W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
503–510
INVESTIGATION INTO THE THERMAL DEFORMATION
BEHA VIOR AND THE ESTABLISHMENT OF A CONSTITUTIVE
RELATIONSHIP FOR Mn-Cr-Ni-Mo STEEL
RAZISKA V A TERMI^NE DEFORMACIJE IN IZDELA V A
KONSTITUTIVNEGA MODELA ZA JEKLO VRSTE Mn-Cr-Ni-Mo
Wei Liu
1
, Hongchao Ji
1
*, Weimin Yin
2
, Yubin Chen
2
, Zhiru Dou
2
, Xiaoli Yu
2
1
College of Mechanical Engineering, North China University of Science and Technology, Tangshan, Hebei, 063210,China.
2
China MCC22 Group Corporation LTD., Hebei, Tangshan, 063035, China.
Prejem rokopisa – received: 2024-04-06; sprejem za objavo – accepted for publication: 2024-07-02
doi:10.17222/mit.2024.1150
Isothermal compression tests were conducted on Mn-Cr-Ni-Mo steel at temperatures ranging from 1173 K to 1473 K and strain
rates from 0.01 s
–1
to 10 s
–1
using a Gleeble-3800 thermal simulation tester. Four constitutive models for Mn-Cr-Ni-Mo steel,
namely the Arrhenius model, Fields-Backofen model (F-B), original Johnson-Cook model (J-C), and the improved John-
son-Cook model (mJ-C) were established. A correlation coefficient (R) and the average absolute relative error (AARE) were
employed to evaluate the predictive capability of these four models. Among them, the Arrhenius model exhibited superior accu-
racy in predicting the behavior of Mn-Cr-Ni-Mo steel. It and the isothermal thermal compression finite-element model were im-
ported into Deform-3D software for a numerical simulation, aiming to analyze the distribution law of the equivalent stress field.
A comparison was made between the time-stress data obtained from numerical simulation under different conditions and that
from the isothermal compression tests. The results demonstrate good agreement between the time-stress curves of the numerical
simulation and the experimental measurements, indicating that the established Arrhenius model can effectively simulate the
thermal deformation of Mn-Cr-Ni-Mo steel. These research findings provide valuable fundamental data for simulating the plas-
tic-deformation process of Mn-Cr-Ni-Mo steel.
Keywords: Mn-Cr-Ni-Mo steel, constitutive model, thermal deformation behavior, finite-element analysis
Avtorji v ~lanku opisujejo visokotemperaturne tla~ne preizkuse v temperaturnem obmo~ju med 1173 in 1473K in hitrostih
deformacije med 0,01 s
–1
in 10 s
–1
na jeklu vrste Mn-Cr-Ni-Mo. Preizkuse so avtorji izvajali na termo simulacijskem aparatu
Gleeble-3800. Za izdelavo konstitucijskega modela za izbrano jeklo vrste Mn-Cr-Ni-Mo so uporabili oziroma analizirali
Arrheniusov model, Fields-Backofen model (F-B), originalni Johnson-Cooksov model (J-C), in izbolj{an Johnson-Cook model
(mJ-C). Za ovrednotenje sposobnosti napovedi obna{anja vseh {tirih izbranih modelov so uporabili korelacijski koeficient (R) in
