W. XIN: CARBURIZING-QUENCHING HARDNESS MODEL ANALYSIS AND COMPARISon
9–16
CARBURIZING-QUENCHING HARDNESS MODEL ANALYSIS
AND COMPARISON
MODELNA ANALIZA IN PRIMERJAVA MODELA ZA NAPOVED
TRDOTE JEKLA DOSE@ENE S POSTOPKOM NAOGLJI^ENJA IN
KALJENJA
Wang Xin
Mechanical Engineering College, Henan University of Engineering, Zhengzhou, PR China
Prejem rokopisa – received: 2023-09-23; sprejem za objavo – accepted for publication: 2023-11-13
doi:10.17222/mit.2023.978
Unlike the conventional calculation models of the hardness, a carburizing-quenching hardness model considers the influence of
the carbon content on the phase transformation and hardness. The volume fraction model (VFM) and Jominy curve model
(JCM) for calculating carburizing-quenching hardness of 20CrNi2Mo steel were built in this study, and the models were used to
calculate the hardness of Jominy and gear samples. The hardness results of both models were compared, and the simulation re-
sults were verified with the corresponding test results. The results show that the hardness values obtained with both models have
a certain calculation accuracy. But due to considering the influence of residual austenite (RA) on the hardness, the simulation
accuracy of the VFM was better for the low Jominy distance and the hardened case, while the simulation accuracy of the JCM
was better for the large Jominy distance and the low-carbon martensite region; the calculation of the latter is more convenient
and its accumulated error is small.
Keywords: carburizing-quenching, Jominy distance, carbon content, hardness
V modelu za napoved trdote jekla z naoglji~enjem in kaljenjem je potrebno izvesti primerjavo s konvencionalnimi modeli za
izra~un trdote, kiupo{tevajo vsebnost ogljika in fazne transformacije. V tem ~lanku avtor opisuje izdelavo modela volumskega
dele`a (VFM, angl.: Volume Fraction Model) in modela Jominy krivulj (JCM, angl.: Jominy curve model) za izra~un trdote
jekla za zobnike vrste 20CrNi2Mo po naoglji~enju in kaljenju. Modela je nato avtor verificiral z Jominyjevim preizkusom in na
vzorcih zobnikov. Med seboj je primerjal oba modela. Rezultati primerjav med obema modeloma so pokazali dokaj dobro
natan~nost izra~unov. Toda zaradi upo{tevanja vpliva vsebnosti zaostalega austenita (RA, angl.: residual austenite) na trdoto je
bila natan~nost simulacije z VFM modelom bolj{a pri manj{ih Jominy oddaljenostih od hlajene ploskve preizku{anca. Bolj{a
natan~nost JCM modela pa je bila pri ve~jih Jominy oddaljenostihod hlajene povr{ine preizku{ancev in pri podro~jih z manj{o
vsebnostjo ogljika v martenzitu. Tako se je izkazalo, da je ta model kumulativno ugodnej{i za izra~unavanje trdote in ima v
celoti manj{o napako.
Klju~ne besede: naoglji~enje in kaljenje, Jominy razdalja, vsebnost ogljika, trdota
1 INTRODUCTION
Carburization-quenching can improve the surface
hardness and wear resistance of a sample. It is very im-
portant to obtain the core and surface hardness distribu-
tion of parts exposed to carburizing-quenching.
1,2
In or-
der to avoid time-consuming process optimization, it is
important to build an accurate hardness prediction
model. At present, there are mainly two models for hard-
ness calculation, namely the VFM and JCM.
3–5
The VFM
can simulate a nonhomogeneous microstructure field,
and confirm the hardness of the microstructure. Doane et
al.
6
indicated that the method is only suitable when the
carbon content is lower than 0.5 %. Wood et al.
7
pre-
dicted the quenching hardness by simulating the temper-
ature and microstructure fields of a steel cylinder, taking
the hardness of RA as being equal to the hardness of
martensite. Yuan et al.
8
summed up some formulas for
the calculation of martensite hardness, and predicted the
hardness by simulating the martensite transformation.
Zhang et al.
9
predicted the carburizing-quenching hard-
ness by considering the effects of RA and the carbon
content on the martensite hardness. But this method can
only show the hardness of a discrete point. The distribu-
tion of the hardness field cannot be presented as a shaded
display and easily extracted. Schwenk et al.
