DESIGN OF A SPLIT HOPKINSON PRESSURE BAR

Transcription

1 DESGN OF A SPT HOPKNSON PESSUE BA Felpe Glln Goup of Sold Mechnc nd Sucul mpc Unvey of São Pulo São Pulo SP - Bzl felpe.glln@pol.up..s. Bch mpc eech Cene Unvey of vepool UK 123@lv.c.uk Mcílo Alve Goup of Sold Mechnc nd Sucul mpc Unvey of São Pulo São Pulo SP - Bzl mlve@up. Ac. Th ppe m o clfy ome mpon pec ou he degn of Spl Hopknon Peue B (SHPB) ppu. Th mchne ued o on he e-n elon dffeen hgh n e. The ledy well-elhed heoy of he SHPB peened nd he mjo vle fo he degn of SHPB e houghly exploed. A udy ce hown, hghlghng he mjo vle one need o del wh n ode o popely degn he SHPB. Keywod. Hopknon, elc wve, mel chcezon, hgh n e. 1. noducon The cenfc communy h lwy ough o model he ehvo of he dffeen knd of exng mel. Elc nd elo-vco-plc mel model wee developed nd hve een exenvely ued n he decpon of he ehvou of mel. All hee model need o ely on pmee h hould lwy e meued o expemenl ppu hve een exenvely developed. Among hee g, he donl enle e mchne he mjo equpmen fo meuemen of qu-c mel popee, whle he Spl Hopknon Peue B (SHPB) ecme he chef expemenl echnque o on dynmc mel popee. The Spl Hopknon Peue B expemenl echnque, lo known Kolky B, w nvened y Kolky n The ppu con of pecmen, he eed mel, ndwched eween wo elc (ee Fg. 1). A gh-velng compeve e pule geneed n he npu. When he pule eche he -pecmen nefce, plly nmed hough he pecmen nd plly efleced. The efleced nd nmed pule e meued y n gge loced n he npu nd oupu. The ecoded gnl cn e ued n he d nly o deemne he n hoy of he pecmen. The compeve pule geneed y he mpc of ke gn he npu. The ke uully cceleed y g gun peclly developed fo h pupoe. Fo ddonl del he ede efeed o Zuk e l. (1983). Snce nvenon, he SHPB h undegone evel modfcon. w dped o do enle e (ndholm e l., 1968), oon e (Bm e l., 1999), mulneou compeon nd oon (ew e l., 1973), fcue dynmc (Klepczko, 1979), mong ohe von. Hdng (198) peen evel of hee von he SHPB h undegone ove he me. n pcul, Nem-Ne e l. (1991) developed novel echnque o he SHPB, whch llow he execuon of compeon e followed y enon, llowng he nly of he Buchnge effec unde hgh n e. The echnque hown n Nem-Ne e l. (1991) llow lo he execuon of dynmc ecovey expemen, n whch he pecmen ujeced o pe-gned e pule nd hen ecoveed whou ddonl lodng fo po-e mcoucue nly. Te wh he SHPB unde condon of hgh empeue hve een lo epoed (Mulle, 1972). Meng e l. (23) h ued wve epon echnque o educe me-hfng dnce eween -pecmen nefce nd n gge on, heefoe mnmng wve dpeon nd enuon n he SHPB e. The employed wve epon echnque con of wo-pon n meuemen mehod le o pl he ncden nd efleced wve. A p of pogm o chceze mel ehvou unde dynmc condon, SHPB eng degned n he uho nuon, whch demnded fo mhemcl nly of he wve popgon n, peened n econ 2, 3 nd 4 of h cle. Secon 5 wll del wh he d nly of he vou gnl geneed n dynmc e ung SHPB. The numenon equed n ypcl expemen wll e deced n econ 6, wh econ 7 peenng ce udy o mee ome degn conn.

