In this section, the plate structure comprises piezoelectric materials for the top and bottom layers and FG-CNTRC for the middle layer, as shown in Figure 1. Four CNT distributions along the thickness direction of the FG-CNTRC layer are considered including uniformly distributed (UD) and functionally graded-V (FG-V), FG-O, and FG-X. In the UD pattern, the CNT is uniformly distributed along the thickness direction. In the FG-V distribution type, CNT is found in abundance on the top of the plate while poorly distributed in the bottom. Moreover, the CNT is abundant in the middle of the plate while poorly distributed in the top and bottom in the FG-O distribution type. In contrast, the CNT is abundant in both the top and the bottom of the plate while poorly distributed in the middle in the FG-X distribution type. The CNT volume fraction of each layer is expressed as V C N T ∗ ( z ) = V C N T                     ( UD ) , (1) V C N T ∗ ( z ) = 2 ( 1 − 2 | z | h ) V C N T    ( FG - O ) , (2) V C N T ∗ ( z ) = 4 | z | h V C N T             ( FG - X ) , (3) V C N T ∗ ( z ) = ( 1 + 2 z h ) V C N T      ( FG - V ) , (4) V C N T = w C N T w C N T + ( ρ C N T / ρ M ) − ( ρ C N T / ρ M ) w C N T , (5) where w C N T is the mass fraction of CNT; ρ C N T and ρ M are the densities of the CNT and polymeric matrix, respectively; V C N T is the overall CNT volume fraction of the multilayer CNTRC plate; and h is the total thickness of the plate along z direction.

According to the rule of mixtures (Shen, 2009), Young’s modulus and the shear modulus of the CNTRC plate are expressed as E 11 = η 1 V C N T E C N T 11 + V M E M , (6) η 2 E 22 = V C N T E C N T 22 + V M E M , (7) η 3 G 12 = V C N T G C N T 12 + V M G M , (8) where E C N T 11 , E C N T 22 , and G C N T 12 represent Young’s modulus along the longitudinal and transverse directions, and the shear modulus of CNT, E M , and G M represent Young’s modulus and the shear modulus of the matrix material; η 1 , η 2 , and η 3 are the CNT efficiency parameters along the longitudinal, transverse, and shear directions, respectively, and the volume fractions of composite component V C N T and V M satisfy V CNT + V M = 1 . The calculated formulae of Poisson’s ratio and density are shown as follows: ν 12 = V C N T ν C N T 12 + V M ν M , (9) ρ = V C N T ρ C N T + V M ρ M . (10)

In the piezoelectric-integrated FG-CNTRC plate with four distribution types of CNT, the core layer is made of FG-CNTRC material. The schematic sketch of the piezoelectric-integrated FG-CNTRC plate is illustrated in Figure 1. The length and width of the plate are denoted by a and b , respectively, and the total thickness of the plate is h = h c + 2 h p with core layer thickness h c and piezoelectric layer thickness h p .

According to the FSDT, the displacement fields are defined as u ( x , y , z , t ) = u 0 ( x , y , t ) + z ϕ x ( x , y , t ) , (11) v ( x , y , z , t ) = v 0 ( x , y , t ) + z ϕ y ( x , y , t ) , (12) w ( x , y , z , t ) = w 0 ( x , y , t ) , (13) where u 0 ( x , y , t ) , v 0 ( x , y , t ) , and w 0 ( x , y , t ) are displacements of the mid-plane along x , y , z directions at the time of t, respectively, and ϕ x ( x , y , t ) and ϕ y ( x , y , t ) denote the transverse normal rotations about y and x axes, respectively. The strain displacement relation is expressed as [ ε x ε y γ x y ] = [ ε x 0 ε y 0 γ x y 0 ] + z [ κ x κ y κ x y ] = [ ∂ u 0 ∂ x ∂ v 0 ∂ y ∂ u 0 ∂ y + ∂ v 0 ∂ x ] + z [ ∂ ϕ x ∂ x ∂ ϕ y ∂ y ∂ ϕ x ∂ y + ∂ ϕ y ∂ x ] = ε 0 + z κ, (14) [ γ y z γ x z ] = [ ϕ y + ∂ w 0 ∂ y ϕ x + ∂ w 0 ∂ x ] = γ 0 . (15)

