Volume Issue 2 MODEL AND NUMERICAL ANALYSIS OF HARDENING PROCESS PHENOMENA FOR MEDIUM-CARBON STEEL

A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume 52 27 Issue 2 A. BOKOTA, A. KULAWIK MODEL AND NUMERICAL ANALYSIS OF HARDENING PROCESS PHENOMENA FOR MEDIUM-CARBON STEEL MODEL I ANALIZA NUMERYCZNA ZJAWISK PROCESU HARTOWANIA STALI ŚREDNIOWĘGLOWEJ Ths paper refers to numercal modelng of thermal phenomena, phase transformatons n sold state and mechancal phenomena occurrng durng hardenng process of steel element (C45). In the algorthm heat transfer equaton, equlbrum equatons and macroscopc model of phase transformatons bass of CCT dagrams are used. Couplng between basc phenomena of hardenng process s consdered, n partcular the nfluence of latent heat on the temperature, and also thermal, structural and plastc strans the transformaton nduced plastcty n the model of mechancal phenomena s taken nto account as well. Heat conductvty equaton s used to estmate the temperature feld n the process of heatng and coolng. Ths equaton s solved by fnte element method n G a l e r k n formulaton. Feld of stresses and strans are obtaned from solutons of fnte element method equatons of equlbrum n ncrement form. The nfluence of the temperature on materal propertes s also taken nto account. To calculate of plastc strans the H u b e r-m s e s condton wth sotropc enhancement s used. The method of calculatng the phase transformaton durng heatng appled by the authors uses data from the contnuous heatng dagram (CHT). Because start and fnsh of phase transformatons strongly depends on the rate of heatng or holdng n specfed temperature the dynamcs curves A c1 and A c3 are used. The homogenzaton lne of austente determnes the end of heatng. The volume fracton of austente durng hgh rate of heatng s determned by modfed K o s t n e n-m a r b u r g e r equaton. The volume fractons of phase that emerge durng coolng s determned by A v r a m equaton. The nfluence of austensaton temperature on the knetcs of transformatons s taken nto account. To calculate the ncrease of martenste content Kostnen-Marburger formula s used. The numercal model was mplemented n the Borland C++Bulder 5. Envronment. Usng presented model, smulaton of hardenng for the cubc steel element s made. The element was heated by superfcal source, and then cooled n water. The obtaned results confrmed correctness of developed model and numercal algorthms. Praca dotyczy modelowana numerycznego zjawsk ceplnych, przeman fazowych w stane stałym oraz zjawsk mechancznych towarzyszących procesom hartowana elementów ze stal średnowęglowej (C45). W algorytme wykorzystano równane przewodzena cepła, równana równowag oraz makroskopowy model przeman fazowych oparty na wykresach CTPc. Zostały uwzględnone sprzężena pomędzy podstawowym zjawskam procesu hartowana, a w szczególnośc: wpływ cepła przemany fazowej na temperaturę, a w modelu zjawsk mechancznych uwzględnono oprócz odkształceń termcznych, strukturalnych plastycznych równeż odkształcena transformacyjne. Do wyznaczana pól temperatury w procese nagrzewana chłodzena wykorzystano równane różnczkowe przewodzena cepła. Równane to rozwązano metodą elementów skończonych w sformułowanu G a l e r k n a. Pola naprężeń odkształceń uzyskano z rozwązana metodą elementów skończonych równań równowag w forme przyrostowej. Stałe termofzyczne uzależnono od temperatury. Odkształcena plastyczne wyznacza sę stosując warunek H u b e r a-m s e s a ze wzmocnenem zotropowym. W modelu przeman fazowych nagrzewana wykorzystuje sę dane z wykresu CTPa. Poneważ początek konec przemany slne zależy od prędkośc nagrzewana bądź wytrzymana w określonej temperaturze, zastosowano