Geometrically Non-Linear Analysis of Composite Laminated Plates Subjected to Low-Velocity Impact

Transcription

1 Iteratoal Joural of Composte Materals, (6B): -9 DOI:.59/s.cmaterals..4 Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact Smo Wag,*, Xuq Zhag, Ygshu Zhag, Che yu Zhag Departmet of Aeroautcal ad Automotve Egeerg, Loughborough Uversty, Loughborough LE TU, UK Tayua Uversty of Techology, 79 West Ygze Street, Tayua 4, Shax, Cha Doosa Babcock Eergy Lmted, Porterfeld Road, Refrew PA4 8DJ, UK Abstract A B-sple fte strp model s developed the cotext of a layer-wse plate theory for aalysg the geometrcally o-lear traset respose of lamated composte plates subjected to trasverse low-velocty mpact. To smplfy the complcated cotact aalyss, a Hertz-type cotact law has bee corporated to the fte strp (FS) model for accoutg for the cotact behavour. The model cludes the geometrcal o-learty through use of vo Karma's o-lear stra-dsplacemet relatoshp. The resultg o-lear dyamc problem s solved usg the Newmark tme-steppg scheme together wth Newto-Raphso terato. Several umercal applcatos are descrbed ad a close comparso s foud betwee the results calculated through the preset model ad the exstg aalytcal ad expermetal results. Keywords Lamated Composte, Low-velocty Impact, Geometrcal o-learty, Traset, B-sple Fte St rp. Itroducto Lamated composte structures have played a mportat role arcraft, vehcles ad may other demadg applcatos because of ther hgh stregth-to-weght ad hgh stffess-to-weght ratos. As t s well kow, however, these structures are very susceptble to low-velocty mpact damage, whch ca be troduced durg maufacture ad servce. Impact duced damage ca have a sgfcat effect o the stregth, stablty ad relablty of the structures. Therefore, great cocer has bee receved o the low-velocty mpact of the structures[]. For uderstadg low-velocty mpact respose of composte structures, several aalytcal approaches have bee used by a umber of researchers. These approaches vary from smple mathematcal models, such as sprg-mass models ad eergy-balace models, to more complcated dyamc aalyss of the mpact evets. I sprg-mass models, the structure s represeted by a assemblage of sprgs ad masses. The low-velocty mpact evet s smulated by a dscrete system wth a few degrees of freedom. These kds of model are geerally used to estmate the mpact force hstory for the mpact evets whch the structures behave quas-statcally. Capro et al[] used a sgle degree of freedom model to predct elastc mpact respose of a small glass cloth-polyester pael due to a drop * Correspodg author: S.Wag@lboro.ac.uk (Smo Wag) Publshed ole at Copyrght Scetfc & Academc Publshg. All Rghts Reserved weght mpact. I ther model, the pael s represeted as a lear sprg of stffess k, whch correspods to the statc rgdty of the pael at the mpact pot. Smlar models are also used by others[,4,5]. Sh vaku mar et al[6] suggested a two degree of freedom model, whch the mpactor ad the plate are treated as two masses, ad four sprgs are used to represet the cotact stffess ad the bedg, shear ad memb rae rg dtes of the plate. Bucell et al[7] emp loyed the same model to study the respose of composte plates to mpacts. I the two-degrees of freedom model of Sjoblom et al[8], the plate ad cotact rgdty s modelled separately usg two sprgs. Toh et al[9,] also used a two-degrees of freedom model to predct the mpact force. Gog ad La m[] proposed a mproved two-degrees of freedom sprg-mass model by mplemetg the structural dampg to determe the cotact force betwee the target ad strker durg mpact. Aother kd of frequetly used smple models for the mpact evets whch the structures behave quas-statcally are eergy-balace models[6,]. I these models, t s assumed that the velocty of the mpactor becomes zero whe the structure reaches ts ma xmu m deflecto. Eergy-balace models allow drect estmate of the maxmum cotact force, cosderg the coservato of eergy wthout havg to compute the etre tme hstory. Whlst they are useful uderstadg the ma features of low-velocty mpact evets, the above-metoed smple models are adequate accoutg for the dyamc ature of the composte structures uder the low-velocty mpact. For uderstadg the tato ad propagato of the low-velocty mpact damage as well as the teracto betwee the damage ad plate dyamc respose, dyamc

