IMECE LOW-VELOCITY GLOBAL-LOCAL IMPACT RESPONSE OF SMART COMPOSITE AND SANDWICH COMPOSITE PLATES WITH PIEZOELECTRIC TRANSDUCERS

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1 Proceedgs of the ASME 214 Iteratoal Mechacal Egeerg Cogress & Exposto IMECE214 November 14-2, 214, Motreal, Quebec, Caada IMECE LOW-VELOCITY GLOBAL-LOCAL IMPACT RESPONSE OF SMART COMPOSITE AND SANDWICH COMPOSITE PLATES WITH PIEZOELECTRIC TRANSDUCERS Theofas S. Plagaaos Departmet of Mechacal Egeerg Natoal Techcal Uversty of Athes Athes, Greece Evagelos G. Papadopoulos Departmet of Mechacal Egeerg Natoal Techcal Uversty of Athes Athes, Greece ABSTRACT Hgher-order layerwse pezoelectrc lamate mechacs are preseted for predctg the low-velocty mpact respose of prste composte ad sadwch composte plates wth pezoelectrc trasducers. The preset formulato eables predcto of the global (temporal varato of mpact force, deflecto, stra ad sesory potetal) ad local throughthcess (dstrbuto of dsplacemet, stress ad stra) mpact respose of plates wth pezoelectrc layers or patches. Its ehaced capabltes clude effcecy terms of computatoal cost, sce the system matrces are reduced by meas of a Guya scheme or by usg the egevectors, thus leadg to a plate-mpactor system cotag a sgle or two deflecto ampltudes per vbrato mode, depedg o cosderato of trasverse compressblty. The trasfer of the plate-mpactor system to state-space eables vestgato of the feasblty of real-tme actve cotrol towards mpact force reducto by usg output feedbac cotrol laws. INTRODUCTION Smart sadwch plates wth composte faces ad foam core, as well as embedded pezoelectrc actuators ad sesors, combe the superor mechacal propertes of sadwch structures, such as, hgh flexural stffess to mass rato, wth the capablty to motor the structural respose o-ste realtme ad to adapt ther respose accordg to selected cotrol crtera. At hgh loadg rates, such as the case of mpact loadg, damage may occur the form of matrx cracs ad terfacal delamatos, whch cases of low-velocty mpact may be vsble. Thus, predcto ad motorg of the global dyamc respose ad the local through-thcess stress feld, s essetal order to eep both desg ad fucto wth approprate safety lmts. Moreover, the formulato of computatoally effcet methodologes for predctg the plate-mpactor structural system s respose s crucal order to develop realstc real-tme actve cotrol algorthms ad relevat applcatos by usg the pezoelectrc actuators. Extesve lterature revews o mpact o composte structures have bee coducted, amog others, by Abrate [1], Cha ad Zhu [2] ad Catwell ad Morto [3]. O the bass of the ematc assumptos used to predct the respose of the mpacted structure, the exstg models may be dvded to two ma categores: () mass-sprg models (Shvaumar et al. [4]; Olsso [5]) ad () full cotuum models based o eergy equlbrum equatos. I both categores, oe of the ey ssues regardg the accuracy of the predcted mpact respose s the cotact law betwee plate ad mpactor, whch ca be Hertza, elasto-plastc wthout/wth plastc detato [6], or may tae to accout damage effects, such as core crushg [7]. Aalytcal solutos for composte plates subjected to lowvelocty mpacts have bee developed amog others by Chrstoforou ad Ygt [8] o the bass of Krchhoff s plate theory ematcs. Fte elemet solutos for predctg the low-velocty mpact respose of composte plates have bee reported amog others by Su ad Che [9], who developed a quadratc Lagrage elemet based o Resser-Mdl ematcs ad a expermetally determed o-lear detato. As far as sadwch plates subjected to low-velocty mpacts are cocered, hgher-order sgle-layer models have bee developed amog others by Yag ad Qao [1]. Icard ad Ferrero [11] reported a refed plate elemet based o global-local 3-D layerwse ematc assumptos, cosdered materal degradato ad predcted damage ad throughthcess dstrbutos of trasverse dsplacemet ad terlamar shear stress. The dea of embeddg pezoelectrc sesors to composte structures order to detect mpact locato ad recostruct the cotact force tme-profle was reported the late 9 s by Tracy ad Chag [12], ad Seydel ad Chag [13], ad has bee elaborated for the desg of realtme motorg etwors by Par et al. [14], ad Lu ad 1 Copyrght 214 by ASME

