On the Role of Kinematics in Constructing Predictive ...

TMS (The Minerals, Metals & Materials Society),

On the Role of Kinematics in Constructing Predictive Models of Austenite Decomposition Mark T. Lusk1 , Wei Wang2 , Xiaoguang Sun3 , Young-Kook Lee4 1

2

Materials Science Program, Division of Engineering Colorado School of Mines Golden, Colorado USA 80404

Technology Center Research and Development Department Baoshan Iron and Steel Co., LTD 1 Kedong Rd, Baoshan District Shanghai, China 201900 3

Department of Computing and Information Technology Fudan University Shanghai, China, 200433 4

Department of Metallurgical Engineering Yonsei University Seoul, Korea 120-749 Abstract

Computational models of austenite decomposition should provide predictions of external shape and internal microstructure as functions of time and temperature. This information may be used directly, or it can be sent to other simulators in order to make predictions about material working properties. It is often the case that such decomposition models are constructed using kinematic information—i.e. data that relate external shape change to the internal microstructural state. These models are based on the supposition that a differential shape change can be related to a differential microstructural change given that the base state is known. This is a purely kinematic link that does not depend on the rate of decomposition. A starting point is often to estimate shape change using a weighted average of the lattice parameters of each phase present, and this is adopted in the present work. Based on a review of the literature values for lattice parameters associated with low carbon steels, an optimal set of lattice values is obtained. These thermophysical functions are then used in a new forward fitting algorithm to obtain kinetic parameters for an internal state variable model of austenite decomposition. The key idea here is that the kinetic parameters are optimized so as to match dilatometry data without the need to back out any phase fraction data prior to fitting. The model is applied to a class of industrial steels to demonstrate the accuracy of the lattice parameters and the viability of the new forward fitting methodology. Introduction Models of austenite decomposition can be used to track the phase fraction of product phases as a function of time and temperature. The progress of isothermal, diffusive transformations, for example, can be approximated using the Johnson-Mehl-Avrami-Kolmorgorov

(JMAK) equation [1, 2, 3]: ϕ = 1 − exp (k tn )

(1)

where k and n are empirical parameters and ϕ is the product phase fraction. This equation can also serve as a basis for considering non-isothermal transformation through an appeal to the Rule of Additivity as established by Avrami [4], extended by Cahn [5] and restricted by Lusk and Jou [6]. The JMAK equation can be re-written in the rate form ϕ• = ν(T ) f (ϕ)

(2)

where ν(T ) is the effective mobility as a function of temperature and the dot (•) signifies differentiation with respect to time. This is the form taken in a large number of recent internal state variable (ISV) formulations of austenite decomposition with one equation of this form for each product phase [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Such models are appealing because their differential form allows them to be coupled naturally to governing equations for the thermal and mechanical aspects of a given process, and they are easily solved with minimal computational resources. A differential approach can also be used in place of the Koistinen-Marburger [18] or Skrotzki [19] equations in order to track the athermal formation of martensite [6, 20]. This admits the simulate of martensite formation that is burst-like and exhibits sigmoidal growth with decreasing temperature [21]. Algebraic expressions like the Koistinen-Marburger and Skrotzki equations are also difficult to combine with diffusive kinetics to capture the competing formation of martensite and diffusive products (e.g., Edwards and Kennon [22] showed that the lower bainite 1% curve could extend below Ms temperature for 1.44C-0.38Mn-0.17Cr-0.02Ni steel where isothermal martensite was also observed). This developing interest in differential models for phase fraction kinetics is consistent with current demands for quantitatively accurate predictive tools that best take advantage of available computer resources and increasingly precise experimental data. A framework which allows general differential models to be developed for both diffusive and martensitic transition kinetics would, in fact, allow for greater flexibility in generating quantitatively accurate predictive tools and would enable the kinetics to be coupled naturally with the thermal and mechanical aspects of a given process. This is especially true when an internal state variable approach is also used to model inelasticity [23, 16]. Such differential models may also be more easily adapted to be consistent with fully micromechanical simulations of a given process [24]. Internal state variable simulations of austenite decomposition typically rely on dilatometry data in order to fit many of the model parameters. While dilatometric measurement has been used to determine the start and end temperatures of phase transformation for at least sixty years [25, 26], it is the ability to continuously monitor dimensional change that makes the approach particularly useful in fitting model parameters. This is because the model prediction can be compared with experimental data at each data collection time and the parameters optimized so as to minimize the discrepancy. Critical to this approach is the ability to convert the dilatometry data to phase fraction so that it can be compared with model predictions. For isothermal transformations that go to completion, it is often assumed that the normalized transformation strain is equal to the product phase fraction [27], but this does not account for the effect of carbon redistribution during transformation. Moreover, it is not clear how this approach can be extended to nonisothermal transformations. This has motivated a number of efforts to make quantitative links between product phase fraction and dilatometric strain. The first theoretical analysis of this sort that we are aware of is that of Onink et al. [28] who considered lattice dilation caused by changes of temperature and carbon

