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Accepted Manuscript Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method (CS-DSG3) K Nguyen-Quang, H Dang-Trung, V Ho-Huu, H Luong-Van, T NguyenThoi PII: DOI: Reference: S0263-8223(16)32476-X http://dx.doi.org/10.1016/j.compstruct.2017.01.006 COST 8139 To appear in: Composite Structures Received Date: Revised Date: Accepted Date: November 2016 29 December 2016 January 2017 Please cite this article as: Nguyen-Quang, K., Dang-Trung, H., Ho-Huu, V., Luong-Van, H., Nguyen-Thoi, T., Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method (CS-DSG3), Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct 2017.01.006 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method (CS-DSG3) K Nguyen-Quang1,4, H Dang-Trung1,2, V Ho-Huu3, H Luong-Van4, T Nguyen-Thoi1,2,* Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Faculty of Civil Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam E-mail addresses: nguyenquangkha.ce@gmail.com (K Nguyen-Quang); dangtrunghau@tdt.edu.vn (H DangTrung); vinh.ho.h@gmail.com (V Ho-Huu); lvhai@hcmut.edu.vn (H Luong-Van); nguyenthoitrung@tdt.edu.vn (T Nguyen-Thoi) Abstract A cell-based smoothed discrete shear gap method (CS-DSG3) based on the first-order shear deformation theory was recently proposed for static and dynamics analyses of Mindlin plates In this paper, the CS-DSG3 is extended for analysis and active vibration control of the functionally graded material (FGM) plates integrated with piezoelectric sensors and actuators In the piezoelectric FGM plates, the properties of core material are assumed to be graded through the thickness by the power law distribution while the electric potential is assumed to be a linear function through the thickness of each piezoelectric sub-layer A closed-loop control algorithm based on the displacement and velocity feedbacks is used to control static deflection and active vibration of piezoelectric FGM plates Several numerical examples are conducted to demonstrate the reliability and accuracy of the proposed method compared to other available numerical results Keywords: Cell-based smoothed discrete shear gap method (CS-DSG3); FGM plates; sensor/ actuator layers; displacement/ velocity feedback control algorithm; active deflection/ vibration control Introduction Being first discovered in Sendai by a group of Japanese scientists in 1984, functionally graded materials (FGMs) have been then developed rapidly around the world [1–7] The FGMs are usually manufactured by combining metals and ceramics following a certain distribution rule of the material fraction This enables FGMs to inherit the best properties of these two materials including: (1) low thermal conductivity and high thermal resistance from * Corresponding author Tel.: +84 933 666 226 E-mail addresses: nguyenthoitrung@tdt.edu.vn (T Nguyen-Thoi) ceramics and (2) durability and high loading resistance from metals Subsequently, various progress has also been developed over the last decade for the active vibration suppression and shape control of FGM plates by using bonded or embedded piezoelectric materials One of the essential features of piezoelectric materials is the ability of transformation between mechanical energy and electric energy When a structure integrated with the piezoelectric material is deformed, the piezoelectric material generates an electric charge, which is termed the direct piezoelectric effect In contrast, when an electric field is applied, it produces mechanical deformation in the structure due to the converse piezoelectric effect [8] As a result, if the force that causes deformation of the structure is controlled appropriately, the vibration of the structure may be suppressed adequately Therefore, the integration of FGM plates with piezoelectric materials to give active lightweight smart structures has attracted the considerable interest of researchers in various industries such as automotive sensors, actuators, transducers and active damping devices, etc Due to these attractive properties, various numerical methods have been proposed to simulate the behavior of piezoelectric FGM plates He et al [9] investigated the shape and vibration control of FGM plates integrated with sensors and actuators In this work, a finite element formulation based on classical plate theory (CPT) was presented Liew et al [10] developed finite element formulations to study the behaviors of FGM plates containing sensors/actuators patches under environments subjected to a temperature gradient, using linear piezoelectric theory and first-order shear deformation theory (FSDT) Balamurugan and Narayanan [11] used the nine-node piezo-laminated plate finite element incorporating the FGM material model and the electromechanical coupling constitutive relations of the piezoelectric