1995 rigid projectile

8/2/2019 1995 Rigid Projectile

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Pe r gam onInt. J. Impact Enyno Vol. 16, No. 5/6, pp. 801-831 , 1995Copyright ~ ' 1995 Elsevier Science LtdPrinted in G reat Britain. All rights reserved0734 -743 X(9 5)00 019- 4 0734 743X/95 $9.50+ 0.00

P E N E T R A T I O N O F A R I G I D P R O J E C T I L E I N T OA N E L A S T I C - P L A S T I C T A R G E T O F F I N I T E T H I C K N E S SA. L. YARIN, M. B. R U B I N a nd I. V. R O I S M A N

F acu l ty o f Mec h an ica l E n g in ee r in g , T ech n io n , I s r ae l I n s t i t u t e o f T ech n o lo g y , 3 2 00 0 Ha i f a , I s rae l

(Received 29 Sep t ember 1994 ; in rev ised for m 2 Janu ary 1995)

S u m m a r y - - T h i s p a p e r c o n s i d e r s t h e p r o b l e m o f n o n - s t e a d y p e n e t r a t i o n o f a r ig i d p ro j e c ti l e in t o a ne l a s t i c -p l a s t i c t a rg e t o f f i ni t e t h ick n ess . A sp ec if ic b lu n t p ro j ec t i l e sh ap e in t h e fo rm o f an o v o id o fRan k in e i s u sed b ec au se i t co r r e sp o n d s to a r easo n ab ly s im p le v e loc i ty fi e ld w h ich ex ac t ly sa ti s fi e s t h eco n t in u i ty eq u a t io n an d th e co n d i t i o n o f im p en e t r ab i l i t y o f t h e p ro j ec t i l e . T h e t a rg e t r eg io n i ssu b d iv id ed in to an e l a s t i c r eg io n ah ead o f t h e p ro jec t i l e w h ere t h e s t r a in s a r e a s su m ed to b e sm a l l , an da r i g id -p l a s t i c r eg io n n ea r t h e p ro j ec t il e w h ere t h e s t r a in s can b e a rb i t r a r i l y l a rg e . Us in g th e ab o v em en t io n ed v e lo c i ty f i e ld , t h e m o m en tu m eq u a t io n i s so lv ed ex ac t ly i n b o th t h e e l a s t i c an d th er ig id -p la s t ic reg ion s to f ind express ions fo r the p re ssu re an d s t ress f ields. The effects o f the f ree f ron tan d r ea r su r f aces o f t h e t a rg e t (w h ich is p re su m ed n o t t o b e t o o th in ) an d th e sep a ra t io n o f t h e t a rg e tm a te r i a l f ro m th e p ro j ec t il e a r e m o d e led ap p ro x im a te ly , an d th e fo rce ap p l i ed to t h e p ro j ec t il e i sca l cu l a t ed an a ly t i ca l ly . A n e q u a t io n fo r p ro j ec t il e m o t io n i s o b t a in ed w h ich i s so lv ed n u m er i ca l ly .A l so , a u se fu l s im p le an a ly t i ca l so lu t io n fo r t h e d ep th o f p en e t r a t i o n o r t h e r e s id u a l v e lo c i ty i sd ev e lo p ed b y m a k in g ad d i t i o n a l en g in ee r in g ap p ro x im a t io n s . Mo reo v er , t h e so lu t io n p ro ced u rep resen ted in t h i s p ap e r p e rm i t s a s t r a ig h t fo rw ard ap p ro x im a te g en e ra l i za t io n to acco m m o d a tea p ro j ec t il e w i th a rb i t r a ry sh a p ed t i p . T h eo re t i ca l p r ed i c t i o n s a r e co m p are d w i th n u m ero u s ex p e r i -m e n t a l d a t a o n n o r m a l p e n e t r a t i o n i n m e t a l t a r g e ts , a n d t h e a g r e e m e n t o f th e t h e o r y w i t h e x p e r i m e n t si s g o o d e v en th o u g h n o e m p i r i ca l p a ram e te r s a r e u sed . A l so , s im u la t io n s fo r co n ica l an d h em isp h e r i -ca l t i p sh ap es i n d i ca t e t h a t t h e ex ac t sh ap e o f t h e p ro j ec t i le t i p d o es n o t s ig n i fi can t ly i n f luen ce th ep red ic t i o n o f i n t eg ra l q u an t i t i e s l i k e p en e t r a t i o n d ep th an d r e s id u a l v e lo c i ty .

1 . I N T R O D U C T I O NA c o m p r e h e n s i v e r e v i e w [ 1 ] o f t h e s t a t e - o f- t h e -a r t o f t h e o r y a n d e x p e r i m e n t s o n p e n e t r a t i o nprob lem s docu m ent s sc i en t if i c i n t e r es t f r om the b eg inn ing o f the 19 th cen tu ry up un t il 1978.M o r e r e c e n t ly , a l a rg e n u m b e r o f th e e x p e r i m e n t a l d a t a w a s c o l l ec t e d in [ 2 ].S e v e r al a n a l y t ic a l a p p r o a c h e s t o t h e p e n e t r a t i o n p r o b l e m h a v e b e e n p r o p o s e d a n d w ill b ed i s c u s s ed p r e s e n tl y . T h e h y d r o d y n a m i c a p p r o a c h n e g le c t s th e y i e ld s t re n g t h o f b o t h t h eta rge t an d the p ro jec t i l e mate r i a l s ( o r on ly the t a rge t m ate r i a l w i th the p ro jec t i l e be ing r ig id )and i s va l id fo r r e l a t ive ly h igh im pac t v e loc it i es . Un de r these cond i t ion s bo th the t a rge t andpro jec t i l e can be mode led as inv i sc id f lu ids wi th ine r t i a l r e s i s t ance to de fo rmat ion . Th i sa l lo w s o n e t o d e v e l o p a h y d r o d y n a m i c t h e o r y o f p e n e t r a t i o n o f t h e t y p e c o n s i d e r e d i n [ 3 - 5 ] .T h e m o d i f i e d h y d r o d y n a m i c a p p r o a c h c o n s i d e r s a r i g i d ( o r r i g i d - p l a s t i c ) p r o j e c t i l ep e n e t r a t in g a r i g i d - p l a s t i c t a r g e t w h i c h i s m o d e l e d i n a v e r y s i m p l e o n e - d i m e n s i o n a l m a n n e rb y a d d i n g p l a s t i c te r m s i n t h e B e r n o u ll i e q u a t i o n [ 6 - 8 ] p r o p o r t i o n a l t o t h e y ie l d s t re n g t h s o fthe t a rge t and p ro jec t i l e .A n o t h e r a p p r o a c h u s e d t o s o l v e th e p r o b l e m o f p e n e t r a t i o n o f a r i g id p r o j e c ti le i n t oa r ig id -p las t i c t a rge t i s based o n pos tu la t ing a genera l fo rm o f the fo r ce app l i ed to thepro jec t i l e wi th emp i r i ca l cons ta n t s [1 , 9 ] . Fo r exam ple , th i s f o r ce i s a ssu me d to cons i s t o fa B ernou l l i -l i ke t e rm w hich m ode l s ine r t i a l e f fec t s and i s qua dra t i c in ve loc i ty , and a t e rmwhich i s i ndepe nde n t o f ve loc i ty and m ode l s the e f f ec t o f pe r f ec t -p l as t ic i ty y i e ld ing theP o n c e l e t e q u a t i o n . T h e s o l u t i o n o f th e e q u a t i o n o f m o t i o n o f t h e p r o j ec t i le w i t h t h e f o r c eg iven by the Po nce le t express ion y ie lds the f ami l ia r f o rmula fo r the pen e t r a t ion dep th P [1 ]

P = c 1 In(1 + c2U~), (1 )w h e r e U o is t h e i m p a c t v e l o c i ty a n d c l , c 2 a r e t w o c o n s t a n t s . T h i s a p p r o a c h a l lo w s o n e t oc o n s i d e r r e la t i v el y c o m p l i c a t e d p h e n o m e n a s u c h a s o b l i q u e p e n e t r a t io n . H o w e v e r , i t in -v o l v e s t w o e m p i r i c a l c o n s t a n t s w h i c h n e e d t o b e m e a s u r e d f o r e a c h c o m b i n a t i o n o f t a r g e tand p ro jec t i l e .80 1


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802 A.L . Yarin e t a l .

An othe r appro ach a l so cons ide r s a r ig id p ro jec t i le pene t r a t ing a r ig id -p las t i c t a rge t bu t i te m p l o y s a o n e - d i m e n s i o n a l c a l c u l a t i o n o f fo r c e s a p p l i e d t o t h e p r o j e c t il e b y a w o r k / e n e r g yba lance . Som e examples o f t h i s one-d ime ns iona l f o r ce ca l cu la t ion a r e g iven in [10] w herea c o m b i n e d m o m e n t u m a n d e n e r g y b a l a n c e is e m p l o y e d f o r a p l u g g i n g m o d e o f p e r f o r a ti o n ,a n d i n [ 1 1 - 1 3 ] w h e r e a g e n e r a li z e d e q u a t i o n f o r t h e f o rc e is p r o p o s e d .A n o t h e r a p p r o a c h p r e s u m e s t h a t t h e n o r m a l s t re s s es a p p l ie d t o t h e p r o j e c t il e s u r fa c e m a yb e a p p r o x i m a t e d b y t h e s o l u t i o n s o f t h e o n e - d i m e n s i o n a l p r o b l e m s o f e x p a n s i o n o f s p h e r ic a lo r cy l ind r i ca l cav i t i e s [ 14, 15 ]. Then by in t eg ra t ing the su r f ace t r ac t ion one ca n de te rm ine ther esu l t an t f o r ce app l i ed to the p ro jec t i le , wh ich i s u sed in so lv ing the equa t ion o f m ot ion [ 15 ].H e r e i t s h o u l d b e e m p h a s i z e d t h a t t h is a p p r o a c h is b a s e d o n t h e re c o g n i t i o n t h a t t h e s h e a rs t r esses app l i ed to the p ro jec t i l e su r f ace a r e neg l ig ib l e r e la t ive to the norm al s t resses .T h e n e x t g r o u p o f a n a l y t ic a l m o d e l s e m p l o y s a t w o - d i m e n s i o n a l c a l c u l a t i o n o f t h e f o r c esimposed on a r ig id (o r de fo rmab le , e rod ing) p ro jec t i l e by a r ig id -p las t i c t a rge t . In th i sa p p r o a c h h y d r o d y n a m i c s o f a n i n v is c id f l ui d is u s e d t o p r o p o s e a n a p p r o x i m a t e v e l o c i ty fie ldin the t a rge t . Th e ve loc i ty f ie ld i s con s t ruc ted separa t e ly in severa l t a rge t dom ains , w i th eachof them be ing essen t i a lly one-d imens ion a l . M oreo ver , i n each dom ain the ve loc i ty f ie ld i sp r e s u m e d t o b e i r r o t a t io n a l a n d h e n c e d e r i v a b l e f r o m a v e l o c i t y p o t e n t ia l . T h e c o m p o s i t etwo -d ime ns ion a l ve loc i ty f ie ld i s t hen o b ta in ed b y match ing the one-d im ens iona l f ie lds us ingcer t a in geomet r i ca l cons t r a in t s .T h e o r i g in s o f t h is l a t te r a p p r o a c h d a t e b a c k t o [ 1 6 - 1 8 ] . T h i s m e t h o d w a s s u c c e s s fu l lyexp lo i t ed us ing an in t eg ra l work - r a t e ba lance to p r ed ic t pene t r a t ion : o f a r ig id b lun tcy l ind r ica l p ro jec t i l e i n [18] ; o f a de fo rm ab le com press ib l e p ro jec t i l e i n [19] ; and o fa d e f o r m a b l e e r o d i n g p r o j e c t i l e i n [ 2 0 ] . F u r t h e r m o r e , t h is a p p r o a c h a l l o w s o n e t o d r a s t ic a l l yimp rove the charac te r i za t ion o f the ve loc i ty f ie ld in the p l as ti c t a rge t [2 1 -2 3] . S imi l a r i deaswere a l so p roposed in [24] to ca l cu la t e the ve loc i ty and co r r espond ing s t r ess f i e lds ina r ig id -p las t i c ma te r i a l f o r t he cases o f s t ead y pen e t r a t ion o f r igid too l s wi th c i r cu la r o rspher i ca l t i p shapes .Pen e t r a t ion o f a r ig id p ro jec t il e i n to an e l as t i c -p las t i c t a rge t has a l so been cons ide redu s i n g n u m e r i c a l m e t h o d s t o s o l v e t h e p a r t ia l d i f fe r e nt ia l e q u a t i o n s o f m o m e n t u m i n t h eta rge t m ate r i a l. F o r e xam ple the wo rk in [25 , 26 ] uses the f in it e e l emen t me tho d to so lve fo rs t eady - s t a t e pene t r a t ion . In p r inc ip le , u s ing numer ica l wav e code s i t i s poss ib l e to cons ide rn o n - s t e a d y - s t a t e p e n e t r a t io n o f a d e f o r m a b l e p r o je c t i le i n to a d e f o r m a b l e t ar g e t. H o w e v e r ,the th r ee -d imen s iona l na tu r e o f ob l iqu e pene t r a t ion s ti ll poses an ex t r eme ly d i f fi cu l tn u m e r i c a l p r o b l e m .T h e i d e a t h a t u n d e r c e r ta i n c o n d i t i o n s t h e f l o w o f a n e l a s t i c - v i s c o p l a s t ic m e t a l c a n b eapp rox im ated by the f low o f a v i scoe las t ic l i qu id ha s been r eexam ined r ece n t ly in [27] . Inpar t i cu la r , i t has bee n show n tha t t he t enso r i a l s t r uc tu r e o f cons t i tu t ive equa t ions fo r a c lasso fe l as t i c -v i sco p las t i c meta l s i s i den t ica l t o tho se fo r a c l ass o f v i scoe las t i c l i qu ids . A l so , i t hasb e e n s h o w n [ 2 7 ] t h a t f o r a s m a l l v a l u e o f t h e D e b o r a h n u m b e r ( w h i c h is ch a r a c t e ri s ti c o fp e n e t r at i o n ), t h e c o n s t i tu t i v e e q u a t i o n f o r a n e l a s t i c - v is c o p l a s t i c m e t a l m a y b e r e d u c e d t otha t charac te r i z ing a r ig id -v i scop las t i c o r r ig id -p las t i c mate r i a l . Consequen t ly , i t i s no ts u r p r is i n g t h a t h y d r o d y n a m i c - t y p e a p p r o x i m a t i o n s o f t h e v e l o c i ty fie ld i n t h e t a r g e t a r esuccess fu l i n the ana lys i s o f pen e t r a t ion p rob lem s .T h e m a i n o b j e c t iv e o f t h e p r e s e n t p a p e r i s to d e v e l o p a n a p p r o x i m a t e s o l u t i o n o f t h ep r o b l e m o f n o n - s t e a d y n o r m a l p e n e t r a t i o n o f a r i g id p r o j e c t il e in t o a n e l a s t i c - p l a s t ic t a r g e to f f in it e th i ckness . The ca l cu la t ion i s based on a fu l ly two-d im ens io na l ve loc i ty f ie ld fo r thew h o l e t a r g e t d o m a i n w h i c h re p r e s e n ts a n a t u r a l e x t e n s i o n o f t h e a p p r o a c h p r o p o s e d i n[ 1 8 - 2 2 ] . T h e t a r g e t m a t e r ia l i s p r e s u m e d t o b e t h i c k e n o u g h t h a t t h e e f fe ct o f d e f o r m a t i o n o fthe f ree su rf ace can be neg lec ted in the p r e d ic t ion o f in t eg ra l quan t i t i e s l i ke pen e t r a t ion dep thand r es idua l ve loc i ty . Spec i f i ca l ly , we cons ide r a cy l ind r i ca l po la r coord ina te sys t em wi thco ord ina tes {r, 0 , z} , ba se v ecto rs {e , e0, ez}, and a f ixed or igin a t the in i t ia l p osi t io n of thef ron t su r f ace o f a t a rge t w i th th i ckness H ( see F ig . 1 ). The t ip o f t he p ro jec t i l e is l oca ted b yz = x ( t ) . I n t h e f o l l o w i n g a n al y s is w e d e r iv e a n e q u a t i o n o f m o t i o n o f th e f o r m

