Jacob Fish and Kamlun Shek 1 Departments of Civil and Mechanical Engineering Rensselaer Polytechnic Institute Troy, NY Abstract.

Transcription

1 Finit dformation plasticity basd on th additiv split of th rat of dformation and hyprlasticity Jacob Fish and Kamlun Shk 1 Dpartmnts of Civil and Mchanical Enginring nsslar Polytchnic Institut Troy, NY 1180 Abstract Finit dformation plasticity formulation basd on additiv split of rat of dformation and hyprlasticity is prsntd. This approach is valid for finit lastic and plastic strains, whil rndring th choic and numrical intgration of objctiv strss rats suprfluous as th rsults ar automatically objctiv. For small lastic strains our mthod rducs to th classical hypolastic-corotational formulation providd that th Dins objctiv strss rat is mployd, whil in th absnc of inlastic dformation it coincids with th hyprlastic formulation. Th validity of th modl has bn xamind on four tst problms and th numrical rsults wr found to b in good agrmnt with ithr th xact solution or xprimntal data. 1.0 Introduction Th propr formulation of finit dformation lastoplastic kinmatics, lastic rspons and th flow rul has bn a subjct of considrabl conjctur. On of th major difficultis stms from th coupling btwn lastic and plastic proprtis. It is wll known that initially lastically isotropic matrial may bcom lastically anisotropic du to plastic flow, which mans that Hlmholtz fr nrgy dnsity is not only a function of lastic dformation but has som dpndncy on th plastic flow. Fortunatly, for modrat plastic dformation (up to 30% [1][13]), dislocations and othr lattic dfcts causd by plastic flow, hav a ngligibl ffct on th dformation of crystal lattic which govrns lastic constants. This has bn ralizd by many practitionrs, who obsrvd that lastic constants ar not apprciably affctd by manufacturing procsss involving plastic forming, nor has th lastic dformation a profound influnc on th plastic flow. Finit lastoplastic kinmatics is anothr issu of considrabl dbat. For xampl, basd on th nrgy consrvation principl, Nmat-Nassr [6] has shown that th rat of dformation, d, dcomposs additivly as d d d + p (1) for finit lastic () and plastic (p) rat of dformation providd that th strain incrmnts ar dfind with rspct to th sam rfrnc configuration. Grn and Naghdi [10], hav argud that an additiv dcomposition of Lagrangian strain, E 1. Currnt addrss: Th Goodyar Tir and ubbr Company, D/431A, Akron, OH

2 E E E + p () into lastic-plastic componnts is supportd by solid thrmodynamic principls. L [0][1] advocatd a thory basd on multiplicativ dcomposition of th dformation gradint, F F F p F (3) whr th lastic dformation gradint, F, is obtaind by indpndntly unloading infinitsimal volum lmnts of th body into an intrmdiat configuration dfind by this collction of unloadd lmnts. It is important to raliz that th thr thoris ar fundamntally diffrnt, i.., if on, for xampl, adopts multiplicativ dcomposition, thn it is a trivial xrcis to show that nithr additiv dcomposition holds, and vic vrsa. In an attmpt to rconcil ths diffrnt kinmatical assumptions, E [11][31] and d [6] hav bn intrprtd as mixd lasto-plastic dformation tnsors. Othr kinmatical splits can b found in th litratur. Nmat-Nassr advocatd an additiv split of th dformation gradints [6], as wll as th multiplicativ dcomposition, whr th ordr of plastic and lastic dformation gradints in (3) is intrchangd [5]. Kim and Odn [19], on th othr hand, suggstd dcomposing th strtch tnsor additivly. Evn though thr is no consnsus with rspct to th most favorabl finit lastoplastic kinmatics, numrical algorithms dvlopd in th past 15 yars hav bn primarily focussing on th following two approachs: Catgory 1: Multiplicativ hyprlastic plasticity This class of mthods is basd on multiplicativ lasto-plastic kinmatics (3), hyprlasticity and th xistnc of Hlmholtz fr nrgy dnsity govrnd by ithr lasto-plastic dformation [7][19][3][5][8][9][34][35][36][39], or lastic dformation only [][1]. Th lattr, as wll as an additiv split of th Hlmholtz fr nrgy dnsity [7][8], ar computationally attractiv as thy prmit computation of strsss basd on th lastic dformation only. Catgory : at-additiv hypolastic plasticity This approach hings on th additiv dcomposition of th rat of dformation (1), hypolasticity, and objctiv strss rats. Spcial car is xrcisd in th intgration of rat constitutiv quations to prsrv objctivity [4][15][17][7]. Evn though ths typ of mthods ar vry attractiv from th computational point of viw [1][3], thy ar limitd to small lastic strains (but larg rotations) for which th hypothsis of hypolasticity is valid. Morovr, thy suffr from a somwhat adhoc choic of objctiv strss rats. Nvrthlss, th rat-additiv hypolastic approach is appropriat for most of th nginring matrials including mtals, whr lastic strains rmain small and thus diffrncs in th formulation of lastic rspons hav littl or no ffct on th computd solution. On th