2 Chattopadhyay [15]. The actve cotrol of mpact respose of composte plates by meas of pezoelectrc layers towards the mmzato of cotact force has bee studed by Saravaos ad Chrstoforou [16], who developed a aalytcal soluto based o frst-order shear ematcs for the composte lamate ad a lear layerwse through-thcess approxmato of the electrc potetal. Yet, the local throughthcess mpact respose of composte ad sadwch composte plates wth pezoelectrc passve/actve trasducers the case of low-velocty mpact has ot bee studed. Moreover, the predcto of the global mpact respose (mpact force, deflecto at mpact pot, electrc potetal at sesors) s based o the soluto of the full plate-mpactor system. Ths approach leads to reduced sutablty for real-tme cotrol applcatos due to large matrx szes ad thus creased computatoal effort. The objectve of the preset wor s to descrbe a mpact mechacs methodology capable of a computatoally effcet predcto of the global ad local through-thcess respose of prste composte ad sadwch composte plates wth pezoelectrc actuators ad sesors. Moreover, a prelmary study o actve cotrol of the force developed durg mpact of a medum mass mpactor o a composte plate s preseted. THEORETICAL FORMULATION The followg paragraphs preset the tegrated mpact mechacs methodology, startg at the geeral pezoelectrc or composte ply level ad arrvg to the coupled plate-mpactor structural system state-space. Basc Physcal Assumptos The methodology s based o the followg physcal assumptos: The mpact eergy s suffcetly low, such as o materal damage s duced by the mpact evet. The mpact s elastc, thus there s o loss of eergy the form of heat. The lamate ples are perfectly boded together throughout the mpact evet ad subsequet vbrato. Geerally, mpacts may be categorzed to low- ad hghvelocty oes o the bass of the overall structural respose [2]. I the preset formulato t s assumed that the mpact durato s sgfcatly loger tha the travel tme of the waves to the boudary, ad s eough for the plate to respod [1]. A practcal low-velocty threshold has bee provded by Catwell ad Morto [3] o the bass of avalable expermetal techques, as beg 1 m/s. The mpact veloctes studed the curret wor are up to 3 m/s. Goverg Materal Equatos I geeral, the lamate layers cludg the pezoelectrc or composte ad foam ples are assumed to exhbt lear pezoelectrc behavour. I the followg formulato, dsplacemets ad electrc potetal ad all other varables arsg from these (stras, stresses, etc.) are tme-depedet. The ply costtutve equatos the atural coordate system Oxyz (Fg. 1) have the form: E T CjSj- em Em S m emj j mm m D S E where, j=1,,6 ad m=1,,3; σ ad S j are the mechacal stress ad egeerg stra, respectvely, vectoral otato; E m s the electrc feld vector; D m s the electrc dsplacemet vector; C j s the elastc stffess tesor; e mj s the pezoelectrc tesor arsg from the pezoelectrc charge tesor ad the stffess tesor; ad ε mm s the electrc permttvty tesor of the materal. Superscrpts E ad S dcate a costat electrc feld, ad stra codtos, respectvely. The above equatos may ecompass the behavour of both a off-axs homogezed fbrous pezoelectrc ply ad a passve composte ply (e mj =). The electrc feld vector E m s the gradet of the electrc potetal φ alog the atural coordate system vectors x,y,z: E m m / x m (2) I the curret wor, through-thcess polarzed pezoelectrc trasducers are cosdered. Through-Thcess Plate Kematcs A typcal composte or sadwch composte lamate wth pezoelectrc trasducers s subdvded to dscrete layers as show schematcally Fg. 1. Fg. 1: Typcal sadwch pezoelectrc composte lamate cofgurato aalyzed wth -dscrete layers. Dscrete layers. Assumed dsplacemet ad electrc potetal dstrbuto each layer. (1) 2 Copyrght 214 by ASME

