Highlights

• Bézier extraction based IGA approach is successfully implemented for the nonlinear staticand dynamic analyses of the FG plate reinforced by GPLs and integrated with piezoelectriclayers.

• The combination of two porosity distribution types and three GPL dispersion patterns alongthe thickness direction of the FG plate is presented.

• The influences of the porosity coefficients, weight fractions of GPLs and the external elec-trical voltage are investigated.

• A constant displacement and velocity feedback control approaches are adopted to activecontrol the responses of the plate structures with piezoelectric sensors and actuators.

• Numerical results demonstrate the efficiency and reliability of the proposed approach.

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Analysis and control of geometrically nonlinear responses ofpiezoelectric FG porous plates with graphene platelets

reinforcement using Bézier extraction

Nam V. Nguyena, Lieu B. Nguyenb, Jaehong Leec, H. Nguyen-Xuand,∗

aFaculty of Mechanical Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City, VietnambFaculty of Civil Engineering, University of Technology and Education Ho Chi Minh City, Vietnam

cDepartment of Architectural Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, Seoul 143-747, SouthKorea

dCIRTech Institute, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Vietnam

Abstract

In this study, we propose an effective numerical approach to analyse and control geometricallynonlinear responses for the functionally graded (FG) porous plates reinforced by graphene platelets(GPLs) integrated with piezoelectric layers. The basis idea is to use isogeometric analysis (IGA)based on the Bézier extraction and theC0-type higher-order shear deformation theory (C0-HSDT).By applying the Bézier extraction, the original Non-Uniform Rational B-Spline (NURBS) controlmeshes can be transformed into the Bézier elements which allow us to inherit the standard nu-merical procedure like the finite element method (FEM). The mechanical displacement field isapproximated based on the C0-HSDT whilst the electric potential is assumed to be a linear func-tion through the thickness of each piezoelectric sublayer. The FG plate contains the internal poresand GPLs dispersed in the metal matrix either uniformly or non-uniformly according to variousdifferent patterns along the thickness of plate. In addition, to control dynamic responses, twopiezoelectric layers are perfectly bonded on the top and bottom surfaces of the FG plate. Thegeometrically nonlinear equations are solved by the Newton-Raphson iterative procedure and theNewmark’s time integration scheme. The influences of the porosity coefficients, weight fractionsof GPLs as well as the external electrical voltage on the geometrically nonlinear behaviours ofthe plates with different porosity distributions and GPL dispersion patterns are evidently investi-gated through numerical examples. Then, a constant displacement and velocity feedback controlapproaches are adopted to active control the geometrically nonlinear static as well as the dynamicresponses of the FG porous plates, where the effect of the structural damping is considered, basedon a closed-loop control with piezoelectric sensors and actuators.

Keywords: Piezoelectric materials, FG porous plate, Graphene platelets reinforcement, Bézierextraction, Nonlinear dynamic, Active control

∗Corresponding authorEmail address: ngx.hung@hutech.edu.vn (H. Nguyen-Xuan )

Preprint submitted to Elsevier March 1, 2019

1. Introduction

Nowadays, the demand for high performance structures with superior mechanical propertiesand chemical stability in engineering applications has been significantly increased. With these cel-lular structures, the porous materials whose the excellent properties such as lightweight, excellentenergy absorption, heat resistance has been extensively employed in various fields of engineeringincluding aerospace, automotive, biomedical and other areas [1–5]. However, the existence of in-ternal pores leads to significant reduction in the structural stiffness [6]. In order to overcome thisshortcoming, the reinforcement with carbonaceous nanofillers such as carbon nanotubes (CNTs)[7, 8] and graphene platelets (GPLs) [9, 10] into the porous materials is an excellent and practicalchoice to strengthen their mechanical properties. More importantly, this reinforcement aims alsoto maintain their potential for lightweight structures [11, 12]. In comparison with CNTs, GPLshave demonstrated great potentials to become a good candidate for reinforcement [13, 14] sinceGPLs have superior mechanical properties, a lower manufacturing cost, a larger specific surfacearea and two-dimensional geometry. In order to increase the performance of structure, the func-tionally graded (FG) porous structures reinforced by GPLs have been proposed in the literature toobtain the desired mechanical properties by modifying the sizes, the density of the internal poresin different directions as well as the dispersion patterns of GPLs [15–17]. In terms of numericalanalysis, large number of investigations have been conducted to study the influences of the internalpores and GPLs on the behaviours of structures under various different conditions. Kitipornchaiet al. [18] and Chen et al. [19] examined the free vibration, elastic buckling and the nonlinear freevibration, postbuckling behaviours of the FG porous beams reinforced with GPLs, respectivelybased on the Timoshenko’s beam theory and Ritz method. Yang et al. [20] utilized the first-ordershear deformation plate theory (FSDT) and Chebyshev-Ritz method to study the uniaxial, biax-ial, shear buckling and free vibration of the FG porous plates reinforced with GPLs uniformly ornon-uniformly distributed in the metal matrix. Based on the isogeometric analysis (IGA), Li et al.[21] analysed the static, free vibration and buckling of the FG porous plates reinforced by GPLsusing both first- and third-order shear deformation plate theories. Based on combination of theGalerkin method and the fourth-order Runge-Kutta approach, Li et al. [22] studied the nonlinearvibration and dynamic buckling of the sandwich FG porous plate reinforced by GPL resting onWinkler-Pasternak elastic foundation.

On the other hand, the piezoelectric materials have also been extensively applied to build upadvanced smart structures for modern industrial products. One of the excellent and essential fea-tures of these materials is the ability of transformation between the electrical and mechanicalenergy which is known as the piezoelectric effect and the converse phenomenon [23]. Regard-ing the analysis for the plate structures integrated with piezoelectric layers, a lot of studies havebeen conducted to predict their behaviours in the literature [24–29]. In addition, the piezoelectricFG carbon nanotubes reinforced composite plates (FG-CNTRC) also attracted remarkable atten-tion of researchers. Alibeigloo [30], [31] investigated the static and the free vibration analysesof FG-CNTRC plate as well as cylindrical panel embedded in thin piezoelectric layers using thethree-dimensional theory of elasticity. Using FSDT and von Kármán strain assumptions, Rafieeet al. [32] investigated the nonlinear parametric instability of initially imperfect the piezoelectric

3

FG-CNTRC plates under a combination of the electrical and thermal loadings. Then, Sharma etal. [33] investigated the active vibration control of FG-CNTRC rectangular plates with piezo-electric sensor and actuator layers using FEM based on FSDT. Selim et al. [34] studied the freevibration behaviour and active vibration control of FG-CNTRC plates with piezoelectric layersusing element-free IMLS-Ritz model based on Reddy’s higher-order shear deformation theory.Nguyen-Quang et al. [35] studied the dynamic response of laminated CNTRC plates integratedwith piezoelectric layers using IGA and HSDT. Recently, Malekzade et al. [36] employed thetransformed differential quadrature method for the free vibration analysis of FG eccentric annularplates reinforced with GPLs and integrated piezoelectric layers.

It is known that the different basis functions are applied for approximation of the geometriesand solutions in the framework of traditional FEM which leads to errors in the computationalprocess. To improve the accuracy of solutions as well as reduce computational costs, the IGA[37] which employs non-uniform rational B-splines (NURBS) basis as shape functions had beendiscovered. The main idea of IGA is fulfilled by using the same basis functions to describe thegeometry model and to approximate the solution field. The IGA has successfully been appliedto various fields of engineering and science. In comparison with the standard FEM, the NURBSbased IGA provides better accuracy and reliability for various engineering problems, especially forones with complex geometries which are reported in the literature. The basic and review of IGAare presented in the established literature [38, 39]. Nevertheless, the implementation of NURBSbased IGA approach is not often easy as their basis functions are not confined to a unique elementbut span over the entire domain instead. To overcome these obstacles, Borden et al. [40] proposedthe Bézier extraction which represents the NURBS basis function in the form of Bernstein polyno-mials basis defined over C0 continuous isogeometric Bézier element. By incorporating Bernsteinpolynomials which are similar to the Lagrangian basis functions as basis function in Bézier ex-traction, the implementation of IGA becomes analogous to the traditional FEM. As a result, theIGA approach can easily be embedded in most existing FEM codes while its advantages are stillkept naturally.

