L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
91–97
MODELING AND SIMULATION BASED ON THE CONSTITUTIVE
EQUATION OF 25Cr2Ni4MoV STEEL FOR A SUPER-LARGE
NUCLEAR-POWER ROTOR
MODELIRANJE IN SIMULACIJA KONSTITUTIVNE ENA^BE
JEKLA 25Cr2Ni4MoV ZA ZELO VELIK ROTOR V JEDRSKI
ELEKTRARNI
Li-yan Ye, Yue-wen Zhai*, Le-yu Zhou, Peng Jiang
Beijing Research Institute of Mechanical and Electrical Technology, Science and Technology Development and Innovation Center,
18 Xueqing Road, Haidian District, Beijing 100083, P.R. China
Prejem rokopisa – received: 2019-06-26; sprejem za objavo – accepted for publication: 2019-08-18
doi:10.17222/mit.2019.136
The constitutive equation of the 25Cr2Ni4MoV steel for a super-large nuclear-power rotor was studied using compression
experiments within a temperature range of 1373–1523 K and a strain rate range of 0.001–0.1 s
–1
on a Gleeble-1500
thermal-mechanical simulation tester. Considering the application of the constitutive equation in a finite-element (FE) software,
the linear-interpolation method incorporated in the FE software and the strain-compensated Arrhenius model were used to
predict the flow stress of the 25Cr2Ni4MoV steel. It was found that the Arrhenius model was stronger in estimating the flow
behavior compared to the linear-interpolation method. By means of user subroutines, the Arrhenius model was integrated into
FE software DEFORM to simulate the isothermal compression. The comparison between the simulated and theoretical results
confirmed the validity of the Arrhenius model.
Keywords: 25Cr2Ni4MoV, strain-compensated Arrhenius model, linear interpolation, high-temperature flow behavior
Konstitutivno ena~bo za jeklo 25Cr2Ni4MoV, ki se uporablja za izdelavo zelo velikega rotorja jedrske elektrarne, so avtorji
prispevka analizirali z uporabo eksperimentalnih podatkov tla~nih preizkusov, dobljenimi v temperaturnem obmo~ju med
1373 K in 1523 K, in v podro~ju hitrosti deformacij med 0,001 s
–1
in 0,1 s
–1
, na termomehanskem simulatorju Gleeble 1500.
Upo{tevajo~ uporabo konstitucijske ena~be v programskem orodju na osnovi metode kon~nih elementov (FE) v katerega so
uvedli metodo linearne interpolacije in uporabili deformacijsko kompenzirani Arrheniusov model za krivulje te~enja jekla
25Cr2Ni4MoV. Ugotovili so, da je Arrheniusov model bolj{i za oceno te~enja, kot metoda linearne interpolacije. V uporabni{ki
podprogram orodja DEFORM-FE so integrirali Arrheniusov model za simulacijo izotermne kompresije (obremenjevanja pod
tlakom). Primerjava med na simulatorju dobljenimi in teoreti~nimi rezultati je potrdila veljavnost Arrheniusovega modela.
Klju~ne besede: jeklo 25Cr2Ni4MoV, deformacijsko kompenzirani Arrheniusov model, linearna interpolacija, obna{anje
materiala med te~enjem (tla~no plasti~no deformacijo) pri visokih temperaturah
1 INTRODUCTION
The material-development history of super-large
nuclear-power rotors includes 34CrNi3Mo,
2%~4%NiCrMoV, 30Cr2Ni4MoV and 25Cr2Ni4MoV.
1–4
With a decrease in the carbon and impurity content,
25Cr2Ni4MoV has a good balance of toughness,
strength and corrosion resistance in a severe service
environment. Therefore, 25Cr2Ni4MoV has a great
potential for super-large nuclear-power rotors. A super-
large nuclear-power rotor is forged with a hydraulic
press at a high temperature from a large cast ingot of
about 650 t, which is known for its high cost, small
amount and long production cycle, and for which it is
difficult to obtain the optimum hot-forming process
parameters through a practical production. A finite-
element simulation provides us with an effective tool for
obtaining the optimum process parameters by predicting
the deformation behavior of metals and alloys; it does
not only guide the practical production, but can also
decrease the cost. The constitutive equation, which plays
an important role in predicting the flow stress, selecting
the press load and designing the forming process, is
crucial in an FE simulation. However, the research about
the constitutive equation of 25Cr2Ni4MoV is seldom
reported.
Considering the application of a constitutive equation
in a finite-element analysis, the phenomenological con-
stitutive model is widely used to predict the flow stress at
elevated temperatures due to its fewer material constants,
which can be obtained with uniaxial hot-compression
tests.
5
The strain-compensated Arrhenius model is one
kind of the commonly used phenomenological model. L.
XuandL.Chen
6
reported on the strain-compensated
Arrhenius model that exhibited accuracy when
predicting the flow stress of the 25Cr3Mo3NiNb steel. Y.
Zhang and H. L. Sun
7
established the strain-compensated
Arrhenius model for the Cu-Zr-Nd alloy. L. Chen and G.
Q. Zhao
8
found two appropriate Arrhenius-type consti-
tutive equations for the 7005 aluminum alloy. Y. F. Li
Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97 91
UDK 519.876.2:544:669.15:502.21:621.039 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 54(1)91(2020)
*Corresponding author's e-mail:
yeliyan2006@sina.com (Yue-wen Zhai)