3 othr hand, for som polymrs xhibiting significant lastic and plastic dformation of comparabl magnitud, a diffrnt tratmnt is rquird. Th primary objctivs of th prsnt manuscript ar thrfold: (i) In an attmpt to capitaliz on th gnrality of th multiplicativ-hyprlastic approach and th computational fficincy of th rat-additiv-hypolastic approach, w prsnt a hybrid formulation basd on th additiv split of th rat of dformation and hyprlasticity. Such an approach, rfrrd hraftr as th rat-additiv-hyprlastic approach, rmains valid for finit lastic and plastic strains, whil rndring th choic and numrical intgration of objctiv strss rats suprfluous as th rsults ar automatically objctiv. (ii) W show that in th cas of small lastic strains th rat-additiv-hyprlastic formulation rducs to th hypolastic-corotational formulation [4][15][18], providd that th Dins [5] objctiv strss rat is mployd. Furthrmor, our formulation rducs to th hyprlastic formulation in th absnc of inlastic dformation. This suggsts that th magnitud of lastic and plastic dformation (charactrizd by an appropriat norm) can b usd as an indicator for appropriat modl slction. (iii) Th validity of th modl will b xamind for th following four problms: (a) Axial tnsion-rigid body rotation of a hyprlastic-plastic bar for which th xact solution can b asily obtaind, (b) th classical simpl shar problm [4], which rvals spurious oscillatory shar strss bhavior for crtain typs of objctiv strss rats, (c) th torsion problm of th hollow cylindr [3] for which th axial strains hav bn obsrvd to b two ordrs of magnitud smallr than thos of shar strains, and (d) th axisymmtric xpansion of an lasto-plastic thick-walld cylindr for which th xact solution has bn rportd in [6]. Th manuscript is organizd as follows. Sction xamins hyprlasticity and hypolasticity within th framwork of corotational formulation and shows that for infinitsimal lastic strtchs th two thoris coincid for a crtain choic of corotational fram. Strss intgration schms and th drivation of th consistnt tangnt stiffnss matrix for th rat-additiv-hyprlastic approach ar prsntd in Sction 3. Numrical xampls conclud th manuscript..0 Hypolasticity vrsus hyprlasticity As a prlud to th finit dformation plasticity incorporating additiv split and hyprlasticity, w start by summarizing th basic hypolasticity and hyprlasticity quations with th intnt of driving th rlation btwn th two formulations..1 Hypolastic corotational formulation Considr two nighboring particls of th body situatd at points H and Q in th undformd configuration such that th unit vctor, dnotd as B, is pointing from Q to H as 3

4 shown in Figur 1. Th particls originally positiond at points H and Q mov to th positions h and q, rspctivly, with th corrsponding unit vctor b. Th rotation btwn th two unit vctors can b writtn as whr is orthogonal such that T I, and I rprsnts an idntity matrix. Th rat of rotation is givn by b B b B W b. whr W 1 Not that W coincids with th spin tnsor if b is alignd along on of th principal dirctions of th rat of dformation tnsor. W now focus on th corotational fram which transforms with rspct to th undformd configuration as x X + constant. A family of th corotational Cauchy strss tnsor, dnotd as σ, can b dfind as x σ T σ (4) Th rat of follows from th linarization of (4) σ σ T σ + T σ + T σ T W T σ + T σ + T σ W T σ or σ σ T (5) whr σ is th so calld objctiv strss rat givn by σ σ W σ σ W T σ can b intrprtd as a rat of σ xprssd in th currnt fram. Sinc σ and σ oby transformation ruls for scond ordr tnsors, thy ar considrd objctiv strss rat masurs. Not σ dos not oby transformation ruls of scond ordr tnsors, and thrfor, it is not objctiv. 4

5 For hypolastic matrials th constitutiv quation is givn by σ L : d (6) whr L is th lasticity fourth ordr tnsor and d is th rat of dformation dfind as d {} l s l v x and v rprsnts th vlocity fild; l dnots th spatial vlocity gradint, and {} s is a symmtrization oprator. Substituting (5) into (6), prmultiplying and postmultiplying th rsult with yilds T and σ T ( L : d) Lt d dnot th st of corotational rat of dformation tnsors dfind as d T d thn th hypolastic constitutiv quation (6) in th corotational fram can b xprssd as σ L : d (7) whr L in th componnt form is givn as L ijkl ai bj ck dl L abcd (8) It can b sn that th form of th constitutiv quations in th corotational fram is idntical to that in small dformation thory. Not that for isotropic matrials th constitutiv proprtis ar rotation indpndnt and thus L. Clarly th constitutiv quation (7) in th corotational fram is not uniqu. First, various choics of rotations,, will ultimatly yild diffrnt matrial rsponss. Scondly, what is a propr choic of L? Can it b chosn th sam as in small dformation thory? Ths qustions ar addrssd in th nxt sction, whr th choic of and L is mad to maintain consistncy with th notion of hyprlasticity.. Hyprlastic corotational formulation Th constitutiv quation for a hyprlastic solid can b writtn in th following form L 5

