Modelling of Lo Velocity Impct mge in Lminted Composites J. Lee*, C. Soutis*, P. T. Cutis nd C. Kong** *Aeospce Engineeing, The Univesity of Sheffield, Sheffield S 3J, UK efence Science nd Technology Lbotoy, UK **Aeospce Engineeing, Chosun Univesity, Kngju, South Koe ABSTRACT In this study simple model is developed tht pedicts impct dmge in composite lminte voiding the need of the time-consuming dynmic finite element method (FEM). The nlyticl model uses non-line ppoximtion method (Ryleigh-Ritz) nd the lge deflection plte theoy to pedict the numbe of filed plies nd dmge e in qusi-isotopic composite cicul plte (xisymmetic poblem) due to point lod t its cente. It is ssumed tht the defomtion due to sttic tnsvese lod is simil to tht occued in lo velocity impct. It is found tht the model, despite its simplicity, is in good geement ith FE pedictions nd expeimentl dt fo the deflection of the composite plte nd gives good estimte of the numbe of filed plies due to fibe bekge. The pedicted dmge zone could be used ith fctue mechnics model developed by the second investigto nd co-okes to clculte the compession fte impct stength of such lmintes. This ppoch could sve significnt unning time hen comped to FE solutions.. INTROUCTION The dmge cused to cbon fibe composite stuctues by lo velocity impct, nd the esulting eduction in compession fte impct (CAI) stength hs been ell knon fo mny yes [,] nd is of pticul concen to the eospce industy, both mility nd civil. An impct dmge site in composite lminte contins delmintions, mtix ccking nd fibe fctue. A model to pedict the dmge e tking into ccount ll these fctos is complex nd could tke consideble time nd funding to develop. The poblem could be simplified by mking some ssumptions bout the ntue of the impct dmge. Sevel methods [3-8] hve been developed bsed on the concept of mismtch of bending stiffness, enegy elese te nd bending stin nd use pmete tht is diectly elted to the dmge stte. This dmge pmete cn be clculted using mesuble dt fom the impct event nd/o mteil nd geometic popeties of the lminte. Pedicting dmge using such pmete cn be suitble fo peliminy design nlysis, s the clcultion time could be considebly shote thn 3- detiled FE modelling. Most of these models hve minly focused on pedicting delmintion nd mtix ccking due to lo velocity impct but fe hve delt ith fibe bekge. The im of this ok is to develop n impct dmge model fo peliminy design nlysis tht voids the need of dynmic finite element nlysis of the stuctue. In this ok simple non-line ppoximtion method (Ryleigh-Ritz method) is folloed to pedict the impct dmge size due to fibe bekge. This esult could then be used s bsic input dt fo pedicting the compessive fte impct (CAI) stength ith the Soutis-Fleck model [9]. A commecil pogmme, Mple VI is used to un the pesent model nd FE nlysis togethe ith expeimentl dt to evlute its ccucy.. IMPACT AMAGE MOELLING The impct dmge model is bsed on the concept tht the lo velocity impct esponse is simil to the defomtion due to sttic concentted ltel lod [] nd tht hen plte is subjected to such ltel loding, the expessions fo the deflection of both isotopic nd composite pltes hve the sme fom []. In the model, by neglecting the ineti foces of the plte, the poblem is educed to sttic equivlent one. By consideing degded stiffness in the plte ith incesing lods, idelized dmge ccumultion is modelled using the Ryleigh- Ritz method togethe ith the pinciple of vitul displcement (PV). In ddition, enegy is coelted to foce nd defomtion by consideing the lod-deflection eltionship nd