In the context of plate theories, there is a great deal of theories that have been introduced anddeveloped to estimate the responses of plate structures under different conditions. While CPT orKirchhoff-Love shows its drawbacks in the analysis of thick plates, FSDT which is capable of boththin and thick plates requires an appropriate shear correction factor. To overcome these shortcom-ings, several higher-order plate theories (HSDTs) have been proposed in the literature [41–45].Nevertheless, these HSDTs require the C1-continuity of the generalized displacement field whichleads to the second-order derivatives of deflection in the stiffness formulation. Therefore, severalC0-continuous elements were proposed [46–48].

As previously mentioned, most of the studies mainly focused on studying the plates integratedwith piezoelectric layers which address only the core layer composed of FGM or FG-CNTRC. Fur-thermore, the geometrically nonlinear static and dynamic analyses of the piezoelectric FG platesunder various loading types are still somewhat limited. In this study, in order to fill the existinggap in the literature, the geometrically nonlinear static and transient responses of piezoelectric

4

FG plates which have the core layer composed of FG porous materials reinforced by GPLs usingBézier extraction via IGA and C0-type HSDT. More importantly, the active control of the geo-metrically nonlinear static and dynamic responses of the FG porous plates with the effect of thestructural damping based on a closed-loop control with piezoelectric sensors and actuators is inves-tigated. By using the Bézier extraction, the IGA preserves the element structure, which allows theIGA approach to integrate conveniently into the existing FEM routine. For material distribution,the core layer is constituted by the combination of two porosity distributions and three dispersionpatterns of GPLs along the thickness plate, while the piezoelectric layer is perfectly bonded onthe both top and bottom surfaces of plate. The Newmark’s integration scheme incorporation withNewton-Raphson iterative procedure is utilized for the geometrically nonlinear static and dynamicanalyses. Then, some verification investigations are also conducted to prove the accuracy and sta-bility of the present method. The influence of some specific parameters such as different porositydistributions, porosity volume fractions and GPL dispersion patterns, input voltages on the non-linear behaviours of plate is addressed and discussed in detail through various numerical examples.

The outline of this paper is as follows. Section 2 provides the material models, the variationaland approximate formulations of the piezoelectric FG porous plates reinforced by GPLs based onC0-type HSDT. Meanwhile, Section 3 describes the active control algorithm. Section 4 presentsthe numerical examples for the geometrically nonlinear static and transient analyses as well as theactive control of the piezoelectric FG porous plates reinforced with GPLs before some concludingremarks are given in Section 5.

2. Theoretical formulations

2.1. Material models of the FG porous plate reinforced with GPLsWe consider a FG plate model whose core layer is made of metal foams reinforced by GPLs

with piezoelectric layers as depicted in Fig. 1. The length, width and total the thickness of piezo-electric FG porous plate are defined as a, b and h = hc + 2hp, respectively, in which hc and hp arethe thicknesses of the porous core and the piezoelectric layers, respectively. The porous core layerof plate is constituted by combining of two different porosity distribution types and three GPLdispersion patterns along the thickness direction of plates which are depicted in Fig. 2, respec-tively. The material properties including the Young’s modulus, shear modulus and mass densitythrough the thickness of the porous core layer corresponding to two porosity distribution types canbe expressed as

E (z) = E1 [1− e0λ (z)] ,G (z) = E (z) / [2 (1 + ν (z))] ,ρ (z) = ρ1 [1− emλ (z)] ,

(1)

where

λ (z) =

{cos (πz/hc), Porosity distribution 1cos (πz/2hc + π/4), Porosity distribution 2

(2)

in which E1 and ρ1 denote the maximum values of Young’s modulus and mass density in thethickness direction of the porous core layer, respectively. Meanwhile, the coefficient of porosity

5

e0 is determined bye0 = 1− E ′2/E ′1. (3)

where E ′1 and E′2 stand for the maximum and minimum values of Young’s modulus for the porous

core layer without GPLs, as shown in Fig. 2. Based on Gaussian Random Field (GRF) scheme[49], the mechanical properties of closed-cell cellular solids can be given as

E (z)

E1=

(ρ (z) /ρ1 + 0.121

1.121

)2.3for

(0.15 <

ρ (z)

ρ1< 1

). (4)

Then, the mass density coefficient em in Eq. 1 can be determined as

em =1.121

(1− 2.3

√1− e0λ (z)

)λ (z)

. (5)

Also according to the closed-cell GRF scheme [50], Poisson’s ratio ν(z) is determined by

ν (z) = 0.221p′ + ν1(0.342p′2 − 1.21p′ + 1

), (6)

where ν1 represents the Poisson’s ratio of metal without internal pores with p′ is given as

p′ = 1.121(

1− 2.3√

1− e0λ (z)). (7)

The volume fraction of GPLs varies along the thickness direction of plate for three dispersionpatterns which are illustrated in Fig. 2 can be given as

VGPL =

Si1 [1− cos (πz/hc)], Pattern ASi2 [1− cos (πz/2hc + π/4)], Pattern BSi3, Pattern C

(8)

where Si1,Si2 and Si3 are the maximum values of GPL volume fraction, in which i = 1, 2 corre-spond to two porosity distributions. The weight fraction of GPLs is related to its volume contentwhich are given as follows

ΛGPLρmΛGPLρm + ρGPL − ΛGPLρGPL

×∫ hc/2−hc/2

[1− emλ (z)]dz =∫ hc/2−hc/2

VGPL [1− emλ (z)]dz. (9)

The effective Young’s modulus of porous core layer reinforced with GPLs without internalpores is determined by the Halpin-Tsai micromechanics model [51, 52] as

E1 =3

8

(1 + ξLηLVGPL1− ηLVGPL

)Em +

5

8

(1 + ξWηWVGPL

1− ηWVGPL

)Em, (10)

in which

ξL =2lGPLtGPL

, ξW =2wGPLtGPL

, ηL =(EGPL/Em)−1

(EGPL/Em)+ξL, ηW =

(EGPL/Em)−1(EGPL/Em)+ξW

, (11)6

where wGPL, lGPL and tGPL are dimensions of GPLs including the average width, length andthickness, respectively; Meanwhile, EGPL and Em are the Young’s modulus of GPLs and metalmatrix, respectively. Finally, ρ1 and ν1 denote the mass density and Poisson’s ratio of the GPLsreinforced porous metal matrix can be determined based on the rule of mixture [53]

ρ1 = ρGPLVGPL + ρmVm, (12)

ν1 = νGPLVGPL + νmVm, (13)

where the mechanical properties for GPLs and metal matrix are denoted with subscript symbolsGPL and m, respectively. Meanwhile, VGPL and Vm = 1 − VGPL denote the volume fraction ofGPLs and metal matrix, respectively.

2.2. Weak form of the governing equationsThe governing equations of motion for the piezoelectric FG plate can be obtained by applying

the Hamilton’s variational principle [54] which can be expressed as follows

δ

∫ t2t1

Ldt = 0, (14)

in which t1 and t2 denote the starting and finish time values, respectively. Meanwhile, L is thegeneral energy functional which contains the summation of the kinetic energy, the strain energy,the dielectric energy and the external work is expressed as follows

L =1

2

∫Ω

(ρu̇T u̇− σTε+ DTE

)dΩ+

∫Γs

uT fsdΓs−∫

Γφ

φqsdΓφ+∑

uTFp−∑

φQp, (15)

where ρ denotes the mass density; u and u̇ represent the mechanical displacement and velocityfield vectors; φ represents the electric potential; Meanwhile, fs and Fp denote the external me-chanical surface and concentrated load vectors; qs and Qp indicate the external surface and pointcharges, respectively; Γs and Γφ denote the external mechanical and the electrical loading surface,respectively.

Then, the variational form for the equations of motion can be expressed as follows∫ t2t1

∫Ω

(ρüδuT − σT δε+ DT δE

)dΩdt+

∫ t2t1

∫Γs

fsδuTdΓsdt−

∫ t2t1

∫Γφ

qsδφdΓφdt

+∫ t2t1

∑Fpδu

Tdt−∫ t2t1

∑Qpδφdt = 0.