2 2. Mhemcl modelng The c equon n whch he pncple of he SHPB ed cn e oned when umng h he e wve vel hough he pecmen f enough o h he popgon me nevl mll when comped o he ol e me. Th llow fo evel eflecon o occu n he end of he eed pecmen o h cn e umed h he pecmen peen unfom e of e nd n. Moeove, he hypohe of unxl e e mpled, whch cn e guneed hough he ue of lucn (nomlly MoS 2 ) eween nd pecmen. Fnlly, umed h he ee nd veloce n he pecmen end e nmed hough he ncden (npu ) nd nme (oupu ) whou dpeon. Adopng he me mel nd co econl e fo oh npu nd oupu, mple expeon fo e, n nd n e n he pecmen e now oned. ε Specmen npu ε ε Oupu Fgue 1. Wokng pncple of he SHPB. Conde Fg. 1, n whch he ncden, efleced nd nmed pule, ε, ε nd ε, e hown. The dplcemen, u, he end of he pecmen e gven y (Zuk e l., 1983 nd Johnon, 1973) u = 1 cε1d (1) nd u = 2 cε2d, (2) whee c = E ρ he velocy of he elc wve n he nd he ucp 1 nd 2 efe, epecvely, o he lef nd gh end of he pecmen. Equon (1) nd (2) cn e ewen funcon of he ncden, efleced nd nmed pule: ( ε ε ) u1 = c d (3) nd = 2 c ε d. (4) u Oeve h n h cle, compeve ee nd n hve pove gn. n vew of n umed unfom e of e nd n n he pecmen hckne, n, ε, gven y he expeon elow: u1 u2 ε =, (5) whee epeen he pecmen lengh. Suung Eq. (3) nd (4) no Eq. (5) he n he pecmen ujec o gven y: c ε = ( ε ε ε ) d. (6) The foce cng on he pecmen end e oned fom:

3 ( ε + ε ) P1 = EA nd (7) P = EAε, 2 whee E nd A e he elc modulu nd he co econl e of he ncden nd nme, epecvely. Snce he pecmen n equlum, P 1 = P 2, o h, fom Eq. (6) nd (7), follow: nd ε +ε = ε (8) c ε = ( ε ε ε ε ) d. (9) The e, n nd n e cng on he pecmen cn hen e clculed y he followng equon: 2c ε = ε d, (1) A σ = E ε, (11) A 2c ε& = ε, (12) whee A he co econl e of he pecmen. Noe h one need o know he elc modulu, deny, co-econ e nd he pecmen geomey, n ode o fully chceze he mel epone o dynmc lod povded he efleced nd nmed wve cn e meued. Thee wve e nowdy meued wh nen ecode, whch cpue he n guge gnl fxed n he (ee econ 6). hould e kep n mnd, poned ou y Zuk e l. (1983), h he ee, n nd n e clculed y he ove expeon epeen only vege vlue. 3. Equon fo he degn of he SHPB Se, n nd n e whn he pecmen e he mo mpon equemen o e ken no ccoun when degnng SHPB. Thee vle e conneced wh he mel o e eed, e.g., eel, non-feou mel, cemc nd wh he n e nge med. Thee equemen e cloely conneced o ome degn vle he lengh of he e pule, he level of e n he, he co econl e of nd pecmen, he mpc velocy, mong ohe. v v = Ske Befoe mpc npu c c v v v v = Ske Afe mpc npu Fgue 2. Se pule geneon. Conde Fg. 2 n whch he coxl mpc eween he ke nd npu cn e oeved. Befoe he conc eween he ode, he ke vel wh velocy v whle he npu y e. Dung he mpc, he foce

4 cng on he ke nd n he common nefce equl nd o he velocy, v ( < v < v, fe mpc). f σ nd σ e he ee geneed n he ke nd npu, epecvely, he followng expeon vld: A σ = Aσ, (13) whee A he ke co econl e. The ee σ nd σ e eled wh he velocy n he common nefce 1 ccodng o ( v v ) σ =ρc (14) nd σ =ρc v, (15) whee ρ nd ρ e he dene nd c nd c e he veloce of he elc wve n he ke nd n he npu. Suung Eq. (14) nd (15) n Eq. (13) one h: βv v =, (16) 1 +β whee Aρc β =. (17) Aρ c Suuon of he ove equon n Eq. (14) nd (15) llow he clculon of he e n he ke nd npu : nd σ ρcv = 1+β ρc βv σ = 1+β. (19) (18) The ke nd he npu emn n conc unl he pule geneed n he ke eflec fom end enle pule, velng owd he conc nefce. The me, p, needed y he pule o eun o he conc nefce equl o: p 2 =, c whee he lengh of he ke. Theefoe, he pule lengh, p, geneed n he ncden gven y: p c = c p = 2. (2) c whch deemnn pmee of he n level n he pecmen, hown elow. The e pule e n he ncden y he ke mpc eche he pecmen nd plly efleced nd plly nmed. The neny of he nmed pule, σ, whch degn vle, mu gunee o lod he pecmen wh pecfc e level, σ. Accodng o Eq. (11), he e nmed gven y A σ = σ. (21) A The efleced pule, σ, degn vle popoonl o he n e n he pecmen (ee Eq. (12)). On un, he n e lo degn equemen o h E σ = ε&. (22) 2c 1 Elc mpc eween wo peened lengh y Johnon (1973).

