Buckling analysis of piezoelectric composite plates using NURBSbased isogeometric finite elements and higher-order shear deformation theory

Transcription

1 Proceedng of the 3 rd nternatonal Conference on Fracture Fatgue and Wear, pp , 014 Bucklng analy of pezoelectrc compote plate ung NURBSbaed ogeometrc fnte element and hgher-order hear deformaton theory P. Phung-Van 1,,*, M. Abdel-Wahab, Loc V. ran 1 and H. Nguyen-Xuan 3 1 Dvon of Computatonal Mechanc, on Duc hang Unverty, Vetnam Department of Mechancal Contructon and Producton, Faculty of Engneerng and Archtecture, Ghent Unverty, Belgum 3 Department of Mechanc, Unverty of Scence, VNU - HCMC, Vetnam *Correpondng author: phuc.phungvan@ugent.be Abtract: h paper further explot the utlty and robutne of ogeometrc Analy (GA) together wth Hgher-order Shear Deformaton heory (HSD) for bucklng analy of pezoelectrc compote plate. n the compote plate, the mechancal dplacement feld approxmated accordng to the HSD model ung NURBS-baed ogeometrc element. hee acheve naturally any dered degree of contnuty through the choce of the nterpolaton order, o that the method ealy fulfl the C 1 -contnuty requrement of the HSD model. he electrc potental aumed to vary lnearly through the thckne for each pezoelectrc ub-layer. he accuracy and relablty of the propoed method verfed by comparng t numercal predcton wth thoe of other avalable numercal approache. Keyword: ogeometrc Analy (GA); compote plate; pezoelectrcty; enor and actuator 1 NRODUCON he ntegraton of compote plate wth pezoelectrc materal to obtan actve lghtweght mart tructure ha attracted a conderable nteret for varou applcaton uch a automotve enor, actuator, tranducer and actve dampng devce. Due to the attractve properte of pezoelectrc compote tructure, varou numercal method have been propoed to model and mulate ther behavour. For tatc and free vbraton analy, Yang and Lee [1] howed that the early work on tructure wth pezoelectrc layer could lead to ubtantal error n the natural frequence and mode hape. Km et al. [] valdated the Fnte Element (FE) model of a mart cantlever plate through comparon wth experment. Wllberg et al. [3] tuded a three-dmenonal pezoelectrc old model ung ogeometrc fnte element. For vbraton control, ome theore ntegrated wth varou numercal method have been propoed and the three mot popular theore are the Clacal Lamnaton heory (CL), the Frt-order Shear Deformaton heory (FSD), and the Hgher-order Shear Deformaton heory (HSD). n the CL, whch baed on the aumpton of Krchhoff plate theory, the nterlamnar hear deformaton neglected. Hwang and Park [4], Lam et al. [5] reported control algorthm baed on clacal negatve velocty feedback control and the FE method whch were formulated baed on the dcrete Krchhoff quadrlateral element. n the FSD, a contant tranvere hear deformaton aumed through the entre thckne of the lamnate and hence tre-free boundary condton are volated at the top and bottom urface of the panel. Mlazzo and Orlando [6] tuded free vbraton analy of mart lamnated thck compote plate. Phung-Van et al. [7] extended the cell-baed moothed dcrete hear gap method to tatc, free vbraton and control of pezoelectrc compote plate. n both CL and FSD theore, a hear correcton factor requred to enure the tablty of the oluton. n order to mprove the accuracy of tranvere hear tree and to avod the ntroducton of hear correcton factor, the HSD baed on the FE method ha been propoed to tudy pezoelectrc plate [8,9]. t worth mentonng that the HSD requre at leat C 1 -contnuty of generalzed dplacement due to the preence of ther econd-order dervatve n the tffne formulaton. h a ource of dffculty n tandard fnte element featurng C 0 nter-element contnuty. A t emerge from the above revew, the avalable tude have focued on the dynamc analy of pezoelectrc compote plate ung the FE method, the moothed FE method, etc. h paper am at further contrbutng to the dynamc analy of pezoelectrc compote plate ung an 134

