Robe NEVIERE, Fancine STANKIEWICZ SNPE Cene de Recheches du BOUCHET BP, 97 Ve-le-Pei FRANCE Andé PFIFFER CELERG Eablissemen de S MEDARD BP, 65 S MEDARD-en-JALLES FRANCE. INTRODUCTION Due o a quie complex molecula sucue, he mechanical behaviou of solid popellan gains may hadly be heoeically fomalized in a consiuive law and hen implemened fo a houghou numeical descipion. I is hus believed ha a decisive impovemen of mechanical inegiy assessmen of popellan sucues may be expeced fom any descipion ha would avoid he exac calculaion of he sess fields. Wih his sandpoin a mehod is poposed o assess he sevice life of solid popellan gains which is based on he analysis of he sain fields calculaed duing he diffeen specific loading cases of he sevice life. The cene poin of he mehod is a genealized sain cieion which is capable o educe he hee dimensional oupus of a finie elemens calculaion (sains ensos) ino an uneasily equivalen defomaion. This uneasily equivalen defomaion is hen a consisen quaniy which may be compaed o expeimenal esuls which ae obviously moe easily gaheed fo his common and convenien sain field of uniaxial ension.. INDUSTRIAL BACKGROUND The mechanical inegiy assessmen of popellan gains is buil up on a numeical calculaion in which he essenial and legiimae assumpion efes o he incompessible behaviou of he modelized solid. Fo solid popellans, his assumpion is jusified fisly by he expeimenal evidence of such a behaviou (a leas up o limied sain ampliudes) and also by he fac ha hese calculaions ae he majoan efeence o analyse he simulaions of he calculaed sucues. A second specific feaue of he popellan gain numeical simulaions lies in he boundaies condiions specified in ems of imposed displacemens boh duing cooling afe casing and duing he opeaional mission (inenal pessue of combusion). Fom he combinaion of hese boundaies condiions and of an incompessible maeial behaviou, i esuls ha he displacemen field of his axisymeic mechanical poblem is only a he second ode dependan on he model implemened (in fac he soluion depends mainly on he volumeic behaviou assumpion). The poposed descipion is hus eniely based on a sain analysis consuced on he assumpion of incompessibiliy wih he objecive o allow he sucue numeical calculaions esuls o be compaed by a simple mehod o uniaxial ensile ess expeimenal infomaion.. ULTIMATE STRAINS OF PROPELLANTS A ypical expeimenal esponse of a HTPB solid popellan o muliaxial sain loadings is depiced schemaically by figue. I may be deduced fom hese esponses o uniaxial ension, equibiaxial ension and simple shea, ha he popellan mechanical behaviou is songly sain field dependan and Pape pesened a he RTO AVT Specialiss Meeing on Advances in Rocke Pefomance Life and Disposal, held in Aalbog, Denmak, -6 Sepembe, and published in RTO-MP-9. RTO-MP-9 6 -
especially in ems of ulimae sains ampliude. Since he popellan maeial is songly viscoelasic, hese facue sains ae ae and empeaue dependan bu i will be consideed houghou his pape as an expeimenal evidence ha he aio beween hese muliaxial facue sains is independen of he expeimenal condiions. Sess EquiBiaxial Tension e.5% EE TU Uniaxial Tension e 5% Simple Shea e 45% E/ E TU sain l l Figue : Typical Response of Solid Popellan o Thee Diffeen Sain Fields. Fom his expeimenal sa poin, he objecive is o descibe he sain field dependence of he maeial ulimae sains in a genealized appoach keeping in mind he consain of a simple indusial usable mehod. The basis of his mehod efes o he incompessible behaviou assumpion fo finie defomaion ampliudes which may be wien as he volume consevaion:.. wih,, he pincipal componens of he displacemen gadien maix, F ~, chosen as he defomaion measue. This abiay choice of he sain measue is jusified by he expeimenal convenience of he Piola-Lagange coninuum mechanics descipion (he laboaoy esuls ae given as l l and T A whee l and AO ae he unseched dimensions of he sample and T he cuen foce). Wih hese definiions, fo he diffeen sain fields depiced in fig., he chaaceisic sain maix, F ~, fo each field is: Uniaxial ension Equibiaxial ension F ~ F ~ Fo simple shea he measued popeies ae no in he pincipal axis: F ~ γ 6 - RTO-MP-9