ssuming tht the mximum stins occu t the mximum deflection hen ll impct kinetic enegy hs been bsobed by elstic defomtion nd dmge. Let us conside clmped cicul plte s shon in Fig.. Tnsvese point lod P is pplied t its cente. The plte is divided into R N egions, hich hve diffeent degded stiffness t ech egion, due to fibe dmge. eflection is finite nd geometicl nonlineity is consideed. In ode to simplify the poblem futhe, the folloing ssumptions e intoduced. Though-thickness she flexibility nd in-plne displcements e neglected. The plte hs n xisymmetic defomtion though thickness. Kichhoff s hypothesis is ssumed to be vlid even in the neighbouhood of the fibe dmge edges. Also, in ode to detemine fibe dmge of dius fo qusi-isotopic plte, it is ssumed tht the mximum filue stin (ε ) long the fibe diection cn be used s dmge citeion. Then by ssuming xisymmetic defomtion, e mke this citeion fit fo ll dil stin (ε ). Hoeve the plte is only qusi isotopic so this is n ppoximtion fo bending stins. Fo membne stins, unifom though thickness, e cn use this citeion fo ε knoing tht it is lso tue fo (ε ) nd (ε 5 ). () (b) Fig.. Cicul plte ith fibe dmge dius,, K, ---, nd N t ech ply The clmped boundy condition hs been chosen since e e modelling locl impct zone, the edge of hich ill be tngentil to the globl displcement. The boundy conditions t the plte boundy nd the plte cente e itten s () d hee nd denote the tnsvese displcements of the undmged plte t its cente nd boundy edge, espectively. Oing to the Kichhoff s hypothesis, the continuity conditions of displcements t the fibe dmge edge,, K, ---, nd N of ech ply e simply itten s k k, d d,, d k k d,, d d () d d d d d d k k hee nd d e the tnsvese displcements of the undmged nd the dmged potion t the k th ply, espectively. In Fig., R to R N denote the egions, hich hve diffeent stiffness. Z k is distnce of the bottom sufce of the k th ply mesued fom the middle plne of the plte. When deiving the elstic enegy expessions nd deling ith fibe dmge es tht e cicul in shpe, it is convenient to descibe thei geomety in tems of pol co-odintes []. Membne stetching enegy (U s ) nd bending enegy (U B ) pe unit e in the pol coodinte system e: (3) U s A cos θ A sin θ ( A A ) sin θ cos θ 8 U B J cos θ K sin θ L sin θ d ()
hee A ij is the lminte extensionl stiffness mtix. Pmetes J, K nd L e s follos: (5) L J K ( ) ( hee ij is the lminte bending stiffness. The totl membne stetching nd bending enegy of cicul plte of dius is obtined by integting ove the e of the plte. The totl intenl enegy is obtined by dding the totl bending enegy eqution to the totl membne stetching enegy eqution. Thus, the totl enegy U T is expessed s d U T A d d d B d ) C d d ( d (8) hee: A π ( 3 3 ), B π ( 6 ) nd C ( 3A 3A A A ) 8 ) (6) (7) π. 3 The ok W done by the extenl foce P is simply given s poduct of foce nd displcement hee the point lod is pplied. W P() (9) In ode to obtin the tnsvese displcement, the Ryleigh-Ritz method pplied to PV (the pinciple of vitul displcement) is dopted to hve n ppoximted solution. The displcement function fo fully clmped boundy conditions is ssumed s [] This function stisfies the boundy condition Eq. () s n dmissible function. When equting the totl intenl enegy Eq. (8) to the extenl enegy Eq. (9) (Equilibium Stte U T W ), e cn poduce genel eqution obtined fo pedicting the finite tnsvese displcement s follos: d A d d d B d d log C d d () P() () Substituting displcement function Eq. () into Eq. (), the totl intenl enegy nd extenl ok using PV e finlly expessed in tems of vitul displcement s: 6 56 3 U T W A C P 56 3 6 C A () 8 8 P hee P is the point lod t the cente of the cicul plte nd is the displcement t. The oot of Eq. () gives the displcement of the elstic cicul plte t its cente nd hen is inseted into displcement function Eq. (), the displcement t ny point long the dil diection is deived. In ode to solve fo, the pogm Mple VI is used [3].. Pediction of Impct mge In ode to pedict dmge e, the mximum filue stin citeion is dopted. Fo simplicity, it is ssumed tht ply dmge occus if ny dil stin vlue (ε ) long the dius T exceeds its ultimte men stin beteen tensile stin vlue ( ε ) nd compessive stin C vlue ( ε ). It is lso ssumed tht the ply dmge hs cicul shpe of dius due to the xisymmetic out-of-plne displcement field. The mximum stin citeion is fomulted belo: ε (3) ε T / C
