Activation energy for isothermal martensite in ferrous alloys

Actu mater. Vol. 45. No. 2. pp. 651-662. 1997 Copyright 10 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454197 $17.00 + 0.00

Pergamon PI1 S1359-6454(96)00186-3

ACTIVATION

ENERGY FOR ISOTHERMAL IN FERROUS ALLOYS A. BORGENSTAM

Division

of Physical

Metallurgy,

and M. HILLERT

Department of Materials Science Stockholm. Sweden (Rrwiwd

MARTENSITE

and Engineering,

KTH,

S-100 44

13 MaI, 1996)

Abstract-The experimental information on isothermal c( martensite in ferrous alloys is reviewed. From the kinetics one can clearly distinguish between three groups of alloys yielding isothermal martensite. The first group contains high alloy steels with a low M, temperature. They form isothermal martensite with a temperature dependence corresponding to a very low activation energy, possibly 7 kJ/mol. The second

group contains steels with an appreciable amount of carbon. Its rate of formation of isothermal martensite can be explained by assuming that it is triggered by submicroscopic plates of bainite formed with a rate controlled by carbon diffusion. The third group contains Fe-Ni alloys with up to about 25% Ni. There the temperature dependence corresponds to an activation energy of about 80 kJ/mol. It is proposed that their transformation is related to the transformation causing plateau II in experiments with very rapid cooling. a transformation which has oreviouslv been proposed to be related to the formation of bainite. Copright OF’1997 Acta Metallurgicd Inc.

INTRODUCTION

A recent survey of the driving force for the start of athermal a martensite in ferrous alloys [1] indicated that it may depend mainly on the temperature. It seems that the effect of the various alloying elements on the martensite starting temperature, MS, is mainly controlled by their effect on the difference in Gibbs energy between the b.c.c. and f.c.c. phases and not so much on solution hardening which could be quite different for different alloying elements. It would thus seem interesting also to examine the effect of alloying elements on isothermal c( martensite in an attempt to identify the main mechanism of that phenomenon. A recent review by Thadhani and Meyers [2] indicates that interest in the kinetics of isothermal martensite formation has mainly centred around the shape of the isothermal transformation curve and explanations of its sigmoidal shape. Many such models have been developed, which is in contrast to the relatively few attempts to explain the shape of the C-curve which will be the subject of the present work. Only the rate of the initial transformation will be discussed. Ideally, one should then consider the start of the transformation, e.g. represented by the 1% transformation curves, but it seems that those curves are rather sensitive to the method of measurement and, in general, 5% transformation curves were thus chosen whenever available. In particular, it seems difficult to identify the start of the isothermal transformation if it is preceded by the formation of some athermal martensite. The present work will thus concentrate on the information on isothermal martensite above the MS temperature. Unfortunately, the definition of the A4, tempera651

ture is not always clear. Many authors give an M, temperature, defined by the start of martensite formation during continuous cooling at a rate that martensite to form may allow some “isothermal” during the cooling. When formed in that way it is sometimes called “anisothermal” martensite [3]. Anyway, in the present work MS will be defined as the starting temperature of athermal martensite, the type of martensite that cannot be suppressed, except possibly at extremely rapid cooling. The upper temperature limit for isothermal martensite, including anisothermal martensite, will be denoted MS,. In a previous study [4] it was argued that it would be useful also to define an Mg temperature, the upper temperature limit where martensite can grow if it is already nucleated, and it was argued that Mg falls above M,. If that is the case, there is a possibility that isothermal martensite can form above M, but below M, if there is an efficient isothermal mechanism of nucleation. M,, may thus fall between Mg and M,. In the following examination of the kinetics of isothermal martensite the information will be divided into three groups: (1) high alloy steels with M, below room temperature, (2) Fe-C alloys and plain carbon steels with M, above room temperature and (3) nickel alloys with M, above room temperature. OCCURRENCE HIGH

OF ISOTHERMAL ALLOY STEELS,

MARTENSITE GROUP 1

IN

When studying the dimensional stability of a hardened steel with 1.07% C. Averbach et al. [5] in 1948 observed a slow decomposition of the retained austenite at temperatures as low as O’C. They proposed that it was caused by isothermal formation

652

BORGENSTAM

and HILLERT:

ACTIVATION FOR ISOTHERMAL MARTENSITE IN ALLOYS

of martensite. In 1949 Averbach and Cohen [6] examined this phenomenon in more detail and proposed that the decomposition of retained austenite below about 120°C is due to isothermal martensite. They regarded this reaction as a “residual effect of the main hardening reaction” and also used the term “after-effect” because they only observed isothermal martensite in specimens that already contained athermal martensite. In 1948 Kurdjumov and Maksimova [7] also reported isothermal martensite but at a higher carbon content, 1.6%, or at high alloy contents: 6% Mn at 0.6% C and 23% Ni, 3.4% Mn without carbon. However, it was not quite clear whether this isothermal martensite formed in the presence of athermal martensite or not. The M, they gave is probably M,,. In any case, they did observe C-curves for isothermal martensite with nose temperatures below room temperature. In 1950 Kulin and Cohen [8] tried to reproduce the results of Kurdjumov and Maksimova with an Fe-8% Mn-0.6% C steel but found that the transformation to martensite was essentially complete above the temperature range of Kurdjumov and Maksimova’s isothermal measurements. In 1951 Das Gupta and Lement [3] investigated isothermal formation of martensite in a 15% Cr0.7% C steel below room temperature and concluded that it is always preceded by some athermal transformation. That was also the result obtained by Machlin and Cohen [9] in 1952 when studying a steel with 29.5% Ni. However, about the results of Kurdjumov and Maksimova they believed on theoretical grounds that “it is possible that isothermal nucleation will occur even though no athermal martensite had previously formed”, but they had not themselves found the combination of factors allowing this to happen. Experimental data for formation of isothermal martensite in alloyed steels will in the following be discussed separately for Fe-Ni-Mn and Fe-Cr-Ni, and only information on isothermal martensite formed above M, will be included.

