© Strojni{ki vestnik 46(2000)11/12,718-731 © Journal of Mechanical Engineering 46(2000)11/12,718-731 ISSN 0039-2480 ISSN 0039-2480 UDK 669.14:621.785:519.86 UDC 669.14:621.785:519.86 Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01) Analiza ob~utljivosti toplotne obdelave jekel A Sensitivity Analysis of the Heat Treatment of Steel Milan Batista - Franc Kosel Prispevek obravnava matematični model toplotne obdelave podeutektoidnih ogljikovih jekel. Pri izračunu zaostalih napetosti, ki so posledica faznih premen in temperaturnih sprememb, sta v modelu upoštevana tako kinetika faznih prehodov kakor termoelasto-plastične konstitutivne enačbe. Izdelani računalniški progam je bil preverjen na že objavljenih rezultatih. Z uporabo analize občutljivosti je ocenjena napaka v izračunanih zaostalih napetostih na temelju ocenjenih napak podatkov o lastnostih materiala. © 2000 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: obdelave jekel, obdelave toplotne, analize občutljivosti, modeli matematični) This paper presents a mathematical model of the heat treatment of hypoeutectoid carbon steel. In the model, the kinetics of phase changes and a thermo-elasto-plastic constitutive relation have been applied to calculate the residual stresses resulting from phase changes and temperature variations. The computer code has been verified for internal consistency with previously published results. The sensitivity analysis has been applied to predict errors in the residual stresses from the estimated errors in the material data. © 2000 Journal of Mechanical Engineering. All rights reserved. (Keywords: steel treatment, heat treatment, sensitivity analysis, mathematical models) 0 UVOD Matematično modeliranje toplotne obdelave jekel je bilo predmet številnih raziskav v zadnjih desetletjih [10]. Te raziskave so pokazale, da se da toplotna obdelava analizirati z mehaniko kontinuov in naslednjimi predpostavkami: - reološki model kontinua je termoelastoplastični material; - model mora vključevati deformacije zaradi strukturnih sprememb in preoblikovalno plastičnost; - enačba prevoda toplote mora vsebovati člen, ki popisuje toploto fazne premene; - za popis difuzijskih faznih premen se uporablja Avramijeva enačba; - za obravnavo austenitno-martenzitne transformacije se uporablja Koistinen-Marburgerjeva enačba; - vse lastnosti materiala so linearne funkcije prostorninskih deležev posameznih faz. S temi predpostavkami je proces toplotne obdelave določen kot deterministični mehanski model kontinua. S tem modelom se da ob predpostavki, da so znani podatki o mehanskih 0 INTRODUCTION Mathematical modeling of the heat treatment of steel was intensively investigated over the last two decades [10]. It was shown that the problem of heat treatment can be analyzed by using the theory of con-tinuum mechanics with the following assumptions: - a continuum rheologic model is thermo-elasto-plastic; - in the mechanical model the structural phase deformation and transformation plasticity have to be included; - the equation of heat transfer is added to the part which represents the heat of the phase transformation; - the Avrami equation is used for treating the ki-netics of the diffusion-controlled transformation of phases; - Koistinen-Marburger’s equation is used for treat-ing the austenite-martensite transformation; - all the properties of the continuum are linear func-tions of the volume fractions of the phases. With these assumptions, the heat-treating process is defined by a deterministic mechanical model of continuum. Using this model, and provided that the data of the mechanical properties and bound- 2 jgnnatäüllMliBilrSO | | ^SSfiflMlGC | stran 718 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis lastnostih in ob znanih robnih pogojih izračunati zaostale napetosti. Poglavitni namen tega prispevka je oceniti napako izračunanih zaostalih napetosti na podlagi ocene napak vhodnih podatkov o materialu. V ta namen je bil izdelan matematični model toplotne obdelave. Na temelju modela je bil izdelan računalniški program, ki izračuna razvoj temperature, strukturnih sprememb in zaostalih napetosti pri ohlajanju neskončnega valja, izdelanega iz podeutektoidnega ogljikovega jekla. Računalniški program je bil primerjalno testiran z rezultati objavljenimi v literaturi. V nadaljevanju so bili izračunani koeficienti občutljivosti za valje premera 10 mm, 30 mm in 50 mm izdelanih, iz izbranega materiala. Na podlagi koeficientov občutljivosti smo ocenili vplivnost vhodnih podatkov in nadalje, na podlagi ocenjenih napak vhodnih podatkov smo ocenili še napako v izračunu zaostalih napetosti. 1 MATEMATIČNI MODEL Matematični model toplotne obdelave mora vključevati izračun temperature, strukture in napetosti. V tem poglavju podajamo pregled osnovnih enačb, ki so vključene v model. 1.1 Opis materiala Jeklo, ki je izpostavljeno toplotni obdelavi, obravnavamo kot zmes N sestavin. Te so: austenit ferit, perlit, bainit in martenzit.