dynamczne krzywe A c1 A c3. Lna homogenzacj austentu determnuje konec nagrzewana. Do szacowana przyrostu fazy austentycznej przy dużych szybkoścach nagrzewana wykorzystano zmodyfkowany wzór K o s t n e n a-m a r b u r g e r a. Objętoścowe udzały faz, mających mejsce podczas chłodzena, szacuje sę wzorem A v r a m e g o. Przyrost udzału martenzytu wyznacza sę natomast zależnoścą K o s t n e n a-m a r b u r g e r a. Aplkacja będąca mplementacją modelu numerycznego została wykonana w środowsku Borland C++Bulder 5.. Wykorzystując opracowany model wykonano symulację hartowana kostk stalowej. Element ten nagrzewano powerzchnowo, a następne chłodzono wodą. Uzyskane wynk potwerdzają poprawność zbudowanego modelu algorytmów numerycznych. INSTITUTE OF MECHANICS AND MACHINE DESIGN, CZĘSTOCHOWA UNIVERSITY OF TECHNOLOGY, 42-2 CZĘSTOCHOWA, 73 ST. DĄBROWSKIEGO STR., POLAND

338 1. Introducton Phase transformatons of steel n sold state are basc element for technology of heat treatment. These processes are hard to expermental observaton and mathematcal descrpton. There s a large number of parameters whch have marked nfluence on the hardenng process concernng both methods of heatng and coolng. Sets of parameter have nfluences on hardenng determnate progress n the numercal method, whch are tools developments for modern process engneer. Rapd growth of processor capacty and creaton modern research nstruments, many types of numercal models of phase transformatons were developed. These methods can be dvded nto the followng types: a) macroscopc, based on Fe 3 -C and CCT dagrams, and b) mcroscopc, analyzed growth of ndvdual grans of phase on the bass of carbon dffuson and balance of energy [1, 2, 3, 4]. One of the most mportant problems n hardenng process modelng s takng nto account mutual nteractons of thermal phenomena, phase transformatons and mechancal phenomena. In mathematcal descrpton relatons between proper models are very complcated, partcularly when feedback s consdered. In numercal model relatonshps of ths type demand the mult-level recurrent algorthms or severe lmtatons of the tme step (comp. Fg. 1). Numercal analyss of heat treatment processes s very mportant problem. Many desgner studos workng for the ndustry (not only metallurgcal) one facng ths problem. However exstng mathematcal and numercal models do not take nto account many mportant factors. For full analyss of heat treatment process the feld of temperature, knetcs of phase transformatons, tthermal and structural stresses should be ncorporated. 2. Model of thermal phenomena To determne of the temperature feld, the partal dfferental equaton descrpton for unsteady heat transfer s used: (λ T) cρ T t +q V =, (2.1) where: λ = λ (T) s the thermal conductvty coeffcent, c = c(t) s the thermal capacty, ρ s the densty and q V = q V (x α, t) s the volumetrc heat source comng from latent heat of transformatons, x α s the vector locaton of consdered pont. Durng heat treatment of steels, phase transformatons generate consderable amount of heat. Partcularly, ths phenomenon n transformaton of austente to pearlte s notceable. In presented model latent heat of transformaton as volumetrc heat source n heat transfer equaton s ntroduced. η q V = H ρ, (2.2) t Fg. 1. The flow dagram of the model where: H [J/kg] s the heat of transformaton, η s the volumetrc ncrement of phase. Changes of temperature, whch are caused by latent heat of phase transformatons, are most vsble durng coolng process of hgh carbon steel [5, 6]. In lterature of specfc heat of phase transformaton the models, heat of transformaton for partcular phase (dependent of the temperature and chemcal composton) are descrbed [5]. In presented paper the results of analyss of Fe-C-Mn system and constant value of enthalpy for austente-martenste transformaton are used [5, 7]. Ths soluton s approxmated by the cubc polynomals: H γ α (T) =.641561T 3.593473214T 2 + 245.2464286T 145423.357 [J/kg] H γ P (T) =.9338T 3 1.5366T 2 + 475.9625T 212728.5714 [J/kg] H γ B (T) =.775416T 3.81671671T 2 + 365.16714T 17675.35 [J/kg] H γ M (T) = 8.25 1 4 [J/kg], (2.3)