[ M + A (x, ~'~)].~ = B ( x , k ) Y c 2 + C ( x , k ) , (2 )


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P e n e t r a t i o n o f a r i g id p r o j e c t il e i n t o a n e l a s t i c - p l a s t i c t a r g e t o f f i n it e t h i c k n e s s 8 03H

i e r U 0 0) is the in i t ia l v eloc i ty of the project i le . In th is so lut ion x(t) is a m o n o t o n i c a l l ydecreas ing func t ion and the E qn (2 ) ceases to be va l id when ~ van i shes .As in [18- 22 ] , t he ve loc i ty f ie ld in the t a rge t i s t aken to be i r ro t a t iona l and hence i sde r ivab le f rom a ve loc i ty po ten t i a l . Th i s a l low s us to cons id e r a spec i fi c b lun t p ro jec t i l e shapewhich co r r esp ond s to a r ea son ab ly s imple ve loc i ty f ie ld tha t exac t ly sa ti s fi e s the equ a t ion o fcon t inu i ty and the cond i t ion o f imp ene t r ab i l i t y o f t he p ro jec ti l e . The t a rge t mate r i a l i sassum ed to be incompres s ib l e and the ta rge t r eg ion i s d iv ided in to an e l as t ic r eg ion ahead o fthe p ro jec t i l e where the s t r a ins r emain in f in i t e s imal , and a r ig id -p las t i c r eg ion near thepro jec t i l e wh ere the s tr a ins can be a rb i t r a r i ly l a rge . Fu r the rm ore , t he r ig id -p las t i c r espon sei s t aken to be r a t e - indep enden t . Us ing th i s ve loc i ty fi eld, we inc lude in t e r t i a l e f f ec ts i nS e c t io n 2 a n d s o l v e t h e m o m e n t u m e q u a t i o n e x a c t l y i n b o t h t h e e la s t ic a n d r i g i d - p l a s t icr eg ions to de te rm ine an express ion fo r the p r essu re f ie ld . Then , t he e f fec t s o f t he f ree f ron t andr e a r s u r fa c e s o f th e t a r g e t a r e m o d e l e d i n a n a p p r o x i m a t e m a n n e r , a n d t h e d e c e l e ra t i n g fo r c eapp l i ed to the p ro jec t i l e by the t a rge t i s ca l cu la t ed in Sec t ion 3 by ana ly t i ca l ly in t eg ra t ingover the r e l evan t po r t ion o f the p ro jec t i l e su r face . The r esu l ting equ a t ion o f m ot io n o f thepro jec t i l e t akes the fo rm o f (2) . Th i s equ a t ion was so lved num er ica l ly in Sec t ion 5 and thet h e o r e ti c a l p r e d ic t i o n s w e r e c o m p a r e d w i t h n u m e r o u s e x p e r im e n t s . W e e m p h a s i z e t h a t t h ea g r e e m e n t i s g o o d e v e n t h o u g h t h e f o r m u l a t i o n h a s n o e m p i r i ca l p a r a m e t e r s .A s impl i f ied ap prox ima te so lu t ion o f (2 ) g iven in Sec t ion 4 y i e lds the fo l lowing ana ly t i ca le x p r e s s io n f o r t h e p e n e t r a t i o n d e p t h P

P , ~ \ 6 7 r Y R ~ J [

whe re p , Y and p a r e the dens i ty , t he y i e ld s tr eng th in un iax ia l t ens ion a nd the shear m odu lusof the t a rge t ; R ~ i s t he r ad ius o f the p ro jec t i l e , and U o i s i ts imp ac t ve loc i ty . Fo r the case o ft a rge t pe r fo ra t ion the r es idua l ve loc i ty o f t he p ro jec t i l e U r i s a l so ca l cu la t ed an a ly t i ca l lyw i t h in t h e f r a m e w o r k o f th e s i m p l if ie d a p p r o x i m a t e s o l u t i o n o f S e c t io n 4 . M o r e o v e r , i nthe r eg ion o f va l id i ty o f t h is ap prox ima te so lu t ion , P as g iven by (4 ) and U , a r e in gooda g r e e m e n t w i th b o t h t h e r e s u lt s o f t h e n u m e r i c al s o l u t i o n a n d w i t h n u m e r o u s e x p e r i m e n t a lda ta .


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80 4 A. L. Yarin e t a l .Table 1 . A br ie f summ ary of var ious parameters an d theequa t ions where they a re def inedPa ra me te r Eq u a t io n Pa ra me te r Eq u a t io nf l (39b) x* (31b)(44a)f2 (39c) ,~* (31 a)( 4 4 b )f3 (39d) x** (34)(44c)f4 (39e) :~** (60b)(44d)/4 (64c) x*** (57)r* (42a) f*** (57)3~* (42b) 0 b (68)3/~3 (41 a,b) ~ ( 11 b)/~4 (48) $ ~ (27b)/~5 (45a) ~-2 (29b )(40)/~6 (47) ~2 (30b)(36)/~L (46a) ~ (24f)(30d) ~-1 (27c)2 (30c) ~2 (29c)

Fur th e rm ore , t he p r esen t ap pro ach i s genera l ized in Sec t ion 6 fo r ana lyz ing p ro jec t il e swi th genera l t i p shapes . Th i s is accomp l i shed by us ing the m etho d o f s ingu la ri t ie s to ca l cu la t ethe ve loc i ty f ie ld . S imula t ions fo r con ica l and hem ispher i ca l t i p shapes ind ica te tha t t he exac tshape o f the p ro jec t i le t i p does no t s ign if i can tly in f luence the p r e d ic t ion o f in t eg ra l qu an t i t i e sl ike pene t r a t ion dep th and r es idua l ve loc i ty .Sec t ion 2 desc r ibes the ana lys i s o f t he t a rge t . In o rde r to a id the r eader , Tab le 1 p rov ide sa b r i e f s u m m a r y o f v a r i o u s p a r a m e t e r s a n d t h e e q u a t i o n s w h e r e t h e y a r e d e fi n e d. S e c t io n3 deve lops the equa t io n o f m ot io n fo r the p ro jec t il e , and Sec t ion 4 desc r ibes an eng ineer ingapprox imat ion which s impl i f i e s th i s equa t ion and y ie lds an ana ly t i ca l express ion fo r thep e n e t r a t i o n d e p t h o r t h e r e s i d u a l v e lo c i ty . S e c t i o n 5 p re s e n t s a n u m b e r o f e x a m p l e s w h i c hs h o w g o o d c o m p a r i s o n o f t h e t h e o r e t ic a l p r e d i c t i o n s w i t h e x p e r i m e n t a l d a t a . S e c t io n 6br i e f ly desc r ibes a genera l i za t ion fo r the case o f a p ro jec t i l e wi th genera l t i p shape andd i scusses s imula t ions o f p ro jec t i le s wi th con ica l and hem ispher i ca l shap ed t ip s . F ina l ly ,Sec t ion 7 d i scusses the m ain c onc lus ion s o f th is w ork .T h e m o s t i m p o r t a n t r e s u l t s o b t a i n e d i n t h e p r e s e n t w o r k a r e g i v e n b y e q u a t i o n s ( 4 9 )su pp lem en ted by (52) an d (53) as well a s b y (27), (30), (31), (36), (39), (40), (42), (44) an d (48) fornum er ica l ca l cu la t ion o f the p ro jec t i le mot ion . Al so , an ap prox ima te an a lys i s y ie lds thesimpl i f ied analyt ica l ex press ions (4) , (62) - (66) and (68) .

2. A N A L Y S I S O F T H E T A R G E TIn the p r esen t sec t ion a b lun t p ro jec t i l e i n the fo rm o f an o vo id o f Ran k ine i s cons id e redand the a sso ciate d velo ci ty f ie ld is used to calcu late the s t ress f ie ld in the target . Specif ically ,the t a rge t mate r i a l i s a ssumed to have a cons tan t dens i ty p and to r emain incompress ib l e .T h i s m e a n s t h a t t h e c o n s e r v a ti o n o f m a s s a n d t h e b a l an c e o f m o m e n t u m e q u a t io n s m a y b ewr i t t en in the fo rms

div v = 0, p~, = - Vp + div e' , (5a, b)w h e r e t h e g r a d i e n t o p e r a t o r V a n d t h e d i v e r g en c e o p e r a t o r d i v a r e d e f i n ed w i t h r e s p e c t t othe p r esen t p os i t ion o f a mate r i a l po in t , v i s t he ab so lu te ve loc i ty o f t he m ate r i a l po in t i n thet a rge t , and the C auc hy s t r ess a h as been sep ara t ed in to a p r essu re p and i t s dev ia to r i c pa r t ty ',


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P e n e t r a t i o n o f a r i g id p ro j e c t i le i n t o a n e l a s t i c -p l a s t i c t a rg e t o f f i n i te t h i c k n e s s 8 0 5

s u c h t h a t~ = - p l + o ' , a ' . l = 0 . (6 a, b )

