Dilatometric analysis of austenite decomposition considering the effect of non-isotropic volume change
Dong-Woo Suh
a,*, Chang-Seok Oh
a, Heung Nam Han
b, Sung-Joon Kim
aaECO-Materials Research Center, Korea Institute of Machinery and Materials, 66 Sangnam, Changwon, Kyungnam 641-010, Republic of Korea
bSchool of Materials Science and Engineering, Seoul National University, Seoul 151-742, Republic of Korea Received 11 September 2006; received in revised form 7 November 2006; accepted 3 December 2006
Available online 12 February 2007
Abstract
One of the important assumptions underlying conventional dilatometric analysis is that volume changes isotropically during phase transformations. However, the volume change does not in fact occur isotropically and thus dilatation data contain non-isotropic con- tributions. In the present study, we expand the conventional analysis model to take into account the effect of non-isotropic volume change. The contribution of the non-isotropic effect to the dilatation data is quantified and is implemented in the analysis model. Dila- tometric analysis is conducted on measured dilatation data of low-carbon steels to validate the suggested model. The phase fractions evaluated with the model show a good agreement with those obtained by metallographic analysis. It can be shown that considering the effect of non-isotropic volume change is critical for quantitative dilatometric analysis.
2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Steels; Dilatometry; Transformation plasticity; Non-isotropic volume change
1. Introduction
Dilatometric analysis is a useful technique for the study of solid-state phase transformations in ferrous alloys[1–6].
When a phase transformation occurs with an accompany- ing volume change, the dilatometric curve provides infor- mation on the change in atomic volume due to the transformation as well as on the thermal expansion charac- teristics. Therefore, by extracting the transformation- induced atomic volume change from the dilatometric curve and interpreting it with an analysis model, the fraction of individual phases involved in the transformation can be determined as a function of time or temperature.
The conventional method of calculating the phase frac- tion from the dilatometric curve is the so-called ‘‘lever rule’’ [1]. The dilatometric curve shows linear thermal expansion characteristics in the temperature range where no transformation occurs. In the lever rule, two linear seg-
ments of a dilatometric curve are extrapolated as shown in Fig. 1, and the fraction of the transformed phase at a given temperature is evaluated by the relative position of the measured dilatometric curve between the extrapolated lines. In principle, the lever rule is only valid for the case in which partitioning of alloy elements does not occur dur- ing the phase transformation.
During the austenite decomposition into ferrite in low- carbon steels, carbon is ejected from the ferrite due to its solubility limit and enriched in the untransformed austen- ite. The carbon enrichment increases the atomic volume of austenite by stretching the austenite lattice, which makes the dilatometric curve of austenite deviate from linearity as illustrated in Fig. 1. However, the lever rule cannot con- sider such non-linear dilatation of austenite caused by car- bon enrichment.
Several authors have independently suggested dilato- metric analysis models that consider the carbon redistribu- tion during the austenite to ferrite transformation [3–6].
They converted the measured dilatation data into the aver- age atomic volume of the specimen, and then calculated the
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2006.12.007
* Corresponding author. Tel.: +82 55 280 3373; fax: +82 55 280 3599.
E-mail address:dongwoo@kmail.kimm.re.kr(D.-W. Suh).
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phase fraction by analyzing the average atomic volume with the lattice parameters of austenite, ferrite and cement- ite formulated as functions of temperature and solute car- bon content.
One of the most important assumptions in the previ- ously suggested analysis models is that the volume changes isotropically during the transformation. This means that the measured dilatation data is assumed to reflect a one- dimensional change originating from the isotropic volume change associated with the transformation. In general, however, the volume change does not occur isotropically in dilatometric specimens during phase transformation [7–10]. Fig. 2shows the dilatometric curve of low-carbon steel undergoing a thermal cycle consisting of continuous heating and cooling. The change of specimen length would not be found after the thermal cycle if the volume change associated with the phase transformation were isotropic.
But the permanent strain appearing in dilatometric curve indicates that the specimen length is changed by the ther-
mal cycle, which implies that the volume change with the transformation has non-isotropic characteristics.
When the volume change originating from the transfor- mation occurs non-isotropically, the measured dilatation data include the effect of non-isotropic volume change. The observed dilatation during the transformation consists of the contribution of both isotropic and non-isotropic volume change as shown inFig. 3. In view of the fact that dilatometry monitors only the length change, the implementation of the non-isotropic contribution into the dilatometric analysis model is important for quantitative evaluation of the phase fraction from the measured dilatation data.