2 4 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact aalyss s ofte eeded for accurate predcto of low-velocty mpact behavour of the composte lamates. I addto, t s the case that for may stuatos the resposes of the structures low-velocty mpact evets caot be vewed as quas-statc oes. Therefore, t s very mportat to have a sght to dyamc respose of the mpacted structures. Dyamc aalyss of low-velocty mpact respose of lamated compostes geerally volves the global dyamc respose of the lamates ad the local cotact betwee the mpactor ad the structure. To model the cotact, oe of the possbltes s based o a formulato of the jot cotact problem for the system of mpactor-target. Ths s tghtly coected wth the partcular umercal method to be appled, for stace, fte elemet method (FEM), fte dfferece method, or other method based o some varatoal prcples. I FEM aalyss of the mpact cotact problem, cotact elemets are used to model the lamate ad the mpactor[,4,5,6,7,8,9]. The soluto of the cotact problem may be acheved usg several cotact algorthms such as the Lagrage multpler method[] ad the pealty method[9]. The Lagrage multpler method has the advatage of eforcg the exact costrats, but duces addtoal parameters whch substatally elarge the overall sze of the equatos to be solved. The pealty method s relatvely smple ad does ot requre addtoal equatos. Wth cotact algorthms, the cotact state s detected at each tme step ad the cotact costrats are mposed to the cotactg odes/elemets oce cotact occurs. Sce cotact s a o-lear problem, full aalyss of the mpact cotact for the system of mpactor-target s a complex ad tme-cosumg procedure. Hece, smplfcato of the cotact problem s ofte made through use of the cotact laws mpact aalyss of composte lamates. Yag ad Su[] have proposed a cotact law for cotact betwee a sphere ad a composte lamate based o statc detato tests. Ths cotact law accouts for permaet detato after uloadg cycles ad uses dfferet relatoshps betwee the cotact force ad the detato for loadg, uloadg ad reloadg processes. Ta ad Su[] further studed the uloadg ad reloadg process ad proposed a modfed verso of the cotact law. Expermetal observatos[,] cofrm that loadg rate effects durg low-velocty mpact of compostes are sgfcat. Ths suggests that statcally determed cotact laws may be used for mpact aalyss of composte structures. The above-metoed cotact laws developed by Su ad hs co-workers have gaed extesve use aalysg the dyamc respose of composte lamates to mpact. It should be oted that these cotact laws do ot gve the cotact stress dstrbuto uder the detor. Determato of the cotact stress dstrbuto has to appeal to aalytcal method. I addto, these cotact laws have ot accouted for damage effect o fo rce-detato relatoshp. Wu ad Shyu[4] showed that the cotact pheomeo s dfferet small ad large detato stages due to occurrece of lamate damage. I the small detato stage where the plate s tact, the chage of lamate stackg sequece has a sgfcat effect o the force-detato relatoshp. Beyod the small detato stage, damage occurs ad the detato sprg s stffeed as a result of matrx crack ad delamato damage of the lamates. The cotact behavour durg low-velocty mpact s very smlar to that a statc test. Because of the complexty of the mpact problem, closed-form exact solutos exst oly for smple cases. I most stuatos, approxmate aalytcal ad/or umercal methods have to be adopted. I cojucto wth cotact laws, dyamc aalyses of low-velocty mpact of composte lamates have bee carred out aalytcally by a umber of researchers. Su ad Chattopadhyay[5] studed a smp ly-supported orthotropc plate subjected to cetral mpact usg the frst order shear deformable plate theory (FSDPT) developed by Whtey ad Pagao[6]. Dobys[7] used the same method but replaced the cocetrated mpact load by a uform patch pressure to avod trasverse shear force sgularty at the cotact pot. Qa ad Swaso[8] examed two soluto techques whch were based o seres expaso. Oe of them was based o a Raylegh-Rtz approach wth umercal tegrato tme, ad the other was a aalytcal approach usg Laplace trasformato of the goverg dfferetal equatos, but requrg a learsato of the cotact law. Carvalho ad Soares[9] studed a smp ly-supported composte plate subjected to a mpact load utlsg the techque of Fourer seres expaso for the soluto of the dyamc plate equatos. Comparso was made betwee the umercal soluto techque based o Newmark tegrato method ad the aalytcal formulato usg the Laplace trasform techque. Perso ad Vazr[] preseted a Fourer seres soluto that retas the frequeces assocated wth rotary erta effects. A double Fourer seres expaso ad the Tmosheko small-cremet method were used by Ambur et al[,] for determg the traset respose of smply supported, rectagular lamated plates subjected to mpact loads. Hetzer[] studed the teracto of a mpactor ad a clamped, asotropc plate at low-veloctes by assumg a seres expaso for the plate deflecto. Large deflecto was take to accout ad the cotact law was used. It s well kow that FEM s oe of the most powerful tools of soluto structural aalyss. Besdes D or D fte elemet aalyses of the jot mpactor-target system[-9,4-6], may researchers have employed the FEM cojucto wth cotact laws for the aalyses of composte lamates due to low-velocty mpact. Su ad hs collaborators[,7,8] used statc detato laws ad FEM based o FSDPT to aalyse the mpact resposes of composte lamates. The tally stressed composte lamates were studed [7] ad dyamc large deflecto was cosdered [8]. Wu ad Chag[9] developed a D fte elemet code for traset dyamc aalyss of lamated composte plates due to trasverse foreg object mpact whch a cotact law s corporated. Combed wth falure crtera ths D fte elemet code has bee used