- 128°C instead of -47°C. On quenching to - 196°C no martensite was formed and thus it was finally confirmed that isothermal martensite can form without any prior athermal martensite. In 1955 this was again shown by Shih et al. [ll] who studied two steels with 23.2% Ni, 3.62% Mn and 0.016% C and one steel with 5.24% Mn and 1.10% C. They claimed that athermal martensite was avoided (but there may have been anisothermal martensite). For the Ni-Mn steel they presented C-curves for formation of isothermal martensite between -90 and - 196°C with a nose at about -140°C and an J4, temperature below - 196°C (Fig. 1). In 1971 Pati and Cohen [12] presented isothermal transformation curves in the temperature range - 80 to - 196°C in an alloy with 24.20% Ni, 2.98% Mn, 0.017% C and 0.001% N. Ghosh and Raghavan in 1986 [13] presented curves between 77 and 203 K in an alloy with 23.2% Ni, 2.8% Mn and 0.009% C. These results are also included in Fig. 1. In 1993 Kakeshita et al. [14] presented isothermal transformation curves for an Fe alloy with 24.9% Ni and 3.9% Mn, and a C-curve for 0.3% isothermal martensite is included in Fig. 1. They reported that no athermal martensite formed during cooling to 4.2 K. The transformation curves did not show any incubation time, as previously reported curves did, and they rather resembled curves in alloys where athermal martensite is present. In several works, isothermal transformation curves at temperatures only above the nose temperature have been reported and the following ones are included in Fig. 1. In 1965 Raghavan and Entwisle [15] studied isothermal martensite in an alloy with 25.9% Ni, 1.94% Mn, 0.025% C and 0.004% N. They found that the isothermal transformation curves are very sensitive to the grain size. In 1968 Entwisle [ 161 presented isothermal transformation

_

L

Fe-Ni-h4n

Another attempt to confirm the results by Kurdjumov and Maksimova was made by Cech and Hollomon [lo] in 1953 using an Fe-Ni-Mn alloy. They presented C-curves for isothermal formation of and martensite at temperatures between -79 - 197°C in an alloy with 22.94% Ni, 3.73% Mn and an impurity content of 0.05% C and 0.015% N (Fig. 1). Unfortunately, the time of formation at the nose temperature is quite uncertain since the cooling rate was not high enough to prevent formation of anisothermal martensite, i.e. isothermal martensite formed during cooling to the holding temperature. Up to 3% martensite was formed during the cooling. They confirmed the main results of Kurdjumov and Maksimova, but found a nose temperature of

Time, s Fig.

1. Isothermal

formation of martensite alloys with very low M,.

in Fe-Ni-Mn

BORGENSTAM

Fig. 2. Isothermal

and HILLERT:

formation of martensite alloys with very low M,.

ACTIVATION

in Fe-C%Ni

curves in an alloy with 22.59% Ni, 3.33% Mn and 0.016% C. In 1972 Peters et al. [17] presented curves between -60 and -8OC in an alloy with 25.9% Ni, 2.0% Mn and 0.022% C. The curve from their work represents 10% isothermal martensite. Neither Raghavan and Entwisle, Entwisle, Pati and Cohen, Peters et al. nor Ghosh and Raghavan reported M, temperatures. The alloys used in these works have similar compositions to the alloys used in the studies by Cech and Hollomon, Shih et al. and Kakeshita et al., which all reported M, < -196-C. It may therefore be concluded that no athermal martensite was present prior to the isothermal holding in these studies either.

FOR ISOTHERMAL

MARTENSITE

IN ALLOYS

653

site in an Fe alloy with 16.93% Cr, 8.18% Ni and 0.11% C. They found two C-curves, the upper one caused by the formation of E and c( martensite and the lower one only by a martensite. In Fig. 2 the lower C-curve for formation of 4% c( martensite is included. The M, temperature for this alloy is - 190’C and it is surprising that their transformation curves showed no incubation time except at - I 10 and - 166 ‘C. They mentioned that three kinds of martensite were observed in samples held at the lower nose temperature and explained this by formation during cooling, but it was not clearly stated if this is the case for the C-curves included in Fig. 2. In 1971 Jones and Entwisle [21] presented C-curves for an alloy with 25.7% Ni, 2.95% Cr, and 0.006% C (Fig. 2). The temperature for the maximum transformation rate was - 13O’C and the first burst was observed at - 150-C. The experimental value at - 130’C is not included in Fig. 2 since the isothermal transformation curve was displaced towards shorter times in a strange way compared with curves for other temperatures. In 1995 Holmquist et al. [22] presented C-curves for isothermal martensite formation in an alloy with 12.0% Cr, 0.3% Mn, 9.0% Ni, 4.0% MO, 0.9% Ti, 2.0% Cu and C + N < 0.05%. No athermal martensite was detected on quenching to - 196 C. The C-curve representing 20% is included in Fig. 2. The M, temperature reported by Kuhn and Speich is much higher than other M, temperatures reported for similar alloys. It seems as if their M, is actually M,,, measured because of an insufficient cooling rate.

Fe-Cr-Ni In 1952 Kulin and Speich [18] studied a steel with 14.38% Cr, 9.06% Ni, 1.21% Mn, 0.008% C and 0.034% N and found indications of the possibility of isothermal martensite formation at temperatures above the “true IV,“. They presented transformation curves at temperatures between -1.5 and -196’C and reported an M, temperature at -2O’C, which is above the nose temperature (Fig. 2). In 1964 Lagneborg [19] studied isothermal transformation in a steel with 18.2% Cr, 7.5% Ni, 0.99% Mn, 0.08% C and 0.015% N and obtained C-curves with a nose at about - 150°C. M, was reported as - 5OC, but that was probably meant as the maximum temperature of isothermal martensite, M,,. The isothermal transformation curves indicate that the true M, is somewhere between - 150 and - 196’C. In this alloy both E and a martensite was observed and it is not clearly stated if F martensite was formed during the isothermal transformation. nor if any martensite was formed during cooling to the holding temperature. The C-curve for 3% isothermal martensite is included in Fig. 2. In 1966 Imai et al. [20] presented C-curves for isothermal transformation to both c( and c marten-

METHOD

OF ANALYSING INFORMATION GROUP 1 STEELS

FROM

It is generally assumed that the kinetics of isothermal martensite is governed by the rate of nucleation because the observed rate of growth is very high. It is thus common to employ nucleation theory. It seems that nucleation theory was first applied to martensite by Fisher et al. [23] in 1949 but without any mention of isothermal martensite. Also in 1949, the activation energy of a critical nucleus of martensite was considered by Cohen et al. [24]. They mentioned isothermal martensite and proposed that its nucleation depends on some relaxation phenomenon. Kurdjumov and Maksimova [25] in 1950 seems to be the first to discuss nucleation theory in connection with isothermal martensite. The ideas have developed over the years and a recent review was given by Raghavan [26]. However, the main idea has remained the same. A very simple treatment will be sketched here; for more details reference is made to the review by Raghavan [26]. Classical nucleation theory for homogeneous nucleation in its simplest form yields the following

654

BORGENSTAM and HILLERT:

ACTIVATION FOR ISOTHERMAL MARTENSITE IN ALLOYS

for the rate of nucleation: - AG*/kT) = N.v.exp( - 16rco3V2,/3PkT)

dn/dt = N.v.exp(

(1)

where N is the number of possible nucleation sites, v is a frequency factor, AG* is the energy barrier, o is the specific surface energy, vmis the molar volume, F is the driving force for the phase transformation and could be denoted - AG,, k is the Boltzmann constant and T is the absolute temperature. To a first approximation F = - AG, = A&( T - To) where To is the temperature where F starts from zero. AS, is the entropy of transformation at To. We thus get the following form of equation: dn/dt = A.exp( - B/(T - T#T)

of chemical reactions has been modelled for more than 100 years and more recently the approach has been generalized by Glasstone et al. [28] under the name of absolute reaction rate theory. Figure 3 illustrates the case of regularly spaced energy barriers of height Q. The driving force is represented by the slope of the three parallel lines, and the effective barrier for the two opposing reactions are shown with vertical lines, AG&, and AC;,,,. Their magnitudes depend on the shape of the curve and different shapes have been tried. However, if the driving force, F, is small relative to Q, then one simply gets for the rate of reaction, v

(2)

which will predict a C-curve for the inverse of the rate (which represents time of transformation) with its nose at T = T0/3. When analysing experimental C-curves for isothermal martensite one usually tries to estimate N and v and then evaluate AG* from the rate close to To, i.e. close to the horizontal asymptote of the C-curve at MS,. It would seem natural also to use the lower parts of the C-curve but in all practical cases it is evident that the nose temperature is much higher than To/3 and equation (2) is not applicable. It is thus necessary to modify the theory and it is common to consider heterogeneous nucleation. Such treatments will reduce AG* and N drastically and, in addition, it is even possible that AG* goes to zero at some temperature [27]. The shape of the curve will then change completely. It still starts from an infinite time at a horizontal asymptote at MSi but it goes to very short times at a temperature which may be identified as MS, the start of athermal martensite formation. Even though it is doubtful that this case has ever been observed, it is common to analyse experimental information above the nose in this way. There are at least two major difficulties. The modelling of the active heterogeneity and the critical nucleus are very uncertain and so are the estimates of N and v. When using nucleation theory in this way to interpret the kinetics of isothermal martensite, one evaluates the activation barrier from each experimental temperature and obtains a strong temperature dependence which reproduces the shape of the C-curve. The strong temperature dependence is necessary in order to explain the high nose temperature. However, it yields no explanation of the shape unless one can interpret the temperature dependence of the activation barrier in a convincing way. A completely different approach to the kinetics of isothermal martensite will now be attempted based on the experimental C-curve. It will be assumed that the time it takes for a nucleus of martensite to develop is governed by an activated process that, in principle, could go forth and back and the net rate is given by the difference in rate. This is how the rate

v=k.[exp(

-v)-exp(

---)I

E k(F/RT).exp(

- Q/RT)

(3)

In principle, it should be easy to evaluate F as a function of temperature. However, the thermodynamic descriptions of the iron-base systems are not very reliable at low temperatures. A simpler approach will thus be taken here. It will simply be assumed that F is proportional to To - T, where F is assumed to start from zero at To, the temperature where the time of formation of isothermal martensite goes to infinity, i.e. Msi. The constant of proportionality is equal to the negative of the entropy of transformation. Thus, F/T will be assumed to be proportional to (M,i/T - 1) which gives D = l/t z k/R(M,,/T

- l).exp( - Q/RT)

(4)

where t is the time of formation of a certain low fraction of isothermal martensite and k is a constant of proportionality. The quantity t+M,,/T - 1) will thus be plotted logarithmically vs - l/Tin an attempt to extract the activation energy, Q.

Progress of reaction + Fig. 3. Illustration

of effect of driving energy.

force on activation

BORGENSTAM and HILLERT:

ACTIVATION FOR ISOTHERMAL

MARTENSITE

655

IN ALLOYS

Mm-

I

500-

P

F 400 300 -

2001 100 t.(M,TT-l), s

R&kliffemd Ro11ason(30): 40.779bC

~l.o7%c

smith et al. (33): e 1.168c ~*ga$amdo (35): v l.lO%C +1:45%c ~l.so%c

I

I

I

1

I

I

10’

Id

Id

104

ld

106

Time, s

Fig. 4. Information from Fig. I, replotted in order to evaluate low temperature asymptote.

Fig. 6. Information on isothermal formation of martensite and lower bainite in Fe-C alloys.

It could be mentioned that the approximation in equation (3) is reasonable also at low temperatures and it makes no big difference if equation (3) is treated more exactly.

the point at the highest temperature of each experimental curve falls on the chosen line. At lower temperatures the straight line represents all the data surprisingly well. From a single experimental C-curve it would not have been possible to evaluate the straight line with any confidence. but Figs 4 and 5 give a strong indication that the rate controlling mechanism at low temperatures is essentially the same for all the steels and its temperature dependence can be evaluated from the straight line. This yields the surprisingly low activation energy of about 7 kJ/mol. When trying to find the rate limiting mechanism for isothermal martensite in this group of steels, it thus seems that one should look for a process with a very low activation energy. It can hardly be a process involving individual movements of atoms. It would seem more promising to compare it with the activation energy for slip or twinning although it may be difficult to see how such processes can take such a long time. At higher temperatures the fit of data to the straight line is less convincing and cannot be used to support equation (4). However. the main purpose of equation (4) was to help analyse the data below the nose of the C-curves. It could be suggested here that the pronounced peak in the line of Cech and Hollomon might be the result of the uncertain measurement of the time for their nose temperature. since some isothermal martensite was formed during the cooling, so-called anisothermal martensite.

KINETICS

OF

ISOTHERMAL MARTENSITE GROUP 1 STEELS

IN

The information in Figs 1 and 2 was thus modified by multiplying the time by (M,,/T - 1) and plotting it logarithmically vs -l/T for the Fe-Ni-Mn and Fe-Cr-Ni systems, (Figs 4 and 5). In order to do this, M,, had to be chosen in some way since M,, has not been determined experimentally in the various alloys. It was first chosen as the highest reported temperature for isothermal transformation to martensite in each alloy. A straight line was then chosen to fit the data for both Fe-Ni-Mn and Fe-Cr-Ni. The value of M,, was then changed so that

2m

OCCURRENCE OF ISOTHERMAL MARTENSITE IN Fe-C ALLOYS AND PLAIN CARBON STEELS, GROUP 2

Fig. 5. Information from Fig. 2. replotted in order to evaluate low temperature asymptote. The selected line is the same as in Fig. 4.