sensors/actuators to investigate the active control of piezoelectric FGM plates Ebrahimi and Rastgoo [12] presented a nonlinear free vibration of a thin annular FGM plate integrated with two uniformly distributed actuator layers on the top and bottom surfaces based on Kirchhoff plate theory by analytical solutions Ahmad Gharib [13] also developed an analytical solution for analyzing deflection control of FGM beams with embedded piezoelectric sensors and actuators by using FSDT Saidi et al [14] presented another analytical approach for free vibration analysis of moderately thick functionally graded rectangular plates coupled with piezoelectric layers Liu et al [15,16] presented active vibration control of laminated composite beams and plates containing distributed sensors and actuators based on CPT and the radial point interpolation method (RPIM) Shakeri and Mirzaeifar [17] proposed a general finite element formulation based on the layerwise theory for static and dynamic analysis of thick FGM plates with piezoelectric layers Aryana et al [18] used a finite element formulation based on CPT and an efficient method based on firstand second-order approximations in Taylor expansion to modify dynamic characteristics of piezoelectric FGM plates Hosseini-Hashemi et al [19] studied a 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers Recently, Selim et al [20] proposed a Reddy’s higher-order shear deformation theory with the element-free IMLS-Ritz method for active vibration control of piezoelectric FGM plates Phung-Van et al [21] presented a generalized shear deformation theory in combination with isogeometric approach for nonlinear transient analysis of FGM plates under thermo-electromechanical loads In general, it can be seen from the above mentioned literature review that there has been only a few published papers carried out to study behaviors of piezoelectric FGM plates in terms of deformable characteristic, stress distribution and vibration characteristics In addition, the use of simple linear plate elements such as three-node triangular Mindlin plate elements for analysis of piezoelectric FGM plates is somewhat still limited In the other front of the development of numerical methods, Liu et al [22] have integrated the strain smoothing technique [23] into the FEM to create a series of smoothed FEM (S-FEM) such as a cell/element-based smoothed FEM (CS-FEM) [24], a node-based smoothed FEM (NS-FEM) [25], an edge-based smoothed FEM (ES-FEM) [26] and a facebased smoothed FEM (FS-FEM) [27] Each of these S-FEM has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems Several earlier and related works of the S-FEM models have been provided in Refs [28–32] Among these S-FEM models, the CS-FEM [24,32–34] shows some interesting properties in the solid mechanics problems Extending the idea of the CS-FEM to plate structures, Nguyen-Thoi et al [35] have recently formulated a cell-based smoothed discrete shear gap method (CS-DSG3) which belongs to the group of simple three-node triangular Mindlin plate elements In the CS-DSG3, each triangular element is divided into three subtriangles, and in each sub-triangle, the stabilized DSG3 is used to compute the strains and avoid the transverse shear locking Then the cell-based strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles The numerical results showed that the CS-DSG3 is free of shear locking and achieves the high accuracy compared to others existing elements It has been successfully extended to analyze various plate and shell problems such as flat shells [36], stiffened plates [37], composite and sandwich plates [38], piezoelectric composite plates [39], plates resting on viscoelastic foundation subjected to moving loads [40,41], cracked plates and shells [42,43], and some other extensions [44–46], etc Based on the above considerations, in this paper, the CS-DSG3 is extended to investigate the static response, free vibration and dynamic control of piezoelectric FGM plates The mechanical properties of FGM plates are assumed to vary across the thickness of the plates by a simple power rule of the volume fractions of the constituents The electric potential is assumed to be a linear function through the thickness of each piezoelectric sub-layer A closed-loop control algorithm based on the displacement and velocity feedbacks is used to control static deflection and active vibration of piezoelectric FGM plates Equilibrium equation is derived from the principle of virtual displacements based on FSDT The accuracy and reliability of the proposed method are verified by comparing its numerical results with those of other available numerical approaches The static analysis of piezoelectric FGM plate is investigated with different voltages and boundary conditions The numerical results are presented in both