Theoretical Development of an Orthotropic Elasto-Plastic Generalized Composite Material Model

Transcription

1 3 th Intrnational LS-DYNA Usrs onfrnc Sssion: Arosac hortical Dvlomnt of an Orthotroic Elasto-Plastic Gnralizd omosit Matrial Modl Robrt K. Goldbrg and Klly S. arny NASA Glnn Rsarch ntr, lvland O Paul Du Bois Gorg Mason Univrsity, airfax VA anio offarth, Josh arrington and Subramaniam Rajan Arizona Stat Univrsity, m AZ Gunthr Blanknhorn Livrmor Softwar chnology ororation, Livrmor A Abstract h nd for accurat matrial modls to simulat th dformation, damag and failur of olymr matrix comosits is bcoming critical as ths matrials ar gaining incrasd usag in th arosac and automotiv industris. Whil thr ar svral comosit matrial modls currntly availabl within LS-DYNA, thr ar svral faturs that hav bn idntifid that could imrov th rdictiv caability of a comosit modl. o addrss ths nds, a combind lasticity and damag modl suitabl for us with both solid and shll lmnts is bing dvlod and is bing imlmntd into LS-DYNA as MA_3. A ky fatur of th imrovd matrial modl is th us of tabulatd strss-strain data in a varity of coordinat dirctions to fully dfin th strss-strain rsons of th matrial. o dat, th modl dvlomnt fforts hav focusd on crating th lasticity ortion of th modl. h sai-wu comosit failur modl has bn gnralizd and xtndd to a strain-hardning basd orthotroic matrial modl with a non-associativ flow rul. h cofficints of th yild function, and th strsss to b usd in both th yild function and th flow rul, ar comutd basd on th inut strss-strain curvs using th ffctiv lastic strain as th tracking variabl. h cofficints in th flow rul ar comutd basd on th obtaind strss-strain data. h dvlod matrial modl is suitabl for imlmntation within LS-DYNA for us in analyzing th nonlinar rsons of olymr comosits. Introduction As comosit matrials ar gaining incrasing us in aircraft comonnts whr imact tolranc is imortant (such as th turbin ngin fan cas), th nd for accurat matrial modls to simulat th dformation, damag and failur rsons of olymr matrix comosits undr imact conditions is bcoming mor critical. Within LS-DYNA [], thr ar svral matrial modls availabl for alication to th analysis of comosits. or xaml, th hang-hang failur modl [] is utilizd in MA_ and MA_54. In ths modls, combinations of ratios of strsss to failur strngths ar utilizd to rdict fibr or matrix basd failur. h rsons is assumd b linar lastic, with limitd caability to simulat th nonlinar shar rsons. In MA_ th failur is assumd to b brittl, whil in MA_54 th comosit lastic constants -

2 Sssion: Arosac 3 th Intrnational LS-DYNA Usrs onfrnc ar slctivly rducd basd on th failur mod, and a gradual unloading is rmittd until ultimat lmnt failur is rachd. In MA_58, a continuum damag modl dvlod by Matznmillr t al [3] is mloyd, whr th initiation and accumulation of damag is assumd to b th rimary drivr of th nonlinarity in th comosit rsons. h failur strsss and strains of th matrial in ach of th coordinat dirctions ar scifid by th usr, and th matrial strss-strain curvs ar aroximatd basd on this data. h original vrsion of th matrial modl assums a matrial rsons that is indndnt of strain rat. owvr, in MA_58 a modifid vrsion of th modl is mloyd which incororats strain rat dndnc into th matrial rsons []. A viscolastic Prony sris basd on shar moduli is usd to modify th comutd strsss. h strain rat dndnc in all of th coordinat dirctions is assumd to b idntical. MA_6 is an additional continuum damag mchanics modl availabl within LS-DYNA [4]. A strss to strngth ratio aroach in a varity of coordinat dirctions is usd to scify th bginning of ithr fibr or matrix basd failur. Aroriat failur mod basd damag functions ar thn usd to comut th rduction in lastic moduli in ach of th coordinat dirctions. In MA_9, basd on th ODAM modl [5], a strain vrsus failur strain ratio basd aroach is usd to rdict th initiation of