So far, we have mainly considered the formation of isothermal martensite below room temperature, but in the Fe-C system isothermal martensite has been observed above room temperature. TTT (TimeTemperature-Transformation diagrams for formation of isothermal martensite in Fe-C will be

656

BORGENSTAM and HILLERT:

ACTIVATION FOR ISOTHERMAL MARTENSITE IN ALLOYS

presented in two different figures: Fig. 6 including information from synthetic Fe-C alloys and Fig. 7 including information from plain carbon steels. Already by 1949 Howard and Cohen [29] had observed an increasing rate of isothermal transformation of austenite as the M, temperature was approached in a plain carbon steel with 1.35% C and an M, of 104°C (see Fig. 7). They dismissed the idea that it could be due to isothermal martensite and proposed that it was instead caused by an increased rate of the diffusional transformation. The phenomenon was thus regarded as a “swing back” of the bainite curve, a name proposed later by Radcliffe and Rollason [30] who also dismissed the idea of isothermal martensite as an explanation. Radcliffe and Rollason also presented C-curves for a plain carbon steel with 0.65% C and M, at 293°C and for two Fe-C alloys with 0.77% C and M, at 264”C, and 1.07% C and MS at 196°C (Figs 6 and 7). The same phenomenon was observed by Schaaber [31] in 1955 and he believed that it was caused by isothermal martensite. Vasudevan et al. [32] observed a similar effect in a plain carbon steel with 0.97% C but they interpreted the isothermal product formed below MS as bainite. Their C-curve for 1% transformation is included in Fig. 7. Finally, in 1959 Smith et al. [33] made in situ observations on an Fe-1.16% C alloy, and found an isothermal transformation product that “snapped into view” at 188°C after 1300 s and at 18 1“C after 900 s, and did not grow further. This was regarded as evidence of isothermal martensite formed above MS because MS was at 175°C. Their curve for 10% isothermal martensite is included in Fig. 6. In 1970 Edwards and Kennon [34] presented C-curves for a plain carbon steel with 1.44% C and M, at 93°C. They presented C-curves for 1% transformation to two different isothermal products, the lower curve representing isothermal martensite and the upper one lower bainite (Fig. 7). Recently this phenomenon has been studied very carefully by Oka and Okamoto [35]

\

‘x_ ---Hmv8~J~dC!d1a1(29) ----V~udevmetd.(32) . . . . . . . . . .. Rddiffe d Mh m-

‘\

---___ I> (jo)

----~EdtwdsandKumon(34) ---Huhlpcll(39) --ohmaietd.(44)

200’ ld

I 10’

I ld

I ld

I 104

I 16

I 106

l-he. s

Fig. 7. Information on isothermal formation of martensite and lower bainite in plain carbon steels.

in four high carbon alloys, 0.85, 1.10, 1.45 and 1.80% C (Fig. 6). At low carbon contents the nose temperature is below MS but at higher carbon contents it is above. The reported MS temperatures in all these studies are included in Figs 6 and 7, represented by a black dot on the respective line. The kinetics of lower bainite will first be examined as a background for the analysis of the kinetics of isothermal martensite in group 2. METHOD

OF

ANALYSING BAINITE

INFORMATION

ON

It is the firm belief of the present authors that the edgewise growth rate of bainite, including lower bainite, is mainly governed by diffusion of carbon away from the edge of a growing ferrite plate. In order to compare TTT diagrams for different steels it is thus convenient first to eliminate the effect of the carbon content. This will be done here by applying Hillert’s modification of Zener’s equation for lengthening of a plate [36], which may be a rough approximation but sufficient for the present purpose: 21_ D AxmatnrN D(x~‘~- x0)2.- RT ,. AXtransE X0 8OV,

(5)

where x0 is the initial mole fraction of carbon in the steel and it is usually much larger than the carbon content of the growing ferrite in bainite. Thus, Axtransf could be approximated by x0. Furthermore, xYbis the mole fraction of carbon in y (i.e. austenite) at the interface to CI (i.e. ferrite). It varies much with temperature and the extrapolation from experimental information at higher temperatures is uncertain and depends on the particular thermodynamic model chosen for the solution of carbon in f.c.c. Fe. It also varies with the alloy content and in a manner that depends on what kind of conditions are established at the moving interface. Equilibrium information on XY’from = the binary Fe-C system can be taken from the SGTE (Scientific Group Thermodata Europe) solution database [37] and shows a strong increase at decreasing temperature. The diffusion coefficient D varies with temperature and with x7. Thus, the variation of D with temperature is even more uncertain. In order to reduce the uncertainty in a comparison between different steels one may thus study the rate v not directly but as vx’/T which will then be interpreted roughly as Dfxy’” - x0)*R/8aV,,, in the hope that this combination of quantities does not vary with the steel composition. The TTT diagrams do not give lengthening rates of bainite plates but the time for formation of various fractions of bainite. It will be assumed that the time to form a certain low fraction is proportional to the lengthening rate and we shall thus plot log[t.T/x’] against - 1/T. INFORMATION

ON BAINITE

There are many papers giving TTT diagrams for steels and many compilations have been published.

BORGENSTAM

&

and HILLERT:

ACTIVATION FOR ISOTHERMAL MARTENSITE IN ALLOYS

14-

14-

16-

16-

18-

18& 6 20: 0 22-

c’ 20: b 22242628-

y.

4

3: FbO.97%C-2.918Cr Fmm Ref. 41: 4: Fe-I.O4%w.O%Cr 5: Fe-I.O5%C-5.7%Cr ohmd et al. (44) 6: Fe-LOIBC-l.Ol%cr t t I

5

6

I

2426-

I

I

8 9 Iog(t.T/xq

1

1

10

II

g___---

12

Fig. 8. Information on isothermal formation of lower bainite Cr steels. The straight line is chosen to represent the low temperature asymptote to all the information in Figs X-12.

For the present examination only a small fraction of the available TTT diagrams for bainite formation has been considered. Information has been taken from a number of careful studies of TTT diagrams for synthetic Fe-C alloys and plain carbon steels. In order to explore the effect of alloying elements, several diagrams will be taken from two papers by Hultgren [38, 391, the reason being that it was regarded essential to be able to compare different TTT diagrams, which is safer if they are determined in the same way. Hultgren presented a number of C-curves for Fe-C-X alloys and from these works C-curves for Fe-C-Cr are included in Fig. 8, for Fe-C-Mn in Fig. 9 and for Fe-C-Mn-Si, Fe-C-MO and Fe-C-Ni in Fig. 10. A C-curve presented by Hultgren for a plain carbon steel is included in Fig. 7. The curves taken from Hultgren represents 1% transformation. In 1944 Russell and McGuire [40] presented C-curves for an Fe-0.53% C-4.05% Mn steel. Their curve for isothermal formation of bainite is included in Fig. 9. From Supplement to the Atlas of Isothermal Transformation Diagrams [41], published in 1953, two C-curves for the formation of bainite in Fe-C-Cr steels are included in Fig. 8. From Atlas zur Wiirmebehandlung der Stiihle II in 1954156 [42] a C-curve for a high tungsten steel is included in Fig. 10. In 1967 Rao and Winchell [43] presented isothermal transformation curves for Fe10.2 at.% Ni alloys with 0.12 to 0.75 at.% C at 383 and 400°C. Points for the highest and lowest carbon content at 400°C are included in Fig. 10. From Ohmori et al. [44] a C-curve for formation of bainite for a 0.58% C plain carbon steel is included in Fig. 7 and a C-curve for an Fe-l% C-l% Cr steel is included in Fig. 8, the curves representing the start of transformation.