wih γ U he ecoded displacemen divided by he sample hickness fom which he fis pincipal l sain may be deduced by: γ + γ + 4. When each of hese equaions ae epesened and pojeced upon he plane ( i, j ), appeas a sysem of cuves which descibes a solid concave suface of evoluion ceneed on he poin (,,) defining he fee sain sae as shown on figue. To give a moe visual picue of he siuaion depiced i may be compaed o an open umbella which veex is meged o he poin (,,), he shaf goes along he issecix ( ) and each of he ibs descibes a paicula sain pah..5 i j i j j. i j i j j Uniaxial Tension Equibiaxial ension Simple Shea i.75.65.9.5 i Figue : Defomaion Pahs and Facue Poins. When each muliaxial facue sain is placed in his plane, appeas clealy he egula figue depiced on figue which is obviously descibing an ellipse. This plo may be guessed o be he esul of he inesecion of a plane pependicula o he space issecix ( ) wih he umbella solid suface. To demonsae he exisence of his feaue i is poposed o pefom a change in he efeence axis. Choosing he fis pincipal axis of his new efeence meged wih he issecix, he choice fo he wo emaining axis is abiay bu mus fom a caesian co-odinaes sysem. The effecive sysem adoped heeafe is shown on figue. RTO-MP-9 6 -
u v Figue : Refeence Axis Sysem Roaion. This new se of axis is of couse associaed wih a oaion maix each componen of which being he diecion coefficiens of he new axis sysem. If he classical opeaions of co-odinaes oaion ae hen applied o he esuls of figue and if he pojecions of he ansfomed esuls ae pefomed upon planes boh paallel and nomal o he issecix, one obains especively figues 4a and 4b..76.5 v.75.74. u UnT EqT SSh u.7 -.5..5 -.5 -.5..5 a) paallel pojecion b) pependicula pojecion Figue 4: Pojecions of he Sain Pahs and Facue Poins in he Tansfomed Co-odinaes Sysem. The analysis of hese pojeced figues confims he emakable following evidences: he sae equaion of incompessibiliy descibes a evoluion suface ceneed on he issecix of he pincipal sains space ( ) he facue poins associaed o diffeen sains fields descibes he inesecion of his suface wih a plane pependicula o he issecix which is compleely deemined by he heigh,, of he locaion inesecion poin along ha issecix. 6-4 RTO-MP-9
As a consequence any hee-dimensional sain field fiing he incompessibiliy equiemen equaion will facue when he inensiy,, measued in his same manne along he issecix will each his ciical heigh,. If i is noiced ha he facue cieion is compleely deemined by his single value of he heigh,, which defines he inesecion plane posiion along he issecix, he descipion of facue may be genealized o any muliaxial sain field which is no involved in he cieion consucion. In his way, evaluaing is a measue of he inensiy of any muliaxial sain fields encouneed in numeic calculaions. Fo convenience, he sain field inensiy is conveed ino an uniaxial equivalen sain of equal inensiy. This is done by he simply solving: ( P + P ) UT P. + P. + P. P. UT +. Wih P ij he oaion maix componens, (,, ) he calculaed componens of he sain enso and UT he equivalen uniaxial sain eseached. 