1 Introduction

Due to the excellent electrical and mechanical properties, piezoelectric materials have been utilized in many engineering fields, e.g., aerospace engineering, medical imaging, and mechanical engineering. The coupled electromechanical properties of piezoelectric materials make them well suited for sensors, actuators, resonant ultrasonic inspection devices, micro piezoelectric power generators, etc.^[1-6]. By combining piezoelectric material technology with nanotechnology, the piezoelectric nanostructure based sensing technology can be developed. Since there are more than 6 000 different types of sensors in an aircraft, the interconnected cabling and maintenance are costly. It is expected that the combination of piezoelectric nanostructure based sensing technology and wireless transmission can provide promising low-cost energy storage solution in aircrafts.

Pan et al.^[7] first reported the ZnO piezoelectric nanostructures on Science. Subsequently, various piezoelectric nanomaterials and nanostructures are concerned and reported^[8-12]. Piezoelectric nanostructures possess significant physical/chemical properties, e.g., thermal properties, electrical properties, and mechanical properties. Therefore, they are considered as ideal candidates for constructing nanodevices^[13-15].

Piezoelectric nanostructures have the dimension ranging from a few nanometers to several hundred nanometers, where the effects of the scale parameters need to be considered. The size effect of piezoelectric nanostructures has been observed in experiments. Using a piezoresponse force microscopy, Zhao et al.^[16] observed that the effective piezoelectric coefficient of ZnO nanowires was much larger than that of the bulk ZnO. Chen et al.^[17] reported size-dependent Young's modulus in ZnO nanowires. They showed that Young's modulus for the nanowires with diameters smaller than 120 nm increased dramatically when the diameter decreased. These studies give important evidences for the size-dependent material properties of piezoelectric nanostructures. Therefore, the size effect must be considered in both theoretical and experimental studies on piezoelectric nanostructures.

The nonlocal elasticity theory incorporating the size effect was proposed by Eringen^[18-20]. Ke and Wang^[21] first extended this nonclassical continuum theory to study the size-dependent mechanical behavior of piezoelectric nanostructures. Wang and Wang^[22] studied the surface and nonlocal effects on the electromechanical coupling behavior of a piezoelectric nanowire. Ansari et al.^[23] investigated the free vibration of piezoelectric nanobeams in the vicinity of postbuckling domain via the nonlocal elasticity theory. Ke et al.^[24] studied the linear thermo-electro-mechanical vibration of piezoelectric cylindrical nanoshells under various boundary conditions. Based on the Kirchhoff plate theory, Jandaghian and Rahmani^[25] investigated the linear vibration of functionally graded piezoelectric nanoplates. Razavi et al.^[26] studied the electromechanical vibration of a functionally graded piezoelectric cylindrical nanoshell. Using Love's thin shell model, Kheibari and Beni^[27] studied the linear vibration of single-walled piezoelectric nanotubes. It is noted that all the above-mentioned studies are concentrated on the linear vibrations of piezoelectric nanostructures. The nonlinear vibration investigation on piezoelectric nanostructures, however, is very limited in the literature. Ke et al.^[28] studied the nonlinear vibration of piezoelectric nanobeams based on the nonlocal elasticity theory. Asemi et al.^[29] analyzed the nonlinear vibration of piezoelectric nanoelectro-mechanical resonators. Ansari and Gholami^[30] investigated the nonlinear free vibration of magneto-electro-thermo-elastic nanoplates. Liu et al.^[31] studied the nonlinear free vibration of rectangular piezoelectric nanoplates resting on the Winkler foundation. All these studies focus on piezoelectric nanoscale beams and plates.

For solving the nonlinear vibration problem of structures, different analytical and numerical methods have been used and proposed. Wang^[32] and Wang and Zu^[33-34] used the harmonic balance method to investigate the nonlinear vibration of functionally graded, piezoelectric, and porous plates. Ding et al.^[35] proposed a simple and efficient method to determine the convergence of the modal truncation of the vibration analysis of continuum on elastic foundations. This facilitates the vibration analysis of complex elastic systems^[36]. Wang et al.^[37] used the multiple-scale method to study the nonlinear vibration of moving plates immersed in fluids.