In the present study, we expand the previous dilatomet- ric analysis model to take into account the effect of non- isotropic volume change. The contribution of non-isotropic volume change to the dilatation data is quantified and implemented in the analysis model. To validate the sug- gested analysis model, we apply the model to the analysis of the measured dilatometric curves of low-carbon steels.
The analysis results are compared with those from metallo- graphic measurement, and the importance of considering the non-isotropic contribution is discussed.
2. Model development
2.1. Previous dilatometric analysis scheme[5,6]
The average atomic volume of a specimen is represented by a linear combination of the atomic volumes of the con- stituent phases as follows:
V ¼X
i
fi Vi ð1Þ
where V is the average atomic volume of the specimen, Viis the atomic volume of phase i, and fiis the volume fraction 0
0 0 1 0
0 8 0
0 6 0
0 4 2 0 0 . 0
4 0 0 . 0
6 0 0 . 0
8 0 0 . 0
0 1 0 . 0
2 1 0 . 0
4 1 0 . 0
dL/L 0(μm/μm)
( e r u t a r e p m e
T oC)
( n i a r t s t n e n a m r e
p ΔT)
Fig. 2. Dilatometric curve of low carbon steel showing length change of specimen after thermal cycle.
e s a h p t n e r a P
x y
n o i t a t a l i d
c i p o r t o s i r e d n u
) f l a h ( e g n a h c e m u l o v
f o n o i t u b i r t n o c
c i p o r t o s i - n o n
e g n a h c e m u l o v
) f l a h (
n o i t a t a l i d d e v r e s b o
) f l a h ( s i x a - x g n o l a
d e m r o f s n a r t
e s a h p
z
Fig. 3. Effect of non-isotropic volume change on measured dilatation data.
e r u t a r e p m e T
Dilatation
A
B
C e t i l r a e p
t r a t s
f o n o i t a t a l i d
h t i w e t i n e t s u a
t n e m h c i r n e n o b r a c n o i t c a r f d e m r o f s n a r t
) C A ( / ) C B (
= e l u r r e v e l y b
Fig. 1. Schematic dilatometric curve during transformation of steel on cooling.
of phase i. Assuming that austenite decomposes into ferrite and pearlite successively on cooling, the volume fractions of ferrite (fa) and pearlite (fp) during the transformation are derived from Eq.(1):
fa ¼ V Vc
Va Vc ð2Þ
fp ¼V Vcþ fa ðVc VaÞ
Vp Vc ð3Þ
where Vc, Vaand Vpdenote the atomic volumes of austen- ite, ferrite and pearlite, respectively.
If a volume change associated with the phase transfor- mation is relatively small and is supposed to be isotropic, the average atomic volume change of specimen (DV) is related to a length change (DLiso) monitored by dilatome- try as follows:
DLiso
L0 ¼1 3DV
V0 ¼1 3 V
V0 1
ð4Þ where L0is the specimen length at a reference temperature from which the relative length change (DLiso) is measured.
V0is the average atomic volume at the reference tempera- ture. From Eq. (4), the measured length change is con- verted into the average atomic volume of the specimen by the following relation:
V ¼ a V0 3DLiso
L0
þ 1
ð5Þ The optimizing factor a, which would be unity for an ideal case, is introduced to compensate a non-ideality orig- inating from the experimental error. Using the lattice parameter of austenite, ferrite and cementite in Table 1 [11–15], Eq.(6) gives the atomic volume of each phase as a function of temperature and carbon content. Then, the phase fraction of ferrite and pearlite can be evaluated from the measured dilatation data by combining Eqs. (2), (5) and (3), (5), respectively.
Vc¼ 1 4
a3c; Va¼ 1 2
a3a
Vp¼ ð1 qÞ Vaþ q Vh; Vh¼ 1 12
ah bh ch
ð6Þ
where q is the phase fraction of cementite in pearlite.
Since the volume change of the transformation was in the previous studies supposed to be isotropic, the measured length change is directly applied to DLisoin Eq. (5). How-
ever, when the volume change associated with the transfor- mation occurs non-isotropically, it is not rigorous to presume that the measured dilatation data is applicable to Eq. (5). For quantitative dilatometric analysis, the con- tribution of the non-isotropic volume change in the mea- sured dilatation data should be compensated for.
2.2. Dilatometric analysis considering the effect of non- isotropic volume change
Fig. 4schematically describes the effect of non-isotropic volume change on the dilatation of low-carbon steel. If the volume change during the transformation occurs isotropi- cally, the dilatation curve would form a closed loop as shown by the dotted line for a thermal cycle consisting of continuous heating and cooling. As mentioned, however, the measured dilatation data includes the contribution of non-isotropic volume change associated with the transfor- mation, which makes the dilatometric curve an open loop.
The contribution of non-isotropic volume change to the dilatation data is designated as a non-isotropic dilatation in the present study. The non-isotropic dilatation can be specified by comparing both dilatometric curves in Fig. 4.
The total non-isotropic dilatation on thermal cycle (DT) consists of the non-isotropic dilatation during austenite formation on heating (D1) and that during austenite decomposition on cooling (D2). From their graphical defi- nitions, D1and D2can be expressed as follows:
D1¼ jobserved dilatation between T1and T2j
jideal dilatation under isotropic volume change between T1and T2j
D2¼ jideal dilatation under isotropic volume change between T3 and T4j
jobserved dilatation between T3 and T4j
Therefore, a positive value of D1indicates that the dila- tation specimen contracts more along the monitored direc- tion than it does under isotropic volume change associated with austenite formation. In the same way, a positive value of D2 indicates that the specimen expands less along the monitored direction than it does under isotropic volume change accompanying austenite decomposition. Note that D1or D2in Fig. 4 indicates an accumulated non-isotropic dilatation as a result of the complete formation or decom- position of austenite. To compensate for the non-isotropic
Table 1
Lattice parameters of austenite, ferrite and cementite[11–15]
Lattice parameter in A˚
Austenite ac= (3.6306 + 0.78 Æ Cc) Æ {1+(24.9–50 Æ Cc) Æ 106Æ(T–1000)}
Ferrite aa= 2.8863 Æ {1 + 17.5 Æ 106Æ(T–800)}
Cementite ah= 4.5234 Æ {1 + (5.311 Æ 106–1.942 Æ 109Æ T+ 9.655 Æ 1012Æ T2) Æ (T – 293)}
bh= 5.0883 Æ {1 + (5.311 Æ 106–1.942 Æ 109Æ T+ 9.655 Æ 1012Æ T2) Æ (T – 293)}
ch= 6.7426 Æ {1 + (5.311 Æ 106–1.942 Æ 109Æ T+ 9.655 Æ 1012Æ T2) Æ (T – 293)}
Ccis the atomic fraction of solute carbon in austenite and T is in K.
dilatation in the measured dilatation data during the trans- formation, it is necessary to consider the evolution of the non-isotropic dilatation with the progress of the transformation.
As indicated in Fig. 4, the non-isotropic dilatation on heating as well as that on cooling should be considered to analyze the overall transformation kinetics during the complete thermal cycle. In the present study, we try to focus on the dilatometric analysis of austenite decomposi- tion, so hereafter the procedure for the implementation of non-isotropic dilatation on cooling is introduced into the dilatometric analysis model. Although the procedure is intended to account for the non-isotropic effect during austenite decomposition, the same procedure can be applied for the implementation of non-isotropic dilatation on heating to analyze the transformation kinetics during austenite formation.
Previous studies have reported that transformation plas- ticity was responsible for the dimensional change of speci- mens undergoing thermal cycling [5,6,9,10].
Transformation plasticity is a deformation phenomenon which is observed when a phase transformation accompa- nying the volume change proceeds under external stress even lower than the yield stress of the material. The widely accepted interpretation of transformation plasticity is that the external stress biases the internal strain originating from the internal stress created during the transformation by volume mismatch between the allotropic phases.
Assuming rigid plastic deformation of the weaker phase, Greenwood and Johnson derived an analytical solution for the permanent strain after completing the transforma- tion as a function of volume mismatch, yield stress of weaker phase and external stress[16]. Schuh and Dunand generalized the Greenwood–Johnson model to formulate the strain rate of the transformation plasticity as the trans- formation progresses[17]. If the weaker phase is deformed through yielding, the strain rate is given by
_e¼5 6 r
rY
DV V
_f ð7Þ
where rYis the yield stress of the weaker phase, r is the external stress, jDV/Vj is the volume mismatch and _f is the time derivative of the transformed phase fraction. In fact, the experimental results on the transformation plastic- ity indicated that the evolution of the permanent strain had a linear relationship with the fraction of the transformed phase[18–20].
