X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
715–723
HOT-DEFORMATION BEHAVIOR AND A MODIFIED
PHYSICALLY BASED CONSTITUTIVE MODEL FOR As-CAST
12 % Cr STEEL DURING HOT DEFORMATION
VRO^A DEFORMACIJA IN MODIFICIRANA PLASTI^NOST –
KONSTITUTIVNI MODEL ZA OBNA[ANJE LITEGA JEKLA Z
12 % Cr MED VRO^O DEFORMACIJO
Xuezhong Zhang, Fei Chen, Yongxing Jiao, Jiansheng Liu
*
School of Materials Science and Engineering, Taiyuan University of Science and Technology, No. 66 Waliu Road,
Wanbailin District, Shanxi, Taiyuan 030024, PR China
Prejem rokopisa – received: 2020-02-17; sprejem za objavo – accepted for publication: 2020-04-28
doi:10.17222/mit.2020.032
The development of ultra-supercritical thermal-power units has a significant effect on energy conservation and emission reduc-
tion. Based on a uniaxial isothermal compression test, the hot-plastic-deformation behavior of the as-cast 12 % Cr ultra-super-
critical rotor steel was studied systematically. The flow stress/strain curves for the as-cast 12 % Cr steel were obtained under the
conditions of a deformation temperature ranging from 1173 K to 1573 K and strain rate ranging from 0.001 to 1 s
–1
. By analyz-
ing the evolution characteristics of the macro-flow stress, the influence of deformation-process parameters such as the strain, de-
formation rate and temperature on the flow stress was studied. A physically based constitutive model was established to describe
the high-temperature flow stress, and the model was modified to improve its accuracy. The average absolute relative error of the
modified model is 12.2 % and the root-mean-square error is 7.0 MPa, so that its prediction accuracy is higher than that of the
unmodified model.
Keywords: 12 % Cr, ultra-supercritical rotor, high-temperature plastic deformation, constitutive model
Razvoj novih ultra-superkriti~nih rotorjev za termo-energetske objekte ima pomemben pomen pri zmanj{anju porabe energije in
onesna`evanja okolja. Avtorji so na osnovi enoosnega izotermi~nega tla~nega preizkusa sistemati~no {tudirali vro~o plasti~no
deformacijo litega jekla z 12 % Cr, namenjenega za ultra-superkriti~ne rotorje. Krivulje te~enja litega jekla z 12 % Cr so dolo~ili
z njegovo deformacijo v temperaturnem obmo~ju med 1173 K in 1573 K ter hitrostih deformacije od 0,001 s
–1
d o1s
–1
.
Analizirali so zna~ilnosti razvoja makronapetosti te~enja in {tudirali vpliv parametrov procesa deformacije kot so deformacija,
hitrost in temperatura deformacije na napetost te~enja. Postavili so fizikalni konstitutivni model za opis visoko temperaturne
napetosti te~enja in ga dodatno modificirali za izbolj{anje njegove to~nosti. Povpre~na absolutna relativna napaka
modificiranega modela je 12,2 % in povpre~ni kvadratni koren napake je 7,0 MPa, kar je precej manj{a napaka kot so jo dosegli
z nemodificiranim modelom.
Klju~ne besede: jeklo z 12 % Cr v litem stanju, ultra-superkriti~ni rotor, visokotemperaturna plasti~na deformacija, konstitutivni
model
1 INTRODUCTION
12 % Cr ultra-supercritical rotor steel is a kind of
high-strength and high-alloy steel that is difficult to de-
form.
1,2
Extensive research work has been carried out on
ultra-supercritical rotor materials and technology by
many scholars.
3–5
The manufacture of ultra-supercritical
rotors usually includes ingot casting, cogging, drawing,
heat treatment, machining and other processes. Espe-
cially in the forging process, the processing procedure
may vary, and the technology is complex. In order to im-
prove the quality of ultra-supercritical rotor steel, the
thermal-deformation behavior and constitutive model of
the 12 % Cr steel need to be studied systematically. It is
necessary to study the high-temperature plastic-deforma-
tion behavior and the constitutive model of the 12 % Cr
steel to optimize the production process.
The plastic-deformation behavior and constitutive
model of the materials have been studied by scholars in-
ternationally. P. Ludwik
6
was the first to propose a
flow-stress model for the materials under plastic defor-
mation, which included the effects of the yield stress and
strain on the stress. Voce proposed a constitutive model,
which mainly considered the effects of the yield stress,
saturation stress, true strain and relaxation strain on the
flow stress.
7
Fields and Bachofen suggested that the
strain rate should also be an important parameter in the
constitutive model.
8
Based on the comprehensive effect
of the temperature and strain rate on the microstructure
evolution, Sellars et al. proposed a mathematical sine hy-
perbolic model.
9,10
U. F. Kocks et al.
11–12
established con-
stitutive models of work-hardening and dynamic-recov-
ery phase based on the dislocation-density equation,
namely a KM and EM dislocation-equation stress model.
G. R. Johnson and W. H. Cook
13
proposed a flow-stress
model based on different deformation temperatures,
Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723 715
UDK 621.78:621.78.015:539.4.016:621.785 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 54(5)715(2020)
*Corresponding author's e-mail:
jianshliu@126.com (Jiansheng Liu)