Če je xk prostorninski deležk-te sestavine, potem velja: ary conditions are known, the residual stresses in a treated element can be calculated. The main aim of this paper was to estimate the error in the calculated residual stresses from the estimated error in the inputed material data. In order to do this a mathematical model of the heat treatment was developed. This model was converted to a computer program which performed the calculation of the ther-mal and structural evaluations and the residual stresses during the cooling of an infinitely long cylinder made of hypoeutectoid carbon steel. The computer code was verified for internal consistency with previously published results. Next, the sensitivity coefficients of the selected material data were calculated for cylin-ders of 10 mm, 30 mm and 50 mm diameter. On the basis of the sensitivity coefficients we estimated the impor-tance of the input data and, in addition, using the esti-mated errors of the material data we determined the error in the calculated residual stresses. 1 MATHEMATICAL MODEL A consistent mathematical model of heat treatment must include thermal, structural and stress calculations. In this section we shall review the es-sential equations which were used in the model. 1.1 Material description The steel used in the heat-treatment pro-cess is considered to be a mixture of N constituents: austenite, ferrite, pearlite, bainite and martensite. If xk is the volume fraction of the kth constituent then: (1). Če je wk masni delež k-te sestavine, potem velja podobno: Similarly, if wk is the mass fraction of the kth con-stituent then: (2). Vsaka sestavina zmesi ima gostoto rk. Če je Each constituent of the mixture has a density rk. If r r gostota zmesi, potem sta prostorninski in masni is the mixture density, then the volume fraction and delež k-te sestavine povezana na naslednji način: the mass of the kth constituent are connected by: rk xk (3). Pri jeklu so razlike med gostotami posameznih sestavin majhne, zato velja ocena wk » xk. Iz (1) do (3) se da izpeljati naslednji zvezi, ki ju bomo uporabili v nadaljevanju: r In the case of steel, the differences between the densities of the constituents are small, so in this case we have wk « xk. On the basis of (1) to (3), the following relations, which will be used later, can be derived: r 1 r N k =1 N t1 r k (4) (5). stran 719 bcšd00 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis 1.2 Elastičnost Če predpostavimo, da je material izotropen, potem so elastične deformacije eie podane z: 1.2 Elasticity If we assume that a material is isotropic, then the elastic strain ei ej is given by: E Pri tem so sij napetostni tenzor, E, n in a Youngov modul, Poissonovo razmerje in koeficient termičnega raztezanja, gk pa dilatacijski koeficient k-te sestavine. Nadalje predpostavimo, da so n in gk konstante, E in a pa linearni funkciji prostorninskih deležev: e e = - [(1 + n) sij - nsmmdij ] + aJdij + L gkxk dij (6). Here sij is the stress tensor, E, n and a are theYoung’s modulus, Poisson’s ratio and thermal expansion coefficient, respectively, and gk is the dilatation coefficient of the kth constituent. We assume that n and gk are constants. E and a are taken as linear functions of the volume fractions of the constituents: N (7) (8), k =1 pri čemer je E(J) temperaturno odvisni Youngov modul k-te sestavine in ak njen razteznostni koeficient, za katerega predpostavimo, da je nespremenljiv. Dilatacijski koeficient gk lahko izračunamo takole: če so deformacije majhne, potem po zakonu ohranitve mase in (4) dobimo: ev=1- r = r0 pričemer je e prostorninska deformacija in r0 gostota zmesi v referenčnem stanju. S primerjavo te enačbe in (6) dobimo za sij = 0 in J = 0: where E(J) is the temperature-dependent Young’s modulus of the kth constituent and ak is its thermal expansion coefficient, which is assumed to be constant. We shall now describe a method for calcu-lating the dilatation coefficients gk. If deformations in the material are small, then from the conservation of mass and equation (4) we obtain: rk -r0 r0 xk (9), where ev is the volume deformation and r0 is the mixture density in a reference state. By comparing this equation with equation (6) at sij = 0 and J = 0 we obtain: 1 r0-rk 3 r0 (10). 1.3 Plastičnost 1.3 Plasticity Prirastek plastičnih deformacij dep izračunamo z uporabo klasične teorije plastičnosti in Missesovega kriterija tečenja: The plastic strain increment dei pj is calculated using the classical theory of plasticity with the Misses yield criterion and the associated flow rule. Thus: F=sij sij - s2f 3 dep = dLps (11) (12), pri čemer je sij deviatorični napetostni tenzor, definiran kot: where sij are the components of the deviatoriv stress tensor given by: 1 s =s - s d ij ij mmij 3 (13) in sf napetost tečenja. Pogoj F < 0 pomeni, da ni plastičnega tečenja, pogoj F = 0 pa, da je. Če združimo (11) in (12), dobimo: and sf is the flow stress. The condition F < 0 means that there is no plastic flow and F = 0 means that plastic flow takes place. Combining of (11) and (12) gives: dLp 3 dep 2s (14) pri čemer je dep dejanski prirastek plastičnih deformacij, podan z: where de p is the effective plastic strain increment given by: 2 jgnnatäüllMliBilrSO | | ^SSfiflMlGC | stran 720 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis dep -dep dep 3 i j i j (15). Predpostavimo, da je napetost tečenja podana z linearnim zakonom utrjevanja: We assume that the flow stress is given by the linear hardening rule: s=s+ Hep f (16), pri čemer je sY napetost tečenja in H koeficient utrjevanja. Nadalje predpostavimo, da sta napetost tečenja in koeficient utrjevanja linearni funkciji prostorninskih deležev posameznih sestavin: where sY is the yield stress and H is the strain-hardening coefficient. In addition, we assume that the yield stress and the strain-hardening coefficient are linear functions of the volume fractions of constituents: N N (17) (18). Pri tem sta sY, k in Hk napetost tečenja in koeficient utrjevanja k-te sestavine. 