Recently, Jaramillo and Lusk[10]observed non-isotro- pic dilatation during a thermal cycle without externally applied stress. They suggested that even if the dilatometric specimen was not subjected to external stress, the radial thermal gradient of the dilatometric specimen induced the apparent stress responsible for the transformation plastic- ity. Therefore, it is reasonably assumed that the evolution of non-isotropic dilatation has a linear relationship with the transformed fraction in the present study.
For austenite decomposition on cooling, the non-isotro- pic dilatation (DLnon-iso) is
DLnon-iso¼ ðfaþ fpÞ D2; ð8Þ
where fa and fp are the volume fractions of ferrite and pearlite, respectively. InFig. 4, the non-isotropic dilatation is defined by the difference between the isotropic dilatation (DLiso) and the measured one (DLexp). Thus the isotropic dilatation during the transformation is formulated as DLiso¼ DLexpþ DLnon-iso¼ DLexpþ ðfaþ fpÞ D2 ð9Þ
Since DLisoin Eq.(9)is the isotropic dilatation compen- sated for the effect of non-isotropic volume change, it can be applied to Eq.(5).
For ferrite transformation, Eqs.(5) and (9)make
V ¼ a V0 3DLiso
L0 þ 1
¼ a V0 3DLexpþ 3fa D2
L0
þ 1
¼ A faþ B A¼ a V0 3D2
L0
; B¼ a V0 3DLexp
L0 þ 1
ð10Þ
Combining Eq.(10)with Eq.(2) generates a non-linear equation with respect to the fraction of ferrite.
For pearlite transformation V ¼ a V0 3DLiso
L0
þ 1
¼ a V0 3DLexpþ 3ðfaþ fpÞ D2
L0 þ 1
¼ A faþ A fpþ B
ð11Þ
From Eqs.(11) and (3), the pearlite fraction during the aus- tenite decomposition is given by
fp ¼A faþ B Vcþ fa ðVc VaÞ
Vp Vc A ð12Þ
e r u t a r e p m e T
Dilatation
c i p o r t o s i - n o n
c i p o r t o s i
Δ1
T1 T2
T3 T4
ΔT=Δ1+Δ2
Fig. 4. Hypothetical dilatometric curve representing non-isotropic dilata- tion upon thermal cycle.
If the austenite decomposition is regarded as successive transformations of ferrite and pearlite, the criterion for the initiation of the pearlite transformation has to be estab- lished. In the present study, the pearlite transformation is assumed to start when the precipitation of cementite in aus- tenite becomes thermodynamically possible[21]. As the fer- rite transformation proceeds, carbon enriches the remaining austenite. When the carbon content in austenite reaches the extended Acm line in the phase diagram, the pearlite transformation is supposed to be initiated. Here, the Acm line is the temperature–composition line in the phase diagram, which indicates the chemical composition of austenite in equilibrium with the cementite.
2.3. Quantification of non-isotropic dilatation on cooling (D2)
Using the suggested dilatometric analysis model, the phase fraction during austenite decomposition can be eval- uated as long as the non-isotropic dilatation on cooling (D2) is known. In what follows, the procedure for quantifi- cation of non-isotropic dilatation from the dilatometric curve is presented.
Fig. 5shows the definition of non-isotropic dilatation on cooling. Note that the austenite is taken as the reference state to define D2. Let T3and T4represent arbitrary tem- peratures before and after the austenite decomposition.
The solid line represents the measured dilatometric curve, while the broken line indicates the ideal dilatometric curves under the isotropic volume change. The non-isotropic dila- tation after the completion of the transformation (D2) is given by the difference between the ideal dilatation (DL2) and the measured dilatation (DLf2) between T3and T4.
The ideal dilatation (DL2) between T3and T4has the fol- lowing relationship to the atomic volume and the length of the specimen at each temperature:
VðT4Þ ¼ a V ðT3Þ 3DL2
LðT3Þþ 1
; ð13Þ
where V(T) and L(T) are the average atomic volume and the length of specimen at temperature T, respectively.
If the phase fractions at T3and T4are known, which give the average atomic volume of specimen at each tempera- ture, the ideal dilatation between T3 and T4(DL2) can be evaluated from Eq. (13), and therefore the non-isotropic dilatation on cooling (D2) can be quantified.
Since T3refers to an austenite single-phase region, the phase fraction is obvious and the average atomic volume is simply calculated. However, the average atomic volume at T4is not straightforward to evaluate because the frac- tion of ferrite and pearlite after completing the transforma- tion might be different from the equilibrium value depending on the cooling rate during the transformation.