damag in a sublaminat of th comosit. Sarat damag accumulation and modulus rduction functions basd on failur mod and coordinat dirction ar imlmntd in a continuum damag mchanics formulation. In MA_, damag accumulation functions basd on currnt strains, damag initiation strains and failur strains ar usd to rduc th lastic moduli of th comosit in ach of th coordinat dirctions in a damag mchanics aroach []. In MA_6, which is basd on a damag and failur modl dvlod by Pinho t al [6,7], sarat modls for fibr tnsion failur, fibr kinking failur, matrix tnsil failur and matrix comrssiv failur ar dvlod, whr diffrnt functional forms ar usd for ach of th failur critria (for xaml, th matrix comrssion failur critrion is basd on an xtnsion of th Mohr-oulomb failur critrion). h various failur critria ar combind togthr using fractur mchanics concts to dvlo a constitutiv modl. In MA_6, basd on a modl dvlod by amonho, t al [8,9], an nrgy aroach is utilizd to gnrat damag functions in various coordinat dirctions within th contxt of a continuum damag mchanics formulation. Whil all of th matrial modls discussd abov hav bn utilizd with som lvl of succss in modling th imact rsons of olymr comosits, thr ar som aras whr th rdictiv caability can b imrovd. In gnral, th xisting modls oftn rquir significant a riori knowldg of th damag and failur such that thir us as rdictiv tools can b limitd. Whil ths modls gnrally assum that th nonlinar rsons of th comosit is du to ithr dformation mchanisms (lasticity) or damag mchanisms, an imrovd modl should hav th caability to simulat th actual matrial bhavior in which th matrial nonlinarity is du to a combination of both dformation and damag mchanisms. h inut to th currnt matrial modls gnrally consists of oint-wis rortis that lad to curv fit aroximations to th matrial strss-strain curvs. his ty of aroach lads to ithr modls with only a fw aramtrs, which rovid a crud aroximation at bst to th actual strss-strain curv, or to modls with many aramtrs which rquir a larg numbr of comlx tsts to charactriz. An imrovd aroach would b to us tabulatd data from a wll-dfind st of xrimnts to accuratly dfin th comlt strss-strain rsons of th matrial. urthrmor, many of th xisting modls ar only suitabl for us with two-dimnsional shll lmnts, which cannot catur th through-thicknss rsons which might b significant in imact alications. Idally, a fully thr-dimnsional formulation suitabl for us with solid lmnts would b -

3 3 th Intrnational LS-DYNA Usrs onfrnc Sssion: Arosac availabl, along with th shll lmnt formulation. In addition, incororating th ffcts of strain rat is only ossibl in a fw of th xisting modls, and in thos modls, th ffcts of strain rat on th matrial rsons ar assumd to b th sam in all of th matrial coordinat dirctions. o bgin to addrss ths nds, a multi-institution consortium has bn formd to dvlo and imlmnt a nw comosit matrial modl within LS-DYNA, which will b imlmntd as MA_3. urrntly, th rimary focus of th ffort has bn on th dvlomnt and tsting of th orthotroic macroscoic thr-dimnsional lasticity modl. As will b discussd in mor dtail in th following sctions of this ar, th commonly usd sai-wu comosit failur critrion [] has bn gnralizd and xtndd to a strain-hardning modl with a nonassociativ flow rul. A similar aroach has bn usd by Sun and hn [], in which a gnral quadratic lastic otntial function was gnralizd into a lasticity modl suitabl for us with comosits. h cofficints of th yild function for th nw comosit modl ar dtrmind basd on tabulatd strss-strain curvs in th various normal and shar dirctions, along with slctd off-axis curvs. h non-associativ flow rul is usd to comut th comonnts of th lastic strain along with th ffctiv lastic strain. h strsss that ar usd in th flow rul ar comutd basd on th inut tabulatd strss-strain curvs. Systmatic rocdurs hav bn dvlod to comut th various cofficints in th yild function and