Nonlinear Dyn DOI 10.1007/s11071-016-3085-6 ORIGINAL PAPER P. Phung-Van · Loc V. Tran · A. J. M. Ferreira · H. Nguyen-Xuan · M. Abdel-Wahab Received: 29 December 2015 / Accepted: 10 September 2016 © Springer Science+Business Media Dordrecht 2016 Abstract We present a generalized shear deformation theory in combination with isogeometric (IGA) P. Phung-Van (B)· L. V. Tran Department of Electrical Energy, Systems and Automation, Faculty of Engineering and Architecture, Ghent University, Zwijnaarde, Belgium e-mail: phuc.phungvan@ugent.be approach for nonlinear transient analysis of smart piezoelectric functionally graded material (FGM) plates. The nonlinear transient formulation for plates is formed in the total Lagrange approach based on the von Kármán strains, which includes thermo-piezoelectric effects, and solved by Newmark time integration scheme. The electric potential through the thickness of each piezoelectric layer is assumed to be linear. The material properties vary through the thickness of FGM according to the rule of mixture and the Mori–Tanaka schemes. Various numerical examples are presented to demonstrate the effectiveness of the proposed method. ised 2 A. J. M. Ferreira Faculdade de Engenharia, Universidade do Porto, Porto, Portugal H. Nguyen-Xuan (B) Center for Interdisciplinary Research in Technology (CIRTech), HUTECH University, 700000 Ho Chi Minh City, Vietnam e-mail: ngx.hung@hutech.edu.vn Rev 1 Pro of Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads H. Nguyen-Xuan Department of Architectural Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 143-747, Republic of Korea M. Abdel-Wahab (B) Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam e-mail: magd.abdelwahab@tdt.edu.vn e-mail: Magd.abdelwahab@ugent.be M. Abdel-Wahab Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam M. Abdel-Wahab Soete Laboratory, Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde 903, 9052 Zwijnaarde, Belgium TYPESET DISK LE 4 5 6 7 8 9 10 11 12 13 14 Keywords Isogeometric analysis (IGA) · Nonlinear transient analysis · Sensors and actuators · Thermoelectro-mechanical load · FGM plates 17 1 Introduction 18 A new class of non-homogeneous composites known as functionally graded materials (FGMs) was firstly proposed by Koizumi [1]. Material properties are continuous and smoothly change from one surface to the other along thickness direction. These materials are capable of withstanding severe high temperature gradients, while maintaining structural integrity [1]. Piezoelectric material is also an intelligent material class, which has coupled electrical and mechanical properties. Important features of piezoelectric materials can be seen in the transformations of mechanical energy into electri- 123 Journal: 11071 MS: 3085 3 CP Disp.:2016/9/26 Pages: 16 Layout: Medium 15 16 19 20 21 22 23 24 25 26 27 28 29 P. Phung-Van et al. 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 fiber-reinforced composite using FSDT was presented by Panda and Ray [22]. The based HSDT finite element formulations for geometrically nonlinear analysis of functionally graded piezoelectric plates were also reported in Refs. [23–25]. For naturally discrete systems, D’Annibale et al. [26–28] examined the stationary response of piezoelectric control [26] and linear stability of piezoelectric-controlled discrete mechanical systems [27,28]. Recently, analysis of piezoelectric composite plates subjected to electro-mechanical loads using isogeometric analysis (IGA) and HSDT was studied [29]. However, nonlinear transient analysis and effect of the thermal environment on deflection of the smart plates have not taken into account in previous work. Besides, there have been few investigations about nonlinear transient for the smart FGM plates using the generalized shear deformation theory. Isogeometric analysis (IGA) was proposed by Hughes et al. [30] with combining between computeraided design (CAD) and finite element analysis (FEA). The basic functions of IGA are the same as those of CAD (most notably NURBS or T-Splines). One of the features of IGA is that IGA naturally satisfies the C 1 -continuity of plates based on the HSDT assumptions. In the past few years, IGA has been successfully applied to various fields. Cazzani et al. [31–33] studied curved beam using IGA. Cuomo et al. [34– 36] used B-spline interpolation for Kirchhoff–Love space rods [34,35] and for the analysis of cracked bodies [36]. Besides, a high-continuity finite element model for two-dimensional elastic structures [37] and for three-dimensional elasticity [38] was investigated. Particularly relevant to this paper is the study of structural vibrations and the development of shell and plate isogeometric elements [39–46]. However, the literatures mentioned above have not considered geometrically nonlinear transient responses. So far, there are few published materials related to geometrically nonlinear analysis using IGA for composite plates based on FSDT [47,48] and HSDT [49], solid shell [50], continuum shell [51] and Euler–Bernoulli beam [52]. Apparently, there are no researches on geometrically nonlinear transient using isogeometric analysis for the FGM or piezoelectric FGM plates based on the generalized shear deformation theory. This paper thus aims to fill this research gap by using IGA based on the generalized shear deformation theory for geometrically nonlinear transient analysis of the piezoelectric FGM plates. The nonlinear formulation for plates based on Pro of 32 cal energy when the plate is subjected mechanical loading, and the contrary phenomenon can be seen in the transformations of electrical energy into mechanical deformations, when the plate is subjected to voltages. And this phenomenon is known as piezoelectric effect and the converse phenomenon [2]. Hence, the integration of FGM and piezoelectric offers a considerable interest in many practical applications, such as microelectro-mechanical systems (MEMS) and smart material systems, especially in the medical and aerospace industries [3]. Because of their coupled thermal, electrical and mechanical properties, numerous