strains and strain rates. F. J. Zerilli and R. W. Armstrong
established a constitutive model, which considered the
effects of strain hardening, strain-rate hardening and
thermal softening on the flow behavior of a metal.
14
Y. V.
R. K. Prasad used a neural network to predict the
flow-stress curve of medium-carbon steel, and the results
showed that the neural-network model had a good pre-
dictive effect.
15
Y. C. Lin et al.
16
built a strain-compensa-
tion model for the 42CrMo steel based on the Sellars
model.
16
Laasraoui and Jonas studied the deformation
behavior of low-carbon steel through a single-channel
hot-compression test and established a two-stage physi-
cally based constitutive model of work-hardening/dy-
namic-recovery stage and dynamic-recrystallization
stage based on an EM dislocation-equation stress model
and dynamic-recrystallization dynamics model, respec-
tively.
17,18
F. Chen et al.
19
studied the high-temperature
flow law of typical large forging materials and estab-
lished a two-stage high-temperature physically-based
constitutive model of the 30Cr2Ni4MoV steel.
19
Our predecessors did a lot of research on constitutive
models, and the physically-based constitutive model,
based on microscopic mechanisms was recognized by
many scholars both in terms of the theoretical basis and
model accuracy. However, the constitutive model of the
as-cast 12 % Cr ultra-supercritical rotor steel is rarely
studied. Based on the isothermal uniaxial compression
test, the flow-stress curve of the as-cast 12 % Cr steel
was obtained. By analyzing the effects of different defor-
mation parameters on the deformation behavior of steel,
the physically-based constitutive model was established
and modified. The modified model improves the accu-
racy of the traditional physically-based constitutive
model. It is useful for the optimization of the ultra-super-
critical rotary-forging process.
2 EXPERIMENTAL PART
The material used for the test is the as-cast 12 % Cr
ultra-supercritical rotor steel, whose chemical composi-
tion is shown in Table 1. The sample was taken from a
12 % Cr ingot. The test sample was a cylinder with 8
mm in diameter and 12 mm in height. In order to study
the thermal-deformation behavior of the 12 % Cr steel,
in accordance with the hot-compression diagram in Fig-
ure 1, a Gleeble-1500D thermal/force simulation test
machine was used for the uniaxial isothermal compres-
sion test. During the test, the sample was heated to
1473 K at a rate of 10 Ks
–1
and kept at this temperature
for 2 min. The sample was then heated or cooled to the
deformation temperature at a rate of 5 Ks
–1
and kept at
this temperature for 1 min to eliminate the temperature
gradient. Next, the sample was compressed to a true
strain of 0.7. The temperature ranged from 1173 K to
1573 K, with an interval of 50 K. The strain rates were
(0.001, 0.01, 0.1 and 1) s
–1
.
Table 1: Chemical composition of the as-cast 12 % Cr steel (w/%)
CC rS iSPM nN iM o
0.109 11.193 0.06 0.0041 0.038 0.52 0.768 1.22
3 RESULTS AND DISCUSSION
3.1 Influence of deformation parameters on the
flow-stress curve
Figure 2 shows the flow-stress curves of the as-cast
12 % Cr steel at different temperatures and strain rates.
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
716 Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723
Figure 2: True-stress curves for different temperatures and strain
rates: a) 0.001 s
–1
; b) 1437 K
Figure 1: Schematic presentation of hot-deformation tests for the
as-cast 12 % Cr steel