T / C hee ε denotes dil stin t ech ply nd ε is the ultimte men stin beteen unidiectionl tensile nd compessive filue stin. In the non-line cse, the k th ply dil stin eqution is expessed s the combintion of membne stetching dil stin nd bending dil stin, i.e., k d () ε ε S ε B Z k d d hee ε S is the membne stetching dil stin nd ε B is the bending dil stin. Z k is distnce of the bottom sufce of the k th ply mesued fom the plte s middle plne, Fig. (b).. mge Accumultion Pocess Sttegy fo dmge ccumultion is bsed on the degded stiffness egion of the cicul plte, s shon in Fig. (b). When cetin lod is pplied to the plte, the extent of dmge i.e.,, 3, ---, nd N t ech ply is detemined using the mximum stin citeion Eq. (3) nd lso numbe of filed plies. Bsed on these filed plies, the plte is divided into N egions, hich hve diffeent degded membne nd bending stiffnesses, see Fig. (b). Thus, in Fig. (b), egion R shos the most degded stiffness egion nd egion R N epesents the undmged egion. Consideing Kichhoff s hypothesis, hich is the continuity condition of displcement t the dmge edge, the totl stin enegy Eq. (8) is expessed s follos: U T ( ) A B C d 3 ( ) A B C d L L ( ) A k B k C k d L L ( ) A N B N C N d hee A, B nd C e degded membne nd bending stiffness deduced fom Eq. (8) t egion R hich is situted beteen nd long dius, see Fig. (b). A, B nd C e degded membne nd bending stiffness pmetes deduced fom Eq. (8) t egion R, hich is situted beteen nd 3 long dius. A k, B k nd C k e degded membne nd bending stiffnesses estimted by Eq. (8) t egion R k, hich is situted beteen nd long dius. A N, B N nd C N e sme s elstic membne nd bending stiffnesses of Eq. (8) t egion R N, hich is situted beteen nd plte dius,, long dius becuse thee e no filed plies. In ode to clculte the degded membne nd bending stiffnesses, the mteil popeties of the filed plies t ech egion e eliminted using the clssicl lminte plte theoy. The bove Eq. (5) expesses the degded intenl stin enegy of the dmged plte t cetin pplied lod. Agin, equting Eq. (5) to the extenl ok Eq. (9) nd clculting the equtions like in the elstic plte cse, the degded plte deflection nd the popgted dmge e e obtined. This pocedue is epeted in piece-ise non-line itetion pocedue beyond initil dmge. Finlly, impct enegy is coelted to foce nd defomtion by consideing the lod-deflection eltionship of the dmged plte. 3. MOEL SET-UP FOR VALIATION 3. Finite Element Modelling fo Nonline Sttic nd ynmic Anlysis A finite element nlysis s used to exmine the ccucy of the pesent nlysis fo the tnsvese displcement of n elstic cicul plte fo vious vlues of pplied lod P. The mteil modelled s n IM7/855 multidiectionl 3mm thick cbon fibe composite cicul plte ith ly-up [5/-5/9/] 3s. The though-thickness (Z-diection) displcements ee clculted using FE77 []. A to-dimensionl nonline sttic nd dynmic finite element nlysis ee pefomed. A unifomly distibuted lod of KN on cicul e (dimete: mm) fo the nonline sttic (5)