I: Fe-O.S3%4XO5%Mn Hultgnn (38): 2: Fe-a.wtC-1.89%Mn 3: Fe-1M%GZAl%Mn 4: Fe.O.52%C-3.2396Mn 5: Fe-O.5I%C-4.12%Mn 6: tWW9%C-5.2O%Mn 7: Fe+.63%C-!.O5%Mn,

4

7

6

5

log Fig.

9. Information

651

,

8

9

IO

(t,T/x’)

on isothermal formation bainite in Mn steels.

of lower

KINETICS OF LOWER BAINITE AND ISOTHERMAL MARTENSITE IN GROUP 2 In order to facilitate comparison with lower bainite in alloy steels, the information on lower bainite and isothermal martensite for group 2 in Figs 6 and 7 was modified and replotted as for bainite (Figs 11 and 12). A single line was then selected in order to describe the bainite formation at low temperatures in all systems presented in Figs 8-12; the straight line and its slope corresponds to an activation energy of 80 kJ/mol. The lower parts of the C-curves for bainite transformation in the Fe-C alloys and the plain carbon steels in Figs 11 and 12 can be represented surprisingly well with this single line. The curves for the alloy steels in Figs 8-12 seem to be displaced in parallel, as the same slope is indicated. Of course, one could expect individual differences due to differences in composition, but the parallel displacement is surprisingly small. This can only be explained by

1: Fe-O.18%C-l.2I%Mn-O.35%Si 3: PeA54%C-O.82%Mo 5: Fd).27%C-3.38%Ni-O.l8%hh -0.718Cr-O.S3%MnG?6%Si

4

5

6

7

8

9

10

log (t.T/x”) Fig.

10. Information on isothermal formation bainite in various steels.

of lower

658

BORGENSTAM and HILLERT:

4

5

6

7 8 lo8 (t.T/x”)

ACTIVATION FOR ISOTHERMAL

9

10 log k

Fig. 11. Information from Fig. 6 replotted in order to evaluate a common low temperature asymptote for lower bainite. The information falling below the asymptote concerns isothermal martensite.

assuming that the conditions at the x/y interface is much closer to paraequilibrium than orthoequilibrium [45]. Otherwise, the effect of different alloy contents should have been much stronger. In order to be noticed experimentally it is necessary for isothermal martensite to form at shorter times than is represented by the selected straight line. It would have been extremely interesting to evaluate a straight line representing the low temperature asymptote for isothermal martensite, but the formation of athermal martensite makes that impossible. Nevertheless, the impression from Figs 11 and 12 is that this asymptote could be roughly parallel to the asymptote selected for lower bainite, but displaced 1 or 2 orders of magnitude to shorter times. It is thus tempting to suggest that the

10,

I

1

1

9

10

dU

4

5

6

7 8 log (t-T/x”)

MARTENSITE IN ALLOYS

Fig. 12. Information from Fig. 7 replotted in order to evaluate a common low temperature asymptote for lower bainite. The information falling below the asymptote concerns isothermal martensite.

Fig. 13. Comparison of slope of straight line in Figs 8-12 with slope of a line representing temperature dependence of a reaction controlled by C diffusion. For the calculation of D, the average value of XYin the concentration profile was taken as half the equilibrium value, x+.

formation of isothermal martensite in these alloys is closely related to the formation of lower bainite, e.g. by the following mechanism. Suppose the C-curves for bainite represent the formation of bainite units of a detectable size, say 10 pm in length. Assuming constant growth rate and a negligible incubation time, one can then estimate that the bainite units were about 0.1 pm long at the time when isothermal martensite starts to form. It thus seems possible that the isothermal formation of martensite units in these alloys may have been triggered by bainite units when they reached a size of about 0.1 pm. In order to test if the slope of the straight line drawn in Figs 8-12 is representative of a reaction controlled by carbon diffusion, one should examine the temperature dependence of D(x”” - x0)*R/8aVm. As an approximation x0 was neglected in comparison with XY’and ~ R/&TV, was treated as a constant; .xY’~ was taken here as the equilibrium value. Since D depends strongly on the carbon content of y, the result is very sensitive to the choice of the average value for xY in the concentration profile. The evaluation was made here by taking half the equilibrium value, XY’“,and using an expression for @(xc) due to Agren [46]. In Fig. 13 the calculated line for l/D’(x~~“)*is compared with the slope of the straight line in Figs 8-12. The lines have been displaced horizontally in order to coincide as well as possible. It is evident that the slope of the straight line can be explained by assuming that the rate of formation of bainite is governed by diffusion of carbon away from the advancing edge of an tl plate. OCCURRENCE OF ISOTHERMAL MARTENSITE IN Fe-Ni ALLOYS, GROUP 3 There is much information site in Fe-Ni alloys with

on isothermal martenlow carbon contents.

BORGENSTAM and HILLERT: .

-

ACTIVATION FOR ISOTHERMAL

---.Yeo(50): -----Kmnk0(51) , ,,.,,. 1 , “1”1

-Tslmki et al. (47) Moireye” * al. (48)

mi’~~~,~,~“““*:“i

tlI(4)

1

Time, s Fig. 14. Isothermal formation of martensite in Fe-Ni alloys with very low C contents. The arrows marked II and IV show the temperatures of plateaux II and IV at the same Ni contents.

However, the information is quite confusing difficult to find useful data.

and it is

Complete C-curves have only been reported by Tsuzaki et al. [47] in three Fe-Ni alloys with 14.99, 18.08 and 23.46% Ni and low carbon contents, < 0.005%, Nos l-3 in Fig. 14. MS is reported to be just above the isothermal curves, but it could equally be A4,, since the cooling rate used in the M, determinations is very low, only 0.20 K/s. Moiseyev et al. [48], Raghavan and Cohen [49], Yeo [50] and Korenko [51] have all observed isothermal martensite above MS. However, at lower temperatures the transformation becomes too rapid to be measured experimentally or the transformation

-

-

Tsumki et al. (47)

?I/ ..,..

ul . . .....~ . . ._y

10-2

10“

loo 10’ ‘.@iJr-1). s

ld

ld

Fig. 15. Information from Fig. 14 for less than 25% Ni, replotted in order to evaluate a common low temperature asymptote.