and Z. H. Wang
9
proposed an Arrhenius-type constitutive
equation for the V-5Cr-5Ti alloy. A. Abbasi Bani and A.
Z. Hanzaki
10
studied Arrhenius-type equations for the
Mg-6Al-1Zn alloy. However, integrating a strain-com-
pensated Arrhenius model into the FE software was
seldom reported.
Three methods can be used to input a constitutive
equation into the FE software. The first one involves the
use of the coefficients and constants of the material
model provided by the FE software. The second one
requires the definition of the flow stress presented in the
tabular data format. The third one involves the building
of flow-stress models using user-defined flow-stress
subroutines. However, regarding the first method, the
flow-stress models supported by the FE software are not
applicable to all metals and alloys. For the second
method, the stress-strain curves at different deformation
conditions are obtained with linear interpolation based
on the data table, which means that the accuracy of the
material model is not guaranteed. Therefore, the third
method based on programming the relationships between
stress, strain, strain rate and temperature with the FE
software using a user routine is recommended, as it does
not only simulate the constitutive equation that is not
available in the material library of the FE software, but
can also accurately predict the flow behavior of metals
and alloys.
In this paper, the hot-deformation behavior of
25Cr2Ni4MoV was investigated using compression tests
with different temperatures and strain rates. Both the
strain-compensated Arrhenius model and linear-inter-
polation method were used to predict the flow stresses,
and the prediction precision of the models was assessed
using standard statistical parameters. By integrating the
Arrhenius model into the FE software, hot compressions
were simulated.
2 EXPERIMENTAL PART
The chemical composition of the 25Cr2Ni4MoV
steel is shown in Table 1. Compression specimens with a
diameter of 8 mm and length of 12 mm were machined.
The materials were provided by Sinomach-he Co, Ltd
and the initial grain size was 24.22 μm. Before the
compression, the specimens were first heated to 1523 K
at a heating rate of 10 K/s and dwell time of 5 min to
L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
92 Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97
Table 1: Chemical composition of steel 25Cr2Ni4MoV (in mass fractions, (w/%)
C Mn P S Si Ni Cr Mo V Cu Fe
F+0.3 0.1~0.3 +0.015 +0.015 +0.012 2.0~2.5 2.15~2.45 0.6~0.85 +0.12 +0.17 Bal.
Figure 1: True-stress-strain curves obtained at different temperatures: a) 1373 K, b) 1423 K, c) 1473 K, d) 1523 K

