The Development of a Three-Dimensional Material Point Method Computer Simulation Algorithm for Bullet Impact Studies

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Bruce Flowers
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1 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( The Develomen of a Three-Dmensonal Maeral on Mehod omuer Smulaon lgorhm for ulle Imac Sudes M.J. onnolly, E. Maldonado and M.W. Roh Dearmen of hyscs Unversy of Norhern Iowa edar Falls, Iowa US Receved November, 8 cceed March 7, 9 STRT The wo-dmensonal Maeral on Mehod (MM algorhm oulned by hen and rannon has been exended o hree dmensons. The develomen of he code s dscussed as well as alcaons for smulang bulle mac on bologcal and non-bologcal sysems. I. INTRODUTION The sudy of bulle macs on varous sysems s a currenly acve research feld whose resuls have moran meanng for he healh and safey of eole. Due esecally o he desrucve naure of bulle nsul, comuer smulaons rove o be very useful n undersandng and redcng he behavor of sysems nvolved n such macs. I s of arcular morance o sudy sysems whose comonens are elasc and can fal. Ths nvolves he bulle and arge beng comosed of eher a fne elemen grd or acual arcles. In order o sudy maeral deformaon and falure, we chose he laer scenaro. One sgnfcan challenge n smulang mac wh sysems comosed of arcles s ha nernal forces need o be communcaed hroughou he obec when deforms. In order o acheve such communcaon s normally hough ha a collson algorhm beween he arcles consung he sysem s requred whch, n hree dmensons s que me consumng. There exss, however a wo-dmensonal (D Maeral on Mehod (MM algorhm evaluaed by han and rannon [] whch allows modelng of elasc behavor and falure wh no conac algorhm, as dscussed laer. The urose of hs work s o develo a hree-dmensonal (3D MM algorhm and o demonsrae s uly for mac smulaons. II. THE MTERI OINT METHOD We began wh he D Maeral on Mehod (MM algorhm oulned and evaluaed by Z. hen and R. rannon []. MM s advanageous because doesn' requre a comlex me consumng algorhm o smulae collsons or maeral falure. Insead, MM mas he mass and momenum of grous of arcles n he smulaon o a background grd and ulzes he conservaon equaon of mass as well as conservaon of momenum o ulmaely advance he sysem hrough me. The use of he background grd also avods enanglemen assocaed wh oher conac based algorhms. To begn wh, he nal condons for he smulaon nclude a collecon of arcles wh nal osons { { x }, veloces v } and boundary condons chosen so as o reflec he moran hyscs of he acual sysem as much as ossble. Nex, he background grd s defned, where comuaonal sace s searaed no a defned number of grd cells n whch nodes are laced n each corner of each cell. The mass M of each arcle a me s 33

2 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( maed o he nodes of he grd cells based on he shae funcon N ( resulng n he mass m of node a me : x, m N M N (x. ( The same s done for he momenum (Mv of he arcles:. ( The D rogram uses recangular cells o dvde u sace, wh for nodes one a each verex of he square. Shae funcons hen are used o rovde weghed orons of a arcle s mass and momenum o he grd nodes. Our shae funcons N ( x are calculaed n he followng way, usng egh verces of a box. The coordnaes of he box conanng a arcle are normalzed so ha hey ake on exreme values of or a verces of he box, as shown n Fg.. So f a arcular box has a verex a (x,y,z and anoher verex a (x + x,y + y,z + z across s dagonal, he normalzed coordnaes are ( x x x ( y y y ( z z z (3 and he egh shae funcons used n he arcle o node mangs are calculaed from he normalzed coordnaes as shown n Table so ha hey are equal o uny for a verex (node where he arcle s a and s zero for a gven node f he arcle s a any oher node. Once he mangs n equaons ( and ( s comleed hen he forces a each node mus be calculaed. The nernal force (f arses from elasc deformaon of he obec and s calculaed as Fgure. secon of he MM grd. The gray arcle s n he mddle of s box and so s roeres are oroned ou o he box verces wh weghng funcons bu he whe arcle has he enrey of s roeres assgned o he verex s sng on. ( f n N G ( x s M. (4 The exernal force (f mlemened s hen calculaed a each node f ex c M gm (5 noe ha gravy can be exlcly aken no accoun n equaon 5. Then he nodal acceleraons are obaned usng f a m where he oal force s gven by f n f f ex are hen udaed n me. The node momena ( mv ( mv f, (6 and subsequenly he knemac varables assocaed wh each arcle n he smulaon are udaed usng node o arcle mangs: 34