1.4 Preoblikovalna plastičnost Za izračun prirastka preoblikovalne plastičnosti detp uporabimo model, ki so ga predlagali Here, sY,k and Hk correspond to the yield stress and the strain-hardening coefficient of the kth constituent, respectively. v [11] in [15]: pri čemer je: detp N dLtp=32Kk(1-xk)dx k=2 1.4 Transformation Plasticity For the calculation of the transformation plasticity strain increment deitjp we applied the model proposed in [11] and [15]: dLtpsij (19), where (20) in Kk konstante, ki jih je treba določiti s preskusi. 1.5 Prevod toplote and Kk are constants which must be determined ex-perimentally. 1.5 Heat Conduction Temperaturno polje dobimo z rešitvijo The temperature field is calculated by solv- enačbe prevoda toplote: ing the basic equation of heat conduction: 0J) + Ndlkxk (21), a 3J dt dx pri čemer so t čas, J temperatura, c specifična toplota, where t is time, J is the temperature, c is the heat l koeficient prevoda toplote in lk latentne toplote capacity, l is the coefficient of thermal conductivity premen. and lk are the enthalpies of transformation. Temperaturno odvisna c in l vzamemo kot Both c and l are temperature dependent and linearni funkciji prostorninskih deležev sestavin: are taken as linear functions of the volume fractions of the constituents: N c = %ck(J)xk N (22) (23), pri čemer sta ck(J ) in lk(J) specifična toplota in toplotna prevodnost sestavine k. Enačbo (21) rešujemo skupaj s konvekcijskim robnim pogojem: and ck(J) and lk(J) the heat capacity and the thermal conductivity for phase k. Equation (21) is solved together with convection, boundary condition: -l Jn = hJ-Je ) (24), I isfinHi(s)bJ][M]ifln;?n 00 stran 721 I^HSSTTIMIDC M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis pri čemer je h konvekcijski koeficient, ki je lahko temperaturno odvisen in Je temperatura okolice. 1.6 Fazne premene Opazujmo fazni prehod faze m v fazo n. V ta namen prepišimo (1) v obliko: xm +xn in uvedimo novo spremenljivko z, definirano kot: z= where h is the convection heat-transfer coefficient, which may be temperature dependent and Je is the environmental temperature. 1.6 Phase Transformations We can consider the phase transformation from, say, phase m to phase n. For this purpose we write equation (1) as: 1-Xxk (25) and then introduce a new variable z defined by: xn 1-Xxk (26), ki jo bomo uporabili za opis napredovanja premene. Pri izotermnih pogojih se da heterogena premena opisati z Avramijevo enačbo ([5],[6] in [16]): which will be used to describe the extent of the transformation. Under isothermal conditions a heteroge-neous solid-state transformation can be described by the Avrami equation ([5],[6] and [16]): pri čemer je z = 1-exp [-(bt)n] where b ( J ) = K0 ( JE-J ) nexp (27), (28). K0 in Q sta konstanti, n je Avramijev eksponent, ki je odvisen od geometrijske oblike rastočih kristalnih zrn in je temperaturno neodvisen [6], T je absolutna temperatura, JE pa ravnotežna temperatura premene. Za neizotermne pogoje se da Avramijeva enačba zapisati v obliki: K0and Q are constants, n is the Avrami exponent, which is dependent on grain growth geom-etry and can be taken as temperature independent [6], T is the absolute temperature, and JE is the equi-librium temperature of the transformation. For nonisothermal conditions the Avrami equation can be written in the form: dz dt nb(J)(1-z)[-ln(1-z)]n (29), ki jo dobimo, če iz (27) izločimo čas t. Ko je n konstanta, se da K0 in Q dobiti iz diagramov TTT takole: čas t0, ki je potreben za premeno z0 pri dani temperaturi T, je iz (27): which is obtained by eliminating t from (27). When n is constant, K0 and Q can be deter-mined from TTT diagrams in the following way. The time t0 for a fixed amount of transformation z0 at a given temperature T is, from (27): K0 eT 1 (30). (JE -J)n Če sta (tm ,Jm) koordinati ekstrema na krivulji If (t ,Jm ) are coordinates of the nose in the TTT, potem je v tej točki dt/dJ = 0. Iz tega pogoja in TTT curve tWdt/dJ = 0 at this point. From these (30) izhaja: conditions and (30) it follows that: n Q= nTm2 Tm JE -Jm Ko poznamo Q,lahko iz (30) izračunamo K: By knowing Q we can calculate K : 1 K0=tm(JE-Jm)neTm[ln(1-z0)Y (31). (32). Martenzitna premena je odvisna le od temperature, zato ima v tem primeru kinetični zakon obliko z = (J). Iz te enacbe je prirastek premene dz =/(J)dJ. A martensitic transformation depends only on temperature, therefore the kinetic law is, in this case, given by z = f(J). From this equation the increment of 2 jgnnatäüllMliBilrSO | | ^@©^ifW]D[lC | stran 722 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis Iz osnovnega kinetičnega zakona lahko izrazimo J kot funkcijo z , zato je dz = g(z)dJ. Ko je premena končana, tj. ko je z = 1, mora biti g(z) = 0. Najpreprostejša funkcija, ki ustreza temu pogoju, je linearna, torej: transformation is dz = f’(J)dJ. From the kinetic law we can express J as a function of z , hence dz = g(z)dJ. When the transformation is completed i.e. when z = 1, we must have g(z) = 0. The simplest equation com-patible with this requirement is linear, namely: Z integracijo dobimo: dz = kM(1-z)dJ (33). By integrating this equation we obtain: z = 1-exp[-kM (J-Ms)] (34), pri čemer je M temperatura, pri kateri se začne martenzitna premena. Enačba (34) je identična Koinstinen-Marburger empirični enačbi [10]. Za večino jekel ima konstanta kM vrednost 0,011. 