339 where: H γ α s the heat of austente to ferrte transformaton, H γ P s the heat of austente to pearlte transformaton, H γ B s the heat of austente to bante transformaton, H γ M s the heat of austente to martenste transformaton, T s the temperature n C. Latent heat of transformaton has nfluence especally on temperature changes and knetcs of phase transformatons n the volumetrc hardenng, whereas t s nsgnfcant n transformatons durng the surface hardenng (Fg. 2). 15 9 6 3 coolng curve, wth H pearlte, wth H coolng curve, wthout H pearlte, wthout H 1 2 3 4 5 Tme [s] 1..8.6.4.2. Phase content Fg. 2. Influence of latent heat of transformaton on knetc of phase transformaton n the volumetrc hardenng, the mddle node of cubcal steel element The equaton completed boundary and ntal condtons, and n fnte element method of G a l e r k n formulaton s solved [8, 9, 1]. 3. Model of phase knetcs durng heatng The proposed model of phase transformatons durng contnuous heatng uses data from CHT dagram [11,12]. Ths dagram presents decompostons of phases nto austente, whch s formed durng heatng process. Because start and fnsh of phase transformaton strongly depend on heatng rate or tme of holdng, dynamcal curves A c1 and A c3 are used. To defne end of heatng the curve of homogenzaton of austente s used (Fg. 3). Above ths curve can be observed non-advsable growth grans of austente. In every step the fracton of new phase s calculated on the bass of knetcs of the transformaton modeled accordng to J o h n s o n-m e h l-a v r a m laws. Emprcal A v r a m equaton, derved usng an assumpton of monophase of materal, for heatng process has the form [13]: η γ (T, t) = 1 exp ( b (T) t n(t)), (3.1) where: η γ s a volume fracton of formng austente, b and n are functons whch depend on temperature as well as on start (t s )andfnsh(t f ) tmes of transformaton [9]. 115 11 15 1 95 9 85 8 75 7 A c3 A c1 15 1 53 1 5 t t s t f heatng rate [ o C/s] heatng curve curve of austente homogenzaton 1 1 1 1 1 Tme [s] Fg. 3. The analyss of the dagram CHT, steel C45 95 9 85 8 75 7 eq. JMA eq. KM Exper. -1.E-3-5.E-4.E+ 5.E-4 1.E-3 1.5E-3 Stran Fg. 4. The strans durng austente formaton for heatng rate 1 C/s n (T) = 6.12733/ ln ( ) tf (T), b (T) =.15. (3.2) t s (T) t n(t) s To determnate the ncrement of austente volume, partcularly for hgh rate of heatng, the modfed Kostnen-Marburger equatonsused(comp. Fg. 4) [3, 9]:

34 η γ (T, t) = 1 exp ( ( k γ Tsγ T )), k γ = 4,6517, T sγ T f γ (3.3) where: T sγ s start temperature of austente formaton, T f γ s estmated fnsh temperature of transformaton depended on rate of heatng. 4. Model of phase knetcs durng coolng The volume fractons η (.) (T, t) growth durng tme of coolng s calculated by A v r a m equatons [9, 1, 13]. η () (T, t) = mn η (%), η γ η j j (1 exp ( (4.1) b (T) t n(t))), where: η j s volume fracton of phase formed durng coolng process, η (%) s fnal fracton of -phase estmaton of bass on CCT dagrams consdered steel. In presented model knetcs of phase transformaton are determnate by A v r a m equatons. In paper the model, whch determnate knetcs of ndvdual phases wth the use of splne functons s appled. If the phase transformaton s frst n coolng process and tme of ts start s known, then fnsh tme of