated with open-circuit and closed-circuit piezoelectric layers. In this table, geometrical characteristics of the host panel are the same with those used in Table 4, but three different ratios of $h_p/h$ are considered. Volume fraction of CNT is chosen as $V_{\mathrm{CNT}}^{*}=0.17$ , and FG-X pattern of CNTs is considered. Various boundary conditions, including classical and elastic ones, are covered in this table. It is seen that influence of piezoelectric thickness on frequencies for various boundary conditions is not the same. For instance, in CCCC and DDDD plates, when thickness ratio changes from 0.05 to 0.15, the fundamental frequency increases. Also, it is observed that increasing piezoelectric layer thickness decreases the fundamental frequency of CFFF, CFCF, CFE¹F, CFE²F, and CFE³F plates, while the third and fourth frequencies increase as the piezoelectric layers get thicker.

Table 6.

Influence of piezoelectric layer thickness on first four natural frequencies (in Hz) for FG-CNTRC conical panels integrated with piezoelectric layers with the open-circuit or closed-circuit electrical boundary conditions ( $R_0=1m$ , $L=2m$ , $h/R_0=0.05$ , $\alpha =30\circ$ , and ${\theta }_0=90\circ$ ; $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern is considered; $K_0={10}^3{D}_m$ ).

Boundary conditions	$h_p/h$	Closed circuit				Open circuit
		$f_1$	$f_2$	$f_3$	$f_4$	$f_1$	$f_2$	$f_3$	$f_4$
DDDD	0.05	118.633	135.023	171.376	258.230	122.910	136.988	181.301	274.219
	0.10	128.916	142.674	194.705	283.841	135.008	145.551	207.870	293.184
	0.15	136.191	147.050	210.878	286.834	143.458	150.491	226.021	296.214
CCCC	0.05	190.353	213.656	273.481	336.923	195.800	221.863	286.407	354.061
	0.10	201.479	225.732	295.774	378.035	209.464	237.125	313.236	390.437
	0.15	209.808	232.901	311.659	377.313	219.409	246.207	331.587	391.869
CFFF	0.05	25.301	31.636	46.064	75.148	25.431	32.410	46.667	78.892
	0.10	24.722	33.343	48.438	87.317	24.917	34.230	49.447	91.912
	0.15	24.386	34.202	50.061	95.523	24.632	35.081	51.382	100.322
CFCF	0.05	137.037	139.173	171.638	182.058	137.717	140.282	173.613	187.791
	0.10	130.810	134.821	173.616	193.899	132.034	136.636	176.841	202.282
	0.15	127.173	132.759	174.918	203.355	128.844	135.086	179.013	213.425
CFE1F	0.05	135.319	137.791	168.106	181.116	136.004	138.865	170.028	186.861
	0.10	128.453	132.868	168.730	192.566	129.701	134.624	171.876	200.964
	0.15	124.372	130.358	169.051	201.760	126.086	132.608	173.063	211.845
CFE2F	0.05	135.975	138.328	143.897	157.763	136.732	139.407	143.910	160.148
	0.10	129.555	133.449	141.117	157.690	130.906	135.232	141.146	161.511
	0.15	125.816	130.877	138.945	158.068	127.643	133.182	138.994	162.861
CFE3F	0.05	112.850	114.760	152.141	159.116	113.539	115.786	154.238	163.800
	0.10	109.226	112.850	155.943	171.450	110.307	114.461	159.050	178.130
	0.15	107.111	112.222	158.133	180.742	108.465	114.234	161.845	188.636

Figure 3 shows the variation of the lowest three frequency parameters $\hat{\omega }=\omega L_0^2/h\sqrt{{\rho }^m/E^m}$ of an FG-X CNTRC conical shell integrated with closed-circuit piezoelectric layers against the subtended angle ${\theta }_0$ . Various mechanical boundary conditions, including DDDD, CCCC, CFCF, CFE¹F, CFE²F, and CFE³F, are considered in this case. The geometrical sizes are the same as those used in Table 4, except for the subtended angle ${\theta }_0$ . CNT volume fraction $V_{\mathrm{CNT}}^{*}=0.17$ is considered. As observed from Figure 3, when the panel is subjected to DDDD and CCCC boundary conditions, the frequency parameters of the panel generally decrease and then remain stable with the increase in the subtended angle, except for some small fluctuations in the curves of the DDDD boundary condition. For CFCF, CFE¹F, and CFE³F boundary conditions, the curves of the first three frequency parameters have the same trend: the first frequency parameter curve increases and then remains stable, whereas the second frequency parameter first decreases and then fluctuates, and finally tends to be stable. For the third frequency parameter, the curve first climbs up, then declines, and finally tends to be stable with some small fluctuations. When the panel is subjected to CFE²F boundary conditions, the different trends of the second and the third frequency parameter curves are observed compared with those of CFCF, CFE¹F, and CFE³F boundary conditions. These differences may be due to the change in the boundary spring stiffness in the circumferential direction. Moreover, a common trend has been observed in all cases of Figure 3: there are always two curves that converge to the same value with the increase in the subtended angle. This may be because the symmetrical modes appear when the conical panel is closed in the circumferential direction ⁴⁴ and as the subtended angle increases and approaches gradually to 360°, the open panel tends to be circumferential-closed.

