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Constitutive Model of Warm Deformation Behavior of Medium Carbon Steel
JOURNALOFIRON ANDSTEELRESEARCH,INTERNATIONAL２ ０ １ ６,２ ３( ９):９ ４ ０ Ｇ ９ ４ ８
Con s t i t u t i v eMod e lo fWa rm De f o rma t i onBe ha v i o ro f Me d i um Ca r bonS t e e l Hong Ｇ b i nLI１ ２ , Yun Ｇ l iFENG２ , TaoYAN３ , En Ｇ l i nYU１ ,
( i ve r s i t １．Na t i ona lEng i ne e r i ngRe s e a r chCen t e rf o rEqu i tandTe chno l ogyo fCo l dS t r i l l i ng,YanshanUn y, pmen pRo Qi nhuangdao０６６００４,Hebe i,Ch i na; ２．Co l l egeo fMe t a l l u r r r t hCh i naUn i ve r s i t fSc i enc eand gyandEne gy,No yo Te chno l ogy,Tangshan０６３００９,Hebe i,Ch i na; ３．Equ i tManu f a c t u r i ngCo l l ege,Hebe iUn i ve r s i t f pmen yo Eng i ne e r i ng,Handan０５６０３８,Hebe i,Ch i na) Ab s t r a c t:Thec omp r e s s i vebehav i o r so fmed i umc a r bons t e e lspe c imenswe r ei nve s t i t edove rawi der angeo ft emＧ ga r a t u r e sands t r a i nr a t e sus i ngaGl e eb l e Ｇ ３５００t he rmo Ｇ s imu l a t i onma ch i ne．Ther e su l t sshowt ha tt hef l ows t r e s si n Ｇ pe c r e a s edwi t hs t r a i na tf i r s t,andt heng r adua l l c r e a s eda f t e rr e a ch i ngape akva l ue．Thef l ows t r e s ss o f t en i ngr a t e yde a tah i t r a i nr a t ewa sl a r rt hant ha ta tal ows t r a i nr a t e．Thee f f e c t so fde f o rma t i onhe a t i ngandf r i c t i ononf l ow ghs ge s t r e s swe r eana l z ed．Anewf r i c t i onc o r r e c t i onme t hod,whe r e i nt hee f f e c to fs t r a i nonf r i c t i ona lc oe f f i c i en twa sc on Ｇ y , s i de r ed wa se s t ab l i shedhe r e．Thes t r e s s e sr e v i s edbyt henew me t hodde v i a t edf r omt heme a su r eds t r e s s e swi t hi n Ｇ c r e a s i ngs t r a i n．Me anwh i l e,t heappa r en tf r i c t i ona lc oe f f i c i en tva r i a t i onl aw wi t hs t r a i nwa sob t a i ned．Thef r i c t i ona l c oe f f i c i en ti nc r e a s eda st hes t r a i ni nc r e a s edandt hens l i t l c r e a s eda f t e rma i n t a i n i ngac ons t an tva l ue．Thes t r e s s gh yde wa sc o r r e c t edbyc ons i de r i ngde f o rma t i onhe a t i ng．Thea c cur a cyo ft het empe r a t ur ecor r e c t i onme t hodwa sve r i f i ed us i ngaspe c i a lexpe r imen t．Ther e su l t so ft heve r i f i c a t i onexpe r imen tshowt ha tt het empe r a t u r ec o r r e c t i on me t hod exh i b i t edagooda c cu r a cyi nc a l cu l a t i ngt heva r i a t i ono fs t r e s sc aus edbyde f o rma t i onhe a t i ng．Ac ons t i t u t i vemode l c ons i de r i ngs t r a i nwa sp r opo s edhe r et ode s c r i bet hede f o rma t i onbehav i o r s．Compa r edwi t hexpe r imen t a lda t a,t he mod i f i edc ons t i t u t i vemode lexh i b i t edagooda c cu r a cya st oc ons t i t u t i vec o r r e l a t i on． Ke r d s:f r i c t i onc o r r e c t i on;t empe r a t u r ec o r r e c t i on;de f o rma t i onhe a t i ng;f l ows t r e s s;c ons t i t u t i vemode l ywo
Du r i ngap l a s t i cde f o rma t i on p r o c e s s,t he mi Ｇ [ ] c r os t ruc t u r a levo l u t i ono fa l l oysi sc omp l ex１Ｇ３ ． I ti s we l l Ｇknownt ha tt het he rmo Ｇme chan i c a lpa r ame t e r s wi l la f f e c tt he mi c r o s t ruc t u r a levo l u t i on,f l ow be Ｇ [ ] hav i o randp r ope r t i e so fa l l oys４Ｇ６ ．Fu t he rmo r e,t he f l ow behav i o ri sa l s oi n f l uenc ed by t he ma t e r i a l r ope r t i e s．Att hei n i t i a ls t age o fde f o rma t i on,a p wo rk Ｇ ha r den i ngbehav i o ro c cu r r ede spe c i a l l o ra l Ｇ yf l oyswi t hl ame l l a rs t ruc t u r eo rl ows t a ck i ngf au l ten Ｇ [] e rgy,sucha seu t e c t o i ds t e e landt i t an i um a l l oy７ ． Thedynami cr e c ove r e c r t a l l i z a t i onhaves i Ｇ yandr ys g n i f i c an te f f e c t si ns o f t en i ng;howeve r,t he wo rk ha rden i ng wi l ll e adt os t r a i nha r den i ng．The r e f o r e, t het he rma lp l a s t i cf o rmi ngbehav i o ri sac ompe t i t i ve r oc e s sbe twe en wo rk ha r den i ng anddynami cso f Ｇ p
t en i ng me chan i sms．Thus,i ti sne c e s s a r odeve l op yt asu i t ab l econs t i t u t i veequa t i ont ode s c r i bet hecomＧ l excompe t i t i vep r oc e s s． p Thei so t he rma ls t r e s s Ｇ s t r a i nda t aa r er equ i r ed [ ,] f o rt hecons t i t u t i vebehav i o rana l i s８ ９ ．Howeve r, ys whent hes t r a i nr a t ei s mo r et hanac e r t a i nva l ue, －１ sucha s０ ８s ,t hea c t ua lt empe r a t ur eo faspe c i Ｇ
menmayva r i a t ef r omt hep r e s e tt empe r a Ｇ yanddev t ur eduet ot hep l a s t i cde f o rma t i onhe a t i nggene r a t ed [ ] f r om p l a s t i cwo r k１０ ． I na dd i t i on,t hei n t e r f a c i a lf r i c t i on i si n e v i t ab l edu r i ng d e f o rma t i onb e twe e nt hes e c ime n p andd i e s．Event houghl ubr i c a t i oni sus edt o mi n i Ｇ mi z et hei n t e r f a c i a lf r i c t i on,t hee f f e c to fl ubr i c a t i on ont hei n t e r f a c i a lf r i c t i oni sno ta lwaysve r s Ｇ ygoode c i a l l i ngl a r f o rma t i ons．Be c aus et hef r i c Ｇ pe ydur gede
Founda t i onI t em: I t em Spons o r edbyNa t i ona lNa t ur a lSc i enc eFounda t i ono fCh i na ( ５１４７４０９２);I r onandS t e e lJ o i n tFundso fHebe iPr ov i nc eo f Ch i n a( E２０１５２０９２４３);I nno v a t i onT e am Le a d i ngT a l e n ti nUn i ve r s i t i e sCu l t i va t i onP l ano fHebe iPr ov i nc eo fCh i na ( LJRC００７) B i o r aphy: Hong Ｇ b i nLI,Ma s t e r; E Ｇma i l: l i hongb i nxp＠１２６ c om; Re c e i v e dDa t e:De c embe r３０,２０１５ g Co r r e s i ngAu t ho r:En Ｇ l i nYU,Do c t o r,Pr o f e s s o r; E Ｇma i l:yuen l i n＠ysu edu cn pond
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
９４１
t i on wi l lc aus ei nhomogeneousde f o rma t i on o ft he [ ] spe c imens１１ ,s omee r r o r so c cu ri nt hec a l cu l a t i on o fcons t i t u t i vemode lpa r ame t e r s．The r e f o r e,Evans [ [ １２] １３] , andScha rn i ng Me nge ta l a swe l la sEb r ah imi [ ] and Na a f i z adeh１４ p r e s en t ed wayst or emovet he j e f f e c to fi n t e r f a c i a lf r i c t i on ont he me a su r edf l ow s t r e s s．Ba s edont her e su l t so fRe f s [ １２, １５],Mo Ｇ [ １６] s t a f a e iand Ka z emi n e z h a d c o r r e c t e dt h ef l ows t r e s s,
duc t edonaGl e eb l e Ｇ ３５００t he rmo Ｇme chan i c a ls imu l a Ｇ t o ra tf ourd i f f e r en tde f o rma t i ont empe r a t ur e s( ５５０, ６００,６５０,and７００ ℃ )andf i ved i f f e r en ts t r a i nr a t e s －１ ( , , , , ) , ０ ００１ ０ ０１ ０ １ １ and１０s r e spe c t i ve l y． Thespe c imens we r ecomp r e s s edt oas t r a i no f０ ８ andhe a t e dt ot hed e s i e df o rmi ngt empe r a t ur ea ta gn －１ , he a t i ngr a t eo f１０℃ s t he nhe l df o r３mi nt oe l imi Ｇ n a t et het he rma lg r ad i en tbe f o r ede f o rma t i on．The compa r edt hed i f f e r enc ei nf l ow s t r e s sc o r r e c t edby f l ows t r e s scur ve swe r ea c i r edau t oma t i c a l l i ng qu yus [ [ １２] １３] , Evansand Scha rn i ng and Meng e ta l and t het he rmo Ｇme chan i c a ls imu l a t o r．Toob t a i nt hede Ｇ co r r e c t edt hes t r e s so ft head i aba t i che a t i nge f f e c t． f o rmedmi c r o s t r u c t u r e,t hes e c ime n swe r eimme d i a t e Ｇ p [ ] [ ] L i ne ta l １７ a nd Zh a ng e ta l １８ c o r r e c t e dt h ef l ow l nco l dwa t e ra f t e rcomp r e s s i on． yquenchedi s t r e s sba s edont her e su l t sr e r t e di nRe f [ １４]．The po ２ Re s u l t sandDi s c u s s i on abov e Ｇmen t i oned me t hodsr equ i r eaf r i c t i onf r a c t i on t oco r r e c tt hef l ow s t r e s s．Howeve r,t hef r i c t i on ２ １ Ana l s i so ff l ows t r e s sa ndmi c r o s t r u c t u r a le v o l u t i o n y [ １９] f r a c t i onva r i e swi t hs t r a i n andi sd i f f i cu l tt ocon Ｇ F i ２showst hef l ows t r e s s e so fmed i um c a r Ｇ g f i rma c cu r a t e l r e f o r e,i ti sne c e s s a r oc o r r e c t bons t e e ldur i ng wa rm de f o rma t i on．Theva l ue so f y．The yt t hes t r e s scu r vei nt e rmso fde f o rma t i onhe a t i ngand t hes t r e s s e swe r eobv i ous l f f e c t edbyt hede f o rma Ｇ ya i n t e r f a c i a lf r i c t i onwh i l ec ons i de r i ngt hes t r a i n． t i ont emp e r a t u r eands t r a i nr a t e．Thes t r e s si n c r e a s e d Thes t udya imst oempha s i z et hene c e s s i t f wi t hi nc r e a s i ngs t r a i nr a t eandde c r e a s i ng de f o rma Ｇ yo , f l ows t r e s sc o r r e c t i onsf o rmed i umc a rbons t e e land t i ont empe r a t ur e．I n gene r a l t heva r i a t i onsi nt he t oe s t ab l i shac ons t i t u t i ver e l a t i ont ode s c r i bewa rm f l ows t r e s swi t ht empe r a t ur eands t r a i nr a t ewe r ea t Ｇ de f o rma t i onbehav i o rbyc ons i de r i ngt hec ompens a Ｇ t r i bu t edt ot hee vo l u t i ono fd e f o rma t i onme c han i sms． t i ono ft hes t r a i nr a t e． Att he beg i nn i ng o f de f o rma t i on,t hef l ow s t r e s si nc r e a s edr ap i d l e a ched a max imum; y andr