3 Iteratoal Joural of Composte Materals, (6B): -9 5 by Wu ad Sprger[4], Cho ad Chag[4], F ad Sprger[4,4], ad Scarpo et al[44] for falure aalyses of mpacted lamates. Cho ad Chag[45-47] developed a D FEM for studyg the mpact damage of lamated composte beam resultg from the le-loadg mpact. Daves ad Zhag et al[48-5] vestgated mpact-duced damage usg FEM based o Mdl s plate theory combed wth a cotact law[]. The FEM based o FSDPT ad a Hertza-type detato law was employed by Hu et al[6] for aalysg the traset respose of composte lamates wth mu ltple delamatos subjected to low-velocty mpact. It appears that the majorty of prevous aalytcal ad umercal aalyses are lmted to s mall-deflecto behavour. Although such lear aalyses are practcal ad useful for may mpact problems of composte structures, t s ofte the case that the geometrc o-learty (GNL) has very sgfcat effects o the mpact respose. I ths paper, a layer-wse B-sple fte strp model developed [5] s exteded through troducto of tme dmeso for the GNL traset aalyss of lamated composte plates subjected to trasverse low-velocty mpact of sphercal object wth the cotext of a layer-wse plate theory[5]. It s oted that [54,55] a B-sple FSM has bee used for traset aalyss of lamated composte plates subjected to dyamc loads based o the frst order shear deformable plate theory (FSDPT). I the preset study, the geometrcal o-learty s take to cosderato by use of vo Karma's o-lear stra-dsplacemet relatoshp. The dyamc problem s solved usg the Newma rk t me -steppg scheme the tme doma ad the soluto of the resultg o-lear equatos s sought wth Newto-Raphso terato. The mpact problem the small-deflecto regme s brefly dscussed as well. Several umercal applcatos are preseted. It s oted that materal damage ad delamato are ot dealt wth the curret work.. Problem Descrpto ad Fudametals A m, v h x B F gure. T rasverse mpact o a rect agular plat e x x Cosder a lamated rectagular plate wth arbtrary lay-ups subjected to trasverse low-velocty mpact of a sphercal object as show Fgure. The mpactor s of radus r s, mass m ad tal velocty v. It s assumed that the vbratos of the elastc mpactor ca be eglected. A orthogoal Cartesa co-ordate system x (=,,) s used ths paper... Basc Plate Equat os I the preset study, the layer-wse plate theory proposed by Reddy[5] s adopted for represetato of dsplacemet behavour the plate. Through the thckess drecto, the lamated plate s dvded to a desred umber, N, of umercal layers, whch ca be less tha, equal to, or greater tha the umber of physcal layers. The assumed dsplacemets ( µ α, u ) at a geeral pot of the lamate, wth t as the tme dmeso, take the form as what follows. uα ( xβ, x, t) = uα ( xβ, t) + Ψ ( x u ( x, x, t) = u ( x, t) β β ) u α ( x β, t) where the usual Cartesa dcal otato s adopted. The Greek subscrpts, whch take values ad, ad subscrpt refer to x, x, x drectos, respectvely. Repeated dces mply the summato coveto. Superscrpt, raged from to N, s related to the odes through the thckess. α ad u deote the three dsplacemet compoets of the referece plae (x =). Ψ are pecewse cotuous fuctos ad defed as ~ ~ k Ψ = R ~ Ψ () k where s raged from to N ad k ~ from to N+. ~ k ~ Ψ s defed terms of lear Lagrage terpolato fuctos k x x k k N+ [, ] [, ] k k x x x x x x x () k ( ) k Ψ x = + x x k k + N+ x [ x, x ] [ x, x ] k + k x x N + x [ x, x ] ad R ~ k s defed as = k < m or = k m m m+ R = x / x k = k (4) = m others Here m s the umber of the layer at whch the referece plae s. The resultg dsplacemet cofgurato s show Fgure.. () u

4 6 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact uα = uα + Ψ uα u = u x u m N N x N x N + N-th layer h/ x m+ m x m-th layer x α h/ x x st layer x m m ( u m+ α x F gure. I-plae dsplacemet cofgurato of layerwse theory ) The expressos of the stra-dsplacemet relatoshp ca be gve by substtuto of the dsplacemet feld Eq. the Gree's expressos for -plae o-lear stras ad eglectg hgher order terms a maer cosstet wth vo Karma's assumptos. They are: εαβ = εαβ + Ψ εαβ (5) γα = γα + Ψ,γα where ε αβ = ( u α, β + uβ, α ) + u, αu, β ε αβ = ( u α, β + u β, α ) (6) γα = u, α γα = uα The stra eergy of the lamate cosdered ca be obtaed as a tegral over the referece plae area S. The result s: U = ( N N Q Q ) dxdx S αβεαβ + αβεαβ + α γα + α γ (7) α where k Nαβ A B αβγω αβγω εγω = k k Nαβ B αβγω D ε αβγω γω k Q α Aαβ Bαβ γ β = k k (8) Q α B αβ D γ αβ β N αβ, N αβ, Q α, ad Q α are geeralsed stresses per ut legth. A αβγω, B αβγω, etc, are the stffess coeffcets of the lamate whch are defed as ( A k h k αβγω, B αβγω, D αβγω ) = Q (,, ) dx h αβγω Ψ ΨΨ k h k ( Aαβ, Bαβ, Dαβ) = Q (,,,,,) dx h α β Ψ Ψ Ψ (9) where Q αβγω ad Q α β deote the trasformed reduced stffess coeffcets. Q αβγω possess symmetry the dces α ad β, γ ad ω, ad the pars of αβ ad γω. The smlar symmetrcal property also apples to Q α β... Fte Strp Approx mato The whole plate s modelled wth a umber of fte strps alog the crosswse x -drecto. Each strp s further parttoed logtudally to q sectos. A typcal dvdual fte strp elemet (quadratc strp) s show Fgure. Over the strp each of the fudametal