4. SOLID PROPELLANT GRAIN INTEGRITY ASSESSMENT To give an oveview of he poenial applicaions of his cieion o solid popellan gains mechanical design, an inegiy assessmen evaluaion is pefomed fo a finocyl popellan sucue which is he gain of a acical ocke moo. The geomey of such a sucue is depiced on figue 5. One will assume he case of a elaively cold empeaue opeaional condiion (le say -4 C o moe pecisely θ- C) and a quie high inenal pessue of MPa. The calculaion solves he wo loading cases of hemal cooling and pessuisaion in he same poblem and akes ino accoun fo he sucue siffness. The popellan is assumed o be incompessible and is epoduced in he calculaion by a simple neo-hookeen behaviou wih a modulus C (he shea modulus) fixed a MPa. The value of his modulus has no impoan influence on he esul in em of displacemen field (as menioned ) since only he sain field is equied fo he mechanical inegiy assessmen. The hemal and pessue induced sains ae exaced fom he numeical simulaion pocess in a seies of poins expeced o be ciical owads facue. Geneally hese poins ae locaed in cene of he inenal boe and a he fee and head of he slo. Figue 5: Typical Geomey of a Tacical Missile Popellan Gain. The esuls of his calculaion in ems of pincipal sains duing his simplified sevice life ae shown fo example in he middle of he cenal boe on figue 6. RTO-MP-9 6-5
.5 hemal loading Middle boe sain pessuisaion Geen's sain (l).5.95 Radial Hoop Longiudinal.85 5 5 loading (calculaion sep) Figue 6: Pincipal Sains duing he Sevice Life of a Tacical Moo. If aenion is paid o hese esuls, one may obseve ha he sain fields in hese poins ae fa away fom uniaxial ension and i is hus difficul o compae he pincipal values of his enso diecly o an expeimenal obsevaion. The pesened sain cieion inoduces he lacking ools o ovepass his poblem and o coninue he mechanical design of he sucue. Though he compaison o isohemal facue daa is heefoe accessible since he uniaxial equivalen sain has been evaluaed, i is pefeed o ake ino accoun fo he viscoelasic naue of he facue behaviou a his sage. This specific feaue of he mechanical behaviou of popellan leads o a coninuously vaying value of he facue sain as he empeaue is deceased. The figue 7 shows, as an example, he mase cuve of he facue sain fo a HTPB solid popellan as a funcion of he educe invese sain ae. &.a T log(facue sain in %)..5. owads he glass ansiion.5. -8. -.. log(/.at) Figue 7: Mase Cuve of he Facue Sain of an HTPB Popellan. 6-6 RTO-MP-9
An oiginal way o ea his difficuly is o cumulae a sevice life vaiable defined as he aio of each incemen of defomaion applied a he cuen empeaue and sain ae o he coesponding facue value fo his empeaue and sain ae []. The esuling definiion of he sevice life vaiable is hus: d D... ( θ, &,P ) To check he peinence of his appoach fo he hemal loading, a seies of uniaxial ension ess o which a coninuous cooling is supeposed ae pefomed. The maeial esponse o his paicula es is shown on figue 8 fo diffeen couples of sain ae and cooling aes. Also shown on his figue is he coesponding se of isohemal esponses. 4 Supeposed cooling Isohemal -5 C θ -6 C θ -54 C Sess (MPa) -4 C θ -9 C - C -5 C + C θ +5 C +6 C 5 5 5 Sain (%) Figue 8: Response of an HTPB Popellan o Tension wih Supeposed Cooling. These expeimenal esuls demonsae fisly he vey paicula feaue of he maeial esponse o hese loading condiions and secondly he sensible aleaion of he ulimae popeies in em of sain which obviously should be accouned fo in he sucue design owads facue. The validaion of he sain vaiable concep is he esul of he confonaion of he cumulaed vaiable wih he aio of he ensile ae o he cooling ae since hese wo popeies ae consan duing he ess. d &.d dθ θ&.d D d &. dθ D dθ θ& ( θ) θ& ( θ) ( θ) & and These elaionships povide a facue empeaue pedicion which is compaed fo consan aes aios o he expeimenal esuls as shown on figue 9. & θ & RTO-MP-9 6-7
8 & ( C/%) θ & Facue Poins Model Pedicion 6 4 limiing value of he cumulaed sevice life vaiable - -75-5 -5 5 5 75 θ ( C) inceasing cumulaed vaiable pediced facue empeaue Figue 9: Cumulaed Sevice Life Vaiable fo Uniaxial Tes wih Supeposed Cooling. Fo he pessue loading case, he same appoach is used bu, his ime, i should be aken ino accoun fo he pessue sensiive maeial sain capabiliy. Sill as an example, he facue sain fo he specific sain ae and empeaue of his paicula gain fiing condiion is shown on figue as a funcion of pessue. The benefic effec of he pessue on maeial popeies have been he objec of an exended wok in popellan indusy [,] and no deails will be given hee, bu i is poined ou ha a convenien way o descibe he pessue influence is o inoduce a pessue effec coefficien. This coefficien is defined as he aio of he sain (o sess) capabiliy a pessue P o is coesponding value a ambien pessue (his value is sain ae and empeaue dependan). Pessue effec coefficien..5..5. 