１ Exp e r imen t a l
Theexpe r imen t a lma t e r i a lwa sa med i um c a r Ｇ bons t e e lwi t hchemi c a lc ompo s i t i onso fC０ ４６,S i ０ ２３,Mn０ ７２,P０ ０３,S０ ０３,Ni０ ０２,Cr０ ０２, Cu０ ０３,and Feba l anc e( i n ma s s％ )．Cy l i nd r i c a l
t hent heme chan i sm o fso f t en i ngde c r e a s edwi t hi n Ｇ c r e a s i ngt empe r a t ur eandde c r e a s i ngs t r a i nr a t e．The
spe c imenswi t had i ame t e ro f１０mmandahe i to f gh １５ mm we r ema ch i nedf r om aho tr o l l edba rwi t ha d i ame t e ro f１３ mm．Thei n i t i a l mi c r o s t ruc t u r ei s showni nF i １,wh i chshowst ha tt heda rka r e ai s g a r l i t e( P)andt hewh i t ea r e ai sf e r r i t e( F)．The pe , me ang r a i ns i z ei sabou t２０μm me a su r edbyt hei n Ｇ t e r c ep tme t hod．Wa rmcompr e s s i ont e s t swe r econ Ｇ
F i １ I n i t i a lmi c r o s t r u c t u r eo fme d i umc a r bon g s t e e lf o rc omp r e s s i ont e s t
F i ２ F l ows t r e s so fme d i umc a r bons t e e lme a s u r e d g du r i ngwa rmc omp r e s s i on
９４２
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
s t r e s scu r ve sf o rah i t r a i nr a t eandl owt empe r a Ｇ ghs t ur eexh i b i t ed add i t i ona lf l ow s o f t en i ng a f t e rt he aks t r e s s．Th i si si nc on t r a s t wi t ht he gene r a l pe e f f e c tru l eo ft empe r a t u r eands t r a i nr a t eont hef l ow s t r e s s． I ti sc aus edbyt he mi c r o s t ruc t u r a lva r i a t i on o f med i um c a rbon s t e e l．No rma l med i um c a rbon s t e e li sc ompo s edo ftwo pha s e s:po l lf e r r i t e ygona andl ame l l a rc emen t i t e．Du r i ngde f o rma t i on,t hel a Ｇ [ ] me l l a rc eme n t i t eb e nd s,f r a c t u r e so rd e c ompo s e s２０ , wh i l et hef e r r i t er e c ove r so rr e c r t a l l i z e s．The s e ys va r i a t i onsi nt hetwopha s e sl e adt ot hede c r e a s eo f , t hef l ows t r e s s．Atah i t r a i nr a t e dynami cr e Ｇ ghs cove r l lo c cu ri nf e r r i t eandt hec emen t i t e wi l l y wi bendo rf r a c t u r e．Th i si sj us tano t he rc emen t i t edy Ｇ
nami csphe r o i d i z a t i onme chan i smt ha ti sdemons t r a Ｇ [ ] t edbyRa s t ega r ie ta l ２１ ． I tc aus e st hes t r e s st ode Ｇ c r e a s ebeyondt hepe aks t r e s swi t hah i t r a i nr a t e． ghs , Atal ows t r a i nr a t et hedynami cr e c r t a l l i z a t i ono f ys f e r r i t eoc cur r ed,andc emen t i t ede compos ed,f o rmＧ i ngs eve r a lsma l lpa r t i c l e s．Thepa r t i c l e swou l dp i n t hed i s l oc a t i onso rg r a i nbounda r i e sdur i ngde f o rma Ｇ t i on．Thepa r t i c l e sp l ayedt her o l eo fs t r eng t hen i ng, andhenc e,t heso f t en i ngo ft hes t r e s swa sno tobv i Ｇ , ous t houghdynami cr e c r t a l l i z a t i ono ff e r r i t eoc Ｇ ys cur r eda tal ow s t r a i nr a t e．Cemen t i t ef r a c t ur eand ( de compos i t i on we r e obs e r ved s i ng l ea r r ow i n F i ３( a)and３( b)),andf e r r i t edynami cr e c r t a l Ｇ gs ys l i z a t i onoc cur r ed ( doub l ea r r owi nF i ３( b))． g
( a)６００°C,１０s－１; ( b)６００°C,０ ０１s－１．
F i ３ Cemen t i t ef r a c t u r eandd e c ompo s i t i ona td i f f e r en ts t r a i nr a t e s g
２ ２ Fr i c t i onc o r r e c t i ono ff l ows t r e s sc u r v e As chema t i cd i ag r am o ft he de f o rma t i oni s showni nF i ４． g
( a)Be f o r ec omp r e s s i on; ( b)Af t e rc omp r e s s i on．
F i ４ S ch ema t i cd i a r amo fs e c imenc omp r e s s i ont e s t g g p
Du r i ngt hep l a s t i cde f o rma t i on,f r i c t i oni suna Ｇ vo i dab l aus edbyt her e l a t i ve mo t i onbe twe ent he yc spe c imensandd i e s．Thef r i c t i onbe twe ent hespe c i Ｇ mensandanv i l swou l di n f l uenc et hede f o rma t i ono f me t a l s． I nt h i sexpe r imen t,t hef r i c t i onbe twe enspe c i Ｇ
mensandanv i l sa f f e c t edt hes t r i c a lde f o rma Ｇ ymme t i ono ft hespe c imen．Thes t r a i n Ｇ s t r e s sda t ar e co rded byt het e s t i ngs t emcon t a i nedt hei n f l uenc eo ff r i c Ｇ ys , t i on．Thus t hef l ows t r e s sa c i r edbyt e s t i ngsys Ｇ qu t e rmshou l dbe mod i f i edbye l imi na t i ngt hef r i c t i on i n f l uenc e．Thef l ows t r e s sc anbemod i f i edus i ngt he [ ] me t hodr e r t e di nRe f s １２ Ｇ １４ ．Thef o l l owi nge a Ｇ po qu t i on shavebe enus edi nRe f [ １２]: σ０ ( σ＝ １) １＋ ( ２/３ ３) m( R０/h０) exp( ３ ε/２) andi nRe f [ １３]:
σ０ ( σ＝ ２) １＋ ( ２mRe/３ ３h) whe r e, σi st hef r i c t i onco r r e c t ed; σ０i st heme a sur ed ; ; f l ows t r e s s e s εi st het rues t r a i n mi st hef r i c t i ona l coe f f i c i en t;h０i st hei n i t i a lhe i to fspe c imens;hi s gh t hespe c imenshe i i ngde f o rma t i on;Rei san ghtdur equ i va l en tr ad i us;andR０i st hei n i t i a lr ad i us． [ ] Ebr ah imiand Na a f i z adeh１４ expr e s s edt hei n Ｇ j t e r f a c i a lf r i c t i onco r r e c t edequa t i ona sf o l l ows:
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
σ０C２ ( σ＝ [ ( ) ３) ２ exp C －C －１] whe r e,C i st hec o r r e c t i onc oe f f i c i en to fi n t e r f a c i a l , : f r i c t i on wh i chi sexp r e s s eda s ( C ＝２m/h ４) ０ ５ ( Re＝R０/( ５) h０/h) [ , ] , I nRe f s １２ １３ m wa sc ons i de r edt obe０ ２． I nRe f [ １４],m wa seva l ua t edus i ngt hef o l l owi ng : expr e s s i on ( Re/h) b ( ６) m ＝ ( ２ ４/ ３)－ ( b/３ ３) ΔRh b＝４ ReΔh ΔR ＝R M －RT
( ７) ( ８)
h０ ( ９) RT ＝ ３ R２ R２M ０ －２ h ( Δh＝h０ －h １０) whe r e,R M i st he max imumr ad i us;RT i st hehe ad r ad i us;andbi st hebu l kc oe f f i c i en t． F i ５showst ha tt hemax imum d i ame t e rchan Ｇ g sdu r i ngt hec omp r e s s i ont e s t;henc e,i ti sne c e s Ｇ ge s a r oc o r r e c tt hef r i c t i onus i ngt her e a l Ｇ t imeR M yt andt hehe i tdu r i ngde f o rma t i on． gh Du r i ngt hepr o c e s so fde f o rma t i on,t hei n t e r f a Ｇ c i a lf r i c t i oni sa f f e c t edbyt hede f o rma t i on．Ther e a l Ｇ t ime ma x imum r a d i u s wa s me a s u r e du s i nga Gl e e b l e Ｇ ３５００C ＧGauges ens o r．Theva l ueo fR M i sshowni n F i ５． I tc anbes e ent ha tt hed i f f e r enc eo fR M unde r g , d i f f e r en tc ond i t i onsi si ns i i f i c an t． I nt h i sc a s et he gn e f f e c t so ft hef o rmi ngt empe r a t u r eands t r a i nr a t eon RM c anbeneg l e c t ed．
９４３
F i ５ Va r i a t i oni nRM du r i ngd e f o rma t i on g
anbec a l Ｇ Ac co rd i ngt oEq ( ６),t heva l ueo fm c cu l a t edus i ngR M andh．Theva l ue so fC andm a r e showni nF i ６．Theva l ueo fC i nc r e a s ed wi t ht he g sshowni n s t r a i n．Thef i t t i ngf o rmu l af o rC andm i n i t i a l l n Ｇ Eq s ( １１)and ( １２)．Theva l ueo fm i yi c r e a s edt oape akva l ue ( ０ ２５)t henr ema i nedcon Ｇ , s t an t．Af t e rt ha ti tde c r e a s eds l i t l t hi nc r e a s Ｇ gh y wi i ngs t r a i n．Atas t r a i no f０ ８,t heva l ueo fm wa s ０ １９andshowedl i t t l ed i f f e r enc ewi t ht heva l ueus ed [ , ] i nRe f s １２ １３ ．Theconc en t r a t i ono fl ubr i c an tde Ｇ c r e a s edwi t hi nc r e a s i ngs t r a i n,andt hef r i c t i ona lco Ｇ [ ] e f f i c i en ti nc r e a s ed１９ ．Thei nc r e a s eo ff r i c t i ons t r e s s wa sc aus ed by t hei nc r e a s e si n bo t hf r i c t i on and s t r e s swi t hs t r a i n．Onc et hef r i c t i ons t r e s sexc e eded , t hey i e l ds t r e s so ft hespe c imen t her e l a t i veg l i de be twe ent hespe c imensandd i e ss t opped．Thec l i n Ｇ y de rp r o f i l ei sshowni nF i ７,andt hef r i c t i ona lcoe f Ｇ g ( ) f i c i en tob t a i nedbyEq １２ de c r e a s ed．