5 Iteratoal Joural of Composte Materals, (6B): -9 7 dsplacemet quattes, u, ad ca be approxmated as a fucto of multplcatve type, whch q+k B-sple fuctos of degree k are used the logtudal x -drecto ad smple polyomals of degree the crosswse x-drecto. Mathematcally, the fudametal dsplacemet quattes are gve as u u u u α = N I Φ Φ Φ d d d Φ d I α I I I () Here d α, d ad d α are colum matrces of geeralsed dsplacemet parameters assocated wth, u ad, respectvely. The row matrces Φ ad Φ are modfed B-sple fucto bases of order k ad k- the logtudal x -drecto. N I are polyomal fuctos of degree the crosswse x -drecto ad here Lagraga shape fuctos are used. The superscrpt, raged from to N, s related to the odes through the thckess. The captal superscrpt I, ragg from to, deotes the umber of a referece le the crosswse x -drecto of the strp. N,u A x x b uα = uα + Ψ uα u = u x F gure. A typcal layerwse fte strp elemet. Goverg Equatos of Moto I the absece of dampg, the goverg equatos of the plate moto ca be gve through use of Hamlto's prcple terms of stra eergy U, ketc eergy T ad potetal eergy W as M d + Kd = Fc + f () where M ad K deote the mass matrx ad the effectve stffess matrx, respectvely, ad K are fuctos of d, d ad h f are colum matrces of geeralsed dsplacemet parameters ad geeralsed force due to appled loadg, F s mpact cotact force, c s a colum matrx, resultg from the product of the polyomals the x -drecto ad the B-sple fuctos the x -drecto at the mpactg pot of the plate, ad c T d defes the deflecto of the referece plae of the plate at mpact pot. The relatoshp betwee the cotact force F ad the detato depth α s assumed as[] / F = k c α whe loadg (a) F 5 / = F m [( α α ) /( α m α )] / = F m [( α α ) /( α m α )] F whe uloadg(b) whe reloadg (c) where k c s a costat. F m deotes the maxmum cotact force just before uloadg, α m s the detato correspodg to F m, ad α s the permaet detato durg the loadg/uloadg cycle. k c ca be determed by expermet or smply calculated from the modfed Hertz cotact coeffcet proposed by Su[] as 4 r [( ν ) / E + / ] s s s kc = E () here r s, ν s ad E s are the radus, the Posso's rato ad the Youg's modulus of the mpactor, respectvely, ad E s the trasverse modulus of elastcty of the plate. Eq. ca be expressed a geeral form as q F = α α ) (4) k s ( where k s ad q are the cotact coeffcet ad a costat, respectvely. It s otced that for the loadg, uloadg ad reloadg process α, k s ad q may be dfferet. The detato depth α s the dfferece of the dsplacemet of the mpactor ad the deflecto of the referece plae of the plate at mpact pot, so that where / ( / k ) q + α = c T s v t s d m F (5) s = t t Fdtdt ( = α) (6) Eqs. ad 5 together wth approprate tal codtos defe the preset mpact dyamc problem. It s here assumed that d, d add are kow at tme t = ad F, s, s are equal to zero at tme t =. 4. Problem Soluto I the tme doma the mpact problem defed Eqs. ad 5 ca be solved wth a few of dfferet approaches. I the preset vestgato the soluto of the problem s sought usg the popular Newmark tme tegral algorthm. Suppose that the tme perod cocered s dvded to a umber of equal tme tervals Δ t ad cosder the satsfacto of Eqs. ad 5 at tme () t ad wrte Md + Kd = F c + f (7a)

6 8 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact / q T ( F / k s ) + α = v ( + ) s c d wth approxmatos for d d s s = d = d = s = s m d, d + d + ( β ) d + ( β ) d + β d (7b) s ad s as + + s + ( β ) F + + ( β ) F + β F β d β F (8a) (8b) where the parameters β ad β ca be chose as such that good approxmato propertes to the algorthm are yelded. I ths study the costat-average-accelerato verso of Newmark method s used, ad cosequetly, β =β =/. The algorthms yeld ucodtoal stablty for lear problems at least. Belytschko ad Schoeberle[56] use the Newmark-β method for a olear structural dyamcs problem. Ther proof of ucodtoal stablty wth β =.5 usg a eergy method apples oly to olear materal propertes but they state that umercal results show ucodtoal stablty wth geometrc olearty also. Substtuto of Eq.8 to Eq.7 leads to a set of o-lear equatos whch the ukows are d ad F. The o-lear equatos ca be wrtte terms of resdual forces as R = Md + K( d + d + ( β) d + βd ) F c f = T / q r = c d + ( F / k s ) + ( s + s + ( β ) F m + α v ( + ) = + β F (9a) )(9b) To seek the soluto of the o-lear problem defed Eq.9, the Newto-Raphso terato scheme s here adopted. It s oted that, to the frst order, Eq.9 ca be approxmated as + R R + ( M + β K T ) δd δf c = (a) where δd T r + β c δd + ktδf = (b ) + r K ( Kd = ) T d d = d / q ( F ) kt = + β / q qk s m + + = d d δf = F F () () Here superscrpt deotes the terato couter. K T s the symmetrc taget stffess matrx evaluated at terato. k t s defed wth the codto of + > F. δd ad δ F + are cremets, or teratve correctos, of the ukows d ad F at terato, respectvely, ad they ca be gve from Eq. as T r δd = ( M + βkt + βcc ) ( R + c) kt kt () T δf = ( r + βc δd ) + k t Through the use of Eq. wth the correctos gve Eq., the terato procedure results mproved soluto for the ukows d ad F. The procedure s repeated utl the followg covergece crtera are satsfed T δd δd T d d ε =. + < ε ad δf F + < ε (4) where 5s used ths work. Up to ow the mpact dyamc trasets are cocered wth geometrc o-learty ad the soluto procedure s descrbed detals as above. For comparso, the mpact problem s also vestgated wth the cotext of small deflecto ths paper. I fact there exsts o dfferece betwee the geometrc o-lear ad lear aalyses of the mpact problem the temporal approxmato. Wth a tme step, however, the soluto procedure used for the lear traset respose s dfferet from that descrbed above for the o-lear oe, ad s here dscussed below. I the case of geometrc lear aalyses of the mpact problem, the o-lear stra terms are gored the stra-dsplacemet relatoshp. As a result, the stffess matrx K becomes depedet of the plate dsplacemet. From Eq.9a the followg equatos ca be gve ad ( M + ( c, f β K)( d K( d c, + d d f ) = + ( β ) d c f = F d + d )) (5) d (6) Eqs.5 ad 6 together wth Eq.9b defe the preset mpact problem terms of small deflecto. It s obvous that Eq.5 defes two sets of lear algebrac equatos wth c respect to ukows d f ad d. These two sets of lear equatos ca be solved smultaeously sce they share the same coeffcet matrx. The soluto procedures volve tragular decomposto of the coeffcet matrx ad the forward elmato ad back substtuto. Most of the computg efforts are expeded the procedure of tragular decomposto of the coeffcet matrx durg the soluto of the equatos. Notg the coeffcet matrx s

