Abdel Wahab, Magd

Nguyen-Xuan, Hung

Phung Van, Phuc

Tran Vinh, Loc

Ghent University Academic Bibliography

 International Journal of Fracture Fatigue and Wear, Volume 3 265  Proceedings of the 4th International Conference On  Fracture Fatigue and Wear, pp. 265-271, 2015 AN EFFICIENT COMPUTATIONAL APPROACH FOR NONLINEAR ANALYSIS OF SMART PIEZOELECTRIC PLATES  P. Phung-Van1, Loc V. Tran1, M. Abdel-Wahab1, H. Nguyen-Xuan2 1Department of Mechanical Construction and Production, Faculty of Engineering and Architecture, Ghent University, Belgium  2Faculty of Mathematics & Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam   Abstract: In this paper, a simple and effective formulation based on isogeometric finite elements (IGA) for geometrically nonlinear analysis of FGM plates integrated with sensors and actuators is investigated. The nonlinear formulation is based on a generalized five-parameter displacement field of higher order shear deformation (HSDT) and the von Kármán strains, which includes thermo-piezoelectric effects. 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 by the rule of mixture and the Mori–Tanaka scheme. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of available other numerical results. Keywords: geometrically nonlinear, isogeometric analysis (IGA), functionally graded material plates (FGM), sensors and actuators.  1 INTRODUCTION The integration of piezoelectric and FGM has many practical applications such as smart material systems, the medical, micro-electromechanical systems (MEMS) and aerospace industries [1], etc. Because of properties of thermo-electro-mechanical coupling, many methods have been proposed He et al. [2] developed a finite element model based on variational principle and linear piezoelectricity theory. The active control of FGM integrated with piezoelectric sensors and actuators was studied by Liew et al. [3]. The behaviour of a FGM piezoelectric plate in thermal environments using HSDT was investigated in Refs. [4,5]. Yang et al. [6] examined the nonlinear thermo-electro-mechanical bending response of piezoelectric FGM plates. The FE formulations based HSDT for nonlinear analysis of functionally graded piezoelectric plates were also reported in [7]. Recently, isogeometric analysis (IGA) of the piezoelectric composite plates using HSDT was studied by Ref. [8]. However, nonlinear transient analysis has not taken into account for their previous work. So far, there are few published materials related to geometrically nonlinear analysis using IGA for composite plates. Apparently, there are no study on nonlinear analysis based on isogeometric analysis using the generalized shear deformation theory for the piezoelectric FGM plates. So, this paper hence tries to fill this gap by using IGA based on the generalized shear deformation theory for geometrically nonlinear transient analysis of the piezoelectric FGM plates. The nonlinear formulation is based on a generalized higher order shear deformation (HSDT) and the von Kármán strains, which includes thermo-piezoelectric effects. 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 by the rule of mixture and the Mori–Tanaka scheme. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of available other numerical results. 2 THE GENERALIZED HSDT FOR SMART PIEZOELECTRIC PLATES 2.1 The piezoelectric FGM model A sandwich plate shown in Figure 1a has the volume fraction of ceramic and metal phase across thickness as follows 1( ) , ( ) 1 ( )2ncc mczV z V z V zh = + = −   (1)    266  where c and m refer to the ceramic and metal, respectively; 2 3[ , ]cz z z∈ and hc = z3 – z2 is thickness of core which are displayed in Figure 1b. The material constituents of piezoelectric FGM can be obtained as:   (a) (b) Fig. 1 (a) Configuration of a piezoelectric FGM plate; (b) The sandwich plate with piezoelectric skins and FGM core  The material properties including Young’s modulus (E), Poisson’s ratio (ν) and density (ρ) based on the mixture rule are defined by: where Pc, Pm represent the individual properties of material of ceramic and metal. To consider the interactions among the constituents, the Mori-Tanaka scheme is used in this paper.  