Vibration characteristics of piezoelectric functionally graded carbon nanotube-reinforced composite doubly-curved shells

This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube (CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton’s principle. Navier’s solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.


Introduction
Functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials are a new generation of composite materials in which carbon nanotubes (CNTs) are designed purposefully to grade with specific rules along with desired directions within an isotropic matrix. Thanks to the outstanding properties of FG-CNTRC and electromechanical properties of piezoelectric materials, the complicated FG-CNTRC structures with integrated piezoelectric layers (PFG-CNTRC) offer great potential for use in engineering, advanced aerospace, medicine, military, and automotive structural applications. The PFG-CNTRC structures are applied as smart structures to control the free vibration, suppress the forced vibration, decrease the deflection and stresses, delay the buckling, decrease the post-buckling deflection, flutter control, The panel with constant principal curvatures is referred to as an orthogonal curvilinear coordinate system (x, y, z). The length, width, and two radii of principal curvatures of the middle surface of the panel are denoted by a, b, and R x and R y , respectively. The thickness of the core and each piezoelectric layer are also marked by h c and h p , respectively.
As shown in Fig. 2, for each layer, five types of distributions of CNT are considered. UD represents the uniform distributions, and the other four types of functionally graded distributions of CNT are denoted by FG-A, FG-V, FG-X, and FG-O. where in which, w CNT is the mass fraction of the CNT, ρ m and ρ CNT are mass densities of the matrix and CNT, respectively. By using the extended rule of the mixture, the effective material properties of each FG-CNT layer can be expressed as follows [28] : where E CNT 11 , E CNT 22 , and E m are Young's moduli of CNT and matrix; G CNT 12 and G m are the shear modulus of CNT and matrix; η 1 , η 2 , and η 3 are CNT efficiency parameters; V m (z) and V CNT (z) are volume fractions of matrix and CNT with the relation of V CNT (z) + V m (z) = 1; ν CNT 12 and ν m are Poisson's ratios of CNT and matrix.

Approximation on mechanical displacement
The displacements field for doubly curved shell based on the four-variable shear deformation refined theory is given by the following equations [29] : w(x, y, z, t) = w b (x, y, t) + w s (x, y, t), where u 0 and v 0 are the in-plane displacements in the directions of x and y; w b and w s represent the bending and shear components of the transverse displacement, respectively. The polynomial shape function is used as f (z) = z − 1 8 + 3 2 z h 2 [26] . The strains at any point in the shell space associated with the displacement field in Eq. (4) can be obtained by applying the fundamental kinematic relations of a 3D body in curvilinear coordinates as follows [30] : where Vibration characteristics of PFG-CNTRC doubly-cured shells 823

Approximation on electric potential
In this work, the electric potential distribution in the transverse direction of each piezoelectric layer is approximated by the linear function as follows [31][32][33] : The electric field is related to electric potential by the following relation: where ∇ denotes the gradient operator. Thus, the components of the electric field are obtained as

Constitutive equations
The stress-strain relation for the kth CNT layer is expressed as [34]  in whichQ k ij (z) are transformed material constants expressed in terms of material constants [35] , where Q c ij (z) are the plane stress-reduced stiffnesses defined in terms of the engineering constants in the material axes of the layer, For the kpth piezoelectric layer (kp = top, bottom), the stress components and electric displacement relations are given as [30,34,[36][37] in which, the elastic constants for the piezoelectric layer [36][37] are defined as where C kp ij are stiffness coefficients for the piezoelectric layers; e ij is the electromechanical coupling matrix; p ij is the dielectric permittivity matrix; E is the electric field, and D is the electrical displacement in the piezoelectric layer.