last phase (t f )s calculated by functon based on predcton the volume fracton and fnsh tme of frst transformaton (analyss of CCT dagrams): ( ) A (ln (ts ) ln (t)) t f (t,η %,t s )=exp t s. (4.2) B (η % ) If the phase transformaton s last and tme of ts fnsh s known, then start tme of frst transformaton (t s ) s determnated by equaton: ( ) ( ) ( ) B(η % ) A A B(η % ) t s t,η%,t f =, (4.3) t B(η %) t f where: η % s estmated volume fracton of the last phase. Whereas, f the phase transformaton takes place between another, then start and fnsh tme of transformaton bass on tme of end and volume fracton prevous transformaton, and predctons tme of end and volume fractons consdered phase s determned: ( ) t s t1,η (1%), t 2,η (2%) = = N (η (1%),η (2%)) A t B(η (1%)) exp ( A ln (t 1 2 )) B (η (1%)). (4.4) The coeffcents A(), B() and N() are calculated usng followng formulas: N ( ( )( ) A η (1%),η (2%) = B ( ) + 1 η (2% ) B ( η (1% ) ) A B ( η (1% ) ) ) + 1 (4.5) A = exp ln ln ( ) 1 η ( ) f ln (1 ln (1 η s ), B (η η% ) %) = ln. ln (1 η s ) (4.6) Among consderable factors decdng n prmary measure about course of transformaton are stress condtons. In ths model modfcaton of equatons descrbng knetcs of phase transformatons are ntroduced [11,14,15]. Influence of the mechancal phenomena on changes of materal mcrostructure s taken nto account. Ths phenomenon s especally notceable durng martenste transformaton. Ths effect descrbes as dependent the martenste start temperature on normal and tangental stresses (comp. Fg. 5). Stress [MPa] 3 2 1-1 -2-3 Ms (σ x ) Ms (τ xy ) 3 4 5 6 7 Fg. 5. Influence of stresses (normal σ x and tangental τ xy ) on martenste start temperature (M s ) The phase transformaton durng the hgh-rate coolng (transformaton of austente to martenste) s determned by classc form of K o s t n e n-m a r b u r g e r equaton (for T < M s ) [3, 11, 14, 15]: η M (T, t) = η γ η M ( ( ( ))) 1 exp k MS T + A M σ ef f + B M σ a, k =.1537, (4.7)

341 where: σ a = (σ ) /3 average stress, σ ef f = 3/2 ( σd σ D) effectve stress, σ D devatorc stresses, A M and B M coeffcents for C45 steel are equal to A M = 1.25 1 6 [K/Pa], B M =.75 1 6 [K/Pa] [11]. Accordng to eq. 4.7 and Fg. 5 t can be notced, that temperature M s lowers when average stress s less than zero, otherwse M s rases. Effectve stress always ncreases temperature M s, so the austente martenste transformaton s accelerated. The ncrease of sotropc stran resultng from the temperature and phase transformaton (dε Tph = dε T + dε ph ) durng heatng and coolng s descrbed by formula [9]: dε T = α (T) η dt, dε ph = ε ph (T) dη (4.8) where: α s a thermal expanson coeffcents for phase, ε ph = δv /(3V) s a structural expanson coeffcents for transformaton. Expermental research and numercal smulatons dlatometrc curves, assgn these values (thermal and structural expanson coeffcents). For the C45 steel they have the values: 2.178, 1.534, 1.534, 1.171 and 1.36 ( 1 5 )[1/K]and 1.986, 1.534, 1.534, 4. and 6.5 ( 1 3 ) for austente, ferrte, pearlte, bante and martenste respectvely [9]. The knetcs of phase transformatons durng coolng strongly depend on austensaton temperature, therefore model, whch explots two CCT dagrams for C45 steel (88 C and 15 C austensaton temperature, Fg. 6), was developed [16, 17]. Dsplaced dagrams wth respect to each other are used to determne ntermedate dagrams usng lnear approxmaton [9]. Fg. 6. The CCT dagrams of the C45 steel for dfferent austentsaton temperatures: a) 88 C [16], b) 15 C [17] t j (T Aust ) = γ Aust t 15 j + (1 γ Aust ) t 88 j T j (T Aust ) = γ Aust T 15 j + (1 γ Aust ) T 88 j (4.9) η (%) j (T Aust ) = γ Aust η(%) 15 j + (1 γ Aust) η(%) 88 j. Parameter γ Aust s descrbed by formula [9]: γ Aust = for T max < 88 C γ Aust = T max 88 15 88 for 88 C < T max < 15 C γ Aust = 1 for T max > 15 C. (4.1) The presented numercal model for numercal smulatons of strans durng phase transformaton of heatng and coolng for dfferent austensaton temperatures (Fg. 7) s used. Fg. 7. Influence of austentsaton temperature on the dlatometrc curves. (Expermental researches and numercal smulatons)