methods on a wide range of topics related to FGMs and piezoelectric FGM have been proposed. Praveen and Reddy [4] studied nonlinear response of functionally graded ceramic– metal plates using finite element method based on a first-order shear deformation plate theory (FSDT). Zhao and Liew [5] also used FSDT combining the element-free kp-Ritz method to investigate geometrically nonlinear analysis of functionally graded plates (FGPs). Geometrically nonlinear analysis of FGM plates subjected to thermal–mechanical load was studied [6]. In the nonlinear formulation, smoothed finite element method (S-FEM) based on the C0-type highorder shear deformation plate theory (C0-HSDT) and the von Kármán strains were presented. Higher-order shear deformation theories (HSDTs) [7–10] were devised to solve nonlinear behavior of FGM composite structures. In addition, some equivalent theories were also studied to analyze FGM composite plates [11–15]. For piezoelectric FGM plates, a finite element model based on variational principle and linear piezoelectricity theory was developed by He et al. [16] and Liew et al. [17] for investigating the active control of FGM integrated with piezoelectric sensors and actuators. Reddy and Cheng [18] used the transfer matrix formulation and the asymptotic expansion to propose a 3D asymptotic solution for smart FGM plates. The nonlinear frequencies of a FGM plate with piezoelectric layers in thermal environments using HSDT were examined by Huang and Shen [19]. The nonlinear thermo-electro-mechanical bending response of piezoelectric FGM plates was investigated by Yang et al. [20]. Butz et al. [21] developed geometrically and materially nonlinear formulation based on Timoshenko beam theory using finite element method for a threedimensional piezoelectric beam. Nonlinear analysis of smart FGPs integrated with a layer of piezoelectric ised 31 Rev 30 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 Nonlinear transient isogeometric analysis 128 129 130 131 132 133 134 135 136 137 138 the von Kármán strains is based on the total Lagrange method and solved by Newmark time integration associated with the iteration methods. The electric potential of each piezoelectric layer is assumed linearly through the thickness of each piezoelectric layer. The material properties of FGM are assumed to vary through the thickness according to the rule of mixture and the Mori– Tanaka schemes. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical results. 141 2.1 The piezoelectric FGM model 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 A sandwich plate shown in Fig. 1a is made of one core that is graded from ceramic to metal and two outside skins that are piezoelectric. In the core layer, the volume fraction of ceramic and metal phase across thickness is described as follows [53]: Vc (z) =  zc 1 + 2 hc n , Vm (z) = 1 − V (z) (1) where c and m refer to the ceramic and metal, respectively; z c ∈ [z 2 , z 3 ] and h c = z 3 – z 2 is the thickness of core, which is shown in Fig. 1b. The material constituents of piezoelectric FGM can be obtained as: Vc (z) = 1, h c ∈ [z 1 , z 2 ] for botton skin   zc n 1 + , h c ∈ [z 2 , z 3 ] for core Vc (z) = 2 hc Vc (z) = 1, h c ∈ [z 3 , z 4 ] for top skin Vm (z) = 1 − Vc (z) (2) The material properties including Young’s modulus (E), Poisson’s ratio (ν) and density (ρ) based on the mixture rule are defined by: (3) where Pc and Pm represent the individual material properties of ceramic and metal. To consider the interactions among the constituents, the Mori–Tanaka scheme [54] is used in this paper by calculating the effective bulk and shear moduli as follows: Ke − Km Vc ; = c −K m Kc − Km 1 + Vm K mK+4/3µ m Vc µe − µm = m µc − µm 1 + Vm µµcm−µ + f1 TYPESET DISK LE 160 161 162 163 164 165 166 m (9K m +8µm ) where f 1 = µ6(K . The Young’s modulus and m +2µm ) Poisson’s ratio are now expressed by: 168 3K e − 2µe 9K e µe ; νe = 3K e + µe 2(3K e + µe ) (5) Temperature distributions of the bottom surface and top surface of piezoelectric FGM model are assumed to be constant. The temperature variation along the thickness is obtained by solving the one-dimensional steady heat state equation that is given by:   dT d k(z) =0 (6) − dz dz with boundary conditions T = Ttop at z = h/2 169 170 171 172 173 174 175 176 T = Tbot at z = −h/2 (7) where Ttop and Tbot are the top and bottom surface temperatures, respectively, and k(z) represents the coefficient of thermal conductivity at z position and is also expressed similar to Eq. (1) 2.2 The generalized higher-order shear deformation theory for piezoelectric FGM plates In the piezoelectric FGM plates, there are two field variables including a mechanical displacements field CP Disp.:2016/9/26 Pages: 16 Layout: Medium 167 177 123 Journal: 11071 MS: 3085 159 (4) Ee = Rev 142 P = Pc Vc (z) + Pm Vm (z) ised 140 2 The generalized higher-order shear deformation theory for piezoelectric FGM plates 139 Pro of Fig. 1 a Configuration of a piezoelectric FGM plate; b the sandwich plate with piezoelectric skins and FGM core 178 179 180 181 182 183 184 185 186 P. Phung-Van et al. 188 189 190 191 192 193 194 195 196 197 198 199 200 and an electrical field that need to be approximated. In this paper, the electrical field is assumed independently on each layer and the mechanical displacements are approximated by the generalized higher-order shear deformation theory and expressed as follows [55–57]: ∂w u(x, y, z) = u 0 (x, y) − z + f (z)βx (x, y) ∂x ∂w + f (z)β y (x, y) v(x, y, z) = v0 (x, y) − z ∂y w(x, y, z) = w(x, y) (8) where u 0 , v0 , βx , β y and w are displacement variables. The function f (z) is a continuous function through the 5 3 plate thickness and is chosen as f (z) = 87 − 2 hz 2 + 2 hz 4 [58]. For a bending plate, the Green’s strain vector can be presented by: ⎫ ⎧ ⎪ ∂u + 1 ∂u 2 + 1 ∂v 2 + 1 ∂w 2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂x 2 ∂x 2 ∂x 2 ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ∂u ∂v ∂w 1 1 ∂v 1 ⎪ ⎪ ⎬ ⎨ ∂y + 2 ∂y + 2 ∂y + 2 ∂y 203 204 205 206 207 208 209 210 211 212 213 (9) Using the Von Karman assumptions [59,60], which imply that derivatives of u and v are small and noting that w is independent of z, Eq. (9) can be rewritten as: (10) f ′ (z) is the derivative of the f (z) function and ⎤ ⎡ 2 ⎤ w,x u 0,x 1 2 ⎦ + ⎣ w,y ⎦ = εL + εN L ; εm = ⎣ v0,y 2 u 0,y + v0,x 2w,x y ⎤ ⎡ ⎤ ⎡ βx,x w,x x ⎦; κ1 = − ⎣ w,yy ⎦ ; κ2 = ⎣ β y,y βx,y + β y,x 2w,x y   βx κs = (11) βy ⎡ in which the nonlinear component can be expressed as: εN L = ⎡ ⎤ w 0 1 ⎣ ,x w,x 0 w,y ⎦ w,y 2 w,y w,x      ε̄ σ c −eT = e g E D  = 1 Aθ θ 2 (12) TYPESET DISK LE 214 215 216 where σ and ε̄ = [ε γ]T are the stress and strain vectors, respectively, D is the dielectric displacement, e is the piezoelectric constant matrix, g denotes the dielectric constant matrix and E is the electric field vector that is defined as: 221 E = −grad φ (14) 222 in which φ is the electric potential field and c is the elasticity matrix and defined as: 224 ⎡ A ⎢B c=⎢ ⎣N 0 BN CF FH 0 0 ⎤ 0 0 ⎥ ⎥ 0 ⎦ DS (15) where 217 218 219 220 223 225 226 Ai j , Bi j , Ci j , Ni j , Fi j , Hi j  h/2 = 1, z, z 2 , f (z), z f (z), f 2 (z) Q i j dz, 227 228 −h/2 i, j = 1, 2, 6  h/2  ′ 2 DiSj = f (z) G i j dz, i, j = 4, 5 229 (16) 230 −h/2 231 ⎡ ⎤ 1 νe 0 Ee ⎣ ⎦; 1 0 ν e 1 − νe2 1 0 0 2 (1 − νe )   Ee 10 G= 2(1 + νe ) 0 1 Q= 232 (17) 233 3 The piezoelectric FGM formulation based on NURBS basic functions 235 3.1 Knot vector and NURBS basic functions 236   A knot vector
= ξ1 , ξ2 , ..., ξn+ p+1 is defined as a sequence of parameter values ξi ∈ R, i = 1, ..., n + p. A B-spline basis function is C ∞ continuous inside a knot span and C p−1 continuous at a single knot. 123 Journal: 11071 MS: 3085 (13) in which ε = εm + zκ1 + f (z)κ2 γ = f ′ (z)κs where  ised 202 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Rev 201 ⎫ ⎧ εx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ε yy ⎪ ⎬ ε = γx y = ∂u ∂v ∂u ∂u ∂v ∂v ∂w ∂w ⎪ ⎪ ⎪ ∂y + ∂x + ∂x ∂y + ∂x ∂y + ∂x ∂y ⎪ ⎪ ⎪ γ ⎪ ⎪ ⎪ x z ⎪ ⎪ ∂u + ∂w + ∂u ∂u + ∂v ∂v + ∂w ∂w ⎪ ⎪ ⎩ ⎭ ⎪ ∂x ∂ x ∂z ∂ x ∂z ∂ x ∂z γ yz ⎪ ∂z ⎪ ⎩ ∂v + ∂w + ∂u ∂u + ∂v ∂v + ∂w ∂w ∂z ∂y ∂ y ∂z ∂ y ∂z ∂ y ∂z The material behavior of piezoelectric FGM is expressed as [61,62]: Pro of 187 CP Disp.:2016/9/26 Pages: 16 Layout: Medium 234 237 238 239 240 Nonlinear transient isogeometric analysis 241 242 243 244 245 Using Cox–de Boor algorithm, the univariate Bspline basis functions Ni, p (ξ ) are defined in Ref. [63] for the corresponding knot vector starting with order p = 0: Ni,0 (ξ ) =  1 if ξi ≤ ξ < ξi+1 0 otherwise  (18) Pro of Fig. 2 B-splines basic functions: a univariate quadratic; b univariate cubic 3.2 Mechanical displacements 268 The displacement field u of the plate using NURBS basic function is approximated as: 270 uh (ξ, η) = m×n  (22) R I (ξ, η)d I 269 271 I =1 246 For p + 1, the basis functions are obtained from: 247 Ni, p (ξ ) = 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 ised 249 (19) From the tensor product of basis functions in two parametric dimensions ξ and η with two  knot vectors
=  ξ1 , ξ2 , ..., ξn+ p+1 and H = η1 , η2 , ..., ηm+q+1 , the two-dimensional B-spline basis functions are obtained as: N A (ξ, η) = Ni, p (ξ ) M j,q (η) (20) An example of B-spline basis is illustrated in Fig. 2.  Using two open knot vectors
= 0, 0, 0, 51 , 25 , 35 ,    3 4 1 1 3 5 , 5 , 1, 1, 1 and H = 0, 0, 0, 0, 4 , 2 , 4 , 1, 1, 1, 1 , the two sets of univariate quadratic and cubic B-splines are plotted in Fig. 2a and b, respectively. To present exactly some conic sections, e.g., circles, cylinders and spheres, non-uniform rational B-splines (NURBS) need to be used. Being different from Bspline, each control point of NURBS has an additional value called an individual weight ζ A > 0: Rev 248 ξ − ξi Ni, p−1 (ξ ) ξi+ p − ξi ξi+ p+1 − ξ + Ni+1, p−1 (ξ ) ξi+ p+1 − ξi+1 R A (ξ, η) = N A (ξ, η) ζ A m×n  T  is the vector of where d I = u 0I v0I βx I β y I w I degrees of freedom associated with the control point I , and R I is the shape function as defined in Eq. (21). Substituting Eq. (22) into Eqs. (10)–(12), the strains can be rewritten as: (21) N A (ξ, η) ζ A ε̄ = [ε γ]T = m×n  B LI I =1 where B LI = which ⎡ R I,x m ⎣ BI = 0 R I,y ⎡ 0 ⎣ Bb1 = − 0 I 0 ⎡ 0 0 ⎣ Bb2 = 0 0 I 0 0  0 0 s BI = 0 0  BmI 0 R I,y R I,x 0 0 0 0 0 0 0 0  1 NL dI + BI 2 T  b1 BI 0 0 0 0 0 0 T  b2 BI T  s BI ! T T 267 And the B-spline function is recovered as the individual weight of control point is constant. ⎤  w I,x 0 0 L ⎣ ⎦ 0 w BN (d) = I,y I 0 w I,y w I,x 278 279 280 281 282 (24) TYPESET DISK LE 283 284 0 0 R I,x R I,y 0 0 0 0  g = Aθ B I 123 Journal: 11071 MS: 3085 276 , in L is calculated by: and B N I ⎡ 275 277 A=1 266 273 274 (23) ⎤ 0 0⎦, 0 ⎤ R I,x x 0 0 R I,yy 0 0 ⎦ , 2R I,x y 0 0 ⎤ R I,x 0 0 R I,y ⎦ , R I,y R I,x  0 RI 0 RI 272 CP Disp.:2016/9/26 Pages: 16 Layout: Medium P. Phung-Van et al. (25) 285 287 288 289 290 291 292 293 294 295 296 297 298 299 312 Kuu =    1 (B L + B N L )T c B L + B N L d ; 2 313 Kuφ =  (B L )T eT Bφ d 314 Kφφ =  BφT pBφ d ; 3.3 Electric potential field The electric potential field of each piezoelectric layer is approximated through the thickness as [64]: φ i (z) = Rφi φi (26) where Rφi is the shape functions for the electric potential, which is defined in Eq. (21) with p = 1, and φi is the vector containing the electric potentials at the and bottom surfaces of the ith sublayer: φi = topi−1  i φ φ (i = 1, 2, ...., n sub ) in which n sub is the number of piezoelectric layers. For each piezoelectric sublayer element, values of electric potential are assumed to be equal at the height along the thickness [61]. The electric field E in Eq. (14) can be rewritten as: 315 Pro of 286 where  Muu = ÑT mÑd ; f =  q̄0 R̄d ; 316 C̄ = α R M̄ + β R K̄ (31) in which α R , β R are Rayleigh damping factors, q̄0 is a  uniform load, R̄ = 0 0 0 0 R I ;and m is defined by: ⎤ ⎡ I1 I2 I4 m = ⎣ I2 I3 I5 ⎦ , I4 I5 I7 (I1 , I2 , I3 , I4 , I5 , I7 )  h/2 = ρ 1, z, z 2 , f (z), z f (z), f 2 (z) dz (32) E = −∇Rφi φi = −Bφ φi 302 In this work, the piezoelectric constant matrix e and the dielectric constant matrix g are defined as follows [64]: 303 ⎡ ⎤ ⎤ p11 0 0 0 0 0 0 d15 0 e = ⎣ 0 0 0 d15 0 0 ⎦ ; g = ⎣ 0 p22 0 ⎦ 0 0 p33 d31 d32 d33 0 