2 ogeometrc approach baed on Non-Unform B-Splne (NURBS) ba functon. n partcular, we how that a HSD formulaton fulfllng C 1 -contnuty requrement ealy acheved n the ogeometrc framework. ogeometrc analy (GA) ha been recently propoed by Hughe et al. [10] wth the orgnal objectve to tghtly ntegrate Computer Aded Degn (CAD) and FE analy. GA make ue of the ame ba functon typcally ued n the CAD envronment (mot notably NURBS or -Splne) to decrbe the geometry of the problem exactly a t produced from CAD a well a to approxmate the oluton feld for the analy. h paper explot further the advantage of a NURBS-baed ogeometrc approach for bucklng analy of lamnated compote plate ntegrated wth pezoelectrc enor and actuator ung the HSD theory. n the compote plate, the mechancal dplacement feld approxmated accordng to the HSD model ung NURBS-baed ogeometrc element. hee acheve naturally any dered degree of contnuty through the choce of the nterpolaton order, o that the method ealy fulfl the C 1 -contnuty requrement of the HSD model. he electrc potental aumed to vary lnearly through the thckne for each pezoelectrc ub-layer. he accuracy and relablty of the propoed method verfed by comparng t numercal predcton wth thoe of other avalable numercal approache. WEAK FORM AND FEM FORMULAON FOR PEZOELECRC COMPOSE PLAE.1 Lnear pezoelectrc conttutve equaton he lnear pezoelectrc conttutve equaton can be expreed a c e ε D e g E (1) and ε are the tre and tran vector; D and E are delectrc dplacement and electrc vector; c the elatcty matrx; e the pezoelectrc contant matrx and g denote the delectrc contant matrx. he Galerkn weak form of the governng equaton of pezoelectrc tructure can be derved by ung Halmnton varatonal prncple [4], whch can be wrtten a ( ) L ρu u ε + D E + uf q d + u Fp Qp 0 () u and u are the mechancal dplacement and velocty; the electrc potental; the mechancal load and pont load; q, Q p are the urface charge and pont charge.. Approxmaton on the mechancal dplacement feld..1 Governng equaton for a thrd-order hear deformaton theory model f and F are p Accordng to the thrd-order hear deformaton theory propoed by Reddy [11], the dplacement of an arbtrary pont n the plate are expreed by ( ) u u + zβ + cz β + w ; v v + zβ + cz ; w w (3) x x, x 0 y 0, w and β β x y are the membrane dplacement, the deflecton of the md-plane and the rotaton of the md-plane around y-ax and x ax, repectvely. t the thckne of the plate; c 4/3t and the varable u [ u v ] he tran are thu expreed by the followng equaton 135

3 εxx u0, x βx, x β + w x, x 0, xx 3 3 ε ε v z β zc β w z z yy 0, y y, y y, y 0, yy ε κ (4) γ u + v β + β β + β + w xy 0, y 0, x x, y y, x x, y y, x 0, xy γxz β + w x 0, x β + w x 0, x z 3c z γ + + β + w β + w (5) yz y 0, y y 0, y From Hooke law and the lnear tran gven by Eq. (4) and (5), the tre computed by D 0 0 Dγ p c (6) p and are the n-plane tre component and hear tre; D and D are materal contant matrce gven n the form of A B E A B D B D F, D B D E F H, h/ ( A B DEF H) ( ) h/ h/ 4 ( A, B, D ) ( 1, z, z ) Q dz j h/,,,,, 1, z, z, z, z, z Q dz (, j 1,, 6) j, j 4, 5 (7).. NURBS-baed novel compote plate formulaton Ung the NURBS ba functon [10], the dplacement feld u of the plate approxmated a u h m n ( ξ, η) N ( ξ, η) d (8) [ u v w β β ] x x d the vector of degree of freedom aocated wth the control pont. Subttutng Eq. (8) nto Eq. (4) and (5), the n-plane and hear tran can be rewrtten a: ( ) ( ) ( ) ( ) ( ) m n m b b 0 1 ε B B B B B d A 1 ε κ (9) N , x N, x N N, xx, x m b1 b B 0 N 0 0 0, B N, B c 0 0 0,, y N N, y, yy, y N N N N 0 0 N N N B, y, x, y, x, xy, y, x 0 0N N 0 0 0N N 0, N 0, x 1, x 3c N B N, y N, y (10) 136