5 5 Pessue (MPa) HTPB -4 C HTPB -4 C Exponenial Ajus. Figue : Pessue Effec Coefficien fo HTPB Popellan (-4 C - - sec - ). To complee he mechanical inegiy assessmen of he gain, he pessue effec coefficien has o be fied wih an exponenial adjusmen: 6-8 RTO-MP-9
A he em of hese effos he popellan gain mechanical inegiy is accessible since we have: he sain ensos calculaed along he sevice life in he ciical locaions an equivalen sain cieion o educe hese sains o an uniaxial equivalen a maeial capabiliy model aking ino accoun fo he sevice life loading condiions a cumulaive sevice life vaiable o quanify he impac of each sevice life individual loading on he mechanical inegiy of he maeial. These diffeen ools ae gaheed o poduce he plo of figue, on which appeas as a funcion of he loading pah in he middle of he boe, he maeial capabiliy and he equivalen sain in he sense of he poposed cieion. Is also eminded on his figue he convenional equivalen sains in common applicaion fo mechanical inegiy assessmen in SNPE official design mehod [4]. Namely and suppoed by a somehow diffeen analysis, a Sassi equivalen sain is used fo he hemal loading case and a Von Mises equivalen sain (Von Mises sess divided by he calculaion modulus) fo he pessue loading case. I should be emphasized ha he inconvenien of his double definiion of he equivalen sain leads o disconinuous safey coefficien and ha, as a song benefi, his poblem is ovepassed by he poposed mehod. Noa: he ocaedic sesses ae he fis and second invaian especively of he spheic and deviaoic componens of he sess enso. The ocaedic pessue is: and he ocaedic shea (in em of global enso ems) τ oc σ oc ( σ + σ + σ )..[( σ σ ) + ( σ σ ) + ( σ σ ) ] he coesponding sains ae he value of hese sesses divided by he modulus. 5 Middle boe sains equivalen sain (%) 5 5 5 hemal loading pessuisaion Maeial capabiliy Sassi sain Von Mises sain Poposed cieion 5 5 loading Figue : Sucual Loading and Maeial Capabiliy Necessay fo Mechanical Assessmen in Tems of Equivalen Uniaxial Sains (Pessuisaion Condiions -4 C and MPa). Noa: he dop in he Sassi equivalen sain is due o he pesence of he pessue in he equivalen sain definiion (see able ). RTO-MP-9 6-9
I should be noed on his figue ha he sain cieion is vey close o Sassi cieion in he hemal loading phase and exhibis a compaable slope o he Von Mises cieion in he pessuisaion phase. A convenien manne o quanify he diffeences inoduced in he safey coefficien calculaion by his new mehod is o compae he safey coefficiens ha would have been calculaed by he classical Sassi-Von Mises appoach, o hose issued fom an idenical definiion of he safey coefficien bu including he sain cieion and finally o he complee sevice life vaiable appoach. To allow a clea lecue of he obained figues in able which compaes all he mehods, he definiions involved in hei calculaion ae eminded in he following synopsis: Table : Synopsis of he Diffeen Safey Coefficien Calculaions Loading case Themal pessue load measue STASSI Sassi éq. sain STASSI Sain σoc + σoc + τoc E Von Mises éq. sess Von MISES Sess σ MISES 9xτoc Cσ emaks Capabiliy measue Safey coefficien definiion facue sain a final θ and oveall cooling ime: a θ a θ STASSI hemal sess fee facue sain a fiing θ, P and pessuisaion ime: σ a σ σ θ MISES,P a θ Sain Cieion Mehod Complee sevice Loading case Themal pessue life vaiable mehod equivalen sain equivalen sain load measue TH TH+P - TH Cumulaed equivalen sain TH+P Capabiliy measue Safey coefficien definiion facue sain a final θ and oveall cooling ime: a θ TH a θ hemal sain fee facue sain a fiing θ, P and pessuisaion ime: a θ,p hemal sain fee,p a TH+ P TH boh load cases facue sain evoluion as a funcion of educed sain ae and pessue: D a θ,p d θ ( θ, &,P...) 6 - RTO-MP-9