povpre~no absolutno relativno napako (AARE; angl.: average absolute relative error). Med izbranimi modeli se je Arrheniusov
model za jeklo vrste Mn-Cr-Ni-Mo izkazal z najbolj{o sposobnostjo napovedi in najbolj{o natan~nostjo. Ta model in izotermi~ni
tla~ni deformacijski model na osnovi kon~nih elelmentov (FEM; angl.: finite element model) so avtorji vnesli v programsko
orodje Deform-3D z namenom analize in porazdelitve zakona obna{anja ekvivalentnega napetostnega polja. Avtorji so
primerjali pri razli~nih pogojih dobljene eksperimentalne podatke ~as-napetost s podatki dobljenimi z numeri~no simulacijo.
Rezultati analiz so pokazali dobro ujemanje med krivuljami ~as-napetost numeri~nih simulacij z eksperimentalnimi meritvami.
Pri~ujo~e raziskave so potrdile, da lahko z Arrheniusovim modelom u~inkovito simuliramo termi~no deformacijo izbranega
jekla vrste Mn-Cr-Ni-Mo. Te raziskave po mnenju avtorjev prestavljajo tudi temeljne in kvalitetne podatke za simulacijo
procesa plasti~ne deformacije Mn-Cr-Ni-Mo jekel.
Klju~ne besede: Mn-Cr-Ni-Mo jeklo, konstitutivni model, termi~na deformacija, analiza s pomo~jo metode kon~nih elementov
1 INTRODUCTION
As one of the most vulnerable components in the ball
mill, the ball mill liner is affixed to the inner side of the
cylinder to safeguard it against direct impact and grind-
ing from both mill balls and materials.
1–5
The conven-
tional approach for manufacturing ball-mill liners pri-
marily involves casting processes; however, this method
is susceptible to defects that can compromise part perfor-
mance and service life.
6
With the increasing scale and
prevalence of ball mills, there are higher demands on the
mechanical properties and the longevity of liner parts.
7
Therefore, we propose utilizing a die-forging forming
process to enhance the performance of these parts.
Typically, high manganese steel, alloy white iron,
and medium-to-low alloy wear-resistant steel are
employed as traditional lining materials;
8
however, these
materials are mainly suitable for casting processing
rather than forging processing. In order to meet both the
mechanical properties required for ball-mill liner materi-
als and machinability suitable for forging processing, we
have modified the alloying elements and composition
based on 40Cr to develop a special Mn-Cr-Ni-Mo steel
specifically tailored for forging processes. This type of
steel not only fulfills the working requirements for
ball-mill liners, but also exhibits excellent mechanical
properties and machinability.
The process of thermal deformation is accompanied
by the evolution of the microstructure, necessitating an
Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510 503
UDK 669.14:620.172.2 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 58(4)503(2024)
*Corresponding author's e-mail:
jihongchao@ncst.edu.cn (Hongchao Ji)

