A New Interference Approach for Ballistic Impact into Stacked Flexible Composite Body Armor

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1 5h AIAA/ASME/ASCE/AHS/ASC Suues, Suual Dynams, and Maeals Confeene<b>17h 4-7 May 9, Palm Sngs, Calfona AIAA A New Inefeene Aoah fo Balls Ima no Saked Flexble Comose Body Amo S. Legh Phoenx 1 and A. Kad Yavuz Conell Unvesy, Ihaa, NY, 14853, USA and Pankaj K. Powal 3 Indan Insue of Tehnology, Mumba 476, Inda In hs sudy, a new nefeene model s develoed o nvesgae he balls esonse of a hybd, wo-layeed, flexble body amo. Sne he sakng ode of layes, whh have dsnly dffeen mehanal oees, has a lage effe on he V 5 lm veloy fo eneaon, we aly he analyss o a wo-laye sysem onssng of fabs of aamd (Kevla9 ) fbe and of ula-hgh moleula wegh and dawn olyehylene (Dyneema ) fbes. Usng gh-ula ylndal ojeles, evous exemenal esuls of Cunnff [1] (usng a smla fbe, Sea nsead of Dyneema ) showed nealy a fao of wo dffeene n he V 5 veloy n he wo ossble sakng odes,.e., Sea vesus Kevla9 as he ske fae. Ths new model exends ou evous fundamenal wok [] by addessng nefeene faos n ems of nsananeous maeal nflow o he ma one usng a onvoluon ove uen loal san aound he ojele edge dung s deeleaon. Ou evous mullaye model [3] gnoed he nefeene ase hough a subsequen wok [4] aoxmaed s effes bu whou nsananeous esonse of maeal nflow o loal ensle san. The uen model s muh moe omlee and odues veloy, san and defomaon hsoes ehe o efoaon o o halng he ojele. In he new model, V 5 veloy dffeenes wh sakng ode and wh laye bondng vesus no bondng ae even lage. Howeve, bak-fae defleons un ou o be mnmally affeed. Calulaons ae vey fas and os-oessng fgues an be obaned n less han a mnue usng MATLAB wh a lao ( MHz Coe Duo CPU Gb RAM). I. Inoduon e develo a new model o nvesgae he balls esonse of a hybd, wo-layeed, flexble body amo, Wsysem, whee he ndvdual layes ae no bonded ogehe and dffe onsdeably n he mehanal oees. Suh sysems ae used n body amo fo ubl safey offes and seuy esonnel as well as n alne ok doos, heloes and he doos of auomobles and lgh uks. In one aula sysem onssng of an aamd (Kevla9 ) fab and a fab of ula-hgh moleula wegh and dawn olyehylene (Dyneema o Sea ) was found exemenally by Cunnff [1] ha he sakng ode of he wo layes has a vey lage effe on he V 5 lm veloy (a sasal veloy heshold fo efoaon obaned fom seveal ess). I was hyoheszed ha lang he sffe and lghe fbous olyehylene laye on he ske fae led o an nefeene effe wheeby he naually oung ma one of he olyehylene laye lashed wh ha of he Kevla laye. Ths would esul n ma one nefeene and a shf n load fom he Kevla9 laye o he o olyehylene laye (Dyneema o Sea ). The man goal of he esen ae s wofold: Fs we nodue a new veson of he sngle laye model ha esonds nsananeously, n ems of maeal nflow veloy, o he hangng enson aound he ojele edge as deeleaes. Seond, we aly he new veson o he wo-laye, nefeng sysem n ode o sudy analyally he negave effes of suh nefeene on V 5 efomane and bak-fae defleon 1 Pofesso, Deamen of Theoeal and Aled Mehans, Conell Unvesy, Ihaa NY Reseah Assoae, Deamen of Theoeal and Aled Mehans, Conell Unvesy, Ihaa NY Asssan Pofesso, Deamen of Cvl Engneeng, Indan Insue of Tehnology, Mumba 476, Inda. 1 Coygh 9 by he, In. All ghs eseved.

2 Fgue 1. Key quanes n he analyss fo boh he non-nefeng (o) and nefeng (boom) ases. A. Poblem Geomey. The geomey of he oblem of ma no wo saked layes s shown n Fgue 1 along wh seveal key quanes. We assume he ona beween he wo layes s fonless wh no bondng beween hem. The analyss s fs efomed on a un oblem assumng an nal ojele veloy, V V, mmedaely afe ma, and a onsan membane san ove me gh a he ojele edge,,,,, 1,. Ths suaon aually oesonds o a sef ae of ojele deeleaon and ofle of deeasng veloy, V, ha s afal. Howeve, he un soluon esuls an hen be adaed o he moe geneal and naual ase of a ojele deeleang fom membane enson aound he ojele umfeene ang a one angle,, elave o he hozonal lane of he membane. In hs ase haens ha, nally neases n me,, bu eahes a maxmum value befoe deeasng o zeo as he ojele sos. Noe ha n he analyss, V s aken as he ojele veloy mmedaely afe nsananeous momenum ansfe n he ona egon beween he ojele and he wo-laye membane. Po o ona, he ojele veloy wll be a slghly lage value han V deendng on he ao of he ojele mass, M, o he mass of he wo-laye membane egon whn he ona le, as s easly alulaed fom onsevaon of loal momenum. When seakng of quanes n ems of maeal oodnaes we mean ha he quanes ae wh efeene o he ognal adal loaon,, of a maeal on of nees. Ths dsnon s aulaly moan when dsussng one wave veloy, whehe n ems of he veloy wh ese o he ognal oson of he maeal ales o wh ese o he uen loaon of he maeal ons as vewed fom he gound, alled gound oodnaes. Fo he wo layes denoed, 1,, we defne as he maeal densy, h as he hkness, and E as he ensle modulus of laye, and we also have he ensle wave-seeds, a, E. Vaous ohe quanes ae nvolved as shown n Fgue 1, wh he undesandng hey all deend on me, and some also deend on (.e., n maeal oodnaes). In aula, when nefeene exss beween layes hee s he ommon quany,, whh s he ansvese one wave seed elave o gound (.e., n gound oodnaes), bu he n-flow veloes of he layes, u, n maeal oodnaes ae oenally dffeen as ae he ansvese one wave seeds,, n maeal oodnaes. Also we le be he laye san n maeal oodnaes. In addon o, we le u, and, be