10
calculated
the hardness depth for spur or bevel gears, considering
all physical aspects of the carburizing and quenching
process with the VFM. During the simulation, geometric
approximation of gears ensured the accuracy of carburiz-
ing and hardness calculations.
The JCM predicts the hardness based on the Jominy
hardness curve and Jominy cooling curve.
11
Ko et al.
12
predicted the hardness of an A16061 steel cylinder in the
quenching process with the QFA (quench factor analy-
sis). The predicted hardness was in a good agreement
with the experimental hardness. But this model only cal-
culated the hardness of a discrete point, and the extrac-
tion of the result was inconvenient. Kianezhad et al.
13
im-
proved the QFA method, based on the work by Rometsch
Materiali in tehnologije / Materials and technology 58 (2024) 1, 9–16 9
UDK 669.14:621.785.5 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 58(1)9(2024)
*Corresponding author's e-mail:
haiyang630@163.com (Wang Xin)

et al.,
14
experimentally proving a higher efficiency of the
new method. Moreover, Kianezhad et al.
15
predicted the
hardness of quenched steel parts using the QFA and
ANNs (artificial neural networks), and pointed out that
the accuracy of the QFA was limited to the dominant
structure. Wang et al.
16
summed up some formulas for
the Jominy hardness. These formulas were only applica-
ble to a carbon content of less than 0.56 %. Wang et al.
11
built an improved JCM for the carburizing-quenching
process, but did not consider the influence of RA on the
quenching hardness. The above studies mostly focused
on a hardness prediction model, and there is little analy-
sis or comparison of both models in terms of the princi-
ple and effectiveness. In the present study, the VFM and
JCM of 20CrNi2Mo steel were built. Both models were
used for the hardness calculation of Jominy and gear
samples using DEFORM_V11. The corresponding test
results were utilized to verify the simulation results, and
the simulation accuracy of both models was analysed
and discussed.
2 EXPERIMENTAL PROCEDURE
Standard Jominy and three-tooth gear samples were
prepared for the experiment and analysis, as shown in
Figure 1. The experiment material was 20CrNi2Mo
steel, and its composition is shown in Table 1. A quarter
of the Jominy model and a half of a single tooth model
were built as symmetric models. The mesh of the Jominy
model in the length direction was refined. The carburiz-
ing process for both the Jominy and gear samples was
the same. The carburizing was carried out in two stages,
boost and diffusion, where the temperature was 930 °C
while the carbon potentials and times were 1.15 for 22 h
and 0.85 for 5.75 h. The carbon content was tested with
an X-350A spectrometer. The carburized Jominy sam-
ples were subjected to the Jominy test where the heating
temperature was 820 °C. Finally, the hardness of 12
points on the two axial planes were tracked and mea-
sured within 60 mm from the water-cooled end. The
quenching temperature of the gear was 830 °C; the gear
was immersed in quenching oil at 60 °C for 0.5 h. The
hardness within 60 mm was tracked and tested in the ad-
dendum and dedendum.
Table 1: Composition of 20CrNi2Mo
Element C Mn Ni Si Cr Mo
w/% 0.20 0.47 1.80 0.16 0.53 0.24
3 VFM AND JCM
Carbon diffusion can be described with Fick’s law,
according to which the transfer coefficient is 0.0001123
mm/s when the temperature is 930 °C.
17
In addition to
considering the influence of the temperature and carbon
content, with the calculation of the diffusion coefficient
(D), the effect coefficient of alloying elements is also
considered to ensure the calculation accuracy,
18,19
as
shown below:
DC
C
RT
=−
−− ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ 00047 16
37000 6600
. exp( . )exp
()
⎟ ×
×
++
−−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 0013 0013 0040
0055 0014
...
..
Mn Mo Cr
Si Ni
(1)
where T and C denote the temperature and carbon con-
tent, and each element denotes its mass percentage, as
shown in Table 1. R is the gas constant.
The VFM hardness is calculated with the linear mix-
ing principle,
20
as follows:
HRC V HRC
ii e
=
∑
(2)
where HRC
e
is the Rockwell hardness of an element
while V
i
and HRC
i
are the volume fraction and
Rockwell hardness of the phase, respectively. The VFM
W. XIN: CARBURIZING-QUENCHING HARDNESS MODEL ANALYSIS AND COMPARISon
10 Materiali in tehnologije / Materials and technology 58 (2024) 1, 9–16
Figure 1: Jominy and gear sample

calculates the volume fraction and hardness of each ele-
ment and then estimates the hardness based on the linear
mixing principle. The volume fraction calculation is di-
vided into the diffusive transformation and martensite
transformation in the carburizing-quenching process.