0 0 (28) 301 ⎡ 305 306 307 310 311 322 ⎤ 0 0 00 RI 0 0 0 ⎦ ; 0 RI 0 0 ⎤ ⎡ 0 0 0 RI 0 Ñ3 = ⎣ 0 0 0 0 R I ⎦ 000 0 0 (33) 324 325 326 3.4 Governing equations of piezoelectric FGM plates 4.1 Iteration time 328 The governing equations for piezoelectric FGM plates can be written by:       d̈ Muu 0 d Kuu Kuφ + φ 0 0 K K φ̈ φu φφ " #$ % " #$ % " #$ % " #$ % We now consider the discretized system of equations for nonlinear transient problem, which is similar to that of Eq. (29). For the dynamic analysis, the Newmark method [65] is used in this paper. At initial time, t = 0, displacements, velocities, accelerations are set zero, we aim to find a new state, the first and second derivative of displacements, at (m + 1)t, using the following formulations: q̈ = K̄ q   f ⇔ M̄q̈ + K̄q = f̄ q "#$% (29) f̄ 309 321 327 M̄ 308 320 4 Nonlinear transient solution Rev 304 319 323 ⎧ ⎫ ⎡ RI ⎨ Ñ1 ⎬ Ñ = Ñ2 , Ñ1 = ⎣ 0 ⎩ ⎭ 0 Ñ3 ⎤ ⎡ 0 0 R I,x 0 0 Ñ2 = − ⎣ 0 0 R I,y 0 0 ⎦ ; 00 0 00 ised (27) 318 −h/2 and 300 317 1 1 (qm+1 − qm ) − q̇m − βt 2 βt For Rayleigh damping case, Eq. (29) can be rewritten as: q̈m+1 = M̄q̈ + C̄q̇ + K̄q = f̄ q̇m+1 = q̇m + t (1 − γ )q̈m + γ t q̈m+1 (30) 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium   1 − 1 q̈m 2β (34) (35) 329 330 331 332 333 334 335 336 337 338 339 Nonlinear transient isogeometric analysis 340 341 342 343 where the default values of β = 0.25 and γ = 0.5 are used as in Ref. [66]. Substituting Eq. (34) into Eq. (29), the following equation is obtained: &i+1 & & qm+1 − i qm+1 & & & < tol &i qm+1 & (42) 5 Numerical results 345 346 347 348 K̃m+1 qm+1 = f̃m+1 where K̃m+1 and f̃m+1 can be defined as: 1 M̄ K̃m+1 = K̄m+1 + βt 2  1 1 q̇m f̃m+1 = f̄m+1 + M̄ qm + 2 βt βt    1 − 1 q̈m + 2β (37) 353 4.2 Iterative method 351 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 In nonlinear analysis, the residual force, ϕ, is introduced to present errors of the approximation and tends to zeros during each iteration. From Eq. (36), the residual force at time step (m + 1)t, ϕm+1 , can be defined as follows: ϕm+1 = K̃m+1 qm+1 − f̃m+1 (38) To make unbalance residual force if i qm+1 is an approximate trial solution at the ith iteration, an improved solution, i+1 qm+1 , can be introduced as: i+1 qm+1 = i qm+1 + q Rev 350 (39) where q is the incremental displacement and calculated as [68]: q = −i ϕm+1 /KT (40) in which KT is called tangent stiffness matrix and computed following to the Newton–Raphson method [67]: 369 370 371 372 373 In this section, many numerical examples are investigated and compared to other available numerical methods to show the accuracy and effectiveness of the present method. Table 1 presents the properties of the piezoelectric FGM plates, including elastic properties (E), Poisson’s ration (ν), mass density (ρ), piezoelectric coefficients (d), electric permittivities ( p), coefficient of thermal expansion (α) and thermal conductivity (k). The boundary conditions used are: simply supported (S), clamped (C) or free (F) edges. The symbol, CFSF, represents clamped, free, simply supported and free edges. 387 5.1 Free vibration and static analyses of piezoelectric FGM plates 389 ised 352 Note that, the effective stiffness K̂m+1 in Eq. (36) is dependent on the displacements qm+1 . Therefore, the Newton–Raphson method [67] is used in this work and presented in the next section. 349 375 (36) Pro of 344 KT = ∂ ϕ(i q)/∂q (41) At each time step, Eq. (39) is repeated until the error of the displacements between two consecutive iterations becomes small and less then tolerance error: A square Al2O3/Ti-6Al-4V plate (400 mm × 400 mm) is considered and shown in Fig. 3. The thickness of the plate, h, is 5 mm and thickness of each piezoelectric layer, h_pie, is 0.1 mm. The reference solution using finite element method (FEM) with cubic Hermite shape functions was studied by He et al. [16]. First, a comparison for natural frequencies of the CCCC and SSSS plate with various volume fracture exponents is plotted in Fig. 4. It can be seen that the results using IGA match well with those of the reference solution [16] for both SSSS and CCCC cases. We can see that the frequencies of the CCCC plate are larger than those of the SSSS plate. This is because the stiffness of the CCCC plate is higher than that of the SSSS plate. Next, we now consider a CFFF plate subjected to a uniform load q = 100 N/m2 . Centerline deflections of the CFFF plate subjected to uniform load and actuator voltages V = 0 and 40 are shown in Fig. 5. It can be observed that with the increase in volume fraction exponent n, the deflection of the plate decreases. Besides, the deflection decreases for increasing input voltage. Similar results were obtained in Ref. [16]. Thus, we can see that in order to control the deflection of the piezoelectric FGM plate, we can apply equal-amplitude voltages at the bottom and top piezoelectric actuator layers. Figure 6 shows the centerline deflection of the plate 123 Journal: 11071 MS: 3085 374 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 376 377 378 379 380 381 382 383 384 385 386 388 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 P. Phung-Van et al. Properties Al Al2 O3 E (GPa) 70 320.24 63.0 151.9 105.7 ν 0.3 0.26 0.30 0.3 0.2981 ρ (kg/m3 ) 2707 3750 7600 3000 4429 k(W/mK)  α ×10−6 /C 204 10.4 – 2.09 18.1 d31= d32 (m/V) G1195N ZrO2 -2 Ti-6Al-4V 23 7.2 – 10 6.9 – – 2.54e–10 – – – – – – – 15.3e–9 – – d15 (m/V) – p11 (F/m) – – – – 15.3e–9 – – – 15.0e–9 – – ised p22 (F/m) p33 (F/m) Pro of Table 1 Material properties of piezoelectric and FGM materials Fig. 3 Square plate models and their discretization: a simply supported plate; b clamped plate; c meshing of 13 × 13 cubic elements 423 5.2 Nonlinear transient analysis of piezoelectric FGM plates GPa, G 12 = G 23 = G 13 = 10.5 GPa, ν = 0.25, ρ = 800 kg/m3 , length L = 250 mm and thickness h = 5 mm. The normalized central deflection, w̄ = w/ h, of linear and nonlinear analyses is plotted in Fig. 7. We can see that deflection responses of the present method match very well with those of finite strip method (FSM) [69]. In addition, magnitudes of nonlinear