Non-Linear Analysis of Interlaminar Stresses in Composite Beams with Piezoelectric Layers

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1 7TH ITERATIOA OFEREE O OMPOSITE SIEE AD TEHOOGY on-inar Analysis of Intrlaminar Strsss in omosit Bams with Piolctric ayrs MASOUD TAHAI 1, AMIR TOOU DOYAMATI 1 Dartmnt of Mchanical Enginring, Faculty of Enginring, Frdowsi Univrsity of Mashhad, Mashhad, Iran. -mail: mtahani@frdowsi.um.ac.ir Dartmnt of Mchanical Enginring, Faculty of Enginring, Islamic Aad Univrsity, Mashhad Branch, Mashhad, Iran Abstract A third-ordr shar dformation bam thory (TSDBT) is usd to analy intrlaminar strsss in gnrally laminatd comosit bams with iolctric actuators subjctd to lctrical loading. Th non-linar strain-dislacmnt rlations in th von- Kàrmàn sns ar usd to study th ffct of gomtric non-linarity on intrlaminar strsss. Th quilibrium quations ar obtaind by using th rincil of minimum total otntial nrgy and thn solvd actly. Th numrical rsults clarly indicat th singular bhavior of intrlaminar normal and shar strsss nar th nds of th laminatd bam. Ky words iolaminatd bams, intrlaminar strsss, von-kàrmàn non-linarity 1. Introduction A smart structur with iolctric actuators has drawn a considrabl intrst in structural rformanc control du to th could mchanical and lctrical rortis of iolctric matrials [1,]. During th ast two dcads, th intrlayr strsss and failur mchanism of adativ structurs hav causd incrasingly attntion of ractical nginrs and rofssional rsarchrs. It is wll known that th stiffnss mismatch btwn adjacnt lis in laminatd comosits causs intrlaminar strss concntrations nar th dgs of a laminat whr w hav matrial discontinuity (.g., s [3 6]). Th could iolctric analysis of th fr dg ffct was rformd by Davi and Milao [7], with using boundary lmnt formulation. Artl and Bckr [8] and Zhn and Wanji [9], using th finit lmnt mthod, analyd th influnc of iolctric couling on intrlaminar strsss and lctric fild nar th dgs of iolctric comosit laminats. To th tnt of th author s knowldg, thr is no analytical solution of intrlaminar strsss in iolaminatd bams with considring gomtric non-linarity in th von-kàrmàn sns. Hr, in th rsnt work, w ar intrstd to study th natur of th strss fild nar

2 on-linar Analysis of Intrlaminar Strsss in omosit Bams with Piolctric ayrs: Tahani, M. and Tolu Donyamati, A. th nds of a iolaminatd bam. To this nd, th gomtric non-linarity mrging from th strain-dislacmnt rlations ar includd in th analysis.. Thortical Formulation It is assumd that a laminatd bam is constructd with iolctric layrs. Th bam is subjctd to lctrical loading. Th dislacmnt fild using a third-ordr shar dformation bam thory may b rrsntd as: u 1( 3 3, y, ) u ψ ϕ η, u (, y, ) v, u (, y, ) w( ) ψ (1) whr u 1, u, and u 3 ar th dislacmnts in th, y, and dirctions, rsctivly, of a matrial oint initially at (, y, ) in th undformd bam. Substituting Eqs. (1) into th von- Kàrmàn non-linar strain-dislacmnt rlations, th following rsults will b obtaind [1]: ε ε κ κ κ, ε ε, γ ε, ε γ, γ ε κ κ () y 6 y y whr 1 1 ( w ), κ1 ψ, κ1 φ, κ1 η, ε3 ψ, ε6 v, ε5 ψ w, κ5 φ ψ, κ5 ε u 3η (3) 1 A rim in Eqs. (3) indicats an ordinary drivativ with rsct to. Using th rincil of minimum total otntial nrgy, quilibrium quations can b obtaind in trms of strss rsultants as [11]: d d, dr d d d, y d dw dq,, d d d d dp R, 3S d d dm d Q, (4) Th strss rsultants in Eqs. (4) ar dfind as: h / / / 3 h h σ σ y σ y σ h/ h/ h/ (5) (, M,, P ) (1,,, ) d, ( Q, R, S ) (1,, ) d, (, ) (, ) d Bcaus th width of th bam is small comaring to its lngth, it is assumd that σ y σ y [11], so th rducd io-lastic constitutiv law of th kth orthotroic iolctric lamina is utilid as [5]: σ σ y σ y ε ε γ y E E y E, σ 55 γ (6) whr { σ } k and {} ε k ar th strss and strain vctors, { E } k is th lctric fild vctor,