The former is calculated by the KJMA model based on
TTT curves.
21,22
Considering the influence of the carbon
content, all the TTT curves are calculated by JMatPro,
as shown in Figure 2. With an increase in the carbon
content, the TTT curves of bainite and ferrite shift to the
right and the transformation temperature decreases. The
carbon content has little influence on the pearlite trans-
formation.
Martensite transformation is calculated with the
Koistinen-Marburger equation,
23,24
as shown in Equation
(3), wherein the martensite transformation temperature
(Ms) under different carbon contents is calculated with
Equation 4.
25
VM s T
M
=− − − 1 exp( ( ))  (3)
M
S
= 548-440C-(26Mn+14Si+14Ni+11Cr+9Mo) (4)
Here,V
M
is the martensite content while is taken as
0.011.
The hardness of pearlite, ferrite, bainite and marten-
site with a carbon content of less than 0.5 % is calculated
based on the Maynier equation,
9
as shown below.
The Vickers hardness of ferrite and pearlite (HV
F-P
)is
HV
(F-P)
= 42+223C+53Si+30Mn+12.6Ni+7Cr+19Mo+
(10-19Si+4Ni+8Cr+130V)logV
(F-P)
(5)
The Vickers hardness of bainite (HV
B
)is
HV
B
= -323+185C+330Si+153Mn+65Ni+144Cr+
191Mo +(89+53C-55Si-22Mn-10Ni-20Cr-
33Mo)logV
B
(6)
The Vickers hardness of martensite HV
M
is
HV
M
= 127+949C+27Si+11Mn+8Ni+16Cr+21logV
M
(7)
V
F-P
, V
B
and V
M
are critical cooling rates of each
phase, respectively. The calculating equations are as fol-
lows:
lgV
(F-P)
= 6.36-(0.43C+0.49Mn+0.78Ni+0.26Cr+
0.301P
A
) (8)
lgV
B
= 10.17-(3.8C+1.07Mn+0.7Ni+0.57Cr+
1.58Mo+0.00321P
A
) (9)
lgV
M
= 9.81-(4.62C+1.1Mn+0.54Ni+0.5Cr+0.6Mo+
0.001831P
A
) (10)
W. XIN: CARBURIZING-QUENCHING HARDNESS MODEL ANALYSIS AND COMPARISon
Materiali in tehnologije / Materials and technology 58 (2024) 1, 9–16 11
Figure 2: TTT curves of the diffusive transformation Figure 3: Jominy hardness curve and cooling curve

Here, P
A
is equal to the product of the heating time
and heating temperature. The calculated Vickers hard-
ness of each phase can be converted to Rockwell hard-
ness. The hardness of RA is equal to that of ferrite.
The martensite hardness with the carbon content of
more than 0.5 % is only related to the carbon content,
26
calculated with Equation (11).
HRC
C
C
C
M
=
+
+
−+
4 213
01 32
21 5
3
3
..
. (c > 0.5 %) (11)
Jominy hardness curve and Jominy cooling curve are
obtained with the JCM. Each cooling time corresponds
to a Jominy distance in the Jominy cooling curve, and
each Jominy distance corresponds to a Jominy hardness
in the Jominy hardness curve; the cooling time and
Jominy hardness mutually correspond based on the two
curves. So when the cooling time of an element is calcu-
lated, the corresponding Jominy hardness is the hardness
value of the element. Due to the carbon gradient in the
carburizing case, the JCM should consider the influence
of the carbon content on the hardness. So it should calcu-
late the Jominy hardness curve for different carbon con-
tents. The Jominy hardness curve of 20CrNi2Mo steel
with the initial carbon content (0.2 %) was obtained
based on the experiment result.
27
For the other carbon
content, the hardness of Jominy distance within 40 mm
is calculated with the USS-Atlas formula,
11
and the re-
sidual Jominy hardness is obtained with JmatPro. The
Jominy cooling time is the representative cooling time of
most alloy steels, that is, the cooling time of 820–500 °C
(t
8/5
).
28
The t
8/5
at different locations of the Jominy sam-
ple was obtained based on the temperature field of the
Jominy sample. The Jominy hardness and cooling curve
are shown in Figure 3. With the increasing carbon con-
tent, the Jominy hardness is also gradually improved.
