VERIFICATION OF FRAME INDIFFERENCE FOR COMPLICATED NUMERICAL CONSTITUTIVE MODELS

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1 ECTC Pocdig ASME Ealy Ca Tchical Coc otd y ASME Ditict E ad ivity o Akaa Suppot Povidd y th ASME Old Guad ad th Committ o Ealy Ca Dvlopmt Mach -Apil,, Fayttvill, A VEIFICATION OF FAME INDIFFEENCE FO COMPLICATED NMEICAL CONSTITTIVE MODELS Kiha Kamoala *, cca M. Bao *. *Dpatmt o Mchaical Egiig, ivity o tah. 5 S. Ctal Campu D., Salt Lak City, T 84. ABSTACT Th picipl o matial am idic qui patial t to otat with th matial, wha c t mut iitiv to otatio. Ttig o a claical uiaxial tai polm with uppod otatio val that a vy commo appoach to tog icmtal octivity tak i iit lmt cod to atiy am idic (amly wokig i a appoxat u-otatd am ail thi plitic tt. A mo complicatd viicatio xampl i cotuctd ad o th mthod o mauactud olutio (MMS which ivolv th am chaact o loadig at all poit, povidig a ma to tt ay olia-latic aitaily aiotopic cotitutiv modl. INTODCTION Th picipl o matial am idic (PMFI qui that i a domd matial i otatd, th th patial tactio ad t hould otat alog with it wha th c t mut iitiv to th otatio. A viw o matial am idic i alady ptd i []. Thi cocpt i dit om ai idic. Th dic tw ai idic ad am idic i th iitial coiguatio. Figu pt a pl ha ud ai otatio ad uppod otatio. d ai otatio, oth th iitial coiguatio ad th domd coiguatio a otatd wha ud uppod otatio, oly th domd coiguatio i otatd. Th PMFI dmad that th t o domatio i Fig..( hould am a Fig..(a xcpt otatd appopiatly. PMFI do ot guaat th accuacy o th matial modl. It i o o th phyical picipl ttd to chck matial modl o coitcy ud uppod otatio ad talatio. Viicatio & Validatio o dvlopmt cod with complicatd umical cotitutiv modl i potat o ttig th accuacy ad out o th mthod []. Viicatio i a poc that i ud to valuat th coct o th olutio o th govig quatio i th cod. Validatio i a quality cotol poc o talihig vidc that th quatio thmlv povid a accptal dciptio o ality with pct to itdd quimt, ad i do y compaig with th xptal data. Th MMS i a accptd tadad o viicatio ttig i th citiic commuity that ha ud xtivly i luid mchaic [4], ut i aly dmotatd i olid mchaic cau o th icad mathmatical complxity. MMS i a poc o dtmiig th xtal ody oc quid to achiv a p-dcidd domatio aalytically. Th th cod i viid y uig it with th computd ody oc ad dmotatig that th p-dcidd domatio i achivd. Thi pap will it ocu o PMFI wh a uiaxial tai polm with uppod otatio i ttd with ad without a am idit cotitutiv modl. A aalytical olutio o th plt poil cotitutiv modl (lia laticity i uicit to dmotat that a commo appoxatio ud i plmtatio o tog icmtal octivity [6] ult i ailu to atiy th PMFI. Suqutly a MMS appoach will ud to cotuct a mo complicatd viicatio xampl o lag domatio ad lag otatio o a thick vtical am that will v a a ma to tt aitaily olia aiotopic latic cotitutiv modl o ultaou ai ad am idic. Figu : (a Bai otatio ( Suppod otatio. Copyight ASME

