A Simple-FSDT-Based Isogeometric Method for Piezoelectric Functionally Graded Plates

: An efficient isogeometric analysis method (IGA) based on a simple first-order shear deformation theory is presented to study free vibration, static bending response, dynamic response, and active control of functionally graded plates (FGPs) integrated with piezoelectric layers. Based on the neutral surface, isogeometric finite element motion equations of piezoelectric functionally graded plates (PFGPs) are derived using the linear piezoelectric constitutive equation and Hamilton’s principle. The convergence and accuracy of the method for PFGPs with various mechanical and electrical boundary conditions have been investigated via free vibration analysis. In the dynamic analysis, both time-varying mechanical and electrical loads are involved. A closed-loop control method, including displacement feedback control and velocity feedback control, is applied to the static bending control and the dynamic vibration control analysis. The numerical results obtained are accurate and reliable through comparisons with various numerical and analytical


Introduction
Functionally graded materials (FGMs) [1] have been used extensively in the aerospace, nuclear power industries, biomedical field, and other applications [2][3][4] for their superior thermo-mechanical properties, such as low thermal conductivity and high thermal resistance. Classified as smart materials, piezoelectric materials are also known for their piezoelectric effects [5] (used as sensors) and converse piezoelectric effects (used as actuators). Unique electro-mechanical coupling characteristics of piezoelectric materials enable them to be applied in the fields of health monitoring and structural control. Accordingly, by fabricating (or embedding) piezoelectric materials on the surface (inside) of functionally graded beams, plates, shells, and other structures, they can respond rapidly according to their physical characteristics and shape changes, when the external environment changes.
Piezoelectric functionally graded plates (PFGPs) are one of the basic structural forms of piezoelectric functionally graded structures. Traditional numerical methods of PFGPs consume a mass of computational meshes based on the conventional three-dimensional elastic theory framework. To overcome these shortcomings, scholars combined various numerical methods with equivalent single layer theory [6] for predicting the behaviors of PFGPs [7,8]. The finite element method (FEM) has always been one of the popular methods to study PFGPs. Ray et al. [9] and Loja et al. [10] presented investigation for the static response analysis of PFGPs. He et al. [11] employed the piezoelectric materials to investigate the dynamic vibration control of functionally graded plates (FGPs) by using the classical plate theory (CPT). Similarly, Liew et al. [12,13] studied the active control of FGPs with distributed piezoelectric materials under thermal loads based on the first-order shear deformation theory (FSDT). Aryana et al. [14] proposed a method to identify the most sensitive design variables that affect the dynamic characteristics of the structure. Using a cell-based smoothed discrete shear gap method, Nguyen-Quang et al. [15] studied the static and dynamic control analysis of FGPs bonded with piezoelectric actuators and sensors. In this method, each parent triangle element is divided into three sub-triangle elements, and the discrete shear gap method is used in each subtriangle element to eliminate the shear-locking effects [16]. Then, the strain smoothing technology is applied in the whole parent triangle element to solve the defect of rigidity of triangle elements in FEM. By using the higher-order shear deformation theory (HSDT), Fakhari et al. [17,18] analyzed the nonlinear free and forced vibration, as well as nonlinear vibration control of PFGPs in thermal environments. Recently, a generalized C 0 -type HSDT polygonal finite element method was presented by Nguyen et al. [19] for investigating the active control of smart, functionally graded metal foam plates reinforced by graphene platelets. To avoid the decrease in the calculation accuracy [20,21] caused by element distortion in the FEM, various scholars also attempted to use the mesh-free method to analyze the behaviors of PFGPs. Using the element-free Galerkin method, Dai et al. [22] investigated the static, active control analysis of FGPs with surface-bonded piezoelectric materials in thermal environments. The stability of PFGPs subjected to distributed thermo-electro-mechanical loads was studied by Chen et al. [23]. The radial point interpolation method was implemented to investigate the geometrically nonlinear response of PFGPs under mechanical and electrical loads by Hossein et al. [24]. Considering the stability of the plate in vibration control, Selim et al. [25] studied the active control of two types of PFGP structures. They pointed out that for the structure of two piezoelectric layers distributed symmetrically on the upper and lower surfaces of the FGPs, due to the stretching-bending coupling effect, the velocity feedback control in the active vibration control is unstable when the gradient index is in the range of 0 < n < ∞.