tabular and graphical forms For dynamic vibration control, the effect of various types of load, and the influence of feedback control gain on static and dynamic response are also studied The remainder of this paper is outlined as follows Section describes the weak form of governing equations and finite element formulation for FGM plates related to static, free vibration and dynamic control problems In section 3, the active control analysis is presented Section presents numerical examples to verify the reliability and efficiency of the present method Finally, some conclusions are drawn in section Galerkin weak form and finite element formulation for piezoelectric FGM plates In this section, the Galerkin weak form and finite element formulation for piezoelectric FGM plates are established via a variational formulation [47,48] The piezoelectric FGM plate is assumed to be perfectly bonded, elastic and orthotropic in behavior [49], with small strains and displacements [50], and the deformation taken place under isothermal conditions In addition, the piezoelectric sensors/actuators are made of homogeneous and isotropic dielectric materials [51], and high electric fields as well as cyclic fields are not involved [52] Based on these assumptions, a linear constitutive relationship [53] can be employed for the static and dynamic analysis of the piezoelectric FGM plates 2.1 Formulation of functionally graded material Functionally graded materials are often composed of a mixture of two distinct material phases: ceramic and metal Ceramic can resist high thermal load because its thermal conductivity is low while the metal component can maintain flexibility of structure under the high-temperature gradient The material properties are assumed to be graded through the thickness by the power law distribution expressed as P ( z ) = ( Pm − Pc )V f ( z ) + Pc (1) in which V f ( z ) is volume fraction function defined by n 1 z V f ( z) =  +  , ≤ n ≤ ∞ 2 h (2) where subscript m and c refer to the metal and ceramic constituents, respectively; z is the thickness coordinate and varies from − h / to h / ; n ≥ is the power law index; P( z ) denotes material properties such as Young’s modulus ( E ) , mass density ( ρ ) and Poisson’s ratio (ν ) The volume fraction variation with n is shown in Fig When n = , the plate is Volume fraction V c fully metal, and when n → ∞ , the homogeneous ceramic plate is retrieved, respectively Fig Variation of the volume fraction against the non-dimensional thickness 2.2 Linear piezoelectric constitutive equations The linear piezoelectric constitutive equations can be expressed as  σ  c -eT   ε    D =    e g  E  (3) where σ and ε are the stress and strain vectors; D and E are the dielectric displacement and electric field vectors; c is the elasticity matrix displayed in section 2.4.1; e is the piezoelectric constant matrix and g denotes the dielectric constant matrix displayed in section 2.5 In addition, the electric field vector E is related to the electric potential field φ by using a gradient vector [54] as E = −gradφ (4) 2.3 Galerkin weak form of the governing equations The Galerkin weak form of the governing equations of piezoelectric structures can be derived by using Hamilton’s variational principle [55] which can be written as δL=0 (5) where L is the general energy functional which describes a summation of kinetic energy, strain energy, dielectric energy and external work, and is written in the form of (6) where u and are the mechanical displacement and velocity; φ is the electric potential; fs and Fp are the mechanical surface loads and point loads; qs and Q p are the surface charges and point charges In the variational form of Eq (5), the mechanical displacement field u and electric potential field φ are unknown functions To solve these unknowns numerically, it is necessary to use efficient numerical methods to approximate the mechanical displacement field and electric potential field In the present work, the CS-DSG3 [35] is used to approximate the mechanical displacement field of piezoelectric FGM plates Additionally, a linear constitutive relationship is also employed [53] for the analysis of the piezoelectric FGM plates, and the formulation for each field will be presented separately 2.4 Approximations of the mechanical displacement field 2.4.1 FGM plate model based on FSDT Consider a FGM plate under bending deformation as shown in Fig The neutral surface of the plate is chosen as the reference plane that occupies a domain Ω ⊂ R The displacement field according to the Reissner-Mindlin model based on FSDT [56] can be expressed by u ( x, y , z ) = u ( x , y ) + ( z − z ) β x ( x, y ) v( x, y, z ) = v0 ( x, y) + ( z − z0 )β y ( x, y ) (7) w( x, y, z ) = w( x, y ) where z0 is the distance between the mid and neutral surface defined by h /2 ∫ z0 = E ( z ) zdz − h /2 h /2 ∫ (8) E ( z )dz − h /2 and u0 , v0 , w are the displacements of the neutral plane of the plate; β x , β y are the rotations of the neutral plane around the y - and x -axes, respectively, with the positive directions defined in Fig Fig The FGM plate and positive directions of the displacements u, v, w and two rotations β x , β y The linear strains can be given by  ε x   u0, x   β x,x         ε y  =  v0, y  + ( z − z0 )  β y , y  = ε + ( z − z0 ) κ γ  u + v  β + β  y, y   xy   0, x 0, y   x, x γ xz   w, x + β x   = =γ γ yz   w, y + β y  (9) (10) For elastic and isotropic FGM plates, the constitutive relation between stresses and strains can be written as σ xx   Q11 Q12 σ    yy  Q21 Q22  τ xy  = Q61 Q62 τ   0  xz    τ yz   Q16 Q26 Q66 0 0 Q55 Q45 where the material constants are given by  ε xx     ε yy     γ xy   Q54  γ xz    Q44  γ yz  (11) Q11 = Q22 = E ( z) E ( z )ν ( z ) ; Q12 = Q21 = −ν ( z ) −ν ( z ) E( z) Q16 = Q61 = Q26 = Q62 = Q45 = Q54 = 0; Q44 = Q55 = Q66 = 2(1 +ν ( z )) (12) Substituting Eqs (9) and (10) into Eq (11), the stress is computed by (13) where ε p = [ ε T κ ] ; σ p and τ are the in-plane stress and shear stress components, respectively; D and Ds are material constant matrices given in the form of D D= m B h /2 B ; Ds = ks ∫ Qij dz; i, j = 4,5 Db  − h/ (14) in which h /2 Dmij = h /2 ∫ Bij = Qij dz; − h /2 ∫ ( z − z0 )Qij dz − h/ (15) h /2 Dbij = ∫ ( z − z0 ) Qij dz; (i, j = 1, 2, 6) − h /2 Note that the parameter ks in Eq (14) aims to ensure a more accurate approximation of the shear stress The procedure of evaluating shear correction factors is presented in the reference [57] 2.4.2 FEM formulation for FGM plates Now, by discretizing the bounded domain Ω of a FGM plate into Ne finite elements such that Ω = ∪ Ne=e1 Ω e and Ω i ∩ Ω j = ∅ , i ≠ j , the finite element solution of the FGM plate is expressed as u = u v w β x h  Ni 0 Nn  T β y  = ∑  i =1  0  0 0 Ni 0 Ni 0 0 Ni 0 0  di = Nd  0 N i  (16) where Nn is the total number of nodes of the problem domain discretized; Ni is the shape function at the i th node and d i = ui vi T β yi  is the displacement vector of the β xi wi nodal degrees of freedom of u h associated to the i th node The membrane, bending and shear strains can be then expressed in matrix form as ε = ∑ Bim di ; κ = ∑ Bib di ; γ = ∑ Bsi di i i i (17) where  Ni , x  B =  Ni , y  m i Ni , y Ni , x 0 0  0 0 ; 0  0 0 N i , x  B = 0 0 0 0 N i , y  b i 0 N i , x B si =   0 Ni , y   Ni, y  ; N i , x  Ni  N i  (18) in which N i , x and N i , y are the first derivatives of the shape functions in the x - and y directions, respectively 2.4.3 Brief description of the CS-DSG3 formulation for FGM plates Firstly, the CS-DSG3 is developed by incorporating the CS-FEM [22,24] with the original DSG3 element [58] Details of the CS-DSG3 formulation can be found in [35,37–39,59] In the CS-DSG3, each triangular element Ωe is divided into three sub-triangles ∆1 , ∆ and ∆ by connecting the central point O of the element to three field nodes as shown in Fig Then, in each sub-triangle ∆ j ( j = 1, 2, 3) , the strain fields are calculated similarly to that by the discrete shear gap method (DSG3) to give the membrane, bending and shear strains in each sub-triangle B m∆ j , B b∆ j and B s∆ j , respectively Finally, the cell-based strain smoothing operation in the CS-FEM is applied to give the smoothed strain fields of the element Ωe as follows: (19) where d e = ui Ωe ; , and vi wi β xi T β yi  (i = 1, 2,3) is the nodal displacement vector of element are the smoothed strain gradient matrices, respectively, given by 0.5 0.4 Normalized thickness z/t 0.3 0V 0.2 40V 0.1 20V -0.1 -0.2 -0.3 -0.4 -0.5 -1.5 -1 -0.5 Stress 0.5 1.5 (a/2, b/2) n = 0.5 Normalized thickness z/t Normalized thickness z/t n=0 x n=5 n=∞ Fig 10 The stress profiles σ x (MPa) through the thickness of the cantilevered piezoelectric FGM plate under a uniform loading and different input actuator voltages 4.3 Dynamic vibration control of a piezoelectric FGM plate We now consider a simply supported piezoelectric FGM plate with both length and width set at 200 mm , as shown in Fig 11(a) The thickness of FGM core layer is mm , while the thickness for each piezoceramic layer is 0.1 mm The material properties of the plate are the same as those in section 4.2 The FGM plate is also discretized by 12 ×12 × uniform threenode triangular elements as shown in Fig 11(b), and the power law exponent is taken by n =2 21 (a) (b) Fig 11 (a) The square piezoelectric FGM plate model; (b) A discretization 12 ×12 × uniform three-node triangular elements Fig 12 Effect of the displacement feedback control gain Gd on the static deflection of the simply supported plate under a uniformly distributed load First, we study the control of the static deflection Fig 12 shows the