3 he nflow veloy and san, esevely, n laye a he oson of he one wave-fon, denoed, n maeal oodnaes. In he ase of nefeene, he hee veloes fo eah laye ae elaed by 1, u, (1) sne he ansvese wave s movng n sehed maeal, aouned fo by he fao 1,, and he seond em aouns fo he fa ha he one-wave fon s aually avellng n maeal movng owads he ogn (as seen fom he gound). II. Analyss fo he Un Poblem Assumng Fxed San a he Pojele Edge We fs onsde he behavo of a adally oagang enson wave n laye, and subje o he fxed san,,,,,, a adus,. Ths un oblem does no ye efle he veal defleon of he membane by he ojele no he hange n hozonal dslaemen and veloy ha develos a loaons whn he one wave,,, hus evenng fuhe adal moon owads he ogn. Howeve, he assoaed san ofle s oe fo ou uoses sne s wen n ems of maeal oodnaes. Noe ha n he aual oblem eaed lae, he ensle wave avels moe han an ode of magnude fase han he one wave, whh n un s smla n magnude o he ojele veloy. Thus nea nsananeous adjusmens an ou n membane san beween he ojele adus,, and he one adus,,, as hough govened by quas-sa equlbum. Lae, hs wll allow us o ea, seaaely, he defomaon whn he one egon bu esevng he enson wave behavo beyond he one wave fon. Fom hs un soluon, he nflow dslaemen ofle n laye 1, s a, u,,, ln, a, he san ofle s, u,,,, a, (3) and he nflow veloy ofle s u, u, a,,,, a, a, (4) whh we noe s aually ndeenden of adal oson,. Ousde hs egon hese quanes ae zeo. A he ojele edge,, he above elaons yeld and a, u,,, ln 1 (6),,, u, a,,, (7) a, esevely, and a he one wave-fon, and,, hese elaons yeld a, u,,, ln (8),,,, (9), 3 () (5)

4 u, a,,, (1) a, whee agan we noe he ndeendene wh ese o,. A. Change n Radal Lengh Comably. As s dsussed lae, n he fomaon of he one wave eesenng he veal defleon of he membane by he ojele, a key equemen n he nefeng layes s omably n he hange-n-lengh fom he ensle wave-fons a L a, bak o he ojele ma le of adus,, sne beyond ha wave-fon he maeal s undsubed, and nsde boh maeals follow he same geome ah o he ojele edge. Fo laye, he hange n lengh due o he san dsbuon of he enson wave s defned as l, and fom Eq. (3) we oban a, a, 1 a, l, d,, d,, ln (11) Fo hange-n-lengh omably we mus have s l l l (1) 1 so ha l,, ln 1, 1, (13), whee a, a (14),1,1,, Eqs. (5) hough (14) ollase o hose fo he oesondng 1D ase ovded ha he enson wave has aveled only a small dsane omaed o he ojele adus,.e., L a, fo a, s only slghly lage han. Fom a Taylo sees exanson of he logahm n Eq. (13),.e., ln 1z zz 3 z 3 we fnd ha,1a,1,a,,,1 and, (15) o,1,,1 a,,,, a,1 as (16) Howeve, fo longe mes, whee he enson wave has aveled a lage dsane,.e., L a,, we aoxmaely have,1 ln, ln,,1 and, (17),1, I an be seen mmedaely ha o sasfy he hange n lengh omably ondon he sans a he ojele edge anno smulaneously sasfy boh Eqs. (16) and (17), and hus, mus be allowed o adjus wh me, whh s fundamenally dffeen fom wha ous n he 1D model of a wo-laye ae. Howeve, as wll be shown lae, uns ou ha an effeve se of sans and veloes an be defned suh ha he wo laye sysem an be eaed as a sngle laye un oblem, whee he effeve san a he ojele edge s fxed n me. Thus we allow he ndvdual laye sans a he ojele edge o modeaely adjus ove me, and we use he noaon,, 1, o efle hs adjusmen. Thus by Eq. (17), hange n lengh omably means we mus have,1 ln,1 ln ln,1 1 as ln ln ln,,, 4 (18)