Although the FEM and mesh-free methods have achieved considerable success in the analysis of piezoelectric smart structures, there continues to be strong interest in new numerical methods. Recently, an isogeometric analysis (IGA) method was proposed by Hughes et al. [26]. Because of the advantages of high-order continuity and simple meshing, Phung-Van et al. [27] employed IGA to study the nonlinear dynamic response analysis of PFGPs under thermo-electro-mechanical loads. Similar studies of smart piezoelectric composite plates were developed by the same authors in the literature [28,29]. Along the same line, Nguyen et al. [30] and Nguyen-Quang et al. [31] presented developments of isogeometric analysis of piezoelectric, functionally graded porous plates reinforced by graphene platelets, and laminated carbon nanotube-reinforced composite plates bonded with piezoelectric materials, respectively.
The above studies on the dynamic analysis of piezoelectric smart structures are focused on the dynamic response of structures under mechanical loads. Few people pay attention to the dynamic response of the structures under time-varying voltage loads. However, many piezoelectric smart structures, such as piezoelectric motors [32] and robots [33], realize their motions precisely by utilizing the dynamic response under voltage loads. In addition, for the PFGPs, the presence of additional electric potential fields will result in more complicated mathematical modelling. Hence, a low-cost, efficient numerical method without loss of accuracy is extremely important. Most current approaches used CPT, FSDT, and HSDTs for the analysis of PFGPs. In these plate theories, CPT has the least amount of calculation because it has only three unknowns, but it is only valid for the thin plates due to the shear deformation effects being ignored [34]. The computational cost of FSDTs and HSDTs is more expensive than CPT because they have five or more unknowns. It is worth noting that Thai et al. [35,36] proposed a new simple first-order shear deformation theory (S-FSDT) with only four unknowns. Compared with FSDTs and HSDTs, the decrease of unknowns makes S-FSDT have a certain advantage in computational efficiency. Moreover, because S-FSDT is derived from FSDT, it is suitable for both thin and thick plates. The accuracy and validity of this theory for analyzing the static bending, buckling, free vibration [37], and geometrically nonlinear responses [38] of FGPs have been demonstrated.
In summary, the main motivations of this paper are: (i) there are no studies on isogeometric analysis for the analysis of PFGPs based on the simple first-order shear deformation theory, and the present study can fill this research gap; (ii) the S-FSDT-based IGA method has only four unknowns, and it is easily applied to the open-source IGA frameworks; (iii) the investigation of dynamic response of the structures under time-varying voltage loads can provide a more comprehensive understanding of the mechanical behavior of piezoelectric smart structures. Therefore, in this paper, we propose an approach based on S-FSDT and the NURBS-based isogeometric analysis for analyzing the mechanical behavior of the PFGPs. The mechanical displacement field and electric potential field in PFGPs are approximated using the NURBS basis functions. Isogeometric finite element equations of PFGPs are derived through the linear piezoelectric constitutive equation and Hamilton's variational principle. The dynamic response of PFGPs under mechanical loads and voltage loads are studied by using the Newmark-β direct integration method. Additionally, the static and dynamic closed-loop control are used for controlling the shapes and vibration of the plates.

Mathematical Model
The piezoelectric functionally graded plate is shown in Figure 1 with the size of

Functionally Graded Materials
The material properties of functionally graded plates along the thickness direction can be described as The volume fraction of metal materials is defined as where the subscript symbols m and c represent the metal and ceramic, respectively; c P and m P are the corresponding material properties, such as mass density ( ρ ), Young's modulus ( E ), and various other properties; n is the gradient index. Figure 2 shows the variation of the composition of the FGP with n . It is seen that as n increases, the less metal components in the plate.