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 and an electrical field that need to be approximated. The mechanical displacements are approximated by the generalized higher order shear deformation theory and expressed as follows  where u0, v0, βx, βy and w are displacement variables. The function f(z) is a continuous function through the plate thickness and chosen as ( ) arctan(2 / )f z h z h z= −  as in Thai et al. [9] For a bending plate, the Green’s strain vector can be presented by 1 12 2ji k kijj i i juu u ux x x xε ∂∂ ∂ ∂= + +  ∂ ∂ ∂ ∂  (5) The material behaviour of piezoelectric FGM is expressed as following [10] T −   =         σ c eD Ee gε (6) where σ  and T[ε γ]ε =  are the stress and strain vectors, respectively; D is the dielectric displacement; e is the piezoelectric constant matrix and g denotes the dielectric constant matrix; E is the electric field vector that is defined as gradφ= −E  (7) in which φ  is the electric potential field; and c is the elasticity matrix and defined as 1 2 2 33 41( ) 1, [ , ]; ( ) , [ , ] for core2( ) 1, [ , ]; ( ) 1 ( )ncc c c ccc c m czV z h z z V z h z zhV z h z z V z V z = ∈ = + ∈  = ∈ = − (2) ( ) ( )c c m mP PV z P V z= +  (3) 0 0/ ( ) ; / ( ) ;x yu u z w x f z v v z w y f z w wβ β= − ∂ ∂ + = − ∂ ∂ + =  (4)    267  S   =   A B N 0B C F 0cN F H 00 0 0 D (8) where ( )[ ]/ 2 2 22/212/ 2 2/ 21 0, , , , , 1, , , ( ), ( ), ( ) 1 0 d , , 1,2,610 01 0( ) d , , 4,50 12(1 )eeheij ij ij ij ij ij ehehS eij heEA B C N F H z z f z zf z f z z i jED f z z i jννννν−−−  = = −    ′= = +  ∫∫ (9) And the piezoelectric constant matrix e and the dielectric constant matrix g are defined      15 1115 2231 32 33 330 0 0 0 0 0 00 0 0 0 0 ; 0 00 0 0 0 0d pd pd d d p      = =         e g  (10) 3 THE SMART PIEZOELECTRIC PLATE FORMULATION BASED ON NURBS BASIC FUNCTION Using Cox-de Boor algorithm, the univariate B-spline basis functions ( ),i pN ξ  are defined in Les Piegl and Wayne (1997) on the corresponding knot vector start with order p = 0  ( ) 1,01 if0 otherwisei iiNξ ξ ξξ +≤ < =    (11) as p + 1 the basis functions are obtained from ( ) ( ) ( )1, , 1 1, 11 1 i pii p i p i pi p i i p iN N Nξ ξξ ξξ ξ ξξ ξ ξ ξ+ +− + −+ + + +−−= +− − (12) To present exactly some conic sections, e.g., circles, cylinders, spheres, etc., non-uniform rational B-splines (NURBS) need to be used. Being different from B-spline, each control point of NURBS has an additional value called an individual weight 0Aζ > .  And the B-spline function is recovered as the individual weight of control point is constant. 3.1 Mechanical displacements The displacement field u of the plate using NURBS basic function is approximated as ( ) ( )1, ,m nhI IIRξ η ξ η×== ∑u d  (14) where 0 0TI I I xI yI Iu v wβ β =  d is the vector of degrees of freedom associated with the control point I, and RI is the shape function as defined in Eq. (13). The strains can be expressed as 11[ ]2m nT L NLI I II×= = = +  ∑ε γ B B dε  (15) where ( ) ( ) ( ) ( )1 2 TT T T TL m b b sI I I I I =   B B B B B , in which ( ) ( )( )1,,,A AA m nA AANRNξ η ζξ ηξ η ζ×==∑ (13)    268  , , ,1 2, , ,, , , , ,0 000 00 00 000 000000 000 00 00 000 0 ,000 0000 002 00 000I x I xx I xIm b b sI I y I I yy I I y III y I x I xy I y I xR R RRR R RRR R R R R           = = − = =                  B ,B ,B B (16) and NLIB is calculated by ,,,,, ,0 0 0 0 0( ) 0 0 0 0 0I xI xNL gI I y II yI y I xwRwRw wθ   = =     B d A B    (17) 3.2 Electric potential field The electric potential field of each piezoelectric layer is approximated through the thickness as     ( )i i iz φφ = R φ  (18) where iφR  is the shape functions for the electric potential which is defined in Eq. (13) with p = 1, and iφ  is the vector containing the electric potentials at the top and bottom surfaces. The electric field E in Eq. (7) can be rewritten as [10]     i i iφ φ= −∇ = −E R Bφ φ  (19) 3.3 Governing equations of smart piezoelectric plates The governing equations for piezoelectric FGM plates can be written by   (20) where    (21) in which 0q  is a uniform load; [ ]0 0 0 0 ;IR=R  m is defined by  ( ) ( )1 2 4 / 2 2 22 3 5 1 2 3 4 5 7 / 24 5 7, , , , , , 1, , , ( ), ( ), ( ) dhhI I II I I I I I I I I z z f z zf z f z zI I Iρ−  = =   ∫m  (22) and              (23) 4 NUMERICAL EXAMPLES 4.1 Free vibration analysis Consider a square Al2O3/Ti-6Al-4V plate (400mm × 400mm) with the thickness of the plate, h, is 5mm and thickness of each piezoelectric layer, h_pie, is 0.1mm. Figure 2 displays frequencies of the CCCC and SSSS plate. We can see that the frequencies of the present method match well with those of He et al. (2001) and those of the CCCC plate are larger than those of the SSSS plate.    