5 so ha he ao of he wo sans mus evenually aoah uny, ahe han he ao of ensle wave-seeds, efleed n Eq. (16). Founaely, he san onvoluon of un soluons efomed lae ems auomaally handlng of suh san dffeenes beween layes. Fo he nflow veloy, and agan gnong he laeal defleon effes fomng he one wave, we had Eqs. (4), (7) and (1), and agan usng a Taylo sees exanson n he me eod afe ma, a he one wave-fon we have 1 u, a,,, a,,, 1, 1, 1,,,, (19) Thus fo sho mes we see fom Eq. (19) ha he wo n-flow veloes ae he same,.e., u,1 u, a,1,,1,,1 and,1 () o u,1 1 u, as (1) Howeve, fo muh longe mes we see ha 1 a,,,,,, u, a,,,,, 1, and 1, () so ha boh deay adly o zeo n nvese ooon o me,. We see, howeve, ha Eqs. (1) and () anno be sasfed smulaneously, and ha he nflow veloy ao ends o follow he ao of he sans. Howeve, as we have jus seen, he san a he ojele edge mus adjus self as me ogesses so ha he san ao evenually sasfes Eq. (19). Thus elang,, n Eq. () wh, agan esuls n u,1,1 u 1, as (3),, The wo esuls, Eqs. (1) and (3) sugges ha he nflow veloes wll eman aoxmaely he same ove me a he one wave-fon (and whn he one egon), hough by Eq. () boh veloes adly deay o zeo so he absolue dffeene qukly vanshes. Ths equvalene an also be agued fom he eseve ha he nflow, dslaemens, u, a l gven by Eq. (14). Howeve n geneal (and whou onsdeng laeal defleon o fom he one) hee s vey lle dffeene beween he nflow veloes evaluaed a vesus hose evaluaed a, n hs wolaye veson of he un oblem, and n he sngle laye un oblems, Eq. (8) omaed wh Eq. (11) shows hee s no dffeene a all. and gven by Eq. (6), ae he same fo all me, sne hey ae he same as B. Fuhe Evaluaon of he San Balane a he Pojele Edge. As was aleady noed, an moan onsequene of nefeene s ha s no ossble fo a ojele veloy ofle, V,, o be hosen ha yelds un soluons fo eah laye esevng boh lengh omably and he same san a he ojele edge fo all mes. Thus he un soluon vewon eques modfaons o aommodae suh hanges ove me, howeve modes. Thee ae wo man egmes of me: () a sho me egme oesondng o unequal laye sans and gowh of he enson wave u o abou one ojele damee and omable wh he 1D soluon, and () a muh longe me egme, whee he laye sans aoah equaly when he enson wave fon has gown o beome lage omaed o he ojele adus. Whle hs seond egme evenually domnaes behavo n he un oblem, he fs egme deemnes nal values. In he aual oblem, he queson ases as o whehe he fs egme an domnae behavo a mes muh lage han, and even lage han suggesed by he un oblem esuls. Ths s oenally an ssue beause, n he aual oblem, he sans,,, nally nease que adly unl hey eah a eak value a mes muh lage han,, and he soluon of he aual oblem s suued as Duhamel onvoluon negal of me-delayed un soluons of magnude d afe he nal se o,, gh afe ma. Thus he fs egme ould, 5

6 onevably nfluene behavo unl he eak san has been eahed, afe whh all quanes deay o zeo as he ojele sos. Ths oenal nfluene has been nvesgaed by sudyng lengh hanges assoaed wh ma no a sngle membane laye whee he oe alulaon fo lengh hange ove me n he aual oblem would be a onvoluon on Eq. (14) wh ese o hangng san n me, gvng s l lun lun sds (4), whee l s he un soluon hange n lengh gven by Eq. (13). De subsuon hus gves un s l s ds (5),ln 1 ln 1 On he ohe hand a smlfed alulaon of de subsuon of san fo, no l yelds l lun ln 1 (6), Fgue shows he evolvng shfs n he san ao fom he nal ao a,1 a, 1.5 down o 1. a long dmensonless mes (nomalzed n ems of he laye wh he fases ensle wave seed). The sold lne s he alulaed value fom usng he onvoluon and he doed lne s a smlfed eesenaon of he ao of he sans anaed n eah laye as gven by, ln 1 ln 1,1,1, whee a he exemes n me we have, a,1 a,,,,1 1,, un (7) (8) Fgue. Assessmen of hanges of he ao of sans n he wo layes ove me. Two ases ae onsdeed n Fgue, a vey hgh fab o ojele aeal densy ao and a low fab o ojele aeal densy ao. The ao R l l l was found o be faly lose o uny ove he ange fom ma o well as he on whee he maxmum san ous. Ths same behavo s seen fo he full ange 6