2 LASTNOSTI MATERIALA V tej raziskavi smo za material izbrali podeutektoidno ogljikovo jeklo, ker je za ta material v literaturi dostopnih dovolj podatkov, s katerimi lahko zgradimo analitični model materiala. V nadaljevanju bomo uporabili oznake A,B,M,P in W za austenit, bainit, martenzit, perlit in cementit. Simbole C, Si in Mn bomo uporabili za masne deleže ogljika, silicija in mangana v jeklu. 2.1 Youngov modul in Poissonovo število Za ogljikova jekla se da Youngov modul perlita izračunati po naslednjem obrazcu: where Ms is the martensitic starting temperature. Equa-tion (34) is identical with the Koinstinen-Marburger empirical formula [10]. The constant kM is equal to 0.011 in most steels. 2 MATERIAL PROPERTIES As a target material in this investigation we chose hypoeutectoid carbon steel because there were enough data available in the literature to construct a material model as well as the analytical formulas. In the following we use the indices A,B,M,P and W for austenite, bainite, marten-site, pearlite and cementite, respectively. We will also use the symbols C, Si and Mn to denote the weight percent of carbon, silicon and manganese in the steel. 2.1 Young’s Modulus and Poisson’s Ratio For carbon steel the Young’s modulus of pearlite is calculated by: EP 209,3 -0,076 J ±1,62 GPa (35). Ta obrazec je dobljen na podlagi regresijske analize podatkov, ki jih je podal [7]. Predpostavljamo, da ta obrazec velja prav tako za bainit in martenzit. Obrazec, ki podaja Youngov modulu austenita, smo dobili iz podatkov, ki jih navaja [22] in ima obliko: This formula was obtained on the basis of a regression analysis from the data [7]. We assume that the same formula holds for bainite and marten-site. The formula for the Young’s modulus of austen-ite was obtained from data reported by [22] and has the form: EA 200,2-0,08J±0,32 GPa (36). Predpostavljamo, da ima Poissonovo število vrednost 0,3 za vse sestavine. 2.2 Razteznostni koeficient V literaturi ni zaslediti enotnih podatkov za vrednost razteznostnih koeficientov, zato smo vzeli naslednje vrednosti [10]: aA = 22-106 K-1 aB =13-106 K-1 aM = 12-106 K-1 aP = 14-106 K-1 The Poisson’s number is assumed to be con-stant and is equal to 0.3 for all constituents. 2.2 Coefficient of Expansion There is no single value in the literature for the expansion coefficient of the different phases so in the present model we adopted the following values [10]: (37). I igfinHi(s)bJ][M]ifln:7n 00 stran 723 I^HSSTTIMIDC M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis Iz podatkov v literaturi ocenjujemo, da so podane vrednosti znotraj območja ±1 • 10-6 K-1. From the data published in the literature we conclude that all the above values are within the range ±110-6 K- 2.3 Preoblikovalna deformacija Večina znanih del uporablja za dilatacijske koeficiente eksperimentalne vrednosti iz [10]. V tem delu bomo dilatacijske koeficiente izračunali na temelju kristalografskih podatkov iz preglednice 1 in (10). Iz kristalografskih podatkov izračunamo linearizirane gostote sestavin jekla: rA =8156 rb =7897 rm =7676 rF =7897 Preglednica 1. Kristalografski podatki Table 1. Crystallographic data 2.3 Transformation Strain Most previous studies used experimental values for the dilatation coefficients [10]. We have chosen to calculate these values on the basis of the crystallographic data in table 1 and equation (10). From crystallographic data the following linearised densities of the steel constituents are obtained: -216 C kg/m3 kg/m3 kg/m3 -248 C kg/m3 (38). Faza Phase Tip Type fcc ort Fe C Lattice parameters [Ä] Vir Source Austenit Austenite 4 - a = 3,5735 + 0,0316C [19] Cementit Cementite 12 3 a = 4,5234 b = 5,0883 c = 6,7426 [4] Ferit Ferrite bcc 2 0 a = 2,8664 [4] Martenzit Martensite bct 2 - a = 2,8664 – 0,013C c = 2,8664 + 0,116C [4] Iz (4) je gostota perlita: 1 rP From (4) we have for the pearlite density: 0,12 0,88 rW rF (39), iz katere dobimo, če vstavimo vrednosti iz (38): Podobno dobimo gostoto zmesi ferita in perlita: which gives, by substituting values from (38), the following value: = 7861 kg/ (40). For the ferrite-pearlite mixture we then obtain, by a similar procedure, the result: rP+F=7897-45 C kg/m 3 (41). Če vzamemo za referenčno strukturo jekla zmes ferita in perlita, dobimo na podlagi (10): If we take the mixture of ferite-perlite as a steel reference structure then from (10) we obtain: Prav tako predpostavimo, da je gB = 0. g A = -0,0109 +0,0072C (42). g M = 0,0086C We also assume that gB = 0. 2 jgnnatäüllMliBilrSO | | ^SSfiflMlGC | stran 724 r M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis 2.4 Meja tečenja in koeficient utrjevanja 2.4 Yield Strength and Strain-Hardening Coefficient Iz podatkov, ki sta jih objavila [12] in [10], From the data published by [12] and [10] we dobimo naslednje obrazce za mejo plastičnega tečenja: derive the following formulas for the yield stresses: 123-0,1J-0,01CJ±1,3 MPa 434-0,64J + 0,77CJ±34,4 MPa 491 + 757C-1,02CJ±26,1 MPa 445 + 1375C±30,4 MPa (43) in koeficient plastičnega utrjevanja: and the coefficient of strain hardening: HA = 45,3-0,04J±0,63x102 MPa HB=5,88 + 226,2CJ±19,7x102 MPa HF = 29,5 + 0,33CJ±41,7x102 MPa HM=179 + 1,3CJ±29,9x102 MPa (44). 