(16)

In this study, the linear constitutive relationships of the FG porous plate reinforced by GPLswith the piezoelectric layers can be presented as follows [55][

σD

]=

[c −eTe g

] [ε̄E

], (17)

7

in which ε̄ = [ε, γ]T and σ represent the strain and stress vectors, respectively; D denotes thedielectric displacement vector and e represents the stress piezoelectric constant matrix; g indicatesthe dielectric constant matrix; Meanwhile, the electric field vector E which is calculated followingthe electric potential field φ can be defined as [56]

E = −gradφ = −∇φ. (18)

And the material constant matrix c is defined as

c =

A B L 0 0B G F 0 0L F H 0 00 0 0 As Bs

0 0 0 Bs Ds

, (19)in which

(A,B,G,L,F,H) =∫ hc/2−hc/2 (1, z, z

2, f(z), zf(z), f 2(z))Qbijdz,

(As,Bs,Ds) =∫ hc/2−hc/2 (1, f

′(z), f ′2(z))Qsijdz,(20)

in which

Qb =Ee

1− ν2e

1 νe 0νe 1 00 0 1−νe

2

, Qs = Ee2(1 + νe)

[1 00 1

]. (21)

where Ee and νe are the effective Young’s modulus and Poisson’s ratio, respectively.

2.3. Approximation of the mechanical displacement field2.3.1. C0-type higher-order shear deformation theory

Considering a plate carrying a domain V = Ω × (−h2, h

2), in which Ω ∈ R2. Based on the

higher-order shear deformation theory [57], the displacement field at an arbitrary point in the platecan be presented as follows

u(x, y, z) = u0(x, y) + zu1(x, y) + f(z)u2(x, y), (22)

where

u =

uvw

,u0 =

u0v0w0

, u1 = −

w0,xw0,y

0

,u2 =

θxθy0

, (23)in which u0, v0, w0, θx and θy are the displacement components in the x, y, z directions and therotation components in the y- and the x- axes, respectively. Meanwhile, the subscript symbols xand y denote the derivatives of any function with respect to x and y directions, respectively; andf(z) is a function of the z−coordinate which is defined to describe the shear strains and stressesalong the thickness of plate, as is listed in [58]. In this work, the famous third-order functionproposed by Reddy is utilized as f(z) = z − 4z3

3h2[59].

8

In order to avoid the high order derivations in approximate formulations and convenientlyimpose the boundary conditions, additional assumptions are formulated as follows

w0,x = βx, w0,y = βy. (24)

Then, substituting Eq. 24 into Eq. 23, one obtains

u0 =

u0v0w0

,u1 = −

βxβy0

,u2 =

θxθy0

. (25)It can be seen that the compatible strain fields which are presented in Eq. 25 only require the

C0-continuity of the generalized displacements. Therefore, this theory is called as the C0-typehigher-order shear deformation theory (C0-HSDT).

The Green strain vector of a bending plate can be expressed in compact form as follows

εij =1

2

(∂ui∂xj

+∂uj∂xi

+∂uk∂xi

∂uk∂xj

). (26)

Employing the von Kármán assumptions, the strain-displacement relations can be rewritten as

ε ={εxx, εyy,γxy

}T= ε0 + zκ1 + f(z)κ2, (27a)

γ = {γxz, γyz}T = εs + f ′(z)κs, (27b)

where

ε0 =

u0,xv0,y

u0,y + v0,x

+ 12

w2,xw2,y

2w,xy

= εL0 + εNL0 ,κ1 = −

βx,xβy,y

βx,y + βy,x

,κ2 =

θx,xθy,y

θx,y + θy,x

,ε0 =

{w0,x − βxw0,y − βy

},κs =

{θxθy

}(28)

where the nonlinear strain component is expressed as follows

εNL0 =1

2

w,x 00 w,yw,y w,x

{ w,xw,y

}=

1

2ΘΛ. (29)

2.3.2. Isogeometric analysis based on Bézier extraction of NURBS2.3.2.1. B-spline and NURBS basis functions. In one dimensional (1D) space, the B-spline basisfunctions can be expressed by a set of knot vector in the parametric space which is defined byΞ = {ξ1, ξ2, ..., ξn+p+1}, where (i = 1, ...n + p) denotes the knot index. Meanwhile n and p are

9

the number of basis functions and the polynomial order, respectively. For a given knot vector Ξ,the B-spline basis functions are defined according to recursive form

Ni,0 (ξ) =

{1, if ξi ≤ ξ ≤ ξi+10, otherwise , for p = 0, (30)

Ni,p (ξ) =ξ − ξiξi+p − ξi

Ni,p−1 (ξ) +ξi+p+1 − ξξi+p+1 − ξi+1

Ni+1,p−1 (ξ) , for p > 0. (31)

Then, B-spline curves can be determined by taking a linear association of B-spline basis func-tions and the control points Pi (i = 1, 2, ..., n) as

T (ξ) =n∑i=1

PiNi,p (ξ). (32)

In two dimensional (2D) space, the B-splines basis functions can be also obtained by takinga tensor product of two basis functions in 1D space. Similarly, the B-splines surfaces are alsoexpressed by

S (ξ, η) =n∑i=1

m∑j=1

Pi,jNi,p (ξ)Mj,q (η) = PTN (ξ, η) . (33)

in which Ni,p and Mj,q represent the basis functions with orders p and q in the ξ and η directionscorresponding with the knot vectors Ξ = {ξ1, ξ2, ..., ξn+p+1} and H = {η1, η2, ..., ηm+q+1}, re-spectively.

Due to B-splines basis functions are limited the ability to exactly description some conic shapessuch as circles, cylinders, ellipsoids and spheres, the NURBS have been introduced based on theB-spline and a set of weights. Accordingly, the NURBS basis functions can be expressed as

Ri,j (ξ, η) =Ni,p (ξ)Mj,q (η)wi,j

n∑̂i=1

m∑̂j=1

Nî,p (ξ)Mĵ,q (η)wî,ĵ

, (34)

where wi,j represents the weight values. Then, the NURBS surfaces can be determined as follows

S (ξ, η) =n∑i=1

m∑j=1

Ri,j (ξ, η)Pi,j. (35)

2.3.2.2. Bézier extraction of NURBS. The major purpose of Bézier extraction is to instead theNURBS basis functions by the C0-continuous Bernstein polynomial basis functions defined overBézier elements which have the similar element structure with standard FEM. By using the Bern-stein polynomial as the basis function in Bézier extraction, the IGA approach is straightforwardlyperformed as well as can be integrated in most available FEM structures. It is well known that

10

the B-spline basis function of pth order has Cp−k continuity across each element, in which k rep-resents the multiplicity of knots in the knot value. Therefore, the C0-continuity can be obtainedby inserting the new knots into the B-spline basis function until k = p. Accordingly, a new knotξ̄ ∈ [ξk, ξk+1] with (k > p) is inserted into the original knot vector Ξ = {ξ1, ξ2, ..., ξn+p+1}. As aresult, a new set of control points are obtained and expressed as follows [37]

P̄i =

P1,αiPi+(1−αi)Pi−1Pn

i = 1,1 < i < n+ 1,i = n+ 1,

(36)

where

αi =

1 1 ≤ i ≤ k − p,ξ̄−ξi

ξi+p−ξi k − p+ 1 ≤ i ≤ k,0 i ≥ k + 1,

(37)

in which Pi and P̄i are the original and new control points, respectively.