content on hypo-eutectoid steels. The analysis is only valid for isothermal decomposition of austenite into ferrite and cementite, although Li et al. [29] expanded its application to hyper-eutectoid steels. Kop et al. [30] analyzed continuous cooling transformation with a consideration of dynamic carbon enrichment during ferrite transformation. However, the application is limited to ferritic and pearlitic or bainitic reactions under continuous cooling conditions. As an aside, they did not use consistent units in their calculation and their calculated lengths were larger than the experimental data. The authors ascribed this error to the plastic deformation due to phase transformation and arbitrarily enlarged the measured length using a fitting constant. Recently Zhao et al. [31] used a similar method to analyze dilatometric results of continuous cooling, but their model has the same limitations as that of Kop et al. These and other links between phase fraction and dilatometric strain are reviewed as part of the current study, and a table is constructed for the lattice parameters that we have adopted for our own research. Lattice parameters are used to convert dilatometric data to phase fraction, and model predictions are compared to this converted data. This amounts to solving an inverse problem prior to fitting model parameters. We propose that it is more reasonable to use the lattice parameters to convert the simulated phase fractions to strain and to compare this data directly with the dilatometry measurements. This obviates the need to perform a differential inversion of the dilatometric data. Optimization then results in a simulation that most closely approximates the observed dimensional changes. The conversion between phase fraction and strain is then a part of the optimization process as opposed to being a pre-processing step, and this allows some optimization of the strain-to-phase conversion as well. For instance, critical temperatures are often used to delineate temperature ranges over which product phases can form, but these temperatures are based on equilibrium measurements and may need to be corrected for nonisothermal conditions. In the same vein, it is necessary to make an assumption about which phase is forming in order to convert dilaometry data to phase fraction, but this is a kinetic issue that should be a part of the ISV model. Finally, the forward fitting methodology allows the transformation model to be coupled to thermo-mechanical simulation. Under the assumption of homogeneous material properties and specimen conditions, this allows fitting to be undertaken under loaded conditions. The forward fitting approach can also be used, though, to fit kinetic parameters even when the specimen is known to exhibit spatial variation in properties and phase provided the thermo-mechanical simulator captures spatial variation as well. After a literature survey of the links between phase fraction and strain, we demonstrate the forward fitting algorithm by applying it to a class of low carbon steels. Lattice Parameters In order to be able to make a quantitative link between dilatometer measurement and product phase fraction the lattice parameters of each phase must be known precisely. This has been the subject of significant activity of late and we will only attempt to highlight these efforts. In lattice parameter estimates found in the literature, three different measures of the carbon content in steel are used: weight percent denoted [C%]; atomic percent denoted [Cc ]; and, carbon atoms per 100 iron/substitutional alloy lattice sites [C]. These metrics

are equivalent as is clear from the two relations given below: [Cc ] =

[C%] k P

(3)

(Wi /AWi )

i=1

[C] =

100[Cc ] 100 − [Cc ]

Here Wi is the wt.% of chemistry element i, AWi is the atomic weight of element i, and k is the total number of elements. Austenite Austenite Lattice Parameter at Room Temperature The austenite lattice parameter depends upon both chemistry and temperature. The literature data on the austenite lattice parameter of pure iron at room temperature in Table 1 was obtained by extrapolating the lattice parameter measured from retained austenite in Fe-C martensitic steels to zero carbon content [32, 33, 34, 35, 36, 37]. In this summary, room temperature is assumed to be 25 ◦ C because the exact temperature pertaining to the lattice parameter data have not been reported quantitatively in most cases. Table 1. Austenite lattice parameter of pure iron at room temperature (nm) obtained by extrapolating the lattice parameter measured from retained austenite of steels to zero carbon content. *This value was obtained by a least square fitting of the other data in this table. aγ (nm) Reference 0.3548 [32] 0.3580 [33] 0.3543 [34] 0.3546 [35] 0.3477 [36] 0.3553* [37] The significant scattering of the austenite lattice parameters in Table 1 is most likely attributable to the fact that measurements were performed on post-quench, retained austenite that was heavily deformed. The measurement might be improved if the austenite was unstrained, and this can be accomplished in two ways. Some researchers have measured the austenite lattice parameters of highly alloyed steels (e.g. Fe-Mn alloys and austenitic stainless steels) for which austenite is stable at room temperature. They then extrapolate this data to zero alloying element content. Others have directly measured the lattice parameter of annealed austenite at elevated temperatures then extrapolate this data to room temperature. The austenite lattice parameters obtained through these techniques are summarized in Tables 2 and 3, respectively.

Table 2. Austenite lattice parameter of pure iron at room temperature (nm) from measurements of highly alloyed steels for which austenite is stable at room temperature. These measurements are then extrapolated to zero alloying element content. Material aγ (nm) at 25 ◦ C Test Temperature ( ◦ C) Reference Fe-Mn alloys 0.35618 25 [38] Fe-Mn alloys 0.35730 25 [39] Fe-Ni alloys 0.35553 25 [40] stainless steels 0.35770 25 [41] pure iron 0.35671 907, 977 [42] Table 3. Austenite lattice parameter of pure iron at room temperature (nm) obtained from annealed austenite of pure iron at elevated temperatures. *We have calculated this lattice parameter based on the volume per iron atom in FCC pure iron measured by Hume-Rothery [48]. Material aγ (nm) at 25 ◦ C Test Temperature ( ◦ C) Reference Pure iron 0.35671 907, 977 [42] Pure iron 0.35773 916, 1388 [43] Pure iron 0.35817 912, 1255 [44] Pure iron 0.35916 950, 1361 [45] Pure iron 0.36007 920, 1070 [46] Pure iron 0.35654* 910, 1388 [48] On the whole, the lattice parameters of unstrained austenite in Tables 2 and 3 are greater than those of deformed austenite of Table 1. Note that the latest results by Onink et al. [42] using neutron diffraction, which can probe a larger volume of material than X-ray diffraction, show good agreement with Hume-Rothery’s value [48]. Dependence of austenite lattice parameter on carbon content As with the nominal value of the austenite lattice parameter, the linear carbon effect on the lattice has been measured for both strained and unstrained austenite. The carbon coefficient measured from strained austenite, as reported by a host of investigators, is summarized in Table 4. Table 4. The dependence of austenite lattice parameter on carbon content. *Cheng et al.’s value was obtained by the least square fitting through the other data in Table 4. kc ( nm/at.%C) Reference 9.807 [32] 6.910 [33] 11.077 [38] 10.815 [34] 10.375 [35] 10.027 [36] 10.5* [37]