4 .3 Approxmaton of the electrc potental feld n each ub-layer, a lnear electrc potental functon aumed through the thckne a [1]: ( z) N (11) N the vector of the hape functon for the electrc potental, and electrc potental at the top and bottom urface of the -th ub-layer. the vector contanng the For each pezoelectrc ub-layer element, the electrc feld E n Eq. (1) can be rewrtten a [13]: E N B (1).4 Elementary governng equaton of moton he fnal form of equaton of bucklng wrtten n the followng form K ω Mu K λ cr K g u (13) ( ) 0 and ( ) 0 Muu 0 Kuu Ku d F M,, 0 0 K u Ku K Q (14) n whch K B cb d ; K B e B d ; K B pb d ; M N mnd uu u u u u uu (15) whch B [ m b b ] u B B B B B ; m defned by c4 m c c4 c5 c 7 h/ ( ) ρ ( ),,,,, 1, z, z, z, z, z dz h/ (16) and N, x 00 N N x xy K ( ) d, g B NB B g 0 g g N N (17), y 00 N N xy y and ω, λ are the natural frequency and the crtcal bucklng value, repectvely. cr n th work, the contant gan G d of the dplacement feedback control ued to couple the nput actuator voltage vector a and the output enor voltage vector a G a d 137

5 he global tffne matrx can be rewrtten [7]: K K K K K (18) * 1 uu + Gd u u 3 NUMERCAL RESULS n th ecton, frt, we verfy the accuracy and effcency of the propoed ogeometrc element for analyzng the natural frequence of the pezoelectrc compote plate. We conder a quare fve-ply pezoelectrc lamnated compote plate [pe/0/90/0/pe] n whch pe denote a pezoelectrc layer. he plate mply upported and the thckne to length rato of each compote ply t/a 1/50. he lamnate confguraton nclude three layer of Graphte/Epoxy (Gp/Ep) wth fber orentaton of [0/90/0]. wo contnuou PZ-4 pezoelectrc layer of thckne 0.1t are bonded to the upper and lower urface of the lamnate. wo et of electrc boundary condton are condered for the nner urface of the pezoelectrc layer ncludng: (1) a cloed-crcut condton n whch the electrc potental kept zero (grounded); and () an open-crcut condton n whch the electrc potental reman free (zero electrc dplacement). able 1 how the dmenonle frt natural frequency of the pezoelectrc compote plate wth mehng of 8 8. n th tudy, the ogeometrc element ue the HSD wth only 5 dof per control pont whle Ref [15] ue the layerwe theory and Ref [8] ue HSD wth 11 dof per node. t een that the reult gven by the ogeometrc formulaton are lghtly lower than the analytcal oluton [16], however the error are le than 5%. We oberve that the ogeometrc reult are table n both a cloed-crcut condton and an opencrcut condton mlarly to the analytcal oluton [16], whle thoe of Ref [15,8] are very dfferent for a cloed-crcut condton and an open-crcut condton. able 1 Dmenonle frt natural frequency of the pezoelectrc compote plate [pe/0/90/0/pe] Method Mehng Degree of freedom (DOF) f ωa Cloed crcut ( t ρ ) / Open crcut GA (5 dof per control pont) FE layerwe [15] Q9 - HSD (11 dof per node) [8] Q9 - FSD (5 dof per node) [8] Ref [16] (a) he bucklng load Fg. 1 Model of a 5-ply pezoelectrc compote plate (b) Frt three bucklng mode 138

