Background problem The following preliminary coding exercise will be helpful in completing this weeks homework. Let 3 Y * * 2 e. 2 (1 + ν) 3(1 2 ν)

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1 E4: Computtonl mthods n Structurl nd Sold Mchncs Homwork 9: Explct Dynmcs: modlng dynmc ductl rctur Du Wd ov 8, 5 School o Engnrng Brown Unvrsty Bckground problm Th ollowng prlmnry codng xrcs wll b hlpul n compltng ths wks homwork. Lt / s * p * s (, p) = + q cosh q ( + q ) Y Y * Whr q, q, q,, Y r constnts. Lt * E * E σ = σ p= p v ( + ν) ( ν) whr σ * *, p, E, ν r constnts. Wrt MATLAB cod to solv th ollowng qutons or εv by wton-rphson trton (wrt your own wton-rphson loop, don t us th mtlb quton solvrs. Th gol s to gt th wton loop to work so you cn trnsr t nto your Fortrn cod) / σ 9 p t σ φ φ φ m F (, v) = + φ = / v 9 p t p φ φ φ m F (, v) = + φ = σ whr ε, m, t r constnts. Us th ollowng vlus or th constnts (you cn chck wth othr vlus * * too but not tht solutons only xst or φσ (, p ) > ): E =, ν =., Y = 8, =., m= 5, q =.5, q =., q =.5, * * * σ =.5, = 85, p =, t =. Th wton trtons rqur rptdly solvng th ollowng lnr qutons or corrctons d, d v F F v d F = F F d v F v Th hrd prt o ths problm s clcultng nd typng n th drvtvs o F corrctly (not tht φ s uncton o, v so th drvtvs o F contn scond drvtvs o φ ). You nd to b xtrmly orgnzd n both your clcultons nd dsgnng th cod to mk th stps trnsprnt. Fortuntly s long s th trtons convrg sucntly rpdly (4-5 trtons r typcl) you cn somtms gt wy wth w smll rrors, s long s F s corrctly clcultd!

2 In th rmndr o ths homwork you wll mplmnt n xplct dynmc smulton o ductl rctur n polycrystlln mtl, usng th mous Gurson plstcty modl, whch ccounts or nuclton nd growth o vods n mtl. You cn nd short dscusson o th Gurson modl hr, nd som mor dtl o th modl tht wll b mplmntd hr n th ABAQUS thory mnul. Implct dynmcs wth nt strns: Bcus ductl rctur gnrlly nvolvs lrg plstc strns, w must mplmnt nt strn vrson o th xplct dynmc nt lmnt mthod. Th procdur ollowd hr s somwht smlr to ABAQUS/Explct, xcpt tht w wll us th Jumnn rthr thn Grn-ghd msur o strss rt ( mor xtnsv dscusson o strss rt msurs nd th drnc btwn thm cn b ound hr) Th nt-strn vrson o xplct dynmcs s bsd on th prncpl o vrtul powr v td LdV + ρ dv tdv = t V V A whr τ s th Krchho strss; δ v s vrtul vlocty ( tst uncton) nd δl =δv / yj s th vrtul vlocty grdnt. Snc w r solvng mtl plstcty problm w ntrpolt dsplcmnt nd vrtul vlocty lds usng th nt-strn vrson o th B-br mthod, whch ylds systm o qutons or th cclrtons b d u M b = R + F dt b d mj M b = dv R tmj[ Fkl ] d = m + dv y V j n y y V Whr n = or D problm nd n= or D problm. E4FEA ntgrts ths qutons wth rspct to tm usng wmrk schm, wth lumpd mss mtrx, β = nd β =. Ths gvs th ollowng lgorthm: At tm t= th vlocts v () nd cclrtons () r ntlzd (th usr cn spcy th vlocty; th cclrtons r zro). Th tm ncrmntton loop procds s ollows: t. Clcult u = t v () t + () t ; mpos ny prscrbd dsplcmnts. b. Clcult R ( u ) k