MARTENSITE IN ALLOYS

659

kinetics is changed to burst martensite and complete C-curves are not reported. The curves are included in Fig. 14. Isothermal martensite below MS has also been reported in several works [9,52-551, but here the isothermal martensite transformation might be influenced by the presence of athermal martensite and is therefore not included in Fig. 14. Figure 14 indicates that in the Fe-Ni system there are two separate groups of transformation, one containing the results reported by Tsuzaki et al. and Moiseyev et al. in the temperature range 400-7Ob K, and the other containing results around 300 K reported by Raghavan and Cohen, Yeo and Korenko. Temperatures representing plateaux II and IV, taken from the lines drawn in a previous review by the present authors [l] for the Fe-Ni system, are marked in Fig. 14 for the alloys in the first group. These are plateaux for the thermal arrest temperatures on very rapid cooling. KINETICS

OF

ISOTHERMAL GROUP 3

MARTENSITE

IN

Figure 15 gives the information from Tsuzaki et al. in Fig. 14, modified in the same way as for group 1. high alloy steels. MS, was chosen in the same way as described for group 1 steels. A straight line is chosen to describe the three curves in an attempt to determine the activation energy. However, it should be realized that this line is actually based on alloys 2 and 3 only, because the information on alloy 1 is limited to a very narrow range of temperature and it falls close to the line in Fig. 15 simply because the MS, value has been adjusted to accomplish that. Admittedly, the information may be too limited to justify the construction in Fig. 15, but it is interesting to note that it results in a low temperature asymptote with a slope represented by an activation energy around 80 000 J/mol, which is much larger than the activation energy determined for group 1, high alloy steels, about 7000 J/mol. The value of 80 000 J/mol is comparable to the value for lower bainite and isothermal martensite in the carbon containing alloys of group 2 given by the straight line in Figs 8-12. However, in that case it was concluded that the rate controlling mechanism is the diffusion of carbon and here we should look for a different mechanism because these alloys have a very low carbon content. If the Fe-Ni information were plotted in the same way, i.e. by dividing by .P. it would fall many orders of magnitude to the right. When looking for the mechanism operating in these Fe-Ni alloys, a hint may be given by the fact that the group of curves at about 300 K in Fig. 14 behaves in a completely different way; unfortunately, however, that information is limited to the upper parts of the C-curves and does not even extend down to the nose. It is quite possible that these alloys belong to group 1, high alloy steels. These Fe-Ni alloys all contain about 29% Ni except for one of

660

BORGENSTAM

and HILLERT:

ACTIVATION FOR ISOTHERMAL MARTENSITE IN ALLOYS

Yeo’s alloys which had 24.9% Ni and 1.58% Ti. The alloys studied by Tsuzaki et al. had 14.99, 18.08 and 23.46% Ni. Figure 14 gives the impression that their alloys transform by a process that is depressed to lower temperatures and longer times as the Ni content is increased and may have moved outside the right-hand side of the diagram for 29% Ni. Instead, another transformation appears at lower temperatures and shorter times. Supposing the latter transformation is the isothermal formation of martensite, it seems as an interesting possibility that the transformation studied by Tsuzaki et al. is related to the transformation causing plateau II at even lower Ni contents. Thus, the positions of the plateaux observed on very rapid cooling are marked to the right in Fig. 14. The information on plateau II comes from Mirzayev et al. [56] by interpreting some results as due to plateau II [57], although they were originally thought to be due to plateau I. Even with that modification, the information available on plateau II extends to only about 15% Ni. It is thus valuable to be able to compare with the results of Moiseyev et al. obtained for 12% Ni. They found a double C-curve with its upper nose at about 30 K above its plateau II, which may be compared to the nose observed by Tsuzaki et al. for 14.99% Ni, falling at about 30 K above its plateau II. This may be taken as support for the suggestion that Tsuzaki et al. did not really study a martensitic transformation but a transformation related to plateau II. However, it is disturbing that the transformation studied by Moiseyev et al. is considerably slower than the one studied by Tsuzaki et al. It should be noted that the M, temperatures they reported may simply represent the upper temperature limit of their transformation, whatever kind of transformation it is. It is interesting to note that the distance between the nose and plateau IV, which supposedly represents M,, decreases from about 150 K at 12%, to 100 K at 14.99% Ni, about 50 K at 18.08% Ni and about 20 K at 23.46% Ni. This may explain why this transformation disappears at higher Ni contents. The second nose indicated by the results of Moiseyev et al. may be related to plateau IV and may represent isothermal martensite. In order to test the idea that the transformation studied in Fig. 15 is related to the transformation causing plateau II in experiments with very rapid cooling, one should also compare the rates of the transformations. In pure Fe, plateau II occurs at 720°C and can be prevented by cooling rates of 110 000, 54 000 and < 8000 K/s at grain sizes of 7, 60 and 150 pm [58]. It is not evident how one can use these values to estimate the time of isothermal transformation in pure Fe. However, it has been reported [59] for an Fe-0.31% C alloy, which has a bainite nose at 550°C and 0.3 s, that the transformation can be prevented by a cooling rate of 1000 K/s but not at 800 K/s. The three cooling rates given above may thus give transformation times of pure Fe

of about 0.003, 0.006 and > 0.03 s, respectively. If these values were plotted in Fig. 14 they would compare favourably with a curve drawn at the values for the noses on the three C-curves. It may not be justified to take this result as support for the idea that the transformation studied by Tsuzaki et al. was not isothermal martensite but a transformation related to bainite, but at least it does not contradict that idea. DISCUSSION AND SUMMARY In the present work an attempt has been made to rationalize the wealth of information on isothermal martensite in ferrous alloys. It has been necessary to neglect many of the fine details. For example, the effect of grain size has not been taken into account. It has also been necessary to compare C-curves for different amounts of martensite, but 5% has been preferred whenever available. Attention has been concentrated on the formation of isothermal martensite above M, in order to avoid complicating effects, but in some cases it has been necessary to add information below M, in order to get a more complete picture. In several cases so-called anisothermal martensite has been reported to form during cooling to the experimental temperature. Information from such experiments has been included in order to cover a wider range of alloy compositions. Finally, it should be emphasized that different austenitizing treatments have been used in different studies and that could have had an effect on the results. In view of such factors one should not expect close agreement between results from different studies, but it seems reasonable to expect that it should be possible to identify general tendencies. The main result of the present work is the conclusion that the information on isothermal martensite actually concerns three different phenomena and they occur in three groups of alloy. It should be emphasized here that only so-called a martensite was examined and E occurring in high Mn steels, was martensite, excluded. It is generally agreed that the difficult step in the formation of martensite is the nucleation. For athermal martensite it is well established that growth occurs at high speed and it is usually assumed that the same holds for all isothermal martensite, although the experimental evidence is less complete. If this is true, the kinetics of isothermal martensite will also be controlled by the nucleation process and the difference in kinetic behaviour between different groups of alloys should then be caused by different mechanisms of growth during the formation of a nucleus capable of growing at high speed. One may distinguish between different kinds of martensite according to their microscopic appearance (e.g. lath and plate martensite) or crystallographic features. Such differences were not considered in the present work because it was presumed that they do