However, when the carbon content is greater than
0.63 %, the Jominy hardness of the low Jominy distance
is slightly different. So when the cooling time and car-
bon content of an element are calculated, the correspond-
ing Jominy hardness in Figure 3 is the hardness of the
element.
4 RESULTS AND DISCUSSION
4.1 Carbon content distribution
The simulated results of the carbon content for the
Jominy sample are shown in Figure 4. The maximum
carbon content is located on the surface of the sample,
with a value of 0.85 %. The experiment result for the
maximum carbon content on the surface was 0.97 %,
which was significantly higher than the simulation. The
material had a certain saturated carbon concentration,
and under a high carbon potential during the boost stage,
a certain amount of carbides occurred on the surface, re-
sulting in a higher surface carbon content. The SEM mi-
crograph of the sample surface in Figure 4c shows an
obvious distribution of carbide particles. However, the
carburizing simulation did not consider the carbide; it
only considered the carbon diffusion in austenite, so the
experiment results for the surface were significantly
higher than the simulated results. But when the distance
from the surface was greater than 0.5 mm, the measured
and simulated results were in good agreement, and the
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12 Materiali in tehnologije / Materials and technology 58 (2024) 1, 9–16
Figure 4: Carbon content of the Jominy sample: a) shade display, b) carbon content distribution curve, c) SEM micrograph
Figure 5: Carbon content of the gear

maximum difference between the simulated and mea-
sured results was 0.04 %.
The simulated results for the carbon content on the
gear are shown in Figure 5. The maximum carbon con-
tent is located on the surface of the gear, and the maxi-
mum value is 0.748 %. Due to convex and concave
shapes, the addendum region has the highest carbon con-
tent. The distance from the surface to the carbon content
of 0.35 % defines the hardened area, which is about
2–3 mm at the addendum and dedendum.
4.2 Phase transformation results
The phase transformation results for the carburized
Jominy sample are shown in Figure 6. Martensite trans-
formation mainly occurs in the J5 (distance from the wa-
ter-cooled end) region. The maximum martensite content
is 98.5 %, and there is also RA with a maximum content
of 17.5 %. Bainite transformation mainly occurs in the
J(5-16) region. In addition, due to the increase in the car-
bon content on the surface, the temperature of bainite
transformation decreases, and bainite occurs on the sur-
face in the region above J16, which corresponds to the
TTT curve from Figure 2. Ferrite occurs in the core in
the region above J16. As shown in Figure 7, the surface
microstructure of the Jominy sample is mostly lower
bainite, observed with an optical microscope, which is
basically consistent with the simulated results. The phase
transformation results of the gear are shown in Figure 8.
Low-carbon and high-carbon martensite occur in the
core and surface of the gear, respectively. In addition,
there is still some RA on the surface. The content of RA
at the addendum is the highest (14 %). The surface and
core microstructures in Figure 9 are basically consistent
with the simulation results.
4.3 Hardness results
Figure 10 shows the hardness shaded display results
and hardness curves of both models. The VFM hardness
was calculated based on the carbon content and the con-
tent of each phase. The carbon contents at different
W. XIN: CARBURIZING-QUENCHING HARDNESS MODEL ANALYSIS AND COMPARISon
Materiali in tehnologije / Materials and technology 58 (2024) 1, 9–16 13
Figure 7: Surface microstructure of the Jominy sample
Figure 8: Phase transformation of the gear
Figure 6: Phase transformation of the Jominy test

points and the phase transformations at different Jominy
distances are different, so the hardness is also different.
Martensite transformation mainly occurs in the J5 re-
gion. The hardness of the J5 region is the highest; the
maximum hardness is 62.3 HRC, followed by the hard-
ness in the J(5-16) region and on the surface in the re-
gion above J16 where the bainite transformation mainly
occurs. The low hardness occurs in the residual region
where the pearlite and ferrite transformations occur, and
the minimum hardness is 17.1 HRC. Meanwhile, due to
a different carbon content and phase transformation,
there is a certain hardness gradient from the surface to
the centre at the same Jominy distance. The JCM hard-
ness is calculated based on the t
8/5
and carbon content,
which are different in the regions with different Jominy
distances, so the hardness is also different. The wa-
ter-cooled end has the minimum cooling time and a high
carbon content, so the hardness is the largest and the
maximum hardness is 64.3 HRC. At the same Jominy
distance, the surface has a lower cooling time and a
higher carbon content, so the hardness of the surface is
higher. From the surface to the core, the cooling time in-
creases and the carbon content decreases gradually, so
the hardness also exhibits a certain gradient.