5 n ode o fy he equemen expeed n Eq. (21) nd (22), he ncden pule h vel hough he npu A E σ = σ + ε&, (23) A 2c oned ung Eq. (8), (21) nd (22). Now, f expeon (19) uued no Eq. (23), he mpc velocy, v, degn vle cn e oned funcon of he degn equemen σ e ε&, ccodng o v 1+β A E σ + ρcβ A 2c = ε&. (24) Th equon llow he clculon of he ke velocy o gven n e nd e level n he pecmen cheved. The l degn equemen o e fed he mxmum defomon o whch he pecmen ujeced. Accodngly o Eq. (1), he n n he pecmen decly popoonl o he neny of he e pule nd o he me duon of he ncden pule. Aumng h he efleced pule conn, he duon me of he pule, p, gven y p ε = 2c ε. (25) whch, ogehe wh Eq. (2), gve elon eween he lengh of he pule (degn vle) nd he equed pecmen n: p ε =. (26) 2 ε Accodng o he convenonl SHPB echnque, he lengh of he npu nd oupu mu e gee hn he lengh of he gee pule h wll e nmed y hem. Th equemen necey o vod upepoon of he e pule wh own eflecon n he n gge on. The wo-pon n meuemen echnque ued n (Meng e l., 23) llow gnfcn educon of he pce occuped y he. The ze of he ke necey o on he equed n cn e deemned fom he pule lengh (ee Eq. (2)). The pevou nly peume h he ke, ncden nd nme wok n he elc egme, heefoe mu e checked n he degn phe f he ee e elow he yeld lm of he mel. The equon peened hee e uffcen fo he clculon of he mn degn vle of he SHPB. 4. Clculon n he ccl opeonl condon The SHPB h wo ccl opeonl condon: e hgh n e nd e low n e. n he f ce, he mxmum n e h cn e eched depend on he mxmum e uppoed y he wh no yeld. Becue of h, he mel o e choen fo he conucon of he mu hve hgh yeld engh. Tnum, whoe yeld e ppoxmely eween 1 o 12 MP, mgng eel, ool eel o he eed lloy eel e mel h mee h equemen. The mxmum n e of he e cn e lo nceed hough he educon of he pecmen lengh (ee Eq. 12). Unfounely he educon of he pecmen ounded y he lmon mpoed y mnufcung. Some eeche ue o do he e wh hn hee compeon pecmen, mng nceng he n e nd conequenly educng he neny necey fo he e pule. Alo, fo mll elon /d (whee d he pecmen dmee), he fcon n he pecmen end led o non-unxl e e, whch could nvlde he d. woh o emnd h he equon deved ove wee oned fom he hypohe h he e e n he pecmen w unxl. S e l. (1991) ued enon veon of he SHPB o nvege he nfluence of he o /d on he eul. They howed h he lengh o dmee o ffeced he eul only when /d w mlle hn 1.6. Fo lge vlue, he eul wee pcclly ndnguhle. n he ce of low n e e, he lmon mpoed y he lengh of he. A ledy d, f he convenonl SHPB e-up doped, he lengh of he lge pule h cn e nmed hould e mlle hn he lengh of he npu nd oupu. Howeve, f he e-up uggeed y Meng e l. (23) ued, hen gnfcn educon of he lengh cn e cheved. Theefoe, when he degne defne he ze of he, he mxmum lengh of he e pule uomclly e.