Then, the Bézier extraction operator can be determined by using the new set of knots{ξ̄1, ξ̄2, ..., ξ̄n+1

}as follows [40, 60]

Cj =

α1 1− α2 0 . . . 00 α2 1− α3 0 . . . 00 0 α3 1− α4 0 . . . 0...0 . . . 0 αn+j−1 1− αn+j

. (38)

Applying the Bézier extraction operator Cj , a new Bézier control points Pb associated with Bern-stein polynomial basis can be determined as follows [61]

Pb = CTP, (39)

where the whole Bézier extraction operator C is defined as

C =∏n

j=1Cj. (40)

It should be noted that the geometries will not change after inserting a new knot into the origi-nal knot vector. As a result, the B-spline surface is also obtained based on Bernstein polynomialsand Bézier control points as

S (ξ, η) =n∑i

m∑j

Bi,j (ξ, η) Pbi,j =

(Pb)T

B (ξ, η) , (41)

in which the 2D Bernstein polynomials B (ξ, η) in terms of parametric coordinates ξ and η aredefined recursively as

Bi,j,p (ξ, η) =14

(1− ξ) (1 + η)Bi,j−1,p−1 (ξ, η) + 14 (1− ξ) (1− η)Bi,j,p−1 (ξ, η) +14

(1 + ξ) (1− η)Bi−1,j,p−1 (ξ, η) + 14 (1 + ξ) (1 + η)Bi−1,j−1,p−1 (ξ, η) ,(42)

11

whereB1,1,0 (ξ, η) = 1, Bi,j,p (ξ, η) = 0 (i, j < 1 or i,j> p + 1) . (43)

From Eq. 33 and Eq. 41, yields the following relation(Pb)T

B (ξ, η) = PTN (ξ, η) . (44)

According to Eq. 39 for 2D, the B-spline basis functions in Eq. 44 can be rewritten based onBernstein polynomials as follows

N (ξ, η) = CB (ξ, η) , (45)

Based on Eq. 45, the NURBS basis functions can be presetned by Bernstein polynomials asfollows

R (ξ, η) =W

W (ξ, η)N (ξ, η) =

W

W (ξ, η)CB (ξ, η) , (46)

where W denotes the diagonal matrix of the local NURBS weights. Meanwhile, the weight func-tions W (ξ, η) are expressed with the Bernstein basis functions as follows

W (ξ, η) =(CTw

)TB (ξ, η) =

(wb)T

B (ξ, η) , (47)

where w and wb are the weights for the NURBS and Bézier, respectively. The relation of Béziercontrol points and NUBRS ones is described by

Pb =(Wb)−1

CTWP. (48)

2.3.2.3. Bézier extraction of NURBS for FG porous plate formulations. Based on the Bézier ex-traction of NURBS, the mechanical displacement field u(ξ, η) of the FG porous plate can beapproximated as follows

u (ξ, η) =m×n∑A

ReA (ξ, η)dA, (49)

in which n×m represents the number of basis functions. Meanwhile, ReA (ξ, η) denotes a NURBSbasis function which is presented in Eq. 46; dA =

{u0A v0A w0A βxA βyA θxA θyA

}T isthe vector of the nodal degrees of freedom associated with the control point A.

By substituting Eq. 49 into Eq. 28, the in-plane and shear strains can be expressed as

[ε,γ]T =m×n∑A=1

(BLA +

1

2BNLA

)dA, (50)

12

in which BLA =[

B1A B2A B

3A B

s1A B

s2A

]T , whereB1 =

RA,x 0 0 0 0 0 00 RA,y 0 0 0 0 0RA,y RA,x 0 0 0 0 0

, B2 = − 0 0 0 RA,x 0 0 00 0 0 0 RA,y 0 0

0 0 0 RA,y RA,x 0 0

,

B3 =

0 0 0 0 0 RA,x 00 0 0 0 0 0 RA,y0 0 0 0 0 RA,y RA,x

,Bs1 =

[0 0 RA,x −RA 0 0 00 0 RA,y 0 −RA 0 0

], Bs2 =

[0 0 0 0 0 RA 00 0 0 0 0 0 RA

],

(51)

and BNLA = ΘΛ, in which

Θ =

wA,x 00 wA,ywA,y wA,x

, Λ = [ 0 0 RA,x 0 0 0 00 0 RA,y 0 0 0 0

]. (52)

2.4. Approximation of the electric potential fieldBy discretizing the piezoelectric layer into finite sublayers along the thickness, the electric

potential field on each layer is then approximated. Accordingly, in each sublayer, the electricpotential variation is considered to be linear and is approximated through the thickness as follows[62]

φi(z) = Riφφi, (53)

where Riφ is the shape function of the electric potential function which is determined in Eq. 46with p = 1. Meanwhile, φi =

[φi−1, φi

]with (i = 1, 2, ..., nsub) denotes the electric potentials

at the top and bottom surfaces of the sublayer, where nsub represents the number of piezoelectricsublayers.

In each sublayer element, the values of the electric potentials are estimated to equal at the sameheight according to z− direction [55]. Therefore, the electric potential field E for each sublayerelement which is presented in Eq. 18 can be expressed as follows

E = −∇Riφφi = −Bφφi, (54)

whereBφ =

{0 0 1

hp

}T. (55)

13

Finally, the stress piezoelectric constant matrix e, the strain piezoelectric constant matrix kand the dielectric constant matrix g can be determined by [62]

e =

0 0 0 0 e150 0 0 e15 0e13 e31 e33 0 0

,k = 0 0 0 0 k150 0 0 k15 0k31 k32 k33 0 0

,g = p11 0 00 p22 0

0 0 p33

,(56)

2.5. Governing equation of motionBy substituting Eqs. 49 and 54 into Eq. 16, the final form of the elementary governing equation

can be obtained and expressed as follows [62][Muu 0

0 0

] [d̈

φ̈

]+

[Kuu KuφKφu −Kφφ

] [dφ

]=

[fQ

], (57)

where

Kuu =∫

Ω(BL + BNL)

Tc(BL + 1

2BNL)dΩ, Kφφ =

∫Ω

BTφgBφdΩ,

Kuφ =∫

Ω(BL)

TẽTBφdΩ, Kφu = K

Tuφ, Muu=

∫Ω

ÑTmÑdΩ, f =∫

Ωq̄0N̄dΩ,

(58)

in whichẽ = [ eTm ze

Tm f (z) e

Tm e

Ts f

′ (z) eTs ],

Ñ = [ N0 N1 N2 ]T , N̄ =[

0 0 RA 0 0 0 0],

(59)

where

em =

0 0 00 0 0e31 e32 e33

, es = 0 e15e15 0

0 0

,N0 = RA 0 0 0 0 0 00 RA 0 0 0 0 0

0 0 RA 0 0 0 0

,

N1 = −

0 0 0 RA 0 0 00 0 0 0 RA 0 00 0 0 0 0 0 0

, N2 = 0 0 0 0 0 RA 00 0 0 0 0 0 RA

0 0 0 0 0 0 0

(60)

and

m =

I1 I2 I4I2 I3 I5I4 I5 I6

(61)in which the mass inertia terms Ii with (i = 1 : 6) are given as

(I1, I2, I3, I4, I5, I6) =

∫ h/2−h/2

ρ(z)(1, z, z2, f(z), zf(z), f 2(z)

)dz. (62)

14

Since the electric field E exists only according to the z direction, Kuφ in Eq. 58 can be rewrittena

Kuφ =

∫Ω

((B1)T

eTmBφ + z(B2)T

eTmBφ + f(z)(B3)T

eTmBφ

)dΩ. (63)

Now, substituting the second equation into the first one of Eq. 57, one obtains

Muud̈ +(Kuu + KuφK

−1φφ

Kφu

)d = F + KuφK

−1φφ

Q. (64)

3. Active control analysis

In this section, a piezoelectric FG porous plate, as depicted in Fig. 3, is considered for theactive control the static and dynamic responses of the FG plates. Whereas the bottom layer is apiezoelectric sensor labeled with the subscript s, the top layer represents a piezoelectric actuatordenoted with the subscript a. The combination between the displacement feedback control [55],which helps the piezoelectric actuator to generate the charge, and the velocity feedback control[54][63][64], which can provide a velocity component based on an appropriate electronic circuit,is utilized in this study. Furthermore, a consistent method [64][65] which can predict the dynamicresponses of piezoelectric FG plate is also applied. Two constant gains Gd and Gv of the displace-ment and velocity feedback control, respectively, are adopted in order to couple the input actuatorvoltage vector φa and the output sensor voltage vector φs as follows [64]

φa = Gdφs +Gvφ̇s. (65)

Assuming without any the external charge Q, the generated potential from the piezoelectric sensorlayer can be obtained from the second equation of Eq. 57

φs =[K−1φφ

]s

[Kφu

]sds, (66)

and the sensor charge resulted due to the deformation is determined by

Qs = [Kφu]sds. (67)

which can be understood that when the FG plate structures deform, the electric charges are gener-ated and gathered in the sensor layer because of the piezoelectric effect. After that, these electriccharges are amplified based on a closed loop control in order to convert into the voltage signalbefore being sent and applied to the actuator layer. Due to the converse piezoelectric effect, thestrains and stresses of structures are formed which can be applied to actively control the dynamicresponse of the FG porous plate.