Cheng et al. [37] studied the dependence of the austenite lattice parameter on carbon content for unstrained austenite. Though they did not carry out experiments to measure the austenite lattice parameters of Fe-C steels, they combined the value of 0.3573 nm [39] for FCC iron lattice parameter at room temperature, which was measured by extrapolating the lattice parameter data of Fe-Mn austenite to zero manganese content, with data for FeC specimens measured at elevated temperatures where the specimens are fully austenitic [49] . The resulting carbon coefficient is 7.50 nm/at.%C. A similar result was obtained by Dyson and Holmes [41], who examined the effects of alloy elements on the austenite lattice parameter in austenitic stainless steels. They reported a carbon coefficient of 6.50 nm/at.%C. Ledbetter and Austin [50] also measured the austenite lattice parameter in austenitic stainless steels with different carbon and nitrogen contents. They reported a carbon coefficient of 7.83 nm/at.%C and a nitrogen coefficient of 8.61 nm/at.%N. Most recently, Onink et al. [42] directly measured the lattice parameters of unstrained austenite of Fe-C steels at elevated temperatures using neutron diffraction. Their result is 7.83 nm/at.%C, which is very close to the values reported by Cheng et al. [37] and Ledbetter and Austin [50]. Analytical approach for carbon dependence of austenite lattice parameter Ledbetter and Austin [50] made a simple lattice model to explain the lattice expansion of austenite due to the addition of interstitial atoms. They assumed that an interstitial atom occupies only an octahedral site in FCC unit cell and that both host and interstitial atoms are rigid and incompressible. √ In that study, the FCC iron atomic radius was taken to be RF e = 2aγ /4 = 0.1271 nm. For carbon, Pauling’s tetrahedral-covalent radius of 0.0771 nm was used [51]. However, a correction must be implemented to account for a change in coordination number from CN=4 to CN=6. When a diamond cubic lattice (CN=4) transforms to a simple cubic lattice (CN=6), the ratio of atomic radius is: RCN=6 2 = √ = 1.155 RCN=4 3

(4)

This gives a corrected carbon radius of 0.0890 nm. A higher coordination number means that fewer valence electrons contribute to binding, thus there is less tight binding between adjacent atom pairs. Using these atomic radii, it is straightforward to calculate the unit cell dimensional change due to interstitial atoms in the FCC octahedral hole: aγ,F e Rc RF e NF CC

= = = =

Vγ,F e = Vγ,F eC = ∆V V

=

∆L = L

0.36306 nm 0.089 nm √ 2aγ /4 = 0.128361 nm 4 (2 (Rc + RF e ))3 nm3 NF CC a3γ,F e nm3 NF CC (Vγ,F eC − Vγ,F e ) Cγ0 Vγ,F e (1 − Cγ0 ) ¶1/3 µ ∆V −1 1+ V

(5)

The parameter NF CC is the number of iron atoms in a FCC unit cell, Vγ,F e is the volume per iron atom without a carbon atom in an octahedral site, Vγ,F eC is the volume per iron atom with a carbon atom in an octahedral site, and Cγ0 is the atomic percent of carbon in austenite. The final step in this analytical estimate for the carbon coefficient kc of the austenite lattice parameter, was to make a linear fit for aγ,F e × ∆L as a function of atomic percent of L carbon. The carbon coefficient was found to be kc = 8.892 × 10−4 nm/at.%Cγ0 . In order to compare their estimate for carbon dependence with an experimentally obtained value, Ledbetter et al. used the equations above with the austenite lattice parameter of measured by Onink et al. [42] to obtain a carbon coefficient of kc = 7.83 × 10−4 nm/at.%Cγ0 . The calculated value of kc falls within the range of the previous experimental results and is in between the value measured from strained austenite and the value measured from unstrained annealed austenite. This calculated kc value is a little larger than that of Onink et al. [42] who fitted the lattice parameter data as a function of total carbon content in the specimens which was determined by chemical analysis. Strictly speaking, though, the actual carbon content dissolved in the austenite matrix should be used for fitting. This means that the amount of carbon atoms segregated to defects in austenite such as grain boundaries and dislocations should be subtracted from total carbon content. Krauss [52] reported that even during quenching, a considerable amount of carbon atoms segregated to austenite grain boundaries and caused quench embrittlement—particulary in steels with a carbon content greater than 0.5 wt.%C. If the segregation of carbon atoms had been taken into account in the work of Onink et al, they would have obtained larger carbon coefficient that would be, presumably, closer to the analytical value of Ledbetter el al. Dependence of austenite lattice parameter on temperature The table below shows the linear thermal expansion coefficients of austenite, β γ for pure iron and steels and their references. Table 5. Austenite lattice parameter of pure iron at room temperature (nm) obtained by extrapolating the lattice parameters measured from retained austenite of steels to zero carbon content. *We have calculated this lattice parameter based on the volume per iron atom in FCC pure iron measured by HumeRothery [48]. Steel βγ Iron Iron Iron Iron Iron* Iron Fe-Mn-Si-C Fe-Ni-Si-C Fe-Ni-Si-Cr-Mn-Mo-V-C Fe-Mn-Si-C Fe-Cr-Mn-Si-C

(×10−6 ◦ C−1 ) Reference 23.6 [43] 21.5 [44] 20.3 [45] 19.5 [46] 23.2* [48] 23.0 [53] 21.2 [54] 18.4 [54] 18.0 [54] 21.0 [55] 18.9 [56]

As indicated in this table, it is standard practice to assume that the thermal expansion coefficient for austenite is not a function of carbon content. However, Onink et al. [42] found

that this parameter does change with carbon and fitted the measured thermal expansion coefficients of Fe-C steels as a function of carbon content. The result is: β γ = 24.9 − 0.5

(6)