7 of aal membane o ojele aeal densy aos. Ths ndaes ha, as he san bulds u o he on of maxmum san, he delay effe n he onvoluon s small and only vey lae n he slowng of he ojele s an effe modesly noeable. In any even he ogessve shf n san ao s moan o he oveall esonse and anno be gnoed. The onvoluon fomulaon used o ea he oblem s aable of aommodang abaly dffeng san hsoes and aos n he ndvdual layes. Howeve, he analyss s gealy smlfed usng Eq. (7) sne we wll be able o defne effeve aamees suh ha he oblem edues o eamen of an effeve sngle laye fom whh ndvdual san hsoes and ohe quanes an be alulaed subsequenly. One neesng feaue s ha he effeve ensle wave-seed, whh we denoe as â, wll aually hange slghly ove me. C. Pulley Analogy fo Cone Wave-Seed Deemnaon. To exlly deemne he vaous model sans, dslaemens and veloes we mus sefally deemne how he one wave veloy deends on he ndvdual laye sans,,, 1,, a he one wave-fon. The alulaon s an nsananeous alulaon n san, so ales o he ase of me-deenden san vaaon a he ojele edge and one-wave fon. We onsde he ensons and lnea denses n layes 1 and and he maeal foes ha ou as he one wave fon asses by and he maeal suddenly hanges veloy deon fom hozonal nflow o nealy veal moon a he same seed as he ojele. Fo hs uose we use he analogy of a wo hn bels, of hkness h 1 and h, avellng a onsan veloy ove a small ulley of adus,, and wh one on o of he ohe suh ha hee s a ona essue beween he wo as hey ass aound he ulley, hus mmkng he nefeene. We hen deemne a al veloy suh ha he ona essue beween he nnemos bel (he lowe laye) and he small ulley self aually shnks zeo (sne he ulley s vual and does no aually exs o suo load). Ths al veloy, oesondng o, 1,, n he sehed maeal, s he al seed of he maeal assng ove he ulley o yeld a vanshng ona foe wh he ulley. Fuhemoe, we wll fnd ha he al veloy s ndeenden of he ulley adus,, and so s he oal neave foe a he one wave fon ang along a lne bseng he wa-aound angle,. A smle geome analyss of he enfugal foes ang ove a small wa-aound angle nemen, d, shows ha he ona foe e un lengh, N, fo laye s 1, 1, d h h d Nd F (9) whee we have aouned fo he slgh do n mass e un lengh due o he sehng of he maeal. Sne F he,, 1,, (3) we have N he h 1, 1, (31),, To have vanshng ona essue ono he ulley (whh doesn exs n he aual oblem) we mus have N N (3) 1 Subsung Eq. (31) no Eq. (3) esuls n 1h11 1,1 h 1, he 1 1,1 he, 1 1 Sne a E we an ewe Eq. (33) as, 1 1 and sne 1 1,1, we have h h ha h a (34) 1 1,1 1, 1 1,1,1,, 1,1, (33) 7

8 , 1 1ha 1,1,1 ha,, h h ,1, (35) D. Cone Wave-Seed Behavo Immedaely Afe Ima. Rgh afe ma, when a, we have Eq. (16) o,,1 a,1,, a, and an assume,,. We aly subss o sgnfy quanes mmedaely afe ma, and have,,,, by Eq. (1), n whh ase a a. Also,,1,1,,, 1 1 u u u, u a (36),1,,1,,,,,1,,,,,1,,1 whee equaly efles he lm as. Thus Eq. (35) beomes, 1,, a,,, a,11h1 a,h h 1 h 1 1 1,,1,, Ths suaon has been nvesgaed n he ase of a wo-laye ae sysem n 1D whee has been found ha an effeve se of aamees an be nodued, as desbed n Aendx A. The effeve aamees, ĉ, ˆ, and â, ae gven by ˆ ˆ (38),1,,1,,, h 1 a h a h h,1 1 1,,1,,, 1 1,,1,, whh s auaely aoxmaed by a,11h1 a,h (4) 1h1 h sne he negleed sans ae small and no oo dffeen n value. Also a a ˆ (41),1,,,1,, ˆ a Mos moan we fnd ha Eq. (37) edues o ˆ ˆ 1 ˆ An addonal esul needed fom he 1D model s he omably of he flow of sehed maeal no he one egon wh he gowh of he lengh (hyoenuse) sannng fom he one wave-fon o he ojele edge. The elevan esul s 1 1 a V, 1, (43),,,,,,,,, whh by Eqs. (38) o (4) an be wen n ems of he effeve aamees as 1 ˆ 1 ˆ ˆ ˆ ˆ V (44) Squang boh sdes and manulang he esul leads o a quada ha an be faoed o yeld V ˆ ˆ a1 1, 1, ˆ 1 ˆ (37) (39) (4) (45) 8

9 Elmnang ĉ usng Eq. (4) and manulang he esul gves ˆ 13 1 V ˆ ˆ 41 3 Ths s used o esablsh nal ondons on he effeve san, ˆ, uon ma. Rahe han solve Eq. (46) numeally, we an develo an auae aoxmaon hough eaon. A fs aoxmaon fo he san on ma s 1 V ˆ ˆ and a seond, moe auae aoxmaon s V 1 ˆ ˆ a ˆ ˆ 1 1 ˆ 1 V A hd and exemely auae aoxmaon ha we use s (46) (47a) (47b) V 1 ˆ ˆ a ˆ ˆ 1 ˆ 1 V One ˆ s known, he sang san values gh afe ma ae smly (47) ˆ, ˆ (48),,1,, a,1 a E. Cone Wave-Seed Behavo a Longe Tmes afe Ima. The ase of longe mes has been eaed n Aendx B. Thee s shown ha an effeve se of aamees, ĉ, ˆ, ˆ, and â an agan be defned o edue he wo-laye ma oblem wh nefeene o one of ma no a sngle laye. Agan, ndvdual laye sans and ohe quanes an be alulaed fom he esuls. The mos ual quany s he effeve ensle wave-seed 1ha 1,1 ha, ln 1a,1 ln 1a, 1 1 â ln 1 ln 1 h h whh, unfounaely s an ml equaon ha would eque numeal soluon. Howeve, n Aendx C we show ha a vey auae aoxmaon esuls fom usng â n lae of â on he gh-hand sde and fuhemoe yelds he oe lms a sho and long mes. Sefally, hs aoxmaon s ln 1 ln 1 1ha 1,1 ha, ln 1a,1 ln 1a, 1h1 h We also have a key esul, namely, ˆ ˆ 1 ˆ (49a) (49b) (5) 9