achieve a homogeneous austenitic microstructure, then
cooled down to the deformation temperature at a rate of
10 K/s and dwell time of 1 min. In order to minimize the
effect of friction, graphite sheets used as the lubricant
material were applied to both ends of the specimens.
Hot-compression tests with four different temperatures
(1373, 1423, 1473, and 1523) K and five strain rates
(0.001, 0.005, 0.01, 0.05, 0.1) s
–1
at a true strain of 90 %
were performed on a Gleeble-1500 thermal-mechanical
simulator.
3 RESULTS AND DISCUSSION
3.1 Flow behavior
Figure 1 shows a number of stress-strain curves
obtained during the compression tests. It can be found
that the flow stress increases with the increased strain
rate when the deformation temperature is fixed, or that
the flow stress decreases with the increased deformation
temperature when the strain rate remains unchanged. For
most deformation conditions, stress-strain curves have
the characteristics of a dynamic recrystallization (DRX)
with a single peak. In the initial stage of the forming
process, the flow stress rapidly reaches the peak value.
This is because the work-hardening rate induced by
dislocation intersections and pileups is higher than the
softening rate induced by dynamic recovery (DRV).
Then the flow stress decreases due to the dominant
softening rule of DRX. Finally, the stress reaches a
steady state when the equilibrium of work softening and
work hardening is obtained. At deformation conditions
of 1373 K/0.1 s
–1
and 1423 K/01 s
–1
, the stress keeps
increasing or remains constant, indicating a dynamic
recovery as well as a dynamic recrystallization, lasting
until the end of the deformation process.
3.2 Strain-compensated Arrhenius-type constitutive
model
3.2.1 Establishment of the constitutive equation
For the high-temperature deformation, the Arrhenius
model and Zener-Hollomon (Z) model are widely used to
describe the relationship between the peak stress,
deformation temperature and strain rate.
11–13
Z
Q
RT
A
A
A
n
=
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =
<
>
 exp
(.
exp( )( . )   
 
1
2
08
12
pp
pp
[] sinh( ) 
p
n
1
⎧ ⎨ ⎪ ⎩ ⎪ (1)
Here,
 is the strain rate. A, A
1
and A
2
are the material
constants independent of the temperature. Q is the hot-
deformation activation energy. R is the gas constant. T is
the absolute temperature of deformation.  p
is the peak
stress.  ( =  /n) and n
1
are the constants.
By taking the logarithm of both sides, Equation (1)
becomes:
ln ln
 ln ln ( .
ln ( . )
ln
Z
Q
RT
An
A =+=
<
+>   
 
1
2
08
12
pp
pp
[] An +
⎧ ⎨ ⎪ ⎩ ⎪ 1
ln sinh( ) 
p
(2)
L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97 93
Figure 2: Variations of material constants of: a)  ,b )n,c )Q and d) ln A with strain

Stress exponent n can be calculated from the slope of
ln
 versus ln  p
. Constant  is the slope of ln
 versus  p
n
1
is the slope of ln
 versus ln(sinh(
p
). ln A can be cal-
culated from the intercept of ln Z versus ln(sinh(
p
).
Table 2: Coefficients of the 8
th
-order polynomial function for  , n, Q,
ln A
 nQ ln A
0.04463 7.17284 333377.48 24.64146
–0.34769 –28.40935 2.27008E6 142.70483
3.11804 78.24198 –2.78735E7 –1930.30145
–15.31759 –90.63266 1.36559E8 9676.35721
46.09771 37.47988 –3.37958E8 –24041.8169
–86.77108 0 4.50276E8 31935.3452
98.89885 0 –3.08756E8 –21769.95705
–62.01835 0 8.56192E7 5994.97466
16.34468 0 0 0
The Arrhenius model mentioned above does not
consider the influence of the strain. However, by taking
the same solution process as explained with Equations
(1) and (2), different material constants ( ,n,Q,ln A)
were obtained at different strains.
14–16
Hence, the strain-
compensated Arrhenius-type constitutive model is
applied to describe the flow stress, assuming that the
material constants ( ,n,Q,ln A) are the polynomial
functions of the strains. The flow curves with strain rates
of (0.001, 0.005, 0.01 and 0.1)s
–1
are used to calculate
the strain-compensated Arrhenius model. The relation-
ships between the material constants and strains of
25Cr2Ni4MoV are shown in Figure 2. Using the poly-
nomial fitting, the constitutive equations are obtained as
given in Equation (3). The coefficients of the eighth-
order polynomial are listed in Table 2.
nnnnnnnnnn =++++++++
=+
012345678
01


