1 Introduction

Since piezoelectric materials with deformation can create voltages, they can be used in the construction of sensors. Nanocomposites reinforced by carbon nanotubes with a high strength to density ratio have also many applications in structures. The responses of piezoelectric materials and cylindrical shells have been widely studied.

Loy et al.^[1] analyzed an isotropic homogenous cylindrical shell based on the generalized differential quadrature (GDQ) method, and investigated the effects of different boundary conditions on the natural frequency. Loy et al.^[2] used the strain-displacement relationship and Rayleigh-Ritz method to analyze the cylindrical shells made of functionally gradient material (FGM), and investigated the effects of the configurations of the constituent materials and volume fractions on the natural frequency. Bhangale and Ganesan^[3] considered the free vibration of magneto-electro-elastic cylindrical shells, and used the finite element model to solve the governing equations of motion. Annigeri et al.^[4] studied the vibrations of cylindrical shells under magnetic and electric fields and simply supported boundary conditions, and investigated the effects of piezomagnetism on the natural frequency of the cylindrical shell. Zheng et al.^[5] extended the application of active constrained layer damping (ACLD) to the control of the vibration amplitude of a cylindrical shell, and investigated the dynamic behavior of the cylindrical shell with the ACLD treatments such as response amplitude and natural frequency. Alashti and Khorsand^[6] considered the differential quadrature method (DQM) to study FGM cylindrical shells with piezoelectric layers, and studied the effects of different parameters on the displacement and stress of the cylindrical shells. Barzoki et al.^[7] analyzed the effects of partially filled poly ethylene foam core on the torsional buckling of an isotropic, simply supported piezoelectric polymeric cylindrical shell subjected to combined electro-thermo-mechanical effects by use of the Hamilton principle. Shen and Xiang^[8] investigated the large amplitude vibration behavior of nanocomposite cylindrical shells in thermal environment, and showed that the nonlinear to linear frequency ratios of the carbon nanotube-reinforced composite (CNTRC) shells increased when the temperature increased in most cases. Malekzadeh and Heydarpour^[9] used the centrifugal and Coriolis force to analyze the free vibration of rotating functionally gradient cylindrical shells and archive the equations of motion and the related boundary conditions, and studied the effects of different parameters, e.g., the angular velocity, on the amplitude response and natural frequency. Shen and Xiang^[10] presented the postbuckling of nanocomposite cylindrical shells under combined loadings based on the higher order shear deformation shell theory. Wattanasakulpong and Ungbhakorn^[11] presented the bending, buckling, and vibration behaviors of CNTRC beams, and showed the effects of various parameters such as spring constant factors, carbon nanotube volume fraction on deflection, critical buckling load, and natural frequency. Sun et al.^[12] studied the vibration of rotating cylindrical shells with the Sanders shell theory, and investigated the effects of centrifugal and Coriolis forces under different boundary conditions. Han et al.^[13] used the Love thin shell theory and the Hamilton principle to study the stability of a cylindrical thin shell with periodically time-varying rotating speeds, and showed that increasing the constant rotating speed greatly enhanced the instability regions. Based on Sander's shear deformation theory, Hosseini-Hashemi et al.^[14] obtained the critical speed of a rotating functionally gradient moderately thick cylindrical shell, and showed the effects of different boundary conditions, centrifugal speeds, and material properties of the cylindrical shell on the natural frequency. Jafariet al.^[15] analyzed the nonlinear vibration of functionally gradient cylindrical shells with piezoelectric layers with the Lagrange equations under the assumption of the Donnell nonlinear shallow-shell theory, and showed the effects of excitation force and applied voltage on the vibration behavior of the cylindrical shell. Shen and Yang^[16] investigated the nonlinear vibration response of hybrid composite cylindrical shells, and used a boundary layer theory and perturbation technique to obtain the linear and nonlinear frequency responses of the hybrid composite shells. Lin and Xiang^[17] used the Hamilton principle to present the vibration analysis of nanocomposite beams, and used the p-Ritz method to solve the equation of motion. Ke et al.^[18] studied the nonlocal vibration of piezoelectric cylindrical nanoshells with the Love thin shell theory, and used the DQM to achieve the numerical solutions of piezoelectric nanoshells under various boundary conditions. Beni et al.^[19] analyzed the Navier type solution to study the free vibration of functionally gradient cylindrical nanoshells. Based on the Reddy shell theory and imperfect FGM, Nguyen et al.^[20] presented the nonlinear dynamic response and vibration of thick cylindrical shells on the elastic foundation. Rouzegar and Abad^[21] used the four-variable refined plate theory to study the vibration behavior of a sandwich plate with the Hamilton principle and the Maxwell equation. Li and Pan^[22] analyzed the bending and vibration behaviors of functionally gradient micro plates. Nguyen et al.^[23] analyzed the thermo-electro-mechanical nonlinear vibration of imperfect FGM with thick double curved shallow shells embedded the piezoelectric actuators. Arani et al.^[24] investigated the nonlinear vibration behavior of a nanobeam coupled with a piezoelectric nanobeam with the strain gradient theory, and revealed that increasing the external electric voltage led to the decrease in the dimensionless frequency of the nanobeam. Nguyen^[25] studied the effects of the temperature, material, and geometrical properties on the elastic foundation in the analysis of the nonlinear dynamic behavior of imperfect functionally gradient cylindrical shells with the Reddy's shell theory. Fereidoon et al.^[26] presented the nonlinear vibration of the viscoelastic embedded nanosandwich structures containing a double walled carbon nanotube (DWCNT) integrated with two piezoelectric layers, and indicated that when the surface effect increased, the frequency and critical velocity of fluid increased. Based on the third-order shear deformation theory (TSDT), Mohammadimehr et al.^[27] analyzed the electro elastic solution of sandwich plates with functionally gradient core and composite face sheets made of piezoelectric layers. With the classical laminated plate theory (CLPT), Nasihatgozar et al.^[28] studied the buckling response of piezoelectric cylindrical nanocomposite panels. Based on the high order sandwich plate theory (HSAPT), Mohammadimehr and Mostafavifar^[29] studied the vibration of sandwich plates with a transversely flexible core and nanocomposite face sheets. Based on the higher order shear deformation theory, Nguyen^[30] investigated the dynamic response of functionally gradient cylindrical sandwich shells. Zhang et al.^[31] studied the vibration of functionally gradient nanocomposite triangular plates, and studied the effects of the volume fraction of the carbon nanotube, the isosceles triangular angle, and the carbon nanotube distribution type on the vibration behavior of the plates. Zhang et al.^[32] investigated the large deformation behavior of functionally gradient nanocomposite triangular plates under transversely distributed loads. Based on the meshless method, Lei et al.^[33] investigated the nonlinear vibration of a functionally gradient nanocomposite laminated plate. Liew et al.^[34] examined experimentally the damping and mechanical properties of cementitious nanocomposite structures.