3 I the case of composte plates, the dsplacemet feld assumed through the thcess of the lamate s based o a 2- D hgher-order layerwse formulato (HLPT 2-D), whch approxmates -plae dsplacemets ad electrc potetal by pecewse lear, parabolc ad cubc fuctos of the dscrete layer thcess, whle matag dsplacemet cotuty across dscrete layer boudares [17]. For the sadwch structures, the core compressblty effects are tae to accout by applyg a smlar through-thcess approxmato o the trasverse dsplacemet (HLPT 3-D). I ths cotext, the ematc assumptos tae the form: Both Composte ad Sadwch Plates u v x(x,y) 3( ) x(x,y) 4( ) y(x,y) 3( ) y(x,y) 4( ) 1 z z 1 z 2 (x,y) 3( ) (x,y) 4( ) (x, y, ) U (x, y) ( ) U (x, y) ( ) (x,y, ) V (x,y) ( ) V (x,y) ( ) (x,y, ) (x,y) ( ) (x,y) ( ) Compostes Plates (HLPT 2-D) w (x,y, ) w (x,y) (4) Sadwch Plates (HLPT 3-D) w z(x,y) 3( ) z(x,y) 4( ) (x, y, ) W (x, y) ( ) W (x, y) ( ) where u ad v are the -plae dsplacemets, w s the trasverse dsplacemet, superscrpts =1,, deote dscrete layer ad mdsurface, ad ζ s the local thcess coordate of layer defed such that ζ = at the mddle of the dscrete layer, ζ =1 ad ζ =-1 at the top ad the bottom, of the dscrete layer, respectvely. 1, 2 are lear ad 3, 4 are quadratc, cubc terpolato fuctos, respectvely, through the thcess of the layer. U, V, W, U +1, V +1, W +1 ad z, +1 z are dsplacemets ad electrc potetal at the bottom ad top of the dscrete layer, effectvely descrbg exteso ad rotato, ad electrc potetal at the termals, respectvely, of the layer, ad w s the trasverse dsplacemet at the mdplae. The terms x, y, z,, x, y, z, are ampltudes of quadratc ( ) ad cubc ( ) varatos of dsplacemets (subscrpt x, y ad z) ad electrc potetal (subscrpt φ) through the thcess of the dscrete layer. The cotrbutos of these hgher-order varatos to the -plae dsplacemet ad electrc potetal dstrbuto through the thcess of the dscrete layer vash at ts top ad bottom terfaces. (3) (5) Lamate Eergy The stffess ad mass matrces of the sadwch composte plate are derved o the bass of Hamlto s prcple: where A deotes the mdplae (Fg. 1), u s the vector of all degrees of freedom of the lamate arsg from the ematc assumptos (3)-(5), are the tractos at the boudary surface Γ, δh L ad δk L are the varatos of the electromechacal ad etc eergy of the lamate per ut area, expressed as: T T L j 1 z H (( S ) ( E ) Dj )dz (7) L 1 T u u 2 1 z K (( ) )dz (8) where deotes dscrete layer, =1,,6 ad j=1,2,3, ad S, u are mechacal stra ad dsplacemet vectors. Combato of equatos (6)-(8) wth the stra-dsplacemet relatos, the electrc feld equato (2) ad the costtutve equato (1) yelds the lamate stffess, pezoelectrc ad electrc permttvty matrces. I the case of the HLPT 2-D, terlamar shear stress compatblty s mposed through the thcess of the lamate [17], leadg to elmato of 2+2, hgher-order elastc varables. I the case of the HLPT 3-D the out of plae stress compatblty s wealy mataed through the equatos of moto (6). I-Plae Approxmato of Elastc-Electrc Varables Before proceedg wth tegrato alog the plate s mdsurface, as dctated by equato (6), a -plae approxmato of dsplacemets ad electrc potetal of the lamate should be mplemeted. I the case of a Rtz-type aalytcal soluto, the frst order electromechacal varables are approxmated by Fourer seres expasos, whch are expressed a geeral form as, m j j j j j j j j u j v j j w j W (x,y) W (x,y) m U (x,y) U (x,y) m V (x, y) V (x, y) m (x,y) (x,y) (6) (9) 3 Copyrght 214 by ASME

4 where m, deote modes alog x ad y, respectvely, ad X j are approxmato fuctos of each varable dcated by superscrpt, whch are derved by satsfyg the arbtrary boudary codtos. The hgher-order terms of dsplacemet ad electrc potetal are approxmated accordgly o the bass of the frst-order varable they refer to. The mpact load s assumed to act trasversely at the mpact pot ad s approxmated usg the approxmato fuctos of the trasverse dsplacemet. The geeral form of the above approxmatos cludes Naver-type solutos applcable for cross-ply smply-supported plates. I the case of fte elemet approxmatos, the trasverse dsplacemet s approxmated by mplemetg ether C 1 - cotuous Hermta shape fuctos, as madated by the explct mposto of terlamar shear stress compatblty HLPT-2D, or C -cotuous lear Lagrage shape fuctos HLPT-3D. The -plae elastc varables ad the electrc potetal terms are approxmated usg lear Lagrage shape fuctos. The fte elemet soluto yelds the plate modal matrces, whch are reduced ad fed to the plate structural subsystem, as explctly descrbed the followg sectos. Plate Modal Matrces Substtutg the expressos for lamate electromechacal ad etc eergy (7) ad (8) to the goverg equatos of moto (6) ad tag to accout the stra-dsplacemet relatos ad the -plae