On Methods for Determining the Dependence of Changes in the Properties of Hardened Steels on Temperature-Time Exposure During Tempering
Abstract Generated abstract
This paper critiques the Hollomon and Jaffe time temperature parameter for describing changes in hardened steel properties during tempering, arguing that its assumptions about activation energy and process constants are theoretically and practically problematic. The authors propose using the classical Arrhenius relation instead, treating activation energy as a conditional quantity and constructing superposed curves or nomograms from experimental hardness data plotted against logarithmic time and reciprocal absolute temperature. Examples for carbon steels show that the method can combine tempering data over broad ranges, estimate conditional activation energies, and reveal cases where these energies vary with tempering conditions. The approach is presented as a more transparent basis for extrapolating tempering regimes, comparing processes, and selecting treatments to obtain specified property combinations.
Full Text
ON METHODS FOR DETERMINING THE DEPENDENCE OF CHANGES IN THE PROPERTIES OF HARDENED STEELS ON TEMPERATURE–TIME EFFECTS DURING TEMPERING
Yu. M. Lozinskii, I. T. Tikhonov
(Submitted by the scientific seminar of the Department of Metal Technology)
To determine the dependence of the properties of hardened steels on the tempering temperature and its duration, Hollomon and Jaffe [1] proposed the so-called parametric equation
\[ P = T(C+\lg t), \tag{1} \]
where
\(P\) is a parameter having a constant value for tempering treatments with different temperatures and durations, after which the hardened steel acquires definite properties, for example hardness;
\(T\) is the tempering temperature in °K;
\(t\) is the tempering duration;
\(C\) is a constant selected in constructing the parametric curve in accordance with the minimum scatter of the experimental data.
In deriving equation [1], the authors named proceeded from the assumption that the hardness of steel after tempering is related to its temperature and duration by the following functional dependence:
\[ H = f_1\left(te^{-\frac{Q}{RT}}\right). \tag{2} \]
It was found, moreover, that in turn
\[ Q=f_2(H), \tag{3} \]
\[ te^{-\frac{Q}{RT}}=t_0. \tag{4} \]
where \(t_0\) is a constant depending only on the properties of the hardened steel.
Equations (3) and (4) give
\[ Q=(\ln t-\ln t_0)RT=f_2(H), \]
\[ H=f[T(C+\lg t)]. \tag{5} \]
Hence, for constant values of hardness \(H=\text{const}\), equation (1) follows, where \(C=-\lg t_0\).
The parametric equation (1) has become widely used abroad in studies of the behavior of hardened stee-
... steels in the course of their tempering, for example, [2, 3, 4]; however, despite this, its use seems to us unjustified. Indeed, the quantity \(Q\) in equation (2) corresponds to the activation energy of the process and therefore cannot be a function of hardness, that is, of the level of the process; thus equation (5) is theoretically unjustified. On the other hand, the quantity \(t_0\) in equation (4) is regarded as a process constant, whereas theoretically it is precisely this quantity that should change as its level changes. Equation (4), which serves as the basis for obtaining the Hollomon–Jaffe parametric dependence, is incorrect, since from equation (2) it is evident that hardness is a function of the left-hand side of equation (4); consequently, the quantity \(t e^{-Q/RT}\) cannot be constant when hardness changes. In addition to theoretical shortcomings, the parametric equation (1) also causes substantial practical difficulties associated with the choice of the constant \(C\), whose value may vary within rather wide limits, from 10 to 20. As a result, on parametric curves one often observes a considerable scatter of points corresponding to experimental data [3]. Moreover, the features of parametric equation (1) do not even make it possible to identify the cause of the deviations that appear, for example, if at some stage of tempering there is a change in the process leading to a change in the properties of the hardened steel and, consequently, the activation energy changes. On the basis of the foregoing, it seems more rational to us, for solving the problem posed, to use the classical Arrhenius equation. According to this equation, for a process proceeding to a definite stage or level,
\[ t e^{-\frac{U}{RT}} = B, \tag{6} \]
where
\(T\) — process temperature, °K;
\(t\) — duration of the process;
\(U\) — activation energy;
\(B\) — a quantity that changes with a change in the level of the process.
In tempering hardened steels, the constant \(U\) can be spoken of only as a conditional activation energy [5], while the quantity \(B\), depending on the level attained by the tempering process, in the case of studying the hardness of tempered steels may be considered as \(B=f(H)\).
The change in the properties of tempered steels is the result of complex processes; therefore one cannot expect the conditional activation energy to retain a constant value over a wide range of tempering temperatures and durations. However, as will be shown below, the proposed method makes it possible to take these features into account as well.
For the case of two tempering regimes with different temperatures and durations, as a result of which the hardened steel acquires a definite property, for example hardness, in accordance with equation (6) we may write
\[ t_1 e^{-\frac{U}{RT_1}} = t_2 e^{-\frac{U}{RT_2}} \tag{7} \]
and, after taking logarithms,
\[ \lg t_1 - \lg t_2 = \frac{U}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)\lg e . \tag{8} \]
Consequently, if we have two curves constructed from experimental data and corresponding to the change in some property of steel as a function of the logarithm of the duration of tempering at two temperatures \(T_1\) and \(T_2\), then, provided that the value \(U\) remains constant, one curve must be shifted relative to the other along the logarithmic time axis by a constant amount, since in this case \(\lg t_1-\lg t_2=\mathrm{const}\). Similarly, if the experimental curves are plotted in coordinates of the steel property after tempering versus the quantity reciprocal to its absolute temperature, then again the curves in the indicated case may be superposed on one another by a simple shift, since for tempering treatments of different durations \(t_1\) and \(t_2\) the condition
\[ \frac{1}{T_1}-\frac{1}{T_2}=\mathrm{const} \]
will be satisfied.