The stress components of the kth layer in the FG-CNTRC plate are calculated by the constitutive relation as [ σ x ( k ) σ y ( k ) τ x y ( k ) τ y z ( k ) τ x z ( k ) ] = [ Q 11 ( k ) Q 12 ( k ) 0 0 0 Q 21 ( k ) Q 22 ( k ) 0 0 0 0 0 Q 66 ( k ) 0 0 0 0 0 Q 44 ( k ) 0 0 0 0 0 Q 55 ( k ) ] [ ε x ( k ) ε y ( k ) γ x y ( k ) γ y z ( k ) γ x z ( k ) ] , (16) where Q 11 ( k ) = E 11 ( k ) 1 − υ 12 ( k ) υ 21 ( k ) ,, Q 22 ( k ) = E 22 ( k ) 1 − υ 12 ( k ) υ 21 ( k ) , Q 12 ( k ) = υ 21 ( k ) E 11 ( k ) 1 − υ 12 ( k ) υ 21 ( k ) , Q 21 ( k ) = υ 21 ( k ) E 11 ( k ) 1 − υ 12 ( k ) υ 21 ( k ) Q 44 ( k ) = G 23 ( k ) , Q 55 ( k ) = G 13 ( k ) , and Q 66 ( k ) = G 12 ( k ) . The shear modulus G 13 ( k ) is equal to G 12 ( k ) , whereas G 23 ( k ) is equal to 1.2 G 12 ( k ) (Shen, 2014).

Axial forces and bending moments are expressed by the following equations ( N , M ) = ∑ k = 1 N L ∫ z k z k + 1 ( 1 , z ) σ d z , (17) Q ¯ = ∑ k = 1 N L ∫ z k z k + 1 k s σ d z , (18) where N L is the total layer number of the plate and the shear correction factor k s = 5 / 6 . Substituting Eq. 16 into Eqs 17, 18, the axial force and bending moments are expressed as N = [ N x N y N x y ] = ∑ k = 1 N L [ Q 11 ( k ) Q 12 ( k ) Q 16 ( k ) Q 21 ( k ) Q 22 ( k ) Q 26 ( k ) Q 61 ( k ) Q 62 ( k ) Q 66 ( k ) ] ∫ z k z k + 1 [ σ x ( k ) σ y ( k ) τ x y ( k ) ] d z , (19) M = [ M x M y M x y ] = ∑ k = 1 N L [ Q 11 ( k ) Q 12 ( k ) Q 16 ( k ) Q 21 ( k ) Q 22 ( k ) Q 26 ( k ) Q 61 ( k ) Q 62 ( k ) Q 66 ( k ) ] ∫ z k z k + 1 [ σ x ( k ) σ y ( k ) τ x y ( k ) ] z d z , (20) Q ¯ = [ Q y Q x ] = ∑ k = 1 N L [ Q 44 ( k ) 0 0 Q 55 ( k ) ] ∫ z k z k + 1 k s [ τ y z ( k ) τ x z ( k ) ] d z , (21) N = [ N x N y N x y ] = [ A 11 A 12 A 16 A 21 A 22 A 26 A 61 A 62 A 66 ] [ ε x 0 ε y 0 γ x y 0 ] + [ B 11 B 12 B 16 B 21 B 22 B 26 B 61 B 62 B 66 ] [ κ x κ y κ x y ] , (22) M = [ M x M y M x y ] = [ B 11 B 12 B 16 B 21 B 22 B 26 B 61 B 62 B 66 ] [ ε x 0 ε y 0 γ x y 0 ] + [ D 11 D 12 D 16 D 21 D 22 D 26 D 61 D 62 D 66 ] [ κ x κ y κ x y ] , (23) Q ¯   = [ Q y Q x ] = k s [ A 44 0 0 A 55 ] [ γ y z γ x z ] . (24)