3 . Clcult ( t+ t) = ( R + F )/ M + = = + 4. Updt v ( t t) v ( t) t ( t t) 5. Updt u ( t t) u () t u Ths procdur s lrdy codd n E4FEA (th cod s n xplct_dynmc_stp.9). You only nd b to wrt cod to clcult R ( u ), through th usr subroutn. k To mplmnt ny nw mtrl modl, you must clcult th Krchho strss τ mj[ Fkl ] gvn b dsplcmnt ncrmnt u, nd hnc clcult th nodl orcs R ( uk). Onc τ mj[ Fkl ] hs bn b ound, you cn nd R ( u ) usng th usul mthod. k In ths homwork, th strss wll b clcultd rom th dsplcmnts usng th consttutv rltons or porous plstcty modl known s th Extndd Gurson modl. Ths s vry smlr to th rt-dpndnt vscoplstc consttutv quton dscussd n clss, xcpt tht t lso ccounts or th nuclton, growth nd colscnc o vods. Summry o th Gurson modl: Followng th stndrd procdur n nt strn plstcty, w dcompos th dormton grdnt nto lstc nd plstc prts, whch thn llows us to dcompos th totl strn rt nto lstc nd plstc prts p F = F F k kj = sym( F kfkj ) W = skw( F kfkj ) p = + Th plstc strn rt s rltd to th Krcho strss by φ < m D p φ φ τ φ = + δ φ > σ σ p () φ φ + σ 9 p whr Y s th yld strss o th mtrx (w nglct strn hrdnng, so Y s constnt), ε,m r two mtrl constnts ( chrctrstc strn rt nd th strss xponnt), nd / s * p * = + q cosh q ( + q ) Y Y D D D kk kk τ = τ τ δ / s = t t / p = t / Hr, q, q, q r thr mtrl proprts, nd * s uncton tht qunts th loss o strngth rsultng rom vod growth. It s rltd to th volum rcton o vods n th mtrl V by V V < c * F c = c + ( V c) c < V < F () F c F V > F

4 Hr, c, F r two mtrl prmtrs tht rprsnt th crtcl vod volum rcton whn vods strt to colsc, nd whn th mtrl ultmtly ls ntrly, nd q+ q q F = q Th volum rcton o vods s zro n th ntl mtrl. As th mtrl s plstclly dormd, vods nuclt nd grow, whch cuss V to ncrs. It s govrnd by dv p dmtrx mtrx = ( V ) kk + xp () dt dt s s p whr, s, ε r mtrl proprts controllng th rt o vod nuclton wth plstc strn (vods nuclt rpdly whn th strn rchs ε ), nd ε mtrx s th totl ccumultd plstc strn n th mtrx, whch volvs ccordng to / d mtrx m σ = + + p dt V σ 9 p σ p Th lstc strn rt s rltd to th so clld co-rottonl strss rt by E ν τ = + kkδ + ν ν whr dt t = + tkwkj Wktkj dt (or mor dtld dscusson o rt orms o consttutv qutons or lstc-plstc sold s hr) Implmntng th dynmc rctur/plstcty modl rqurs thr stps: () Gvn th dsplcmnt ncrmnt u nd th dsplcmnt u t th strt o th stp, you must clcult th strn ncrmnt ε, th spn ncrmnt W nd th rotton ncrmnt R () Gvn ε, R, th strss, nd stt vrbls t th strt o th ncrmnt, clcult th strss nd stt vrbls t th nd o th ncrmnt d mj () Clcult th lmnt rsdul orcs R = τmj[ Fkl ] d m + dv y V j n y y Ths stps r dscrbd n mor dtl blow, wth rough cod tmplt:

5 Loop ovr th ntgrton ponts:. Clcult th ncrmnt n dormton grdnt snc ths s th chng n F) F = u x j ( t+ t/) = δ + + x j (not thr s no dntty ddd,. Clcult th md-pont dormton grdnt F ( u u ) nd th ssoctd ( t+ t/) Jcobn J= dt( F ). Comput th contrbuton rom th currnt ntgrton pont to th volum vrgd Jcobn nd th vrg volumtrc strn ncrmnt η = JdV V l V l + η = J L dv L = F F η V l V l ( t t/) kk k kj 4. Comput th contrbuton rom th currnt ntgrton pont to th volum vrgd sptl shp uncton drvtvs ( t+ t/) = J dv = F y k ηv l y V y xk l 5. Comput th contrbuton rom th currnt ntgrton pont to th lmnt volum. End ntgrton pont loop Dvd th volum vrgd quntts by th lmnt volum Loop ovr th ntgrton ponts scond tm. Clcult th ncrmnt n dormton grdnt. Clcult th md-pont dormton grdnt F = u x j ( t+ t/) = δ + + x j F ( u u ). Comput th corrctd vlocty grdnt ncrmnt ( t+ t/) ( t+ t/) L = Fk Fkj + ( η Flk Fkl ) δ / 4. Comput th strn nd spn ncrmnts ε = ( L + L )/ W = ( L L )/ j j t+ t 5. Clcult th rotton ncrmnt ssoctd wth th spn, whch s gvn by R = d + W R dt. Th ntgrl cn b pproxmtd usng md-pont rul WI ( + R) RR = W R I + W W R ( I ) I+ k kj t