2.5 Koeficienti preoblikovalne plastičnosti 2.5 Transformation Plasticity Coefficient Na temelju poskusov, opisanih v [18], smo From the experimental work of [18] the follow- vzeli naslednje vrednosti konstant K: ing values for the constant Kk were adopted: KB=4,18-10-5 MPa-1 Km=5,08-10-5 MPa-1 KP 4,18-10-5 MPa- (45). 2.6 Specifična toplota Podatki za regresijsko analizo specifične toplote so vzeti iz [9]. Iz teh podatkov je specifična toplota perlita: 2.6 Specific Heat Capacity The data for the regression analysis of the specific heat capacity were taken from [9]. From this we obtain the following formula: 3,76 + 0,3C + 6,210-6J2±0,15 MJ/m 3 K (46). Predpostavimo, da ta obrazec velja tudi za bainit in martenzit. Za specifično toploto austenita smo uporabili obrazec: We take this formula to be valid also for bainite and martensite. For the austenite heat capac-ity we use the formula: cpA dobili smo ga iz podatkov v [22]. 2.7 Toplotna prevodnost 4,152 + 8,410-4 J M J m 3 K (47), which was obtained from data published in [22]. 2.7 Thermal Conductivity Regresijsko formulo za toplotno prevodnost perlita smo dobili iz podatkov, objavljenih v [3] in [8]: The regression formula for thermal conduc-tivity was obtained from published data in [3] and [8]: lP = 66,2 - 37,9C-0,049J ±1,7 4 W/m K (48). Predpostavimo, da ta obrazec velja tudi za bainit. Za martenzit smo vzeli vrednost lM =30±5 W/mK, kakor jo predlaga [10]. Iz podatkov, objavljenih v [22], smo dobili naslednji obrazec za izračun toplotne prevodnosti austenita: We assume that the above formula is also valid for bainite. For martensite we take the thermal conductivity to be lM 30±5W/mK, as suggested by [10]. From the data of [22] we obtain the following formula for the thermal conductivity of austenite: lA=15 + 0,01J W/mK (49). I isfinHi(s)bJ][M]ifln;?n 00 stran 725 I^HSSTTIMIDC M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis 2.8 Fazne temperature Za izračun ravnotežnih temperatur, ki se uporabljajo v (28), smo uporabili naslednje empirične obrazce: 2.8 Transformation temperatures For the calculation of the equilibrium tem-peratures which are used in (28) we chose the follow-ing empirical formulae: Bs M 723-10,7Mn + 29,1Si±10 0C 910-203C + 44,7S±10 0C 830-270C-90Mn±25 0C 539 -423 C-30,4Mn ±25 0C (50). Obrazce za A1, A3 in M je podal [1], temperaturo začetka tvorjenja bainita Bs pa [21]. 3 POSTOPEK REŠEVANJA Opisani model je bil uveden v računalniški program, ki omogoča izračun temperature, strukture in zaostalih napetosti pri hlajenju neskončnega valja, izdelanega iz podeutektoidnega jekla. Temperaturno polje se računa na temelju implicitne metode končnih razlik, napetosti pa z metodo zaporednih približkov, ki jo podaja [17]. Vhodni podatki so: - premer valja - kemična sestava jekla - začetna in končna temperatura procesa ohlajanja - konvekcijski koeficient - koordinate ekstremnih točk na diagramu TTT Izhodni podatki so: - porazdelitev zaostalih napetosti in struktur - temperaturni in napetostni potek v osi in na površini valja 4 PRESKUS MODELA Kot prvi primer smo obravnavali valj s premerom 60 mm, izdelan iz 0,43% ogljikovega jekla, gašenega v vodi s temperature 870oC. Ta primer je teoretično in eksperimentalno obdelan v [13] in [14] in prav tako teoretično v [20]. Iz objavljenih podatkov smo ocenili, da mora biti koeficient prestopa toplote blizu 25 kW/ m2K. Omenimo, da tudi s tako visokim konvekcijskim koeficientom nismo dobili take hitrosti ohlajanja osi valja, kakor jo navaja [13]. Kljub temu pa so izračunane zaostale napetosti v skladu z eksperimentalnimi vrednostmi, kakor se vidi s slike 1. Kot drug primer so obravnavali valje s premeri 10 mm, 30 mm, 50 mm in 100 mm izdelane iz 0,44% ogljikovega jekla. Porazdelitev zaostalih napetosti za take valje eksperimentalno obravnava [2], teoretično pa [23]. Vsi valji so gašeni v vodi na 20 oC s temperaturo 850 oC. Vrednosti konvekcij-skega koeficienta smo vzeli med 3200 W/m2K in The formulae for A1, A3 and Ms were given by [1], and for bainite the start temperature Bs was given by [21] 3 SOLUTION METHOD The described model was converted into a computer program which performed the calculation of ther-mal and structural evaluations and the internal stresses during cooling of an infinitely long cylinder made from hypoeutectoid carbon steel. The temperature field was calculated using the implicit finite-difference method and the stress evaluation wasbased on the successive approximation method described by [17]. The input data for the program are: - cylinder diameter, - chemical composition of the steel, - process start and end temperature, - heat convection coefficient, - coordinates of the extreme point of the TTT curves. The output of the program is: - residual stress and structure distribution, - temperature-time and axial stress-time evaluation on the surface and centere of the cylinder. 