In this paper, the nonlinear vibration of piezoelectric nanoshells resting on an elastic foundation, subjected to thermo-electro-mechanical loads, is first proposed. The Donnell nonlinear shell theory and the nonlocal elasticity theory are used to model the system. The governing equations and boundary conditions are developed by using the Hamilton principle. The Galerkin method together with the multiple-scale method is used to approximately analyze the nonlinear vibration of the piezoelectric nanoshells.

2 Preliminaries 2.1 Nonlocal elasticity theory for piezoelectric materials

In Eringen's nonlocal elastic theory^[20], the stress field at a reference point in an elastic continuum depends not only on the strain at that point but also on the strains at all other points. The nonlocal elasticity theory can well explain some phenomena related to the scales of atoms and molecules, e.g., high-frequency vibration and wave scattering. The most general form of the constitutive relation for nonlocal elasticity involves an integral over the whole body. The basic equations for uniform and non-local piezoelectric solids can be written as follows^{[18, 24]}:

(1)

(2)

(3)

where i, j, k, l =1, 2, 3. σ_ij, ε_ij, D_i, E_i, and u_i represent the components of the stress, the strain, the electric displacement, the electric field, and the displacement, respectively. c_ijkl, e_kij, s_ik, β_ij, p_i, and ρ are the components of the elasticity tensor, the piezoelectric tensor, the dielectric tensor, the thermal modulus tensor, the pyroelectric vector, and the mass density, respectively. ΔT denotes the temperature change. is the electric potential. α₀(|x'-x|, e₀ a/l) is the nonlocal kernel function. e₀a/l is the scale parameter. e₀a is the scale coefficient revealing the size effect on the response of nanostructures. e₀ is a material constant determined experimentally or approximated by matching the dispersion curves of the plane waves with those of the atomic lattice dynamics. a and l represent the internal and external characteristic lengths of the nanostructures, respectively. x' represents the coordinate of any material point in the area except x. |x'-x| represents the Euclidean distance.

Equivalent differential forms can be used to represent the overall constitutive relation as follows^[19]:

(4)

(5)

where is the Laplace operator.

2.2 Nonlocal piezoelectric cylindrical nanoshell model

Consider a piezoelectric cylindrical nanoshell composed of PZT-4 and resting on an elastic foundation. Figure 1 shows the geometry of the nanoshell with the length L, the middle-surface radius R, and the thickness h. The elastic medium is stimulated by employing the Winkler-Pasternak foundation model. Additionally, the nanoshell is subjected to an electric potential (x, θ, z, t) and a uniform temperature change ΔT. u(x, θ, t), v(x, θ, t), and w(x, θ, t) are the displacements of the points in the middle plane of the piezoelectric nanoshell along the x-, θ-, and z-axes, respectively.

According to the Kirchhoff-Love hypothesis, the displacement fields are^[38]

(6)

(7)

(8)

where t is the time, and u₁ (x, θ, z, t), u₂ (x, θ, z, t), and u₃ (x, θ, z, t) are the displacements of an arbitrary point of the nanoshell along the x-, θ-, and z-axes, respectively.

Based on Donnell's nonlinear shell theory, the strain components ε_xx, ε_θθ, and γ_xθ at an arbitrary point of the nanoshell are^[39]

(9)

where ε_{x, 0}, ε_{θ, 0}, and γ_{xθ, 0} represent the middle-surface strains, k_x, k_θ, and k_xθ are the curvature and torsion of the middle surface, and z is the distance of the arbitrary point of the nanoshell from the middle surface.

The corresponding expressions in the above equation are

(10)

(11)

Following Wang^[40], the distribution of the electric potential along the thickness of the piezoelectric nanoshell is assumed as follows:

(12)

where β=π/h, Φ(x, θ, t) is the spatial and time variation of the electric potential in the x- and θ-directions, and V₀ represents the initial external electric potential applied to the piezoelectric nanoshell.