The fraction of ferrite and pearlite at T4is closely related to the amount of non-isotropic dilatation because the phase fraction determines the average atomic volume, V(T4), and conversely the amount of non-isotropic dilata- tion affects the final fraction of ferrite and pearlite. This means that compatibility between the phase fraction and the amount of non-isotropic dilatation should be main- tained for reliable dilatometric analysis. By making use of the correlation, the phase fraction at reference temperature T4together with the amount of non-isotropic dilatation is quantified via the following procedure.
(i) Assume the fraction of ferrite and pearlite at T4to be equilibrium value (f(i)).
(ii) Evaluate D2from f(i), and then implement D2into the dilatometric analysis model to return the phase frac- tion at T4(f(i+1)).
(iii) Compare f(i+1)with f(i)by checking the convergence criterion j(f(i+1)/f(i)) – 1j 6 e (e = 0.001 was adopted for the present study).
(iv) Repeat (ii) and (iii) with replacing f(i)by f(i+1) until convergence criterion is satisfied.
3. Experimental procedure
Table 2 shows the chemical composition of the alloys used here for dilatometric measurement. The ingots of steel A and B were prepared by vacuum induction melting and hot-rolled into plates of 30 and 20 mm thick, respectively.
For homogenization of initial microstructure, the plates were heat-treated at 1300C for 2 h followed by air-cooling to room temperature and then reheated to 950C followed by furnace cooling. Cylindrical dilatometric specimens (3 mm (diameter)· 10 mm (length, L)) were machined
Dilatation
Δ2
ΔLf2
ΔL2
T3 T4
d e r u s a e m
c i p o r t o s i
Fig. 5. Definition of non-isotropic dilatation on cooling.
Table 2
Chemical composition of investigated alloys (wt.%)
C Mn Si P S Fe
Steel A 0.055 1.3 0.29 0.03 0.007 Bal.
Steel B 0.144 1.1 0.11 0.01 0.006 Bal.
from the plates with the longitudinal direction (L) parallel to the rolling direction.
The dilatometric measurements were performed using a quench dilatometer (Dilatronic III, Theta Inc.) which heats the specimen with an induction coil and detects the length change with a linear variable displacement transducer (LVDT). The dilatometric specimens were heated to 950C at a rate of 1 C s1 and held for 1 min followed by cooling to room temperature at a rate of 1C s1. For steel B, different cooling rates of 0.3 or 5C s1were also adopted to investigate the effect of cooling rate on the transformation behavior and the corresponding dilatomet- ric analysis results.
The microstructure after the thermal cycle was observed by light microscopy. The specimens for metallographic analysis were prepared using a standard method with 2%
nital etchant. The fraction of ferrite and pearlite was mea- sured using quantitative image analysis software.
4. Results and discussion 4.1. Dilatometric curves
Fig. 6 shows the measured dilatometric curves of the investigated low-carbon steels. Note that the dilatometric curves are shifted along y-axis to coincide with each other in the austenite region. Fig. 6a shows the dilatometric curves of steel A measured during repeated thermal cycling.
The dilatometric curves show similar behavior on each ther- mal cycle, but they exhibit some deviation along both the temperature and the length scales. The shift of the dilato- metric curve along the length scale is ascribed to the differ- ence in magnitude of the non-isotropic effect, which is discussed later. The shift along the temperature scale, on the other hand, is believed to be more dependent on exper- imental conditions, such as the condition of the thermocou- ple attachment during the thermal cycle. Hence it is necessary to average the analysis results from the dilatomet- ric curves measured under the same heat treating schedule in order to obtain a reliable analysis of the transformation
behaviors. The dilatometric curves of steel A show open loops, which indicate that the specimen length is decreased after each thermal cycle, possibly because of non-isotropic dilatation during the transformation. Fig. 6b shows the dilatometric curves of steel B with different cooling rates.
It is found that the austenite decomposition starts at lower temperatures with increasing cooling rate, but a change of specimen length after the thermal cycle still occurs regard- less of cooling rate.
The specimen length was measured with a micrometer before and after each thermal cycle and compared with the length change obtained from the dilatometric curve.
Fig. 7shows the reductions in specimen length measured with the micrometer and from the dilatometric curve.