( a)Re l a t i onbe twe enC andε; ( b)Re l a t i onbe twe enm andε．
F i ６ Re l a t i onb e twe enC,m ,andε g
( C ＝－０ ６５４ ε－０ １８０ ε２ １１) ２ ３ m ＝－０ １０９－５ １６２ ε－３１ １５５ ε －９８ ４９４ ε－ ( １６８ ７１９ ε４ －１４６ ６０２ ε５ －５０ ５５７ ε６ １２) Thec o r r e c t edr e su l t sus i ng d i f f e r en t me t hods a r eshowni nF i ８．Thed i f f e r enc ebe twe enc o r r e c Ｇ g
t edandunco r r e c t edcur ve sg r adua l l nc r e a s ed wi t h yi [ １６] s t r a i n ． I ti squ i t econs i s t en twi t ht hea c t ua ls i t ua Ｇ t i ono facomp r e s s i ont e s t．Atas t r a i no f０ ８,t he s t r e s sco r r e c t edby Eq ( ３)wi t ht her e a l Ｇ t ime R M wa sequa lt ot heone sco r r e c t edbyEq ( ３)wi t ht he
９４４
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
F i ７ S imp l er ep r e s en t a t i ono fup t u rn i ng g
F i ８ Compa r i s ono fme a s u r e dandc o r r e c t e dc u r v e s g
f i na lR M ．Bu tf o rt hewho l ep r o c e s s,t hes t r e s sco r Ｇ ( ) r e c t edby Eq ３ wi t ht her e a l Ｇ t imeR M wa s mo r e cons i s t en twi t ht her e a ls t r e s st h a nt h a tw i t ht h ef i n a l
R M ．The pr ed i c t edf r i c t i on l e s ss t r e s sc a l cu l a t ed by Eq ( ２)wa sl e s st hant ha tc a l cu l a t edbyEq ( １)．The s t r e s sc a l cu l a t edbyEq ( １)i sc l os et ot ha tob t a i ned ( ) byEq ３ wi t ht her e a l Ｇ t imeR M ．Thef r i c t i onf a c t o r wa scons i de r edt obe０ ２i nRe f s [ １２, １３],bu tQi an [ ] e ta l １０ cons i de r edt ha tt hef r i c t i onf a c t o rwa s０ ３． , I nf a c tt hef r i c t i onf a c t o rva r i eddur i ngt hef o rmi ng． Fo r t una t e l hef r i c t i onf a c t o rc anbededuc edf r om y,t Eq ( ６)by me a sur i ngt her e a l Ｇ t imeva l ueo fR M ． ２ ３ An a l s i so ft emp e r a t u r ev a r i a t i o nd u r i n e f o rma t i o n y gd Mos to ft heene r st r ans f o rmedi n t ohe a t gy wa wh i l eaf r a c t i ono ft heene r ss t o r edi nt hespe c Ｇ gywa [ ２２] imensi nt hef o rm o fd i s t o r t i on ene r ft he gy ．I , s t r a i nr a t ei sh i ri tc aus e sani nc r e a s ei ns t r e s s ghe andade c r e a s ei nt hede f o rma t i ont ime;henc e,mos t o ft hede f o rma t i onhe a tc anno tber emovedf r omt he spe c imens．Thes amp l et empe r a t ur et heni nc r e a s e s duet ot hede f o rma t i on he a t i ng．I nt h i sc a s e,t he t empe r a t ur eva r i a t i onmaybes i i f i c an t． gn , Fo rt hecomp r e s s i ont e s t s t het empe r a t ur ei s showni nF i ９．Thet empe r a t ur ei nc r e a s edwi t hi n Ｇ g c r e a s i ngs t r a i nandde c r e a s i ngde f o rma t i ont empe r a Ｇ t ur e．Fur t he rmo r e,t het empe r a t ur eva r i a t i ona ta s t r a i nr a t eo f０ １s－１ wa sve r l landcou l dbei Ｇ ysma g
F i ９ Cu r v e so ft emp e r a t u r ev a r i a t i ondu r i ngd e f o rma t i on g [ ] no r ed１６ ,a sshowni nF i ９( b)． g An Ar rhen i us Ｇ t t i oni sa lwaysus edt o ypeequa de s c r i bet het he rma lde f o rma t i onbehav i o ro ft hea l Ｇ l oys．Thet empe r a t u r eva r i a t i ondu r i ngde f o rma t i on i sco r r e c t edbyus i ng Ar rhen i usequa t i onandZene r Ｇ [ ] Ho l l omi npa r ame t e ra sf o l l ows２３ : æ ö çQ ÷ ( ÷÷ Z ＝̇ １３) εexpçç R è Tø ( Z ＝A１σn１ ( α σ＜１ ２) １４) ( ( Z ＝A２exp( σ) α σ＞１ ２) １５) β n ( ) ( ) ( Z ＝As i nhα σ Fo ra l lσ １６) whe r e,A１ ,A２ ,A ,n１ ,n andβ a r e ma t e r i a l spa Ｇ
̇ r ame t e r s;αi st hes t r e s smu l t i l i e r( α＝β/n１ ); s εi p ; t hes t r a i nr a t e Qi st heappa r en ta c t i va t i onene r gy; Ri st heun i ve r s a lga scons t an t;andT i st heabso Ｇ l u t et empe r a t ur e． By subs t i t u t i ng Eq ( １３)i n t o Eq s ( １４)and ( １５)andt hent ak i ngt hel oga r i t hmandr e a r r ang i ng Eq s ( １４)and ( １５),t hef o l l owi ngequa t i onsc anbe ob t a i ned: é æ öù ê ΔT ç Q ÷ú ê x ç ÷ ú ( Δσ＝σ e α σ＜１ ２) ( １７) pç êê n T ＋ΔT )÷ø úúû ë è １RTT ( Q æç ΔT ö÷ ÷ Δσ＝ çç ＋ΔT ÷ø è βR T
( α σ＞１ ２) ( １８)
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
Ac c o r d i ngt oRe f [ ２４],αi sabou t０ ００３,and i ti se a s oge tt ha ti ft hes t r e s si sg r e a t e rt han３５０ yt MPa．Eq ( １７)i ssu i t ab l ef o rde f o rma t i onc ond i t i on －１ －１ －１ , , , sucha s７００ ℃ １０s o r１s and６５０ ℃ １s ． Ot he rwi s e,Eq ( １８) i ssu i t ab l ef o rde f o rma t i onconＧ d i t i onss u c ha s５５０℃ ,１０s－１o r１s－１ ,６００℃ ,１０s－１ o r１s－１ ,and６５０ ℃ ,１０s－１ ．Thepa r ame t e rc a l cu l a Ｇ t i onpr o c e s si sshowna sf o l l ows． Ba s edont he me a su r edda t a,as t r a i no f０ ４ wa st aken a s an c a s et o de s c r i bet he c a l cu l a t i on oc e s so ft hema t e r i a lc ons t an t s．Us i ngt hena t ur a l pr ( ) ( ) , l oga r i t hmf o rEqs １４ and １５ t hef o l l owi ngr e Ｇ l a t i onc anbeob t a i ned: ̇ ( l n nA１ ＋n１l σ ε＝l n １９) ̇ ( ε＝l nA２ ＋β σ ２０) l n Then,subs t i t u t i ng t hef l ow s t r e s so fs t r a i n , ０ ４ wh i ch wa s ob t a i ned du r i ng t he c omp r e s s i on t e s t,ands t r a i nr a t ei n t oEq s ( １９)and ( ２０),i tc an tt her e l a t i onbe twe ent hef l ow s t r e s sands t r a i n ge , r a t ej us ta sshowni nF i １０． g ̇ εandl I ti sobv i oust ha tt hel i ne so ft heσＧ l n n σＧ
̇ l n εc anbec ons i de r edt obepa r a l l e l,andt hen１andβ
９４５
equa lt ot hes l opo ft hepa r a l l e ll i ne s,r e spe c t i ve l y． Cons i de r i ngt hee f f e c to ft empe r a t ur eonn１ andβ, t a i nedbyl i ne a rf i t t i ng t heme anva l ue so fn１andβob －１ , me t hoda r e１０ ５６７and０ ０３５ MPa r e spe c t i ve l y． The r e f o r e,αi scons i de r edt obe０ ００３３． Fo ra l lt hes t r e s sl eve l s,Eq ( １５)c anbed i f f e r Ｇ , en t i a t edandt hus d l n[ s i nh( α σ)] ( ２１) Q ＝R n d( １/T ) ̇ d( l n ε) ( n＝ { [ ( )]} ２２) σ dl ns i nhα Bysubs t i t u t i ngt het empe r a t ur e s,s t r a i nr a t e s ( ) ( ) ands t r e s s e s a tas t r a i no f０ ４ i n t oEq s ２１ and ( ２２),t her e su l t s we r eob t a i nedanda r eshowni n
F i １１． g Theva l ue so fn and Q a r ec a l cu l a t edt o be ７ ６７７and３８４ ８１０kJmo l－１ ,r e spe c t i ve l y． Thed a t awi t hc o r r e c t e dt emp e r a t u r e sa r es howni n F i １２．F i １２( a)showst her e su l t sco r r e c t edf o r g g t empe r a t ur e．Thed i f f e r enc ebe twe enr e su l t sco r r e c Ｇ t edbyEq s ( １７)and ( １８)i sve r l i t;t hus,t he ys gh e f f e c to fus i ng d i f f e r en tequa t i onsi sneg l i i b l e．I n g t h i sr e s e a r ch,Eq ( １８)i sus edt oco r r e c tf o rt hei n Ｇ
̇ ̇ ( a)l n σＧ l n ε; ( b)σＧ l n ε．
F i １０ Re l a t i onb e twe ent h es t r a i nr a t eandf l ows t r e s s g
F i １１ Ca l c u l a t i on so fQ andn g
９４６
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
( a)１０s－１; ( b)Compa r i s onbe twe enme a sur edandc o r r e c t eds t r e s s e s．
F i １２ Cu r v e so fs t r e s sc o r r e c t e df o rd e f o rma t i onh e a t i ng g
f l uenc eo fde f o rma t i onhe a t i ng． F i １２( b)showst he g compa r i s onbe twe ent h et emp e r a t u r ec o r r e c t e ds t r e s s ( ) ba s edonEq １８ andt he me a su r eds t r e s swi t hi n Ｇ c r e a s i ngt empe r a t u r ef r om５５０t o７００ ℃ du r i ngde Ｇ f o rma t i onwi t has t r a i nr a t eo f０ ０１s－１ ．Thec o r r e c Ｇ t eds t r e s sva l ue sc o i nc i de we l l wi t ht he me a sur ed s t r e s s．
２ ４ Con s t i t u t i v ee t i on so ff l ows t r e s sf o rme d i um qua c a r bons t e e l Ba s edont hes t r e s sc o r r e c t edf o rf r i c t i onand de f o rma t i on he a t i ng,t he p r o c e s so fe s t ab l i sh i ng cons t i t u t i veequa t i onsi ss imi l a rt os e c t i on２ ２． Ac c o r d i ngt ot hec a l cu l a t i onr e su l t s,t hev a l u e s o fα,n１ a ndβa r e０ ００３５,９ ２９４,a nd０ ０３３ MPa－１ ,
r e spe c t i ve l l ue so fn and Q a r e６ ６７３and y．Theva ３５３ ５６５kJmo l－１ ,r e spe c t i ve l y． Tak i ngt hel oga r i t hmo fbo t hs i de so fEq ( １６),