As shown in Figure 2a, under the same strain rate, the
peak stress reaches the minimum when the temperature
is 1573 K. The peak stress decreases with the rise in the
temperature. It can be seen from the figure that the tem-
perature has a significant effect on the flow stress. There-
fore, the flow stress decreases obviously with the in-
crease in the deformation temperature. This is because
the higher the deformation temperature, the smaller is
the bonding force among the metal atoms so that the va-
cancy diffusion, dislocation climbing and crossed-slip re-
sistance are reduced, and the number of dynamic-rec-
rystallization nuclei is lower. At the same time, the metal
may, at higher temperatures, also develop a new slip sys-
tem, which can make plastic deformation easy, resulting
in a stress reduction. In addition, a higher temperature
reduces the activation energy required for the dynamic
recovery and dynamic recrystallization of austenite,
making the dynamic recovery and dynamic recrys-
tallization more likely to occur. As softening mecha-
nisms, they can eliminate work hardening, reducing the
flow stress. Besides, at a higher deformation tempera-
ture, dynamic softening requires less deformation.
As shown in Figure 2b, at the same temperature,
when the strain rate is 0.001s
–1
, the peak stress is the
lowest, while the peak stress increases with the strain
rate. This is because under a high deformation rate, the
slipping of dislocation can be hindered by the increase in
the dislocation density. The occurrence of dynamic
recrystallization is limited by the nucleation time. So,
larger deformation is required for a full recrystallization
and a large flow stress.
3.2 Establishment of the Arrhenius equation
The deformation of metal at high temperatures is a
process of thermal activation. The activation energy of
thermal deformation is a physical quantity, which repre-
sents the difficulty of thermal deformation. During hot
deformation, the influence of the Zener-Hollomon (Z)
parameters, deformation temperature and strain rate on
the flow stress of the materials can be expressed with the
Arrhenius equation proposed by Zener and Hollomon:
20
 exp
(. )
exp( )(
'
  
 
Q
RT
A
A
n
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =
<
1
2
08
[]
>
⎧ ⎨ ⎪ ⎩ ⎪ 12 .)
sinh( (for all) A
n

(1)
where
 is the strain rate (s
–1
), Q is the thermal deforma-
tion activation energy (J/mol), R is the universal gas
constant (8.314 J/(mol·K)), T is the absolute tempera-
ture (K),  is the flow stress (MPa), while A
1
,A
2
,A, ,
 , n’ and n are the material constants. Power functions
apply to low-stress levels, while exponential functions
describe high-stress levels.
21
The hyperbolic sine func-
tion used in all the cases is obtained by combining the
two functions. By taking the natural logarithms of both
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723 717
Figure 4: Relationships between[] ln sinh( 
p
and ln  or 1000/T:a)
[] ln sinh( 
p
– ln  ;b)[] ln sinh( 
p
– 1000/T
lnsinh p
Figure 3: Relationships with the peak stress and strain rate: a) ln  –
 p; (b) ln  –  p

sides of Equation (2), the following expression can be
obtained:
ln
 ln ' ln ( . )
ln ( . )    
  +=
<
+>
Q
RT
An
A
1
2
0
12
8
[] (for all) ln ln sinh( An +
⎧ ⎨ ⎪ ⎩ ⎪ 
(2)
The thermal deformation activation energy is charac-
terized by the corresponding peak stress  p
. The peak
stress is obtained with the flow-stress curve. The values
of  , n’, n and Q are obtained with the linear fitting of
the experimental data. And they can be calculated with
the average slope of the oblique lines from Figures 3 and
4. The relationship between ln
 and  p
at a constant tem-
perature is shown in Figure 3a. The relationship be-
tween and ln p
is shown in Figure 3b. In general, the
value of  is determined by  and n’that is  =  /n’. The
relationships between ln
 ,[] ln sinh( 
p
and 1000/T are
shown in Figure 4. According to the above procedure,
the values of A,  ,  , n’, n and Q are 6.0569×10
17
,
0.0233, 0.1600, 6.8659, 3.2237 and 451765.36J·mol
–1
,
respectively. Therefore, the parameter Z can be expressed
with Equation (3) and the peak stress can be expressed
with Equation (4) after the transformation.
[]
Z
RT
=
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =
=×
 exp
.
. sinh( .
 