effect of the displacement feedback control gain Gd on the static deflection of a simply supported piezoelectric FGM plate subjected to a uniformly distributed load of 100 N/m It is seen that when the displacement feedback control gain Gd becomes larger, the deflection becomes smaller, as expected This phenomenon is similar to that in RPIM [16] This is because when the plate is deformed by an external force, electric charges are induced and collected in the 22 sensor layer Subsequently, the charges are amplified through a closed loop control as shown in Eq (30) and then converted into the open circuit voltage The converted signal is then fed back into the distributed actuator Finally, a control force due to the converse piezoelectric effect is formed to repress the static response of the FGM plate 10-5 Without control Gv=0 Control gain Gv=2e-4 Deflection w (m) -1 -2 -3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) Fig 13 Effect of the velocity feedback control gain Gv on the dynamic response of the simply supported plate Next, we investigate the active vibration control of the simply supported plate using integrated sensors and actuators In vibration control, the upper piezoelectric layer serves as actuators, whereas the lower one acts as sensors The responses of the plate are controlled through the dynamic velocity feedback control algorithm and a closed loop It is assumed that the initial condition of the plate is caused by a uniformly distributed load of 100 N/m , and the load is then removed For simplicity’s sake, the modal superposition technique is used, the first six modes are considered in the modal space analysis and the initial modal damping ratio for each mode is assumed to be 0.8% The transient response of the FGM plate is solved by the Newmark- β direct integration method [67], and the parameters α and β are taken to be 0.5 and 0.25, respectively All the calculations for transient response are performed using a time step of 0.0005s Fig 13 shows the transient response of center point A of the piezoelectric FGM plate by using the velocity feedback gain It can be seen that when the gain Gv is equal to zero (without control), the response decreases with respect to time due to the structural damping In addition, by increasing the velocity feedback gain, the transient response is further suppressed and the amplitude of the center point deflection of the 23 plate decreases faster, as expected This is because the active damping becomes stronger, as shown in Eq (36) Similarly, as seen from Fig 14, the sensor voltage response of the 156th element (as shown in Fig 11) attenuates faster when the control gain Gv increases Next, the corresponding piezoelectric actuator responses are also presented in Fig 15 It is clear that s Sensor voltage Sensor voltage s (Volt) (Volt) the actuator voltage increases as the gain value increases, as shown in Eq (30) Actuator voltage Actuator voltage a a (Volt) (Volt) Fig 14 Figures of piezoelectric sensor response for the simply supported plate Fig 15 Figures of piezoelectric actuator response for the simply supported plate Lastly, we study the active vibration control of a fully clamped (CCCC) piezoelectric FGM plate with the same material properties and dimensions of length and width as the former plate, but the thickness of core FGM layer in this case is taken to be 20 mm while the thickness of each piezoelectric layer is mm In this work, the plate is subjected to sinusoidally distributed transverse loads expressed as follows: π x  π y  q = q0 sin   sin   F (t )  a   b  where F (t ) is defined as 24 (40)  1 ≤ t ≤ t1 Step load   t > t1 0   1 − t / t ≤ t ≤ t 1   Triangular load F (t ) =   t > t1   sin(π t / t1 ) ≤ t ≤ t1 Sinusoidal load   t > t1  Explosive blast load e−γ t  (41) Values of force F in which q0 = 10 N/m , γ = 330 s-1 and F (t ) is plotted as shown in Fig 16 Fig 16 Types of load: step load, triangular load, sinusoidal load and explosive blast load The transient response of center point A and the sensor/actuator voltage response of the 156 th element as shown in Fig 11 are investigated and presented in figures from Fig 17 to Fig 24 Again, it can be seen that the transient and sensor voltage responses with control are smaller than those without control while the voltage response of actuator increases when control gain increases Additionally, the results from figures also show that the actuator voltage equals zero as the plate does not oscillate, as shown in Eq (30) Next, we consider the particular case in which the plate is subjected a sinusoidal load that includes an increased load phase and a decreased load phase As can be seen, the deflection of point A decreases when the control gain Gv increases significantly, but when the load is removed or the plate is in free vibration state, the plate