342 5. Model of mechancal phenomena The heat treatment process of steel elements causes a sudden change n stress felds. Ths change s the result of dfferent types of strans. In the presented model, except elastc (ε e ), thermal (ε T ), structural (ε ph ) and plastc (ε pl ) strans, the transformatons plastcty (ε tp )aretaken nto accounts as well (ε = ε e + ε T + ε ph + ε pl + ε tp ) [8, 9, 11]. The consttutve relatons are assumed n the form: σ = D ε e + D ε e, (5.1) where: σ s tensor of stress, D s tensor of materal constants. In presented model ncrement of plastc stran s determned by nonsothermc plastc flow law: ε pl = λ 3σD (5.2) 2 σ n whch scalar plastc multpler s calculated by (sotropc hardenng): λ = 2 σ 3σD( D ( ε ε T ε ph ε tp) +Ḋε e) 2 σh σ TṪ 9σ D( Dσ D), +4κ σσ D ef (5.3) where: κ s the hardenng modulus, HT σ softenng modulus descrbed by: s the thermal H σ T = σ T + κ T εpl ef f. (5.4) The actual level of effectve stress ( σ) s depended on temperature and composton of phase: σ T, η,ε pl ef f = σ T, η +κ(t)εpl ef f, (5.5) where: σ s the yeld pont, ε pl ef f s the effectve plastc stran. The yeld pont depends on volume fractons of phase, s completed by nonlnear nfluence of hard phase (martenste) [11]: σ T, η = η σ (T) + f (η M ) σ M (T), (5.6) M where: σ, σ M are the yeld ponts of partcular phases [18], values of functon f = f (η M ) s calculated followng: ( f (η M ) = η M ζ + 2 (1 ζ) ηm (1 ζ) ηm) 2, ζ = 1.383 σ γ (T) / σ M (T). (5.7) Transformatons nduced plastcty ε tp are non-reversble changes observed durng phase transformatons occur under loadng (n state of stress). In the models based on Greenwood-Johnson mechansm transformatons nduced plastcty s descrbed by [9, 16, 19]: ε tp 3 = 2 K F(η ) σd σ η (5.8) where: K = 5 δv 6 V s a parameter depended on change of volume durng growth of phase, F (η ) = ln (η ) (Leblond [11, 19]) s a functon whch determne knetcs of transformaton plastcty. 6. Expermental research The expermental research concernng the phase transformaton durng contnuous heatng and coolng for carbon steel was made. The objects of research were specmens made of rolled round ron from C45 steel (norm PN-93/H-8419). The expermental research was done on two thermal cycle smulators: SMITWELD TCS 145 (perpendcular specmens measurng 11 11 55 mm, wth center hole φ4, for hgh coolng rate), whch s the property of Insttute of Mechancs and Machne Desgn of Techncal Unversty of Częstochowa and dlatometer DIL85 (cylndrcal specmens φ4/2 1 mm and for the rate of coolng 5 C/s φ4/3 1 mm), whch s on equpment of Insttute for Ferrous Metallurgy of Glwce. In the expermental research, phase transformatons durng the thermal cycles of steel 45 exposed to quck heatng and coolng wth dfferent rates (heatng - 1 C/s, coolng 1, 2, 3, 5, 1, 2 and 25 C/s), were nvestgated. The research dealt wth: phase transformatons of the austente steel C45 durng the smulated hardenng cycle (dlatometrc tests), determnaton of thermal and structural expanson coeffcents of the ndvdual phases of the steel, the nfluence of the austensaton temperature on knetcs of transformatons. The results obtaned from expermental analyss were appled n numercal model, and some were used as a comparatve materal to verfy the model (comp. Fg. 8).