The stress and moment resultants are obtained as follows: N = A ε 0 + Bκ , (25) M = B ε 0 + Dκ , (26) Q ¯   =   k s K γ 0 . (27)

The in-plane stiffness matrix, coupled bending–stretching matrix, and bending–stiffness matrix are expressed as ( A i j , B i j , D i j ) = ∑ k = 1 N L ∫ z k z k + 1 Q i j ( k ) ( 1 , z , z 2 ) d z ,   ( i ,   j = 1 ,   2 ,   6 ) , (28) K i j = ∑ k = 1 N L ∫ z k z k + 1 Q i j ( k ) d z .   ( i ,   j = 4 ,   5 ) . (29)

The linear constitutive relations of the piezoelectric-integrated plate are expressed as σ = Sε − e T E , (30) D = eε + ΞE , (31) where σ , ε , D, and E represent the stress, strain, electric displacement, and electric field intensity vectors, respectively; S , e , and Ξ indicate the elastic, piezoelectric, and dielectric constant matrices.

The electric variables need to satisfy Maxwell’s static electricity equation, where the divergence for electric displacements vanishes at any arbitrary point through the piezoelectric actuator and sensor layer thickness, that is, ∫ − h / 2 − h p − h / 2 ( ∇ D ) d z + ∫ h / 2 h / 2 + h p ( ∇ D ) d z . (32)

The electric potential is approximately assumed as a combination of cosine and linear variation (Wang, 2002) φ ˜ ( x , y , z ) = − cos ( β z ) φ ( x , y ) + 2 z V 0 h , (33) where β = π / h , φ ( x , y ) is the electric potential in the mid-plane and V 0 is the external electric voltage. The electric field intensity vector is expressed as (Ke et al., 2015; Barati and Zenkour, 2016) E = [ E x E y E z ] = − [ ∂ φ ˜ / ∂ x ∂ φ ˜ / ∂ y ∂ φ ˜ / ∂ z ] . (34)

The piezoelectric and dielectric constant matrices are defined as e = [ 0 0 0 0 e 15 0 0 0 e 24 0 e 31 e 32 e 33 0 0 ] , (35a) Ξ= [ Ξ 11 0 0 0 Ξ 22 0 0 0 Ξ 33 ] . (35b)

The strain energy of the plate is obtained as U = 1 2 ∫ Ω ( σ T ε− D T E ) d Ω = ∫ Ω ( 1 2 ε T Sε− ε T e T ε− 1 2 E T ΞE ) d Ω . (36)

The kinetic energy is obtained as V = 1 2 ∫ Ω ∑ k = 1 N L ∫ z k z k + 1 ρ ( k ) ( u ˙ 2 + v ˙ 2 + w ˙ 2 ) d z d Ω = 1 2 ∫ Ω [ I 0 ( u ˙ 0 2 + v ˙ 0 2 + w ˙ 0 2 ) + 2 I 1 ( u ˙ 0 ϕ ˙ x + v ˙ 0 ϕ ˙ y ) + I 2 ( ϕ ˙ x 2 + ϕ ˙ y 2 ) ] d Ω , (37) ( I 0 , I 1 , I 2 ) = ∑ k = 1 N L ∫ z k z k + 1 ρ ( k ) ( 1 , z , z 2 ) d z , (38) where ρ ( k ) represents the mass density of the kth layer in the FG-CNTRC plate. The external work is described as W = ∫ Ω ( u T f s − φ ˜ T q s ) d Ω , (39) where f _s represents the external mechanical surface load and q _s represents the external surface charge. The total energy function of the piezoelectric-integrated FG-CNTRC plate is expressed as L = U − V − W . (40)