is suppressed more slowly compared to the case controlled by lower gain value Therefore, depending on the specific circumstances, the value of control gain can be designed to satisfy an expectation such as 25 controlling displacement or oscillation time or even both Finally, it should be noted that since piezoelectric layers have their own breakdown voltages, the gain value could not be Deflection w (m) increased indefinitely Fig 17 Transient response of the fully clamped plate subjected to step load 15 Without control Gv=0 Control gain Gv=1e-4 Control gain Gv=2e-4 (Volt) s Sensor voltage Control gain Gv=1e-4 Control gain Gv=2e-4 10 0 -1 -5 -2 Forced vibration -10 -3 Forced vibration Free vibration -4 Forced vibration -15 0.5 1.5 2.5 Time (s) 3.5 4.5 10 -3 0.5 1.5 2.5 3.5 4.5 Time (s) Fig 18 Piezoelectric sensor and actuator responses of the fully clamped plate subjected to step load 26 10 -3 Deflection w (m) Fig 19 Transient response of the fully clamped plate subjected to triangular load Fig 20 Piezoelectric sensor and actuator responses of the fully clamped plate subjected to triangular load 27 Deflection w (m) Fig 21 Transient response of the fully clamped plate subjected to sinusoidal load Fig 22 Piezoelectric sensor and actuator responses of the fully clamped plate subjected to sinusoidal load 28 Deflection w (m) Fig 23 Transient response of the fully clamped plate subjected to explosive blast load Fig 24 Piezoelectric sensor and actuator responses of the fully clamped plate subjected to explosive blast load Conclusions The paper presents an extension of the CS-DSG3 using three-node triangular elements for the static, free vibration analyses and dynamic control of FGM plates integrated with piezoelectric sensors and actuators On the basis of first-order shear deformation theory, a model of piezoelectric FGM plate is formulated by using the Galerkin weak-form formulation The electric potential is assumed to be a linear function through the thickness of each piezoelectric sublayer A closed-loop control algorithm based on the displacement and velocity feedbacks is used to control static deflection and active vibration of piezoelectric 29 FGM plates Through the present formulation and numerical results, we can withdraw several following points: (i) The present CS-DSG3 only uses three-node linear triangular elements, so it is easily generated automatically for complicated geometry domains In addition, the obtained results by the CS-DSG3 show that this technique is free of shear locking for both thin and thick piezoelectric FGM plates (ii) Although the CS-DSG3 only uses five degrees of freedom at each vertex node and first-order shear deformation theory, it still gives the results which agree well with those obtained by the classical plate theory for thin FGM plates, or by higher-order shear deformation theory and analytical solution for thick FGM plates (iii) Two effective schemes are presented in this paper for static shape control of the FGM plates The first scheme is to use an input voltage with opposite signs applied across the thickness of two piezo-layers The second scheme is to adjust a displacement control gain Gd in order to reduce static deflection of the plate to the desired tolerance Finally, for vibration control, the dynamic response demonstrates the effectiveness of the velocity feedback control algorithm in a closed loop If the velocity control gain Gv is designed approximately, the vibration of the plates can be depressed adequately as expected Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.99-2014.11 30 References [1] Fukui Y, Yamanaka N Elastic analysis for thick-walled tubes of functionally graded material subjected to internal pressure JSME Int Journal Ser 1, Solid Mech Strength Mater 1992;35:379–85 [2] Obata Y, Noda N Unsteady thermal stresses in a functionally gradient material plateAnalysis of one-dimensional unsteady heat transfer problem JSME Trans 1993;59:1090–6 [3] Tani J, Liu G-R SH surface waves in functionally gradient piezoelectric plates JSME Int Journal Ser A, Mech Mater Eng 1993;36:152–5 [4] Liu GR, Tani J Surface waves in functionally gradient piezoelectric plates J Vib Acoust 1994;116:440–8 [5] Obata Y, Noda N Optimum material design for functionally gradient material 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piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method (CS-DSG3) K Nguyen-Quang1,4,... for analysis and active vibration control of the functionally graded material (FGM) plates integrated with piezoelectric sensors and actuators In the piezoelectric FGM plates, the properties of. .. analyses and dynamic control of FGM plates integrated with piezoelectric sensors and actuators On the basis of first-order shear deformation theory, a model of piezoelectric FGM plate is formulated