nlysis is pplied t the cente of the cicul plte (plte dimete: mm) nd the plte is fully clmped t the plte edge. The dimetes fo the loding e nd the cicul plte e the sme s those used in the expeimentl test. In this ppoch, sufce qudiltel 8-node elements ee used fo nonline sttic nd dynmic nlyses ith 396 elements. A typicl finite element mesh fo both nlyses is illustted in Fig.. In the dynmic nlysis, the impcto is modelled s lumped mss nd intects ith the cente of tget plte ith the contct defomtion govened by the ell-knon nonline Hetzin contct ls [5,6]. Fo the pesent model, the constnts fo indenttion ls used fo cbon/epoxy plte fully clmped in Ref. 5 ee pplied. They e k.3 x 6 N/cm.5, n.5, q.5, β.9 nd α p.7 x cm (n, q, β e dimensionless). Fig.. Finite element mesh fo nonline sttic nd dynmic nlysis of the cicul plte In ode to compe fibe dmge development clculted by the pesent model using the dmge ccumultion pocess, the pogessive filue nlysis poposed by Chng [7] ithin FE77 s used. 3. mge Pediction Model Fibe dmge modelling t ply level in mm nd 3mm thick composite cicul pltes (Cytec Fibeite IM7/855) s pefomed. The pltes ee mm in dimete, fully clmped nd the mteil popeties nd stcking sequences e given belo: Mteil Popeties: E 55 GP, E GP, ν.3, G.5 Gp, Ultimte tensile filue stin, ε T.5 %, Ultimte compessive filue stin, ε C.%, Stcking Sequences: [5/-5//9] s nd [5/-5//9] 3s. It is ssumed tht the fist ply (top ply) hs ledy filed due to the contct foce of the impcto. The dmged egion is egded s being equivlent to the impcto dius, 6mm. Estimted dil tensile nd compessive stins (ε ) e comped to the filue vlue of ε T nd ε C, espectively fo detemining the extent of the fibe dmge t ech ply.. RESULTS The theoeticl esults fo undmged plte deflection, stins, dmge e nd dmged plte deflection pedicted fom the pesent model nd FE77 e comped ith the expeimentl esults [8] mesued fom qusi-sttic nd impct tests fo 3mm thick IM7/855 [5/-5//9] 3s cbon fibe composite cicul plte. Fo the nonline dynmic nlysis (FE77), the lminte is impcted by 5.5kg impcto t velocity of.5 m/s, geneting n impct enegy of 7.8 J. Also, fo the nonline dynmic pogessive filue nlysis, the lminte is loded ith the sme impct mss t diffeent velocities, geneting impct enegy levels of beteen 5 nd 6 J.
. Un-degded eflection Compison beteen Model, FE nd Expeimentl Results In Fig. 3 compisons of theoeticl nd expeimentl foce-displcement esults e shon; the displcement is mesued t the cente of the plte. The pedictions fom the ppoximte solution (pesent model) nd numeicl solution fo nonline sttic nd dynmic nlysis (FE77) e in excellent geement. Thee is, hoeve, smll discepncy beteen pedictions nd mesuements in the figue. This might be cused by diffeence beteen the boundy condition pescibed in the theoeticl simultion nd tht elized in the expeiment. Fig. 3. Foce-centl deflection cuves fo 3mm thick IM7/855 [5/-5//9] 3s cicul plte Fo clmped lminte, in the theoeticl clcultion, the boundies e stictly estined fom tnsltion nd ottion. Hoeve, in the expeiment, it is not possible to completely pevent the lminte edges fom some movement. Thus, the theoeticl nlysis tends to povide n ove-estimted flexul stiffness comped to the expeiment, hence esulting in highe pek foce [9,]. The CPU time spent on the sttic nlysis by using the pesent model is educed immensely, poducing n ccute pediction fo the mximum impct foce nd deflection hen comped ith the dynmic finite element computtion. Fo this lminte, the CPU time fo the sttic nlysis is only five seconds comped to 6 minutes fo the dynmic FE computtion.. Stin Compison Beteen Model nd Expeimentl Results Fig. () nd (b) sho the nlyticl nd mesued sttic bending stins t the top nd bottom sufce of 3mm thick IM7/855 [5/-5//9] 3s cicul plte. Stin guges ee ttched on the top sufce t mm nd mm y fom the cente nd the bottom fce, t the cente nd mm y. () Bending stin t the top sufce (b) Bending stin t the bottom sufce Fig.. Compison beteen pedicted nd mesued stins: () t the top sufce nd (b) bottom sufce of 3mm thick IM7/855 [5/-5//9] 3s cicul plte.