A l s o , I d e n o t e s t h e u n i t t e n s o r, t h e n o t a t i o n a . b d e n o t e s t h e u s u a l s c a l a r p r o d u c t b e t w e e n t w ovec to r s a , b , and the no ta t ion A" B = t r (AB T) deno tes the inner p ro du c t b e twe en two s econdo r d e r t e n s o r s A , B .In the fo l lowing so lu t ion we de te rm ine a ve loc i ty fi eld tha t e xac t ly sa ti s fi e s the co n t inu i tyE qn (5a ) and the con d i t ion o f imp ene t r ab i l i t y a t t he su r f ace o f the p ro jec t i le . A t the ins t an ttha t t he p ro jec t i l e t ouches the t a rge t ' s f r on t su r f ace the t a rge t mate r i a l beg ins to de fo rmelas ti ca lly . Ho we ver , a t som e po in t du r ing the pene t r a t ion p rocess the t a rge t mate r i a l beg inst o d e f o r m p l a s t ic a l ly a n d a n e l a s t i c - p l a s t ic b o u n d a r y p r o p a g a t e s a w a y f r o m t h e t i p o f t h epro jec t i l e . In the e l as t i c r eg ion the s t r a ins r emain smal l whereas in the p l as t i c r eg ion thes t r a ins can be ve ry la rge . F ur the rmo re , f o r s impl i c i ty we assum e tha t t he mate r i a l r e spo nse inthe p l as t i c r eg ion i s r a t e - insens i t ive and r ig id -p las t i c so tha t t he dev ia to r i c s t r ess may bea p p r o x i m a t e d b y

a ' = D, (7)( D " D ) 1 /2wh ere Y i s t he con s tan t y i e ld s t r ess in un iax ial t ens ion and D i s t he sym met r i c pa r t o f t heve loc i ty g r ad ien t . A l so , i n the e l as t ic r eg ion w e assum e the m ate r i a l r e spo nse i s li nea r e l as t icand i so t rop ic so tha t f o r i sochor i c m ot ion the dev ia to r i c s t ress is re l a t ed to the l i nea r s t r a ins

by the express iono ' = 2 / ~ , (8)

where /~ i s t he cons tan t shear modu lus . Here , we wi l l deve lop express ions fo r the p r es -su re in bo th the p l as t i c and e l as t i c r eg ions tha t ex ac t ly sa ti s fy the equa t ion o f mo t ion (5b).A l t h o u g h t h e e q u a t i o n s o f m o t i o n a r e s a t is f ie d e x a ct ly , th e b o u n d a r y c o n d i t i o n s a s s o c i a t e dwi th the f r ee f ron t and r ea r su r f aces o f t he t a rge t and the f r ee su r f ace tha t dev e lops nearthe p ro jec t i l e wi l l be sa t i s f i ed approx imate ly . In o rder to impose these boundary cond i -t i o ns a n d t o d e t e r m i n e t h e l o c a t i o n o f t h e e l a s ti c - p l a s ti c b o u n d a r y i t is c o n v e n i e n t t o d i v id et h e r e g i o n i n to f o u r p a r t s s e p a r a t e d b y t h e b o u n d a r i e s z = :z 1 = - H a s s o c i a t e d w i t hthe t a rge t ' s r ea r su rf ace ; z 2 assoc ia t ed w i th the e l as t i c -p las t i c bou nda ry ; z 3 assoc ia t edwi th the beg inn ing o f the r eg ion o f con tac t b e twe en the t a rge t and the p ro jec ti l e; andz4 assoc ia t ed wi th the s epara t ion l ine o f the t a rge t mate r i a l f r om the p ro jec t i le su r f ace ( seeFig. 2).

1 0 . 0

5 . 0 -

~ 0 . 0 -

- 5 . 0 .

- 1 0 . 0- 1 5 . 0

E l a s t i c - P l a s t i c B o u n d a r yF r o n t - " ~ Z I. , P r IS u r f a c eR e a r __ ~ . _ . ~S u r f a c e ~ . . .

Z - ' - - H z z z 41 2 3

- 1 0 . 0 - 5 . 0 0 .0 5 .0Fig . 2 . Def in i t i on o f t he ax i a l l oca t io n of : t he t a rge t ' s re ar surfac e (z 0 a nd f ront su rface (z = 0) ; t hee l a s t i c -p l a s t i c b o u n d a r y ( z2 ); t h e p ro j e c t i le t i p (z 3) ; a n d t h e s e p a r a t i o n l i n e o f t h e t a rg e t m a t e r i a l f r o m

the pro j ec t i l e surface (z4) .


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806 A.L . Yarin et al.V e l o c i t y f i e l d a n d p r o j e c t i l e s h a p e

I t i s we ll know n tha t fo r ax isymm etr ic i r ro ta t io na l m ot ion , the ve loc i ty f ie ld i s de te r m inedby a po ten t ia l func t ion ~b(r,z , t ) such tha tv = V ~ b . ( 9 )

T h e n , t h e c o n t in u i t y E q n ( 5a ) a n d t h e b a l a n c e o f m o m e n tu m ( 5b ) re d u c e t oV 2 ( ~ = O , V [ p {~-~t + 1V~b. V~b } + p ] = div 6' . (10a, b)

Here , we take the p ro jec t i le to be an ovo id o f Ran k ine because the assoc ia ted ve loc i ty f ieldw h ic h s at is fi es t h e c o n d i t i o n o f im p e n e t r a b i l i t y is k n o w n to b e c h a r a c t e r i z e d b y a c o m b in a -t ion o f a s ing le source an d un i fo r m f low. Spec if ica lly, the ov o id i s a b ody o f revo lu t ion o fleng th L w hose la te ra l su r face i s de f ined by r = R [ 2 8 - 3 0 ] w h e r e- R 2 + ~ -- 2 )1 /2 +1 = 0 , f o r - < ~ < - + L , ( | la )

~ = z - x - 2 ( l l b )In (11) Ro~ i s a con s tan t con t ro l l ing the ra d ius o f the p ro jec t i le and ~ i s mea sured re la t ive toa m ate r ia l po in t in the p ro jec t i le ( see F igs 1 and 3 ). I t can be show n tha t the su r face de f ined by(1 la ) can be expressed in the equ iva len t pa ram etr ic fo rm s

2 R 2 _ R 2= E(R ) = 2(R2 _ R E ) l / 2 , (12a)r = R ( 0 = ~- + ( 4 2 + 2 R 2 ) u2

No w, th e ve loc i ty po ten t ia l ~b, the rad ia l com po nen t v and ax ia l com po ne n t v= o f the ve loc i tyf ie ld , wh ich sa t i s fy the con t inu i ty Eqn (10a) , and the cond i t ion tha t ( l la ) [o r 02) ] i s am ate r ia l su r face ( i. e. the p ro jec t i le is im pene t rab le ) a re g iven by= + x (~2 + r2)1/2 ,

l . F 1(13a)

(13b,c)

1.51 . 0 -0 . 5 -0 . 0 -

- 0 . 5 -

' i i 11 ,- 1 . 0, , , I , , , , I , , , ,

i , i i i i i 1 , , ~ 10 . 0 1 . 0~ / R

2 .0Fig. 3. A ctualscaledshape of the projectile ip.


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Penetrationof a rigid projectile nto an elastic-plastic arge t of finite hickness 807Before p roce ed ing fu r the r i t is wo r th d iscuss ing the mot iv a t ion fo r us ing the abo ve ve lo -c i ty field. In th i s reg ard , i t i s no ted tha t the pe r fo ra t ion p rocess o f a p la te i s d iv ided in tos e v e ra l s ta g e s i n c lu d in g : d y n a m ic p e n e t r a t i o n , b u lg e f o r m a t io n , b u lg e a d v a n c e m e n t , p lu gfo rm at ion , and ex i t [18]. F or re la t ive ly th ick duc t i le me ta l p la tes , wh ich a re o f in te res ti n t h e p r e s e n t s t u d y , t h e d o m in a t i n g s t a g e o f l o n g e s t d u r a t i o n i s t h e d y n a m ic p e n e t r a t i o ns tage. C onsequ en t ly , it is un l ike ly tha t the o the r te rm ina l s tages (wi th a cum ula t ive dura t i on

shor te r tha n tha t o f the pene t ra t ion s tage) can s ign i fican t ly e f fect the p re d ic t ion o f in teg ra lq u a n t i ti e s l i k e p e n e t r a t i o n d e p th a n d r e s id u al v e lo c it y . M o r e o v e r , t h e g e n e r a l p a t t e r n o f t h eve loc i ty f ie ld dur ing the s tages o f bu lge fo rmat ion and advancement does no t d i f fe rs ign i f ican t ly f rom the ve loc i ty f ie ld dur ing dy nam ic pene t ra t ion even th oug h d i f fe ren tfo rm ulae a re used fo r the i r desc r ip t ions [18] . Also , no te th a t fo r su f f ic ien tly th ick ta rge tsth e d y n a m ic p e n e t r a t i o n s t a g e w h ic h i s m o d e l e d h e r e is t h e o n ly s t a g e o f th e p e n e t r a t i o nprocess .S imple geom etr ica l reason ing de m and s tha t the ve loc i ty f ie ld shou ld sa t i sfy the con d i t iono f im p e n e t r a b i l i ty a t t h e p r o j ec t il e s u r fa c e a n d d e g e n e r a t e t o w a r d s a u n i f o r m f lo w f a r f r o m i t.Co nseque n t ly , the fu l ly two -d im ens ion a l ve loc i ty field (13b ,c ) , wh ich sat is fies these tw oc o n d i t i o n s , i s e x p e c te d t o a d e q u a t e ly m o d e l t h e k in e m a t i c s o f t h e t a r g e t m a te r i a l d u r in g t h ed y n a m ic p e n e t r a t i o n s t a ge o f a r e l at i ve ly th i c k t a r g e t.The po ten t ia l ( i r ro ta t iona l ) chara c te r o f the ve loc i ty f ield (13b ,c ) , l ike the po ten t ia lcha rac te r o f a l l o the r v e loc i ty f ie lds used in the l i te ra tu re [15, 18 -23] , does n o t p red ic t thede ta i led e f fec ts o f shea r ing nea r th e p ro jec t i le su r face . Never the less , p red ic t ions o f in teg ra lquan t i t ie s l ike pene t r a t ion dep th and res idua l ve loc i ty us ing th is ve loc i ty field a re expec ted tobe goo d becau se the dece le ra t ing fo rce o f the ta rge t i s dom ina te d by the norm al s tressesapp l ied to the p ro jec t i le su r face and no t by the shear s t resses. T h is fac t was c lea r ly reco-gn ized in the c lass ica l work s [14 , 16, 31] a nd i s con t inu a l ly exp lo i ted to con s t ruc t p o ten t ia lve loc i ty f ie lds app ropr ia te fo r ana ly t ica l m ode ls o f the p ene t ra t ion p rocess [1 5-20 , 32 , 33].Al l o f these c i ted ve loc i ty f ie lds a re based on the m ode ls o f expan s ion o f cy l indr ica l o rspher ica l cav i t ie s , wh ich mode l the main k inemat ics respons ib le fo r the deve lopment o fnor m al s t resses. M oreov er , these c i ted ve loc i ty f ie lds do no t m ode l de ta i l s o f the s l ip p rocessnea r the su r face o f the p ro jec t i le which i s re spons ib le fo r the dev e lopm ent o f shea r s t ressthere .Th e r igid-p l as t ic region o f the target

Usin g the ve loc i ty f ie ld (13b ,c ) i t can then be show n tha t the ra te o f de fo rm at ion tensorD b e c o m e sD = - - 4 ( ~ 2 - { - r2)S/2 [(~2 _ 2r2)e, e, + (~2 + r2)e0 eo + (r 2 _ 2~2)ez @ ez

- - 3r~(er e z + ez er)], (14)wh ere the sym bol deno tes the usua l tensor p roduc t . Thus , reca l l ing tha t i s non-po s i t ive i tcan a l so be show n tha t wh eneve r the p ro jec t i le is m oving (~ < 0 ), the dev ia to r ic s t re ss (7)cor re spon d ing to the ve loc i ty field (13b ,c ) m ay be wr i t ten in the fo rm

O' 3(~ 2 ~_ r2 ) [ ( ~ 2 - - 2 r 2 ) e , e , + ( ~ Z + r Z ) e o e o + ( r 2 - - 2 ~ 2 ) e z e z- - 3r~(e, ez + e, @ e,)].

F u r th e r m o r e , it c a n b e s h o w n u s in g ( 15 ) t h a td i v a ' = V - Y l n k R 2 ] j .

