Hygro-thermo-mechanical bending of FG piezoelectric plates using quasi-3D shear and normal deformations theory

A simple quasi-3D sinusoidal shear and normal deformations theory for the hygro-thermo-mechanical bending of functionally graded piezoelectric (FGP) plate is developed under simply-supported edge conditions. The governing equations are deduced based on the principle of virtual work. The exact solutions for FGP plate are obtained. The current study investigates the effect of some parameters, like piezoelectricity, hygrothermal parameter, gradient index and electric loading on the mechanical and electric displacements, electric potential and stresses. They are explored analytically and numerically presented and discussed in detail. The numerical results clearly show the effect of piezoelectric and hygrothermal parameter on the FGP plate.


INTRODUCTION
Piezoelectric materials are great widely used as smart structures in different aerospace applications, because they can generate voltage, drive microelectronics directly and store charge. The effect of piezoelectricity is clear through the linear electromechanical interaction between electrical and mechanical states. Piezoelectric materials are used to detect stresses and deformations. Also, these materials can exchange output electrical potential and input mechanical energy. Zenkour (2014a, b) proposed an analytical solution describing the hygro-thermo-elastic responses of piezoelectric inhomogeneous hollow cylinders where the significance of influence of several parameters was investigated. He assumed that the piezoelectric nanoplate is simply-supported under an external electric voltage as well as a biaxial force and a uniform temperature change. The thermo-electro-mechanical free vibration of piezoelectric nanoplates based on the classical theory and nonlocal theory has been investigated by Liu et al. (2013). On the other hand, the thermo-electromechanical vibration of the piezoelectric rectangular nanoplate under different boundary conditions formed by using the nonlocal theory and the first-order shear deformation theory has been studied by Ke et al. (2015). Moreover, in the last decade, the analysis of static and dynamics for plates and shells made of piezoelectric materials have led to increase research attempts. In fact, the known theory for studying the mechanical behavior of rectangular plates is Kirchhoff classical plate theory (CPT). Jandaghian et al. (2013Jandaghian et al. ( , 2014 presented exact solutions for the transient bending of a circular rectangular plate surface bounded by two piezoelectric layers by using CPT. Furthermore, Huang and Yu (2006) discussed the piezoelectricity model under the effect of piezoelectric surface layers. Consequently, a piezoelectric material under and efficiency of the theory by comparing the obtained results with those computed using several other theories available in the literature.

Properties of FGP plate
This article presents a simply-supported rectangular plate of length , width and uniform thickness ℎ made of FGP material. The FGP plate is subjected to applied voltage at upper and lower surfaces, elevated temperature ( , , ) as well as a moisture exposure ( , , ) and sinusoidally distributed transverse load ( , ), as shown in Fig. 1. Material properties of FGP plates are assumed to vary through thickness according to a power law distribution. Effective material properties ( ) such as Young's modulus, moisture expansions and thermal expansions are supposed to vary continuously in the depth direction according to a power-law. Suppose that the FGP plate is made by mixing two different material phases, for instance, ceramic and metal, and are expressed as where and are the properties of the metal and ceramic, respectively, and denotes the non-negative gradient index.
The material properties of the exponentially graded piezoelectric (EGP) plates are supposed to follow the exponential law and is expressed as (Zenkour 2007): where = 0 is metal property and = 0 e is the ceramic one.

Quasi-3D sinusoidal theory of FGP plates
According to the extended quasi-3D sinusoidal plate theory (Zenkour 2007), the displacement components and electric potential of such theory are given as: where , and are the displacements of mid-plane along the axes , and respectively, 1 , 2 denote rotational displacement about , axes, respectively, 3 is additional displacement to show the inclusion of normal deformation, and ( , ) is the electric potential on the mid-plane. The displacement component ( , ) of the current theory is select based on the following assumptions: the axial in-plane and transverse displacements are partitioned into bending and shear components as the bending component likes to that given by the CPT; while the shear component gives rise to the sinusoidal variations of shear strains and hence to shear stresses through the thickness of the plate, in such a way that shear stresses are vanish on the top and bottom surfaces of the plate. Then, the displacement field can be obtained by using Eq. (3) in the form: where ( ) = − 1 sin( ) and ( ) = 1 − ′ ( ) = cos( ). Based on Eq. (5), the number of unknown functions is only six. The linear strains with the displacements in Eq. (5) are Coupled elastic and electrical field equations due to (Tiersten 1969) are given as where , , and are the stress tensors, electric displacement vector, strain and electric field components, respectively; , and are elastic, piezoelectric and dielectric coefficients, respectively. The fourth-order stiffness tensor may be written as a tensor of second order (as symmetric matrix ) with 11→1; 22→2; 33→3; 23, 32→4; 13, 31→5; and 12, 21→6. The corresponding stress-strain relations accounting for piezoelectric and thermal effects can be written as: The non-zero components of electric field ( 1 , 2 , 3 ) are defined as: The generalized temperature field variation and moisture exposure through the FGP plate is assumed as where and are the thermal and moisture loads.