investigation into the formation mechanism of hot pro-
cessing for novel materials. The constitutive model, a
mathematical representation reflecting macroscopic ma-
terial properties, plays a pivotal role in predicting mate-
rial flow stress and offers a reliable, theoretical founda-
tion for practical production.
9–12
Commonly employed
constitutive models encompass the Arrhenius model,
13
Johnson-Cook model,
14
Zerili-Armstrong model,
15
artifi-
cial neural network (ANN) approach,
16
Fields-Backofen
model,
17
and physics-based constitutive models .
18
Gao et
al. developed an Arrhenius constitutive model for a
solid-solution-strengthened Ni-Cr-Fe alloy, comprehen-
sively investigating the microstructure evolution and op-
timizing the hot-working diagram.
13
Ji et al. employed
the improved Johnson-Cook constitutive model to char-
acterize the flow-stress equation of a 22MnB5 alloy, in
conjunction with the hot-working diagram, and derived
the appropriate range for the hot-working process.
14
Gurusamy et al. refined the Zerilli-Armstrong model and
established a constitutive model for Inconel718, a
nickel-based superalloy.
15
Cai et al. developed an artifi-
cial neural network (ANN) model to predict the stress,
dynamic recrystallization (DRX) fraction, and DRX
grain size of 33Cr23Ni8Mn3N heat-resistant steel.
16
Jia
et al. employed the Fields-Backofen constitutive model
to characterize the thermal deformation behavior of
as-cast AZ31B magnesium alloy and proposed a refined
version of the Fields-Backofen constitutive model.
17
Currently, there is a lack of research on the thermal
deformation behavior and constitutive relationship of
Mn-Cr-Ni-Mo steel, specifically for the forging process
of ball-mill liners. In this study, we conducted an isother-
mal compression test on Mn-Cr-Ni-Mo steel using the
Gleeble-3800 thermal simulation tester. A precise consti-
tutive model was developed to accurately describe the
mechanical behavior of Mn-Cr-Ni-Mo steel, thereby es-
tablishing a solid foundation for comprehensive predic-
tion of its thermal flow deformation and properties.
2 ISOTHERMAL COMPRESSION EXPERIMENT
2.1 Test materials
The test material is a specially fabricated Mn-Cr-
Ni-Mo steel for ball-mill liners, designed to undergo the
forging process based on a 40Cr alloy with modified al-
loying elements and their respective concentrations. Its
elemental composition was determined using a direct
reading spectrometer and XRD energy spectrum analy-
sis, as presented in Table 1. The preparation process is
as follows:
The steel is melted to obtain molten steel, followed
by the addition of alloying elements for compositional
analysis and fine tuning to meet the quality requirements
of chemical composition. Subsequently, the molten steel
is poured into a 200-mm-diameter mold for casting. Af-
ter cooling, the resulting billet is heated in a resistance
furnace at 1100 °C and maintained at this temperature
for 1 h. Then, the forging process is conducted to trans-
form it into a long material with a required section size
of 50 mm × 50 mm, as specified by the laboratory.
Finally, it is cooled to room temperature.
2.2 Experimental design
The material is extracted from the central region of
the cross-section of the prepared Mn-Cr-Ni-Mo steel rod,
ensuring that the sample’s orientation aligns with that of
the rod. The Mn-Cr-Ni-Mo steel was machined into cy-
lindrical specimens with a diameter of 10 mm and a
height of 15 mm using an electric spark wire-cutting ma-
chine for isothermal compression testing. Subsequently,
the cut sample ends were polished to achieve smoothness
before conducting the isothermal thermal compression
test on a Gleeble-3800 thermal simulation testing ma-
chine. The test procedure involved: (1) Applying a layer
of graphite sheet at both ends of the sample to minimize
friction between the sample and indenter; (2) Heating the
sample to the specified temperature at a rate of
100 °C/min using current, followed by holding it for
3 min to ensure complete austenization within the mate-
rial; (3) Performing isothermal compression tests at dif-
W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
504 Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510
Table 1: Chemical composition of test materials (%)
Element C Si Mn P S Cr Ni Mo Fe
Content (w/%) 0.432 0.53 1.179 0.0077 0.0035 0.775 0.516 0.179 Balance
Figure 1: True stress strain curve of Mn-Cr-Ni-Mo steel: a) 1173 K, b) 1273 K, c) 1373 K, d) 1473K