(15)

(16)T h i s s im p le r e s u lt i n d i c a te s t h a t t h e b a l a n c e o f m o m e n tu m ( 1 0b ) c a n b e s o lve d e x a c t l y i n t h e


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808 A.L. Yarin e t a l .r i g i d - p l a s t i c r e g i o n t o o b t a i n a n e x p r e s s i o n f o r t h e p r e s s u r e o f t h e f o r m

p = f(t) + p22 [ R2~4( 2 + r2)3/2 32(~ 2 + rZ)2 JR ~- p :~ R ~ [ 4 ( ~ 2 ~ r 2 ) , / 2 ] Y l n { ~ 2 + r 2 )

\ R ~ / '(17)

w h e r e f(t ) i s a fu n c t i o n o f t i m e o n l y w h i c h m u s t b e d e t e r m i n e d b y a b o u n d a r y c o n d i t i o n .I n o r d e r t o d e t e r m i n e t h e s t r e s s a c t i n g o n t h e s u r f a ce o f t h e p r o j e c ti l e , w e n o t e t h a t t h e u n i tn o r m a l n ( o u t w a r d n o r m a l f r o m t h e p r o je c t il e ), t h e u n it t a n g e n t v e c t o r r , a n d t h e e le m e n t o fa r e a d a m a y b e e x p r e s s e d i n t h e f o r m s

/~(3 - 2/~2)e, - 2(1 - R 2 ) 3 / 2 e zn = (;4 -- 3/~2) 1/22(1 - /~2 )3 /2e , + /~( 3 - 2 /~2)ez

(4 - 3/~2) 1/z= Ly6-- j R dRdO,

w h e r e / ~ i s t h e n o r m a l i z e d v a l u e o f t h e r a d i u s ( 1 2 b )

(18a)

(18b)(18c)

y [ - ( - 8 + 9 / ~ 2 ) ]- p + L j ,

r [

Y%0 = - P + ~-,[- -- 211~(!"~ J~2)1/2 ]

c r = Y (4 - 3/~ z) J 'p = f ( t ) + p 2 2 I ( 1 - 3 R 2 )( 1 - R 2 ) ] [ ( 1 - / ~ 2 ) ' / 2 - [2 - pY~R~ ~- J + r ln [4 ( 1 - /~2 ) 3 . (20f)

S i n c e w e h a v e n o t i m p o s e d a n y b o u n d a r y c o n d i t i o n o n t h e s l ip v e l o c it y o r t h e s h e a r s t re s sa t t h e p r o jec t i l e su r f ace we d o n o t ex p ec t t h a t t h e v a lu e o f t h e sh ea r s t r e s s a ,~ p r ed ic t ed b y (2 0)wi l l b e v e r y accu r a t e . I n f ac t , i t c an b e seen f r o m ( 2 0) th a t s in ce a.~ i s n o n - p o s i t i v e , t h e sh ea rs t r e s s ap p l i ed to th e p r o jec t i l e su r f ace ac t s in th e n eg a t iv e z d i r ec t io n , w h ich seem s p h y s ica l lyin co r r ec t b ecau se i t t en d s to acce le r a t e th e p r o jec t i l e i n s t ead o f d ece le r a t e i t. O n th e o th e rh a n d , t h e d i a g o n a l c o m p o n e n t s a., , %0, o~ o f s t re s s a r e p r e d o m i n a n t l y d e t e r m i n e d b y th ei~ inema t ic s o f t h e f lo w p as t t h e p r o jec t i l e . C o n seq u en t ly , s in ce we h av e sa t i s f i ed th e co n d i -t io n o f imp en e t r a b i l i t y o f t h e p r o jec t i l e an d th e b a la n ce l aws ( 10 ) ex ac t ly , we ex p ec t t h e sed i a g o n a l c o m p o n e n t s o f s t re s s to b e r e a s o n a b l y a c c u r a t e ( in c o n f o r m i t y w i t h t h e id e a s o f[1 4 , 1 6, 3 1 ] ) . F o r th e se r ea so n s , we wi l l ig n o r e th e sh e a r s t r e s s co m p o n e n t a ,~ in th e su b -s e q u e n t a n a l ys i s .

O n c e a n a p p r o p r i a t e b o u n d a r y c o n d i t i o n i s i m p o s e d t o d e t e r m i n e t h e fu n c t io n f(t ) in theex p r es s io n f o r t h e p r e ssu r e ( 1 7) an d ( 2 0f ), t h e v a lu e o f t h e s t r e s s o n th e b o u n d a r y o f t h ep r o jec t i l e (2 0 ) w i l l b e k n o w n . T h i s m ean s th a t t h e ax ia l f o r ce F wh ich d ece le r a t e s th ep r o j e c ti l e c a n b e d e t e r m i n e d b y i n te g r a t i n g t h e a p p r o p r i a t e e x p r e s s i o n fo r th e t r a c t io n v e c t o rap p l i ed to th e p r o jec t i l e su r f ace .

(20a)(20b ,c)

(20d,e)

= o . , n n + aooeoeo+ a ~ r + o . , ( n + ~ n ) ,

Now, using the expressions (6a) , (11) , (17) , (18) and (19) we may evaluate (15) on the sur -f ace o f t h e p r o jec t i l e an d ex p r e ss th e r e su l t in g s t r e s s in t e r ms o f t h e v ec to r s { n, e0, z} to d ed u cet h a t

R/~ = - - . ( 1 9)R ~


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Penetration of a rigid projectile into an elastic-plastic target of finite thickness 809E l a s ti c r e g i o n o f th e t a r g e t

N e x t , w e d e t e r m i n e a s o l u t i o n i n t h e e l a st i c r e g i o n o f th e t a r g e t w h e r e t h e s t r a i n s r e m a i ns m a l l . T h e v e l o c i t y f i e ld ( 13 ) i s a l s o a s s u m e d t o a p p r o x i m a t e t h e m o t i o n i n t h e e l a s t i c r e g i o nb e c a u s e i t h a s t h e m a i n p h y s i c a l f e a t u r e s t h a t t h e v e l o c i t y v a n i s h e s a s r >> R ~ o r z - ~ - ~ .A l s o , i t w i ll b e s h o w n t h a t t h e r e s u l t s b a s e d o n t h i s a s s u m p t i o n a r e c o n s i s t e n t w i t h r e s u l t s o ft h e p u n c h p r o b l e m [ 3 1 ].

U s i n g t h i s a s s u m p t i o n t h e i n f in i t e si m a l s t r a in e c a n b e d e t e r m i n e d b y i n t e g r a t i n g t h ee q u a t i o n sOg(r, z, t)- - - D . ( 2 1 )g t

w h e r e t h e s t r a i n e a d m i t s t h e r e p r e s e n t a t i o n se = g ( r , z , t ) = ~ . (r , ( ( z , x ( t ) ) = C z ( r, z - x ( t ) - R ~ ) ,

2(22)

a n d t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o t i m e i s u s e d i n s t e a d o f t h e m a t e r i a l d e r i v a -t i v e b e c a u s e t h e d e f o r m a t i o n s a re a s s u m e d t o b e i n f i n i t e s im a l in t h e e l a s t ic re g i o n . N o t i n gt h a t

O g ( r , z , t ) . O ~ ( r , ~ )- - = x ( 2 3 )Ot g~ '

w e m a y u s e (1 4) , ( 21 ) a n d i n t e g r a t e ( 2 3) s u b j e c t t o t h e i n i t ia l c o n d i t i o n t h a t ~ v a n i s h e s a t t h ei n i t ia l t i m e o f i m p a c t w h e n x (0 ) v a n is h e s . T h i s p r o c e d u r e y i e l d s t h e e x p r e s s i o n s= e r ~ e , e , + e o o e o eo + e=e~ e~ + e~ (e, ez + e~ e,) , (24a)

R 2 [ ~ ( ~ 2 + 2 r 2 ) ( ( ( 2 + 2 r 2 ) ]e,, = ~ (~2 + r2)3/2 -} ((2 + r Z ) a / 2 , (24b)CO0 = ~r 2 (~2 +r 2) 1/ 2 ((2 + r 2 ) 1 / 2 , (24c )

~ = -~ - (~2 + r 2 ) 3 / 2 ( ( 2 ~ _ ( - r2 ) 3 / 2 , (24d)= R ~ r ~ 1 1 ]er~ 4 L( ~ 2 q.- r2) 3/2 ((2 q_-r2)3/2 , (24e)

RoC( = z - - - (24f)2 '

w h e r e ~ i s g i v e n (1 l b ). N o w , u s i n g t h e e x p r e s s i o n ( 8) f o r t h e s t r e s s a n d n o t i n g t h a t d i v Dv a n i s h e s , i t f o l l o w s t h a t i n t h e e l a s t ic r e g i o n

d iv ~ ' = 2p d iv e = 0 , (25 )s o t h a t t h e e q u a t i o n o f m o t i o n ( 1 0b ) y i e l d s a n e x p r e s s i o n f o r th e p r e s s u r e o f t h e f o r m

P = g ( t ) + P 2 2 1 4 ( ~ . 2 + r 2 ) 3 / 2 32( ~2+ r2) 2 4 (~ 2~ r2 ) 1/2- ' (26)w h e r e 9 ( t) is a n o t h e r f u n c t i o n o f t i m e t o b e d e t e r m i n e d i n t h e e l a s t i c r e g i o n .B o u n d a r y a n d m a t c h in g c o n d i t i o n s

A l t h o u g h w e c a n n o t s a t is f y t h e b o u n d a r y c o n d i t i o n t h a t t h e r e a r s u r f a ce o f t h e t a r g e t iss t re s s f re e p o i n t w i s e u s i n g t h e a s s u m e d v e l o c i t y fi e ld (1 3b ,c ), w e c a n a p p r o x i m a t e t h i sc o n d i t i o n b y r e q u i r i n g t h e s t r e s s t~ = t o v a n i s h a t r - - 0 a n d z = z 1 = - H . T h u s , i t f o l l o w s f r o m


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810 A.L. Yarin e t a l .

(8 ), (24) , a n d (26 ) t ha t fo r t he e l a s t i c so lu t i on19 (t ) = - p x 2 4 ~ 1 1 - p ~ R ~ o - , (27a)32(~

H x 1 H 12- (27b,c )~ - R oo R ~, 2 ' ~ - R ~ 2

G ive n t h i s v a lue o f 0 ( t ) , t he s t r e s s i n t h e e l a s t i c r e g ion i s de t e r m ine d b y Eq ns (6.) , (8) , (24 ) , a n d( 26 ). T h i s a p p r o x i m a t i o n o f t h e b o u n d a r y c o n d i t i o n a t t h e r e a r t a r g e t s u r f a c e is n o t e x p e c t e dt o b e t o o r e s t r i c t i v e b e c a u s e t h e s t r e s s o z : a t t h e r e a r t a r g e t s u r f a c e r e m a i n s r e l a t i v e l y s m a l l f o rr > 0 u n t i l t h e p r o j e c t i l e c o m e s c l o s e t o t h e r e a r s u r f a c e .

N e x t , w e d e t e r m i n e t h e f u n c t i o n f ( t ) i n ( 17 ) b y a m a t c h i n g c o n d i t i o n a t t h e e l a s t i c - p l a s t i cb o u n d a r y . T o t h is e n d , w e n o t e t h a t t h e e l a s ti c - p la s t ic b o u n d a r y is d e te r m i n e d b y t h e v a l u eso f r a n d z w h i c h c a u s e t h e e l a s t i c s o l u t i o n t o s a t is f y t h e y i e l d c o n d i t i o n t h a t

G ( r , z , x ) 3 ' ' 1/2[ ~ , ~ ' ~ ] - Y = # [ 6 ~ ' ~ . ] t i 2 - Y = 0 . ( 2 8 )F o r a n e x a c t s o l u t i o n i t i s n e c e s s a r y to r e q u i r e c o n t i n u i t y o f s u r f a c e t r a c t i o n s p o i n t w i s e o nt h e e l a s t i c - p l a s t i c b o u n d a r y . H o w e v e r , f o r t h e a p p r o x i m a t e v e l o c i t y f ie ld w e o n l y r e q u i r ec o n t i n u i t y o f t h e a x i a l s te s s G = a t t h e i n t e r s e c t i o n z = z 2 o f t h e e l a s t i c - p l a s t i c b o u n d a r y w i t ht h e r = 0 a x is . T h u s , f o r a g i v e n v a l u e o f x ( t ) w e c a n e v a l u a t e ( 28 ) a t r = 0 a n d z = z 2 t o o b t a i na n e q u a t i o n o f t h e f o rm

[~z2 - 1 _ 4 r- ~ , ( 2 9 a )_ z 2 x 1 ~ - 2 z 2 1 Y x

2 ' R 2 (29b , c )fo r - H ~< z2 ~< x ~< 0. (2 9d )

A l t e r n a t i v e l y , w e c a n s u b s t i t u t e ( 2 9 c ) i n t o ( 2 9 a ) a n d s o l v e th e r e s u l t f o r x t o d e d u c e t h a t/ ~2 \1/ 2

/ 4 Y ' ~ ' /2 ~- f 4 Y " l ' /2 x

w h e r e t h e n o r m a l i z e d v a r i a b l e s (3 0 b ,c ,d ) h a v e b e e n i n t r o d u c e d t o s i m p l if y E q n ( 30 a ). T h i sa p p r o x i m a t i o n o f t h e b o u n d a r y c o n d i t i o n a t t h e e l a st i c- p l a s ti c b o u n d a r y is n o t e x p e c t ed t ob e t o o s e v e r e b e c a u s e t h e s h a p e o f t h e e l a s t i c - p l a s t ic b o u n d a r y a h e a d o f t h e p r o j e c t il e isr e l a t i v e l y f l a t i n t h e r a n g e r ~< R~o [ ( s e e F i g . 1 3 ( c) ]. M o r e o v e r , t h e s e t y p e s o f a p p r o x i m a t i o n sa r e c o m m o n i n t h e s o l u t i o n o f p l as t i c it y p r o b l e m s u s i n g a p p r o x i m a t e v e l o c i ty fi el d s ( e.g . s eet h e p e n e t r a t i o n p r o b l e m i n [ 3 4 ] ) .