Using Eq. (12), the electric field components E_i (i=x, θ, z) can be written as follows^[41]:

(13)

(14)

For the piezoelectric cylindrical nanoshell, the nonlocal constitutive relationship of Eqs. (4) and (5) can be given by^[42-43]

(16)

(17)

(18)

(19)

(20)

(21)

Thereinto, , and are defined as follows:

(22)

The strain energy Π_s of the piezoelectric cylindrical nanoshell can be written as follows:

(23)

in which the resultant forces and moments can be, respectively, calculated as follows:

(24)

(25)

The kinetic energy Π_k is expressed as follows^[44]:

(26)

where , and the rotatory inertia term is neglected due to its slight effect.

Moreover, the work Π_F1 done by the external forces can be given by

(27)

where (N_Tx, N_Tθ) and (N_Ex, N_Eθ) are the thermal and electrical forces induced by the uniform temperature rise ΔT and the uniform external electric potential V₀, respectively, and

(28)

The work Π_F2 done by the nonlinear Winkler-Pasternak foundation can be calculated as follows:

(29)

where k_w is the Winkler foundation parameter, and k_p is the Pasternak foundation parameter.

Using Hamilton's principle

(30)

and substituting Eqs. (23), (26), (27), and (29) into Eq. (30), we can obtain the following differential equations of motion:

(31)

(32)

(33)

(34)

where

(35)

The corresponding boundary conditions are

(36)

(37)

(38)

(39)

(40)

(41)

where n_x and n_θ denote the directional cosines of the outward unit normal to the boundaries of the midplane.

From Eqs. (16)-(21), we obtain

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

where

Substituting Eqs. (42)-(50) into Eqs. (31)-(34) yields

(51)

(52)

(53)

(54)

where

(55)

(56)

It is assumed that the electric potential is equal to zero at both ends of the nanoshell. Then, the simply supported boundary condition can be written as follows:

(57)

3 Solution procedure

In this study, we consider that the piezoelectric cylindrical nanoshell is simply-supported at both ends. Therefore, the displacement functions can be written as follows:

(58)

(59)

(60)

(61)

where m is the number of the axial half waves, and n is the number of the circumferential waves. u_mn(t), v_mn(t), w_mn(t), and Φ_mn(t) are displacement amplitude components.

By substituting Eqs. (58)-(61) into Eqs. (51)-(54) and then utilizing the Galerkin method, we can obtain the following ordinary differential equations:

(62)

(63)

(64)

(65)

where the coefficients G_{i, j} are given in Appendix A.

In Eqs. (62) and (63), the inertia terms and can be neglected due to their insignificant effects compared with the transverse inertia term. Thus, by solving Eqs. (62), (63), and (65) with respect to u_mn, v_mn, and Φ_mn and then inserting the results in Eq. (64), we have

(66)

where

(67)

(68)

(69)

From Eq. (66), the linear circular frequency can be obtained as follows:

(70)

The initial conditions are set as follows:

(71)

where w_max denotes the maximum of w_mn(t).

For the convenience of calculation, let us introduce the following dimensionless variables:

(72)

Applying Eq. (72) into Eqs. (66) and (71) yields

(73)

(74)

where

Then, we use the multiple-scale method^[45-46] to solve Eq. (73). First, let us introduce the scaled time as follows:

(75)

where ε is a small dimensionless parameter, and is used as a bookkeeping device.

The time derivatives can be written in terms of T_n as follows:

(76)

where

(77)

The response is the function of different scaled time, and thus can be written as follows:

(78)

Substituting Eqs. (76) and (78) into Eq. (73) and equating the coefficients of like powers of ε, we obtain

(79)

(80)

(81)

The solution of Eq. (79) can be assumed as follows:

(82)

Substituting Eq. (82) into Eq. (80) gives

(83)

To eliminate the secular terms in Eq. (83), should be satisfied. Thus, we have

(84)

Introducing Eqs. (82) and (84) into Eq. (81) yields

(85)

The solvability condition of the above equation requires

(86)

In order to solve the above equation, we seek the solution in the form of

(87)

Utilizing Eq. (87) in Eq. (86) and separating the real and imaginary parts, we obtain

(88)

The integral of Eq. (88) gives

(89)

Thus, Eq. (87) can be rewritten as follows:

(90)

Substituting Eq. (90) into Eq. (78) yields

(91)

where ω_NL is the normalized nonlinear frequency of the piezoelectric cylindrical nanoshell.