ferent strain rates until reaching a maximum deformation
of 60 % for each specimen; and (4) Air-cooling the
tested samples to room temperature after completion.
2.3 Experimental result
After analyzing the data obtained from the isothermal
hot-compression test using Origin, a data-processing
software, we can obtain the high-temperature rheological
curve of Mn-Cr-Ni-Mo steel, as depicted in Figure 1.
3 CONSTITUTIVE MODEL FOR MN-CR-NI-MO
STEEL IS ESTABLISHED
3.1 Arrhenius model and parameter determination
based on strain compensation
3.1.1 Reckoning the Arrhenius peak-stress constitutive
model
The Arrhenius model is widely employed to elucidate
the correlation between strain rate, deformation tempera-
ture, and flow stress during high-temperature deforma-
tion, effectively capturing the characteristic initial rise
followed by a subsequent decline in the stress-strain
curve.
19
 exp
exp exp   
 =
−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −
⎛ ⎝ ⎜ ⎞ ⎠ A
Q
RT
A
Q
RT
1
2
m
, < 0.8
[]
⎟ −
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ , >1.2
, for all

  A
Q
RT
n
sinh exp
⎪ ⎪ ⎪ (1)
where R is the molar gas constant, Î is the strain rate, T
is the deformation temperature, Q is the activation en-
ergy of hot deformation, and A,  ,  , m, and n are the
material constants.
When the deformation temperature remains constant,
combine equation (1) and deform it to obtain:
m
n
T
T
=
⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =
⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =
∂
∂
∂
∂
∂
∂
ln
 ln
ln
 ln
ln
 ln(sin
      h(
⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ T
(2)
The peak stress under different deformation condi-
tions was determined from the stress-strain curve de-
picted in Figure 1. Linear regression analysis using Ori-
gin software was performed separately on rows ln Î –
ln  p
and ln Î –  p
, yielding the results shown in Fig-
ure 2a and Figure 2b, respectively. It can be observed
that the slopes of the fitted lines are essentially identical,
with an average value of 7.140233. Similarly, fitting the
data in Figure 2b gives us  as 0.083055 and  as
0.011632 derived from  /m’s values. Likewise, when
maintaining a constant strain rate, a fitting line for ln Î –
ln [sinh(
p
)] is plotted (Figure 2c). By averaging the
slopes of these fitting lines under various deformation
conditions, n = 5.0101675 is obtained. Plotting ln
[sinh(
p
)] against the reciprocal temperature (1/T) and
performing linear regression analysis yields a slope
W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510 505
Figure 2: Fitting of relevant parameters: a) ln Î –l n p
linear fitting, b) ln Î –  p
linear fitting, c) ln Î –l n[sinh(
p
)] linear fitting, d) ln
[sinh(
p
)] –1 / T linear fitting, e) ln Z – n ln [sinh(
p
)] linear fitting

whose average value is Q/Rn1 = 7942.156; substituting
respective values for R and n gives Q = 330826.8 J/mol
as the activation energy for deformation.
The Zener-Hollomon parameter, also referred to as
the Z parameter, can be utilized to elucidate the correla-
tion among flow stress, strain rate, and deformation tem-
perature in a scholarly manner.
Z
Q
RT
=
⎛ ⎝ ⎜ ⎞ ⎠ ⎟  exp  (3)
Combined with equation (1), the above equation and
the deformation can be obtained:
[] ln ln ln sinh( ) ZA n =+  (4)
The scatter plots of ln Z and ln [sinh(
p
)] are de-
picted under various deformation conditions, followed
by line fitting as illustrated in Figure 2e. The intercept of
the fitted line corresponds to the value of ln A, with A be-
ing equal to 1.71683 · 10
12
.
By substituting all the parameters into the equation,
the Arrhenius peak stress constitutive model of
Mn-Cr-Ni-Mo steel can be derived as follows:
[]
 . sinh( . exp
. .
 =× − 1 71683 10 0 011632
330826
12
5 0101675
p
8
831
330826 8
1
0 011632
.
 exp
.
.
l
T
Z
RT
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =⋅ −
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =
  p
n
..
.ZZ
1 71683 10 1 71683 10
12
1
5 0101675
12
×
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +
×
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2
5 0101675
1
2
1
.
+
⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪ ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (5)
3.1.2 Reckoning the Arrhenius model based on strain
compensation
The constitutive model of peak stress can solely pre-
dict the maximum deformation resistance of the material,
without being able to anticipate the variation in flow
stress throughout the entire deformation process. To en-
hance the predictive accuracy of the Mn-Cr-Ni-Mo
steel’s constitutive model, strain is incorporated into the
prediction model, resulting in a comprehensive constitu-
tive model that accounts for strain. In this study, several
strain variables are selected within a range 0.05
–
0.8 with
an interval of 0.05. A flow-stress prediction model is es-
tablished for each strain, and the material parameters ob-
tained in each model are fitted. The accuracy of fitting is
found to be the highest when using a seventh-degree
polynomial. The relationship between material parame-
ters and dependent variables can be accurately repre-
sented by a seventh-degree polynomial through fitting
the experimental results. Consequently, this polynomial
enables the calculation of material parameters for any
given set of dependent variables. Subsequently, by plug-
ging in each deformation condition into equation (3), the
Z parameter can be calculated for each condition.
Finally, by substituting the obtained material parameters
and corresponding Z parameters into equation (4) for a
true stress calculation, the corresponding true stress
value of Mn-Cr-Ni-Mo steel under any strain variable
can be determined.
3.1.3 Verification of the Arrhenius model with strain
compensation
The equation of the Arrhenius model with strain
compensation for Mn-Cr-Ni-Mo steel is presented below,
and the corresponding prediction results are illustrated in
Figure 3.
a=+−+−
++
0 01435 0 00686 03275 2 24316
7 00132 1
....
.

 
 143629 9 47502 314204
356216 72 75132
...
..