The reduction in specimen length displayed on the dilato- metric curve is 8.6–13 lm for steel A and 4.6–6.4 lm for steel B. It is found that the amount of length change is not the same on each thermal cycle even after the repeated thermal cycles for steel A. As mentioned above, the length change after one thermal cycle is affected by the total amount of non-isotropic dilatation during the transformations on heating and cooling, and it is believed that transformation plasticity plays an important role in the evolution of non-isotropic dilatation. According to studies on the transformation plasticity, directional move- ment of the phase boundary during the transformation can generate the non-isotropic strain[16]. This means that the radial thermal gradient in a cylindrical dilatometric specimen [10], which encourages the propagation of the phase front along the radial direction, can give rise to non-isotropic dilatation. In real situations, the movement of the phase boundary during the transformation will be complicated, and the propagation of the phase front will not be the same even for the same thermal history. This phenomenon is thought to be responsible for the differ- ences in the magnitude of the length change after repeated thermal cycling.
The changes in specimen length obtained from the dila- tometric curves are comparable with those measured by the dilatometer; however, it is found that the dilatometric
0 0 0 1 0 0 9 0 0 8 0 0 7 0 0 6 0 0 5 6 0 0 . 0
9 0 0 . 0
2 1 0 . 0
5 1 0 . 0
dL/L 0(m/m)
( e r u t a r e p m e
T oC)
1 e l c y c
2 e l c y c
3 e l c y c
0 0 0 1 0 0 9 0 0 8 0 0 7 0 0 6 0 0 5 6 0 0 . 0
9 0 0 . 0
2 1 0 . 0
5 1 0 . 0
dL/L0(m/m)
( e r u t a r e p m e
T oC)
3 . 0 oC/s 1oC/s 5oC/s
a b
Fig. 6. Measured dilatometric curves during thermal cycle. (a) Steel A. (b) Steel B.
curve consistently shows larger reductions in specimen length compared to the micrometer measurements. Similar results were reported by other researchers[9], but the rea- sons for this discrepancy are not clearly understood at present.
4.2. Microstructures
Fig. 8shows the microstructures of dilatometric speci- mens before and after the thermal cycles. The microstruc- ture of steel A is not noticably changed by the thermal cycle with a cooling rate of 1C s1. For steel B that is sub- jected to different cooling rate, it is found that the ferrite grains are refined by the increase in cooling rate. Table 3 shows the pearlite fraction after the thermal cycle measured by quantitative image analysis. The pearlite fraction of steel A is close to the equilibrium fraction. The pearlite fraction of steel B is also comparable to the equilibrium fraction when the cooling rate is 0.3C s1, but it is grad- ually increased with the acceleration of cooling rate. When the cooling rate is 5C s1, the measured pearlite fraction is increased by 4% compared with the equilibrium fraction.
This increase is believed to be associated with the thermo- dynamic aspects of cementite precipitation in austenite, and will be discussed later.
4.3. Dilatometric analysis with the developed model
Using the dilatometric model expanded to considering non-isotropic dilatation, the measured dilatometric curves were analyzed to evaluate the phase fraction. As mentioned above, iterative calculation of the phase fraction is per- formed until the phase fraction after completing the trans- formation and the non-isotropic dilatation (D2) converge on such values that are compatible with each other.
As indicated inFig. 9, the amounts of non-isotropic dila- tation upon cooling (D2) are calculated to be 4.9, 5.0 and 6.3 lm for steel A, and 3.9, 3.7 and 4.7 lm for steel B.
The positive value of non-isotropic dilatation on cooling
indicates that the actual increase in specimen length associ- ated with austenite decomposition is less than that expected under ideal isotropic volume expansion. This is possibly related to the radial thermal gradient in the cylindrical dila- tometric specimen which induces the transformation plas- ticity through the non-random propagation of the phase front during the transformation. Fig. 10 demonstrates two extreme cases of two-dimensional shape change during austenite decomposition according the interaction between the phases. When both phases do not interact with each other during the transformation, the shape change indicates an isotropic expansion as shown in Fig. 10b. However, as indicated by Greenwood and Johnson in their seminal work on transformation plasticity [16], if we consider the interaction between both phases by confining the deforma- tion to the weaker phase, the shape change of the weaker phase along the interface in restricted by the harder phase.
Then it will display a non-isotropic characteristic, as shown in Fig. 10c. If the direction of the movement of the phase front is completely random during the transformation, the global shape strain becomes isotropic for either case inFig. 10, but when the movement of phase front becomes non-random – due to the radial thermal gradient in the cylindrical specimen – the overall transformation strain has non-isotropic characteristics, and the increase associ- ated with the transformation is less along axial direction than that expected under ideal isotropic expansion. This finding is consistent with the positive value of non-isotropic dilatation on cooling found in the present study.
Fig. 9a shows the dilatometric analysis results of steel A.