7 Iteratoal Joural of Composte Materals, (6B): -9 9 depedet of tme cremets, the tragular decomposto of the coeffcet matrx s performed ahead of the tme terato the preset methodology ad each tme cremet oly the forward elmato ad back substtuto are volved. Wth ths strategy, the computer efforts ca be greatly reduced. Substtuto of Eq.6 to Eq.9b leads to a o-lear equato wth respect to F, ad aga, the Newto-Raphso terato techque s adopted for the soluto of the o-lear equato. The teratve correctos for F ca be wrtte as where c k t ad F = k t r T + c d δ (7) c r are the same as those gve before ad d are evaluated from Eq Numercal Applcatos The above aalyss procedure has bee mplemeted a -house computer software. It ca be used for mpact problems ad, eglectg mpact-related terms, for plate traset respose uder dyamc loadg. Numercal tests have bee coducted to verfy the valdty of the preset computatoal model, whch clude a seres of examples of composte lamate plates uder low-velocty mpact. The selected examples of the mpact problem refer to two typcal kds of low-velocty mpact evets,.e. s mall mpact mass wth relatvely hgh speed ad large mass wth low speed. I all the applcatos, the quadratc Lagraga shape Cotact force (N) fuctos are appled the crosswse x -drecto. Uless otherwse specfed, the smply supported boudary codtos are defed such that the plate s smply supported for out-of-plae behavour whlst the -plae movemet of the sdes s costraed the taget drecto but allowed the ormal drecto. For the mpact problems cosdered, the cotact law defed Eq.a s used for both the loadg ad uloadg processes. The detals are preseted the followg subsectos. 5.. Impact of a[/9//9/]s Plate Cosder the mpact problem volvg cetral mpact of a smply supported square T/94 graphte/epoxy [/9//9/] s plate by a steel object wth a sphercal mpact tp of dameter.7mm. The plate has dmesos of.69 mm. The materal propertes of a T/94 graphte/epoxy lama are gve as E = GPa, E = 7.9 GPa, ν =. G =G = G = 5.5GPa, ρ = 58 kg/m Two cases of mpact evets are cosdered here,.e., mpact by a 7.5g mass wth a velocty of m/s ad by a 9.6g mass wth a velocty of m/s, respectvely. I the preset study, the cotact law defed Eq.a s smply employed for the whole cotact processg. Sce the lamate s th oe umercal layer s used wth shear correcto factor k 55 =.8875 ad k = , evaluated by the method Referece[6] Case Impact by a 7.5g mass of m/s t= -6 Sec t= -6 Sec t= -6 Sec t= -6 Sec Half plate, 5 quadratc strps, sectos k s =5644 N/mm / Tme (Sec) x -4 (a) Cotact force

8 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact.5.4 Cetral dsplacemet (mm) t= -6 Sec t= -6 Sec t= -6 Sec t= -6 Sec Half plate, 5 quadratc strps, sectos k s =5644 N/mm / Tme (Sec) x - (b) Cetral deflecto F gure 4. Covergece wth respect to tme step for the problem of a[/9//9/] s plate mpacted by a 7.5g mass of m/s Cotact force (N) e=, q=4 e=, q=6 e=, q= e=5, q= e=, q= e=, q= t= -6 Sec Tme (Sec) x -4 (a) Cot act force

9 Iteratoal Joural of Composte Materals, (6B): Cetral dsplacemet (mm)... e=, q=4 e=, q=6 e=, q= -. e=5, q= e=, q= -. e=, q= t= -6 Sec Tme (Sec) x - (b) Cetral deflecto F gure 5. Covergece wth respect to plate mesh for the problem of a[/9//9/] s plate mpacted by a 7.5g mass of m/s Numercal tests are carred out o the mpact problem of the[/9//9/] s lamated plate by a 7.5g mass wth a velocty of m/s. The example chose here s the oe studed by Su ad Che[7], ad Qa ad Swaso[8]. The covergece of tme tegrato s frst studed. Takg advatage of symmetry ths th plate geometry, half of the plate (crosswse drecto of the strp) s modelled usg fve quadratc strps the crosswse drecto. Each strp has te sectos the logtudal drecto ad oe umercal layer through the thckess. The cotact coeffcet k c s evaluated usg Eq., whch s equal to 5644 N/mm.5. The tme cremets are take as µs, µs, µs ad µs. Fgures 4a ad 4b show the cotact force hstory ad the cetral deflecto of the plate, respectvely. It ca be see that all the chose tme cremets gve stably coverged solutos. As the tme cremet decreases, covergece behavour ca be observed. It s oted that the results for cotact force ad deflecto hstores obtaed usg a tme cremet of µs are farly reasoable. The results obtaed by usg a tme cremet of µs are almost detcal to those usg a tme cremet of µs, as well as. µs (ot preseted the fgure), wth plottg accuracy. The predcted maxmum cotact forces are. N, 9. N, 9. N, ad 9.4 N, respectvely, whe usg tme steps of µs, µs µs ad µs, respectvely. Next, the covergece wth respect to the plate mesh s vestgated. Half of the plate s modelled usg a umber, e, of quadratc strps wth dfferet secto umber q as () e=, q=4, () e=, q=6, () e=, q=, (4) e=5, q=, (5) e=, q=, ad (6) e=, q=. The tme cremet s take as µs. The cotact coeffcet k c s equal to 5644 N/mm.5. Fgures 5a ad 5b show the covergece of cotact force ad the cetral deflecto hstores wth respect to the plate mesh. A coverged tred ca be observed the fgures as the umber of the strp ad sectos creases. It s foud that use of coarse mesh yelds reasoable deflecto predcto but uacceptable estmato of the cotact force. Both the cotact force ad deflecto hstores calculated by usg 5 strps wth sectos are early as good as those usg strps wth sectos. The follo wg set of umercal tests s carred out to study the effect of the cotact coeffcets. For the same mpact problem, Su ad Che[7] meshed oe quarter of the plate wth 8 8 e-ode quadrlateral fte elemets ad used a cotact law as Eq. wth k c = 4468 N/mm.5, whereas Qa[8] ad Swaso appled the Raylegh-Rtz techque ( modes) ad corporated the cotact law as Eq.a wth the cotact area chagg at each tme step ad k c = 654 N/mm.5. I the preset study, the cotact law defed Eq.a s used wth a seres of cotact coeffcets, k c = 4468 N/mm.5 (the same as Su ad Che for the loadg processg), k c = 666 N/mm.5 ad k c = 5644 N/mm.5 (evaluated from Eq.). Fgure 6 shows the comparso of the cotact force ad cetral deflecto hstores. I the preset FSM aalyss, quadratc strps wth 4 sectos are used to mesh the whole plate ad the tme cremet s take as µs. The predcted maxmum cotact forces are.8 N, 97. N, ad 87. N, respectvely, whe usg cotact coeffcets of 4468 N/mm.5, 666 N/mm.5 ad 5644 N/mm.5, respectvely. It s oted that the cotact force ad deflecto hstores predcted by the preset FSM usg cotact coeffcet of 5644 N/mm.5 (close to that of Qa