Ferrite A simplifying assumption that is typically made is that ferrite is free of carbon so that the lattice parameter for ferrite does not change with carbon concentration and is only a function of temperature. The following formula for the ferrite lattice parameter is from Onink et al. [42] and is based on ferrite lattice parameters measured in the temperature range of 800 to 1200 K: aα = 0.28863 [1 + β α (T + 273 − 800)] β α = 1.75 × 10−5 × 10−5 ◦ C−1

(7)

Here as throughout this paper, the temperature is in degrees Celsius ( ◦ C) and the lattice parameter units are nanometers. An alternate expression, with carbon dependence, is offered by Bhadeshia et al. [47]: £ ¤ aα = 0.288634 1 + 1.59 × 10−5 (T + 273 − 800) (8) " # (0.28664 − 0.000279[Cc ])2 (0.28664 + 0.002496[Cc ]) − 0.286443 × 1+ 3 × 0.286642 The values 0.28609 nm and 0.28633 nm for ferrite lattice parameter at room temperature calculated using Equations (7) and (8) compare well with both the value 0.28610 nm obtained by Roberts [32] from the changes in lattice parameters of martensite with zero carbon content at room temperature and the value 0.28600 nm from the measured volume per iron atom as a function of temperature by Hume-Rothery [48]. The JCPDS card 6-0696 gives the lattice parameter of pure B.C.C. iron as 0.28664 nm. The ferrite lattice parameter at various temperatures calculated with Equation (7) shows good agreement with literature data measured using pure iron (Fig. 7 in reference [42]), but we have adopted the formula of Equation (8) so as to include the admittedly small dependence on carbon content. Cementite Takahashi and Bhadeshia [27] used the following formulae of cementite lattice parameters using cementite lattice parameters at room temperature and the ferrite thermal expansion coefficient β α because they did not have experimental data for this parameter: aθ bθ cθ βα

= = = =

0.451 × (1 + β α (T − 298)) nm 0.508 × (1 + β α (T − 298)) nm 0.673 × (1 + β α (T − 298)) nm 1.1826 × 10−5 ◦ C−1

(9)

However, the ferrite thermal expansion coefficient is much less than other literature values including that used by Onink et al. and given in Equation (7). An alternate set of cementite parameter is proposed by Onink et al. obtained by extrapolating room temperature lattice parameters to elevated temperatures using the temperature dependent mean expansion formulae derived by Stuart and Ridley [57]:

aθ bθ cθ βθ

= = = =

0.45234 × (1 + β θ (T − 298)) nm 0.508 × (1 + β θ (T − 298)) nm 0.673 × (1 + β θ (T − 298)) nm 1.1826 × 10−5 ◦ C−1

(10)

As an aside, Stuart and Ridley’s thermal expansion coefficient [57] is less than the value of 1.1826 × 10−5 ◦ C−1 obtained by Lifshitz [53]. For either set of lattice parameters, the unit cell volume is given by Vθ = aθ bθ cθ . We offer a third formula for Vθ that is a linear fit to the data generated by Reed and Root [58]: Vθ = 0.1593333 + 7.83333 × 10−6 T (11) and it is this formula that was used in the fitting algorithm developed in this work because it provided a better match with the 51xx dilatometry data used in our study. Martensite Martensite lattice parameter without carbon at room temperature The lattice parameter of unalloyed martensite (BCC) is substantially the same as ferrite (BCC), and can be obtained by extrapolating the lattice parameters of carbon steels to zero carbon content. Roberts [32] collected many previous values showing the effects of carbon content on the martensite lattice parameter and expressed a best fit to all of the data in following equations: aα0 = 0.28610 − 0.0002898[Cc ] nm cα0 = 0.28610 + 0.0025855[Cc ] nm

(12)

Some researchers [53, 60] used the lattice parameter of unalloyed martensite (BCC) of 0.28664 nm in JCPDS card 6-0696 to fit the martensite lattice parameter as a function of carbon content. As mentioned in ferrite lattice parameter, Roberts’ lattice parameter of unalloyed martensite of 0.28610 nm is very close to not only Hume-Rothery’s value of 0.28600 nm but also Onink’s value of 0.28609 nm. The latter value is the most recent experimental data using neutron diffraction at elevated temperatures. Dependence of martensite lattice parameter on carbon content Both Cheng et al. [59] and Jack [60] adopted 0.28664 nm in JCPDS card 6-0696 as the lattice parameter of pure BCC iron, fitted some literature data as Roberts did, and proposed as follows, respectively: aα0 = 0.28664 − 0.00002[Cc ] nm cα0 = 0.28610 + 0.00004[Cc ] nm

(13)

aα0 = 0.28664 − 0.0002898[Cc ] nm cα0 = 0.28610 + 0.00247[Cc ] nm

(14)

Equations (12), (13) and (14) show good agreement with one another. In our study, though, Roberts’ equation (12), showing coincidence to ferrite lattice parameter measured by Onink et al. [42], is used for the calculation of martensitic transformation strain.