6 S Ψ Ψ C (9) whr Ψ is th Hlmholtz fr nrgy dnsity function; rprsnts th right strtch tnsor and T ; C is th Grn dformation tnsor whr C ; S dnots th scond Piola-Kirchhoff strss tnsor which is rlatd to th Cauchy strss, σ, as S JF 1 σ F T (10) whr F rprsnts th dformation gradint and J is th Jacobian which is dfind as th dtrminant of. In th following w focus on th mmbr of th corotational family of strsss by slcting th rotation tnsor from th polar dcomposition F. Thus σ T σ J Ψ (11) Taking th matrial tim drivativ of (11) givs σ J 1 Ψ d Ψ dt Ψ dj Ψ dt (1) To linariz th right strtch tnsor w rcall and dfining th following corotational masurs: d T d, w T w, Ω T Ω, Ω T, whr w {} l a is an antisymmtric part of l, th rat of th right strch tnsor can b xprssd as: Th scond trm in (1) can b writtn as l F F 1 T + 1 T ( d + w Ω ) or d 1 { }s (13) d ---- Ψ Ψ dt : d Ψ dt : ( F T d F) Ψ : ( d ) (14) Substituting (13), (14) with J Jtr( d ) into (11) yilds σ L : d + { h } s d ω tr( d )I (15) whr ω w Ω ; L and h in th componnt form ar givn as 6

7 L ijkl ( ) dt( ) ip jq kr Ψ ls pq rs h ij ( ) dt( ) Ψ ik jl kl It should b notd that for small lastic dformation, I, th following holds Ψ O( I) J O( 1) h O( I) and hnc (15) can b approximatd as σ 4 Ψ :d (16) Comparing (7) and (16), w conclud that in th cas of small lastic dformation (but arbitrary rotations), th hyprlastic and hypolastic formulations ar idntical providd that L 4 Ψ and For mor dtails on th corotational formulation s [4][37][38]. 3.0 Finit dformation plasticity basd on th additiv split of th rat of dformation and hyprlasticity 3.1 Basic assumptions In dvloping finit dformation lasto-plastic quations th primary critria w shall adopt ar as follows: i. Th finit dformation lasto-plastic rat rlations should coincid with hyprlasticity in th absnc of inlastic dformation. ii. W postulat that th strss in finit dformation plasticity may b drivd from Hlmholtz fr nrgy dnsity [7][8]. Thrfor, th corotational Cauchy strss can b writtn as σ dt( ) Ψ (17) 7

8 analogous to (11). In othr words, w assum that in a finitly dforming lasto-plastic solid, th objctiv rlations ar govrnd by (15), with th only xcption that various strain and strain rat masurs, such as d, ar rplacd by thir lastic countrparts,, whras rotations ar associatd with lastic dformation only, i.., d w w (18) iii. Finally, w will adopt an additiv split of th rat of th dformation tnsor d d d + p (19) whr th lft subscripts and p dnot lastic and plastic dformation, rspctivly. 3. Govrning quations As prviously hypothsizd th strss in a finitly dforming lasto-plastic solid is drivd from th lastic strain dnsity function Ψ. Thus th rat of corotational Cauchy strss can b writtn, similar to (15), in trms of th lastic strtch and th lastic corotational rat of dformation d as follows σ L ( ) : d h ( ) + { } s d ω tr( d )I. (0) Sinc plastic strains ar a nonlinar function of strsss, it is ncssary to intgrat th constitutiv quations along th prscribd loading path in ordr to obtain th currnt strss stat. In this sction w prsnt a simpl, computationally fficint implicit procdur for th lasto-plastic strss updats. For simplicity of prsntation, w adopt an lasto-plastic matrial with isotropic lastic proprtis obying von Miss yild function with a linar combination of isotropic and kinmatic hardning [15]. Considr th yild function of th following form φ σ Y (1) in which Y is th yild strss of th matrial in th uniaxial tst which volvs according to th hardning laws assumd; σ is th ffctiv strss dfind as σ 3 --ξ : P : ξ ξ σ α α corrsponds to th cntr of th yild surfac in th corotational dviatoric strss spac. Evolution of corotational back strsss, α T α, is assumd to follow th kinmatic hardning rul. For von Miss plasticity, P is a projction oprator which transforms a symmtric scond ordr tnsor from non-dviatoric spac to dviatoric spac, i.., 8