approxmatos of dsplacemets ad electrc potetal (9), the plate structural subsystem dscrete form s bult for each mode par : PP M uu u uu K K u u P PP PP P K u K PA A q (t) K u P PA A D (t) K (1) where superscrpts P ad A deote passve (sesory) ad actve pezoelectrc layers, ad T l l w,u,v, x, y T P P l l z,, u (11) where =1,, ad l=1,,-1 are the plate modal elastc ad electrc varable vectors the case of the HLPT 2-D. Smlar vectors are derved the case of the HLPT 3-D, whch addtoally cota lear ad hgher-order terms of trasverse dsplacemet, as well as, -plae hgher-order terms elmated HLPT 2-D. The vector q cotas the exterally appled loads per ut area, whle D s the vector of exterally appled charges. I the absece of exteral charge sources, the structural subsystem s codesed as, A A uu u cu u q ce M K (t) K (12) where, -1 T PP PP PP cu uu u u K K K K K -1 A PP PP PA PA ce u u K K K K K (13) (14) Reducto of Plate Modal Matrces As dcated by Eqs. (11) ad (12), the sze of the mass ad stffess matrces of the plate deped o the throughthcess dscretzato. For a plate dscretzed throughthcess by dscrete layers, Eq. (11) yelds 4+1 or 9+3 depedet elastc varables the case of a HLPT 2-D or HLPT 3-D Rtz-type aalytcal soluto, respectvely. These elastc varables determe the sze of the mass ad stffess matrx equato (12). Cosderg that for predctg the dyamc respose of a plate subjected to a pot mpact the plate-mpactor system should be solved for a large amout of mode pars at each tme step, the layerwse through-thcess dscretzato would lead to mass ad stffess matrces of large sze ad respectve computatoal cost. I order to reduce ths cost ad eable mplemetato of the methodology to realtme cotrol applcatos, whle retag the formato regardg the through-thcess respose, approprate reducto techques should be appled o the plate subsystem. Two types of reducto are mplemeted the curret formulato: () A Guya reducto scheme the case of Rtztype Naver solutos ad () reducto by meas of the modal vectors the case of other boudary codtos ad fte elemet approxmatos. The dea behd both reducto techques s to select a prmary (depedet) elastc varable ad express t as a fucto of the reduced (depedet) oes. Sce the mpact force s assumed to act purely the z- drecto (Fg. 1) ad thus bedg vbrato modes are prmarly excted, the trasverse dsplacemet was selected as the depedet varable both reducto schemes. I the case of the Guya reducto appled to the HLPT 2- D Naver soluto, the modal vector s related to the trasverse dsplacemet as, w U V u u T w d u T (15) l x u l y 4 Copyrght 214 by ASME

5 where T s the modal trasformato vector, whch arses from cosderato of a statc load case for depedet (superscrpt ) ad depedet (superscrpt d) elastc varables: K Kd u q (16) K d d Kdd u I ths cotext, the plate stffess matrx for each mode par s reduced as: r T K T K T cu (17) The plate mass matrx s reduced a smlar maer. The plate matrces reducto procedure yelds per mode par a sgle term for stffess ad mass, respectvely. I the HLPT 3- D Naver soluto, the reducto techque s mplemeted a smlar maer by selectg the top ad bottom plate face trasverse dsplacemet as depedet varables, thus yeldg per mode par two terms for stffess ad mass, respectvely, whereas the rest 9+1 depedet varables are expressed as a fucto of top ad bottom trasverse dsplacemet. A detaled descrpto of the mplemetato of Guya reducto techque ca be foud [19]. I the cases of other boudary codtos tha smplesupports ad o cross-ply lamatos, the fte elemet soluto s mplemeted ad the reduced modal matrces are derved by multplcato of the plate structural matrces wth the modal vectors as, T M uu V[M] V (18) T K uu V[K] V The pezoelectrc matrces appearg equatos (12)-(14) are reduced a smlar maer. Both reducto techques lead to the formulato of the reduced subsystem of the plate, whch has the followg form: r r w M w K w q (t) (19) Plate-Impactor Cotact Force I the preset methodology, mpactors wth hem-sphercal tp are assumed. Lear cotact laws are mplemeted order to reta low computatoal cost ad facltate mplemetato to real-tme cotrol applcatos. I the case of composte plates, the lear elasto-plastc cotact law proposed by Ygt ad Chrstoforou [8] s cosdered. Durg mpact at a