Y ie ld ing f i r s t oc c u r s a t t he t ip o f t he p r o j e c t i l e ( z = x , ~-2 = - 1 /2 ) w h e n ~ t a ke s t h e va lue Y*( a n d x t a k e s t h e v a l u e x * ) g i v e n b y

Z ~ I / 2 1 ~ - - ~ \ - ~ ) , ( 31 a)2 \ 3 , ) 1 - 3 . )

Ro ~ \4Y) 1 6 \ 3 / i ) 'w h e r e w e h a v e u s e d t h e f ac t th a t f o r m o s t m e t a l s Y / # i s le ss th an ab o ut 0 .01. Th us , fo r .,7 ~< . f*t h e e l a s t i c - p l a s t i c b o u n d a r y o c c u r s a h e a d o f th e p r o j e c ti le . A l t e rn a t i v e ly , w e m a y u s e (2 9 c)


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P e n e t r a t i o n o f a r i g i d p ro j e c t i l e i n t o a n e l a s t i c -p l a s t i c t a rg e t o f f i n i t e t h i c k n e s s 8 11a n d ( 30 ) t o d e d u c e t h a t

v , )2 + x= \ 3 . ). . . ~ , ( 3 2 )s o t h a t x c a n b e w r i t t e n a s a f u n c t i o n o f z z o f t h e f o r m

- - -R -~ 1 - ( 4 y , ] ( l _ z 2 , ] 2 112 (33)R ~ 1 + \ 3 / x ) \ 2 R o o JI t f o l l o w s t h a t t h e e l a s t i c - p l a s t i c b o u n d a r y r e a c h e s t h e r e a r t a r g e t s u r f a c e w h e n z 2 = - Ha n d x = x * * , w h e r e

j> JN o t i c e a l s o f r o m (3 0a ), t h a t ~ a p p r o a c h e s n e g a t i v e i n f i n i ty w h e n ~ 2 a p p r o a c h e s n e g a t i v eun i ty . Thu s , fo r l a rge va lues o f [xl the s o lu t io n (30a) y ie lds

~ z ~ - 1 , ~ 2 = - \ ~ j . (35a,b)F o r t h e g e n e r a l c a se , t h e v a l u e o f ~ z a s s o c i a te d w i t h t h e e l a s t i c - p la s t i c b o u n d a r y c a n b ed e t e r m i n e d a n a l y t i c a l l y a s a f u n c t i o n o f ~ b y r e w r i t in g ( 30 a) a s a q u a r t i c e q u a t i o n o f th e f o r m

~2 + 2~23 _~_ 3~2~2~2 2X ~2 - ~2 = 0. (36)U s i n g D e s c a r t e ' s r u l e o f si g n s r 3 5 ] a n d t h e f a c t t h a t ~ is n e g a t i v e it c a n b e s h o w n t h a t ( 36 )h a s o n l y o n e r e a l n e g a t i v e r o o t , a n d t h a t t h i s r o o t c o r r e s p o n d s t o t h e e l a s t i c - p l a s t i cb o u n d a r y .Fu r th erm ore , u s ing the exp res s ions (8) , (24) , (26 ), (27) , an d (29 ) , the ax ia l s t r es s in the e las t i cr e g i o n a t t h e e l a s t ic - p l a s t i c b o u n d a r y m a y b e e x p r e s se d i n t h e f o r m

a z = ( O ' z 2 ' t ) = - - P 2 4 ~ 4 ~ 3 2 ~ - 2 4 + - p S ~ R ~ 4? 2 4~ 1-? (37)

O n t h e o t h e r h a n d , w e c a n u s e (6 ), ( 15 ), a n d ( 17 ) t o e v a l u a t e t h e a x i a l s t re s s i n t h e p l a s t i c r e g i o na t t h e e l a s t i c - p l a s ti c b o u n d a r y t o o b t a i n

2 1 2 2a==(O,z2, t ) = - f ( t ) - p 2 [ 4 ~ 3 ~ ' ~ - 2 4 } - P ' f R ~ [ ~ z } + Y [ l n ~ 2 - 3 ] " ( 3 8 ,S i n c e o u r m a t c h i n g c o n d i t i o n re q u i r e s th e a x i a l s t re s s t o b e c o n t i n u o u s a t t h e e l a s t ic p l a s t i cb o u n d a r y w e m a y e q u a t e ( 37 ) a n d ( 3 8) to o b t a i n f ( t ) i n t h e f o r m

f ( t ) = pY R~ f l ( t) - p22 f z (t ) + Y f 3( t) - p f 4( t) , (39a)1 [ 1 2_1_141 (3 9b ,c )f , ( t ) = 4 ~ 1 ' f 2 ( t ) = 4 ~ 2 3 '

f3( t ) = In ~~, / 4 ( t ) = ~ [ - ~ - ~ 1 . (39d,e)In v iew o f the r e s u l t (31) it m ay be s een t ha t fo r .-7" < ,-7 ~ 0 the t a rg e t m ate r ia l on the r = 0a x i s r e m a i n s e l a s ti c . H o w e v e r , s i nc e f o r m o s t m e t a l s Y/'p i s l es s tha n ab ou t 0 .01 i t fo l lows f rom( 3 0 ) a n d ( 3 l ) t h a t I x * / R = l i s l e s s t h a n a b o u t 0 . 0 0 1 . F o r t h i s r e a s o n w e a s s u m e t h a t t h ee la s t i c -p las t i c b ou nd ar y i s loca te d a t th e t ip o f the p ro je c t i l e un t i l .-7 i s l es s tha n .-7* s o th a t f ( t )


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812 A. L, Ya rin et al.i s d e t e r m in ed b y ( 3 9) w i th

f o r < ( 4 0 )Th is ca u ses a n eg l ig ib l e e r r o r f o r v e r y sma l l v a lu es o f Ix ]. No w, f o r ~ le s s th an ~* an d g r ea te rth an so m e c r i t ic a l v a lu e , t h e q u a n t i ty ~'2 i s d e t e r m in ed b y th e so lu t io n o f ( 3 6 ). Th i s so lu t io n i sv a l id u n t i l ~ -2 eq u a l s t h e v a lu e ~-1 an d th e e l a s t i c - p l a s t i c b o u n d a r y r each es th e r ea r su r f ace o ft h e t ar g e t. O n c e t h is o c c u r s w e c a n d e t e r m i n e f ( t ) d i r ec t ly b y r eq u i r in g th e ax ia l s t r e s s azz in(38) a t r = 0 to van ish wh en ~-2 is rep l ace d b y ~-1. Ho w ev er , the resu l ts o f s im ula t ions sugg estth a t i t i s a b e t t e r ap p r o x ima t io n to r eq u i r e th e av e r ag e ax ia l s t r e s s o n th e r ea r su r f ace in th ep la s t i c r eg io n to v an i sh .D e p en d in g o n th e th i ck n ess o f t h e t a r g e t an d th e v a lu e o f th e imp ac t v e lo c i ty U o , t h e t i p( z = x ) o f t h e p r o jec t i l e m ay r each th e r ea r su r f ace o f t h e t a r g e t . I f we n eg lec t t h e m o t io n( b u lg in g ) o f th e t a r g e t ' s r ea r su r f ace ( wh ich i s p r ac t i ca l ly in s ig n i f ican t f o r r e l a tiv e ly th i ckt a r g et s ) th e n t h i s o c c u rs w h e n x = - H . F u r t h e r m o r e , i f t h e p e n e t r a t i o n p r o c e s s c o n t i n u e sth en p a r t o f t h e p r o jec t i l e t ip p r o t r u d es f r o m th e t a r g e t . To mo d e l t h i s p r o cess i t i s co n v en ien tto d e f in e th e r ad iu s R 3 an d th e n o r ma l i zed r ad iu s /~a o f t h e p r o jec t i l e a t t h e f i rs t p o in t o fc o n t a c t o f t h e p r o j e c ti l e w i t h t h e t a r g e t m a t e r i al b y t h e f o r m u l a e

R3I1~3 ~---R-~ = 0 for -- H ~


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Penetration of a rigid projectile into an ela stic-plastic target of finite thicknesst h e n o r m a l i z e d r a d iu s /~ L a t t h e e n d o f th e p r o j e c t il e

[ - - ~ ~ - 1 1 / 2 '~ -L - R ~ o c 2 '~ L = + (~ '~ L+ 2 ) / 2 L 1813

(46a,b)a n d t h e n o r m a l i z e d r a d iu s /~ 6 o f t h e f i rs t p o in t ( s m a l le s t v a lu e o f /~ 6 ) a t w h ic h t h e n o r m a lstress a . , in (20b) vanishe s

""(/~6) = - f ( t ) - p x 2 [ ( 1 - 3/~2)(1/ ~ ) ] 2I pR~(B~Z - K2)l/2q+ L ( - M +/ I~ C) ] [ (1 ~ - 1 - Y ln [4( 1 - /~ 2 ) ]

+ [_ 3 ~ - ~ J = O ' (47)w h e r e u s e h a s b e e n m a d e o f th e e q u a t i o n o f m o t io n ( 2) t o e l im in a t e t h e t e r m ~ i n t e r m s o f2 an d th e func t ions A, B, C w hich wi ll be de te rm ined la te r [ see (53) ]. T hen the sep ara t ion l ineo f t h e t a r g e t m a te r i a l f r o m th e p r o j e c ti l e su r f ac e is d e t e r m in e d b y t h e n o r m a l i z e d r a d iu s /~ 4def ined by R 4 = M in [R 5, R L, R6]. (48)

3. E Q U A T I O N O F M O T I O N O F T H E P R O J E C T I L ET h e e q u a t i o n o f m o t io n o f t h e p r o je c t il e in t h e a x ia l d i r e c t i o n is o b t a in e d f r o m r ig id b o d yd y n a m ic s a n d m a y b e w r i t t e n in t h e f o r m

MY = F , (4%)-- 'L, -- (49b)= 2 (1 ~2~3/2_ (1 ~z~1/2"'L, + (1 -2 1/2 '

- - R L )wh ere M is the m ass o f the p ro jec t i le , pp is i ts densi ty , an d F is the re sulta nt axial force act ingin the po s i t ive e z d i rec t ion . Th is fo rce can be ca lcu la ted by in teg ra t ing the t rac t ion vec to rt app l ied by the ta rg e t to the p ro jec t i le , wh ich is g iven by

t = ~ n = a . . ( / ~ ) n + a . ~ ( / ~ R . ( 5 0 )In v iew of the d iscuss ion in the la s t sec t ion , we neg lec t the e f fec t o f a , , on the va lue o fF . Also ,we reca l l tha t the no rm al s t ress a . . i s com press ive over the reg ion /~3 ~< /~ ~


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814 A.L. Yarin e t a l .I n t h e a b o v e w e h a v e a s s u m e d t h a t l im i ts o f i n t e g r a ti o n / ~ 3 a n d / ~ 4 m a y d e p e n d o n x a n d ~ s oth e fu n c t i o n s A, B , C a l so d ep en d o n x an d ~ . Fu r th e rmo re , t h e i n t eg ra l s i n (5 2 ) may b eev a lu a t ed an a ly t i ca l l y t o o b t a in

h = r ~ p R 3 []{(11 _ _ / ~ 2 ) 3 / 2 3 . . . .1 1 ~ 2 ) 3 / 2 } f a ( t ) ( R ] /~2)], (53a)B : - ~ p R ~ [/~.(1 --/~2)2 _ /~(1 - - 2 2_ R3 ) _ f2(t)(/~24 _/ ~2 )] , (53b)C = r t y R 2 [ ( /~2 - / ~ 2) In ( 4 ) - (1 _ /~2) in ( 1 _ /~42) + (1 _ R3) ln (- 2 _ R 3-2

[ ~ J + f3 (t)( /~ 2 _ R3 ) _ 7 z # R 2 f a ( t ) ( ~ 2 _ ~ 2 ) . (53c)N o te t h a t A i s o f t en re fe r red t o a s t h e v i r tu a l mass o f t h e t a rg e t m a te r i a l a s so c i a t ed wi thth e p ro j ec t il e . Giv e n v a lu es fo r # an d Y , t h e c r i t i c a l v a lu e 2 " i s d e t e rm in ed b y (31 ). Th e n ,wh en Y* < f f ~< 0 , t h e fu n c t i o n s f l , f2 , f3 , an d f4 a re g iv en b y (3 9 b -e ) wi th ~2 sp ec i fi edb y (40 ); wh e reas wh en f f ~< if* , t h e fu n c t i o n s fa , f 2 , f3 , an d f4 a re g iv en b y (3 9 b -e ) wi th ~-2sp ec i f ied b y t h e so lu t i o n o f (3 0b ) an d (36 ) u n t i l t h e e l a s t i c -p l a s t i c b o u n d a ry reach es t h e re a rsur fac e w ith ~-2 = ~-1 [w ith ~-1 spec ified by (27b)] . W he n :~ ~< 2" an d the ela st i c- pla st i cb o u n d a r y h a s r e a c h e d t h e r e a r s u r f a c e s o t h a t r ~ d e t e r m i n e d b y ( 4 2 ) i s g r e a t e r t h a n z e r o ,t h e fu n c t i o n s f l , f2 , f3 , an d f4 a re g iv en b y (4 4 a -d ) wi th /~3 sp ec i f i ed b y (4 1) an d /~4 sp ec i f i edby (48).