9 1 1 P ijkl -- ( δ ik δ jl + δ il δ jk ) --δ 3 ij δ kl For simplicity w adopt th associativ flow rul pd λ P : ξ () and th hardning volution law [15] in th contxt of isotropic, homognous, lastoplastic mdia. A scalar, matrial dpndnt paramtr β whr 0 β 1 is usd as a masur of th proportion of isotropic and kinmatic hardning with λ as a plastic paramtr dtrmind by th consistncy condition. Accordingly, th volution of Y and α can b xprssd in th following rat forms: α --λ 3 Y ( 1 β)hp : ξ βh p d (3) (4) whr pd is th corotational ffctiv plastic rat of dformation dfind as pd -- d 3 p : d p (5) Whil β 0 rfrs to pur isotropic hardning, β 1 is mrly th widly usd Ziglr-Pragr kinmatic hardning rul [33] for mtals without isotropic hardning. H is a hardning paramtr dfind as th ratio btwn ffctiv strss rat to ffctiv plastic rat of dformation. 3.3 Strss updat procdur In a typical load stp, th nw configuration can b xprssd as a sum of th configuration at th prvious load stp n and th displacmnt incrmnt u : x n x + u whr th lft suprscript rfrs to th load stp count with bing th currnt stp. Subsquntly w omit th lft suprscript for th currnt stp, such that all variabls without th lft suprscripts rfr to th currnt load stp. Incrmnts of th rat of dformation and spin tnsors ar intgratd using th midpoint rul to obtain th scond ordr accuracy: 9

10 d d t u x s w w t u x a (6) whr 1 n x -- ( x + x) Th rotation incrmnt Ω of Ω which yilds Ω t is also valuatd using th midpoint intgration Ω Ω t n ( ) T Th corotational incrmnts of d and ω ar givn as d T d (7) ω T ( w Ω) (8) Applying th backward Eulr schm to th scond quation in (13), w obtain th lastic incrmnt of th rat of dformation as 1 d -- ( ) (9) whr. Solving (9) for yilds n { n ( I d ) 1 } s (30) Intgrating () - (4) using th backward Eulr schm givs pd n pd + λp : ξ (31) α n α λp ( β)h : ξ 3 (3) Y n Y + βh p d (33) Furthr substituting d d d p and (31) into (30), yilds n I d { ( + d + λp : ξ ) 1 } s n p (34) 10

11 sing backward Eulr intgration schm for σ in (0) yilds σ n σ + L ( ) : d +{ h ( ) } s (35) whr d ω tr( d )I (36) Th procss is trmd lastic if σ Y λ 0 < 0 othrwis th procss is plastic, which is th focus of th subsqunt drivation. Subtracting (3) from (35) w arriv at th following rsult g ξ Q : f 0 (37) in which f 1 ( β)h Q I I λp 1 3 n ξ + L : d + { h } s (38) (39) It should b notd that Q in (37) is a function of ξ and λ. Onc λ is found, ξ can b solvd using Nwton s mthod as ξ k ( + 1) ξ k g ( ) I I Q : f 1 ξ ξ k ( ) (40) whr f ξ f : f ξ d : d p d : d ξ p (41) and f L : d h s (4) λ{ n (( I d ) 1 P ( I d ) 1 )} s ξ (43) 11

12 f L + { h ( I I) } d s d I I p d d λp p ξ (44) Th valu of λ is obtaind by satisfying th consistncy condition which assurs that at th nd of th currnt load stp,, th strss stat lis on th yild surfac. To this nd (37) and (33) ar substitutd into th yild function (1), φξ (, Y) 0, which producs a nonlinar quation for λ. A standard Nwton s mthod is applid to solv for λ : λ λ ( k + 1) λ ( k) φ φ λ ( k) (45) whr th right suprscript in parnthsis rprsnts th itration count. It can b shown that th drivativ φ λ rquird in (45) can b writtn as φ λ ξ σ : P : ξ βhy λ 3 (46) in which ξ λ 1 ( β)h f I I Q: : ξ λ :p f 1 d Q:P:ξ Th convrgd valu of λ is thn substitutd into (37) from whr ξ can b computd. Onc λ and ξ ar known, can b calculatd using (34). Corotational strss and intrnal variabls ar found from (31) to (33) and (35). Strss and back strss ar thn calculatd from σ σ T and α α T. mark: An altrnativ strss updat procdur can b mployd. Aftr obtaining th xprssion for in (34), th corotational Cauchy strss dirctly follows from (17). Con- squntly, plastic paramtr λ is calculatd using th Nwton mthod. Analogous to (39), w hav f dt( ) Ψ n α Numrical xprimnts indicat that th two procdurs ar comparabl in trms of computational complxity and accuracy. 1