response are smaller than those of linear response and wavelength of the nonlinear response also changes comparing to that of linear case. 424 5.2.1 An orthotropic plate 5.2.2 Geometrically nonlinear analysis 417 418 419 420 421 422 425 426 427 428 429 under thermo-electro-mechanical load. It can be seen that the deflection changes when the plate is subjected to mechanical, thermal, electrical, thermo-mechanical, electro-mechanical or thermo-electro-mechanical load. This is because of the effect of piezoelectric converse, thermal converse or thermo-piezoelectric converse. Rev 416 This example aims to verify the accuracy of the present method for geometrically nonlinear transient analysis. A SSSS square plate under a uniform loading of q0 = 1 MPa is considered. Material properties and the geometry are given E 1 = 525 GPa, E 2 = 21 In this example, geometrically nonlinear analysis of the piezoelectric FGM plates subjected to mechanical, thermo-mechanical and thermo-electro-mechanical load is investigated. The square piezoelectric FGM plate has length L = 1, thickness of FGM layer 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 Nonlinear transient isogeometric analysis 446 447 448 449 450 451 452 453 454 Rev ised Pro of Fig. 4 First lowest eight natural frequencies of the simply supported (SSSS) and clamped (CCCC) piezoelectric FGM plate with the different volume fraction exponents h FGM = L/20 and thickness of each piezoelectric layer h piezo = h FGM /10. The boundary condition of the plate is SSSS. For the plate subjected to mechanical load (parameter load q̄ = qo ×105 ), Figs. 8 and 9 illustrate the effect of volume fraction exponent n to deflection of piezoelectric FGM (Al2 O3 /Ti6Al4V) and (Al/ZrO2 − 2) plates under mechanical load, respectively. It is again confirmed that the magnitude of deflection of nonlin- ear analysis is smaller than that of linear analysis. With the piezoelectric FGM (Al2 O3 /Ti6Al4V) plate, when volume fraction exponent increases, the deflection of the plate, shown in Fig. 8, decreases. In contrast to the piezoelectric FGM (Al/ZrO2 − 2) plate, the deflection of the plate plotted in Fig. 9 increases when n increases. Next, the effect of temperature on nonlinear deflection of the plate under thermo-mechanical load with n = 5 is shown in Fig. 10. It is observed that the behav- 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 455 456 457 458 459 460 461 462 463 Fig. 7 Displacement of the plate under step uniform load ised Fig. 5 Centerline deflection of the CFFF piezoelectric FGM plate subjected to mechanical and electro-mechanical load Pro of P. Phung-Van et al. 464 465 466 467 468 469 470 471 472 473 474 475 476 Rev Fig. 6 Centerline deflection of the CFFF piezoelectric FGM plate under thermo-electro-mechanical load ior of deflection subjected to thermo-mechanical load is different from the pure mechanical loading. When the mechanical load is zero, the deflection of the plate is not zero. This is because of the thermal expansion phenomenon. Also, the deflection decreases correspondingly to the increase in the temperature. We next study deflection of the plate under thermoelectro-mechanical load. The temperature at the ceramic surface is held Tc = 300 ◦ C and temperature at the metal surface is set at Tm = 20 ◦ C. Figure 11 plots effect of parameter input voltage (φ = V × 103 / p33 ) on deflection of the plate with n = 100. We can see that under the piezoelectric effect, the deflection of Fig. 8 Effect of volume fraction exponent n on deflection of piezoelectric FGM (Al2 O3 /Ti6Al4V) plates under mechanical load the plate is converse when parameter input voltage becomes larger. Further, Fig. 12 shows the effect of volume fraction exponent n on deflection of piezoelectric FGM (Al/ZrO2 − 2) plates under thermo-electromechanical load. Again, we can see that the deflection decreases when n increases. 482 5.2.3 Geometrically nonlinear transient 483 This example is a piezoelectric FGM plate subjected to sinusoidally distributed transverse loads, the CCCC square plate has length L = 0.2, thickness of FGM layer h FGM = L/10 and thickness of each piezoelectric 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 477 478 479 480 481 484 485 486 487 Fig. 11 Effect of input voltage on deflection of piezoelectric FGM (Al/ZrO2 − 2) plates with n = 100 subjected to thermoelectro-mechanical load Rev ised Fig. 9 Effect of volume fraction exponent n on deflection of piezoelectric FGM (Al/ZrO2 − 2) plates subjected to mechanical load Pro of Nonlinear transient isogeometric analysis Fig. 10 Effect of temperature on deflection of piezoelectric FGM (Al/ZrO2 − 2) plates with n = 5 under thermo-mechanical load 488 489 490 491 492 493 layer h piezo = h FGM /10. The FGM layer is made of Al/ZrO2 − 2. The sinusoidally distributed transverse load is expressed as follows: q = q0 sin πy πx sin F(t) L L (43) Fig. 12 Effect of volume fraction exponent n on deflection of piezoelectric FGM (Al/ZrO2 − 2) plates under thermo-electromechanical load ⎡ 1 0 ≤ t ≤ t1 Step load ⎢ 0 t > t1 ⎢ ⎢ 1 − t/t1 0 ≤ t ≤ t1 ⎢ Triangular load t > t1 F(t) = ⎢ ⎢0 ⎢ sin(πt/t1 ) 0 ≤ t ≤ t1 ⎢ Sinusoidal load ⎣ 0 t > t1 −γ t e Explosive blast load (44) in which q0 = 4 × 108 Pa, γ = 330 s−1 and F(t) is plotted in Fig. 13. where 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 494 495 496 P. Phung-Van et al. 0.5 0 0.8 0.4 0.3 0.2 0.1 n=0 0 0.6 0 1 Explosive blast load n=5 1.5 2 Time 0.4 x 10 -3 Fig. 16 Effect of volume fraction exponent n on nonlinear transient responses of piezoelectric FGM plates subjected to sinusoidal load Triangular load 0.2 =1 0.5 Pro of Value of force (F ) Deflection Step load 1 Sine load 0 1 0.5 1 1.5 2 Time x 10 -3 Fig. 13 Types of load: step, triangular, sinusoidal and explosive blast n=1 1 1.5 2 x 10 n=1 0.5 -3 0.4 0.2 0 Damping (n = 0) Damping (n = 1) 0.5 1 Damping (n = 5) 1.5 Time 2 x 10 -3 Fig. 18 Effect of damping on nonlinear transient responses of piezoelectric FGM plates subjected to step load 0 0.5 1 1.5 1 Without damping (n = 5) 2 x 10 -3 Fig. 15 Effect of volume fraction exponent n on nonlinear transient