45176536
60569 10 233
17
p
3 2237 .
(3)
 p
=×
−−
4292 32817 10
1 5 0 3102
.s i n h(. )
.
Z (4)
3.3 Modelling of the work hardening/dynamic recovery
stage
The physically based constitutive model based on the
microstructure evolution takes the dislocation changes
during hot deformation into account and can accurately
predict the flow stress under different conditions. During
the work hardening/dynamic recovery stage, the harden-
ing and softening mechanisms are work hardening (WH)
and dynamic recovery (DRV), respectively. Since the
variation in the dislocation density is affected by both
work hardening and dynamic recovery, the evolution
model of dislocation density can be expressed with
Equation (5):
22
d
d
   =− kk
12
(5)
where d /d represents the change in the rate of disloca-
tion density caused by the strain,  represents the dislo-
cation density, k
1
is the work-hardening coefficient and
k
2
is the dynamic-recovery coefficient.
During the thermal-deformation process, the relation-
ship between the flow stress of materials and dislocation
density can be expressed as:
11
 
WH
= b (6)
where  is the material constant, μ is the shear modulus
and b is the Burgers vector.
The derivative of Equation (6) can be obtained as fol-
lows:
d
d
WH
  
 =
b
2
(7)
Synthesizing Equations (5) and (7), the work-harden-
ing rate can be expressed as:

          
d
d
d
d
d
d
WH WH
WH
=⋅ =
=
−
=
− bk bk bk k
12 12
22
(8)
When  = 0, then
 
sat
=
bk
k
1
2
(9)
Where  sat
denotes the saturation stress. According to
Equation (8), it can be obtained that:
d
d2 d
WH WH
WH

    
=
− bk k
12
(10)
By integrating Equation (10), it can be obtained that:

  
k
bk k C
2
12 1
ln( ) −+
WH
(11)
With an equivalent transformation, we obtain the fol-
lowing:
   bk k C
k
12 2
2
2
−=−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ WH
exp (12)
When = 0, then  = 0 and substitute  = 0 in Equa-
tion (12), we obtain:
Cb k
 
1
(13)
Substituting Equations (9) and (13) into Equation
(12), the work hardening/dynamic recovery model can be
obtained:
   
 
WH
sat sat
=
−−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =
=− −
bk bk
k
k
k
11
2
2
2
2
2
exp
exp
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (14)
When building a material constitutive model, the
work-hardening rate  and flow stress  curve can indi-
rectly reveal the evolution of the microstructure during
hot deformation to determine the value of the character-
istic stress. The work-hardening schematic diagram is
shown in Figure 5. When the material begins to deform,
the work-hardening rate decreases rapidly with the in-
crease in the stress. When the flow stress reaches the
critical stress value  c
, material dynamic crystallization
occurs. When the stress reaches the peak stress  p
, the
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
718 Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723

work-hardening rate is 0. With the increase in the dy-
namic-recrystallization fraction, the work-hardening rate
is negative. When dynamic recrystallization, dynamic re-
covery and work hardening are balanced, the work-hard-
ening rate becomes 0 again.
The saturation stress of the material can be obtained
from the relationship diagram of the work-hardening rate
and flow stress, and the dynamic-recovery coefficient k
2
can be calculated from Equation (14). Dynamic-response
coefficient k
2
can usually be expressed with parameter Z
and its mathematical relation is shown with Equation
(15):
kCZ
n
23
1
= (15)
Based on the relationship between lnk
2
and lnZ and
the linear-regression method, coefficients C
3
and n
1
can
be solved, and Equation (15) can be expressed as:
kZ
2
0 06485
4560078 =
−
.
.
(16)
Peak stress  p
on the stress/strain curve can be char-
acterized with parameter Z, while saturation stress  sat
can be characterized with peak stress  p
. In the form of
linear regression, the relationship between the saturation
stress and peak stress can be expressed as:

sat p
=108 . (17)
To sum up, the constitutive model for 12 % Cr ultra-
supercritical rotor steel during work hardening/dynamic
recovery period can be represented with the following
model:
    
 WH sat sat
Z = exp(451765.
=− − exp(
 k
2
36 ) /RT
kZ
2
0 06485
4560078 =
−
.
.
sat p
 =108 .
p
=×
−−
4292 32817 10
15 0 3 1
.s i n h(.
.
Z
02
)
⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ (18)
3.4 Modeling of the dynamic-recrystallization stage
In general, during the dynamic recrystallization
stage, the dynamic-recrystallization kinetics can be ex-
pressed with the Avrami equation:
23
X
dd
c
p
c
=− −
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥ 1e x p ( )  
  (19)
where X
d
is the dynamic-recrystallization percentage,  d
and k
d
are the material constants and  c
is the critical
strain. Dynamic-recrystallization percentage can also be
expressed as:
24
X
d
WH
sat ss
c
=
−
−
≥


 () (20)
where  WH
is the flow stress in the period of work hard-
ening and dynamic recovery,  ss
is the steady-state
stress and  sat
is the saturation stress. Combined with
Equations (19–20), the flow stress in the dynamic-
recrystallization period can be expressed with the fol-
lowing equation:
  