t hef o l l owi ngequa t i onc anbeob t a i ned ( l nZ ＝l nA ＋nl n[ s i nh( α σ)] ２３) , Ba s edont heexpe r imen t a lr e su l t st heva l ue so f r et hei n t e r c ep tands l opeo fonl nAＧ l n l nA andn a [ s i nh( α σ)]p l o t,r e spe c t i ve l heva l ueo fA y．Thus,t １９ c anbee a s i l a l cu l a t edt obe１ ３３８×１０ ． yc I ns imi l a r ways,t heva l ue so f ma t e r i a lcon Ｇ Q ,A ,n,andα)o ft hec ons t i t u t i veequa t i on s t an t s( we r ec ompu t edunde rd i f f e r en tde f o rma t i ons t r a i ns wi t h i nt her angeo f０ ０５－０ ８０i ni n t e r va l so f０ ０５． Ther e l a t i onbe twe enQ ,A ,n, α,andt hes t r a i nf o r , med i um c a rbons t e e lc an be po l a lf i t t ed a s ynomi showni nF i １３． g Asc anbes e eni nF i １３( a),t hea c t i veene r g gy , i n i t i a l l c r e a s edwi t hs t r a i nr e a chedi t smi n imum yde va l uea tas t r a i no f０ ４５, i n c r e a s e dupt o３６２ ８６k J mo l－１ a tas t r a i no f０ ７,andf i na l l d e c r e a s e dt o y －１ ３５７ ３２kJ mo l a tas t r a i no f０ ８．Theva l ue sa r e l e s st hant ha tr epo r t edf o rmed i umc a rbons t e e lwi t h
F i １３ Va r i a t i on so fQ ( a),α ( b),n ( c),and g A( d)wi t hs t r a i nba s e donpo l a lf i t t i ng ynomi
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
ama r t ens i t emi c r o s t ruc t u r e２４ ．Th i sr e f l e c t st hef a c t t ha ta i rc oo l ed med i um c a rbon s t e e lha sa be t t e r wo rkab i l i t hanquenchedmed i umc a rbons t e e l．Ac Ｇ yt ( ) , co rd i ngt oF i １３ b t heva l ueo fα i n i t i a l l Ｇ g y de c r e a s edt ot hemi n imumva l uea tas t r a i no f０ １,and i nc r e a s eds t e ad i l t hi nc r e a s i ngs t r a i n, l i ngi nt he ywi y －１ r ange o f０ ００３３ － ０ ００３８ MPa ．Re f e r r i ng t o F i １３( c),t heva l ueo fni n i t i a l l c r e a s edwi t hi n Ｇ g yde c r e a s i ng s t r a i n,and r e a ched t he mi n imum,t hen s l i t l nc r e a s ed．Theva l ue so ft he s epa r ame t e r so f gh yi t hecons t i t u t i vemode la r es imi l a rt ot heda t ar epo r Ｇ [ ] t edbyL ie ta l ２４ ． anbe Ac c o r d i ngt oEq ( １６),t hef l ows t r e s sσc opo s eda saf unc t i ono ft heZene r ＧHo l l omon pa Ｇ pr , r ame t e r．The r e f o r et hep r opo s edc ons t i t u t i vemode l c anbesumma r i z eda sf o l l ows: [ ]
ö２ n éæ ù１ ２ üï ê çZ ÷ ú １ ìïï æçZ ö÷１ n ï ( níïï çç ÷÷ ＋ êêê çç ÷÷ ＋１úúú ýïï σ＝ l ２４) α î èA ø ë èA ø û þ Q＝３７３ ６５９＋１９０ ９２２ ε＋５６２ ３７０ ε２－１２３ ９５６ ε３－ /
/
/
９４７
ε４ －６６２８ ε５ －３ ３６７０ ８５５ ４７５ ６４６×１０３ε６ ( ２５)
００４＋０ ０２１ ε＋０ １５８ ε２ ＋０ ε３ ＋ α＝０ ５６６ ( １ ０４２ ε４ ＋０ ９４８ ε５ ＋０ ３３７ ε６ ２６) ２ ３ n＝１１ ２１３＋７１ ５０１ ε＋４７７ ５３５ ε ＋１５８９ ２１８ ε＋ ( ２７４６ ６３４ ε４ ＋２３５４ ４７２ ε５ ＋７９１ ０８２ ε６ ２７) A ＝６ １３０×１０１９ ＋５ ５６×１０２０ε＋１ ５０×１０２０ε２ － ２２ ３ ２２ ４ １ ２４６×１０ ε －４ ７６×１０ ε －６ ７１× ( １０２２ε５ －３ ２９×１０２２ε６ ２８)
２ ５ Ve r i f i c a t i ono fde v e l ope dc on s t i t u t i v ee t i on s qua Tot e s tt heve r a c i t ft hedeve l opedcons t i t u Ｇ yo t i veequa t i on,t hec a l cu l a t edf l ow s t r e s sda t a wa s compa r ed wi t h me a sur emen t sove rt heen t i r et emＧ r a t ur eands t r a i nr a t er ang ei nv e s t i a t e d,a ss hown pe g i nF i １４． g Thec a l cu l a t edf l ows t r e s s e sp r ov i deagoodap Ｇ r ox ima t i ont ot heco r r e c t edva l ue s．Toeva l ua t et he p a c cur a c ft hede f o rma t i onc on s t i t u t i v ee a t i on,t he yo qu
F i １４ Compa r i s onb e twe enf r i c t i on Ｇ c o r r e c t e dandc a l c u l a t e df l ows t r e s s e s g
e r r o ra nda v e r a ea b s o l u t er e l a t i v ee r r o r( AARE),be Ｇ g twe ent hec a l cu l a t edf l ow s t r e s s( σc )andf r i c t i on co r r e c t edf l ows t r e s s( σe)c a nb ec a l c u l a t e da sf o l l ows: N
R２ ＝
∑( Ei －E )( Pi －P )
i＝１ N
N
∑( Ei －E )∑ ( Pi －P )
i＝１
２
i＝１
( ２９)
１ N Ei －Pi ( Rave＝ ∑ ×１００％ ３０) Ni＝１ Ei whe r e,E i st heexpe r imen t a lda t a;P i st hec a l cu Ｇ l a t edva l uebyt hedeve l opedcons t i t u t i veequa t i ons; r et heme anva l ue so fE andC ,r E andPa e spe c t i ve Ｇ ; l st henumbe ro fda t a． y andN i Ac co rd i ngt oEq s ( ２９)and ( ３０),t hemax imum
９４８
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
R２ wa s０ ９９８ and t he AARE wa s０ ５４７％ ,a s showni nF i １５．Th i si nd i c a t e st ha tt he p r opos ed g cons t i t u t i veequa t i onp r ov i de sana c cu r a t ep r ed i c t i on o ft hes t r e s sf o rmed i umc a rbons t e e l．
F i １５ Co r r e l a t i onb e twe enc o r r e c t e dandp r e d i c t e d g f l ows t r e s sdu r i ngc omp r e s s i ont e s t
３ Conc l u s i on s ( １)Thef l owcu r ve sa th i rs t r a i nr a t e sex Ｇ ghe h i b i t edanadd i t i ona ls o f t en i ngt endenc t i t e y．Cemen f r agmen t swe r eobs e r vedt ode f o rmi nt hes amp l e sa t , h i h e rs t r a i nr a t e s a n df e r r i t ed n a m i cr e c r s t a l l i Ｇ g y y z a t i ona swe l la sc emen t i t ede c ompo s i t i on we r eob Ｇ s e r ved． ( ２)Thef l ow s t r e s s wa ss i i f i c an t l n f l u Ｇ gn yi enc edbyde f o rma t i onhe a t i ngandf r i c t i on．Thede Ｇ f o rma t i onhe a t i ngc o r r e c t ed me t hodve r i f i edbyex Ｇ r imen t a lr e su l t sexh i b i t edagooda c cu r a c nc a l Ｇ pe yi , cu l a t i ngt heva r i a t i oni ns t r e s s．Thus anewf r i c t i on me t hod c ons i de r i ng s t r a i n wa se s t ab l i shed．The e f f e c to fs t r a i nont hef r i c t i onf a c t o rwa sobv i ous． ( ３)Ther e l i ab i l i t ft hec o r r e c t edc ons t i t u t i ve yo mode lwa seva l ua t edbys t a t i s t i c a lana l i s．Theco r Ｇ ys r e c t edc on s t i t u t i v emod e lp r o v i d e da na c c u r a t ee s t ima t e o ft hes t r e s s．The r e f o r e,t hedeve l opedc ons t i t u t i ve equa t i onsa r er e l i ab l ef o rf i n i t ee l emen tana l i sf o r ys wa rmde f o rma t i onc ond i t i ons．
Re f e r enc e s: [ １] YC．L i n,X M．Chen,Ma t e r．De s． ３２ ( ２０１１)１７３３ Ｇ １７５９． [ ２] P．Wan a r a,M．J aha z i,H．Mona a t i,S．Yue,J P． Imma r i Ｇ j j t e r． Sc i．Eng．A３９６ ( ２００５)５０ Ｇ ６０． geon,Ma [ ３] A．Momen i,K．Dehghan i,GR． Ebr ah imi,S．Ka z emi,Me t a l l． Ma t e r．Tr ans．A４４ ( ２０１３)５５６７ Ｇ ５５７６． [ ４] H．Mi r z adeh,A．Na a f i z adeh,I S I JI n t． ５３ ( ２０１３)６８０ Ｇ ６８９． j [ ５] C．J．Sh i,WM．Mao,X G．Chen,Ma t e r．Sc i．Eng．A ５７１ ( ２０１３)８３ Ｇ ９１． [ ６] F．Yi n,L．Hua,HJ．Mao,X H．Han,D S．Qi an,R． Zhang, Ma t e r．De s． ５５ ( ２０１４)５６０ Ｇ ５７３． [ ７] Y C．L i n,K K．L i,HB．L i,J．Ch e n,X M．Ch e n,D X．We n, Ma t e r．De s． ７４ ( ２０１５)１０８ Ｇ １１８． [ ８] R．Goe t z,S．Semi a t i n,J．Ma t e r．Eng．Pe r f o rm．１０ ( ２００１) No ６,７１０ Ｇ ７１７． [ ９] L．L i,J．Zhou,J．Dus z c z t e r．Pr o c e s s．Te chno l． １７２ yk,J．Ma ( ２００６)No ３,３７２ Ｇ ３８０． [ n t．J．Pr e s s． X．Wang,I １０] L Y．Qi an,G．Fang,P．Zeng,L Ve s s e l sP i １２３ Ｇ １３３ ( ２０１５)４３ Ｇ ４５． p． [ １１] GW．L i u,Y．Han,Z Q．Sh i,J P．Sun,D N．Zou,GJ． Qi ao,Ma t e r．De s． ５３ ( ２０１４)６６２ Ｇ ６７２． [ t e r． Sc i．Te chno l． １７ ( ２００１) １２] R W． Evans,P J． Scha r n i ng,Ma No ８,９９５ Ｇ １００４． [ R．Ni e,Ma t e r． L．L i,HM．L i,H．Huang,Z １３] G．Meng,B Sc i．Eng．A５１６ ( ２００９)１３１ Ｇ １３７． [ １４] R．Ebr ah imi,A．Na a f i z adeh,J．Ma t e r． Pr o c e s s．Te chno l． １５２ j ( ２００４)No ２０,１３６ Ｇ １４３． [ １５] A．Gho l amz adeh,A．Ka r imiTahe r i,Me ch．Re s． Commun i c a Ｇ t i ons３６ ( ２００９)No ２,２５２ Ｇ ２５９． [ １６] MA．Mo s t a f a e i,M．Ka z emi ne zhad,Ma t e r．Sc i．Eng．A５３５ ( ２０１２)２１６ Ｇ ２２１． [ １７] YC．L i n,YC．Xi a,X M．Chen,MS．Chen,Compu t．Ma Ｇ t e r． Sc i． ５０ ( ２０１０)No ４,２２７ Ｇ ２３３． [ J． C． Zhang,T Y．Wang,C．Yu,J １８] J Q． Zhang,HS．Di,X Ma,Ma t e r．De s． ４４ ( ２０１３)２５４ Ｇ ３６４． [ １９] YP．L i,H．Ma t sumo t o,A．Ch i ba,Me t a l l．Ma t e r．Tr ans．A ４０ ( ２００９)No ４,９８２ Ｇ ９９０． [ ２０] L．S t o r o eva,D．Ponge,R．Ka spa r,D．Ra abe,Ac t a Ma t e r． j ５２ ( ２００４)２２０９ Ｇ ２２２０． [ ２１] H．Ra s t ega r i,A．Ke rmanpur,A．Na a f i z adeh,D．Po r t e r,M． j Soman i,J．Al l oysComp． ６２６ ( ２０１５)１３６ Ｇ １４４． [ ２２] C X．Wang,F X．Yu,D Z． Zhao,X．Zhao,L．Zuo,Ma t e r． Sc i．Eng．A５７７ ( ２０１３)７３ Ｇ ８０． [ ２３] C． Zene r,H．Ho l l omon,J．App l． Phys． １５ ( １９４４)No １,２２ Ｇ ３２． [ ２４] Q．L i,T S．Wang,HB．L i,YW．Gao,N．L i,T F． J i ng, J． I r onS t e e lRe s． I n t． １７ ( ２０１０)No ５,３４ Ｇ ３７．