6 whr I s th dntty mtrx. 6. Clcult th Krcho strss t th nd o th ncrmnt τ ( n+ ) nd clcult th vlus o th updtd mtrl stt vrbls or th currnt ntgrton pont (do ths nsd subroutn s blow or th stps) 7. Clcult th contrbuton to th lmnt orc vctor rom th currnt ntgrton pont T R = B τ dv whr B cn b ssmbld s + + y y y y y y y y y y y + tc y y y y y y y = y y y y y y V B Ths s th sm B-br mtrx you usd rlr, xcpt tht th drvtvs r wth rspct to y nstd o x. End ntgrton pont loop Strss updt lgorthm Th strss s updtd usng th pproch dscussd n clss or smll-strn plstcty problms (som o you ( ) ( ) ( ) my hv mplmntd smll strn plstcty modl n HW6 lrdy). Gvn ε, t n, ε n, V n, w ( ) ( ) ( ) wsh to dtrmn n +, n + t, n εmtrx V +, s ollows:. Comput th lstc prdctors or th dvtorc nd hydrosttc strss * E Dn * ( n) E S = D +DR τ D R p = τ kk + D + n ( n) Dn ( n) ( n) D =D D kkδ / τ = τ τ kk. Clcult / * * s * p * = + q cosh q ( + q ) Y Y ( ) k kl jl kk s * * * = SS / mtrx

7 V V < V V V ( n) ( n) c * F c ( n) ( n) = c + ( c) c < < F F c ( n) F > F I φ <, th ncrmnt s lstc, nd th strss ollows mmdtly s n+ * * τ = S + p δ. I φ >, solv th ollowng two qutons or th mgntuds o th dvtorc nd volumtrc plstc strns, (usng wton-rphson trton you dd ths n MATLAB lrdy) v / σ 9 p t σ φ φ φ m + φ = / v 9 p t p φ φ φ φ m + = σ whr φ s computd usng th vod volum rcton t th strt o th ncrmnt, togthr wth th ollowng xprssons or p nd σ * E * E σ = σ p= p v ( + ν) ( ν) 4. Th strss t th nd o th ncrmnt cn thn b clcultd rom * n+ * E S * E τ = S + p * v δ + n σ ( n) 5. Fnlly, th vod volum rcton nd mtrx strn cn b updtd s ( n+ ) ( n) = + mtrx mtrx mtrx ( n) n+ ( n) mtrx mtrx V = + ( V ) xp( v) + xp s s p < / mtrx = t m s ( n) + + p > V s 9 p s p Implmntton ots To mplmnt n xplct dynmc lmnt n E4FEA, you must wrt two usr-subroutns () routn to comput th lmnt rsdul orc (thr s n xmpl n th l_lnlst_dbsc.9 l). Ths routn must lso comput vlus or th updtd mtrl stt vrbls. Th subroutn rgumnts r shown blow. subroutn l_lnlst_dbsc_dynmc(lmn, lmnt_dntr, n_nods, nod_proprty_lst, & n_proprts, lmnt_proprts,lmnt_coords, lngth_coord_rry, & do_ncrmnt, do_totl, lngth_do_rry, & n_stt_vrbls, ntl_stt_vrbls, updtd_stt_vrbls,lmnt_rsdul,lmnt_dltd)! Output vrbls