pot (x,y ), the cotact force F s assumed to vary learly wth the local detato, whch s defed as the relatve dstace betwee the mpactor posto ad the face deflecto, whch HLPT 2-D s assumed to be costat through-thcess o the bass of the ematc assumptos (4): w w (x,y ),w w (x,y ),w w (x,y ) y F(x,y ) (2) where w s the vertcal dstace of the mpactor (modelled as a pot mass) from the plate s surface posto just before mpact (Fg. 2) ad y s the cotact stffess, whch depeds o mpactor radus ad elastc propertes of mpactor ad plate materal [6]. O the bass of eq. (2), the smulato of a loweergy mpact of a steel sphere o a composte plate cludes two dstct mpact states: () aggregato plate ad mpactor moto are coupled ad () dsaggregato plate ad mpactor move depedetly. The trgger for swtchg betwee these two dstct states s the relatve dstace betwee mpactor posto ad plate mdsurface. Fg. 2: Schematc represetato of the learzed cotact model [8] mplemeted durg mpact o a composte plate. I the case of sadwch composte plates, learzed cotact laws the form of equato (2) are adopted, wth trasverse dsplacemet of the mpacted face stead of mdplae dsplacemet w. Plate-Impactor Structural System The plate-mpactor structural system s formulated by combg the plate subsystem (19) wth the cotact force equato (2), the goverg equato of moto of the mpactor, m w (t) F (x,y,t) (21) ad the expresso of trasverse modal load per ut area q by meas of Fourer seres terms, whch the case of smple supports s, 4F (x, y ) m q x, y s xs y LL x y Lx L y (22) Thus, the coupled plate-mpactor system s formulated tme doma as, w (t) w (t) w(t) Ms Ks w(t) (23) The above system s fally trasferred to state-space order to facltate real-tme cotrol applcatos: A B Cx A x x+ (24) y where [A], [B], [C] are the system, put ad output matrx, respectvely ad 5 Copyrght 214 by ASME

6 T x = w,w,w,w (25) s the vector of state varables. Depedg o the case study, the output varables vector y may clude sesory electrc potetals, mechacal dsplacemets, stras ad stresses or related tme dervatves, such as the electrc curret desty reported the actve cotrol case the Results secto. RESULTS AND DISCUSSION I the followg paragraphs predctos of the curret mpact mechacs methodology are compared wth Rtz-type aalytcal solutos for composte plates wthout/wth pezoelectrc layers, ad the effect of mpactor mass o the respose s quatfed. The global respose of composte plates wth pezoelectrc sesory patches s studed for the case of eccetrc mpact. The local through-thcess mpact respose of sadwch composte plates wth faces cludg pezoelectrc layers s vestgated. Fally, prelmary studes are coducted for actvely cotrollg mpact force composte plates wth pezoceramc layers. The electromechacal propertes of the materals cosdered are lsted Table 1. Table 1: Electromechacal Propertes of Materals Cosdered Materal Propertes Graphte/Epoxy Foam PIC 181 Mass Propertes ρ (g/m 3 ) Elastc Propertes E 11 (GPa) E 22 (GPa) E 33 (GPa) G 23 (GPa) G 13 (GPa) G 12 (GPa) ν ν ν Pezoelectrc Propertes d 31 (1-12 m/v) d 32 (1-12 m/v) d 33 (1-12 m/v) d 15 (1-12 m/v) d 24 (1-12 m/v) Delectrc Propertes ε 11 (1-12 Farad/m) ε 22 (1-12 Farad/m) ε 33 (1-12 Farad/m) Composte Plate wth Pezoelectrc Trasducers A [(/9) 2 /] S Graphte/Epoxy square composte plate was studed as a bechmar case ad the effects of boudary codtos, mpactor mass ad tal velocty were vestgated. The plate had edge legth of a=.2 m ad thcess aspect rato of a/h=74, whereas each composte ply had a thcess of.135 mm. The preset HLPT 2-D formulato s valdated wth a aalytcal soluto based o frst-order shear lamate theory ematcs [16]. As a frst valdato, the plate was mpacted upwards at the cetre by a steel sphere havg a mass of m =8.537 g ad a tal velocty at cotact v =3. m/s. The cotact stffess had a value of y =6.65e6 N/m [8], as arsg from yeld stregth of a typcal Graphte/Epoxy composte materal. A sgle dscrete layer was used to model the plate through-thcess ad 11x11 bedg modes were tae to accout for both lamate theores. Fg. 3 shows predcted global mpact respose for smply-supported (S-S) ad clamped-clamped (C-C) boudary codtos. I the former case the plate modal matrces have bee derved by a Rtz-type soluto, whereas