 =−−−−
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ WH sat ss d
c
p
()e x p 1
k
d
⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ≥ () 
c
(21)
According to Equation (21),  ss
,  p
,  c
,  d
and k
d
need
to be determined. Peak stress  p
can be obtained from the
stress/true strain curve and peak stress can be expressed
in the exponential form of the Zener-Hollomon parame-
ter:
 p
=kZ
m
3
(22)
where k
3
and m are material constants. The peak stress
can be expressed with the function of parameter Z as
follows:
 p
= 00008783
0 1593
.
.
Z (23)
Generally, dynamic recrystallization occurs before
the flow stress reaches the peak stress. There is a com-
plex functional relationship between the critical strain
and deformation temperature, strain rate, temperature
and recrystallization thermal-activation energy. Some
scholars believe that the critical strain of a material and
the peak strain satisfy the mathematical relationship
shown in Equation (24):
25,26

cp
=08 . (24)
In addition, steady-state stress  ss
can be obtained
from the stress/strain curve. The relationship between the
steady-state stress and the peak stress can be expressed
as:

ss p
= 09551 . (25)
In order to obtain the values of k
d
and  d
, Equation
(19) can be rewritten as:
[] ln ln( ) ln ln −−=+
−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1Xk
dd d
c
p
 
 (26)
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723 719
Figure 5: Relationship between the work-hardening rate and flow
stress

According to Equation (26), parameters k
d
and  d
,
can be calculated with linear regression. After the mathe-
matical calculation, k
d
= 1.13 and  d
= 0.58.
Therefore, the constitutive model of the 12 % Cr al-
loy steel during dynamic recrystallization can be ex-
pressed as follows:
    

WH sat sat
DRX WH sat
=− −
=−−
exp(
(
k
2
ss
c
p
)e x p. (
.
10 5 8
113
−−
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ 
 
  ≥
=
−
c
sa
Z = exp(451765.36 )
)
 .
.
/RT
kZ
2
0 06485
4560078
tp
p
p
=
=×
=
−−
108
42 92 32817 10
0000
1 5 0 3102
.
.s i n h(. )
.
.
   Z
8783
08
09551
0 1593
Z
.
.
.


cp
ss p
=
=
⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (27)
3.5 Model validation
Based on Equations (18) and (27), the comparison
between experimental data and predicted data of the con-
stitutive model under different conditions is shown in
Figure 6. As can be seen, the constitutive model can ac-
curately predict the flow stress. In addition, in order to
compare the results obtained from the constitutive model
with the experimental data more directly, the reliability
of the constitutive model is evaluated according to Equa-
tions (28–29)
AARE
n
i
n
=
−
×
=
∑
1
100
1

 ep
e
% (28)
RMSE
n
i
n
=−
=
∑
1
2
1
 
ep
) (29)
where  e
is the experimental flow stress,  p
is the pre-
dicted flow stress and n is the total number of data. The
results show that the average absolute relative error
(AARE) is 14.6 % and the root mean square error
(RMSE) is 19.3 MPa. This indicates that the constitutive
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
720 Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723
Figure 6: Comparison between the experimental and predicted flow stress at different strains for strain rates of: a) 0.001 s
–1
,b )0 . 0 1s
–1
,
c) 0.1 s
–1
,d)1s
–1