8 () th usul routn to projct strsss nd othr ld vrbls (g th vod volum rcton) to th nods. Th Gurson mtrl hs mtrl proprts: E, ν, Y,, mq,, q, q,,, s, c, F Th mtrl hs th ollowng stt vrbls: strss componnts τ, th mtrx strn ε mtrx nd th vod volum rcton V. Th vlus o ths vrbls nd to b stord, nd updtd, or t ch ntgrton pont n th lmnt. In E4FEA, ll stt vrbls or n lmnt r stord n two -D vctors: th usr subroutn wll provd th vlus o stt vrbls t th strt o th stp n vctor clld ntl_stt_vrbls; your usr-subroutn must rturn thr vlus t th nd o th stp n vrbl clld updtd_stt_vrbls. You cn us ny storg schm you lk or your vrbls s long s thy r consstnt. For xmpl, n my cod I xtrct th stt vrbls or th rst ntgrton pont s strss = ntl_stt_vrbls(:6)! Strss t strt o ncrmnt mtrx = ntl_stt_vrbls(7) V = ntl_stt_vrbls(8) Th dt or th scond ntgrton pont strts t ntl_stt_vrbls(9), nd so on. It s bst to sprt th clcultons nto lmnt oprtons (clcultng shp uncton drvtvs, dormton msurs, tc) nd mtrl oprtons (clcultng th strss). Thr r vrous wys to do ths, but th bsc d s to wrt subroutn tht clcults th updtd strss nd updtd stt vrbls or just on ntgrton pont, gvn vlus o th stt vrbls, th strn ncrmnt, nd th rotton ncrmnt, wth subroutn o th orm subroutn gurson(lmnt_proprts,n_proprts,n_stt_vrbls,ntl_stt_vrbls, & updtd_stt_vrbls,dstrn,drot,strss) us Typs us PrmIO us Globls, only : TIME, DTIME mplct non ntgr, ntnt( n ) ntgr, ntnt( n ) rl (prc), ntnt( n ) rl (prc), ntnt( n ) rl (prc), ntnt( n ) rl (prc), ntnt( n ) :: n_proprts :: n_stt_vrbls :: lmnt_proprts(n_proprts) :: ntl_stt_vrbls(n_stt_vrbls) :: dstrn(,) :: drot(,) rl (prc), ntnt( out ) :: strss(6) rl (prc), ntnt( out ) :: updtd_stt_vrbls(n_stt_vrbls) ot tht n_stt_vrbls hr s th numbr o stt vrbls or OE ITEGRATIO POIT, not ll o thm, nd you must slc th vctor tht stors th ull st o vrbls or ll th ntgrton ponts to pss th corrct st to ths subroutn. Th condton φ > s bst chckd numrclly s φ > ps, whr ps s smll numbr ( -8 ) to vod dvd by zro rrors. Dn Th Elmnt Utlts modul n E4FEA dns uncton to comput DRkτkl D Rjl strss = rottsymvc(strss,dr) Hr dr s x orthogonl mtrx nd strss s vctor contnng componnts o symmtrc tnsor s= [s,s,s,s,s,s]. Both E4FEA nd ABAQUS llow lmnts to b dltd durng n xplct dynmc computton, to smult mtrl lur. In E4FEA ths s ccomplshd by sttng th

9 lmnt_dltd vrbl to.tru. nsd th usr-lmnt subroutn. You could dlt lmnts whn V > F or ll th ntgrton ponts n th lmnt. You r codng nd runnng rly sophstctd FEA modl n ths homwork, so dbuggng mght tk bt o work. Th ollowng procdur wll tst your cod n stgs:. Run your cod wth only two lmnts (g wth th msh n Gurson_d_dynmc.n), nd prmtr vlus tht wll produc n lstc mtrl, g E =, ν =., Y = 8, ε =., m= 5, q =.5, q =., q =.5, =, ε.5, s =.5, c =.5, F =.5 Us mss dnsty ρ = nd tm-stp o. unts. You cn pply ny snsbl boundry condtons. Your cod should rproduc th rsults n lnr_lstc_dynmc_d.n (wth th sm boundry condtons).. Run your lmnt tst wth plstcty, but no vod nuclton, g. E =, ν =., Y =.8, ε =., m= 5, q =.5, q =., q =.5, =, ε.5, s =.5, c =.5, F =.5 Chck th convrgnc o th wton trtons n th plstcty modl thy should convrg tr -4 trtons. I ths dos not hppn thr s probbly n rror n th drvtvs you r usng n th wton-rphson loop (but hopully you cught ny problms n th prlmnry MATLAB cod you wrot, so ths prt should work.. I stp () works rpt th lmnt tst wth vod nuclton try prmtrs E =, ν =., Y =.8, ε =., m= 5, q =.5, q =., q =.5, =.5, ε =.5, s =.5, c =.5, F =.5 (supprss lmnt dlton or ths tst). Chck th convrgnc o your wton trtons. In ths smulton you should s th strss n th lmnts drop s th mtrl ls. 4. I tst () works, you cn try runnng th smulton shown n th gur. An nput l clld notch_rctur_dynmc.n hs bn provdd or ths purpos. You my nd to chng th ordr o th mtrl proprty dnton to b consstnt wth your cod. 5. Optonl: you r curous you cn mk your own rctur spcmn nd (vrtully) brk t. Th smulton wll run str you compl nd run your cod n rls mod nstd o dbug mod.