Governing equation
Hamilton's principle is applied herein to obtain the equations of motion of laminated PFG-CNTRC doubly curved shell panels, where δU is the variation of strain energy, and δK is the variation of kinetic energy. The expression of variation of strain energy for the PFG-CNTRC panels is Substituting Eq. (5), Eq. (9), Eq. (13), and Eq. (14) into Eq. (17), we can get the variation strain energy δU as , and (Q xs , Q ys ) are stress resultants given in detail in Appendix A. The variation of kinetic energy for the PFG-CNTRC panels is given as where The governing equations can be obtained by substituting the variation of strain energy and kinetic energy from Eq. (18) and Eq. (19) into Eq. (16) and then set the coefficients of δu 0 , δv 0 , δw b , δw s , δϕ t , and δϕ b to be zero, where 3 Solution procedure In this study, two sets of simply supported boundary conditions named cross-ply (SS-1) and angle-ply (SS-2) laminates shown in Table 1 are considered.
Following the Navier solution procedure, the following expansion displacements for u 0 , v 0 , w s , w b , φ t , and φ b are chosen to satisfy the boundary conditions given in Table 1. Here, u mn , v mn , w b mn , w s mn , ψ t mn , and ψ b mn are arbitrary parameters to be determined, α = mπ a , β = nπ b , and m, n denote the numbers of haft-waves in the x-and y-directions, respectively. By substituting the expansion displacement functions in Table 2 into Eq. (25), the following matrix form can be obtained: where m ij and k ij are given in Appendices B1 and B2 for SS-1 and SS-2 boundary conditions, respectively. Equation (27) can be rewritten in the short form as where M uu is the mass matrix, K uu is the elastic matrix, K uφ is the piezoelectric matrix, and K φφ is the permittivity matrix. Substituting the solution of the second equation into the first equation in Eq. (24), we obtain a short form of the equation for vibration characteristics of FG-CNTRC shell panels integrated with piezoelectric layers, For free vibration, Eq. (29) reduces to an eigenvalue problem by setting (u) = (û)e iωt Equation (30) is associated with the natural frequencies of an open-circuit (Opc) PFG-CNTRC doubly curved shell panels (see Fig. 3(a)). For the closed-circuit (Clc) (see Fig. 3(b)) condition, both the upper and lower piezoelectric layers are grounded cause the electric displacement disappears from Eq. (28). Hence, we obtain the following eigenvalue problem for the Clc boundary condition: The eigensolution of free vibration for PFG-CNTRC-DCP can be obtained using a general eigenvalue approach.  Unless mentioned otherwise, the material properties of PFG-CNTRC doubly curved shell panels are given in Table 3. PmPV [15] CNT [15] Al 2 O 3 [32] PZT-4 [32] PZT-5A [15] ρ/(kg · m −3 ) 1 150 1 400 3 800 7 500 7 750 Also, the CNT efficiency parameters η j (j = 1, 2, 3) associated with a given volume fraction V * CNT are [15] : Firstly, the fundamental frequencies of the simply supported isotropic doubly curved shell panel with surface-bonded piezoelectric layers for different geometric parameters are presented in Table 4. The Opc and Clc of electric boundary conditions are also considered. The shell It should be noted that the results reported by Sayyaadi et al. [32,38] were based on the HSDT. The excellent agreement between the results shows the accuracy of the current approach.

Example 2
Secondly, the free vibration of the laminated cross-ply and angle-ply PFG-CNTRC plates with simply supported boundary conditions is investigated. The fundamental frequencies are given in Table 5 for different CNT volume fractions, CNT distribution types, and electrical boundary conditions (EBC). Properties of the plate are set equal to a = b = 0.4 m, h = a/20, and h p = h/10. The substrate is made of a multi-laminate of armchair SWCNT, and piezoelectric layers are PZT-5A. It can be seen that the present results agree well with those by Nguyen-Quang et al. [15] based on the isogeometric approach and the HSDT. The maximum difference is only 1. Parametric studies are carried out to understand the effects of material properties, geometric parameters, laminate configurations, and electrical boundaries on the free vibration responses of PFG-CNTRC doubly curved shell panels. The substrate of the panel is made of armchair SWCNT, and two piezoelectric layers are PZT-5A. Tables 6-8 list the fundamental frequencies of the CYL, spherical panel (SPH), and hyperbolic paraboloid (HPR) with various inlet parameters. It is observed from these tables that the distribution types of CNT have a significant effect on the stiffness of the panel. In detail, the FG-O panel has the lowest value of frequencies, while the FG-X panels have the highest ones. This conclusion is compatible with the conclusion of other researchers in the related studies in the literature. In three forms of doubly-curved shell panels, the HPR shell panels have the lowest frequencies, and the SPH shell panels have the highest frequencies. These results may become from the fact that SHP has a curvature effect, while the HPR has both positive and negative curvature that neutralize the effect of each other. These tables also reveal that the Opc of electrical boundary conditions always have higher frequencies than the Clc with all other parameters. Figure 4 depicts the effect of distribution types of CNT on the fundamental frequencies of cross-ply PFG-CNTRC doubly curved shell panels for different R x /R y ratio in case of the Opc condition. It is observed that the previous conclusions regarding the CNT distribution types are   Figure 4 also shows that the value of the fundamental frequencies is minimum at the ratio R x /R y = 1. This observation, maybe because of the curvature effect, is suppressed when the panels have the same value of positive and negative curvatures. Moreover, the percentage change of frequency (f PC ) of the SPH panel is shown in Fig. 5, where f PC is defined as [39]