2 NOMENCLATE σ D a * Cauchy t at t tp Symmtic pat o th vlocity gadit at t tp otatio matix at t tp Body oc vcto Acclatio vcto Matial dity Sttch Lam modulu Sha modulu SINGLE ELEMENT TEST Th cocpt o am idic i illutatd uig a claical igl lmt tt [7]. Th tt polm i dcid a ollow: Th lmt udgo uiaxial tai (alog th x-axi om t t= to t=. Thi domd coiguatio udgo uppod otatio o 9 dg (aout th z-axi om t t= to t=. Th xact olutio duig th cod lg i giv y: wh σ co[ ] i[ ] A σ L T σ ( i[ ] co[ ], L, σ i th u-otatd t, A i th axial t ad L i th latal t. Thu σ A co [ ] L i [ ] ( A L i[ ] co[ ] ( A L i[ ] co[ ] co i [ ] L [ ] A L Figu. how th plot o omalizd t compot v. t o th aov mtiod aalytical olutio. Thi how that, although th picipal t do ot chag duig otatio, th t compot mut vay to accout o otatio o th picipal dictio o t. Th itah computatioal amwok [8] at th ivity o tah wa ud o all th umical ulatio how i thi pap. Th hypolatic cotitutiv modl i th itah amwok wa cho to u th tt polm. Th Cauchy t σ i govd y om uctio o th ymmtic pat o th vlocity gadit D. Thi ca ptd a: σ σ g( D ( Sic thi omulatio iclud o octiv at o Li taomatio to a u-otatd coiguatio, it i wll kow to violat am idic. Figu. how th modl pdictio o th t (omalizd y.6 t th pak t v. t. Duig th it t itval wh th Figu : Nomalizd t compot v. t o th aalytical olutio lmt udgo uiaxial tai, th plot how coct lia po o th, ad compot o t with th ha compot ig zo. Duig th cod t itval, howv, wh th domd lmt udgo uppod otatio, th ymmtic pat o th vlocity gadit i zo. c th valu o all th t compot mai cotat ad qual to that o th valu at th d o th it t itval. Thi val that thi cotitutiv modl i ot l coitt ud uppod otatio ad tho ail th am idic tt. To atiy am idic, val cotitutiv modl i th itah amwok ud a appoxatd u-otatd am. A lawd chma, ummaizd i Fig. 4., had a ollow: St ad ymmtic pat o th vlocity gadit a iitializd at th giig o th tp. Th t ad ymmtic pat o th vlocity gadit dd a iput to th cotitutiv modl w uotatd uig th pola otatio at th d o th tp. Th updatd t comig out o th cotitutiv modl wa -otatd uig th am otatio (i.., at th d o th tp. Th tt polm wa u o o o th cotitutiv modl i itah that qui th hot cod to apply th modl i a uotatd am to atiy am idic. Copyight ASME

3 zo, th cotitutiv modl coctly pdictd that th uotatd t maid uchagd. Tho, th valu o σ ad σ a qual. Sic th t i u-otatd y o dig it to th cotitutiv modl ad th otatd y th am amout at comig out o th cotitutiv modl, th valu o patial Cauchy t do ot chag a it hould duig thi itval. σ Figu : Nomalizd t compot v. t o hypolatic cotitutiv modl Figu 5: Nomalizd t compot v. t o th cotitutiv modl that u th u-otatio chma a i Fig. 4. Figu 4: Flawd t updat algoithm Figu 5. how th omalizd t compot v. t plot o thi modl wh th lawd chma o Fig. 4 wa ud i th hot cod. Th ult val that th cotitutiv modl aild thi plitic tt o am idic. Th modl hav a xpctd o th it t itval wh th lmt i ud uiaxial tai. Duig th cod t itval, ic th ymmtic pat o th vlocity gadit i c th valu o all th t compot mai cotat ad qual to that o th valu at th d o th it t itval. ducig th t tp to a vy mall valu will ot olv th polm cau o th u o th am otatio to o oth opatio, which pli a oouly cotat t gadl o th t tp. To ctiy thi polm, th ollowig chag w mad to th u-otatio chma Th t ad ymmtic pat o th vlocity gadit a u-otatd uig. Th t output om th cotitutiv modl i otatd uig. Figu 6. how that thi coctd icmtal tog octivity algoithm atii th picipl o matial am idic cau th t ow coctly otat with th domd matial. Th ollowig ovatio ca mad om Fig. 6.: At th d o th cod t itval, th valu o i qual to th valu o at th d o th it t itval. At th d o th cod t itval, th valu o i qual to th valu o at th d o th it t itval. Duig th cod t itval, th valu o i zo cau it th out o pla t. c, th Copyight ASME

4 valu o mai cotat ad i qual to that o th valu at th d o th it t itval. Duig th cod t itval, th ha a pak valu o itval. at th hal-t o th cod Th motio i govd y th quatio: σ a ( wh a i th acclatio ad i th ody oc. I th MMS, th domatio i p-dcidd o that acclatio i kow. Moov, o a latic matial modl, th t i a kow uctio o domatio, which allow th t Figu 7: Nomalizd ptatio dpth v. t compaio Figu 6: Nomalizd t compot v. t o th cotitutiv modl that u th chagd u-otatio chm Tho, thi tt val that, o th polm ivolvig lag matial otatio, dit pola otatio to d to ud. Ptatio ulatio, o xampl, ivolv lag matial domatio ad otatio. A ptatio ulatio wa u i itah with ad without th chag to th t u-otatio ad wa compad. Figu 7. how that omalizd ptatio dpth v. t wa ot igiicatly povd y th am idic ug coctio, ut wok i udway to a th iluc o th chag o matial po, uch a damag alog th ptatio chal, that i xpctd to mo itiv to am idic o. METOD OF MANFACTED SOLTIONS Th Mthod o mauactud olutio i a ytmatic poc o viyig th dvlopmt cod y uig thm with th aalytically computd xtal ody oc ad dmotatig th p-dcidd domatio. A pl -D xampl o MMS i alady availal [9] which i cotuctd ad o []. Thi mthod had uccully ud o viicatio ttig i th citiic commuity icludig luid hat ta [], luid-tuctu itactio [] ad xtivly i luid mchaic. Thi ctio will ocu o aalytically dtmiig th xtal ody oc quid o lag domatio ad lag otatio o a thick vtical am uig thi tchiqu. Thi polm will hav th am chaact o loadig at all itat o t a dcid i th igl lmt ctio. At all poit, th ody will udgo a uiaxial tai ud uppod otatio. Figu 8. how th aphot o th domatio o dig am i t. Figu 8: Saphot o th domatio i t. divgc to coidd a kow. Tho, th goal i to id th uctio ad utitut th ult ito Eq. ( to id th ody oc (which, i th MMS, th i ud a th ocig uctio i th cod to i th pdictio cov th p-dcidd domatio. Idicial otatio o th ody oc i th aov quatio i giv y 4 Copyight ASME