the latter case by a fte elemet -plae approxmato. Excellet agreemet s observed betwee predctos of the curret methodology ad [16], whereas the plate yelds a slghtly more complat respose tha [16] due to accurate capturg of terlamar shear effects. mpact force (N) mpact force (N) x1-4 2.x x1-4 1.x1-4 5.x x x x x1-4 HLPT 2-D, S-S FSPT [16], S-S HLPT 2-D, C-C. 1.x1-4 2.x1-4 3.x1-4 4.x1-4 5.x1-4. w HLPT 2-D, S-S w HLPT 2-D, S-S w FSPT [16], S-S w FSPT [16], S-S w HLPT 2-D, C-C w HLPT 2-D, C-C. 1.x1-4 2.x1-4 3.x1-4 4.x1-4 5.x1-4 Fg. 3: Valdato of curret methodology predctos for the case of a small mass mpact o a [(/9) 2 /] S Gr/Ep plate for two sets of boudary codtos: mpact force, plate dsplacemet ad mpactor posto. However, the developed method the sze of the reduced plate-mpactor system was 74x74, whereas the full system, such as the case of the FSPT, would have a sze of 362x362 for equal modes state space. The requred computatoal 6 Copyrght 214 by ASME

7 tme o a dual core processor (3.6 GHz, 6 MB) for the reduced plate mpactor system was 1.5% of that of the full system. I the case of fte elemet approxmatos, the system sze would deped o the mesh dscretzato, however t s expected to be far beyod the matrx sze the curret method. Overall, the developed method could effcetly capture the wave-cotrolled mpact respose of the composte plate (local respose occurrg pror to reflecto of waves from the boudares, as descrbed [5]), as well as, the mpact chatterg observed after t=.328 ms. As a secod valdato,.2 mm thc pezoceramc (pzt-4 from Morga-Matroc Ic. [16]) layers were cosdered at top ad bottom of the smply-supported composte lamate. The plate was modelled usg 3 dscrete layers through the thcess. A cotact stffess y =1.234e7 N/m [16] was mplemeted. The plate was assumed to be ht upwards at the cetre by a sphercal mpactor of relatvely large mass m =5. g wth tal velocty v =1. m/sec. Fg. 4 shows predcted temporal varato of the mpact force usg HLPT 2-D, whch s very good agreemet wth predctos of [16]. Due to the large mass of the mpactor a quas-statc mpact respose s observed (term defed [8]), meag that the respose s domated by the erta of the mpactor, whle the plate s vbrato s eglgble. The valdty of the latter commet s llustrated Fg. 4, where the eergy equlbrum durg mpact s show. As far as computatoal effort s cocered, the developed method leads to a platempactor system of sze 74x74 state space, as the prevous case study, ad to smlar computatoal ga, whereas the soluto of a full system of sze 938x938 would have bee requred the case of o reducto. The attachmet of pezoelectrc patches o the plate faces s a more practcal cofgurato towards realstc applcatos. Moreover, mpact does geerally ot occur at the cetre of the plate. Thus, the mpact respose of the smply-supported plate was addtoally studed for a mpact of a sphercal mpactor of mass m =.1 g ad tal velocty v =1 m/s at pot x (a/8,a/8) the case that four pezoceramc patches wth dmeso 1x1x.25 mm are symmetrcally attached o each plate face (Fg. 5). 2 HLPT 2-D FSPT [16] 16 mpact force (N) x1-3 5.x x1-3 1.x x x1-2 eergy (J) x1-3 6.x1-3 9.x x x1-2 total cotact sprg elastc (plate) electrc (plate) etc (mpactor) etc (plate) Fg. 4: Valdato of curret methodology predctos for the case of a 5. g mass mpact o a [pzt-4/(/9) 2 /] S plate: mpact force, eergy equlbrum durg mpact evet. Fg. 5: Smply-supported composte plate wth pezoelectrc patches. Appled boudary codtos are reported at the plate edges. The cluso of the patches does practcally ot affect the plate s dyamc respose terms of modal frequeces, stffess ad mass, thus they were ot modelled ad the electrc potetal (sesory sgal) was predcted at the cetral pot of the patch wthout beg averaged, by meas of the mechacal stras ad the secod costtutve equato (1). Predcted trasverse dsplacemet at the mpact pot ad the cetre of the patches are preseted Fg. 6. It s terestg that durg aggregato betwee plate ad mpactor the trasverse dsplacemet ear the mpact pot exceeds the deflecto predcted at the mpact locato. As expected, the sgal of the sesor beg earest to the mpact locato s tally hgher tha the sgals of the other sesors, as show Fg Copyrght 214 by ASME