Effect of CNT distribution types
It can be seen from Fig. 5 that among four CNT distribution types, only FG-X shows the positive of f PC , while others show the negative of f PC . These results indicate that only the FG-X panel has higher stiffness than the UD panel, while UD is the simplest distribution type. Figure 6 indicates that the fundamental frequencies of the PFG-CNTRC doubly curved shell panels strongly increase with the increase of CNT volume fractions. The panels are set by a/h = 20, a/b = 1, R x /a = 5, FG-X in Opc condition and [p/0 • /90 • /0 • /p] of configuration.

Effect of electrical boundary conditions
The natural frequency of laminated cross-ply FG-CNTRC doubly curved shell panels couped with closed and open piezoelectric layers are shown in Fig. 7. The parameters of the panels in this example are set by a/h = 20; a/b = 1; R x /a = 5; FG-X; V * CNT = 28%. Figure 7 shows that the FG-CNTRC panels couped with the Opc always vibrate with the higher value of frequencies compared to the FG-CNTRC panels couped with the Clc because the Opc converts electric potential to mechanical energy while the Clc does not.

Effect of curvature of shell panels
The effect of radii of curvature on the natural frequencies of the SPH, CYL, and HPR shell panels are shown in Figs. 8 to 10, respectively. It can be seen from these figures that with the increase of R x /a ratio, the frequencies of SPH and CYL decrease while the frequencies of HPR increase. The frequencies of all three types of shell panels are approximately equal to those of the plate with the corresponding input parameters when the value of R x /a ratio reaches 20. This observation once again confirms that the opposite curvature will reduce the stiffness of the shells. 4.2.6 Effect of piezoelectric layer thickness In order to investigate the effect of piezoelectric layer thickness on the free vibration response of the composite shell panel, the variation of percentage difference in natural frequency β is defined as [40] β = ω with piezoelectric layer − ω without piezoelectric layer ω without piezoelectric layer × 100%. Figures 11, 12, and 13 present the percentage difference in natural frequency β versus the h p /h ratios for different forms of the panel, distribution types of CNT, and CNT volume fractions. It is observed from these figures that the HPR panel has a higher value of β than CYL and SPH panel, the panel reinforced with lower CNT volume fractions has a higher value of β, among five CNT distribution types, the FG-O panel has the highest value of β while the FG-X has lowest one. It can be concluded that the piezoelectric effects are more effective in the case of smaller stiffness panels rather than the case of greater stiffness ones. Furthermore, these figures also indicate that increasing the thickness of piezoelectric layers from zero to a specific value leads to a decrement of β. It can be explained by the fact that the piezoelectric material has a higher mass density and lower elastic moduli compared with the core material.
After that specific value of h p /h, the percentage difference in natural frequency β increases by the increment of piezoelectric layers thickness because the electromechanical coupling effect increases with an increase in h p /h and should increase the value of β. The electromechanical coupling effect is lower than the combined effects of the increase in the mass density, and the decrease in the stiffness results in negative values of β for values of h p /h.

Conclusions
In this paper, an analytical solution based on the four-variable shear deformation refined theory is developed for carrying out the free vibration of the laminated functionally graded nanotube-reinforced composite doubly curved shell panels with surface-bonded piezoelectric layers. Comparison studies validate the accuracy of the model. Numerical results are provided to explore the effects of the CNT volume fraction, the CNT distribution type, the thickness of the piezoelectric layers, laminate configurations, and mechanical and electrical boundary conditions on the natural frequencies of the hybrid panels. Through the present formulation and numerical results, some conclusions can be drawn. (i) The volume fraction of CNT has substantial effects on dynamic responses of PFG-CNTRC panels. (ii) The natural frequencies of PFG-CNTRC panel with Opc electrical boundary condition are always higher than those with the Clc case when all other inlet parameters are the same. (iii) CNT reinforcements distributed close to top and bottom are more efficient than those distributed near the midplane for increasing the stiffness of the panels.
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