Thermo-Electro-Mechanical Vibrations of Porous Functionally Graded Piezoelectric Nanoshells

In this work, we aim to study free vibration of functionally graded piezoelectric material (FGPM) cylindrical nanoshells with nano-voids. The present model incorporates the small scale effect and thermo-electro-mechanical loading. Two types of porosity distribution, namely, even and uneven distributions, are considered. Based on Love’s shell theory and the nonlocal elasticity theory, governing equations and corresponding boundary conditions are established through Hamilton’s principle. Then, natural frequencies of FGPM nanoshells with nano-voids under different boundary conditions are analyzed by employing the Navier method and the Galerkin method. The present results are verified by the comparison with the published ones. Finally, an extensive parametric study is conducted to examine the effects of the external electric potential, the nonlocal parameter, the volume fraction of nano-voids, the temperature rise on the vibration of porous FGPM cylindrical nanoshells.


Introduction
Piezoelectric materials are characterized by the excellent coupling between the electric and mechanical fields. Applying mechanical load to piezoelectric materials generates an electric field, while putting piezoelectric materials in an electric field creates mechanical strain in them. This two-way property has made piezoelectric materials ideal for making actuators and sensors [1][2][3][4]. Besides, the two-way action of turning mechanical energy to electric energy and vice versa has made piezoelectric materials useful in resonant ultrasonic inspection and micro/nano piezoelectric power generators [5][6][7].
Unfortunately, there are some deficiencies such as low resistance to external loads, creeping in high temperature, and high stress concentration in homogeneous piezoelectric materials. In order to eliminate these problems, functionally graded piezoelectric materials (FGPMs) were proposed. The concept of functionally graded materials was first proposed in the 1980s [8]. Functionally graded materials are generally composed of two different materials, and are characterized by continuous variations in both mechanical properties and material composition in one or more dimension(s). Likewise, FGPMs are generally composed of two different piezoelectric materials. They have many advantages such as multifunctionality, ability to control deformation, and minimization or removal of stress. Hence, FGPMs have received wide engineering applications [9][10][11][12][13]. In FGPMs, owing to the technical issues, nano-voids or porosities may occur within materials. It is reported that a considerable number of nanopores appeared in the functionally graded material during the preparation process

Nonlocal Elasticity Theory for FGPMs
In Eringen's nonlocal elastic theory [19][20][21], nonlocal constitutive equations are written as [19,32]: in which i, j, l, k = 1, 2, 3; ε ij , σ ij , u i , E i and D i denote the components of the strain, stress, displacement, electric field, and electric displacement, respectively; e kij , c ijkl , p i , β ij and s ik represent the components of the piezoelectric tensor, elasticity tensor, pyroelectric vector, thermal modulus tensor and the dielectric tensor, respectively; denotes the mass density; Φ and ∆T are the electric potential and temperature change, respectively; α 0 (|x − x|, e 0 a/l e ) is the nonlocal kernel function; e 0 a/l e represents the scale parameter; x' represents all material point coordinates except x point in the area; |x − x| represents the Euclidean Distance. Equivalent differential forms can be used to represent the overall constitutive relation as follows [20]: in which ∇ 2 is the Laplace Operator.

Nonlocal Porous FGPM Cylindrical Nanoshell Model
Consider a porous FGPM cylindrical nanoshell composed of PZT-5H and PZT-4. Figure 1 shows the geometry of the nanoshell with the thickness h, the middle-surface radius R and the length L. The FGPM nanoshell is supposed to contain nano-voids that disperse evenly (FGPM-I) or unevenly (FGPM-II) along the thickness direction. Additionally, the nanoshell is subjected to a uniform temperature change ∆T and electric potential Φ(x, θ, z, t). U(x, θ t), V(x, θ, t) and W(x, θ, t) are displacements of points at the middle plane of the shell in x-, θ-and z-axes directions, respectively.
For the FGPM-I nanoshell, the general material properties are given by [37] where z is the distance from the mid-surface of the FGPM cylindrical nanoshell; P 4 and P 5H are material properties of PZT-4 and PZT-5H, respectively; α is the porosity volume fraction. Therefore, the elastic constants c ij , the piezoelectric constants e ij , the mass density ρ, and the dielectric constants s ij of the FGPM-I nanoshell can be expressed as: For the FGPM-II nanoshell, on the other hand, the material properties in Equations (9)- (12) can be replaced by [38] According to the Kirchhoff-Love hypothesis, the displacement fields are [39]: in which t is time, and u(x, θ, z, t), v(x, θ, z, t) and w(x, θ, z, t) are the displacements of an arbitrary point along the x-, θ-and z-axes, respectively. Using Love's first approximation shell theory, the strain-displacement relations can be written as [40]: Following Wang [41], the distribution of electric potential along the thickness of the FGPM nanoshell is assumed as: in which β = π/h; V 0 represents the initial external electric voltage applied to the FGPM nanoshell; Φ(x, θ, t) represents the spatial and time variation of the electric potential in the x-direction and θ-direction.
Using Equation (23), the electric field components E i are given by For the porous FGPM cylindrical nanoshell, the nonlocal constitutive relationship (5) and (6) can be given by [42,43] in which ∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂(Rθ) 2 ; c ij , e ij , s ij , β 11 , β 22 The strain energy Π s of the porous FGPM cylindrical nanoshell is expressed as follows: Substituting Equations (20)- (22) and Equations (24)- (26) into Equation (29) gives in which the resultant forces and the moments can be respectively calculated as The kinetic energy Π k is given by: −h/2 ρ(z)dz, and the rotatory inertia term is neglected due to its slight impact. Moreover, the work Π F done by external forces can be written as: in which (N Ex ,N Eθ )and (N Tx ,N Tθ ) are the electrical and thermal forces induced by the uniform external electric voltage V 0 and uniform temperature rise ∆T, respectively. They are given by Using Hamilton's principle [44,45]: and applying Equations (29), (33), and (34), it yields the governing equations: The corresponding boundary conditions are: where n x and n θ denote the direction cosines of the outward unit normal to the boundaries of the mid-plane. From Equation (27), we obtain the following equations: where Substituting Equations (48)- (56) into Equations (37)-(40) gives The electric potential at both ends of the FGPM nanoshell is assumed to be zero. Then, the associated boundary conditions are expressed as for a simply-supported end, and for a clamped end.