13 3.4 Consistnt linarization Whil intgration of th constitutiv quations affcts th accuracy of th solution, th formation of th tangnt stiffnss matrix consistnt with th intgration procdur is ssntial to maintain th quadratic rat of convrgnc if on is to adopt th Nwton mthod for th solution of th global nonlinar systm of quations [30]. Drivation of th consistnt tangnt is obtaind from th linarization of th incrmntal constitutiv quation σ n σ + L ( ) : d +{ h ( ) } s (47) Taking th matrial tim drivativ of (47), (3) and (33) yilds: L σ : : d L : d h : + h s (48) 1 ( β)h α λ P : ξ 3 ( + λp : ξ ) (49) Y βh σ λ ξ λ σ : P : ξ (50) whr d d λ P : ξ λp : ξ (51) d 1 + ω --tr( d )I n ( I d ) 1 { d ( I d ) 1 } s T : d (5) (53) and T in (53) is dfind as T ijkl n ir ( δrk d ) 1 ( δ d ) 1 + n lj jr ( δrk d ) 1 ( δ d ) 1 li rk lj Dtails of th linarization of d and ω consistnt with th intgration procdurs dscribd in th prvious sction ar givn in th Appndix. Th xprssions for d and ω can b symbolically xprssd as rk li d T d : l ω T ω : l (54) whr and ar fourth ordr tnsors drivd in th Appndix ((88) and (89)). T d T ω 13

14 In th cas of dviatoric plasticity tr( d) tr( d ) sinc tr( p d) 0 (55) and thus (5) can b simplifid as d + T : l whr T ijkl Substituting quations (51) - (56) into (48) yilds σ A : d + Ã : l Tω 1 ijkl --δ ij T d ppkl (56) (57) whr A ijkl L 1 ijkl L ijpq h ip h jp -- ( δ il h jk + δ jl h ik ) T mnkl d pq mn pj mn pi mn Ãijkl Subtracting (49) from (57) yilds 1 --T mnkl ( δ in h jm + δ jn h im ) ξ ( A : T d + Ã) : l A : λ ξ + ( + λξ ) (58) whr A A I ( β)h I 3 : P Thrfor, ξ can b writtn xplicitly as ξ T l : l + λ T ξ : ξ (59) in which T l ( I I λa) 1 : ( A : T d + Ã) T ξ ( I I λa) 1 : A In ordr to liminat λ from (48), w substitut (50) and (59) into th linarizd form of th consistncy condition (1) φ σ ξ : P : ξ Y 0 which yilds 14

15 λ c ξ : P : T l : l c 4βHY ξ λβh : P : T ξ : ξ 1 (60) Substituting (54), (59) and (60) into (51) yilds d T d : l (61) whr Td λp : Tl c{ ξ :P:( I I+ λt ξ )} ( T l :P:ξ ) T d Th consistnt linarization of is obtaind by substituting (61) into (57) σ σ Finally, substituting (6) and yilds Dˆ : l whr Dˆ G : l into σ A : T d + Ã σ T T σ σ (6) T σ D : l (63) whr D is th consistnt stiffnss tangnt oprator dfind as D ijkl im Dˆ mnkl jn im G nmkl σ nj σ im G mnkl jn (64) 4.0 Numrical xampls In this sction w prsnt numrical xampls to study th accuracy of th proposd formulation. Th B -bar approach [14][16] is mployd to handl th isochoric natur of plastic flow. In th following xampls, th isotropic lastic strain nrgy dnsity function is chosn as Ψ Ψ λ -- ( 4 mm 3) µ ( nn 3) + -- ( δ mn mn )( mn δ mn ) whr λ and µ ar Lam constants. 4.1 otating-strtching shaft In this first xampl, w compar th classical hypolastic basd plasticity approach to th proposd formulation for th rotating-strtching shaft problm [7]. W considr an isotropic shaft subjctd to a motion whos dformation gradint is givn by 15

16 F cosθt sinθt γt 0 0 sinθt cosθt t (65) Th motion dfind by (65) givs ris to a triaxial strss stat whr th principal axis is alignd with th rotating x 1 -axis. W choos θ π and γ 1, so that at t 1, th shaft is alignd with th global x 1 -axis, and th corrsponding axial componnt of th Lagrangian strain rachs th valu of 3. Thr matrials with Young s modulus corrsponding to soft, mdium and stiff bhavior wr slctd to study th diffrncs btwn th hypolastic and hyprlastic formulations. Th following matrials proprtis wr chosn: Young s modulus ( E) 500, 3000, 0000; Poison ratio ( ν) 0.3; Y 1000; H 0; β 1. It can b asily shown that prior to yilding th corotational Cauchy strss as obtaind with th two modls is givn as σ hypo E ln( 1 + t) 1 + ν (66) and σ hypr E t( + t) { ( 1 ν) ( 1 + t) ν} ( + ν) ( 1 ν) ( 1 + t) (67) Th two corotational ffctiv strsss ar plottd vrsus th axial componnt of Lagrangian strain (Figur ) for th thr matrials considrd. For th stiffst matrial (E0000) th maximum lastic strtch rachs th valu of and no significant diffrncs btwn th two formulations is obsrvd. Howvr, in th cas of E3000, th axial strain corrsponding to th onst of yilding in th hypolastic approach is.6 tims highr than that in th prsnt hyprlastic approach. For softr matrials th diffrnc btwn th two approachs bcoms vn mor profound. 4. Simpl shar problm 11 Considr th classical shar problm dpictd in Figur 3. Th dformation gradint is chosn as F 1 t t