th flow rul basd on th tabulatd inut data. An imortant oint to not is that th dvlod matrial law is not limitd to th analysis of unidirctional lis alon. Altrnativly, th matrial modl is mant to b a fully gnralizd modl suitabl for us with any comosit architctur (unidirctional, laminatd or txtil). In th following sctions of this ar, a dtaild drivation of th yild function and th flow law ar rsntd. h rocdurs to b usd to charactriz th matrial constants in th yild function and th flow law ar also discussd in dtail. Dtails of th numrical imlmntation of th matrial modl within LS-DYNA and dscritions of vrification and validation studis that wr conductd using th dvlod matrial modl ar availabl in a comanion ar []. Matrial Law Drivation A gnral quadratic thr-dimnsional orthotroic yild function basd on th sai-wu failur modl is scifid as follows, whr --3 rfr to th rincial matrial dirctions. f a () In th yild function, σ ij rrsnts th strsss and ij ar cofficints that vary basd on th currnt valus of th yild strsss in th various coordinat dirctions. By allowing th cofficints to vary, th yild surfac volution and hardning in ach of th matrial dirctions can b rcisly dfind. h valus of th normal and shar cofficints can b dtrmind by simlifying th yild function for th cas of unidirctional tnsil and comrssiv loading in ach of th coordinat dirctions along with shar tsts in ach of th shar dirctions, with rsults as shown blow. -3

4 Sssion: Arosac 3 th Intrnational LS-DYNA Usrs onfrnc a () In th abov quation, th strsss ar th currnt valu of th yild strsss in th normal and shar dirctions (dtrmind using rocdurs to b discussd blow), whr th surscrit rrsnts th tnsil yild strss, and th surscrit rrsnts th absolut valu of th comrssiv yild strss. o dtrmin th valus of th off-axis cofficints (which ar rquird to catur th strss intraction ffcts), th rsults from 45 off-axis tsts in th various coordinat dirctions can b usd. or xaml, th strsss in th local matrial axis systm rsulting from a uniaxial tnsil tst of a [45 ] comosit (for a unidirctional comosit), or any uniaxial tnsil tst conductd at 45 in th - lan from th longitudinal () matrial axis for a multi-ly laminatd or txtil comosit, can b comutd to b th following by th strss transformation quations [] (3) In th abov quation, σ 45 is th structural lvl currnt uniaxial yild strss rsulting from th off-axis tnsil tst, and th rmaining strsss ar th matrial axis systm basd strsss. By substituting this quation into Equation () and solving, th following xrssion can b obtaind for th off-axis constant (4) whr all of th trms ar as dfind arlir. By using similar rocdurs involving 45 rotations in th -3 lan and th -3 lan, similar xrssions for th constants 3 and 3 can b dtrmind. A non-associativ flow rul is usd to comut th volution of th comonnts of lastic strain. h lastic otntial for th flow rul is as shown blow h (5) whr σ ij ar th currnt valus of th strsss and ij ar indndnt cofficints, which ar assumd to rmain constant. Procdurs for dtrmining th valus of th cofficints will b discussd latr in this ar. h lastic otntial function in Equation (5) is usd as follows in a -4

5 3 th Intrnational LS-DYNA Usrs onfrnc Sssion: Arosac flow law, whr th usual normality hyothsis from classical lasticity [3] is assumd to aly and th variabl λ is a scalar lastic multilir. h 3 h 3 h 3 3 h h h 66 3 h (6) In th abov quation, ε ij ar th comonnts of lastic strain. or th shar comonnts, tnsorial, not nginring, strains ar usd. By xamining ratios of th comonnts of lastic strain, th normal and off-axis cofficints ij in th lastic otntial function can b rlatd to lastic Poisson s ratios as follows, which as will b shown latr in this ar, is usful in dvloing rocdurs to charactriz th valus of th cofficints (7) An imortant oint to not is that whil th yild function can accommodat diffrncs btwn th tnsil and comrssiv rsonss of th comosit, th flow