JOURNALOFIRON ANDSTEELRESEARCH,INTERNATIONAL２ ０ １ ６,２ ３( ９):９ ４ ０ Ｇ ９ ４ ８
Con s t i t u t i v eMod e lo fWa rm De f o rma t i onBe ha v i o ro f Me d i um Ca r bonS t e e l Hong Ｇ b i nLI１ ２ , Yun Ｇ l iFENG２ , TaoYAN３ , En Ｇ l i nYU１ ,
( i ve r s i t １．Na t i ona lEng i ne e r i ngRe s e a r chCen t e rf o rEqu i tandTe chno l ogyo fCo l dS t r i l l i ng,YanshanUn y, pmen pRo Qi nhuangdao０６６００４,Hebe i,Ch i na; ２．Co l l egeo fMe t a l l u r r r t hCh i naUn i ve r s i t fSc i enc eand gyandEne gy,No yo Te chno l ogy,Tangshan０６３００９,Hebe i,Ch i na; ３．Equ i tManu f a c t u r i ngCo l l ege,Hebe iUn i ve r s i t f pmen yo Eng i ne e r i ng,Handan０５６０３８,Hebe i,Ch i na) Ab s t r a c t:Thec omp r e s s i vebehav i o r so fmed i umc a r bons t e e lspe c imenswe r ei nve s t i t edove rawi der angeo ft emＧ ga r a t u r e sands t r a i nr a t e sus i ngaGl e eb l e Ｇ ３５００t he rmo Ｇ s imu l a t i onma ch i ne．Ther e su l t sshowt ha tt hef l ows t r e s si n Ｇ pe c r e a s edwi t hs t r a i na tf i r s t,andt heng r adua l l c r e a s eda f t e rr e a ch i ngape akva l ue．Thef l ows t r e s ss o f t en i ngr a t e yde a tah i t r a i nr a t ewa sl a r rt hant ha ta tal ows t r a i nr a t e．Thee f f e c t so fde f o rma t i onhe a t i ngandf r i c t i ononf l ow ghs ge s t r e s swe r eana l z ed．Anewf r i c t i onc o r r e c t i onme t hod,whe r e i nt hee f f e c to fs t r a i nonf r i c t i ona lc oe f f i c i en twa sc on Ｇ y , s i de r ed wa se s t ab l i shedhe r e．Thes t r e s s e sr e v i s edbyt henew me t hodde v i a t edf r omt heme a su r eds t r e s s e swi t hi n Ｇ c r e a s i ngs t r a i n．Me anwh i l e,t heappa r en tf r i c t i ona lc oe f f i c i en tva r i a t i onl aw wi t hs t r a i nwa sob t a i ned．Thef r i c t i ona l c oe f f i c i en ti nc r e a s eda st hes t r a i ni nc r e a s edandt hens l i t l c r e a s eda f t e rma i n t a i n i ngac ons t an tva l ue．Thes t r e s s gh yde wa sc o r r e c t edbyc ons i de r i ngde f o rma t i onhe a t i ng．Thea c cur a cyo ft het empe r a t ur ecor r e c t i onme t hodwa sve r i f i ed us i ngaspe c i a lexpe r imen t．Ther e su l t so ft heve r i f i c a t i onexpe r imen tshowt ha tt het empe r a t u r ec o r r e c t i on me t hod exh i b i t edagooda c cu r a cyi nc a l cu l a t i ngt heva r i a t i ono fs t r e s sc aus edbyde f o rma t i onhe a t i ng．Ac ons t i t u t i vemode l c ons i de r i ngs t r a i nwa sp r opo s edhe r et ode s c r i bet hede f o rma t i onbehav i o r s．Compa r edwi t hexpe r imen t a lda t a,t he mod i f i edc ons t i t u t i vemode lexh i b i t edagooda c cu r a cya st oc ons t i t u t i vec o r r e l a t i on． Ke r d s:f r i c t i onc o r r e c t i on;t empe r a t u r ec o r r e c t i on;de f o rma t i onhe a t i ng;f l ows t r e s s;c ons t i t u t i vemode l ywo
Du r i ngap l a s t i cde f o rma t i on p r o c e s s,t he mi Ｇ [ ] c r os t ruc t u r a levo l u t i ono fa l l oysi sc omp l ex１Ｇ３ ． I ti s we l l Ｇknownt ha tt het he rmo Ｇme chan i c a lpa r ame t e r s wi l la f f e c tt he mi c r o s t ruc t u r a levo l u t i on,f l ow be Ｇ [ ] hav i o randp r ope r t i e so fa l l oys４Ｇ６ ．Fu t he rmo r e,t he f l ow behav i o ri sa l s oi n f l uenc ed by t he ma t e r i a l r ope r t i e s．Att hei n i t i a ls t age o fde f o rma t i on,a p wo rk Ｇ ha r den i ngbehav i o ro c cu r r ede spe c i a l l o ra l Ｇ yf l oyswi t hl ame l l a rs t ruc t u r eo rl ows t a ck i ngf au l ten Ｇ [] e rgy,sucha seu t e c t o i ds t e e landt i t an i um a l l oy７ ． Thedynami cr e c ove r e c r t a l l i z a t i onhaves i Ｇ yandr ys g n i f i c an te f f e c t si ns o f t en i ng;howeve r,t he wo rk ha rden i ng wi l ll e adt os t r a i nha r den i ng．The r e f o r e, t het he rma lp l a s t i cf o rmi ngbehav i o ri sac ompe t i t i ve r oc e s sbe twe en wo rk ha r den i ng anddynami cso f Ｇ p
t en i ng me chan i sms．Thus,i ti sne c e s s a r odeve l op yt asu i t ab l econs t i t u t i veequa t i ont ode s c r i bet hecomＧ l excompe t i t i vep r oc e s s． p Thei so t he rma ls t r e s s Ｇ s t r a i nda t aa r er equ i r ed [ ,] f o rt hecons t i t u t i vebehav i o rana l i s８ ９ ．Howeve r, ys whent hes t r a i nr a t ei s mo r et hanac e r t a i nva l ue, －１ sucha s０ ８s ,t hea c t ua lt empe r a t ur eo faspe c i Ｇ
menmayva r i a t ef r omt hep r e s e tt empe r a Ｇ yanddev t ur eduet ot hep l a s t i cde f o rma t i onhe a t i nggene r a t ed [ ] f r om p l a s t i cwo r k１０ ． I na dd i t i on,t hei n t e r f a c i a lf r i c t i on i si n e v i t ab l edu r i ng d e f o rma t i onb e twe e nt hes e c ime n p andd i e s．Event houghl ubr i c a t i oni sus edt o mi n i Ｇ mi z et hei n t e r f a c i a lf r i c t i on,t hee f f e c to fl ubr i c a t i on ont hei n t e r f a c i a lf r i c t i oni sno ta lwaysve r s Ｇ ygoode c i a l l i ngl a r f o rma t i ons．Be c aus et hef r i c Ｇ pe ydur gede
Founda t i onI t em: I t em Spons o r edbyNa t i ona lNa t ur a lSc i enc eFounda t i ono fCh i na ( ５１４７４０９２);I r onandS t e e lJ o i n tFundso fHebe iPr ov i nc eo f Ch i n a( E２０１５２０９２４３);I nno v a t i onT e am Le a d i ngT a l e n ti nUn i ve r s i t i e sCu l t i va t i onP l ano fHebe iPr ov i nc eo fCh i na ( LJRC００７) B i o r aphy: Hong Ｇ b i nLI,Ma s t e r; E Ｇma i l: l i hongb i nxp＠１２６ c om; Re c e i v e dDa t e:De c embe r３０,２０１５ g Co r r e s i ngAu t ho r:En Ｇ l i nYU,Do c t o r,Pr o f e s s o r; E Ｇma i l:yuen l i n＠ysu edu cn pond
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
９４１
t i on wi l lc aus ei nhomogeneousde f o rma t i on o ft he [ ] spe c imens１１ ,s omee r r o r so c cu ri nt hec a l cu l a t i on o fcons t i t u t i vemode lpa r ame t e r s．The r e f o r e,Evans [ [ １２] １３] , andScha rn i ng Me nge ta l a swe l la sEb r ah imi [ ] and Na a f i z adeh１４ p r e s en t ed wayst or emovet he j e f f e c to fi n t e r f a c i a lf r i c t i on ont he me a su r edf l ow s t r e s s．Ba s edont her e su l t so fRe f s [ １２, １５],Mo Ｇ [ １６] s t a f a e iand Ka z emi n e z h a d c o r r e c t e dt h ef l ows t r e s s,
duc t edonaGl e eb l e Ｇ ３５００t he rmo Ｇme chan i c a ls imu l a Ｇ t o ra tf ourd i f f e r en tde f o rma t i ont empe r a t ur e s( ５５０, ６００,６５０,and７００ ℃ )andf i ved i f f e r en ts t r a i nr a t e s －１ ( , , , , ) , ０ ００１ ０ ０１ ０ １ １ and１０s r e spe c t i ve l y． Thespe c imens we r ecomp r e s s edt oas t r a i no f０ ８ andhe a t e dt ot hed e s i e df o rmi ngt empe r a t ur ea ta gn －１ , he a t i ngr a t eo f１０℃ s t he nhe l df o r３mi nt oe l imi Ｇ n a t et het he rma lg r ad i en tbe f o r ede f o rma t i on．The compa r edt hed i f f e r enc ei nf l ow s t r e s sc o r r e c t edby f l ows t r e s scur ve swe r ea c i r edau t oma t i c a l l i ng qu yus [ [ １２] １３] , Evansand Scha rn i ng and Meng e ta l and t het he rmo Ｇme chan i c a ls imu l a t o r．Toob t a i nt hede Ｇ co r r e c t edt hes t r e s so ft head i aba t i che a t i nge f f e c t． f o rmedmi c r o s t r u c t u r e,t hes e c ime n swe r eimme d i a t e Ｇ p [ ] [ ] L i ne ta l １７ a nd Zh a ng e ta l １８ c o r r e c t e dt h ef l ow l nco l dwa t e ra f t e rcomp r e s s i on． yquenchedi s t r e s sba s edont her e su l t sr e r t e di nRe f [ １４]．The po ２ Re s u l t sandDi s c u s s i on abov e Ｇmen t i oned me t hodsr equ i r eaf r i c t i onf r a c t i on t oco r r e c tt hef l ow s t r e s s．Howeve r,t hef r i c t i on ２ １ Ana l s i so ff l ows t r e s sa ndmi c r o s t r u c t u r a le v o l u t i o n y [ １９] f r a c t i onva r i e swi t hs t r a i n andi sd i f f i cu l tt ocon Ｇ F i ２showst hef l ows t r e s s e so fmed i um c a r Ｇ g f i rma c cu r a t e l r e f o r e,i ti sne c e s s a r oc o r r e c t bons t e e ldur i ng wa rm de f o rma t i on．Theva l ue so f y．The yt t hes t r e s scu r vei nt e rmso fde f o rma t i onhe a t i ngand t hes t r e s s e swe r eobv i ous l f f e c t edbyt hede f o rma Ｇ ya i n t e r f a c i a lf r i c t i onwh i l ec ons i de r i ngt hes t r a i n． t i ont emp e r a t u r eands t r a i nr a t e．Thes t r e s si n c r e a s e d Thes t udya imst oempha s i z et hene c e s s i t f wi t hi nc r e a s i ngs t r a i nr a t eandde c r e a s i ng de f o rma Ｇ yo , f l ows t r e s sc o r r e c t i onsf o rmed i umc a rbons t e e land t i ont empe r a t ur e．I n gene r a l t heva r i a t i onsi nt he t oe s t ab l i shac ons t i t u t i ver e l a t i ont ode s c r i bewa rm f l ows t r e s swi t ht empe r a t ur eands t r a i nr a t ewe r ea t Ｇ de f o rma t i onbehav i o rbyc ons i de r i ngt hec ompens a Ｇ t r i bu t edt ot hee vo l u t i ono fd e f o rma t i onme c han i sms． t i ono ft hes t r a i nr a t e． Att he beg i nn i ng o f de f o rma t i on,t hef l ow s t r e s si nc r e a s edr ap i d l e a ched a max imum; y andr