−+
=+ n    


+−
−+ −+
3518 929
8781 12276 28 904035 2
.
. . 72962
296555 830762 3 25 10
7819
7
.
.
  !   
Q=+ +×−
5700 106 10 76182000 2 26 10
266568
87


+× − +×
=
..
ln . A 4 3839359 1530 47
348145 4388 43
++ −
−+
..
..
     
 4 2908 96 7921565
1
1

  
−+
=
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +
..
ln
/
Z
A
Z
A
n
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +
⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ =⋅
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2
12
1
/
/
 exp
n
Z
Q
RT
 (6)
3.2 Parameter determination of F-B model, J-C model
and mJ-C model
Additionally, we have also established the F-B
model, J-C model and mJ-C model for Mn-Cr-Ni-Mo
steel and determined the corresponding parameters. The
W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
506 Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510
Figure 3: Comparison of Arrhenius model with strain compensation predicted values with experimental values: a) Î =1 0s
–1
;b )Î=1s
–1
;c )Î =
0.1 s
–1
;d )Î = 0.01 s
–1

constitutive models are presented below, while the pre-
diction results are illustrated in Figure 4, Figure 5 and
Figure 6.
F-B:

 !
=− + +
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⋅
⋅
65154 20 014
1091321
..l o g
 T
! 
!
 log
..
 T
T
⋅
+ 0 0001002 0 000147
(7)
J-C:

"
=+ + −
−
(. . ) ( . l n  )(
.* .
75 31 5 609061 1 018052 1
0 4021 0 6
T
3
)

() / ()
*
 # 
"
 =
=− −
⎧ ⎨ ⎪ ⎩ ⎪TTTTT
rmr
(8)
mJ-C:
 
   =+−+
++
(. . .
.) (.
9687675 1088253 327 447
2114949 1 01
[]
8402
0 00537 0 000305
ln
 )
exp ( . . ln
 )( )
  "
"
⋅
⋅−+ − TT
r
(9)
4 ACCURACY EVALUATION OF THE
CONSTITUTIVE MODEL OF Mn-Cr-Ni-Mo
STEEL
To provide a more precise explanation of the predic-
tive accuracy of the equation, correlation coefficient (R)
and average absolute relative error (AARE) were em-
ployed to evaluate the predictive capability of these four
models. The following equation is employed:
R
Ei E Pi Pi
i
N
Ei E Pi Pi
i
N
i
=
−−
−−
=
= =
∑
∑
() ()
()()

 
1
22
1 1
N
∑
(10)
AARE
N
Ei Pi
Ei
i
N
(%) =
−
=
∑
1
1

 (11)
where,  Ei
and  E
represent the test value and the aver-
age value of the test value respectively, MPa;  Pi
and  Pi
represent the predicted value and the average value of
the predicted value respectively, MPa; N indicates the
total number of data.
A smaller AARE (%) value indicates a closer prox-
imity between the two sets of data, while a larger R value
signifies a stronger correlation. The AARE values and R
values of the four constitutive models are presented in
Table 2. From the results depicted in the table, it is evi-
dent that the Arrhenius model exhibits robust predictive
capabilities, whereas the F-B model demonstrates rela-
tively limited forecasting abilities.
W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510 507
Figure 4 Comparison of F-B model predicted values with experimental values: a) Î =1 0s
–1
,b )Î =1s
–1
,c )Î = 0.1 s
–1
,d )Î = 0.01 s
–1
Figure 5: Comparison of J-C model predicted values with experimental values: a) Î =1 0s
–1
,b )Î =1s
–1
,c )Î = 0.1 s
–1
,d )Î = 0.01 s
–1
Figure 6: Comparison of mJ-C model predicted values with experimental values: a) Î =1 0s
–1
,b )Î =1s
–1
,c )Î = 0.1 s
–1
,d )Î = 0.01 s
–1