law, du to its lack of linar trms, cannot mak such a distinction. owvr, introduction of a linar trm would mak th lastic Poisson s ratio, and thus th cofficints in th flow law, highly dndnt on th strss which would most likly lad to rratic bhavior of th modl. Anothr considration is that th cofficints of th flow law could b assumd to vary basd on th currnt strss and strain stat, but that would rquir dvloing volution conditions on th flow law, which would b vry difficult to dtrmin basd on xrimntal data. Givn th flow law, th rincial of th quivalnc of lastic work [3] can b usd to dtrmin xrssions for th ffctiv strss and ffctiv lastic strain. By taking th vctor roduct of th strss tnsor and th ffctiv lastic strain tnsor, on nds u with th lastic otntial function multilid by th lastic multilir λ. Sinc by th rincial of th -5

6 Sssion: Arosac 3 th Intrnational LS-DYNA Usrs onfrnc quivalnc of lastic work th vctor roduct of th ffctiv strss and th ffctiv lastic strain must qual th roduct of th ffctiv strss and th ffctiv lastic strain, on can conclud that th lastic otntial function h can b dfind as th ffctiv strss and th lastic multilir λ can b dfind as th ffctiv lastic strain. his conct is xrssd mathmatically in th quation shown blow W h : h (8) : whr σ is th ffctiv strss and ε is th ffctiv lastic strain. o comut th currnt valu of th yild strsss ndd for th yild function, th common ractic in lasticity constitutiv quations is to us analytical functions to dfin th volution of th strsss as a function of th comonnts of lastic strain (or th ffctiv lastic strain). Altrnativly, in th dvlod modl tabulatd strss-strain curvs ar usd to track th yild strss volution. h usr is rquird to inut twlv strss vrsus lastic strain curvs. Scifically, th rquird curvs includ uniaxial tnsion curvs in ach of th normal dirctions (,,3), uniaxial comrssion curvs in ach of th normal dirctions (,,3), shar strss-strain curvs in ach of th shar dirctions (-, -3 and 3-), and 45 dgr off-axis tnsion curvs in ach of th -, -3 and 3- lans. h 45 dgr curvs ar rquird in ordr to rorly catur th strss intraction ffcts. By utilizing tabulatd strss-strain curvs to track th volution of th dformation rsons, th xrimntal strss-strain rsons of th matrial can b caturd xactly without any curv fit aroximations. h rquird strss-strain data can b obtaind ithr from actual xrimntal tst rsults or by aroriat numrical xrimnts utilizing stand-alon cods. urrntly, only static tst data is considrd. utur fforts will involv adding strain rat and tmratur dndnt ffcts to th comutations. or crtain comosit architcturs, th numbr of strss-strain curvs that nd to b inut can b simlifid. or xaml, for unidirctional comosits, th normal rsons in th 3 dirction can b assumd to b idntical to th normal rsons in th dirction du to th transvrs isotroy of th matrial. Likwis, th shar rsons in th -3 dirction can b assumd to b idntical to th shar rsons in th - dirction. Similarly, th 45 dgr off-axis curv in th -3 lan can b assumd to b idntical to th 45 dgr off-axis curv in th - lan. Du to th transvrs isotroy of a unidirctional comosit, on can also dmonstrat that for th cas of a unidirctional comosit th 45 dgr off-axis curv in th -3 lan is aroximatly qual to th normal strss-strain curv in th (or 3 dirction). or othr comosit architcturs with significant symmtry (such as two-dimnsional lain or satin wav comosits), a crtain dgr of simlification in th strss-strain data that is rquird to b inut is also ossibl. o track th volution of th dformation rsons along ach of th strss-strain curvs, th ffctiv lastic strain is chosn to b tracking aramtr. Using th numrical rocdur dscribd brifly latr in this ar and in mor dtail in a comanion ar [], th ffctiv lastic strain is comutd for ach tim/load st. h strsss for ach of th tabulatd inut curvs corrsonding to th currnt valu of th ffctiv lastic strain ar thn usd to comut th yild function cofficints. o nabl this tracking, ach of th inut strss-strain curvs nds to b convrtd into a lot of strss vrsus ffctiv lastic strain. o nabl this convrsion, th rincial of th quivalnc of lastic work is usd again to rlat th -6