Dependence of martensite lattice parameter on temperature Data for the linear thermal expansion coefficient of martensite, β m , is relatively sparse, and this may be due to the difficulty in avoiding tempering. Lifshitz [53] reported that the linear expansion coefficient of martensite is β m = 1.15 × 10−5 ◦ C−1 . Cheng et al. [61] measured the dilatational change of martensite specimen during heating at 10 ◦ C/min from -196 to 480 ◦ C with an 1.13 wt.%C steel. Before heat-up, the specimen was immersed into liquid nitrogen and had 6 volume percent of austenite after liquid nitrogen quenching. Because they carried out this experiment to investigate the tempering behavior, the dilatometric data in the tempering temperature range was of interest and the linear thermal expansion coefficient of martensite was not calculated. We therefore picked two sets of dilatometric and temperature data before and after temperature range in which the shrinkage of the specimen occurred due to tempering effect, and then calculated the linear thermal expansion coefficient of martensite using initial specimen length and retained austenite volume fraction. While the value of was β m = 0.8446 × 10−5 ◦ C−1 at the temperatures below about 15 ◦ C, it was 1.47657 × 10−5 ◦ C−1 at the temperatures above about 380 ◦ C. The lower temperature value is not affected by tempering and is about half of that of pure ferrite measured by Onink et al. [42] of 1.75 × 10−5 ◦ C−1 . This makes sense because the low temperature value is associated with a large concentration of defects in the martensite. The high temperature value is much larger because it measured after tempering is complete and the density of defects has defects; it is still lower than expansion coefficient of pure ferrite because of the lower thermal expansion coefficient of cementite that formed during tempering. Lifshitz’s value of 1.15 × 10−5 ◦ C−1 is very close to the average of these two values (1.16 × 10−5 ◦ C−1 ). As an additional check on his estimate, Lifshitz quenched an SAE 4120 specimen in liquid nitrogen to obtain 100% martensite and followed this with a rapid heat-up from 298 to 423 K within 1 second using a dilatometer. It is thought that there was little tempering effect during heat-up because of the rapid heating rate and low maximum temperature. The value obtained for the thermal expansion coefficient of martensite was β m = 1.156×10−5 ◦ C−1 — the value adopted for our study. Lattice Parameters Adopted Table 6. The lattice parameters adopted used in the forward fitting algorithm. Vol. per Lattice Parameters (nm) and Unit Cell Volumes Atom Fe Phase (nm3 ) (nm3 ) Aust.

Ferr. Cem.

aγ = (0.363086 + 7.52 × 10−4 [C]) × {1 + (T + 273 − 1000) (24.92 + 0.61[C]) × 10−6 } aαh= 0.288634 [1 + 1.59 × 10−5 (T + 273 − 800)] i 2 (0.28664+0.002496[Cc ])−0.286443 × 1 + (0.28664−0.000279[Cc ])3×0.28664 2 Vθ = 0.15933 + 7.833333 × 10−6 T

cm = (1 + km ) (0.28664 + 0.00256[Cc ]) km = 1.156 × 10−5 (T − 25)

[53]

[47]

(fit of data from [58])

am = (1 + km ) (0.28664 − 0.00028[Cc ]) Mart.

[62]

vα =

a3γ 4

vα =

a3α 2

vθ =

Vθ 12

[32, 59] [32, 59]

vm =

a2m cm 2

Critical Temperatures A thermodynamic algorithm developed by Kirkaldy and Venugopalan [63] was used to generate tables of Ae3 , Ae1 , and Acm temperatures as functions of carbon content for each alloy studied. Although algebraic fits are available [64], the thermodynamic algorithm is also easy to implement and is valid over a broader range of chemistry. A quadratic fit was then made to the variation in these critical temperatures with carbon, and the analytic expressions were used to construct a function for the equilibrium volume fraction of ferrite as a function of carbon and temperature. The bainite start temperature is taken to be: Bs = 656.0 − 57.7 C − 35.0 M n − 75.0 Si − +15.3 Ni − 34.0 Cr − 41.2 Mo + abs + bbs C

(15)

Note that two additional constants abs and bbs have been added to the usual expression. These constants are optimized as part of the forward fitting process to be discussed subsequently. The martensite start temperature is given by the Andrews product formula: Ms = 512.0 − 453.0 C − 16.9 N i + 15.0 Cr − 9.5 Mo + 217.0 C 2 − 71.50 C M n − 67.60 C Cr − 7.50 Si

(16)

Kinetic Equations The quench model is a mathematical simulation of phase transformations that occur in low alloy steels on cooling. The routine tracks the atomic fraction of austenite, ferrite, pearlite, bainite, and martensite as a function of time and temperature [15, 16, 20]. A state variable formula of a single-phase transformation can be written as: ϕ• = ν(T )ϕα (1 − ϕ)β ϕ(0) = 0.0001

(17)

where ϕ is the atomic fraction of the product phase whose time rate of change (denoted by a dot on the top) depends on the current atomic fraction and temperature T . The nonnegative parameters α and β control the shape of the atomic fraction versus time curve which is usually a sigmoidal curve unless α or β becomes zero. According to above evolution equation, the atomic fraction ϕ increases from 0.0001 to 1.0. Given α, β and ν(T ), the above equation can be solved to obtain the atomic fractions of parent and product phases with arbitrary thermal profile T (t). Diffusive Kinetics The state variable approach has been extended to include multi-phase decomposition of austenite, ϕa , to three diffusive phases: ferrite, ϕf , pearlite, ϕp , and bainite, ϕb . The final equations for predicting microstructure are expressed by a set of ordinary differential equations:

ϕ•f ϕ•p ϕ•b

(

® β ­ α ν f (T )ϕf f ϕa f ϕf,eq − ϕf , Bs < T < Ae3 = 0, otherwise ½ ¡ ¢β −1 a ν p (T )ϕpp ϕa 1 − ϕp p , Bs < T < Ae1 = 0, otherwise ½ ¡ ¢a ν b (T ) ϕb + ϕEn,b b ϕa (1 − ϕb )β b −1 (ϕb,stasis − ϕb ), = 0, otherwise

(18)