6 Equon (26) how h he mxmum pecmen n (degn equemen) decly popoonl o he pule lengh. f he e o e done hgh n e, he efleced pule ε lge nd heefoe he pule lengh cn e mll. n h opeonl condon, he mxmum equed pecmen n ely eched. Bu when he n e low, he efleced pule h educed neny nd heefoe he lengh p mu e lge o enue h he equed n eched. n genel, fo low n e, he mxmum pecmen n cnno e lge, ecue hgh n eque lge vlue of p, omeme mpccl due o he vlle oom fo he SHPB nllon. Equon (26) could gve he fle mpeon h he educon of he pecmen lengh would mply n educon of he necey pule lengh, u h no ue. Fo conn n e, n leon n mple n n equl chnge n he efleced pule neny nd heefoe he con of oh effec would cncel ech ohe. 5. D nly The d nly of he convenonl SHPB e-up con mply n he negon of he ecoded ε gnl nd n olvng Eq. (1) o (12) n ode o on he e, n nd n e o whch he pecmen w ujeced dung he e. The e-up popoed y Meng e l. (23), whch mke ue of wo-pon n meuemen echnque h, n ddon o he d nly of he convenonl SHPB, ome pecfc opeon equed y uch echnque. n he new e-up, wo n gge e plced on he npu cloe o ech ohe nd ne -pecmen nefce (ee Fg. 3). Th mnmze wve dpeon nd enuon n he SHPB e nd llow educon of he lengh. A he meuemen of n hppen n wo dffeen pon, pole o epe he upepoed ncden nd efleced pule. Fgue 3. gngn dgm fo longudnl wve n he ncden (Meng e l., 23). The wve epon echnque ed on followng elon: () =ε ( ) ε, (27) nd () =ε ( ) ε (28) ( ) c = = =, (29) Ι Ι whee ε () nd ε () e he ncden n pule, ε, pon nd ; () ε () ε nd e he efleced n pule, ε, pon nd ; nd e he me when ε eche pon nd ; nd e he me when ε eche pon nd. Equon (27) nd (28) e only vld when he pule hpe chnge neglgle. The lgohm o epe he ncden nd efleced pule : () =ε ( ) ε, (3) () =ε () ε () =ε () ε ( ) ε, (3)

7 nd () =ε ( ) ε (3c) () =ε () ε () =ε () ε ( ) ε, (3d) whee ε () nd ε () e he n meued y he n gge on loced nd. Th lgohm ued wh he condon deced elow: zeo. ) Fo ) Fo. =ε = ε < : The pule doe no ech pon nd, heefoe ε () =ε () =ε () =ε () =ε () =ε () = < = c) Fo < : The pule eche pon u no, heefoe () () ε whle he ohe emn equl : The ncden pule pe hough nd, nd he efleced pule h no eched n gge ye. ε () =ε () ε, ε () =ε () = ε nd ε () =ε () = d) Fo () nd ε () <. : The efleced pule pe hough, wh cue he upepoon of ε nd ε. Thu, ε e gven epecvely y (3) nd (3). e) Fo y (3c) nd (3d). : Wve upepoon occu n gge nd heefoe () nd ε () The ncden nd efleced pule e oned fom () nd ε () ε e clculed epecvely ε fe condeng he phe dffeence. Once ε nd ε wee clculed, he emnng d emen excly equl o he convenonl SHPB e-up. 6. numenon A ypcl SHPB numenon ue n gge, n ocllocope o nen ecode, n eleconc nego o n opeonl mplfe, gge ccu, powe upply nd velocy meuemen yem. n he convenonl SHPB lyou couple of n gge plced n he mddle of he npu nd nohe couple plced n he mddle of he oupu. Ech n gge of couple loced dmeclly oppoe de of he. Th