’#$ "%
      

’#$ "%
 +++++++
=+++
2345678
012
QQQQQQQQQQ
AAAA A
345678
0123


’#$ "%
’
+++++
=++++ ln AAAAA
45678

#$ "%
++++
(3)
3.2.2 Prediction of stress-strain curves
Theoretical values and the experimental values at
different strains are shown in Figure 3. It can be seen
that the theoretical results show good agreement with the
experimental results. In order to quantitatively evaluate
the accuracy of the developed constitutive model, the
correlation coefficient (R) and average absolute relative
error (AARE) are calculated with the expressions below.
R
PPEE
PP EE
ii i
n
i i
n
i i
n
=
−−
−−
=
==
∑
∑∑
() ()
()()
1
2
1
2
1
(4)
AARE
n
PE
E
ii
i
i
n
=
−
=
∑
100
1
% ()
(5)
Here, P
i
is the predicted value and P is the mean
value of P
i
. E
i
is the experimental value and E is the
mean value of E
i
. n is the number of data employed in
the investigation. The strain range is from 0.05 to 0.5
with an interval of 0.05 and from 0.5 to 0.9 with an
interval of 0.1. A comparison of the predicted values and
experimental results for all the deformation conditions is
L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
94 Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97
Figure 3: Comparison of the experimental and predicted flow stress at different temperatures: a) 1373 K, b) 1423 K, c) 1473 K, d) 1523 K

shown in Figure 4. It should be mentioned that the
maximum average absolute relative error is 6.7 % at the
deformation conditions including a temperature of
1373 K and a strain rate of 0.1 s
–1
. This is because the
flow-stress curve of this deformation condition does not
show the single peak of dynamic recrystallization, while
the other flow curves do. The correlation coefficient (R)
of all the deformation conditions is 0.987 and the aver-
age absolute relative error (AARE) is 5.18 %, showing
that the constitutive model has a good predictability of
the flow stress for the 25Cr2Ni4MoV steel.
3.3 Comparison of the Arrhenius model and linear-
interpolation method
As the strain-compensated Arrhenius model is not
provided by the FE software, the tabular-data format or
user-defined flow-stress subroutine can be used to input
the flow curves. The method of the tabular-data format
allows the calculation of the flow curves at different
deformation conditions using linear interpolation based
on the data table. For example, for a certain deformation
temperature,  0.1
and  0.01
represent the flow stress at
strain rates of 0.1 s
–1
and 0.01 s
–1
, respectively. Accord-
ing to the linear interpolation, 0.05 equals 0.556 times
0.001 plus 0.444 times 0.1. Hence, to compute the flow
stress at a strain rate of 0.05 s
–1
at a certain temperature,
Equation (6) is employed using linear interpolation.
17
 0.05
= 0.556σ
0.01
+ 0.444 0.1
(6)
Figure 5 shows the comparison of the flow stress
predicted by the strain-compensated Arrhenius model
and the linear-interpolation method at the strain rate of
0.05 s
–1
. It is obvious that the strain-compensated Arrhe-
nius model is much better for predicting the flow stress
than the linear-interpolation method, especially when
predicting a flow stress lower than the peak stress. The
average absolute relative errors (AARE) for the Arrhe-
nius model and linear-interpolation method are 3.62 %
and 7.55 %, respectively. The correlation coefficients (R)
for the Arrhenius model and interpolation-linear method
are 0.9890 and 0.9806, respectively. Therefore, consider-
ing the application of flow curves in the FE simulation
for the 25Cr2Ni4MoV steel, inserting the strain-compen-
sated Arrhenius model into the FE software with the
user-defined flow-stress subroutine should be chosen as
the more accurate method than the linear-interpolation
method.
3.4 FE simulation and verification
To apply the constitutive equation to the numerical
simulation, a FEM code was developed based on the
constitutive model to simulate the isothermal com-
pression with FE software DEFORM. The geometry
model of the workpiece was the same as the compression
test mentioned above. The die velocity was 0.1 mm/s and
0.5 mm/s and the friction factor was 0.3. The tracking of
the values of the effective strain, effective strain rate and
effective stress at the center of the workpiece during the
isothermal compression is shown in Figure 6. Figure 7
shows that at the deformation condition of 1373 K–
0.1 mm/s with a die stroke of 5.4 mm, the effective
strain, effective strain rate and effective stress at the
center of the workpiece are 0.874, 0.0226 s
–1
and
L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97 95
Figure 4: Correlation between the experimental and predicted flow-
stress data
Figure 6: Tracking point
Figure 5: Comparison of the experimental and predicted flow stress at
the strain rate of 0.05 s
–1
with different temperatures: a) 1373 K,
b) 1423 K, c) 1473 K, d) 1523 K

