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Accepted Manuscript Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory B.A Selim, L.W Zhang, K.M Liew PII: DOI: Reference: S0263-8223(16)31107-2 http://dx.doi.org/10.1016/j.compstruct.2016.07.059 COST 7659 To appear in: Composite Structures Received Date: Accepted Date: July 2016 22 July 2016 Please cite this article as: Selim, B.A., Zhang, L.W., Liew, K.M., Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.07.059 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 Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory B.A Selim1, L.W Zhang2,*, K.M Liew1,3,* Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China College of Information Science and Technology, Shanghai Ocean University, Shanghai 201306, China City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, China Abstract Literature researching the active vibration control of functionally graded material (FGM) plates with piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) using any of the element-free methods does not exist To the best of the authors’ knowledge, this paper is the first to use Reddy’s HSDT with the element-free IMLS-Ritz method to investigate this problem In this study, seven mechanical degrees of freedom (DOF) and one additional electrical DOF are considered for each node of the discretized domain The natural frequency results of two FGM plates with top and bottom piezoelectric layers are compared with the literature in terms of various electrical and mechanical boundary conditions, volume fraction exponent (n) and dimension ratios, with obvious agreement Furthermore, parametric studies are performed, for the first time, to study the effects of mechanical boundary conditions, n value, FGM plate thickness-to-width ratio and piezoelectric layer thickness to FGM plate thickness ratio on the natural frequency increment between open and closed circuit conditions For the purpose of active vibration control, a constant velocity feedback approach is utilized The effectiveness of two proposed positions, of piezoelectric sensor and actuator layers, to control the vibration of FGM plates is investigated Keywords: Functionally graded material; Piezoelectric materials; Smart structures; Vibration control; Reddy’s third-order shear deformation theory; Mesh-free method ∗ Corresponding authors Email address: zlvwen@hotmail.com (L.W Zhang), kmliew@cityu.edu.hk (K.M Liew) Introduction In the mid-1980s, functionally graded material (FGM) was introduced for the first time [1] FGM is characterized by spatial variation in material properties as it is composed of at least two different components with a gradual changing of volume fraction along at least one direction [2] The purpose of this concept is to combine the best properties of two or more constituents Additionally, it helps to eliminate the interface problems commonly found in composite materials in order to achieve a smoother stress distribution [3] Moreover, smart materials, including piezoelectric materials, have drawn increasing attention from several researchers due to their applicability to active control of structures He et al [4] utilized the finite element method (FEM) and the classical plate theory (CPT) for the shape and vibration control of FGM plates with integrated piezoelectric sensors and actuators Liew et al used FEM with CPT [5] and the first-order shear deformation theory (FSDT) [6], [7] for the active control of FGM plates with integrated piezoelectric sensors and actuators subjected to a thermal gradient Ootao and Tanigawa [8] studied a three-dimensional transient piezothermoelasticity problem for FGM rectangular plate bonded to a piezoelectric layer due to partial heat supply Reddy and Cheng [9] provided three dimensional solutions for FGM plates with piezoelectric actuator layers, utilizing transfer matrix and asymptotic expansion techniques Based on the element-free Galerkin method and FSDT, Dai et al [10] presented an active shape control and dynamic response suppression model for FGM plates containing distributed piezoelectric sensors and actuators Ray and Sachade [11] studied the static analysis of FGM plates integrated with a layer of piezoelectric fiber reinforced composite (PFRC) material using FEM and FSDT Based on HSDT and von Kármán-type equations, Huang and Shen [12] investigated the nonlinear vibration and dynamic response of an FGM plate with surface-bonded piezoelectric layers in thermal environments utilizing an improved perturbation technique Fakhari and Ohadi [13] studied the nonlinear vibration control of FGM plates with integrated piezoelectric sensors and actuator layers under a thermal gradient and transverse