17 Only lastic bhavior is considrd. Matrial constants considrd ar: E10000, ( ν) 0.3. Th corotational ffctiv strss as obtaind with th corotational hypolastic approach mploying Jaumann and Dins rats ar compard to th prsnt approach in Figur 4. It can b sn that for small dformation only, t < 0.4, th thr approachs produc similar lastic strtchs Th two hypolastic modls hav a similar bhavior up to t < 1.5, at which point th Jaumann formulation rsults in a wll known oscillatory rspons. 4.3 Torsion of thick-walld cylindr Th gomtry of th cylindr was chosn as: lngth, innr radius 1, outr radius. Matrial paramtrs considrd wr: E 1000, ν 0.3, Y 1, H 1000, β 1. All th dgr of frdoms at on nd wr fixd, whras at th othr nd appropriat displacmnts wr prscribd to simulat fr-nd torsion. Th finit lmnt msh containing nod brick lmnts is shown in Figur 5. Th axial strain vrsus th shar dformation is dpictd in Figur 6. It can b sn that th maximum xtnsion is just 0.3% whil th tub is rotatd by 90 dgrs. This agrs wll with xprimntal obsrvations indicating that th axial lngth changs during th fr-nd torsion of hollow cylindrs of FCC and BCC mtals at room tmpratur ar usually small for finit rotation []. 4.4 Axisymmtric xpansion of thick-walld cylindr Axisymmtric xpansion of a thick-walld cylindr is on of th most popular bnchmark for validating finit plasticity formulations (s for xampl [8]). Th configuration of th cylindr is shown in Figur 7. W considr a cylindr with innr radius of 10 and outr radius of 0 units. This problm is solvd using 0 4-nod bi-linar axisymmtric lmnts. Th matrial paramtrs considrd ar: E 11050, ν 0.454, Y 0.5, H 0, β 1. Ths valus wr chosn so as to rplicat rigid-plastic bhavior and to allow comparison with th xact solution obtaind in [6]. In Figur 8 w show th rlationship btwn th innr radius and intrnal prssur, and in Figur 9 w show th σ rr profil vrsus position rlativ to th innr radius. It can b sn that numrical rsults ar in good agrmnt with th xact solution. 5.0 Discussion Finit dformation plasticity modl basd on th postulat that th corotational Cauchy strss can b drivd from th Hlmholtz fr nrgy dnsity as 17

18 σ dt( ) Ψ (68) has bn dvlopd. Equation (68) is similar to th constitutiv modl originally proposd by L [0] with only xcption that σ dt( ) is rplacd by S and by F. Th major diffrnc, howvr, is in kinmatical assumption: th modl of L mploys th multiplicativ dcomposition [0] whras in th prsnt manuscript th additiv split of th rat of dformation and th absnc of plastic spin ar postulatd. Th modl has bn validatd on four tst problms as th numrical rsults wr found to b in good agrmnt with ithr th xact solution or xprimntal data. From a numrical standpoint, th proposd formulation has svral advantags. On on hand it mploys plasticity formulation similar to that for small dformation thory, but on th hand, th us of hyprlastic constitutiv modl rndrs th choic and numrical intgration of objctiv strss rats ntirly suprfluous as th rsults ar automatically objctiv. Evn though w hav drivd an xplicit xprssion for th tangnt moduli consistnt with th updat stratgy, th high computational cost associatd with th consistnt tangnt valuation might not b justifid. In particular, this might b th cas whn multilvl mthods ar mployd as linar solvrs within th Nwton mthod [9]. Thus in a gnral purpos implmntation an approximation of th consistnt tangnt moduli or th us of quasi- Nwton mthod might b a bttr choic. frncs 1 ABAQS Thory Manual, Vrsion 5.4, Hibbit, Karlson & Sornsn, Inc., S. N. Atluri, An ndochronic approach and othr topics in small and finit dformation computational lasto-plasticity, Finit Elmnt Mthods for Nonlinar Problms, Europ-S Symposium, Trondhim, Norway 1985, ds. Brgan, Bath, Wundrlich, Springr, Brlin Hidlbrg, T. Blytschko, An ovrviw of smidiscrtization and tim intgration procdurs, Computational Mthods for Transint Analysis, (ds., T. Blytschko and T. J.. Hughs), Elsvir Scinc Publishrs, pp.1-95, T. Blytschko and B. J. Hsih, Nonlinar transint finit lmnt analysis with convctd coordinats, Intrnational Journal of Numrical Mthods in Enginring, 7, J. K. Dins, On th analysis of rotation and strss rat in dforming bodis, Acta Mch., 3, pp. 17-3, (1979). 6 D. Durban, Larg strain solution for prssurizd lasto-plastic tubs, Journal of Applid Mchanics, 46, A. Ev and B. D. ddy, Th variational formulation and solution of problms of finit-strain lastoplasticity basd on th us of a dissipation function, Intrnational Journal of Numrical Mthods in Enginring, 37, pp ,