BORGENSTAM

and HILLERT:

ACTIVATION

that nucleation stage but develop during the rapid growth. In the group of high alloy steels, the MS temperature is very low and there is a wide range of temperatures above M, where isothermal martensite can be studied. The main ambition in the present work was to evaluate the temperature dependency of the rate controlling mechanisms by finding the low temperature asymptote to the C-curves. The experimental information on each alloy is too limited to give any accuracy to such an evaluation. However, by examining information from all alloys in this group it is possible to see a general tendency. The slope of the single asymptote, chosen to represent all the alloys, corresponds to an activation energy of 7 kJ/mol. Even though this value is very uncertain, the analysis clearly indicates that the rate controlling mechanism has a very low activation energy and can hardly depend on any diffusion process. It is more probable that dislocation movements are involved. In any case, the low activation energy serves to identify this type of isothermal formation of martensite. In carbon containing alloys one has observed isothermal martensite close to MS but not always above it. In this case it is even more difficult to evaluate the low temperature asymptote. The examination thus started with an evaluation of the asymptote for lower bainite which is observed in all these alloys. However, for each alloy the information is too limited and it was necessary again to combine information from many alloys. In doing this it was assumed that the time of formation depends on the rate of diffusion of carbon away from the advancing edge of an c(plate and the experimental time was thus divided by the carbon content of the steel. It was then possible to choose a single asymptote for the Fe-C alloys and the plain carbon steels. Even information from high alloy steels with carbon fall surprisingly close to the same asymptote and support the slope chosen for the asymptote. The slope of the asymptote is close to what one could expect from a diffusion controlled process. No attempt was made to account for the absolute values of the time of formation, only its temperature dependence. The reason is that there are large uncertainties in the calculation of the growth rate of bainite at low temperatures. A major difficulty is caused by the strong composition dependence of the diffusion coefficient of carbon in the 7 phase. Another difficulty is the unknown shape of the edge of the bainite plate at low temperatures. When comparing the time of formation of isothermal martensite in carbon containing alloys with the chosen asymptote, there is a strong impression of a close relationship. It thus seems natural to propose that isothermal martensite in these alloys is triggered by the formation of submicroscopic plate-like bainite, still too small to be recognized as bainite. If this is correct then the temperature dependence for isothermal martensite should be

not concern

FOR ISOTHERMAL

MARTENSITE

IN ALLOYS

661

roughly the same as for lower bainite and corresponds to an activation energy of about 80 kJ/mol. This is very much higher than the value obtained for the first group of alloys and from a practical point of view it is not difficult to distinguish between the two cases. It is generally agreed that the athermal formation of martensite is limited by the difficult process of nucleation. Any factor promoting the nucleation may thus give martensite above MS that is athermal or isothermal depending on how the nucleation is promoted. However, there should always be an upper temperature limit, M,, above which no martensite can form because it cannot grow there even if it is already nucleated. In a recent attempt to determine M, in an Fe-C alloy with 1.62% C it was observed that athermal martensite can indeed grow above M,. at least about 8 K above M, [4]. The information on isothermal martensite discussed in the present report indicates that M, in high alloy steels may be as much as 150 or 200 K above MS. For Fe-C alloys, Oka and Okamoto studied the distance between M, and M,,. which must certainly be below M,. They reported a temperature difference of 90 K at 1.8% C but it decreased to zero at about 1% C. This will certainly affect the possibility of studying isothermal martensite without severe interaction from athermal martensite in the range of lower carbon contents. However. it does not necessarily have any consequence for the present proposal that isothermal martensite is triggered by submicroscopic plates of lower bainite. The third group of alloys are binary Fe-Ni alloys with up to about 25% Ni. The information for each alloy is much too limited to allow a low temperature asymptote to be evaluated. However, an asymptote can perhaps be found if it is assumed that all the alloys have a common asymptote, an assumption which seems reasonable in view of the results from the other groups of alloys. The slope of an asymptote, chosen in this way, corresponds to an activation energy of about 80 kJ/mol. This is similar to the slope of a line calculated for the escape of carbon from the edge of an x plate by diffusion, also about 80 kJ/mol. However, that mechanism should give a very high growth rate due to the low carbon contents in the Fe-Ni alloys. An interesting possibility is that the transformation in these alloys is not strictly martensitic but is related to the transformation yielding plateau II on very rapid cooling of alloys with lower alloy contents. It has previously [57] been proposed that this transformation is crystallographitally related to the formation of bainite. If the isothermal transformation studied by Tsuzaki et ul. is not isothermal formation of lath martensite. as they claimed. then one might even doubt that it is a transformation with rapid growth, controlled by slow nucleation. One should be open to the possibility that it is a transformation with gradual growth and related to bainite. Comparison of Fig. 14 with Fig. 6

indicates that the growth rate would then be about

662

BORGENSTAM and HILLERT:

ACTIVATION FOR ISOTHERMAL MARTENSITE IN ALLOYS

100 times faster than for bainite. Since the carbon content is so low, < 0.005%, one may interpret this as the limiting growth rate of bainite obtainable by decreasing the carbon content to a very low value. That rate would be controlled by the mobility of the cr/v interface at the edge of an K plate. However, it should be emphasized that it is surprising that this mechanism should have an activation energy so close to the one found for bainite in alloys with carbon, in both cases evaluated as 80 kJ/mol. Isothermal formation of martensite in FeNi alloys with 28.5% Ni or more may be of the same types as the one observed in high alloy steels with a very low M,. Finally, it should be mentioned that the information on the Fe-Ni alloys, denoted 2 and 3 in Fig. 14, would fall well in line with the information on Fe-C alloys in Fig. 6. Thus, one should consider quite a different possibility, namely that isothermal martensite in alloys of group 2 and 3 form by the same mechanism, but then it should be a mechanism that is not sensitive to the carbon content. Acknowledgement-This work was supported by a special grant from the Royal Institute of Technology and NUTEK.