4. A N E N G I N E E R I N G A P P R O X I M A T I O NTh e e q u a t i o n o f m o t io n o f th e p ro j ec t i l e (49 ), su p p l em en ted b y t h e ex p re ss io n s fo r t h ed ece l e ra t i n g fo rce (5 2 a ) an d (5 3 ) , may b e i n t eg ra t ed fo r t h e g en e ra l c a se o n ly n u mer i ca l l y .H o w e v e r , f o r s o m e i m p o r t a n t p a r t i c u l a r c as e s, r e a s o n a b l e a p p r o x i m a t i o n s c a n b e i n t r o d u c -e d w h i c h s i m p l if y t h e e q u a t i o n o f m o t i o n t o a f o r m t h a t c a n b e i n t e g r a t e d a n a l y ti c a ll y .Th i s ap p ro x ima te an a ly s i s d ev e lo p ed b e lo w y i e ld s an a ly t i ca l ex p re ss io n s fo r t h e p en e -t ra t i o n d ep th wh en t h e t a rg e t i s n o t p e r fo ra t ed an d t h e re s id u a l v e lo c i t y wh en t h e t a rg e t i sp e r fo ra t ed .Th e an a ly s i s o f t h i s p ap e r a s s u mes t h a t t h e p ro j ec t i l e rema in s r i g id . Co n seq u en t ly , fo ra me ta l p ro j ec t i l e p en e t ra t i n g a m e ta l t a rg e t t h i s a s su m p t io n l imi ts t h e imp ac t v e lo c i t y to b el es s t h an ab o u t 1 .5 k m/ s . T h e re su l t s o f p re l im in a ry s im u la t i o n s u s in g imp ac t v e lo c it i e s i n t hi sran g e i n d i ca t ed t h a t t h e re s is t i n g fo rce F rem a in s rea so n ab ly co n s t an t wh en a p ro j ec t il e

p en e t r a t e s a semi - in f in i t e t a rg e t. Fo r t h i s s i t u a t i o n i t i s a g o o d ap p ro x ima t io n t o se t /~3 eq u a lto zero a nd /~ 4 equ al to un i ty in (53). Furt he rm or e , fo r a sem i- inf in i te ta rge t , ~-1 an d (-1 in(39 ) ap p ro a ch n eg a t i v e i n f i n it y , so t h a t t h e ex p re ss io n s (5 3) red u c e t o

1 3A = A ~ = ] l t p R ~ , B = 0, (54a,b)C 2 5 in ~-2]. (54c)r c Y R o ~ [ ~ ln(4) +

I n g en e ra l , t h e v a lu e o f ~-2 a s a fu n c t i o n o f x can b e d e t e rm in ed b y u s in g (3 0 b) an d so lv in g (3 6) .Al t e rn a t i v e ly , we can ap p ro x im a te t h e so lu t i o n o f (3 0 a ) b y n o t i n g t h a t fo r sma l l v a lu es o f l ~2[(30a) yields2 z. t55)

F u r t h e r m o r e , s i n c e th e e x a c t s o l u t i o n o f ( 30 a) r e q u i r e s ~ 2 t o a p p r o a c h n e g a t i v e u n i t y w h e nap p ro ach es n eg a t i v e i n f i n i t y , t h i s su g g es t s t h e ap p ro x ima t io n[ 23~ ~2/3

2:~ ~2 ~ fo r :~ ~< ~* < O, (56a ,b)w h i c h h a s t h e c o r r e c t a s y m p t o t i c f o r m f o r s m a l l v a lu e s o f lx l a n d a p p r o a c h e s t h e c o r r e c t l im i tfo r l a rg e v a lu es o f I x I. M o re sp ec if i ca l ly , i t wi ll b e sh o wn l a t e r i n F ig . 9 t h a t t h e ap p ro x ima teso lu t ion (56a) i s qu i te c lose to the exact express ion (30a) .


15/31

Penetration of a rigid projectile nto an elastic plastic arge t of finite hickness 815U s i n g ( 5 4 c ) a n d t h e a p p r o x i m a t e s o l u t i o n ( 5 6 a ) i t f o l l o w s t h a t t h e f u n c t i o n C p a s s e st h r o u g h z e r o a n d b e c o m e s p o s i ti v e w h e n x = x * * * w i t h

, l- , ( , 4 r , ] ~ / ~ / . i s 7 )

R ~ ~ 1 - ~ t - ~ , , / JkV r - iW i t h t h e h e l p o f ( 3 1 b ) a n d t h e f a c t t h a t Y /#


16/31

816 A .L. Yarin et al:z r ( M + A ~ ) f 4 Y ' ~ I / 2 ] u2x = L \ ~ - ~ - ~ , ~ - ~ } j ~ , (64b)

( 4 Y ~ u2 H (64c)

[ { M + A . )O , = L \ d v , , ( 6 4 d )~ ( # } = ( # * * * - - # ) I n { 4 5 1 9 4 3 - --~ )

+ ~ . + 2 x * * * I n L .I _ ~ ) j . ( 6 4 e )E q u a t io n ( 6 3) i s u s e d t o d e t e r m in e e i t h e r t h e d e p th o f p e n e t r a t i o n o r t h e r e s id u a l v e lo ci ty .W h e n th e t a r g e t is th i c k e n o u g h , t h e d e p th o f p e n e t r a t i o n P = - x i s d e t e r m in e d b y s o lv in g(63) wi th ~ = 0. Al te rna t ive ly , wh en th e ta rge t i s pe r fo ra ted , the res idua l ve loc i ty U , i sde te rm ined by (63d) and (64d).N ot ice tha t fo r deep p ene t ra t ion [ t~ l >> 1 ] in to a semi- in f in ite ta rge t the fun c t ion t~(~7)m a yb e a p p r o x i m a t e d b y

3#

s o t h a t t h e d e p th o f p e n e t r a t i o n d e t e r m in e d b y (6 3b ) is g iv en b y~ = ( 4 Y ~ " 2 P = 02

\ 3 / ~ ] R ~ l n i 4 S / 9 { 3 # ' ~ ' (66)} . ' ' \ 4 Y / I Jo r b y ( 4 ) i n d im e n s io n a l f o r m . T h e a n a ly t i c a l e x p r e s s io n ( 4 ) c a n b e c o m p a r e d w i th t h es o lu t i o n o f t h e P o n c e l e t E q n (1 ) w h ic h h a s tw o e m p i r i c a l c o n s t a n t s c ~ a n d c 2. N o w i t isim p o r t a n t t o e m p h a s i z e t h a t o u r a s s u m p t io n t h a t t h e p r o je c t il e r e m a in s r ig id is o n ly v a li dw h e n t h e im p a c t v e lo c it y U o is b e lo w a c r it i ca l v a lu e . F o r m a n y p r a c t ic a l s i t u a ti o n s w h e na m eta l p ro jec t i le pene t ra tes a m eta l ta rg e t th i s causes the va lue o f c 2U 2 to b e m u c h s m a l l e rt h a n u n i t y s o th a t ( 1 ) m a y b e a p p r o x im a te d b y

P ~ c l c 2 U 2. (67)A l th o u g h (4 ) a n d ( 6 7 ) b o th p r e d i c t t h a t t h e p e n e t r a t i o n d e p th i s a q u a d r a t i c f u n c t i o n o fimpac t ve loc i ty , i t i s impor tan t to emphas ize tha t the ana ly t ica l fo rmula (4 ) expresses thep e n e t r a t i o n d e p th i n t e r m s o f m e a s u r a b l e p h y s i c a l p r o p e r t i e s o f t h e p r o j e c t il e a n d t h e t a r g e tw i th o u t t h e u s e o f a n y e m p i r i c a l c o n s t a n t s .The ba l l is t ic l imi t i s de f ined as th e m in im um va lue U b o f the im pac t ve loc i ty U o whichcauses the p ro jec t i le to pe r fo ra te the ta rge t . Us ing the ana ly t ica l so lu t ion (63d) we mayde te rm ine the ba l l i s tic l imi t by se t t ing U o - - U b and U, = 0 to ob ta in the express ion

/~2 = ~(~ ** ) + (~** +/4)t~(~**). (68)La te r i t wi ll be show n in F ig . 14 tha t the p red ic t ions o f the ba l l i s tic limi t de te rm ined by thea p p r o x im a te a n a ly t i c a l s o lu t i o n ( 6 8 ) c o m p a r e w e l l w i th c a l c u l a t i o n s o f t h e m o r e g e n e r a lequa t ions in Sec t ion 3 .

5. C O M P A R I S O N W I T H E X P E R I M E N T S , A N D D I S C U S S IO NU s in g t h e f o r m u la t i o n d e s c r ib e d a t t h e e n d o f S ec t i o n 3 w e d e v e lo p e d a c o m p u te r p r o g r a mto num er ica l ly in teg ra te Eq n (2 ) o r (49a), sub jec t to the in i tia l cond i t io ns (3) wi th U o be ing theim p a c t v e lo ci t y. W h e n th e im p a c t v e lo c i ty U o is s m a l l e n o u g h o r t h e t a r g e t t h i c k n e s s H i sl a r g e e n o u g h th e p r o j e c t il e a t t a in s a m a x im u m d e p th o f p e n e t r a t i o n P w h ic h is g iv en b y t h e


17/31

P e n e t r a t i o n o f a r i g i d p r o j e c ti l e i n t o a n e l a s t i c - p l a s t i c t a r g e t o f f i n it e t h i c k n e s s 8 1 7

T a b l e 2 . A l is t o f e x p e r i m e n t s , r e f e re n c e s , a n d r e l e v a n t m a t e r i a l a n d g e o m e t r i c a l p r o p e r t i e s, t T h e m a t e r i a l w a s t h es a m e a s t h a t u s e d i n E x p 1 , b u t t h e y i e ld s t r e s s w a s n o t g i v e n . S T h e m a s s e s o f t h e p r o j e c t i le s r a n g e d b e t w e e n 2 4 a n d

2 5 g . T h e m a t e r i a l p a r a m e t e r s c o r r e s p o n d t o t h e m i d d l e s o i l l a y e rP r o j e c t i l e T a r g e t

M L R ~ p Y /~ HE X P R e f. S h a p e ( g) ( r am ) ( r am ) ( k g / m 3 ) ( M P a ) ( G P a ) ( m m )

1 1 1 O g i v e 2 4 . 8 8 2 . 9 3 . 5 5 2 7 1 0 3 9 6 2 8 o o2 1 1 S p h e r e 2 3 . 3 7 4 . 7 3 . 5 5 2 7 1 0 3 9 6 2 8 o c3 1 1 C o n e 2 3 . 8 8 1 . 8 3 . 5 5 2 7 1 0 3 9 6 t 2 8 o c4 3 6 S p h e r e 1 1 .8 7 3 . 7 2 . 5 4 2 7 1 0 3 9 6 t 2 8 o c5 3 6 S p h e r e 2 3 . 4 7 4 . 7 3 . 5 5 2 7 1 0 3 9 6 t 2 86 3 6 S p h e r e 1 2 .1 3 9 .1 3 . 5 5 2 7 1 0 3 9 6 t 2 8 o c7 3 7 O g i v e 1 1 0 - - 1 0 .0 7 8 0 0 8 6 0 8 3 8 08 3 7 O g i v e 1 1 0 - - 1 0.0 7 8 0 0 9 2 0 8 3 4 09 3 7 O g i v e 1 1 0 - - 1 0 .0 7 8 0 0 1 0 8 0 8 3 2 0

10 38 C on e 24 .5 :~ 81 .8 3 .55 2710 305 28 25 .41 1 3 9 C o n e 7 8 . 0 1 4 6 . 0 4 . 7 6 5 2 7 1 0 3 0 5 2 8 2 5 . 41 2 4 0 O g i v e 2 3 , 1 0 0 6 7 4 4 7 . 6 1 8 6 0 1 0 0 .0 5 o o

T a b l e 3 . A l is t o f c o m p u t a t i o n s c o r r e l a t e d w i t h t h e e x p e r i m e n t s d e s c r i b e d i n T a b l e 2A N A L C h a n g e s f o r t h e c a l cu l a t io n s , w i t h t h e o t h e rC O M P E X P p a r a m e t e rs o f t h e c o r r e sp o n d i n g e x p e r im e n t b e in g f i x e d

1 1 - -2 43 5 - -4 6 - -5 76 7 H = o o7 8 - -8 8 H = ~9 9 - -

1 0 9 H = o o11 10 - -12 11 - -1 3 1 H = 0 . 0 6 m14 1 H = 0 .04 m1 5 1 U o = 0 . 4 k m / s ; H = 0 . 0 2 - 0 . 1 m1 6 1 U o = 0 . 6 k m / s ; H = 0 . 0 2 - 0 . 1 m1 7 1 U o = 0 . 7 k m / s ; H = 0 . 0 2 - 0 . 1 m1 8 1 0 - -1 9 1 0 R ~ = 7 . 0 m m2 0 1 0 Y = 5 0 0 M P a2 1 1 0 Y = 1 0 0 0 M P a ; / t = 8 3 G P a ; p = 7 8 0 0 k g / m 3 ( S te e l t a rg e t }2 2 1 2 ' U o = 0 . 2 8 , 0 . 5 0 , 1 . 0 k m / s2 3 2 H = o o2 4 3 H = o o

m a x i m u m v a l u e o f I x l e = Ixlmax- (69)O n t h e o t h e r h a n d , i f t h e imp ac t ve loc i ty i s la rge enoug h then th e p ro jec t i le wil l ex i t t h e t a r g e twit h a residual v e loc i t y U r.