In this article, the free vibration and static bending of a rotating cylindrical sandwich shell will be analyzed with the consideration of nanocomposite core and piezoelectric layers subjected to magnetic field based on the first-order shear deformation theory (FSDT). The governing equations and the corresponding boundary conditions are established through the Hamilton principle and the Maxwell equation. The obtained results show the effects of the applied voltage, which is related to the nanocomposite core, the intensity of magnetic field, the volume fraction of carbon nanotube, and the temperature gradient, on the critical angular velocity. The critical angular velocity of the nanocomposite sandwich cylindrical shell is obtained. The results show that the mechanical properties, e.g., Young's modulus and thermal expansion coefficient, for the carbon nanotube and the matrix are functions of temperature, and the magnitude of the critical angular velocity can be adjusted by changing the applied voltage.

2 Governing equations of motion for a sandwich cylindrical shell

Consider a sandwich cylindrical shell with the average radius R, the core thickness h, the piezoelectric layer h_p, the length L, and a constant angular velocity Ω (see Fig. 1).

2.1 Displacement field of the cylindrical shell

Based on the FSDT, we can express the displacement field of the cylindrical shell as follows^[19]:

(1)

(2)

(3)

where U(x, θ, t), V(x, θ, t), and W(x, θ, t) are considered as the neutral axis displacements, and ψ_x (x, θ, t) and ψ_θ (x, θ, t) denote the rotations of a transverse normal.