10 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact ad Swaso) are bascally the same as those of Qa ad Swaso usg the Raylegh-Rtz method. Use of cotact coeffcets 4468 N/mm.5, the same value as used by Su ad Che, gves a agreed predcto for the deflecto hstory to that of Su ad Che, but a slghtly hgher maxmum cotact force tha that of Su ad Che. Sce sple fuctos are the smoothest pecewse polyomals, they possess the bleded advatages of smooth aalytcal fuctos ad versatle polyomals. As fewer degrees of freedom are requred for the same accuracy, the preset FSM s much more computatoally effcet comparso wth the covetoal FEM whch covetoal polyomals are used. More detals ca be foud authors earler work[5, 54, 55]. 5 5 FEM, Su ad Che Raylegh-Rtz, Qa ad Swaso Preset FSM, e=, q=4, t=µ, whole plate k s =4468 N/mm / k s =666 N/mm / k s =5644 N/mm / Cotact force (N) Tme (Sec) x -4 (a) Cotact force.45.4 Cetral dsplacemet (mm) FEM, Su ad Che. Raylegh-Rtz, Qa ad Swaso Preset FSM, e=, q=4, t=µ, whole plate.5 k s =4468 N/mm / k s =5644 N/mm / Tme (Sec)..4.6 x - (b) Cetral deflecto F gure 6. Comparso of predcted cotact force ad cetral deflecto hstores of a[/9//9/] s plate mpacted by a 7.5g mass of velocty m/s

11 Iteratoal Joural of Composte Materals, (6B): -9 Cotact force (N) e=5, q= e=5, q= e=, q= e=, q=4 t= -6 Sec Tme (Sec) x -4 (a) Cotact force.4. Normalsed cetral dsplacemet e=5, q= e=5, q= e=, q= e=, q=4 t= -6 Sec Tme (Sec)..4.6 x - (b) Cetral deflecto F gure 7. Covergece wth respect to plate mesh for the problem of a[/9//9/] s plate mpacted by a 9.6 g mass of velocty m/s Case Impact by a 9.6g mass of m/s Cosder the[/9//9/] s lamated plate mpacted by a 9.6g mass wth a velocty of m/s. I the preset study, the cotact coeffcet s take as k c = N/mm.5, the same value as that used by Che ad Su[8] for loadg process. The covergece of the cotact force ad cetral deflecto hstores wth respect to plate mesh s show Fgure 7. The results are obtaed usg tme step of µs ad quadratc strps wth the meshes () e=5, q=, () e=5,

12 4 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact q=, () e=, q=, ad (4) e=, q=4. It ca be observed that all the meshes gve a close predcto the cetral deflecto hstory ad the maxmum cotact force. However, stable results for the cotact force hstory are oly acheved by usg quadratc strps wth sectos at least. Note that the case example the use of 5 quadratc strps ad sectos gves stable ad good results for the cotact force hstory. Ths dcates that the covergece behavour s affected by the teracto of the plate ad the mpactor. The prevous aalyss s performed usg the defaulted type of smply supported boudary codtos as descrbed at the begg of ths secto,.e., the -plae movemet of the sdes s costraed the taget drecto but allowed the ormal drecto. Besdes the defaulted oe, aother type s also cosdered of smply supported boudary codtos whch all the -plae movemet of the sdes s costraed. I ths aalyss half of the plate s meshed to quadratc strps wth 4 sectos. Fgure 8 shows comparso betwee the results for the cotact force ad plate deflecto obtaed from use of the two types of smply supported boudary codtos as well as the lear aalyss. I the fgure, the defaulted type of smply supported boudary codtos s deoted as BC whlst the other oe s deoted as BC. The fgure also gves the results for the same mpact problem obtaed from o-lear FEM aalyss by Che ad Su[8] ad from a commercal FEM package, LS-DYNAD, by Lu ad Dag[6]. Che ad Su obtaed ther results by meshg a quarter of the plate wth 8 8 e-ode quadrlateral fte elemets ad used the cotact law as gve Eq., whereas Lu ad Dag performed ther aalyss usg a 4 by 4 mesh. The 4 boudary codtos of BC were used by all of them. I Fgure 8. t s show that whe usg the same boudary codtos the preset results compare well wth those obtaed from the FEM aalyss[8] ad the LS-DYNA D aalyss[6]. For the cotact force hstory, a very close agreemet s foud Fgure 8a betwee the preset o-lear FSM aalyses wth two dfferet type boudary codtos ad lear FSM aalyss. It s ot strage, sce the plate deflectos are less tha half of the plate thckess durg the cotact processg as show Fgure 8b, ad therefore, the geometrcal o-learty has a qute sgfcat effect. However, t s clearly show Fgure 8b that predcted plate deflecto resposes from the preset o-lear FSM aalyses wth use of BC ad BC ad lear FSM aalyss are dfferet from oe aother. 5.. Quas-sotropc Plate Subjected to Drop-weght Impact Here cosder the example of a steel drop-weght mpact o a[45//-45/9] s quas-sotropc plate of AS4/5 graphte-epoxy whch was studed by Ambur et al[]. The dmesos of the expermetal specmes were 54 mm log ad 7 mm wde ad the thckess of a sgle layer was.7 mm. The specmes were smply supported o all four edges ad mpacted o the plate cetre at mpact-damage-tato threshold eergy level of.6779j (6.-lb). The materal propertes of AS4/5 are gve as E = 7.8 GPa, E = 9. GPa, ν =., G =G = 6. GPa, G =.5GPa, ρ = 57 kg/m FEM, BC, Che ad Su LS-DYNAD, BC, Lu ad Dag Preset FSM, e=, q=4, t=µ, half plate Lear Nolear, BC Nolear, BC Cotact force (N) Tme (Sec) x -4 (a) Cotact force