Ms < T < Bs

where ϕf,eq the temperature dependent equilibrium atomic fraction of ferrite. The dot (•) signifies differentiation with respect to time and < . > is a McCalley bracket that gives the value of the argument if it is greater than zero and gives zero otherwise. The expression, ϕEn,b := γ b (ϕf + ϕp ), reflects the influence of existing ferrite and pearlite on the rate of formation of bainite. In our diffusive model, both ferrite and bainite have limits to the maximum atomic fractions. When the temperature is between ferrite start temperature Ae3 and pearlite start temperature Ae1 , the equilibrium atomic fraction of ferrite ϕf,eq can be estimated by the lever rule using Ae3 curve and initial carbon content. When temperature is below Ae1 , ϕf,eq is dependent on the kinetics of the cementite and can be estimated by the lever rule using Ae3 and Acm curves extrapolated below Ae1 and initial carbon content. For bainite, there is a stasis atomic fraction ϕb,stasis corresponding to the incomplete bainitic transformation. It is well known that, for high alloy steels, there is usually a gap in TTT diagram between the ferrite and bainite "C" curves. For low alloy steels, the overlapping makes ferrite C curve block the incomplete bainite transformation region, but that region should still be there theoretically. In our model, the stasis effect is modeled by an increasing maximum atomic fraction of bainite from 0 (at bainite start temperature) to a maximum value (less than 1 of course) linearly with decreasing temperature below the bainite start temperature. In particular, the our routine uses ¸ · τ b1 − T ϕb,stasis (T ) = min λb1 , , T < τ b1 (19) λb2 The parameter λb1 sets the maximum atomic fraction that bainite can achieve, and λb2 sets the temperature range over which the stasis atomic fraction goes from zero to its maximum value. The mobility terms, ν i (T ), have the following form ( h ³ ´i2 T −τ ν f0 exp [− (ω f1 + ω f2 gf (T )) gf (T )] , gf (T ) = ln τ f 2f 1 , T > τ f1 ν f (T ) = (20) 0, otherwise ( h ³ ´i2 p1 ν p0 exp [− (ω p1 + ω p2 gp (T )) gp (T )] , gp (T ) = ln T −τ , T > τ p1 τ p2 ν p (T ) = 0, otherwise ( h ³ ´i2 ν b0 exp [− (ω b1 + ω b2 gf (T )) gb (T )] , gb (T ) = ln τ b1τ b2−T , T < τ b1 ν b (T ) = 0, otherwise with coefficients ν i0 , ω f1 and ω f 2 non-negative. These mobility functions are non-negative and are bell shaped with maximum values at T = τ f1 + τ f 2 , T = τ p1 + τ p2 , and T = τ b1 − τ b2 , respectively. The coefficients ω i1,2 are used to adjust the asymmetry of the mobility dependence on temperature.

Carbon Dependence In situations where the kinetic model is to be used for parts with a variation in carbon content, it is useful to make the kinetic parameters themselves vary with carbon content in the austenite. However, having parameters that depend on the dynamic carbon content in the austenite allows the simulator to explicitly account for carbon enrichment and depletion during the quench. We therefore make the following parameters linear functions of dynamic carbon concentration (LFDC): αi , β i , ω i1 , ω i2 , τ i1 , τ i2 , γ b , γ m , λb1 , and λb2 . For instance, αf = aαf + bαf [Cc ] . (21) Martensite Kinetics The martensite transformation is assumed to be athermal but is written and solved in a rate form: ¡ ¢α ϕ•m = ν m ϕm + ϕEn,m m (1 − ϕm )βm −1 ϕa T • U (Min(Tmin − Ms ) − T ) (22) Here U is the Heavyside step function: ½ 0, x ≤ 0 U(x) = 1 x>0

(23)

The temperature Tmin is the lowest temperature attained, and αm , β m , and ν m are algebraic functions of carbon level: αm = αm0 + αm1 [Cc ] β m = β m0 + β m1 [Cc ] ν m = ν m0 + ν m1 [Cc ] + ν 2m2 [Cc ] + ν 3m3 [Cc ]

(24)

The expression, ϕEn,m = γ m (ϕf + ϕp + ϕv ), reflects the influence of existing diffusive phases on the rate of formation of martensite. A Forward Fitting Methodology With the lattice parameters and austenite decomposition model in place, the next step is to fit the 59 kinetics parameters using experimental dilatometry data. Dilatometry sets of time, temperature and linear strain are read into a Fortran code. The code uses the time and temperature as input for the quench simulator to estimate the phase fractions at each time step. The lattice parameters, and a simple rule of mixtures, is then used to generate a prediction for the linear strain as a function of time. The least mean square difference between measured and predicted dilatometry is taken as an objective function for each data set. The sum of these objective functions is then minimized by a commercial optimization routine [65] by varying the 59 kinetic parameters. The optimization software was set to use a combination of simplex, genetic, exhaustive search, hypercube, factorial, orothogonal matrix and Monte Carlo algorithms and we have used it with success on a number of optimization projects in the past. Application The forward fitting algorithm was applied to a set of 51xx alloys with an ASTM grain size of 9. The alloys were made by Republic Steel in special 100-pound heats and have a chemistry in wt. % shown in Table 7.

Figure 1: Temperature profiles for the 18 tests used to fit the 51xx kinetic parameters. When referring to individual tests for each carbon level, the tests are in order of decreasing cooling rate. Table 7. 51xx chemistry. Mn Si Ni Cr Mo 0.84 0.22 0.15 0.8 0.04 Three carbon levels were considered: 0.23, 0.39, and 0.60 wt. %. An MMC axial dilatometer was used, and all experiments was performed at Oak Ridge National Laboratories. The cylindrical specimens, with a length of 8mm and a diameter of 3mm, were machined with a ferrite/pearlite microstructure. While maintaining a vacuum, all specimens were induction heated to 900 ◦ C in 30 seconds, held for 5 minutes and then rapidly quenched using Helium to room temperature. After this settling run, specimens were reheated at the same rate, held again for 5 minutes, and then cooled at the rate prescribed for each test. Eighteen continuous cooling tests were used to fit the kinetic parameters—9 tests of the low carbon material, 6 tests for the medium carbon material, and 4 tests for the high carbon material. The cooling curves are shown in the figure below. The 9 martensite parameters were fitted first by considering only the most rapid quenches for each alloy. These parameters were then held fixed while the remaining 50 parameters were fitted using all 18 data sets. A final optimization run was then made wherein all 59 parameters could be optimized at once. Each iteration of the optimization routine ran in approximately 0.3 seconds. The 59 fitting parameters are provided in Tables 8 and 9. aνf aωf1 aωf2 aτ f 1 aτ f 2 aαf aβf bνf bωf 1 bωf 2