4 MODEL VERIFICATION For the first example we took a 60-mm-diam-eter cylinder of 0.43% carbon steel quenched from 870oC into water. This example was analyzed theo-retically and experimentally by [13] and [14] and also theoretically by [20]. From the published data we estimated the value of the coefficient of heat transfer to be near 25 kW/m2K. It should be mentioned that even with such a high value for the coefficient of convection heat transfer we could not obtain the cooling speed at the centre of the cylinder which was reported by [13]. Neverthe-less, the calculated residual stresses were in good agreement with experimental values, as can be seen from Fig.1. As a second example we considered cylinders of diameters 10 mm, 30 mm, 50 mm and 100 mm made from 0.44% carbon steel. The axial residual stress distribution for these cylinders was investigated experimentally by [2] and theoretically by [23]. All the cylinders were quenched into water at 20 oC from a temperature of 850 oC. For the coefficient of convective heat transfer we 2 isnnatäirjllMliBilrSO | | ^SsFÜWEIK | stran 726 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis MPa 1000 800 — 600 ------ 400 -------- 200------ 0 -------- -200 '------ -400 ------ -600 -------- -800 — I -1000 -------- -1200 0 5 10 15 20 25 30 mm polmer / radius Sl 1. Zaostale napetosti v gasenem valju premera 60 mm Fig. 1. Residual stress distribution in a 60 mm-diameter quenched cylinder 5700 W/m2K Prav tako smo izvedli izračun s konvekcijskim koeficientom, ki je temperaturno odvisen in katerega odvisnost je iz podatkov [12]. took values for the constants between 3200 W/m2K and 5700 W/m2K. We also performed calculations with the temperature-dependent coefficient of convective heat transfer, which was calculated from the experimental data given by [12] h = 1029 + 63l9-0,14l92+0,7510-4l93 W/m2K (51). Na slikah 2 do 5 so poleg izračunanih osnih napetosti podane tudi eksperimentalne vrednosti, ki so jih podali [2] in [23]. Kakor se vidi iz teh slik, se izračunane vrednosti za valje s premeri 30 mm, 50 mm in 100 mm dobro ujemajo z eksperimentalnimi vrednostmi. Za valj premera 10 mm so vrednosti zaostalih napetosti, ki jih podaja [2] negativne, tiste, ki jih podaja [23], pa so pozitivne, zato se nismo mogli odločiti, ali so izračunane vrednosti prave. 5 ANALIZA OBČUTLJIVOSTI Z namenom, da določimo zanesljivost modela, smo izvedli analizo občutljivosti. Kot izhodni parameter modela smo vzeli osne zaostale napetosti na osi in robu valja. Za nadzor smo vzeli 36 materialnih parametrov. Če z a označimo parameter, potem je njegov relativni koeficient občutljivosti definiran z: In figures 2 to 5 the calculated residual axial stresses are shown together with the experimental values given by [2] and [23]. As can be seen from these figures the calculated residual stresses for the cylinders of diameter 30 mm, 50 mm and 100 mm are in very good agreement with the experimental values. For the 10 mm-diameter cylinder the value of the residual stress given by [2] is negative and that given by [23] is positive, so we could not conclude that the calculated values are correct. 5 SENSITIVITY ANALYSIS In order to estimate the accuracy of the model we performed a sensitivity analysis. As an output parameter from the model we took the axial residual stress on the axis of the cylinder and on its boundary. For the control parameters we took 36 material parameters. If we denote these parameters as ak then the relative sensitivity coefficient is defined by: s da (52). Izračun koeficientov občutljivosti smo izvedli za valje s premerom 10 mm, 30 mm in 50 mm. Parcialne odvode v (52) smo izračunali numerično. Podatki in rezultati izračuna so podani v preglednici 2. We carried out the calculation of the sensi-tivity coefficients for cylinders of diameter 10 mm, 30 mm and 50 mm. The partial derivatives in (52) were performed by numerical differentiation The data and the results of the calculation are shown in Table 2. I isfinHi(s)bJ][M]ifln;?n 00 stran 727 I^HSSTTIMIDC M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis 300 MPa 100 0 — -100 -300 ---------Calculated with h=5700 [W/m2K] ¦ Measured [2] • Measured [23] ' / // S I ---------Izračun z h=5700 [W/m2K] --------Izračun z h=h(J) ¦ Izmerjeno [2] • Izmerjeno [23] ^y' _ _ _ ..--¦'' polmer / radius Sl. 2. Osne zaostale napetosti v gasenem valju premera 10 mm Fig. 2. Axial residual stress distribution in a quenched 10 mm diameter cylinder MPa 800 600 400 200 0 -200 -400 -600 -800 -1000 _ _ _ —^^r -^ ^s. -v ---------Calculated with h=5700 [W/m2K] .....Calculated with h=4000 [W/m2K] --------Calculated with h=h(j) V\ X. K 2 .....Izračun z h=4000 [W/m2K] --------Izračun z h=h(j) *\\-.; • Izmerjeno [23] \ V 0 10 15 polmer / radius Sl. 4. Osne zaostale napetosti v gasenem valju premera 50 mm Fig. 4. Axial residual stress distribution in a quenched 50 mm diameter cylinder Iz preglednice 2 vidimo, da so v skoraj vseh primerih najvplivnejši parametri model temperaturni razteznostni koeficient austenita aA, dilatacija austenita gA in martenzita g, ekstremna točka na krivulji TTT za perlit (0> ) in bainit (/ ,B ) in nazadnje temperaturi A in B. Prav tako je model nekoliko občutljiv za konvekcijski koeficient h in masni delež ogljika C. Iz povedanega sklepamo, da je model zelo občutljiv za kinetiko prehoda austenita v perlit in bainit. V primeru valja s premerom 10 mm, ko se skoraj celoten valj spremeni v martenzit, je model tudi zelo občutljiv za parametre kinetične enačbe (34). Po drugi strani pa je model razmeroma neobčutljiv za mejo plastičnega tečenja in druge mehanske in fizikalne