3 7TH ITERATIOA OFEREE O OMPOSITE SIEE AD TEHOOGY and [ ] k and [ ] k ar th mchanical comlianc and iolctric matrics of th kth layr. It is assumd that th voltag is only alid in th dirction (i.., E Ey ). Uon substitution of Eqs. () into Eqs. (6) and th subsqunt rsults into Eqs. (5), th gnralid strss rsultants will b obtaind which can b rrsntd as follows: A B D E A A M Q A B D ε 5, R B55 D55 E55 κ 5 P 1 S D55 E55 F55 κ 5 y A B D E A A ε M B D E F B B κ D E F G D D κ1 P E11 F11 G11 H11 E13 E 16 κ1 A13 B13 D13 E13 A33 A ε 3 y ε (7) whr ( A, B (, M, D, E, F,, P ), G h, H ) h h ( k) (1,, 3,, h 4, 5, 6 )d, 3 k k (1,,, ) 13 E d, 33 E d, y h h h h k E d (8) In ordr to obtain th act solutions of Eqs. (4), th first and scond quations in (4) ar intgratd with rsct to to yild and y y whr and y ar constants of intgration to b found by imosing som conditions which will b discussd latr. t, w solv th aformntiond quations for ε 1 and ε 6 to obtain: ε Bˆ κ Dˆ κ Eˆ κ Aˆ ε ˆ, ε Bˆ κ Dˆ κ Eˆ κ Aˆ ε ˆ (9) y In Eqs. (9) th cofficints ar dfind as follows: ˆ B A B A, ˆ D A D A, ˆ E A E A, ˆ A A A A B D E A Δ Δ Δ Δ ˆ B A B A, ˆ D A D A, ˆ E A E A, ˆ A A A A B D E A Δ Δ Δ Δ ˆ ( y y ) ( ) A16 A66 Δ, ˆ y ( y y ) ( ) A11 A16 Δ (1) whr Δ A A A. With (9), Eqs. (3) and (7) ar substitutd into Eqs. (4) to yild: δw: ( A ) w A ψ B ψ B φ 3D η Z δψ : A55( ψ w ) a1ψ aψ a3φ B55 φ a4η 3D55 η ˆ ˆ δψ : B w bψ D ψ b ψ b φ b η A A y δφ : B ( ψ w ) cψ c ψ 4D φ cφ c η 6E η δη : 3 D ( ψ w ) d ψ d ψ 6E φ d φ c η 9F η (11)

4 on-linar Analysis of Intrlaminar Strsss in omosit Bams with Piolctric ayrs: Tahani, M. and Tolu Donyamati, A. whr th cofficint a, b, c, and d i ( i 1,,3,4 ) in Eqs. (11) ar givn by: i i i a B Bˆ B Bˆ D, a B Aˆ B Aˆ B B, a B Dˆ B Dˆ E, a B Eˆ B Eˆ F, b B B A Bˆ A Bˆ, b A A Aˆ A Aˆ, b3 D55 D13 A ˆ ˆ 13 D11 A D16, b4 3E55 E13 A 13 E ˆ ˆ 11 A E 16 c D Bˆ D Bˆ E, c D Aˆ D Aˆ D D, c D Dˆ D Dˆ F, c D Eˆ D Eˆ G, d E Bˆ E Bˆ F, d E Aˆ E Aˆ E 3 E, d3 E11 Dˆ 11 E ˆ ˆ 16 D16 G11, d4 E11 E11 E16 Ê16 H11 (1) In this ar, th following siml and clamd suorts at th nds of th bam ar usd: Siml suort (S): u v M P R w lamd (): u v ψ φ η ψ w (13) Eqs. (11) can b solvd analytically for any sts of th boundary conditions in trms of unknown constants and. Thn w will us two mor conditions to find th final y solutions. For aml, for th S-S and - boundary tys w hav u v at ± (s Eqs. (13)) which will allow us to find and nd, Eqs. (9) ar intgratd with rsct to from to to gt: ˆ y in a trial and rror rocss. To this 1 1 ( Bˆ Dˆ κ Eˆ κ Fˆ ε w ) d, ˆ ( Bˆ κ Dˆ κ Eˆ κ F ε ) 11κ y d (14) ˆ Finally, by making th solutions of Eqs. (11) to satisfy (14) in a trial and rror rocss, w will obtain th act valus of and y. 3. umrical rsults Hr, th numrical rsults ar obtaind for a simly suortd bam with hight h.1 m and lngth 3h. Th numrical rsults ar comard with thos obtaind by a first-ordr shar dformation thory. Each lamina is assumd to b of th sam thicknss (t) and is idalid as an orthotroic matrial with th mchanical [1] and lctrical rortis [] as follows: E 13 GPa, E E 1.8 GPa, G G 5.56 GPa, G 3.86 GPa ν ν.4, ν d d /, d 35 1 /, d d 6 1 / (15) Piolctric layrs ar subjctd to voltags E a and - E a. Th cntral dflction w was urosfully chosn to b qual to.5h whr th non-linarity ffct is significant (s Fig. 1(a)). Fig. 1(b) shows th dflction of th S-S bam. Also Figs. (a) and (b) show th distri-