7 3 th Intrnational LS-DYNA Usrs onfrnc Sssion: Arosac unidirctional lastic strain incrmnt to th ffctiv lastic strain incrmnt. or xaml, for th cas of unidirctional loading in th dirction th ffctiv lastic strain incrmnt (and intgratd currnt total lastic strain) can b comutd as follows d d d d ij ij d hd d d hd d (9) whr σ is th unidirctional strss in th loading dirction, dε is th lastic strain incrmnt in th loading dirction and dε is th incrmnt in ffctiv lastic strain. Not that in a gnral cas th incrmnt in ffctiv lastic strain can b comutd as th incrmntal ara undr th strss-lastic strain curv dividd by th currnt valu of th ffctiv strss. A grahical rrsntation of rocss is shown blow in igur. E d h Usr rovidd load curvs ar tru strss vrsus tru lastic strain Intrnally stord is tru strss vrsus ffctiv lastic strain igur : onvrsion of strss vrsus lastic strain curvs to strss vrsus ffctiv lastic strain curvs. o comut th volution of th ffctiv lastic strain, a numrical algorithm basd on th radial rturn mthod is mloyd. Givn th rvisd valu of th ffctiv lastic strain, th yild strss valus and th ovrall strss stat of th matrial can b udatd. Dtails of th numrical imlmntation ar rovidd in a comanion ar [], but som of th ky thortical fundamntals of th algorithm ar givn blow. h standard lastic constitutiv quation is usd to comut th rvisd strsss, whr th flow law is alid for th comutation of th lastic strains. -7

8 Sssion: Arosac 3 th Intrnational LS-DYNA Usrs onfrnc h : : () In th abov quation, is th standard lastic stiffnss matrix, ε is th total strain, and th rmaindr of th trms ar as dfind rviously. o comut th ffctiv lastic strain rat, th consistncy condition is alid in combination with th lastic constitutiv quation f f f q q f h f dq f : : q d f : f h f dq : q d () whr f is th yild function dfind in Equation () and q is th vctor of yild strsss basd on th inut strss-strain curvs. q () o comut th drivativ of th q vctor with rsct to th ffctiv lastic strain λ, a chain rul ty of comutation is carrid out. dq dq d (3) d d d h drivativ of th yild strss vctor with rsct to lastic strain is th instantanous slo of th strss vrsus lastic strain curv from ach of th inut strss-strain curvs. h drivativ of th lastic strain with rsct to th lastic multilir λ can b comutd from th flow law for ach of th scializd unidirctional (or off-axis) cass rrsntd by th inut strss-strain curvs. Not that for th off-axis cass th flow law must first b writtn in th matrial axis systm and thn th lastic strains must b transformd into th structural axis systm in ordr to comut th ror drivativ of th lastic strain with rsct to th lastic multilir. A scant itration aroach is usd in combination with th radial rturn mthod to comut th scific valu of th ffctiv lastic strain rat []. -8 haractrization of low Law officints h cofficints in th flow rul nd to b charactrizd basd on data obtaind from xrimntal strss-strain curvs for th comosit. Scifically, data from th xrimnts lading to th inut strss-strain curvs must b utilizd. h data can b obtaind ithr from actual xrimnts or numrical xrimnts. or xaml, if on knows th mchanical