6 Next, we conder a pezoelectrc compote plate under axal compreon. he bucklng load parameter ( ) λ λ ρ of pezoelectrc wth boundary condton: mply upported and clamped plotted cr cr a / t n Fg. 1a. t can be een that the bucklng load for clamped plate hgher than that for mply upported plate, a expected. h becaue the tffne of the clamped plate tffer. Next, the effect of the contant gan of the dplacement to the bucklng load dplayed n Fg. 1b. he reult how that the bucklng load ncreae when the contrant gan ncreae. 4 CONCLUSONS h paper preent a mple and effectve approach baed on the combnaton of GA and HSD for the bucklng analye of compote plate ntegrated wth pezoelectrc enor and actuator. n the pezoelectrc compote plate, the mechancal dplacement feld approxmated accordng to the HSD ung ogeometrc element baed on NURBS and featurng at leat C 1 -contnuty, a the electrc potental aumed to vary lnearly through the thckne for each pezoelectrc ub-layer. he accuracy and relablty of the propoed method verfed by comparng t numercal predcton wth thoe of other avalable numercal approache. 5 REFERENCES [1] S.M. Yang, Y.J. Lee, nteracton of tructure vbraton and pezoelectrc actuaton, Smart Materal and Structure, 3, , [] J. Km, V.V. Varadan, V.K. Varadan, X.Q. Bao, Fnte element modellng of a mart cantlever plate and comparon wth experment, Smart Materal and Structure, 5, , [3] C. Wllberg, U. Gabbert, Development of a three-dmenonal pezoelectrc ogeometrc fnte element for mart tructure applcaton, Acta Mechanca, 3(8), , 01. [4] W.C. Hwang, H.C. Park, Fnte element modellng of pezoelectrc enor and actuator, AAA Journal, 31, , [5] K.Y. Lam, X.Q. Peng, G.R. Lu, J.N. Reddy, A fnte-element model for pezoelectrc compote lamnate, Smart Materal and Structure, 65, 83 91, [6] A. Mlazzo, C. Orlando, An equvalent ngle-layer approach for free vbraton analy of mart lamnated thck compote plate, Smart Materal and Structure, 1, , 01. [7] P. Phung-Van,. Nguyen-ho,. Le-Dnh, H. Nguyen-Xuan, Statc and free vbraton analye and dynamc control of compote plate ntegrated wth pezoelectrc enor and actuator by the cellbaed moothed dcrete hear gap method (CS-FEM-DSG3), Smart Materal and Structure,, 09506, 013. [8] M.F.C. Vctor, A.A.G. Mara, S. Afzal, M.M.S. Crtóvão, A.M.S. Carlo, C.V.M. Franco, Modellng and degn of adaptve compote tructure, Computer Method n Appled Mechanc and Engneerng, 185, , 000. [9] J.N. Reddy, On lamnated compote plate wth ntegrated enor and actuator, Engneerng Structure 1, , [10].J.R. Hughe, J.A. Cottrell, Y. Bazlev, ogeometrc analy: CAD, fnte element, NURBS, exact geometry and meh refnement, Computer Method n Appled Mechanc and Engneerng, 194(39-41), , 005. [11] J.N. Reddy, A mple hgher-order theory for lamnated compote plate, Journal of Appled Mechanc, 51, , [1] S.Y. Wang, A fnte element model for the tatc and dynamc analy of a pezoelectrc bmorph, nternatonal Journal of Sold and Structure, 41, , 004. [13] S.Y. Wang, S.. Quek, K.K. Ang, Vbraton control of mart pezoelectrc compote plate, Smart Materal and Structure, 10, ,

7 [14] M.F.C. Vctor, A.A.G. Mara, S. Afzal, M.M.S. Crtóvão, A.M.S. Carlo, C.V.M. Franco, Modellng and degn of adaptve compote tructure, Computer Method n Appled Mechanc and Engneerng, 185, , 000. [15] D.A. Saravano, P.R. Heylger, D.A. Hopkn, Layerwe mechanc and fnte element for the dynamc analy of pezoelectrc compote plate, nternatonal Journal of Sold and Structure, 34, , [16] P. Heylger, D.A. Saravano, Exact free-vbraton analy of lamnated plate wth embedded pezoelectrc layer, Journal of the Acoutcal Socety of Amerca, 98, ,