5 wh i i i ai (4 i th i th compot o divgc o th Cauchy t. Th dig o th am o hight ad a B i paamtizd y a t-vayig paamt [t] qual to th agl o th top uac o th am. Th copodig mappig om a iitial poitio to domd poitio x i co x (5 x i A ca cod y dict utitutio, th domatio gadit dcompod ito otatio ad ttch, i F computd y th ollowig quc o calculatio wh i th agl o otatio at th matial poit o itt, i th amout o ttch i th -dictio, i th otatio to ad i th ttch to. t] [ t] [, co[ ] i[ ] i[ ] co[ ],, Th ollowig ctio will div th cotiutio o th oc om th matial acclatio ad om th t divgc. Th th valu ca utitutd i Eq. (4 to valuat th ody oc. Foc cotiutio om th divgc o th t. I th MMS, w aum that w hav all th iomatio i Eq. ( xcpt th ody oc. It i poil (.g. y uig a igl-lmt modl div to otai all t compot a a uctio o uiaxial tai ttchig i th -dictio WITOT OTATION. Suppo, o xampl, th cotitutiv modl i th ollowig pl Noooka modl: (6 J dt[ ] Computig th u-otatd t o th pl Noooka modl, cau that modl i a iotopic modl, it ha axial ad latal t compot a th oly ozo uctio o uiaxial ttch. To apply thi mauactud olutio to a aitay, pottially aiotopic ad v mo olia, latic cotitutiv modl, th upcomig aalyi pum oly that th po uctio o all compot o t ud uiaxial tai i th -dictio a kow; i.., (8 [ ] [ ] [ ] σ [ ] [ ] [ ] (9 [ ] [ ] [ ] Th Cauchy t σ i computd uig T σ σ ( wh i th otatio to. Now w d th divgc o th Cauchy t. Sutitutig Eq. ( i th idicial om o th t divgc giv i ( x x By th poduct ul, x i ( x ( x ( x ( ( callig om Eq. (9 that σ dpd oly o th ttch, whil Eq. (6 how that th pola otatio dpd oly o th otatio agl, uig chai ul ( d( ( x d x ad ( d( x d x (4 * Log[ J ] T σ I [ F F I] (7 J J * wh i th Lam modulu, i th ha modulu, J i th Jacoia, F i th domatio gadit ad I i th * idtity to. W hav cho to Lam modulu cau i alady did to th amout o ttch i th -dictio. Fo uiaxial tai i th -dictio, th otatio i I, ad tho th domatio gadit i F, ad it Jacoia i : ig th xpio o om Eq. (6, wh d d A i[ ] co[ ] co[ ] i[ ] A 5 Copyight ASME

6 Tho Eq. ( com ( x callig that Not that A [ ] d x x t ip pm d F Tho Eq. (5 com Thi pli that x d a ( d Aippm a x ( d App a x a a a (5 (6 (7 (8 Th xpio o gadit o otatio xpd i Eq. (7 ad Eq. (8 a utitutd i Eq. ( to giv i i x A ip pm d ( x A p a p a d a a (9 Multiplyig oth id o th Eq. (9 uig iq ad pliyig uig th act that i othogoal, w hav q iq i A qm q ( q x A a a ( call that th u-otatd t dpd oly o th ttch, ad dpd o. Tho, y chai ul, ( q d( q d( q d p ( x d x d x Eq. ( ca till ducd to ( q d( q d pt x d p p t ( ig th diitio o, ad A ad utitutig Eq. ( i Eq. (, w hav q d q Aq Aq q d Thu, u-otatd oc i d d d d d d ( (4 Th oc cotiutio om th divgc o t aov apply v to aiotopic cotitutiv modl. Fo th pcial ca o a iotopic cotitutiv modl, th u-otatd t will diagoal, givig d d (5 Th a th u-otatd oc compot. To apply th i a calculatio o th ody oc o th mauactud olutio, th actual oc vcto ca computd y callig that wh. Tho, d d (6 ad co[ ] i[ ] (7 Fo th Noooka cotitutiv modl ud i thi pap, th valu o com out to * [ t] Log [ t] [ t] [ t] [ ] (8 t I gal, th oc vcto o a aiotopic latic matial will iclud a agula compot o that, (9 Foc cotiutio om th matial acclatio. Dtmiig th cotiutio o oc om th matial acclatio i lativly ay cau w alady hav th latio tw x ad. Acclatio i ply 6 Copyight ASME