responses of piezoelectric FGM plates subjected to triangular load 504 2 x 10 Without damping (n = 5) n=5 Time 503 1.5 0.6 Rev Deflection n=0 -0.5 0 0.8 -3 -0.2 0 1 501 1 n=5 Deflection 0.5 Figures 14, 15, 16 and 17 show the effect of volume fraction exponent on nonlinear transient response of the plate subjected to the step, triangular, sinusoidal and explosive blast load, respectively. It can be observed that when n increases, the deflection increases. Figures 18, 19 and 20 show effect of damping on nonlinear transient responses of piezoelectric FGM plates subjected to step load, triangular load and explosive Deflection 0 Fig. 14 Effect of volume fraction exponent n on nonlinear transient responses of piezoelectric FGM plates subjected to step load 502 0.5 Fig. 17 Effect of volume fraction exponent n on nonlinear transient responses of piezoelectric FGM plates subjected to explosive blast load 0 Time 500 0 ised Deflection 0.5 n=0 499 n=5 Time -0.5 498 n =1 0.5 -0.5 0 1 497 n=0 Deflection 0 Damping (n = 5) 0.5 0 Damping (n = 0) Damping (n = 1) -0.5 0 0.5 1 Time TYPESET DISK LE 2 x 10 -3 Fig. 19 Effect of damping on nonlinear transient responses of piezoelectric FGM plates subjected to triangular load blast load, respectively. It turns out that deflection of the plate decreases under damping effect. Next, transient responses of linear, undamped nonlinear and nonlinear with damping of the plate under the step, triangular, 123 Journal: 11071 MS: 3085 1.5 CP Disp.:2016/9/26 Pages: 16 Layout: Medium 505 506 507 508 Nonlinear transient isogeometric analysis 1 0.8 Without damping (n = 5) Damping (n = 5) Deflection 0.4 0.2 0 Damping (n = 0) -0.2 0 1 1.5 2 Time x 10 Linear Nonlinear with damping 0 -0.5 0 0.4 0.8 Free vibration 1.2 1.6 2 Time 0.8 x 10 -3 Nonlinear with damping 1.2 1.6 Forced vibration Free vibration 0 Al/ZrO2-2 AL2O3/Ti6AL4V -0.5 -1 0 0.5 1 1.5 2 0 Deflection Forced vibration Free vibration 0.5 1.2 1.6 x 10 Rev Deflection Nonlinear Linear 0.4 0.2 0 Nonlinear with damping Forced vibration 0.4 0.8 Time Free vibration 1.2 1.6 2 x 10 -3 Fig. 23 Transient responses of linear, undamped nonlinear and nonlinear with damping of piezoelectric FGM plates subjected to sinusoidal load with n = 0.5 512 0 -0.5 2 0 -3 sinusoidal and explosive blast load are studied and illustrated in Figs. 21, 22, 23 and 24, respectively. Again, it can be seen that nonlinear response has lower central deflection and higher frequency than that of the lin- 0.4 0.8 1.2 1.6 Time 2 x 10 -3 Fig. 26 Nonlinear transient responses of the plate under triangular load with n = 10 0.6 Deflection 0.8 0.6 511 Free vibration 0.5 Forced vibration 0.4 Fig. 22 Transient responses of linear, undamped nonlinear and nonlinear with damping of piezoelectric FGM plates subjected to triangular load with n = 0.5 510 Al2O3/Ti6Al4V Al/ZrO2-2 Nonlinear with damping ised Deflection Linear Time 509 -3 1 Nonlinear -0.2 0 x 10 Fig. 25 Nonlinear transient responses of the plate under step load with n = 10 1 -0.5 0 -3 0.5 Time Fig. 21 Transient responses of linear, nonlinear and nonlinear with damping of piezoelectric FGM plates subjected to step load with n = 0.5 2 x 10 Fig. 24 Transient responses of linear, undamped nonlinear and nonlinear with damping of piezoelectric FGM plates subjected to explosive blast load with n = 0.5 1 0.5 Forced vibration Linear 0.4 Time Deflection Deflection Nonlinear -0.5 0 -3 Fig. 20 Effect of damping on nonlinear transient responses of piezoelectric FGM plates subjected to explosive blast load 1 0 Nonlinear Damping (n = 1) 0.5 0.5 Pro of Deflection 0.6 Forced vibration 0.4 Free vibration 0.2 0 Al2O3/Ti6Al4V Al/ZrO2-2 -0.2 0 0.4 0.8 1.2 Time 1.6 2 x 10 -3 Fig. 27 Nonlinear transient responses of the plate under sinusoidal load with n = 10 ear response, and deflection of the plate shows damped oscillations under damping effect. Lastly, Figs. 25, 26, 27 and 28 plot nonlinear response of the Al/ ZrO2 − 2 and Al2 O3 /Ti6Al4V plate subjected to the step, triangular, sinusoidal and explosive blast load, respectively. 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 513 514 515 516 517 P. Phung-Van et al. Deflection Al/ZrO2-2 Al2O3/Ti6Al4V 0.5 0 -0.5 0 0.4 0.8 1.2 1.6 2 Time x 10 -3 Fig. 28 Nonlinear transient responses of the plate under explosive blast load with n = 10 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 6 Conclusions We have addressed an efficient computational approach based on the generalized shear deformation model and IGA for geometrically nonlinear transient analysis of piezoelectric functionally graded plates (FGPs). The electric potential is assumed linearly through the thickness of each piezoelectric layer. The material properties of FGM are assumed to vary through the thickness by the rule of mixture and the Mori–Tanaka schemes. From the present formulation and numerical results, some significant remarks are emphasized as follows: Acknowledgments H. Nguyen-Xuan would like to thank the Alexander von Humboldt Foundation for granting the Georg Forster Research Award. References ised 519 We can see that the deflection of the Al/ ZrO2 -2 plate is larger than that of the Al2 O3 /Ti6Al4V plate. (1) In the analysis process, the thermal, electrical and mechanical loadings are considered and a two-step procedure is developed including a step of calculating the temperature field along the thickness of the plate and a step of analyzing the geometrically nonlinear behavior of the plate under thermo-electromechanical loadings. (2) A new function f (z) through the plate thickness is used for the piezoelectric FGM plates. (3) In static and free vibration analyses, the results of the present method showed high accuracy and effectiveness compared to other available numerical methods. (4) The nonlinear formulation for plates is formed in the total Lagrange approach based on the von Kármán strains, which includes thermo-piezoelectric effects, and solved by Newmark time integration associated with the iteration methods. Geometrically nonlinear transient of the piezoelectric FGM plates is developed. Rev 518 (5) For the comparison between the nonlinear and linear behaviors, nonlinear response has the lower central deflection and the higher frequency than those of the linear response. Wavelength of the nonlinear response also changes compared to that of linear case. Because of thermal expansion phenomenon, the deflection of the plates is upward and decreases when temperature increases. (6) Effects of various parameters and damping on the geometrically nonlinear responses of the piezoelectric FGP plate have been also investigated. (7) Numerous numerical examples have been given to show the accuracy and reliability of the present method for the geometrically nonlinear responses of the piezoelectric FGM plate by comparing its numerical solutions with those of other available numerical results. Pro of 1 1. Koizumi, M.: The concept of FGM. Ceram. Trans. Function. Graded Mater. 