Figure 3.

Influence of subtended angle on first three frequency parameters $\omega L^2/h\sqrt{{\rho }^m/E^m}$ for FG-CNTRC conical panels integrated with piezoelectric layers with the closed-circuit electrical boundary conditions.

Figure 4 shows the variation of the first three frequency parameters $\hat{\omega }=\omega L_0^2/h\sqrt{{\rho }^m/E^m}$ of a piezoelectri FG-X CNTRC conical panel subjected to the elastically restrained boundary conditions against the spring stiffness coefficient, $K_0$ . Three elastic boundary conditions, CFE¹F, CFE²F, and CFE³F, are considered in this case. The piezoelectric layers are closed circuit. The geometrical sizes are the same as those used in Table 4, and the CNT volume fraction is set to $V_{\mathrm{CNT}}^{*}=0.17$ . As shown in Figure 4, the frequency parameters are almost maintained at a level when the stiffness coefficient is smaller than $10D_m$ . As it further increases, the frequency parameters increase rapidly, and when it is beyond ${10}^7{D}_m$ , the frequency parameters gradually tend to a constant. Further study shows that the obtained constant values are equal to the first three frequency parameters of such panel subjected to CFCF boundary conditions (they are 44.595, 45.963, and 59.189, respectively). It can be proved that the proposed boundary spring technique is capable of simulating the panel subjected to the classical boundary by setting the boundary spring stiffness coefficients to a proper value. In our study, for example, to make the given displacement component in the boundary be zero, the corresponding boundary spring stiffness coefficient in the boundary is set to be equal to ${10}^8{D}_m$ .

Figure 4.

Effects of the given spring stiffness coefficient on the first three frequency parameters $\omega L^2/h\sqrt{{\rho }^m/E^m}$ of FG-CNTRC conical panels integrated with closed-circuit piezoelectric layers subjected to elastic boundary conditions.

Table 7 presents the effect of the temperatures on the first four natural frequencies of FG-CNTRC conical panel integrated with open-circuit and closed-circuit piezoelectric layers subjected to various boundary conditions. In this table, except for some temperature-dependent material properties, other material properties and geometrical sizes of the hybrid panels are consistent with those used in Table 6. Three values of the temperature are considered in this case, and they are set to 300, 500, and 700 K, respectively. As expected, the increase in the temperature results in the decrease in the frequencies of the hybrid panel under various boundaries. This trend is due to the decrease in the elastic stiffness of the CNT, matrix, and piezoelectric layers with the increase in the temperature. Similar to the conclusions of the previous tables for various boundary conditions, closed-circuit hybrid panels have lower frequencies compared with those of open circuit, and the panels with CCCC boundary conditions have the highest natural frequencies.

Table 7.

Influence of the temperature on first four natural frequencies (in Hz) for FG-CNTRC conical panels integrated with piezoelectric layers with the open-circuit or closed-circuit electrical boundary conditions ( $R_0=1m$ , $L=2m$ , $h/R_0=0.05$ , $h_p/h=0.1$ , $\alpha =30\circ$ , and ${\theta }_0=90\circ$ ; $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern is considered; $K_0={10}^3{D}_m$ ).