１ Exp e r imen t a l
Theexpe r imen t a lma t e r i a lwa sa med i um c a r Ｇ bons t e e lwi t hchemi c a lc ompo s i t i onso fC０ ４６,S i ０ ２３,Mn０ ７２,P０ ０３,S０ ０３,Ni０ ０２,Cr０ ０２, Cu０ ０３,and Feba l anc e( i n ma s s％ )．Cy l i nd r i c a l
t hent heme chan i sm o fso f t en i ngde c r e a s edwi t hi n Ｇ c r e a s i ngt empe r a t ur eandde c r e a s i ngs t r a i nr a t e．The
spe c imenswi t had i ame t e ro f１０mmandahe i to f gh １５ mm we r ema ch i nedf r om aho tr o l l edba rwi t ha d i ame t e ro f１３ mm．Thei n i t i a l mi c r o s t ruc t u r ei s showni nF i １,wh i chshowst ha tt heda rka r e ai s g a r l i t e( P)andt hewh i t ea r e ai sf e r r i t e( F)．The pe , me ang r a i ns i z ei sabou t２０μm me a su r edbyt hei n Ｇ t e r c ep tme t hod．Wa rmcompr e s s i ont e s t swe r econ Ｇ
F i １ I n i t i a lmi c r o s t r u c t u r eo fme d i umc a r bon g s t e e lf o rc omp r e s s i ont e s t
F i ２ F l ows t r e s so fme d i umc a r bons t e e lme a s u r e d g du r i ngwa rmc omp r e s s i on
９４２
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
s t r e s scu r ve sf o rah i t r a i nr a t eandl owt empe r a Ｇ ghs t ur eexh i b i t ed add i t i ona lf l ow s o f t en i ng a f t e rt he aks t r e s s．Th i si si nc on t r a s t wi t ht he gene r a l pe e f f e c tru l eo ft empe r a t u r eands t r a i nr a t eont hef l ow s t r e s s． I ti sc aus edbyt he mi c r o s t ruc t u r a lva r i a t i on o f med i um c a rbon s t e e l．No rma l med i um c a rbon s t e e li sc ompo s edo ftwo pha s e s:po l lf e r r i t e ygona andl ame l l a rc emen t i t e．Du r i ngde f o rma t i on,t hel a Ｇ [ ] me l l a rc eme n t i t eb e nd s,f r a c t u r e so rd e c ompo s e s２０ , wh i l et hef e r r i t er e c ove r so rr e c r t a l l i z e s．The s e ys va r i a t i onsi nt hetwopha s e sl e adt ot hede c r e a s eo f , t hef l ows t r e s s．Atah i t r a i nr a t e dynami cr e Ｇ ghs cove r l lo c cu ri nf e r r i t eandt hec emen t i t e wi l l y wi bendo rf r a c t u r e．Th i si sj us tano t he rc emen t i t edy Ｇ
nami csphe r o i d i z a t i onme chan i smt ha ti sdemons t r a Ｇ [ ] t edbyRa s t ega r ie ta l ２１ ． I tc aus e st hes t r e s st ode Ｇ c r e a s ebeyondt hepe aks t r e s swi t hah i t r a i nr a t e． ghs , Atal ows t r a i nr a t et hedynami cr e c r t a l l i z a t i ono f ys f e r r i t eoc cur r ed,andc emen t i t ede compos ed,f o rmＧ i ngs eve r a lsma l lpa r t i c l e s．Thepa r t i c l e swou l dp i n t hed i s l oc a t i onso rg r a i nbounda r i e sdur i ngde f o rma Ｇ t i on．Thepa r t i c l e sp l ayedt her o l eo fs t r eng t hen i ng, andhenc e,t heso f t en i ngo ft hes t r e s swa sno tobv i Ｇ , ous t houghdynami cr e c r t a l l i z a t i ono ff e r r i t eoc Ｇ ys cur r eda tal ow s t r a i nr a t e．Cemen t i t ef r a c t ur eand ( de compos i t i on we r e obs e r ved s i ng l ea r r ow i n F i ３( a)and３( b)),andf e r r i t edynami cr e c r t a l Ｇ gs ys l i z a t i onoc cur r ed ( doub l ea r r owi nF i ３( b))． g
( a)６００°C,１０s－１; ( b)６００°C,０ ０１s－１．
F i ３ Cemen t i t ef r a c t u r eandd e c ompo s i t i ona td i f f e r en ts t r a i nr a t e s g
２ ２ Fr i c t i onc o r r e c t i ono ff l ows t r e s sc u r v e As chema t i cd i ag r am o ft he de f o rma t i oni s showni nF i ４． g
( a)Be f o r ec omp r e s s i on; ( b)Af t e rc omp r e s s i on．
F i ４ S ch ema t i cd i a r amo fs e c imenc omp r e s s i ont e s t g g p
Du r i ngt hep l a s t i cde f o rma t i on,f r i c t i oni suna Ｇ vo i dab l aus edbyt her e l a t i ve mo t i onbe twe ent he yc spe c imensandd i e s．Thef r i c t i onbe twe ent hespe c i Ｇ mensandanv i l swou l di n f l uenc et hede f o rma t i ono f me t a l s． I nt h i sexpe r imen t,t hef r i c t i onbe twe enspe c i Ｇ
mensandanv i l sa f f e c t edt hes t r i c a lde f o rma Ｇ ymme t i ono ft hespe c imen．Thes t r a i n Ｇ s t r e s sda t ar e co rded byt het e s t i ngs t emcon t a i nedt hei n f l uenc eo ff r i c Ｇ ys , t i on．Thus t hef l ows t r e s sa c i r edbyt e s t i ngsys Ｇ qu t e rmshou l dbe mod i f i edbye l imi na t i ngt hef r i c t i on i n f l uenc e．Thef l ows t r e s sc anbemod i f i edus i ngt he [ ] me t hodr e r t e di nRe f s １２ Ｇ １４ ．Thef o l l owi nge a Ｇ po qu t i on shavebe enus edi nRe f [ １２]: σ０ ( σ＝ １) １＋ ( ２/３ ３) m( R０/h０) exp( ３ ε/２) andi nRe f [ １３]:
σ０ ( σ＝ ２) １＋ ( ２mRe/３ ３h) whe r e, σi st hef r i c t i onco r r e c t ed; σ０i st heme a sur ed ; ; f l ows t r e s s e s εi st het rues t r a i n mi st hef r i c t i ona l coe f f i c i en t;h０i st hei n i t i a lhe i to fspe c imens;hi s gh t hespe c imenshe i i ngde f o rma t i on;Rei san ghtdur equ i va l en tr ad i us;andR０i st hei n i t i a lr ad i us． [ ] Ebr ah imiand Na a f i z adeh１４ expr e s s edt hei n Ｇ j t e r f a c i a lf r i c t i onco r r e c t edequa t i ona sf o l l ows:
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
σ０C２ ( σ＝ [ ( ) ３) ２ exp C －C －１] whe r e,C i st hec o r r e c t i onc oe f f i c i en to fi n t e r f a c i a l , : f r i c t i on wh i chi sexp r e s s eda s ( C ＝２m/h ４) ０ ５ ( Re＝R０/( ５) h０/h) [ , ] , I nRe f s １２ １３ m wa sc ons i de r edt obe０ ２． I nRe f [ １４],m wa seva l ua t edus i ngt hef o l l owi ng : expr e s s i on ( Re/h) b ( ６) m ＝ ( ２ ４/ ３)－ ( b/３ ３) ΔRh b＝４ ReΔh ΔR ＝R M －RT
( ７) ( ８)
h０ ( ９) RT ＝ ３ R２ R２M ０ －２ h ( Δh＝h０ －h １０) whe r e,R M i st he max imumr ad i us;RT i st hehe ad r ad i us;andbi st hebu l kc oe f f i c i en t． F i ５showst ha tt hemax imum d i ame t e rchan Ｇ g sdu r i ngt hec omp r e s s i ont e s t;henc e,i ti sne c e s Ｇ ge s a r oc o r r e c tt hef r i c t i onus i ngt her e a l Ｇ t imeR M yt andt hehe i tdu r i ngde f o rma t i on． gh Du r i ngt hepr o c e s so fde f o rma t i on,t hei n t e r f a Ｇ c i a lf r i c t i oni sa f f e c t edbyt hede f o rma t i on．Ther e a l Ｇ t ime ma x imum r a d i u s wa s me a s u r e du s i nga Gl e e b l e Ｇ ３５００C ＧGauges ens o r．Theva l ueo fR M i sshowni n F i ５． I tc anbes e ent ha tt hed i f f e r enc eo fR M unde r g , d i f f e r en tc ond i t i onsi si ns i i f i c an t． I nt h i sc a s et he gn e f f e c t so ft hef o rmi ngt empe r a t u r eands t r a i nr a t eon RM c anbeneg l e c t ed．
９４３
F i ５ Va r i a t i oni nRM du r i ngd e f o rma t i on g
anbec a l Ｇ Ac co rd i ngt oEq ( ６),t heva l ueo fm c cu l a t edus i ngR M andh．Theva l ue so fC andm a r e showni nF i ６．Theva l ueo fC i nc r e a s ed wi t ht he g sshowni n s t r a i n．Thef i t t i ngf o rmu l af o rC andm i n i t i a l l n Ｇ Eq s ( １１)and ( １２)．Theva l ueo fm i yi c r e a s edt oape akva l ue ( ０ ２５)t henr ema i nedcon Ｇ , s t an t．Af t e rt ha ti tde c r e a s eds l i t l t hi nc r e a s Ｇ gh y wi i ngs t r a i n．Atas t r a i no f０ ８,t heva l ueo fm wa s ０ １９andshowedl i t t l ed i f f e r enc ewi t ht heva l ueus ed [ , ] i nRe f s １２ １３ ．Theconc en t r a t i ono fl ubr i c an tde Ｇ c r e a s edwi t hi nc r e a s i ngs t r a i n,andt hef r i c t i ona lco Ｇ [ ] e f f i c i en ti nc r e a s ed１９ ．Thei nc r e a s eo ff r i c t i ons t r e s s wa sc aus ed by t hei nc r e a s e si n bo t hf r i c t i on and s t r e s swi t hs t r a i n．Onc et hef r i c t i ons t r e s sexc e eded , t hey i e l ds t r e s so ft hespe c imen t her e l a t i veg l i de be twe ent hespe c imensandd i e ss t opped．Thec l i n Ｇ y de rp r o f i l ei sshowni nF i ７,andt hef r i c t i ona lcoe f Ｇ g ( ) f i c i en tob t a i nedbyEq １２ de c r e a s ed．