llow cncelng ny endng effec peen n he due o mpefec lgnmen eween, pecmen nd ke. n he e-up deced n (Meng e l., 23) n ddonl n gge couple mu e plced n he ncden. The ocllocope o he nen ecode ued o ecod he n gnl meued y he n gge on. The ε gnl cn e neged ehe y n eleconc nego o n opeonl mplfe. The eul of h negon gnl popoonl o he pecmen n, whch cn e npu o he X-x of ecodng numen. The ε gnl npu n he Y-x. Th pocedue llow dec vulzon of he pofle of he e-n cuve of he mel. The ecoded gnl cn lo e mnpuled n compue n ode o pefom he opeon deced y Eq. (1) o (12). The velocy meuemen yem nended o deemne he ke mpc velocy. The wo mo common meuemen echnque employed e me nevl coune oced o ehe mgnec pck-up o phoocell nd lgh ouce. The gge ccu ued o conol he moun of d ecoded. wok n uch wy h only one complee lod even egeed. Von fom he ypcl SHPB numenon deced ove cn e found n he leue. Mulle (1972) h ued cylnde condene gge o meue he n pule. Thee gge e enve o he dl ufce dplcemen of he. Accodng o Mulle, he ue of hee devce w eenl nce he we n gge ecme dmged nd unelle unde hock condon. He lo ued fle newok conng of uned conducnce-cpcnce fle nd low p ence-cpcnce fle n ode o enue he Pochhmme-Chee ocllon nd elmne noe. ndholm e l. (1968) ued coxl cpcnce gge o meue he pecmen dl n whch, oced wh he meued efleced pule, ε, llow he clculon of he dynmc Poon o. The dl n meuemen cn e lo ued o check he xl n meuemen oned y he gge on he npu. ew e l. (1973) dece he numenon necey fo xl e: mulneou oon nd compeon. S e l. (1991) hve ued wo full Wheone dge wh wo 35 Ω cve gge ech o deec flexul wve componen n wo pependcul decon on he oupu. Th meuemen echnque w le o mono ny endng momen nmed hough he pecmen. 7. Sudy ce Th econ m o how how ypcl SHPB degn would e developed wh he d of he heoy peened n econ 2, 3 nd 4. e conde h SHPB ppu hould e degned ccodng o he equemen deced

8 follow. The pecmen o e eed wll e mde of mellc mel. The gee n e equed 5 1. A h n e, he SHPB mu e le o n he pecmen up o 5% mxmum e of 15 MP. The lowe n e doped fo he e 5 1. n h e condon he mxmum pecmen n nd e e epecvely 2% nd 15 MP. The f ep n he degn he defnon of he mel nd dmee of ke nd nd he defnon of he pecmen geomey (lengh nd co econl e). f dffeen pecmen geomee wee o e ued n he e, hen he degn mu e done fo he ccl geomey,.e., he one wh gee lengh nd co econl e. n h udy ce, ume h he ke nd e mde of hgh engh eel wh dmee of 25.4 mm. The pecmen geomey 1 mm n lengh nd 5 mm n dmee. Once hee choce e mde nd he degn equemen e known, he emnng degn vle cn e deemned n ghfowd mnne mply y olvng he equon fom econ 3. A menoned n econ 4, he SHPB h wo ccl opeonl condon: mxmum nd mnmum n e e. Accodngly, he degn hould e uch h he equemen of oh condon e fed. The hgh n e e deemne he gee e pule o e nmed hough he. lo elhe he mxmum mpc velocy nd he mxmum enegy equed fo he ke cceleon. On he ohe hnd, he low n e