lastnosti. Na temelju koeficientov občutljivosti smo ocenili relativno napako izračunanih zaostalih napetosti po obrazcu: 600 MPa 200 -200 -400 20 25 mm -600 -800 -1000 "" -~'-. ^^x ^N. — d with h=5700 [W/m2K] d with h=4 000 [W/m2K] d with h=h(J) -- - - Calculate - Calculate 1 Measured [23] \\ — Izračun z h=5700 [W/m2K] ^ vi -- - Izračun z h=h(J) Izmerjeno [2] > 0 2 4 6 8 10 Sl. 3. Osne zaostale napetosti v gasenem valju premera 30 mm Fig. 3. Axial residual stress distribution in a quenched 30 mm diameter cylinder 800 MPa -200 -400 -600 -800 :' ~ ~ "' ^x ¦^v_ - --------- Calculate d with h =5 700 [W/m2K] --------Calculate d with h =3 200 [W/m2K] ¦ Measured [2] \ --------- Izračun z h=5700 [W/m2K] ¦ Izmerjeno [2] ^v 0 10 20 30 40 polmer / radius mm Sl. 5. Osne zaostale napetosti v gasenem valju premera 100 mm Fig. 5. Axial residual stress distribution in a quenched 100 mm diameter cylinder From table 2 we can see that the most impor-tant parameters in the model are, in almost all cases: the thermal expansion of austenite aA; the dilatation of aus-tenite gA and martensite gM; the extreme points on the TTT curve for prearlite (tP,Pm) and bainite (tB,Bm);and finally, the temperatures A3 and Bs. Also, the model is only moderately sensitive to the coefficient of convec-tive heat transfer (h) and the weight percent of carbon. From the above we can conclude that the model is very sensitive to the kinetics of phase change from austenite to pearlite and bainite. In the case of the 10-mm-diameter cylinder, where almost all the cyl-inder is transformed to martensite, the model is also very sensitive to parameters in the martensite kinetic equation (34). On the other hand, the model is rela-tively insensitive to yield stresses and other mechani-cal and physical data. On the basis of sensitivity coefficients we can estimate the relative error of the residual stress with the formula: :Tsiei 2 (53), 2 jgnnatäüllMliBilrSO | | ^§§FirWEI& | stran 728 400 200 -200 mm 600 400 200 0 5 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis Preglednica 2. Podatki o materialu in izračunani koeficienti občutljivosti za valje premera 10, 30 in 50 mm Table 2. Material data and calculated sensitivity coefficients for cylinders with diameter 10, 30 and 50 mm d par. c P cA l P W/mK l A W/mK 15 lM l PA l MA EP EA GPa 200 n - aA 10-4 K-1 22 1 -3,971 aB aP -4 -1 10 K aM 10-4 K-1 12 10-3 8,50 KM sYB sYA sYM Jf kM Bm Pm tB t P A3 Bs Ms Si enota unit MJ/m3K 3,76 MJ/m3K W/mK MJ/m3 MJ/m3 GPa 209 1,26 0,646 1,680 1,654 1,227 1,034 0,129 10-3 10-5 MPa-1 10-5 MPa-1 MPa MPa MPa Js oC oC h W/m2K 5700 - oC oC C - Mn - vrednost value 4,15 66,20 25 630 660 0,30 0,01 0,171 0,894 0,711 0,501 0,402 0,033 13 14 10,7 4,18 5,08 sYP MPa 434 34 -0,006 491 123 445 850 20 460 560 1,3 1 oC 910 830 oC 539 0,66 - 0,22 0,07 območje range ± 0,15 -0,379 0,133 0,980 0 0,2 1,74 0,573 1,74 30 -0 30 0,32 1 1 1 0,714 0,22 0,7 0,1 0,1 34 2 20 5 700 20 20 0,1 0,1 10 25 25 0,1 10 mm središče center -0,844 -0,857 -0,607 -0,577 0,144 -0,455 -0,126 -0,030 -4,280 -2,127 -0,198 -0,833 -0,021 -0,279 ,065 -0,257 -0,090 -0,013 -0,013 -0,002 -0,002 -0,188 -0,138 -0,099 -1,017 0,049 -0,052 -0,044 -0,206 4,165 5,397 -0,021 -0,003 -0,062 0,315 1,080 1,330 -5,260 -0,388 površina surface 0,678 -0,644 -0,074 0,313 -5,223 -0,218 1,188 4,857 6,385 0,004 -0,049 -0,023 -0,597 30 0 0 0 0 0 0,196 0,849 1,126 -4,186 -1,150 0,057 -0,046 0,045 30 mm središče center 0,573 0,011 0,420 -0,188 8,452 -0,678 -0,132 0,042 -6,629 -0,217 0,005 0,068 -1,441 0,011 0,0005 3,793 3,181 0,110 0,539 0,002 0,086 0,260 -0,816 4,150 0,296 0,44 0,02 1,340 1,547 1,816 0,537 2,080 1,155 površina surface ,053 -0,424 -0,082 0,042 -0,063 0,363 1,740 -0,634 0,742 0,214 -0,059 -1,483 -0,320 -0,218 -0,742 0,005 0,005 0 0 0,143 0,012 -2,809 -0,228 -0,145 -0,112 -0,246 -0,093 -0,153 2,017 1,228 0,812 2,018 0,571 1,523 3,190 2,082 -0,435 4,791 -0,520 2,272 4,819 3,693 5,547 2,918 2,111 4,460 0,922 -1,338 -3,552 -2,590 0,445 -3,801 -0,601 -1,829 1,156 0,391 0,349 -0,145 0,237 -0,052 0,177 50 mm središče center -0,271 -0,448 0,007 0,063 -0,188 10,86 -4,572 -0,008 -7,453 -0,385 0,794 0,062 0,013 1,687 -0,030 0,110 0,054 0,779 4,359 0,024 -0,014 0,047 površina surface -0,110 -0,348 -0,003 -0,242 0,147 0,078 0,002 0 -0,113 0,008 0,128 0,459 -0,012 0,323 -2,265 0,524 -3,749 0,559 -0,041 stran 729 + g M g A K B M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis pri čemer je ei relativna napaka i tega parametra. Z uporabo podatkov iz preglednice 2 smo izračunali oceno relativne napake izračunanih zaostalih napetosti. Rezultati so podani v preglednici 3. where ei is the relative error of ith parameter. By us-ing data from Table 2 the estimated relative error in the residual stresses are calculated. The results are shown in Table 3. Preglednica 3. Ocenjene relativne napake osnih napetosti Table 3. Estimated relative errors of the axial residual stresses d [mm] 10 0,11 0,11 Center 30 0,11 0,07 50 0,13 0,05 Površina Surface 6 SKLEP Izdelan je bil matematični model toplotne obdelave in zbrani vsi potrebni podatki. Model smo uspešno preskusili na dveh primerih izračuna zaostalih napetosti, za katere so znani eksperimentalni rezultati. Nadalje smo izvedli analizo občutljivosti modela, s katero smo dobili kolikostno oceno vplivnosti posameznih podatkov v materialu. Prav tako je bila ocenjena relativna napaka v izračunu zaostalih napetosti. Na podlagi tega so glavne ugotovitve: 1. model toplotne obdelave je izredno občutljiv za podatke, ki so vključeni v kinetičnih enačbah; 2. model je prav tako pomembno občutljiv za temperaturni razteznostni koeficient austenita in dilatacijski koeficient prehoda med austenitom in martenzitom; 3. relativna napaka pri izračunu zaostalih napetosti je ocenjena na 13%. 