13 Iteratoal Joural of Composte Materals, (6B): Normalsed cetral dsplacemet FEM, BC, Che ad Su LS-DYNAD, BC, Lu ad Dag Preset FSM, e=, q=4, t=µ, half plate Lear Nolear, BC Nolear, BC Tme (Sec) x - (b) Cetral deflecto F gure 8. Comparso of predcted cetral deflecto hstory of a[/9//9/] s plate mpacted by a 9.6 g mass of velocty m/s Nolear FSM o Expermetal data [] Preset FSM, 5.4-mm-dam tp Preset FSM,.7-mm-dam tp Cotact force (N) 5 5 Lear FSM Tme (Sec) (a) Cotact force hstory

14 6 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact Lear FSM Preset model 5.4-mm-dam tp.7-mm-dam tp Cetral dsplacemet (mm) 5 4 Nolear FSM Tme (Sec) (b) Cetral deflecto of the plate F gure 9. Comparso of mpact respose of a[45//-45/9] s plate to drop-weght Nolear FSM Preset FSM Impact mass.8 kg Impact mass.59 kg Cotact force (N) 5 5 Lear FSM Tme (Sec) for mass.8 kg.44 for mass.59 kg (a) Cotact force hstory

15 Iteratoal Joural of Composte Materals, (6B): Lear FSM Preset FSM Impact mass =.8 kg Impact mass =.59 kg Cetral dsplacemet (mm) 5 4 Nolear FSM Tme (Sec) for mass.8 kg.44 for mass.59kg (b) Cetral deflecto of the plate Fgure. Comparso of mpact respose of a[45//-45/9] s plate to masses.8kg ad.59kg at a gve mpact eergy of.6779j I the preset vestgato, the plate model s used wth the dmesos 4. mm log by 4. mm wde v ew of the fact that the supports are located from the actual specme boudares[]. The whole plate s modelled usg quadratc strps wth 4 sectos ad through the plate thckess oe umercal layer s used sce the lamate s qute th. The mass of the mpactor s take as.8 kg, whlst two dfferet values, 5.4 mm ad.7 mm, of mpact-tp dameters are cosdered for comparso. The calculatos are performed usg a tme step of. ms ad the cotact coeffcet evaluated from Eq.. The results of cotact force are show Fgure 9a compared wth the expermetal data ad the cetral deflecto of the plate Fgure 9b. It ca be foud that the geometrcally o-lear ad lear aalyses gve marked dfferet predctos for the mpact respose cludg cotact force hstory, plate deflecto ad cotact durato. The cotact force predcted by the preset geometrcally o-lear aalyss s closely compared wth those from expermet whlst the lear aalyss gves a poor predcto of the cotact force. Ths dcates that geometrc o-learty has sgfcat effects o the mpact respose ad t has to be cluded the aalyss. It s observed that use of dfferet mpact tp dameters the preset calculato results oly slghtly dfferet predcto for the cotact respose. I other words, the total cotact force ad the plate deflecto are sestve to cotact coeffcet. The s maller maxmum cotact force correspods to the bgger dameter of mpact tp but however, ths may be utrue for mpact evets other tha the preset specfc case. I a effort to uderstad the effect of the mpact mass ad velocty, aalyss s also performed usg a mpact mass of.59 kg wth a.7-mm-dameter mpact tp but the mpact eergy s stll.6779j. Usg dfferet tme scales the results are compared wth those obtaed prevously for a mpact mass.8 kg of.7-mm-dameter mpact tp Fgure. I the fgure, the tme scale for.8 kg mass s a ut whlst the tme scale for.59 kg mass s equal to the rato of mometa for.8 kg mass ad.59 kg,.e.,. It s show that durg the mpact cotact perod the results, both cotact force ad plate cetral-deflecto, correspodg to those two masses are a close agreemet whe usg the tme scales metoed. Ths dcates that for a specfed drop-weght mpact evet, the maxmum cotact force ad the maxmum plate deflecto at the mpact posto deped o mpact eergy level but the cotact perod s domated by the mpact mass ad a loger cotact perod arses from a bgger mpact mass. 6. Cocludg Remarks A Layerwse B-sple FSM has bee developed for aalysg the geometrcally o-lear traset respose of lamated composte plates to trasverse low-velocty mpact. To smplfy the complcated cotact aalyss, a Hertz-type cotact law has bee corporated to the fte strp model for accoutg for the cotact behavour. The soluto of the o-lear mpact problem s sought usg Newmark tme tegrato scheme cojucto wth Newto-Raphso terato. The geometrcally lear