The components of the strain field are obtained by the assumption as follows^[19]:

(4)

(5)

(6)

(7)

(8)

2.2 Constitutive equations for nanocomposite core and piezoelectric layers

The linear constitutive equations for the nanocomposite core of sandwich cylindrical shell are expressed as follows^[8]:

(9)

In the above equation, Q₁₁, Q₁₂, Q₂₂, Q₆₆, Q₄₄, and Q₅₅ are the stiffness components of the uniformly distributed nanocomposite core, and are defined by^[8]

E₁₁ and E₂₂ are Young's moduli, G₁₂ is the shear modulus, and υ₁₂ is Poisson's ratio, which are defined by^[8]

(10)

(11)

(12)

(13)

(14)

(15)

where E₁₁^CN, E₂₂^CN, G₁₂^CN, α_xx^CN, and α_θθ^CN are Young's moduli, shear modulus, and thermal expansion coefficient of the carbon nanotube, and E^m and G^m are Young's modulus and shear modulus of the matrix. V_CN and V_m are the volume fractions of the carbon nanotube and the matrix, respectively.

The relationships between the stress and strain for the piezoelectric layers are written as follows^[6]:

(16)

(17)

where (D_xx, D_θθ, D_zz)^T is the electric field intensity vector, (E_x, E_θ, E_z)^T is the electric displacement vector, e₃₁ and e₁₅ are the piezoelectric coefficients, μ₁₁, μ₂₂, and μ₃₃ are the dielectric coefficients, and P₁, P₂, and P₃ are the pyroelectric constants. By ignoring the in-plane electric intensities E_x and E_θ and only considering the electric field through the shell thickness^[15], we have

and

(18)

2.3 The Hamilton principle

The equations of motion for the sandwich cylindrical shell can be derived with the Hamilton principle as follows^[27]:

(19)

where δT and δU are the variations of the kinetic energy and the strain energy, respectively.

2.3.1 Variations of the kinetic energy

The variation of the kinetic energy of the cylindrical shell is expressed based on Eq. (20) as follows^[13]:

(20)

where

(21)

2.3.2 Variations of the strain energy

With the variational method, we can express the strain energy as follows^[15]:

(22)

2.3.3 Variations of the strain energy for a rotating cylindrical shell

Due to this hoop tension, we can derive the variation of the strain energy for a rotating cylindrical shell as follows^[14]:

(23)

2.3.4 Variations of the resultant initial thermal force

The resultant initial thermal force is considered as follows:

(24)

(25)

The variation strain energy of the initial thermal force is considered as follows:

(26)

2.3.5 Variations of the work done by the Lorentz and external force

The electro-dynamic Maxwell equations for the sandwich cylindrical shell can be written as follows^[25]:

(27)

(28)

(29)

where U, H, and η_L are the displacement field (u, v, w), the magnetic field, and the magnetic permeability, respectively. If the magnetic field (0, 0, H_x) is applied in the length of the sandwich cylindrical shell, we can obtain the Lorentz force by Eqs. (27) and (28) as follows:

(30)

(31)

(32)

Then, the variation of the work done by the Lorentz and external force can be calculated as follows:

(33)

2.4 Equations of the rotating sandwich cylindrical shell

The equations of the rotating sandwich cylindrical shell with the consideration of nanocomposite core and piezoelectric layers subjected to the magnetic field by use of the Hamilton principle can be derived as follows:

(34)

(35)

(36)

(37)

(38)

(39)

where the resultant force and moment are defined by

(40)

(41)

(42)

(43)

Substituting Eqs. (16)-(18) into Eqs. (40)-(43) yields

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

where the coefficients are defined in Appendix A.

Substituting Eqs. (44)-(52) into Eqs. (34)-(39) yields the governing equations of motion for the rotating sandwich cylindrical shell as follows:

(53)

(54)

(55)

(56)

(57)

3 Navier's type solution for the sandwich cylindrical shell

The analytical solutions for a simply supported rotating sandwich cylindrical shell with the consideration of the nanocomposite core and piezoelectric layers are obtained with the Navier solution technique. The independent variables U, V, W, ψ_x, ψ_θ, ϕ, and P are extended as follows^[19]:

(58)

(59)

(60)

(61)

(62)

(63)

(64)

where m and n represent the wavenumbers in the axial and circumferential directions, respectively, and

(65)

Substituting Eqs. (58)-(64) into the equations of motion yields

(66)

where the mass and stiffness matrices are presented in Appendix B.