The phase evolution during the austenite decomposition as well as the transformation start temperature of ferrite and pearlite is consistently evaluated from the dilatometric curves by repeated thermal cycles. InFig. 9a, symbols rep- resent the phase fraction after completing the transforma- tion measured by the metallographic analysis. The phase fraction evaluated by dilatometric analysis shows a good agreement with results from the metallographic analysis.
This demonstrates that the suggested dilatometric analysis 3
2 1
0 0 . 0
5 0 . 0
0 1 . 0
5 1 . 0
0 2 . 0
Δ T/L 0(%)
e l c y c f o r e b m u n
r e t e m o t a li D
r e t e m o r c i M
5 1
3 . 0 0 0 . 0
5 0 . 0
0 1 . 0
5 1 . 0
Δ T/L 0(%)
( e t a r g n il o o
c oC/s)
r e t e m o t a li D
r e t e m o r c i M
a b
Fig. 7. Change of specimen length after thermal cycle. (a) Steel A. (b) Steel B.
Fig. 8. Microstructures of dilatometric specimens. (a) Steel A. (b) Steel B.
Table 3
Phase fraction of pearlite after the thermal cycle
Steel A Steel B
Cooling rate (C s1) 1 0.3 1 5
Measured fraction 0.075 ± 0.017 0.214 ± 0.015 0.225 ± 0.025 0.246 ± 0.015
Equilibrium fraction 0.071 0.204
model gives a reliable phase fraction during austenite decomposition into ferrite and pearlite.
Fig. 9b shows the dilatometric analysis results of steel B.
Acceleration of the cooling rate lowers the transformation start temperature. The pearlite fraction after completing the transformation increases with faster cooling rate, which is comparable with the metallographic results.Fig. 9c com- pares in detail the pearlite fractions evaluated by dilatomet- ric analysis with those obtained from the metallographic analysis. It is found that the final pearlite fraction accord- ing to the dilatometric analysis is in excellent agreements with the results of the metallographic analysis.
The increase of the final pearlite fraction with the accel- eration of the cooling rate is possibly related to the thermo- dynamic criterion for the precipitation of cementite in austenite. Considering the equilibrium between austenite and cementite, the carbon content of austenite in equilib- rium with cementite is decreased as the temperature is low- ered. This suggests that the pearlite transformation can be triggered with less carbon enrichment in austenite due to the decrease in the transformation temperature. Because the decomposition of austenite into ferrite proceeds at a
) e t i r r e f ( e s a h p r e k a e w e
s a h p r e d r a h
) e t i n e t s u a (
y r a d n u o b e s a h p f o n o i t c e r i d g n i v o M
y r a d n u o b e s a h p y b t p e w s e r o f e b ) a (
y r a d n u o b e s a h p y b t p e w s r e t f a ) b (
) n o i t c a r e t n i o n (
y r a d n u o b e s a h p y b t p e w s r e t f a ) c (
e c a f r e t n i
g n o l a e s a h p r e k a e w f o e g n a h c e p a h s (
) e s a h p r e d r a h y b d e n i a r t s n o c s i
Fig. 10. Effect of the direction of phase front movement on the shape change: (a) before swept by phase boundary (b) after swept by phase boundary (no interaction) (c) after swept by phase boundary (shape change of weaker phase along interface is constrained by harder phase).
0 0 9 0
0 8 0
0 7 0
0 6 0 . 0
2 . 0
4 . 0
6 . 0
8 . 0
0 . 1
( 1 e l c y
c 2=4.9) ( 2 e l c y
c 2=5.0) ( 3 e l c y
c 2=6.3)
Phase fraction
( e r u t a r e p m e
T oC)
e t i n e t s u e a
t i r r e f d i o t c e t u e - o r p
e t il r a e p
0 0 8 0 5 7 0 0 7 0 5 6 0 0 6 0 5 5 0 . 0 2 . 0 4 . 0 6 . 0 8 . 0 0 . 1
Phase fractions
( e r u t a r e p m e
T oC)
( e r u t a r e p m e
T oC)
3 .