10 One he effeve san ˆ s found fom he effeve sngle-laye alulaon, he ndvdual sans an be found fom an adaaon of Eq. (7), namely,, ln 1 ˆ, 1, ln 1 a Also, gh afe ma we have ˆ a ˆ ˆ fom Eq. (4). sang one wave-seed, III. fom Eq. (4) and 1 (51) ˆ ˆ ˆ fom Eqs. (47a-), as well as he Soluon Aoah fo Vayng Pojele Veloy and Loal San The soluon aoah fo he ase of vayng ojele veloy unde deeleang foes fom he wo-laye membane sysem, and wh nal veloy, V, gh afe ma, an be adaed fom he same oblem fo a sngle membane laye. All ha s neessay s o use he effeve oey aamee se, ˆ ˆ and â, ĉ, and he nal values as jus desbed. Key equaons fom he D membane analyss ae gven as follows: We le he dslaemen be, V s ds (5) and he effeve one wave-fon oson be ˆ ˆ d (53) whh s he effeve oson of he one wave fon n maeal oodnaes whee ĉ s gven by Eq. (5). Also we denoe û as he nflow dslaemen a he one wave-fon and û as he nflow veloy (owads he ma egon) of he maeal ons a he one wave (maeal oodnaes). Then he one waveseed wh ese o gound s ˆ1 ˆ ˆ u (54) The one angle wh ese o gound s gven by ˆ uˆ () sn and os (55) ˆ uˆ () ˆ uˆ () and fom onsdeaons of he gowh ae of he one sde (hyoenuse), we have he D, sngle-laye based esul ˆ 1 ˆ os uˆ sn V 1 os We also have he nsananeous behavo ˆ ˆ ˆ (57) and fo he one wave-fon adus we also have usng he D sngle-laye esul deved n Aendx C ha whee 3 3 ˆ 3 ˆ sds 1 ˆ ˆ (58) Tunng o Eqs. (54) hough (57) we an deve ˆ 1 ˆ (59) (56)

11 1 os uˆ sn V 1 os Squang Eq. (6) and faong he quada n we oban (6) 14 1 ˆ (61) and by Eq. (57) we oban ˆ ˆ 14 1 Consdeng now he nflow dslaemen, a Duhamel onvoluon negal. Fom Eq. (8) we oban ˆ û, and veloy û ˆ ˆ ˆˆ ln ln ˆ ds Howeve usng Eq. (49) and (66) we an ewe a key quany n Eq. (69) as, 11 (6), we use he un soluons develoed eale n a d ˆ s a s s u ds (63) and fom Eq. (1) ˆ d ˆs s u ˆ ds 1 (64) ˆ ds 1ass whee n Eq. (63), he oson of evaluaon s he uen one-wave fon, ˆ, eseve of he san hsoy. Thus boh û and û deend on he san hsoy, d ˆ s ds, s. To alulae he deeleaon ofle of he ojele, dv d, we mus onsde he essng foe aled by he fab and he key elaon s Fmembane Eh 1 1,1 Eh, sn dv (65) M 1h1h, d Leng Eh, M 1h1ha, (66) h, 1, M h h we we Eq. (66) as 1 1 dv a,1 a,,1,1,, d sn whee he nal ondon s V V. Howeve by Eqs. (9), (51) and (57) and he fa,1, ln 1 ˆ, 1, so ha, ln 1 a, dv a ln 1 a ln 1 d,1,,1 ˆ, ln 1a,1 ln 1a, sn (67) we have (68) (69)

12 Thus we le a a ˆ ˆ ln 1 ln 1 a,1,,1, ln 1a,1 ln 1 a, a,1 ln 1 ˆ ln 1 h a a a,h M h h M h h 1 1 ln 1 a,1 1 1 ln 1a, h h M h h h h ˆ 1 1 M 1h1 h and he key dffeenal equaon edues o dv d ˆ ˆ sn wh he nal ondon V V. Inegang Eq. (7) we oban (7) (71) (7) V V ˆ ˆ ssn s ds (73) A numeal soluon an be develoed nemenally n small me ses as was done fo he sngle D membane oblem. The nal values ae gven by Eqs. (4), (4) and (47a-) as well as V,, u ˆ ˆ a and V and V os, sn V V whee,1,,1 uˆ (74) = ˆ 1 ˆ ˆ (76) A. Non-Inefeng Two-Laye Behavo and Bonded Two-Laye Behavo. Fo he ase of non-nefeng behavo (obaned by evesng he wo-laye sakng), he layes an be eaed seaaely. In he above, a sngle laye s eeved smly by makng he hkness h of he ohe laye zeo. The key ondon govenng oulng s ha he eaon foes fom eah laye mus sum o gve he ommon deeleaon, dv d,.e., dv M 1h1 h Eh 1 1,1 sn1 Eh, sn, (77) d Leng h a,,, 1, (78) M 1h1h whee a,11h1 a,h (79) 1h1 h we we Eq. (77) as (75) 1