8 trasverse dsplacemet (m) electrc potetal (Volt) 4.x1-4 3.x1-4 2.x1-4 1.x x x x x1-3 2.x1-3 3.x1-3 4.x1-3 5.x1-3 mpact pot sesor 1 sesor 2 sesor 3 sesor x1-3 2.x1-3 3.x1-3 4.x1-3 5.x1-3 sesor 1 sesor 2 sesor 3 sesor 4 Fg. 6: Impact respose of a eccetrcally ht [(/9) 2 /] S Gr/Ep plate wth pezoceramc patches: trasverse dsplacemet, sesory sgals at the mpacted face patches. Sadwch Composte Plate wth Pezoelectrc Layers The mpact respose of a square [PIC 181//foam] s sadwch composte plate of thcess aspect rato a/h = 31 was studed. The plate cossted of two surface attached pezoelectrc layers, each havg a thcess of.2 mm, Graphte/Epoxy faces of 2 mm each ad a 15 mm PVC foam. The plate was mpacted upwards at the cetre of the bottom face by a mass of m =.25 g havg a tal velocty of 1 m/s. A cotact stffess y =1.234e7 N/m [16] was assumed. The HLPT 3-D formulato was used for predctg the global/local mpact respose of the plate. The plate was modelled usg fve dscrete layers throughthcess, amely oe for each materal sublamate, resultg to 46 deflecto-depedet DOF per mode ad a reduced structural system of sze 258x258 state-space for 15x15 modes, whereas soluto of a system of sze 6146x6146 would be requred the case of a full structural system of a ureduced correspodg aalytcal soluto. Fg. 7 shows the predcted global mpact respose of the plate. trasverse dsplacemet (m) mpact force (N) x1-3 5.x x x x x x1-3 electrc potetal (Volt). 1.x1-3 2.x1-3 3.x1-3 4.x1-3 5.x1-3. HLPT 3-D. 1.x1-3 2.x1-3 3.x1-3 4.x1-3 5.x bottom sesory layer bottom face top face mpactor x1-3 2.x1-3 3.x1-3 4.x1-3 5.x1-3 (c) Fg. 7: Global mpact respose of a sadwch composte plate wth pezoceramc layers: mpact force, trasverse dsplacemet, (c) electrc potetal at sesory layers. I Fg. 8 the local through-thcess stress respose s preseted, at pots of maxmum values at the tmestep correspodg to maxmum mpact force. The predcted stress dstrbutos reveal the beeft of mplemetg hgher-order layerwse ematc assumptos, sce pecewse throughthcess varatos up to secod order ca be effcetly captured usg a mmum umber of dscrete layers. The latter s a major advatage compared to lear layerwse lamate theores, whch would requre a large umber of dscrete layers ad thus degrees of freedom order to capture such throughthcess stress profles. O the other had, the lac of explct mposto of shear stress compatblty equatos the case of 8 Copyrght 214 by ASME

9 the HLPT 3-D leads to o-zero terlamar shear stresses at the free faces, although they ted to get to zero. z/h z/h at pot (a/2,a/2) 2 at pot (a/2,a/2) x1 8-5.x x1 7 1.x plae ormal stress (Pa) x1 5 2.x1 5 3.x1 5 4.x1 5 5.x1 5 terlamar shear stress (Pa) 4 at pot (a/2,a) 5 at pot (a,a/2) Fg. 8: Local through-thcess mpact respose of a sadwch composte plate wth pezoceramc layers: -plae stress, terlamar shear stress. Prelmary study o Actve Cotrol A prelmary feasblty study was coducted regardg the actve cotrol of the mpact respose of a [(/9) 2 //(/9) 2 /p P /p A ] composte plate towards mpact force reducto. Two pezoelectrc layers were assumed to be attached to the bottom face of the composte plate studed above. Ther terface was grouded ad the.25 mm thc outer layer was cofgured to act as a actuator, whereas the.125 mm thc er layer provded the sesory sgals. The output of the state-space system of equato (24) was assumed to be the electrc curret desty (curret per ut area), meag that t would be possble to measure the electrc curret flow through practcally short-crcuted pezoelectrc termals of the sesory layer. The plate was mpacted at ts cetre by a mpactor havg a mass of m =.5 g ad a tal velocty v =1 m/s, ad 3x3 bedg vbrato modes were tae to accout. The mpact soluto was programmed Smul order to facltate realzato of actve cotrol a plaed expermetal cofgurato. A proportoal (P) output feedbac cotrol law wth ga p =1 was mplemeted, whereas the desred output value was zero, meag that o curret should flow through the pezoelectrc sesor termals. Fg. 9 llustrates the passve (ope-loop) ad actve (closed-loop) global respose. The curret mpact case yelds mpact chatterg ad maxmum force after the frst aggregato betwee plate ad mpactor. By meas of the actve cotrol, the force s reduced ad the