0 oC/s( 2=3.9) 1oC/s(
2=3.7) 5oC/s( 2=4.7)
e t i r r e f d i o t c e t u e - o r p
e t i n e t s u a
e t il r a e p
a
c
b
0 0 8 0 5 7 0 0 7 0 5 6 0 0 6 0 5 5 0 . 0
1 . 0
2 . 0
3 . 0
Phase fractions
3 . 0 oC/s 1oC/s 5oC/s
e t il r a e p
Fig. 9. Phase fractions evaluated with the suggested model compared with metallographic analysis results. (a) Steel A. (b) Steel B. (c) Pearlite fraction of steel B.
lower temperature range under faster cooling rates, the critical carbon content of austenite for the start of pearlite transformation will be reached at a lower ferrite fraction under higher-speed cooling, which results in the increase in the final fraction of pearlite. In fact, since austenite transforms to bainite or martensite over a critical cooling rate, the increase in the pearlite fraction by accelerating the cooling rate is limited to a range in which austenite is decomposed into ferrite and pearlite.
The agreement of the dilatometric analysis results with the metallographic ones in Fig. 9c is primarily attributed to the implementation of thermodynamic criterion for the pearlite transformation. However, it is noted that the fer- rite transformation kinetics should be evaluated before- hand in order to determine the pearlite transformation start point on thermodynamic grounds because the ferrite fraction at a certain temperature affects the carbon content in the remaining austenite. As mentioned, the thermody- namic criterion compares the carbon content in the remain- ing austenite with the critical carbon content for the precipitation of cementite at a given temperature to esti- mate whether or not the pearlite transformation is initi- ated. Therefore, it is believed that an accurate evaluation of the ferrite fraction during austenite decomposition, as well as a reliable criterion for the pearlite transformation, is important to evaluate the fraction of each phase involved in the transformation. In the present study, the non-isotro- pic dilatation is considered to evaluate quantitatively the evolution of the phase fraction during austenite decompo- sition. In the following section, we will discuss the enhance- ment of the analysis results on the transformation kinetics by incorporating the effect of non-isotropic dilatation into the dilatometric analysis.
4.4. Importance of considering non-isotropic dilatation in dilatometric analysis
Fig. 11 compares the dilatometric analysis results in which the non-isotropic dilatation is either taken into
account or ignored. When the non-isotropic dilatation is not considered in the dilatometric analysis (D2= 0), the evaluated phase fractions indicate that austenite still remains even at low temperatures at which austenite decomposition is complete.Fig. 11demonstrates that it is difficult to obtain phase fractions consistent with the metal- lographic analysis results without considering non-isotro- pic dilatation.
This anomaly originates from the fact that the observed increase in specimen length associated with the austenite decomposition is less than that expected for ideal isotropic volume expansion due to the presence of non-isotropic dilatation. Several attempts have been made in earlier stud- ies to resolve this problem, but as far as the present authors are aware, none of them explicitly considers the effect of non-isotropic dilatation.
Fig. 11 also compares the present dilatometric analysis results with those obtained using the lever rule. The figure indicates that the starting and completion temperature of austenite decomposition from the lever rule are comparable with those from the proposed analysis. However, it is noted that the overall transformation kinetics of ferrite and pearl- ite according to the lever rule does not agree well with the present dilatometric analysis. Although the transformation kinetics from the lever rule show some agreement with the proposed analysis results at the early stages of austenite decomposition, there is an inconsistency that increases as the transformation proceeds. As shown inFig. 11, the dif- ference in the transformation kinetics between the two methods becomes greatest when the remaining austenite starts to transform into pearlite, and the inconsistency can be clearly indicated by considering the pearlite trans- formation starting temperature. Based on the measured pearlite fraction in Table 3, the pearlite transformation start temperature is evaluated to be around 703C for steel A and 685C for steel B by the lever rule. These are dis- tinctly higher than the pearlite start temperatures evaluated with the proposed dilatometric analysis, which are around 667C for steel A and 663 C for steel B.
0 0 8 0 5 7 0 0 7 0 5 6 0 0 6 0 5 5 0 . 0
2 . 0
4 . 0
6 . 0
8 . 0
0 . 1
Phase fractions
( e r u t a r e p m e
T oC)
Δ2=3.7 Δ2=0.0
e l u r r e v e l
e t i r r e f d i o t c e t u e - o r p
e t i n e t s u a
e t il r a e p
0 0 9 0 5 8 0 0 8 0 5 7 0 0 7 0 5 6 0 0 6 0 . 0
2 . 0
4 . 0
6 . 0
8 . 0
0 . 1
Phase fraction
( e r u t a r e p m e
T oC)
Δ2=4.9 Δ2=0.0
e l u r r e v e l
e t i r r e f d i o t c e t u e - o r p
e t i n e t s u a
e t il r a e p
a b
Fig. 11. Effect of non-isotropic dilatation on dilatometric analysis results. (a) Steel A. (b) Steel B.