5 7TH ITERATIOA OFEREE O OMPOSITE SIEE AD TEHOOGY butions of th in-lan strss σ and transvrs shar strsss σ, rsctivly, in th [ /9 ] s σ, σ, σ σ, σ, σ E and [ ][ ][ d ]. and [ /3 ] s bams. It is notd that a 31 Fig. 1 (a) Variation of th cntr dflction of th bam vrsus th lctrical load and (b) dflction of th S-S bam Fig. (a) Through th thicknss distribution of in-lan strss σ and (b) distribution of shar strss σ along midlan ( ) of th bam Fig. 3 Distributions of intrlaminar normal strss σ (a) along th middl lan and (b) along th first intrfacs of [ /9 ] s and [ /3 ] s bams

6 on-linar Analysis of Intrlaminar Strsss in omosit Bams with Piolctric ayrs: Tahani, M. and Tolu Donyamati, A. Fig. 3 illustrats th distributions of non-dimnsional intrlaminar normal strss along th middl lan and along th first intrfacs (i.., along /9 and /3 intrfacs) of [ /9 ] s and [ /3 ] s bams whn w is qual to.5h. σ 4. onclusions In this study a third-ordr shar dformation bam thory, with thicknss strtching, is dvlod to study th intrlaminar strsss in iolaminatd comosit bams subjctd to lctrical loads. Th non-linar strain-dislacmnt rlations ar usd to study th ffct of th gomtric non-linarity on th intrlaminar strsss. Th quilibrium quations ar solvd actly. Th singular bhavior of th intrlaminar strsss in th boundary-layr rgion is obviously sn in th numrical rsults. Also th rsults indicat that th magnituds of th out-of-lan strsss in cross-ly iolaminatd bam [ /9 ] s ar largr than thos of angl-ly bam [ /3 ] s. Rfrnc 1. Rddy, J..: On laminatd comosit lats with intgratd snsors and actuators. Eng. Struct. 1, (1999). Blandford, G.E., Tauchrt, T.R., Du, Y.: Slf-straind iothrmolastic comosit bam analysis using first-ordr shar dformation thory. omos. 3, (1999) 3. Kant, T., Swaminathan, K.: Estimation of transvrsly intrlaminar strsss in laminatd comosits a slctiv rviw and survy of currnt dvlomnts. omos. Struct. 49, () 4. Kauria, S.: An fficint could thory for multilayrd bams with mbddd iolctric snsory and activ layrs. Int. J. Solids Struct. 38, (1) 5. Rddy, J..: Mchanics of laminatd comosit lats and shlls. R Prss, w York (4) 6. Yao,.Q., Zhang, J.G., u,. and ai, M.O.: onlinar tnsion and bnding of iolctric laminatd lat undr larg alid fild actuation. Smart Mat. Struct. 13, (4) 7. Davi, G., Milao, A.: Strss and lctric filds in iolctric comosit laminats. Elctronic Journal of Boundary Elmnts, Btq 1, 43-5, () 8. Artl, J., Bckr, W.: ould and uncould analyss of iolctric fr dg ffct in laminatd lats. omos. Struct. 69, (5) 9. Zhn, W., and Wanji,.: Rfind triangular lmnt for laminatd lastic iolctric lats. omos. Struct. 78, (7) 1. Rddy, J..: Enrgy rincils and variational mthods in alid mchanics. John Wily and Sons, Inc., w Jrsy, () 11. Tahani, M.: Analysis of laminatd comosit bams using layrwis dislacmnt thoris. omos. Struct. 79, (7)

AS 5850 Finite Element Analysis

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