In terms of the active control analysis for the CNTRC plate, the FEM is an efficient numerical technique among the community of laminated composite beams, plates, and shells due to its robustness in handling engineering practical problems with complex geometry and complicated condition (Steiger and Mokrý, 2015). The calculation amount of the FEM increases with the increase of element number, and an appropriate element number is needed to satisfy the requirements of computational accuracy and efficiency. The plate is discretized into a certain number of quadrilateral elements, and the approximate displacement functions of arbitrary quadrilateral elements are expressed as [ u 0 h v 0 h w 0 h ϕ x h ϕ y h ] = ∑ I = 1 4 ψ I ( x , y ) I 5 d I = ∑ I = 1 4 ψ I d I , (41) ψ 1 ( x , y ) = 1 4 ( 1 − ξ ) ( 1 − η ) , (42) ψ 2 ( x , y ) = 1 4 ( 1 + ξ ) ( 1 − η ) , (43) ψ 3 ( x , y ) = 1 4 ( 1 − ξ ) ( 1 + η ) , (44) ψ 4 ( x , y ) = 1 4 ( 1 + ξ ) ( 1 + η ) , (45) d I = [ u I , v I , w I , ϕ xI , ϕ yI ] T . (46)

The membrane strain, bending strain, and shear strain matrix of the plate are presented as ε 0 = ∑ I = 1 4 B I m d I ,    κ  = ∑ I = 1 4 B I b d I ,   γ 0 = ∑ I = 1 4 B I s d I , (47) where B I m = [ ψ I , x 0 0 0 0 0 ψ I , y 0 0 0 ψ I , y ψ I , x 0 0 0 ] , (48) B I b = [ 0 0 0 ψ I , x 0 0 0 0 0 ψ I , y 0 0 0 ψ I , y ψ I , x ] , (49) B I s = [ 0 0 ψ I , x 0 ψ I 0 0 ψ I , y ψ I 0 ] . (50)

Substituting Eqs 36–39, 41, 47 into Eq. 40, the governing equations of motion are obtained as   [ M u u 0 0 0 ] [ d ¨ φ ¨ ] + [ K u u K u ϕ K ϕ u − K ϕ ϕ ] [ d φ ˜ ] = [ F Q ] , (51)

where the detailed expressions of matrices are shown in Appendix A.

The FEM is used to analyze the free and forced vibration control of the piezoelectric-integrated FG-CNTRC plate. Three boundary conditions are considered including simply supported (S), fully clamped (C), and free (F) edge. The simply supported boundary condition is expressed as v 0 = w 0 = ϕ y = 0 ,   ( x = 0 ,   a ) . (52) u 0 = w 0 = ϕ x = 0 ,   ( y = 0 ,   b ) . (53)

The fully clamped boundary condition is expressed as u 0 = v 0 = w 0 = ϕ x = ϕ y = 0. (54)

Since the electric field intensities E only depend on the component along the z direction, the matrix K u ϕ can be simplified as K u ϕ = ∫ Ω ( B m T e m T B ϕ + z B b 1 T e m T B ϕ ) d Ω . (55)

Substituting Eq. 55 into Eq. 51, the governing equations of motion are obtained as follows: M u u d ¨ + ( K uu + K u ϕ K ϕ ϕ − 1 K ϕ u ) d = F + K u ϕ K ϕ ϕ − 1 Q . (56)

When the external charge Q of the piezoelectric-integrated FG-CNTRC plate is equal to zero, under the action of initial disturbance, the potential generated by the direct piezoelectric effect on the sensor layer is φ ˜ s = [ K ϕ ϕ − 1 ] s [ K ϕ u ] s d s , (57) where the sensor charge is determined by Q s = [ K ϕ u ] s d s . (58)

The actuator voltage φ ˜ a is expressed as φ ˜ a = G v φ ˙ s , (59) where the constant G v represents the velocity feedback control gain. The magnitude of the actuator layer charge is obtained by substituting Eqs 57, 59 into Eq. 51 as Q a = [ K ϕ u ] a d a − G v [ K ϕ ϕ ] a [ K ϕ ϕ − 1 ] s [ K ϕ u ] s d s , (60) where Q a represents the actuator charge and the motion equation is expressed as M d ¨ + ( C a + C R ) d ˙ + Kd = F, (61) K = K u u + G v [ K u ϕ ] a [ K ϕ ϕ − 1 ] s [ K ϕ u ] s , (62) C a = G v [ K u ϕ ] a [ K ϕ ϕ − 1 ] s [ K ϕ u ] s , (63) where C R is the Rayleigh damping matrix.