moto of plate ad mpactor becomes smoother. It should be oted that the realstc applcablty of the curret cotrol attempt s stll uder cosderato ad wll be proved by meas of measuremets, plaed to be coducted a expermetal cofgurato developed at the CSL Lab (Fg. 1). trasverse dsplacemet (m) mpact force (N) error (A/m 2 ) x x1-3 1.x1-3 8.x1-4 6.x1-4 4.x1-4 2.x x x x x x1-3 2.x1-3 3.x1-3 4.x1-3 5.x ope loop closed loop. 1.x1-3 2.x1-3 3.x1-3 4.x1-3 5.x1-3 plate, ope loop mpactor, ope loop plate, closed loop mpactor, closed loop x1-3 2.x1-3 3.x1-3 4.x1-3 5.x1-3 closed loop (c) Fg. 9: Actve cotrol of mpact force a composte plate wth pezoceramc layers: mpact force, trasverse dsplacemet, (c) error to desred value of electrc curret desty. 9 Copyrght 214 by ASME

10 [6] Chrstoforou, A. P., Ygt, A. S., ad Majeed, M., 213, Low-Velocty Impact Respose of Structures wth Local Plastc Deformato: Characterzato ad Scalg, ASME J. Comput. Nol. Dy., 8(1), pp [7] Olsso, R., ad McMaus, H. L., 1996, Improved Theory for Cotact Idetato of Sadwch Paels, AIAA J., 34(6), pp [8] Chrstoforou, A. P., Ygt, A. S., 1998, Characterzato of Impact Composte Plates, Comp. Struct., 43(1), pp [9] Su, C. T., ad Che, J. K., 1985, O the Impact of Itally Stressed Composte Lamates, J. Comp. Mater., 19(6), pp [1] Yag, M., ad Qao, P., 25, Hgher-Order Fg. 1: Expermetal cofgurato for mpact testg. Impact Modellg of Sadwch Structures wth Flexble Core, It. J. Solds Struct., 42(2), pp. SUMMARY The preset tegrated hgher-order layerwse mechacs [11] Icard, U., ad Ferrero, L., 29, Impact Aalyss methodology eables predcto of the coupled electromechacal mpact respose of composte ad sadwch Elemet wth Stra Eergy Updatg, Comp. of Sadwch Compostes Based o a Refed Plate composte plates wth pezoelectrc trasducers. Its major Struct., 89(1), pp advatage les ts computatoal effcecy, as t leads to a [12] Tracy, M., ad Chag, F. K., 1998, Idetfyg plate-mpactor system of mmum sze, whereas t may Impacts Composte Plates wth Pezoelectrc accurately capture the local through-thcess respose o the Stra Sesors, Part I: Theory, J. Itel. Mat. Syst. bass of ts sophstcated ematcs. A prelmary study o the Str., 9(11), pp feasblty of actve reducto of mpact force was coducted; [13] Seydel, R, ad Chag, F. K., 21, Impact however, the cotroller applcablty must be expermetally Idetfcato of Stffeed Composte Paels: I. tested. Ths wll be the subject of subsequet wor. System Developmet, Smart Mater. Struct., 1(2), pp ACKNOWLEDGMENTS [14] Par, J., Ha, S., ad Chag, F. K., 29, The research leadg to the results preseted ths wor has receved fudg from the People Programme (Mare Cure Actos) of the Europea Uos Seveth Framewor Programme (FP7/27-213) uder REA grat agreemet o REFERENCES [1] Abrate, S., 1997, Localzed Impact o Sadwch Structures wth Lamated Facgs, ASME Appl. Mech Rev., 5(2), pp [2] Cha, G. B., ad Zhu, S., 211, A Revew of Low- Velocty Impact o Sadwch Structures, J. Mater. Des. Appl., 225, pp [3] Catwell, W. J., ad Morto, J., The mpact resstace of composte materals a revew, Compostes, 22(5), [4] Shvaumar, K. N., Elber W., ad Illg, W., 1985, Predcto of Impact Force ad Durato due to Low-Velocty Impact o Crcular Composte Lamates, ASME J. Appl. Mech., 52(3), pp [5] Olsso, R., 23, Closed Form Predcto of Pea Load ad Delamato Oset uder Small Mass Impact, Comp. Struct. 59(3), Motorg Impact Evets Usg a System- Idetfcato Method, AIAA J., 47(9), pp [15] Lu, Y., ad Chattopadhyay, A., 213, Low- Velocty Impact Damage Motorg of a Sadwch Composte Wg, J. Itel. Mat. Syst. Str., 24(17), pp [16] Saravaos, D. A., ad Chrstoforou, A.P., 22, Impact Respose of Adaptve Pezoelectrc Lamated Plates, AIAA J., 4(1), pp [17] Plagaaos, T. S., ad Saravaos, D. A., 28. Coupled Hgh-Order Layerwse Lamate Theory for Sadwch Composte Plates wth Pezoelectrc Actuators ad Sesors, Proc. 19 th Iteratoal Coferece o Adaptve Structures Techologes (ICAST), Ascoa CH. [18] Dugudj, J., 1988, Smple Expressos for Hgher Vbrato Modes of Uform Euler Beams, AIAA J., 26(8), pp [19] Plagaaos, T. S., ad Papadopoulos E. G., 214, Low-Eergy Impact Respose of Composte ad Sadwch Composte Plates wth Pezoelectrc Sesory Layers, It. J. Solds Struct., 51(14), pp Copyrght 214 by ASME