General formulations
The principle of virtual work is stated as � � ( 11 11 + 22 22 + 33 33 + 23 23 + 13 13 + 12 12 )d where is the sinusoidally distributed load of the plate. Substituting Eqs. (6), (7) and (10) 3 : 1 + 2 − 33 = 0, where the resultant components are defined as in which and are the stress resultants and stress couples, are additional higher-order stress couples given according to ( ), and and 33 are the transverse and normal shear stress resultants.

Dimensionless quantities
In this section, the numerical results for the effect of hygrothermal and mechanical loading on the EGP and FGP plates by using quasi-3D sinusoidal plate theory are thoroughly discussed. The top surface of the plates is ceramic-rich while the bottom surface of the plate is metal-rich. The elastic coefficients for the present plate are given by , ( = 4,5,6).
Now, the EGP consisted of Aluminum/Alumina is considered. Young's modulus for aluminum is 70 GPa while for alumina is 380 GPa. Also, Poisson's ratio for both equals 0.3 (Al Khateeb and Zenkour 2014). Also, the FGP plates consisted of Titanium/Zirconia are considered. The material properties are given by (Zidi et al. 2014) Metal ( The non-dimensional form of the deflection and stresses parameters of hygro-thermo-mechanical bending are Hygro-thermo-mechanical bending of FG piezoelectric plates using quasi-3D shear and normal deformations theory    Table 6 illustrates the comparison of dimensionless deflection and stresses of Titanium/Zirconia FGP rectangular plate under sinusoidally load.

Effect of gradient index
Some numerical examples for the simply-supported bending of FGP rectangular plate are provided based on quasi-3D sinusoidal plate theories. The results of the EG plates given in Table 1 are the same as those of the solution of Al- Khateeb and Zenkour (2014), as well as results of the EGP plates in other Tables 2-5. The dimensionless deflection � decreases as increases. The normal stresses � 11 and � 22 decrease as k decreases, while the transverse shear stress � 13 increases. It is to be noted in Table 6 that the dimensionless deflection � decreases as and /ℎ increase. The normal stresses � 11 and � 33 increase as /ℎ and increase. If we neglect the hygrothermal effect in Table 6 we will get the same results as those in Zenkour and Hafed (2019).
1/3 0.  Table 6: The dimensionless deflection of and stresses in FGP plates ( 0 = 100, � 1 = � 3 = 0, ̅ 1 = ̅ 3 = 0, � 2 = 100, ̅ 2 = 2, 0 = −0.5).     Figure 2 illustrates the change of normalized transverse deflection � with the thickness ratio /ℎ of the FGP plate when / = 2 and 0 = 1. The dimensionless deflection increases as /ℎ increases and decreases. The metallic plate gives the highest deflection. Figures 3 and 4 illustrate the change of normalized normal stresses � 11 and � 33 with the thickness ratio /ℎ in the FGP plate when / = 2, 0 = 1. Both stresses increase as /ℎ and increase. The metallic plate gives the smallest normal stresses. Figure 5 shows the change of electric displacement � 3 vs the thickness ratio /ℎ of the FGP rectangular plate. The electric displacement increases as /ℎ increases for homogeneous metallic plate while the electric displacement FGP rectangular plates decreases as /ℎ increases. Also, � 3 decreases as increases.  Figures 6 and 7 illustrate the through the thickness distributions of the longitudinal and transverse normal stresses � 11 and � 33 in the FGP plate. The longitudinal normal stress � 11 is tensile at the top surface and compressive at the bottom surface of the FGP plate. The longitudinal normal stress � 11 increases as decreases in the region −0.25 ≤ ̅ ≤ 0.3 and vice versa outside this region. However, the transverse normal stress � 33 is always tensile through the thickness of the FGP plate. Also, the transverse normal stress � 33 increases as decreases in the region −0.22 ≤ ̅ ≤ 0.26 and vice versa outside this region. Figure 8 shows the change of normalized transverse deflection � through the thickness of the FGP plate. The dimensionless deflection � may be independent of the thickness of the FGP plate. The metallic plate gives the highest deflection. Otherwise, the deflection decreases as increases. Figure 9 illustrates the through the thickness distributions of the tangential shear stress � 12 of the FGP plate. The tangential shear stress � 12 is tensile at the bottom surface and compressive at the top surface of the FGP plate. The tangential shear stress � 12 increases as increases in the region −0.28 ≤ ̅ ≤ 0.34 and vice versa outside this region.  In what follows, we will discuss the effect of the transverse hygrothermal parameters � 3 , ̅ 3 and the external electric voltage 0 on the bending response of FGP plates. Figures 10-17 are studied based on the following five cases: Figures 10-13 establish the change of �, � , � 11 and � 3 , respectively, with the thickness ratio /ℎ for different values of � 3 , ̅ 3 and 0 . The dimensionless deflection �, electric potential � and longitudinal normal stress � 11 are increasing with the increase in � 3 , ̅ 3 and 0 while the dimensionless electric displacement � 3 is decreasing. The dimensionless deflection � and electric displacement � 3 are directly increasing with the increase in the side-to-thickness ratio /ℎ. However, the electric potential � and longitudinal normal stress � 11 are no longer decreasing as /ℎ and they are increasing again to get their maximum values for higher values of /ℎ. Figure 11: Dimensionless electric potential � (0.5,0.5,0) vs the side-to-thickness ratio /ℎ of the FGP plate ( / = 2, = 1).