In or d e r to e xam ine the ac c ur ac y o f the num e r i c a l so l u t i on o f Se c t i on 3 and the appr ox i -m a t e s o l u t i o n d e v e l o p ed i n S e c t i o n 4 w e h a v e c o m p a r e d c o m p u t a t i o n s w i t h a n u m b e r o fexper im ents wi t h projec t i l es of d i f ferent shapes , length s , rad i i , masses and targets w i thd i f ferent mater ial propert ies and th icknesses . A l i s t of these exper im ents , toge ther wi t h theassoc ia t e d r efe re nce s and r el evan t m ate ria l an d ge om e t r i c prope r t i e s i s r e cor d e d in Tab l e 2 .A l s o , T a b l e 3 p r o v i d e s a l i s t o f c o m p u t a t i o n s w h i c h i s c o r re la t e d w i t h t h e e x p e ri m e n t s


18/31

818 A.L . Yarin e t a l .0.500.40'

~ 0.30.

0.200.100.00

0.2

. . . j . * . l . . . I . . . I , , . I . . . 1 , . I1 . . .

o EXP 1 o /~ / / /+ E X P 2n EXP 3 /C O M P1 ~ + + ~ o

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8~ k n v s )Fig. 4. Semi-infinite target. C om paris on of the com puta tion w ith experimental results for threedifferent projectile tips; ogive (EXP 1); sphere (EXP 2); and cone (EXP 3).

descr ibed in Tab le 2 . Fo r con ven ience in p r esen t ing th i s da ta in the fo l lowing figu res we usea n u m b e r i n g s y s t e m w h i c h i n d i c a t e s t h a t f o r e x a m p l e E X P 2 c o r r e s p o n d s t o e x p e r i m e n tn u m b e r 2 , C O M P 1 c o r r e s p o n d s to c o m p u t a t i o n n u m b e r 1, a n d A N A L 1 4 c o r r e s p o n d s t ot h e a p p r o x i m a t e a n a l y s i s o f S e c t i o n 4 w i t h th e v a l u e s a s s o c ia t e d w i th C O M P 1 4. W ee m p h a s i z e t h a t i n p e r fo r m i n g t h e c a l c u l a t i o n s w e h a v e m e r e l y u s e d th e p u b l i s h e d v a l u e s o ft h e m a t e r ia l p r o p e r t i e s a n d h a v e m a d e n o a t t e m p t t o a d j u s t m a t e ri a l p a r a m e t e r s t o m a t c h t h eexper imen ta l da t a .F igures 4 and 5 compare the computa t iona l r esu l t s wi th exper imen t s fo r a semi - in f in i t et a rge t . F igu re 4 show s tha t t he e f f ec t o f the shape o f the p ro jec t i l e t i p i s r e la t ive ly small .H o w e v e r , t h e c o m p u t a t i o n s e e m s to c o m p a r e b e t t e r w i t h t h e o g i v e p r o j e c ti le o f E X P1 b e c a u s e t h e o g i v e is a c lo s e r a p p r o x i m a t i o n t o t h e o v o i d o f R a n k i n e t h a n t h e s p h e r e o f E X P2 o r t h e c o n e o f E X P 3 . F i g u r e 5 s h o w s t h e e f fe c t o f c h a n g i n g t h e m a s s a n d d i m e n s i o n s o f t h epro jec t i l e wh i l e keep ing the t i p shape con s tan t . F igure 6 show s the e ff ec t on the pen e t r a t iondep th o f ta rge t s o f d i f fe r en t fin i te th i cknesses an d F ig . 7 show s the e f f ec t o f chang ing the m assand d ime ns ions o f the p ro jec t i l e f o r the sam e t a rge t t h i ckness . F igure 6 a l so show s tha t t hee f fec t o f the f ree r ea r t a rge t su r f ace is on ly f e l t wh en the p ro jec t i l e t i p com es c lose to the r ea rs u r fa c e b e c a u s e t h e p r e d i c t i o n s o f th e m a i n p e n e t r a t i o n s t a g e o f th e p r o j e ct i le a r e n e a r l y t h esam e fo r f in it e and in fin i te th i ckness t a rge t s . Th i s su ppo r t s t he va l id i ty o f t he ap prox ima teb o u n d a r y c o n d i t i o n u s e d a b o v e a t t h e r e a r t a r g e t s u rf a ce . F ig u r e 8 (a ) s h o w s t h e p r o p a g a t i o nof the p ro jec t i l e t i p and the e l as t i c -p las t i c bo un da ry as func t ions o f t ime , and F ig . 8 (b ) show sthe d r a g fo r ce fo r f in i te and semi - in f in i t e t a rge t s . N o t i ce f rom F ig . 8(b) tha t t he d r ag fo r cer e m a i n s r e a s o n a b l y c o n s t a n t u n t il t h e p r o j e c ti l e t ip l e a v e s th e t a r g e t a n d t h e s u r f a c e a r e a o fthe p ro jec t i l e i n con ta c t w i th the t a rge t be g ins to sh r ink r ap id ly . A l so , no t i ce f rom F ig . 8(b)t h a t t h e d r a g f o r c e e x h i b it s a k i n k w h e n t h e e l a s t ic - p l a s t ic b o u n d a r y r e a c h e s th e t a r g e t ' s r e a rs u r fa c e a n d t h e f o r m o f th e s t r es s -f re e b o u n d a r y c o n d i t i o n i s c h a n g e d t o t h a t d e s c r i b e d b y(43).F i g u r e 9 s h o w s t h a t t h e a p p r o x i m a t e s o l u t i o n ( 56 ) o f th e l o c a t i o n o f th e e l a s t i c - p l a s t icb o u n d a r y c o m p a r e s w e l l w i t h t h e e x a c t s o l u t i o n o b t a i n e d b y s o l v i n g (3 6) . T h i s s u g g e s t s th a tt h e a p p r o x i m a t e a n a l y t ic a l f o r m u l a s d e v e l o p e d i n S e c t io n 4 m a y p r o d u c e r e a s o n a b l ya c c u r a t e p r e d i c t i o n s o f b o t h p e n e t r a t i o n d e p t h a n d r e s id u a l v e lo c i ty . F i g u r e 1 0 s u p p o r t s t h i sc o n c l u s i o n b e c a u s e i t s h o w s t h a t t h e p e n e t r a t i o n d e p t h p r e d i c t e d b y ( 66 ) c o m p a r e s w e l l w i t ha n u m b e r o f e x p e r im e n t s .A d d i t i o n a l f ig u r es e x a m i n e v a r i o u s p r e d i c t i o n s o f t h e n u m e r i c a l c o m p u t a t i o n s o f S e c t i o n 3and the ana ly t i ca l so lu t ion o f Sec t ion 4. For exam ple , F ig . 11 show s tha t t he p r ed ic t ions o ft h e c o m p u t a t i o n s a n d t h e a n a l y t i c a l s o l u t io n a r e v e r y c l o s e a n d t h a t t h e e f fe c t o f th e f i n it etarge t th ickn ess i s not s ign if icant for th is case. Th e ef fect of the f in i te targe t th ick ness i sexp lo re d mo re ca r e fu l ly in F ig . 12 and i s aga in o bse rv ed to be no t t oo s ign i fi can t excep t nea rthe ball ist ic l imit .


19/31

Penetration of a rigid projectile nto an elastic-plastic arge t of finite thickness 8190 . 2 0

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0 . 0 5

0 . 0 0 , , , i , , , i , , , ~ , , , i , , ,0 . 2 0 .4 0 . 6 0 .8 1 . 0 1 . 2

u o (k .m/s)Fig. 5. Semi-infinite arget. Com parisonof the computationwith experimentalresults or projectileswith the sam e tip shape but differentmasses and dimensions.

F i g u r e 1 3 s h o w s t h e c o m p u t a t i o n a l p r e d i c t i o n s o f t h e e l a s ti c - p l a s t ic b o u n d a r y f o rp e n e t r a t i o n i n t o a s e m i - in f i ni t e t a r g e t a n d f o r d i ff e r e n t v a l u e s o f t h e p e n e t r a t i o n d e p t h . F o rc o n v e n i e n c e w e h a v e p l o t t e d t h e v a l u e s { f , ~ , ~ o f { r , z , 4 } n o r m a l i z e d b y t h e p r o j e c t i l ed i m e n s i o n R oo. F i g u r e s 1 3 (a ,b ) s h o w t h e e l a s t i c - p l a s t i c b o u n d a r y p r o p a g a t i n g r e l a t i v e t o t h ef i x e d s p a t i a l c o o r d i n a t e 2 , w h e r e a s F i g . 1 3 ( c ) s h o w s t h e e l a s t i c - p l a s t i c b o u n d a r y r e l a t i v e t ot h e c o o r d i n a t e ~- m o v i n g w i t h t h e p r o j e c ti l e . T h e v e r t i c a l l in e s i n F i g . 1 3 (c ) i n d i c a t e t h el o c a t i o n o f t h e t a r g e r s f r o n t s u r fa c e . N o t i c e f r o m F i g . 1 3 (c ) t h a t t h e e l a s t i c - p l a s ti c b o u n d a r y


20/31

8 2 0 A .L . Ya r in e t a l .0 . 0 60 . 0 50 . 0 40 . 0 30 . 0 20 . 0 10 . 0 0

0 . 2

, , I I , I ,

( a )' o EXP 7 ~ .- - c o M P s / f . -. . . . . . C O M P 6 . , ~ . "(H= ""

I I ' I

0 . 4 0 . 6 0 .8U o (km/s)

.0

0 . 0 60 . 0 50 . 0 40 . 0 30 . 0 20 . 0 10 . 0 0

0 , 2

, , , I , , , I , i I I I i

( b ) /o E X P 8 /C O M P 7 ~ . ""

. . . . . . C O M P 8 . / . - " "

I ' ' I 1 '

0 . 4 0 . 6 0 . 8Uo (km/s)

1 .0

0 . 0 60 . 0 50 . 0 4

= . , 0 .03O . 0 20 . 0 10 . 0 0

o E X P 9 IiIC O M P 9. . . . . . C O M P i 0 [( H = * * ) ]

O 0 0 . - "

i i I(c )

I I I0 . 2 0 . 4 0 . 6 0 . 8 1 . 0U (kin/s)

Fig . 6 . Fin i te ta rge t . Co mp arison o f the com puta t io n wi th exper imenta l resul ts for the sam eprojectile p ene tratin g targets o f different thicknesses. The co mp uta tion s for the finite thickness targetsa re show n by the so lid l ines (CO M P 5 , 7 , 9) , whereas those show n by the dashed l ines (COM P 6 , 8 , 10)correspo nd to a sem i- inf in i te a rge t wi th the o ther param eters of EXP 7 , 8 , 9 be ing f ixed .

q u i c k l y p r o p a g a t e s a h e a d o f th e p r o j e c ti l e a n d a t t a i n s a s t e a d y s t a t e s h a p e r e la t iv e t o t h ep r o j e c t il e , w i t h t h e c o o r d i n a t e ~ - a p p r o a c h i n g t h e v a l u e ~-2 i n (3 5b ) a t t h e i n t e r s e c t i o n o f t h ee l a s t i c -p l a s t i c b o u n d a r y w i t h t h e r = 0 a x is .

F i g u r e 1 4 s h o w s t h a t b o t h t h e c o m p u t a t i o n s a n d t h e a n a l y t i c a l s o l u t i o n p r e d ic t r e a s o n -a b l y c l o s e v a l u e s f o r t h e n o r m a l i z e d b a l l i st i c li m i t . I n t h i s f i g u r e t h e v a l u e s a s s o c i a t e d w i t h t h eana lys i s were de te rm ined us ing Eqns (60a) , (64c) and (68 ) .


21/31

P e n e t r a t i o n o f a r i g id p r o j e c t il e i n t o a n e l a s t i c - p l a s t i c t a r g e t o f f i n it e t h i c k n e s s 8 21

2 . 0 0

1 . 5 0

~ ' ~ ' 1 , 0 00 . 5 0

(a)

0 , 0 0 , , , / ' , l , t , , i , , , , i , r , ,0 . 0 0 . 5 1 . 0 1 .5 2 . 0

, , 1 , , I I = , , , I * ,

o E X P 1 0

u (~s)

1 .0 0 = , , J , l( b )0 . 8 0 o E X P 11

" ~ 0 . 6 0

~ 0 . 4 0

0 . 2 0

0 . 0 0 ' ' ' i . . . . . .0 . 2 0 . 4 0 . 6 0 . 8 . 0

u 0 (~Jnls)Fig. 7. Finite target (H =0.254m).Comparisonofthecomputationwithexperimentalresults for two

p r o j e c ti l e s w i th t h e s a m e t ip s h a p e b u t d i f f e re n t d i m e n s i o n s .