Substituting the initial conditions into Eq. (74), we obtain

(92)

The results are presented as the nonlinear frequency ratio calculated as follows:

(93)

where

4 Results and discussion

For the purpose of examining the validity of the present analysis, the linear natural frequencies of a macroscopic isotropic thin cylindrical shell are calculated and compared with the literature in Table 1. The material and geometric parameters of the shell are as follows:

The results show good agreement.

The second comparison is related to a nanoshell incorporating the size effect and piezoelectric effect. Table 2 lists the comparison of the linear natural frequencies of a PZT-4 piezoelectric nanoshell with

The material properties of PZT-4 are shown in Table 3. It is found that the present results agree well with those given by Ke et al.^[24], bespeaking the validity of the present study.

In order to validate the present nonlinear analysis, we consider anisotropic homogeneous cylindrical shell to make a comparison of nonlinear results in Table 4. The following parameters are used:

The comparison study shows that the results obtained from the present method are in good agreement with the existing ones. The small differences between the present results and those of Raju and Rao^[50] and Rafiee et al.^[51] might be due to the different shell theories and different solving processes.

In what follows, we will analyze the nonlinear vibration of the piezoelectric cylindrical nanoshell shown in Fig. 1. The material properties of the nanoshell are displayed in Table 3. If not specified, the following parameters of the nanoshell are considered:

Figure 2 shows the variations of the linear natural frequency against the circumferential wave number of the piezoelectric nanoshell. It can be found that the natural frequency of the nanoshell initially decreases and then increases when the circumferential wave number increases. The fundamental natural frequency occurs at n=5. Therefore, the mode (m=1, n=5) corresponds to the lowest natural frequency of the piezoelectric nanoshell. This mode is chosen as a representative mode in the following discussion.

Figure 3 illustrates the nonlinear frequency ratio versus the dimensionless vibration amplitude of the piezoelectric nanoshell for different nonlocal parameters. Obviously, the nonlinear frequency ratio increases with the increase in the dimensionless vibration amplitude. The nanoshell exhibits a hardening behavior. At a given vibration amplitude, a larger nonlocal parameter leads to smaller linear and nonlinear frequencies but a higher nonlinear frequency ratio. The reason is that the nonlocal effect tends to decrease the stiffness of the nanoshell and hence decrease the values of the linear and nonlinear frequencies. Moreover, the nonlinear frequency drops faster than the linear frequency, which results in the increase in the nonlinear frequency ratio.

Figure 4 presents the effects of the external electric potential on the nonlinear frequency ratio of the piezoelectric nanoshell. As we can see, the positive/negative potential decreases/increases the linear and nonlinear frequencies of the piezoelectric nanoshell. This is due to the fact that the axial compressive and tensile forces are generated in the nanoshell by the applied positive and negative potentials, respectively. At a given vibration amplitude, it is found that a change in the external electric potential from -0.000 2 V to 0.000 2 V leads to the increase in the nonlinear frequency ratio.

Figure 5 shows the frequency ratio versus the external electric potential for different dimensionless vibration amplitudes of the piezoelectric nanoshell, where the frequency ratio is defined as the non-classical nonlinear frequency to its classical counterpart. Since the classical nonlinear frequency is calculated without the consideration of the nonlocal effect, i.e., e₀a=0, the frequency ratio increases with the increase in the external electric potential. This characteristic indicates that, at a larger external electric potential, the size effect on the vibration characteristics of the piezoelectric nanoshell is more notable.

The effects of the temperature change on the nonlinear frequency ratio of the piezoelectric nanoshell are depicted in Fig. 6. It can be found that when the temperature change increases, the linear and nonlinear frequencies decrease accordingly. The reason is that a larger temperature change results in a reduction in the nanoshell stiffness, and hence leads to lower linear and nonlinear frequencies of the nanoshell. Additionally, at a given vibration amplitude, the nonlinear frequency ratio increases with the increase in the temperature change. Comparing the effects of the external electric potential and temperature change on the nonlinear vibration behaviors (see Figs. 4 and 6), we find that the external electric potential has a greater effect than the temperature change from the numerical point of view. This phenomenon can be explained because the coefficients of the thermal force N_T and the electrical force N_E have the relationship , as deduced above.