By substituting Eqs. 65 and 66 into the second equation in Eq. 57, one obtains

Qa =[Kφu

]ada −Gd

[Kφφ

]a

[K−1φφ

]s

[Kφu

]sds −Gv

[Kφφ

]a

[K−1φφ

]s

[Kφu

]sḋs. (68)

15

Then, substituting Eq. 68 into Eq. 64 yields

Md̈ + Cḋ + K∗d = F, (69)

in whichK∗ = Kuu +Gd

[Kuφ

]a

[K−1φφ

]s

[Kφu

]s, (70)

and C is the active damping matrix which is expressed as

C = Gv[Kuφ]a[K−1φφ

]s

[Kφu

]s. (71)

Considering the effect of the structural damping, Eq. 69 can be rewritten as

Md̈ + (C + CR)ḋ + K∗d = F, (72)

in which CR denotes the Rayleigh damping matrix which is defined based on a linear associationbetween M and Kuu as follows

CR = αRM + βRKuu, (73)

where αR and βR are Rayleigh damping coefficients that can be defined from experiments. In thisstudy, the procedure in order to define the Rayleigh damping coefficients was reported in [66].

4. Numerical examples

In this study, the Newton-Rapshon iterative procedure [67] is employed to obtain the solutionsof the nonlinear problems. Accordingly, the iterations, where the solutions of current time step canbe obtained based on the solutions of previous time step, are repeated until the solutions converge.For the geometrically nonlinear dynamic analysis of the FG plate under various dynamic loadings,which the equations of dynamic problem depend on both the time domain and unknown displace-ment vector, the Newmark’s integration scheme [68] is selected. In all numerical examples, thePZT-G1195N piezoelectric is employed and perfectly bonded on the top and bottom surfaces ofthe FG plate structure as well as ignored the adhesive layers.

4.1. Validation analysisIn this section, various numerical studies regarding the geometrically nonlinear static and dy-

namic analyses of the isotropic as well as the piezoelectric FG square plates are carried out inorder to demonstrate the accuracy and stability of the present approach. Firstly, a fully clamped(CCCC) isotropic square plate is considered to show the validity of the present formulation forthe geometrically nonlinear analysis. The plate is subjected to uniformly distributed load whilethe width-to-thickness ratio (a/h) is taken equal to 100. The material properties of plate areE = 3 × 107 psi and ν = 0.316. In this example, the normalized central deflection and loadparameter can be defined as w = w/h and P = q0a4/(Eh2), respectively. Table 1 presents thenormalized central deflections of the isotropic square plate which are compared with those of the

16

Levy’s analytical solution [69], Urthaler and Reddy’s mixed FEM using FSDT [70] and Nguyenet al. based on IGA and refined plate theory (RPT) [71]. As can be observed that the proposedresults are in good agreement with the existing analytical solution as well as other approximateresults.

Next, in order to verify the accuracy of the proposed approach for the geometrically nonlineartransient analysis, a fully simply supported (SSSS) orthotropic square plate subjected to uniformload with q0 = 1.0 MPa is conducted in this example. The material properties and the geometryof plate are considered as follows: Young’s modulus E1 = 525 GPa, E2 = 21 GPa, shear modulusG12 = G23 = G13 = 10.5 GPa, Poisson’s ratio ν = 0.25, mass density ρ = 800 kg/m3, lengthof the plate L = 0.25 m and thickness h = 5 mm. Fig. 4 depicts the geometrically nonlineartransient response of the square plate subjected to uniform load. It can be seen that the presentresults are in an excellent agreement with those obtained from the finite strip method, which wasreported by Chen at al. [72].

Last but not least, a cantilever piezoelectric FG square plate is exhaustively presented todemonstrate the accuracy and validity of the present method for the static analysis of the FGplates integrated with piezoelectric layers. The FG plate which is bonded by two piezoelectric lay-ers on both the upper and the lower surfaces is made of aluminum oxide and Ti-6A1-4V materialswhose material properties are given in Table 2. In this study, the rule of mixture [53] is utilizedto describe the distribution of the ceramic and metal phases in core layer. The plate has a sidelength a = b = 0.4 m while the thickness of the FG core layer and each piezoelectric layer arehc = 5 mm and hp = 0.1 mm, respectively. The cantilever piezoelectric FG plate is subjected tosimultaneously a uniformly distributed load with q0=100 N/m2 and various input voltage values.The centerline linear deflections of the piezoelectric FG square plate are plotted in Fig. 5 while thetip node deflections are also listed in Table 3 with various material index n. The results which aregenerated from the proposed method are compared with those reported in [73] using a cell-basedsmoothed discrete shear gap method (CS-DSG3) based on FSDT. It can be observed that the re-sults obtained by the present formulation generally agree well with the reference solutions.

In the next part, investigations into the geometrically nonlinear static and dynamic responsesof the piezoelectric FG porous square plate reinforced by GPLs will be presented.

4.2. Geometrically nonlinear static analysisFirstly, the geometrically nonlinear static analysis of a piezoelectric FG plate subjected to uni-

form load with parameter load q = q0 × 103 is addressed. A SSSS piezoelectric FG square platewhich is made of aluminum oxide and Ti-6A1-4V has a side length a = b = 0.2 m, thickness ofFG core layer hc = 2 mm and thickness of each piezoelectric layer hp = 0.1 mm. Fig. 6 illustratesthe influence of the material index n on the normalized linear and nonlinear central deflections ofthe piezoelectric FG plates under mechanical load. As can be seen that, by increasing the materialindex n, the deflection of the piezoelectric FG plate decreases gradually. The largest deflectionis obtained when material index n = 0, where the plate consists only of Ti-6A1-4V leads to the

17

decrease in the bending stiffness. Furthermore, the values of the central deflection of the geomet-rically nonlinear analysis are always smaller than that of the linear one and this difference reduceswith the increase of the material index.

Next, a SSSS piezoelectric FG plate with porous core layer which is constituted by combiningof two porosity distribution types and three GPL dispersion patterns, respectively, is consideredin this example. The piezoelectric FG plate is subjected to sinusoidally distributed load whichis defined as q = q0sin(πx/a)sin(πy/b) in which q0 = 1.0 MPa. The plate has a side lengtha = b = 0.4 m, thickness of the FG porous core layer hc = 20 mm and thickness of each piezoelec-tric layer hp = 1 mm. In this study, the copper is chosen as the metal matrix whose material prop-erties are given in Table 2 while the dimensions of GPLs are lGPL = 2.5 µm, wGPL = 1.5 µm,tGPL = 1.5 nm. Fig. 7 examines the influence of the porosity coefficients on the nonlinear de-flection of the piezoelectric FG porous plate with GPL dispersion pattern A (ΛGPL = 1.0 wt. %)for two porosity distribution types, respectively. It can be observed that an increase of the porositycoefficients leads to the increase of the nonlinear deflection of the FG porous plate since the higherdensity of internal pores in material yields the reduction stiffness of plate structures. In addition,Fig. 8 depicts the effect of the weight fraction and the GPL dispersion patterns on the nonlineardeflection of the piezoelectric FG porous plate with e0 = 0.2 and two porosity distribution types,respectively. It can be observed that the effective stiffness of the FG porous core layer is greatlystrengthened when adding a small amount of GPLs (ΛGPL = 1.0 wt. %) into metal matrix as ev-idenced by decreasing the nonlinear deflection of the FG plate. More importantly, the reinforcingeffect of GPLs also depends significantly on the dispersion of GPLs in material matrix. Accord-ingly, with the same weight fraction of GPLs, the dispersion pattern A, where GPLs are dispersedsymmetric through the midplane of porous core layer, achieves the smallest nonlinear deflectionwhile the asymmetric dispersion pattern B provides the largest one. For further illustration, Fig. 9depicts the variation of the nonlinear deflection of the piezoelectric FG porous plate reinforced byGPLs which is constituted by porosity distribution 1 and three different GPL dispersion patternscorresponding to parameter load 10, respectively.