19 8. A. Ev, B. D. ddy and. T. ockafllar, An intrnal variabl thory of lastoplasticity basd on th maximum plastic work inquality, Quart. Appl. Math., 48, pp , J. Fish and K. L. Shk, Computational Aspcts of Incrmntally Objctiv Algorithms for Larg Dformation Plasticity, Intrnational Journal of Numrical Mthods in Enginring, in print (1998). 10 A. E. Grn and P. M. Naghdi, A gnral thory of an lasto-plastic continuum, Arch. at. Mch. Anal., 18, A. E. Grn and P. M. Naghdi, Som marks on Elastic-Plastic Dformation at Finit Strain, Int. J. Engng. Sci., 9, pp , Hill, Th Mathmatical Thory of Plasticity, p. 3. Clarndon Prss, Oxford, J. V. Howard and S. L. Smith, cnt dvlopmnts in tnsil tsting, Proc. oy. Soc. London, A107, pp , T. J.. Hughs, Gnralization of slctiv intgration procdurs to anisotropic and nonlinar mdia, Intrnational Journal of Numrical Mthods in Enginring, 15, T. J.. Hughs, Numrical implmntation of constitutiv modls: at-indpndnt dviatoric plasticity, in S. Nmat-Nassr,. J. Asaro and G. A. Hgmir, ditors, Thortical Foundation for Larg Scal Computations for Nonlinar Matrial Bhavior, Martinus Nijhoff Publishrs, T. J.. Hughs, Th Finit Elmnt Mthod: Linar Static and Dynamic Finit Elmnt Analysis, Prntic-Hall, T. J.. Hughs and J. Wingt, Finit rotation ffcts in numrical intgration of rat constitutiv quations arising in larg dformation analysis, Intrnational Journal of Numrical Mthods in Enginring, 15, G. C. Johnson and D. J. Bammann, A discussion of strss rats in finit dformation problms, Sandia pt., SAND8-881, S. J. Kim and J. T. Odn, Finit lmnt analysis of a class of problms in finit lastoplasticity basd on th thrmodynamic thory of matrials of typ N, Computr Mthods in Applid Mchanics and Enginring, 53, pp.77-30, E. H. L, Elastic-plastic dformations at finit strains, Journal of Applid Mchanics, 36, E. H. L, Som commnts on lastic-plastic analysis, Int. J. Solids Structurs, 17, pp , P. Majors and E. Krmpl, Commnts on inducd anisotropy, th Swift ffct, and finit dformation inlasticity, Mchanics sarch Communications, 1, B. Moran, M. Ortiz and C. F. Shih, Formulation of implicit finit lmnt mthods for multiplicativ finit dformation plasticity, Intrnational Journal of Numrical 19

20 Mthods in Enginring, 9, L. C. Nagtgaal and J. E. D Jong, Som computational aspcts of lastic-plastic larg strain analysis, Intrnational Journal of Numrical Mthods in Enginring, 17, pp , S. Nmat-Nassr, Dcomposition of strain masurs and thir rats in finit dformation lastoplasticity, Int. J. Solids Structurs, 15, pp , S. Nmat-Nassr, On Finit dformation lasto-plasticity, Int. J. Solids Structurs, 18, pp , M. M. ashid, Incrmntal Kinmatics for Finit Elmnt Applications, Intrnational Journal for Numrical Mthods in Enginring, 36, pp , J. C. Simo, A framwork for finit strain lastoplasticity basd on maximum plastic dissipation and th multiplicativ dcomposition. Part II: Computational aspcts, Computr Mthods in Applid Mchanics and Enginring, 68, J. C. Simo and M. Ortiz, A unifid approach to finit dformation lasto-plastic analysis basd on th us of hyprlastic constitutiv tnsor, Computr Mthods in Applid Mchanics and Enginring, 49, J. C. Simo and. L. Taylor, Consistnt tangnt oprators for rat-indpndnt lastoplasticity, Computr Mthods in Applid Mchanics and Enginring, 48, A. V. Skachnko and A. N. Sporykhin, On th additivity of tnsors of strains and displacmnts for finit lastoplastic dformations, PMM, Vol. 41, pp , H. Swift, Lngth changs in mtals undr torsional ovrstrain, Enginring, 163, pp. 53, H. Zilgr, A modification of Pragr s hardning rul, Quartrly of Applid Mathmatics, 17, A.M. Cuitino and M.Ortiz, A matrial-indpndnt mthod for xtnding strss updat algorithms from small-strain plasticity to finit plasticity with multiplicativ kinmatics, Eng. Comp., Vol. 9, pp , A.L. Etrovich and K.J.Bath, A hyprlastic basd larg strain lasto-plastic constitutiv formulation with combind isotropic-kinmatic hardning using logarithmic strsss and strain masurs, Intrnational Journal for Numrical Mthods in Enginring, Vol. 30, pp , J.C. Simo, Algorithms for static and dynamic multiplicativ plasticity that prsrv th classical rturn mapping schms of th infinitsimal thory, Computr Mthods in Applid Mchanics and Enginring, Vol. 98, pp , J.C. Simo and T.J.. Hughs, Computational Inlasticity, Springr Vrlag, Nw York, J.C. Simo and K.S.Pistr, marks on rat constitutiv quations for finit dformation problms: computational implications, Computr Mthods in Applid Mchanics 0