In the pesent model, the nonline bending stins ee clculted fom Eq. () tht ccounts fo membne stetching nd bending stins. Tensile stin is mesued nd pedicted t mm y fom cente of the top sufce. This is cused by the membne stetching effect y fom the cente of the top sufce. In genel the nlyticl nd expeimentl stins e in good geement..3 mge Ae Pedictions in 3mm Thick Cicul Plte Fo the pediction of the dmge e popety degdtion model needs to be dopted. In the pesent nlysis, it is ssumed tht the ovell elstic popeties (E nd E, G nd ν ) of the dmged egion in ech filed ply e zeo. It is lso ssumed tht the popeties of the undmged e of the filed ply e educed to.5 E nd. (E, G nd ν ) due to stiffness cused by the fibe bekge in the neighbouing egion. The educed ovell elstic moduli of the dmged egion in the plte e estimted by the clssicl lminte plte theoy. In this dmge nlysis the mximum pplied lod is 8 kn nd it is incesed by kn incements ith ech itetion. Fibe dmge is initited t the plte cente t n pplied lod of 7 kn nd popgtes to the plte edge ith incesing lod. mge is found in plies (loded top ply),, 3,, 3 nd. The centl deflection of the degded plte is. mm nd the educed stiffness popeties clculted by the lminte plte theoy e: E x 39.7 GP, E y.6 GP, G xy 5. GP, ν xy.9 nd ν yx.3. (egded Elstic Moduli: E x 9.8 GP, E y 9.8 GP, G xy 7.5 GP, ν xy.3 nd ν yx.3) () Though-thickness (Z- plne) fibe dmge (b) Centl deflection vesus pplied foce Fig. 5. () Though-thickness (Z- plne) fibe dmge in 3mm thick [5/-5//9] 3s cicul plte; pplied lod 8 kn nd (b) centl deflection vesus pplied foce. Fig. 5() illusttes fibe bekge t ech ply fo n pplied lod of 8 kn. With incesing pplied lod the plte deflection t its cente inceses, Fig. 5(b) due to degded stiffness popeties. In Fig. 5(b) the centl deflection of the undmged nd dmged cicul plte fo diffeent pplied lods is pesented. The cuves stt to divet t lod of 7 kn nd the diffeence beteen the cuves becomes significnt t highe lods due to intoduced fibe dmge. Fig. 6. Pek contct foce ginst impct enegy, pedicted nd mesued vlues, fo clmped IM7/855 [5/-5//9] 3s cicul pltes of mm dimete.
The e unde the foce-deflection cuve fo the dmged plte cn be coelted to impct enegy by the folloing eqution: Impct Enegy Wok mv F δ d (6) Fig. 6 shos the qusi-sttic esponse obtined fom the pesent model using the dmge ccumultion pocess descibed elie (Eq. (5)) nd comped to the qusi-sttic nd impct test esults. The pek contct foce is plotted s function of impct enegy; the enegy bsobed being clculted by integting the e unde the lod-displcement cuve, Fig. 5(b) on the bsis of Eq. (6). It cn be seen tht t given enegy the nlyticlly pedicted pek foce is in good geement the mesued sttic nd impct test esults.. mge Ae Compison beteen Model, FE nd Expeimentl Results The mximum fibe bekge lengths pedicted fom the pesent simple model nd FE77 (nonline sttic nd dynmic nlyses) nd mesued fom impct tests e plotted vesus the pek foce nd impct enegy in Fig. 7() nd (b), espectively. Fom these figues, the pek foce nd enegy equied to initite fibe bekge cn be obtined, nmely 7 kn nd 8 J fom the simple nlysis, 9 kn nd J fom the nonline sttic FE nlysis, 6. kn nd 5 J fom the nonline dynmic FE nlysis nd 9.7 kn nd 7.8 J fom the impct test. The fibe bekge length pedicted fom the pesent model is shply incesed fte n pplied lod of bout kn, esulting in significnt discepncy beteen the model nd the FE77 esults. This is cused by the onset of fibe dmge cused by the membne stetching effect, Fig. 5(). The pedicted fibe bekge lengths fom the finite element nlyses e in good geement. The limited impct test esults gee esonbly ell ith the theoy but futhe testing is equied. () Fibe dmge length vesus pek foce (b) Fibe dmge length vesus impct enegy Fig. 7. Impct dmge in n IM7/855 ([5/-5//9] 3s cicul plte. () Fibe dmge length vesus pek foce nd (b) Fibe dmge length vesus impct enegy 5. ISCUSSION uing impct, the dmge of plte consists minly of