343 1 Cool. rate 1 o C/s Exper. 1 Cool. rate 3 o C/s Exper. 8 6 4 Model 8 6 4 Model 2 2 1 8 6 4 2 -.3.1.5.9.13.17 Stran ε Tph Cool. rate 5 o C/s Exper. Model -.3.1.5.9.13.17 Stran ε Tph 1 8 6 4 2 -.3.1.5.9.13.17 Stran ε Tph Cool. rate 2 o C/s Exper. Model -.3.1.5.9.13.17 Stran ε Tph Fg. 8. Expermental and smulated dlatometrc curves: a) 1 C/s, b) 3 C/s, c) 5 C/s, d) 2 C/s 7. Numercal smulaton The numercal smulaton of the hardenng process was made for the cubc steel element (dmensons.35.35.35 [m]). Heatng was modeled by superfcal heat source (Neumann condton), takng q n = 1 6 [W/m 2 ]. After obtaned maxmum temperature equal 1673 K n corners (temperature n the mddle of cube n ths tme was 961 K, A c1 = 158 K), the element was cooled n water of temperature 333 K (6 C). Smulaton of coolng was fnshed after levelng cube and water temperatures. The temperature dependence of convectve heat transfer coeffcent was descrbed as a pecewse lnear functon. Values of these coeffcents were determned n the expermental research [2]. In presented smulaton of stresses was assumed, that values of Young s and tangental modulus were constant, value of tangental modulus was equal to: E t =.1 E. Statc yeld pont was determned from values of expermental research from paper [18]. These values were approxmated by pecewse lnear functon. The analyss of numercal smulaton ndcated that hardenng element, whch was made from C45 steel, was not full hardened (see Fgs. 9 and 1). Because the water temperature was ncreased and the hardenablty of steel was modest, the through hardenng has not occurred. For ths hardenng the maxmum level of effectve stresses were 5 [MPa] and the value of maxmal effectve plastc strans was equal.8 (comp. Fgs. 11 13).

344 Fg. 9. The knetcs of phase transformatons and temperature: a) n the mddle node, b) n the corner node Fg. 1. The feld of martenste: a) cross-sectons, b) external surface Fg. 11. The strans on external surface: a) the structural strans, b) the effectve transformatons plastcty

345 Fg. 12. The effectve stresses: a) cross-sectons, b) external surface Fg. 13. The effectve plastc strans: a) cross-sectons, b) external surface 8. Conclusons On the bass of the presented numercal and expermental research, the followng conclusons are drawn: The results of the numercal model of phase transformatons are n good conformty wth the results of the expermental researches (Fg. 8). The thermal and structural expanson coeffcents of the ndvdual phases of the C45 steel determned from the dlatometrc tests allow to calculate the strans of heat-treated steel elements. Developed 3D model allows to smulate the hardenng processes for free forms of machne elements. Takng nto account latent heats of transformaton change n great agreement the knetcs of phase durng hardenng (Fg. 2). The calculaton of volumetrc ncrease of austente for hgh rate of heatng from Avram equaton s not accurate, therefore the modfed Kontnen-Marburger equaton s used (Fg. 4). The knetcs of phase transformatons durng coolng strongly depends on austensaton temperature. Ths phenomenon n the presented model s taken nto account. If more dagrams CCT for dfferent heatng parameters are used, then presented model wll be more accurate. The results of numercal smulaton should be verfed expermentally. REFERENCES [1] J.B. L e B l o n d, J. D e v a u x, A new knetc model for ansothermal metallurgcal transformatons n steels ncludng effect of austente gran sze, Acta Metallca 32, 137-146 (1984). [2] B. Radhakrshnan, T. Zachara, Smulaton of a curvature-drven gran growth by usng a modfed Monte Carlo algorthm, Metallurgcal and Materals Transactons 26, 167-18 (1995). [3] D.P. K o s t n e n, R.E. M a r b u r g e r, A general equaton prescrbng the extent of the autente-martenste transformaton n pure ron-carbon alloys and plan carbon steels, Acta Metallca 7, 59-6 (1959).