54.6 MPa, respectively. At the deformation condition of
1523 K–0.1 mm/s with a die stroke of 5.4 mm, the
effective strain, effective strain rate and effective stress
are 0.865, 0.0227 s
–1
and 29.2 MPa, respectively. The
comparison between the theoretical results and simulated
results is given in Figure 8. It can be seen that the
theoretical values are nearly equal to the simulated
values with a stress range of 17–70 MPa. The average
absolute relative error (AARE) of the simulated and
theoretical results is 0.046 %. The theoretical results
show good agreement with the experimental results, as
stated above. Hence, it can be deduced that the simulated
results predict well the flow behavior of the
25Cr2Ni4MoV steel.
4 CONCLUSIONS
To perform a simulation of a forging process for the
25Cr2Ni4MoV steel, a set of compression tests was
carried out to investigate the flow behavior of the
25Cr2Ni4MoV steel. Conclusions were obtained as
follows:
• An Arrhenius model considering the influence of the
strain was established to describe the flow stress. The
comparison between the experimental result and
theoretical result indicates that the strain-com-
pensated Arrhenius model predicts the flow behavior
of the material well.
L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
96 Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97
Figure 8: Comparison between the simulated and theoretical flow stress at different die velocities: a) 0.1 mm/s, b) 0.5 mm/s
Figure 7: Distribution of effective strain, effective strain rate and effective stress at a die stroke of 5.4 mm: a) 1373 K–0.1 mm/s, b) 1523 K–0.1
mm/s

• The strain-compensated Arrhenius model and
linear-interpolation method were used to predict the
flow stress at a strain rate of 0.05 s
–1
. The average
absolute relative errors (AARE) for the Arrhenius
model and linear-interpolation method were 3.62 %
and 7.55 %, respectively. Hence, the strain-com-
pensated Arrhenius model is more accurate than the
linear-interpolation method.
• The application of the strain-compensated Arrhenius
model in FE software DEFORM with a user-defined
material subroutine confirms that the developed
model is capable of predicting the flow stress and
utilizing in the FE simulation software.
• The above research results provide an important basis
for forging simulations involving the 25Cr2Ni4MoV
steel for super-large nuclear-power rotors.
Acknowledgement
The authors appreciate the financial support from the
National Science and Technology Major Project of China
(Grant No. 2018ZX04044001).
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L.-Y. YE et al.: MODELING AND SIMULATION BASED ON THE CONSTITUTIVE EQUATION OF 25Cr2Ni4MoV STEEL ...
Materiali in tehnologije / Materials and technology 54 (2020) 1, 91–97 97