Navier Procedure
For the porous FGPM cylindrical nanoshell with simply supported-simply supported (SS-SS) boundary condition, analytical solutions of the free vibration problem can be obtained utilizing Navier's method. For this purpose, the following displacement functions which satisfy the SS-SS boundary condition are introduced: W(x, θ, t) = W mn sin mπx L cos(nθ)e iωt (65) where U mn , V mn , W mn and Φ mn represent the displacement amplitude components; m and n are mode numbers; ω represents the natural circular frequency of the porous FGPM nanoshell. Substituting Equations (63)-(66) into Equations (57)-(60), the following equation can be obtained      q 11 q 12 q 13 q 14 q 21 q 22 q 23 q 24 q 31 q 32 q 33 q 34 q 41 q 42 q 43 q 44 The elements in the above matrix are given in the Appendix A. Equation (67) gives the characteristic equation for the natural frequencies of the porous FGPM cylindrical nanoshell. To obtain a nontrivial solution, the determinant of the coefficient matrix must be set to zero.

Inserting Equations (68)-(71) in Equations
in which the matrices M and K are the mass matrix and stiffness matrix of the porous FGPM cylindrical nanoshell, respectively. To find the non-zero solutions, the determinant of the coefficient matrix must be equal to zero. Then, natural frequencies of FGPM nanoshells with nano-voids can be determined [47][48][49][50][51][52][53][54].

Results and Discussion
For examining the validity of the present analysis, the comparison is performed on natural frequencies of a PZT-4 piezoelectric cylindrical nanoshell. Tables 2-4 Table 5. It is found that the present results match those given by Ke et al. [31] very well, bespeaking the validity of the present study.  Table 5. Material properties of PZT-4 and PZT-5H [31,55].

