research efforts have focused on the formation kinetics of ferrite, pearlite, bainite and martensite. However, tempering can have a large influence on distortion ...
An Internal State Variable Model for the Low Temperature Tempering of Low Alloy Steels Mark T. Lusk1, Young-Kook Lee1, Herng-Jeng Jou2, William H. Elliott3, Gerard M. Ludtka3 1. Materials Science Program, Division of Engineering, Colorado School of Mines, Golden, CO, U.S. 80401 2. Questek Innovations LCC, 1801 Maple Avenue, Evanston, IL, U.S 60201 3. Oak Ridge National Laboratory, Metals & Ceramics Division, Mechanical Properties Group, Bldg. 4500-S, MS6155, 1 Bethel Valley Road, P.O. Box 2008, Oak Ridge, TN 37831-6155 Abstract: An internal state variable (ISV) framework is used to predict the rate and degree of strain relaxation associated with the stage I, low temperature tempering of martensite. A single variable tracks the change in volume between the tempered an untempered structures. The rate at which this occurs is taken to be a function of martensite fraction, temperature, degree of tempering, and carbon concentration. Experimental dilatometry data is used to both fit and validate the model. Key Words: Tempering; Model; Quenching; Phase Transformations
Introduction Quantitatively accurate predictions of quench-induced distortion and residual stress rely, in part, on the ability to predict the microstructural evolution throughout the quench process. In low alloy steels, current research efforts have focused on the formation kinetics of ferrite, pearlite, bainite and martensite. However, tempering can have a large influence on distortion and residual stress, and efforts to account for this lag behind the modeling of the primary phase transitions. As a first step towards accounting for tempering, we have developed an internal state variable model that uses a single variable to track the degree to which a martensitic structure tempers. Attention is restricted to the low temperature tempering, so-called tempering stage I. According to the early work of Cohen and his colleagues[1], tempering at low temperatures below about 250°C is commonly referred to as the first stage of tempering. In this stage of tempering, the carbon supersaturation of the as-quenched (AQ) martensite is relieved by the precipitation of fine transition carbides. These carbides have been identified as epsilon carbide with a hexagonal structure[2] or eta carbide with an orthorhombic structure[3], and have been shown by transmission electron microscopy to form in linear clusters of particles 2 to 5 nm in size[3]. The transition carbide precipitation has an influence on the mechanical properties of tempered martensite and simultaneously causes distortion and residual stress change. Although there are extensive literatures concerning the effects of tempering on microstructures, hardness, impact toughness, and tensile properties, the information about the strain relaxation, distortion, and residual stress during tempering still lacks. In order to give perspective to the way in which model is to be used, a brief review is first provided of the transformation kinetics model that we have previously developed.
1. Transformation Kinetics Model Phase transformation kinetics are modeled using an internal state variable framework wherein the volume fraction of each product phase is tracked as a function of time. Phase volume fractions are denoted by Φ , with subscripts of A, F, P, B, and M referring to austenite, ferrite, pearlite, bainite, and martensite on this and other symbols. Time is given as t, temperature as T, Carbon wt. % by C, and derivatives with respect to time • are denoted by Φ . The following equations are then solved simultaneously to obtain the volume fraction of each phase for arbitrary cooling profiles.
Φ A = 1 − ΦF − ΦP − ΦB •
α
Φ F = ν F (T )Φ F F (1 − Φ F ) β F −1 Φ A (Φ F ,equil (T ) − Φ F ), Φ F (0) = Φ F 0 •
(1.1)
α
Φ P = ν P (T )Φ P P (1 − Φ P ) β P −1 Φ A , Φ P (0) = Φ P 0 •
Φ B = ν B (T )(Φ B + Φ eB )α B (1 − Φ B ) β B −1 Φ A (Φ B , stasis (T ) − Φ B ), Φ B (0) = Φ B 0 d ΦM = dT
0, T > M s
ν M (C )(Φ M + Φ eM )α
M
(C )
(1 − Φ M ) β M (C )−1 (1 − Φ M − Φ F − Φ P − Φ B ), T ≤ M s
Note that the diffusive phase mobilities(νF, νP, νB) are functions of temperature while the martensite mobility is a function of carbon. This set of equations comprises the Customized Kinetics Model in the software package DANTE. Parameters in the mobility and exponent functions are fitted with combinations of isothermal and continuous cooling dilatometry data. A kinetics data file is generated for each material. In DANTE, this set of equations is solved, at every node point, in full coupling with the thermal and mechanical equations. The present investigation seeks to add a tempering equation to this kinetics set.
2. Tempering Dilatometry The tempering effect to be captured is best explained by considering dilatometry data gathered for a specific low alloy steel. Fig. 1 shows several dilatometry traces derived by rapidly quenching and re-heating one of our test materials with different tempering temperatures. During the re-heat, carbon leaves the martensite and forms fine transition carbides. This carbide precipitation is measured as material shrinkage in the dilatometer. Note that for this specimen the heating rate is sufficiently slow that the material starts to temper before reaching the tempering temperatures.
Fig. 1. Four dilatometry traces for heat-up and hold of a specimen that is primarily martensite.
Our goal, in this work, is to predict the sort of tempering dilatation shown in Fig. 1 as a function of material, time, and temperature. In order to develop a useful parameter for tracking the strain associated with tempering, although there may be a little differences in microstructure, transformation and thermal strains between bainite and tempered martensite, it is assumed that a lower bound on the shrinkage is the size of the specimen if it were a completely bainitic structure at the tempering temperature. The rate of progress of tempering will then be characterized by a strain ratio:
ΦT (T0 ) =
Martensite Strain - Dilatometer Strain Martensite Strain - Bainite Strain
(2.1)
This is illustrated in Fig. 2, where the ΦT at temperature T0 is obtained for the dilatometer strain at point P by
ΦT (T0 ) =
d1 Φ M (T0 ) d2
(2.2)
Fig. 2. This schematic shows how the tempering fraction field is obtained from dilatometer data.
Though this term will be referred to as a tempering fraction, one must be careful to understand what is really being measured. ΦT is a subset of the total martensite present and is intended to measure the degree to which the martensite has tempered. In this definition, though, there is no way to separate amount of total martensite from degree to which that martensite has undergone strain relaxation; for example, a given specimen would have the same value of ΦT as one with half the martensite but twice the strain relaxation. The tempering fraction is not a measure of how much martensite has undergone complete strain relaxation. A series of dilatometry experiments were performed in order to obtain the tempering fraction as a function of time, temperature and material chemistry. This data was used to fit the tempering kinetics model described in the following section. As will be seen, the model has parameters that depend on carbon level but not on any other chemistry variations. This was motivated by dilatometry results that were obtained for five materials with the same carbon level but different concentrations of Nickel (0.0 to 0.5 wt. %), Chromium (0.5 to 1.44 wt. %), Molybdenum (0.05 to 0.4 wt. %) and Manganese (0.38 to 0.85 wt. %). The results, for an isothermal tempering hold at 152°C, are shown in Fig. 3. The similarity of kinetics observed in tempering stage I has motivated the simplifying assumption to consider only variations in carbon level.
Fig. 3. Tempering fraction as a function of time for 5 different materials having the same carbon level. The difference in tempering kinetics was deemed sufficiently small to disregard chemistry changes other than carbon. This data was obtained for isothermal holds at 152°C.
3. An Internal State Variable Model for Tempering Consistent with the ISV framework used in the phase transition and plasticity models, the tempering fraction, ΦT , is assumed to vary at a rate that depends only on temperature, the amount of martensite, and degree to which that martensite has been tempered: •
ΦT =ν (T ) ΦαT (C ) (Φ M − ΦT ) β (C )
ν (T ) =
Exp (− 0,
Q(C ) ) (TC − T ) n , R (T + 273) T > TC
T ≤ Tc
(3.1)
α (C ) = α 0 + α1C + α 2C
2
β (C ) = β 0 + β1C + β 2C 2 Q(C ) = Q0 + Q1C + Q2C 2 This tempering model requires that 11 constants be fitted, and this is done using dilatometry data collected for several different carbon levels and temperatures. Rapid quenching followed by liquid Nitrogen immersion was used to obtain an initial martensite volume fraction close to 1.0. The hold temperatures were obtained in less than 1 second in an attempt to prevent any significant amount of tempering from occurring during the heat up. Once the tempering data were obtained, each set was reduced to approximately 30 pairs of time and tempering fraction. A previously developed differential fitting routine, developed at the Colorado School of Mines, was then used to fit the 11 tempering parameters. This fitting routine accepts all of the tempering data sets at once and adjusts the tempering parameters so as to obtain an optimal fit to all of the dilatometry data.
4. Goodness of Fit and Predictive Capability of the Tempering Model The tempering model was numerically implemented as a user defined subroutine in DANTE. Fig. 4 gives a comparison of the model predictions and the some of the experimental data used to fit the model. This is not a predictive test and only serves as an indicator of the goodness of fit.
Fig. 4. Comparison of model predictions for isothermal holds with the dilatometry data used to fit the tempering model. All four curves are for the same material.
The model was then used to make predictions about the rate of strain relaxation in specimens for which there a significant amount of retained austenite (~10 to 20 vol. %). The results, shown in Fig. 5, suggest that the approach can be useful even when there are phases other than martensite present.
Fig. 5. Comparison of model predictions with experimental data not used to fit the model. This data set was obtained by quenching specimens to room temperature followed by a rapid temperature rise to the hold temperatures listed in the figure. The retained austenite volume fractions are approximately 0.1 and 0.2 for the two materials shown.
The tempering model was fit using isothermal hold data, and so it is natural to explore its predictive capability under nonisothermal conditions. Specimens were rapidly quenched to room temperature and subsequently heated, at a slow rate, up to an isothermal hold. The heating was sufficiently slow that nearly half of the tempering occurred under nonisothermal conditions. In order to compare simulation and experiment, the experimental temperature profile was used as input into the differential code. The results are shown in Fig. 6 alongside a test associated with a very rapid rise to the same hold temperature. The tempering model does a reasonable job at capturing both sets of data. Results of this sort suggest that this model might be used under general heat-treatment temperature profiles.
Fig. 6. Comparison of model prediction with experimental data for an experiment not used to fit the model. In the figure, the data associated with the liquid Nitrogen quench is one of the sets used to fit the model parameters. The second data set was obtained by quenching to room temperature, followed by a slow (~20 second) heat-up to the hold temperature. Nearly half of the martensite strain relaxation occurs before the hold temperature is reached. The model prediction used the experimental temperature profile to make the prediction shown.
Since this tempering model run in full coupling with the rest of the phase transition kinetics equations, it is possible to simulate auto-tempering phenomena that occur during quenching. This is illustrated in Fig. 7, where the temperature profile shown is inputted into the kinetics/tempering model. The quenching is slow enough that bainite forms before the martensite, and the cooling is then interrupted at a temperature below Ms but sufficiently high
that sub-Ms bainite continues to form in competition with the tempering of martensite formed between Ms and the hold temperature. ****put Ms & hold temp.***
Fig. 7. Model predictions of microstructural fractions for an interrupted quench that results in auto-tempering. Note that the martensite is tempering at the same time that sub-Ms bainite is forming. The cooling profile is too rapid to result in ferrite or pearlite formation even though these phases are included in the kinetics routine.
5. Conclusions The simple ISV tempering model that has been proposed is able to provide quantitatively accurate predictions of strain relaxation in tempering stage I for a class of low alloy steels. It was found that the model parameters should depend on Carbon concentration but not on other alloying elements. The paradigm couples easily with an existing ISV model for phase transitions and can be solved under arbitrary cooling/heating conditions. Our intent is to further validate this approach in order to understand its temperature and time limitations as well as to quantify its predictive power when applied to auto-tempering. ACKNOWLEDGEMENTS We gratefully acknowledge our program sponsors. This research was conducted as part of the National Center for Manufacturing Science Heat-Treatment/Distortion Project. The Oak Ridge National Laboratory also sponsored this investigation through their MPLUS program. References [1] C.S. Roberts, B.L. Averbach and M. Cohen: Trans. ASM, 45 (1953) 576-604.
[2] K.H. Jack: J. Iron Steel Inst., 169 (1951) 26-36. [3] Y. Hirotsu and S. Nagakura: Acta Metall., 20 (1972) 645-55