Boundary conditions	$T(K)$	Closed circuit				Open circuit
		$f_1$	$f_2$	$f_3$	$f_4$	$f_1$	$f_2$	$f_3$	$f_4$
DDDD	300	128.916	142.674	194.705	283.841	135.008	145.551	207.870	293.184
	500	121.767	133.493	183.786	271.309	128.174	136.526	197.652	280.362
	700	114.220	123.699	172.200	258.196	120.991	126.916	186.884	262.083
CCCC	300	201.479	225.732	295.774	378.035	209.464	237.125	313.236	390.437
	500	190.809	212.765	278.270	357.927	199.116	224.774	296.703	373.068
	700	179.606	198.985	259.652	335.153	188.253	211.724	279.207	354.654
CFFF	300	24.722	33.343	48.438	87.317	24.917	34.230	49.447	91.912
	500	23.795	31.759	45.807	82.279	23.996	32.692	46.870	87.169
	700	22.852	30.103	42.977	76.906	23.058	31.090	44.108	82.153
CFCF	300	130.810	134.821	173.616	193.899	129.701	134.624	171.876	200.964
	500	126.264	130.008	164.650	184.263	125.444	130.036	163.721	191.875
	700	121.725	125.179	155.276	174.191	121.201	125.435	155.192	182.387
CFE1F	300	128.453	132.868	168.730	192.566	130.906	135.232	141.146	161.511
	500	124.196	128.261	160.470	183.129	126.306	127.830	130.463	153.861
	700	119.968	123.657	151.844	173.268	112.469	122.039	125.652	145.903
CFE2F	300	129.555	133.449	141.117	157.690	81.963	130.589	134.302	156.971
	500	125.024	127.723	128.652	149.877	70.568	126.154	129.536	149.718
	700	112.444	120.677	123.820	141.744	56.798	121.720	124.714	142.208
CFE3F	300	109.226	112.850	155.943	171.450	110.307	114.461	159.050	178.130
	500	102.244	105.767	145.240	160.757	103.441	107.485	148.649	167.674
	700	93.918	97.404	133.160	148.968	95.329	99.309	137.033	156.153

The first four mode shapes of the hybrid conical panels subjected to various boundary conditions are given in Figure 5. The material and geometrical parameters are consistent with those used in Table 7. As can be seen from Figure 5, the mode shapes of different boundary conditions are quite different. Compared to Figure 5(d)–(g), it is found that the elastically restrained edge considerably affects the mode shapes of the hybrid panel, and different types of the elastically restrained edges also result in different mode shapes.

Figure 5.

First four mode shapes of piezoelectric FG-CNTRC conical panel with various boundary conditions ( $R_0=1m$ , $L=2m$ , $h/R_0=0.05$ , $h_p/h=0.1$ , $\alpha ={30}^{\circ }$ , and ${\theta }_0={90}^{\circ }$ ; $V_{\mathrm{CNT}}^{*}=0.17$ and FG-X pattern is considered; $K_0={10}^3{D}_m$ ): (a) CCCC, (b) DDDD, (c) CFFF, (d) CFCF, (e) E¹FE¹F, (f) E²FE²F, and (g) E³FE³F.

Conclusion

Adopting the Rayleigh–Ritz procedure with the Chebyshev polynomial basis function and the boundary spring technique, free vibrational characteristics of FG-CNTRC conical panels integrated with two piezoelectric layers and subjected to elastically restrained boundary condition are investigated in this study. Material properties of the host panel are temperature-dependent and obtained according to the extended rule of mixtures. Solution method of this research is suitable for arbitrary classical and elastic mechanical boundary conditions. Besides, both the open-circuit and closed-circuit boundary conditions are considered for the top and bottom piezoelectric layers of the hybrid structure. The excellent convergence and accuracy of the presented method are verified by the convergence and comparison studies. Parametrical studies are conducted to explore the influences of volume fraction of CNT, distribution type of CNTs, geometrical parameters, temperatures, and electrical and mechanical boundary conditions on the natural frequencies of the hybrid conical panel.

It is found out that increasing the volume fraction of CNT increases the natural frequency of the hybrid panel. In addition, among the four possible graded types of the CNTs, FG-X type of CNT distribution results in higher frequency parameter and FG-O type has the lowest one. Numerical results also reveal that in general, the panel with open-circuit boundary conditions has higher fundamental frequency in comparison with those with closed-circuit electrical boundary conditions. It was observed that with the semi-vertex angle decreased or the thickness increased of the conical panel, the fundamental frequency of the hybrid panel is increased. Besides, the influence of the thickness of piezoelectric layers on natural frequencies of the hybrid panel is not the same and also depends on the boundary conditions. Also, the stiffness coefficients of the elastically restrained boundary conditions have a remarkable effect on the natural frequencies, and the increase in the temperature results in the decrease in the frequencies of the hybrid panel.