6 CONCLUSIONS We have developed a mathematical model for heat treatment and collected the necessary data needed for practical calculations. We successfully tested the model in two examples for which the ex-perimental data were available. We then performed a sensitivity analysis of the model which resulted in a quantitative determination of the relative importance of the various material data. The relative error in the calculated residual stresses was also estimated. The main conclusions are as follows: 1. the model of heat treatment is very sensitive to the data which are used in the kinetic equations; 2. the model is also very sensitive to the austenite thermal expansion coefficient and the dilatation of austenite and martensite; 3. the calculated relative error for the residual stresses, using data published in the literature, are within 13%. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] 7 LITERATURA 7 REFERENCES Andrews, K.W. (1965) Empirical formulae for the calculation of some transformation temperatures. J. Iron and Steel Inst, 203, 721-727. Büchler, H., A.Rose (1969) Representation of the origin of internal stresses in work pieces of steel by means of T-T-T diagrams. Arch. Eisenhüttenwes., 40, 411-423. Bungardt, K., W.Spyra (1965) Wärmeleitfähigkeit unlegierter und legierter Stähle und Legierungen bei Temperaturen zwischen 20 und 700 C. Arch. Eisenhüttenwes., 36, 257-267. Cheng, L., Brakman, CM., Korevaar, B.M., EJ.Mittermejer (1988) The tempering of iron-carbon martensite; dilatometric and calorimetric analysis. Metall. Trans A, 19A, 2415-2426. Burke, J. (1965) The kinetics of phase transformations in metals. Pergamon, London. Christian, JW. (1965) The theory of transformations in metals and alloys. Pergamon Press, Oxford. Date, E.H.F. (1969) Elastic properties of steels. J. Iron and Steel Inst., 207, , 988-991. Davies, J., P.Simpson (1979) Induction Heating Handbook, McGraw-Hill, London. Esser, H., E.Friderich (1941) Die wahre spezifische Wärme von reinem Eisen und Eisen-Kohlenstoff-Legierungen von 20 bis 1100. Arch. Eisenhüttenwes., 17, 617-626. Fletcher, A.J. (1989) Thermal stress and strain generation in heat treatment. Elsevier Applied Science, London. Giusti, J. (1981) Ph.D. Thesis, University of Paris VI 2 jgnnatäüllMliBilrSO | | ^SSfiflMffiC | stran 730 M. Batista - F. Kosel: Analiza ob~utljivosti - A Sensitivity Analysis [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] Hildenwall, B., T.Ericsson (1978) Hardenability concepts with applications to steel, D.Done and D.Kirkaldy (eds), AIME, 579-606. Inoue, T., K.Tanaka (1975) An elastic-plastic stress analysis of quenching when considering a transformation. Int.J.mech.Sci., 17, 361-367. Inoue, T., Nagaki, S., Kishino, T., M. Monkawa (1981) Description of transformation kinetics, heat conduc-tion and elastic-plastic stress in the course of quenching and tempering of some steels. Ing.Arch., 50, 315-327. Leblond, J.B., Mottet, G., Devaux, J., J.C.Devaux (1985) Mathematical models of anisothermal phase transformations in steels, and predicted plastic behaviour. Mater. Sci. Technol., 1, 815-822. Lindemblom, B.E., Höglund, L., C. Andersson (1971) Computer simulation of hardening. J. Iron and Steel Inst., 958-961. Mendelson, A. (1968) Plasticity theory and application. MacMilan, New York. Nagasaka, Y, Brimacombe, J.K., Hawbold, E.B., Samarasakera, I.V., Hernandet-Morales, B., S.E.Chidac (1993) Mathematical model of phase transformations and elasto-plastic stress in the water spray quenching of steel bars. Metallurgical Transactions A, 24a, 795-808 Ridley, N, H.Stuart (1970) Partial molar volumes from high temperature lattice parameters of iron-carbon austenites. Met. Sci. 4,. 219-221. Sjöström, S. (1983) Berechung der Abschreckeigenspannungen in Stahl. Proc. Int. Conf. Eigenspannung-Entstehung-Berechnung-Messung-Bewertung, Karlsruhe, Deutsche Geselschaft für Metallkunde, 155-185. Steven, W., A.G. Haynes (1956) The temperature of formation of martenzite and bainite in low-alloy steels. J. Iron and Steel Inst., 349-359. Yu, HJ., Wolfstieg, U., E.Macherauch (1978) Calculation of residual stresses with special finite element program. Arch. Eisenhüttenwes., 49, 499-503. Yu, HJ., Wolfstieg, U., E.Macherauch (1980) On the influence of the diameter on the residual stresses in oil and water quenched steel cylinders. Arch. Eisenhüttenwes., 51, 195-200. Naslova avtorjev: Dr. Milan Batista Fakulteta za pomorstvo in promet Univerze v Ljubljani Pot pomorščakov 4 6320 Portorož Prof.dr. Franc Kosel Fakulteta za strojništvo Univerza v Ljubljani Aškerčeva 6 1000 Ljubljana Authors’ Addresses: Dr. Milan Batista Faculty of Maritime Studies and Transport University of Ljubljana Pot pomorščakov 4 6320 Portorož, Slovenia Prof.Dr. Franc Kosel Faculty of Mechanical Eng. University of Ljubljana Aškerčeva 6 1000 Ljubljana, Slovenia Prejeto: Received: 10.10.2000 Sprejeto: Accepted: 20.12.2000 stran 731 bcšd00