Table 2: AARE and R of four constitutive models
Item Arrhenius F-B J-C m J-C
AARE (%) 7.1286 22.2 12.1 10.4
R 0.985803 0.883 0.965234 0.975347
The established strain compensation Arrhenius model
of Mn-Cr-Ni-Mo steel yields predicted values that
closely match the experimental measurements, indicating
a relatively high prediction accuracy for the proposed
model equation in this study. The AARE and R values
were further improved to 7.13 % and 0.986, respectively.
The calculated values of the F-B model for Mn-Cr-
Ni-Mo steel exhibit excellent agreement with the experi-
mental data prior to reaching peak stress. As depicted in
Figure 4, this model solely captures the strain-hardening
stage of Mn-Cr-Ni-Mo steel. However, experimental re-
sults reveal a recrystallization softening phenomenon
during thermal deformation, which cannot be accurately
described by the F-B model. The AARE and R values
were 22.2 % and 0.883, respectively. The J-C model for
Mn-Cr-Ni-Mo steel offers the advantage of requiring
fewer parameters, being easily obtainable from limited
experiments, and simple to implement. However, it also
has several limitations. The described thermal softening
behavior exhibits a rough linearity but deviates to some
extent from the actual test measurements due to assum-
ing strain hardening, strain-rate strengthening, and ther-
mal softening as independent phenomena that can be
separated from each other. This assumption overlooks
the coupling effect of strain, strain rate, and temperature
on a material’s rheological behavior. The values of
AARE and R were determined to be 12.1 % and 0.965
respectively. The mJ-C model for Mn-Cr-Ni-Mo steel in-
corporates the coupling relationship between
temperature and strain rate, thereby addressing certain
limitations of its original counterpart, while maintaining
better predictive capabilities. The AARE and R values
are 10.4 % and 0.975, respectively.
The correlation between the four constitutive models
and the test results is illustrated in Figure 7. The accu-
racy of the model’s prediction relies on the proximity of
data points to the solid line, with a higher concentration
indicating a stronger correlation between the model and
test values. It can be observed that the Arrhenius model
effectively characterizes the hot-deformation behavior of
Mn-Cr-Ni-Mo steel.
5 CONSTITUTIVE MODEL VERIFICATION OF
Mn-Cr-Ni-Mo STEEL
The thermal deformation behavior of Mn-Cr-Ni-Mo
steel was investigated at various deformation tempera-
tures and strain rates using Deform-3D finite-element
software. A well-fitting constitutive model was em-
ployed to numerically simulate the isothermal thermal
shrinkage test under different conditions. Based on the
precision comparison presented in Table 2, the Arrhenius
model exhibited strong correlation with experimental
data obtained at different strain rates. By utilizing
ABSOFT and FORTRAN programming languages, the
finite-element user sub-routine was compiled to establish
an EXE file, which enabled the input of the model into
the finite-element software. Subsequently, a comparison
between the time-stress curve obtained from Deform-3D
W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
508 Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510
Figure 7: Correlation between constitutive model and test results; a) Arrhenius; b) F-B; c) J-C ; d) mJ-C
Table 3: Hot-forging process parameters of ball-mill liner
Argument Value
Top die speed (mm/s)
V 10
= –10 x + 150, V
1
=– x +1 5
V
0.1
= –0.1 x + 1.5, V
0.01
= –0.01 x + 0.15 (x is the travel)
Initial temperature of test element (K) 1173, 1273, 1373, 1473
Initial die temperature (K) 1173, 1273, 1373, 1473
Ambient temperature (K) 1173, 1273, 1373, 1473
Heat-transfer coefficient (N/s/mm/K) 1
Convection coefficient (N/s/mm/K) 0.02
Friction coefficient 0.3
Number of blank grids 150000
Die material AISIH13
Test-piece material Mn-Cr-Ni-Mo steel