16 8 Smo Wag et al.: Geometrcally No-Lear Aalyss of Composte Lamated Plates Subjected to Low-Velocty Impact traset aalyss of the mpact problem has bee dscussed brefly as well. Wth the eglect of mpact-related terms, the developed procedure s capable of aalysg plate trasets uder dyamc loadg. The capablty of the model has bee used a few applcatos cludg the mpact respose aalyss of a plate subjected to relatvely-hgh ad low-velocty mpact. It should be oted that the mpact eergy both cases are low to avod materal damage ad delamato whch are ot dealt the preset work. It has bee dcated that the covergece behavour of the mpact dyamc problem s affected by the teracto of the plate ad the mpactor. Comparso of the results obtaed from the geometrcally o-lear aalyss has bee made wth those of small-deflecto soluto ad/or avalable results gaed from fte elemet calculato ad expermet. Ths comparso demostrates the valdty of the aalyss procedure as well as the effect of geometrc o-learty o plate respose ad cotact force. The results show that for a specfed drop-weght mpact evet, the mpact eergy level domates the maxmum cotact force ad the maxmum plate deflecto at the mpact posto but the mpact mass domates the mpact cotact durato. REFERENCES [] Abrate, S., 99, Impact o lamated composte materals, Appl Mech Rev., 44(4),55-9. [] Capro, G., Crvell, V. I. ad Ilo, A. D., 984, Elastc behavour of composte structures uder low velocty mpact, Compostes, 5(), -4. [] Rotem, A., 988, Resdual Flexural Stregth of FRP Composte Specmes Subjected to Trasverse Impact Loadg, SAMPE Joural, 4(), 9-5. [4] Alderso, K. L., Evas, K. E., 99, Dyamc aalyss of flamet woud ppe udergog low velocty mpact, Composte Scece ad Techology, 45, 7-. [5] Ml, F., Necb, B.,, Impact behavor of cross-p ly lamated composte plates uder low veloctes, Composte Structures, 5, [6] Shvakumar, K. N., Elber, W., Illg, W., 985, Predcto of low-velocty mpact damage th crcular lamates, AIAA Joural, (), [7] Bucell, R. B., Nusmer, R. J., Koury, J. L., Respose of composte plates to quas-statc mpact evets. I: O Bre TK. ASTM STP, 99, p [8] Sjoblom, P. O., Hartess, J. T., Cordell, T. M., 988, O Low-velocty mpact testg of composte materals, Joural of Composte Materals,, -5. [9] Toh, S. L., Gog, S. W., Shm, V. P. W., 995, Traset stresses geerated by low velocty mpact o orthotropc lamated cyldrcal shells. Composte Structures,, -8. [] Gog, S. W., Shm, V. P. W., Toh, S. L., 996, Cetral ad ocetral ormal mpact o orthotropc composte cyldrcal shells, AIAA Joural, 4(8), [] Gog, S. W., Lam, K. Y.,, Effects of structural dampg ad stffess o mpact respose of layered structure, AIAA Joural, 8(9), [] Robso, P., Daves, G. A. O., 99, Impactor mass ad specme geometry effects low velocty mpact of lamated compostes, It. J. Impact Eg., (), [] Collombet, F., Bo, J., Latallade, J. L., 996, A three-dmetoal modellg of low velocty mpact damage composte lamates, It. J. Numer. Meth. Egg., 9, [4] Bo, J.,Collombet, F., Latallade, J. L., Numercal modellg of cotact for low velocty mpact damage composte lamates. I: Alabad MH, Brebba CA. Cotact Mechacs, st Iteratoal Coferece o Cotact Mechacs, Southampto, July 99, p [5] Baerjee, R., Numercal smulato of mpact damage composte lamates. Proceedgs of the Amerca Socety for Compostes 99; p [6] Lu, D., Dag, X., Testg ad smulato of lamated compostes subjected to mpact loadg. I: Bucell RB. Composte Materals: Fatgue ad Fracture, 7th Volume, ASTM STP, Amerca Socety for Testg ad Materals, 998, p7-84. [7] Luo, R. K., Gree, E. R., Morrso, C. J., 999, Impact damage aalyss of composte plates, Iteratoal Joural of Impact Egeerg,, [8] Gree, E. R. Morrso, C. J., Luo, R. K.,, Smulato ad expermetal vestgato of mpact damage composte plates wth holes, Joural of Composte Materals, 4(6), 5-5. [9] Goo, N. S., Km, S. J., 997, Dyamc cotact aalyss of lamated composte plates uder low-velocty mpact, AIAA Joural, 5(9), [] Yag, S. H., Su, C. T., Idetato law for composte lamates. composte materals: Testg ad Desg (6th Coferece), ASTM STP 787, 98, p [] Ta, T. M., Su, C. T., 985, Use of statc detato laws the mpact aalyss of lamated composte Plates, Joural of Appled Mechacs, 5, 6-. [] Lesser, A. J., Flppov, A. G., 99, Ketcs of damage mechacs lamated compostes, It. SAMPE Symp. Ad Exhbto, 6 (Pt.), [] Srvasa, K., Jackso, W. C., Hkle, J. A., 99, Respose of composte materals to cow velocty mpact, It. SAMPE Symp. Ad Exhbto, 6 (Pt.), [4] Wu, E., Shyu, K., 99, Respose of composte lamates to cotact loads ad relatoshp to low-velocty mpact, Joural of Composte Materals, 7(5), [5] Su, C. T., Chattopadhyay, S., 975, Dyamc respose of asotropc lamated plates uder tal stress to mpact of a mass, Joural of Appled Mechacs, 4, [6] Whtey, J. M., Pagao, N. J., 97, Shear deformato

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