The convergence and comparison study of the piezoelectric-integrated plate under three different boundary conditions is studied. The length and width of the piezoelectric-integrated plate are a = b = 400 mm , and the thicknesses of the core layer and each piezoelectric layer are h c = 5 mm mm and h p = 0.1 mm , respectively. Both the top and bottom layers consist of the G1195-N piezoelectric material, and the core layer is the aluminum oxide (Ti-6Al-4V) material. The Ti-6Al-4V core layer is isotropic with Young’s modulus E = 105.7 Gpa , Poisson’s ratio v = 0.2981 , and density ρ = 4429 kg / m 3 , and the material parameters of G1195-N piezoelectric layers are E = 63 Gpa , v = 0.3 , ρ = 7600 kg / m 3 , e 31 = e 32 = 22.86 C / m 2 , e 15 = e 24 = 0 , and Ξ 33 = 1.5 × 10 − 8 F / m . The convergence study of the piezoelectric-integrated plate under CFCF, CCCC, CFFF, and SSSS boundary conditions is demonstrated in Figure 2. It is observed that fundamental frequency declines as the number of elements increases. When the element number exceeds 20 × 20, the fundamental frequency approaches a stable value gradually, which verifies the convergence of the FEM numerical simulation.

The first six natural frequencies of the piezoelectric-integrated plate under the CCCC boundary condition are presented in Table 1 and are compared with those obtained based on the CLPT (He et al., 2001), FSDT (Kiani, 2016a), and HSDT (Selim et al., 2016). It is observed that the maximum relative errors with the FSDT and HSDT are 1.28 and 0.22%, respectively, and the CLPT is less than 2.5%, which is possibly because the element number in He et al. (2001) is 8 × 8 and did not give a completely converged value of the frequency (Askari Farsangi et al., 2013). In Table 2, the maximum relative error of the first six frequencies for the piezoelectric-integrated plate is about 0.28% under the SSSS boundary condition, which proves the accuracy of the numerical method and the applicability of different boundary conditions. Figures 3, 4 depict the first six vibration modes of CFFF and SSSS piezoelectric-integrated plates.

In this section, CNT and poly methyl methacrylate (PMMA) are used as the reinforcement and matrix materials of the core layer. Unless otherwise stated, the geometrical and material parameters of the piezoelectric-integrated FG-CNTRC plate are listed as follows: Young’s modulus, Poisson’s ratio, and density of PMMA are 3.52 Gpa , 0.34, and 1150 kg / m 3 , respectively. The efficiency parameters of different CNT volume fractions are η 1 = 0.137 and η 2 = 1.022 for V CNT = 0.12 , η 1 = 0.142 and η 2 = 1.626 for V CNT = 0.17 , and η 1 = 0.141 and η 2 = 1.585 for V CNT = 0.28 (Kiani, 2016b). In addition, η 3 is equal to 0.7 η 2 , which is obtained from molecular dynamics (MD) simulations (Shen and Zhang, 2010). The shear modulus G 13 is equal to G 12 , whereas G 23 is equal to 1 .2 G 12 (Shen, 2011). The lead zirconate titanate (PZT-5A) piezoelectric material is selected as the sensor and actuator layer with Young’s modulus, Poisson’s ratio, and density of 63 GPa, 0.3, and 7750 kg/m³, respectively, and the piezoelectric constants e 31 = e 32 = 6.1468 C/m 2 .