e gve nfomon ou he mnmum lengh of nd he longe ke h mu e ued. The eul conned n T. 1 nd 2 wee oned fom he oluon of Eq. (18) nd (21) o (26). The enegy equed fo he ke cceleon, E kn, gven y he knec enegy conned n he ke, whch funcon of he mpc velocy nd ke dmenon. Tle 1. Hgh n e e. σ [MP] ε [-] ε& [ 1 ] σ [MP] σ [MP] σ [MP] σ [MP] 15 5% v [m/] E kn [J] [m] [m] [m] p [m] p [] Tle 2. ow n e e. σ [MP] ε [-] ε& [ 1 ] σ [MP] σ [MP] σ [MP] σ [MP] 15 2% v [m/] E kn [J] [m] [m] [m] p [m] p [] A cn e een n T. 1, he mel o e choen fo he nd ke mu hve yeld engh hghe hn MP. A mxmum enegy of 1.44 kj mu e yeld o ccelee he ke. Th nfomon ued n le ge fo he degn of he g gun. Fo exmple, f he vlle lengh fo he ke cceleon 2 m, hen he fng peue mu e le 14.2 (1.42 MP). Tle 2 how h he ncden nd nme mu hve le 2 m. The longe ke o e ued n he e h 1 m n lengh. Shoe ke e doped n hgh n e e. Oeve h fo he e condon of T. 1 he necey ke lengh only.26 m. Even hoe ke cn e ued f deed. we o mplemen he equon of econ 3 n compue pogm. n h wy, ey o nlyze wh would hppen wh he degn vle f ohe dmee nd mel wee choen fo he nd ke. The nfluence of pecmen geomey cn e deemned y he me mehod. 8. Concluon The SHPB eng echnque llow compeon, enon nd oon e o e pefomed fo od nge of n e. Te wh n e vyng fom 1 o 1 1 wee commonly epoed n he leue. oom lmon, he necey of hgh engh mel fo he conucon of he nd hgh-peue g-gun poe dffcule fo e hghe n e. w hown h he degn vle of SHPB cn e deemned n ghfowd mnne mply y olvng he e of equon peened n econ 3. The d emen of h e que mple. The equed numenon cn e mde mple nd even o mpon mel ehvou cn e oned. The udy ce h hown ypcl SHPB degn. w pole o hve n de of e vlue nd dmenon of componen. The neeed ede hould lo e ome ohe von n ode o elze he nfluence of ech degn vle nd degn equemen. 9. efeence Bm, M.N. nd Pnc, N.,1999, Hgh Sn e Effec on he Sn of Alloy Seel, Jounl of Mel Poceng Technology, 92-93, pp

9 Hdng, J.,198, Teng Technque Hgh e of Sn, O.U.E.. epo, No. 138/8, pp Johnon, W., 1973, mpc Sengh of Mel, Ed. Edwd Anold, pp Klepczko, J.,1979, Applcon of he Spl Hopknon Peue B o Fcue Dynmc, n. Phy. Conf. Se., No. 47: Chpe 2, pp ew, J.. nd Goldmh, W., 1973, A Bxl Spl Hopknon B fo Smulneou Toon nd Compeon, ev. Sc. num., Vol.44, No. 7, pp ndhom, U.S. nd Yekley,.M.,1968, Hgh Sn-e Teng: Tenon nd Compeon, Expemenl Mechnc, Vol.8, No. 1, pp Meng, H. nd, Q.M.,23, An SHPB Se-up wh educed Tme-Shf nd Peue B engh, nenonl Jounl of mpc Engneeng, Vol.28, pp Mulle, T.,1972, Hgh Sn e Behvo of on nd Nckel, Jounl Mechncl Engneeng Scence, Vol.14, No.3, pp Nem-Ne, S., c, J.B. nd Se, J.E.,1991, Hopknon Technque fo Dynmc ecovey Expemen, Poc.. Soc. ond., Vol. 435 A, Ge Bn, pp S, G.H. nd Gl, A.,1991, A Dec-enon Spl Hopknon B fo Hgh Sn-e Teng, Expemenl Mechnc, Sepeme, pp Zuk, J.A., Nchol, T., Swf, H.F., Gezczuk,.B. nd Cun, D..,1983, mpc Dynmc, Ed. John Wley & Son, pp