model can effectively predict the flow behavior of the
12 % Cr steel.
3.6 Model modification
The physically based constitutive model established
on the basis of the mechanism of microstructure evolu-
tion has high accuracy, but it can be seen from this paper
and other scholars’ papers that the error of the predicted
value of this model at the initial stage of deformation is
relatively large.
27,28
It can be seen that the error of the
model mainly appears in the dynamic-recovery stage.
The main reason for this error is the fact that the consti-
tutive model is established on the premise that the dy-
namic-recovery coefficient is related to parameter Z. So,
the model is based on the presumption that the dy-
namic-recovery coefficient k
2
is only related to the defor-
mation temperature and strain rate. However, according
to Equation (14), it can be seen that the dynamic-recov-
ery coefficient is not only related to the deformation tem-
perature and strain rate but also to the strain; this is the
reason for the large error of the constitutive model at the
initial stage of deformation. As a result, the model is
modified in this study.
According to Equation (14), the relationship between
the dynamic-recovery coefficient and strain under differ-
ent deformation conditions can be obtained. After
smoothing, the relationship curve for the dynamic-recov-
ery coefficient k
2
and strain is obtained, as shown in Fig-
ure 7. According to the curve morphology from this fig-
ure, dynamic-recovery coefficient k
2
can be modified as
follows:
kK K
K
21 3
10
2
=× +
 (30)
where K
1
, K
2
, K
3
are the model coefficients.
The mathematic relation of dynamic-recovery coeffi-
cient k
2
and strain was fitted with a custom nonlinear
function in the Origin software to obtain model coeffi-
cients under different conditions. The relationships be-
tween coefficients K
1
, K
2
and ln Z are shown in Figure 8.
Through linear fitting, we can obtain:
KZ b
1
1100 =− + .l n (31)
KZ
2
07294 456575 =− .l n. (32)
where b is the intercept of the fitted line. The intercept
is obviously related to the strain rate, so it can be ob-
tained using linear fitting with the strain rate, therefore:
KZ
1
1100 2689 60334 =− + + .l n .l n
 .  (33)
K
3
is the dynamic-recovery coefficient in the
steady-state stage. According to the characteristics of the
function of Equation (30), the value of K
3
is consistent
with the dynamic-recovery coefficient before the modifi-
cation, namely:
KZ
3
0 06485
4560078 =−
−
.
.
(34)
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723 721
Figure 8: Relationship between lnZ and K
1
, K
2
Figure 7: Relationship between k
2
and the strain

To sum up, the modified physically based constitutive
model of the as-cast 12 % Cr steel is as follows:


    
=<
=≥
=− −
WH c
DRX c
WH sat sat
()
()
exp( k
2


 DRX WH sat ss
c
p
=−−−−
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ()e x p. 10 5 8
113
2
456
.
 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ =
Z = exp(451765.36 )  /RT
k .
.l n .l n
 .
.
.
0078
1100 2689 60334
072
0 06485
1
2
Z
KZ
K
−
=− + +
=
 94 456575
456 0078
108
42
3
0 06485
ln .
.
.
.
.
Z
KZ
−
=
=
=
−

 sat p
p
92 32817 10
0 0008783
1 5 0 3102
0 1593
sinh ( . )
.
.
.
−−
×
=
=
Z
Z   p
c
08
0 9551
.
.
 
p
ss p
=
⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (35)
Based on Equation (35), the comparison between ex-
perimental data and predicted data is shown in Figure 9.
After the error analysis, the average absolute relative er-
ror (AARE) is 12.2 % and the root mean square error
(RMSE) is 7.0 MPa. It can be seen that the modified
constitutive model can further reduce its error, especially
the root mean square error, which is reduced by about
63.7 %.
4 CONCLUSIONS
Based on a uniaxial isothermal compression test, the
stress/strain curve of the as-cast 12 % Cr ultra-supercriti-
cal rotor steel was established. Through an in-depth anal-
ysis of the macroscopic flow-stress characteristics of this
steel, a physically based constitutive model based on the
microstructure mechanism was established. The main
conclusions are as follows:
1) According to the Arrhenius equation, the basic
thermodynamic parameters of the as-cast 12 % Cr ul-
tra-supercritical rotor steel were calculated. And the acti-
vation energy of the thermal deformation of the steel is
451.765 kJ/mol.
X. ZHANG et al.: HOT-DEFORMATION BEHAVIOR AND A MODIFIED PHYSICALLY BASED CONSTITUTIVE MODEL ...
722 Materiali in tehnologije / Materials and technology 54 (2020) 5, 715–723
Figure 9: Comparison between the experimental and predicted flow stress at different strains for strain rates of:a) 0.001 s
–1
, b) 0.01 s
–1
,
c) 0.1 s
–1
,d)1s
–1

2) There are two kinds of flow-stress curves for the
as-cast 12 % Cr ultra-supercritical rotor steel, namely,
the flow-stress curves of the work hardening/dynamic re-
covery and dynamic recrystallization. The softening
mechanisms of the two flow-stress curves are different,
including dynamic recovery and dynamic recrystal-
lization.
3) Based on the two flow-stress curves of the 12 %
Cr steel, a two-stage physically based constitutive model
was established and the model was modified. The aver-
age absolute relative error (AARE) of the modified model
is 12.2 % and the root-mean-square error (RMSE)is7.0
MPa. The accuracy of the model is improved compared
with the physically based constitutive model before the
modification.
Acknowledgments
The work was sponsored by the National Natural Sci-
ence Foundation of China (51775361), the Shanghai
Dianji University, and the Shanghai Research Center of
Engineering Technology for Large Parts Thermal Manu-
facturing.
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