( a)Re l a t i onbe twe enC andε; ( b)Re l a t i onbe twe enm andε．
F i ６ Re l a t i onb e twe enC,m ,andε g
( C ＝－０ ６５４ ε－０ １８０ ε２ １１) ２ ３ m ＝－０ １０９－５ １６２ ε－３１ １５５ ε －９８ ４９４ ε－ ( １６８ ７１９ ε４ －１４６ ６０２ ε５ －５０ ５５７ ε６ １２) Thec o r r e c t edr e su l t sus i ng d i f f e r en t me t hods a r eshowni nF i ８．Thed i f f e r enc ebe twe enc o r r e c Ｇ g
t edandunco r r e c t edcur ve sg r adua l l nc r e a s ed wi t h yi [ １６] s t r a i n ． I ti squ i t econs i s t en twi t ht hea c t ua ls i t ua Ｇ t i ono facomp r e s s i ont e s t．Atas t r a i no f０ ８,t he s t r e s sco r r e c t edby Eq ( ３)wi t ht her e a l Ｇ t ime R M wa sequa lt ot heone sco r r e c t edbyEq ( ３)wi t ht he
９４４
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
F i ７ S imp l er ep r e s en t a t i ono fup t u rn i ng g
F i ８ Compa r i s ono fme a s u r e dandc o r r e c t e dc u r v e s g
f i na lR M ．Bu tf o rt hewho l ep r o c e s s,t hes t r e s sco r Ｇ ( ) r e c t edby Eq ３ wi t ht her e a l Ｇ t imeR M wa s mo r e cons i s t en twi t ht her e a ls t r e s st h a nt h a tw i t ht h ef i n a l
R M ．The pr ed i c t edf r i c t i on l e s ss t r e s sc a l cu l a t ed by Eq ( ２)wa sl e s st hant ha tc a l cu l a t edbyEq ( １)．The s t r e s sc a l cu l a t edbyEq ( １)i sc l os et ot ha tob t a i ned ( ) byEq ３ wi t ht her e a l Ｇ t imeR M ．Thef r i c t i onf a c t o r wa scons i de r edt obe０ ２i nRe f s [ １２, １３],bu tQi an [ ] e ta l １０ cons i de r edt ha tt hef r i c t i onf a c t o rwa s０ ３． , I nf a c tt hef r i c t i onf a c t o rva r i eddur i ngt hef o rmi ng． Fo r t una t e l hef r i c t i onf a c t o rc anbededuc edf r om y,t Eq ( ６)by me a sur i ngt her e a l Ｇ t imeva l ueo fR M ． ２ ３ An a l s i so ft emp e r a t u r ev a r i a t i o nd u r i n e f o rma t i o n y gd Mos to ft heene r st r ans f o rmedi n t ohe a t gy wa wh i l eaf r a c t i ono ft heene r ss t o r edi nt hespe c Ｇ gywa [ ２２] imensi nt hef o rm o fd i s t o r t i on ene r ft he gy ．I , s t r a i nr a t ei sh i ri tc aus e sani nc r e a s ei ns t r e s s ghe andade c r e a s ei nt hede f o rma t i ont ime;henc e,mos t o ft hede f o rma t i onhe a tc anno tber emovedf r omt he spe c imens．Thes amp l et empe r a t ur et heni nc r e a s e s duet ot hede f o rma t i on he a t i ng．I nt h i sc a s e,t he t empe r a t ur eva r i a t i onmaybes i i f i c an t． gn , Fo rt hecomp r e s s i ont e s t s t het empe r a t ur ei s showni nF i ９．Thet empe r a t ur ei nc r e a s edwi t hi n Ｇ g c r e a s i ngs t r a i nandde c r e a s i ngde f o rma t i ont empe r a Ｇ t ur e．Fur t he rmo r e,t het empe r a t ur eva r i a t i ona ta s t r a i nr a t eo f０ １s－１ wa sve r l landcou l dbei Ｇ ysma g
F i ９ Cu r v e so ft emp e r a t u r ev a r i a t i ondu r i ngd e f o rma t i on g [ ] no r ed１６ ,a sshowni nF i ９( b)． g An Ar rhen i us Ｇ t t i oni sa lwaysus edt o ypeequa de s c r i bet het he rma lde f o rma t i onbehav i o ro ft hea l Ｇ l oys．Thet empe r a t u r eva r i a t i ondu r i ngde f o rma t i on i sco r r e c t edbyus i ng Ar rhen i usequa t i onandZene r Ｇ [ ] Ho l l omi npa r ame t e ra sf o l l ows２３ : æ ö çQ ÷ ( ÷÷ Z ＝̇ １３) εexpçç R è Tø ( Z ＝A１σn１ ( α σ＜１ ２) １４) ( ( Z ＝A２exp( σ) α σ＞１ ２) １５) β n ( ) ( ) ( Z ＝As i nhα σ Fo ra l lσ １６) whe r e,A１ ,A２ ,A ,n１ ,n andβ a r e ma t e r i a l spa Ｇ
̇ r ame t e r s;αi st hes t r e s smu l t i l i e r( α＝β/n１ ); s εi p ; t hes t r a i nr a t e Qi st heappa r en ta c t i va t i onene r gy; Ri st heun i ve r s a lga scons t an t;andT i st heabso Ｇ l u t et empe r a t ur e． By subs t i t u t i ng Eq ( １３)i n t o Eq s ( １４)and ( １５)andt hent ak i ngt hel oga r i t hmandr e a r r ang i ng Eq s ( １４)and ( １５),t hef o l l owi ngequa t i onsc anbe ob t a i ned: é æ öù ê ΔT ç Q ÷ú ê x ç ÷ ú ( Δσ＝σ e α σ＜１ ２) ( １７) pç êê n T ＋ΔT )÷ø úúû ë è １RTT ( Q æç ΔT ö÷ ÷ Δσ＝ çç ＋ΔT ÷ø è βR T
( α σ＞１ ２) ( １８)
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
Ac c o r d i ngt oRe f [ ２４],αi sabou t０ ００３,and i ti se a s oge tt ha ti ft hes t r e s si sg r e a t e rt han３５０ yt MPa．Eq ( １７)i ssu i t ab l ef o rde f o rma t i onc ond i t i on －１ －１ －１ , , , sucha s７００ ℃ １０s o r１s and６５０ ℃ １s ． Ot he rwi s e,Eq ( １８) i ssu i t ab l ef o rde f o rma t i onconＧ d i t i onss u c ha s５５０℃ ,１０s－１o r１s－１ ,６００℃ ,１０s－１ o r１s－１ ,and６５０ ℃ ,１０s－１ ．Thepa r ame t e rc a l cu l a Ｇ t i onpr o c e s si sshowna sf o l l ows． Ba s edont he me a su r edda t a,as t r a i no f０ ４ wa st aken a s an c a s et o de s c r i bet he c a l cu l a t i on oc e s so ft hema t e r i a lc ons t an t s．Us i ngt hena t ur a l pr ( ) ( ) , l oga r i t hmf o rEqs １４ and １５ t hef o l l owi ngr e Ｇ l a t i onc anbeob t a i ned: ̇ ( l n nA１ ＋n１l σ ε＝l n １９) ̇ ( ε＝l nA２ ＋β σ ２０) l n Then,subs t i t u t i ng t hef l ow s t r e s so fs t r a i n , ０ ４ wh i ch wa s ob t a i ned du r i ng t he c omp r e s s i on t e s t,ands t r a i nr a t ei n t oEq s ( １９)and ( ２０),i tc an tt her e l a t i onbe twe ent hef l ow s t r e s sands t r a i n ge , r a t ej us ta sshowni nF i １０． g ̇ εandl I ti sobv i oust ha tt hel i ne so ft heσＧ l n n σＧ
̇ l n εc anbec ons i de r edt obepa r a l l e l,andt hen１andβ
９４５
equa lt ot hes l opo ft hepa r a l l e ll i ne s,r e spe c t i ve l y． Cons i de r i ngt hee f f e c to ft empe r a t ur eonn１ andβ, t a i nedbyl i ne a rf i t t i ng t heme anva l ue so fn１andβob －１ , me t hoda r e１０ ５６７and０ ０３５ MPa r e spe c t i ve l y． The r e f o r e,αi scons i de r edt obe０ ００３３． Fo ra l lt hes t r e s sl eve l s,Eq ( １５)c anbed i f f e r Ｇ , en t i a t edandt hus d l n[ s i nh( α σ)] ( ２１) Q ＝R n d( １/T ) ̇ d( l n ε) ( n＝ { [ ( )]} ２２) σ dl ns i nhα Bysubs t i t u t i ngt het empe r a t ur e s,s t r a i nr a t e s ( ) ( ) ands t r e s s e s a tas t r a i no f０ ４ i n t oEq s ２１ and ( ２２),t her e su l t s we r eob t a i nedanda r eshowni n
F i １１． g Theva l ue so fn and Q a r ec a l cu l a t edt o be ７ ６７７and３８４ ８１０kJmo l－１ ,r e spe c t i ve l y． Thed a t awi t hc o r r e c t e dt emp e r a t u r e sa r es howni n F i １２．F i １２( a)showst her e su l t sco r r e c t edf o r g g t empe r a t ur e．Thed i f f e r enc ebe twe enr e su l t sco r r e c Ｇ t edbyEq s ( １７)and ( １８)i sve r l i t;t hus,t he ys gh e f f e c to fus i ng d i f f e r en tequa t i onsi sneg l i i b l e．I n g t h i sr e s e a r ch,Eq ( １８)i sus edt oco r r e c tf o rt hei n Ｇ
̇ ̇ ( a)l n σＧ l n ε; ( b)σＧ l n ε．
F i １０ Re l a t i onb e twe ent h es t r a i nr a t eandf l ows t r e s s g
F i １１ Ca l c u l a t i on so fQ andn g
９４６
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
( a)１０s－１; ( b)Compa r i s onbe twe enme a sur edandc o r r e c t eds t r e s s e s．
F i １２ Cu r v e so fs t r e s sc o r r e c t e df o rd e f o rma t i onh e a t i ng g
f l uenc eo fde f o rma t i onhe a t i ng． F i １２( b)showst he g compa r i s onbe twe ent h et emp e r a t u r ec o r r e c t e ds t r e s s ( ) ba s edonEq １８ andt he me a su r eds t r e s swi t hi n Ｇ c r e a s i ngt empe r a t u r ef r om５５０t o７００ ℃ du r i ngde Ｇ f o rma t i onwi t has t r a i nr a t eo f０ ０１s－１ ．Thec o r r e c Ｇ t eds t r e s sva l ue sc o i nc i de we l l wi t ht he me a sur ed s t r e s s．
２ ４ Con s t i t u t i v ee t i on so ff l ows t r e s sf o rme d i um qua c a r bons t e e l Ba s edont hes t r e s sc o r r e c t edf o rf r i c t i onand de f o rma t i on he a t i ng,t he p r o c e s so fe s t ab l i sh i ng cons t i t u t i veequa t i onsi ss imi l a rt os e c t i on２ ２． Ac c o r d i ngt ot hec a l cu l a t i onr e su l t s,t hev a l u e s o fα,n１ a ndβa r e０ ００３５,９ ２９４,a nd０ ０３３ MPa－１ ,
r e spe c t i ve l l ue so fn and Q a r e６ ６７３and y．Theva ３５３ ５６５kJmo l－１ ,r e spe c t i ve l y． Tak i ngt hel oga r i t hmo fbo t hs i de so fEq ( １６),