34, 3–10 (1993) 2. Wang, Z., Chen, S., Han, W.: The static shape control for intelligent structures. Finite Elem. Anal. Des. 26, 303–314 (1997) 3. Takagi, K., Li, J.F., Yokoyama, S., Watanabe, R.: Fabrication and evaluation of PZT/Pt piezoelectric composites and functionally graded actuators. J. Eur. Ceram. Soc. 23, 1577– 1583 (2003) 4. Praveen, G.N., Reddy, J.N.: Nonlinear transient thermo elastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35, 4457–4476 (1998) 5. Zhao, X., Liew, K.M.: Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method. Comput. Methods Appl. Mech. Eng. 198, 2796– 2811 (2009) 6. Phung-Van, P., Nguyen-Thoi, T., Luong-Van, H., LieuXuan, Q.: Geometrically nonlinear analysis of functionally graded plates using a cell-based smoothed three-node plate element (CS-MIN3) based on the C0-HSDT. Comput. Methods Appl. Mech. Eng. 270, 15–36 (2014) 7. Aliaga, J.W., Reddy, J.N.: Nonlinear thermoelastric analysis of functionally graded plates using the third-order shear deformation theory. Int. J. Comput. Eng. Sci. 5, 753–779 (2004) 8. Phung-Van, P., Nguyen-Thoi, T., Bui-Xuan, T., Lieu-Xuan, Q.: A cell-based smoothed three-node Mindlin plate element (CS-FEM-MIN3) based on the C0-type higher-order shear deformation for geometrically nonlinear analysis of laminated composite plates. Comput. Mater. Sci. 96, 549–558 (2015) 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 Nonlinear transient isogeometric analysis 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 Pro of 606 23. Behjat, B., Khoshravan, M.R.: Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Compos. Struct. 94, 874–882 (2012) 24. Yiqi, M., Yiming, F.: Nonlinear dynamic response and active vibration control for piezoelectric functionally graded plate. J. Sound Vib. 329, 2015–2028 (2010) 25. Vahid, F., Abdolreza, O., Peyman, Y.: Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment. Compos. Struct. 93, 2310–2321 (2011) 26. D’Annibale, F., Rosi, G., Luongo, A.: Piezoelectric control of Hopf bifurcations: A non-linear discrete case study. Int. J. Non-Linear Mech. 80, 160–169 (2016) 27. D’Annibale, F., Rosi, G., Luongo, A.: Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces. Meccanica 50(3), 825– 839 (2015) 28. D’Annibale, F., Rosi, G., Luongo, A.: On the failure of the “Similar Piezoelectric Control” in preventing loss of stability by nonconservative positional forces. Zeitschrift für angewandte Mathematik and Physik 66(4), 1949–1968 (2015) 29. Phung-Van, P., De Lorenzis, L., Chien, Thai H., AbdelWahab, M., Nguyen-Xuan, H.: Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Comput. Mater. Sci. 96, 496–505 (2015) 30. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005) 31. Cazzani, A., Malagu, M., Turco, E.: Isogeometric analysis of plane curved beams. Math. Mech. Solids (2014). doi:10. 1177/1081286514531265 32. Cazzani, A., Malagu, M., Turco, E., Stochino, F.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21, 182–209 (2016) 33. Cazzani, A., Malagu, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28, 139–156 (2016) 34. Greco, L., Cuomo, M.: B-Spline interpolation of KirchhoffLove space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013) 35. Greco, L., Cuomo, M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014) 36. Cuomo, M., Contraffatto, L., Greco, L.: A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci. 80, 173–188 (2014) 37. Aristodemo, M.: A high continuity finite element model for two-dimensional elastic structures. Comput. Struct. 21, 987– 993 (1985) 38. Bilotta, A., Formica, G., Turco, E.: Performance of a highcontinuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26, 1155–1175 (2010) 39. Thai, H., Chien, Zenkour, A.M., Abdel-Wahab, A., NguyenXuan, H.: A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sand- ised 605 9. Reddy, J.N., Kim, J.: A nonlinear modified couple stressbased third-order theory of functionally graded plates. Compos. Struct. 94, 1128–1143 (2012) 10. Shankara, C.A., Iyegar, N.G.R.: A C0 element for the free vibration analysis of laminated composite plates. J. Sound Vib. 191, 721–738 (1996) 11. Nguyen-Thoi, T., Rabczuk, T., Lam-Phat, T., Ho-Huu, V., Phung-Van, P.: Free vibration analysis of cracked Mindlin plate using an extended cell-based smoothed discrete shear gap method (XCS-DSG3). Theor. Appl. Fract. Mech. 72, 150–163 (2014) 12. Luong-Van, H., Nguyen-Thoi, T., Liu, G.R., Phung-Van, P.: A cell-based smoothed finite element method using threenode shear-locking free Mindlin plate element (CS-FEMMIN3) for dynamic response of laminated composite plates on viscoelastic foundation. Eng. Anal. Bound. Elem. 42, 8–19 (2014) 13. Phung-Van, P., Nguyen-Thoi, T., Luong-Van, H., ThaiHoang, C., Nguyen-Xuan, H.: A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise deformation theory for dynamic response of composite plates resting on viscoelastic foundation. Comput. Methods Appl. Mech. Eng. 272, 138–159 (2014) 14. Phung-Van, P., Thai, C.H., Nguyen-Thoi, T., Nguyen-Xuan, H.: Static and free vibration analyses of composite and sandwich plates by an edge-based smoothed discrete shear gap method (ES-DSG3) using triangular elements based on layerwise theory. Compos. B: Eng. 60, 227–238 (2014) 15. Phung-Van, P., Luong-Van, H., Nguyen-Thoi, T., NguyenXuan, H.: A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the C0-type higher-order shear deformation theory for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle. Int. J. Numer. Meth. Eng. 98(13), 988–1014 (2014) 16. He, X.Q., Ng, T.Y., Sivashanker, S., Liew, K.M.: Active control of FGM plates with intergrated piezoelectric sensors and actuators. Int. J. Solids Struct. 