13 dv d,1,1 sn1,, sn (8) wh he nal ondon V V. Inegaon gves V V,1,1 ssn1 s,, ssn sds (81) Fo he ase when he layes ae bonded ogehe one may ea he sysem as a one-laye sysem usng effeve oees gven by E eff Eh Eh h h , eff h1 h h1 h, heff h1 h, The sans ae he same n boh layes,.e., and 13 a E eff,eff (8) eff,1,,1,, and he falue san s he smalles of he ndvdual sans, unless he ohe laye an suo he oal enson one he weakes laye fals. IV. Dmensonless Famewok fo Numeal Soluon I s onvenen o efame he oblem n ems of vaous dmensonless quanes n ode o make he alulaons moe ansaen. We le be dmensonless me defned by ˆ, ˆ ˆ a (83) whee ˆ s he me equed fo he effeve ensle wave fon o avel a dsane equal o he ojele adus, usng he nal effeve ensle wave-seed value, â defned by Eq. (4). (Usng â n he nomalzaon ovdes a onvenen salng fo all he esuls.) Fo he one wave-fon, he nomalzed veloy and dslaemen ae, esevely, ˆ ˆ C (84) and ˆ ˆ ˆ ˆ s R ds 1 C d ˆ a (85) Nomalzed quanes desbng he vaous sans and san ae ae ˆ ˆ, ˆ ˆ R and, ˆ (86) and ˆ ˆ (87) Nomalzed quanes fo he ojele veloy and dslaemen ae V ˆ V, (88) and ˆ d (89) Nomalzed quanes fo he n-flow dslaemen and veloy ae uˆ ˆ U uˆ U, U and U, 1 (9) ˆ ˆ ˆ a Nomalzng he one wave-seed n gound oodnaes, Eq. (5) wh Eq. (57) beomes

14 C 1 R whee ˆ (91b) dely efles he mldly hangng effeve ensle wave-seed ove me. Also Eq. (58) beomes R d 1,, (91a) (9) ˆ so ha The one angle s e-aameezed as sn 1 os Also fom Eqs. (59) o (6) we have whee R R,U( ) 1, (93) 14 1 (94) ˆ,U 1os os ( ) sn Fuhemoe Eq. (7) beomes d ˆ 1h1 h sn, ˆ d M h h Fo he nal ondons we an eavely solve ,, 1 1, as was done n Eqs. (47a-). Also as well as ˆ V a, sn 1 1,, U U, 1 and U, 1,, (95) (96) V (97) U (98), (99), os 1 sn V. Alaon o Kevla /Sea Hybd Sysem We have aled he esulng new model o hybd mul-laye oblems wh and whou nelaye bondng and nefeene among les. Fo hs uose we have evsed he well known exemenal sudy on sysem effes n a Kevla9 /Sea hybd sysem ognally ublshed by Cunnff [1] and suded n ou mos een wok [4]. The mehanal oees of he wo maeals wee Kevla : Sea :.9 Kg m Ad,K (aeal densy), Ad,S.18 Kg m (aeal densy), 3 K 1, 45 Kg m (maeal densy), K,yan 79 GPa 3 S 97 Kg m (maeal densy), ES,yan 1 GPa E 14

15 The ojele adus and mass wee.76 mm and 16 gans (1.4 g), esevely. In he model he modulus fo eah laye s E E,yan beause of he yan ossngs o fom an soo shee. The ensle wave veloes ae, esevely, a,k E K K 5, 19 and a,s E S S 7,865 m/s, so wh he hghe Sea modulus and lowe densy, he ensle wave seeds dffe subsanally. I should also be noed ha n Cunnff s sudy, he aeal densy ao of oal membane o ojele,.184, s an ode of magnude lowe han n yal body amo, so he maxmum exemenal V 5 veloes ~ m/s ae elavely low omaed o eal body amo. Fgue 3. Laye san evoluon eded n Powal and Phoenx [4] fo exemenal sysem of Cunnff [1] (assumng Sea modulus of 1 GPa) showng sgnfan dffeenes n san aens deendng on whehe he layes ae nefeng o non-nefeng. Ima veloy s 14 m/s. Cunnff showed ha evesng he ode of Sea and Kevla layes had a lage effe on he V 5 eneaon veloy, and he wos ase esuled fom lang he Sea laye fs, n lage a beause of nefeene effes of he muh hghe ensle wave-seed n he Sea laye as omaed o he Kevla laye and he muh lowe maeal densy. The esuls fom ou evous modelng wok [4], whh used eoave san subsuon, onfmed Cunnff s fndngs. Fgue 3, shows some esuls fom ha veson, whee K s nomalzed gowh of he one wave edge (nomalzed by he ojele adus). As a benhmak es of he new model and ode wh he Duhamel onvoluon we ean he alulaons on he Cunnff hybd sysem [1]. Addonally we nvesgaed he effe n Cunnff s sysem of some eoed balls sffenng effes seen n Sea by Pevosek e al. [5] and Dyneema (vae ommunaon fom manufaue, DSM) wheeby he ensle modulus a balls loadng aes s yally a muh hghe value, GPa, ahe han he 1 GPa value used by Cunnff, based on enson ess a san aes of.5/mn. In hs ase he Sea ensle wave seed beomes a,s ES S 1,153 m/s, aoxmaely double ha fo he Kevla9 laye Fgues 4 and 5 omae vaous ases: () bonded layes, () non-nefeng laye (.e., Kevla9 laye fs), () nefeng layes (.e., Sea laye fs) and all one maeal a he same aeal densy ao. As seen n Fgues 4 and 5 omaed o Fgue 3, he new veson of he model gves esuls faly onssen wh he evous veson [4], alhough he dffeenes beween nefeng and non-nefeng ases ae sgnfanly moe onouned wh he new model, and he sans ae slghly hghe fo he new model. The dffeenes also nease makedly when he hghe Sea modulus, GPa, s used ahe han 1 GPa. Ths means ha he V5 veloy of eneaon wll dffe wdely when laye sans eah he falue sans. Addonally, usng nondmensonal me (nomalzed n ems of me, a ˆ, whee â s an effeve sysem ensle wave seed on ma) we see n Fgue 5 ha he ojele veloy deay ofles ae susngly vey lose among he vaous ases of oulng, dese he fa ha he ndvdual laye sans dffe sgnfanly. 15