The frequency ratio versus the temperature change for different dimensionless vibration amplitudes of the piezoelectric nanoshell is displayed in Fig. 7. It is observed that the increasing temperature change leads to the increase in the frequency ratio. This characteristic is quite obvious when the dimensionless vibration amplitude is large, showing that the nonlocal effect is quite significant when the dimensionless vibration amplitude is large.

The nonlinear frequency ratio versus the dimensionless vibration amplitude of the piezoelectric nanoshell for different Winkler foundation parameters is plotted in Fig. 8. It can be found that when the Winkler foundation parameter increases, the linear and nonlinear frequencies increase, whereas the nonlinear frequency ratio decreases.

Figure 9 illustrates the effects of the Winkler foundation parameter on the frequency ratio of the piezoelectric nanoshell. One can find that with the increase in the Winkler foundation parameter, the frequency ratio decreases. This indicates that the nonlocal effect is weaker when the foundation parameter becomes larger. In addition, at a given Winkler foundation parameter, the nonlocal effect is more pronounced when the dimensionless vibration amplitude is larger.

Figure 10 presents the nonlinear frequency ratio versus the dimensionless vibration amplitude of the piezoelectric nanoshell for different Pasternak foundation parameters. It is found that when the Pasternak foundation parameter increases, the linear and nonlinear frequencies increase while the nonlinear frequency ratio decreases. Figure 11 shows the effects of the Pasternak foundation parameter on the frequency ratio of the piezoelectric nanoshell. It can be found that the frequency ratio decreases with the increase in the Pasternak foundation parameter, which means that the Pasternak foundation can weaken the nonlocal effect.

The effects of the length-to-radius ratio on the nonlinear frequency ratio are presented in Fig. 12. At a given vibration amplitude, the nonlinear frequency ratio increases when the length-to-radius ratio increases. Figure 13 shows the frequency ratio versus the length-to-radius ratio of the piezoelectric nanoshell for different dimensionless vibration amplitudes. It is found that when the length-to-radius ratio increases, the frequency ratio increases initially, and then tends to keep constant.

5 Conclusions

The nonlinear free vibration of piezoelectric cylindrical nanoshells resting on a Winkler-Pasternak foundation, subjected to thermal and electrical loads, is investigated. The theoretical model is established based on Donnell's nonlinear shell theory and Eringen's nonlocal elasticity theory. The nonlinear governing equations and boundary conditions are derived by using Hamilton's principle. With the Galerkin method, the partial differential equations are discretized into a set of nonlinear ordinary differential equations. Afterwards, the method of multiple scales is used to approximately analyze the nonlinear free vibration of the piezoelectric nanoshells. The results are summerized as follows:

(ⅰ) Larger vibration amplitudes lead to higher nonlinear frequency ratios, and higher nonlocal parameters lead to lower linear and nonlinear frequencies but higher nonlinear frequency ratios.

(ⅱ) A change in the external electric potential from a negative value to a positive one leads to the increase in the nonlinear frequency ratio, and the nonlinear frequency ratio increases with the increase in the temperature change. Moreover, when the external electric potential and the temperature change increase, the nonlocal effects on the vibration characteristics of the piezoelectric nanoshells increase.

(ⅲ) The nonlinear frequency ratio decreases when the length-to-radius ratio decreases. The nonlocal effect is quite significant at small length-to-radius ratios, but does not change notably when the length-to-radius ratio reaches a certain value.

(ⅳ) The Winkler-Pasternak foundation results in the increase in the linear and nonlinear frequencies but the decrease in the nonlinear frequency ratio of the piezoelectric nanoshells. Moreover, the Winkler-Pasternak foundation can weaken the nonlocal effect on the piezoelectric nanoshells.

Appendix A