38, 1641–1655 (2001) 17. Liew, K.M., He, X.Q., Ng, T.Y., Kitipornchai, S.: Finite element piezothermo-elasticity analysis and active control of FGM plates with integrated piezoelectric sensors and actuators. Comput. Mech. 31, 350–358 (2003) 18. Reddy, J.N., Cheng, Z.Q.: Three-dimensional solutions of smart functionally graded plates. J. Appl. Mech. ASME 68, 234–241 (2001) 19. Huang, X.L., Shen, H.S.: Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. J. Sound Vib. 289, 25–53 (2006) 20. Yang, J., Kitipornchai, S., Liew, K.M.: Non-linear analysis of thermo-electro-mechanical behavior of shear deformable FGM plates with piezoelectric actuators. Int. J. Numer. Meth. Eng. 59, 1605–32 (2004) 21. Butz, A., Klinkel, S., Wagner, W.: A geometrically and materially non-linear piezoelectric three-dimensional-beam finite element formulation including warping effects. Int. J. Numer. Meth. Eng. 76, 601–635 (2008) 22. Panda, S., Ray, M.C.: Nonlinear analysis of smart functionally graded plates integrated with a layer of piezoelectric fiber reinforced composite. Smart Mater. Struct. 15, 1595– 1604 (2006) Rev 603 604 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 P. Phung-Van et al. 40. 725 726 727 728 41. 729 730 731 732 42. 733 734 735 736 737 43. 738 739 740 44. 741 742 743 744 745 45. 746 747 748 749 46. 750 751 752 753 47. 754 755 756 48. 757 758 759 49. 760 761 762 763 764 50. 765 766 767 768 769 770 771 51. 52. Oliver, W., Utz, W., Bernd, S.: Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations. Nonlinear Dyn. 72, 813–835 (2013) 53. Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Method Eng. 47, 663–684 (2000) 54. Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973) 55. Soldatos, K.P.: A transverse shear deformation theory for homogenous monoclinic plates. Acta Mech. 94, 195–220 (1992) 56. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29, 745–752 (1991) 57. Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos. Struct. 89, 94–101 (2009) 58. Nguyen-Xuan, H., Thai, H., Chien, Nguyen-Thoi, T.: Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory. Compos. B. Eng. 55, 558–574 (2013) 59. Fung, Y.C.: Foundation of Solid Mechanics. Prentice Hall, New Jersey (1965) 60. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, Oxford (2004) 61. Wang, S.Y., Quek, S.T., Ang, K.K.: Vibration control of smart piezoelectric composite plates. Smart Mater. Struct. 10, 637–644 (2001) 62. Tiersten, H.F.: Linear Piezoelectric Plate Vibrations. Plenum press, New York (1969) 63. Les Piegl, Wayne Tiller: The NURBS book. Springer 2nd edition, Germany (1997) 64. Phung-Van, P., Nguyen-Thoi, T., Le-Dinh, T., NguyenXuan, H.: Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3). Smart Mater. Struct. 22, 095026 (2013) 65. Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. ASCE 85, 67–94 (1959) 66. Reddy, J.N.: Geometrically nonlinear transient analysis of laminated composite plates. AIAA J. 21, 621–629 (1983) 67. Bathe, K.J.: Finite Element Procedures. Prentice-Hall, New Jersey (1996) 68. Pica, A., Wood, R.D., Hinton, E.: Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation. Comput. Struct. 11, 203–215 (1980) 69. Chen, J., Dawe, D.J., Wang, S.: Nonlinear transient analysis of rectangular composite laminated plates. Compos. Struct. 49, 129–139 (2000) Pro of 724 wich plates based on isogeometric analysis. Compos. Struct. 139, 77–95 (2016) Thai, H., Chien, Kulasegaram, S., Loc, Tran V., NguyenXuan, H.: Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach. Compos. Struct. 141, 94–112 (2014) Nguyen, V.P., Kerfriden, P., Brino, M., Bordas, S.P.A., Bonisoli, E.: Nitsche’s method for two and three dimensional NURBS patch coupling. Comput. Mech. 53, 1163–1182 (2013) Thai, H., Chien, Ferreira, A.J.M., Abdel-Wahab, A., Nguyen-Xuan, H.: A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis. Acta Mech. 227(5), 1225–1250 (2016) Nguyen, V.P., Nguyen-Xuan, H.: High-order B-splines based finite elements for delamination analysis of laminated composites. Compos. Struct. 102, 261–275 (2013) Phung-Van, P., Abdel-Wahab, M., Liew, K.M., Bordas, S.P.A., Nguyen-Xuan, H.: Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory. Compos. Struct. 123, 137–149 (2015) Loc, Tran V., Phung-Van, P., Lee, J., Abdel-Wahab, A., Nguyen-Xuan, H.: Isogeometric analysis for nonlinear thermomechanical stability of functionally graded plates. Compos. Struct. 140, 655–667 (2016) Nguyen-Thanh, N., Kiendl, J., Nguyen-Xuan, H., Wüchner, R., Bletzinger, K.U., Bazilevs, Y., Rabczuk, T.: Rotation free isogeometric thin shell analysis using PHT-splines. Comput. Methods Appl. Mech. Eng. 200, 3410–3424 (2011) Kapoor, H., Kapania, R.: Geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates. Compos. Struct. 94, 3434–3447 (2012) Le-Manh, T., Lee, J.: Postbuckling of laminated composite plates using NURBS-based isogeometric analysis. Compos. Struct. 109, 286–293 (2014) Phung-Van, P., Nguyen, L.B., Tran, L.V., Dinh, T.D., Thai, C.H., Bordas, S.P.A., Abdel-Wahab, M., Nguyen-Xuan, H.: An efficient computational approach for control of nonlinear transient responses of smart piezoelectric composite plates. Int. J. Non-Linear Mech. 76, 190–202 (2015) Hosseini, S., Joris, J.C., Remmersa, Clemens, V., Verhoosela, René de Borst.: An isogeometric solid-like shell element for non-linear analysis. Int. J. Numer. Method Eng. 95, 238–256 (2013) Hosseini, S., Joris, J.C., Remmers, Clemens, V., Verhoosel, René de Borst.: An isogeometric continuum shell element for non-linear analysis. Comput. Methods Appl. Mech. Eng. 271, 1–22 (2014) ised 723 Rev 722 123 Journal: 11071 MS: 3085 TYPESET DISK LE CP Disp.:2016/9/26 Pages: 16 Layout: Medium 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819