Tables 3, 4 indicate the effects of CNT volume fraction and distribution type on the first three frequencies for the piezoelectric-integrated FG-CNTRC plate under CCCC and SSSS boundary conditions, respectively. In these two cases, the numerical result of natural frequency increases as the CNT volume fraction increases. In addition, FG-X has the highest natural frequency among the four CNT distributions of FG-CNTRC plates, and the FG-O CNTRC plate has the lowest one, which matches the data from the existing literature consistently (Thanh et al., 2018). This phenomenon indicates that the volume fraction and distribution type of CNTs affect the stiffness of the piezoelectric-integrated FG-CNTRC plate significantly.

The frequency difference ratio between the piezoelectric-integrated FG-CNTRC plate and core plate is generally adopted to quantify the effect of the piezoelectric layer thickness on fundamental frequency. Figure 5 demonstrates the effect of the piezoelectric layer thickness on the fundamental frequency for the piezoelectric-integrated FG-CNTRC plate. The frequency difference ratio is positive when V CNT = 0.12 , which indicates that the fundamental frequency of the piezoelectric-integrated FG-CNTRC plate is higher than that of the core plate, and the effect of electromechanical coupling on the fundamental frequency exceeds the combined effect of mass density and elastic modulus. When V CNT = 0.28 , the frequency difference ratio initially decreases and then increases with the increase of thickness for the piezoelectric layer, which demonstrates that the fundamental frequency is related to the combined effect of mass density, elastic modulus, and electromechanical coupling. Moreover, it should be noticed that the electromechanical coupling effect is enhanced by increasing the thickness of the piezoelectric layer, which leads to the increment of fundamental frequency for the piezoelectric-integrated plate (Abad and Rouzegar, 2019). When the thickness of the piezoelectric layer reaches a certain value, the effect of electromechanical coupling exceeds the combined effect of density and stiffness, which is consistent with the works done by Askari Farsangi and Saidi (2012); Askari Farsangi et al. (2013).

The active control of dynamic vibration is studied for the piezoelectric-integrated FG-CNTRC plate with two PZT-5A piezoelectric layers. The velocity feedback control algorithm is used to control the vibration of the piezoelectric-integrated FG-CNTRC plate. Figure 6 illustrates the results under SSSS boundary condition with different velocity feedback gains. When the feedback gain is equal to 0.001, the deflection of plate midpoint converges to zero after a time period of 0.25 s, and the period is shortened to 0.1 s when the feedback gain is 0.005. The vibration response of the piezoelectric-integrated FG-CNTRC plate attenuates rapidly while the velocity feedback gain increases. In addition, it should be emphasized that the feedback control gain is restricted because of the failure voltage of the piezoelectric material.

The effects of boundary condition and dynamic load on the vibration response are investigated for the piezoelectric-integrated FG-CNTRC plate. Figures 7, 8 present the deflection response of the piezoelectric-integrated plate subject to step load and sine excitation under the CFCF boundary condition. In the time period t = 0 to t = 0.05 s , the FG-CNTRC plate is subjected to the time-dependent distributed transverse load, which is forced vibration initially; afterward, the load is withdrawn at t = 0.05 s , and the forced vibration is transformed into free vibration. It is also illustrated that the deflection of the piezoelectric-integrated FG-CNTRC plate decreases with the increase of CNT volume fraction, and the decay rate increases significantly with the increase of velocity feedback control gain. The deflection response of the piezoelectric-integrated plate subject to sine excitation and step load under SSSS boundary condition is shown in Figures 9, 10. These figures illustrate that the vibration attenuates faster with the increase of velocity feedback control gain, and the addition of CNTs leads to significant improvements of plate stiffness, which is compatible with the previous results.

The amplitude–frequency response curve for the piezoelectric FG-CNTRC plate is shown in Figure 11. The vibration responses in the frequency domain are investigated for various feedback gains. With the increase of velocity feedback control gain, the active damping of the piezoelectric FG-CNTRC plate increases and the resonant amplitude decreases, which depicts that the velocity feedback control gain plays a major role in the process of vibration attenuation.