The combination influences of two porosity distribution types and three GPL dispersion pat-terns on the nonlinear deflection of the piezoelectric FG porous plate with ΛGPL = 1.0 wt. % ande0 = 0.4 is also investigated. As evidently depicted in Fig. 10, for all the considered associations,the combination between the porosity distribution 1 and the GPL dispersion pattern A obtains thebest reinforcing performance in the geometrically nonlinear static analysis of the piezoelectric FGporous plate. This indicated that the plate structures, where the internal pores are distributed on themidplane and GPLs are dispersed around the top and bottom surfaces, can provide the optimumreinforcement.

4.3. Geometrically nonlinear dynamic analysisIn this part, the geometrically nonlinear dynamic responses of a CCCC piezoelectric FG porous

plate reinforced by GPLs are studied. The dimensions and the material properties of the FG plate

18

are the same previous example. The plate is assumed to be subjected to time-dependent sinu-soidally distributed transverse loads which are expressed as follows q = q0sin(πx/a)sin(πy/b)F (t),where F (t) is defined as

F (t) =

{10

0 ≤ t ≤ t1,t > t1,

Step load{1− t/t10

0 ≤ t ≤ t1,t > t1,

Triangular load{sin (πt/t1)

00 ≤ t ≤ t1,

t > t1,Sinusoidal load

e−γt, Explosive blast load

(74)

in which q0 = 100 MPa, γ = 330s−1 and the time history F (t) is plotted in Fig. 11.

Fig. 12 illustrates the influence of the porosity coefficient on the nonlinear transient re-sponse of the piezoelectric FG porous plate with porosity distribution 1 and dispersion patternA (ΛGPL = 1.0 wt. %) under step and sinusoidal loads, respectively. It can be seen that byincreasing the porosity coefficients, the amplitude of the transverse deflection of the FG porousplate can be increased while the period of motion does not seem to affect. It can be concludedthat the presence of porosities in core layer of the FG plate reduces the capacity of itself againstexternal excitation. Furthermore, Fig. 13 demonstrates the influence of the weight fraction andthe dispersion pattern of GPLs on the nonlinear transient response of the piezoelectric FG porousplate with e0 = 0.2 and porosity distribution 2 corresponding to triangular and explosive blastloads, respectively. As expected, smaller magnitude of the deflection can be obtained when theweight fraction of GPLs in metal matrix increase. Again, the dispersion of GPLs into the metalmatrix also affects the reinforcing performance of structure that dispersion pattern A provides thesmallest magnitude of the deflection.

Next, the combination influences of various porosity distribution types and the GPL disper-sion patterns on the nonlinear dynamic response of the piezoelectric FG plate is also examinedand indicated in Fig. 14. For this specific example, the porous core layer of the piezoelectricplate has the porosity coefficient e0 = 0.4 and the GPL weight fraction ΛGPL = 1.0 wt. %.As clearly demonstrated in Fig. 14, the combination between the porosity distribution 1 and theGPL dispersion pattern A always provides the best reinforcement as evidenced by obtaining thesmallest amplitude of the deflection. Moreover, the dynamic responses of the linear and nonlinearof the FG porous plate with porosity distribution 2 (e0 = 0.3) and the GPL dispersion patternC (ΛGPL = 1.0 wt. %) under triangular and sinusoidal loads are also considered and depictedin Fig. 15. As can be observed, the geometrically nonlinear responses generally obtain smallermagnitudes of the deflection and periods of motion.

4.4. Static and dynamic responses active controlIn this section, the active control for the static and dynamic responses of the FG porous plate

reinforced by GPLs using integrated sensors and actuators is investigated. Firstly, the active con-19

trol for the linear static responses of a SSSS FG plate which is subjected to a uniformly distributedload with q0=100 N/m2 is investigated to perform the accuracy of the proposed approach. TheFG plate composed of Ti-6A1-4V and aluminum oxide materials with material index n = 2 hasthe side length a = b = 0.2 m while thickness of core FG layer and each piezoelectric layerare taken to be 1 mm and 0.1 mm, respectively. Fig. 16 illustrates the linear static deflectionsof the FG plate with various the displacement feedback control gains Gd. As can be observedthat the present results agree well with the reference solution which is reported in [73] who em-ployed the CS-DSG3 based on FSDT. As expected, when the displacement feedback control gainGd increases, the linear static deflection of the FG plate decreases. Furthermore, the active controlfor the linear dynamic responses of the FG plate is also investigated based on a constant velocityfeedback control algorithm Gv and a closed loop control. In this specific example, the FG plateis initially subjected to a uniform load q0=100 N/m2 and then the load is suddenly removed. Inthis study, the modal superposition is adopted in order to reduce the computational cost and thefirst six modes are considered in the modal space analysis, while the initial modal damping ratiofor each mode is assumed to be 0.8 %. Fig. 17 shows the linear dynamic responses of the centraldeflection of the FG plate. The results which are generated from present method agree well withthe reference solution [73].

Next, the active control for the nonlinear static responses of the SSSS FG porous plate rein-forced with GPLs is further investigated in this part. The FG plate consisting of combined theporosity distribution 1 and GPL dispersion pattern A, which provides the best structural perfor-mance, is selected to study. The material properties of the FG porous plate are the same in Section4.2. The plate has a side length a = b = 0.4 m, thickness of the FG porous core layer hc = 20 mmand thickness of each piezoelectric layer hp = 1 mm under sinusoidally distributed load which isdefined as q = q0sin(πx/a)sin(πy/b) with q0 = 1.0 MPa. Fig. 18 depicts the nonlinear staticdeflection of the FG porous reinforced by GPLs with the porosity coefficient e0 = 0.4 and theGPL weight fraction ΛGPL = 1.0 wt. % corresponding to various displacement feedback controlgains. As can be observed that the deflection of the FG porous plate decreases significantly whenthe displacement feedback control gain increase.

In the last example, the active control for the geometrically nonlinear dynamic responses ofthe CCCC FG porous plate reinforced by GPLs is conducted. The plate has the both length andwidth set the same at 0.2 m with the thickness of core layer hc = 10 mm and each piezoelectriclayer hp = 0.1 mm. The FG plate with the porosity distribution 1 (e0 = 0.4) and dispersionpattern A (ΛGPL = 1.0 wt. %) is subjected to sinusoidally distributed transverse loads which arethe same as those in Section 4.3. Fig. 19 illustrates the nonlinear dynamic responses of the centraldeflection of the FG plate corresponding to various the velocity feedback control gains Gv. It canbe observed that when the control gain Gv is equal to zero corresponding to without control case,the nonlinear dynamic response of the FG porous plate still attenuates with respect to time sincethe effect of the structural damping is considered in this study. More importantly, the geometricallynonlinear dynamic response can be suppressed more faster in the case controlled by higher velocityfeedback control gain values. As a result, depending on the specific cases, the responses of theFG porous plate structures including deflection, oscillation time or even both can be controlled to

20

satisfy an expectation by designing an appropriate value for the velocity feedback control gain. Itshould be noted that the feedback control gain values could not be increased without limit sincepiezoelectric materials have their own breakdown voltage values. In addition, Fig. 20 depicts theinfluence of the velocity feedback control gain Gv on the linear and nonlinear responses of theCCCC FG porous square plate subjected to step load. As expected, the geometrically nonlineardynamic responses provide smaller magnitudes of the deflection and periods of motion.