tnsvese ccking, delmintion, nd fibe bekge due to both concentted foce tht induces high though-thickness stesses nd lge deflections, hich induce bending nd membne stetching hen the deflection exceeds the plte thickness. The cuent model only tkes into ccount fibe bekge. Fibe bekge lengths ee pedicted successfully t ech ply bsed on the mximum stin filue citeion. The pedicted fibe dmge lengths e in esonble geement ith the numeicl esults (FE77) nd impct test esults. Fom the esults of the filue pttens of the 3mm thick cicul pltes clculted fom the pesent model, it is identified tht dmge is initited fom the plte cente but then s the lod is incesed ne dmge e initites hich foms n nnulus. The inne nd oute dmge ing incese in e s the lod inceses, Fig. 5(). All these filue pttens cn be explined by the mount tht the plte deflects due to ltel pplied lod. The non-line stin eqution, Eq. (), consists of membne stetching stin tem nd bending stin tem. The bending
stin tem is impotnt in the line cse hee the plte deflection does not exceed the plte thickness. In the 3mm thick plte ith n initil dmge lod of 7 kn the deflection is.mm, hich does not exceed the plte thickness nd the ovell stin is due to bending (line cse). As the pplied lod inceses the plte deflection becomes lge thn the plte thickness nd the totl stin is dominted by the membne stin component (stetching). This is simil to the behviou of the mm thin plte hee t lod of.5 kn extensive locl dmge occus nd the deflection of 6.75mm exceeds the plte thickness by moe thn six times. The membne stetching effect, Fig. 8(), domintes the ovell nonline stin. In Fig. 8() dil bending stin distibution, dil membne stin distibution nd totl dil stin distibution e plotted ginst the plte dius ( 5mm) developed in the top nd bottom plies hen the pplied lod is kn. Significnt tensile stins due to stetching e developed in the top ply futhe y fom its cente tht cn intoduce significnt tensile fibe bekge. Fo plte tht is ltelly defomed the deflection is invesely popotionl to bending stiffness (bending stiffness pidly inceses becuse it is function of t 3 hee t is the plte thickness)). Fig. 8(b) illusttes the dil stin distibutions long the dius ( 5mm) t the top nd bottom plies of the 3mm thick plte t n pplied lod of 8kN. Fig. 8(b) indictes tht the bending stins e quite significnt ound the centl egion hile membne stins e eltively smll. The stetching effect get bigge t highe lods, Fig. (), cusing tensile dmge in the fom of fibe bekge y fom the cente (7-mm). () mm thick cicul plte, pplied lod kn (b) 3mm thick cicul plte, pplied lod 8 kn Fig. 8. Rdil stin distibutions long dius t the top nd bottom ply of the mm thick cicul plte () nd 3mm thick cicul plte (b) It ppes tht the cuent nlyticl solution cn sve significnt time in pedicting the defomtion nd fibe dmge e in composite pltes unde lo velocity impct, comped ith numeicl solutions (FE). The deflection of the cicul plte shon in Fig. 5(b) s clculted by the cuent model in five seconds comped to minutes tken by nonline FE sttic nlysis nd 6 minutes fo the nonline FE dynmic nlysis. 6. CONCLUING REMARKS In ode to pedict the numbe of filed plies nd the totl fibe dmge e intoduced in n xisymmetic composite cicul plte by tnsvese lod t its cente, n nlyticl model s developed using simple non-line ppoximtion method (Ryleigh-Ritz method) nd the pinciple of vitul ok. As filue pediction sttegy fo exmining the fibe dmge t ech ply, the mximum stin filue citeion s dopted togethe ith the clssicl lminte plte theoy. The pesent model gees ell ith the finite element nlysis fo the deflection of the undmged elstic plte nd pedicts successfully the numbe of filed plies nd dmge e due to fibe bekge in the shpe of cicle. Fom the theoeticl nlysis, fibe dmge t ech ply of the plte shos to types of filue pttens. When the plte is quite thin (mm