Mechanical Displacement and Strain Based on S-FSDT
For the simple first-order shear deformation theory, the unknowns are reduced from five to four using two assumptions [36]: first, the transverse displacement 0 ( , ) w x y in the standard FSDT is split into bending and shear parts, that is, = + 0 b s w w w ; second, the rotations x β and y β are expressed using the partial derivatives of the transverse bending components, that is, The neutral surface is introduced to avoid the stretching-bending coupling effect. Hence, the displacement formulation of the S-FSDT can be written as In which, 0 u and 0 v represent the displacements of the neutral plane of the plate in x and y directions, respectively; x β and y β are the rotation variables; 0 z is the distance between the mid and neutral surface, which can be denoted as The geometric strains ε  can be given as with

Constitutive Relationship
For the functionally graded plates, the constitutive relationship between the geometric stresses and strains can be denoted as where with in which For the piezoelectric layers, the linear piezoelectric constitutive equation [18] is given by where σ p is the stress; D is the electrical displacement; g is the permittivity constant matrix; and e is the piezoelectric stress constant matrix. Q p is the elastic constant of piezoelectric materials with the similar form of Q f .
Only the electric field component in the z direction is considered. Hence, the electric field E is calculated by The stress resultants in the PFGPs are expressed as where with where k is the shear correction factor and it is set to be 5/6, and since the introduction of neutral surface [39], 0 The forces and bending moments generated by the electric field can be calculated by

Nurbs-Based Isogeometric Analysis
In this section, the displacements and electric potential are approximated by using the NURBS basis functions. Then, the isogeometric finite element motion equations are derived through the variational form of the PFGPs' motion equations.

Approximation of Mechanical Displacement
The mechanical displacement can be approximated as where = is the basis function defined in (20).
The strain functions can be achieved by substituting (21) into (6) and (7), which yields where

Approximation of the Electric Potential Field
The electric potential is assumed to vary linearly along the thickness [41], expressed as where φ k is the generalized nodal electric potential vector.

Isogeometric Finite Element Motion Equations
The variational form of the equations of motion for the plate can be obtained by Hamilton's principle where L is the total energy, including kinetic energy T , potential energy U , and external work ext where u is the displacement; u  is velocity; q s denotes the surface charge; f s is the surface loads.
By substituting (8), (12), (22), (23), and (25) into (27), the governing equations of PFGPs can be denoted as and in which In (31), z f is the transverse mechanical surface load; E q is the charge density on the surface of piezoelectric layers.
where U is the electric potential applied on piezoelectric layers. Eliminating the potential φ , (29) is rewritten as in which Introducing Rayleigh damping, (35) is rewritten as in which R α and R β are Rayleigh damping factors that can be confirmed through [42].

Closed-Loop Control
In this section, the static and dynamic closed-loop control of the FGPs by using piezoelectric sensors and actuators are studied. As shown in Figure 3, the upper piezoelectric layer serves as an actuator, and the lower piezoelectric layer serves as a sensor. The sensor generates electric charges because of the piezoelectric effect when the structure deforms. Then, these electric charges are amplified as the input voltage of the actuator. On account of the converse piezoelectric effect, the actuator exerts a force opposite to the external mechanical loads to suppress the static bending or dynamic vibration of the structure, achieving the purpose of active control.  According to the prior analysis, (29b) can be rewritten as φus φφs s φs where the subscripts a and s represent the terms associated with the actuator and sensor, respectively. Using the feedback control law [43], a φ is expressed as where d G and v G are the displacement and velocity feedback control gain, respectively.
For the sensor layer, it is assumed that there is no external charge f φs , ignoring the converse piezoelectric effect. From (40), the voltage generated on the sensor layer can be defined as From (41), (42), and (39), the charge of the actuator layer can be derived by From (43) and (37), the governing equation is rewritten as in which and the active damping matrix C is

Numerical Results
The order of the NURBS basis functions should be equal to or greater than two to satisfy the requirement of 1 C continuity in approximate formulations [44]. Hence, the orders of the 2D NURBS basis functions in this study are set to be three ( 3 p q = = ). For convenience, the plate's mechanical boundary conditions are simplified as S, C, and F, where S represents the simply supported edge, with C and F representing the clamped edge and free edge, respectively. The dynamic responses and active vibration analysis of the plate are calculated via the Newmark-β direct integration method.