5. Conclusions

In this study, the IGA based on the Bézier extraction and the C0-HSDT was successfullywas presented for the geometrically nonlinear static and dynamic responses for FG porous plateswith GPLs reinforcement and integration with piezoelectric layers. The equations of motion werederived based on the C0-HSDT in conjunction with von Kármám strain assumptions. Whereas themechanical displacement field is approximated using the C0-HSDT based on the Bézier extractionof NURBS, the electric potential field was considered as a linear function through the thickness ofeach piezoelectric sublayer. Two porosity distributions and three dispersion patterns of GPLs withvarious related parameters were exhaustively carried out through numerical examples. The controlalgorithms based on the constant displacement and velocity feedbacks were utilized to controlthe geometrically nonlinear static and dynamic responses of the FG porous plate reinforced withGPLs. Through the present numerical results, several major remarks can be drawn:

• By applying Bernstein polynomials as basis functions in the Bézier extraction, the IGAapproach can easily be integrated into most existing FEM structures while its advantagesare maintained effectively.

• After adding a small amount of GPLs into the metal matrix, the stiffness of the structuresis significantly improved while an increase of the porosity coefficients leads to the decreaseof the reinforcing effect. Furthermore, the distribution of porosities and GPLs in metalmatrix also affect significantly the reinforcing performance of the structures. For all thecombinations, the association between the porosity distribution type 1 with internal poresdistributed on the midplane and the GPL dispersion pattern A, where GPLs are dispersedaround the top and bottom surfaces, obtained the best reinforcing performance.

• For geometrically nonlinear static responses control of the FG porous plates, two effectivealgorithms are considered including the input voltage control with opposite signs appliedacross the thickness of two piezoelectric layers and the displacement feedback control al-gorithm. In addition, the dynamic response of the FG porous plate can be expectantly sup-pressed based on the effectiveness of the velocity feedback control algorithm.

• Finally, the combination advantages of both the porous architecture and GPL reinforcementinto material matrices is a good choice to provide the advanced ultra-light high-strengthstructures in engineering.

21

Acknowledgement

The support provided by RISE-project BESTOFRAC (734370)H2020 is gratefully acknowl-edged.

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b

a

0q

Piezoelectric layer

Piezoelectric layer

FG porous with GPLsch

ph

ph

hx

y

z

Figure 1: Configuration of a piezoelectric FG porous plate reinforced by GPLs.

25

x

z

'

1E

'

1E

'

2E

Porosity distribution 1

2

ch

2

ch−

x

z

'

1E

'

2E

Porosity distribution 2

2

ch

2

ch−

(a) Porosity distribution types

1iS

1iS

0

z

GPLV

2

ch

2

ch−

Pattern A

2iS

0

0

z

GPLV

2

ch

2

ch−

Pattern B

3iS

0

z

GPLV

2

ch

2

ch−

Pattern C

(b) Dispersion patterns of GPLs.

Figure 2: Porosity distribution types and dispersion patterns of GPLs [18].

Sensor output

Actuator input

ControllerFG porous with GPLs

Actuator layer

Sensor layer

Figure 3: A schematic diagram of a FG porous plate with integrated piezoelectric sensors and actuators.

26

0 0.5 1 1.5 2Time (second) #10-3

-0.5

0

0.5

1

1.5

2

2.5

3

Norm

alize

dde.

ection,w

FSMPresent

Figure 4: Normalized nonlinear transient central deflection of a square orthotropic plate under the uniform load.

27

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4x axis distance (m)

-2.5

-2

-1.5

-1

-0.5

0

Cen

terlin

ede.

ection,w

(m)

#10-4

CS-DSG3 (0V)Present (0V)CS-DSG3 (20V)Present (20V)CS-DSG3 (40V)Present (40V)

(a) n = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4x axis distance (m)

-2.5

-2

-1.5

-1

-0.5

0

Cen

terlin

ede.

ection,w

(m)

#10-4

CS-DSG3 (0V)

Present (0V)

CS-DSG3 (20V)

Present (20V)

CS-DSG3 (40V)

Present (40V)

(b) n = 0.5

Figure 5: Centerline linear deflections of the cantilever piezoelectric FG plate under the uniform loading and variousactuator input voltages with n = 0 and n = 0.5.

-20 -15 -10 -5 0Load parameter

-0.5

-0.4

-0.3

-0.2

-0.1

0

Norm

alize

dde.

ection,w

n = 0 (Linear)n = 0 (Nonlinear)n = 0:5 (Linear)n = 0:5 (Nonlinear)n = 2:0 (Linear)n = 2:0 (Nonlinear)n = 10 (Linear)n = 10 (Nonlinear)

Figure 6: Effect of the material index n on the linear and nonlinear central deflections of the piezoelectric FG plateunder the mechanical load.

28

-20 -15 -10 -5 0Load parameter

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Norm

alize

dde.

ection,w

e0 = 0:0e0 = 0:2e0 = 0:4e0 = 0:6

(a) Porosity distribution 1

-20 -15 -10 -5 0Load parameter

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Norm

alize

dde.

ection,w

e0 = 0:0

e0 = 0:2

e0 = 0:4

e0 = 0:6

(b) Porosity distribution 2

Figure 7: Effect of the porosity coefficients on the nonlinear deflection of the piezoelectric FG porous square platewith GPL dispersion pattern A and ΛGPL = 1.0 wt. %.

29

-20 -15 -10 -5 0Load parameter

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Norm

alize

dde.

ection,w

$GPL = 0 wt%

Pattern A ($GPL = 1 wt%)

Pattern B ($GPL = 1 wt%)

Pattern C ($GPL = 1 wt%)

(a) Porosity distribution 1

-20 -15 -10 -5 0Load parameter

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Norm

alize

dde.

ection,w

$GPL = 0 wt%Pattern A ($GPL = 1 wt%)Pattern B ($GPL = 1 wt%)Pattern C ($GPL = 1 wt%)

(b) Porosity distribution 2

Figure 8: Effect of the weight fractions and dispersion patterns of GPLs on the nonlinear deflection of the piezoelectricFG porous square plate with e0 = 0.2.

30

e0

-0.1283

$(%)

-0.2418

-0.2034

-0.1597

-0.251 0.6

0.40.5

-0.2

0.2

Norm

alize

dde.

ection,w

0 0

-0.15

-0.12

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

(a) Pattern A

e0

-0.1871

-0.1538

-0.2418

-0.2034

$(%)

-0.250.61

0.40.5

0.2

-0.2

Norm

alize

dde.

ection,w

00

-0.15

-0.24

-0.23

-0.22

-0.21

-0.2

-0.19

-0.18

-0.17

-0.16

(b) Pattern B

-0.1838

-0.1515

-0.2418

e0-0.2034

$(%)

-0.250.61

0.40.5

0.2

-0.2

Norm

alize

dde.

ection,w

00

-0.15

-0.24

-0.23

-0.22

-0.21

-0.2

-0.19

-0.18

-0.17

-0.16

(c) Pattern C

Figure 9: Effect of the porosity coefficients and weight fractions of GPLs on the nonlinear deflection of piezoelectricFG porous square plate for porosity distribution 1 and different GPL dispersion patterns.

31

-20 -15 -10 -5 0Load parameter

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Norm

alize

dde.

ection,w

Dist. 1 and Pattern ADist. 1 and Pattern BDist. 1 and Pattern CDist. 2 and Pattern ADist. 2 and Pattern BDist. 2 and Pattern C

Figure 10: Effect of the porosity distributions and GPL dispersion patterns on the nonlinear deflection of the piezo-electric FG porous square plate with e0 = 0.4 and ΛGPL = 1.0 wt. %.

0 1 2 3 4 5 6Time (second) #10-3

0

0.2

0.4

0.6

0.8

1

Load

fact

or

Step loadTriangular loadSinusoidal loadExplosive blast load

Figure 11: Time history of load factor.

32

0 1 2 3 4 5 6Time (second) #10-3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Norm

alize

dde.

ection,w

e0 = 0:0e0 = 0:2e0 = 0:4e0 = 0:6

Forced vibration Free vibration

(a) Step load

0 1 2 3 4 5 6Time (second) #10-3

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Norm

alize

dde.

ection,w

e0 = 0:0e0 = 0:2e0 = 0:4e0 = 0:6

Forced vibration Free vibration

(b) Sinusoidal load

Figure 12: Effect of the porosity coefficients on the nonlinear dynamic responses of the CCCC piezoelectric FGporous plate with GPL dispersion pattern A and ΛGPL = 1.0 wt. %.