To obtain the natural frequency for Eq. (59), the determinant of the coefficient matrix should be equal to zero. The dimensionless natural frequency of the sandwich plate is defined as follows:

(67)

where (P(x, θ) = P₀). Therefore, we have

(68)

The matrix form of the bending equations for a sandwich cylindrical shell is written as follows:

(69)

The dimensionless deflection of the sandwich cylindrical shell is defined by

(70)

4 Numerical results and discussion

In this section, we will analyze the vibration and static bending of the rotating sandwich cylindrical shell with the consideration of the nanocomposite core and piezoelectric layers subjected to thermal and magnetic fields. The effects of various parameters such as the angular velocity, the geometrical parameters, the volume fraction of the carbon nanotube, the applied voltages on the inner and outer piezoelectric layers, and the magnetic fields on the natural frequency and deflection are investigated.

The used materials in the matrix and resin of the core are polymethylmethacrylate (PMMA) and single-walled carbon nanotubes (SWCNTs) (the length and thickness of the tubes are 9.26 nm and 0.067 nm, respectively, and the average radius of the tubes is 0.68 nm). The relationships between the mechanical properties of the matrix and resin with temperature are expressed as follows:

(71)

(72)

(73)

(74)

(75)

(76)

(77)

Tables 1 and 2 present the efficiency parameters of the carbon nanotube for various volume fractions and the electromechanical properties of the piezoelectric layers, respectively.

4.1 Validation of results

To validate the present analysis, we compare the results for the simply supported isotropic homogeneous cylindrical shell with those of Loy et al.^[1-2] (see Table 3) and the results for the simply supported nanotube-reinforced composite cylindrical shell with the results of Shen and Xiang^[10] (see Table 4). The results show that the present study agrees with those in the literature.

From Tables 3 and 4, it can be seen that the results of the present study agree well with the other results^{[1-2, 10]}.

4.2 Free vibration of sandwich cylindrical shell

The natural frequencies of the cylindrical sandwich shell are obtained by use of Eq. (59). In Fig. 2, we can see the effects of the applied voltages on the dimensionless fundamental natural frequency of the sandwich cylindrical shell. If the applied voltages are symmetric with respect to the nanocomposite core (V₁=100 V and V₂=-100 V), the natural frequency will be greater than that when no voltage is applied to the sandwich cylindrical shell. The applied asymmetric voltages (V₁=100 V and V₂=100 V) also decrease the natural frequency of the sandwich cylindrical shell. The dependence of the gradient dimensionless frequency on temperature in the BNN is more than that in the PZT-A4. It is seen that when the temperature change is enhanced, the stiffness of the microstructures decreases, which will lead to a decrease in the natural frequency of the sandwich cylindrical shell. When the two voltages (V₁ and V₂) are positive, the stiffness of the micro sandwich shell structures will decrease similarly with the temperature change.

Figure 3 illustrates the effects of the angular velocity on the natural frequency by use of the FSDT. It can be seen that when the angular velocity increases, the dimensionless natural frequency enhances. Figure 4 demonstrates the graph of the dimensionless natural frequency versus the gradient temperature for magnetic field with different intensities. From the figure, we can see that, the dimensionless frequency reduces when the intensity of the magnetic field increases. It is noted that the relationship between the intensity of the magnetic field and the stiffness of the sandwich structure is similar to that between the temperature change and the applied voltages. Moreover, with the consideration of the applied voltage and intensity of the magnetic field, the resonance phenomenon in the rotating sandwich cylindrical shell with the nanocomposite core can be controlled.