Convergence and Verification Studies
The convergence of the present method is verified via the calculation of the natural frequencies of a SSSS square PFGP. The dimensions of the PFGP as depicted in Figure 1 are: a = b = 400 mm,  Table 1.
The four different control meshes of 10 × 10, 12 × 12, 16 × 16, and 18 × 18 are shown in Figure 4. Table 2 lists the lowest 10 natural frequencies of the plate. It is seen that the natural frequencies calculated with the 16 × 16 mesh match well with the analytical solutions [45]. Furthermore, the effects on the natural frequency of each mode are not significant when the mesh level is greater than 16 × 16. As a result, the 16 × 16 mesh and cubic NURBS basis functions are applied to all numerical subsequent examples. Figure 5 shows the first six mode shapes of the PFGP with a gradient index 2 n = .

Free Vibration Analysis
For investigating the effects of the electrical boundary conditions on natural frequencies of the PFGP, a SSSS PFGP with different ratios of f h a and p f h h is taken into consideration for comparison. In the closed circuit, two piezoelectric layers are both grounded, while in the open circuit, the electric potential remains free, which implies that there is no electric displacement. The materials of PFGP are Al/Al2O3 and PZT-4, and the material parameters of PZT-4 are defined in [45]. Note that, in this example, 0 n = means that the FGP consists only of ceramic (Al2O3). Conversely, n = ∞ means that the FGP is an isotropic metal (Al) plate. Table 3 lists the natural frequencies of the plate with different dimension scales. Besides, Table 4

Static Bending Analysis
A cantilevered (CFFF) square PFGP with the same materials and dimensions as those in Section 4.1 is used for analyzing the static bending responses.
First, the centerline deflection of the plate under a uniformly distributed load of −100 N/m 2 is displayed in Figure 6. The deflection obviously decreases when n increases due to the increase in the elastic modulus value E(z) of the FGP.
Next, the centerline deflection of the plate under a 10 V voltage is plotted in Figure 7. This situation involves the open-loop control of the plate, in which both two piezoelectric layers act as actuators. In this situation, the upper piezoelectric layer is polarized in the direction of the applied electric field, and the lower piezoelectric layer is polarized in the opposite direction of the applied electric field. In the case of the converse piezoelectric effect, the upper piezoelectric layer contracts, while the lower piezoelectric layer extends along the direction of the length.
The centerline deflection of the plate under a uniformly distributed load of −100 N/m 2 and different input electric voltages is shown in Figure 8. It can be observed that the deflection decreases as the input voltages increase. Meanwhile, Table 5 lists the tip deflection values of the plate. The results in this approach are in good agreement with the solutions in [15].  We also investigated the bending response of PFGPs which are subjected to the voltage range 0-60 V with different mechanical boundary conditions. Figure 9 provides the tip deflection or central node deflection of the plate with 2 n = . It is observed that the deflections increase linearly with the increase in the voltage. For the CFFF plate, the deflection is positive, while for the other boundary conditions, the deflections are negative. This phenomenon is due to the upper piezoelectric layer being shortened and the lower piezoelectric layer being elongated due to the converse piezoelectric effect. At the same time, the change of mechanical boundary conditions also results in the change of the deflection direction of the plate.