W ith r e g a r d t o t h e f o r m u la t i o n o f t h e c o m p u ta t i o n s o f S e c ti o n 3 w e n o t e t h a t i n a d d i t i o nto the m ass o f the p ro jec t i le M in (49) , the ine r t ia o f the ta rge t en te rs the ana lys is th ro ugh theexpress ions fo r A and B in (52a) , a s we l l a s in the de te rmina t ion o f the pos i t ion o f thesepara t ion l ine f rom the so lu t ion o f (47) . Howev er , the appro x im at ions o f Sec t ion 4 ind ica teth a t f o r a m e ta l p r o j ec t i le p e n e t r a t i n g a m e ta l t a r g e t a n d f o r im p a c t v e lo c ii ie s b e lo w a b o u t1 .5 km/s , the ine r t ia l e ffec ts o f the ta rge t can be m ode le d s imply by tak in g A con s tan t andneg lec t ing B. Al tho ugh th e co m puta t ions o f Sec t ion 3 wou ld p red ic t s ign i f ican t ine r t ia leffects for h igher impact velocit ies , i t is no longer possible to ignore the e i-osion of thep r o j e c t i l e a s i t p e n e t r a t e s t h e m e ta l t a r g e t . H o w e v e r , b y c o n s id e r in g t h e p e n e t r a t i o n o fa m eta l p ro jec t i le in to a so i l t a rge t w i th a re la t ive ly low y ie ld s t reng th i t i s poss ib le to ob se rvesignif icant inert ia l effects while s t il l ens urin g that the projec t i le rem ains r ig id . F or e xam ple,we used the m ate r ia l p roper t ie s o f so il g iven in [40] to pe r fo rm the ca lcu la t io i i s p resen ted inFig. 15 for a sem i-inf ini te soil targe t . Fi gur e 15(a) show s the de cele rat io n of the pro ject i le asa func t ion o f t ime (which ag rees re la t ive ly we l l wi th the exper imen ta l da ta o f [40]) andFig. 15(b) shows the resu l t ing pen e t ra t io n de p th , each fo r d i f fe ren t imp ac t ve loc it ie s. Inpart icular , notice that the inert ia l effects for soi l targets become signif ie~ti t for impactve loc it ie s a s low as a bou t 0 .5 km/s . These ine r t ia l e ffec ts man i fes t themse lves i ti the d ev ia t ionof : f f rom a con s tan t a s w e ll a s in the la rge d i f fe rences be twe en the comput ia t i0 f i s o f Sec t ion 3and the ana ly t ica l so lu t ion o f Sec t ion 4 [ see F ig. 15 (b ) ].


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8 2 2 A . L . Y a r i n e t a l .0 , 0 20.00-

-0.02

,~E -0 .0 4N- 0 . 0 6- 0 . 0 8

- 0 . 1 0 ,0 .0

( a) C O M P ] 4 ~ - - F r o n t S u r f a c e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E l a s t i c - P la s t i c B o u n d a r y /!P r o j e c t i l e T i p

1 ' I I ' ' ' I ' ' ' I ' 1 ' I ' ' ' I ' ' '2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 1 0 0 . 0 1 2 0 . 0t ( I . tsec)

8 0 . 0 ' ' ' l ' ' ' l = l i l = = l i l ' ' 1 ' ' '( b ) . . . . . . . . . . .6 0 . 0

O .0 - - C O M P 14( H = 0 . 0 4 m )

- 2 0 . 0 ' ' I ' ' ' I ' . ' I ' ' ' i . . . I ' ' '0 . 0 ' 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 1 0 0 . 0 1 2 0 . 0t (p~seaz)

, - - , 4 0 . 0Zu . 2 0 . 0

Fig . 8 . Fin i t e t a rge t (H = 0 .04 m) . Loca t ions of the pro jec t i l e t ip and the e las t i c -p las t i c boun dary aswell as plots of the drag force for f ini te thickne ss and semi-infinite targets . Ap art from the targe tt hi ck n es s, t h e o t h e r g e o me t r ic a l a n d m a t e r ia l p a r a me t e r s o f C O M P 1 a n d C O M P 1 4 a r e t h e sa me .- 0 . 5 , , . I , , , i , , , i , , , i , , , i- o . 6 . . . . . E X A C T I

i(36) [- O . 7 . - A P P R O X I M A T E ]

- 0 . 9- 1 . 0- 1 . 1 , ' i ' ' ' I ' ' I ' ' I . . . . . .- ' 0 . 0 - 8 . 0 - 6 . 0 - 4 . 0 - 2 . 0 0 . 0 2 .0

Fig. 9. Com parison of the exact (36) and appro xim ate (56) expressions for the elast ic-plast ic boundary.

6. P E N E T R A T I O N O F P R O J E C T I L E O F R E V O L U T I O N W I T HA R B I T R A R Y S H A P E D T I PT o g e n e r al i ze t h e a b o v e t h e o r y f o r t h e c a s e o f a p r o j e c t il e o f re v o l u t i o n w i t h a r b i t r a r ys h a p e d t i p i t i s n e c e s s a r y t o c o n s t r u c t t h e p o t e n t i a l f l o w p a s t s u c h a b o d y . O n e o f t h e s e v e r a lm e t h o d s f o r c o n s t r u c t i n g t h e d e s i r e d v e l o c i t y fi e l d i s th e m e t h o d o f s i n g u l a r it i e s [ 2 8 , 2 9, 4 1 ] .T h e v e l o c i t y f i el d a s s o c i a t e d w i t h t h e o v o i d o f R a n k i n e i s th e s i m p l e s t o f s u c h f i e l d s b e c a u s e i t


23/31

Penet ra t ion of a r ig id pro jec t i l e in to an e las t i c -p las t i c t a rge t of f in i te th ickness 8231 5 . 0 , , , , l , , , , I , , , ,/

- - ( 6 6 a )o E X P 1 / o[ ] E X P 2

1 0 . 0 . x E X P 4 ~ . ~A E X P 5~ o E X P 6 ~5 0 1 ~ ~ ~

0 . 0 . . . . I . . . . I . . . .0 . 0 5 . 0 1 0 . 0 1 5 . 0

0 2 / I n [ 4 5 / 9 ( 3 1 a / 4 Y ) ]0

Fig . 10. Co mp ar i son of the ana ly t i ca l approxim at ion (66) of the norma l ized depth o f pene t ra t ionwi th exper imen ta l da ta .

0 . 0 70 . 0 6 .0 . 0 5 ,

_ o . o 4 .

= ,- , 0 . 0 30 . 0 2 .0 .010 . 0 00.00

. . . . I . . . . I . . . . t . . . . I . . . . I . . . .

(a ) H = 0.06 m , ) . ." C O M P 3 / /

- - - A N A L 13 / /. . . . . . C O M P ! i S ' /2 Y

. . . . I . . . . I . . . . I . . . . I . . . . I . . . .0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0U o ( k n d s )

1 .00 . 8 -0 . 6 -

0 . 4 -

0 . 2 -0 .00.40

, , I , , , I , I ,

(b) H = 0 . 0 6 /

- - - C O M P 13

0 , 6 0 0 . 8 0 1 .O0U (kin/s) 1 . 2 0

0 . 0 70 . 0 60 . 0 5

. ~ 0 . 0 4~" 0 . 0 3

0 . 0 20 .010.0 0 ~ " r".0 0

. * , * I , , , , I l l , l l , l l , l l . l l l l l . ,

(c) H = 0.04 m- - CO M P 1 4 ."- - - ANAL 1 4 ~ . ". . . . . . C O M P 1 ~ . ' "

I I I I I . . . .0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0u o ( k . e s )

1 .00 . 6 -

~ ' 0 . 6 -.

~ 0.4- "0 . 2 -0 .0

0 . 4 0

(d) H ~

- - - C O M P 14i . . . . ; T i ? ? , ' i . . .0 . 6 0 0 . 8 0 1 . 0 0 1 . 2 0U o ( k i n / s )

Fig . 1 1 . C o mp a r i s o n o f th e p r e d i c t io n s o f t h e c o mp u t a t i o n s ( CO MP ) o f th e e q u a t i o n s i n Se c t i o n 3wi th the an a ly t i ca l so lu t ion (AN AL ) of Sec t ion 4 for two ta rge t s of fin i t e th ickness . The d ot t ed l inesshow the resul ts for a semi-infini te target with the other parameters being f ixed.

r eq u i r e s o n ly a s in g le so u r c e - ty p e s in g u la r i ty . M o r e g en e r a l ly , w i th in th e co n tex t o f t h i sm e t h o d i t i s p o s s i b l e t o s a t is f y t h e i m p e n e t r a b i l it y c o n d i t i o n a t t h e s u r f a c e o f a n a r b i t r a r yb o d y o f r e v o l u t i o n b y i n t r o d u c i n g a d d i t i o n a l s i n gu l a ri ti e s . T h e d i s t r i b u t e d s i n gu l a ri t ie s m a yb e e i th e r so u r ces , s in k s , d o u b le s , o r v o r t ex r in g s . H e r e , we co n s id e r t h e s imp le ca se o fa d i s t r ib u t io n o f so u r ce - s in k s in g u la r i t i e s a lo n g th e ax i s o f t h e b o d y .


24/31

8 2 4 A . L . Y a r i n e t a l .0 . 7 00 . 6 0 -0 . 5 0

~ 0 . 4 0 .

. r . i . i I ~ i i I i i

(a) U = 0.4 knVs- - C O M P 15. . . . . . A N A L 15

~ 0 3 0 " .

o . 2 o ~ N ~ i0 . 1 00 . 0 0 ~ ~ . . ~ . , .

0 . 0 2 0 0 , 0 4 0 0 0 6 0 0 . 0 8 0 0 . 1 0 0H (m)

0 . 0 6g~ " 0 . 0 4

0. 10 . = , , , J , , , t , . , t , , ,(b) U = 0.4 knv's

0 . 0 8 - - C O M P 15. . . . . . A N A L 15

. . . . . . . . . . . . . . . . . . . . . .

0 . 0 2

0 . 0 0 I ' ' ' I ' ' I ' ' ' I ' ' '0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2H (m)

0 . 7 00 . 6 0 .O . S O .

~ 0 . 4 0 .

~ 0 . 3 0 "0 . 2 0 '0 . 1 0 0 . 0 0

O . 0 2 O

, t , I i = I , i . I I I

(c) U = 0.6 km/s~ COMP 16

ANAL 16

0 . 0 4 0 0 , 0 6 0 0 . 0 8 0 0 . 1 0 0H (m )

AE

0 . 1 0

0 . 0 8 -

0 , 0 6 -

0 . 0 4 -

0 . 0 2 -

, I . . . I = i t I i i = I * l ,

(d) V =0 .6 knV~. , -- - . . . . . . . . . . . . . . . .

- - C O M P 1 6. . . . . . A N A L 16

0 . 0 0 . . . ~ . . . , . . ; ~ . . . ~ . , ,0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 .1 0 0 . 1H (m)

0 .6 0 - ~ ~ km/s0 . 5 0 -

~ 0 . 4 0 -

~ 0 . 3 0 - " ' " ,C O M P 1 7 " , N N0 . 2 0 -

ot.oo " . . . . : A L l 7 , ," i~0 . 0 2 0 0 . 0 4 0 0 . 0 6 0 0 . 0 8 0 0 . 1 0 0

H ( m )

0. 10 , , . i , , ; I , i , i , , i i , , ,M ( f ) o o = 0 . 7 k ~0 .0 8 _ v - . . . . . . . . . . . .

COMP 17. . . . . . A N A L 17

0 . 0 2 -

0.00 ' ' ' ~ ' ; ~ ' ' ~ ' ' ' i ' , ,0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2

H ( m )

0 . 0 6 .g0 . 0 4 -

Fig . 1 2 . Co mp a r i s o n o f th e p r e d i c ti o n s o f t h e c o mp u t a t i o n s ( CO MP} o f th e e q u a t i o n s i n Se c t i o n 3wi th the ana ly t i ca l so lu i ion (A NA L) of Sec t ion 4 for a rang e o f ta rge t th icknesses.

Us ing th is m et ho d th e ve loc i ty po te n t ia l ~b i s g iven by~b(r, ~, t) = ~ ( t ) R ~ f ~ L - ~ ' / ~ F q ( f l )q - - J - ~ / " ~ L l - ( ~ - n ~ + : ] ~ / ~ J d O ' (70a)

= z - x ( t ) - c< , (70b)whe re ~ i s the Veloci ty of the b od y in the posi t ive e z d i rec t ion , R ~ i s a para m ete r charac ter i s t icof the bod y 's rad ius , L i s the leng th of the body , the t ip of the bod y i s loca te d a t ~ = - ct (wi th0 < ~ < L) , a r id q is the d is t r ibu t io n of s i t ig i ila r it ies . O iven a pro jec t i le sh ape chara c ter izedby the rad iu s r = R() i t ma y be sho wn tha t the d is t r ib u t ion q of sourCe-s ink s ingular i t ieswh ich en SUres imp en e t rab i l i t y i s d e t e rm in ed b y t h e F re d h o im in t eg ra l eq Ua t io n o f t h e


25/31

1 0 0 . 0

8 0 . 0 :6 0 . 0 L

4 0 . 0 -

2 0 . 0 -

0 . 0 :

1 I i a I J l( a )L l ' ' l t l ' l l l l , ,

(b )- ~

0 .5- x / R

1 ,10f l 0 0

1 5 . 0

1 0 . 0 -

5 . 0 -

0 . 0 -

- 5 . 0 -

- 1 0 . 0 -

- 1 5 . 0- 2 0 . 0 " , , , ' , . , i , , ' ' ' ' I . . . . I

1995 rigid projectile

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