The effects of the thickness of the core and the thickness of the piezoelectric layer (h/h_p) (thickness ratio) on the dimensionless natural frequency are plotted in Fig. 5. It is seen that the dimensionless natural frequency of the sandwich cylindrical shell is quite sensitive to h/h_p. It is seen that, when this ratio increases, the dimensionless frequency increases, the stiffness of the microstructures increases, and thus the natural frequency increases. Figure 6 discusses the effects of the volume fraction on the dimensionless natural frequency. For the two material piezoelectric layers, i.e., PZT-Z4 and BNN, the dimensionless frequency increases when the volume fraction increases because of the increase in the stiffness of the sandwich cylindrical shell.

The intersections of the bisector graphs with the angular velocity curves are plotted in Figs. 7-10. Figure 7 shows that, when the intensity of the magnetic field increases, the critical angular velocity of the sandwich cylindrical shell decreases. Figure 8 shows that, if the applied voltages are symmetric with respect to the nanocomposite core (V₁=100 V and V₂=-100 V), the critical angular velocity will be greater than that when no voltage is applied to the sandwich cylindrical shell. Figure 9 shows that, when the volume fraction of the carbon nanotube increases, the critical angular velocity increases. Figure 10 discusses the effects of the gradient temperature on the critical angular velocity. It is clear that, when the gradient temperature increases, the critical angular velocity decreases because of the decreasing stiffness of the sandwich cylindrical shell.

The effects of the angular velocity on the deflection is shown in Table 5. It is found that, when the magnitude of the angular velocity increases, the maximum deflection decreases. It can be seen from Fig. 11 that the applied voltages on the sandwich cylindrical shell lead to an increase in the deflection of the sandwich structure. The deflection of the sandwich cylindrical shell with symmetric applied voltages is more than that with asymmetric applied voltages. Figures 12 and 13 show that, when the intensity of the magnetic field and gradient temperature increase, the maximum deflection increases. Moreover, when the intensity of the magnetic field and gradient temperature decrease, the stiffness of the microstructure decreases, which leads to an increase in the maximum deflection and a decrease in the natural frequency.

5 Conclusions

In this article, we analyze the static bending and vibration behaviors of a rotating sandwich cylindrical shell with the consideration of the nanocomposite core and piezoelectric layers subjected to thermal and magnetic fields. The FSDT of shells is used to model the displacement field of the sandwich cylindrical shell. With the variational method, the governing equations of motion for the sandwich cylindrical shell are obtained. The Navier type solution is used to derive the dimensionless deflection and natural frequency of the sandwich cylindrical shell, and the mechanical properties, e.g., Young's modulus and thermal expansion coefficient, for the carbon nanotube and matrix are expressed as functions of temperature.

The results show that, if the applied voltages are symmetric with respect to the nanocomposite core, i.e., V₁=100 V and V₂=-100 V, the natural frequency and the critical angular velocity will be larger than that when no voltage is applied to the sandwich cylindrical shell. If the applied voltages are asymmetric, i.e., V₁=100 V and V₂=100 V, the natural frequency of the sandwich cylindrical shell will decrease.

Moreover, when the angular velocity, the volume fraction of the carbon nanotube, and the thickness ratio (h/h_p) increase, the dimensionless natural frequency will increase. When the applied voltages are symmetric with respect to the nanocomposite core, the dimensionless natural frequency and critical angular velocity will increase. When the applied voltages are asymmetric with respect to the nanocomposite core, the dimensionless frequency and critical angular velocity will decrease. When the intensity of the magnetic field and gradient temperature increase, the boundary stability decreases. The increase in the applied voltages, which are either symmetric or asymmetric, can increase the maximum deflection of the sandwich cylindrical shell. When the magnitude of the angular velocity increases, the maximum deflection decreases, and vice versa. When the intensity of the magnetic field and the gradient temperature increase, the maximum deflection increases.

Appendix A

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

(A13)

(A14)

(A15)

(A16)

(A17)

(A18)

(A19)

(A20)

(A21)

(A22)

Appendix B

(B1)

(B2)

(B3)

(B4)

(B5)

(B6)

(B7)

(B8)

(B9)

(B10)

(B11)

(B12)

(B13)

(B14)

(B15)

(B16)

(B17)

(B18)

(B19)

(B20)

(B21)

(B22)

(B23)

(B24)

(B25)

(B26)

(B27)

(B28)

Acknowledgements The authors would like to thank the referees for their valuable comments. Also, they are thankful to the University of Kashan for supporting this work.