Dynamic analysis
A SSSS square FGP (Al/Al2O3) with 1 n = is used for demonstrating the validity of the dynamic analysis with the S-FSDT-based IGA. The length and thickness of the plate are 200 mm and 10 mm. Four different distributed transverse forces are defined as Step load  Figure 10. Apparently, the results of this study coincide with those in [46].
Next Introducing Rayleigh damping, Figure 11 shows the transient deflections of the plate. It is observed that the vibration amplitude and period of motions decrease with the increase in the gradient index n due to the strengthening of stiffness of the plate.  Next, we analyze the dynamic response of PFGPs with 2 n = under the time-varying electric loads. As shown in Figure 12, the magnitude of the voltage amplitude 0 U will affect the vibration amplitude of the plate. Due to the effect of structural damping, the central vibration amplitude of the plate tends to be stable after a period of time. In this example, the time step is set to be 1 2π (20 ) w , in which 1 w is the first natural frequency (rad/s) of the plate.

Active Control Analysis
A square plate and a clamped circular plate are applied for the static bending control analysis and dynamic vibration control analysis of the FGPs through piezoelectric actuators and sensors. The materials of the FGPs and piezoelectric layers in this section are Ti-6Al-4V/aluminum oxide and PZT-G1195N. 2. Circular plate A clamped circular PFGP with 2 n = is also considered. The thicknesses of the plate and each piezoelectric layer are 0.3 mm and 0.05 mm. The radius R is 50 mm. Figure 14 shows the deformation of the plate under a mechanical load of −100 N/m 2 . We can see that the deformation of the circular plate can also be effectively controlled by d G .

Dynamic Vibration Control
1. Square plate First, we assume that the former square CFFF PFGP is initially subjected to a downward uniformly distributed load of −100 N/m 2 , then the load is instantaneously removed, resulting in the motion from the initial displacement. The piezoelectric stress constants 31 e and 32 e are 6.1468 C/m 2 , and the Rayleigh damping R C of the plate is not considered in this example.
If the stretching-bending coupling effect exists, the active damping C ( not a symmetric positive definite matrix [25,47]. As shown in Figure 15, the dynamic vibration control of plate structures with piezoelectric layers symmetrically distributed on the upper and lower surfaces may be unstable. The main purpose of introducing the neutral surface in this paper is to solve the instability of dynamic vibration control for this sort of plate structure. Figure 15. Tip deflection of a cantilevered plate with 1 n = by using a geometric middle surface. Figure 16 shows the transient deflection response at the tip of the PFGP with 0 n = and n = ∞ by using the neutral surface. It is clear that the vibration response attenuates faster when the control gain v G increases and the results match well with the mesh-free method presented by Selim et al. [25]. It is noteworthy that the stretching-bending coupling effect does not exist when 0 n = and n = ∞ . Similarly, Figure 17 depicts the transient deflection of the PFGP with 1 n = and 15 n = , and we can see that the dynamic vibration control effect is still stable.  Figure 18. Similarly, in both the forced vibration state and free vibration state, the oscillation is significantly suppressed when the control gain v G increases.

Circular plate
The circular plate in the section of static bending control is used for further investigation. With the action of an initial uniformly distributed load of −100 N/m 2 , the time of vibration attenuation to the weakest is investigated in Figure 19. It is observed that the decay time decreases with the increase of n, and the larger the value of v G , the faster the vibration disappears.
Based on the above analysis, we can conclude that the static deformation and the vibration of FGPs can be effectively controlled by using the displacement feedback control gain d G and the velocity feedback control gain v G . By using this feature, controllers can be designed and optimized according to the requirements of different applications to control both the displacement and oscillation time.

Conclusions
In this study, an isogeometric analysis method (IGA) based on a simple first-order shear deformation theory (S-FSDT) was used for investigating the free vibration, static bending response, dynamic response, and active control analysis of FGPs with surface-bonded piezoelectric actuators and sensors. Through some numerical examples, several major points can be drawn as follows:  The isogeometric finite element motion equations of piezoelectric functionally graded plates (PFGPs) based on the simple first-order shear deformation theory can be derived easily, due to one unknown saved in S-FSDT.  From some comparison studied of free vibration and static bending analyses of piezoelectric functionally graded plates, it can be obtained that although one unknown is saved in simple first-order shear deformation theory, the S-FSDT-based IGA method is still effective and accurate.