7 a x t Th ial ult o th acclatio vcto com out to a a a ( wh a a i co [ t ] [ t ] co [ t ] [ t ] [ t ] [ t ] [ t ] [ t ] [ t ] ' [ t ] '' [ t ] [ t ] [ t ] i [ t ] [ t ] [ t ] co[ ] i[ ] ad i[ ] co[ ] ' [ t ] '' [ t ] [ t ] ( ( Dity i giv y th ollowig quatio: ( Sutitutig Eq. (8, (, (, ad ( ito Eq. (4 giv th compot o th total ody oc quid o thi domatio: wh (4 a, a. CONCLSIONS Thi pap dcid th cocpt o am idic tatig with a claical igl lmt tt ud uiaxial tai with uppod otatio. ult o thi mial tt vald that a vy commo appoach tak y iit lmt cod to atiy am idic (amly wokig i a appoxat u-otatd am o which otatio duig th icmt i plicitly glctd y uig oly a igl othogoal to o all u-otatio opatio duig th tp ail thi plitic tt. With thi tt it wa cod that i a matial udgo otatio, th dit pola otatio d to ud. Alo a w mauactud olutio wa ptd o a lag domatio ad lag otatio o a dig am. Thi polm tt ay gally olia ad aiotopic latic cotitutiv modl ultaouly o ai idic ad am idic. It pt a a ca o mauactud olutio that ca applid to aitay cotitutiv modl, which wa poil cau th polm wa digd to put all matial poit i th am chaact o domatio (uiaxial tai o vaiou ititi with uppod otatio. ACKNOWLEDGMENTS W would lik to ackowldg th Schlumg tchology copoatio, pcially D. J Guilky, o uppotig th wok. W would alo lik to thak th mm o CSM (computatioal olid mchaic goup at th ivity o tah o thi uppot. EFEENCES [].Chal G. Spzial, A viw o Matial Fam- Idic i Mchaic, Appl. Mch. v. 5, 489 (998, DOI:.5/.99. [].Liu, I-Shih. Futh mak o Euclida octivity ad th picipl o matial am-idic. Cotiuum Mchaic ad Thmodyamic, 5 May. [].Thack, B.., Th ol o Nodtmi i Viicatio ad Validatio o Computatioal Solid Mchaic Modl. liaility & out Dig i Automotiv Egiig, SAE It., SP-76, No. --5, SAE Wold Cog, Dtoit, MI, -6 Mach. [4].Kamiz Salai ad Patick Kupp. Cod Viicatio y Mthod o Mauactud Solutio, SANDIA EPOT, Ju. [5].L Schw. Mthod o Mauactud Solutio: Dmotatio. Documt/MMS-Dmo-Sp.pd (Augut [6].ahid MM, Icmtal kimatic o iit lmt applicatio, Itatioal oual o umical mthod i giig 99; 6 (: [7].Joh oio, A igl lmt tt. ComputMthod i Applid Mchaic ad Egiig, Volum 7, Iu, Fuay 976, Pag 9-, ISSN , DOI:.6/45-785(769-8 [8].J Guilky, Todd ama, Juti Luit, Joh Schmidt, Jmy Thoock, J. Davio d St. Gmai, Siddhath Shaka, Joph Pto, Cao Bowl. itah guid, SCI Ititut Tchical pot, 9. [9].B. Ba, Mthod o mauactud olutio, (Octo 6. [].. C. Bata ad. Q. Liag, Fiit dyamic domatio o mat tuctu, Computatioal Mchaic,, 47 48, 997. [].. C. Bata d B. M. Lov, Multical aalyi o adiaatic ha ad i tugt havy alloy paticulat compoit, Itatioal Joual o Multical Computatioal Egiig, 4(, [].Bu, T. A, Dvlopmt o a gy olia thmal adiatio diuio viicatio polm. 7 Copyight ASME

8 Taactio o th Amica Nucla Socity 95, [].Tmlay, D., Eti, S. & Pllti, D. Cod viicatio ad th mthod o mauactud olutio o luid-tuctu itactio polm. I 6th AIAA Fluid Dyamic Coc, vol., pp Sa Facico, CA. 6 8 Copyight ASME