9 3 th Intrnational LS-DYNA Usrs onfrnc Sssion: Arosac rortis and strss-strain curvs for th fibr and matrix constitunts, numrical analyss can b conductd using ithr high fidlity finit lmnt analysis (whr th fibr and matrix ar modld as distinct constitunts) or analytical tools such as th NASA Glnn dvlod micromchanics cod MA/GM [4] to obtain numrical aroximations to th actual comosit xrimntal data. irst, th rocdurs usd to obtain th rquird cofficints for th simlifid cas of a unidirctional carbon fibr basd olymr matrix comosit will b discussd. Aftrward, a gnralizd rocdur to obtain th rquird constants for a gnral comosit will b rsntd. or a unidirctional carbon fibr comosit, as discussd by Sun and hn [] a rasonabl aroximation to mak is that du to th linar bhavior of th carbon fibr th lastic strain in th fibr dirction (th dirction) is qual to zro for all valus of strss. By xamining th scond xrssion in Equation (6), on can conclud that for a gnral strss stat th only way th lastic strain in th dirction can b zro is for th cofficints, and 3 to all b idntically zro. Nxt, sinc th transvrs tnsion rsons of th comosit can dislay a crtain dgr of nonlinarity, a rasonabl aroximation to mak is that for th cas of a unidirctional load in th dirction th ffctiv strss is qual to th alid transvrs tnsion strss σ. By simlifying th lastic otntial function g (shown in Equation (5)) for th cas of a unidirctional transvrs tnsion strss, and by stting this trm (along with th transvrs tnsil strss) qual to th ffctiv strss, on can show that th valu of th cofficint must qual on. Sinc a unidirctional comosit is transvrsly isotroic, on can st th valu of to also qual on. Givn ths valus, using th dfinition of th lastic Poisson s ratio υ 3 shown in Equation (7), on can find that th cofficint 3 quals th ngativ of th lastic Poisson s ratio υ 3. Sinc th flow surfac cofficints ar assumd to b constant, an avrag valu of th lastic Poisson s ratio must b dtrmind from th xrimntal and/or numrical tst data from a unidirctional transvrs tnsion tst. o dtrmin th in-lan shar cofficint, utilizing a rocdur similar to that dvlod by Sun and hn [] for thir modl, th lastic otntial function g givn in Equation (5) and th lastic strain dfinition givn in Equation (6) can b simlifid for th cas of in-lan shar loading in th dirction, yilding th following xrssions. g.5.5 (4) Starting from th xrimntal strss vrsus lastic strain lot for th cas of in-lan shar loading, using th xrssions in Equation (4) th valu of can b otimizd to rovid th closst ossibl match with th ovrall ffctiv strss vrsus ffctiv strain curv, slctd for th cas of a unidirctional carbon fibr comosit to b th transvrs tnsion strss vrsus lastic strain curv. Du to th transvrs isotroy of th unidirctional comosit, th valu of th cofficint 66 can b st qual to th valu of th cofficint. o dtrmin th valu of th 55 cofficint, a similar rocdur can b carrid out for th cas of shar loading in th 3 dirction. Altrnativly, du to th transvrs isotroy of th comosit, by comuting th ffctiv strss for th cas of off-axis loading in th -3 lan, an isotroic ty of rlation can b dtrmind for th cofficint 3-9

10 Sssion: Arosac 3 th Intrnational LS-DYNA Usrs onfrnc 55 3 (5) whr υ 3 is th lastic Poisson s ratio in th 3 dirction, obtaind from th matrial data. Similar rocdurs ar usd for th cas of a gnral comosit with no idntifid symmtris. An xaml of a gnral comosit would b a triaxially braidd comosit which is modld as a smard, homognous matrial. Anothr xaml would b a gnralizd laminatd comosit modld as a smard, homognous matrial. or th cas of a gnral comosit, a unidirctional tnsion tst in th longitudinal () dirction is chosn as th baslin cas. By utilizing Equation (5) and stting th lastic otntial function and th unidirctional longitudinal strss qual to th ffctiv strss, th valu of th cofficint can b found to qual on. Basd on th rlations givn in Equation (7), th valu of th cofficint can b found to qual th ngativ of th lastic Poisson s ratio ν, and th valu of th cofficint 3 can b found to b qual to th ngativ of th lastic Poisson s ratio ν 3. As bfor, th valu of th lastic Poisson s ratio usd in th comutations is th avrag valu of th aramtr dtrmind from th xrimntal and/or numrically obtaind data. In a similar fashion, th valus of th cofficints, 3 and can b comutd (6) o comut th valus of th cofficints, 55, and 66, th sam rocdur that was utilizd to comut G for th cas of a unidirctional carbon fibr