Material PZT-4 PZT-5H
Elastic In the following sections, free vibration of the porous FGPM cylindrical nanoscale shell shown in Figure 1 is performed; the material properties of the nanoshell are displayed in Table 5. If not specified, the following parameters are used: In Tables 6-8, the variation of natural frequency of the FGPM-I nanoshell against the circumferential wave number is represented for different porosity volume fractions and different boundary conditions, where N = 20. Among them, α = 0 corresponds to the FGPM cylindrical nanoshell without nano-voids. The results reveal that the natural frequency decreases as the porosity volume fraction increases. With the increase of the circumferential wave number, it is seen that the natural frequency decreases first and then increases. In addition, under the same condition, the SS-SS porous FGPM nanoshell has the lowest natural frequency while the C-C one has the highest natural frequency. This is because the end support is the weakest (in terms of stiffness) for the SS-SS FGPM nanoshell and the strongest for the C-C one. Under the SS-SS boundary condition, it is seen that the minimum natural frequency occurs at n = 3. Therefore, the fundamental frequency of the SS-SS FGPM nanoshell corresponds to mode (m = 1, n = 3). In the next studies, the SS-SS FGPM nanoshell is taken as an example and the mode (1,3) is chosen as a representative mode. Natural frequency against the radius-to-thickness ratio for different porosity volume fractions is plotted for the FGPM-I nanoshell in Table 9. As the radius-to-thickness ratio increases, one can see that the natural frequency decreases initially and then increases; the frequency does not change monotonously with the radius-to-thickness ratio. Table 9. Variation of natural frequency ω(GHz) against the radius-to-thickness ratio R/h for different porosity volume fractions of FGPM-I nanoshell (n = 3, L = 300 h, N = 20).  Figure 2 presents the effect of temperature change on the natural frequency of the FGPM-I nanoshell. The natural frequency decreases with the increase of temperature change. This is due to the fact that the larger temperature change results in a reduction in the stiffness and hence leads to the lower natural frequency of the porous FGPM nanoshell.  Table 10 illustrates the natural frequency against the circumferential wave number for different power-law indexes of the FGPM-I nanoshell. The natural frequencies of the FGPM nanoshell decreases with the increase of the power-law index. Additionally, it is seen that the fundamental natural frequency occurs at mode (m = 1, n = 3), which does not change with the power-law index.  Figure 3 gives the variation of the natural frequency against the length-to-radius ratio for different power-law indexes. As a whole, it is observed that the natural frequency is quite susceptive to the length-to-radius ratio when this ratio is small; the frequency drops quickly as the length-to-radius ratio increases of the porous FGPM nanoshell. However, when L/R > 15, the natural frequency is no longer sensitive to the length-to-radius ratio change.  Figure 4 presents the variation of natural frequency against the radius-to-thickness ratio for different power-law indexes. The natural frequency decreases first and then increases as the radius-to-thickness ratio increases. A trend can also be observed that the natural frequency decreases gradually with the increase of the power-law index. The variation of the natural frequency against external electric potential V 0 for different power-law indexes is presented in Figure 5. Here, N = 0 corresponds to the cylindrical nanoshell made of pure PZT-4. As we can see, the natural frequency is quite sensitive to the applied external electric voltage. The natural frequency decreases as the voltage changes from V 0 = −0.0002 V to 0.0002 V. The reason is that the axial and circumferential compressive and tensile forces are generated in the porous FGPM nanoshells by the applied positive and negative voltages, respectively. Thereinto, the applied positive voltage reduces the nanoshell stiffness but the negative voltage increases the nanoshell stiffness.  Table 11 presents the variation of nonlocal parameter against natural frequency of the FGPM-I nanoshell. One can see that the frequency decreases gradually with the increasing nonlocal parameter. This is because the nonlocal effect tends to decrease the stiffness of the nanoshell and hence decreases the natural frequency. This phenomenon was also found in nano-beams and nano-plates [56][57][58]. Table 11. Variation of natural frequency ω(GHz) against the circumferential wave number n for different nonlocal parameter e 0 a of FGPM-I nanoshell.  In order to reveal the porosity type effect, the natural frequency against the circumferential wave number for FGPM-I and FGPM-II nanoshells is plotted in Figure 6. One can see that the natural frequency of the FGPM-II nanoshell is close to that of the FGPM-I one at small circumferential wave number. However, the natural frequency of the FGPM-II nanoshell becomes higher than that of its FGPM-I counterpart with the rise of circumferential wave number. The difference between them gets more and more obvious as the circumferential wave number increases further.  Table 12 gives the natural frequencies of FGPM-I and FGPM-II nanoshells for various porosity volume fractions. One can find that the larger nano-void volume fraction leads to the lower natural frequency of the FGPM-I nanoshell, while it leads to the higher natural frequency of the FGPM-II nanoshell. Therefore, it can be concluded that the porosity distribution type has a notable impact on vibration characteristics of FGPM nanoshells.

Conclusions
In this work, free vibration of porous FGPM nanoshells subjected to thermal and electrical loads is studied in the framework of Love's shell theory and nonlocal elasticity theory. Size-dependent governing equations and boundary conditions are obtained based on Hamilton's principle. Then, natural frequencies of the porous FGPM cylindrical nanoshells are obtained via the Navier method as well as the Galerkin method. The following conclusions were drawn: (1) The fundamental natural frequency of the porous FGPM nanoshell decreases initially and then increases as the radius-to-thickness ratio increases. Furthermore, the fundamental frequency decreases with the rise of the length-to-radius ratio; especially, the frequency changes notably when the length-to-radius ratio is small; (2) Applying positive voltage decreases the stiffness while applying negative voltage increases the stiffness of the porous FGPM cylindrical nanoshell. Furthermore, the temperature rise results in a reduction in the stiffness. In addition, the larger power-law index leads to the lower natural frequencies of the porous FGPM cylindrical nanoshell; (3) The nonlocal parameter has a softening effect on the free vibrations of the porous FGPM nanoscale shells; (4) The Galerkin solution procedure is an alternative method, which can give numerical results with satisfactory accuracy; (5) Increasing the porosity volume fraction has a different effect on the natural frequencies of the FGPM-I and FGPM-II nanoshells, which shows that the porosity distribution type plays a notable role on vibration characteristics of the FGPM nanoscale shells.

Conflicts of Interest:
The authors declare no conflict of interest.