3 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( Node oordnaes Shae Funcon (,, (-(-(- (,, (-(- (,, (- (,, (-(- v a x Nn Nn f m N ( x ( mv N ( x m x v (7 (8 (9 (,, (-(- (,, (- (,, (,, (- Table. Exressons for he egh shae funcons used o ma arcle roeres o each of he egh verces of he grd cube n whch resdes. The new arcle knemac varables n eqs. (7-9 are maed back ono he nodes o oban udaed nodal veloces. Now because he smulaed maeral(s may have deformed n beng udaed, he elasc roeres of he sysem mus be aken no accoun. The rae of sran ensor e s now calculaed from he velocy gradens: e ' ' x x. Here he rmed varables reresen velocy comonens and, exlcly n 3D we have e u x v v y x u w z x u v y x v y u u y y u w z x v w z y w z ( Subsequenly, he 3D elasc sress sran relaonsh s ulzed o calculae he s s udaed sress ensor, where s. Dfferen kl s kl yes of sran n 3D are vsualzed n Fg. and sress s shown n Fg. 3. There are several secal cases of he elasc ensor as well as sran ensor whch wll be dscussed. The algorhm hen reurns o mang arcle roeres o he grd (eq. and s reeaed unl a desred me lm s kl reached. III. SEI SES OF THE ESTIITY TENSOR There form of he elasc ensor s relaed o he symmery roeres of he maeral beng modeled [3]. In he mos general case, he maeral we are dealng wh may have no lanes of symmery a all. In ha case he elasc ensor requres unque elemens o descrbe he elasc 35

4 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( 36 Fgure. Dfferen yes of maeral deformaon. Normal comresson or exenson (lef resuls n a volume change for he maeral and s relaed o dagonal elemens of he sran ensor; shear sran (rgh conserves volume and relaes o off-dagonal elemens []. resonse of he maeral. The ensor s necessarly symmerc and so he lower dagonal block s omed bu mled: U T S R Q O N M K J I H G F E D. ( Now suose he maeral has one lane of symmery. I s ermed monoclnc and he number of elemens of he elasc ensor needed are now only 3. Ths arses from he fac ha he maeral roeres do no change uon a mrror reflecon abou he (x,y lane so he sresses are dencal for he srans and T ', where. The resul s U S Q O K H G F. ( n orhoroc maeral has 3 lanes of symmery and hen he elasc ensor s furher reduced o nne elemens: U S H G. (3 Now furher assume ha he maeral has 3 lanes of symmery and one s soroc. Then only fve ndeenden consans are needed

5 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( 37 Fgure 3. Vsualzaon of maeral sress. [3] The e are un ouward normals o he cube of maeral and he elemens of he sress ensor σ are he sresses n drecon acng on he face of he cube sanned by he angen vecos T ( normal o he un vecors e wh coordnae x = consan. (. (4 Now suose ha he maeral s enrely soroc. Then he number of consans reduces o wo: ( ( (. (5

6 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( 38 In our smulaons we assume soroc meda and so he elasc ensor akes for form of equaon (5 and he sress-sran relaonsh can be exressed exlcly n erms of relevan maeral roeres: xz yz xy z y x zx yz xy z y x E ( (. (6 Here E s he elasc modulus and s osson s rao. In he case where a flud s beng modeled, sheer forces are comleely absen and he elasc ensor s wren as. (7 SEI SES OF STRIN There are also mes when he geomery of srucures beng consdered has a coordnae whose behavor can be gnored [4]. Ths s usually he case, for examle, when he dmensons of an obec on one dmenson are much larger han n wo ohers. The resuls s lane sran, where he sran ensor akes he form D, (8 and he corresondng sress ensor s wren as e d c b a. (9 For ceran symmeres here may be anlane sran where h g f c. ( Noe for anlane sran ha he resonse of he maeral s urely shear. SIMUTIONS The 3D smulaons develoed here were valdaed by monorng oal mechancal energy conservaon of wo colldng elasc sheres. s shown n Fgure 4, he algorhm conserves oal mechancal energy and ncely descrbes he collson que well. The frs maor alcaon of he 3D MM mehod s bulle mac on a muscle / bone comose srucure. Fgure 5 shows a relmnary smulaon where a roecle nsuls a smulaed muscle / bone comlex. The MM mehod has also been used o smulae bulle mac on

7 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( Fgure 4. Knec energy (snusodal curve sarng a uer lef, elasc oenal energy (snusodal curve sarng a lower lef and oal mechancal energy (horzonal lne a o for he wo-shere collson valdaon run. Fgure 5. Sequence s o lef o boom rgh. Snashos from a relmnary muscle/bone mac smulaon. The bulle eners a lef; muscle ssue s dark gray and bone s whe; all maerals have dfferen elasc roeres. 39

$MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( Fgure 6. ulle mac on ceramc / Kevlar comose ves for varous fracons of maeral breakng ons.$

8 MERIN JOURN OF UNDERGRDUTE RESERH VO. 9, NOS. & 3 ( Fgure 6. ulle mac on ceramc / Kevlar comose ves for varous fracons of maeral breakng ons. ceramc/kevlar comose vess; a ycal run s shown below where he sensvy of he resuls o he maerals elasc breakng ons s examned. REFERENES. Z hen, RM rannon, Sanda Naonal aboraores (SND-48,. Mahemacs of Sran and Sran Rae, eded by sbørn Søylen, Norwegan Unversy of Scence and Technology: h://folk.nnu.no/soylen/sranrae/mahemahcs/ 3. Infnesmal Sran Theory, Wkeda arcle: h://en.wkeda.org/wk/sran_ensor 4. Hgh Srengh omoses by cha Rusmee, webage Unversy of Uah, Dearmen of Mechancal Engneerng: h:// moses.hml 4

Solution in semi infinite diffusion couples (error function analysis)

Solution in semi infinite diffusion couples (error function analysis) Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of