comosit is usd, only all thr shar loading cass (σ, σ 3, and σ 3 ) must b usd. - onclusions A gnralizd lasticity modl suitabl for us in olymr comosit imact simulations has bn dvlod. h lasticity modl will b art of a gnralizd lasticity and damag modl which will b imlmntd into LS-DYNA as MA_3. h yild function is basd on th sai-wu comosit failur modl, and a suitabl nonassociativ flow rul was dfind. Mthods of utilizing tabulatd strss-strain data to track th volution of th yild strsss as a function of th ffctiv lastic strain hav bn dvlod. h lastic constitutiv quation can b usd in combination with th consistncy condition to comut th ffctiv lastic strain. In a comanion ar [], furthr dtails of th numrical algorithm ar rovidd along with xamls of vrification and validation studis which hav bn rformd to xamin th accuracy and fficincy of th analytical modl. utur fforts will involv th dvlomnt of an quivalnt damag modl, to b mloyd in combination with th lasticity modl, which will also b basd on utilizing tabulatd strssstrain data to track th volution of damag in th comosit. Elmnt failur will also b

11 3 th Intrnational LS-DYNA Usrs onfrnc Sssion: Arosac incororatd within th modl. In addition, strain rat and tmratur ffcts will b addd to th analysis caability. Ovrall, whn comltd th comosit modl as imlmntd in MA_3 will rovid significant imrovmnts to th stat of th art in th modling of th imact rsons of olymr matrix comosits. Acknowldgmnts Authors offarth, arrington and Rajan gratfully acknowldg th suort of th dral Aviation Administration through Grant #-G- ntitld omosit Matrial Modl for Imact Analysis, William Emmrling, chnical Monitor. Rfrncs [] allquist, J.: LS-DYNA Kyword Usr s Manual, Vrsion 97. Livrmor Softwar chnology ororation, Livrmor, A, 3. [] hang,.-k.; and hang, K.-Y.: A Progrssiv Damag Modl for Laminatd omosits ontaining Strss oncntrations, Journal of omosit Matrials, Vol., , 987. [3] Matznmillr, A.; Lublinr, J; and aylor, R.L.: A constitutiv modl for anisotroic damag in fibrcomosits, Mchanics of Matrials, Vol.,. 5-5, 995. [4] Yn,..: "Ballistic Imact Modling of omosit Matrials," 7th Intrnational LS-DYNA Usrs onfrnc, Livrmor Softwar chnology ororation,. [5] Williams, K.V.; Vaziri, R.; and Poursarti, A.: A hysically basd continuum damag mchanics modl for thin laminatd comosit structurs, Intrnational Journal of Solids and Structurs, Vol. 4,. 67-3, 3. [6] Pinho, S..; Iannucci, L.; and Robinson, P.: Physically-basd failur modls and critria for laminatd fibrrinforcd comosits with mhasis on fibr kinking: Part I: Dvlomnt, omosits: Part A, Vol. 37, , 6. [7] Pinho, S..; Iannucci, L.; and Robinson, P.: Physically-basd failur modls and critria for laminatd fibrrinforcd comosits with mhasis on fibr kinking: Part II: E imlmntation, omosits: Part A, Vol. 37, , 6. [8] Maimi, P.; amanho, P.P.; Mayugo, J.A.; and Davila,.G.: A continuum damag modl for comosit laminats: Part I onstitutiv modl, Mchanics of Matrials, Vol. 39, , 7. [9] Maimi, P.; amanho, P.P.; Mayugo, J.A.; and Davila,.G.: A continuum damag modl for comosit laminats: Part II omutational imlmntation and validation, Mchanics of Matrials, Vol. 39, , 7. [] Danil, I.M; and Ishai, O.: Enginring Mchanics of omosit Matrials. Oxford Univrsity Prss, Nw York, 6. [] Sun,..; and hn, J.L.: A Siml low Rul for haractrizing Nonlinar Bhavior of ibr omosits, Journal of omosit Matrials, Vol. 3,. 9-, 989. [] offarth,.; arrington, J.; Rajan, S.; Goldbrg, R.; arny, K.; DuBois, P.; Blanknhorn, G.: Vrification and Validation of a hr-dimnsional Gnralizd omosit Matrial Modl, 3 th Intrnational LS-DYNA Usrs onfrnc, Livrmor Softwar chnology ororation, 4. [3] Khan, A.S.; and uang, S.: ontinuum hory of Plasticity. John Wily and Sons, Nw York, 995. [4] Bdnarcyk, B.A.; and Arnold, S.M.:. MA/GM 4. Usr s Manual - Kywords Manual, NASA/M- -77/VOL, National Aronautics and Sac Administration, Washington, D..,. -