21 and Enginring, Vol. 46, pp , G. Wbbr and L. Anand, Finit dformation constitutiv quations and tim intgration procdur for isotropic, hyprlastic-viscoplastic solids, Computr Mthods in Applid Mchanics and Enginring, Vol. 79, pp.173-0, Appndixs A. Linarization of and To valuat d and ω apparing in quations (51) and (5) consistnt with th intgration schm mployd w first focus on th consistnt linarization of. and Sinc th consistnt tangnt oprator is calculatd at th nd of load stp,, our first task consists of xprssing and in trms of th vlocity gradint at. For a typical tim t, th vlocity gradint may b writtn as t l t F t F 1 t t T + t t t 1 t T (69) t T Pr-multiplying (69) with and post-multiplying it with F givs t t T t t t l F T t t + t (70) Th rlation btwn t and t l can b obtaind by subtracting th transpos of (70) from th abov quation t t G : t l (71) whr t G is a fourth ordr tnsor with th componnts givn as t Gijkl t t ir t js t is jr ( ) 1 ( ) t t kr F t ls t ks F lr (7) Hnc, may b writtn as G : l (73) 1

22 In ordr to driv th xprssion for l and l, w us th following rlation btwn + v d l x + x x x x dt -- l x n 1 l Thus, can b writtn in trms of as n G : l (74) whr G ijkl 1 -- G ijkm xl x m (75) B. Linarization of d and w W start by taking th matrial tim drivativ of th gradint of th displacmnt incrmnt with rspct to th position vctor at th mid-stp (s quation (6)): d dt u v n x u d n x x n x x n x dt x (76) Th scond trm in th right hand sid of (76) can b writtn as d dt n n x x d x x x x dt n x x n (77) Combining (76) and (77) givs d dt u v u d x x x x x dt n x x n (78) Equation (78) can b furthr simplifid by xploiting th following rlation

23 d x dt n d x n x d x + x x dt n x dt v n x which aftr substitution into (78) yilds n d dt u u I v x x x (79) Equation (79) can b rcast into th following form: d dt u M : l x n xi whr M ijkl x l x x k j (80) by utilizing th following quality 1 I -- u x n 1 x -- u x x x + 1 n Th final xprssions for d and w can b obtaind by substituting (80) into th dfinition of d and w : d u 1 d dt M : l whr M ijkl -- ( M x ijkl + M jikl ) s (81) w d u 1 dt Mˆ : l whr Mˆ jikl -- ( M x ijkl M jikl ) a (8) C. Linarization of d and ω Aftr consistntly linarizing,, d and w in th Appndixs A and B, w now procd with th linarization of d and ω. Considr quations (7), (8) and d T d 3

24 ω T ( w Ω) Ω n ( ) T Taking th matrial tim drivativ of th abov quations yilds ω d T d T d + T ( w Ω ) T ( w Ω) + Ω T n T + ( ) (83) (84) (85) Combining (73), (74), (8) and (85) rsults in th following rlation w Ω M : l (86) whr Mijkl Mˆ ijkl G imkl ( )G jmkl Substituting (74), (81), (8) and (86) into (83), (84) yilds jm im n im d T d : l ω T ω : l (87) whr Td ijkl ri M rskl sj + d rs ( ri G sjkl + sj G rikl) (88) Tω ijkl ri M rskl sj + ω rs ( ri G sjkl + sj G rikl) (89) 4

25 x 3 X 3 Q dx BdX H tim t X x u q u+du h dx bdx tim t+ t x X x 1 X 1 FIGE 1. Dfinition of dformation gradint Corotational ffctiv strss σ E0000 E3000 E500 Lagrangian strain E 11 FIGE. Corotational ffctiv strss for th rotating-strtching shaft problm 5

26 x t 0 t > 0 x 1 FIGE 3. Simpl shar in x 1 dirction Corotational ffctiv strss σ Prsnt Hypolastic Dins Hypolastic Jaumann Tim t FIGE 4. Corotational ffctiv strss for th simpl shar problm 6

27 FIGE 5. Finit lmnt msh for th torsion problm Extnsion Shar strain FIGE 6. Torsion of a hollow cylindr 7

28 Axis Prscribd innr radius Initial msh FIGE 7. Gomtry and msh of th thick-walld cylindr Intrnal prssur Innr radius FIGE 8. Intrnal prssur vrsus innr radius 8

29 85 adial strss σ rr Innr radius 0 Position rlativ to innr radius FIGE 9. adial strss vrsus position rlativ to innr radius 9