1456. 20.05 337.3 207.2 443.7 1.178 19.82 347.2 77.05 816.7

bτ f 1 bτ f 2 bαf bβf aνp aωp1 aωp2 aτ p1 aτ p2 aαp

3.61 3.62 0.1751 50.08 19.14 0.0608 0.398 282.99 13.21 0.504

aβp bνp bωp1 bωp2 bτ p1 bτ p2 bαp bβp abs bbs

12.60 27.36 0.0143 0.0900 147.41 20.44 0.1429 -1.105 45.26 -33.19

Table 8. Ferrite and pearlite kinetic parameters. The units of each parameters follow directly from the equations in which they appear.

aνb aωb1 aωb2 aτ b1 aτ b2 aαb aβb aλb1 aλb2 aγb

1.180 8.787 8.550 8.940 400.50 0.00869 5.198 2.019 142.0 -3.747

bνb bωb1 bωb2 bτ b1 bτ b2 bαb bβb bλb1 bλb2 bγb

1.1846 14.48 37.57 2.421 23.89 0.2465 -3.143 1.844 88.15 -5.039

ν m0 ν m1 ν m2 ν m3 αm0 αm1 β m0 β m1 γm

7728. 2199. 596.5 315.1 33.74 1.730 243.3 28.43 2.000

Table 9. Bainite and martensite kinetic parameters. The units of each parameters follow directly from the equations in which they appear.

The optimized kinetic parameters were used to generate the strain predictions shown in Figures 2, 3 and 4. Also shown in these plots is the original dilatometry data. The simulator does a reasonable job of reproducing the experimental strains. In this sense, the model predictions are only a measure of goodness of fit. Conclusions A forward fitting methodology has been introduced for fitting the kinetic parameters of internal state variable models of austenite decomposition. Rather than convert dilatometry data to phase fraction for the purpose of fitting, the simulator uses lattice parameters and critical temperatures to produce strain data. Parameter fitting is accomplished by an iterative process that minimizes the difference between experimental and simulated strain measurements. The new method was applied to a class of 51xx steels in order to develop a set of carbon dependent kinetic parameters, and the results indicate that the method is viable. The forward fitting method relies on a precise knowledge of the lattice parameters for each phase as a function of temperature and dynamic carbon content. Any effect of defect structures (grain boundaries, dislocations) on lattice parameters is not accounted for in this algorithm. An implicit way of dealing with such defects is to include a fitting term on the lattice parameters themselves to be optimized along with the kinetic parameters. We have implemented this procedure and do obtain a better fit to the dilatometric data as a result. However, a primary motivation for basing our routine on lattice parameters was to bring more thermo-physical information into ISV modeling, so the lattice parameter fitting was de-activated for the results shown. The forward fitting approach could be generalized so that the transformation simulator is coupled to a thermo-mechanical simulator in order to account for elastic and plastic deformation induced during the austenite decomposition. This could be further generalized to solve partial differential equations for the thermomechanical-kinetic state in situations where the specimen is known to exhibit a spatial variation in its temperature and/or stress. Acknowledgements We gratefully acknowledge the key role played by Dr. Gerard M. Ludtka and Mr. William H. Elliott, Jr., of Oak Ridge National Laboratory, who performed all of the dilatometry experiments associated with this work and whose insights throughout this investigation were extremely valuable to us.

Figure 2: Comparison of model predictions with dilatometry data for a 5123 steel. Ae3 , Ae1 , Bs , and Ms are shown in dashed lines. All predictions for all three carbon levels were created using a single set of model parameters.

Figure 3: Comparison of model predictions with dilatometry data for a 5139 steel. Ae3 , Ae1 , Bs , and Ms are shown in dashed lines. All predictions for all three carbon levels were created using a single set of model parameters.

Figure 4: Comparison of model predictions with dilatometry data for a 5139 steel. Ae3 , Ae1 , Bs , and Ms are shown in dashed lines. All predictions for all three carbon levels were created using a single set of model parameters.

References [1] W.A. Johnson and R.F. Mehl, Trans. AIME 135 416 (1939) [2] M. Avrami, J. Chem. Phys. 7 1103 (1939) [3] A.N. Kolmorgorov, Izv. Akad. Nauk. SSSR 3 355 (1937) [4] M. Avrami, J. Chem. Phys. 8 212 (1940) [5] J. Cahn Acta Metall. 4 572 (1956) [6] M.T. Lusk, and H.-J. Jou, Metall. Trans. 28A(2) 287 (1997) [7] G.R. Speich and R.M. Fisher, Recrystallization, Grain Growth and Textures, ASM International, Ohio (1966) [8] J. Cahn, Trans. Am. Inst. Min. Engrs. 239 610 (1967) [9] T. R´eti, M. Gergely, and P. Tardy, Mats. Sci. and Tech. 3 365 (1987) [10] R. Vandermeer, R. Masumura, and B. Bath, Acta Metall. Mater. 39 383 (1991) [11] K. Hanawa and T. Mimura, Metall. Trans. 15A 1147 (1984) [12] E.-S. Lee and Y. Kim, Acta Metall. Mater. 38 1669 (1990) [13] N. Luiggi and A. Betancourt, Metall. Trans. 25B 917 (1994) [14] N. Luiggi and A. Betancourt, Metall. Trans. 25B 927 (1994) [15] M.T. Lusk, G. Krauss and H.-J. Jou, J. de Phys. IV Colloque C8-5 279 (1995) [16] D. Bammann, V. Prantil, A. Kumar, J. Lathrop, D. Mosher, M. Lusk, H.-J Jou, G. Krauss, W. Elliott, G. Ludtka, W. Dowling, D. Nikkel, T. Lowe, and D. Shick, Proceedings of the 2nd International Conference on Quenching and the Control of Distortion, eds. G. Totten, M. Howes, S. Sjöström, and K. Funatani 367 (1996) [17] T. R´eti, I. Felde, L. Horv´ath, R. Kohlh´eb, and T. Bell, Heat Treat. of Met. 1 11 (1996) [18] D. Koistinen and R. Marburger, Acta Metall. Mater. 7 59 (1959) [19] B. Skrotzki, J. de Physique IV Colloque C8-1 367 (1991) [20] M.T. Lusk and Y.-K. Lee, 7th Intl. Sem. IFHT (International Federation for Heat Treatment and Surface Engineering) Budapest, Hungary: Hungarian Scientific Society of Mechanical Engineering, Budapest, Hungary 1027 (1999) [21] R. Grange and H. Stewart, Trans. AIME 167 467 (1946) [22] R. Edwards and N. Kennon, J. Aust. Inst. Met. 15 201 (1970) [23] D. Stouffer and L. Dame, Inelastic Deformation of Metals: Models, Mechanical Properties, and Metallurgy, John Wiley & Sons, Inc. (1996) [24] H.-J. Jou and M.T. Lusk, Phys. Rev. B 55 8114 (1997) [25] R.N. Gillmor, Trans. A.S.M. Dec 1377 (1942)