The indicated regularities make it possible to obtain superposed curves that can be used for extrapolation when selecting tempering regimes over a wide range of their temperatures and durations, and also to determine the values of the conditional activation energy on the basis of experimental data. Figure 1 gives a superposed curve showing the change in hardness of hardened steel with 0.35% C as a function of the logarithm of the duration of its tempering at 350°C. The curve was constructed from the data of [6] and combines the results obtained after four tempering treatments at temperatures of 350, 450, 550, and 650°C with durations from 0.01 to 1000 hours. In this case, as Fig. 1 shows, a sufficiently good superposition of all the experimental data into one curve is achieved at a single value of the conditional activation energy \(U=56400\ \text{cal/mol}\). As a result, the relationship between the temperatures and durations of tempering treatments after which the steel
Fig. 1. Dependence of the hardness of hardened steel with 0.35% C on the duration of tempering at 350°C. The superposed curve was constructed from data [6] on the change in hardness after tempering treatments of duration from 0.01 to 1000 hours at different temperatures: ○ — 350; + — 450; △ — 550; × — 650°C.
acquires the specified hardness, can be determined by the following equation:
\[ t = 1.42 \cdot 10^{-20} \cdot \tau_0 e^{\frac{56400}{RT}}, \]
where
\[ \tau_0 \]
is the tempering duration for which the combined curve was constructed, in this case tempering at 350°C.
If, within the range of the tempering regimes studied, the value of the apparent activation energy does not remain constant, then, in order to generalize the experimental data, a nomogram may be constructed in coordinates of the logarithm of the tempering duration and the value reciprocal to its absolute temperature. Fig. 2 shows such a nomogram, constructed from data [1] for hardened steel with 0.31% C, tempered at 200, 250, 300, 400, 500, 600, and 700°C with holding times varying from 0.025 to 24 hours. It is easy to see that the lines on the nomogram, constructed from the experimental data, have different
Fig. 2. Dependence of the hardness of hardened steel with 0.31% C on the duration and temperature of tempering. The nomogram was constructed from data [1] on the change in hardness after tempering for durations from 0.025 to 24 hours, at temperatures ranging from 200 to 650°C
slopes and bends, which is a consequence of the change in the value of the apparent activation energy of the tempering process, approximately from 45200 to 78600 cal/mol.
A comparison of the two methods for determining the dependence of the properties of hardened steels on the temperature and duration of tempering shows that the first method, using the parametric equation (1), is substantially inferior to the second, based on the usual Arrhenius equation. The method based on the exponential Arrhenius dependence makes it possible to note the following advantages:
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From the slope of the lines corresponding to constant values of a property, in coordinates “logarithm of tempering duration—quantity reciprocal to its absolute temperature,” it is easy to determine the numerical values of the conditional activation energy of the tempering process.
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The overall picture of changes in the processes occurring during the tempering of hardened steels can be used to solve problems of rational alloying, for example, of heat-resistant steels.
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It makes it possible to outline optimal tempering regimes for obtaining in hardened steel a definite set of properties, for example, a combination of high strength and impact toughness [7].
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It makes it possible to generalize experimental data after tempering under various regimes without the need to select any constants (the constant $C$ in the parametric equation).
REFERENCES
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J. H. Hollomon and Jaffe. Time-temperature Relations in Tempering Steel. “Transactions of the American Institute of Mining and Metallurgical Engineers,” v. 162, 1945, p. 233—248.
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K. Bungardt, G. Hoch, O. Mülders. Einfluss der Wärmebehandlung auf die Zugfestigkeit und Kerbschlagzähigkeit von Warmarbeitsstählen, “Stahl und Eisen,” n. 16, 1955, S. 1035—1046.
-
T. Nishimura, J. Shinyama. On the Tempering Behaviorz of W—Cr—V Steels, “Tetsu-to-hagane,” v. 49, No. 10, 1963, p. 1539—1541.
-
T. Nishimura, J. Shinyama. On the Tempering Behaviorz of 5 Cr—Mo—W—V and 5 Cr—Mo—W Steels. “Tetsu-to-hagane,” v. 50, No. 4, 1964.
-
V. I. Arkharov. On clarifying the concepts “activation energy” and “elementary act” for the solid state. Journal of Technical Physics, vol. 24, issue 3, 1954, pp. 375—387.
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Edgar Bain. The Influence of Alloying Elements on the Properties of Steel. Metallurgizdat, 1945.
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Yu. M. Lozinskii, I. T. Tikhonov. Investigation of the interrelation between changes in the mechanical properties of steel during tempering. Izvestiya Vuzov, “Ferrous Metallurgy,” No. 2, 1971, pp. 113—116.