Wih hese diffeen design mehod he following figues fo he safey coefficien ae gaheed in able ogehe wih he values of he loadings and he coesponding capabiliy. Table : Safey Coefficien Figues fom Diffeen Mehods Middle boe Slo fee Slo head hemal pessue hemal pessue hemal pessue Sandad capabiliy 8.5% 7.88MPa 8.5% 7.88MPa 8.5% 7.88MPa mehod loading.%.8mpa 4.7% 4.48MPa 5.%.8MPa sandad Sc safey coef..7.79.94.75.9.7 Sain capabiliy 8.5% 5.5% 8.5% 5.5% 8.5% 5.5% mehod loading.%.8% 5.% 5.% 4.% 4.6% sandad Sc safey coef..7.55.86.5..4 Cumulaed sevice life vaiable safey coef..7.69.85.4..5 As could be expeced fom he diffeen definiions, and fom he esuls of figue, he safey coefficiens figues ae vey close whaeve he mehod used as long as he hemal loading is concened. The values fo he pessuisaion loading calls fo moe commens he fis of which being ha in he sandad mehod he safey coefficien figue is highe fo his loading case han fo he hemal one. This is obviously due o he coefficien definiion disconinuiy beween his wo loading cases and may lead o incoheence in he inepeaion of hese figues since one would expec he safey coefficien o be a deceasing funcion duing he sevice life. When he safey coefficien is evaluaed in ems of sains only his poblem vanishes bu he esuling figues ae ahe low fo he cold empeaue pessue loading specially in he fee of he slo (Sc.5). This figues ae due o once again o he choice in he secuiy coefficien mehod since he sain capabiliy fo his case is he von Mises sess (7.88MPa including he pessue effec) divided by he modulus which fo his high sain ae and cold empeaue is 5MPa. The las ange of safey coefficien in he able is he esul of he poposed mehod which gives coninuously deceasing values as he loading case is applied which magniude ae judged as saisfacoy since his paicula ocke moo neve shown facue fo cold empeaue fiing condiions. I is of couse of pincipal concen o check now on analog moos design owads facue he pedicive poenial of he poposed appoach. 5. CONCLUSIONS An oiginal mehod is poposed o assess he mechanical inegiy of popellan gain in acical solid ocke moos applicaions wih a special aenion paid o he cold fiing loading condiions. The mehod is eniely buil up on a homogeneous analysis of he boe sains which is suppoed by he inoducion of a muliaxial sain failue cieia. This cieia allows he educion of he mulidimensional sain enso which esuls of he numeical simulaions ino an uniaxial equivalen sain compaable o expeimenal daa. Insead of a discee evaluaion a he end of each loading case (hemal and pessue), he safey coefficien definiion is a cumulaed compaison of he acual sain applied o he insananeous value of he sain capabiliy of he maeial. As a esul of his definiion, he sevice life inegiy assessmen is measued by a coninuously deceasing value which is expeced o give a moe convenien inepeaion. RTO-MP-9 6 -
I should be menioned ha, hough aacive, he mehod lack of a houghou validaion on analog moos designed owads facue in cold fiing condiions. 6. ACKNOWLEDGEMENTS This sudy as been financially and scienifically suppoed by he Diecion Généale de l Amemen (DGA) and especially he SPNuc sevice. 7. REFERENCES [] J.P. NOTTIN Mechanical Behaviou of Solid Popellans duing Tensile Tes wih Tempeaue. P. RACIMOR Poc. AIAA/ASME 5 h Join Populsion Conf. MONTEREY, CAL, (989), AIAA Pape n 89-645. [] Y. TRAISSAC Mechanical Behaviou of a Solid Composie Popellan duing Moo Igniion. J. NINOUS Rubbe Chem. and Tech. vol.68 n pp.46-57 (995). R. NEVIERE [] R.J. FARRIS The Influence of Vacuoles Fomaion on he Response and Failue of Filled Elasomes. Tans. of he Sociey of Rheology :, pp.5-4 (968). [4] A. DAVENAS Solid Popellan Technology, ed. MASSON (989). 6 - RTO-MP-9