t hef o l l owi ngequa t i onc anbeob t a i ned ( l nZ ＝l nA ＋nl n[ s i nh( α σ)] ２３) , Ba s edont heexpe r imen t a lr e su l t st heva l ue so f r et hei n t e r c ep tands l opeo fonl nAＧ l n l nA andn a [ s i nh( α σ)]p l o t,r e spe c t i ve l heva l ueo fA y．Thus,t １９ c anbee a s i l a l cu l a t edt obe１ ３３８×１０ ． yc I ns imi l a r ways,t heva l ue so f ma t e r i a lcon Ｇ Q ,A ,n,andα)o ft hec ons t i t u t i veequa t i on s t an t s( we r ec ompu t edunde rd i f f e r en tde f o rma t i ons t r a i ns wi t h i nt her angeo f０ ０５－０ ８０i ni n t e r va l so f０ ０５． Ther e l a t i onbe twe enQ ,A ,n, α,andt hes t r a i nf o r , med i um c a rbons t e e lc an be po l a lf i t t ed a s ynomi showni nF i １３． g Asc anbes e eni nF i １３( a),t hea c t i veene r g gy , i n i t i a l l c r e a s edwi t hs t r a i nr e a chedi t smi n imum yde va l uea tas t r a i no f０ ４５, i n c r e a s e dupt o３６２ ８６k J mo l－１ a tas t r a i no f０ ７,andf i na l l d e c r e a s e dt o y －１ ３５７ ３２kJ mo l a tas t r a i no f０ ８．Theva l ue sa r e l e s st hant ha tr epo r t edf o rmed i umc a rbons t e e lwi t h
F i １３ Va r i a t i on so fQ ( a),α ( b),n ( c),and g A( d)wi t hs t r a i nba s e donpo l a lf i t t i ng ynomi
I s sue９ Cons t i t u t i veMode lo fWa rm De f o rma t i onBehav i o ro fMed i um Ca r bonS t e e l
ama r t ens i t emi c r o s t ruc t u r e２４ ．Th i sr e f l e c t st hef a c t t ha ta i rc oo l ed med i um c a rbon s t e e lha sa be t t e r wo rkab i l i t hanquenchedmed i umc a rbons t e e l．Ac Ｇ yt ( ) , co rd i ngt oF i １３ b t heva l ueo fα i n i t i a l l Ｇ g y de c r e a s edt ot hemi n imumva l uea tas t r a i no f０ １,and i nc r e a s eds t e ad i l t hi nc r e a s i ngs t r a i n, l i ngi nt he ywi y －１ r ange o f０ ００３３ － ０ ００３８ MPa ．Re f e r r i ng t o F i １３( c),t heva l ueo fni n i t i a l l c r e a s edwi t hi n Ｇ g yde c r e a s i ng s t r a i n,and r e a ched t he mi n imum,t hen s l i t l nc r e a s ed．Theva l ue so ft he s epa r ame t e r so f gh yi t hecons t i t u t i vemode la r es imi l a rt ot heda t ar epo r Ｇ [ ] t edbyL ie ta l ２４ ． anbe Ac c o r d i ngt oEq ( １６),t hef l ows t r e s sσc opo s eda saf unc t i ono ft heZene r ＧHo l l omon pa Ｇ pr , r ame t e r．The r e f o r et hep r opo s edc ons t i t u t i vemode l c anbesumma r i z eda sf o l l ows: [ ]
ö２ n éæ ù１ ２ üï ê çZ ÷ ú １ ìïï æçZ ö÷１ n ï ( níïï çç ÷÷ ＋ êêê çç ÷÷ ＋１úúú ýïï σ＝ l ２４) α î èA ø ë èA ø û þ Q＝３７３ ６５９＋１９０ ９２２ ε＋５６２ ３７０ ε２－１２３ ９５６ ε３－ /
/
/
９４７
ε４ －６６２８ ε５ －３ ３６７０ ８５５ ４７５ ６４６×１０３ε６ ( ２５)
００４＋０ ０２１ ε＋０ １５８ ε２ ＋０ ε３ ＋ α＝０ ５６６ ( １ ０４２ ε４ ＋０ ９４８ ε５ ＋０ ３３７ ε６ ２６) ２ ３ n＝１１ ２１３＋７１ ５０１ ε＋４７７ ５３５ ε ＋１５８９ ２１８ ε＋ ( ２７４６ ６３４ ε４ ＋２３５４ ４７２ ε５ ＋７９１ ０８２ ε６ ２７) A ＝６ １３０×１０１９ ＋５ ５６×１０２０ε＋１ ５０×１０２０ε２ － ２２ ３ ２２ ４ １ ２４６×１０ ε －４ ７６×１０ ε －６ ７１× ( １０２２ε５ －３ ２９×１０２２ε６ ２８)
２ ５ Ve r i f i c a t i ono fde v e l ope dc on s t i t u t i v ee t i on s qua Tot e s tt heve r a c i t ft hedeve l opedcons t i t u Ｇ yo t i veequa t i on,t hec a l cu l a t edf l ow s t r e s sda t a wa s compa r ed wi t h me a sur emen t sove rt heen t i r et emＧ r a t ur eands t r a i nr a t er ang ei nv e s t i a t e d,a ss hown pe g i nF i １４． g Thec a l cu l a t edf l ows t r e s s e sp r ov i deagoodap Ｇ r ox ima t i ont ot heco r r e c t edva l ue s．Toeva l ua t et he p a c cur a c ft hede f o rma t i onc on s t i t u t i v ee a t i on,t he yo qu
F i １４ Compa r i s onb e twe enf r i c t i on Ｇ c o r r e c t e dandc a l c u l a t e df l ows t r e s s e s g
e r r o ra nda v e r a ea b s o l u t er e l a t i v ee r r o r( AARE),be Ｇ g twe ent hec a l cu l a t edf l ow s t r e s s( σc )andf r i c t i on co r r e c t edf l ows t r e s s( σe)c a nb ec a l c u l a t e da sf o l l ows: N
R２ ＝
∑( Ei －E )( Pi －P )
i＝１ N
N
∑( Ei －E )∑ ( Pi －P )
i＝１
２
i＝１
( ２９)
１ N Ei －Pi ( Rave＝ ∑ ×１００％ ３０) Ni＝１ Ei whe r e,E i st heexpe r imen t a lda t a;P i st hec a l cu Ｇ l a t edva l uebyt hedeve l opedcons t i t u t i veequa t i ons; r et heme anva l ue so fE andC ,r E andPa e spe c t i ve Ｇ ; l st henumbe ro fda t a． y andN i Ac co rd i ngt oEq s ( ２９)and ( ３０),t hemax imum
９４８
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２３
R２ wa s０ ９９８ and t he AARE wa s０ ５４７％ ,a s showni nF i １５．Th i si nd i c a t e st ha tt he p r opos ed g cons t i t u t i veequa t i onp r ov i de sana c cu r a t ep r ed i c t i on o ft hes t r e s sf o rmed i umc a rbons t e e l．
F i １５ Co r r e l a t i onb e twe enc o r r e c t e dandp r e d i c t e d g f l ows t r e s sdu r i ngc omp r e s s i ont e s t
３ Conc l u s i on s ( １)Thef l owcu r ve sa th i rs t r a i nr a t e sex Ｇ ghe h i b i t edanadd i t i ona ls o f t en i ngt endenc t i t e y．Cemen f r agmen t swe r eobs e r vedt ode f o rmi nt hes amp l e sa t , h i h e rs t r a i nr a t e s a n df e r r i t ed n a m i cr e c r s t a l l i Ｇ g y y z a t i ona swe l la sc emen t i t ede c ompo s i t i on we r eob Ｇ s e r ved． ( ２)Thef l ow s t r e s s wa ss i i f i c an t l n f l u Ｇ gn yi enc edbyde f o rma t i onhe a t i ngandf r i c t i on．Thede Ｇ f o rma t i onhe a t i ngc o r r e c t ed me t hodve r i f i edbyex Ｇ r imen t a lr e su l t sexh i b i t edagooda c cu r a c nc a l Ｇ pe yi , cu l a t i ngt heva r i a t i oni ns t r e s s．Thus anewf r i c t i on me t hod c ons i de r i ng s t r a i n wa se s t ab l i shed．The e f f e c to fs t r a i nont hef r i c t i onf a c t o rwa sobv i ous． ( ３)Ther e l i ab i l i t ft hec o r r e c t edc ons t i t u t i ve yo mode lwa seva l ua t edbys t a t i s t i c a lana l i s．Theco r Ｇ ys r e c t edc on s t i t u t i v emod e lp r o v i d e da na c c u r a t ee s t ima t e o ft hes t r e s s．The r e f o r e,t hedeve l opedc ons t i t u t i ve equa t i onsa r er e l i ab l ef o rf i n i t ee l emen tana l i sf o r ys wa rmde f o rma t i onc ond i t i ons．
Re f e r enc e s: [ １] YC．L i n,X M．Chen,Ma t e r．De s． ３２ ( ２０１１)１７３３ Ｇ １７５９． [ ２] P．Wan a r a,M．J aha z i,H．Mona a t i,S．Yue,J P． Imma r i Ｇ j j t e r． Sc i．Eng．A３９６ ( ２００５)５０ Ｇ ６０． geon,Ma [ ３] A．Momen i,K．Dehghan i,GR． Ebr ah imi,S．Ka z emi,Me t a l l． Ma t e r．Tr ans．A４４ ( ２０１３)５５６７ Ｇ ５５７６． [ ４] H．Mi r z adeh,A．Na a f i z adeh,I S I JI n t． ５３ ( ２０１３)６８０ Ｇ ６８９． j [ ５] C．J．Sh i,WM．Mao,X G．Chen,Ma t e r．Sc i．Eng．A ５７１ ( ２０１３)８３ Ｇ ９１． [ ６] F．Yi n,L．Hua,HJ．Mao,X H．Han,D S．Qi an,R． Zhang, Ma t e r．De s． ５５ ( ２０１４)５６０ Ｇ ５７３． [ ７] Y C．L i n,K K．L i,HB．L i,J．Ch e n,X M．Ch e n,D X．We n, Ma t e r．De s． ７４ ( ２０１５)１０８ Ｇ １１８． [ ８] R．Goe t z,S．Semi a t i n,J．Ma t e r．Eng．Pe r f o rm．１０ ( ２００１) No ６,７１０ Ｇ ７１７． [ ９] L．L i,J．Zhou,J．Dus z c z t e r．Pr o c e s s．Te chno l． １７２ yk,J．Ma ( ２００６)No ３,３７２ Ｇ ３８０． [ n t．J．Pr e s s． X．Wang,I １０] L Y．Qi an,G．Fang,P．Zeng,L Ve s s e l sP i １２３ Ｇ １３３ ( ２０１５)４３ Ｇ ４５． p． [ １１] GW．L i u,Y．Han,Z Q．Sh i,J P．Sun,D N．Zou,GJ． Qi ao,Ma t e r．De s． ５３ ( ２０１４)６６２ Ｇ ６７２． [ t e r． Sc i．Te chno l． １７ ( ２００１) １２] R W． Evans,P J． Scha r n i ng,Ma No ８,９９５ Ｇ １００４． [ R．Ni e,Ma t e r． L．L i,HM．L i,H．Huang,Z １３] G．Meng,B Sc i．Eng．A５１６ ( ２００９)１３１ Ｇ １３７． [ １４] R．Ebr ah imi,A．Na a f i z adeh,J．Ma t e r． Pr o c e s s．Te chno l． １５２ j ( ２００４)No ２０,１３６ Ｇ １４３． [ １５] A．Gho l amz adeh,A．Ka r imiTahe r i,Me ch．Re s． Commun i c a Ｇ t i ons３６ ( ２００９)No ２,２５２ Ｇ ２５９． [ １６] MA．Mo s t a f a e i,M．Ka z emi ne zhad,Ma t e r．Sc i．Eng．A５３５ ( ２０１２)２１６ Ｇ ２２１． [ １７] YC．L i n,YC．Xi a,X M．Chen,MS．Chen,Compu t．Ma Ｇ t e r． Sc i． ５０ ( ２０１０)No ４,２２７ Ｇ ２３３． [ J． C． Zhang,T Y．Wang,C．Yu,J １８] J Q． Zhang,HS．Di,X Ma,Ma t e r．De s． ４４ ( ２０１３)２５４ Ｇ ３６４． [ １９] YP．L i,H．Ma t sumo t o,A．Ch i ba,Me t a l l．Ma t e r．Tr ans．A ４０ ( ２００９)No ４,９８２ Ｇ ９９０． [ ２０] L．S t o r o eva,D．Ponge,R．Ka spa r,D．Ra abe,Ac t a Ma t e r． j ５２ ( ２００４)２２０９ Ｇ ２２２０． [ ２１] H．Ra s t ega r i,A．Ke rmanpur,A．Na a f i z adeh,D．Po r t e r,M． j Soman i,J．Al l oysComp． ６２６ ( ２０１５)１３６ Ｇ １４４． [ ２２] C X．Wang,F X．Yu,D Z． Zhao,X．Zhao,L．Zuo,Ma t e r． Sc i．Eng．A５７７ ( ２０１３)７３ Ｇ ８０． [ ２３] C． Zene r,H．Ho l l omon,J．App l． Phys． １５ ( １９４４)No １,２２ Ｇ ３２． [ ２４] Q．L i,T S．Wang,HB．L i,YW．Gao,N．L i,T F． J i ng, J． I r onS t e e lRe s． I n t． １７ ( ２０１０)No ５,３４ Ｇ ３７．