thick in the cuent nlysis), fibe dmge s poduced by the effect of geometicl nonlineity, i.e. membne stetching effect due to lge deflection nd the mximum non-line effect occued t cetin inne egion of ech ply displying n ovell nnul dmge shpe. Fo thick cicul plte (3mm thick), fibe dmge initited fom the cente of ech ply nd popgted to the plte edge. In this cse, the dmge s dominted by bending effect since the plte deflection did not exceed the plte thickness. Hoeve, t lge pplied lod, hen the ltel deflection exceeded the plte thickness, fibe dmge stted to initite t cetin inne egion fom the top ply like the cse of the mm thick plte. Bsed on the numbe of filed plies in the plte, degded elstic moduli ee obtined using the clssic lminte plte theoy nd the ovell dmge e s estimted. This could be used ith fctue mechnics bsed model [] to pedict the compessive fte impct (CAI) stength of ny symmetic composite lminte. In ddition, the simplified model pesented hee cn sve significnt unning time, comped ith FE non-line sttic o dynmic solutions. ACKNOWLEGEMENT This ok s cied ith the finncil suppot of the Stuctul Mteils Cente, QinetiQ, Fnboough, UK. The uthos e gteful fo mny useful discussions ith Pofesso G. A. O. vies of the eptment of Aeonutics, Impeil College London, UK. Refeences. Whitehed R.S., ICAF Ntionl Revie, Pis, My, 985. pp -6. Geszczuck L.B., mge in Composite Pnels due to Lo Velocity Impct, Impct ynmics, Ed. Zuks Z.A. J. Wiley. 98. 3. Liu,., Impct-induced elmintion A Vie of Bending Stiffness Mismtching, Jounl of Composite Mteils, 988, (July), pp. 67-69.. Fuoss, E., Stznicky, P. V. nd Poon, C., Pediction of Impct Induced elmintion in Composite Pltes, Advnced Composite Lettes, 99, 3 (3), pp. 93-96. 5. Mui, G. B. nd Guynn, E. G., Anlysis of elmintion Goth fom Mtix Ccks in Lmintes Subjected to Bending Lods, Composite mteils: Testing nd esign (Eighth Confeence), ASTM STP 97, 988, pp. 3-339. 6. Suemsu, H. nd Mjim O., Multiple elmintions nd Thei Seveity in Cicul Axisymmetic Pltes Subjected to Tnsvese Loding, Jounl of Composite Mteils, 996, 3 (), pp. -53. 7. Suemsu, H. nd Mjim O., Multiple elmintions nd Thei Seveity in Nonline Cicul Pltes Subjected to Concentted Loding, Jounl of Composite Mteils, 998, 3 (), pp. 3-. 8. Fuoss, E., Stznicky, P. V. nd Poon, C., Effects of Stcking Sequence on The Impct Resistnce in composite Lmintes. Pt : Pediction Method, Composite Stuctues, 998, (), pp. 77-86. 9. Soutis, C. nd Fleck, N. A., Sttic Compession Filue of Cbon Fibe T8/9C Composite Plte ith Single Hole, Jounl of Composite Mteils, 99, (My), pp. 536-558.. Sjobolm P. O., J. T. Htness nd T. M. Codell, On Lo Velocity Impct Testing of Composite Mteils, Jounl of Composite Mteils, 988, (Jn.), pp. 3-5.. S. R. Lee nd C. T. Sun, On The Appent Bending Isotopy in Clmped Elliptic Composite Lmintes, Jounl of Composite Mteils, 995, 9 (), pp. 6-6.. Timoshenko nd Woinosky-Kiege, Theoy of Pltes nd Shells, McG-Hill Book Compny, nd Ed., 959. 3. Mongn, M. B., Mple 6: Pogmming Guide, Wteloo, Ont., Wteloo Mple,.. Hitchings., FE77 Use Mnul Vesion.9, Impeil College, Aeonutics, 995. 5. Khoo, S. W., Lo Velocity Impct of Composite Stuctues, Phd Thesis, Univesity of London, ecembe, 99 6. Wtson, S. A., The Modelling of Impct mge in Kevl-Reinfoced Epoxy Composite Stuctues, Ph Thesis, Univesity of London, Novembe, 99. 7. Chng, F. K. nd Chng, K. Y., A Pogessive mge Model fo Lminted Composites Contining Stess Concenttions, Jounl of Composite Mteils, 987, (Sept.), pp. 83-85. 8. Lee, J., Compessive Behviou of Composite Lmintes befoe nd fte Lo Velocity Impct, Ph Thesis, Impeil College London, UK, 3. 9. vies, G. A. O. nd Zhng, X., Impct mge Pediction in Cbon Composite Stuctues, Intentionl Jounl of Impct Engineeing, 995, 6 (), pp. 9-7.. Liou, W. J., Tseng, C. I. nd Cho, L. P., Stess Anlysis of Lminted E-glss Epoxy composite Pltes Subject to Impct ynmic Loding, Computes & Stuctues, 996, 6 (), pp. -.. Soutis C. nd Cutis P. T., Pediction of The Post-Impct Compessive Stength of CFRP Lminted Composites, RA/SMC Tech. Repot. 95. Nov. 995.