0 1 2 3 4 5 6Time (second) #10-3

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Norm

alize

dde.

ection,w

$GPL = 0:0 wt:%

Pattern A ($GPL = 1:0 wt:%)

Pattern B ($GPL = 1:0 wt:%)

Pattern C ($GPL = 1:0 wt:%)

Forced vibration Free vibration

(a) Triangular load

0 1 2 3 4 5 6Time (second) #10-3

-1

-0.5

0

0.5

Norm

alize

dde.

ection,w

$GPL = 0:0 wt:%

Pattern A ($GPL = 1:0 wt:%)

Pattern B ($GPL = 1:0 wt:%)

Pattern C ($GPL = 1:0 wt:%)

(b) Explosive blast load

Figure 13: Effect of the weight fractions and dispersion patterns of GPLs on the nonlinear dynamic responses of theCCCC piezoelectric FG porous square plate with porosity distribution 2 and e0 = 0.2.

33

(a) Step load (b) Sinusoidal load

Figure 14: Effect of the porosity distributions and GPL dispersion patterns on the nonlinear dynamic responses of theCCCC piezoelectric FG porous square plate with e0 = 0.4 and ΛGPL = 1.0 wt. %.

0 1 2 3 4 5 6Time (second) #10-3

-1.3

-1

-0.5

0

0.5

0.8

Norm

alize

dde.

ection,w

Linear

Nonlinear

Forced vibration Free vibration

(a) Triangular load

0 1 2 3 4 5 6Time (second) #10-3

-0.8

-0.6

-0.4

-0.2

0

0.2

Norm

alize

dde.

ection,w

LinearNonlinear

Forced vibration Free vibration

(b) Sinusoidal load

Figure 15: Linear and nonlinear dynamic responses of the CCCC piezoelectric FG porous square plate with porositydistribution 2 (e0 = 0.3) and dispersion pattern C (ΛGPL = 1.0 wt. %).

34

0 0.05 0.1 0.15 0.2x axis distance (m)

-3

-2.5

-2

-1.5

-1

-0.5

0

Cen

terlin

ede.

ection,w

(m)

#10-5

Present

Ref.

Gd = 30

Gd = 10

Gd = 5

Gd = 0

Figure 16: Effect of the displacement feedback control gain Gd on the linear static responses of the SSSS platesubjected to uniformly distributed load.

0 0.1 0.2 0.3 0.4 0.5Time (second)

-3

-2

-1

0

1

2

3

De.

ection,w

(m)

#10-5

Present (without control Gv = 0)

CS-DSG3 (without control Gv = 0)

Present (control gain Gv = 2e! 4)

CS-DSG3(control gain Gv = 2e! 4)

Figure 17: Effect of the velocity feedback control gain Gv on the linear dynamic response of the SSSS FG squareplate.

35

-20 -15 -10 -5 0Load parameter

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Norm

alize

dde.

ection,w

Gd = 0Gd = 5Gd = 10Gd = 30

Figure 18: Effect of the displacement feedback control gain Gd on the nonlinear static responses of the SSSS FGporous plate with porosity distribution 1 (e0 = 0.2) and GPL dispersion pattern A (ΛGPL = 1.0 wt. %).

36

0 1 2 3 4 5 6Time (second) #10-3

-1.5

-1

-0.5

0

0.5

1

Norm

alize

dde.

ection,w

Without control Gv = 0

Control gain Gv = 2e! 3

Control gain Gv = 5e! 3

Forced vibration Free vibration

(a) Step load

0 1 2 3 4 5 6Time (second) #10-3

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Norm

alize

dde.

ection,w

Without control Gv = 0

Control gain Gv = 2e! 3

Control gain Gv = 5e! 3

Forced vibration Free vibration

(b) Triangular load

0 1 2 3 4 5 6Time (second) #10-3

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Norm

alize

dde.

ection,w

Without control Gv = 0

Control gain Gv = 2e! 3

Control gain Gv = 1e! 1

Forced vibration Free vibration

(c) Sinusoidal load

0 1 2 3 4 5 6Time (second) #10-3

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Norm

alize

dde.

ection,w

Without control Gv = 0

Control gain Gv = 2e! 3

Control gain Gv = 5e! 3

(d) Explosive blast load

Figure 19: Effect of the velocity feedback control gain Gv on the nonlinear dynamic responses of the CCCC FGporous square plate subjected to dynamic loadings.

37

0 1 2 3 4 5 6Time (second) #10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Norm

alize

dde.

ection,w

Linear (Gv = 0)

Nonlinear (Gv = 0)

Linear (Gv = 3e ! 3)

Nonlinear (Gv = 3e ! 3)

Forced vibration Free vibration

Figure 20: Effect of the velocity feedback control gain Gv on the linear and nonlinear dynamic responses of the CCCCFG porous square plate subjected to step load.

Table 1: Normalized central deflection w of CCCC isotropic square plate under the uniform load with a/h = 100.

P Present Analytical [69] MXFEM [70] IGA-RPT [71]17.79 0.2348 0.237 0.2328 0.236538.3 0.4663 0.471 0.4738 0.469263.4 0.6873 0.695 0.6965 0.690895.0 0.8983 0.912 0.9087 0.9024

134.9 1.1016 1.121 1.1130 1.1060184.0 1.2960 1.323 1.3080 1.3008245.0 1.4875 1.521 1.5010 1.4926318.0 1.6728 1.714 1.6880 1.6784402.0 1.8492 1.902 1.8660 1.8552

38

Table 2: Material properties of the core and piezoelectric layers.

Properties Core layer Piezoelectric layerTi-6A1-4V Aluminum oxide Copper GPLs PZT-G1195N

Elastic propertiesE11 (GPa) 105.70 320.24 130 1010 63.0E22 (GPa) 105.70 320.24 130 1010 63.0E33 (GPa) 105.70 320.24 130 1010 63.0G12 (GPa) − − − − 24.2G13 (GPa) − − − − 24.2G23 (GPa) − − − − 24.2ν12 0.2981 0.26 0.34 0.186 0.30ν13 0.2981 0.26 0.34 0.186 0.30ν23 0.2981 0.26 0.34 0.186 0.30Mass densityρ (kg/m3) 4429 3750 8960 1062.5 7600Piezoelectric coefficientsk31 (m/V) − − − − 254× 10−12k32 (m/V) − − − − 254× 10−12Electric permittivityp11 (F/m) − − − − 15.3× 10−9p22 (F/m) − − − − 15.3× 10−9p33 (F/m) − − − − 15.3× 10−9

Table 3: Tip node deflection of the cantilever piezoelectric FGM plate subjected to the uniform load and various inputvoltages (×10−4 m).

n Method Input voltages (V)0 20 40

n = 0 Present -2.5437 -1.3328 -0.1229CS-DSG3 [73] -2.5460 -1.3346 -0.1232

n = 0.5 Present -1.6169 -0.8418 -0.0667CS-DSG3 [73] -1.6199 -0.8440 -0.0681

n = 5 Present -1.1233 -0.5808 -0.0382CS-DSG3 [73] -1.1266 -0.5820 -0.0375

n =∞ Present -0.8946 -0.4608 -0.0271CS-DSG3 [73] -0.8947 -0.4609 -0.0271

39

1 Introduction2 Theoretical formulations2.1 Material models of the FG porous plate reinforced with GPLs2.2 Weak form of the governing equations2.3 Approximation of the mechanical displacement field2.3.1 C0-type higher-order shear deformation theory2.3.2 Isogeometric analysis based on Bézier extraction of NURBS

2.4 Approximation of the electric potential field2.5 Governing equation of motion

3 Active control analysis4 Numerical examples4.1 Validation analysis4.2 Geometrically nonlinear static analysis4.3 Geometrically nonlinear dynamic analysis4.4 Static and dynamic responses active control

5 Conclusions