[26] A.L. Christenson, E.C. Nelson, and C.E. Jackson, Transactions of the A.I.M.E. 162 606 (1945) [27] M. Takahashi and H.K.D.H. Bhadeshia, Journal of Materials Science Letters 8 477 (1989) [28] M. Onink, et al., Zeitschrift fur Metallkunde 87(1) 24 (1996) [29] C.-M. Li, F. Sommer, and E.J. Mittemeijer, Zeitschrift fur Metallkunde 92(1) 2 (2001) [30] T.A. Kop, J. Sietsma, and S. van der Zwaag, Journal of Materials Science 36 519 (2001) [31] J.Z. Zhao, C. Mesplont, and B.C. de Cooman, ISIJ International 41(5) 492 (2001) [32] C.S. Roberts, Trans. AIME, 1953, Feb. 203 (1953) [33] F. Wever et al., Arch. Eisen. 21 367 (1950) [34] K. Honda and Z. Nishiyama, Sci. Rpts. Tohoku. Imp. Univ. Ser. 1 21 299 (1932) [35] J. Mazur, Nature 66 828 (1950) [36] W.J. Wrazej, Nature 163 212 (1949) [37] L. Cheng, et al., Metall. Trans. A 19A 2415 (1988) ..

..

[38] E. Ohman, Zeitschrift fur Physikalische Chemie 8 81 (1930) [39] A.R. Troiano and F.T. McGuire, Trans. AIME 31 340 (1943) [40] H.M. Ledbetter and R.P. Reed, Mater. Sci. Eng. 5 341 (1969/1970) [41] D.J. Dyson and B. Holmes, J. of Iron and Steel Institute May 469 (1970) [42] M. Onink, C.M. Brakman, F.D. Tichelaar, E.J. Mittemeijer, and S.van der Zwaag, Scripta Mater. 29 1011 (1993) [43] Z.S. Basinski, W. Hume-Rothery, F.R.S. Sutton, and A.L. Sutton, Proc. Roy. Soc. London A 229 459 (1955) [44] H.J. Goldschmidt, Adv. X-ray Analysis vol. 5, Plenum Press, New York 191 (1962) [45] R. Kohlhaas, Ph. Dunner, N. Schmitz-Pranghe, and Z. Angew, Phys. 23 509 (1990) [46] A.T. Gorton, G. Bitsianes, T.L. Joseph, Trans. AIME 233 519 (1965) [47] H.K.D.H. Bhadeshia, et al., Materials Science and Technology 7 686 (1991) [48] Hume-Rothery, The Structure of Alloys of Iron, Pergamon Press (1966) [49] N. Ridley, H. Stuart, and L. Zwell, Trans. AIME 245 1834 (1969) [50] H.M. Ledbetter and M.W. Austin, Mater. Sci. Tech. 3 101 (1987) [51] L. Pauling, The Nature of the Chemical Bond, Ithaca, NY, Cornell University Press (1960)

[52] George Krauss, Iron Steel Inst. Jpn. Int. 35 349 (1995) [53] B.G. Lifshitz, Physical Properties of Metals and Alloys, Mashgiz, Moscow 257 (1956) [54] H.K.D.H. Bhadeshia, J. de Phys. 43 C4-443 (1982) [55] S. Denis, S. Sjostrom, and A. Simon, Metall. Trans. A 18A 1203 (1987) [56] S.S. Babu and H.K.D.H. Bhadeshia, Mater. Sci. & Tech. 6 1005 (1990) [57] H. Stuart and N. Ridley, J. of Iron and Steel Institute 711 (1996) [58] R.C. Reed and J.H. Root, Scripta Mater. 38(1) 95 (1998) ..

[59] L. Cheng, A. Bottger, Th.H. de Keijser, E.J. Mittemeijer, Scr. Met. & Mater. 4 509 (1990) [60] D.H. Jack, Mater. Sci. Eng. 11 1 (1973) [61] L. Cheng, C.M. Brakman, B.M. Korevaar, and E.J. Mittemeijer, Metall. Trans. A 19A 2415 (1988) [62] M. Onink, et al., Scripta Metallurgica et Materialia 29(8) 1011 (1993) [63] J.S. Kirkaldy and D. Venugopalan, Phase Transformations in Ferrous Alloys, Eds. A.R. Marder and J.I. Goldstein, pp. 125—148, The Matallurgical Society of AIME, NY (1984) [64] Y.-K. Lee and Mark T. Lusk, Metall. Trans. A 30A 2325 (1999) [65] Epogy Software, Synaps Inc., Atlanta, Georgia, http://www.synaps-inc.com