REFERENCES 1. A. Borgenstam and M. Hillert, unpublished work (1996). 2. N. N. Thadhani and M. A. Meyers, Prog. Muter. Sci. 30, 1 (1986). 3. S. C. Das Gupta and B. S. Lement, Trans. ACME, J. Metals 191, 727 (1951). 4. A. Borgenstam, M. Hillert and J. Agren, Acta metall. mater. 43, 945 (1995). 5. B. L. Averbach, M. Cohen and S. G. Fletcher, Trans. ASM 40, 728 (1948). 6.

B. L. Averbach and M. Cohen, Trans. ASM 41, 1024 (1949). 7. G. V. Kurdjumov and 0. P. Maksimova, Dokl. Akad. Nauk SSSR 61,83 (1948). 8. S. A. Kulin and M. Cohen, Trans. AIME, J. Metals 188, 1139 (1950). 9. E. S. Machlin and M. Cohen, Trans. AIME, J. Metals 194,489

(1952).

10. R. E. Cech and J. H. Hollomon, Trans. AIME, J. Metals 197, 685 (1953).

11. C. H. Shih, B. L. Averbach and M. Cohen, Trans. AZME, J. Metals 203, 183 (1955). 12. S. R. Pati and M. Cohen, Actu metall. 19, 1327 (1971). 13. G. Ghosh and V. Raghavan, Mater. Sci. Engng. SO,65 (1986). 14. T. Kakeshita, K. Kuroiwa, K. Shimizu, T. Ikeda, A. Yamagishi and M. Date, Mater. Trans. JIM 34, 415 (1993). 15. V. Raghavan and A. R. Entwisle, iron Steel Inst., Spec. Rep. 93, 30 (1965). 16. A. R. Entwisle, MetalSci. J. 2, 153 (1968). 17. C. T. Peters, P. Bolton and A. P. Miodownik, Acta metall. 20, 881 (1972). 18. S. A. Kulin and G. R. Speich, Trans. AZME, J. Metals 194, 258 (1952). 19. R. Lagneborg, Actu metall. 12, 823 (1964). 20. Y. Imai, M. Izumiyama and K. Sasaki, Sci. Rep. Res. Inst. Tohoku Univ. 18, 39 (1966). 21. W. K. C. Jones and A. R. Entwisle, Met. Sci. J. 5, 190

(1971). 22. M. Holmquist, J.-O. Nilsson and A. Hultin Stigenberg, Scripta metall. Mater. 33, 1367 (1995).

23. J. C. Fisher, J. H. Hollomon and D. Turnbull, Metals Trans. 185, 691 (1949). 24. M. Cohen, E. S. Machlin and V. G. Paranjpe, in Thermodynamics in Physical Metallurgy, pp. 242-270.

ASM, Cleveland OH (1949). 25. G. V. Kurdjumov and 0. P. Maksimova, Dokl. Akad. Nauk SSSR 73, 95 (1950). Tribute to Morris Cohen 26. V. Raghavan, in Martensite-A (edited bv G. B. Olson and W. S. Owen). DD. 197-225.

ASM International, Cleveland. OH (1992): ’ 27. G. Ghosh and G. B. Olson, Acta MetaN. Mater. 42,337l

(1994). 28. S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes. McGraw-Hill. New York (1941). 29. R. T. Howard and M. Cohen, Trans. AZmE l?S, 384 (1949). 30. S. V. Radcliffe and E. C. Rollason, J. Iron Steel Inst.

191, 56 (1959). 31. 0. Schaaber, Trans. AIME, J. Metals 203, 559 (1955). 32. P. Vasudevan, L. W. Graham and H. J. Axon, J. Iron Steel Inst. 190, 386 (1958). 33. M. F. Smith, G. R. Speich and M. Cohen, Trans. AIME 215, 528 (1959). 34. R. H. Edwards and N. F. Kennon, J. Austra. Inst. Metals 15, 203 (1970). 35. M. Oka and H. Okamoto, Metall. Trans. 19A, 447

(1988). 36. h. Hillert, Metall. Mater. Trans. 25A, 1957 (1994). 37. SGTE solution database with Thermo-Calc extensions,

Dept. of Materials Science and Engineering, Royal Institute of Technology, Stockholm, Sweden. 38. A. Hultaren. Trans. ASM 39. 915 (1947). 39. A. H&gren, Jernkontorets knnaler 135, 403 (1951). 40. J. V. Russell and F. T. McGuire, Trans. ASM 33, 103 (1944). 41. Supplement to the Atlas of Isothermal Transformation Diagrams, United States Steel Corporation, Pittsburgh,

PA (1953). 42. Atlas zur

Wiirmebehandlung der Stiihle II, Verlag Stahleisen M. B. H. Dusseldorf. (1954156). 43. M. M. Rao and P. G. Winchell; Trans: A>ME 239,956 (1967).

44. Y. Ohmori, H. Ohtsubo, K. Georgima and N. Maruvama. Mater. Truns. JIM 34. 216 (1993). 45. M. Hfilert and J. Agren, in Advances in Ph;?se Transitions (edited by J. D. Embury and G. R. Purdy), pp. l-19. Per amon Press, Oxford (1988). 46. J. R gren, Sciptu metall. 20, 1507 (1986). 47. K. Tsuzaki, T. Fukiage, T. Maki and I. Tamura, Mater. Sci. Forum, 56-58, 229 (1990). 48.

A. N. Moiseyev, L. I. Izyumova, M. P. Usikov and E. I.

Estrin, Phys. Met. Metall. 51, 137 (198?). 49. V. Raghavan and M. Cohen, Metall. Trans. 2, 2409 (1971). 50. R. B. G. Yeo, Trans. AZME 224, 1222 (1962). 5 1. M. K. Korenko, PhD Thesis, Massachusetts Institute of

Technology (1973). 52. A. V. Anadaswaroop and V. Raghavan, Scripta metall. 3, 221 (1969). 53. C. A. V. de A Rodrigues, C. Prioul and L. Hyspecka, Metall. Trans. 15A, 2193 (1984). 54. V. M. Yershov and N. L. Oslon, Phys. Met. Metall. 27, 164 (1969). 55. J. Mrovec and M. Jakesova, Scrtpta metall. 12, 1091 (1978). 56. D. A. Mirzayev, 0. P. Morozov and M. M. Shteynberg, Phys. Met. Metall. 36, 99 (1973). 57. A. Borgenstam and M. Hillert, Metall. Mater. Trans. 27A, (1996). 58. 0. P. Morozov, D. A. Mirzayev and M. M. Shteynberg, Phvs. Met. Metall. 34, 114 (1972). 59. 0. P. Morozov, Phys. Met. Metall. 57, 129 (1984).