mechanical loads using FEM, HSDT and von Kármán-type equations Fakhari et al [14] studied the nonlinear free and forced vibration behavior of FGM plate with piezoelectric layers in a thermal environment under thermal, electrical and mechanical loads utilizing FEM, HSDT and von Kármán-type equations too Based on the differential reproducing kernel (DRK) interpolation, Wu et al [15] presented a meshless collocation method for the three-dimensional coupled analysis of simply-supported, doubly curved FGM piezo-thermo-elastic shells under a thermal load Farsangi and Saidi [16] presented an analytical Levy type approach for the free vibration analysis of moderately thick FGM plates with piezoelectric layers using FSDT Loja et al [17] studied the static and free vibration behavior of FGM plate with piezoelectric skins using B-spline finite strip element models based on FSDT and HSDT Using Hamilton's principle, Maxwell's equation and the Navier method, Rouzegar and Abad [18] presented an analytical solution for the free vibration analysis of an FGM plate with piezoelectric layers by employing the four-variable refined plate theory Among the available plate theories in the literature, CPT, FSDT and Reddy’s higherorder shear deformation theory (HSDT) are the most popular CPT is only suitable for thin plates as it completely neglects the shear effect However, the shear effect becomes significant as the plate becomes thicker and ignoring it leads to erroneous outcomes Although FSDT incorporates the shear effect, it necessitates the usage of a shear correction factor to satisfy zero shear conditions on the plate’s top and bottom surfaces, which is a shortcoming To overcome the drawbacks of CPT and FSDT, Reddy proposed his HSDT in 1984 [19] which identifies the displacement field by a third order function in terms of the thickness coordinate Both CPT and FSDT are considered to be special cases of HSDT, wherein the shear effect is considered and there is no necessity to use any shear correction factor [2], [19] Furthermore, HSDT is valid for thin, moderately thick and thick plates On the other hand, the usage of the moving least-square (MLS) approximation is considered to represent a turning point, resulting in substantial improvements in the element-free method A powerful property of the MLS approximation is that its continuity is related to the continuity of the weight function Accordingly, highly continuous approximations can be obtained by choosing the appropriate weight function [20] However, MLS approximation still has some limitations, such as the fact that the resulting system of algebraic equations may be ill3 conditioned In such cases, there are no mathematical techniques to investigate whether the system of algebraic equations is ill-conditioned before it is solved Consequently an accurate solution for the system of algebraic equations may not be found/correctly found The improved moving least-square (IMLS) approximation was proposed to overcome these drawbacks of MLS in the construction of shape function by utilizing a weighted orthogonal function [21]–[25] Through reviewing the literature, it is obvious that there exist no studies on the free vibration and active vibration control of functionally graded material (FGM) plates with piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) with any of the element-free methods This paper investigates the free vibration analysis as well as the active vibration control of FGM plates with piezoelectric layers using Reddy’s HSDT in association with the element-free IMLS-Ritz method [26]–[31] Functionally graded material (FGM) plates In this paper, the FGM plate is assumed to have two constituents as it is composed of metallic and ceramic materials The effective properties alter throughout the thickness direction, as follows: = − + , (1) where is the effective material property of the FGM plate (i.e., the modulus of elasticity or is the volume fraction of the metallic constituent of the FGM plate which can be expressed density) and the subscripts m and c denote the metallic and ceramic constituents, respectively as: 2 + ℎ , = 2ℎ (2) in which ℎ is the height of the FGM plate and n is the volume fraction exponent 0 ≤ ≤ ∞ At any level (z) of the plate thickness, the relationship between the volume fraction constituents of the metallic and ceramic parts is expressed by: m + = (3) Fig shows the geometry and coordinate system of a rectangular FGM plate with top and bottom piezoelectric layers Theoretical formulation 3.1 Governing equations for Reddy’s HSDT model Based on Reddy’s HSDT [19], the displacement field can be expressed as: , , = , + ∅ , + ! " #∅ + ' , , = ' , + ∅( , + ! " #∅( + %, , = % , , $% & , $ $% & , $ (4) (5) (6) where , ' , % are displacement components along the , , directions, respectively, of a point on the plate’s mid-plane z = 0 ∅ and −∅( are the rotations about the and axes, respectively, and the constant ! = − "+ - in which ℎ is the total thickness of the plate Supposing that / = ∅ + rewritten as: 012 * , and /( = ∅( + 012 0( as reported in [32], Eqs (4)-(6) can be , , = , + ∅ , + ! " / , ' , , = ' , + ∅( , + ! " /( , %, , = % , (7) (8) (9) Therefore, the basic mechanical field variables (mechanical degrees of freedom) for each node based on the current formulation are , ' , % , ∅ , ∅( , / and /( The in-plane strains can be expressed as: where = [5 5(( / ( ]7 = 3 + 8! + " 89 , (10) $/ $∅ ? $0 ? < ? < < $ $ ; > $ ; > ; > ; ; $/ > $∅ > $'0 > ; > > , 82 = ; 30 = ; > , 81 = ;; $ $ $ > ; > ; > ;$∅ > ; > $ $' $∅ $/ 0> $/ > ; 0+ > ; + ; + $ = : $ $ = : $ $ = : $ (11) @ = [/(A / A ]7 = 3B + 8B , (12) However, the out-of-plane shear strains are given by: where $% 0? ; D 3 = ; , = > / E ;∅ + $%0 > : $ = (13) The linear constitutive relations are expressed by: J N < 11 ;N12 J = ;; HJ L ; G G FJ K ; : IJ M G G N12 0 ? 5 M N22 0 >I G G > / N66 0 > , H L 0 N44 > G/ G > / 0 N55 = F K (14) where the material constants are given by: N!! = !TU SSU , N99 = !TU U , N!9 = !TU-S R S- -S R S- -S U RSS S- U-S , NVV = W!9 , N** = W9" , NXX = W!" , where Y!! and Y99 are the effective Young’s moduli in the principal material coordinate W!9 , W!" and W9" are the shear moduli and '!9 and '9! are Poisson’s ratios 3.2 Linear constitutive equations of piezoelectric composite plates In the present study, the piezoelectric layers are assumed to be perfectly bonded to the host FGM plates The constitutive relationships of the FGM plate with the piezoelectric layers are given as follows: Z = [3 − \7 ] , (15) ^ = \3 + _] , (16) where Z, 3, ] and ^ denote the stress, strain, electric field and electric displacement vectors, respectively [, \ and _ denote the material constant, the piezoelectric constant and the dielectric constant matrices, respectively The electric field vector E is given as follows: ] = −grad d = −∇ d , (17) where d is the electric potential difference vector across the piezoelectric layer assuming that there is an electrical DOF for each node The material constant matrix [ is given in the form of: f g h > , ^B > iB = (18) in which fkl , gkl , ^kl , ]kl , ikl , jkl = m f , ^ , i = m ℎp ⁄2 ℎp ⁄2 −ℎp /2 n1, , 2 , 3 , 4 , 6 o Nkl d, k, l = 1, 2, 6, n1, 2 , 4 o Nkl d, k, l = 4, 5 −ℎp /2 \ and _ are given as follows: (19) (20) s\s = t 31 0 14 15 v11 0 0 24 25 u , _ = t v22 u 0 v33 32 0 (21) 3.3 Total energy functional The plate potential energy is expressed by: where 1 1 w = m # Z7 − ^7 ]& xy = m # 37 [3 − 37 \7 ] − ]7 _]& xy , 2 z z (22) = {3 (23) 8! 89 3B 8B |7 The plate kinetic energy is expressed by: +, ⁄9 }= mm ~ + ' + % x xy , z T+,/9 (24) where ~ is the mass density and , ' , % are the components of the mechanical velocity along the , , directions, respectively The external work is given as: = m 7 B − d7 B xy + z ! i − ! d7 , (25) where and i are the external mechanical surface loads and point loads, respectively Meanwhile, and are the external surface charges and point charges, respectively Hence, the plate total energy function can be expressed as: = w − } − (26) 3.4 Two-dimensional IMLS shape function Utilizing the weighted orthogonal basis functions, a detailed formulation of the IMLS approximation is presented in [33], [34] A summary of the construction of the IMLS shape function is illustrated here Let u(x) be the function of the field variable defined in the domain Ω The approximation of u(x) at point x is denoted by uh (x) and this trial function is expressed as: ℎ = k = , k=1 k (27) where p(x) is a vector of basis functions that commonly consist of monomials of the lowest order to ensure minimum completeness, m is the number of terms of the monomials and a(x) is a vector of coefficients given as: = n0 , 1 , … , o , (28) = 1, , = 3, k k , (29) = n1, , , , 2 , 2 o = 6, xpk k (30) which are functions of x For a two-dimensional problem, the following basis can be chosen: or The local approximation at x, as described by Lancaster and Salkauskas [35], is: = ℎ , k = , k=1 k (31) is the point of the local approximation of x To obtain the local approximation of the where function u(x), the difference between the local approximation uh(x) and the function u(x) has to be minimized using a weighted least-squares method Here, we define a function: = ! = ! % − [+ , − ]9 % − [ ! · − ]9 , (32) where % − is a weight function with a domain of influence, and (I=1, 2,…, n) are the nodes with domains of influence which cover the point x