Similarly, due to different phase transformations, car-
bon contents and cooling times, a certain hardness gradi-
ent appears between the surface and the core of the gear.
The maximum hardness of the gear in the VFM is 59.2
HRC and it is 63.5 HRC in the JCM. In summary, the
carbon contents of both models are the same. The VFM
must couple the results of the temperature and phase
transformation, and then it calculates the hardness with
the Maynier equation and Equation (11). The JCM deter-
mines the hardness only on the basis of the temperature
result; this calculation process is more suitable. The
maximum hardness obtained with the JCM is greater
than that obtained with the VFM, regardless of the gear
or Jominy sample. In Figure 3, the Jominy hardness in-
creases with the increase in the carbon content; however,
there is some RA during the quenching, which reduces
the hardness. The JCM does not consider the influence
of RA on the hardness, so the JCM hardness is higher
W. XIN: CARBURIZING-QUENCHING HARDNESS MODEL ANALYSIS AND COMPARISon
14 Materiali in tehnologije / Materials and technology 58 (2024) 1, 9–16
Figure 9: a) Surface and b) core microstructure of the gear
Figure 10: a) Hardness shaded display of the Jominy sample and
b) the gear; c) hardness distribution curves of the Jominy sample and
d) the gear

than that of the VFM in the regions on the surfaces of the
Jominy and gear samples.
Compared with the experiment, the hardness values
simulated by both models were generally consistent with
the measured results, and the maximum error between
the VFM and test hardness was 8.8 HRC. It showed that
both models have a certain calculation accuracy. As the
JCM did not consider the influence of RA on the hard-
ness, the JCM hardness was higher than the test results in
the curves of the region below J10, and the maximum er-
ror was 5 HRC, while the VFM hardness was basically
in good agreement with the test results. In the region
above J10, the simulation accuracy of the JCM was
good, while the VFM hardness was smaller than the test
result. The VFM needed to couple the results of the tem-
perature and phase transformation; there were certain er-
rors in the calculation of the diffusive phase transforma-
tion and Maynier equation. The JCM hardness could
only be obtained based on the temperature field results,
and the accumulated error was small, so the accuracy of
the JCM was good. In the 3 mm hardness curve, the dif-
ference between both models was relatively small,
mainly because the carbon content at this depth was 0.3
% and the hardness was less affected by the carbon con-
tent.
Similarly, taking 50 HRC as the boundary of
case-hardening, the maximum error between the VFM
hardness and the test results was only 3 %, and the accu-
racy of the VFM was higher than that of the JCM for
case-hardening. The JCM hardness was higher than the
test results, and the maximum error was 4.3 HRC. Out-
side the hardened case, the simulation accuracy of the
VFM was higher than that of the JCM. This shows that
the JCM hardness is more accurate than the hardness of
low-carbon martensite calculated with the Maynier equa-
tion for the core, while the VFM hardness is more accu-
rate for the hardened case.
5 CONCLUSIONS
(1) Unlike the conventional hardness models, a car-
burizing-quenching hardness model considers the influ-
ence of the carbon content on the phase transformation
and hardness. In the VFM, the hardness of each phase
was calculated with the Maynier equation and high-car-
bon martensite hardness equation, while the phase trans-
formations were calculated with the KJMA and
Koistinen-Marburger equations for different carbon con-
tents. Jominy curves were obtained with an experiment,
USS-Atlas formula and JmatPro for different carbon
contents included in the JCM.
(2) By comparison with the test results, the VFM and
JCM have a certain calculation accuracy. Due to consid-
ering the influence of RA on the hardness, the simulation
accuracy of the VFM for the low Jominy distance and
the hardened case was better than that of the JCM. The
maximum errors of the JCM were 5 HRC for the region
below J10 and 4.3 HRC for the hardened case.
(3) Since only the temperature field needed to be cal-
culated, the calculation of the JCM was more convenient
and the accumulated error was small. The simulation ac-
curacy of the JCM for the large Jominy distance and the
low-carbon martensite region was better. Due to the lim-
ited calculation accuracy of the diffusive phase transfor-
mation and Maynier equation, the maximum errors of
VFM were 8.8 HRC for the large Jominy distance and
3 % for the low-carbon martensite region.
Acknowledgment
This work was supported by the Doctoral Cultivation
Fund of the Henan University of Engineering
(D2020004) and Science and Technology Research Pro-
ject of the Henan Province (232102220041).
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