346 [4] Y.J. L a n, D.Z. L, Y.Y. L, Modelng austente decomposton nto ferrte at dfferent coolng rate n low-carbon steel wth cellular automaton method, Acta Materala 52, 1721-1729 (24). [5] K.J. L e e, Characterstcs of heat generaton durng transformaton n carbon steels, Scrpta Materala 4, 735-742 (1999). [6] J.L. L e e, J.K. C h e n y, Y.T. P a n, K.C. H s e h, Evaluaton of transformaton latent heat n C-Mn steels, ISIJ Internatonal 39, 281-287 (1999). [7] E.P. S l v a, P.M.C.L. P a c h e c o, M.A. S a v, On the thermo-mechancal couplng n austente-martenste phase transformaton related to the quenchng process, Internatonal Journal of Solds and Structures 41, 1139-1155 (24). [8] M. K l e b e r, Fnte element method n nonlnear contnuum mechancs, Warszawa-Poznań, (1985), (n Polsh). [9] A. K u l a w k, Numercal analyss of thermal and mechancal phenomena durng hardenng processes of the 45 steel, PhD Thess, Częstochowa (25), (n Polsh). [1] A. K u l a w k, A. B o k o t a, The analyss of mutual nfluence of selected phenomena of hardenng process for the C45 steel, KomPlasTech 1, 231-238 (25) (n Polsh). [11] H.J.M. G e j s e l a e r s, Numercal smulaton of stresses due to sold state transformatons. The smulaton of laser hardenng, Thess Unversty of Twente, The Netherlands (23). [12] J. O r l c h, A. R o s e, P. W e s t, Atlas zur Wärmebehandlung von Stähle, III Zet Temperatur Austentserung Schaublder, Verlag Stahlesen MBH, Düsseldorf (1973). [13] M. A v r a m, J. Chem. Phys. 7, 113 (1939). [14] J. R o n d a, G.J. O l v e r, Consstent thermo-mechano-metallurgcal model of welded steel wth unfed approach to dervaton of phase evoluton laws and transformaton-nduced plastcty, Comput. Methods Appl. Mech. Engrg. 189, 361-417 (2). [15] B. C h e n, X.H. P e n g, S.N. N o n g, X.C. L a n g, An ncremental consttutve relatonshp ncorporatng phase transformaton wth the applcaton to stress analyss, Journal of Materals Processng Technology 122, 28-212 (22). [16] F. W e v e r, A. R o s e, Alas zur Wärmebehandlung von Stähle, Verlag Stahl Esen MBH, Düsseldorf (1954). [17] F. W e v e r, A. R o s e, Atlas zur Wärmebehandlung von Stähle, I Zet Temperatur Umwandlungs Schaublder, Verlag Stahl Esen MBH, Düsseldorf (1961). [18] M. C o r e t, A. C o m b e s c u r e, A mesomodel for the numercal smulaton of the multphasc behavor of materals under ansothermal loadng (applcaton to two low-carbon steels), Internatonal Journal of Mechancal Scences 44, 1947-1963 (22). [19] L. T a l e b, N. C a v a l l o, F. W a e c k e l, Expermental analyss of transformaton plastcty, Internatonal Journal of Plastcty 17, 1-2 (21). [2] C.H. G ü r, A.E. T e k k a y a, Numercal nvestgaton of non-homogeneous plastc deformaton n quenchng process, Materals Scence and Engneerng A319-321, 164-169 (21). Receved: 1 March 27.