16 Fgue 4. Laye san evoluon eded fo hybd sysem of Cunnff (assumed Sea modulus of 1 GPa) showng sgnfan dffeenes n san aens deendng on exen of laye oulng. Smla ondons o Fgue 3 a ma veloy 14 m/s. Fgue 5. Laye san evoluon eded fo exemenal sysem of Cunnff (Sea modulus of GPa) showng muh lage dffeenes n san aens omaed o hose n Fgue. V 14 m/s. 16

17 Fgue 6. Laye san evoluon eded fo sysem of Cunnff (assumng Sea modulus of GPa) bu wh 1 layes eah of Sea and Kevla hus allowng a hghe ma veloy of 43 m/s o aheve he same san levels as n Fgue 5. Fgue 6 shows esuls fo he ase of 1 layes eah of Sea and Kevla (4 n oal) hus allowng a hghe ma veloy V 43 m/s o aheve he same san levels as n Fgue 5. Mos skng s he fa ha smla elave san magnudes ae seen n Fgue 6 as omaed o hose n Fgue 5 exe ha dffeene n laye sans n he nefeng ase s even moe onouned n Fgue 6. Noe ha he san onenaon s no as hgh by vue of he whole deeleaon oess haenng abou a fao o 1 soone n me dese he hghe ma veloy. Based on Fgues 5 and 6 hee s omse of effeve salng faos fo smla sysems bu ooonally neasng numbes of layes. VI. Conludng Commens The mos moan esul s ha we wee able o develo effeve sysem oees fo a hybd sysem allowng o be lagely exessed n ems of a sngle laye oblem, hus gealy smlfyng he numbe of ndeenden aamees needed, bu allowng fo dealed alulaon of ly sans fom he effeve oey soluon deendng on ue maeal wave-seed aos. Noe ha effeve aamees ae onsdeably dffeen among ases of layes songly bonded ogehe, vesus nefeng layes, vesus non-nefeng layes, and vesus sysems based jus one of he maeal a he same oveall fab aeal densy. Fuue wok nvolves exendng he model o saked baxal fabs (ohoo n lane wh low shea modulus of smla maeals wh onasng mehanal oees. Addonally we ae eang he ase of small a-gas beween layes and he eduon n V 5 veloy ha esuls as he sang neases. Aendx A: Defnon of Effeve Quanes Immedaely Afe Ima Consdeng behavo mmedaely afe ma we have,,,, so ha he ulley analogy esul gves a,11h1 a,h, 1,, a,,,, 1, (A1) 1h1 1,,1 h 1,, We onsde he ossbly of defnng effeve values gh afe ma as, ĉ, ˆ and â wheeby we oban a key mahemaal elaonsh as ous n he sngle-laye oblem,.e., 17

18 ˆ1 ˆ ˆ ˆ a1 ˆ (A) By Eq. (15) n he man ex we had a,1,,1 a,,,, so we le â ˆ a a (A3),1,,1,,, Also sne 1 1,,1 1,, Fnally we le gh afe ma we le ˆ ˆ 1 1 (A4) 1,,1 1 ˆ a,11h1 a,h h 1 h 1 1 1,1, In ode fo hs se-u o ovde effeve quanes, â, ĉ and unquely. Fom Eqs. (A3) and (A5) we we ˆ (A5), we mus be able o solve fo eah of hem, a h a h a h a h ˆ a,1 1 1,,1 1 1,,1,1 1h1,,1 h,, 1h1,,1 h,, and mosng a ommon denomnao and usng he fa ha a,1,1 a,, we an show 1 1 1h1 1,,1 h 1,, h 1 a h a h a h a h,1 1 1,1,,1 1 1,,1,,,,,, a h a h h,1 1 1,,1,,, 1 1,,1,, Then fom Eqs. (A1), (A3), (A4) and (A5) we onfm he valdy of Eq. (A) yeldng he well known fom ˆ ˆ 1 ˆ (A8) Then he laye sans, gh afe ma, ae obaned as ˆ a, 1, (A9),,, Aendx B: Defnon of Effeve Paamees a Inemedae and Long Tmes We onsde he develomen of effeve oees n ode o use he sngle laye analyss o ea ma no wo nefeng layes. We eall Eq. (7) fom he man ex and dese onvoluon (me delay) effes and a, we have exemely long me we evenually have. Fom Eq. (7) and ln 1,,1,1,1, ln 1,,1, Howeve,1, sne 1 1,1 1,,,,1 and boh,1 and, ae small (well below he falue sans whh ae <.5) and aually aoah eah ohe n magnude. Thus ln 1 ln 1,,,1,1 (B) We eall a key esul fom he ulley analogy, Eq. (37), and dese o onsu a smla effeve suue. Defne and ˆ 1,1 ˆ 1 1 (B3) (A6) (A7) (B1) 18