269  4.2 Nonlinear analysis 4.2.1 An orthotropic plate This example aims to verify the accuracy of the present method for geometrically nonlinear analysis. Consider a SSSS square plate subjected to a uniform loading of q0 = 1 MPa. Material properties and the geometry are given E1 = 525 GPa, E2 = 21 GPa, G12 = G23 = G13 = 10.5 GPa, ν = 0.25, ρ = 800 kg/m3, length L = 250 mm, thickness h = 5 mm. The normalized central deflection, /w w h= , is shown in Figure 3. It can be observed that deflection responses of IGA match well with those of finite strip method (FSM) [11].    n = 0 n = 0.5 Fig. 2 The first lowest eight natural frequencies of the simply supported (SSSS) and clamped (CCCC) piezoelectric FGM plate with the different volume fraction exponents 4.2.2 A smart piezoelectric plate Now we consider the square piezoelectric FGM plate with length L = 1, hFGM = L/20 and hpiezo = hFGM/10. The boundary condition of the plate is SSSS.  Fig. 3 Displacement of the plate under step uniform load  Figure 4a displays the effect of volume fraction exponent n to deflection of piezoelectric FGM (Al/ZrO2-2) plates under mechanical load (parameter load 510oq q= × ). It is seen that magnitude of deflection of nonlinear analysis is smaller than that of linear analysis. Next, effect of temperature to nonlinear deflection of the plate under thermo-mechanical load with n = 5 is displayed in Figure 4b. It is observed that the behaviour 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 thermal expansion phenomenon. Also, the deflection decreases correspondingly to the increase of the temperature.    270  5 CONCLUSIONS The paper presents a simple and effective formulation based on isogeometric finite elements (IGA) for geometrically nonlinear analysis of FGM piezoelectric plates. The nonlinear formulation is based on a generalized five-parameter displacement field of higher order shear deformation (HSDT) and the von Kármán strains, which includes thermo-piezoelectric effects. 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 by the rule of mixture and the Mori–Tanaka scheme. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of available other numerical results.   6    REFERENCES [1] K. Takagi, JF. Li, S. Yokoyama, R. Watanabe, Fabrication and evaluation of PZT/Pt piezoelectric composites and functionally graded actuators, Journal of the European Ceramic Society, 23, 1577–1583, 2003 [2]  XQ. He, TY. Ng, S. Sivashanker, KM. Liew, Active control of FGM plates with intergrated piezoelectric sensors and actuators, International Journal of Solids and Structures, 38, 1641-1655, 2001 [3]  KM. Liew, XQ. He, TY. Ng, S. Kitipornchai, Finite element piezothermo-elasticity analysis and active control of FGM plates with integrated piezoelectric sensors and actuators, Computational Mechanics, 31, 350–358, 2003. [4]  P. Phung-Van, T Nguyen-Thoi, H Luong-Van, Q Lieu-Xuan, Geometrically nonlinear analysis of functionally graded plates using a cell-based smoothed three-node plate element (CS-MIN3) based on the C0-HSDT, Computer Methods in Applied Mechanics and Engineering, 270, 15-36, 2014.  [5]  XL. Huang, HS. Shen, Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound Vibration, 289, 25–53, 2006. [6]  J. Yang, S. Kitipornchai, KM. Liew, Non-linear analysis of thermo-electro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators. International Journal for Numerical Methods in Engineering, 59, 1605–1632, 2004. [7]  B. Behjat, MR. Khoshravan, Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Composites Structures, 94, 874–882, 2012. [8]  P. Phung-Van, L. De Lorenzis, Chien H. Thai, M. Abdel-Wahab, H. Nguyen-Xuan, Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Computational Materials Science, 96, 496-505, 2015.    271  [9]  Chien H. Thai, AJM. Ferreira, SPA Bordas, T Rabczuk, H. Nguyen-Xuan, Isogeometric analysis of laminated composite and sanwich plates using a new inverse trigometic shear deformation theory, European Journal of Mechanics – A/Solids, 43, 89-108, 2014. [10]  P. Phung-Van, T. Nguyen-Thoi, T. Le-Dinh, H. Nguyen-Xuan, 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 Materials and Structures, 22, 095026, 2013. [11]  J. Chen, DJ. Dawe, S. Wang, Nonlinear transient analysis of rectangular composite laminated plates, Composite Structures, 49, 129-139, 2000. 

An efficient computational approach for nonlinear analysis of smart piezoelectric plates

https://biblio.ugent.be/publication/6879718/file/6879719

An efficient computational approach for nonlinear analysis of smart piezoelectric plates

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography