Calibration Against TTT Data

During component quenching, austenite decomposes into a combination of other phases such as ferrite, pearlite, bainite, and martensite. The final phase composition inside a quenched component depends on the thermal history during cooling, and also on the alloying of the steel itself. The variations in quenching characteristics between steels can be significant, and each steel has to be characterized with respect to how phase transformations occur during a thermal transient. A common way to illustrate the phase transformation characteristics is through transformation diagrams. Two of the most commonly used diagram types are the CCT (continuous cooling transformation) and the TTT (time-temperature transformation) diagrams. In the CCT case, the austenitized material is cooled at varying constant temperature rates, and fractions of the emerging phases formed are plotted as points in time-temperature space. In the TTT case, the material is first rapidly cooled, and than held at a constant temperature. Figure 1 shows a schematic of a TTT diagram, in which a single phase transformation is considered. It is customary to plot a curve denoting the start of formation of the destination phase, in this case when the destination phase reaches a fraction of 1%. In other words, the curve shows the time it takes, at a fixed temperature, to form 1% of the destination phase. To complete the diagram, other curves are typically included. In this example, the 50% and 90% curves are also drawn. When several curves are used, phase transformation models can be readily fitted for the isothermal case.

Figure 1: A TTT diagram.

In this model, TTT diagram data is used to calibrate the Johnson–Mehl–Avrami–Kolmogorov (JMAK) phase transformation model. Using the calibration result, a TTT diagram is computed and compared to the original TTT diagram data.

Model Definition

This model considers a set of experimental TTT data for a single phase transformation. The data is given in Table 1. The table shows, for each temperature, the times it takes to form 1%, 50%, and 90% of the phase.

Table 1: Experimental TTT data.
Temperature (K)	Time to 1% (s)	Time to 50% (s)	Time to 90% (s)
853	320	1100	1530
893	210	735	1020
933	180	620	860
973	210	710	990
1013	420	1450	2000

In order to calibrate a phase transformation model to this experimental data, no geometry is required, and the temperature field can be replaced by a temperature parameter. For each temperature in Table 1, a least-squares problem is solved to find a set of phase transformation model parameters such that the evolution of the phase fraction matches the experimental data.

Phase Transformation

The model only considers a single phase transformation, and it is described by the JMAK model. This model requires three parameters:

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An equilibrium phase fraction ξeq

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A time constant τ

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An Avrami exponent n

In general, all these parameters can be used to calibrate the phase transformation against experimental data, but this model considers a special case where the phase fraction tends towards one, and where the Avrami exponent is considered constant and equal to three. For each temperature under consideration, the goal is to find the value of τ that produces the best least squares fit. The JMAK model is integrated using a time-dependent study step, and the phase fraction of the forming phase (the destination phase) can then be schematically expressed as ξ(t,τ), where t is time, and τ is the time constant.

Optimization

The purpose of the optimization, at a given temperature, is to find the optimal value for the phase transformation model parameter τ such that the phase fraction, that evolves over time, best reproduces the experimental data. An objective function can be expressed as

where the experimental data is given by N points, tk is the time at which ξk of the phase has formed. This objective function is minimized for each temperature to find τ.

Results and Discussion

The purpose of calibrating phase transformation models is to be able to use them to describe phase transformations that occur, for example, during component quenching. In this example, the JMAK phase transformation model was calibrated against TTT data. The result from the calibration procedure is the temperature-dependent parameter τ. The parameter is shown in Figure 2. The figure shows that the value of τ is high towards the ends of the temperature range, and smaller in the center of the range. The parameter τ represents a characteristic time for phase transformation, and the experimental data confirms that the phase transformation is most rapid at the intermediate temperatures. When the calibrated phase transformation model is used to compute a TTT diagram, you can compare the resulting diagram to the experimental (input) data. This is done in Figure 3. By merely using τ as a fitting parameter, the 1%, 50%, and 90% lines are reproduced well. For an even better fit to the experimental data, the Avrami exponent, here considered a constant, could be added as a free parameter of the optimization problem.

The methodology in this example can be used sequentially to calibrate several phase transformation models, one phase transformation at a time. A natural next step, once every relevant phase transformation has been characterized, is to simulate continuous cooling, and produce a CCT diagram. Good agreement between a computed CCT diagram with its experimental counterpart may require adjusting the phase transformation model parameters.

Figure 2: The temperature dependent phase transformation model parameter τ.

Figure 3: Comparison of the computed TTT diagram and experimental data, for the 1%, 50% and 90% lines.

Metal_Processing_Module/Transformation_Diagrams/calibration_against_ttt_data

Modeling Instructions

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Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
xieq	1	1	Equilibrium phase fraction
tau	1[s]	1 s	Time constant
n	3	3	Avrami exponent
Tsw	1[K]	1 K	Temperature sweep parameter
T	1[K]	1 K	Temperature