Figure 12:
Dimensionless normal stress � 11 (0.5,0.5,0.5) vs the side-to-thickness ratio /ℎ of the FGP plate ( / = 2, = 1). Figure 14 illustrates that the electric potential � is tensile in the upper-half plane and compressive in the lower-half plane of the FGP plate. The dimensionless electric potential � is increasing with the increase in � 3 , ̅ 3 and 0 in the upperhalf plane and is decreasing in the lower-half plane of the FGP plate. In Figure 15, the electric displacement � 3 increases as � 3 , ̅ 3 and 0 increases. For case I, � 3 may be independent of the thickness of the FGP plate. However, for other cases � 3 gets its maximum value at the bottom surface of the FGP plate and its minimum value at the top one. Figure 13: Dimensionless of electric displacement � 3 (0.5,0.5,0) vs the side-to-thickness ratio /ℎ of the FGP plate ( / = 2, = 1).

Figure 14:
Dimensionless electric potential � (0.5,0.5, ̅ ) through the thickness of the FGP plate ( / = 2, /ℎ = 10, = 1). Figure 16 shows that the longitudinal normal stress � 11 increases as � 3 , ̅ 3 and 0 decrease. The longitudinal normal stress � 11 is tensile near the top surface for cases I and II and compressive for other cases through-the-thickness the FGP plate. Figure 17 shows that the transverse normal stress � 33 is tensile through-the-thickness of FGP plate and it increases as � 3 , ̅ 3 and 0 increase. It is noticed that � 33 has a little dependency on the thickness of the FGP plate.

CONCLUSIONS
This paper presents analytical solutions for bending of FGP rectangular plates subjected to moisture and thermal loads. The equilibrium equations based on the principle of virtual work, and the analytical closed-form solutions of simply-supported FGP plates are obtained by using Navier's method. The solutions are derived by using a quasi-3D sinusoidal shear and normal deformations theory. The number of primary variables in the present theory is six.  The theory satisfies the boundary conditions on the surfaces of the FGP plate without employing shear correction factors. The present FGP plate is subjected to mechanical, thermal, moisture and electric voltage loadings in its top surface. Non-dimensional displacements and stresses are computed for FGP plates with mixture materials. The responses of stress and displacement of the FGP rectangular plate are analyzed under hygro-thermo-mechanical sinusoidal loadings. Material properties are supposed to be dependent on the moisture and temperature effects, which constantly vary through the thickness of the FGP plate. The dependency of the deflection, stresses, electric potential and electric displacement of the FGP plate on different parameters is discussed. It is concluded that the gradients in material properties and the inclusion of other parameters have important roles to play in locating the response of the FGP plate.