numerical simulation for isothermal thermal compres-
sion test and that acquired through experimental means
was conducted to validate the accuracy of the simulation.
5.1 Establishment of finite-element model
The three-dimensional geometric models of both the
isothermal compression standard test specimen and sim-
plified isothermal compression testing apparatus were
created using Solidworks software. Afterwards, these
models were imported into Deform-3D software as STL
files for pre-processing to establish a numerical simula-
tion finite-element model for conducting isothermal ther-
mal compression tests. The process of constructing this
model mainly consisted of five steps: grid division, mate-
rial assignment, defining boundary conditions, position-
ing the models accurately within it and configuring rele-
vant process parameters accordingly (as shown in
Table 3). Figure 8 depicts a simplified three-dimen-
sional thermal coupling model used in performing such
tests with Deform-3D.
5.2 Analysis of simulation results
According to the numerical simulation results, the
isothermal thermal compression test piece of Mn-Cr-
Ni-Mo steel yielded an equivalent stress cloud diagram,
as depicted in Figure 9. It can be observed that the dis-
tribution of equivalent stress in the test piece is relatively
homogeneous, with most areas exhibiting a difference in
equivalent stress values within 20MPa. The gradual in-
crease in equivalent stress distribution from the center to
the edge position indicates more intense metal flow at
the edge during hot compression compared to that at the
center. The stress distribution in the larger area is consis-
tent with the equivalent stress at the end of the
flow-stress curve, thereby affirming the accuracy of our
numerical simulation.
In order to visually demonstrate the accuracy of nu-
merical simulation in describing isothermal hot compres-
sion tests, four distinctive points were carefully selected
from Mn-Cr-Ni-Mo steel specimens for comprehensive
investigation. The feature points are evenly distributed
from the center to the edge of the test piece. The average
equivalent stress value of the four feature points was de-
termined throughout the simulation process. A
time-equivalent stress curve was plotted and compared
with data obtained from isothermal thermal compression
tests, as depicted in Figure 10. It can be observed that
the Arrhenius model accurately assessed both the mate-
rial’s response characteristics in thermal deformation be-
havior and exhibited good agreement with the experi-
mental values. These results affirm the correctness of the
Arrhenius model for simulating Mn-Cr-Ni-Mo steel’s
thermal deformation.
6 CONCLUSIONS
In this study, the isothermal compression test of
Mn-Cr-Ni-Mo steel for a ball-mill liner suitable for a
forging process was conducted using a Gleeble-3800
thermal simulation testing machine at various tempera-
tures and strain rates. Four constitutive models were es-
tablished, including Arrhenius model, F-B model, J-C
model and mJ-C model. Deform-3D finite-element soft-
ware was utilized to simulate equivalent thermal com-
pression tests. The specific results are as follows:
W. LIU et al.: INVESTIGATION INTO THE THERMAL DEFORMATION BEHA VIOR AND THE ESTABLISHMENT ...
Materiali in tehnologije / Materials and technology 58 (2024) 4, 503–510 509
Figure 9: Equivalent stress cloud diagram of the isothermal thermal
compression test piece of Mn-Cr-Ni-Mo steel: a) Î =1 0s
–1
,b )Î =1
s
–1
,c )Î = 0.1 s
–1
,d )Î = 0.01 s
–1
Figure 10: Comparison of time-equivalent stress curves between test and numerical simulation: a) Î =1 0s
–1
,b )Î =1s
–1
,c )Î = 0.1 s
–1
,
d) Î = 0.01 s
–1

(1) The predictive ability of four constitutive models
was evaluated using the correlation coefficient (R) and
average absolute relative error (AARE). Among them,
the Arrhenius model for Mn-Cr-Ni-Mo steel demon-
strated the highest level of prediction accuracy, AARE
(%) and R are 7.13% and 0.986, respectively. The F-B
model exhibited the lowest level of prediction accuracy,
AARE (%) and R are 22.2 % and 0.883, respectively.
The J-C model ranked third in terms of prediction accu-
racy, followed by the mJ-C model, which ranked second.
A smaller AARE (%) value indicates a closer proximity
between the two sets of data, while a larger R value sig-
nifies a stronger correlation.
(2) The Arrhenius model and isothermal thermal
compression finite-element model were incorporated into
the Deform-3D software for a numerical simulation. The
results demonstrated a strong agreement between the
time-stress curves obtained from both numerical simula-
tion and experimental data, validating the applicability of
the established Arrhenius model in simulating the ther-
mal deformation of Mn-Cr-Ni-Mo steel.
Acknowledgement
This research was financially supported by the Sci-
ence and Technology Project of China Minmetals Corpo-
ration Limited (2021ZXA06).
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