19 ˆ as well as,1 Thus we have ln 1 a ln 1 1 ˆ Usng he fa ha ha,1 ˆ ˆ ha 1 1 1ha 1,1, ln 1a,1 ln 1 a, 1 1,1, ln 1a ln 1 a h h ln 1 ln 1 ha ln 1 a,1 ln 1 a, 1 ˆ 1 1 ˆ 1 1 1,1, h h 1 1,1,,,1 and ˆ ae all small, and eallng, a,, (B4) (B5) (B6), we an smlfy Eq. (B6) o 1h a 1, h ln 1 a,1,1 ln 1,1 a,1 ln 1, a,1 1h1 h Combnng Eqs. (B3) o (B7) wh a esul fom he ulley analogy, Eq. (37), leads o he desed fom ˆ1 ˆ ˆ 1 ˆ (B8) whh yelds he effeve ensle wave seed (n unsehed maeal oodnaes) as ˆ ˆ ˆ 1 (B9) These wo equaons ae of he same fom as n he sngle-laye membane ma oblem. A vey sho mes and long mes, esevely, Eq. (B7) edues o a,11h1 a,h 1h1 h whh s vey lose o Eq. (A7) when sans ae small, and we oban 1ha 1,1 ha, lm (B11) 1h1 h To evaluae he dffeene beween Eqs. (B1) and (B11) we assume a faly exeme bu no uneals ase whee he wo layes have equal hkness, hh1 h, bu maeal 1 s half as dense as maeal,.e., 1. Also he lghe maeal has a ensle wave seed, a,1, we as lage as he ohe, a,1 a, (hus he lghe maeal also has double he sffness). Usng maeal 1 as he efeene maeal, Eqs. (B1) and (B11) gve he lms a h 1 a h.6666a,,1 1 1,1 1 1,1 31h1 1 1,1 1 1,1 a,1 31h1 (B7) (B1) ha ha (B1) These lmng effeve wave seeds dffe by only 6%, dese he vey lage dffeenes n laye oees. Lookng a an nemedae value,,1 1 lose o whee he eak san mgh be eahed n he ma oblem a modeae aeal densy ao, (B7) yelds 19

20 ln 11 a,1 a,1 Subsung he wo lms, a, and a,1.771, no he gh-hand sde of Eq. (B13) gves ˆ fnd ha a a a, whh only dffe by abou 1.5%. Though eaon on Eq. (B7) we,1,1,1 (B13) a. Sne he lowe bound dffes fom he oe value by less han 1% and yelds he oe lms, we use hs bound o alulae he effeve ensle wave veloy â a all mes. Thus by hoosng he aoae aamee se he analyss fo ma no a sngle membane an ea he oblem of ma no wo nefeng layes. The ual quany s he effeve ensle wave-seed aken as 1h a 1, h ln 1 a,1,1 ln 1,1 a,1 ln 1, a,1, h h whh gh afe ma gves Eq. (B1). 1 1 (B14) Aendx C: Cone Wave-Fon Poson vesus San Hsoy We onsde a sngle laye and eall Eqs. (5) and (57) bu aled o a sngle laye. Subsung he lae no he fome, akng he squae-oo, mullyng boh sdes by, and anellng ommon faos we oban he nsananeous equaon a (C1) Howeve we also have he nsananeous esul d d d d whh an be wen as Thus we an we o d d d a d d d a d d Inegang boh sdes yelds s sd s sds (C6) 1 a (C) (C3) (C4) (C5)

21 A ual on s ha sne,max 1. In fa, as gows o,max aoxmaely n ooon. Thus a good aoxmaon n (C6) s o use lef-hand sde. Thus he lef-hand sde beomes s,, gows unde he squae-oo on he,max s d s s d s,,, 1 1,max Thus fom (C6) and (C7) we have he smle esul 3,, (C7) a 3 3,, whh eaanges o 3 1 s ds (C8) 3 3,, 3 a sd 1 (C9) Aknowledgmens The auhos wsh o aknowledge he sonsosh ovded by he US Deamen of Juse hough he Naonal Insue of Juse unde gan 7-DE-BX-K3. Refeenes 1 P. M. Cunnff, An analyss of he sysem effes n woven fabs unde balls ma, Texle Reseah Jounal 6 (199), S. L. Phoenx and P. K. Powal, A new membane model fo he balls ma esonse and V5 efomane of mully fbous sysem, Inenaonal Jounal of Solds and Suues, 4 (3), P.K Powal and S. L. Phoenx, Modelng of sysem effes n balls ma no mul-layeed fbous suues fo sof body amos, Inenaonal Jounal of Faue 135 (5), P. K. Powal and S.L. Phoenx, Effes of laye sakng ode on he V5 veloy of wo-layeed hybd amo sysem, Jounal of Mehans of Maeals and Suues, 3 (8), D.C. Pevosek, H.B. Chn, Y.D. Kwon and J.E. Feld, San ae effes n ulasong olyehylene fbes and omoses, Jounal of Aled Polyme Sene: Aled Polyme Symosum, 47 (1991),